Subgroups — #31914073

Order 18176.

398 conjugacy classes of subgroups.

The subgroup lattice has 320 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#17986115	C₂	1	1	yes	maximal, characteristic
2	C₂×#17985828	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×#17985807	C₂	1	1	yes	maximal, characteristic
4	#22590045	C₂	1	1	yes	maximal, characteristic
5	#22589089	C₂	1	1	yes	maximal, characteristic, complemented
6	#22586417	C₂	1	1	yes	maximal, characteristic, complemented
7	C₇₁×#1382	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×#14556150	C₂²	1	1	yes	characteristic
9	C₂×#14556136	—	2	1	no	—
10	C₂²×C₇₁⋊C₁₆	C₂²	1	1	yes	characteristic
11	#17986115	—	2	2	no	—
12	#17986115	—	2	2	no	—
13	C₂×#14556064	C₂²	1	1	yes	characteristic
14	#17986026	—	2	1	no	—
15	C₂×#14556054	—	2	1	no	—
16	C₂×C₈×Dic₇₁	C₂²	1	1	yes	characteristic
17	#17985828	—	2	2	no	—
18	#17985807	—	2	2	no	—
19	#17985802	C₄	1	1	yes	characteristic, complemented
20	#17985802	C₄	1	1	yes	characteristic, complemented
21	#17985799	C₄	1	1	yes	characteristic, complemented
22	#17985799	C₄	1	1	yes	characteristic, complemented
23	#17985758	—	2	1	no	—
24	C₂²×C₁₆×C₇₁	C₂²	1	1	yes	characteristic
25	C₂×#108×C₇₁	C₂²	1	1	yes	characteristic
26	#427×C₇₁	C₂²	1	1	yes	characteristic
27	C₂×#11765865	—	2	1	no	—
28	#14556150	—	2	1	no	—
29	#14556150	—	2	1	no	—
30	#14556150	D₄	1	2	yes	—
31	#14556150	Q₈	1	2	yes	—
32	C₂×Dic₂₈₄	C₂×C₄	1	1	yes	characteristic
33	C₂×#11765854	—	2	1	no	—
34	C₂×D₅₆₈	C₂×C₄	1	1	yes	characteristic
35	#14556136	—	2	4	no	—
36	#14556136	—	4	2	no	—
37	C₂×#11765849	—	2	1	no	—
38	C₂×#11765848	—	4	1	no	—
39	C₂²×C₇₁⋊C₈	D₄	1	1	yes	characteristic
40	C₂×C₈×D₇₁	—	4	1	no	—
41	C₂×C₄×Dic₇₁	D₄	1	1	yes	characteristic
42	C₂×C₇₁⋊C₁₆	—	2	1	no	—
43	C₂×C₇₁⋊C₁₆	—	2	1	no	—
44	C₂×C₇₁⋊C₁₆	C₂×C₄	1	1	yes	characteristic
45	C₂×C₇₁⋊C₁₆	C₂×C₄	1	1	yes	characteristic
46	#14556064	—	2	1	no	—
47	#14556064	—	2	1	no	—
48	C₄⋊(C₇₁⋊C₈)	—	4	1	no	—
49	C₄⋊(C₇₁⋊C₈)	—	4	1	no	—
50	#14556054	—	2	4	no	—
51	C₈×Dic₇₁	—	2	1	no	—
52	C₈×Dic₇₁	—	2	1	no	—
53	C₂×M₁₆×C₇₁	—	2	1	no	—
54	C₂²×C₈×C₇₁	C₂³	1	1	yes	characteristic
55	#117×C₇₁	—	2	1	no	—
56	C₂×C₁₆×C₇₁	—	2	1	no	—
57	C₂×C₁₆×C₇₁	—	2	1	no	—
58	C₂×C₁₆×C₇₁	—	2	1	no	—
59	#108×C₇₁	—	2	1	no	—
60	#108×C₇₁	D₄	1	2	yes	—
61	#107×C₇₁	C₂×C₄	1	1	yes	characteristic
62	#107×C₇₁	C₂×C₄	1	1	yes	characteristic
63	#106×C₇₁	—	2	1	no	—
64	#11765865	—	2	2	no	—
65	#11765865	—	2	2	no	—
66	#11765865	—	4	1	no	—
67	#11765865	—	4	1	no	—
68	C₂×Dic₁₄₂	—	2	1	no	—
69	C₂×D₂₈₄	—	2	1	no	—
70	C₂×C₂²⋊D₇₁	—	4	1	no	—
71	C₂²×Dic₇₁	—	2	1	no	—
72	C₂×C₄×D₇₁	—	4	1	no	—
73	Dic₂₈₄	—	2	1	no	—
74	Dic₂₈₄	C₄⋊C₄	1	2	yes	—
75	#11765854	—	2	2	no	—
76	#11765854	—	2	2	no	—
77	D₅₆₈	—	2	1	no	—
78	D₅₆₈	C₄⋊C₄	1	2	yes	—
79	#11765849	—	4	1	no	—
80	#11765849	—	4	1	no	—
81	#11765848	—	4	4	no	—
82	C₂×C₇₁⋊C₈	—	2	1	no	—
83	C₂×C₇₁⋊C₈	—	2	1	no	—
84	C₂×C₇₁⋊C₈	—	4	1	no	—
85	C₂×C₇₁⋊C₈	D₈	1	1	yes	characteristic
86	C₂×C₇₁⋊C₈	D₈	1	1	yes	characteristic
87	C₈×D₇₁	—	4	4	no	—
88	C₄×Dic₇₁	—	2	1	no	—
89	C₄×Dic₇₁	—	2	1	no	—
90	C₇₁⋊C₁₆	—	2	1	no	—
91	C₇₁⋊C₁₆	—	2	1	no	—
92	C₂²×C₄×C₇₁	C₂×D₄	1	1	yes	characteristic
93	M₁₆×C₇₁	—	2	2	no	—
94	M₁₆×C₇₁	—	4	1	no	—
95	C₂×C₈×C₇₁	—	4	1	no	—
96	C₂×C₈×C₇₁	C₂²×C₄	1	1	yes	characteristic
97	C₂×C₈×C₇₁	C₂×D₄	1	1	yes	characteristic
98	C₂×C₈×C₇₁	C₂×D₄	1	1	yes	characteristic
99	C₂×C₈×C₇₁	C₂×Q₈	1	1	yes	characteristic
100	C₂×C₈×C₇₁	#41	1	1	yes	characteristic
101	C₂×C₈×C₇₁	#41	1	1	yes	characteristic
102	C₄⋊C₄×C₇₁	—	2	1	no	—
103	C₄⋊C₄×C₇₁	—	2	1	no	—
104	C₂²⋊C₄×C₇₁	—	4	1	no	—
105	C₄²×C₇₁	—	4	1	no	—
106	C₁₆×C₇₁	—	4	1	no	—
107	C₂³×C₇₁	#135	1	1	yes	characteristic
108	C₂²×D₇₁	—	4	1	no	—
109	C₂×C₄×C₇₁	—	4	1	no	—
110	C₂×C₄×C₇₁	—	4	1	no	—
111	C₂×C₄×C₇₁	C₄×D₄	1	1	yes	characteristic
112	C₂×C₄×C₇₁	#120	1	1	yes	characteristic
113	C₂×C₄×C₇₁	C₂²⋊Q₈	1	1	yes	characteristic
114	C₂×C₄×C₇₁	C₂×D₈	1	1	yes	characteristic
115	C₂×C₄×C₇₁	AGL(1,ℤ/8)	1	1	yes	characteristic
116	C₂×C₄×C₇₁	#137	1	1	yes	characteristic
117	C₂²⋊D₇₁	—	4	2	no	—
118	C₂²⋊D₇₁	—	4	2	no	—
119	C₂×Dic₇₁	—	2	2	no	—
120	C₂×Dic₇₁	—	4	1	no	—
121	C₂×Dic₇₁	—	4	1	no	—
122	C₂×Dic₇₁	—	4	1	no	—
123	D₂₈₄	—	2	2	no	—
124	D₂₈₄	—	4	1	no	—
125	C₄×D₇₁	—	4	2	no	—
126	C₄×D₇₁	—	4	2	no	—
127	Dic₁₄₂	—	2	2	no	—
128	Dic₁₄₂	—	4	1	no	—
129	C₈×C₇₁	—	4	2	no	—
130	C₈×C₇₁	C₂×C₄⋊C₄	1	1	yes	characteristic
131	C₈×C₇₁	C₄×D₄	1	1	yes	characteristic
132	C₈×C₇₁	C₂²⋊Q₈	1	1	yes	characteristic
133	C₈×C₇₁	#123	1	1	yes	characteristic
134	C₇₁⋊C₈	—	2	1	no	—
135	C₇₁⋊C₈	—	2	1	no	—
136	C₇₁⋊C₈	—	8	1	no	—
137	C₂²×C₇₁	—	2	1	no	—
138	C₂²×C₇₁	—	2	1	no	—
139	C₂²×C₇₁	C₄×D₈	1	1	yes	characteristic
140	C₂²×C₇₁	#452	1	1	yes	characteristic
141	C₂²×C₇₁	#478	1	1	yes	characteristic
142	C₂×D₇₁	—	4	2	no	—
143	C₂×D₇₁	—	8	1	no	—
144	C₂×D₇₁	—	8	1	no	—
145	C₄×C₇₁	—	8	1	no	—
146	C₄×C₇₁	—	8	1	no	—
147	C₄×C₇₁	#385	1	1	yes	characteristic
148	C₄×C₇₁	#440	1	1	yes	characteristic
149	C₄×C₇₁	#447	1	1	yes	characteristic
150	C₄×C₇₁	#474	1	1	yes	characteristic
151	Dic₇₁	—	4	2	no	—
152	Dic₇₁	—	4	2	no	—
153	#12939	—	71	1	no	maximal, Sylow
154	C₂×C₇₁	—	2	1	no	—
155	C₂×C₇₁	—	2	1	no	—
156	C₂×C₇₁	#1862	1	1	yes	characteristic
157	C₂×C₇₁	#2178	1	1	yes	characteristic
158	C₂×C₇₁	#2179	1	1	yes	characteristic
159	D₇₁	—	8	2	no	—
160	C₂×#443	—	71	1	no	—
161	C₂×#368	—	71	1	no	—
162	C₂×#359	—	71	1	no	—
163	#2138	—	71	1	no	—
164	#1945	—	71	1	no	—
165	#1382	D₇₁	1	1	yes	characteristic, complemented
166	#1374	—	71	1	no	—
167	C₇₁	#12939	1	1	yes	characteristic, complemented, Sylow
168	C₂×#135	—	71	1	no	—
169	C₂×#131	—	142	1	no	—
170	C₂²×C₁₆	—	71	1	no	—
171	C₂²×C₁₆	C₂×D₇₁	1	1	yes	characteristic
172	#443	—	142	2	no	—
173	#443	—	142	2	no	—
174	C₂×#108	—	71	1	no	—
175	C₂×#108	C₂×D₇₁	1	1	yes	characteristic
176	#427	C₂×D₇₁	1	1	yes	characteristic
177	#424	—	142	1	no	—
178	C₂×#104	—	142	1	no	—
179	C₂×C₄×C₈	—	71	1	no	—
180	#368	—	142	2	no	—
181	#359	—	142	2	no	—
182	#358	—	71	1	no	—
183	#358	—	71	1	no	—
184	#357	—	71	1	no	—
185	#357	—	71	1	no	—
186	#339	—	142	1	no	—
187	C₂×#41	—	142	1	no	—
188	#135	—	71	2	no	—
189	#135	—	71	2	no	—
190	#135	—	142	1	no	—
191	#135	—	142	1	no	—
192	C₂×Q₁₆	—	71	1	no	—
193	C₂×SD₁₆	—	142	1	no	—
194	C₂×D₈	—	71	1	no	—
195	#131	—	284	2	no	—
196	#131	—	142	4	no	—
197	C₂×M₁₆	—	2	1	no	—
198	C₂×M₁₆	—	284	1	no	—
199	C₂×M₁₆	—	142	1	no	—
200	C₂²×C₈	—	284	1	no	—
201	C₂²×C₈	—	71	1	no	—
202	C₂²×C₈	C₂²×D₇₁	1	1	yes	characteristic
203	#117	—	2	1	no	—
204	C₂×C₄²	—	71	1	no	—
205	C₂×C₁₆	—	2	1	no	—
206	C₂×C₁₆	—	2	1	no	—
207	C₂×C₁₆	—	2	1	no	—
208	C₂×C₁₆	—	71	1	no	—
209	C₂×C₁₆	—	71	1	no	—
210	C₂×C₁₆	—	142	1	no	—
211	C₂×C₁₆	—	142	1	no	—
212	#108	—	2	1	no	—
213	#108	—	142	1	no	—
214	#108	—	142	1	no	—
215	#108	D₂₈₄	1	2	yes	—
216	#107	C₄×D₇₁	1	1	yes	characteristic
217	#107	C₄×D₇₁	1	1	yes	characteristic
218	#106	—	2	1	no	—
219	C₄⋊C₈	—	284	1	no	—
220	C₄⋊C₈	—	284	1	no	—
221	#104	—	142	4	no	—
222	C₄×C₈	—	142	1	no	—
223	C₄×C₈	—	142	1	no	—
224	#41	—	284	1	no	—
225	#41	—	284	1	no	—
226	#41	—	142	2	no	—
227	#41	—	142	2	no	—
228	C₂×Q₈	—	142	1	no	—
229	C₂×D₄	—	284	1	no	—
230	C₂×D₄	—	142	1	no	—
231	C₂²×C₄	—	284	1	no	—
232	C₂²×C₄	—	142	1	no	—
233	C₂²×C₄	D₄×D₇₁	1	1	yes	characteristic
234	Q₁₆	—	71	2	no	—
235	Q₁₆	—	142	1	no	—
236	SD₁₆	—	142	2	no	—
237	SD₁₆	—	142	2	no	—
238	D₈	—	71	2	no	—
239	D₈	—	142	1	no	—
240	M₁₆	—	2	2	no	—
241	M₁₆	—	4	1	no	—
242	M₁₆	—	284	4	no	—
243	M₁₆	—	284	1	no	—
244	M₁₆	—	284	1	no	—
245	C₂×C₈	—	4	1	no	—
246	C₂×C₈	—	284	4	no	—
247	C₂×C₈	—	284	1	no	—
248	C₂×C₈	—	71	1	no	—
249	C₂×C₈	—	71	1	no	—
250	C₂×C₈	—	142	1	no	—
251	C₂×C₈	—	142	1	no	—
252	C₂×C₈	C₂×C₄×D₇₁	1	1	yes	characteristic
253	C₂×C₈	D₄×D₇₁	1	1	yes	characteristic
254	C₂×C₈	C₂×D₂₈₄	1	1	yes	characteristic
255	C₂×C₈	Q₈×D₇₁	1	1	yes	characteristic
256	C₂×C₈	#11765864	1	1	yes	characteristic
257	C₂×C₈	#11765865	1	1	yes	characteristic
258	C₄⋊C₄	—	2	1	no	—
259	C₄⋊C₄	—	2	1	no	—
260	C₂²⋊C₄	—	4	1	no	—
261	C₄²	—	4	1	no	—
262	C₄²	—	142	1	no	—
263	C₄²	—	142	1	no	—
264	C₁₆	—	4	1	no	—
265	C₁₆	—	142	1	no	—
266	C₁₆	—	142	1	no	—
267	C₂³	—	284	1	no	—
268	C₂³	#14556149	1	1	yes	characteristic
269	Q₈	—	284	1	no	—
270	Q₈	—	142	2	no	—
271	D₄	—	284	2	no	—
272	D₄	—	284	2	no	—
273	D₄	—	284	1	no	—
274	D₄	—	142	2	no	—
275	C₂×C₄	—	4	1	no	—
276	C₂×C₄	—	4	1	no	—
277	C₂×C₄	—	284	1	no	—
278	C₂×C₄	—	284	1	no	—
279	C₂×C₄	—	284	1	no	—
280	C₂×C₄	—	284	2	no	—
281	C₂×C₄	—	284	2	no	—
282	C₂×C₄	—	142	2	no	—
283	C₂×C₄	(C₂²⋊D₇₁)⋊C₄	1	1	yes	characteristic
284	C₂×C₄	#14556098	1	1	yes	characteristic
285	C₂×C₄	#14556106	1	1	yes	characteristic
286	C₂×C₄	D₈×D₇₁	1	1	yes	characteristic
287	C₂×C₄	#14556155	1	1	yes	characteristic
288	C₂×C₄	#14556160	1	1	yes	characteristic
289	C₈	—	4	2	no	—
290	C₈	—	568	1	no	—
291	C₈	—	142	1	no	—
292	C₈	—	142	1	no	—
293	C₈	C₄⋊C₄×D₇₁	1	1	yes	characteristic
294	C₈	C₄×D₂₈₄	1	1	yes	characteristic
295	C₈	#14556108	1	1	yes	characteristic
296	C₈	#14556110	1	1	yes	characteristic
297	C₂²	—	2	1	no	—
298	C₂²	—	2	1	no	—
299	C₂²	—	284	2	no	—
300	C₂²	—	568	1	no	—
301	C₂²	—	568	1	no	—
302	C₂²	#17986079	1	1	yes	characteristic
303	C₂²	#17986164	1	1	yes	characteristic
304	C₂²	#17986298	1	1	yes	characteristic
305	C₄	—	8	1	no	—
306	C₄	—	8	1	no	—
307	C₄	—	284	2	no	—
308	C₄	—	284	2	no	—
309	C₄	#17985873	1	1	yes	characteristic
310	C₄	#17986102	1	1	yes	characteristic
311	C₄	#17986134	1	1	yes	characteristic
312	C₄	#17986278	1	1	yes	characteristic
313	C₂	—	2	1	no	—
314	C₂	—	2	1	no	—
315	C₂	—	568	2	no	—
316	C₂	#22588588	1	1	yes	characteristic
317	C₂	#22590218	1	1	yes	characteristic
318	C₂	#22590223	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	D₈×D₇₁
commutator	C₂×C₈×C₇₁	C₂²×C₄
Frattini	C₂²×C₈	C₂²×D₇₁
Fitting	C₇₁×#1382	C₂
socle	C₂²×C₇₁	C₄×D₈
prime radical	C₂×#14556150	C₂²
cosocle	C₂²×C₈×C₇₁	C₂³
hypercenter	#1382	D₇₁
hypocenter	C₇₁	#12939
LCS	G ▷ C₂×C₈×C₇₁ ▷ C₄×C₇₁ ▷ C₂×C₇₁ ▷ C₇₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄ ◁ C₂²×C₈ ◁ #1382

Complements

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

N	H ≅ G/N	type
2. C₂×#17985828	C₂	semidirect
5. #22589089	C₂	semidirect
6. #22586417	C₂	semidirect
7. C₇₁×#1382	C₂	semidirect
19. #17985802	C₄	semidirect
20. #17985802	C₄	semidirect
21. #17985799	C₄	semidirect
22. #17985799	C₄	semidirect
165. #1382	D₇₁	semidirect
167. C₇₁	#12939	semidirect

Maximal subgroups

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

subgroup	index
C₂×#17986115	2
C₂×#17985828	2
C₂×#17985807	2
#22590045	2
#22589089	2
#22586417	2
C₇₁×#1382	2
#12939	71