Subgroups — C₆₇×#3519 #22557950

Order 8576.

1252 conjugacy classes of subgroups.

The subgroup lattice has 490 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#555×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
2	#554×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
3	#553×C₆₇	C₂	1	2	yes	maximal, complemented
4	#553×C₆₇	C₂	1	2	yes	maximal, complemented
5	#550×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
6	#546×C₆₇	C₂	1	4	yes	maximal, complemented
7	#546×C₆₇	C₂	1	2	yes	maximal, complemented
8	D₄²×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
9	D₄²×C₆₇	C₂	1	2	yes	maximal, complemented
10	#540×C₆₇	C₂	1	2	yes	maximal, complemented
11	#538×C₆₇	C₂	1	4	yes	maximal, complemented
12	#532×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
13	#529×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
14	#525×C₆₇	C₂	1	2	yes	maximal, complemented
15	C₂×#121×C₆₇	C₂	1	2	yes	maximal, complemented
16	#518×C₆₇	C₂	1	2	yes	maximal, complemented
17	C₂×C₄×D₄×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
18	C₂×#41×C₆₇	C₂²	1	2	yes	complemented
19	C₂×#41×C₆₇	C₂²	1	2	yes	complemented
20	C₂²×D₄×C₆₇	C₂²	1	1	yes	characteristic
21	C₂²×D₄×C₆₇	C₂²	1	2	yes	—
22	C₂²×D₄×C₆₇	C₂²	1	2	yes	complemented
23	C₂²×D₄×C₆₇	C₂²	1	4	yes	complemented
24	C₂³×C₄×C₆₇	C₂²	1	2	yes	complemented
25	C₄⋊Q₈×C₆₇	C₂²	1	1	yes	characteristic, complemented
26	#127×C₆₇	C₂²	1	1	yes	characteristic, complemented
27	#127×C₆₇	C₂²	1	2	yes	complemented
28	#126×C₆₇	C₂²	1	2	yes	complemented
29	#126×C₆₇	C₂²	1	4	yes	complemented
30	#125×C₆₇	C₂²	1	1	yes	characteristic, complemented
31	#125×C₆₇	C₂²	1	2	yes	complemented
32	#124×C₆₇	C₂²	1	1	yes	characteristic, complemented
33	#124×C₆₇	C₂²	1	2	yes	complemented
34	#124×C₆₇	C₂²	1	4	yes	complemented
35	#123×C₆₇	C₂²	1	4	yes	complemented
36	#123×C₆₇	C₂²	1	2	yes	complemented
37	#123×C₆₇	C₂²	1	4	yes	complemented
38	#123×C₆₇	C₂²	1	4	yes	complemented
39	C₂²⋊Q₈×C₆₇	C₂²	1	1	yes	characteristic, complemented
40	C₂²⋊Q₈×C₆₇	C₂²	1	2	yes	complemented
41	C₂²⋊Q₈×C₆₇	C₂²	1	4	yes	complemented
42	#121×C₆₇	C₂²	1	1	yes	characteristic
43	#121×C₆₇	C₂²	1	2	yes	complemented
44	#121×C₆₇	C₂²	1	4	yes	complemented
45	#121×C₆₇	C₂²	1	4	yes	complemented
46	#121×C₆₇	C₂²	1	2	yes	complemented
47	#121×C₆₇	C₂²	1	2	yes	complemented
48	#121×C₆₇	C₂²	1	2	yes	complemented
49	#121×C₆₇	C₂²	1	2	yes	complemented
50	#121×C₆₇	C₂²	1	4	yes	complemented
51	#121×C₆₇	C₂²	1	4	yes	complemented
52	#121×C₆₇	C₂²	1	2	yes	complemented
53	#121×C₆₇	C₂²	1	2	yes	complemented
54	#121×C₆₇	C₂²	1	4	yes	complemented
55	#120×C₆₇	C₂²	1	4	yes	complemented
56	#120×C₆₇	C₂²	1	4	yes	complemented
57	#120×C₆₇	C₂²	1	4	yes	complemented
58	C₄×Q₈×C₆₇	C₂²	1	1	yes	characteristic, complemented
59	C₄×D₄×C₆₇	C₂²	1	1	yes	characteristic
60	C₄×D₄×C₆₇	C₂²	1	1	yes	characteristic
61	C₄×D₄×C₆₇	C₂²	1	2	yes	—
62	C₄×D₄×C₆₇	C₂²	1	2	yes	complemented
63	C₄×D₄×C₆₇	C₂²	1	1	yes	characteristic, complemented
64	C₄×D₄×C₆₇	C₂²	1	2	yes	complemented
65	C₄×D₄×C₆₇	C₂²	1	1	yes	characteristic, complemented
66	C₄×D₄×C₆₇	C₂²	1	4	yes	complemented
67	C₄×D₄×C₆₇	C₂²	1	2	yes	complemented
68	C₄×D₄×C₆₇	C₂²	1	4	yes	complemented
69	C₄×D₄×C₆₇	C₂²	1	2	yes	complemented
70	C₄×D₄×C₆₇	C₂²	1	4	yes	complemented
71	C₄×D₄×C₆₇	C₂²	1	2	yes	complemented
72	C₄×D₄×C₆₇	C₂²	1	1	yes	characteristic, complemented
73	C₄×D₄×C₆₇	C₂²	1	2	yes	complemented
74	#117×C₆₇	C₂²	1	2	yes	complemented
75	#117×C₆₇	C₂²	1	2	yes	complemented
76	#117×C₆₇	C₂²	1	2	yes	complemented
77	C₂×C₄⋊C₄×C₆₇	C₂²	1	1	yes	characteristic, complemented
78	C₂×C₄⋊C₄×C₆₇	C₂²	1	2	yes	complemented
79	C₂×C₂²⋊C₄×C₆₇	C₂²	1	2	yes	complemented
80	C₂×C₂²⋊C₄×C₆₇	C₂²	1	4	yes	complemented
81	C₂×C₂²⋊C₄×C₆₇	C₂²	1	4	yes	complemented
82	C₂×C₄²×C₆₇	C₂²	1	1	yes	characteristic, complemented
83	C₂⁴×C₆₇	C₂³	1	2	yes	—
84	C₂⁴×C₆₇	C₂³	1	4	yes	—
85	#41×C₆₇	—	4	4	no	—
86	#41×C₆₇	—	4	4	no	—
87	C₂×Q₈×C₆₇	C₂³	1	1	yes	characteristic
88	C₂×Q₈×C₆₇	C₂³	1	2	yes	—
89	C₂×D₄×C₆₇	—	2	4	no	—
90	C₂×D₄×C₆₇	—	2	8	no	—
91	C₂×D₄×C₆₇	—	4	4	no	—
92	C₂×D₄×C₆₇	—	4	8	no	—
93	C₂×D₄×C₆₇	C₂³	1	2	yes	—
94	C₂×D₄×C₆₇	C₂³	1	4	yes	—
95	C₂×D₄×C₆₇	C₂³	1	1	yes	characteristic
96	C₂×D₄×C₆₇	C₂³	1	2	yes	—
97	C₂×D₄×C₆₇	C₂³	1	4	yes	—
98	C₂×D₄×C₆₇	C₂³	1	2	yes	—
99	C₂×D₄×C₆₇	C₂³	1	1	yes	characteristic
100	C₂×D₄×C₆₇	C₂³	1	2	yes	—
101	C₂×D₄×C₆₇	C₂³	1	4	yes	—
102	C₂×D₄×C₆₇	C₂³	1	2	yes	—
103	C₂×D₄×C₆₇	C₂³	1	4	yes	—
104	C₂×D₄×C₆₇	C₂³	1	4	yes	—
105	C₂×D₄×C₆₇	C₂³	1	4	yes	—
106	C₂×D₄×C₆₇	C₂³	1	1	yes	characteristic
107	C₂²×C₄×C₆₇	—	4	4	no	—
108	C₂²×C₄×C₆₇	C₂³	1	1	yes	characteristic
109	C₂²×C₄×C₆₇	C₂³	1	1	yes	characteristic
110	C₂²×C₄×C₆₇	C₂³	1	1	yes	characteristic
111	C₂²×C₄×C₆₇	C₂³	1	2	yes	—
112	C₂²×C₄×C₆₇	C₂³	1	2	yes	—
113	C₂²×C₄×C₆₇	C₂³	1	2	yes	—
114	C₂²×C₄×C₆₇	C₂³	1	2	yes	—
115	C₂²×C₄×C₆₇	C₂³	1	4	yes	—
116	C₂²×C₄×C₆₇	C₂³	1	2	yes	—
117	C₂²×C₄×C₆₇	C₂³	1	4	yes	—
118	C₂²×C₄×C₆₇	C₂³	1	4	yes	—
119	C₂²×C₄×C₆₇	C₂³	1	2	yes	—
120	C₄⋊C₄×C₆₇	C₂³	1	1	yes	characteristic
121	C₄⋊C₄×C₆₇	C₂³	1	1	yes	characteristic
122	C₄⋊C₄×C₆₇	C₂³	1	2	yes	—
123	C₄⋊C₄×C₆₇	C₂³	1	1	yes	characteristic
124	C₄⋊C₄×C₆₇	C₂³	1	2	yes	—
125	C₄⋊C₄×C₆₇	C₂³	1	4	yes	—
126	C₄⋊C₄×C₆₇	C₂³	1	1	yes	characteristic
127	C₄⋊C₄×C₆₇	C₂³	1	2	yes	—
128	C₄⋊C₄×C₆₇	C₂³	1	2	yes	—
129	C₄⋊C₄×C₆₇	C₂³	1	2	yes	—
130	C₄⋊C₄×C₆₇	C₂³	1	2	yes	—
131	C₄⋊C₄×C₆₇	C₂³	1	4	yes	—
132	C₄⋊C₄×C₆₇	C₂³	1	2	yes	—
133	C₂²⋊C₄×C₆₇	C₂³	1	2	yes	—
134	C₂²⋊C₄×C₆₇	C₂³	1	2	yes	—
135	C₂²⋊C₄×C₆₇	C₂³	1	4	yes	—
136	C₂²⋊C₄×C₆₇	C₂³	1	2	yes	—
137	C₂²⋊C₄×C₆₇	C₂³	1	2	yes	—
138	C₂²⋊C₄×C₆₇	C₂³	1	4	yes	—
139	C₂²⋊C₄×C₆₇	C₂³	1	4	yes	—
140	C₂²⋊C₄×C₆₇	C₂³	1	2	yes	—
141	C₂²⋊C₄×C₆₇	C₂³	1	4	yes	—
142	C₂²⋊C₄×C₆₇	C₂³	1	2	yes	—
143	C₂²⋊C₄×C₆₇	C₂³	1	4	yes	—
144	C₂²⋊C₄×C₆₇	C₂³	1	2	yes	—
145	C₂²⋊C₄×C₆₇	C₂³	1	2	yes	—
146	C₂²⋊C₄×C₆₇	C₂³	1	2	yes	—
147	C₂²⋊C₄×C₆₇	C₂³	1	4	yes	—
148	C₂²⋊C₄×C₆₇	C₂³	1	4	yes	—
149	C₄²×C₆₇	C₂³	1	1	yes	characteristic
150	C₄²×C₆₇	C₂³	1	1	yes	characteristic
151	C₄²×C₆₇	C₂³	1	1	yes	characteristic
152	C₄²×C₆₇	C₂³	1	1	yes	characteristic
153	C₄²×C₆₇	C₂³	1	4	yes	—
154	C₄²×C₆₇	C₂³	1	2	yes	—
155	C₂³×C₆₇	—	2	4	no	—
156	C₂³×C₆₇	—	2	4	no	—
157	C₂³×C₆₇	—	2	8	no	—
158	C₂³×C₆₇	—	4	2	no	—
159	C₂³×C₆₇	—	4	4	no	—
160	C₂³×C₆₇	—	4	4	no	—
161	C₂³×C₆₇	C₂⁴	1	2	yes	—
162	C₂³×C₆₇	C₂⁴	1	2	yes	—
163	C₂³×C₆₇	C₂⁴	1	4	yes	—
164	C₂³×C₆₇	C₂⁴	1	2	yes	—
165	C₂³×C₆₇	C₂⁴	1	1	yes	characteristic
166	Q₈×C₆₇	—	2	2	no	—
167	Q₈×C₆₇	—	4	2	no	—
168	D₄×C₆₇	—	2	4	no	—
169	D₄×C₆₇	—	2	8	no	—
170	D₄×C₆₇	—	2	4	no	—
171	D₄×C₆₇	—	2	8	no	—
172	D₄×C₆₇	—	2	2	no	—
173	D₄×C₆₇	—	2	4	no	—
174	D₄×C₆₇	—	2	2	no	—
175	D₄×C₆₇	—	2	4	no	—
176	D₄×C₆₇	—	4	4	no	—
177	D₄×C₆₇	—	4	4	no	—
178	D₄×C₆₇	—	4	4	no	—
179	D₄×C₆₇	—	4	2	no	—
180	D₄×C₆₇	—	4	4	no	—
181	D₄×C₆₇	#41	1	4	yes	complemented
182	C₂×C₄×C₆₇	—	2	2	no	—
183	C₂×C₄×C₆₇	—	2	2	no	—
184	C₂×C₄×C₆₇	—	2	4	no	—
185	C₂×C₄×C₆₇	—	2	2	no	—
186	C₂×C₄×C₆₇	—	2	4	no	—
187	C₂×C₄×C₆₇	—	2	4	no	—
188	C₂×C₄×C₆₇	—	4	2	no	—
189	C₂×C₄×C₆₇	—	4	2	no	—
190	C₂×C₄×C₆₇	—	4	2	no	—
191	C₂×C₄×C₆₇	—	4	4	no	—
192	C₂×C₄×C₆₇	—	4	4	no	—
193	C₂×C₄×C₆₇	—	4	4	no	—
194	C₂×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
195	C₂×C₄×C₆₇	C₂⁴	1	2	yes	—
196	C₂×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
197	C₂×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
198	C₂×C₄×C₆₇	C₂⁴	1	2	yes	—
199	C₂×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
200	C₂×C₄×C₆₇	C₂⁴	1	4	yes	—
201	C₂×C₄×C₆₇	C₂⁴	1	2	yes	—
202	C₂×C₄×C₆₇	C₂⁴	1	2	yes	—
203	C₂×C₄×C₆₇	C₂⁴	1	2	yes	—
204	C₂×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
205	C₂×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
206	C₂²×C₆₇	—	2	1	no	—
207	C₂²×C₆₇	—	2	1	no	—
208	C₂²×C₆₇	—	2	2	no	—
209	C₂²×C₆₇	—	2	2	no	—
210	C₂²×C₆₇	—	2	2	no	—
211	C₂²×C₆₇	—	2	2	no	—
212	C₂²×C₆₇	—	2	4	no	—
213	C₂²×C₆₇	—	2	2	no	—
214	C₂²×C₆₇	—	2	4	no	—
215	C₂²×C₆₇	—	2	4	no	—
216	C₂²×C₆₇	—	2	2	no	—
217	C₂²×C₆₇	—	2	2	no	—
218	C₂²×C₆₇	—	2	2	no	—
219	C₂²×C₆₇	—	4	8	no	—
220	C₂²×C₆₇	—	8	8	no	—
221	C₂²×C₆₇	C₂×#41	1	4	yes	—
222	C₂²×C₆₇	H(5,𝔽₂)	1	2	yes	—
223	C₂²×C₆₇	C₂⁵	1	1	yes	characteristic
224	C₄×C₆₇	—	2	1	no	—
225	C₄×C₆₇	—	2	2	no	—
226	C₄×C₆₇	—	2	1	no	—
227	C₄×C₆₇	—	2	1	no	—
228	C₄×C₆₇	—	2	2	no	—
229	C₄×C₆₇	—	2	2	no	—
230	C₄×C₆₇	—	2	1	no	—
231	C₄×C₆₇	—	2	2	no	—
232	C₄×C₆₇	—	2	4	no	—
233	C₄×C₆₇	—	2	2	no	—
234	C₄×C₆₇	C₂×#41	1	2	yes	—
235	C₄×C₆₇	H(5,𝔽₂)	1	2	yes	—
236	C₂×C₆₇	—	2	2	no	—
237	C₂×C₆₇	—	2	4	no	—
238	C₂×C₆₇	—	4	2	no	—
239	C₂×C₆₇	—	4	4	no	—
240	C₂×C₆₇	—	4	2	no	—
241	C₂×C₆₇	C₂²×#41	1	1	yes	characteristic
242	C₂×C₆₇	C₂×H(5,𝔽₂)	1	1	yes	characteristic
243	C₂×C₆₇	C₂×H(5,𝔽₂)	1	1	yes	characteristic
244	#3519	C₆₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
245	C₆₇	#3519	1	1	yes	characteristic, direct product factor, Sylow
246	#555	C₂×C₆₇	1	1	yes	characteristic, complemented
247	#554	C₂×C₆₇	1	1	yes	characteristic, complemented
248	#553	C₂×C₆₇	1	2	yes	complemented
249	#553	C₂×C₆₇	1	2	yes	complemented
250	#550	C₂×C₆₇	1	1	yes	characteristic, complemented
251	#546	C₂×C₆₇	1	4	yes	complemented
252	#546	C₂×C₆₇	1	2	yes	complemented
253	D₄²	C₂×C₆₇	1	1	yes	characteristic, complemented
254	D₄²	C₂×C₆₇	1	2	yes	complemented
255	#540	C₂×C₆₇	1	2	yes	complemented
256	#538	C₂×C₆₇	1	4	yes	complemented
257	#532	C₂×C₆₇	1	1	yes	characteristic, complemented
258	#529	C₂×C₆₇	1	1	yes	characteristic, complemented
259	#525	C₂×C₆₇	1	2	yes	complemented
260	C₂×#121	C₂×C₆₇	1	2	yes	complemented
261	#518	C₂×C₆₇	1	2	yes	complemented
262	C₂×C₄×D₄	C₂×C₆₇	1	1	yes	characteristic, complemented
263	C₂×#41	C₂²×C₆₇	1	2	yes	complemented
264	C₂×#41	C₂²×C₆₇	1	2	yes	complemented
265	C₂²×D₄	C₂²×C₆₇	1	1	yes	characteristic
266	C₂²×D₄	C₂²×C₆₇	1	2	yes	—
267	C₂²×D₄	C₂²×C₆₇	1	2	yes	complemented
268	C₂²×D₄	C₂²×C₆₇	1	4	yes	complemented
269	C₂³×C₄	C₂²×C₆₇	1	2	yes	complemented
270	C₄⋊Q₈	C₂²×C₆₇	1	1	yes	characteristic, complemented
271	#127	C₂²×C₆₇	1	1	yes	characteristic, complemented
272	#127	C₂²×C₆₇	1	2	yes	complemented
273	#126	C₂²×C₆₇	1	2	yes	complemented
274	#126	C₂²×C₆₇	1	4	yes	complemented
275	#125	C₂²×C₆₇	1	1	yes	characteristic, complemented
276	#125	C₂²×C₆₇	1	2	yes	complemented
277	#124	C₂²×C₆₇	1	1	yes	characteristic, complemented
278	#124	C₂²×C₆₇	1	2	yes	complemented
279	#124	C₂²×C₆₇	1	4	yes	complemented
280	#123	C₂²×C₆₇	1	4	yes	complemented
281	#123	C₂²×C₆₇	1	2	yes	complemented
282	#123	C₂²×C₆₇	1	4	yes	complemented
283	#123	C₂²×C₆₇	1	4	yes	complemented
284	C₂²⋊Q₈	C₂²×C₆₇	1	1	yes	characteristic, complemented
285	C₂²⋊Q₈	C₂²×C₆₇	1	2	yes	complemented
286	C₂²⋊Q₈	C₂²×C₆₇	1	4	yes	complemented
287	#121	C₂²×C₆₇	1	1	yes	characteristic
288	#121	C₂²×C₆₇	1	2	yes	complemented
289	#121	C₂²×C₆₇	1	4	yes	complemented
290	#121	C₂²×C₆₇	1	4	yes	complemented
291	#121	C₂²×C₆₇	1	2	yes	complemented
292	#121	C₂²×C₆₇	1	2	yes	complemented
293	#121	C₂²×C₆₇	1	2	yes	complemented
294	#121	C₂²×C₆₇	1	2	yes	complemented
295	#121	C₂²×C₆₇	1	4	yes	complemented
296	#121	C₂²×C₆₇	1	4	yes	complemented
297	#121	C₂²×C₆₇	1	2	yes	complemented
298	#121	C₂²×C₆₇	1	2	yes	complemented
299	#121	C₂²×C₆₇	1	4	yes	complemented
300	#120	C₂²×C₆₇	1	4	yes	complemented
301	#120	C₂²×C₆₇	1	4	yes	complemented
302	#120	C₂²×C₆₇	1	4	yes	complemented
303	C₄×Q₈	C₂²×C₆₇	1	1	yes	characteristic, complemented
304	C₄×D₄	C₂²×C₆₇	1	1	yes	characteristic
305	C₄×D₄	C₂²×C₆₇	1	1	yes	characteristic
306	C₄×D₄	C₂²×C₆₇	1	2	yes	—
307	C₄×D₄	C₂²×C₆₇	1	2	yes	complemented
308	C₄×D₄	C₂²×C₆₇	1	1	yes	characteristic, complemented
309	C₄×D₄	C₂²×C₆₇	1	2	yes	complemented
310	C₄×D₄	C₂²×C₆₇	1	1	yes	characteristic, complemented
311	C₄×D₄	C₂²×C₆₇	1	4	yes	complemented
312	C₄×D₄	C₂²×C₆₇	1	2	yes	complemented
313	C₄×D₄	C₂²×C₆₇	1	4	yes	complemented
314	C₄×D₄	C₂²×C₆₇	1	2	yes	complemented
315	C₄×D₄	C₂²×C₆₇	1	4	yes	complemented
316	C₄×D₄	C₂²×C₆₇	1	2	yes	complemented
317	C₄×D₄	C₂²×C₆₇	1	1	yes	characteristic, complemented
318	C₄×D₄	C₂²×C₆₇	1	2	yes	complemented
319	#117	C₂²×C₆₇	1	2	yes	complemented
320	#117	C₂²×C₆₇	1	2	yes	complemented
321	#117	C₂²×C₆₇	1	2	yes	complemented
322	C₂×C₄⋊C₄	C₂²×C₆₇	1	1	yes	characteristic, complemented
323	C₂×C₄⋊C₄	C₂²×C₆₇	1	2	yes	complemented
324	C₂×C₂²⋊C₄	C₂²×C₆₇	1	2	yes	complemented
325	C₂×C₂²⋊C₄	C₂²×C₆₇	1	4	yes	complemented
326	C₂×C₂²⋊C₄	C₂²×C₆₇	1	4	yes	complemented
327	C₂×C₄²	C₂²×C₆₇	1	1	yes	characteristic, complemented
328	C₂⁴	C₂³×C₆₇	1	2	yes	—
329	C₂⁴	C₂³×C₆₇	1	4	yes	—
330	#41	—	4	4	no	—
331	#41	—	4	4	no	—
332	C₂×Q₈	C₂³×C₆₇	1	1	yes	characteristic
333	C₂×Q₈	C₂³×C₆₇	1	2	yes	—
334	C₂×D₄	—	2	4	no	—
335	C₂×D₄	—	2	8	no	—
336	C₂×D₄	—	4	4	no	—
337	C₂×D₄	—	4	8	no	—
338	C₂×D₄	C₂³×C₆₇	1	2	yes	—
339	C₂×D₄	C₂³×C₆₇	1	4	yes	—
340	C₂×D₄	C₂³×C₆₇	1	1	yes	characteristic
341	C₂×D₄	C₂³×C₆₇	1	2	yes	—
342	C₂×D₄	C₂³×C₆₇	1	4	yes	—
343	C₂×D₄	C₂³×C₆₇	1	2	yes	—
344	C₂×D₄	C₂³×C₆₇	1	1	yes	characteristic
345	C₂×D₄	C₂³×C₆₇	1	2	yes	—
346	C₂×D₄	C₂³×C₆₇	1	4	yes	—
347	C₂×D₄	C₂³×C₆₇	1	2	yes	—
348	C₂×D₄	C₂³×C₆₇	1	4	yes	—
349	C₂×D₄	C₂³×C₆₇	1	4	yes	—
350	C₂×D₄	C₂³×C₆₇	1	4	yes	—
351	C₂×D₄	C₂³×C₆₇	1	1	yes	characteristic
352	C₂²×C₄	—	4	4	no	—
353	C₂²×C₄	C₂³×C₆₇	1	1	yes	characteristic
354	C₂²×C₄	C₂³×C₆₇	1	1	yes	characteristic
355	C₂²×C₄	C₂³×C₆₇	1	1	yes	characteristic
356	C₂²×C₄	C₂³×C₆₇	1	2	yes	—
357	C₂²×C₄	C₂³×C₆₇	1	2	yes	—
358	C₂²×C₄	C₂³×C₆₇	1	2	yes	—
359	C₂²×C₄	C₂³×C₆₇	1	2	yes	—
360	C₂²×C₄	C₂³×C₆₇	1	4	yes	—
361	C₂²×C₄	C₂³×C₆₇	1	2	yes	—
362	C₂²×C₄	C₂³×C₆₇	1	4	yes	—
363	C₂²×C₄	C₂³×C₆₇	1	4	yes	—
364	C₂²×C₄	C₂³×C₆₇	1	2	yes	—
365	C₄⋊C₄	C₂³×C₆₇	1	1	yes	characteristic
366	C₄⋊C₄	C₂³×C₆₇	1	1	yes	characteristic
367	C₄⋊C₄	C₂³×C₆₇	1	2	yes	—
368	C₄⋊C₄	C₂³×C₆₇	1	1	yes	characteristic
369	C₄⋊C₄	C₂³×C₆₇	1	2	yes	—
370	C₄⋊C₄	C₂³×C₆₇	1	4	yes	—
371	C₄⋊C₄	C₂³×C₆₇	1	1	yes	characteristic
372	C₄⋊C₄	C₂³×C₆₇	1	2	yes	—
373	C₄⋊C₄	C₂³×C₆₇	1	2	yes	—
374	C₄⋊C₄	C₂³×C₆₇	1	2	yes	—
375	C₄⋊C₄	C₂³×C₆₇	1	2	yes	—
376	C₄⋊C₄	C₂³×C₆₇	1	4	yes	—
377	C₄⋊C₄	C₂³×C₆₇	1	2	yes	—
378	C₂²⋊C₄	C₂³×C₆₇	1	2	yes	—
379	C₂²⋊C₄	C₂³×C₆₇	1	2	yes	—
380	C₂²⋊C₄	C₂³×C₆₇	1	4	yes	—
381	C₂²⋊C₄	C₂³×C₆₇	1	2	yes	—
382	C₂²⋊C₄	C₂³×C₆₇	1	2	yes	—
383	C₂²⋊C₄	C₂³×C₆₇	1	4	yes	—
384	C₂²⋊C₄	C₂³×C₆₇	1	4	yes	—
385	C₂²⋊C₄	C₂³×C₆₇	1	2	yes	—
386	C₂²⋊C₄	C₂³×C₆₇	1	4	yes	—
387	C₂²⋊C₄	C₂³×C₆₇	1	2	yes	—
388	C₂²⋊C₄	C₂³×C₆₇	1	4	yes	—
389	C₂²⋊C₄	C₂³×C₆₇	1	2	yes	—
390	C₂²⋊C₄	C₂³×C₆₇	1	2	yes	—
391	C₂²⋊C₄	C₂³×C₆₇	1	2	yes	—
392	C₂²⋊C₄	C₂³×C₆₇	1	4	yes	—
393	C₂²⋊C₄	C₂³×C₆₇	1	4	yes	—
394	C₄²	C₂³×C₆₇	1	1	yes	characteristic
395	C₄²	C₂³×C₆₇	1	1	yes	characteristic
396	C₄²	C₂³×C₆₇	1	1	yes	characteristic
397	C₄²	C₂³×C₆₇	1	1	yes	characteristic
398	C₄²	C₂³×C₆₇	1	4	yes	—
399	C₄²	C₂³×C₆₇	1	2	yes	—
400	C₂³	—	2	4	no	—
401	C₂³	—	2	4	no	—
402	C₂³	—	2	8	no	—
403	C₂³	—	4	2	no	—
404	C₂³	—	4	4	no	—
405	C₂³	—	4	4	no	—
406	C₂³	C₂⁴×C₆₇	1	2	yes	—
407	C₂³	C₂⁴×C₆₇	1	2	yes	—
408	C₂³	C₂⁴×C₆₇	1	4	yes	—
409	C₂³	C₂⁴×C₆₇	1	2	yes	—
410	C₂³	C₂⁴×C₆₇	1	1	yes	characteristic
411	Q₈	—	2	2	no	—
412	Q₈	—	4	2	no	—
413	D₄	—	2	4	no	—
414	D₄	—	2	8	no	—
415	D₄	—	2	4	no	—
416	D₄	—	2	8	no	—
417	D₄	—	2	2	no	—
418	D₄	—	2	4	no	—
419	D₄	—	2	2	no	—
420	D₄	—	2	4	no	—
421	D₄	—	4	4	no	—
422	D₄	—	4	4	no	—
423	D₄	—	4	4	no	—
424	D₄	—	4	2	no	—
425	D₄	—	4	4	no	—
426	D₄	#41×C₆₇	1	4	yes	complemented
427	C₂×C₄	—	2	2	no	—
428	C₂×C₄	—	2	2	no	—
429	C₂×C₄	—	2	4	no	—
430	C₂×C₄	—	2	2	no	—
431	C₂×C₄	—	2	4	no	—
432	C₂×C₄	—	2	4	no	—
433	C₂×C₄	—	4	2	no	—
434	C₂×C₄	—	4	2	no	—
435	C₂×C₄	—	4	2	no	—
436	C₂×C₄	—	4	4	no	—
437	C₂×C₄	—	4	4	no	—
438	C₂×C₄	—	4	4	no	—
439	C₂×C₄	C₂⁴×C₆₇	1	1	yes	characteristic
440	C₂×C₄	C₂⁴×C₆₇	1	2	yes	—
441	C₂×C₄	C₂⁴×C₆₇	1	1	yes	characteristic
442	C₂×C₄	C₂⁴×C₆₇	1	1	yes	characteristic
443	C₂×C₄	C₂⁴×C₆₇	1	2	yes	—
444	C₂×C₄	C₂⁴×C₆₇	1	1	yes	characteristic
445	C₂×C₄	C₂⁴×C₆₇	1	4	yes	—
446	C₂×C₄	C₂⁴×C₆₇	1	2	yes	—
447	C₂×C₄	C₂⁴×C₆₇	1	2	yes	—
448	C₂×C₄	C₂⁴×C₆₇	1	2	yes	—
449	C₂×C₄	C₂⁴×C₆₇	1	1	yes	characteristic
450	C₂×C₄	C₂⁴×C₆₇	1	1	yes	characteristic
451	C₂²	—	2	1	no	—
452	C₂²	—	2	1	no	—
453	C₂²	—	2	2	no	—
454	C₂²	—	2	2	no	—
455	C₂²	—	2	2	no	—
456	C₂²	—	2	2	no	—
457	C₂²	—	2	4	no	—
458	C₂²	—	2	2	no	—
459	C₂²	—	2	4	no	—
460	C₂²	—	2	4	no	—
461	C₂²	—	2	2	no	—
462	C₂²	—	2	2	no	—
463	C₂²	—	2	2	no	—
464	C₂²	—	4	8	no	—
465	C₂²	—	8	8	no	—
466	C₂²	C₂×#41×C₆₇	1	4	yes	—
467	C₂²	H(5,𝔽₂)×C₆₇	1	2	yes	—
468	C₂²	C₂⁵×C₆₇	1	1	yes	characteristic
469	C₄	—	2	1	no	—
470	C₄	—	2	2	no	—
471	C₄	—	2	1	no	—
472	C₄	—	2	1	no	—
473	C₄	—	2	2	no	—
474	C₄	—	2	2	no	—
475	C₄	—	2	1	no	—
476	C₄	—	2	2	no	—
477	C₄	—	2	4	no	—
478	C₄	—	2	2	no	—
479	C₄	C₂×#41×C₆₇	1	2	yes	—
480	C₄	H(5,𝔽₂)×C₆₇	1	2	yes	—
481	C₂	—	2	2	no	—
482	C₂	—	2	4	no	—
483	C₂	—	4	2	no	—
484	C₂	—	4	2	no	—
485	C₂	—	4	4	no	—
486	C₂	C₂²×#41×C₆₇	1	1	yes	characteristic
487	C₂	C₂×H(5,𝔽₂)×C₆₇	1	1	yes	characteristic
488	C₂	C₂×H(5,𝔽₂)×C₆₇	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₆₇	C₂⁵
commutator, Frattini, cosocle	C₂²	C₂⁵×C₆₇
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₆₇ ◁ G

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

N	H ≅ G/N	type
1. #555×C₆₇	C₂	semidirect
2. #554×C₆₇	C₂	semidirect
3. #553×C₆₇	C₂	semidirect
4. #553×C₆₇	C₂	semidirect
5. #550×C₆₇	C₂	semidirect
6. #546×C₆₇	C₂	semidirect
7. #546×C₆₇	C₂	semidirect
8. D₄²×C₆₇	C₂	semidirect
9. D₄²×C₆₇	C₂	semidirect
10. #540×C₆₇	C₂	semidirect
11. #538×C₆₇	C₂	semidirect
12. #532×C₆₇	C₂	semidirect
13. #529×C₆₇	C₂	semidirect
14. #525×C₆₇	C₂	semidirect
15. C₂×#121×C₆₇	C₂	semidirect
16. #518×C₆₇	C₂	semidirect
17. C₂×C₄×D₄×C₆₇	C₂	semidirect
18. C₂×#41×C₆₇	C₂²	semidirect
19. C₂×#41×C₆₇	C₂²	semidirect
22. C₂²×D₄×C₆₇	C₂²	semidirect
23. C₂²×D₄×C₆₇	C₂²	semidirect
24. C₂³×C₄×C₆₇	C₂²	semidirect
25. C₄⋊Q₈×C₆₇	C₂²	semidirect
26. #127×C₆₇	C₂²	semidirect
27. #127×C₆₇	C₂²	semidirect
28. #126×C₆₇	C₂²	semidirect
29. #126×C₆₇	C₂²	semidirect
30. #125×C₆₇	C₂²	semidirect
31. #125×C₆₇	C₂²	semidirect
32. #124×C₆₇	C₂²	semidirect
33. #124×C₆₇	C₂²	semidirect
34. #124×C₆₇	C₂²	semidirect
35. #123×C₆₇	C₂²	semidirect
36. #123×C₆₇	C₂²	semidirect
37. #123×C₆₇	C₂²	semidirect
38. #123×C₆₇	C₂²	semidirect
39. C₂²⋊Q₈×C₆₇	C₂²	semidirect
40. C₂²⋊Q₈×C₆₇	C₂²	semidirect
41. C₂²⋊Q₈×C₆₇	C₂²	semidirect
43. #121×C₆₇	C₂²	semidirect
44. #121×C₆₇	C₂²	semidirect
45. #121×C₆₇	C₂²	semidirect
46. #121×C₆₇	C₂²	semidirect
47. #121×C₆₇	C₂²	semidirect
48. #121×C₆₇	C₂²	semidirect
49. #121×C₆₇	C₂²	semidirect
50. #121×C₆₇	C₂²	semidirect
51. #121×C₆₇	C₂²	semidirect
52. #121×C₆₇	C₂²	semidirect
53. #121×C₆₇	C₂²	semidirect
54. #121×C₆₇	C₂²	semidirect
55. #120×C₆₇	C₂²	semidirect
56. #120×C₆₇	C₂²	semidirect
57. #120×C₆₇	C₂²	semidirect
58. C₄×Q₈×C₆₇	C₂²	semidirect
62. C₄×D₄×C₆₇	C₂²	semidirect
63. C₄×D₄×C₆₇	C₂²	semidirect
64. C₄×D₄×C₆₇	C₂²	semidirect
65. C₄×D₄×C₆₇	C₂²	semidirect
66. C₄×D₄×C₆₇	C₂²	semidirect
67. C₄×D₄×C₆₇	C₂²	semidirect
68. C₄×D₄×C₆₇	C₂²	semidirect
69. C₄×D₄×C₆₇	C₂²	semidirect
70. C₄×D₄×C₆₇	C₂²	semidirect
71. C₄×D₄×C₆₇	C₂²	semidirect
72. C₄×D₄×C₆₇	C₂²	semidirect
73. C₄×D₄×C₆₇	C₂²	semidirect
74. #117×C₆₇	C₂²	semidirect
75. #117×C₆₇	C₂²	semidirect
76. #117×C₆₇	C₂²	semidirect
77. C₂×C₄⋊C₄×C₆₇	C₂²	semidirect
78. C₂×C₄⋊C₄×C₆₇	C₂²	semidirect
79. C₂×C₂²⋊C₄×C₆₇	C₂²	semidirect
80. C₂×C₂²⋊C₄×C₆₇	C₂²	semidirect
81. C₂×C₂²⋊C₄×C₆₇	C₂²	semidirect
82. C₂×C₄²×C₆₇	C₂²	semidirect
181. D₄×C₆₇	#41	semidirect
244. #3519	C₆₇	direct (N × H)
245. C₆₇	#3519	direct (N × H)
246. #555	C₂×C₆₇	semidirect
247. #554	C₂×C₆₇	semidirect
248. #553	C₂×C₆₇	semidirect
249. #553	C₂×C₆₇	semidirect
250. #550	C₂×C₆₇	semidirect
251. #546	C₂×C₆₇	semidirect
252. #546	C₂×C₆₇	semidirect
253. D₄²	C₂×C₆₇	semidirect
254. D₄²	C₂×C₆₇	semidirect
255. #540	C₂×C₆₇	semidirect
256. #538	C₂×C₆₇	semidirect
257. #532	C₂×C₆₇	semidirect
258. #529	C₂×C₆₇	semidirect
259. #525	C₂×C₆₇	semidirect
260. C₂×#121	C₂×C₆₇	semidirect
261. #518	C₂×C₆₇	semidirect
262. C₂×C₄×D₄	C₂×C₆₇	semidirect
263. C₂×#41	C₂²×C₆₇	semidirect
264. C₂×#41	C₂²×C₆₇	semidirect
267. C₂²×D₄	C₂²×C₆₇	semidirect
268. C₂²×D₄	C₂²×C₆₇	semidirect
269. C₂³×C₄	C₂²×C₆₇	semidirect
270. C₄⋊Q₈	C₂²×C₆₇	semidirect
271. #127	C₂²×C₆₇	semidirect
272. #127	C₂²×C₆₇	semidirect
273. #126	C₂²×C₆₇	semidirect
274. #126	C₂²×C₆₇	semidirect
275. #125	C₂²×C₆₇	semidirect
276. #125	C₂²×C₆₇	semidirect
277. #124	C₂²×C₆₇	semidirect
278. #124	C₂²×C₆₇	semidirect
279. #124	C₂²×C₆₇	semidirect
280. #123	C₂²×C₆₇	semidirect
281. #123	C₂²×C₆₇	semidirect
282. #123	C₂²×C₆₇	semidirect
283. #123	C₂²×C₆₇	semidirect
284. C₂²⋊Q₈	C₂²×C₆₇	semidirect
285. C₂²⋊Q₈	C₂²×C₆₇	semidirect
286. C₂²⋊Q₈	C₂²×C₆₇	semidirect
288. #121	C₂²×C₆₇	semidirect
289. #121	C₂²×C₆₇	semidirect
290. #121	C₂²×C₆₇	semidirect
291. #121	C₂²×C₆₇	semidirect
292. #121	C₂²×C₆₇	semidirect
293. #121	C₂²×C₆₇	semidirect
294. #121	C₂²×C₆₇	semidirect
295. #121	C₂²×C₆₇	semidirect
296. #121	C₂²×C₆₇	semidirect
297. #121	C₂²×C₆₇	semidirect
298. #121	C₂²×C₆₇	semidirect
299. #121	C₂²×C₆₇	semidirect
300. #120	C₂²×C₆₇	semidirect
301. #120	C₂²×C₆₇	semidirect
302. #120	C₂²×C₆₇	semidirect
303. C₄×Q₈	C₂²×C₆₇	semidirect
307. C₄×D₄	C₂²×C₆₇	semidirect
308. C₄×D₄	C₂²×C₆₇	semidirect
309. C₄×D₄	C₂²×C₆₇	semidirect
310. C₄×D₄	C₂²×C₆₇	semidirect
311. C₄×D₄	C₂²×C₆₇	semidirect
312. C₄×D₄	C₂²×C₆₇	semidirect
313. C₄×D₄	C₂²×C₆₇	semidirect
314. C₄×D₄	C₂²×C₆₇	semidirect
315. C₄×D₄	C₂²×C₆₇	semidirect
316. C₄×D₄	C₂²×C₆₇	semidirect
317. C₄×D₄	C₂²×C₆₇	semidirect
318. C₄×D₄	C₂²×C₆₇	semidirect
319. #117	C₂²×C₆₇	semidirect
320. #117	C₂²×C₆₇	semidirect
321. #117	C₂²×C₆₇	semidirect
322. C₂×C₄⋊C₄	C₂²×C₆₇	semidirect
323. C₂×C₄⋊C₄	C₂²×C₆₇	semidirect
324. C₂×C₂²⋊C₄	C₂²×C₆₇	semidirect
325. C₂×C₂²⋊C₄	C₂²×C₆₇	semidirect
326. C₂×C₂²⋊C₄	C₂²×C₆₇	semidirect
327. C₂×C₄²	C₂²×C₆₇	semidirect
426. D₄	#41×C₆₇	semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
#555×C₆₇	2
#554×C₆₇	2
#553×C₆₇	2
#550×C₆₇	2
#546×C₆₇	2
D₄²×C₆₇	2
#540×C₆₇	2
#538×C₆₇	2
#532×C₆₇	2
#529×C₆₇	2
#525×C₆₇	2
C₂×#121×C₆₇	2
#518×C₆₇	2
C₂×C₄×D₄×C₆₇	2
#3519	67

Subgroups — C67×#3519 #22557950

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₆₇×#3519 #22557950