Subgroups — #30875756

Order 17152.

636 conjugacy classes of subgroups.

The subgroup lattice has 384 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#22562686	C₂	1	1	yes	maximal, characteristic
2	#22561148	C₂	1	2	yes	maximal
3	#22560770	C₂	1	2	yes	maximal
4	#22560763	C₂	1	2	yes	maximal
5	#22560328	C₂	1	2	yes	maximal
6	#22560086	C₂	1	2	yes	maximal
7	#22560068	C₂	1	2	yes	maximal
8	#22560062	C₂	1	1	yes	maximal, characteristic
9	C₆₇×#1842	C₂	1	1	yes	maximal, characteristic
10	C₂×C₄×Dic₁₃₄	C₂²	1	1	yes	characteristic
11	C₂×C₄⋊(C₆₇⋊C₈)	C₂²	1	1	yes	characteristic
12	C₂×C₄⋊(C₆₇⋊C₈)	C₂²	1	1	yes	characteristic
13	C₂×#14523954	C₂²	1	2	yes	—
14	C₂×#14523953	C₂²	1	2	yes	—
15	C₂×C₈×Dic₆₇	C₂²	1	2	yes	—
16	#16863302	C₄	1	1	yes	characteristic
17	#16863302	C₄	1	1	yes	characteristic
18	#16863294	C₄	1	1	yes	characteristic
19	#16863294	C₄	1	1	yes	characteristic
20	#16863280	C₄	1	2	yes	—
21	#16863280	C₄	1	2	yes	—
22	C₄×Dic₆₇⋊C₄	C₂²	1	2	yes	—
23	C₄×C₄⋊Dic₆₇	C₂²	1	1	yes	characteristic
24	#16863242	C₂²	1	2	yes	—
25	#16863151	C₂²	1	2	yes	—
26	#16863150	C₂²	1	2	yes	—
27	#16863150	C₂²	1	2	yes	—
28	#16863150	C₂²	1	2	yes	—
29	#16863150	C₂²	1	2	yes	—
30	#16863150	C₂²	1	2	yes	—
31	#16863150	C₂²	1	2	yes	—
32	C₂×C₄⋊C₈×C₆₇	C₂²	1	2	yes	—
33	C₂×C₄⋊C₈×C₆₇	C₂²	1	2	yes	—
34	C₂×C₄⋊C₈×C₆₇	C₂²	1	1	yes	characteristic
35	C₂×C₄⋊C₈×C₆₇	C₂²	1	1	yes	characteristic
36	C₄³×C₆₇	C₂²	1	1	yes	characteristic
37	C₂²×Dic₁₃₄	C₂×C₄	1	1	yes	characteristic
38	C₂²×C₆₇⋊C₈	C₂³	1	2	yes	—
39	C₂²×C₆₇⋊C₈	C₂³	1	2	yes	—
40	C₄×Dic₁₃₄	—	2	2	no	—
41	C₄×Dic₁₃₄	—	2	2	no	—
42	C₂×C₄⋊Dic₆₇	C₂×C₄	1	1	yes	characteristic
43	C₂×C₄⋊Dic₆₇	C₂×C₄	1	1	yes	characteristic
44	C₂×C₄⋊Dic₆₇	C₂×C₄	1	1	yes	characteristic
45	C₂×Dic₆₇⋊C₄	C₂×C₄	1	2	yes	—
46	C₂×Dic₆₇⋊C₄	C₂×C₄	1	2	yes	—
47	C₂×Dic₆₇⋊C₄	C₂×C₄	1	2	yes	—
48	C₂×C₄×Dic₆₇	C₂³	1	2	yes	—
49	C₂×C₄×Dic₆₇	C₂³	1	2	yes	—
50	C₄⋊(C₆₇⋊C₈)	—	2	2	no	—
51	C₄⋊(C₆₇⋊C₈)	Q₈	1	4	yes	complemented
52	#14523954	—	2	4	no	—
53	#14523953	—	4	2	no	—
54	C₈×Dic₆₇	—	4	2	no	—
55	C₆₇⋊#95	C₂×C₄	1	2	yes	—
56	C₆₇⋊#95	C₂×C₄	1	2	yes	—
57	C₆₇⋊#95	C₂×C₄	1	2	yes	—
58	C₆₇⋊#95	C₂×C₄	1	2	yes	—
59	C₆₇⋊#95	C₂×C₄	1	2	yes	—
60	C₆₇⋊#95	C₂×C₄	1	2	yes	—
61	C₆₇⋊#95	C₂×C₄	1	2	yes	—
62	C₂²×C₈×C₆₇	C₂³	1	2	yes	—
63	C₂²×C₈×C₆₇	C₂³	1	2	yes	—
64	C₂×C₄²×C₆₇	C₂×C₄	1	1	yes	characteristic
65	C₂×C₄²×C₆₇	C₂×C₄	1	1	yes	characteristic
66	C₂×C₄²×C₆₇	C₂×C₄	1	1	yes	characteristic
67	C₂×C₄²×C₆₇	C₂×C₄	1	1	yes	characteristic
68	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
69	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
70	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
71	C₄⋊C₈×C₆₇	—	2	4	no	—
72	C₄⋊C₈×C₆₇	—	2	4	no	—
73	C₄⋊C₈×C₆₇	—	4	1	no	—
74	C₄⋊C₈×C₆₇	—	4	1	no	—
75	C₂×Dic₁₃₄	—	2	2	no	—
76	C₂×Dic₁₃₄	—	4	1	no	—
77	C₂×Dic₁₃₄	—	4	1	no	—
78	C₂²×Dic₆₇	C₂²×C₄	1	2	yes	—
79	C₂²×Dic₆₇	C₂²×C₄	1	2	yes	—
80	C₂²×Dic₆₇	C₂²×C₄	1	2	yes	—
81	C₂²×Dic₆₇	C₂²×C₄	1	2	yes	—
82	C₂×C₆₇⋊C₈	—	2	2	no	—
83	C₂×C₆₇⋊C₈	—	2	2	no	—
84	C₂×C₆₇⋊C₈	—	2	2	no	—
85	C₂×C₆₇⋊C₈	—	2	2	no	—
86	C₂×C₆₇⋊C₈	—	2	2	no	—
87	C₂×C₆₇⋊C₈	C₂×Q₈	1	4	yes	—
88	C₄⋊Dic₆₇	—	2	2	no	—
89	C₄⋊Dic₆₇	—	4	1	no	—
90	C₄⋊Dic₆₇	—	4	1	no	—
91	Dic₆₇⋊C₄	—	2	4	no	—
92	Dic₆₇⋊C₄	—	2	4	no	—
93	Dic₆₇⋊C₄	—	4	2	no	—
94	C₄×Dic₆₇	—	2	2	no	—
95	C₄×Dic₆₇	—	2	2	no	—
96	C₄×Dic₆₇	—	2	2	no	—
97	C₄×Dic₆₇	—	2	2	no	—
98	C₂²×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
99	C₂²×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
100	C₂²×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
101	C₂²×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
102	C₂²×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
103	C₂²×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
104	C₂²×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
105	C₂×C₈×C₆₇	—	2	2	no	—
106	C₂×C₈×C₆₇	—	2	2	no	—
107	C₂×C₈×C₆₇	—	2	2	no	—
108	C₂×C₈×C₆₇	—	2	2	no	—
109	C₂×C₈×C₆₇	—	2	2	no	—
110	C₂×C₈×C₆₇	—	2	2	no	—
111	C₄²×C₆₇	—	2	2	no	—
112	C₄²×C₆₇	—	2	1	no	—
113	C₄²×C₆₇	—	2	1	no	—
114	C₄²×C₆₇	—	4	1	no	—
115	C₄²×C₆₇	—	4	1	no	—
116	C₄²×C₆₇	—	4	1	no	—
117	C₄²×C₆₇	C₂×Q₈	1	2	yes	—
118	C₄²×C₆₇	#41	1	2	yes	—
119	C₄²×C₆₇	#41	1	2	yes	—
120	C₄²×C₆₇	#41	1	2	yes	—
121	C₂³×C₆₇	C₂³×C₄	1	1	yes	characteristic
122	C₂×C₄×C₆₇	—	2	1	no	—
123	C₂×C₄×C₆₇	—	2	1	no	—
124	C₂×C₄×C₆₇	—	2	1	no	—
125	C₂×C₄×C₆₇	—	2	1	no	—
126	C₂×C₄×C₆₇	—	2	1	no	—
127	C₂×C₄×C₆₇	—	2	1	no	—
128	C₂×C₄×C₆₇	—	2	1	no	—
129	C₂×C₄×C₆₇	—	2	1	no	—
130	C₂×C₄×C₆₇	—	2	1	no	—
131	C₂×C₄×C₆₇	—	2	1	no	—
132	C₂×C₄×C₆₇	—	2	1	no	—
133	C₂×C₄×C₆₇	—	2	1	no	—
134	C₂×C₄×C₆₇	—	2	1	no	—
135	C₂×C₄×C₆₇	#117	1	2	yes	—
136	C₂×C₄×C₆₇	#117	1	2	yes	—
137	C₂×C₄×C₆₇	C₄×Q₈	1	2	yes	—
138	C₂×C₄×C₆₇	C₄×Q₈	1	2	yes	—
139	C₂×C₄×C₆₇	#131	1	2	yes	—
140	C₂×C₄×C₆₇	C₂²×Q₈	1	1	yes	characteristic
141	C₂×C₄×C₆₇	C₂×#41	1	1	yes	characteristic
142	C₂×C₄×C₆₇	C₂×#41	1	1	yes	characteristic
143	C₂×C₄×C₆₇	C₂×#41	1	1	yes	characteristic
144	C₂×C₄×C₆₇	H(5,𝔽₂)	1	1	yes	characteristic
145	C₂×C₄×C₆₇	2₋¹⁺⁴	1	1	yes	characteristic
146	C₂×Dic₆₇	—	2	2	no	—
147	C₂×Dic₆₇	—	2	2	no	—
148	C₂×Dic₆₇	—	2	2	no	—
149	C₂×Dic₆₇	—	2	2	no	—
150	C₂×Dic₆₇	—	2	2	no	—
151	C₂×Dic₆₇	—	2	2	no	—
152	C₂×Dic₆₇	—	2	2	no	—
153	C₂×Dic₆₇	—	2	2	no	—
154	C₂×Dic₆₇	—	2	2	no	—
155	C₂×Dic₆₇	—	2	2	no	—
156	C₂×Dic₆₇	—	2	2	no	—
157	C₂×Dic₆₇	—	2	2	no	—
158	Dic₁₃₄	—	4	4	no	—
159	C₈×C₆₇	—	4	2	no	—
160	C₈×C₆₇	—	4	2	no	—
161	C₆₇⋊C₈	—	2	4	no	—
162	C₆₇⋊C₈	—	4	2	no	—
163	C₂²×C₆₇	C₂×#117	1	1	yes	characteristic
164	C₂²×C₆₇	C₂×C₄×Q₈	1	1	yes	characteristic
165	C₂²×C₆₇	#520	1	1	yes	characteristic
166	C₂²×C₆₇	C₂×#131	1	1	yes	characteristic
167	C₂²×C₆₇	#568	1	1	yes	characteristic
168	C₂²×C₆₇	#568	1	1	yes	characteristic
169	C₂²×C₆₇	#568	1	1	yes	characteristic
170	C₄×C₆₇	—	2	2	no	—
171	C₄×C₆₇	—	2	2	no	—
172	C₄×C₆₇	—	2	2	no	—
173	C₄×C₆₇	—	2	2	no	—
174	C₄×C₆₇	—	2	2	no	—
175	C₄×C₆₇	—	4	1	no	—
176	C₄×C₆₇	#554	1	1	yes	characteristic
177	C₄×C₆₇	#556	1	1	yes	characteristic
178	C₄×C₆₇	#557	1	1	yes	characteristic
179	C₄×C₆₇	#559	1	1	yes	characteristic
180	Dic₆₇	—	4	2	no	—
181	Dic₆₇	—	4	2	no	—
182	Dic₆₇	—	4	2	no	—
183	Dic₆₇	—	4	2	no	—
184	#22824	—	67	1	no	maximal, Sylow
185	C₂×C₆₇	#2355	1	1	yes	characteristic
186	C₂×C₆₇	#2921	1	1	yes	characteristic
187	C₂×C₆₇	#2925	1	1	yes	characteristic
188	C₂×C₆₇	#2961	1	1	yes	characteristic
189	C₂×C₆₇	#2962	1	1	yes	characteristic
190	C₂×C₆₇	#2992	1	1	yes	characteristic
191	C₂×C₆₇	#2995	1	1	yes	characteristic
192	#2295	—	67	1	no	—
193	#1985	—	67	2	no	—
194	#1917	—	67	2	no	—
195	#1916	—	67	2	no	—
196	#1842	D₆₇	1	1	yes	characteristic
197	#1832	—	67	2	no	—
198	#1771	—	67	2	no	—
199	#1768	—	67	2	no	—
200	#1767	—	67	1	no	—
201	C₆₇	#22824	1	1	yes	characteristic, complemented, Sylow
202	C₂×C₄×Q₈	—	67	1	no	—
203	C₂×C₄⋊C₈	—	67	1	no	—
204	C₂×C₄⋊C₈	—	67	1	no	—
205	C₂×C₄⋊C₈	C₂×D₆₇	1	2	yes	—
206	C₂×C₄⋊C₈	C₂×D₆₇	1	2	yes	—
207	C₂×C₄⋊C₈	C₂×D₆₇	1	1	yes	characteristic
208	C₂×C₄⋊C₈	C₂×D₆₇	1	1	yes	characteristic
209	C₂×#97	—	67	2	no	—
210	C₂×#97	—	67	2	no	—
211	C₂×C₄×C₈	—	67	2	no	—
212	#391	—	67	1	no	—
213	#391	—	67	1	no	—
214	#389	—	67	1	no	—
215	#389	—	67	1	no	—
216	#387	—	67	2	no	—
217	#387	—	67	2	no	—
218	C₄×C₄⋊C₄	—	67	1	no	—
219	C₄×C₄⋊C₄	—	67	2	no	—
220	#376	—	67	2	no	—
221	C₄³	C₂×D₆₇	1	1	yes	characteristic
222	#336	—	67	2	no	—
223	#336	—	67	2	no	—
224	#336	—	67	2	no	—
225	#336	—	67	2	no	—
226	#336	—	67	2	no	—
227	#336	—	67	2	no	—
228	#336	—	67	2	no	—
229	C₂²×Q₈	—	67	1	no	—
230	C₂²×C₈	—	67	2	no	—
231	C₂²×C₈	—	67	2	no	—
232	C₂²×C₈	C₂²×D₆₇	1	2	yes	—
233	C₂²×C₈	C₂²×D₆₇	1	2	yes	—
234	C₄×Q₈	—	134	2	no	—
235	C₄×Q₈	—	134	2	no	—
236	C₂×C₄⋊C₄	—	67	2	no	—
237	C₂×C₄⋊C₄	—	67	1	no	—
238	C₂×C₄⋊C₄	—	67	1	no	—
239	C₂×C₄⋊C₄	—	67	2	no	—
240	C₂×C₄⋊C₄	—	67	1	no	—
241	C₂×C₄⋊C₄	—	67	2	no	—
242	C₂×C₄²	—	67	2	no	—
243	C₂×C₄²	—	67	2	no	—
244	C₂×C₄²	C₄×D₆₇	1	1	yes	characteristic
245	C₂×C₄²	C₄×D₆₇	1	1	yes	characteristic
246	C₂×C₄²	C₄×D₆₇	1	1	yes	characteristic
247	C₂×C₄²	C₄×D₆₇	1	1	yes	characteristic
248	C₂×C₄²	C₂²×D₆₇	1	1	yes	characteristic
249	C₂×C₄²	C₂²×D₆₇	1	1	yes	characteristic
250	C₂×C₄²	C₂²×D₆₇	1	1	yes	characteristic
251	C₄⋊C₈	—	2	4	no	—
252	C₄⋊C₈	—	2	4	no	—
253	C₄⋊C₈	—	4	1	no	—
254	C₄⋊C₈	—	4	1	no	—
255	C₄⋊C₈	—	67	4	no	—
256	C₄⋊C₈	—	134	2	no	—
257	#97	—	268	2	no	—
258	#97	—	134	4	no	—
259	C₄×C₈	—	268	2	no	—
260	#95	—	67	2	no	—
261	#95	—	67	2	no	—
262	#95	—	67	2	no	—
263	#95	—	67	2	no	—
264	#95	—	67	2	no	—
265	#95	—	67	2	no	—
266	#95	—	67	2	no	—
267	C₂×Q₈	—	268	1	no	—
268	C₂×Q₈	—	268	1	no	—
269	C₂×Q₈	—	134	2	no	—
270	C₂²×C₄	—	67	2	no	—
271	C₂²×C₄	—	67	2	no	—
272	C₂²×C₄	—	67	2	no	—
273	C₂²×C₄	—	67	2	no	—
274	C₂²×C₄	C₂×C₄×D₆₇	1	1	yes	characteristic
275	C₂²×C₄	C₂×C₄×D₆₇	1	1	yes	characteristic
276	C₂²×C₄	C₂×C₄×D₆₇	1	1	yes	characteristic
277	C₂²×C₄	C₂×C₄×D₆₇	1	1	yes	characteristic
278	C₂²×C₄	C₂×C₄×D₆₇	1	1	yes	characteristic
279	C₂²×C₄	C₂×C₄×D₆₇	1	1	yes	characteristic
280	C₂²×C₄	C₂³×D₆₇	1	1	yes	characteristic
281	C₂×C₈	—	2	2	no	—
282	C₂×C₈	—	2	2	no	—
283	C₂×C₈	—	2	2	no	—
284	C₂×C₈	—	2	2	no	—
285	C₂×C₈	—	2	2	no	—
286	C₂×C₈	—	2	2	no	—
287	C₂×C₈	—	67	4	no	—
288	C₂×C₈	—	134	2	no	—
289	C₂×C₈	—	134	2	no	—
290	C₂×C₈	—	134	2	no	—
291	C₂×C₈	—	134	2	no	—
292	C₂×C₈	—	134	2	no	—
293	C₄⋊C₄	—	268	2	no	—
294	C₄⋊C₄	—	268	1	no	—
295	C₄⋊C₄	—	268	1	no	—
296	C₄⋊C₄	—	134	4	no	—
297	C₄⋊C₄	—	134	2	no	—
298	C₄⋊C₄	—	134	4	no	—
299	C₄²	—	2	2	no	—
300	C₄²	—	2	1	no	—
301	C₄²	—	2	1	no	—
302	C₄²	—	4	1	no	—
303	C₄²	—	4	1	no	—
304	C₄²	—	4	1	no	—
305	C₄²	—	134	2	no	—
306	C₄²	—	134	2	no	—
307	C₄²	—	134	2	no	—
308	C₄²	—	134	2	no	—
309	C₄²	Q₈×D₆₇	1	2	yes	—
310	C₄²	#11761627	1	2	yes	—
311	C₄²	#11761627	1	2	yes	—
312	C₄²	#11761629	1	2	yes	—
313	C₂³	C₂²×C₄×D₆₇	1	1	yes	characteristic
314	Q₈	—	268	4	no	—
315	C₂×C₄	—	2	1	no	—
316	C₂×C₄	—	2	1	no	—
317	C₂×C₄	—	2	1	no	—
318	C₂×C₄	—	2	1	no	—
319	C₂×C₄	—	2	1	no	—
320	C₂×C₄	—	2	1	no	—
321	C₂×C₄	—	2	1	no	—
322	C₂×C₄	—	2	1	no	—
323	C₂×C₄	—	2	1	no	—
324	C₂×C₄	—	2	1	no	—
325	C₂×C₄	—	2	1	no	—
326	C₂×C₄	—	2	1	no	—
327	C₂×C₄	—	2	1	no	—
328	C₂×C₄	—	134	2	no	—
329	C₂×C₄	—	134	2	no	—
330	C₂×C₄	—	134	2	no	—
331	C₂×C₄	—	134	2	no	—
332	C₂×C₄	—	134	2	no	—
333	C₂×C₄	—	134	2	no	—
334	C₂×C₄	—	134	2	no	—
335	C₂×C₄	—	134	2	no	—
336	C₂×C₄	—	134	2	no	—
337	C₂×C₄	—	134	2	no	—
338	C₂×C₄	—	134	2	no	—
339	C₂×C₄	—	134	2	no	—
340	C₂×C₄	#14524001	1	2	yes	—
341	C₂×C₄	#14524001	1	2	yes	—
342	C₂×C₄	#14524008	1	2	yes	—
343	C₂×C₄	#14524008	1	2	yes	—
344	C₂×C₄	#14524049	1	2	yes	—
345	C₂×C₄	C₂×Q₈×D₆₇	1	1	yes	characteristic
346	C₂×C₄	C₂×#11761627	1	1	yes	characteristic
347	C₂×C₄	C₂×#11761627	1	1	yes	characteristic
348	C₂×C₄	C₂×#11761629	1	1	yes	characteristic
349	C₂×C₄	#14524088	1	1	yes	characteristic
350	C₂×C₄	#14524091	1	1	yes	characteristic
351	C₈	—	4	2	no	—
352	C₈	—	4	2	no	—
353	C₈	—	268	2	no	—
354	C₈	—	134	4	no	—
355	C₂²	C₂×#14524001	1	1	yes	characteristic
356	C₂²	C₂×#14524008	1	1	yes	characteristic
357	C₂²	#16863877	1	1	yes	characteristic
358	C₂²	C₂×#14524049	1	1	yes	characteristic
359	C₂²	#16864162	1	1	yes	characteristic
360	C₂²	#16864162	1	1	yes	characteristic
361	C₂²	#16864162	1	1	yes	characteristic
362	C₄	—	2	2	no	—
363	C₄	—	2	2	no	—
364	C₄	—	2	2	no	—
365	C₄	—	2	2	no	—
366	C₄	—	2	2	no	—
367	C₄	—	4	1	no	—
368	C₄	—	268	2	no	—
369	C₄	—	268	2	no	—
370	C₄	—	268	2	no	—
371	C₄	—	268	2	no	—
372	C₄	#16864107	1	1	yes	characteristic
373	C₄	#16864120	1	1	yes	characteristic
374	C₄	#16864121	1	1	yes	characteristic
375	C₄	#16864132	1	1	yes	characteristic
376	C₂	#22563168	1	1	yes	characteristic
377	C₂	#22568846	1	1	yes	characteristic
378	C₂	#22568874	1	1	yes	characteristic
379	C₂	#22569146	1	1	yes	characteristic
380	C₂	#22569152	1	1	yes	characteristic
381	C₂	#22569345	1	1	yes	characteristic
382	C₂	#22569376	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂²×C₄	C₂³×D₆₇
commutator, socle, prime radical	C₂³×C₆₇	C₂³×C₄
Fitting	C₆₇×#1842	C₂
cosocle	C₂²×C₄×C₆₇	C₂⁴
hypercenter	#1842	D₆₇
hypocenter	C₆₇	#22824
LCS	G ▷ C₂³×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂²×C₄ ◁ #1842

Complements

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

N	H ≅ G/N	type
51. C₄⋊(C₆₇⋊C₈)	Q₈	semidirect
201. C₆₇	#22824	semidirect

Maximal subgroups

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

subgroup	index
#22562686	2
#22561148	2
#22560770	2
#22560763	2
#22560328	2
#22560086	2
#22560068	2
#22560062	2
C₆₇×#1842	2
#22824	67