Subgroups — #32152360

Order 18176.

534 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	#22597991	C₂	1	1	yes	maximal, characteristic
2	#22596948	C₂	1	1	yes	maximal, characteristic
3	#22590558	C₂	1	8	yes	maximal, complemented
4	#22590114	C₂	1	1	yes	maximal, characteristic, complemented
5	#22590113	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×#17985815	C₂	1	1	yes	maximal, characteristic, complemented
7	#22589960	C₂	1	1	yes	maximal, characteristic
8	C₂×#362×C₇₁	C₂	1	1	yes	maximal, characteristic, complemented
9	#17986765	—	2	1	no	—
10	C₂×#14556134	C₂²	1	1	yes	characteristic
11	#17986573	—	2	1	no	—
12	C₂×#14556068	C₂²	1	1	yes	characteristic
13	#17986254	C₂²	1	4	yes	—
14	#17986237	—	2	2	no	—
15	#17986047	—	2	1	no	—
16	#17986046	C₂²	1	4	yes	—
17	C₂×#14556064	C₂²	1	1	yes	characteristic
18	#108×D₇₁	—	2	2	no	—
19	#17986037	C₂²	1	1	yes	characteristic
20	#17986008	—	2	1	no	—
21	#17985823	C₂²	1	4	yes	—
22	#17985822	C₂²	1	4	yes	—
23	#17985815	C₂²	1	4	yes	—
24	#17985780	C₂²	1	4	yes	—
25	C₂²×Q₁₆×C₇₁	C₂²	1	1	yes	characteristic
26	C₂×M₃₂×C₇₁	C₂²	1	1	yes	characteristic
27	C₂×#108×C₇₁	C₂²	1	1	yes	characteristic
28	#362×C₇₁	C₂²	1	4	yes	—
29	C₂×#11765865	D₄	1	1	yes	characteristic
30	#14556136	—	4	2	no	—
31	#14556134	—	2	1	no	—
32	#14556134	—	2	1	no	—
33	#14556134	—	2	1	no	—
34	#14556134	—	2	1	no	—
35	#14556134	—	4	2	no	—
36	C₂×#11765849	—	2	1	no	—
37	C₂×#11765849	D₄	1	1	yes	characteristic
38	C₂×#11765848	C₂³	1	2	yes	—
39	M₁₆×D₇₁	—	2	4	no	—
40	C₂×C₈×D₇₁	D₄	1	2	yes	—
41	#14556126	—	4	1	no	—
42	#14556117	—	4	1	no	—
43	#14556111	—	4	2	no	—
44	#14556104	—	2	4	no	—
45	#14556087	—	2	1	no	—
46	#14556068	—	2	1	no	—
47	#14556068	C₂³	1	2	yes	—
48	C₂×C₇₁⋊C₁₆	—	2	1	no	—
49	#14556064	—	2	1	no	—
50	#14556064	C₂³	1	2	yes	—
51	#14556063	—	2	4	no	—
52	#14556062	—	2	1	no	—
53	#14556060	C₂³	1	2	yes	—
54	#14556052	—	2	4	no	—
55	C₂²×Q₈×C₇₁	—	2	1	no	—
56	C₂×Q₁₆×C₇₁	—	2	1	no	—
57	C₂×Q₁₆×C₇₁	—	2	1	no	—
58	C₂×Q₁₆×C₇₁	—	2	1	no	—
59	C₂×Q₁₆×C₇₁	—	2	1	no	—
60	C₂×Q₁₆×C₇₁	—	2	1	no	—
61	C₂×Q₁₆×C₇₁	C₂³	1	2	yes	—
62	C₂×M₁₆×C₇₁	—	2	1	no	—
63	C₂²×C₈×C₇₁	C₂³	1	1	yes	characteristic
64	M₃₂×C₇₁	—	2	1	no	—
65	M₃₂×C₇₁	C₂³	1	2	yes	—
66	C₂×C₁₆×C₇₁	—	2	1	no	—
67	#108×C₇₁	—	2	1	no	—
68	#108×C₇₁	C₂³	1	2	yes	—
69	#11765865	—	2	1	no	—
70	#11765865	—	2	1	no	—
71	#11765865	—	2	1	no	—
72	#11765865	—	2	1	no	—
73	C₂×Dic₁₄₂	SD₁₆	1	1	yes	characteristic
74	C₂×D₂₈₄	SD₁₆	1	1	yes	characteristic
75	C₂×C₂²⋊D₇₁	SD₁₆	1	2	yes	—
76	C₂×C₄×D₇₁	C₂×D₄	1	2	yes	—
77	#11765849	—	2	1	no	—
78	#11765849	—	2	1	no	—
79	#11765849	—	2	2	no	—
80	#11765849	—	4	1	no	—
81	#11765848	—	2	2	no	—
82	#11765848	—	2	2	no	—
83	#11765848	—	4	4	no	—
84	C₂×C₇₁⋊C₈	—	4	1	no	—
85	C₂×C₇₁⋊C₈	C₂×D₄	1	2	yes	—
86	C₈×D₇₁	—	2	2	no	—
87	C₈×D₇₁	—	2	2	no	—
88	C₈×D₇₁	—	4	4	no	—
89	C₄⋊Dic₇₁	—	2	2	no	—
90	Dic₇₁⋊C₄	—	4	4	no	—
91	C₂²⋊Dic₇₁	—	4	1	no	—
92	(C₂×D₇₁)⋊C₄	—	4	4	no	—
93	C₄×Dic₇₁	—	4	1	no	—
94	C₇₁⋊C₁₆	—	2	2	no	—
95	C₂×Q₈×C₇₁	—	2	2	no	—
96	C₂×Q₈×C₇₁	—	4	1	no	—
97	C₂×Q₈×C₇₁	—	4	1	no	—
98	C₂×Q₈×C₇₁	—	4	1	no	—
99	C₂×Q₈×C₇₁	—	4	1	no	—
100	C₂×Q₈×C₇₁	—	4	1	no	—
101	C₂²×C₄×C₇₁	—	4	1	no	—
102	C₂²×C₄×C₇₁	C₂×D₄	1	1	yes	characteristic
103	Q₁₆×C₇₁	—	2	2	no	—
104	Q₁₆×C₇₁	—	2	2	no	—
105	Q₁₆×C₇₁	—	4	1	no	—
106	Q₁₆×C₇₁	—	4	1	no	—
107	M₁₆×C₇₁	—	2	2	no	—
108	M₁₆×C₇₁	—	4	1	no	—
109	C₂×C₈×C₇₁	—	4	1	no	—
110	C₂×C₈×C₇₁	C₂×D₄	1	1	yes	characteristic
111	C₂×C₈×C₇₁	#41	1	1	yes	characteristic
112	C₂×C₈×C₇₁	#41	1	1	yes	characteristic
113	C₂×C₈×C₇₁	#41	1	1	yes	characteristic
114	C₂×C₈×C₇₁	#41	1	1	yes	characteristic
115	C₂×C₈×C₇₁	C₂⁴	1	1	yes	characteristic
116	C₁₆×C₇₁	—	2	2	no	—
117	C₂³×C₇₁	C₂×SD₁₆	1	1	yes	characteristic
118	C₂²×D₇₁	C₂×SD₁₆	1	2	yes	—
119	Q₈×C₇₁	—	4	2	no	—
120	Q₈×C₇₁	—	4	2	no	—
121	Q₈×C₇₁	—	8	1	no	—
122	Q₈×C₇₁	—	8	1	no	—
123	C₂×C₄×C₇₁	—	4	2	no	—
124	C₂×C₄×C₇₁	—	8	1	no	—
125	C₂×C₄×C₇₁	—	8	1	no	—
126	C₂×C₄×C₇₁	#123	1	1	yes	characteristic
127	C₂×C₄×C₇₁	#123	1	1	yes	characteristic
128	C₂×C₄×C₇₁	C₂×SD₁₆	1	1	yes	characteristic
129	C₂×C₄×C₇₁	AGL(1,ℤ/8)	1	1	yes	characteristic
130	C₂×C₄×C₇₁	AGL(1,ℤ/8)	1	1	yes	characteristic
131	C₂×C₄×C₇₁	C₂²×D₄	1	1	yes	characteristic
132	C₂²⋊D₇₁	—	2	2	no	—
133	C₂²⋊D₇₁	—	2	2	no	—
134	C₂×Dic₇₁	—	4	2	no	—
135	C₂×Dic₇₁	C₂×SD₁₆	1	2	yes	—
136	D₂₈₄	—	2	1	no	—
137	D₂₈₄	—	2	1	no	—
138	C₄×D₇₁	—	2	2	no	—
139	C₄×D₇₁	—	2	2	no	—
140	Dic₁₄₂	—	2	1	no	—
141	Dic₁₄₂	—	2	1	no	—
142	C₈×C₇₁	—	4	2	no	—
143	C₈×C₇₁	#123	1	1	yes	characteristic
144	C₈×C₇₁	#123	1	1	yes	characteristic
145	C₈×C₇₁	C₂×#41	1	1	yes	characteristic
146	C₈×C₇₁	C₂×#41	1	1	yes	characteristic
147	C₇₁⋊C₈	—	2	2	no	—
148	C₇₁⋊C₈	—	4	2	no	—
149	C₂²×C₇₁	—	4	1	no	—
150	C₂²×C₇₁	#481	1	1	yes	characteristic
151	C₂²×C₇₁	#481	1	1	yes	characteristic
152	C₂²×C₇₁	C₂²×SD₁₆	1	1	yes	characteristic
153	C₂×D₇₁	—	2	2	no	—
154	C₂×D₇₁	—	4	2	no	—
155	C₄×C₇₁	—	8	2	no	—
156	C₄×C₇₁	#481	1	1	yes	characteristic
157	C₄×C₇₁	#481	1	1	yes	characteristic
158	C₄×C₇₁	C₂×#123	1	1	yes	characteristic
159	C₄×C₇₁	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
160	Dic₇₁	—	2	2	no	—
161	Dic₇₁	—	8	2	no	—
162	#33658	—	71	1	no	maximal, Sylow
163	C₂×C₇₁	—	2	1	no	—
164	C₂×C₇₁	—	2	1	no	—
165	C₂×C₇₁	—	2	1	no	—
166	C₂×C₇₁	C₂×#481	1	1	yes	characteristic
167	D₇₁	—	4	2	no	—
168	#3059	—	71	1	no	—
169	#2916	—	71	1	no	—
170	#2239	—	71	8	no	—
171	#2155	—	71	1	no	—
172	#2155	—	71	1	no	—
173	C₂×#362	—	71	1	no	—
174	C₂×#362	D₇₁	1	1	yes	characteristic, complemented
175	#2115	—	71	1	no	—
176	C₇₁	#33658	1	1	yes	characteristic, complemented, Sylow
177	C₂²×Q₁₆	C₂×D₇₁	1	1	yes	characteristic
178	#568	—	142	1	no	—
179	C₂×#131	—	71	1	no	—
180	#536	—	142	1	no	—
181	C₂×M₃₂	—	71	1	no	—
182	C₂×M₃₂	C₂×D₇₁	1	1	yes	characteristic
183	#470	—	71	4	no	—
184	#467	—	142	2	no	—
185	#430	—	71	4	no	—
186	#430	—	142	1	no	—
187	C₂×#108	—	71	1	no	—
188	C₂×#108	—	142	2	no	—
189	C₂×#108	C₂×D₇₁	1	1	yes	characteristic
190	#427	—	71	1	no	—
191	#419	—	142	1	no	—
192	#365	—	71	4	no	—
193	#365	—	71	4	no	—
194	#362	—	71	4	no	—
195	#362	C₂×D₇₁	1	4	yes	—
196	#349	—	71	4	no	—
197	C₂×#41	—	71	1	no	—
198	C₂²×Q₈	—	2	1	no	—
199	C₂×Q₁₆	—	2	1	no	—
200	C₂×Q₁₆	—	2	1	no	—
201	C₂×Q₁₆	—	2	1	no	—
202	C₂×Q₁₆	—	2	1	no	—
203	C₂×Q₁₆	—	2	1	no	—
204	C₂×Q₁₆	C₂²×D₇₁	1	2	yes	—
205	#131	—	284	2	no	—
206	#131	—	284	2	no	—
207	#131	—	142	1	no	—
208	#131	—	142	1	no	—
209	#131	—	142	1	no	—
210	#131	—	142	1	no	—
211	C₂×M₁₆	—	2	1	no	—
212	C₂×M₁₆	—	71	1	no	—
213	C₂×M₁₆	—	71	2	no	—
214	C₂×M₁₆	—	142	4	no	—
215	C₂×M₁₆	—	142	1	no	—
216	C₂²×C₈	—	71	2	no	—
217	C₂²×C₈	C₂²×D₇₁	1	1	yes	characteristic
218	C₄⋊Q₈	—	284	1	no	—
219	#124	—	284	1	no	—
220	#123	—	284	2	no	—
221	C₂²⋊Q₈	—	142	4	no	—
222	#117	—	142	1	no	—
223	M₃₂	—	2	1	no	—
224	M₃₂	—	71	2	no	—
225	M₃₂	—	142	1	no	—
226	M₃₂	C₂²×D₇₁	1	2	yes	—
227	C₂×C₁₆	—	2	1	no	—
228	C₂×C₁₆	—	142	1	no	—
229	#108	—	2	1	no	—
230	#108	—	71	2	no	—
231	#108	—	142	4	no	—
232	#108	—	142	1	no	—
233	#108	C₂²×D₇₁	1	2	yes	—
234	#107	—	142	1	no	—
235	#106	—	71	2	no	—
236	#103	—	142	4	no	—
237	#41	—	142	1	no	—
238	#41	—	142	1	no	—
239	#41	—	142	1	no	—
240	#41	—	142	1	no	—
241	C₂×Q₈	—	2	2	no	—
242	C₂×Q₈	—	4	1	no	—
243	C₂×Q₈	—	4	1	no	—
244	C₂×Q₈	—	4	1	no	—
245	C₂×Q₈	—	4	1	no	—
246	C₂×Q₈	—	4	1	no	—
247	C₂×Q₈	—	71	1	no	—
248	C₂×D₄	—	71	2	no	—
249	C₂×D₄	—	71	1	no	—
250	C₂²×C₄	—	4	1	no	—
251	C₂²×C₄	—	71	2	no	—
252	C₂²×C₄	D₄×D₇₁	1	1	yes	characteristic
253	Q₁₆	—	2	2	no	—
254	Q₁₆	—	2	2	no	—
255	Q₁₆	—	4	1	no	—
256	Q₁₆	—	4	1	no	—
257	M₁₆	—	2	2	no	—
258	M₁₆	—	4	1	no	—
259	M₁₆	—	284	4	no	—
260	M₁₆	—	284	1	no	—
261	M₁₆	—	142	2	no	—
262	M₁₆	—	142	1	no	—
263	M₁₆	—	142	1	no	—
264	M₁₆	—	142	2	no	—
265	M₁₆	—	142	2	no	—
266	C₂×C₈	—	4	1	no	—
267	C₂×C₈	—	284	4	no	—
268	C₂×C₈	—	284	1	no	—
269	C₂×C₈	—	71	2	no	—
270	C₂×C₈	—	142	2	no	—
271	C₂×C₈	—	142	2	no	—
272	C₂×C₈	D₄×D₇₁	1	1	yes	characteristic
273	C₂×C₈	#11765864	1	1	yes	characteristic
274	C₂×C₈	#11765864	1	1	yes	characteristic
275	C₂×C₈	#11765865	1	1	yes	characteristic
276	C₂×C₈	#11765865	1	1	yes	characteristic
277	C₂×C₈	C₂³×D₇₁	1	1	yes	characteristic
278	C₄⋊C₄	—	284	4	no	—
279	C₄⋊C₄	—	142	2	no	—
280	C₂²⋊C₄	—	284	4	no	—
281	C₂²⋊C₄	—	284	1	no	—
282	C₄²	—	284	1	no	—
283	C₁₆	—	2	2	no	—
284	C₁₆	—	142	2	no	—
285	C₂³	—	71	2	no	—
286	C₂³	SD₁₆×D₇₁	1	1	yes	characteristic
287	Q₈	—	4	2	no	—
288	Q₈	—	4	2	no	—
289	Q₈	—	8	1	no	—
290	Q₈	—	8	1	no	—
291	Q₈	—	142	1	no	—
292	Q₈	—	142	1	no	—
293	D₄	—	142	2	no	—
294	D₄	—	142	2	no	—
295	D₄	—	142	1	no	—
296	D₄	—	142	1	no	—
297	C₂×C₄	—	4	2	no	—
298	C₂×C₄	—	8	1	no	—
299	C₂×C₄	—	8	1	no	—
300	C₂×C₄	—	284	2	no	—
301	C₂×C₄	—	71	2	no	—
302	C₂×C₄	—	142	2	no	—
303	C₂×C₄	—	142	2	no	—
304	C₂×C₄	#14556110	1	1	yes	characteristic
305	C₂×C₄	#14556110	1	1	yes	characteristic
306	C₂×C₄	SD₁₆×D₇₁	1	1	yes	characteristic
307	C₂×C₄	#14556152	1	1	yes	characteristic
308	C₂×C₄	#14556152	1	1	yes	characteristic
309	C₂×C₄	C₂×D₄×D₇₁	1	1	yes	characteristic
310	C₈	—	4	2	no	—
311	C₈	—	284	2	no	—
312	C₈	—	142	2	no	—
313	C₈	#14556110	1	1	yes	characteristic
314	C₈	#14556110	1	1	yes	characteristic
315	C₈	C₂×#11765864	1	1	yes	characteristic
316	C₈	C₂×#11765865	1	1	yes	characteristic
317	C₂²	—	4	1	no	—
318	C₂²	—	284	2	no	—
319	C₂²	—	142	2	no	—
320	C₂²	#17986316	1	1	yes	characteristic
321	C₂²	#17986316	1	1	yes	characteristic
322	C₂²	C₂×SD₁₆×D₇₁	1	1	yes	characteristic
323	C₄	—	8	2	no	—
324	C₄	—	568	2	no	—
325	C₄	—	142	2	no	—
326	C₄	#17986316	1	1	yes	characteristic
327	C₄	#17986316	1	1	yes	characteristic
328	C₄	C₂×#14556110	1	1	yes	characteristic
329	C₄	C₂×#14556152	1	1	yes	characteristic
330	C₂	—	2	1	no	—
331	C₂	—	2	1	no	—
332	C₂	—	2	1	no	—
333	C₂	—	284	2	no	—
334	C₂	C₂×#17986316	1	1	yes	characteristic

	subgroup	G/H
center	C₂	C₂×#17986316
commutator, cosocle	C₂×C₈×C₇₁	C₂⁴
Frattini	C₂×C₈	C₂³×D₇₁
Fitting	C₂×#362×C₇₁	C₂
socle	C₂×C₇₁	C₂×#481
prime radical	C₂×#11765865	D₄
hypercenter	C₂×#362	D₇₁
hypocenter	C₇₁	#33658
LCS	G ▷ C₂×C₈×C₇₁ ▷ C₄×C₇₁ ▷ C₂×C₇₁ ▷ C₇₁
UCS	C₁ ◁ C₂ ◁ C₂²×C₄ ◁ C₂²×C₈ ◁ C₂×#362

Complements

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

N	H ≅ G/N	type
3. #22590558	C₂	semidirect
4. #22590114	C₂	semidirect
5. #22590113	C₂	semidirect
6. C₂×#17985815	C₂	semidirect
8. C₂×#362×C₇₁	C₂	semidirect
174. C₂×#362	D₇₁	semidirect
176. C₇₁	#33658	semidirect

Maximal subgroups

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

subgroup	index
#22597991	2
#22596948	2
#22590558	2
#22590114	2
#22590113	2
C₂×#17985815	2
#22589960	2
C₂×#362×C₇₁	2
#33658	71