Subgroups — #31013497

Order 17152.

498 conjugacy classes of subgroups.

The subgroup lattice has 342 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#22573077	C₂	1	1	yes	maximal, characteristic, complemented
2	#22562440	C₂	1	2	yes	maximal, complemented
3	#22562432	C₂	1	1	yes	maximal, characteristic, complemented
4	#22562366	C₂	1	2	yes	maximal, complemented
5	#22562361	C₂	1	1	yes	maximal, characteristic, complemented
6	#22562261	C₂	1	2	yes	maximal, complemented
7	#22562241	C₂	1	1	yes	maximal, characteristic
8	Q₈×C₆₇⋊C₁₆	C₂	1	1	yes	maximal, characteristic, complemented
9	C₄×#14523986	C₂	1	1	yes	maximal, characteristic, complemented
10	C₄×#14523984	C₂	1	2	yes	maximal, complemented
11	C₆₇×#3396	C₂	1	1	yes	maximal, characteristic, complemented
12	#16864106	—	2	1	no	—
13	C₂×#14523986	C₂²	1	1	yes	characteristic
14	C₂×#14523984	C₂²	1	2	yes	—
15	#16863801	C₂²	1	1	yes	characteristic
16	#16863798	C₂²	1	2	yes	—
17	#16863697	—	2	2	no	—
18	#16863687	—	2	1	no	—
19	C₄×D₅₃₆	C₂²	1	2	yes	—
20	C₄×D₅₃₆	C₂²	1	1	yes	characteristic
21	#16863223	C₂²	1	1	yes	characteristic
22	#16863221	C₂²	1	2	yes	—
23	C₄⋊(C₆₇⋊C₁₆)	C₂²	1	1	yes	characteristic
24	C₄⋊(C₆₇⋊C₁₆)	C₂²	1	2	yes	—
25	#16863200	C₂²	1	1	yes	characteristic
26	#16863200	C₂²	1	2	yes	—
27	#16863197	C₂²	1	2	yes	—
28	#16863197	C₂²	1	2	yes	—
29	#16863197	C₂²	1	2	yes	—
30	#16863196	C₂²	1	1	yes	characteristic
31	C₄×C₆₇⋊C₁₆	C₂²	1	2	yes	—
32	C₄×C₆₇⋊C₁₆	C₂²	1	1	yes	characteristic
33	#559×C₆₇	—	2	1	no	—
34	#495×C₆₇	C₂²	1	2	yes	—
35	#493×C₆₇	C₂²	1	1	yes	characteristic
36	#463×C₆₇	—	2	2	no	—
37	#460×C₆₇	—	2	1	no	—
38	C₈×Q₈×C₆₇	C₂²	1	1	yes	characteristic
39	C₄×Q₁₆×C₆₇	C₂²	1	1	yes	characteristic
40	C₄×D₈×C₆₇	C₂²	1	2	yes	—
41	C₂×D₅₃₆	C₂³	1	1	yes	characteristic
42	#14524043	—	2	1	no	—
43	#14524035	—	2	2	no	—
44	#14524021	—	4	2	no	—
45	#14524019	—	4	1	no	—
46	C₄×D₂₆₈	—	2	1	no	—
47	C₄×D₂₆₈	—	2	2	no	—
48	C₄⋊C₄×D₆₇	—	4	1	no	—
49	#14523986	—	2	2	no	—
50	#14523984	—	2	4	no	—
51	C₂×C₆₇⋊C₁₆	C₂³	1	2	yes	—
52	C₂×C₆₇⋊C₁₆	C₂³	1	1	yes	characteristic
53	C₂×C₆₇⋊C₁₆	C₂³	1	1	yes	characteristic
54	#14523977	—	2	1	no	—
55	#14523977	C₂³	1	1	yes	characteristic
56	#14523977	C₂³	1	2	yes	—
57	#14523975	—	2	2	no	—
58	#14523965	—	2	2	no	—
59	#14523965	—	2	1	no	—
60	C₂×Q₁₆×C₆₇	C₂³	1	1	yes	characteristic
61	C₂×SD₁₆×C₆₇	—	2	2	no	—
62	C₂×D₈×C₆₇	—	2	1	no	—
63	C₂×D₈×C₆₇	C₂³	1	2	yes	—
64	#127×C₆₇	—	2	1	no	—
65	#124×C₆₇	—	2	2	no	—
66	#124×C₆₇	—	4	1	no	—
67	#123×C₆₇	—	4	2	no	—
68	C₄×Q₈×C₆₇	—	2	1	no	—
69	C₄×Q₈×C₆₇	D₄	1	1	yes	characteristic
70	C₄×D₄×C₆₇	—	2	2	no	—
71	C₄×D₄×C₆₇	—	4	1	no	—
72	#107×C₆₇	C₂³	1	1	yes	characteristic
73	C₄⋊C₈×C₆₇	D₄	1	1	yes	characteristic
74	C₄⋊C₈×C₆₇	D₄	1	2	yes	—
75	#103×C₆₇	—	2	1	no	—
76	#102×C₆₇	—	2	2	no	—
77	C₄×C₈×C₆₇	C₂³	1	1	yes	characteristic
78	C₄×C₈×C₆₇	C₂³	1	2	yes	—
79	C₂×Dic₁₃₄	—	4	1	no	—
80	C₂×D₂₆₈	—	2	1	no	—
81	C₂×C₄×D₆₇	—	4	1	no	—
82	C₂×C₄×D₆₇	—	4	2	no	—
83	D₅₃₆	—	2	1	no	—
84	D₅₃₆	#41	1	2	yes	—
85	C₄⋊Dic₆₇	—	2	1	no	—
86	C₄⋊Dic₆₇	—	2	2	no	—
87	C₄⋊Dic₆₇	—	4	1	no	—
88	Dic₆₇⋊C₄	—	4	2	no	—
89	Dic₆₇⋊C₄	—	4	2	no	—
90	(C₂×D₆₇)⋊C₄	—	4	2	no	—
91	(C₂×D₆₇)⋊C₄	—	4	1	no	—
92	C₆₇⋊C₁₆	—	2	2	no	—
93	C₆₇⋊C₁₆	—	2	1	no	—
94	C₆₇⋊C₁₆	—	2	1	no	—
95	C₂×Q₈×C₆₇	—	2	1	no	—
96	C₂×Q₈×C₆₇	D₈	1	1	yes	characteristic
97	C₂×D₄×C₆₇	—	2	2	no	—
98	C₂×D₄×C₆₇	—	4	1	no	—
99	C₂×D₄×C₆₇	—	4	1	no	—
100	C₂²×C₄×C₆₇	—	4	2	no	—
101	Q₁₆×C₆₇	—	2	1	no	—
102	Q₁₆×C₆₇	—	2	1	no	—
103	SD₁₆×C₆₇	—	8	2	no	—
104	D₈×C₆₇	—	2	2	no	—
105	D₈×C₆₇	—	2	2	no	—
106	D₈×C₆₇	—	8	1	no	—
107	C₂×C₈×C₆₇	C₂×D₄	1	2	yes	—
108	C₂×C₈×C₆₇	C₂×D₄	1	1	yes	characteristic
109	C₂×C₈×C₆₇	C₂⁴	1	1	yes	characteristic
110	C₄⋊C₄×C₆₇	—	2	1	no	—
111	C₄⋊C₄×C₆₇	—	4	1	no	—
112	C₄⋊C₄×C₆₇	D₈	1	1	yes	characteristic
113	C₄⋊C₄×C₆₇	D₈	1	2	yes	—
114	C₂²⋊C₄×C₆₇	—	4	2	no	—
115	C₂²⋊C₄×C₆₇	—	4	2	no	—
116	C₂²⋊C₄×C₆₇	—	4	2	no	—
117	C₄²×C₆₇	—	4	1	no	—
118	C₄²×C₆₇	C₂×D₄	1	2	yes	—
119	C₄²×C₆₇	C₂×D₄	1	1	yes	characteristic
120	C₂³×C₆₇	—	4	2	no	—
121	C₂²×D₆₇	—	4	1	no	—
122	Q₈×C₆₇	—	2	2	no	—
123	Q₈×C₆₇	—	4	1	no	—
124	Q₈×C₆₇	—	4	1	no	—
125	D₄×C₆₇	—	4	2	no	—
126	D₄×C₆₇	—	4	2	no	—
127	D₄×C₆₇	—	8	2	no	—
128	D₄×C₆₇	—	16	1	no	—
129	C₂×C₄×C₆₇	—	4	1	no	—
130	C₂×C₄×C₆₇	—	4	1	no	—
131	C₂×C₄×C₆₇	—	16	2	no	—
132	C₂×C₄×C₆₇	C₂×D₈	1	2	yes	—
133	C₂×C₄×C₆₇	C₂×D₈	1	2	yes	—
134	C₂×C₄×C₆₇	C₂×D₈	1	1	yes	characteristic
135	C₂×C₄×C₆₇	C₂×D₈	1	1	yes	characteristic
136	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
137	C₂×Dic₆₇	—	4	1	no	—
138	C₂×Dic₆₇	—	4	2	no	—
139	D₂₆₈	—	2	2	no	—
140	D₂₆₈	—	4	1	no	—
141	C₄×D₆₇	—	8	2	no	—
142	C₄×D₆₇	—	16	2	no	—
143	Dic₁₃₄	—	16	1	no	—
144	C₈×C₆₇	—	2	1	no	—
145	C₈×C₆₇	—	2	2	no	—
146	C₈×C₆₇	C₂×#41	1	1	yes	characteristic
147	C₈×C₆₇	2₋¹⁺⁴	1	1	yes	characteristic
148	C₂²×C₆₇	—	8	2	no	—
149	C₂²×C₆₇	—	8	2	no	—
150	C₂²×C₆₇	—	8	2	no	—
151	C₂²×C₆₇	C₂²×D₈	1	1	yes	characteristic
152	C₂×D₆₇	—	4	2	no	—
153	C₂×D₆₇	—	8	1	no	—
154	C₂×D₆₇	—	8	1	no	—
155	C₄×C₆₇	—	2	2	no	—
156	C₄×C₆₇	—	2	1	no	—
157	C₄×C₆₇	—	2	2	no	—
158	C₄×C₆₇	—	8	1	no	—
159	C₄×C₆₇	—	8	1	no	—
160	C₄×C₆₇	#508	1	2	yes	—
161	C₄×C₆₇	#548	1	1	yes	characteristic
162	C₄×C₆₇	#577	1	1	yes	characteristic
163	Dic₆₇	—	8	1	no	—
164	Dic₆₇	—	8	2	no	—
165	#33928	—	67	1	no	maximal, Sylow
166	C₂×C₆₇	—	16	2	no	—
167	C₂×C₆₇	#3358	1	1	yes	characteristic
168	C₂×C₆₇	C₂×#508	1	1	yes	characteristic
169	C₂×C₆₇	#3415	1	1	yes	characteristic
170	D₆₇	—	8	2	no	—
171	#3396	D₆₇	1	1	yes	characteristic, complemented
172	#3380	—	67	1	no	—
173	#2242	—	67	2	no	—
174	#2240	—	67	1	no	—
175	#2228	—	67	2	no	—
176	#2227	—	67	1	no	—
177	#2210	—	67	2	no	—
178	#2206	—	67	1	no	—
179	Q₈×C₁₆	—	67	1	no	—
180	C₄×SD₃₂	—	67	1	no	—
181	C₄×D₁₆	—	67	2	no	—
182	C₆₇	#33928	1	1	yes	characteristic, complemented, Sylow
183	#559	—	2	1	no	—
184	#554	—	134	1	no	—
185	C₂×SD₃₂	—	67	1	no	—
186	C₂×D₁₆	—	67	2	no	—
187	#500	—	67	1	no	—
188	#499	—	67	2	no	—
189	#495	C₂×D₆₇	1	2	yes	—
190	#493	C₂×D₆₇	1	1	yes	characteristic
191	#478	—	134	2	no	—
192	#476	—	134	1	no	—
193	#463	—	2	2	no	—
194	#460	—	2	1	no	—
195	C₈×Q₈	C₂×D₆₇	1	1	yes	characteristic
196	C₄×Q₁₆	C₂×D₆₇	1	1	yes	characteristic
197	C₄×D₈	—	67	2	no	—
198	C₄×D₈	—	67	1	no	—
199	C₄×D₈	C₂×D₆₇	1	2	yes	—
200	#367	—	67	1	no	—
201	#366	—	67	2	no	—
202	C₄⋊C₁₆	—	67	1	no	—
203	C₄⋊C₁₆	—	67	2	no	—
204	#358	—	67	1	no	—
205	#358	—	67	2	no	—
206	#357	—	67	2	no	—
207	#357	—	67	2	no	—
208	#357	—	67	2	no	—
209	#357	—	67	1	no	—
210	C₄×C₁₆	—	67	2	no	—
211	C₄×C₁₆	—	67	1	no	—
212	C₂×Q₁₆	C₂²×D₆₇	1	1	yes	characteristic
213	C₂×SD₁₆	—	2	2	no	—
214	C₂×D₈	—	2	1	no	—
215	C₂×D₈	—	67	1	no	—
216	C₂×D₈	C₂²×D₆₇	1	2	yes	—
217	C₄⋊Q₈	—	134	1	no	—
218	#127	—	2	1	no	—
219	#125	—	134	2	no	—
220	#124	—	2	2	no	—
221	#124	—	4	1	no	—
222	#123	—	4	2	no	—
223	C₂²⋊Q₈	—	268	1	no	—
224	C₂²⋊Q₈	—	268	2	no	—
225	C₄×Q₈	—	2	1	no	—
226	C₄×Q₈	C₂²⋊D₆₇	1	1	yes	characteristic
227	C₄×D₄	—	2	2	no	—
228	C₄×D₄	—	4	1	no	—
229	C₄×D₄	—	134	1	no	—
230	C₄×D₄	—	134	2	no	—
231	C₂×C₄⋊C₄	—	268	1	no	—
232	SD₃₂	—	134	2	no	—
233	D₁₆	—	134	4	no	—
234	C₂×C₁₆	—	67	1	no	—
235	C₂×C₁₆	—	67	1	no	—
236	C₂×C₁₆	—	67	2	no	—
237	#107	—	67	2	no	—
238	#107	—	67	1	no	—
239	#107	—	134	1	no	—
240	#107	C₂²×D₆₇	1	1	yes	characteristic
241	#106	—	134	2	no	—
242	C₄⋊C₈	C₂²⋊D₆₇	1	1	yes	characteristic
243	C₄⋊C₈	C₂²⋊D₆₇	1	2	yes	—
244	#103	—	2	1	no	—
245	#102	—	2	2	no	—
246	#102	—	134	2	no	—
247	#102	—	134	1	no	—
248	C₄×C₈	C₂²×D₆₇	1	2	yes	—
249	C₄×C₈	C₂²×D₆₇	1	1	yes	characteristic
250	C₂×Q₈	—	2	1	no	—
251	C₂×Q₈	—	268	1	no	—
252	C₂×Q₈	#11761613	1	1	yes	characteristic
253	C₂×D₄	—	2	2	no	—
254	C₂×D₄	—	4	1	no	—
255	C₂×D₄	—	4	1	no	—
256	C₂×D₄	—	134	1	no	—
257	C₂²×C₄	—	4	2	no	—
258	C₂²×C₄	—	268	1	no	—
259	C₂²×C₄	—	268	2	no	—
260	Q₁₆	—	2	1	no	—
261	Q₁₆	—	2	1	no	—
262	SD₁₆	—	8	2	no	—
263	D₈	—	2	2	no	—
264	D₈	—	2	2	no	—
265	D₈	—	8	1	no	—
266	D₈	—	67	2	no	—
267	D₈	—	134	1	no	—
268	C₂×C₈	C₂×C₂²⋊D₆₇	1	2	yes	—
269	C₂×C₈	C₂×C₂²⋊D₆₇	1	1	yes	characteristic
270	C₂×C₈	C₂³×D₆₇	1	1	yes	characteristic
271	C₄⋊C₄	—	2	1	no	—
272	C₄⋊C₄	—	4	1	no	—
273	C₄⋊C₄	—	268	2	no	—
274	C₄⋊C₄	—	268	1	no	—
275	C₄⋊C₄	—	268	2	no	—
276	C₄⋊C₄	—	134	1	no	—
277	C₄⋊C₄	—	134	2	no	—
278	C₄⋊C₄	#11761613	1	1	yes	characteristic
279	C₄⋊C₄	#11761613	1	2	yes	—
280	C₂²⋊C₄	—	4	2	no	—
281	C₂²⋊C₄	—	4	2	no	—
282	C₂²⋊C₄	—	4	2	no	—
283	C₂²⋊C₄	—	268	1	no	—
284	C₂²⋊C₄	—	268	2	no	—
285	C₄²	—	4	1	no	—
286	C₄²	C₂×C₂²⋊D₆₇	1	2	yes	—
287	C₄²	C₂×C₂²⋊D₆₇	1	1	yes	characteristic
288	C₁₆	—	134	2	no	—
289	C₁₆	—	134	1	no	—
290	C₁₆	—	134	1	no	—
291	C₂³	—	4	2	no	—
292	C₂³	—	268	1	no	—
293	Q₈	—	2	2	no	—
294	Q₈	—	4	1	no	—
295	Q₈	—	4	1	no	—
296	Q₈	—	1072	1	no	—
297	D₄	—	4	2	no	—
298	D₄	—	4	2	no	—
299	D₄	—	8	2	no	—
300	D₄	—	268	1	no	—
301	D₄	—	16	1	no	—
302	D₄	—	134	2	no	—
303	C₂×C₄	—	4	1	no	—
304	C₂×C₄	—	4	1	no	—
305	C₂×C₄	—	268	1	no	—
306	C₂×C₄	—	268	2	no	—
307	C₂×C₄	—	16	2	no	—
308	C₂×C₄	—	536	2	no	—
309	C₂×C₄	—	1072	2	no	—
310	C₂×C₄	C₂×#11761613	1	2	yes	—
311	C₂×C₄	C₂×#11761613	1	2	yes	—
312	C₂×C₄	C₂×#11761613	1	1	yes	characteristic
313	C₂×C₄	C₂×#11761613	1	1	yes	characteristic
314	C₂×C₄	C₂²×C₂²⋊D₆₇	1	1	yes	characteristic
315	C₈	—	2	1	no	—
316	C₈	—	2	2	no	—
317	C₈	#41×D₆₇	1	1	yes	characteristic
318	C₈	#14524090	1	1	yes	characteristic
319	C₂²	—	8	2	no	—
320	C₂²	—	8	2	no	—
321	C₂²	—	8	2	no	—
322	C₂²	—	268	2	no	—
323	C₂²	—	536	1	no	—
324	C₂²	—	536	1	no	—
325	C₂²	C₂²×#11761613	1	1	yes	characteristic
326	C₄	—	2	2	no	—
327	C₄	—	2	2	no	—
328	C₄	—	2	1	no	—
329	C₄	—	8	1	no	—
330	C₄	—	8	1	no	—
331	C₄	—	536	1	no	—
332	C₄	—	536	2	no	—
333	C₄	#16863826	1	2	yes	—
334	C₄	#16864053	1	1	yes	characteristic
335	C₄	#16864207	1	1	yes	characteristic
336	C₂	—	16	2	no	—
337	C₂	—	536	2	no	—
338	C₂	#22572840	1	1	yes	characteristic
339	C₂	C₂×#16863826	1	1	yes	characteristic
340	C₂	#22573348	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂²×#11761613
commutator, cosocle	C₂×C₈×C₆₇	C₂⁴
Frattini	C₂×C₈	C₂³×D₆₇
Fitting	C₆₇×#3396	C₂
socle	C₂²×C₆₇	C₂²×D₈
prime radical	#22562241	C₂
hypercenter	#3396	D₆₇
hypocenter	C₆₇	#33928
LCS	G ▷ C₂×C₈×C₆₇ ▷ C₄×C₆₇ ▷ C₂×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂² ◁ C₄×Q₈ ◁ C₈×Q₈ ◁ #3396

Complements

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

N	H ≅ G/N	type
1. #22573077	C₂	semidirect
2. #22562440	C₂	semidirect
3. #22562432	C₂	semidirect
4. #22562366	C₂	semidirect
5. #22562361	C₂	semidirect
6. #22562261	C₂	semidirect
8. Q₈×C₆₇⋊C₁₆	C₂	semidirect
9. C₄×#14523986	C₂	semidirect
10. C₄×#14523984	C₂	semidirect
11. C₆₇×#3396	C₂	semidirect
171. #3396	D₆₇	semidirect
182. C₆₇	#33928	semidirect

Maximal subgroups

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

subgroup	index
#22573077	2
#22562440	2
#22562432	2
#22562366	2
#22562361	2
#22562261	2
#22562241	2
Q₈×C₆₇⋊C₁₆	2
C₄×#14523986	2
C₄×#14523984	2
C₆₇×#3396	2
#33928	67