Subgroups — #31890658

Order 18176.

366 conjugacy classes of subgroups.

The subgroup lattice has 264 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#22597274	C₂	1	1	yes	maximal, characteristic
2	#22587088	C₂	1	1	yes	maximal, characteristic
3	#22586903	C₂	1	2	yes	maximal
4	#22586198	C₂	1	2	yes	maximal
5	C₇₁×#1846	C₂	1	1	yes	maximal, characteristic
6	#17986556	C₂²	1	1	yes	characteristic
7	#17986129	—	2	1	no	—
8	#17986129	C₄	1	2	yes	complemented
9	C₈×Dic₁₄₂	—	2	1	no	—
10	C₈×Dic₁₄₂	C₄	1	2	yes	complemented
11	#17986052	C₂²	1	2	yes	—
12	#17986025	—	2	1	no	—
13	#17986023	C₂²	1	1	yes	characteristic
14	#17985954	—	2	1	no	—
15	#17985752	—	2	1	no	—
16	#17985750	—	2	1	no	—
17	#17985735	—	2	4	no	—
18	C₂×#107×C₇₁	—	2	1	no	—
19	C₂×#106×C₇₁	—	2	1	no	—
20	C₂×C₄×C₈×C₇₁	C₂²	1	1	yes	characteristic
21	#382×C₇₁	C₂²	1	2	yes	—
22	C₂×#11765849	—	2	1	no	—
23	#14556128	Q₈	1	1	yes	characteristic
24	#14556128	Q₈	1	1	yes	characteristic
25	#14556120	—	2	1	no	—
26	#14556107	—	2	1	no	—
27	#14556107	D₄	1	2	yes	—
28	C₄×Dic₁₄₂	—	2	1	no	—
29	C₄×Dic₁₄₂	C₂×C₄	1	2	yes	—
30	#14556087	—	2	1	no	—
31	C₄⋊(C₇₁⋊C₈)	—	2	1	no	—
32	C₄⋊(C₇₁⋊C₈)	—	2	2	no	—
33	C₄⋊(C₇₁⋊C₈)	C₂×C₄	1	2	yes	—
34	Dic₇₁⋊C₈	—	2	2	no	—
35	Dic₇₁⋊C₈	—	2	2	no	—
36	C₂²⋊(C₇₁⋊C₈)	—	2	2	no	—
37	#14556038	—	2	2	no	—
38	C₄×C₇₁⋊C₈	—	2	2	no	—
39	C₈×Dic₇₁	—	2	2	no	—
40	C₂²×C₈×C₇₁	D₄	1	1	yes	characteristic
41	C₂²×C₈×C₇₁	D₄	1	1	yes	characteristic
42	C₂×C₄⋊C₄×C₇₁	—	2	2	no	—
43	C₂×C₄²×C₇₁	C₂³	1	1	yes	characteristic
44	#107×C₇₁	—	4	2	no	—
45	#106×C₇₁	—	4	2	no	—
46	C₄×C₈×C₇₁	C₂×C₄	1	1	yes	characteristic
47	C₄×C₈×C₇₁	C₂×C₄	1	1	yes	characteristic
48	C₄×C₈×C₇₁	D₄	1	1	yes	characteristic
49	C₄×C₈×C₇₁	D₄	1	1	yes	characteristic
50	#95×C₇₁	—	2	2	no	—
51	#95×C₇₁	—	2	2	no	—
52	C₂×Dic₁₄₂	—	2	1	no	—
53	C₂×Dic₁₄₂	C₄⋊C₄	1	2	yes	—
54	#11765849	—	4	1	no	—
55	#11765849	—	4	1	no	—
56	C₂×C₇₁⋊C₈	—	2	2	no	—
57	C₂×C₇₁⋊C₈	—	4	2	no	—
58	C₄⋊Dic₇₁	—	2	1	no	—
59	C₄⋊Dic₇₁	—	2	1	no	—
60	C₄⋊Dic₇₁	—	2	1	no	—
61	C₄⋊Dic₇₁	C₄⋊C₄	1	2	yes	—
62	Dic₇₁⋊C₄	—	2	2	no	—
63	Dic₇₁⋊C₄	—	2	2	no	—
64	C₂²⋊Dic₇₁	—	2	2	no	—
65	C₄×Dic₇₁	—	2	2	no	—
66	C₂²×C₄×C₇₁	—	4	2	no	—
67	C₂²×C₄×C₇₁	C₂×D₄	1	1	yes	characteristic
68	C₂²×C₄×C₇₁	C₂×D₄	1	1	yes	characteristic
69	C₂²×C₄×C₇₁	#41	1	1	yes	characteristic
70	C₂×C₈×C₇₁	—	2	1	no	—
71	C₂×C₈×C₇₁	—	2	1	no	—
72	C₂×C₈×C₇₁	C₂²⋊C₄	1	1	yes	characteristic
73	C₂×C₈×C₇₁	C₂²⋊C₄	1	1	yes	characteristic
74	C₂×C₈×C₇₁	C₂²⋊C₄	1	1	yes	characteristic
75	C₂×C₈×C₇₁	C₂²⋊C₄	1	1	yes	characteristic
76	C₂×C₈×C₇₁	SD₁₆	1	2	yes	—
77	C₂×C₈×C₇₁	Q₁₆	1	2	yes	—
78	C₄⋊C₄×C₇₁	—	4	2	no	—
79	C₄⋊C₄×C₇₁	—	4	2	no	—
80	C₄²×C₇₁	C₂²×C₄	1	1	yes	characteristic
81	C₄²×C₇₁	C₂×D₄	1	1	yes	characteristic
82	C₄²×C₇₁	C₂×Q₈	1	1	yes	characteristic
83	C₄²×C₇₁	#41	1	1	yes	characteristic
84	C₂³×C₇₁	#121	1	1	yes	characteristic
85	C₂×C₄×C₇₁	—	4	1	no	—
86	C₂×C₄×C₇₁	—	4	1	no	—
87	C₂×C₄×C₇₁	—	8	2	no	—
88	C₂×C₄×C₇₁	—	8	2	no	—
89	C₂×C₄×C₇₁	—	8	2	no	—
90	C₂×C₄×C₇₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
91	C₂×C₄×C₇₁	C₂×C₄⋊C₄	1	1	yes	characteristic
92	C₂×C₄×C₇₁	#117	1	1	yes	characteristic
93	C₂×C₄×C₇₁	#121	1	1	yes	characteristic
94	C₂×C₄×C₇₁	C₂²⋊Q₈	1	1	yes	characteristic
95	C₂×C₄×C₇₁	C₂²⋊Q₈	1	1	yes	characteristic
96	C₂×C₄×C₇₁	C₂×SD₁₆	1	1	yes	characteristic
97	C₂×C₄×C₇₁	C₂×Q₁₆	1	1	yes	characteristic
98	C₂×C₄×C₇₁	AGL(1,ℤ/8)	1	1	yes	characteristic
99	C₂×C₄×C₇₁	#137	1	1	yes	characteristic
100	C₂×Dic₇₁	—	2	2	no	—
101	C₂×Dic₇₁	—	2	2	no	—
102	Dic₁₄₂	—	4	2	no	—
103	Dic₁₄₂	—	4	2	no	—
104	C₈×C₇₁	—	2	1	no	—
105	C₈×C₇₁	—	2	1	no	—
106	C₈×C₇₁	#103	1	2	yes	—
107	C₈×C₇₁	#103	1	2	yes	—
108	C₇₁⋊C₈	—	4	2	no	—
109	C₇₁⋊C₈	—	8	2	no	—
110	C₂²×C₇₁	—	2	1	no	—
111	C₂²×C₇₁	—	2	1	no	—
112	C₂²×C₇₁	#380	1	1	yes	characteristic
113	C₂²×C₇₁	#460	1	1	yes	characteristic
114	C₂²×C₇₁	#462	1	1	yes	characteristic
115	C₄×C₇₁	—	2	1	no	—
116	C₄×C₇₁	—	2	1	no	—
117	C₄×C₇₁	—	2	1	no	—
118	C₄×C₇₁	—	2	1	no	—
119	C₄×C₇₁	—	16	2	no	—
120	C₄×C₇₁	C₂×#103	1	1	yes	characteristic
121	C₄×C₇₁	#417	1	1	yes	characteristic
122	C₄×C₇₁	#475	1	1	yes	characteristic
123	C₄×C₇₁	#477	1	1	yes	characteristic
124	Dic₇₁	—	4	2	no	—
125	Dic₇₁	—	4	2	no	—
126	#8675	—	71	1	no	maximal, Sylow
127	C₂×C₇₁	—	2	1	no	—
128	C₂×C₇₁	—	2	1	no	—
129	C₂×C₇₁	#1803	1	1	yes	characteristic
130	C₂×C₇₁	#1812	1	1	yes	characteristic
131	C₂×C₇₁	#1814	1	1	yes	characteristic
132	#2960	—	71	1	no	—
133	#1846	D₇₁	1	1	yes	characteristic
134	#1564	—	71	1	no	—
135	#1518	—	71	2	no	—
136	#1286	—	71	2	no	—
137	C₇₁	#8675	1	1	yes	characteristic, complemented, Sylow
138	#533	—	71	1	no	—
139	#446	—	71	2	no	—
140	#446	—	142	1	no	—
141	C₈×Q₈	—	71	2	no	—
142	C₈×Q₈	—	142	1	no	—
143	#431	—	71	2	no	—
144	C₂×#107	—	2	1	no	—
145	C₂×#106	—	2	1	no	—
146	#424	—	142	1	no	—
147	#423	—	71	1	no	—
148	#405	—	142	1	no	—
149	C₂×C₄×C₈	C₂×D₇₁	1	1	yes	characteristic
150	#382	C₂×D₇₁	1	2	yes	—
151	#335	—	142	1	no	—
152	#334	—	142	1	no	—
153	#328	—	142	4	no	—
154	C₂×M₁₆	—	142	1	no	—
155	C₂²×C₈	C₂²⋊D₇₁	1	1	yes	characteristic
156	C₂²×C₈	C₂²⋊D₇₁	1	1	yes	characteristic
157	C₄⋊Q₈	—	71	1	no	—
158	C₄⋊Q₈	—	71	1	no	—
159	#125	—	142	1	no	—
160	C₂²⋊Q₈	—	71	2	no	—
161	C₂²⋊Q₈	—	142	1	no	—
162	C₄×Q₈	—	71	2	no	—
163	C₄×Q₈	—	142	1	no	—
164	#117	—	142	1	no	—
165	C₂×C₄⋊C₄	—	2	2	no	—
166	C₂×C₄²	C₂²×D₇₁	1	1	yes	characteristic
167	#107	—	4	2	no	—
168	#106	—	4	2	no	—
169	C₄⋊C₈	—	71	2	no	—
170	C₄⋊C₈	—	142	2	no	—
171	C₄⋊C₈	—	142	2	no	—
172	C₄⋊C₈	—	142	2	no	—
173	C₄⋊C₈	—	142	1	no	—
174	C₂²⋊C₈	—	142	2	no	—
175	#97	—	142	2	no	—
176	C₄×C₈	—	142	2	no	—
177	C₄×C₈	—	142	2	no	—
178	C₄×C₈	C₄×D₇₁	1	1	yes	characteristic
179	C₄×C₈	C₄×D₇₁	1	1	yes	characteristic
180	C₄×C₈	D₂₈₄	1	1	yes	characteristic
181	C₄×C₈	D₂₈₄	1	1	yes	characteristic
182	#95	—	2	2	no	—
183	#95	—	2	2	no	—
184	C₂×Q₈	—	71	2	no	—
185	C₂×Q₈	—	142	1	no	—
186	C₂²×C₄	—	4	2	no	—
187	C₂²×C₄	D₄×D₇₁	1	1	yes	characteristic
188	C₂²×C₄	C₂×C₂²⋊D₇₁	1	1	yes	characteristic
189	C₂²×C₄	#11765864	1	1	yes	characteristic
190	M₁₆	—	284	1	no	—
191	M₁₆	—	284	1	no	—
192	C₂×C₈	—	2	1	no	—
193	C₂×C₈	—	2	1	no	—
194	C₂×C₈	—	284	2	no	—
195	C₂×C₈	—	142	2	no	—
196	C₂×C₈	(C₂×D₇₁)⋊C₄	1	1	yes	characteristic
197	C₂×C₈	(C₂×D₇₁)⋊C₄	1	1	yes	characteristic
198	C₂×C₈	(C₂×D₇₁)⋊C₄	1	1	yes	characteristic
199	C₂×C₈	(C₂×D₇₁)⋊C₄	1	1	yes	characteristic
200	C₂×C₈	#11765853	1	2	yes	—
201	C₂×C₈	#11765855	1	2	yes	—
202	C₄⋊C₄	—	4	2	no	—
203	C₄⋊C₄	—	4	2	no	—
204	C₄⋊C₄	—	71	2	no	—
205	C₄⋊C₄	—	142	2	no	—
206	C₄⋊C₄	—	142	2	no	—
207	C₄⋊C₄	—	142	1	no	—
208	C₄⋊C₄	—	142	1	no	—
209	C₄⋊C₄	—	142	1	no	—
210	C₂²⋊C₄	—	142	2	no	—
211	C₄²	—	142	2	no	—
212	C₄²	C₂×C₄×D₇₁	1	1	yes	characteristic
213	C₄²	C₂×D₂₈₄	1	1	yes	characteristic
214	C₄²	Q₈×D₇₁	1	1	yes	characteristic
215	C₄²	#11765866	1	1	yes	characteristic
216	C₂³	#14556099	1	1	yes	characteristic
217	Q₈	—	284	2	no	—
218	Q₈	—	284	2	no	—
219	C₂×C₄	—	4	1	no	—
220	C₂×C₄	—	4	1	no	—
221	C₂×C₄	—	8	2	no	—
222	C₂×C₄	—	8	2	no	—
223	C₂×C₄	—	8	2	no	—
224	C₂×C₄	—	142	2	no	—
225	C₂×C₄	—	142	2	no	—
226	C₂×C₄	C₂×(C₂×D₇₁)⋊C₄	1	1	yes	characteristic
227	C₂×C₄	C₄⋊C₄×D₇₁	1	1	yes	characteristic
228	C₂×C₄	#14556086	1	1	yes	characteristic
229	C₂×C₄	#14556103	1	1	yes	characteristic
230	C₂×C₄	#14556104	1	1	yes	characteristic
231	C₂×C₄	#14556108	1	1	yes	characteristic
232	C₂×C₄	C₂×#11765853	1	1	yes	characteristic
233	C₂×C₄	C₂×#11765855	1	1	yes	characteristic
234	C₂×C₄	#14556154	1	1	yes	characteristic
235	C₂×C₄	#14556159	1	1	yes	characteristic
236	C₈	—	2	1	no	—
237	C₈	—	2	1	no	—
238	C₈	—	284	2	no	—
239	C₈	—	568	2	no	—
240	C₈	#14556051	1	2	yes	—
241	C₈	#14556051	1	2	yes	—
242	C₂²	—	2	1	no	—
243	C₂²	—	2	1	no	—
244	C₂²	#17985856	1	1	yes	characteristic
245	C₂²	#17986196	1	1	yes	characteristic
246	C₂²	#17986206	1	1	yes	characteristic
247	C₄	—	2	1	no	—
248	C₄	—	2	1	no	—
249	C₄	—	2	1	no	—
250	C₄	—	2	1	no	—
251	C₄	—	16	2	no	—
252	C₄	—	284	2	no	—
253	C₄	—	284	2	no	—
254	C₄	C₂×#14556051	1	1	yes	characteristic
255	C₄	#17985998	1	1	yes	characteristic
256	C₄	#17986281	1	1	yes	characteristic
257	C₄	#17986291	1	1	yes	characteristic
258	C₂	—	2	1	no	—
259	C₂	—	2	1	no	—
260	C₂	#22588342	1	1	yes	characteristic
261	C₂	#22588383	1	1	yes	characteristic
262	C₂	#22588391	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	#17985856
commutator	C₄²×C₇₁	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₇₁
Fitting	C₇₁×#1846	C₂
socle	C₂²×C₇₁	#380
prime radical	C₂³×C₇₁	#121
cosocle	C₂×C₄²×C₇₁	C₂³
hypercenter	#1846	D₇₁
hypocenter	C₇₁	#8675
LCS	G ▷ C₄²×C₇₁ ▷ C₂²×C₇₁ ▷ C₇₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #1846

Complements

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

N	H ≅ G/N	type
8. #17986129	C₄	semidirect
10. C₈×Dic₁₄₂	C₄	semidirect
137. C₇₁	#8675	semidirect

Maximal subgroups

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

subgroup	index
#22597274	2
#22587088	2
#22586903	2
#22586198	2
C₇₁×#1846	2
#8675	71