Subgroups — C₂×C₆₇×#3329 #30740400

Order 17152.

2146 conjugacy classes of subgroups.

The subgroup lattice has 560 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#555×C₆₇	C₂	1	1	yes	maximal, characteristic
2	C₂×D₄×Q₈×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
3	C₆₇×#3329	C₂	1	16	yes	maximal, direct product factor
4	C₂×#487×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂×#485×C₆₇	C₂	1	2	yes	maximal, complemented
6	C₂×#477×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂×#468×C₆₇	C₂	1	2	yes	maximal, complemented
8	C₂×#461×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂×#448×C₆₇	C₂	1	2	yes	maximal, complemented
10	C₂×C₄×SD₁₆×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
11	C₂×#436×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×#419×C₆₇	C₂	1	2	yes	maximal, complemented
13	C₂³×Q₈×C₆₇	C₂²	1	2	yes	—
14	C₂²×SD₁₆×C₆₇	C₂²	1	1	yes	characteristic
15	C₂²×SD₁₆×C₆₇	C₂²	1	2	yes	—
16	C₂²×M₁₆×C₆₇	C₂²	1	2	yes	—
17	#555×C₆₇	—	2	4	no	—
18	#555×C₆₇	C₂²	1	8	yes	—
19	D₄×Q₈×C₆₇	—	2	4	no	—
20	D₄×Q₈×C₆₇	C₂²	1	8	yes	complemented
21	C₂×C₄⋊Q₈×C₆₇	—	2	1	no	—
22	C₂×C₄⋊Q₈×C₆₇	C₂²	1	1	yes	characteristic
23	C₂×C₄⋊Q₈×C₆₇	C₂²	1	1	yes	characteristic
24	C₂×#124×C₆₇	—	2	2	no	—
25	C₂×C₂²⋊Q₈×C₆₇	—	2	2	no	—
26	C₂×C₂²⋊Q₈×C₆₇	C₂²	1	2	yes	—
27	C₂×#121×C₆₇	—	2	1	no	—
28	C₂×#121×C₆₇	C₂²	1	2	yes	—
29	C₂×C₄×Q₈×C₆₇	C₂²	1	1	yes	characteristic
30	C₂×C₄×D₄×C₆₇	—	2	1	no	—
31	C₂×C₄×D₄×C₆₇	C₂²	1	1	yes	characteristic
32	C₂×C₄×D₄×C₆₇	C₂²	1	1	yes	characteristic
33	C₂×#117×C₆₇	—	2	1	no	—
34	C₂×#117×C₆₇	C₂²	1	2	yes	—
35	#487×C₆₇	C₂²	1	8	yes	complemented
36	#485×C₆₇	C₂²	1	16	yes	complemented
37	#477×C₆₇	C₂²	1	8	yes	complemented
38	#468×C₆₇	C₂²	1	16	yes	complemented
39	#461×C₆₇	C₂²	1	8	yes	complemented
40	#448×C₆₇	C₂²	1	16	yes	complemented
41	C₄×SD₁₆×C₆₇	C₂²	1	8	yes	complemented
42	#436×C₆₇	C₂²	1	8	yes	complemented
43	C₂×#107×C₆₇	C₂²	1	2	yes	—
44	C₂×#106×C₆₇	C₂²	1	1	yes	characteristic
45	C₂×C₄⋊C₈×C₆₇	C₂²	1	1	yes	characteristic
46	#419×C₆₇	C₂²	1	16	yes	complemented
47	C₂×#103×C₆₇	C₂²	1	2	yes	—
48	C₂×#103×C₆₇	C₂²	1	2	yes	—
49	C₂×#103×C₆₇	C₂²	1	2	yes	—
50	C₂×#103×C₆₇	C₂²	1	2	yes	—
51	C₂×#103×C₆₇	C₂²	1	1	yes	characteristic
52	C₂×#102×C₆₇	C₂²	1	1	yes	characteristic
53	C₂×C₂²⋊C₈×C₆₇	C₂²	1	2	yes	—
54	C₂×C₄×C₈×C₆₇	C₂²	1	1	yes	characteristic
55	C₂²×Q₈×C₆₇	—	2	2	no	—
56	C₂²×Q₈×C₆₇	—	2	4	no	—
57	C₂²×Q₈×C₆₇	—	2	1	no	—
58	C₂²×Q₈×C₆₇	—	2	8	no	—
59	C₂²×Q₈×C₆₇	—	4	4	no	—
60	C₂²×Q₈×C₆₇	C₂³	1	2	yes	—
61	C₂²×Q₈×C₆₇	C₂³	1	8	yes	—
62	C₂²×Q₈×C₆₇	C₂³	1	1	yes	characteristic
63	C₂²×D₄×C₆₇	—	2	2	no	—
64	C₂²×D₄×C₆₇	D₄	1	1	yes	characteristic
65	C₂²×D₄×C₆₇	C₂³	1	1	yes	characteristic
66	C₂³×C₄×C₆₇	—	2	2	no	—
67	C₂³×C₄×C₆₇	—	2	1	no	—
68	C₂³×C₄×C₆₇	C₂³	1	2	yes	—
69	C₂×SD₁₆×C₆₇	—	2	4	no	—
70	C₂×SD₁₆×C₆₇	—	4	4	no	—
71	C₂×SD₁₆×C₆₇	C₂³	1	4	yes	—
72	C₂×SD₁₆×C₆₇	C₂³	1	8	yes	—
73	C₂×M₁₆×C₆₇	—	4	4	no	—
74	C₂×M₁₆×C₆₇	C₂³	1	8	yes	—
75	C₂²×C₈×C₆₇	C₂³	1	1	yes	characteristic
76	C₂²×C₈×C₆₇	C₂³	1	2	yes	—
77	C₂²×C₈×C₆₇	C₂³	1	1	yes	characteristic
78	C₄⋊Q₈×C₆₇	—	2	8	no	—
79	C₄⋊Q₈×C₆₇	—	2	2	no	—
80	C₄⋊Q₈×C₆₇	—	2	2	no	—
81	C₄⋊Q₈×C₆₇	C₂³	1	4	yes	—
82	C₄⋊Q₈×C₆₇	C₂³	1	4	yes	—
83	#124×C₆₇	—	2	16	no	—
84	C₂²⋊Q₈×C₆₇	—	2	4	no	—
85	C₂²⋊Q₈×C₆₇	—	2	16	no	—
86	C₂²⋊Q₈×C₆₇	C₂³	1	8	yes	—
87	#121×C₆₇	—	2	8	no	—
88	#121×C₆₇	—	2	4	no	—
89	#121×C₆₇	C₂³	1	8	yes	—
90	C₄×Q₈×C₆₇	—	2	2	no	—
91	C₄×Q₈×C₆₇	C₂³	1	4	yes	—
92	C₄×D₄×C₆₇	—	2	8	no	—
93	C₄×D₄×C₆₇	—	2	2	no	—
94	C₄×D₄×C₆₇	D₄	1	4	yes	—
95	C₄×D₄×C₆₇	C₂³	1	4	yes	—
96	C₄×D₄×C₆₇	C₂³	1	4	yes	—
97	#117×C₆₇	—	2	4	no	—
98	#117×C₆₇	—	2	8	no	—
99	#117×C₆₇	C₂³	1	8	yes	—
100	C₂×C₄⋊C₄×C₆₇	—	2	1	no	—
101	C₂×C₄⋊C₄×C₆₇	—	2	1	no	—
102	C₂×C₄⋊C₄×C₆₇	—	2	1	no	—
103	C₂×C₄⋊C₄×C₆₇	—	2	1	no	—
104	C₂×C₄⋊C₄×C₆₇	—	2	2	no	—
105	C₂×C₄⋊C₄×C₆₇	D₄	1	1	yes	characteristic
106	C₂×C₄⋊C₄×C₆₇	C₂³	1	1	yes	characteristic
107	C₂×C₄⋊C₄×C₆₇	C₂³	1	2	yes	—
108	C₂×C₄⋊C₄×C₆₇	C₂³	1	1	yes	characteristic
109	C₂×C₂²⋊C₄×C₆₇	—	2	2	no	—
110	C₂×C₂²⋊C₄×C₆₇	—	2	2	no	—
111	C₂×C₂²⋊C₄×C₆₇	—	2	2	no	—
112	C₂×C₂²⋊C₄×C₆₇	—	2	1	no	—
113	C₂×C₂²⋊C₄×C₆₇	D₄	1	2	yes	—
114	C₂×C₄²×C₆₇	—	2	2	no	—
115	C₂×C₄²×C₆₇	—	2	1	no	—
116	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
117	#107×C₆₇	C₂³	1	8	yes	—
118	#106×C₆₇	C₂³	1	4	yes	—
119	C₄⋊C₈×C₆₇	C₂³	1	4	yes	—
120	#103×C₆₇	C₂³	1	8	yes	—
121	#103×C₆₇	C₂³	1	8	yes	—
122	#103×C₆₇	C₂³	1	8	yes	—
123	#103×C₆₇	C₂³	1	8	yes	—
124	#103×C₆₇	C₂³	1	4	yes	—
125	#102×C₆₇	C₂³	1	4	yes	—
126	C₂²⋊C₈×C₆₇	C₂³	1	8	yes	—
127	C₄×C₈×C₆₇	C₂³	1	4	yes	—
128	C₂⁴×C₆₇	—	2	1	no	—
129	C₂⁴×C₆₇	C₂×D₄	1	2	yes	—
130	C₂×Q₈×C₆₇	—	2	1	no	—
131	C₂×Q₈×C₆₇	—	2	8	no	—
132	C₂×Q₈×C₆₇	—	2	2	no	—
133	C₂×Q₈×C₆₇	—	2	4	no	—
134	C₂×Q₈×C₆₇	—	2	2	no	—
135	C₂×Q₈×C₆₇	—	2	16	no	—
136	C₂×Q₈×C₆₇	—	2	4	no	—
137	C₂×Q₈×C₆₇	—	2	4	no	—
138	C₂×Q₈×C₆₇	—	4	8	no	—
139	C₂×Q₈×C₆₇	—	4	8	no	—
140	C₂×Q₈×C₆₇	—	4	16	no	—
141	C₂×Q₈×C₆₇	—	8	2	no	—
142	C₂×Q₈×C₆₇	#41	1	4	yes	—
143	C₂×Q₈×C₆₇	C₂⁴	1	4	yes	—
144	C₂×Q₈×C₆₇	C₂⁴	1	2	yes	—
145	C₂×D₄×C₆₇	—	2	8	no	—
146	C₂×D₄×C₆₇	—	2	4	no	—
147	C₂×D₄×C₆₇	—	2	1	no	—
148	C₂×D₄×C₆₇	—	2	2	no	—
149	C₂×D₄×C₆₇	—	2	2	no	—
150	C₂×D₄×C₆₇	—	8	4	no	—
151	C₂×D₄×C₆₇	C₂×D₄	1	4	yes	—
152	C₂×D₄×C₆₇	C₂⁴	1	2	yes	—
153	C₂²×C₄×C₆₇	—	2	1	no	—
154	C₂²×C₄×C₆₇	—	2	4	no	—
155	C₂²×C₄×C₆₇	—	2	4	no	—
156	C₂²×C₄×C₆₇	—	2	8	no	—
157	C₂²×C₄×C₆₇	—	2	1	no	—
158	C₂²×C₄×C₆₇	—	2	1	no	—
159	C₂²×C₄×C₆₇	—	2	2	no	—
160	C₂²×C₄×C₆₇	—	2	2	no	—
161	C₂²×C₄×C₆₇	—	2	4	no	—
162	C₂²×C₄×C₆₇	—	4	8	no	—
163	C₂²×C₄×C₆₇	—	4	4	no	—
164	C₂²×C₄×C₆₇	C₂×D₄	1	4	yes	—
165	C₂²×C₄×C₆₇	C₂×D₄	1	2	yes	—
166	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
167	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
168	C₂²×C₄×C₆₇	C₂⁴	1	4	yes	—
169	C₂²×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
170	SD₁₆×C₆₇	—	2	8	no	—
171	SD₁₆×C₆₇	—	4	8	no	—
172	M₁₆×C₆₇	—	4	8	no	—
173	C₂×C₈×C₆₇	—	2	2	no	—
174	C₂×C₈×C₆₇	—	2	2	no	—
175	C₂×C₈×C₆₇	—	2	4	no	—
176	C₂×C₈×C₆₇	C₂⁴	1	2	yes	—
177	C₂×C₈×C₆₇	C₂⁴	1	4	yes	—
178	C₂×C₈×C₆₇	C₂⁴	1	2	yes	—
179	C₄⋊C₄×C₆₇	—	2	4	no	—
180	C₄⋊C₄×C₆₇	—	2	4	no	—
181	C₄⋊C₄×C₆₇	—	2	4	no	—
182	C₄⋊C₄×C₆₇	—	2	4	no	—
183	C₄⋊C₄×C₆₇	—	2	8	no	—
184	C₄⋊C₄×C₆₇	—	2	1	no	—
185	C₄⋊C₄×C₆₇	—	2	2	no	—
186	C₄⋊C₄×C₆₇	—	2	1	no	—
187	C₄⋊C₄×C₆₇	C₂×D₄	1	4	yes	—
188	C₄⋊C₄×C₆₇	C₂⁴	1	2	yes	—
189	C₄⋊C₄×C₆₇	C₂⁴	1	2	yes	—
190	C₄⋊C₄×C₆₇	C₂⁴	1	4	yes	—
191	C₂²⋊C₄×C₆₇	—	2	8	no	—
192	C₂²⋊C₄×C₆₇	—	2	8	no	—
193	C₂²⋊C₄×C₆₇	—	2	8	no	—
194	C₂²⋊C₄×C₆₇	—	2	4	no	—
195	C₂²⋊C₄×C₆₇	C₂×D₄	1	8	yes	—
196	C₄²×C₆₇	—	2	8	no	—
197	C₄²×C₆₇	—	2	4	no	—
198	C₄²×C₆₇	C₂×D₄	1	2	yes	—
199	C₄²×C₆₇	C₂⁴	1	2	yes	—
200	C₂³×C₆₇	—	2	4	no	—
201	C₂³×C₆₇	—	2	4	no	—
202	C₂³×C₆₇	—	2	4	no	—
203	C₂³×C₆₇	—	2	2	no	—
204	C₂³×C₆₇	—	4	2	no	—
205	C₂³×C₆₇	—	4	2	no	—
206	C₂³×C₆₇	—	4	2	no	—
207	C₂³×C₆₇	C₂²×D₄	1	4	yes	—
208	C₂³×C₆₇	C₂²×D₄	1	1	yes	characteristic
209	Q₈×C₆₇	—	2	4	no	—
210	Q₈×C₆₇	—	2	2	no	—
211	Q₈×C₆₇	—	2	4	no	—
212	Q₈×C₆₇	—	2	4	no	—
213	Q₈×C₆₇	—	4	16	no	—
214	Q₈×C₆₇	—	4	4	no	—
215	Q₈×C₆₇	—	8	4	no	—
216	Q₈×C₆₇	C₂×#41	1	4	yes	—
217	D₄×C₆₇	—	2	8	no	—
218	D₄×C₆₇	—	2	2	no	—
219	D₄×C₆₇	—	2	2	no	—
220	D₄×C₆₇	—	4	2	no	—
221	D₄×C₆₇	—	8	8	no	—
222	C₂×C₄×C₆₇	—	2	4	no	—
223	C₂×C₄×C₆₇	—	2	4	no	—
224	C₂×C₄×C₆₇	—	2	2	no	—
225	C₂×C₄×C₆₇	—	2	2	no	—
226	C₂×C₄×C₆₇	—	2	8	no	—
227	C₂×C₄×C₆₇	—	2	4	no	—
228	C₂×C₄×C₆₇	—	2	4	no	—
229	C₂×C₄×C₆₇	—	2	2	no	—
230	C₂×C₄×C₆₇	—	2	2	no	—
231	C₂×C₄×C₆₇	—	2	4	no	—
232	C₂×C₄×C₆₇	—	2	4	no	—
233	C₂×C₄×C₆₇	—	4	16	no	—
234	C₂×C₄×C₆₇	—	4	4	no	—
235	C₂×C₄×C₆₇	—	4	2	no	—
236	C₂×C₄×C₆₇	—	4	2	no	—
237	C₂×C₄×C₆₇	—	4	4	no	—
238	C₂×C₄×C₆₇	—	4	8	no	—
239	C₂×C₄×C₆₇	#137	1	4	yes	—
240	C₂×C₄×C₆₇	C₂²×D₄	1	4	yes	—
241	C₂×C₄×C₆₇	C₂²×D₄	1	2	yes	—
242	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
243	C₂×C₄×C₆₇	C₂²×D₄	1	2	yes	—
244	C₂×C₄×C₆₇	C₂×#41	1	2	yes	—
245	C₂×C₄×C₆₇	H(5,𝔽₂)	1	2	yes	—
246	C₂×C₄×C₆₇	C₂⁵	1	1	yes	characteristic
247	C₈×C₆₇	—	2	2	no	—
248	C₈×C₆₇	—	2	2	no	—
249	C₈×C₆₇	—	2	4	no	—
250	C₂²×C₆₇	—	2	4	no	—
251	C₂²×C₆₇	—	2	4	no	—
252	C₂²×C₆₇	—	2	4	no	—
253	C₂²×C₆₇	—	4	8	no	—
254	C₂²×C₆₇	—	4	2	no	—
255	C₂²×C₆₇	—	4	2	no	—
256	C₂²×C₆₇	—	4	2	no	—
257	C₂²×C₆₇	—	8	4	no	—
258	C₂²×C₆₇	#546	1	2	yes	—
259	C₂²×C₆₇	C₂×#137	1	2	yes	—
260	C₂²×C₆₇	#577	1	2	yes	—
261	C₂²×C₆₇	C₂³×D₄	1	1	yes	characteristic
262	C₄×C₆₇	—	2	4	no	—
263	C₄×C₆₇	—	2	4	no	—
264	C₄×C₆₇	—	2	2	no	—
265	C₄×C₆₇	—	4	4	no	—
266	C₄×C₆₇	—	4	2	no	—
267	C₄×C₆₇	—	4	2	no	—
268	C₄×C₆₇	—	4	4	no	—
269	C₄×C₆₇	#546	1	2	yes	—
270	C₄×C₆₇	C₂×#137	1	4	yes	—
271	C₄×C₆₇	C₂²×#41	1	1	yes	characteristic
272	C₄×C₆₇	C₂×H(5,𝔽₂)	1	1	yes	characteristic
273	C₂×#3329	C₆₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
274	C₂×C₆₇	—	4	4	no	—
275	C₂×C₆₇	—	8	2	no	—
276	C₂×C₆₇	#3329	1	4	yes	direct product factor
277	C₂×C₆₇	C₂×#546	1	1	yes	characteristic
278	C₂×C₆₇	C₂²×#137	1	1	yes	characteristic
279	C₂×C₆₇	C₂×#577	1	1	yes	characteristic
280	C₂×#555	C₂×C₆₇	1	1	yes	characteristic
281	C₂×D₄×Q₈	C₂×C₆₇	1	1	yes	characteristic, complemented
282	#3329	C₂×C₆₇	1	16	yes	direct product factor
283	C₂×#487	C₂×C₆₇	1	1	yes	characteristic, complemented
284	C₂×#485	C₂×C₆₇	1	2	yes	complemented
285	C₂×#477	C₂×C₆₇	1	1	yes	characteristic, complemented
286	C₂×#468	C₂×C₆₇	1	2	yes	complemented
287	C₂×#461	C₂×C₆₇	1	1	yes	characteristic, complemented
288	C₂×#448	C₂×C₆₇	1	2	yes	complemented
289	C₂×C₄×SD₁₆	C₂×C₆₇	1	1	yes	characteristic, complemented
290	C₂×#436	C₂×C₆₇	1	1	yes	characteristic, complemented
291	C₂×#419	C₂×C₆₇	1	2	yes	complemented
292	C₆₇	C₂×#3329	1	1	yes	characteristic, direct product factor, Sylow
293	C₂³×Q₈	C₂²×C₆₇	1	2	yes	—
294	C₂²×SD₁₆	C₂²×C₆₇	1	1	yes	characteristic
295	C₂²×SD₁₆	C₂²×C₆₇	1	2	yes	—
296	C₂²×M₁₆	C₂²×C₆₇	1	2	yes	—
297	#555	—	2	4	no	—
298	#555	C₂²×C₆₇	1	8	yes	—
299	D₄×Q₈	—	2	4	no	—
300	D₄×Q₈	C₂²×C₆₇	1	8	yes	complemented
301	C₂×C₄⋊Q₈	—	2	1	no	—
302	C₂×C₄⋊Q₈	C₂²×C₆₇	1	1	yes	characteristic
303	C₂×C₄⋊Q₈	C₂²×C₆₇	1	1	yes	characteristic
304	C₂×#124	—	2	2	no	—
305	C₂×C₂²⋊Q₈	—	2	2	no	—
306	C₂×C₂²⋊Q₈	C₂²×C₆₇	1	2	yes	—
307	C₂×#121	—	2	1	no	—
308	C₂×#121	C₂²×C₆₇	1	2	yes	—
309	C₂×C₄×Q₈	C₂²×C₆₇	1	1	yes	characteristic
310	C₂×C₄×D₄	—	2	1	no	—
311	C₂×C₄×D₄	C₂²×C₆₇	1	1	yes	characteristic
312	C₂×C₄×D₄	C₂²×C₆₇	1	1	yes	characteristic
313	C₂×#117	—	2	1	no	—
314	C₂×#117	C₂²×C₆₇	1	2	yes	—
315	#487	C₂²×C₆₇	1	8	yes	complemented
316	#485	C₂²×C₆₇	1	16	yes	complemented
317	#477	C₂²×C₆₇	1	8	yes	complemented
318	#468	C₂²×C₆₇	1	16	yes	complemented
319	#461	C₂²×C₆₇	1	8	yes	complemented
320	#448	C₂²×C₆₇	1	16	yes	complemented
321	C₄×SD₁₆	C₂²×C₆₇	1	8	yes	complemented
322	#436	C₂²×C₆₇	1	8	yes	complemented
323	C₂×#107	C₂²×C₆₇	1	2	yes	—
324	C₂×#106	C₂²×C₆₇	1	1	yes	characteristic
325	C₂×C₄⋊C₈	C₂²×C₆₇	1	1	yes	characteristic
326	#419	C₂²×C₆₇	1	16	yes	complemented
327	C₂×#103	C₂²×C₆₇	1	2	yes	—
328	C₂×#103	C₂²×C₆₇	1	2	yes	—
329	C₂×#103	C₂²×C₆₇	1	2	yes	—
330	C₂×#103	C₂²×C₆₇	1	2	yes	—
331	C₂×#103	C₂²×C₆₇	1	1	yes	characteristic
332	C₂×#102	C₂²×C₆₇	1	1	yes	characteristic
333	C₂×C₂²⋊C₈	C₂²×C₆₇	1	2	yes	—
334	C₂×C₄×C₈	C₂²×C₆₇	1	1	yes	characteristic
335	C₂²×Q₈	—	2	2	no	—
336	C₂²×Q₈	—	2	4	no	—
337	C₂²×Q₈	—	2	1	no	—
338	C₂²×Q₈	—	2	8	no	—
339	C₂²×Q₈	—	4	4	no	—
340	C₂²×Q₈	C₂³×C₆₇	1	2	yes	—
341	C₂²×Q₈	C₂³×C₆₇	1	8	yes	—
342	C₂²×Q₈	C₂³×C₆₇	1	1	yes	characteristic
343	C₂²×D₄	—	2	2	no	—
344	C₂²×D₄	D₄×C₆₇	1	1	yes	characteristic
345	C₂²×D₄	C₂³×C₆₇	1	1	yes	characteristic
346	C₂³×C₄	—	2	2	no	—
347	C₂³×C₄	—	2	1	no	—
348	C₂³×C₄	C₂³×C₆₇	1	2	yes	—
349	C₂×SD₁₆	—	2	4	no	—
350	C₂×SD₁₆	—	4	4	no	—
351	C₂×SD₁₆	C₂³×C₆₇	1	4	yes	—
352	C₂×SD₁₆	C₂³×C₆₇	1	8	yes	—
353	C₂×M₁₆	—	4	4	no	—
354	C₂×M₁₆	C₂³×C₆₇	1	8	yes	—
355	C₂²×C₈	C₂³×C₆₇	1	1	yes	characteristic
356	C₂²×C₈	C₂³×C₆₇	1	2	yes	—
357	C₂²×C₈	C₂³×C₆₇	1	1	yes	characteristic
358	C₄⋊Q₈	—	2	8	no	—
359	C₄⋊Q₈	—	2	2	no	—
360	C₄⋊Q₈	—	2	2	no	—
361	C₄⋊Q₈	C₂³×C₆₇	1	4	yes	—
362	C₄⋊Q₈	C₂³×C₆₇	1	4	yes	—
363	#124	—	2	16	no	—
364	C₂²⋊Q₈	—	2	4	no	—
365	C₂²⋊Q₈	—	2	16	no	—
366	C₂²⋊Q₈	C₂³×C₆₇	1	8	yes	—
367	#121	—	2	8	no	—
368	#121	—	2	4	no	—
369	#121	C₂³×C₆₇	1	8	yes	—
370	C₄×Q₈	—	2	2	no	—
371	C₄×Q₈	C₂³×C₆₇	1	4	yes	—
372	C₄×D₄	—	2	8	no	—
373	C₄×D₄	—	2	2	no	—
374	C₄×D₄	D₄×C₆₇	1	4	yes	—
375	C₄×D₄	C₂³×C₆₇	1	4	yes	—
376	C₄×D₄	C₂³×C₆₇	1	4	yes	—
377	#117	—	2	4	no	—
378	#117	—	2	8	no	—
379	#117	C₂³×C₆₇	1	8	yes	—
380	C₂×C₄⋊C₄	—	2	1	no	—
381	C₂×C₄⋊C₄	—	2	1	no	—
382	C₂×C₄⋊C₄	—	2	1	no	—
383	C₂×C₄⋊C₄	—	2	1	no	—
384	C₂×C₄⋊C₄	—	2	2	no	—
385	C₂×C₄⋊C₄	D₄×C₆₇	1	1	yes	characteristic
386	C₂×C₄⋊C₄	C₂³×C₆₇	1	1	yes	characteristic
387	C₂×C₄⋊C₄	C₂³×C₆₇	1	2	yes	—
388	C₂×C₄⋊C₄	C₂³×C₆₇	1	1	yes	characteristic
389	C₂×C₂²⋊C₄	—	2	2	no	—
390	C₂×C₂²⋊C₄	—	2	2	no	—
391	C₂×C₂²⋊C₄	—	2	2	no	—
392	C₂×C₂²⋊C₄	—	2	1	no	—
393	C₂×C₂²⋊C₄	D₄×C₆₇	1	2	yes	—
394	C₂×C₄²	—	2	2	no	—
395	C₂×C₄²	—	2	1	no	—
396	C₂×C₄²	C₂³×C₆₇	1	1	yes	characteristic
397	#107	C₂³×C₆₇	1	8	yes	—
398	#106	C₂³×C₆₇	1	4	yes	—
399	C₄⋊C₈	C₂³×C₆₇	1	4	yes	—
400	#103	C₂³×C₆₇	1	4	yes	—
401	#103	C₂³×C₆₇	1	8	yes	—
402	#103	C₂³×C₆₇	1	8	yes	—
403	#103	C₂³×C₆₇	1	8	yes	—
404	#103	C₂³×C₆₇	1	8	yes	—
405	#102	C₂³×C₆₇	1	4	yes	—
406	C₂²⋊C₈	C₂³×C₆₇	1	8	yes	—
407	C₄×C₈	C₂³×C₆₇	1	4	yes	—
408	C₂⁴	—	2	1	no	—
409	C₂⁴	C₂×D₄×C₆₇	1	2	yes	—
410	C₂×Q₈	—	2	1	no	—
411	C₂×Q₈	—	2	8	no	—
412	C₂×Q₈	—	2	2	no	—
413	C₂×Q₈	—	2	4	no	—
414	C₂×Q₈	—	2	2	no	—
415	C₂×Q₈	—	2	16	no	—
416	C₂×Q₈	—	2	4	no	—
417	C₂×Q₈	—	2	4	no	—
418	C₂×Q₈	—	4	8	no	—
419	C₂×Q₈	—	4	8	no	—
420	C₂×Q₈	—	4	16	no	—
421	C₂×Q₈	—	8	2	no	—
422	C₂×Q₈	#41×C₆₇	1	4	yes	—
423	C₂×Q₈	C₂⁴×C₆₇	1	4	yes	—
424	C₂×Q₈	C₂⁴×C₆₇	1	2	yes	—
425	C₂×D₄	—	2	8	no	—
426	C₂×D₄	—	2	4	no	—
427	C₂×D₄	—	2	1	no	—
428	C₂×D₄	—	2	2	no	—
429	C₂×D₄	—	2	2	no	—
430	C₂×D₄	—	8	4	no	—
431	C₂×D₄	C₂×D₄×C₆₇	1	4	yes	—
432	C₂×D₄	C₂⁴×C₆₇	1	2	yes	—
433	C₂²×C₄	—	2	1	no	—
434	C₂²×C₄	—	2	4	no	—
435	C₂²×C₄	—	2	4	no	—
436	C₂²×C₄	—	2	1	no	—
437	C₂²×C₄	—	2	1	no	—
438	C₂²×C₄	—	2	2	no	—
439	C₂²×C₄	—	2	2	no	—
440	C₂²×C₄	—	2	8	no	—
441	C₂²×C₄	—	2	4	no	—
442	C₂²×C₄	—	4	8	no	—
443	C₂²×C₄	—	4	4	no	—
444	C₂²×C₄	C₂×D₄×C₆₇	1	2	yes	—
445	C₂²×C₄	C₂×D₄×C₆₇	1	1	yes	characteristic
446	C₂²×C₄	C₂×D₄×C₆₇	1	4	yes	—
447	C₂²×C₄	C₂×D₄×C₆₇	1	1	yes	characteristic
448	C₂²×C₄	C₂⁴×C₆₇	1	4	yes	—
449	C₂²×C₄	C₂⁴×C₆₇	1	1	yes	characteristic
450	SD₁₆	—	2	8	no	—
451	SD₁₆	—	4	8	no	—
452	M₁₆	—	4	8	no	—
453	C₂×C₈	—	2	2	no	—
454	C₂×C₈	—	2	2	no	—
455	C₂×C₈	—	2	4	no	—
456	C₂×C₈	C₂⁴×C₆₇	1	2	yes	—
457	C₂×C₈	C₂⁴×C₆₇	1	2	yes	—
458	C₂×C₈	C₂⁴×C₆₇	1	4	yes	—
459	C₄⋊C₄	—	2	4	no	—
460	C₄⋊C₄	—	2	4	no	—
461	C₄⋊C₄	—	2	4	no	—
462	C₄⋊C₄	—	2	4	no	—
463	C₄⋊C₄	—	2	8	no	—
464	C₄⋊C₄	—	2	1	no	—
465	C₄⋊C₄	—	2	2	no	—
466	C₄⋊C₄	—	2	1	no	—
467	C₄⋊C₄	C₂×D₄×C₆₇	1	4	yes	—
468	C₄⋊C₄	C₂⁴×C₆₇	1	2	yes	—
469	C₄⋊C₄	C₂⁴×C₆₇	1	2	yes	—
470	C₄⋊C₄	C₂⁴×C₆₇	1	4	yes	—
471	C₂²⋊C₄	—	2	8	no	—
472	C₂²⋊C₄	—	2	8	no	—
473	C₂²⋊C₄	—	2	8	no	—
474	C₂²⋊C₄	—	2	4	no	—
475	C₂²⋊C₄	C₂×D₄×C₆₇	1	8	yes	—
476	C₄²	—	2	8	no	—
477	C₄²	—	2	4	no	—
478	C₄²	C₂×D₄×C₆₇	1	2	yes	—
479	C₄²	C₂⁴×C₆₇	1	2	yes	—
480	C₂³	—	2	4	no	—
481	C₂³	—	2	4	no	—
482	C₂³	—	2	4	no	—
483	C₂³	—	2	2	no	—
484	C₂³	—	4	2	no	—
485	C₂³	—	4	2	no	—
486	C₂³	—	4	2	no	—
487	C₂³	C₂²×D₄×C₆₇	1	4	yes	—
488	C₂³	C₂²×D₄×C₆₇	1	1	yes	characteristic
489	Q₈	—	2	4	no	—
490	Q₈	—	2	2	no	—
491	Q₈	—	2	4	no	—
492	Q₈	—	2	4	no	—
493	Q₈	—	4	16	no	—
494	Q₈	—	4	4	no	—
495	Q₈	—	8	4	no	—
496	Q₈	C₂×#41×C₆₇	1	4	yes	—
497	D₄	—	2	8	no	—
498	D₄	—	2	2	no	—
499	D₄	—	2	2	no	—
500	D₄	—	4	2	no	—
501	D₄	—	8	8	no	—
502	C₂×C₄	—	2	4	no	—
503	C₂×C₄	—	2	4	no	—
504	C₂×C₄	—	2	2	no	—
505	C₂×C₄	—	2	2	no	—
506	C₂×C₄	—	2	8	no	—
507	C₂×C₄	—	2	4	no	—
508	C₂×C₄	—	2	4	no	—
509	C₂×C₄	—	2	2	no	—
510	C₂×C₄	—	2	2	no	—
511	C₂×C₄	—	2	4	no	—
512	C₂×C₄	—	2	4	no	—
513	C₂×C₄	—	4	16	no	—
514	C₂×C₄	—	4	4	no	—
515	C₂×C₄	—	4	2	no	—
516	C₂×C₄	—	4	4	no	—
517	C₂×C₄	—	4	2	no	—
518	C₂×C₄	—	4	8	no	—
519	C₂×C₄	#137×C₆₇	1	4	yes	—
520	C₂×C₄	C₂²×D₄×C₆₇	1	4	yes	—
521	C₂×C₄	C₂²×D₄×C₆₇	1	2	yes	—
522	C₂×C₄	C₂²×D₄×C₆₇	1	1	yes	characteristic
523	C₂×C₄	C₂²×D₄×C₆₇	1	2	yes	—
524	C₂×C₄	C₂×#41×C₆₇	1	2	yes	—
525	C₂×C₄	H(5,𝔽₂)×C₆₇	1	2	yes	—
526	C₂×C₄	C₂⁵×C₆₇	1	1	yes	characteristic
527	C₈	—	2	2	no	—
528	C₈	—	2	2	no	—
529	C₈	—	2	4	no	—
530	C₂²	—	2	4	no	—
531	C₂²	—	2	4	no	—
532	C₂²	—	2	4	no	—
533	C₂²	—	4	8	no	—
534	C₂²	—	4	2	no	—
535	C₂²	—	4	2	no	—
536	C₂²	—	4	2	no	—
537	C₂²	—	8	4	no	—
538	C₂²	#546×C₆₇	1	2	yes	—
539	C₂²	C₂×#137×C₆₇	1	2	yes	—
540	C₂²	#577×C₆₇	1	2	yes	—
541	C₂²	C₂³×D₄×C₆₇	1	1	yes	characteristic
542	C₄	—	2	2	no	—
543	C₄	—	2	4	no	—
544	C₄	—	2	4	no	—
545	C₄	—	4	2	no	—
546	C₄	—	4	4	no	—
547	C₄	—	4	2	no	—
548	C₄	—	4	4	no	—
549	C₄	#546×C₆₇	1	2	yes	—
550	C₄	C₂×#137×C₆₇	1	4	yes	—
551	C₄	C₂²×#41×C₆₇	1	1	yes	characteristic
552	C₄	C₂×H(5,𝔽₂)×C₆₇	1	1	yes	characteristic
553	C₂	—	4	4	no	—
554	C₂	—	8	2	no	—
555	C₂	C₆₇×#3329	1	4	yes	direct product factor
556	C₂	C₂×#546×C₆₇	1	1	yes	characteristic
557	C₂	C₂²×#137×C₆₇	1	1	yes	characteristic
558	C₂	C₂×#577×C₆₇	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂³×C₆₇	C₂²×D₄
commutator, Frattini, cosocle	C₂×C₄	C₂⁵×C₆₇
prime radical	C₂×#555×C₆₇	C₂
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₆₇ ◁ C₂×C₄×D₄×C₆₇ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₂×D₄×Q₈×C₆₇	C₂	semidirect
3. C₆₇×#3329	C₂	direct (N × H)
4. C₂×#487×C₆₇	C₂	semidirect
5. C₂×#485×C₆₇	C₂	semidirect
6. C₂×#477×C₆₇	C₂	semidirect
7. C₂×#468×C₆₇	C₂	semidirect
8. C₂×#461×C₆₇	C₂	semidirect
9. C₂×#448×C₆₇	C₂	semidirect
10. C₂×C₄×SD₁₆×C₆₇	C₂	semidirect
11. C₂×#436×C₆₇	C₂	semidirect
12. C₂×#419×C₆₇	C₂	semidirect
20. D₄×Q₈×C₆₇	C₂²	semidirect
35. #487×C₆₇	C₂²	semidirect
36. #485×C₆₇	C₂²	semidirect
37. #477×C₆₇	C₂²	semidirect
38. #468×C₆₇	C₂²	semidirect
39. #461×C₆₇	C₂²	semidirect
40. #448×C₆₇	C₂²	semidirect
41. C₄×SD₁₆×C₆₇	C₂²	semidirect
42. #436×C₆₇	C₂²	semidirect
46. #419×C₆₇	C₂²	semidirect
273. C₂×#3329	C₆₇	direct (N × H)
276. C₂×C₆₇	#3329	direct (N × H)
281. C₂×D₄×Q₈	C₂×C₆₇	semidirect
282. #3329	C₂×C₆₇	direct (N × H)
283. C₂×#487	C₂×C₆₇	semidirect
284. C₂×#485	C₂×C₆₇	semidirect
285. C₂×#477	C₂×C₆₇	semidirect
286. C₂×#468	C₂×C₆₇	semidirect
287. C₂×#461	C₂×C₆₇	semidirect
288. C₂×#448	C₂×C₆₇	semidirect
289. C₂×C₄×SD₁₆	C₂×C₆₇	semidirect
290. C₂×#436	C₂×C₆₇	semidirect
291. C₂×#419	C₂×C₆₇	semidirect
292. C₆₇	C₂×#3329	direct (N × H)
300. D₄×Q₈	C₂²×C₆₇	semidirect
315. #487	C₂²×C₆₇	semidirect
316. #485	C₂²×C₆₇	semidirect
317. #477	C₂²×C₆₇	semidirect
318. #468	C₂²×C₆₇	semidirect
319. #461	C₂²×C₆₇	semidirect
320. #448	C₂²×C₆₇	semidirect
321. C₄×SD₁₆	C₂²×C₆₇	semidirect
322. #436	C₂²×C₆₇	semidirect
326. #419	C₂²×C₆₇	semidirect
555. C₂	C₆₇×#3329	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂×#555×C₆₇	2
C₂×D₄×Q₈×C₆₇	2
C₆₇×#3329	2
C₂×#487×C₆₇	2
C₂×#485×C₆₇	2
C₂×#477×C₆₇	2
C₂×#468×C₆₇	2
C₂×#461×C₆₇	2
C₂×#448×C₆₇	2
C₂×C₄×SD₁₆×C₆₇	2
C₂×#436×C₆₇	2
C₂×#419×C₆₇	2
C₂×#3329	67

Subgroups — C2×C67×#3329 #30740400

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₆₇×#3329 #30740400