Subgroups — #31890536

Order 18176.

510 conjugacy classes of subgroups.

The subgroup lattice has 310 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#22588497	C₂	1	1	yes	maximal, characteristic, complemented
2	#22588340	C₂	1	2	yes	maximal
3	C₂×#17985752	C₂	1	1	yes	maximal, characteristic, complemented
4	#22586224	C₂	1	2	yes	maximal, complemented
5	C₂×#436×C₇₁	C₂	1	1	yes	maximal, characteristic
6	C₂×#14556060	—	2	1	no	—
7	C₂×#14556060	—	2	1	no	—
8	C₂×C₄⋊(C₇₁⋊C₈)	C₂²	1	2	yes	—
9	C₂×#14556049	—	2	2	no	—
10	#17985869	C₂²	1	2	yes	—
11	#17985761	—	2	2	no	—
12	#17985752	—	2	2	no	—
13	C₂²×M₁₆×C₇₁	—	2	1	no	—
14	C₂×C₄×D₄×C₇₁	C₂²	1	1	yes	characteristic
15	#436×C₇₁	—	2	2	no	—
16	#436×C₇₁	C₄	1	4	yes	complemented
17	C₂×C₄⋊C₈×C₇₁	C₂²	1	1	yes	characteristic
18	C₂×C₂²⋊C₈×C₇₁	—	2	1	no	—
19	C₂×C₄×C₈×C₇₁	C₂²	1	1	yes	characteristic
20	C₂²×C₇₁⋊C₈	—	2	2	no	—
21	C₂×C₄⋊Dic₇₁	—	2	2	no	—
22	C₂×C₄⋊Dic₇₁	—	2	2	no	—
23	C₂×C₄⋊Dic₇₁	—	2	2	no	—
24	#14556060	—	4	2	no	—
25	#14556060	—	4	2	no	—
26	C₄⋊(C₇₁⋊C₈)	—	2	2	no	—
27	C₄⋊(C₇₁⋊C₈)	—	2	2	no	—
28	#14556049	—	4	4	no	—
29	C₂²×D₄×C₇₁	D₄	1	1	yes	characteristic
30	C₂³×C₄×C₇₁	—	2	1	no	—
31	C₂×M₁₆×C₇₁	—	2	4	no	—
32	C₂×M₁₆×C₇₁	—	4	2	no	—
33	C₂×M₁₆×C₇₁	—	4	2	no	—
34	C₂²×C₈×C₇₁	—	2	1	no	—
35	C₂²×C₈×C₇₁	D₄	1	1	yes	characteristic
36	C₂²×C₈×C₇₁	D₄	1	1	yes	characteristic
37	C₄×D₄×C₇₁	—	2	1	no	—
38	C₄×D₄×C₇₁	—	2	1	no	—
39	C₄×D₄×C₇₁	C₂×C₄	1	2	yes	—
40	C₄×D₄×C₇₁	Q₈	1	2	yes	—
41	C₂×C₄⋊C₄×C₇₁	D₄	1	1	yes	characteristic
42	C₂×C₂²⋊C₄×C₇₁	—	2	1	no	—
43	C₂×C₄²×C₇₁	C₂³	1	1	yes	characteristic
44	C₄⋊C₈×C₇₁	—	2	1	no	—
45	C₄⋊C₈×C₇₁	C₂×C₄	1	2	yes	—
46	C₂²⋊C₈×C₇₁	—	2	4	no	—
47	C₄×C₈×C₇₁	C₂×C₄	1	2	yes	—
48	C₄×C₈×C₇₁	D₄	1	2	yes	—
49	C₂²×Dic₇₁	—	4	2	no	—
50	C₂×C₇₁⋊C₈	—	4	2	no	—
51	C₂×C₇₁⋊C₈	—	4	2	no	—
52	C₂×C₇₁⋊C₈	—	4	2	no	—
53	C₄⋊Dic₇₁	—	4	2	no	—
54	C₄⋊Dic₇₁	—	4	2	no	—
55	C₄⋊Dic₇₁	—	4	2	no	—
56	C₄⋊Dic₇₁	—	4	2	no	—
57	C₄⋊Dic₇₁	—	8	2	no	—
58	C₂⁴×C₇₁	—	2	1	no	—
59	C₂×D₄×C₇₁	—	2	1	no	—
60	C₂×D₄×C₇₁	—	2	2	no	—
61	C₂×D₄×C₇₁	—	2	2	no	—
62	C₂×D₄×C₇₁	C₄⋊C₄	1	2	yes	—
63	C₂²×C₄×C₇₁	—	2	1	no	—
64	C₂²×C₄×C₇₁	—	2	2	no	—
65	C₂²×C₄×C₇₁	—	2	2	no	—
66	C₂²×C₄×C₇₁	—	4	1	no	—
67	C₂²×C₄×C₇₁	—	4	1	no	—
68	C₂²×C₄×C₇₁	—	4	1	no	—
69	C₂²×C₄×C₇₁	—	4	1	no	—
70	C₂²×C₄×C₇₁	C₂×D₄	1	1	yes	characteristic
71	C₂²×C₄×C₇₁	C₂×D₄	1	1	yes	characteristic
72	C₂²×C₄×C₇₁	#41	1	1	yes	characteristic
73	M₁₆×C₇₁	—	4	8	no	—
74	C₂×C₈×C₇₁	—	2	1	no	—
75	C₂×C₈×C₇₁	—	2	1	no	—
76	C₂×C₈×C₇₁	—	2	2	no	—
77	C₂×C₈×C₇₁	—	4	1	no	—
78	C₂×C₈×C₇₁	—	4	1	no	—
79	C₂×C₈×C₇₁	C₂²⋊C₄	1	2	yes	—
80	C₂×C₈×C₇₁	C₂²⋊C₄	1	2	yes	—
81	C₂×C₈×C₇₁	D₈	1	2	yes	—
82	C₂×C₈×C₇₁	SD₁₆	1	2	yes	—
83	C₄⋊C₄×C₇₁	—	2	1	no	—
84	C₄⋊C₄×C₇₁	C₄⋊C₄	1	2	yes	—
85	C₂²⋊C₄×C₇₁	—	2	2	no	—
86	C₂²⋊C₄×C₇₁	—	2	2	no	—
87	C₄²×C₇₁	C₂²×C₄	1	1	yes	characteristic
88	C₄²×C₇₁	C₂×D₄	1	1	yes	characteristic
89	C₄²×C₇₁	C₂×Q₈	1	1	yes	characteristic
90	C₄²×C₇₁	#41	1	1	yes	characteristic
91	C₂³×C₇₁	—	2	2	no	—
92	C₂³×C₇₁	—	4	1	no	—
93	C₂³×C₇₁	—	4	1	no	—
94	C₂³×C₇₁	—	4	2	no	—
95	C₂³×C₇₁	—	4	2	no	—
96	C₂³×C₇₁	#121	1	1	yes	characteristic
97	D₄×C₇₁	—	2	4	no	—
98	D₄×C₇₁	—	4	1	no	—
99	D₄×C₇₁	—	4	1	no	—
100	C₂×C₄×C₇₁	—	2	2	no	—
101	C₂×C₄×C₇₁	—	2	2	no	—
102	C₂×C₄×C₇₁	—	4	1	no	—
103	C₂×C₄×C₇₁	—	4	1	no	—
104	C₂×C₄×C₇₁	—	4	2	no	—
105	C₂×C₄×C₇₁	—	4	2	no	—
106	C₂×C₄×C₇₁	—	4	2	no	—
107	C₂×C₄×C₇₁	—	4	2	no	—
108	C₂×C₄×C₇₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
109	C₂×C₄×C₇₁	C₂×C₄⋊C₄	1	1	yes	characteristic
110	C₂×C₄×C₇₁	#117	1	1	yes	characteristic
111	C₂×C₄×C₇₁	#121	1	1	yes	characteristic
112	C₂×C₄×C₇₁	C₂²⋊Q₈	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	C₂×SD₁₆	1	1	yes	characteristic
116	C₂×C₄×C₇₁	AGL(1,ℤ/8)	1	1	yes	characteristic
117	C₂×C₄×C₇₁	AGL(1,ℤ/8)	1	2	yes	—
118	C₂×C₄×C₇₁	#137	1	2	yes	—
119	C₂×C₄×C₇₁	#137	1	1	yes	characteristic
120	C₂×Dic₇₁	—	8	2	no	—
121	C₂×Dic₇₁	—	8	2	no	—
122	C₂×Dic₇₁	—	8	2	no	—
123	C₈×C₇₁	—	2	2	no	—
124	C₈×C₇₁	—	4	2	no	—
125	C₈×C₇₁	#102	1	4	yes	complemented
126	C₇₁⋊C₈	—	8	2	no	—
127	C₂²×C₇₁	—	4	2	no	—
128	C₂²×C₇₁	—	4	2	no	—
129	C₂²×C₇₁	—	4	2	no	—
130	C₂²×C₇₁	—	8	2	no	—
131	C₂²×C₇₁	—	8	2	no	—
132	C₂²×C₇₁	#380	1	1	yes	characteristic
133	C₂²×C₇₁	#459	1	1	yes	characteristic
134	C₂²×C₇₁	#461	1	1	yes	characteristic
135	C₂²×C₇₁	#468	1	2	yes	—
136	C₂²×C₇₁	#471	1	2	yes	—
137	C₄×C₇₁	—	2	1	no	—
138	C₄×C₇₁	—	2	1	no	—
139	C₄×C₇₁	—	4	2	no	—
140	C₄×C₇₁	C₂×#102	1	1	yes	characteristic
141	C₄×C₇₁	#417	1	1	yes	characteristic
142	C₄×C₇₁	#428	1	2	yes	—
143	C₄×C₇₁	#469	1	1	yes	characteristic
144	C₄×C₇₁	#470	1	1	yes	characteristic
145	C₄×C₇₁	#474	1	1	yes	characteristic
146	C₄×C₇₁	#476	1	1	yes	characteristic
147	Dic₇₁	—	16	2	no	—
148	#8650	—	71	1	no	maximal, Sylow
149	C₂×C₇₁	—	8	2	no	—
150	C₂×C₇₁	#1685	1	2	yes	—
151	C₂×C₇₁	#1691	1	2	yes	—
152	C₂×C₇₁	#1802	1	1	yes	characteristic
153	C₂×C₇₁	#1809	1	1	yes	characteristic
154	C₂×C₇₁	#1815	1	1	yes	characteristic
155	C₂×#436	D₇₁	1	1	yes	characteristic
156	#1844	—	71	1	no	—
157	#1802	—	71	2	no	—
158	C₂×#335	—	71	1	no	—
159	#1296	—	71	2	no	—
160	C₇₁	#8650	1	1	yes	characteristic, complemented, Sylow
161	C₂²×M₁₆	—	2	1	no	—
162	C₂×C₄×D₄	C₂×D₇₁	1	1	yes	characteristic
163	#436	—	2	2	no	—
164	#436	Dic₇₁	1	4	yes	complemented
165	C₂×#106	—	142	1	no	—
166	C₂×#106	—	142	1	no	—
167	C₂×C₄⋊C₈	—	71	2	no	—
168	C₂×C₄⋊C₈	C₂×D₇₁	1	1	yes	characteristic
169	C₂×#102	—	142	2	no	—
170	C₂×C₂²⋊C₈	—	2	1	no	—
171	C₂×C₄×C₈	C₂×D₇₁	1	1	yes	characteristic
172	#384	—	71	2	no	—
173	#340	—	142	2	no	—
174	#335	—	142	2	no	—
175	C₂²×D₄	D₂₈₄	1	1	yes	characteristic
176	C₂³×C₄	—	2	1	no	—
177	C₂×M₁₆	—	2	4	no	—
178	C₂×M₁₆	—	4	2	no	—
179	C₂×M₁₆	—	4	2	no	—
180	C₂²×C₈	—	2	1	no	—
181	C₂²×C₈	—	142	2	no	—
182	C₂²×C₈	C₂²⋊D₇₁	1	1	yes	characteristic
183	C₂²×C₈	C₂²⋊D₇₁	1	1	yes	characteristic
184	C₄×D₄	—	2	1	no	—
185	C₄×D₄	—	2	1	no	—
186	C₄×D₄	Dic₁₄₂	1	2	yes	—
187	C₄×D₄	C₂×Dic₇₁	1	2	yes	—
188	C₂×C₄⋊C₄	—	142	2	no	—
189	C₂×C₄⋊C₄	—	142	2	no	—
190	C₂×C₄⋊C₄	—	142	2	no	—
191	C₂×C₄⋊C₄	D₂₈₄	1	1	yes	characteristic
192	C₂×C₂²⋊C₄	—	2	1	no	—
193	C₂×C₄²	C₂²×D₇₁	1	1	yes	characteristic
194	#106	—	284	2	no	—
195	#106	—	284	2	no	—
196	C₄⋊C₈	—	2	1	no	—
197	C₄⋊C₈	—	142	2	no	—
198	C₄⋊C₈	—	142	2	no	—
199	C₄⋊C₈	C₂×Dic₇₁	1	2	yes	—
200	#102	—	284	4	no	—
201	C₂²⋊C₈	—	2	4	no	—
202	C₄×C₈	C₂×Dic₇₁	1	2	yes	—
203	C₄×C₈	C₂²⋊D₇₁	1	2	yes	—
204	C₂⁴	—	2	1	no	—
205	C₂×D₄	—	2	2	no	—
206	C₂×D₄	—	2	1	no	—
207	C₂×D₄	—	2	2	no	—
208	C₂×D₄	C₄⋊Dic₇₁	1	2	yes	—
209	C₂²×C₄	—	2	1	no	—
210	C₂²×C₄	—	2	2	no	—
211	C₂²×C₄	—	2	2	no	—
212	C₂²×C₄	—	4	1	no	—
213	C₂²×C₄	—	4	1	no	—
214	C₂²×C₄	—	4	1	no	—
215	C₂²×C₄	—	4	1	no	—
216	C₂²×C₄	—	284	2	no	—
217	C₂²×C₄	C₂×C₂²⋊D₇₁	1	1	yes	characteristic
218	C₂²×C₄	C₂×D₂₈₄	1	1	yes	characteristic
219	C₂²×C₄	#11765865	1	1	yes	characteristic
220	M₁₆	—	4	8	no	—
221	C₂×C₈	—	2	1	no	—
222	C₂×C₈	—	2	1	no	—
223	C₂×C₈	—	2	2	no	—
224	C₂×C₈	—	4	1	no	—
225	C₂×C₈	—	4	1	no	—
226	C₂×C₈	—	284	2	no	—
227	C₂×C₈	—	284	2	no	—
228	C₂×C₈	—	284	2	no	—
229	C₂×C₈	C₂²⋊Dic₇₁	1	2	yes	—
230	C₂×C₈	C₂²⋊Dic₇₁	1	2	yes	—
231	C₂×C₈	#11765850	1	2	yes	—
232	C₂×C₈	#11765853	1	2	yes	—
233	C₄⋊C₄	—	2	1	no	—
234	C₄⋊C₄	—	284	2	no	—
235	C₄⋊C₄	—	284	2	no	—
236	C₄⋊C₄	—	284	2	no	—
237	C₄⋊C₄	—	284	2	no	—
238	C₄⋊C₄	—	568	2	no	—
239	C₄⋊C₄	C₄⋊Dic₇₁	1	2	yes	—
240	C₂²⋊C₄	—	2	2	no	—
241	C₂²⋊C₄	—	2	2	no	—
242	C₄²	C₂²×Dic₇₁	1	1	yes	characteristic
243	C₄²	C₂×C₂²⋊D₇₁	1	1	yes	characteristic
244	C₄²	C₂×Dic₁₄₂	1	1	yes	characteristic
245	C₄²	#11765865	1	1	yes	characteristic
246	C₂³	—	2	2	no	—
247	C₂³	—	4	1	no	—
248	C₂³	—	4	1	no	—
249	C₂³	—	4	2	no	—
250	C₂³	—	4	2	no	—
251	C₂³	#14556102	1	1	yes	characteristic
252	D₄	—	2	4	no	—
253	D₄	—	4	1	no	—
254	D₄	—	4	1	no	—
255	C₂×C₄	—	2	2	no	—
256	C₂×C₄	—	2	2	no	—
257	C₂×C₄	—	4	1	no	—
258	C₂×C₄	—	4	1	no	—
259	C₂×C₄	—	4	2	no	—
260	C₂×C₄	—	4	2	no	—
261	C₂×C₄	—	4	2	no	—
262	C₂×C₄	—	4	2	no	—
263	C₂×C₄	—	568	2	no	—
264	C₂×C₄	—	568	2	no	—
265	C₂×C₄	—	568	2	no	—
266	C₂×C₄	C₂×C₂²⋊Dic₇₁	1	1	yes	characteristic
267	C₂×C₄	C₂×C₄⋊Dic₇₁	1	1	yes	characteristic
268	C₂×C₄	#14556087	1	1	yes	characteristic
269	C₂×C₄	#14556102	1	1	yes	characteristic
270	C₂×C₄	#14556107	1	1	yes	characteristic
271	C₂×C₄	#14556107	1	1	yes	characteristic
272	C₂×C₄	C₂×#11765850	1	1	yes	characteristic
273	C₂×C₄	C₂×#11765853	1	1	yes	characteristic
274	C₂×C₄	#14556153	1	1	yes	characteristic
275	C₂×C₄	#14556156	1	2	yes	—
276	C₂×C₄	#14556158	1	1	yes	characteristic
277	C₂×C₄	#14556161	1	2	yes	—
278	C₈	—	2	2	no	—
279	C₈	—	4	2	no	—
280	C₈	—	568	2	no	—
281	C₈	#14556049	1	4	yes	complemented
282	C₂²	—	4	2	no	—
283	C₂²	—	4	2	no	—
284	C₂²	—	4	2	no	—
285	C₂²	—	8	2	no	—
286	C₂²	—	8	2	no	—
287	C₂²	#17985859	1	1	yes	characteristic
288	C₂²	#17986194	1	1	yes	characteristic
289	C₂²	#17986204	1	1	yes	characteristic
290	C₂²	#17986245	1	2	yes	—
291	C₂²	#17986262	1	2	yes	—
292	C₄	—	2	1	no	—
293	C₄	—	2	1	no	—
294	C₄	—	4	2	no	—
295	C₄	—	1136	2	no	—
296	C₄	C₂×#14556049	1	1	yes	characteristic
297	C₄	#17986000	1	1	yes	characteristic
298	C₄	#17986041	1	2	yes	—
299	C₄	#17986252	1	1	yes	characteristic
300	C₄	#17986257	1	1	yes	characteristic
301	C₄	#17986279	1	1	yes	characteristic
302	C₄	#17986289	1	1	yes	characteristic
303	C₂	—	8	2	no	—
304	C₂	#22587805	1	2	yes	—
305	C₂	#22587839	1	2	yes	—
306	C₂	#22588340	1	1	yes	characteristic
307	C₂	#22588373	1	1	yes	characteristic
308	C₂	#22588399	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	#14556102
commutator	C₄²×C₇₁	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₇₁
Fitting	C₂×#436×C₇₁	C₂
socle	C₂³×C₇₁	#121
prime radical	C₂²×D₄×C₇₁	D₄
cosocle	C₂×C₄²×C₇₁	C₂³
hypercenter	C₂×#436	D₇₁
hypocenter	C₇₁	#8650
LCS	G ▷ C₄²×C₇₁ ▷ C₂²×C₇₁ ▷ C₇₁
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×#436

Complements

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

N	H ≅ G/N	type
1. #22588497	C₂	semidirect
3. C₂×#17985752	C₂	semidirect
4. #22586224	C₂	semidirect
16. #436×C₇₁	C₄	semidirect
125. C₈×C₇₁	#102	semidirect
160. C₇₁	#8650	semidirect
164. #436	Dic₇₁	semidirect
281. C₈	#14556049	semidirect

Maximal subgroups

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

subgroup	index
#22588497	2
#22588340	2
C₂×#17985752	2
#22586224	2
C₂×#436×C₇₁	2
#8650	71