Subgroups — C₆₇×#17727 #30696978

Order 17152.

940 conjugacy classes of subgroups.

The subgroup lattice has 528 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₆₇×#3511	C₂	1	1	yes	maximal, characteristic
2	C₆₇×#2988	C₂	1	1	yes	maximal, characteristic, complemented
3	C₆₇×#2925	C₂	1	1	yes	maximal, characteristic, complemented
4	C₆₇×#1553	C₂	1	2	yes	maximal, complemented
5	C₆₇×#1552	C₂	1	2	yes	maximal, complemented
6	C₆₇×#1549	C₂	1	1	yes	maximal, characteristic, complemented
7	C₆₇×#1547	C₂	1	1	yes	maximal, characteristic, complemented
8	C₆₇×#1544	C₂	1	2	yes	maximal, complemented
9	C₆₇×#1540	C₂	1	1	yes	maximal, characteristic, complemented
10	C₆₇×#1540	C₂	1	2	yes	maximal, complemented
11	C₂×#332×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
12	#556×C₆₇	—	2	1	no	—
13	#555×C₆₇	C₄	1	2	yes	—
14	#554×C₆₇	C₄	1	2	yes	—
15	#553×C₆₇	C₄	1	4	yes	—
16	#548×C₆₇	—	2	2	no	—
17	#547×C₆₇	—	2	1	no	—
18	#542×C₆₇	—	2	2	no	—
19	#537×C₆₇	—	2	1	no	—
20	#533×C₆₇	C₂²	1	2	yes	—
21	#532×C₆₇	C₂²	1	1	yes	characteristic
22	#532×C₆₇	C₂²	1	2	yes	—
23	C₂×C₄⋊Q₈×C₆₇	C₂²	1	1	yes	characteristic
24	#520×C₆₇	—	2	1	no	—
25	C₄×#41×C₆₇	C₂²	1	1	yes	characteristic
26	C₈×Q₈×C₆₇	—	2	1	no	—
27	C₈×D₄×C₆₇	—	2	1	no	—
28	C₈×D₄×C₆₇	—	2	2	no	—
29	#433×C₆₇	—	2	2	no	—
30	#432×C₆₇	C₂²	1	2	yes	—
31	#431×C₆₇	C₂²	1	2	yes	—
32	#431×C₆₇	C₂²	1	1	yes	characteristic
33	#431×C₆₇	C₂²	1	2	yes	—
34	#423×C₆₇	C₂²	1	1	yes	characteristic
35	C₂×C₄⋊C₈×C₆₇	C₂²	1	1	yes	characteristic
36	C₂×C₄⋊C₈×C₆₇	C₂²	1	1	yes	characteristic
37	C₂×C₄⋊C₈×C₆₇	C₂²	1	2	yes	—
38	#408×C₆₇	—	2	1	no	—
39	#405×C₆₇	—	2	1	no	—
40	#332×C₆₇	C₂²	1	2	yes	—
41	#332×C₆₇	C₂²	1	2	yes	—
42	#332×C₆₇	C₂²	1	1	yes	characteristic
43	#332×C₆₇	C₂²	1	2	yes	—
44	#332×C₆₇	C₂²	1	1	yes	characteristic
45	#332×C₆₇	C₂²	1	2	yes	—
46	#332×C₆₇	C₂²	1	2	yes	—
47	#331×C₆₇	C₂²	1	1	yes	characteristic
48	#331×C₆₇	C₂²	1	2	yes	—
49	#331×C₆₇	C₂²	1	1	yes	characteristic
50	C₂×#41×C₆₇	—	2	1	no	—
51	C₂×#41×C₆₇	—	2	2	no	—
52	C₂×#41×C₆₇	D₄	1	1	yes	characteristic
53	C₂²×Q₈×C₆₇	—	2	1	no	—
54	C₂×M₁₆×C₆₇	—	2	1	no	—
55	C₂²×C₈×C₆₇	—	2	1	no	—
56	C₂²×C₈×C₆₇	—	2	1	no	—
57	C₂²×C₈×C₆₇	—	2	2	no	—
58	C₄⋊Q₈×C₆₇	—	2	1	no	—
59	C₄⋊Q₈×C₆₇	—	2	1	no	—
60	C₄⋊Q₈×C₆₇	C₂×C₄	1	2	yes	—
61	C₄⋊Q₈×C₆₇	C₂³	1	2	yes	—
62	C₄⋊Q₈×C₆₇	C₂³	1	1	yes	characteristic
63	#127×C₆₇	C₂³	1	1	yes	characteristic
64	#126×C₆₇	—	2	4	no	—
65	#125×C₆₇	C₂×C₄	1	4	yes	—
66	#124×C₆₇	—	2	4	no	—
67	#124×C₆₇	—	2	1	no	—
68	#124×C₆₇	—	2	2	no	—
69	#123×C₆₇	—	2	4	no	—
70	C₂²⋊Q₈×C₆₇	—	2	2	no	—
71	C₂²⋊Q₈×C₆₇	—	2	2	no	—
72	C₂²⋊Q₈×C₆₇	—	2	2	no	—
73	C₂²⋊Q₈×C₆₇	—	2	4	no	—
74	#121×C₆₇	—	2	2	no	—
75	#121×C₆₇	—	2	4	no	—
76	#121×C₆₇	—	2	2	no	—
77	#121×C₆₇	—	2	1	no	—
78	#121×C₆₇	C₂×C₄	1	2	yes	—
79	#121×C₆₇	C₂×C₄	1	4	yes	—
80	C₄×Q₈×C₆₇	—	2	2	no	—
81	C₄×Q₈×C₆₇	—	2	2	no	—
82	C₄×Q₈×C₆₇	C₂×C₄	1	1	yes	characteristic
83	C₄×Q₈×C₆₇	D₄	1	1	yes	characteristic
84	C₄×D₄×C₆₇	—	2	4	no	—
85	C₄×D₄×C₆₇	—	2	2	no	—
86	C₄×D₄×C₆₇	—	2	1	no	—
87	C₄×D₄×C₆₇	C₂×C₄	1	4	yes	—
88	C₄×D₄×C₆₇	C₂×C₄	1	2	yes	—
89	C₄×D₄×C₆₇	C₂×C₄	1	2	yes	—
90	C₄×D₄×C₆₇	C₂×C₄	1	1	yes	characteristic
91	C₄×D₄×C₆₇	D₄	1	1	yes	characteristic
92	C₄×D₄×C₆₇	D₄	1	2	yes	—
93	#117×C₆₇	—	2	2	no	—
94	#117×C₆₇	C₂×C₄	1	4	yes	—
95	#117×C₆₇	D₄	1	1	yes	characteristic
96	#117×C₆₇	D₄	1	2	yes	—
97	C₂×C₄⋊C₄×C₆₇	—	2	1	no	—
98	C₂×C₄⋊C₄×C₆₇	—	2	2	no	—
99	C₂×C₄⋊C₄×C₆₇	C₂×C₄	1	2	yes	—
100	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
101	C₂×C₄²×C₆₇	C₂³	1	2	yes	—
102	C₄⋊C₈×C₆₇	—	2	1	no	—
103	C₄⋊C₈×C₆₇	—	2	1	no	—
104	C₄⋊C₈×C₆₇	—	2	2	no	—
105	C₄⋊C₈×C₆₇	—	2	1	no	—
106	C₄⋊C₈×C₆₇	C₂³	1	1	yes	characteristic
107	C₄⋊C₈×C₆₇	C₂³	1	2	yes	—
108	C₄⋊C₈×C₆₇	C₂³	1	1	yes	characteristic
109	C₄⋊C₈×C₆₇	C₂³	1	1	yes	characteristic
110	C₄⋊C₈×C₆₇	C₂³	1	2	yes	—
111	C₄⋊C₈×C₆₇	C₂³	1	1	yes	characteristic
112	C₂²⋊C₈×C₆₇	—	2	2	no	—
113	C₂²⋊C₈×C₆₇	—	2	2	no	—
114	C₂²⋊C₈×C₆₇	—	2	1	no	—
115	C₂²⋊C₈×C₆₇	—	2	2	no	—
116	#97×C₆₇	—	2	2	no	—
117	C₄×C₈×C₆₇	—	2	2	no	—
118	C₄×C₈×C₆₇	—	2	1	no	—
119	C₄×C₈×C₆₇	—	2	2	no	—
120	#41×C₆₇	—	2	2	no	—
121	#41×C₆₇	—	2	2	no	—
122	#41×C₆₇	—	4	2	no	—
123	#41×C₆₇	—	4	2	no	—
124	#41×C₆₇	—	8	2	no	—
125	#41×C₆₇	—	8	2	no	—
126	C₂×Q₈×C₆₇	—	2	2	no	—
127	C₂×Q₈×C₆₇	—	2	1	no	—
128	C₂×Q₈×C₆₇	—	2	2	no	—
129	C₂×Q₈×C₆₇	—	8	1	no	—
130	C₂×Q₈×C₆₇	—	8	1	no	—
131	C₂×Q₈×C₆₇	C₂²⋊C₄	1	1	yes	characteristic
132	C₂×D₄×C₆₇	—	2	2	no	—
133	C₂×D₄×C₆₇	—	2	4	no	—
134	C₂×D₄×C₆₇	—	2	1	no	—
135	C₂×D₄×C₆₇	—	2	1	no	—
136	C₂×D₄×C₆₇	C₂²⋊C₄	1	2	yes	—
137	C₂×D₄×C₆₇	C₂²⋊C₄	1	1	yes	characteristic
138	C₂×D₄×C₆₇	C₂²×C₄	1	2	yes	—
139	C₂²×C₄×C₆₇	—	2	2	no	—
140	C₂²×C₄×C₆₇	—	2	2	no	—
141	C₂²×C₄×C₆₇	—	2	4	no	—
142	C₂²×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
143	C₂²×C₄×C₆₇	C₂²×C₄	1	2	yes	—
144	C₂²×C₄×C₆₇	C₂×D₄	1	2	yes	—
145	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
146	C₂²×C₄×C₆₇	C₂×D₄	1	2	yes	—
147	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
148	M₁₆×C₆₇	—	4	2	no	—
149	C₂×C₈×C₆₇	—	2	2	no	—
150	C₂×C₈×C₆₇	—	2	1	no	—
151	C₂×C₈×C₆₇	—	2	1	no	—
152	C₂×C₈×C₆₇	—	2	1	no	—
153	C₂×C₈×C₆₇	—	2	2	no	—
154	C₂×C₈×C₆₇	—	2	1	no	—
155	C₂×C₈×C₆₇	—	4	2	no	—
156	C₂×C₈×C₆₇	—	8	1	no	—
157	C₂×C₈×C₆₇	—	8	2	no	—
158	C₄⋊C₄×C₆₇	—	2	2	no	—
159	C₄⋊C₄×C₆₇	—	2	4	no	—
160	C₄⋊C₄×C₆₇	—	2	2	no	—
161	C₄⋊C₄×C₆₇	—	2	4	no	—
162	C₄⋊C₄×C₆₇	—	2	2	no	—
163	C₄⋊C₄×C₆₇	—	2	1	no	—
164	C₄⋊C₄×C₆₇	C₂²⋊C₄	1	1	yes	characteristic
165	C₄⋊C₄×C₆₇	C₂²⋊C₄	1	1	yes	characteristic
166	C₄⋊C₄×C₆₇	C₂²⋊C₄	1	2	yes	—
167	C₄⋊C₄×C₆₇	C₂²⋊C₄	1	2	yes	—
168	C₄⋊C₄×C₆₇	C₂²×C₄	1	4	yes	—
169	C₄⋊C₄×C₆₇	C₂²×C₄	1	2	yes	—
170	C₂²⋊C₄×C₆₇	—	2	4	no	—
171	C₂²⋊C₄×C₆₇	—	2	4	no	—
172	C₂²⋊C₄×C₆₇	—	2	2	no	—
173	C₂²⋊C₄×C₆₇	—	2	4	no	—
174	C₂²⋊C₄×C₆₇	—	2	2	no	—
175	C₂²⋊C₄×C₆₇	C₂²⋊C₄	1	2	yes	—
176	C₂²⋊C₄×C₆₇	C₂²⋊C₄	1	2	yes	—
177	C₂²⋊C₄×C₆₇	C₂²⋊C₄	1	2	yes	—
178	C₄²×C₆₇	—	2	4	no	—
179	C₄²×C₆₇	C₂²×C₄	1	2	yes	—
180	C₄²×C₆₇	C₂²×C₄	1	1	yes	characteristic
181	C₄²×C₆₇	C₂×D₄	1	1	yes	characteristic
182	C₄²×C₆₇	C₂×D₄	1	1	yes	characteristic
183	C₄²×C₆₇	C₂×D₄	1	2	yes	—
184	C₄²×C₆₇	C₂×D₄	1	2	yes	—
185	C₄²×C₆₇	C₂⁴	1	1	yes	characteristic
186	C₂³×C₆₇	—	2	2	no	—
187	C₂³×C₆₇	C₂×C₂²⋊C₄	1	2	yes	—
188	C₂³×C₆₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
189	Q₈×C₆₇	—	2	2	no	—
190	Q₈×C₆₇	—	4	1	no	—
191	Q₈×C₆₇	—	4	1	no	—
192	Q₈×C₆₇	—	4	2	no	—
193	Q₈×C₆₇	—	4	2	no	—
194	Q₈×C₆₇	—	8	2	no	—
195	D₄×C₆₇	—	2	4	no	—
196	D₄×C₆₇	—	2	2	no	—
197	D₄×C₆₇	—	4	2	no	—
198	D₄×C₆₇	—	4	4	no	—
199	D₄×C₆₇	—	4	1	no	—
200	D₄×C₆₇	—	4	1	no	—
201	D₄×C₆₇	—	4	1	no	—
202	D₄×C₆₇	—	4	1	no	—
203	D₄×C₆₇	—	8	4	no	—
204	C₂×C₄×C₆₇	—	2	2	no	—
205	C₂×C₄×C₆₇	—	2	2	no	—
206	C₂×C₄×C₆₇	—	2	1	no	—
207	C₂×C₄×C₆₇	—	2	2	no	—
208	C₂×C₄×C₆₇	—	2	1	no	—
209	C₂×C₄×C₆₇	—	2	2	no	—
210	C₂×C₄×C₆₇	—	2	2	no	—
211	C₂×C₄×C₆₇	—	2	4	no	—
212	C₂×C₄×C₆₇	—	4	1	no	—
213	C₂×C₄×C₆₇	—	4	2	no	—
214	C₂×C₄×C₆₇	—	4	4	no	—
215	C₂×C₄×C₆₇	—	8	4	no	—
216	C₂×C₄×C₆₇	—	8	2	no	—
217	C₂×C₄×C₆₇	C₂×C₂²⋊C₄	1	2	yes	—
218	C₂×C₄×C₆₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
219	C₂×C₄×C₆₇	C₂×C₂²⋊C₄	1	2	yes	—
220	C₂×C₄×C₆₇	C₂×C₂²⋊C₄	1	2	yes	—
221	C₂×C₄×C₆₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
222	C₂×C₄×C₆₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
223	C₂×C₄×C₆₇	#135	1	2	yes	—
224	C₂×C₄×C₆₇	#135	1	2	yes	—
225	C₂×C₄×C₆₇	C₂³×C₄	1	1	yes	characteristic
226	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
227	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
228	C₈×C₆₇	—	4	1	no	—
229	C₈×C₆₇	—	4	1	no	—
230	C₈×C₆₇	—	4	2	no	—
231	C₈×C₆₇	—	4	2	no	—
232	C₈×C₆₇	—	4	1	no	—
233	C₈×C₆₇	—	4	1	no	—
234	C₂²×C₆₇	—	2	1	no	—
235	C₂²×C₆₇	—	2	1	no	—
236	C₂²×C₆₇	—	2	2	no	—
237	C₂²×C₆₇	—	2	2	no	—
238	C₂²×C₆₇	—	2	2	no	—
239	C₂²×C₆₇	—	4	2	no	—
240	C₂²×C₆₇	—	4	2	no	—
241	C₂²×C₆₇	—	4	2	no	—
242	C₂²×C₆₇	#416	1	2	yes	—
243	C₂²×C₆₇	C₂²×C₂²⋊C₄	1	1	yes	characteristic
244	C₄×C₆₇	—	2	1	no	—
245	C₄×C₆₇	—	2	2	no	—
246	C₄×C₆₇	—	2	1	no	—
247	C₄×C₆₇	—	2	2	no	—
248	C₄×C₆₇	—	2	2	no	—
249	C₄×C₆₇	—	2	1	no	—
250	C₄×C₆₇	—	4	2	no	—
251	C₄×C₆₇	—	4	4	no	—
252	C₄×C₆₇	#416	1	2	yes	—
253	C₄×C₆₇	C₂×#135	1	1	yes	characteristic
254	C₄×C₆₇	C₂×#135	1	1	yes	characteristic
255	C₄×C₆₇	#576	1	1	yes	characteristic
256	C₄×C₆₇	#577	1	1	yes	characteristic
257	#17727	C₆₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
258	C₂×C₆₇	—	2	2	no	—
259	C₂×C₆₇	—	4	2	no	—
260	C₂×C₆₇	—	8	2	no	—
261	C₂×C₆₇	#2889	1	1	yes	characteristic
262	C₂×C₆₇	C₂×#416	1	1	yes	characteristic
263	C₂×C₆₇	#2897	1	1	yes	characteristic
264	#3511	C₂×C₆₇	1	1	yes	characteristic
265	#2988	C₂×C₆₇	1	1	yes	characteristic, complemented
266	#2925	C₂×C₆₇	1	1	yes	characteristic, complemented
267	#1553	C₂×C₆₇	1	2	yes	complemented
268	#1552	C₂×C₆₇	1	2	yes	complemented
269	#1549	C₂×C₆₇	1	1	yes	characteristic, complemented
270	#1547	C₂×C₆₇	1	1	yes	characteristic, complemented
271	#1544	C₂×C₆₇	1	2	yes	complemented
272	#1540	C₂×C₆₇	1	1	yes	characteristic, complemented
273	#1540	C₂×C₆₇	1	2	yes	complemented
274	C₂×#332	C₂×C₆₇	1	1	yes	characteristic, complemented
275	C₆₇	#17727	1	1	yes	characteristic, direct product factor, Sylow
276	#556	—	2	1	no	—
277	#555	C₄×C₆₇	1	2	yes	—
278	#554	C₄×C₆₇	1	2	yes	—
279	#553	C₄×C₆₇	1	4	yes	—
280	#548	—	2	2	no	—
281	#547	—	2	1	no	—
282	#542	—	2	2	no	—
283	#537	—	2	1	no	—
284	#533	C₂²×C₆₇	1	2	yes	—
285	#532	C₂²×C₆₇	1	1	yes	characteristic
286	#532	C₂²×C₆₇	1	2	yes	—
287	C₂×C₄⋊Q₈	C₂²×C₆₇	1	1	yes	characteristic
288	#520	—	2	1	no	—
289	C₄×#41	C₂²×C₆₇	1	1	yes	characteristic
290	C₈×Q₈	—	2	1	no	—
291	C₈×D₄	—	2	1	no	—
292	C₈×D₄	—	2	2	no	—
293	#433	—	2	2	no	—
294	#432	C₂²×C₆₇	1	2	yes	—
295	#431	C₂²×C₆₇	1	2	yes	—
296	#431	C₂²×C₆₇	1	1	yes	characteristic
297	#431	C₂²×C₆₇	1	2	yes	—
298	#423	C₂²×C₆₇	1	1	yes	characteristic
299	C₂×C₄⋊C₈	C₂²×C₆₇	1	1	yes	characteristic
300	C₂×C₄⋊C₈	C₂²×C₆₇	1	1	yes	characteristic
301	C₂×C₄⋊C₈	C₂²×C₆₇	1	2	yes	—
302	#408	—	2	1	no	—
303	#405	—	2	1	no	—
304	#332	C₂²×C₆₇	1	2	yes	—
305	#332	C₂²×C₆₇	1	2	yes	—
306	#332	C₂²×C₆₇	1	1	yes	characteristic
307	#332	C₂²×C₆₇	1	2	yes	—
308	#332	C₂²×C₆₇	1	1	yes	characteristic
309	#332	C₂²×C₆₇	1	2	yes	—
310	#332	C₂²×C₆₇	1	2	yes	—
311	#331	C₂²×C₆₇	1	1	yes	characteristic
312	#331	C₂²×C₆₇	1	2	yes	—
313	#331	C₂²×C₆₇	1	1	yes	characteristic
314	C₂×#41	—	2	1	no	—
315	C₂×#41	—	2	2	no	—
316	C₂×#41	D₄×C₆₇	1	1	yes	characteristic
317	C₂²×Q₈	—	2	1	no	—
318	C₂×M₁₆	—	2	1	no	—
319	C₂²×C₈	—	2	1	no	—
320	C₂²×C₈	—	2	1	no	—
321	C₂²×C₈	—	2	2	no	—
322	C₄⋊Q₈	—	2	1	no	—
323	C₄⋊Q₈	—	2	1	no	—
324	C₄⋊Q₈	C₂×C₄×C₆₇	1	2	yes	—
325	C₄⋊Q₈	C₂³×C₆₇	1	2	yes	—
326	C₄⋊Q₈	C₂³×C₆₇	1	1	yes	characteristic
327	#127	C₂³×C₆₇	1	1	yes	characteristic
328	#126	—	2	4	no	—
329	#125	C₂×C₄×C₆₇	1	4	yes	—
330	#124	—	2	4	no	—
331	#124	—	2	1	no	—
332	#124	—	2	2	no	—
333	#123	—	2	4	no	—
334	C₂²⋊Q₈	—	2	2	no	—
335	C₂²⋊Q₈	—	2	2	no	—
336	C₂²⋊Q₈	—	2	2	no	—
337	C₂²⋊Q₈	—	2	4	no	—
338	#121	—	2	2	no	—
339	#121	—	2	4	no	—
340	#121	—	2	2	no	—
341	#121	—	2	1	no	—
342	#121	C₂×C₄×C₆₇	1	2	yes	—
343	#121	C₂×C₄×C₆₇	1	4	yes	—
344	C₄×Q₈	—	2	2	no	—
345	C₄×Q₈	—	2	2	no	—
346	C₄×Q₈	C₂×C₄×C₆₇	1	1	yes	characteristic
347	C₄×Q₈	D₄×C₆₇	1	1	yes	characteristic
348	C₄×D₄	—	2	4	no	—
349	C₄×D₄	—	2	2	no	—
350	C₄×D₄	—	2	1	no	—
351	C₄×D₄	C₂×C₄×C₆₇	1	4	yes	—
352	C₄×D₄	C₂×C₄×C₆₇	1	2	yes	—
353	C₄×D₄	C₂×C₄×C₆₇	1	2	yes	—
354	C₄×D₄	C₂×C₄×C₆₇	1	1	yes	characteristic
355	C₄×D₄	D₄×C₆₇	1	1	yes	characteristic
356	C₄×D₄	D₄×C₆₇	1	2	yes	—
357	#117	—	2	2	no	—
358	#117	C₂×C₄×C₆₇	1	4	yes	—
359	#117	D₄×C₆₇	1	1	yes	characteristic
360	#117	D₄×C₆₇	1	2	yes	—
361	C₂×C₄⋊C₄	—	2	1	no	—
362	C₂×C₄⋊C₄	—	2	2	no	—
363	C₂×C₄⋊C₄	C₂×C₄×C₆₇	1	2	yes	—
364	C₂×C₄²	C₂³×C₆₇	1	1	yes	characteristic
365	C₂×C₄²	C₂³×C₆₇	1	2	yes	—
366	C₄⋊C₈	—	2	1	no	—
367	C₄⋊C₈	—	2	1	no	—
368	C₄⋊C₈	—	2	2	no	—
369	C₄⋊C₈	—	2	1	no	—
370	C₄⋊C₈	C₂³×C₆₇	1	1	yes	characteristic
371	C₄⋊C₈	C₂³×C₆₇	1	2	yes	—
372	C₄⋊C₈	C₂³×C₆₇	1	1	yes	characteristic
373	C₄⋊C₈	C₂³×C₆₇	1	1	yes	characteristic
374	C₄⋊C₈	C₂³×C₆₇	1	2	yes	—
375	C₄⋊C₈	C₂³×C₆₇	1	1	yes	characteristic
376	C₂²⋊C₈	—	2	2	no	—
377	C₂²⋊C₈	—	2	2	no	—
378	C₂²⋊C₈	—	2	1	no	—
379	C₂²⋊C₈	—	2	2	no	—
380	#97	—	2	2	no	—
381	C₄×C₈	—	2	2	no	—
382	C₄×C₈	—	2	1	no	—
383	C₄×C₈	—	2	2	no	—
384	#41	—	2	2	no	—
385	#41	—	2	2	no	—
386	#41	—	4	2	no	—
387	#41	—	4	2	no	—
388	#41	—	8	2	no	—
389	#41	—	8	2	no	—
390	C₂×Q₈	—	2	2	no	—
391	C₂×Q₈	—	2	1	no	—
392	C₂×Q₈	—	2	2	no	—
393	C₂×Q₈	—	8	1	no	—
394	C₂×Q₈	—	8	1	no	—
395	C₂×Q₈	C₂²⋊C₄×C₆₇	1	1	yes	characteristic
396	C₂×D₄	—	2	2	no	—
397	C₂×D₄	—	2	4	no	—
398	C₂×D₄	—	2	1	no	—
399	C₂×D₄	—	2	1	no	—
400	C₂×D₄	C₂²⋊C₄×C₆₇	1	2	yes	—
401	C₂×D₄	C₂²⋊C₄×C₆₇	1	1	yes	characteristic
402	C₂×D₄	C₂²×C₄×C₆₇	1	2	yes	—
403	C₂²×C₄	—	2	2	no	—
404	C₂²×C₄	—	2	2	no	—
405	C₂²×C₄	—	2	4	no	—
406	C₂²×C₄	C₂²×C₄×C₆₇	1	1	yes	characteristic
407	C₂²×C₄	C₂²×C₄×C₆₇	1	2	yes	—
408	C₂²×C₄	C₂×D₄×C₆₇	1	2	yes	—
409	C₂²×C₄	C₂×D₄×C₆₇	1	1	yes	characteristic
410	C₂²×C₄	C₂×D₄×C₆₇	1	2	yes	—
411	C₂²×C₄	C₂×D₄×C₆₇	1	1	yes	characteristic
412	M₁₆	—	4	2	no	—
413	C₂×C₈	—	2	1	no	—
414	C₂×C₈	—	2	1	no	—
415	C₂×C₈	—	2	2	no	—
416	C₂×C₈	—	2	1	no	—
417	C₂×C₈	—	2	2	no	—
418	C₂×C₈	—	2	1	no	—
419	C₂×C₈	—	4	2	no	—
420	C₂×C₈	—	8	1	no	—
421	C₂×C₈	—	8	2	no	—
422	C₄⋊C₄	—	2	2	no	—
423	C₄⋊C₄	—	2	4	no	—
424	C₄⋊C₄	—	2	2	no	—
425	C₄⋊C₄	—	2	4	no	—
426	C₄⋊C₄	—	2	2	no	—
427	C₄⋊C₄	—	2	1	no	—
428	C₄⋊C₄	C₂²⋊C₄×C₆₇	1	1	yes	characteristic
429	C₄⋊C₄	C₂²⋊C₄×C₆₇	1	1	yes	characteristic
430	C₄⋊C₄	C₂²⋊C₄×C₆₇	1	2	yes	—
431	C₄⋊C₄	C₂²⋊C₄×C₆₇	1	2	yes	—
432	C₄⋊C₄	C₂²×C₄×C₆₇	1	4	yes	—
433	C₄⋊C₄	C₂²×C₄×C₆₇	1	2	yes	—
434	C₂²⋊C₄	—	2	4	no	—
435	C₂²⋊C₄	—	2	4	no	—
436	C₂²⋊C₄	—	2	2	no	—
437	C₂²⋊C₄	—	2	4	no	—
438	C₂²⋊C₄	—	2	2	no	—
439	C₂²⋊C₄	C₂²⋊C₄×C₆₇	1	2	yes	—
440	C₂²⋊C₄	C₂²⋊C₄×C₆₇	1	2	yes	—
441	C₂²⋊C₄	C₂²⋊C₄×C₆₇	1	2	yes	—
442	C₄²	—	2	4	no	—
443	C₄²	C₂²×C₄×C₆₇	1	2	yes	—
444	C₄²	C₂²×C₄×C₆₇	1	1	yes	characteristic
445	C₄²	C₂×D₄×C₆₇	1	2	yes	—
446	C₄²	C₂×D₄×C₆₇	1	1	yes	characteristic
447	C₄²	C₂×D₄×C₆₇	1	2	yes	—
448	C₄²	C₂×D₄×C₆₇	1	1	yes	characteristic
449	C₄²	C₂⁴×C₆₇	1	1	yes	characteristic
450	C₂³	—	2	2	no	—
451	C₂³	C₂×C₂²⋊C₄×C₆₇	1	2	yes	—
452	C₂³	C₂×C₂²⋊C₄×C₆₇	1	1	yes	characteristic
453	Q₈	—	2	2	no	—
454	Q₈	—	4	1	no	—
455	Q₈	—	4	1	no	—
456	Q₈	—	4	2	no	—
457	Q₈	—	4	2	no	—
458	Q₈	—	8	2	no	—
459	D₄	—	2	4	no	—
460	D₄	—	2	2	no	—
461	D₄	—	4	2	no	—
462	D₄	—	4	4	no	—
463	D₄	—	4	1	no	—
464	D₄	—	4	1	no	—
465	D₄	—	4	1	no	—
466	D₄	—	4	1	no	—
467	D₄	—	8	4	no	—
468	C₂×C₄	—	2	2	no	—
469	C₂×C₄	—	2	2	no	—
470	C₂×C₄	—	2	1	no	—
471	C₂×C₄	—	2	2	no	—
472	C₂×C₄	—	2	1	no	—
473	C₂×C₄	—	2	2	no	—
474	C₂×C₄	—	2	2	no	—
475	C₂×C₄	—	2	4	no	—
476	C₂×C₄	—	4	1	no	—
477	C₂×C₄	—	4	2	no	—
478	C₂×C₄	—	4	4	no	—
479	C₂×C₄	—	8	4	no	—
480	C₂×C₄	—	8	2	no	—
481	C₂×C₄	C₂×C₂²⋊C₄×C₆₇	1	2	yes	—
482	C₂×C₄	C₂×C₂²⋊C₄×C₆₇	1	1	yes	characteristic
483	C₂×C₄	C₂×C₂²⋊C₄×C₆₇	1	2	yes	—
484	C₂×C₄	C₂×C₂²⋊C₄×C₆₇	1	2	yes	—
485	C₂×C₄	C₂×C₂²⋊C₄×C₆₇	1	1	yes	characteristic
486	C₂×C₄	C₂×C₂²⋊C₄×C₆₇	1	1	yes	characteristic
487	C₂×C₄	#135×C₆₇	1	2	yes	—
488	C₂×C₄	#135×C₆₇	1	2	yes	—
489	C₂×C₄	C₂³×C₄×C₆₇	1	1	yes	characteristic
490	C₂×C₄	C₂²×D₄×C₆₇	1	1	yes	characteristic
491	C₂×C₄	C₂²×D₄×C₆₇	1	1	yes	characteristic
492	C₈	—	4	1	no	—
493	C₈	—	4	1	no	—
494	C₈	—	4	2	no	—
495	C₈	—	4	2	no	—
496	C₈	—	4	1	no	—
497	C₈	—	4	1	no	—
498	C₂²	—	2	1	no	—
499	C₂²	—	2	1	no	—
500	C₂²	—	2	2	no	—
501	C₂²	—	2	2	no	—
502	C₂²	—	2	2	no	—
503	C₂²	—	4	2	no	—
504	C₂²	—	4	2	no	—
505	C₂²	—	4	2	no	—
506	C₂²	#416×C₆₇	1	2	yes	—
507	C₂²	C₂²×C₂²⋊C₄×C₆₇	1	1	yes	characteristic
508	C₄	—	2	1	no	—
509	C₄	—	2	2	no	—
510	C₄	—	2	1	no	—
511	C₄	—	2	2	no	—
512	C₄	—	2	2	no	—
513	C₄	—	2	1	no	—
514	C₄	—	4	2	no	—
515	C₄	—	4	4	no	—
516	C₄	#416×C₆₇	1	2	yes	—
517	C₄	C₂×#135×C₆₇	1	1	yes	characteristic
518	C₄	C₂×#135×C₆₇	1	1	yes	characteristic
519	C₄	#576×C₆₇	1	1	yes	characteristic
520	C₄	#577×C₆₇	1	1	yes	characteristic
521	C₂	—	2	2	no	—
522	C₂	—	4	2	no	—
523	C₂	—	8	2	no	—
524	C₂	C₆₇×#2889	1	1	yes	characteristic
525	C₂	C₂×#416×C₆₇	1	1	yes	characteristic
526	C₂	C₆₇×#2897	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₆₇	C₂²×C₂²⋊C₄
commutator	C₂×C₄	C₂³×C₄×C₆₇
Frattini, cosocle	C₄²	C₂⁴×C₆₇
prime radical	C₆₇×#3511	C₂
LCS	G ▷ C₂×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₆₇ ◁ C₄×#41×C₆₇ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₆₇×#2988	C₂	semidirect
3. C₆₇×#2925	C₂	semidirect
4. C₆₇×#1553	C₂	semidirect
5. C₆₇×#1552	C₂	semidirect
6. C₆₇×#1549	C₂	semidirect
7. C₆₇×#1547	C₂	semidirect
8. C₆₇×#1544	C₂	semidirect
9. C₆₇×#1540	C₂	semidirect
10. C₆₇×#1540	C₂	semidirect
11. C₂×#332×C₆₇	C₂	semidirect
257. #17727	C₆₇	direct (N × H)
265. #2988	C₂×C₆₇	semidirect
266. #2925	C₂×C₆₇	semidirect
267. #1553	C₂×C₆₇	semidirect
268. #1552	C₂×C₆₇	semidirect
269. #1549	C₂×C₆₇	semidirect
270. #1547	C₂×C₆₇	semidirect
271. #1544	C₂×C₆₇	semidirect
272. #1540	C₂×C₆₇	semidirect
273. #1540	C₂×C₆₇	semidirect
274. C₂×#332	C₂×C₆₇	semidirect
275. C₆₇	#17727	direct (N × H)

Maximal subgroups

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

subgroup	index
C₆₇×#3511	2
C₆₇×#2988	2
C₆₇×#2925	2
C₆₇×#1553	2
C₆₇×#1552	2
C₆₇×#1549	2
C₆₇×#1547	2
C₆₇×#1544	2
C₆₇×#1540	2
C₂×#332×C₆₇	2
#17727	67

Subgroups — C67×#17727 #30696978

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₆₇×#17727 #30696978