Subgroups — #31890646

Order 18176.

386 conjugacy classes of subgroups.

The subgroup lattice has 306 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄×#14556064	C₂	1	1	yes	maximal, characteristic, complemented
2	#22587096	C₂	1	1	yes	maximal, characteristic, complemented
3	#22586779	C₂	1	2	yes	maximal
4	#22586231	C₂	1	2	yes	maximal, complemented
5	C₇₁×#2965	C₂	1	1	yes	maximal, characteristic
6	C₂×#14556064	—	2	1	no	—
7	#17986023	C₂²	1	2	yes	—
8	C₄×#11765849	C₂²	1	2	yes	—
9	#17985764	—	2	1	no	—
10	#17985750	—	2	1	no	—
11	#17985750	—	2	1	no	—
12	Q₈⋊(C₇₁⋊C₈)	—	2	2	no	—
13	D₄⋊(C₇₁⋊C₈)	—	2	2	no	—
14	C₄×#41×C₇₁	C₂²	1	1	yes	characteristic
15	#446×C₇₁	C₄	1	1	yes	characteristic
16	#446×C₇₁	C₄	1	1	yes	characteristic
17	#436×C₇₁	C₄	1	1	yes	characteristic
18	#436×C₇₁	C₄	1	1	yes	characteristic
19	#435×C₇₁	—	2	1	no	—
20	#435×C₇₁	—	2	1	no	—
21	#431×C₇₁	—	2	1	no	—
22	#431×C₇₁	C₂²	1	1	yes	characteristic
23	C₄×M₁₆×C₇₁	—	2	1	no	—
24	C₂×C₄×C₈×C₇₁	C₂²	1	1	yes	characteristic
25	C₂×#11765849	—	2	2	no	—
26	C₂×#11765849	—	2	2	no	—
27	#14556064	—	2	4	no	—
28	C₄⋊(C₇₁⋊C₈)	—	2	2	no	—
29	C₄⋊(C₇₁⋊C₈)	—	2	2	no	—
30	#14556039	—	2	2	no	—
31	C₄×C₇₁⋊C₈	—	2	2	no	—
32	C₂×#41×C₇₁	Q₈	1	1	yes	characteristic
33	C₂×M₁₆×C₇₁	—	2	1	no	—
34	C₂×M₁₆×C₇₁	—	2	1	no	—
35	C₂²×C₈×C₇₁	D₄	1	1	yes	characteristic
36	C₂²×C₈×C₇₁	D₄	1	1	yes	characteristic
37	C₄×Q₈×C₇₁	C₂×C₄	1	1	yes	characteristic
38	C₄×Q₈×C₇₁	D₄	1	1	yes	characteristic
39	C₄×D₄×C₇₁	—	2	1	no	—
40	C₄×D₄×C₇₁	—	2	1	no	—
41	C₄×D₄×C₇₁	C₂×C₄	1	1	yes	characteristic
42	C₄×D₄×C₇₁	D₄	1	1	yes	characteristic
43	#117×C₇₁	—	2	1	no	—
44	#117×C₇₁	Q₈	1	1	yes	characteristic
45	C₂×C₄²×C₇₁	—	2	1	no	—
46	C₂×C₄²×C₇₁	C₂³	1	1	yes	characteristic
47	C₄⋊C₈×C₇₁	—	2	1	no	—
48	C₄⋊C₈×C₇₁	—	2	1	no	—
49	C₄⋊C₈×C₇₁	C₂×C₄	1	1	yes	characteristic
50	C₄⋊C₈×C₇₁	C₂×C₄	1	1	yes	characteristic
51	C₂²⋊C₈×C₇₁	—	2	1	no	—
52	C₂²⋊C₈×C₇₁	—	2	1	no	—
53	C₂²⋊C₈×C₇₁	—	2	1	no	—
54	#97×C₇₁	—	2	1	no	—
55	#97×C₇₁	—	2	1	no	—
56	C₄×C₈×C₇₁	—	2	1	no	—
57	C₄×C₈×C₇₁	C₂×C₄	1	1	yes	characteristic
58	C₄×C₈×C₇₁	C₂×C₄	1	1	yes	characteristic
59	C₄×C₈×C₇₁	D₄	1	1	yes	characteristic
60	C₄×C₈×C₇₁	D₄	1	1	yes	characteristic
61	#11765849	—	2	4	no	—
62	#11765849	—	4	2	no	—
63	#11765849	—	8	2	no	—
64	C₂×C₇₁⋊C₈	—	4	2	no	—
65	C₂×C₇₁⋊C₈	—	4	2	no	—
66	#41×C₇₁	—	4	2	no	—
67	C₂×Q₈×C₇₁	C₄⋊C₄	1	1	yes	characteristic
68	C₂×D₄×C₇₁	—	2	1	no	—
69	C₂×D₄×C₇₁	C₄⋊C₄	1	1	yes	characteristic
70	C₂²×C₄×C₇₁	—	2	1	no	—
71	C₂²×C₄×C₇₁	—	2	1	no	—
72	C₂²×C₄×C₇₁	—	2	1	no	—
73	C₂²×C₄×C₇₁	C₂×D₄	1	1	yes	characteristic
74	C₂²×C₄×C₇₁	C₂×Q₈	1	1	yes	characteristic
75	C₂²×C₄×C₇₁	#41	1	1	yes	characteristic
76	M₁₆×C₇₁	—	4	2	no	—
77	M₁₆×C₇₁	—	4	2	no	—
78	C₂×C₈×C₇₁	—	2	2	no	—
79	C₂×C₈×C₇₁	—	2	1	no	—
80	C₂×C₈×C₇₁	—	2	1	no	—
81	C₂×C₈×C₇₁	—	4	1	no	—
82	C₂×C₈×C₇₁	C₂²⋊C₄	1	1	yes	characteristic
83	C₂×C₈×C₇₁	C₂²⋊C₄	1	1	yes	characteristic
84	C₂×C₈×C₇₁	C₂²⋊C₄	1	1	yes	characteristic
85	C₂×C₈×C₇₁	C₂²⋊C₄	1	1	yes	characteristic
86	C₄⋊C₄×C₇₁	—	2	1	no	—
87	C₄⋊C₄×C₇₁	—	2	1	no	—
88	C₄⋊C₄×C₇₁	C₄⋊C₄	1	1	yes	characteristic
89	C₄⋊C₄×C₇₁	C₄⋊C₄	1	1	yes	characteristic
90	C₂²⋊C₄×C₇₁	—	2	1	no	—
91	C₂²⋊C₄×C₇₁	—	2	1	no	—
92	C₂²⋊C₄×C₇₁	—	2	1	no	—
93	C₄²×C₇₁	—	2	1	no	—
94	C₄²×C₇₁	—	2	1	no	—
95	C₄²×C₇₁	—	2	1	no	—
96	C₄²×C₇₁	C₂²×C₄	1	1	yes	characteristic
97	C₄²×C₇₁	C₂×D₄	1	1	yes	characteristic
98	C₄²×C₇₁	C₂×D₄	1	1	yes	characteristic
99	C₄²×C₇₁	#41	1	1	yes	characteristic
100	C₂³×C₇₁	—	2	1	no	—
101	C₂³×C₇₁	C₂²⋊Q₈	1	1	yes	characteristic
102	Q₈×C₇₁	—	2	2	no	—
103	D₄×C₇₁	—	2	2	no	—
104	D₄×C₇₁	—	4	2	no	—
105	C₂×C₄×C₇₁	—	2	1	no	—
106	C₂×C₄×C₇₁	—	2	1	no	—
107	C₂×C₄×C₇₁	—	2	2	no	—
108	C₂×C₄×C₇₁	—	2	2	no	—
109	C₂×C₄×C₇₁	—	2	1	no	—
110	C₂×C₄×C₇₁	—	2	1	no	—
111	C₂×C₄×C₇₁	—	2	1	no	—
112	C₂×C₄×C₇₁	—	4	2	no	—
113	C₂×C₄×C₇₁	—	4	1	no	—
114	C₂×C₄×C₇₁	—	4	1	no	—
115	C₂×C₄×C₇₁	—	4	2	no	—
116	C₂×C₄×C₇₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
117	C₂×C₄×C₇₁	C₂×C₄⋊C₄	1	1	yes	characteristic
118	C₂×C₄×C₇₁	#117	1	1	yes	characteristic
119	C₂×C₄×C₇₁	#121	1	1	yes	characteristic
120	C₂×C₄×C₇₁	#121	1	1	yes	characteristic
121	C₂×C₄×C₇₁	C₂²⋊Q₈	1	1	yes	characteristic
122	C₈×C₇₁	—	2	1	no	—
123	C₈×C₇₁	—	2	1	no	—
124	C₈×C₇₁	—	2	1	no	—
125	C₈×C₇₁	—	4	1	no	—
126	C₈×C₇₁	—	4	1	no	—
127	C₈×C₇₁	#104	1	2	yes	—
128	C₇₁⋊C₈	—	4	4	no	—
129	C₇₁⋊C₈	—	8	2	no	—
130	C₂²×C₇₁	—	2	1	no	—
131	C₂²×C₇₁	—	2	1	no	—
132	C₂²×C₇₁	—	2	1	no	—
133	C₂²×C₇₁	—	4	1	no	—
134	C₂²×C₇₁	—	4	1	no	—
135	C₂²×C₇₁	—	4	1	no	—
136	C₂²×C₇₁	#380	1	1	yes	characteristic
137	C₄×C₇₁	—	2	1	no	—
138	C₄×C₇₁	—	2	1	no	—
139	C₄×C₇₁	—	2	1	no	—
140	C₄×C₇₁	—	4	1	no	—
141	C₄×C₇₁	—	4	1	no	—
142	C₄×C₇₁	—	4	1	no	—
143	C₄×C₇₁	C₂×#104	1	1	yes	characteristic
144	C₄×C₇₁	#421	1	1	yes	characteristic
145	C₄×C₇₁	#430	1	2	yes	—
146	C₄×C₇₁	#471	1	2	yes	—
147	#8672	—	71	1	no	maximal, Sylow
148	C₂×C₇₁	—	4	1	no	—
149	C₂×C₇₁	—	8	1	no	—
150	C₂×C₇₁	#1806	1	1	yes	characteristic
151	C₂×C₇₁	#1811	1	1	yes	characteristic
152	C₂×C₇₁	#1815	1	1	yes	characteristic
153	#2965	D₇₁	1	1	yes	characteristic
154	C₄×#108	—	71	1	no	—
155	#1566	—	71	1	no	—
156	#1489	—	71	2	no	—
157	#1299	—	71	2	no	—
158	C₇₁	#8672	1	1	yes	characteristic, complemented, Sylow
159	C₄×#41	C₂×D₇₁	1	1	yes	characteristic
160	#446	Dic₇₁	1	1	yes	characteristic
161	#446	Dic₇₁	1	1	yes	characteristic
162	#436	Dic₇₁	1	1	yes	characteristic
163	#436	Dic₇₁	1	1	yes	characteristic
164	#435	—	2	1	no	—
165	#435	—	2	1	no	—
166	#431	—	2	1	no	—
167	#431	C₂×D₇₁	1	1	yes	characteristic
168	C₂×#108	—	142	1	no	—
169	#423	—	71	2	no	—
170	C₄×M₁₆	—	2	1	no	—
171	C₄×M₁₆	—	71	2	no	—
172	C₂×C₄×C₈	C₂×D₇₁	1	1	yes	characteristic
173	#341	—	142	1	no	—
174	#334	—	142	1	no	—
175	#334	—	142	1	no	—
176	Q₈⋊C₈	—	142	2	no	—
177	D₄⋊C₈	—	142	2	no	—
178	C₂×#41	Dic₁₄₂	1	1	yes	characteristic
179	C₂×M₁₆	—	2	1	no	—
180	C₂×M₁₆	—	2	1	no	—
181	C₂×M₁₆	—	142	2	no	—
182	C₂×M₁₆	—	142	2	no	—
183	C₂²×C₈	C₂²⋊D₇₁	1	1	yes	characteristic
184	C₂²×C₈	C₂²⋊D₇₁	1	1	yes	characteristic
185	C₄×Q₈	D₂₈₄	1	1	yes	characteristic
186	C₄×Q₈	C₂×Dic₇₁	1	1	yes	characteristic
187	C₄×D₄	—	2	1	no	—
188	C₄×D₄	—	2	1	no	—
189	C₄×D₄	D₂₈₄	1	1	yes	characteristic
190	C₄×D₄	C₂×Dic₇₁	1	1	yes	characteristic
191	#117	—	2	1	no	—
192	#117	Dic₁₄₂	1	1	yes	characteristic
193	C₂×C₄²	—	2	1	no	—
194	C₂×C₄²	C₂²×D₇₁	1	1	yes	characteristic
195	#108	—	142	4	no	—
196	C₄⋊C₈	—	2	1	no	—
197	C₄⋊C₈	—	2	1	no	—
198	C₄⋊C₈	—	142	2	no	—
199	C₄⋊C₈	—	142	2	no	—
200	C₄⋊C₈	C₂×Dic₇₁	1	1	yes	characteristic
201	C₄⋊C₈	C₂×Dic₇₁	1	1	yes	characteristic
202	C₂²⋊C₈	—	2	1	no	—
203	C₂²⋊C₈	—	2	1	no	—
204	C₂²⋊C₈	—	2	1	no	—
205	#97	—	2	1	no	—
206	#97	—	2	1	no	—
207	#97	—	142	2	no	—
208	C₄×C₈	—	2	1	no	—
209	C₄×C₈	—	142	2	no	—
210	C₄×C₈	C₂×Dic₇₁	1	1	yes	characteristic
211	C₄×C₈	C₂×Dic₇₁	1	1	yes	characteristic
212	C₄×C₈	C₂²⋊D₇₁	1	1	yes	characteristic
213	C₄×C₈	C₂²⋊D₇₁	1	1	yes	characteristic
214	#41	—	4	2	no	—
215	C₂×Q₈	C₄⋊Dic₇₁	1	1	yes	characteristic
216	C₂×D₄	—	2	1	no	—
217	C₂×D₄	C₄⋊Dic₇₁	1	1	yes	characteristic
218	C₂²×C₄	—	2	1	no	—
219	C₂²×C₄	—	2	1	no	—
220	C₂²×C₄	—	2	1	no	—
221	C₂²×C₄	C₂×C₂²⋊D₇₁	1	1	yes	characteristic
222	C₂²×C₄	C₂×Dic₁₄₂	1	1	yes	characteristic
223	C₂²×C₄	#11765865	1	1	yes	characteristic
224	M₁₆	—	4	2	no	—
225	M₁₆	—	4	2	no	—
226	M₁₆	—	284	2	no	—
227	M₁₆	—	568	2	no	—
228	M₁₆	—	142	4	no	—
229	C₂×C₈	—	2	2	no	—
230	C₂×C₈	—	2	1	no	—
231	C₂×C₈	—	2	1	no	—
232	C₂×C₈	—	4	1	no	—
233	C₂×C₈	—	284	2	no	—
234	C₂×C₈	—	284	2	no	—
235	C₂×C₈	C₂²⋊Dic₇₁	1	1	yes	characteristic
236	C₂×C₈	C₂²⋊Dic₇₁	1	1	yes	characteristic
237	C₂×C₈	C₂²⋊Dic₇₁	1	1	yes	characteristic
238	C₂×C₈	C₂²⋊Dic₇₁	1	1	yes	characteristic
239	C₄⋊C₄	—	2	1	no	—
240	C₄⋊C₄	—	2	1	no	—
241	C₄⋊C₄	C₄⋊Dic₇₁	1	1	yes	characteristic
242	C₄⋊C₄	C₄⋊Dic₇₁	1	1	yes	characteristic
243	C₂²⋊C₄	—	2	1	no	—
244	C₂²⋊C₄	—	2	1	no	—
245	C₂²⋊C₄	—	2	1	no	—
246	C₄²	—	2	1	no	—
247	C₄²	—	2	1	no	—
248	C₄²	—	2	1	no	—
249	C₄²	C₂²×Dic₇₁	1	1	yes	characteristic
250	C₄²	C₂×C₂²⋊D₇₁	1	1	yes	characteristic
251	C₄²	C₂×D₂₈₄	1	1	yes	characteristic
252	C₄²	#11765865	1	1	yes	characteristic
253	C₂³	—	2	1	no	—
254	C₂³	#14556107	1	1	yes	characteristic
255	Q₈	—	2	2	no	—
256	D₄	—	2	2	no	—
257	D₄	—	4	2	no	—
258	C₂×C₄	—	2	2	no	—
259	C₂×C₄	—	2	2	no	—
260	C₂×C₄	—	2	1	no	—
261	C₂×C₄	—	2	1	no	—
262	C₂×C₄	—	2	1	no	—
263	C₂×C₄	—	2	1	no	—
264	C₂×C₄	—	2	1	no	—
265	C₂×C₄	—	4	2	no	—
266	C₂×C₄	—	4	2	no	—
267	C₂×C₄	—	4	1	no	—
268	C₂×C₄	—	4	1	no	—
269	C₂×C₄	C₂×C₂²⋊Dic₇₁	1	1	yes	characteristic
270	C₂×C₄	C₂×C₄⋊Dic₇₁	1	1	yes	characteristic
271	C₂×C₄	#14556087	1	1	yes	characteristic
272	C₂×C₄	#14556102	1	1	yes	characteristic
273	C₂×C₄	#14556102	1	1	yes	characteristic
274	C₂×C₄	#14556107	1	1	yes	characteristic
275	C₈	—	2	1	no	—
276	C₈	—	2	1	no	—
277	C₈	—	2	1	no	—
278	C₈	—	4	1	no	—
279	C₈	—	4	1	no	—
280	C₈	—	284	4	no	—
281	C₈	—	568	2	no	—
282	C₈	#14556055	1	2	yes	—
283	C₂²	—	2	1	no	—
284	C₂²	—	2	1	no	—
285	C₂²	—	2	1	no	—
286	C₂²	—	4	1	no	—
287	C₂²	—	4	1	no	—
288	C₂²	—	4	1	no	—
289	C₂²	#17985859	1	1	yes	characteristic
290	C₄	—	2	1	no	—
291	C₄	—	2	1	no	—
292	C₄	—	2	1	no	—
293	C₄	—	4	1	no	—
294	C₄	—	4	1	no	—
295	C₄	—	4	1	no	—
296	C₄	C₂×#14556055	1	1	yes	characteristic
297	C₄	#17986016	1	1	yes	characteristic
298	C₄	#17986048	1	2	yes	—
299	C₄	#17986262	1	2	yes	—
300	C₂	—	4	1	no	—
301	C₂	—	8	1	no	—
302	C₂	#22588360	1	1	yes	characteristic
303	C₂	#22588382	1	1	yes	characteristic
304	C₂	#22588399	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	#14556102
commutator	C₄²×C₇₁	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₇₁
Fitting	C₇₁×#2965	C₂
socle	C₂²×C₇₁	#380
prime radical	C₂×#41×C₇₁	Q₈
cosocle	C₂×C₄²×C₇₁	C₂³
hypercenter	#2965	D₇₁
hypocenter	C₇₁	#8672
LCS	G ▷ C₄²×C₇₁ ▷ C₂²×C₇₁ ▷ C₇₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #2965

Complements

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

N	H ≅ G/N	type
1. C₄×#14556064	C₂	semidirect
2. #22587096	C₂	semidirect
4. #22586231	C₂	semidirect
158. C₇₁	#8672	semidirect

Maximal subgroups

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

subgroup	index
C₄×#14556064	2
#22587096	2
#22586779	2
#22586231	2
C₇₁×#2965	2
#8672	71