Subgroups — #31888589

Order 18176.

494 conjugacy classes of subgroups.

The subgroup lattice has 196 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄×Dic₇₁⋊C₈	C₂	1	2	yes	maximal
2	C₄×#14556038	C₂	1	1	yes	maximal, characteristic
3	C₂×C₈×C₇₁⋊C₈	C₂	1	1	yes	maximal, characteristic
4	C₇₁⋊#1273	C₂	1	2	yes	maximal
5	C₂×#322×C₇₁	C₂	1	1	yes	maximal, characteristic
6	C₂×Dic₇₁⋊C₈	C₄	1	4	yes	—
7	C₂×#14556038	C₄	1	2	yes	—
8	C₂×#14556038	C₄	1	2	yes	—
9	C₂×C₄×C₇₁⋊C₈	C₂²	1	2	yes	—
10	C₂×C₄×C₇₁⋊C₈	C₂²	1	1	yes	characteristic
11	C₄²×Dic₇₁	C₂²	1	1	yes	characteristic
12	#17985753	C₄	1	4	yes	—
13	C₈×C₇₁⋊C₈	—	2	2	no	—
14	C₂×C₄×C₈×C₇₁	C₂²	1	2	yes	—
15	C₂×C₄×C₈×C₇₁	C₂²	1	1	yes	characteristic
16	#322×C₇₁	—	2	2	no	—
17	C₂²×C₇₁⋊C₈	C₂×C₄	1	2	yes	—
18	C₂²×C₇₁⋊C₈	C₂×C₄	1	4	yes	—
19	C₂×C₄×Dic₇₁	C₂×C₄	1	1	yes	characteristic
20	C₂×C₄×Dic₇₁	C₂×C₄	1	2	yes	—
21	C₂×C₄×Dic₇₁	C₂×C₄	1	2	yes	—
22	C₂×C₄×Dic₇₁	C₂×C₄	1	1	yes	characteristic
23	Dic₇₁⋊C₈	—	2	8	no	—
24	#14556038	—	2	4	no	—
25	#14556038	C₈	1	8	yes	complemented
26	C₄×C₇₁⋊C₈	—	2	2	no	—
27	C₄×C₇₁⋊C₈	—	2	2	no	—
28	C₄×C₇₁⋊C₈	—	2	2	no	—
29	C₂²×C₈×C₇₁	C₂×C₄	1	4	yes	—
30	C₂²×C₈×C₇₁	C₂×C₄	1	2	yes	—
31	C₂×C₄²×C₇₁	C₂³	1	1	yes	characteristic
32	C₄×C₈×C₇₁	—	2	2	no	—
33	C₄×C₈×C₇₁	—	2	2	no	—
34	C₄×C₈×C₇₁	D₄	1	2	yes	—
35	C₄×C₈×C₇₁	Q₈	1	2	yes	—
36	C₂²×Dic₇₁	C₄²	1	1	yes	characteristic
37	C₂²×Dic₇₁	C₄²	1	1	yes	characteristic
38	C₂²×Dic₇₁	C₄²	1	2	yes	—
39	C₂×C₇₁⋊C₈	—	2	2	no	—
40	C₂×C₇₁⋊C₈	—	2	4	no	—
41	C₂×C₇₁⋊C₈	—	2	2	no	—
42	C₂×C₇₁⋊C₈	—	2	4	no	—
43	C₂×C₇₁⋊C₈	C₂×C₈	1	4	yes	—
44	C₂×C₇₁⋊C₈	M₁₆	1	8	yes	—
45	C₄×Dic₇₁	—	2	2	no	—
46	C₄×Dic₇₁	—	2	4	no	—
47	C₄×Dic₇₁	—	2	4	no	—
48	C₄×Dic₇₁	C₂×C₈	1	4	yes	—
49	C₂²×C₄×C₇₁	C₂²×C₄	1	1	yes	characteristic
50	C₂²×C₄×C₇₁	C₂²×C₄	1	2	yes	—
51	C₂×C₈×C₇₁	—	2	4	no	—
52	C₂×C₈×C₇₁	—	2	4	no	—
53	C₂×C₈×C₇₁	—	2	4	no	—
54	C₂×C₈×C₇₁	C₄⋊C₄	1	4	yes	—
55	C₂×C₈×C₇₁	C₂×C₈	1	4	yes	—
56	C₂×C₈×C₇₁	M₁₆	1	4	yes	—
57	C₄²×C₇₁	C₂×D₄	1	1	yes	characteristic
58	C₄²×C₇₁	C₂×Q₈	1	1	yes	characteristic
59	C₄²×C₇₁	#41	1	2	yes	—
60	C₂³×C₇₁	C₂×C₄²	1	1	yes	characteristic
61	C₂×C₄×C₇₁	C₂×C₄⋊C₄	1	1	yes	characteristic
62	C₂×C₄×C₇₁	#117	1	1	yes	characteristic
63	C₂×C₄×C₇₁	C₄×D₄	1	2	yes	—
64	C₂×C₄×C₇₁	C₄×Q₈	1	2	yes	—
65	C₂×C₄×C₇₁	C₂²×C₈	1	2	yes	—
66	C₂×C₄×C₇₁	C₂×M₁₆	1	4	yes	—
67	C₂×C₄×C₇₁	C₂×M₁₆	1	2	yes	—
68	C₂×C₄×C₇₁	#131	1	4	yes	—
69	C₂×Dic₇₁	—	2	2	no	—
70	C₂×Dic₇₁	—	2	2	no	—
71	C₂×Dic₇₁	—	2	1	no	—
72	C₂×Dic₇₁	—	2	1	no	—
73	C₂×Dic₇₁	—	2	2	no	—
74	C₂×Dic₇₁	—	2	2	no	—
75	C₂×Dic₇₁	C₄×C₈	1	4	yes	—
76	C₈×C₇₁	—	4	4	no	—
77	C₈×C₇₁	C₄⋊C₈	1	8	yes	complemented
78	C₇₁⋊C₈	—	2	4	no	—
79	C₇₁⋊C₈	—	2	8	no	—
80	C₂²×C₇₁	C₄×C₄⋊C₄	1	1	yes	characteristic
81	C₂²×C₇₁	C₂×C₄×C₈	1	1	yes	characteristic
82	C₂²×C₇₁	C₄×M₁₆	1	1	yes	characteristic
83	C₂²×C₇₁	C₄×M₁₆	1	2	yes	—
84	C₂²×C₇₁	#405	1	2	yes	—
85	C₄×C₇₁	C₂×C₄⋊C₈	1	2	yes	—
86	C₄×C₇₁	#431	1	2	yes	—
87	C₄×C₇₁	#435	1	4	yes	—
88	C₄×C₇₁	#446	1	4	yes	—
89	Dic₇₁	—	2	4	no	—
90	Dic₇₁	—	4	1	no	—
91	Dic₇₁	—	4	1	no	—
92	#8262	—	71	1	no	maximal, Sylow
93	C₂×C₇₁	C₈×M₁₆	1	2	yes	—
94	C₂×C₇₁	#1453	1	1	yes	characteristic
95	C₂×C₇₁	#1455	1	1	yes	characteristic
96	C₂×C₇₁	C₄×C₄⋊C₈	1	1	yes	characteristic
97	C₂×C₇₁	#1771	1	2	yes	—
98	C₄×C₄⋊C₈	—	71	2	no	—
99	C₄×#97	—	71	1	no	—
100	C₂×#322	D₇₁	1	1	yes	characteristic
101	C₂×C₈²	—	71	1	no	—
102	#1273	—	71	2	no	—
103	C₇₁	#8262	1	1	yes	characteristic, complemented, Sylow
104	C₂×C₄⋊C₈	—	71	4	no	—
105	C₂×#97	—	71	2	no	—
106	C₂×#97	—	71	2	no	—
107	C₂×C₄×C₈	—	71	2	no	—
108	C₂×C₄×C₈	—	71	1	no	—
109	C₂×C₄×C₈	C₂×D₇₁	1	2	yes	—
110	C₂×C₄×C₈	C₂×D₇₁	1	1	yes	characteristic
111	C₄³	—	71	1	no	—
112	#336	—	71	4	no	—
113	#322	—	2	2	no	—
114	C₈²	—	142	2	no	—
115	C₂²×C₈	—	71	4	no	—
116	C₂²×C₈	—	71	2	no	—
117	C₂²×C₈	C₄×D₇₁	1	4	yes	—
118	C₂²×C₈	C₄×D₇₁	1	2	yes	—
119	C₂×C₄²	—	71	1	no	—
120	C₂×C₄²	—	71	2	no	—
121	C₂×C₄²	—	71	1	no	—
122	C₂×C₄²	—	71	2	no	—
123	C₂×C₄²	C₂²×D₇₁	1	1	yes	characteristic
124	C₄⋊C₈	—	142	8	no	—
125	#97	—	71	8	no	—
126	#97	—	142	4	no	—
127	C₄×C₈	—	2	2	no	—
128	C₄×C₈	—	2	2	no	—
129	C₄×C₈	—	142	2	no	—
130	C₄×C₈	—	142	2	no	—
131	C₄×C₈	—	142	2	no	—
132	C₄×C₈	Dic₁₄₂	1	2	yes	—
133	C₄×C₈	C₂²⋊D₇₁	1	2	yes	—
134	C₂²×C₄	—	71	2	no	—
135	C₂²×C₄	—	71	1	no	—
136	C₂²×C₄	—	71	1	no	—
137	C₂²×C₄	C₂×C₄×D₇₁	1	1	yes	characteristic
138	C₂²×C₄	C₂×C₄×D₇₁	1	2	yes	—
139	C₂×C₈	—	2	4	no	—
140	C₂×C₈	—	2	4	no	—
141	C₂×C₈	—	2	4	no	—
142	C₂×C₈	—	71	4	no	—
143	C₂×C₈	—	71	8	no	—
144	C₂×C₈	—	142	2	no	—
145	C₂×C₈	—	142	4	no	—
146	C₂×C₈	—	142	2	no	—
147	C₂×C₈	—	142	4	no	—
148	C₂×C₈	Dic₇₁⋊C₄	1	4	yes	—
149	C₂×C₈	C₈×D₇₁	1	4	yes	—
150	C₂×C₈	#11765848	1	4	yes	—
151	C₄²	—	71	4	no	—
152	C₄²	—	142	4	no	—
153	C₄²	—	142	2	no	—
154	C₄²	—	142	4	no	—
155	C₄²	C₂×C₂²⋊D₇₁	1	1	yes	characteristic
156	C₄²	C₂×Dic₁₄₂	1	1	yes	characteristic
157	C₄²	#11765865	1	2	yes	—
158	C₂³	C₄²×D₇₁	1	1	yes	characteristic
159	C₂×C₄	—	71	4	no	—
160	C₂×C₄	—	142	1	no	—
161	C₂×C₄	—	142	1	no	—
162	C₂×C₄	—	142	2	no	—
163	C₂×C₄	—	142	2	no	—
164	C₂×C₄	—	142	2	no	—
165	C₂×C₄	—	142	2	no	—
166	C₂×C₄	C₂×Dic₇₁⋊C₄	1	1	yes	characteristic
167	C₂×C₄	#14556085	1	1	yes	characteristic
168	C₂×C₄	C₄×C₂²⋊D₇₁	1	2	yes	—
169	C₂×C₄	C₄×Dic₁₄₂	1	2	yes	—
170	C₂×C₄	C₂×C₈×D₇₁	1	2	yes	—
171	C₂×C₄	M₁₆×D₇₁	1	4	yes	—
172	C₂×C₄	C₂×#11765848	1	2	yes	—
173	C₂×C₄	#14556136	1	4	yes	—
174	C₈	—	4	4	no	—
175	C₈	—	142	4	no	—
176	C₈	—	142	8	no	—
177	C₈	Dic₇₁⋊C₈	1	8	yes	complemented
178	C₂²	C₄×Dic₇₁⋊C₄	1	1	yes	characteristic
179	C₂²	C₄×C₈×D₇₁	1	1	yes	characteristic
180	C₂²	C₄×#11765848	1	1	yes	characteristic
181	C₂²	#17985951	1	2	yes	—
182	C₂²	#17985956	1	2	yes	—
183	C₄	—	284	1	no	—
184	C₄	—	284	1	no	—
185	C₄	—	142	4	no	—
186	C₄	C₂×Dic₇₁⋊C₈	1	2	yes	—
187	C₄	#17986050	1	2	yes	—
188	C₄	#17986068	1	4	yes	—
189	C₄	#17986127	1	4	yes	—
190	C₂	#22586619	1	2	yes	—
191	C₂	#22586638	1	1	yes	characteristic
192	C₂	#22586644	1	1	yes	characteristic
193	C₂	C₄×Dic₇₁⋊C₈	1	1	yes	characteristic
194	C₂	#22588216	1	2	yes	—

	subgroup	G/H
center, Frattini	C₂×C₄²	C₂²×D₇₁
commutator	C₂²×C₇₁	C₂×C₄×C₈
Fitting	C₂×#322×C₇₁	C₂
socle, prime radical	C₂³×C₇₁	C₂×C₄²
cosocle	C₂×C₄²×C₇₁	C₂³
hypercenter	C₂×#322	D₇₁
hypocenter	C₇₁	#8262
LCS	G ▷ C₂²×C₇₁ ▷ C₇₁
UCS	C₁ ◁ C₂×C₄² ◁ C₂×#322

Complements

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

N	H ≅ G/N	type
25. #14556038	C₈	semidirect
77. C₈×C₇₁	C₄⋊C₈	semidirect
103. C₇₁	#8262	semidirect
177. C₈	Dic₇₁⋊C₈	semidirect

Maximal subgroups

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

subgroup	index
C₄×Dic₇₁⋊C₈	2
C₄×#14556038	2
C₂×C₈×C₇₁⋊C₈	2
C₇₁⋊#1273	2
C₂×#322×C₇₁	2
#8262	71