Subgroups — #31011128

Order 17152.

598 conjugacy classes of subgroups.

The subgroup lattice has 340 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#22570720	C₂	1	1	yes	maximal, characteristic, complemented
2	#22570636	C₂	1	1	yes	maximal, characteristic
3	#22562439	C₂	1	4	yes	maximal, complemented
4	#22562438	C₂	1	1	yes	maximal, characteristic, complemented
5	#22562437	C₂	1	1	yes	maximal, characteristic, complemented
6	#22562432	C₂	1	1	yes	maximal, characteristic, complemented
7	#22561913	C₂	1	4	yes	maximal, complemented
8	C₄×C₁₆×D₆₇	C₂	1	1	yes	maximal, characteristic, complemented
9	C₆₇×#2240	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂×#14524062	—	2	1	no	—
11	C₂×#14524062	C₂²	1	2	yes	—
12	#16863951	—	2	1	no	—
13	#16863944	—	2	1	no	—
14	C₂×C₁₆×D₆₇	C₂²	1	2	yes	—
15	#16863801	C₂²	1	1	yes	characteristic
16	#16863797	—	2	1	no	—
17	#16863797	C₂²	1	2	yes	—
18	#16863792	—	2	1	no	—
19	#16863779	—	2	1	no	—
20	#16863779	C₂²	1	2	yes	—
21	#16863776	C₂²	1	1	yes	characteristic
22	#16863768	—	2	1	no	—
23	#16863433	—	2	1	no	—
24	#16863433	C₂²	1	2	yes	—
25	C₄×C₈×D₆₇	C₂²	1	1	yes	characteristic
26	#16863201	C₂²	1	4	yes	—
27	#16863199	C₂²	1	4	yes	—
28	#16863197	C₂²	1	4	yes	—
29	C₄×C₆₇⋊C₁₆	C₂²	1	1	yes	characteristic
30	C₁₆×Dic₆₇	C₂²	1	2	yes	—
31	#500×C₆₇	C₂²	1	1	yes	characteristic
32	#493×C₆₇	C₂²	1	1	yes	characteristic
33	#357×C₆₇	C₂²	1	4	yes	—
34	C₄×C₁₆×C₆₇	C₂²	1	1	yes	characteristic
35	C₂×#11761627	—	2	2	no	—
36	#14524062	—	2	8	no	—
37	#14524062	—	4	4	no	—
38	C₂×Dic₂₆₈	—	2	1	no	—
39	C₂×Dic₂₆₈	C₂³	1	2	yes	—
40	C₂×#11761616	—	2	2	no	—
41	C₂×#11761616	—	2	2	no	—
42	C₂×C₈×D₆₇	D₄	1	1	yes	characteristic
43	C₂×C₈×D₆₇	C₂³	1	1	yes	characteristic
44	#14524043	—	2	1	no	—
45	#14524043	—	2	1	no	—
46	#14524034	—	2	2	no	—
47	#14524030	—	2	2	no	—
48	#14524023	—	4	1	no	—
49	#14524019	—	4	1	no	—
50	#14524014	—	4	1	no	—
51	#14524014	—	4	1	no	—
52	Q₈×Dic₆₇	—	4	1	no	—
53	#14524008	—	4	1	no	—
54	D₄×Dic₆₇	—	4	1	no	—
55	D₄×Dic₆₇	—	4	1	no	—
56	#14524001	—	2	2	no	—
57	C₄²×D₆₇	D₄	1	1	yes	characteristic
58	C₂×C₆₇⋊C₁₆	C₂³	1	2	yes	—
59	C₁₆×D₆₇	—	2	4	no	—
60	#14523977	—	2	1	no	—
61	#14523977	C₂³	1	2	yes	—
62	#14523974	—	2	2	no	—
63	#14523974	—	2	2	no	—
64	C₄×C₆₇⋊C₈	D₄	1	1	yes	characteristic
65	C₈×Dic₆₇	D₄	1	1	yes	characteristic
66	C₈×Dic₆₇	C₂³	1	1	yes	characteristic
67	C₂×D₈×C₆₇	—	2	1	no	—
68	C₂×D₈×C₆₇	C₂³	1	2	yes	—
69	C₄⋊Q₈×C₆₇	—	2	1	no	—
70	#127×C₆₇	—	2	1	no	—
71	C₂×C₁₆×C₆₇	C₂³	1	2	yes	—
72	#107×C₆₇	—	2	1	no	—
73	#107×C₆₇	C₂³	1	2	yes	—
74	C₄×C₈×C₆₇	C₂³	1	1	yes	characteristic
75	#11761627	—	4	4	no	—
76	#11761627	—	4	4	no	—
77	C₂×Dic₁₃₄	—	2	2	no	—
78	C₂×Dic₁₃₄	—	4	1	no	—
79	C₂×C₂²⋊D₆₇	—	4	2	no	—
80	C₂²×Dic₆₇	—	4	2	no	—
81	C₂×C₄×D₆₇	D₈	1	1	yes	characteristic
82	C₂×C₄×D₆₇	D₈	1	1	yes	characteristic
83	C₂×C₄×D₆₇	C₂×D₄	1	1	yes	characteristic
84	Dic₂₆₈	—	2	4	no	—
85	Dic₂₆₈	—	4	2	no	—
86	#11761616	—	4	4	no	—
87	#11761616	—	4	4	no	—
88	C₂×C₆₇⋊C₈	C₂×D₄	1	1	yes	characteristic
89	C₂×C₆₇⋊C₈	C₂×D₄	1	1	yes	characteristic
90	C₈×D₆₇	SD₁₆	1	4	yes	—
91	C₈×D₆₇	#41	1	4	yes	—
92	C₄⋊Dic₆₇	—	2	2	no	—
93	C₄⋊Dic₆₇	—	4	1	no	—
94	C₄⋊Dic₆₇	—	4	1	no	—
95	C₄⋊Dic₆₇	—	4	1	no	—
96	C₄⋊Dic₆₇	—	4	2	no	—
97	Dic₆₇⋊C₄	—	4	2	no	—
98	C₂²⋊Dic₆₇	—	4	2	no	—
99	C₂²⋊Dic₆₇	—	4	2	no	—
100	(C₂×D₆₇)⋊C₄	—	4	2	no	—
101	C₄×Dic₆₇	—	4	2	no	—
102	C₄×Dic₆₇	D₈	1	1	yes	characteristic
103	C₄×Dic₆₇	D₈	1	1	yes	characteristic
104	C₄×Dic₆₇	C₂×D₄	1	1	yes	characteristic
105	C₆₇⋊C₁₆	—	2	2	no	—
106	C₂×Q₈×C₆₇	—	4	1	no	—
107	C₂×D₄×C₆₇	—	2	2	no	—
108	C₂×D₄×C₆₇	—	4	1	no	—
109	C₂×D₄×C₆₇	—	4	1	no	—
110	D₈×C₆₇	—	2	4	no	—
111	D₈×C₆₇	—	4	2	no	—
112	C₂×C₈×C₆₇	C₂×D₄	1	1	yes	characteristic
113	C₂×C₈×C₆₇	C₂⁴	1	1	yes	characteristic
114	C₄⋊C₄×C₆₇	—	2	2	no	—
115	C₄⋊C₄×C₆₇	—	4	1	no	—
116	C₄²×C₆₇	C₂×D₄	1	1	yes	characteristic
117	C₁₆×C₆₇	—	2	2	no	—
118	C₂³×C₆₇	—	4	2	no	—
119	C₂²×D₆₇	C₂×D₈	1	1	yes	characteristic
120	Q₈×C₆₇	—	8	2	no	—
121	D₄×C₆₇	—	4	2	no	—
122	D₄×C₆₇	—	4	2	no	—
123	D₄×C₆₇	—	8	2	no	—
124	D₄×C₆₇	—	8	2	no	—
125	C₂×C₄×C₆₇	—	4	2	no	—
126	C₂×C₄×C₆₇	C₂×D₈	1	1	yes	characteristic
127	C₂×C₄×C₆₇	C₂×D₈	1	1	yes	characteristic
128	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
129	C₂²⋊D₆₇	—	8	4	no	—
130	C₂×Dic₆₇	—	4	2	no	—
131	C₂×Dic₆₇	—	4	2	no	—
132	C₂×Dic₆₇	—	8	4	no	—
133	C₂×Dic₆₇	C₂×D₈	1	1	yes	characteristic
134	C₂×Dic₆₇	C₂×D₈	1	1	yes	characteristic
135	C₂×Dic₆₇	C₂×D₈	1	1	yes	characteristic
136	C₄×D₆₇	—	2	2	no	—
137	C₄×D₆₇	—	2	2	no	—
138	C₄×D₆₇	#124	1	2	yes	—
139	C₄×D₆₇	C₂×SD₁₆	1	2	yes	—
140	Dic₁₃₄	—	4	2	no	—
141	Dic₁₃₄	—	4	2	no	—
142	Dic₁₃₄	—	8	2	no	—
143	C₈×C₆₇	C₂×SD₁₆	1	2	yes	—
144	C₈×C₆₇	C₂×#41	1	2	yes	—
145	C₆₇⋊C₈	#124	1	2	yes	—
146	C₆₇⋊C₈	C₂×SD₁₆	1	2	yes	—
147	C₂²×C₆₇	—	8	2	no	—
148	C₂²×C₆₇	—	8	2	no	—
149	C₂²×C₆₇	—	8	2	no	—
150	C₂²×C₆₇	C₂²×D₈	1	1	yes	characteristic
151	C₂×D₆₇	—	2	1	no	—
152	C₂×D₆₇	—	2	1	no	—
153	C₂×D₆₇	#486	1	2	yes	—
154	C₄×C₆₇	—	8	2	no	—
155	C₄×C₆₇	#508	1	2	yes	—
156	C₄×C₆₇	#508	1	2	yes	—
157	C₄×C₆₇	C₂×#124	1	1	yes	characteristic
158	C₄×C₆₇	C₂²×SD₁₆	1	1	yes	characteristic
159	Dic₆₇	—	2	1	no	—
160	Dic₆₇	—	2	1	no	—
161	Dic₆₇	—	8	2	no	—
162	Dic₆₇	—	8	2	no	—
163	Dic₆₇	#486	1	2	yes	—
164	#33697	—	67	1	no	maximal, Sylow
165	C₂×C₆₇	—	16	2	no	—
166	C₂×C₆₇	C₂×#486	1	1	yes	characteristic
167	C₂×C₆₇	C₂×#508	1	1	yes	characteristic
168	C₂×C₆₇	C₂×#508	1	1	yes	characteristic
169	D₆₇	—	2	2	no	—
170	#3160	—	67	1	no	—
171	#3147	—	67	1	no	—
172	#2242	—	67	4	no	—
173	#2241	—	67	1	no	—
174	#2241	—	67	1	no	—
175	#2240	—	67	1	no	—
176	#2240	D₆₇	1	1	yes	characteristic, complemented
177	#2138	—	67	4	no	—
178	C₂×C₄×C₁₆	—	67	1	no	—
179	C₆₇	#33697	1	1	yes	characteristic, complemented, Sylow
180	C₂×#135	—	67	2	no	—
181	C₂×#135	—	134	1	no	—
182	#533	—	134	1	no	—
183	#532	—	134	1	no	—
184	C₂²×C₁₆	—	67	2	no	—
185	#500	—	67	1	no	—
186	#500	C₂×D₆₇	1	1	yes	characteristic
187	#499	—	67	2	no	—
188	#499	—	134	1	no	—
189	#498	—	134	1	no	—
190	#495	—	67	2	no	—
191	#495	—	134	1	no	—
192	#494	—	67	1	no	—
193	#493	C₂×D₆₇	1	1	yes	characteristic
194	#492	—	134	1	no	—
195	#427	—	67	2	no	—
196	#427	—	134	1	no	—
197	C₂×C₄×C₈	—	67	1	no	—
198	#358	—	67	4	no	—
199	#358	—	67	4	no	—
200	#357	—	67	4	no	—
201	#357	C₂×D₆₇	1	4	yes	—
202	C₄×C₁₆	—	67	2	no	—
203	C₄×C₁₆	—	67	1	no	—
204	C₄×C₁₆	C₂×D₆₇	1	1	yes	characteristic
205	C₂×#41	—	134	2	no	—
206	#135	—	268	4	no	—
207	#135	—	134	8	no	—
208	C₂×Q₁₆	—	67	2	no	—
209	C₂×Q₁₆	—	134	1	no	—
210	C₂×SD₁₆	—	134	2	no	—
211	C₂×SD₁₆	—	134	2	no	—
212	C₂×D₈	—	2	1	no	—
213	C₂×D₈	C₂²×D₆₇	1	2	yes	—
214	C₂²×C₈	—	67	1	no	—
215	C₂²×C₈	—	67	1	no	—
216	C₄⋊Q₈	—	2	1	no	—
217	C₄⋊Q₈	—	134	1	no	—
218	C₄⋊Q₈	—	134	1	no	—
219	#127	—	2	1	no	—
220	#125	—	134	2	no	—
221	#124	—	134	2	no	—
222	C₂²⋊Q₈	—	268	1	no	—
223	C₂²⋊Q₈	—	268	1	no	—
224	#121	—	268	1	no	—
225	#121	—	268	1	no	—
226	C₄×Q₈	—	268	1	no	—
227	C₄×Q₈	—	268	1	no	—
228	C₄×D₄	—	268	1	no	—
229	C₄×D₄	—	268	1	no	—
230	#117	—	134	2	no	—
231	C₂×C₄²	—	67	1	no	—
232	C₂×C₁₆	—	67	2	no	—
233	C₂×C₁₆	—	134	4	no	—
234	C₂×C₁₆	C₂²×D₆₇	1	2	yes	—
235	#107	—	2	1	no	—
236	#107	—	67	2	no	—
237	#107	—	134	1	no	—
238	#107	C₂²×D₆₇	1	2	yes	—
239	#106	—	134	2	no	—
240	#106	—	134	2	no	—
241	C₄×C₈	—	67	1	no	—
242	C₄×C₈	—	67	1	no	—
243	C₄×C₈	—	67	1	no	—
244	C₄×C₈	C₂²×D₆₇	1	1	yes	characteristic
245	#41	—	268	4	no	—
246	#41	—	268	4	no	—
247	C₂×Q₈	—	4	1	no	—
248	C₂×Q₈	—	268	1	no	—
249	C₂×Q₈	—	134	2	no	—
250	C₂×D₄	—	2	2	no	—
251	C₂×D₄	—	4	1	no	—
252	C₂×D₄	—	4	1	no	—
253	C₂×D₄	—	268	2	no	—
254	C₂²×C₄	—	268	2	no	—
255	C₂²×C₄	—	67	1	no	—
256	C₂²×C₄	—	67	1	no	—
257	C₂²×C₄	—	67	1	no	—
258	Q₁₆	—	268	2	no	—
259	Q₁₆	—	134	4	no	—
260	SD₁₆	—	268	4	no	—
261	SD₁₆	—	268	4	no	—
262	D₈	—	2	4	no	—
263	D₈	—	4	2	no	—
264	C₂×C₈	—	67	1	no	—
265	C₂×C₈	—	67	1	no	—
266	C₂×C₈	—	67	4	no	—
267	C₂×C₈	—	67	4	no	—
268	C₂×C₈	D₄×D₆₇	1	1	yes	characteristic
269	C₂×C₈	C₂³×D₆₇	1	1	yes	characteristic
270	C₄⋊C₄	—	2	2	no	—
271	C₄⋊C₄	—	4	1	no	—
272	C₄⋊C₄	—	268	2	no	—
273	C₄⋊C₄	—	268	2	no	—
274	C₄⋊C₄	—	268	1	no	—
275	C₄⋊C₄	—	268	1	no	—
276	C₄⋊C₄	—	268	1	no	—
277	C₄⋊C₄	—	134	2	no	—
278	C₂²⋊C₄	—	268	2	no	—
279	C₂²⋊C₄	—	268	2	no	—
280	C₂²⋊C₄	—	268	2	no	—
281	C₄²	—	268	2	no	—
282	C₄²	—	67	1	no	—
283	C₄²	—	67	1	no	—
284	C₄²	—	67	1	no	—
285	C₄²	D₄×D₆₇	1	1	yes	characteristic
286	C₁₆	—	2	2	no	—
287	C₁₆	—	134	2	no	—
288	C₂³	—	4	2	no	—
289	C₂³	—	67	1	no	—
290	Q₈	—	8	2	no	—
291	Q₈	—	268	2	no	—
292	Q₈	—	268	2	no	—
293	Q₈	—	536	2	no	—
294	D₄	—	4	2	no	—
295	D₄	—	4	2	no	—
296	D₄	—	8	2	no	—
297	D₄	—	8	2	no	—
298	D₄	—	536	4	no	—
299	C₂×C₄	—	4	2	no	—
300	C₂×C₄	—	268	2	no	—
301	C₂×C₄	—	268	2	no	—
302	C₂×C₄	—	536	4	no	—
303	C₂×C₄	—	67	1	no	—
304	C₂×C₄	—	67	2	no	—
305	C₂×C₄	—	67	2	no	—
306	C₂×C₄	—	67	1	no	—
307	C₂×C₄	—	67	1	no	—
308	C₂×C₄	—	134	2	no	—
309	C₂×C₄	—	134	2	no	—
310	C₂×C₄	D₈×D₆₇	1	1	yes	characteristic
311	C₂×C₄	D₈×D₆₇	1	1	yes	characteristic
312	C₂×C₄	C₂×D₄×D₆₇	1	1	yes	characteristic
313	C₈	—	67	2	no	—
314	C₈	—	67	2	no	—
315	C₈	SD₁₆×D₆₇	1	2	yes	—
316	C₈	#41×D₆₇	1	2	yes	—
317	C₂²	—	8	2	no	—
318	C₂²	—	8	2	no	—
319	C₂²	—	8	2	no	—
320	C₂²	—	67	2	no	—
321	C₂²	—	134	1	no	—
322	C₂²	—	134	1	no	—
323	C₂²	C₂×D₈×D₆₇	1	1	yes	characteristic
324	C₄	—	8	2	no	—
325	C₄	—	536	2	no	—
326	C₄	—	536	2	no	—
327	C₄	—	67	2	no	—
328	C₄	—	134	1	no	—
329	C₄	—	134	1	no	—
330	C₄	#16863825	1	2	yes	—
331	C₄	#16863827	1	2	yes	—
332	C₄	#124×D₆₇	1	1	yes	characteristic
333	C₄	C₂×SD₁₆×D₆₇	1	1	yes	characteristic
334	C₂	—	16	2	no	—
335	C₂	—	134	2	no	—
336	C₂	#486×D₆₇	1	1	yes	characteristic
337	C₂	C₂×#16863825	1	1	yes	characteristic
338	C₂	C₂×#16863827	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×D₈×D₆₇
commutator, cosocle	C₂×C₈×C₆₇	C₂⁴
Frattini	C₂×C₈	C₂³×D₆₇
Fitting	C₆₇×#2240	C₂
socle	C₂²×C₆₇	C₂²×D₈
prime radical	#22570636	C₂
hypercenter	#2240	D₆₇
hypocenter	C₆₇	#33697
LCS	G ▷ C₂×C₈×C₆₇ ▷ C₄×C₆₇ ▷ C₂×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₄×C₈ ◁ #2240

Complements

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

N	H ≅ G/N	type
1. #22570720	C₂	semidirect
3. #22562439	C₂	semidirect
4. #22562438	C₂	semidirect
5. #22562437	C₂	semidirect
6. #22562432	C₂	semidirect
7. #22561913	C₂	semidirect
8. C₄×C₁₆×D₆₇	C₂	semidirect
9. C₆₇×#2240	C₂	semidirect
176. #2240	D₆₇	semidirect
179. C₆₇	#33697	semidirect

Maximal subgroups

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

subgroup	index
#22570720	2
#22570636	2
#22562439	2
#22562438	2
#22562437	2
#22562432	2
#22561913	2
C₄×C₁₆×D₆₇	2
C₆₇×#2240	2
#33697	67