Subgroups — C₆₇×#34624 #30713875

Order 17152.

2626 conjugacy classes of subgroups.

The subgroup lattice has 624 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#532×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×#529×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
3	C₆₇×#2662	C₂	1	4	yes	maximal, complemented
4	C₆₇×#2661	C₂	1	4	yes	maximal, complemented
5	C₆₇×#2660	C₂	1	1	yes	maximal, characteristic, complemented
6	C₆₇×#2657	C₂	1	1	yes	maximal, characteristic, complemented
7	C₆₇×#2641	C₂	1	1	yes	maximal, characteristic, complemented
8	C₆₇×#2640	C₂	1	1	yes	maximal, characteristic, complemented
9	C₆₇×#2620	C₂	1	2	yes	maximal, complemented
10	C₆₇×#2619	C₂	1	2	yes	maximal, complemented
11	C₆₇×#2392	C₂	1	4	yes	maximal, complemented
12	C₆₇×#2388	C₂	1	4	yes	maximal, complemented
13	C₄×#127×C₆₇	C₂	1	2	yes	maximal, complemented
14	C₄×#124×C₆₇	C₂	1	2	yes	maximal, complemented
15	C₄×#117×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
16	C₂²×#41×C₆₇	C₂²	1	2	yes	—
17	#532×C₆₇	—	2	8	no	—
18	C₂×C₄⋊Q₈×C₆₇	C₂²	1	1	yes	characteristic
19	C₂×#127×C₆₇	C₂²	1	2	yes	—
20	C₂×#127×C₆₇	C₂²	1	1	yes	characteristic
21	#529×C₆₇	—	2	8	no	—
22	C₂×#126×C₆₇	C₂²	1	2	yes	—
23	C₂×#125×C₆₇	C₂²	1	1	yes	characteristic
24	C₂×#124×C₆₇	C₂²	1	2	yes	—
25	C₂×#124×C₆₇	C₂²	1	1	yes	characteristic
26	C₂×#124×C₆₇	C₂²	1	2	yes	—
27	C₂×#123×C₆₇	C₂²	1	2	yes	—
28	C₂×C₂²⋊Q₈×C₆₇	C₂²	1	1	yes	characteristic
29	C₂×#121×C₆₇	C₂²	1	1	yes	characteristic
30	C₂×#121×C₆₇	C₂²	1	2	yes	—
31	C₂×#121×C₆₇	C₂²	1	2	yes	—
32	C₂×C₄×Q₈×C₆₇	C₂²	1	1	yes	characteristic
33	C₂×C₄×D₄×C₆₇	C₂²	1	2	yes	—
34	C₂×C₄×D₄×C₆₇	C₂²	1	1	yes	characteristic
35	C₂×C₄×D₄×C₆₇	C₂²	1	2	yes	—
36	C₂×C₄×D₄×C₆₇	C₂²	1	2	yes	—
37	C₂×C₄×D₄×C₆₇	C₂²	1	8	yes	—
38	C₂×#117×C₆₇	C₂²	1	2	yes	—
39	C₂×#117×C₆₇	C₂²	1	2	yes	—
40	C₂²×C₄²×C₆₇	C₂²	1	1	yes	characteristic
41	#398×C₆₇	C₂²	1	4	yes	—
42	#397×C₆₇	C₂²	1	4	yes	—
43	#396×C₆₇	C₂²	1	8	yes	—
44	#395×C₆₇	C₂²	1	4	yes	—
45	#394×C₆₇	C₂²	1	8	yes	—
46	#394×C₆₇	C₂²	1	8	yes	—
47	#394×C₆₇	C₂²	1	4	yes	—
48	#394×C₆₇	C₂²	1	4	yes	—
49	#393×C₆₇	C₂²	1	4	yes	—
50	#392×C₆₇	C₂²	1	8	yes	—
51	#392×C₆₇	C₂²	1	4	yes	—
52	#392×C₆₇	C₂²	1	4	yes	—
53	#391×C₆₇	C₂²	1	2	yes	—
54	#390×C₆₇	C₂²	1	2	yes	—
55	#390×C₆₇	C₂²	1	2	yes	—
56	#390×C₆₇	C₂²	1	2	yes	—
57	#390×C₆₇	C₂²	1	8	yes	—
58	#388×C₆₇	C₂²	1	8	yes	—
59	#384×C₆₇	C₂²	1	2	yes	—
60	#382×C₆₇	C₂²	1	2	yes	—
61	C₄×C₄⋊C₄×C₆₇	C₂²	1	2	yes	—
62	C₄×C₄⋊C₄×C₆₇	C₂²	1	2	yes	—
63	C₄×C₂²⋊C₄×C₆₇	C₂²	1	2	yes	—
64	C₄×C₂²⋊C₄×C₆₇	C₂²	1	2	yes	—
65	C₄×C₂²⋊C₄×C₆₇	C₂²	1	8	yes	—
66	#376×C₆₇	C₂²	1	1	yes	characteristic
67	#376×C₆₇	C₂²	1	1	yes	characteristic
68	C₄³×C₆₇	C₂²	1	2	yes	—
69	C₂×#41×C₆₇	—	2	8	no	—
70	C₂×#41×C₆₇	—	4	8	no	—
71	C₂²×Q₈×C₆₇	C₂³	1	1	yes	characteristic
72	C₂²×Q₈×C₆₇	C₂³	1	2	yes	—
73	C₂²×D₄×C₆₇	C₂³	1	1	yes	characteristic
74	C₂²×D₄×C₆₇	C₂³	1	2	yes	—
75	C₂²×D₄×C₆₇	C₂³	1	2	yes	—
76	C₂²×D₄×C₆₇	C₂³	1	2	yes	—
77	C₂²×D₄×C₆₇	C₂³	1	8	yes	—
78	C₂²×D₄×C₆₇	C₂³	1	4	yes	—
79	C₂²×D₄×C₆₇	C₂³	1	2	yes	—
80	C₂³×C₄×C₆₇	C₂³	1	2	yes	—
81	C₂³×C₄×C₆₇	C₂³	1	4	yes	—
82	C₂³×C₄×C₆₇	C₂³	1	1	yes	characteristic
83	C₂³×C₄×C₆₇	C₂³	1	1	yes	characteristic
84	C₂³×C₄×C₆₇	C₂³	1	1	yes	characteristic
85	C₄⋊Q₈×C₆₇	—	2	4	no	—
86	#127×C₆₇	—	2	4	no	—
87	#127×C₆₇	—	2	8	no	—
88	#126×C₆₇	—	2	8	no	—
89	#125×C₆₇	—	2	4	no	—
90	#124×C₆₇	—	2	4	no	—
91	#124×C₆₇	—	2	8	no	—
92	#124×C₆₇	—	2	8	no	—
93	#123×C₆₇	—	2	8	no	—
94	C₂²⋊Q₈×C₆₇	—	2	4	no	—
95	#121×C₆₇	—	2	4	no	—
96	#121×C₆₇	—	2	8	no	—
97	#121×C₆₇	—	2	8	no	—
98	C₄×Q₈×C₆₇	—	2	4	no	—
99	C₄×D₄×C₆₇	—	2	8	no	—
100	C₄×D₄×C₆₇	—	2	4	no	—
101	C₄×D₄×C₆₇	—	2	8	no	—
102	C₄×D₄×C₆₇	—	2	8	no	—
103	C₄×D₄×C₆₇	—	4	16	no	—
104	#117×C₆₇	—	2	8	no	—
105	#117×C₆₇	—	2	8	no	—
106	C₂×C₄⋊C₄×C₆₇	C₂³	1	2	yes	—
107	C₂×C₄⋊C₄×C₆₇	C₂³	1	2	yes	—
108	C₂×C₄⋊C₄×C₆₇	C₂³	1	4	yes	—
109	C₂×C₄⋊C₄×C₆₇	C₂³	1	2	yes	—
110	C₂×C₄⋊C₄×C₆₇	C₂³	1	2	yes	—
111	C₂×C₄⋊C₄×C₆₇	C₂³	1	1	yes	characteristic
112	C₂×C₄⋊C₄×C₆₇	C₂³	1	2	yes	—
113	C₂×C₄⋊C₄×C₆₇	C₂³	1	1	yes	characteristic
114	C₂×C₄⋊C₄×C₆₇	C₂³	1	2	yes	—
115	C₂×C₄⋊C₄×C₆₇	C₂³	1	8	yes	—
116	C₂×C₂²⋊C₄×C₆₇	C₂³	1	2	yes	—
117	C₂×C₂²⋊C₄×C₆₇	C₂³	1	2	yes	—
118	C₂×C₂²⋊C₄×C₆₇	C₂³	1	4	yes	—
119	C₂×C₂²⋊C₄×C₆₇	C₂³	1	4	yes	—
120	C₂×C₂²⋊C₄×C₆₇	C₂³	1	4	yes	—
121	C₂×C₂²⋊C₄×C₆₇	C₂³	1	2	yes	—
122	C₂×C₂²⋊C₄×C₆₇	C₂³	1	2	yes	—
123	C₂×C₂²⋊C₄×C₆₇	C₂³	1	2	yes	—
124	C₂×C₂²⋊C₄×C₆₇	C₂³	1	8	yes	—
125	C₂×C₂²⋊C₄×C₆₇	C₂³	1	8	yes	—
126	C₂×C₂²⋊C₄×C₆₇	C₂³	1	8	yes	—
127	C₂×C₂²⋊C₄×C₆₇	C₂³	1	8	yes	—
128	C₂×C₄²×C₆₇	D₄	1	8	yes	complemented
129	C₂×C₄²×C₆₇	C₂³	1	2	yes	—
130	C₂×C₄²×C₆₇	C₂³	1	2	yes	—
131	C₂×C₄²×C₆₇	C₂³	1	4	yes	—
132	C₂×C₄²×C₆₇	C₂³	1	4	yes	—
133	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
134	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
135	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
136	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
137	C₂×C₄²×C₆₇	C₂³	1	2	yes	—
138	#95×C₆₇	C₂³	1	2	yes	—
139	#95×C₆₇	C₂³	1	2	yes	—
140	#95×C₆₇	C₂³	1	2	yes	—
141	#95×C₆₇	C₂³	1	2	yes	—
142	#95×C₆₇	C₂³	1	8	yes	—
143	#95×C₆₇	C₂³	1	8	yes	—
144	C₂⁴×C₆₇	C₂⁴	1	2	yes	—
145	C₂⁴×C₆₇	C₂⁴	1	4	yes	—
146	C₂⁴×C₆₇	C₂⁴	1	1	yes	characteristic
147	#41×C₆₇	—	4	32	no	—
148	C₂×Q₈×C₆₇	—	2	2	no	—
149	C₂×Q₈×C₆₇	—	2	4	no	—
150	C₂×Q₈×C₆₇	—	2	4	no	—
151	C₂×Q₈×C₆₇	—	4	4	no	—
152	C₂×D₄×C₆₇	—	2	4	no	—
153	C₂×D₄×C₆₇	—	2	8	no	—
154	C₂×D₄×C₆₇	—	2	2	no	—
155	C₂×D₄×C₆₇	—	2	4	no	—
156	C₂×D₄×C₆₇	—	2	4	no	—
157	C₂×D₄×C₆₇	—	2	4	no	—
158	C₂×D₄×C₆₇	—	2	4	no	—
159	C₂×D₄×C₆₇	—	2	16	no	—
160	C₂×D₄×C₆₇	—	2	4	no	—
161	C₂×D₄×C₆₇	—	2	4	no	—
162	C₂×D₄×C₆₇	—	4	8	no	—
163	C₂×D₄×C₆₇	—	4	8	no	—
164	C₂×D₄×C₆₇	—	4	4	no	—
165	C₂×D₄×C₆₇	—	4	4	no	—
166	C₂×D₄×C₆₇	—	4	4	no	—
167	C₂×D₄×C₆₇	—	4	8	no	—
168	C₂²×C₄×C₆₇	—	2	4	no	—
169	C₂²×C₄×C₆₇	—	2	4	no	—
170	C₂²×C₄×C₆₇	—	2	4	no	—
171	C₂²×C₄×C₆₇	—	2	8	no	—
172	C₂²×C₄×C₆₇	—	2	8	no	—
173	C₂²×C₄×C₆₇	—	2	16	no	—
174	C₂²×C₄×C₆₇	—	4	2	no	—
175	C₂²×C₄×C₆₇	C₂×D₄	1	4	yes	—
176	C₂²×C₄×C₆₇	C₂×D₄	1	4	yes	—
177	C₂²×C₄×C₆₇	C₂×D₄	1	4	yes	—
178	C₂²×C₄×C₆₇	C₂⁴	1	4	yes	—
179	C₂²×C₄×C₆₇	C₂⁴	1	2	yes	—
180	C₂²×C₄×C₆₇	C₂⁴	1	2	yes	—
181	C₂²×C₄×C₆₇	C₂⁴	1	2	yes	—
182	C₂²×C₄×C₆₇	C₂⁴	1	4	yes	—
183	C₂²×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
184	C₂²×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
185	C₂²×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
186	C₂²×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
187	C₂²×C₄×C₆₇	C₂⁴	1	4	yes	—
188	C₂²×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
189	C₂²×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
190	C₄⋊C₄×C₆₇	—	2	4	no	—
191	C₄⋊C₄×C₆₇	—	2	4	no	—
192	C₄⋊C₄×C₆₇	—	2	4	no	—
193	C₄⋊C₄×C₆₇	—	2	4	no	—
194	C₄⋊C₄×C₆₇	—	2	2	no	—
195	C₄⋊C₄×C₆₇	—	2	4	no	—
196	C₄⋊C₄×C₆₇	—	2	2	no	—
197	C₄⋊C₄×C₆₇	—	2	4	no	—
198	C₄⋊C₄×C₆₇	—	2	8	no	—
199	C₄⋊C₄×C₆₇	—	4	8	no	—
200	C₂²⋊C₄×C₆₇	—	2	4	no	—
201	C₂²⋊C₄×C₆₇	—	2	4	no	—
202	C₂²⋊C₄×C₆₇	—	2	4	no	—
203	C₂²⋊C₄×C₆₇	—	2	8	no	—
204	C₂²⋊C₄×C₆₇	—	2	8	no	—
205	C₂²⋊C₄×C₆₇	—	2	4	no	—
206	C₂²⋊C₄×C₆₇	—	2	4	no	—
207	C₂²⋊C₄×C₆₇	—	2	8	no	—
208	C₂²⋊C₄×C₆₇	—	2	16	no	—
209	C₂²⋊C₄×C₆₇	—	4	8	no	—
210	C₂²⋊C₄×C₆₇	—	4	8	no	—
211	C₂²⋊C₄×C₆₇	—	4	8	no	—
212	C₄²×C₆₇	—	2	4	no	—
213	C₄²×C₆₇	—	2	4	no	—
214	C₄²×C₆₇	—	2	8	no	—
215	C₄²×C₆₇	—	4	4	no	—
216	C₄²×C₆₇	C₂×D₄	1	4	yes	—
217	C₄²×C₆₇	C₂×D₄	1	4	yes	—
218	C₄²×C₆₇	C₂×D₄	1	4	yes	—
219	C₄²×C₆₇	C₂×D₄	1	4	yes	—
220	C₄²×C₆₇	#41	1	8	yes	complemented
221	C₂³×C₆₇	—	2	2	no	—
222	C₂³×C₆₇	—	2	2	no	—
223	C₂³×C₆₇	—	2	2	no	—
224	C₂³×C₆₇	—	2	2	no	—
225	C₂³×C₆₇	—	2	4	no	—
226	C₂³×C₆₇	—	2	4	no	—
227	C₂³×C₆₇	—	2	4	no	—
228	C₂³×C₆₇	—	2	4	no	—
229	C₂³×C₆₇	—	2	4	no	—
230	C₂³×C₆₇	—	2	8	no	—
231	C₂³×C₆₇	—	2	8	no	—
232	C₂³×C₆₇	—	2	4	no	—
233	C₂³×C₆₇	C₂²×D₄	1	2	yes	—
234	C₂³×C₆₇	C₂⁵	1	1	yes	characteristic
235	Q₈×C₆₇	—	4	8	no	—
236	Q₈×C₆₇	—	4	4	no	—
237	D₄×C₆₇	—	4	8	no	—
238	D₄×C₆₇	—	4	32	no	—
239	D₄×C₆₇	—	4	8	no	—
240	D₄×C₆₇	—	4	8	no	—
241	D₄×C₆₇	—	4	16	no	—
242	D₄×C₆₇	—	4	4	no	—
243	D₄×C₆₇	—	4	8	no	—
244	C₂×C₄×C₆₇	—	2	2	no	—
245	C₂×C₄×C₆₇	—	2	4	no	—
246	C₂×C₄×C₆₇	—	2	2	no	—
247	C₂×C₄×C₆₇	—	2	2	no	—
248	C₂×C₄×C₆₇	—	2	4	no	—
249	C₂×C₄×C₆₇	—	2	2	no	—
250	C₂×C₄×C₆₇	—	2	8	no	—
251	C₂×C₄×C₆₇	—	2	8	no	—
252	C₂×C₄×C₆₇	—	2	2	no	—
253	C₂×C₄×C₆₇	—	2	2	no	—
254	C₂×C₄×C₆₇	—	2	2	no	—
255	C₂×C₄×C₆₇	—	2	4	no	—
256	C₂×C₄×C₆₇	—	2	4	no	—
257	C₂×C₄×C₆₇	—	2	4	no	—
258	C₂×C₄×C₆₇	—	2	4	no	—
259	C₂×C₄×C₆₇	—	2	4	no	—
260	C₂×C₄×C₆₇	—	2	8	no	—
261	C₂×C₄×C₆₇	—	2	4	no	—
262	C₂×C₄×C₆₇	—	2	4	no	—
263	C₂×C₄×C₆₇	—	4	4	no	—
264	C₂×C₄×C₆₇	—	4	8	no	—
265	C₂×C₄×C₆₇	—	4	16	no	—
266	C₂×C₄×C₆₇	C₂²×D₄	1	2	yes	—
267	C₂×C₄×C₆₇	C₂²×D₄	1	2	yes	—
268	C₂×C₄×C₆₇	C₂²×D₄	1	2	yes	—
269	C₂×C₄×C₆₇	C₂²×D₄	1	2	yes	—
270	C₂×C₄×C₆₇	C₂²×D₄	1	2	yes	—
271	C₂×C₄×C₆₇	C₂²×D₄	1	2	yes	—
272	C₂×C₄×C₆₇	C₂×#41	1	8	yes	—
273	C₂×C₄×C₆₇	C₂×#41	1	4	yes	—
274	C₂×C₄×C₆₇	H(5,𝔽₂)	1	4	yes	—
275	C₂²×C₆₇	—	2	2	no	—
276	C₂²×C₆₇	—	2	2	no	—
277	C₂²×C₆₇	—	2	2	no	—
278	C₂²×C₆₇	—	4	4	no	—
279	C₂²×C₆₇	—	4	2	no	—
280	C₂²×C₆₇	—	4	4	no	—
281	C₂²×C₆₇	—	4	4	no	—
282	C₂²×C₆₇	—	4	2	no	—
283	C₂²×C₆₇	—	4	4	no	—
284	C₂²×C₆₇	—	4	2	no	—
285	C₂²×C₆₇	—	4	8	no	—
286	C₂²×C₆₇	—	4	8	no	—
287	C₂²×C₆₇	—	4	8	no	—
288	C₂²×C₆₇	C₂³×D₄	1	1	yes	characteristic
289	C₂²×C₆₇	C₂²×#41	1	2	yes	—
290	C₂²×C₆₇	C₂×H(5,𝔽₂)	1	2	yes	—
291	C₂²×C₆₇	#585	1	2	yes	—
292	C₄×C₆₇	—	2	2	no	—
293	C₄×C₆₇	—	2	2	no	—
294	C₄×C₆₇	—	2	2	no	—
295	C₄×C₆₇	—	2	2	no	—
296	C₄×C₆₇	—	2	8	no	—
297	C₄×C₆₇	—	4	2	no	—
298	C₄×C₆₇	—	4	2	no	—
299	C₄×C₆₇	—	4	4	no	—
300	C₄×C₆₇	—	4	2	no	—
301	C₄×C₆₇	—	4	4	no	—
302	C₄×C₆₇	#532	1	4	yes	—
303	C₄×C₆₇	#535	1	4	yes	—
304	#34624	C₆₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
305	C₂×C₆₇	—	4	2	no	—
306	C₂×C₆₇	—	8	4	no	—
307	C₂×C₆₇	—	8	2	no	—
308	C₂×C₆₇	C₂×#532	1	1	yes	characteristic
309	C₂×C₆₇	C₂×#535	1	1	yes	characteristic
310	C₂×C₆₇	#3449	1	1	yes	characteristic
311	C₂×C₆₇	#3460	1	4	yes	—
312	C₂×#532	C₂×C₆₇	1	1	yes	characteristic, complemented
313	C₂×#529	C₂×C₆₇	1	1	yes	characteristic, complemented
314	#2662	C₂×C₆₇	1	4	yes	complemented
315	#2661	C₂×C₆₇	1	4	yes	complemented
316	#2660	C₂×C₆₇	1	1	yes	characteristic, complemented
317	#2657	C₂×C₆₇	1	1	yes	characteristic, complemented
318	#2641	C₂×C₆₇	1	1	yes	characteristic, complemented
319	#2640	C₂×C₆₇	1	1	yes	characteristic, complemented
320	#2620	C₂×C₆₇	1	2	yes	complemented
321	#2619	C₂×C₆₇	1	2	yes	complemented
322	#2392	C₂×C₆₇	1	4	yes	complemented
323	#2388	C₂×C₆₇	1	4	yes	complemented
324	C₄×#127	C₂×C₆₇	1	2	yes	complemented
325	C₄×#124	C₂×C₆₇	1	2	yes	complemented
326	C₄×#117	C₂×C₆₇	1	1	yes	characteristic, complemented
327	C₆₇	#34624	1	1	yes	characteristic, direct product factor, Sylow
328	C₂²×#41	C₂²×C₆₇	1	2	yes	—
329	#532	—	2	8	no	—
330	C₂×C₄⋊Q₈	C₂²×C₆₇	1	1	yes	characteristic
331	C₂×#127	C₂²×C₆₇	1	2	yes	—
332	C₂×#127	C₂²×C₆₇	1	1	yes	characteristic
333	#529	—	2	8	no	—
334	C₂×#126	C₂²×C₆₇	1	2	yes	—
335	C₂×#125	C₂²×C₆₇	1	1	yes	characteristic
336	C₂×#124	C₂²×C₆₇	1	2	yes	—
337	C₂×#124	C₂²×C₆₇	1	1	yes	characteristic
338	C₂×#124	C₂²×C₆₇	1	2	yes	—
339	C₂×#123	C₂²×C₆₇	1	2	yes	—
340	C₂×C₂²⋊Q₈	C₂²×C₆₇	1	1	yes	characteristic
341	C₂×#121	C₂²×C₆₇	1	1	yes	characteristic
342	C₂×#121	C₂²×C₆₇	1	2	yes	—
343	C₂×#121	C₂²×C₆₇	1	2	yes	—
344	C₂×C₄×Q₈	C₂²×C₆₇	1	1	yes	characteristic
345	C₂×C₄×D₄	C₂²×C₆₇	1	1	yes	characteristic
346	C₂×C₄×D₄	C₂²×C₆₇	1	2	yes	—
347	C₂×C₄×D₄	C₂²×C₆₇	1	2	yes	—
348	C₂×C₄×D₄	C₂²×C₆₇	1	2	yes	—
349	C₂×C₄×D₄	C₂²×C₆₇	1	8	yes	—
350	C₂×#117	C₂²×C₆₇	1	2	yes	—
351	C₂×#117	C₂²×C₆₇	1	2	yes	—
352	C₂²×C₄²	C₂²×C₆₇	1	1	yes	characteristic
353	#398	C₂²×C₆₇	1	4	yes	—
354	#397	C₂²×C₆₇	1	4	yes	—
355	#396	C₂²×C₆₇	1	8	yes	—
356	#395	C₂²×C₆₇	1	4	yes	—
357	#394	C₂²×C₆₇	1	8	yes	—
358	#394	C₂²×C₆₇	1	8	yes	—
359	#394	C₂²×C₆₇	1	4	yes	—
360	#394	C₂²×C₆₇	1	4	yes	—
361	#393	C₂²×C₆₇	1	4	yes	—
362	#392	C₂²×C₆₇	1	8	yes	—
363	#392	C₂²×C₆₇	1	4	yes	—
364	#392	C₂²×C₆₇	1	4	yes	—
365	#391	C₂²×C₆₇	1	2	yes	—
366	#390	C₂²×C₆₇	1	2	yes	—
367	#390	C₂²×C₆₇	1	2	yes	—
368	#390	C₂²×C₆₇	1	8	yes	—
369	#390	C₂²×C₆₇	1	2	yes	—
370	#388	C₂²×C₆₇	1	8	yes	—
371	#384	C₂²×C₆₇	1	2	yes	—
372	#382	C₂²×C₆₇	1	2	yes	—
373	C₄×C₄⋊C₄	C₂²×C₆₇	1	2	yes	—
374	C₄×C₄⋊C₄	C₂²×C₆₇	1	2	yes	—
375	C₄×C₂²⋊C₄	C₂²×C₆₇	1	2	yes	—
376	C₄×C₂²⋊C₄	C₂²×C₆₇	1	8	yes	—
377	C₄×C₂²⋊C₄	C₂²×C₆₇	1	2	yes	—
378	#376	C₂²×C₆₇	1	1	yes	characteristic
379	#376	C₂²×C₆₇	1	1	yes	characteristic
380	C₄³	C₂²×C₆₇	1	2	yes	—
381	C₂×#41	—	2	8	no	—
382	C₂×#41	—	4	8	no	—
383	C₂²×Q₈	C₂³×C₆₇	1	1	yes	characteristic
384	C₂²×Q₈	C₂³×C₆₇	1	2	yes	—
385	C₂²×D₄	C₂³×C₆₇	1	1	yes	characteristic
386	C₂²×D₄	C₂³×C₆₇	1	2	yes	—
387	C₂²×D₄	C₂³×C₆₇	1	2	yes	—
388	C₂²×D₄	C₂³×C₆₇	1	2	yes	—
389	C₂²×D₄	C₂³×C₆₇	1	8	yes	—
390	C₂²×D₄	C₂³×C₆₇	1	4	yes	—
391	C₂²×D₄	C₂³×C₆₇	1	2	yes	—
392	C₂³×C₄	C₂³×C₆₇	1	2	yes	—
393	C₂³×C₄	C₂³×C₆₇	1	4	yes	—
394	C₂³×C₄	C₂³×C₆₇	1	1	yes	characteristic
395	C₂³×C₄	C₂³×C₆₇	1	1	yes	characteristic
396	C₂³×C₄	C₂³×C₆₇	1	1	yes	characteristic
397	C₄⋊Q₈	—	2	4	no	—
398	#127	—	2	4	no	—
399	#127	—	2	8	no	—
400	#126	—	2	8	no	—
401	#125	—	2	4	no	—
402	#124	—	2	4	no	—
403	#124	—	2	8	no	—
404	#124	—	2	8	no	—
405	#123	—	2	8	no	—
406	C₂²⋊Q₈	—	2	4	no	—
407	#121	—	2	4	no	—
408	#121	—	2	8	no	—
409	#121	—	2	8	no	—
410	C₄×Q₈	—	2	4	no	—
411	C₄×D₄	—	2	8	no	—
412	C₄×D₄	—	2	4	no	—
413	C₄×D₄	—	2	8	no	—
414	C₄×D₄	—	2	8	no	—
415	C₄×D₄	—	4	16	no	—
416	#117	—	2	8	no	—
417	#117	—	2	8	no	—
418	C₂×C₄⋊C₄	C₂³×C₆₇	1	2	yes	—
419	C₂×C₄⋊C₄	C₂³×C₆₇	1	2	yes	—
420	C₂×C₄⋊C₄	C₂³×C₆₇	1	4	yes	—
421	C₂×C₄⋊C₄	C₂³×C₆₇	1	2	yes	—
422	C₂×C₄⋊C₄	C₂³×C₆₇	1	2	yes	—
423	C₂×C₄⋊C₄	C₂³×C₆₇	1	1	yes	characteristic
424	C₂×C₄⋊C₄	C₂³×C₆₇	1	2	yes	—
425	C₂×C₄⋊C₄	C₂³×C₆₇	1	1	yes	characteristic
426	C₂×C₄⋊C₄	C₂³×C₆₇	1	2	yes	—
427	C₂×C₄⋊C₄	C₂³×C₆₇	1	8	yes	—
428	C₂×C₂²⋊C₄	C₂³×C₆₇	1	2	yes	—
429	C₂×C₂²⋊C₄	C₂³×C₆₇	1	2	yes	—
430	C₂×C₂²⋊C₄	C₂³×C₆₇	1	4	yes	—
431	C₂×C₂²⋊C₄	C₂³×C₆₇	1	4	yes	—
432	C₂×C₂²⋊C₄	C₂³×C₆₇	1	4	yes	—
433	C₂×C₂²⋊C₄	C₂³×C₆₇	1	2	yes	—
434	C₂×C₂²⋊C₄	C₂³×C₆₇	1	2	yes	—
435	C₂×C₂²⋊C₄	C₂³×C₆₇	1	2	yes	—
436	C₂×C₂²⋊C₄	C₂³×C₆₇	1	8	yes	—
437	C₂×C₂²⋊C₄	C₂³×C₆₇	1	8	yes	—
438	C₂×C₂²⋊C₄	C₂³×C₆₇	1	8	yes	—
439	C₂×C₂²⋊C₄	C₂³×C₆₇	1	8	yes	—
440	C₂×C₄²	D₄×C₆₇	1	8	yes	complemented
441	C₂×C₄²	C₂³×C₆₇	1	2	yes	—
442	C₂×C₄²	C₂³×C₆₇	1	2	yes	—
443	C₂×C₄²	C₂³×C₆₇	1	4	yes	—
444	C₂×C₄²	C₂³×C₆₇	1	4	yes	—
445	C₂×C₄²	C₂³×C₆₇	1	1	yes	characteristic
446	C₂×C₄²	C₂³×C₆₇	1	1	yes	characteristic
447	C₂×C₄²	C₂³×C₆₇	1	1	yes	characteristic
448	C₂×C₄²	C₂³×C₆₇	1	1	yes	characteristic
449	C₂×C₄²	C₂³×C₆₇	1	2	yes	—
450	#95	C₂³×C₆₇	1	2	yes	—
451	#95	C₂³×C₆₇	1	2	yes	—
452	#95	C₂³×C₆₇	1	2	yes	—
453	#95	C₂³×C₆₇	1	2	yes	—
454	#95	C₂³×C₆₇	1	8	yes	—
455	#95	C₂³×C₆₇	1	8	yes	—
456	C₂⁴	C₂⁴×C₆₇	1	2	yes	—
457	C₂⁴	C₂⁴×C₆₇	1	4	yes	—
458	C₂⁴	C₂⁴×C₆₇	1	1	yes	characteristic
459	#41	—	4	32	no	—
460	C₂×Q₈	—	2	2	no	—
461	C₂×Q₈	—	2	4	no	—
462	C₂×Q₈	—	2	4	no	—
463	C₂×Q₈	—	4	4	no	—
464	C₂×D₄	—	2	4	no	—
465	C₂×D₄	—	2	8	no	—
466	C₂×D₄	—	2	2	no	—
467	C₂×D₄	—	2	4	no	—
468	C₂×D₄	—	2	4	no	—
469	C₂×D₄	—	2	4	no	—
470	C₂×D₄	—	2	4	no	—
471	C₂×D₄	—	2	16	no	—
472	C₂×D₄	—	2	4	no	—
473	C₂×D₄	—	2	4	no	—
474	C₂×D₄	—	4	8	no	—
475	C₂×D₄	—	4	4	no	—
476	C₂×D₄	—	4	8	no	—
477	C₂×D₄	—	4	8	no	—
478	C₂×D₄	—	4	4	no	—
479	C₂×D₄	—	4	4	no	—
480	C₂²×C₄	—	2	4	no	—
481	C₂²×C₄	—	2	4	no	—
482	C₂²×C₄	—	2	4	no	—
483	C₂²×C₄	—	2	8	no	—
484	C₂²×C₄	—	2	8	no	—
485	C₂²×C₄	—	2	16	no	—
486	C₂²×C₄	—	4	2	no	—
487	C₂²×C₄	C₂×D₄×C₆₇	1	4	yes	—
488	C₂²×C₄	C₂×D₄×C₆₇	1	4	yes	—
489	C₂²×C₄	C₂×D₄×C₆₇	1	4	yes	—
490	C₂²×C₄	C₂⁴×C₆₇	1	4	yes	—
491	C₂²×C₄	C₂⁴×C₆₇	1	2	yes	—
492	C₂²×C₄	C₂⁴×C₆₇	1	2	yes	—
493	C₂²×C₄	C₂⁴×C₆₇	1	4	yes	—
494	C₂²×C₄	C₂⁴×C₆₇	1	2	yes	—
495	C₂²×C₄	C₂⁴×C₆₇	1	1	yes	characteristic
496	C₂²×C₄	C₂⁴×C₆₇	1	1	yes	characteristic
497	C₂²×C₄	C₂⁴×C₆₇	1	1	yes	characteristic
498	C₂²×C₄	C₂⁴×C₆₇	1	1	yes	characteristic
499	C₂²×C₄	C₂⁴×C₆₇	1	4	yes	—
500	C₂²×C₄	C₂⁴×C₆₇	1	1	yes	characteristic
501	C₂²×C₄	C₂⁴×C₆₇	1	1	yes	characteristic
502	C₄⋊C₄	—	2	4	no	—
503	C₄⋊C₄	—	2	4	no	—
504	C₄⋊C₄	—	2	4	no	—
505	C₄⋊C₄	—	2	4	no	—
506	C₄⋊C₄	—	2	2	no	—
507	C₄⋊C₄	—	2	4	no	—
508	C₄⋊C₄	—	2	2	no	—
509	C₄⋊C₄	—	2	4	no	—
510	C₄⋊C₄	—	2	8	no	—
511	C₄⋊C₄	—	4	8	no	—
512	C₂²⋊C₄	—	2	4	no	—
513	C₂²⋊C₄	—	2	4	no	—
514	C₂²⋊C₄	—	2	4	no	—
515	C₂²⋊C₄	—	2	8	no	—
516	C₂²⋊C₄	—	2	8	no	—
517	C₂²⋊C₄	—	2	4	no	—
518	C₂²⋊C₄	—	2	4	no	—
519	C₂²⋊C₄	—	2	8	no	—
520	C₂²⋊C₄	—	2	16	no	—
521	C₂²⋊C₄	—	4	8	no	—
522	C₂²⋊C₄	—	4	8	no	—
523	C₂²⋊C₄	—	4	8	no	—
524	C₄²	—	2	4	no	—
525	C₄²	—	2	4	no	—
526	C₄²	—	2	8	no	—
527	C₄²	—	4	4	no	—
528	C₄²	C₂×D₄×C₆₇	1	4	yes	—
529	C₄²	C₂×D₄×C₆₇	1	4	yes	—
530	C₄²	C₂×D₄×C₆₇	1	4	yes	—
531	C₄²	C₂×D₄×C₆₇	1	4	yes	—
532	C₄²	#41×C₆₇	1	8	yes	complemented
533	C₂³	—	2	2	no	—
534	C₂³	—	2	2	no	—
535	C₂³	—	2	2	no	—
536	C₂³	—	2	2	no	—
537	C₂³	—	2	4	no	—
538	C₂³	—	2	4	no	—
539	C₂³	—	2	4	no	—
540	C₂³	—	2	4	no	—
541	C₂³	—	2	4	no	—
542	C₂³	—	2	8	no	—
543	C₂³	—	2	8	no	—
544	C₂³	—	2	4	no	—
545	C₂³	C₂²×D₄×C₆₇	1	2	yes	—
546	C₂³	C₂⁵×C₆₇	1	1	yes	characteristic
547	Q₈	—	4	8	no	—
548	Q₈	—	4	4	no	—
549	D₄	—	4	8	no	—
550	D₄	—	4	8	no	—
551	D₄	—	4	32	no	—
552	D₄	—	4	16	no	—
553	D₄	—	4	8	no	—
554	D₄	—	4	4	no	—
555	D₄	—	4	8	no	—
556	C₂×C₄	—	2	2	no	—
557	C₂×C₄	—	2	4	no	—
558	C₂×C₄	—	2	2	no	—
559	C₂×C₄	—	2	2	no	—
560	C₂×C₄	—	2	4	no	—
561	C₂×C₄	—	2	2	no	—
562	C₂×C₄	—	2	8	no	—
563	C₂×C₄	—	2	8	no	—
564	C₂×C₄	—	2	2	no	—
565	C₂×C₄	—	2	2	no	—
566	C₂×C₄	—	2	2	no	—
567	C₂×C₄	—	2	4	no	—
568	C₂×C₄	—	2	4	no	—
569	C₂×C₄	—	2	4	no	—
570	C₂×C₄	—	2	4	no	—
571	C₂×C₄	—	2	4	no	—
572	C₂×C₄	—	2	8	no	—
573	C₂×C₄	—	2	4	no	—
574	C₂×C₄	—	2	4	no	—
575	C₂×C₄	—	4	4	no	—
576	C₂×C₄	—	4	16	no	—
577	C₂×C₄	—	4	8	no	—
578	C₂×C₄	C₂²×D₄×C₆₇	1	2	yes	—
579	C₂×C₄	C₂²×D₄×C₆₇	1	2	yes	—
580	C₂×C₄	C₂²×D₄×C₆₇	1	2	yes	—
581	C₂×C₄	C₂²×D₄×C₆₇	1	2	yes	—
582	C₂×C₄	C₂²×D₄×C₆₇	1	2	yes	—
583	C₂×C₄	C₂²×D₄×C₆₇	1	2	yes	—
584	C₂×C₄	C₂×#41×C₆₇	1	8	yes	—
585	C₂×C₄	C₂×#41×C₆₇	1	4	yes	—
586	C₂×C₄	H(5,𝔽₂)×C₆₇	1	4	yes	—
587	C₂²	—	2	2	no	—
588	C₂²	—	2	2	no	—
589	C₂²	—	2	2	no	—
590	C₂²	—	4	4	no	—
591	C₂²	—	4	2	no	—
592	C₂²	—	4	4	no	—
593	C₂²	—	4	4	no	—
594	C₂²	—	4	2	no	—
595	C₂²	—	4	4	no	—
596	C₂²	—	4	2	no	—
597	C₂²	—	4	8	no	—
598	C₂²	—	4	8	no	—
599	C₂²	—	4	8	no	—
600	C₂²	C₂³×D₄×C₆₇	1	1	yes	characteristic
601	C₂²	C₂²×#41×C₆₇	1	2	yes	—
602	C₂²	C₂×H(5,𝔽₂)×C₆₇	1	2	yes	—
603	C₂²	#585×C₆₇	1	2	yes	—
604	C₄	—	2	2	no	—
605	C₄	—	2	2	no	—
606	C₄	—	2	2	no	—
607	C₄	—	2	2	no	—
608	C₄	—	2	8	no	—
609	C₄	—	4	2	no	—
610	C₄	—	4	4	no	—
611	C₄	—	4	2	no	—
612	C₄	—	4	2	no	—
613	C₄	—	4	4	no	—
614	C₄	#532×C₆₇	1	4	yes	—
615	C₄	#535×C₆₇	1	4	yes	—
616	C₂	—	4	2	no	—
617	C₂	—	8	4	no	—
618	C₂	—	8	2	no	—
619	C₂	C₂×#532×C₆₇	1	1	yes	characteristic
620	C₂	C₂×#535×C₆₇	1	1	yes	characteristic
621	C₂	C₆₇×#3449	1	1	yes	characteristic
622	C₂	C₆₇×#3460	1	4	yes	—

	subgroup	G/H
center	C₂²×C₄×C₆₇	C₂⁴
commutator, Frattini, cosocle	C₂³	C₂⁵×C₆₇
socle	C₂³×C₆₇	C₂⁵
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂²×C₄×C₆₇ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₂×#532×C₆₇	C₂	semidirect
2. C₂×#529×C₆₇	C₂	semidirect
3. C₆₇×#2662	C₂	semidirect
4. C₆₇×#2661	C₂	semidirect
5. C₆₇×#2660	C₂	semidirect
6. C₆₇×#2657	C₂	semidirect
7. C₆₇×#2641	C₂	semidirect
8. C₆₇×#2640	C₂	semidirect
9. C₆₇×#2620	C₂	semidirect
10. C₆₇×#2619	C₂	semidirect
11. C₆₇×#2392	C₂	semidirect
12. C₆₇×#2388	C₂	semidirect
13. C₄×#127×C₆₇	C₂	semidirect
14. C₄×#124×C₆₇	C₂	semidirect
15. C₄×#117×C₆₇	C₂	semidirect
128. C₂×C₄²×C₆₇	D₄	semidirect
220. C₄²×C₆₇	#41	semidirect
304. #34624	C₆₇	direct (N × H)
312. C₂×#532	C₂×C₆₇	semidirect
313. C₂×#529	C₂×C₆₇	semidirect
314. #2662	C₂×C₆₇	semidirect
315. #2661	C₂×C₆₇	semidirect
316. #2660	C₂×C₆₇	semidirect
317. #2657	C₂×C₆₇	semidirect
318. #2641	C₂×C₆₇	semidirect
319. #2640	C₂×C₆₇	semidirect
320. #2620	C₂×C₆₇	semidirect
321. #2619	C₂×C₆₇	semidirect
322. #2392	C₂×C₆₇	semidirect
323. #2388	C₂×C₆₇	semidirect
324. C₄×#127	C₂×C₆₇	semidirect
325. C₄×#124	C₂×C₆₇	semidirect
326. C₄×#117	C₂×C₆₇	semidirect
327. C₆₇	#34624	direct (N × H)
440. C₂×C₄²	D₄×C₆₇	semidirect
532. C₄²	#41×C₆₇	semidirect

Maximal subgroups

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

subgroup	index
C₂×#532×C₆₇	2
C₂×#529×C₆₇	2
C₆₇×#2662	2
C₆₇×#2661	2
C₆₇×#2660	2
C₆₇×#2657	2
C₆₇×#2641	2
C₆₇×#2640	2
C₆₇×#2620	2
C₆₇×#2619	2
C₆₇×#2392	2
C₆₇×#2388	2
C₄×#127×C₆₇	2
C₄×#124×C₆₇	2
C₄×#117×C₆₇	2
#34624	67

Subgroups — C67×#34624 #30713875

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₆₇×#34624 #30713875