Subgroups — #31010577

Order 17152.

514 conjugacy classes of subgroups.

The subgroup lattice has 336 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#22569127	C₂	1	1	yes	maximal, characteristic
2	#22562382	C₂	1	4	yes	maximal, complemented
3	#22562377	C₂	1	4	yes	maximal, complemented
4	#22561981	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂×#16863218	C₂	1	1	yes	maximal, characteristic, complemented
6	#22561975	C₂	1	1	yes	maximal, characteristic, complemented
7	#22561944	C₂	1	1	yes	maximal, characteristic
8	C₂×#16863205	C₂	1	1	yes	maximal, characteristic
9	C₆₇×#3163	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂×#14524065	C₂²	1	1	yes	characteristic
11	#16864162	—	2	1	no	—
12	C₂×#14523983	C₂²	1	1	yes	characteristic
13	C₂×#14523983	C₂²	1	1	yes	characteristic
14	#16863519	—	2	2	no	—
15	#16863514	C₂²	1	4	yes	—
16	C₂×#14523979	C₂²	1	1	yes	characteristic
17	#16863414	—	2	1	no	—
18	#16863218	C₂²	1	2	yes	complemented
19	#16863218	C₂²	1	4	yes	complemented
20	#16863218	C₂²	1	2	yes	complemented
21	#16863217	C₂²	1	4	yes	complemented
22	#16863205	C₂²	1	2	yes	—
23	#16863205	C₂²	1	4	yes	—
24	#16863205	C₂²	1	2	yes	—
25	#544×C₆₇	—	2	1	no	—
26	#501×C₆₇	—	2	2	no	—
27	#499×C₆₇	C₂²	1	2	yes	complemented
28	#498×C₆₇	C₂²	1	2	yes	complemented
29	#428×C₆₇	—	2	1	no	—
30	#427×C₆₇	C₂²	1	1	yes	characteristic
31	C₂×#106×C₆₇	C₂²	1	1	yes	characteristic
32	#405×C₆₇	C₂²	1	1	yes	characteristic
33	C₂×#11761628	—	2	1	no	—
34	#14524065	—	2	1	no	—
35	#14524065	—	2	1	no	—
36	#14524065	Q₈	1	2	yes	complemented
37	#14524065	C₂³	1	2	yes	—
38	C₂×Dic₂₆₈	Q₈	1	1	yes	characteristic, complemented
39	C₂×#11761617	—	2	1	no	—
40	C₂×D₅₃₆	Q₈	1	1	yes	characteristic, complemented
41	#14524051	—	4	2	no	—
42	#14524049	—	2	4	no	—
43	C₂×#11761612	—	2	1	no	—
44	M₁₆×D₆₇	—	4	2	no	—
45	#14523983	—	2	1	no	—
46	#14523983	—	2	1	no	—
47	#14523983	C₂³	1	2	yes	—
48	#14523983	C₂³	1	2	yes	—
49	C₂×C₆₇⋊C₁₆	—	2	1	no	—
50	C₂×C₆₇⋊C₁₆	—	2	1	no	—
51	#14523979	—	2	1	no	—
52	#14523979	C₂³	1	2	yes	—
53	#14523971	—	2	4	no	—
54	C₂×M₁₆×C₆₇	D₄	1	1	yes	characteristic
55	C₂²×C₈×C₆₇	C₂³	1	1	yes	characteristic
56	C₄⋊Q₈×C₆₇	—	2	2	no	—
57	C₄⋊Q₈×C₆₇	—	4	1	no	—
58	#125×C₆₇	—	2	2	no	—
59	#125×C₆₇	—	4	1	no	—
60	C₂²⋊Q₈×C₆₇	—	4	2	no	—
61	#117×C₆₇	—	2	1	no	—
62	#117×C₆₇	D₄	1	1	yes	characteristic
63	C₂×C₄⋊C₄×C₆₇	—	2	1	no	—
64	#107×C₆₇	—	2	2	no	—
65	#107×C₆₇	—	2	1	no	—
66	#106×C₆₇	—	2	2	no	—
67	#106×C₆₇	—	2	1	no	—
68	#106×C₆₇	C₂³	1	1	yes	characteristic
69	#106×C₆₇	C₂³	1	2	yes	—
70	#106×C₆₇	C₂³	1	1	yes	characteristic
71	#97×C₆₇	D₄	1	2	yes	—
72	C₄×C₈×C₆₇	C₂³	1	2	yes	—
73	#11761628	—	2	2	no	—
74	#11761628	—	2	2	no	—
75	#11761628	—	4	1	no	—
76	#11761628	—	4	1	no	—
77	C₂×Dic₁₃₄	—	2	1	no	—
78	C₂×D₂₆₈	—	2	1	no	—
79	C₂×C₂²⋊D₆₇	—	4	1	no	—
80	C₂×C₄×D₆₇	—	4	1	no	—
81	Dic₂₆₈	—	2	1	no	—
82	Dic₂₆₈	C₂×Q₈	1	2	yes	—
83	#11761617	—	2	2	no	—
84	#11761617	—	2	2	no	—
85	D₅₃₆	—	2	1	no	—
86	D₅₃₆	C₂×Q₈	1	2	yes	—
87	#11761612	—	2	2	no	—
88	#11761612	—	4	1	no	—
89	#11761611	—	4	4	no	—
90	C₂×C₆₇⋊C₈	—	4	1	no	—
91	C₈×D₆₇	—	4	4	no	—
92	C₆₇⋊C₁₆	—	2	2	no	—
93	C₆₇⋊C₁₆	—	2	2	no	—
94	C₂×Q₈×C₆₇	—	4	2	no	—
95	C₂²×C₄×C₆₇	—	4	1	no	—
96	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
97	M₁₆×C₆₇	SD₁₆	1	2	yes	—
98	M₁₆×C₆₇	SD₁₆	1	2	yes	—
99	C₂×C₈×C₆₇	C₂×D₄	1	1	yes	characteristic
100	C₂×C₈×C₆₇	C₂×D₄	1	2	yes	—
101	C₂×C₈×C₆₇	C₂×Q₈	1	1	yes	characteristic
102	C₂×C₈×C₆₇	C₂×Q₈	1	1	yes	characteristic
103	C₂×C₈×C₆₇	#41	1	1	yes	characteristic
104	C₂×C₈×C₆₇	#41	1	1	yes	characteristic
105	C₂×C₈×C₆₇	C₂⁴	1	1	yes	characteristic
106	C₄⋊C₄×C₆₇	—	2	1	no	—
107	C₄⋊C₄×C₆₇	—	2	2	no	—
108	C₄⋊C₄×C₆₇	—	2	1	no	—
109	C₄⋊C₄×C₆₇	—	2	1	no	—
110	C₄⋊C₄×C₆₇	—	4	2	no	—
111	C₄⋊C₄×C₆₇	—	4	2	no	—
112	C₄⋊C₄×C₆₇	—	4	2	no	—
113	C₄⋊C₄×C₆₇	—	4	1	no	—
114	C₂²⋊C₄×C₆₇	—	2	1	no	—
115	C₂²⋊C₄×C₆₇	—	4	1	no	—
116	C₄²×C₆₇	—	4	1	no	—
117	C₄²×C₆₇	C₂×D₄	1	2	yes	—
118	C₂³×C₆₇	AGL(1,ℤ/8)	1	1	yes	characteristic
119	C₂²×D₆₇	—	4	1	no	—
120	Q₈×C₆₇	—	16	2	no	—
121	C₂×C₄×C₆₇	—	2	2	no	—
122	C₂×C₄×C₆₇	—	4	1	no	—
123	C₂×C₄×C₆₇	—	4	1	no	—
124	C₂×C₄×C₆₇	—	4	2	no	—
125	C₂×C₄×C₆₇	—	16	1	no	—
126	C₂×C₄×C₆₇	C₂²⋊Q₈	1	1	yes	characteristic
127	C₂×C₄×C₆₇	C₂²⋊Q₈	1	1	yes	characteristic
128	C₂×C₄×C₆₇	C₂×SD₁₆	1	1	yes	characteristic
129	C₂×C₄×C₆₇	C₂×SD₁₆	1	1	yes	characteristic
130	C₂×C₄×C₆₇	AGL(1,ℤ/8)	1	1	yes	characteristic
131	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
132	C₂²⋊D₆₇	—	4	2	no	—
133	C₂²⋊D₆₇	—	4	2	no	—
134	C₂×Dic₆₇	—	4	1	no	—
135	D₂₆₈	—	2	2	no	—
136	D₂₆₈	—	4	1	no	—
137	C₄×D₆₇	—	4	2	no	—
138	C₄×D₆₇	—	4	2	no	—
139	Dic₁₃₄	—	2	2	no	—
140	Dic₁₃₄	—	4	1	no	—
141	C₈×C₆₇	C₂²⋊Q₈	1	1	yes	characteristic
142	C₈×C₆₇	C₂²⋊Q₈	1	1	yes	characteristic
143	C₈×C₆₇	C₂×SD₁₆	1	4	yes	—
144	C₈×C₆₇	C₂²×Q₈	1	1	yes	characteristic
145	C₈×C₆₇	C₂×#41	1	1	yes	characteristic
146	C₆₇⋊C₈	—	4	2	no	—
147	C₂²×C₆₇	—	4	1	no	—
148	C₂²×C₆₇	#476	1	1	yes	characteristic
149	C₂²×C₆₇	#476	1	1	yes	characteristic
150	C₂²×C₆₇	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
151	C₂×D₆₇	—	4	2	no	—
152	C₂×D₆₇	—	16	1	no	—
153	C₄×C₆₇	—	4	2	no	—
154	C₄×C₆₇	—	8	1	no	—
155	C₄×C₆₇	—	8	2	no	—
156	C₄×C₆₇	—	8	1	no	—
157	C₄×C₆₇	#476	1	1	yes	characteristic
158	C₄×C₆₇	#476	1	1	yes	characteristic
159	C₄×C₆₇	C₂×C₂²⋊Q₈	1	1	yes	characteristic
160	C₄×C₆₇	C₂²×SD₁₆	1	1	yes	characteristic
161	Dic₆₇	—	4	2	no	—
162	#33638	—	67	1	no	maximal, Sylow
163	C₂×C₆₇	—	2	1	no	—
164	C₂×C₆₇	—	2	1	no	—
165	C₂×C₆₇	—	2	1	no	—
166	C₂×C₆₇	C₂×#476	1	1	yes	characteristic
167	D₆₇	—	8	2	no	—
168	#3163	D₆₇	1	1	yes	characteristic, complemented
169	#2957	—	67	1	no	—
170	#2231	—	67	4	no	—
171	#2230	—	67	4	no	—
172	#2155	—	67	1	no	—
173	C₂×#365	—	67	1	no	—
174	#2153	—	67	1	no	—
175	#2145	—	67	1	no	—
176	C₂×#360	—	67	1	no	—
177	C₆₇	#33638	1	1	yes	characteristic, complemented, Sylow
178	C₂×#135	—	67	1	no	—
179	#568	—	134	1	no	—
180	#544	—	2	1	no	—
181	C₂×M₃₂	—	67	1	no	—
182	C₂×M₃₂	—	67	1	no	—
183	#501	—	2	2	no	—
184	#499	C₂×D₆₇	1	2	yes	complemented
185	#498	C₂×D₆₇	1	2	yes	complemented
186	#444	—	134	2	no	—
187	#443	—	67	4	no	—
188	C₂×#108	—	67	1	no	—
189	#428	—	2	1	no	—
190	#427	C₂×D₆₇	1	1	yes	characteristic
191	C₂×#106	C₂×D₆₇	1	1	yes	characteristic
192	#421	—	134	1	no	—
193	#405	C₂×D₆₇	1	1	yes	characteristic
194	#365	—	67	4	no	—
195	#365	—	67	2	no	—
196	#365	—	67	2	no	—
197	#364	—	67	4	no	—
198	#360	—	67	4	no	—
199	#360	—	67	2	no	—
200	#360	—	67	2	no	—
201	C₂×#41	—	134	1	no	—
202	#135	—	67	2	no	—
203	#135	—	67	2	no	—
204	#135	—	134	1	no	—
205	#135	—	134	1	no	—
206	C₂×Q₁₆	—	67	1	no	—
207	C₂×SD₁₆	—	134	1	no	—
208	C₂×D₈	—	67	1	no	—
209	#131	—	268	2	no	—
210	#131	—	134	4	no	—
211	C₂×M₁₆	—	268	2	no	—
212	C₂×M₁₆	—	134	1	no	—
213	C₂×M₁₆	C₂²⋊D₆₇	1	1	yes	characteristic
214	C₂²×C₈	C₂²×D₆₇	1	1	yes	characteristic
215	C₄⋊Q₈	—	2	2	no	—
216	C₄⋊Q₈	—	4	1	no	—
217	#125	—	2	2	no	—
218	#125	—	4	1	no	—
219	C₂²⋊Q₈	—	4	2	no	—
220	#117	—	2	1	no	—
221	#117	C₂²⋊D₆₇	1	1	yes	characteristic
222	C₂×C₄⋊C₄	—	2	1	no	—
223	M₃₂	—	67	2	no	—
224	M₃₂	—	67	2	no	—
225	M₃₂	—	134	1	no	—
226	M₃₂	—	134	1	no	—
227	C₂×C₁₆	—	134	1	no	—
228	C₂×C₁₆	—	134	1	no	—
229	#108	—	67	2	no	—
230	#108	—	134	1	no	—
231	#107	—	2	1	no	—
232	#107	—	2	2	no	—
233	#106	—	2	1	no	—
234	#106	—	2	2	no	—
235	#106	C₂²×D₆₇	1	2	yes	—
236	#106	C₂²×D₆₇	1	1	yes	characteristic
237	#106	C₂²×D₆₇	1	1	yes	characteristic
238	#104	—	134	4	no	—
239	#97	C₂²⋊D₆₇	1	2	yes	—
240	C₄×C₈	C₂²×D₆₇	1	2	yes	—
241	#41	—	268	1	no	—
242	#41	—	268	1	no	—
243	#41	—	134	2	no	—
244	#41	—	134	2	no	—
245	C₂×Q₈	—	4	2	no	—
246	C₂×Q₈	—	134	1	no	—
247	C₂×D₄	—	268	1	no	—
248	C₂×D₄	—	134	1	no	—
249	C₂²×C₄	—	4	1	no	—
250	C₂²×C₄	—	268	1	no	—
251	C₂²×C₄	C₂×C₂²⋊D₆₇	1	1	yes	characteristic
252	Q₁₆	—	67	2	no	—
253	Q₁₆	—	134	1	no	—
254	SD₁₆	—	134	2	no	—
255	SD₁₆	—	134	2	no	—
256	D₈	—	67	2	no	—
257	D₈	—	134	1	no	—
258	M₁₆	—	268	1	no	—
259	M₁₆	—	268	4	no	—
260	M₁₆	—	134	2	no	—
261	M₁₆	#11761615	1	2	yes	—
262	M₁₆	#11761615	1	2	yes	—
263	C₂×C₈	—	268	1	no	—
264	C₂×C₈	—	268	4	no	—
265	C₂×C₈	C₂×C₂²⋊D₆₇	1	1	yes	characteristic
266	C₂×C₈	C₂×C₂²⋊D₆₇	1	2	yes	—
267	C₂×C₈	Q₈×D₆₇	1	1	yes	characteristic
268	C₂×C₈	Q₈×D₆₇	1	1	yes	characteristic
269	C₂×C₈	#11761629	1	1	yes	characteristic
270	C₂×C₈	#11761629	1	1	yes	characteristic
271	C₂×C₈	C₂³×D₆₇	1	1	yes	characteristic
272	C₄⋊C₄	—	2	1	no	—
273	C₄⋊C₄	—	2	2	no	—
274	C₄⋊C₄	—	2	1	no	—
275	C₄⋊C₄	—	2	1	no	—
276	C₄⋊C₄	—	4	2	no	—
277	C₄⋊C₄	—	4	2	no	—
278	C₄⋊C₄	—	4	2	no	—
279	C₄⋊C₄	—	4	1	no	—
280	C₂²⋊C₄	—	2	1	no	—
281	C₂²⋊C₄	—	4	1	no	—
282	C₄²	—	4	1	no	—
283	C₄²	C₂×C₂²⋊D₆₇	1	2	yes	—
284	C₁₆	—	134	2	no	—
285	C₁₆	—	134	2	no	—
286	C₂³	—	268	1	no	—
287	C₂³	#14524069	1	1	yes	characteristic
288	Q₈	—	268	1	no	—
289	Q₈	—	16	2	no	—
290	Q₈	—	134	2	no	—
291	D₄	—	268	2	no	—
292	D₄	—	268	2	no	—
293	D₄	—	268	1	no	—
294	D₄	—	134	2	no	—
295	C₂×C₄	—	2	2	no	—
296	C₂×C₄	—	4	1	no	—
297	C₂×C₄	—	4	1	no	—
298	C₂×C₄	—	4	2	no	—
299	C₂×C₄	—	268	1	no	—
300	C₂×C₄	—	268	2	no	—
301	C₂×C₄	—	268	2	no	—
302	C₂×C₄	—	16	1	no	—
303	C₂×C₄	#14524019	1	1	yes	characteristic
304	C₂×C₄	#14524019	1	1	yes	characteristic
305	C₂×C₄	C₂×#11761615	1	1	yes	characteristic
306	C₂×C₄	C₂×#11761615	1	1	yes	characteristic
307	C₂×C₄	#14524069	1	1	yes	characteristic
308	C₂×C₄	C₂²×C₂²⋊D₆₇	1	1	yes	characteristic
309	C₈	—	268	2	no	—
310	C₈	#14524019	1	1	yes	characteristic
311	C₈	#14524019	1	1	yes	characteristic
312	C₈	C₂×#11761615	1	4	yes	—
313	C₈	C₂×Q₈×D₆₇	1	1	yes	characteristic
314	C₈	C₂×#11761629	1	1	yes	characteristic
315	C₂²	—	4	1	no	—
316	C₂²	—	268	2	no	—
317	C₂²	—	1072	1	no	—
318	C₂²	#16863683	1	1	yes	characteristic
319	C₂²	#16863683	1	1	yes	characteristic
320	C₂²	C₂×#14524069	1	1	yes	characteristic
321	C₄	—	4	2	no	—
322	C₄	—	8	2	no	—
323	C₄	—	8	1	no	—
324	C₄	—	8	1	no	—
325	C₄	—	268	2	no	—
326	C₄	#16863683	1	1	yes	characteristic
327	C₄	#16863683	1	1	yes	characteristic
328	C₄	C₂×#14524019	1	1	yes	characteristic
329	C₄	C₂²×#11761615	1	1	yes	characteristic
330	C₂	—	2	1	no	—
331	C₂	—	2	1	no	—
332	C₂	—	2	1	no	—
333	C₂	—	536	2	no	—
334	C₂	C₂×#16863683	1	1	yes	characteristic

	subgroup	G/H
center	C₂	C₂×#16863683
commutator, cosocle	C₂×C₈×C₆₇	C₂⁴
Frattini	C₂×C₈	C₂³×D₆₇
Fitting	C₆₇×#3163	C₂
socle	C₂×C₆₇	C₂×#476
prime radical	C₂×#14524065	C₂²
hypercenter	#3163	D₆₇
hypocenter	C₆₇	#33638
LCS	G ▷ C₂×C₈×C₆₇ ▷ C₄×C₆₇ ▷ C₂×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂ ◁ C₂²×C₄ ◁ #405 ◁ #3163

Complements

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

N	H ≅ G/N	type
2. #22562382	C₂	semidirect
3. #22562377	C₂	semidirect
4. #22561981	C₂	semidirect
5. C₂×#16863218	C₂	semidirect
6. #22561975	C₂	semidirect
9. C₆₇×#3163	C₂	semidirect
18. #16863218	C₂²	semidirect
19. #16863218	C₂²	semidirect
20. #16863218	C₂²	semidirect
21. #16863217	C₂²	semidirect
27. #499×C₆₇	C₂²	semidirect
28. #498×C₆₇	C₂²	semidirect
36. #14524065	Q₈	semidirect
38. C₂×Dic₂₆₈	Q₈	semidirect
40. C₂×D₅₃₆	Q₈	semidirect
168. #3163	D₆₇	semidirect
177. C₆₇	#33638	semidirect
184. #499	C₂×D₆₇	semidirect
185. #498	C₂×D₆₇	semidirect

Maximal subgroups

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

subgroup	index
#22569127	2
#22562382	2
#22562377	2
#22561981	2
C₂×#16863218	2
#22561975	2
#22561944	2
C₂×#16863205	2
C₆₇×#3163	2
#33638	67