Subgroups — #31888572

Order 18176.

550 conjugacy classes of subgroups.

The subgroup lattice has 194 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×C₈×Dic₇₁	C₂	1	1	yes	maximal, characteristic
3	C₂×C₈×C₇₁⋊C₈	C₂	1	2	yes	maximal
4	C₇₁⋊#1273	C₂	1	1	yes	maximal, characteristic
5	C₇₁×#1273	C₂	1	1	yes	maximal, characteristic
6	C₂×Dic₇₁⋊C₈	C₄	1	4	yes	—
7	C₂×C₄×C₇₁⋊C₈	C₂²	1	2	yes	—
8	C₂×C₄×C₇₁⋊C₈	C₂²	1	1	yes	characteristic
9	C₂×C₈×Dic₇₁	C₄	1	2	yes	—
10	C₂×C₈×Dic₇₁	C₄	1	2	yes	—
11	C₄²×Dic₇₁	C₂²	1	1	yes	characteristic
12	#17985753	C₄	1	4	yes	—
13	C₈×C₇₁⋊C₈	—	2	4	no	—
14	C₂×C₄×C₈×C₇₁	C₂²	1	2	yes	—
15	C₂×C₄×C₈×C₇₁	C₂²	1	1	yes	characteristic
16	C₂²×C₇₁⋊C₈	C₂×C₄	1	4	yes	—
17	C₂²×C₇₁⋊C₈	C₂×C₄	1	2	yes	—
18	C₂×C₄×Dic₇₁	C₂×C₄	1	1	yes	characteristic
19	C₂×C₄×Dic₇₁	C₂×C₄	1	2	yes	—
20	C₂×C₄×Dic₇₁	C₂×C₄	1	1	yes	characteristic
21	C₂×C₄×Dic₇₁	C₂×C₄	1	2	yes	—
22	Dic₇₁⋊C₈	C₈	1	16	yes	complemented
23	C₄×C₇₁⋊C₈	—	2	2	no	—
24	C₄×C₇₁⋊C₈	—	2	2	no	—
25	C₄×C₇₁⋊C₈	D₄	1	2	yes	—
26	C₄×C₇₁⋊C₈	Q₈	1	2	yes	—
27	C₈×Dic₇₁	—	2	4	no	—
28	C₈×Dic₇₁	C₈	1	8	yes	complemented
29	C₂²×C₈×C₇₁	C₂×C₄	1	2	yes	—
30	C₂²×C₈×C₇₁	C₂×C₄	1	4	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	—	2	2	no	—
35	C₂²×Dic₇₁	C₄²	1	2	yes	—
36	C₂²×Dic₇₁	C₄²	1	1	yes	characteristic
37	C₂²×Dic₇₁	C₄²	1	1	yes	characteristic
38	C₂×C₇₁⋊C₈	—	2	4	no	—
39	C₂×C₇₁⋊C₈	—	2	4	no	—
40	C₂×C₇₁⋊C₈	C₄⋊C₄	1	4	yes	—
41	C₂×C₇₁⋊C₈	C₂×C₈	1	8	yes	—
42	C₂×C₇₁⋊C₈	C₂×C₈	1	4	yes	—
43	C₂×C₇₁⋊C₈	M₁₆	1	4	yes	—
44	C₄×Dic₇₁	—	2	2	no	—
45	C₄×Dic₇₁	—	2	4	no	—
46	C₄×Dic₇₁	C₂×C₈	1	4	yes	—
47	C₄×Dic₇₁	C₂×C₈	1	8	yes	—
48	C₂²×C₄×C₇₁	C₂²×C₄	1	1	yes	characteristic
49	C₂²×C₄×C₇₁	C₂²×C₄	1	2	yes	—
50	C₂×C₈×C₇₁	—	2	2	no	—
51	C₂×C₈×C₇₁	—	2	4	no	—
52	C₂×C₈×C₇₁	—	2	2	no	—
53	C₂×C₈×C₇₁	—	2	4	no	—
54	C₂×C₈×C₇₁	C₂×C₈	1	4	yes	—
55	C₂×C₈×C₇₁	C₂×C₈	1	8	yes	—
56	C₄²×C₇₁	C₂×D₄	1	1	yes	characteristic
57	C₄²×C₇₁	C₂×Q₈	1	1	yes	characteristic
58	C₄²×C₇₁	#41	1	2	yes	—
59	C₂³×C₇₁	C₂×C₄²	1	1	yes	characteristic
60	C₂×C₄×C₇₁	C₂×C₄⋊C₄	1	1	yes	characteristic
61	C₂×C₄×C₇₁	#117	1	1	yes	characteristic
62	C₂×C₄×C₇₁	C₄×D₄	1	2	yes	—
63	C₂×C₄×C₇₁	C₄×Q₈	1	2	yes	—
64	C₂×C₄×C₇₁	C₂²×C₈	1	2	yes	—
65	C₂×C₄×C₇₁	C₂²×C₈	1	4	yes	—
66	C₂×C₄×C₇₁	C₂×M₁₆	1	2	yes	—
67	C₂×C₄×C₇₁	#131	1	4	yes	—
68	C₂×Dic₇₁	—	2	2	no	—
69	C₂×Dic₇₁	—	2	2	no	—
70	C₂×Dic₇₁	—	2	2	no	—
71	C₂×Dic₇₁	C₄×C₈	1	4	yes	—
72	C₂×Dic₇₁	C₄×C₈	1	2	yes	—
73	C₂×Dic₇₁	C₄×C₈	1	4	yes	—
74	C₂×Dic₇₁	C₄×C₈	1	2	yes	—
75	C₈×C₇₁	—	2	4	no	—
76	C₈×C₇₁	—	2	8	no	—
77	C₇₁⋊C₈	—	2	8	no	—
78	C₇₁⋊C₈	C₄⋊C₈	1	8	yes	complemented
79	C₂²×C₇₁	C₄×C₄⋊C₄	1	1	yes	characteristic
80	C₂²×C₇₁	C₂×C₄×C₈	1	2	yes	—
81	C₂²×C₇₁	C₂×C₄×C₈	1	1	yes	characteristic
82	C₂²×C₇₁	C₄×M₁₆	1	1	yes	characteristic
83	C₂²×C₇₁	#405	1	2	yes	—
84	C₄×C₇₁	C₂×C₄⋊C₈	1	2	yes	—
85	C₄×C₇₁	#431	1	2	yes	—
86	C₄×C₇₁	C₈×D₄	1	4	yes	—
87	C₄×C₇₁	C₈×Q₈	1	4	yes	—
88	Dic₇₁	—	2	4	no	—
89	Dic₇₁	—	2	2	no	—
90	Dic₇₁	C₈²	1	4	yes	complemented
91	C₈×C₄⋊C₈	—	71	1	no	maximal, Sylow
92	C₂×C₇₁	C₂×C₈²	1	1	yes	characteristic
93	C₂×C₇₁	C₈×M₁₆	1	2	yes	—
94	C₂×C₇₁	#1450	1	1	yes	characteristic
95	C₂×C₇₁	C₄×C₄⋊C₈	1	1	yes	characteristic
96	C₂×C₇₁	C₈×C₄⋊C₄	1	2	yes	—
97	C₄×C₄⋊C₈	—	71	2	no	—
98	C₄²×C₈	—	71	1	no	—
99	C₂×C₈²	—	71	2	no	—
100	#1273	—	71	1	no	—
101	#1273	D₇₁	1	1	yes	characteristic
102	C₇₁	C₈×C₄⋊C₈	1	1	yes	characteristic, complemented, Sylow
103	C₂×C₄⋊C₈	—	71	4	no	—
104	C₂×C₄×C₈	—	71	2	no	—
105	C₂×C₄×C₈	—	71	1	no	—
106	C₂×C₄×C₈	—	71	2	no	—
107	C₂×C₄×C₈	—	71	2	no	—
108	C₂×C₄×C₈	C₂×D₇₁	1	2	yes	—
109	C₂×C₄×C₈	C₂×D₇₁	1	1	yes	characteristic
110	C₄³	—	71	1	no	—
111	#336	—	71	4	no	—
112	C₈²	—	142	4	no	—
113	C₂²×C₈	—	71	2	no	—
114	C₂²×C₈	—	71	4	no	—
115	C₂²×C₈	C₄×D₇₁	1	4	yes	—
116	C₂²×C₈	C₄×D₇₁	1	2	yes	—
117	C₂×C₄²	—	71	1	no	—
118	C₂×C₄²	—	71	2	no	—
119	C₂×C₄²	—	71	2	no	—
120	C₂×C₄²	—	71	1	no	—
121	C₂×C₄²	C₂²×D₇₁	1	1	yes	characteristic
122	C₄⋊C₈	—	71	16	no	—
123	C₄×C₈	—	2	2	no	—
124	C₄×C₈	—	2	2	no	—
125	C₄×C₈	—	2	2	no	—
126	C₄×C₈	—	71	8	no	—
127	C₄×C₈	—	71	2	no	—
128	C₄×C₈	—	71	2	no	—
129	C₄×C₈	—	142	4	no	—
130	C₄×C₈	—	142	2	no	—
131	C₄×C₈	—	142	2	no	—
132	C₂²×C₄	—	71	1	no	—
133	C₂²×C₄	—	71	2	no	—
134	C₂²×C₄	—	71	1	no	—
135	C₂²×C₄	C₂×C₄×D₇₁	1	1	yes	characteristic
136	C₂²×C₄	C₂×C₄×D₇₁	1	2	yes	—
137	C₂×C₈	—	2	2	no	—
138	C₂×C₈	—	2	4	no	—
139	C₂×C₈	—	2	2	no	—
140	C₂×C₈	—	2	4	no	—
141	C₂×C₈	—	71	4	no	—
142	C₂×C₈	—	71	4	no	—
143	C₂×C₈	—	71	4	no	—
144	C₂×C₈	—	71	8	no	—
145	C₂×C₈	—	142	4	no	—
146	C₂×C₈	—	142	4	no	—
147	C₂×C₈	C₈×D₇₁	1	4	yes	—
148	C₂×C₈	C₈×D₇₁	1	8	yes	—
149	C₄²	—	71	4	no	—
150	C₄²	—	71	8	no	—
151	C₄²	—	142	2	no	—
152	C₄²	—	142	4	no	—
153	C₄²	D₄×D₇₁	1	1	yes	characteristic
154	C₄²	Q₈×D₇₁	1	1	yes	characteristic
155	C₄²	#11765864	1	2	yes	—
156	C₂³	C₄²×D₇₁	1	1	yes	characteristic
157	C₂×C₄	—	71	2	no	—
158	C₂×C₄	—	71	4	no	—
159	C₂×C₄	—	71	2	no	—
160	C₂×C₄	—	71	4	no	—
161	C₂×C₄	—	142	2	no	—
162	C₂×C₄	—	142	2	no	—
163	C₂×C₄	—	142	2	no	—
164	C₂×C₄	C₄⋊C₄×D₇₁	1	1	yes	characteristic
165	C₂×C₄	#14556084	1	1	yes	characteristic
166	C₂×C₄	(C₂²⋊D₇₁)⋊C₄	1	2	yes	—
167	C₂×C₄	#14556093	1	2	yes	—
168	C₂×C₄	C₂×C₈×D₇₁	1	2	yes	—
169	C₂×C₄	C₂×C₈×D₇₁	1	4	yes	—
170	C₂×C₄	M₁₆×D₇₁	1	2	yes	—
171	C₂×C₄	#14556136	1	4	yes	—
172	C₈	—	2	4	no	—
173	C₈	—	2	8	no	—
174	C₈	—	71	8	no	—
175	C₈	—	142	8	no	—
176	C₂²	#17985850	1	1	yes	characteristic
177	C₂²	C₄×C₈×D₇₁	1	1	yes	characteristic
178	C₂²	C₄×C₈×D₇₁	1	2	yes	—
179	C₂²	#17985951	1	1	yes	characteristic
180	C₂²	#17985956	1	2	yes	—
181	C₄	—	71	4	no	—
182	C₄	—	142	2	no	—
183	C₄	—	142	4	no	—
184	C₄	C₄⋊C₈×D₇₁	1	2	yes	—
185	C₄	#17986049	1	2	yes	—
186	C₄	(C₂²⋊D₇₁)⋊C₈	1	4	yes	—
187	C₄	#17986123	1	4	yes	—
188	C₂	C₈²×D₇₁	1	1	yes	characteristic
189	C₂	#22586619	1	2	yes	—
190	C₂	#22586625	1	1	yes	characteristic
191	C₂	#22588188	1	1	yes	characteristic
192	C₂	#22588201	1	2	yes	—

	subgroup	G/H
center, Frattini	C₂×C₄²	C₂²×D₇₁
commutator	C₂×C₇₁	C₂×C₈²
Fitting	C₇₁×#1273	C₂
socle, prime radical	C₂³×C₇₁	C₂×C₄²
cosocle	C₂×C₄²×C₇₁	C₂³
hypercenter	#1273	D₇₁
hypocenter	C₇₁	C₈×C₄⋊C₈
LCS	G ▷ C₂×C₇₁ ▷ C₇₁
UCS	C₁ ◁ C₂×C₄² ◁ #1273

Complements

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

N	H ≅ G/N	type
22. Dic₇₁⋊C₈	C₈	semidirect
28. C₈×Dic₇₁	C₈	semidirect
78. C₇₁⋊C₈	C₄⋊C₈	semidirect
90. Dic₇₁	C₈²	semidirect
102. C₇₁	C₈×C₄⋊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×C₈×Dic₇₁	2
C₂×C₈×C₇₁⋊C₈	2
C₇₁⋊#1273	2
C₇₁×#1273	2
C₈×C₄⋊C₈	71