Subgroups — #31914070

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×#17986116	C₂	1	1	yes	maximal, characteristic
2	C₂×#17985828	C₂	1	1	yes	maximal, characteristic, complemented
3	#22590047	C₂	1	1	yes	maximal, characteristic
4	#22589086	C₂	1	1	yes	maximal, characteristic, complemented
5	#22586436	C₂	1	1	yes	maximal, characteristic, complemented
6	#22586417	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂×#359×C₇₁	C₂	1	1	yes	maximal, characteristic
8	C₂×#14556135	—	2	1	no	—
9	C₂²×C₇₁⋊C₁₆	C₂²	1	1	yes	characteristic
10	#17986116	—	2	2	no	—
11	#17986116	—	2	2	no	—
12	C₂×#14556064	C₂²	1	1	yes	characteristic
13	#17986037	C₂²	1	1	yes	characteristic
14	#17986027	—	2	1	no	—
15	C₂×#14556055	—	2	1	no	—
16	C₂×C₈×Dic₇₁	C₂²	1	1	yes	characteristic
17	#17985828	—	2	2	no	—
18	#17985803	C₄	1	1	yes	characteristic, complemented
19	#17985803	C₄	1	1	yes	characteristic, complemented
20	#17985800	C₄	1	1	yes	characteristic, complemented
21	#17985800	C₄	1	1	yes	characteristic, complemented
22	#17985760	—	2	1	no	—
23	C₂×#135×C₇₁	C₂²	1	1	yes	characteristic
24	C₂²×C₁₆×C₇₁	C₂²	1	1	yes	characteristic
25	C₂×#108×C₇₁	C₂²	1	1	yes	characteristic
26	#359×C₇₁	—	2	2	no	—
27	#14556135	—	2	4	no	—
28	#14556135	—	4	2	no	—
29	C₂×#11765849	—	2	1	no	—
30	C₂×#11765849	—	4	1	no	—
31	C₂²×C₇₁⋊C₈	—	4	1	no	—
32	C₂²×C₇₁⋊C₈	D₄	1	1	yes	characteristic
33	#14556087	—	2	1	no	—
34	C₂×C₄×Dic₇₁	D₄	1	1	yes	characteristic
35	C₂×C₇₁⋊C₁₆	—	2	1	no	—
36	C₂×C₇₁⋊C₁₆	—	2	1	no	—
37	C₂×C₇₁⋊C₁₆	C₂×C₄	1	1	yes	characteristic
38	C₂×C₇₁⋊C₁₆	C₂×C₄	1	1	yes	characteristic
39	#14556064	—	2	1	no	—
40	#14556064	D₄	1	2	yes	—
41	#14556062	C₂×C₄	1	1	yes	characteristic
42	#14556062	C₂×C₄	1	1	yes	characteristic
43	#14556060	—	2	1	no	—
44	Dic₇₁⋊C₈	—	4	1	no	—
45	Dic₇₁⋊C₈	—	4	1	no	—
46	#14556055	—	2	4	no	—
47	C₈×Dic₇₁	—	2	1	no	—
48	C₈×Dic₇₁	—	2	1	no	—
49	C₂×#41×C₇₁	—	2	1	no	—
50	#135×C₇₁	—	2	1	no	—
51	#135×C₇₁	—	2	1	no	—
52	#135×C₇₁	D₄	1	2	yes	—
53	#135×C₇₁	Q₈	1	2	yes	—
54	C₂×Q₁₆×C₇₁	C₂×C₄	1	1	yes	characteristic
55	C₂×SD₁₆×C₇₁	—	2	1	no	—
56	C₂×D₈×C₇₁	C₂×C₄	1	1	yes	characteristic
57	C₂×M₁₆×C₇₁	—	2	1	no	—
58	C₂²×C₈×C₇₁	C₂³	1	1	yes	characteristic
59	C₂×C₁₆×C₇₁	—	2	1	no	—
60	C₂×C₁₆×C₇₁	—	2	1	no	—
61	C₂×C₁₆×C₇₁	—	2	1	no	—
62	#108×C₇₁	—	2	1	no	—
63	#108×C₇₁	—	2	1	no	—
64	C₂²×Dic₇₁	—	2	1	no	—
65	#11765849	—	2	2	no	—
66	#11765849	—	4	4	no	—
67	#11765849	—	4	1	no	—
68	C₂×C₇₁⋊C₈	—	2	1	no	—
69	C₂×C₇₁⋊C₈	—	2	1	no	—
70	C₂×C₇₁⋊C₈	—	4	4	no	—
71	C₂×C₇₁⋊C₈	—	4	1	no	—
72	C₂×C₇₁⋊C₈	D₈	1	1	yes	characteristic
73	C₂×C₇₁⋊C₈	D₈	1	1	yes	characteristic
74	C₄⋊Dic₇₁	—	2	1	no	—
75	C₄⋊Dic₇₁	—	2	1	no	—
76	C₂²⋊Dic₇₁	—	4	1	no	—
77	C₄×Dic₇₁	—	2	1	no	—
78	C₄×Dic₇₁	—	2	1	no	—
79	C₄×Dic₇₁	—	4	1	no	—
80	C₇₁⋊C₁₆	—	2	1	no	—
81	C₇₁⋊C₁₆	—	2	1	no	—
82	#41×C₇₁	—	2	2	no	—
83	#41×C₇₁	—	2	2	no	—
84	#41×C₇₁	—	4	1	no	—
85	#41×C₇₁	—	4	1	no	—
86	C₂×Q₈×C₇₁	—	2	1	no	—
87	C₂×D₄×C₇₁	—	2	1	no	—
88	C₂×D₄×C₇₁	—	4	1	no	—
89	C₂²×C₄×C₇₁	—	4	1	no	—
90	C₂²×C₄×C₇₁	C₂×D₄	1	1	yes	characteristic
91	Q₁₆×C₇₁	—	2	1	no	—
92	Q₁₆×C₇₁	C₄⋊C₄	1	2	yes	—
93	SD₁₆×C₇₁	—	2	2	no	—
94	SD₁₆×C₇₁	—	2	2	no	—
95	D₈×C₇₁	—	2	1	no	—
96	D₈×C₇₁	C₄⋊C₄	1	2	yes	—
97	M₁₆×C₇₁	—	4	1	no	—
98	M₁₆×C₇₁	—	4	1	no	—
99	C₂×C₈×C₇₁	—	4	1	no	—
100	C₂×C₈×C₇₁	C₂²×C₄	1	1	yes	characteristic
101	C₂×C₈×C₇₁	C₂×D₄	1	1	yes	characteristic
102	C₂×C₈×C₇₁	C₂×D₄	1	1	yes	characteristic
103	C₂×C₈×C₇₁	C₂×Q₈	1	1	yes	characteristic
104	C₂×C₈×C₇₁	#41	1	1	yes	characteristic
105	C₂×C₈×C₇₁	#41	1	1	yes	characteristic
106	C₁₆×C₇₁	—	4	1	no	—
107	C₂³×C₇₁	—	4	1	no	—
108	C₂³×C₇₁	#135	1	1	yes	characteristic
109	Q₈×C₇₁	—	2	2	no	—
110	Q₈×C₇₁	—	4	1	no	—
111	D₄×C₇₁	—	2	2	no	—
112	D₄×C₇₁	—	4	2	no	—
113	D₄×C₇₁	—	4	2	no	—
114	D₄×C₇₁	—	4	1	no	—
115	C₂×C₄×C₇₁	—	4	1	no	—
116	C₂×C₄×C₇₁	—	4	2	no	—
117	C₂×C₄×C₇₁	—	4	2	no	—
118	C₂×C₄×C₇₁	C₄×D₄	1	1	yes	characteristic
119	C₂×C₄×C₇₁	#120	1	1	yes	characteristic
120	C₂×C₄×C₇₁	C₂²⋊Q₈	1	1	yes	characteristic
121	C₂×C₄×C₇₁	C₂×D₈	1	1	yes	characteristic
122	C₂×C₄×C₇₁	AGL(1,ℤ/8)	1	1	yes	characteristic
123	C₂×C₄×C₇₁	#137	1	1	yes	characteristic
124	C₂×Dic₇₁	—	2	2	no	—
125	C₂×Dic₇₁	—	4	1	no	—
126	C₂×Dic₇₁	—	4	1	no	—
127	C₂×Dic₇₁	—	4	1	no	—
128	C₂×Dic₇₁	—	4	1	no	—
129	C₈×C₇₁	—	8	1	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	—	4	2	no	—
137	C₂²×C₇₁	—	2	1	no	—
138	C₂²×C₇₁	—	2	1	no	—
139	C₂²×C₇₁	—	4	2	no	—
140	C₂²×C₇₁	—	8	1	no	—
141	C₂²×C₇₁	—	8	1	no	—
142	C₂²×C₇₁	C₄×D₈	1	1	yes	characteristic
143	C₂²×C₇₁	#452	1	1	yes	characteristic
144	C₂²×C₇₁	#478	1	1	yes	characteristic
145	C₄×C₇₁	—	4	2	no	—
146	C₄×C₇₁	#385	1	1	yes	characteristic
147	C₄×C₇₁	#440	1	1	yes	characteristic
148	C₄×C₇₁	#447	1	1	yes	characteristic
149	C₄×C₇₁	#474	1	1	yes	characteristic
150	Dic₇₁	—	4	2	no	—
151	Dic₇₁	—	8	1	no	—
152	Dic₇₁	—	8	1	no	—
153	#12939	—	71	1	no	maximal, Sylow
154	C₂×C₇₁	—	2	1	no	—
155	C₂×C₇₁	—	2	1	no	—
156	C₂×C₇₁	—	8	2	no	—
157	C₂×C₇₁	#1862	1	1	yes	characteristic
158	C₂×C₇₁	#2178	1	1	yes	characteristic
159	C₂×C₇₁	#2179	1	1	yes	characteristic
160	C₂×#443	—	71	1	no	—
161	C₂×#368	—	71	1	no	—
162	C₂×#359	D₇₁	1	1	yes	characteristic
163	#2138	—	71	1	no	—
164	#1945	—	71	1	no	—
165	#1382	—	71	1	no	—
166	#1374	—	71	1	no	—
167	C₇₁	#12939	1	1	yes	characteristic, complemented, Sylow
168	C₂×#135	C₂×D₇₁	1	1	yes	characteristic
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	—	71	1	no	—
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	—	2	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	—	2	1	no	—
188	#135	—	2	1	no	—
189	#135	—	2	1	no	—
190	#135	Dic₁₄₂	1	2	yes	—
191	#135	C₂²⋊D₇₁	1	2	yes	—
192	C₂×Q₁₆	C₄×D₇₁	1	1	yes	characteristic
193	C₂×SD₁₆	—	2	1	no	—
194	C₂×D₈	C₄×D₇₁	1	1	yes	characteristic
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	—	142	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	—	2	1	no	—
214	#108	—	71	2	no	—
215	#108	—	142	1	no	—
216	#107	—	71	1	no	—
217	#107	—	71	1	no	—
218	#106	—	142	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	—	2	2	no	—
225	#41	—	2	2	no	—
226	#41	—	4	1	no	—
227	#41	—	4	1	no	—
228	C₂×Q₈	—	2	1	no	—
229	C₂×D₄	—	2	1	no	—
230	C₂×D₄	—	4	1	no	—
231	C₂²×C₄	—	4	1	no	—
232	C₂²×C₄	—	142	1	no	—
233	C₂²×C₄	D₄×D₇₁	1	1	yes	characteristic
234	Q₁₆	—	2	1	no	—
235	Q₁₆	Dic₇₁⋊C₄	1	2	yes	—
236	SD₁₆	—	2	2	no	—
237	SD₁₆	—	2	2	no	—
238	D₈	—	2	1	no	—
239	D₈	Dic₇₁⋊C₄	1	2	yes	—
240	M₁₆	—	4	1	no	—
241	M₁₆	—	4	1	no	—
242	M₁₆	—	284	1	no	—
243	M₁₆	—	284	4	no	—
244	M₁₆	—	142	2	no	—
245	C₂×C₈	—	4	1	no	—
246	C₂×C₈	—	284	1	no	—
247	C₂×C₈	—	284	4	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×C₂²⋊D₇₁	1	1	yes	characteristic
255	C₂×C₈	C₂×Dic₁₄₂	1	1	yes	characteristic
256	C₂×C₈	#11765864	1	1	yes	characteristic
257	C₂×C₈	#11765864	1	1	yes	characteristic
258	C₄⋊C₄	—	142	1	no	—
259	C₄⋊C₄	—	142	1	no	—
260	C₂²⋊C₄	—	284	1	no	—
261	C₄²	—	284	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	—	4	1	no	—
268	C₂³	#14556149	1	1	yes	characteristic
269	Q₈	—	2	2	no	—
270	Q₈	—	4	1	no	—
271	D₄	—	2	2	no	—
272	D₄	—	4	2	no	—
273	D₄	—	4	2	no	—
274	D₄	—	4	1	no	—
275	C₂×C₄	—	4	1	no	—
276	C₂×C₄	—	4	2	no	—
277	C₂×C₄	—	4	2	no	—
278	C₂×C₄	—	284	1	no	—
279	C₂×C₄	—	284	1	no	—
280	C₂×C₄	—	284	1	no	—
281	C₂×C₄	—	284	1	no	—
282	C₂×C₄	—	142	2	no	—
283	C₂×C₄	(C₂²⋊D₇₁)⋊C₄	1	1	yes	characteristic
284	C₂×C₄	#14556096	1	1	yes	characteristic
285	C₂×C₄	#14556105	1	1	yes	characteristic
286	C₂×C₄	D₈×D₇₁	1	1	yes	characteristic
287	C₂×C₄	#14556152	1	1	yes	characteristic
288	C₂×C₄	#14556157	1	1	yes	characteristic
289	C₈	—	8	1	no	—
290	C₈	—	284	2	no	—
291	C₈	—	142	1	no	—
292	C₈	—	142	1	no	—
293	C₈	C₂×Dic₇₁⋊C₄	1	1	yes	characteristic
294	C₈	(C₂²⋊D₇₁)⋊C₄	1	1	yes	characteristic
295	C₈	#14556105	1	1	yes	characteristic
296	C₈	#14556109	1	1	yes	characteristic
297	C₂²	—	2	1	no	—
298	C₂²	—	2	1	no	—
299	C₂²	—	4	2	no	—
300	C₂²	—	8	1	no	—
301	C₂²	—	8	1	no	—
302	C₂²	#17986079	1	1	yes	characteristic
303	C₂²	#17986158	1	1	yes	characteristic
304	C₂²	#17986301	1	1	yes	characteristic
305	C₄	—	4	2	no	—
306	C₄	—	284	2	no	—
307	C₄	—	568	1	no	—
308	C₄	—	568	1	no	—
309	C₄	#17985871	1	1	yes	characteristic
310	C₄	#17986096	1	1	yes	characteristic
311	C₄	#17986131	1	1	yes	characteristic
312	C₄	#17986277	1	1	yes	characteristic
313	C₂	—	2	1	no	—
314	C₂	—	2	1	no	—
315	C₂	—	8	2	no	—
316	C₂	#22588585	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	C₂×#359×C₇₁	C₂
socle	C₂²×C₇₁	C₄×D₈
prime radical	C₂×#135×C₇₁	C₂²
cosocle	C₂²×C₈×C₇₁	C₂³
hypercenter	C₂×#359	D₇₁
hypocenter	C₇₁	#12939
LCS	G ▷ C₂×C₈×C₇₁ ▷ C₄×C₇₁ ▷ C₂×C₇₁ ▷ C₇₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄ ◁ C₂²×C₈ ◁ C₂×#359

Complements

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

N	H ≅ G/N	type
2. C₂×#17985828	C₂	semidirect
4. #22589086	C₂	semidirect
5. #22586436	C₂	semidirect
6. #22586417	C₂	semidirect
18. #17985803	C₄	semidirect
19. #17985803	C₄	semidirect
20. #17985800	C₄	semidirect
21. #17985800	C₄	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×#17986116	2
C₂×#17985828	2
#22590047	2
#22589086	2
#22586436	2
#22586417	2
C₂×#359×C₇₁	2
#12939	71