Subgroups — #31916021

Order 18176.

338 conjugacy classes of subgroups.

The subgroup lattice has 230 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₄⋊(C₇₁⋊C₁₆)	C₂	1	1	yes	maximal, characteristic
2	C₄×#14556064	C₂	1	1	yes	maximal, characteristic
3	#22586435	C₂	1	4	yes	maximal
4	C₇₁×#3160	C₂	1	1	yes	maximal, characteristic
5	C₂²×C₇₁⋊C₁₆	C₂²	1	2	yes	—
6	C₂×#14556064	C₂²	1	2	yes	—
7	C₄×#11765849	—	2	1	no	—
8	C₄⋊(C₇₁⋊C₁₆)	—	2	1	no	—
9	C₄⋊(C₇₁⋊C₁₆)	—	2	1	no	—
10	#17985764	—	2	1	no	—
11	#533×C₇₁	—	2	1	no	—
12	#500×C₇₁	C₄	1	1	yes	characteristic
13	#500×C₇₁	C₄	1	1	yes	characteristic
14	#499×C₇₁	—	2	1	no	—
15	#499×C₇₁	C₄	1	2	yes	—
16	#498×C₇₁	—	2	1	no	—
17	#427×C₇₁	—	2	1	no	—
18	#427×C₇₁	C₂²	1	2	yes	—
19	C₂×C₄×C₈×C₇₁	C₂²	1	1	yes	characteristic
20	C₂×#11765849	—	2	2	no	—
21	C₂×C₇₁⋊C₁₆	—	2	2	no	—
22	C₂×C₇₁⋊C₁₆	—	2	4	no	—
23	#14556064	—	2	2	no	—
24	#14556064	Q₈	1	4	yes	complemented
25	#14556039	—	4	1	no	—
26	C₄×C₇₁⋊C₈	—	4	1	no	—
27	C₂²×C₈×C₇₁	D₄	1	1	yes	characteristic
28	C₂²×C₈×C₇₁	C₂³	1	1	yes	characteristic
29	C₄⋊Q₈×C₇₁	—	2	1	no	—
30	C₄⋊Q₈×C₇₁	—	2	1	no	—
31	#125×C₇₁	—	2	2	no	—
32	C₂²⋊Q₈×C₇₁	—	4	1	no	—
33	C₂²⋊Q₈×C₇₁	—	4	1	no	—
34	C₄×Q₈×C₇₁	—	4	1	no	—
35	C₄×Q₈×C₇₁	—	4	1	no	—
36	#117×C₇₁	—	2	2	no	—
37	C₂×C₄²×C₇₁	D₄	1	1	yes	characteristic
38	#107×C₇₁	—	2	1	no	—
39	#107×C₇₁	—	2	1	no	—
40	#107×C₇₁	C₂×C₄	1	2	yes	—
41	#107×C₇₁	C₂×C₄	1	2	yes	—
42	#106×C₇₁	—	2	2	no	—
43	#106×C₇₁	—	2	2	no	—
44	C₄×C₈×C₇₁	C₂×C₄	1	1	yes	characteristic
45	C₄×C₈×C₇₁	C₂×C₄	1	1	yes	characteristic
46	C₄×C₈×C₇₁	D₄	1	1	yes	characteristic
47	C₄×C₈×C₇₁	D₄	1	1	yes	characteristic
48	#11765849	—	2	4	no	—
49	#11765849	—	4	2	no	—
50	C₂×C₇₁⋊C₈	—	4	2	no	—
51	C₇₁⋊C₁₆	—	4	2	no	—
52	C₂×Q₈×C₇₁	—	4	1	no	—
53	C₂×Q₈×C₇₁	—	4	1	no	—
54	C₂²×C₄×C₇₁	SD₁₆	1	1	yes	characteristic
55	C₂²×C₄×C₇₁	SD₁₆	1	1	yes	characteristic
56	C₂²×C₄×C₇₁	C₂×D₄	1	1	yes	characteristic
57	C₂×C₈×C₇₁	—	2	2	no	—
58	C₂×C₈×C₇₁	C₂²⋊C₄	1	1	yes	characteristic
59	C₂×C₈×C₇₁	C₂²⋊C₄	1	1	yes	characteristic
60	C₂×C₈×C₇₁	C₂²×C₄	1	1	yes	characteristic
61	C₂×C₈×C₇₁	C₂×D₄	1	1	yes	characteristic
62	C₂×C₈×C₇₁	C₂×Q₈	1	2	yes	—
63	C₂×C₈×C₇₁	#41	1	2	yes	—
64	C₄⋊C₄×C₇₁	—	2	2	no	—
65	C₄⋊C₄×C₇₁	—	2	2	no	—
66	C₄⋊C₄×C₇₁	—	4	2	no	—
67	C₄⋊C₄×C₇₁	—	4	2	no	—
68	C₄⋊C₄×C₇₁	—	4	1	no	—
69	C₄⋊C₄×C₇₁	—	4	1	no	—
70	C₂²⋊C₄×C₇₁	—	4	2	no	—
71	C₄²×C₇₁	—	4	2	no	—
72	C₄²×C₇₁	C₂²⋊C₄	1	1	yes	characteristic
73	C₄²×C₇₁	C₂²⋊C₄	1	1	yes	characteristic
74	C₄²×C₇₁	D₈	1	1	yes	characteristic
75	C₄²×C₇₁	D₈	1	1	yes	characteristic
76	C₂³×C₇₁	C₂×SD₁₆	1	1	yes	characteristic
77	Q₈×C₇₁	—	8	2	no	—
78	Q₈×C₇₁	—	8	2	no	—
79	C₂×C₄×C₇₁	—	2	2	no	—
80	C₂×C₄×C₇₁	—	4	1	no	—
81	C₂×C₄×C₇₁	—	4	2	no	—
82	C₂×C₄×C₇₁	—	4	2	no	—
83	C₂×C₄×C₇₁	#102	1	1	yes	characteristic
84	C₂×C₄×C₇₁	#102	1	1	yes	characteristic
85	C₂×C₄×C₇₁	#102	1	1	yes	characteristic
86	C₂×C₄×C₇₁	#102	1	1	yes	characteristic
87	C₂×C₄×C₇₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
88	C₂×C₄×C₇₁	C₂²⋊Q₈	1	2	yes	—
89	C₂×C₄×C₇₁	C₂×D₈	1	1	yes	characteristic
90	C₂×C₄×C₇₁	#137	1	2	yes	—
91	C₈×C₇₁	—	2	1	no	—
92	C₈×C₇₁	—	2	1	no	—
93	C₈×C₇₁	C₄×Q₈	1	2	yes	—
94	C₈×C₇₁	#124	1	1	yes	characteristic
95	C₈×C₇₁	C₄⋊Q₈	1	1	yes	characteristic
96	C₇₁⋊C₈	—	4	4	no	—
97	C₂²×C₇₁	—	2	1	no	—
98	C₂²×C₇₁	—	2	1	no	—
99	C₂²×C₇₁	C₂×#102	1	1	yes	characteristic
100	C₂²×C₇₁	#475	1	2	yes	—
101	C₄×C₇₁	—	2	1	no	—
102	C₄×C₇₁	—	2	1	no	—
103	C₄×C₇₁	—	2	1	no	—
104	C₄×C₇₁	—	8	2	no	—
105	C₄×C₇₁	—	8	2	no	—
106	C₄×C₇₁	#359	1	2	yes	—
107	C₄×C₇₁	#391	1	1	yes	characteristic
108	C₄×C₇₁	#419	1	1	yes	characteristic
109	C₄×C₇₁	#474	1	2	yes	—
110	#13274	—	71	1	no	maximal, Sylow
111	C₂×C₇₁	—	2	2	no	—
112	C₂×C₇₁	#1988	1	1	yes	characteristic
113	C₂×C₇₁	C₂×#359	1	1	yes	characteristic
114	C₂×C₇₁	#2143	1	1	yes	characteristic
115	#3160	D₇₁	1	1	yes	characteristic
116	C₂×C₄⋊C₁₆	—	71	1	no	—
117	C₄×#108	—	71	1	no	—
118	#1382	—	71	4	no	—
119	C₇₁	#13274	1	1	yes	characteristic, complemented, Sylow
120	#533	—	2	1	no	—
121	C₂²×C₁₆	—	71	2	no	—
122	#500	Dic₇₁	1	1	yes	characteristic
123	#500	Dic₇₁	1	1	yes	characteristic
124	#499	—	2	1	no	—
125	#499	Dic₇₁	1	2	yes	—
126	#498	—	2	1	no	—
127	C₂×#108	—	71	2	no	—
128	#427	—	2	1	no	—
129	#427	C₂×D₇₁	1	2	yes	—
130	C₄×M₁₆	—	142	1	no	—
131	C₂×C₄×C₈	C₂×D₇₁	1	1	yes	characteristic
132	C₄⋊C₁₆	—	142	1	no	—
133	C₄⋊C₁₆	—	142	1	no	—
134	#341	—	142	1	no	—
135	C₂×M₁₆	—	142	2	no	—
136	C₂²×C₈	C₂²⋊D₇₁	1	1	yes	characteristic
137	C₂²×C₈	C₂²×D₇₁	1	1	yes	characteristic
138	C₄⋊Q₈	—	2	1	no	—
139	C₄⋊Q₈	—	2	1	no	—
140	#125	—	2	2	no	—
141	C₂²⋊Q₈	—	4	1	no	—
142	C₂²⋊Q₈	—	4	1	no	—
143	C₄×Q₈	—	4	1	no	—
144	C₄×Q₈	—	4	1	no	—
145	#117	—	2	2	no	—
146	C₂×C₄²	C₂²⋊D₇₁	1	1	yes	characteristic
147	C₂×C₁₆	—	142	2	no	—
148	C₂×C₁₆	—	142	4	no	—
149	#108	—	71	4	no	—
150	#108	—	142	2	no	—
151	#107	—	2	1	no	—
152	#107	—	2	1	no	—
153	#107	C₂×Dic₇₁	1	2	yes	—
154	#107	C₂×Dic₇₁	1	2	yes	—
155	#106	—	2	2	no	—
156	#106	—	2	2	no	—
157	#97	—	284	1	no	—
158	C₄×C₈	—	284	1	no	—
159	C₄×C₈	C₂×Dic₇₁	1	1	yes	characteristic
160	C₄×C₈	C₂×Dic₇₁	1	1	yes	characteristic
161	C₄×C₈	C₂²⋊D₇₁	1	1	yes	characteristic
162	C₄×C₈	C₂²⋊D₇₁	1	1	yes	characteristic
163	C₂×Q₈	—	4	1	no	—
164	C₂×Q₈	—	4	1	no	—
165	C₂²×C₄	#11765853	1	1	yes	characteristic
166	C₂²×C₄	#11765853	1	1	yes	characteristic
167	C₂²×C₄	C₂×C₂²⋊D₇₁	1	1	yes	characteristic
168	M₁₆	—	284	2	no	—
169	M₁₆	—	142	4	no	—
170	C₂×C₈	—	2	2	no	—
171	C₂×C₈	—	284	2	no	—
172	C₂×C₈	C₂²⋊Dic₇₁	1	1	yes	characteristic
173	C₂×C₈	C₂²⋊Dic₇₁	1	1	yes	characteristic
174	C₂×C₈	C₂²×Dic₇₁	1	1	yes	characteristic
175	C₂×C₈	C₂×C₂²⋊D₇₁	1	1	yes	characteristic
176	C₂×C₈	Q₈×D₇₁	1	2	yes	—
177	C₂×C₈	#11765866	1	2	yes	—
178	C₄⋊C₄	—	2	2	no	—
179	C₄⋊C₄	—	2	2	no	—
180	C₄⋊C₄	—	4	2	no	—
181	C₄⋊C₄	—	4	2	no	—
182	C₄⋊C₄	—	4	1	no	—
183	C₄⋊C₄	—	4	1	no	—
184	C₂²⋊C₄	—	4	2	no	—
185	C₄²	—	4	2	no	—
186	C₄²	C₂²⋊Dic₇₁	1	1	yes	characteristic
187	C₄²	C₂²⋊Dic₇₁	1	1	yes	characteristic
188	C₄²	#11765850	1	1	yes	characteristic
189	C₄²	#11765850	1	1	yes	characteristic
190	C₁₆	—	284	2	no	—
191	C₂³	C₂×#11765853	1	1	yes	characteristic
192	Q₈	—	8	2	no	—
193	Q₈	—	8	2	no	—
194	C₂×C₄	—	2	2	no	—
195	C₂×C₄	—	4	1	no	—
196	C₂×C₄	—	4	2	no	—
197	C₂×C₄	—	4	2	no	—
198	C₂×C₄	#14556049	1	1	yes	characteristic
199	C₂×C₄	#14556049	1	1	yes	characteristic
200	C₂×C₄	#14556049	1	1	yes	characteristic
201	C₂×C₄	#14556049	1	1	yes	characteristic
202	C₂×C₄	C₂×C₂²⋊Dic₇₁	1	1	yes	characteristic
203	C₂×C₄	#14556104	1	2	yes	—
204	C₂×C₄	C₂×#11765850	1	1	yes	characteristic
205	C₂×C₄	#14556159	1	2	yes	—
206	C₈	—	2	1	no	—
207	C₈	—	2	1	no	—
208	C₈	—	284	4	no	—
209	C₈	Q₈×Dic₇₁	1	2	yes	—
210	C₈	#14556117	1	1	yes	characteristic
211	C₈	#14556126	1	1	yes	characteristic
212	C₂²	—	2	1	no	—
213	C₂²	—	2	1	no	—
214	C₂²	C₂×#14556049	1	1	yes	characteristic
215	C₂²	#17986281	1	2	yes	—
216	C₄	—	2	1	no	—
217	C₄	—	2	1	no	—
218	C₄	—	2	1	no	—
219	C₄	—	8	2	no	—
220	C₄	—	8	2	no	—
221	C₄	#17985806	1	2	yes	—
222	C₄	#17985906	1	1	yes	characteristic
223	C₄	#17986008	1	1	yes	characteristic
224	C₄	#17986276	1	2	yes	—
225	C₂	—	2	2	no	—
226	C₂	#22589293	1	1	yes	characteristic
227	C₂	C₂×#17985806	1	1	yes	characteristic
228	C₂	#22590068	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₂×#11765850
commutator	C₂×C₈×C₇₁	C₂²×C₄
Frattini	C₂²×C₈	C₂²×D₇₁
Fitting	C₇₁×#3160	C₂
socle	C₂²×C₇₁	C₂×#102
prime radical	C₂³×C₇₁	C₂×SD₁₆
cosocle	C₂²×C₈×C₇₁	C₂³
hypercenter	#3160	D₇₁
hypocenter	C₇₁	#13274
LCS	G ▷ C₂×C₈×C₇₁ ▷ C₄×C₇₁ ▷ C₂×C₇₁ ▷ C₇₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ #3160

Complements

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

N	H ≅ G/N	type
24. #14556064	Q₈	semidirect
119. C₇₁	#13274	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₄⋊(C₇₁⋊C₁₆)	2
C₄×#14556064	2
#22586435	2
C₇₁×#3160	2
#13274	71