Subgroups — #30839039

Order 17152.

774 conjugacy classes of subgroups.

The subgroup lattice has 338 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#22571152	C₂	1	2	yes	maximal, complemented
2	C₂×#16863773	C₂	1	1	yes	maximal, characteristic
3	#22569143	C₂	1	1	yes	maximal, characteristic, complemented
4	#22559694	C₂	1	4	yes	maximal, complemented
5	#22559570	C₂	1	4	yes	maximal, complemented
6	#22558876	C₂	1	2	yes	maximal, complemented
7	C₆₇×#1564	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂²×D₅₃₆	—	2	1	no	—
9	C₂²×D₅₃₆	—	2	1	no	—
10	#16863953	C₂²	1	1	yes	characteristic
11	C₂×#14524040	C₂²	1	2	yes	—
12	#16863773	—	2	1	no	—
13	#16863773	—	2	1	no	—
14	#16863773	C₂²	1	2	yes	—
15	#16863773	C₂²	1	2	yes	—
16	#16863769	—	2	2	no	—
17	#16863743	—	2	2	no	—
18	#16863597	C₂²	1	8	yes	complemented
19	#16863549	—	2	4	no	—
20	#16863526	—	2	1	no	—
21	#16863526	C₂²	1	2	yes	complemented
22	C₈×Dic₁₃₄	—	2	1	no	—
23	C₈×Dic₁₃₄	C₂²	1	2	yes	complemented
24	#16863422	—	2	1	no	—
25	#16863420	C₂²	1	1	yes	characteristic
26	#16863351	—	2	1	no	—
27	#16863139	C₂²	1	8	yes	complemented
28	#431×C₆₇	C₂²	1	2	yes	—
29	C₂×C₄×C₈×C₆₇	C₂²	1	1	yes	characteristic
30	#335×C₆₇	C₂²	1	2	yes	complemented
31	#334×C₆₇	C₂²	1	2	yes	complemented
32	C₂²×D₂₆₈	—	2	2	no	—
33	C₂²×D₂₆₈	—	2	2	no	—
34	C₂²×D₂₆₈	—	2	2	no	—
35	C₂×#11761617	—	2	8	no	—
36	C₂×D₅₃₆	—	2	2	no	—
37	C₂×D₅₃₆	—	2	2	no	—
38	C₂×D₅₃₆	—	2	2	no	—
39	C₂×D₅₃₆	—	2	2	no	—
40	C₂×D₅₃₆	—	4	4	no	—
41	C₂×D₅₃₆	—	4	4	no	—
42	C₂×#11761612	—	2	1	no	—
43	#14524043	D₄	1	1	yes	characteristic
44	#14524043	D₄	1	1	yes	characteristic
45	#14524040	—	2	2	no	—
46	#14524040	—	2	2	no	—
47	#14524040	—	2	2	no	—
48	#14524040	C₂³	1	4	yes	—
49	#14524035	—	2	1	no	—
50	#14524022	—	2	1	no	—
51	#14524022	D₄	1	2	yes	—
52	C₄×Dic₁₃₄	—	2	1	no	—
53	C₄×Dic₁₃₄	C₂³	1	2	yes	—
54	#14524002	—	2	1	no	—
55	C₄⋊(C₆₇⋊C₈)	—	2	1	no	—
56	C₄⋊(C₆₇⋊C₈)	C₂³	1	2	yes	—
57	Dic₆₇⋊C₈	—	2	2	no	—
58	Dic₆₇⋊C₈	—	2	2	no	—
59	#14523965	—	2	8	no	—
60	#14523953	—	2	2	no	—
61	C₈×Dic₆₇	—	2	2	no	—
62	C₂²×C₈×C₆₇	D₄	1	1	yes	characteristic
63	C₂²×C₈×C₆₇	D₄	1	1	yes	characteristic
64	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
65	C₄⋊C₈×C₆₇	C₂³	1	4	yes	—
66	C₂²⋊C₈×C₆₇	—	2	2	no	—
67	C₄×C₈×C₆₇	—	2	2	no	—
68	C₄×C₈×C₆₇	D₄	1	1	yes	characteristic
69	C₄×C₈×C₆₇	D₄	1	1	yes	characteristic
70	C₄×C₈×C₆₇	C₂³	1	1	yes	characteristic
71	C₄×C₈×C₆₇	C₂³	1	1	yes	characteristic
72	C₂³×D₆₇	—	4	2	no	—
73	C₂×Dic₁₃₄	—	2	1	no	—
74	C₂×Dic₁₃₄	C₂×D₄	1	2	yes	—
75	C₂×D₂₆₈	—	2	4	no	—
76	C₂×D₂₆₈	—	2	4	no	—
77	C₂×D₂₆₈	—	2	4	no	—
78	C₂×D₂₆₈	—	4	4	no	—
79	C₂×D₂₆₈	—	4	4	no	—
80	C₂×D₂₆₈	—	4	2	no	—
81	C₂×D₂₆₈	—	4	2	no	—
82	C₂×D₂₆₈	—	4	2	no	—
83	C₂×D₂₆₈	—	8	4	no	—
84	C₂×D₂₆₈	—	8	4	no	—
85	#11761617	—	8	8	no	—
86	D₅₃₆	—	4	4	no	—
87	D₅₃₆	—	4	4	no	—
88	D₅₃₆	—	4	4	no	—
89	D₅₃₆	—	4	4	no	—
90	#11761612	—	4	2	no	—
91	C₂×C₆₇⋊C₈	—	2	2	no	—
92	C₄⋊Dic₆₇	—	2	1	no	—
93	C₄⋊Dic₆₇	—	2	1	no	—
94	C₄⋊Dic₆₇	—	2	1	no	—
95	C₄⋊Dic₆₇	C₂×D₄	1	2	yes	—
96	Dic₆₇⋊C₄	—	2	2	no	—
97	Dic₆₇⋊C₄	—	2	2	no	—
98	C₂²⋊Dic₆₇	—	2	2	no	—
99	C₄×Dic₆₇	—	2	2	no	—
100	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
101	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
102	C₂²×C₄×C₆₇	#41	1	1	yes	characteristic
103	C₂×C₈×C₆₇	—	2	2	no	—
104	C₂×C₈×C₆₇	—	2	4	no	—
105	C₂×C₈×C₆₇	SD₁₆	1	4	yes	complemented
106	C₂×C₈×C₆₇	C₂×D₄	1	1	yes	characteristic
107	C₂×C₈×C₆₇	C₂×D₄	1	1	yes	characteristic
108	C₂×C₈×C₆₇	C₂×D₄	1	1	yes	characteristic
109	C₂×C₈×C₆₇	C₂×D₄	1	1	yes	characteristic
110	C₄²×C₆₇	C₂×D₄	1	1	yes	characteristic
111	C₄²×C₆₇	C₂×D₄	1	1	yes	characteristic
112	C₄²×C₆₇	#41	1	1	yes	characteristic
113	C₄²×C₆₇	C₂⁴	1	1	yes	characteristic
114	C₂³×C₆₇	#121	1	1	yes	characteristic
115	C₂²×D₆₇	—	4	4	no	—
116	C₂²×D₆₇	—	8	4	no	—
117	C₂²×D₆₇	—	8	4	no	—
118	C₂²×D₆₇	—	8	4	no	—
119	C₂×C₄×C₆₇	—	2	2	no	—
120	C₂×C₄×C₆₇	—	2	2	no	—
121	C₂×C₄×C₆₇	#121	1	1	yes	characteristic
122	C₂×C₄×C₆₇	#121	1	1	yes	characteristic
123	C₂×C₄×C₆₇	#121	1	1	yes	characteristic
124	C₂×C₄×C₆₇	C₂×SD₁₆	1	2	yes	—
125	C₂×C₄×C₆₇	AGL(1,ℤ/8)	1	2	yes	—
126	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
127	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
128	C₂×C₄×C₆₇	C₂×#41	1	1	yes	characteristic
129	C₂×Dic₆₇	—	2	2	no	—
130	C₂×Dic₆₇	—	2	2	no	—
131	D₂₆₈	—	4	4	no	—
132	D₂₆₈	—	4	4	no	—
133	D₂₆₈	—	4	8	no	—
134	D₂₆₈	—	8	4	no	—
135	D₂₆₈	—	8	4	no	—
136	D₂₆₈	—	8	4	no	—
137	D₂₆₈	—	8	4	no	—
138	Dic₁₃₄	—	2	4	no	—
139	Dic₁₃₄	—	4	2	no	—
140	C₈×C₆₇	—	2	1	no	—
141	C₈×C₆₇	—	2	1	no	—
142	C₈×C₆₇	—	4	4	no	—
143	C₈×C₆₇	C₂×SD₁₆	1	2	yes	—
144	C₈×C₆₇	C₂×SD₁₆	1	2	yes	—
145	C₆₇⋊C₈	—	4	2	no	—
146	C₂²×C₆₇	—	2	1	no	—
147	C₂²×C₆₇	—	2	1	no	—
148	C₂²×C₆₇	#460	1	2	yes	—
149	C₂²×C₆₇	C₂×#121	1	1	yes	characteristic
150	C₂×D₆₇	—	8	4	no	—
151	C₂×D₆₇	—	8	4	no	—
152	C₂×D₆₇	—	8	4	no	—
153	C₂×D₆₇	—	16	8	no	—
154	C₄×C₆₇	—	2	1	no	—
155	C₄×C₆₇	—	2	1	no	—
156	C₄×C₆₇	#460	1	2	yes	—
157	C₄×C₆₇	#472	1	2	yes	—
158	C₄×C₆₇	C₂²×SD₁₆	1	1	yes	characteristic
159	C₄×C₆₇	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
160	C₄×C₆₇	#576	1	2	yes	—
161	Dic₆₇	—	4	2	no	—
162	Dic₆₇	—	4	2	no	—
163	#19203	—	67	1	no	maximal, Sylow
164	C₂×C₆₇	—	2	2	no	—
165	C₂×C₆₇	C₂×#460	1	1	yes	characteristic
166	C₂×C₆₇	#3058	1	1	yes	characteristic
167	C₂×C₆₇	C₂×#472	1	1	yes	characteristic
168	D₆₇	—	16	4	no	—
169	#3208	—	67	2	no	—
170	C₂×#493	—	67	1	no	—
171	#2960	—	67	1	no	—
172	#1688	—	67	4	no	—
173	#1668	—	67	4	no	—
174	#1564	D₆₇	1	1	yes	characteristic, complemented
175	#1545	—	67	2	no	—
176	C₆₇	#19203	1	1	yes	characteristic, complemented, Sylow
177	C₂²×D₈	—	134	1	no	—
178	C₂²×D₈	—	134	1	no	—
179	#533	—	67	1	no	—
180	C₂×#127	—	67	2	no	—
181	#493	—	67	2	no	—
182	#493	—	67	2	no	—
183	#493	—	134	1	no	—
184	#493	—	134	1	no	—
185	#492	—	134	2	no	—
186	#486	—	134	2	no	—
187	#460	—	67	8	no	—
188	#450	—	134	4	no	—
189	#446	—	67	2	no	—
190	#446	—	134	1	no	—
191	C₈×Q₈	—	67	2	no	—
192	C₈×Q₈	—	134	1	no	—
193	#431	C₂×D₆₇	1	2	yes	—
194	#424	—	134	1	no	—
195	#423	—	67	1	no	—
196	#405	—	134	1	no	—
197	C₂×C₄×C₈	C₂×D₆₇	1	1	yes	characteristic
198	#335	C₂×D₆₇	1	2	yes	complemented
199	#334	C₂×D₆₇	1	2	yes	complemented
200	#331	—	67	8	no	—
201	C₂²×D₄	—	134	2	no	—
202	C₂²×D₄	—	134	2	no	—
203	C₂²×D₄	—	134	2	no	—
204	C₂×SD₁₆	—	134	8	no	—
205	C₂×D₈	—	268	4	no	—
206	C₂×D₈	—	268	4	no	—
207	C₂×D₈	—	134	2	no	—
208	C₂×D₈	—	134	2	no	—
209	C₂×D₈	—	134	2	no	—
210	C₂×D₈	—	134	2	no	—
211	C₂×M₁₆	—	134	1	no	—
212	C₂²×C₈	D₂₆₈	1	1	yes	characteristic
213	C₂²×C₈	D₂₆₈	1	1	yes	characteristic
214	C₄⋊Q₈	—	67	1	no	—
215	C₄⋊Q₈	—	67	1	no	—
216	#127	—	67	4	no	—
217	#127	—	134	2	no	—
218	#127	—	134	2	no	—
219	#127	—	134	2	no	—
220	#125	—	134	1	no	—
221	C₂²⋊Q₈	—	67	2	no	—
222	C₂²⋊Q₈	—	134	1	no	—
223	C₄×Q₈	—	67	2	no	—
224	C₄×Q₈	—	134	1	no	—
225	#117	—	134	1	no	—
226	C₂×C₄²	C₂²×D₆₇	1	1	yes	characteristic
227	C₄⋊C₈	—	67	2	no	—
228	C₄⋊C₈	—	134	2	no	—
229	C₄⋊C₈	—	134	2	no	—
230	C₄⋊C₈	—	134	1	no	—
231	C₄⋊C₈	C₂²×D₆₇	1	4	yes	—
232	#102	—	134	8	no	—
233	C₂²⋊C₈	—	2	2	no	—
234	#97	—	134	2	no	—
235	C₄×C₈	—	2	2	no	—
236	C₄×C₈	—	134	2	no	—
237	C₄×C₈	D₂₆₈	1	1	yes	characteristic
238	C₄×C₈	D₂₆₈	1	1	yes	characteristic
239	C₄×C₈	C₂²×D₆₇	1	1	yes	characteristic
240	C₄×C₈	C₂²×D₆₇	1	1	yes	characteristic
241	C₂⁴	—	268	2	no	—
242	C₂×Q₈	—	67	2	no	—
243	C₂×Q₈	—	134	1	no	—
244	C₂×D₄	—	268	2	no	—
245	C₂×D₄	—	268	2	no	—
246	C₂×D₄	—	268	2	no	—
247	C₂×D₄	—	268	4	no	—
248	C₂×D₄	—	268	4	no	—
249	C₂×D₄	—	536	4	no	—
250	C₂×D₄	—	536	4	no	—
251	C₂×D₄	—	134	4	no	—
252	C₂×D₄	—	134	4	no	—
253	C₂×D₄	—	134	4	no	—
254	C₂²×C₄	D₄×D₆₇	1	1	yes	characteristic
255	C₂²×C₄	C₂×D₂₆₈	1	1	yes	characteristic
256	C₂²×C₄	#11761629	1	1	yes	characteristic
257	SD₁₆	—	536	8	no	—
258	D₈	—	268	4	no	—
259	D₈	—	268	4	no	—
260	D₈	—	268	4	no	—
261	D₈	—	268	4	no	—
262	M₁₆	—	268	2	no	—
263	C₂×C₈	—	2	2	no	—
264	C₂×C₈	—	2	4	no	—
265	C₂×C₈	—	134	2	no	—
266	C₂×C₈	#11761617	1	4	yes	complemented
267	C₂×C₈	C₂×D₂₆₈	1	1	yes	characteristic
268	C₂×C₈	C₂×D₂₆₈	1	1	yes	characteristic
269	C₂×C₈	C₂×D₂₆₈	1	1	yes	characteristic
270	C₂×C₈	C₂×D₂₆₈	1	1	yes	characteristic
271	C₄⋊C₄	—	67	2	no	—
272	C₄⋊C₄	—	134	1	no	—
273	C₄⋊C₄	—	134	1	no	—
274	C₄⋊C₄	—	134	2	no	—
275	C₄⋊C₄	—	134	1	no	—
276	C₄⋊C₄	—	134	2	no	—
277	C₂²⋊C₄	—	134	2	no	—
278	C₄²	—	134	2	no	—
279	C₄²	D₄×D₆₇	1	1	yes	characteristic
280	C₄²	C₂×D₂₆₈	1	1	yes	characteristic
281	C₄²	#11761629	1	1	yes	characteristic
282	C₄²	C₂³×D₆₇	1	1	yes	characteristic
283	C₂³	—	268	4	no	—
284	C₂³	—	536	4	no	—
285	C₂³	—	536	4	no	—
286	C₂³	—	536	4	no	—
287	C₂³	#14524018	1	1	yes	characteristic
288	Q₈	—	268	2	no	—
289	Q₈	—	134	4	no	—
290	D₄	—	268	4	no	—
291	D₄	—	268	4	no	—
292	D₄	—	268	8	no	—
293	D₄	—	536	4	no	—
294	D₄	—	536	4	no	—
295	D₄	—	536	4	no	—
296	D₄	—	536	4	no	—
297	C₂×C₄	—	2	2	no	—
298	C₂×C₄	—	2	2	no	—
299	C₂×C₄	—	134	2	no	—
300	C₂×C₄	—	134	2	no	—
301	C₂×C₄	#14524018	1	1	yes	characteristic
302	C₂×C₄	#14524018	1	1	yes	characteristic
303	C₂×C₄	#14524018	1	1	yes	characteristic
304	C₂×C₄	C₂×#11761617	1	2	yes	—
305	C₂×C₄	#14524071	1	2	yes	—
306	C₂×C₄	C₂×D₄×D₆₇	1	1	yes	characteristic
307	C₂×C₄	C₂²×D₂₆₈	1	1	yes	characteristic
308	C₂×C₄	C₂×#11761629	1	1	yes	characteristic
309	C₈	—	2	1	no	—
310	C₈	—	2	1	no	—
311	C₈	—	4	4	no	—
312	C₈	—	268	2	no	—
313	C₈	C₂×#11761617	1	2	yes	—
314	C₈	C₂×#11761617	1	2	yes	—
315	C₂²	—	2	1	no	—
316	C₂²	—	2	1	no	—
317	C₂²	—	536	4	no	—
318	C₂²	—	536	4	no	—
319	C₂²	—	536	4	no	—
320	C₂²	—	1072	8	no	—
321	C₂²	#16863597	1	2	yes	—
322	C₂²	C₂×#14524018	1	1	yes	characteristic
323	C₄	—	2	1	no	—
324	C₄	—	2	1	no	—
325	C₄	—	268	2	no	—
326	C₄	—	268	2	no	—
327	C₄	#16863597	1	2	yes	—
328	C₄	#16863667	1	2	yes	—
329	C₄	C₂²×#11761617	1	1	yes	characteristic
330	C₄	C₂×#14524071	1	1	yes	characteristic
331	C₄	#16864198	1	2	yes	—
332	C₂	—	2	2	no	—
333	C₂	—	1072	4	no	—
334	C₂	C₂×#16863597	1	1	yes	characteristic
335	C₂	#22569858	1	1	yes	characteristic
336	C₂	C₂×#16863667	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×#14524018
commutator, cosocle	C₄²×C₆₇	C₂⁴
Frattini	C₄²	C₂³×D₆₇
Fitting	C₆₇×#1564	C₂
socle	C₂²×C₆₇	C₂×#121
prime radical	C₂×#16863773	C₂
hypercenter	#1564	D₆₇
hypocenter	C₆₇	#19203
LCS	G ▷ C₄²×C₆₇ ▷ C₂²×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #1564

Complements

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

N	H ≅ G/N	type
1. #22571152	C₂	semidirect
3. #22569143	C₂	semidirect
4. #22559694	C₂	semidirect
5. #22559570	C₂	semidirect
6. #22558876	C₂	semidirect
7. C₆₇×#1564	C₂	semidirect
18. #16863597	C₂²	semidirect
21. #16863526	C₂²	semidirect
23. C₈×Dic₁₃₄	C₂²	semidirect
27. #16863139	C₂²	semidirect
30. #335×C₆₇	C₂²	semidirect
31. #334×C₆₇	C₂²	semidirect
105. C₂×C₈×C₆₇	SD₁₆	semidirect
174. #1564	D₆₇	semidirect
176. C₆₇	#19203	semidirect
198. #335	C₂×D₆₇	semidirect
199. #334	C₂×D₆₇	semidirect
266. C₂×C₈	#11761617	semidirect

Maximal subgroups

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

subgroup	index
#22571152	2
C₂×#16863773	2
#22569143	2
#22559694	2
#22559570	2
#22558876	2
C₆₇×#1564	2
#19203	67