Subgroups — #31887880

Order 18176.

446 conjugacy classes of subgroups.

The subgroup lattice has 190 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#22588085	C₂	1	2	yes	maximal
2	C₄×C₈×Dic₇₁	C₂	1	1	yes	maximal, characteristic
3	C₇₁⋊#1273	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×C₄×C₇₁⋊C₈	C₂²	1	1	yes	characteristic
7	C₂×C₄×C₇₁⋊C₈	C₂²	1	2	yes	—
8	C₂×C₈×Dic₇₁	C₄	1	2	yes	—
9	C₂×C₈×Dic₇₁	C₄	1	2	yes	—
10	C₄²×Dic₇₁	C₂²	1	1	yes	characteristic
11	#17985753	C₄	1	4	yes	—
12	#17985753	C₄	1	4	yes	—
13	C₂×C₄×C₈×C₇₁	C₂²	1	1	yes	characteristic
14	C₂×C₄×C₈×C₇₁	C₂²	1	2	yes	—
15	C₂²×C₇₁⋊C₈	C₂×C₄	1	4	yes	—
16	C₂²×C₇₁⋊C₈	C₂×C₄	1	2	yes	—
17	C₂×C₄×Dic₇₁	C₂×C₄	1	2	yes	—
18	C₂×C₄×Dic₇₁	C₂×C₄	1	2	yes	—
19	C₂×C₄×Dic₇₁	C₂×C₄	1	1	yes	characteristic
20	C₂×C₄×Dic₇₁	C₂×C₄	1	1	yes	characteristic
21	C₄×C₇₁⋊C₈	—	2	1	no	—
22	C₄×C₇₁⋊C₈	—	2	2	no	—
23	C₄×C₇₁⋊C₈	—	2	1	no	—
24	C₄×C₇₁⋊C₈	—	2	2	no	—
25	C₈×Dic₇₁	—	2	4	no	—
26	C₈×Dic₇₁	C₈	1	8	yes	complemented
27	C₂²×C₈×C₇₁	C₂×C₄	1	4	yes	—
28	C₂²×C₈×C₇₁	C₂×C₄	1	2	yes	—
29	C₂×C₄²×C₇₁	C₂³	1	1	yes	characteristic
30	C₄×C₈×C₇₁	—	2	2	no	—
31	C₄×C₈×C₇₁	—	2	2	no	—
32	C₄×C₈×C₇₁	—	2	2	no	—
33	C₂²×Dic₇₁	C₄²	1	1	yes	characteristic
34	C₂²×Dic₇₁	C₄²	1	1	yes	characteristic
35	C₂²×Dic₇₁	C₄²	1	2	yes	—
36	C₂×C₇₁⋊C₈	—	2	2	no	—
37	C₂×C₇₁⋊C₈	—	2	2	no	—
38	C₂×C₇₁⋊C₈	—	2	4	no	—
39	C₂×C₇₁⋊C₈	—	2	4	no	—
40	C₂×C₇₁⋊C₈	—	2	4	no	—
41	C₂×C₇₁⋊C₈	C₂×C₈	1	4	yes	—
42	C₄×Dic₇₁	—	2	4	no	—
43	C₄×Dic₇₁	—	2	4	no	—
44	C₄×Dic₇₁	C₂×C₈	1	4	yes	—
45	C₄×Dic₇₁	M₁₆	1	4	yes	—
46	C₂²×C₄×C₇₁	C₂²×C₄	1	1	yes	characteristic
47	C₂²×C₄×C₇₁	C₂²×C₄	1	2	yes	—
48	C₂×C₈×C₇₁	—	2	2	no	—
49	C₂×C₈×C₇₁	—	2	2	no	—
50	C₂×C₈×C₇₁	—	2	4	no	—
51	C₂×C₈×C₇₁	—	2	4	no	—
52	C₂×C₈×C₇₁	—	2	4	no	—
53	C₂×C₈×C₇₁	C₂×C₈	1	4	yes	—
54	C₄²×C₇₁	#41	1	2	yes	—
55	C₄²×C₇₁	#41	1	1	yes	characteristic
56	C₄²×C₇₁	#41	1	1	yes	characteristic
57	C₂³×C₇₁	C₂×C₄²	1	1	yes	characteristic
58	C₂×C₄×C₇₁	#117	1	1	yes	characteristic
59	C₂×C₄×C₇₁	#117	1	2	yes	—
60	C₂×C₄×C₇₁	#117	1	1	yes	characteristic
61	C₂×C₄×C₇₁	#117	1	2	yes	—
62	C₂×C₄×C₇₁	C₂²×C₈	1	2	yes	—
63	C₂×C₄×C₇₁	C₂×M₁₆	1	2	yes	—
64	C₂×C₄×C₇₁	#131	1	4	yes	—
65	C₂×C₄×C₇₁	#131	1	4	yes	—
66	C₂×Dic₇₁	—	2	2	no	—
67	C₂×Dic₇₁	—	2	2	no	—
68	C₂×Dic₇₁	—	2	2	no	—
69	C₂×Dic₇₁	—	2	2	no	—
70	C₂×Dic₇₁	C₄×C₈	1	2	yes	—
71	C₂×Dic₇₁	C₄×C₈	1	2	yes	—
72	C₂×Dic₇₁	#97	1	4	yes	—
73	C₈×C₇₁	—	2	4	no	—
74	C₈×C₇₁	—	4	4	no	—
75	C₇₁⋊C₈	—	2	4	no	—
76	C₇₁⋊C₈	—	4	4	no	—
77	C₂²×C₇₁	#376	1	1	yes	characteristic
78	C₂²×C₇₁	C₂×C₄×C₈	1	1	yes	characteristic
79	C₂²×C₇₁	C₂×#97	1	1	yes	characteristic
80	C₂²×C₇₁	#405	1	2	yes	—
81	C₂²×C₇₁	#405	1	2	yes	—
82	C₄×C₇₁	#431	1	2	yes	—
83	C₄×C₇₁	#431	1	2	yes	—
84	C₄×C₇₁	#433	1	4	yes	—
85	C₄×C₇₁	#433	1	4	yes	—
86	Dic₇₁	—	2	2	no	—
87	Dic₇₁	—	2	2	no	—
88	Dic₇₁	—	2	4	no	—
89	#8133	—	71	1	no	maximal, Sylow
90	C₂×C₇₁	#1450	1	1	yes	characteristic
91	C₂×C₇₁	#1450	1	1	yes	characteristic
92	C₂×C₇₁	#1452	1	2	yes	—
93	C₂×C₇₁	#1744	1	1	yes	characteristic
94	C₂×C₇₁	#1746	1	2	yes	—
95	#1744	—	71	2	no	—
96	C₄²×C₈	—	71	1	no	—
97	#1273	—	71	2	no	—
98	#1273	—	71	1	no	—
99	#1273	D₇₁	1	1	yes	characteristic
100	C₇₁	#8133	1	1	yes	characteristic, complemented, Sylow
101	C₂×C₄×C₈	—	71	2	no	—
102	C₂×C₄×C₈	—	71	2	no	—
103	C₂×C₄×C₈	—	71	1	no	—
104	C₂×C₄×C₈	—	71	2	no	—
105	C₂×C₄×C₈	C₂×D₇₁	1	1	yes	characteristic
106	C₂×C₄×C₈	C₂×D₇₁	1	2	yes	—
107	C₄³	—	71	1	no	—
108	#336	—	71	4	no	—
109	#336	—	71	4	no	—
110	C₂²×C₈	—	71	2	no	—
111	C₂²×C₈	—	71	4	no	—
112	C₂²×C₈	C₄×D₇₁	1	4	yes	—
113	C₂²×C₈	C₄×D₇₁	1	2	yes	—
114	C₂×C₄²	—	71	1	no	—
115	C₂×C₄²	—	71	1	no	—
116	C₂×C₄²	—	71	2	no	—
117	C₂×C₄²	—	71	2	no	—
118	C₂×C₄²	C₂²×D₇₁	1	1	yes	characteristic
119	C₄×C₈	—	2	2	no	—
120	C₄×C₈	—	2	2	no	—
121	C₄×C₈	—	2	2	no	—
122	C₄×C₈	—	71	8	no	—
123	C₄×C₈	—	142	4	no	—
124	C₄×C₈	—	142	2	no	—
125	C₄×C₈	—	142	1	no	—
126	C₄×C₈	—	142	2	no	—
127	C₄×C₈	—	142	1	no	—
128	C₂²×C₄	—	71	1	no	—
129	C₂²×C₄	—	71	1	no	—
130	C₂²×C₄	—	71	2	no	—
131	C₂²×C₄	C₂×C₄×D₇₁	1	1	yes	characteristic
132	C₂²×C₄	C₂×C₄×D₇₁	1	2	yes	—
133	C₂×C₈	—	2	2	no	—
134	C₂×C₈	—	2	2	no	—
135	C₂×C₈	—	2	4	no	—
136	C₂×C₈	—	2	4	no	—
137	C₂×C₈	—	2	4	no	—
138	C₂×C₈	—	71	4	no	—
139	C₂×C₈	—	142	4	no	—
140	C₂×C₈	—	142	4	no	—
141	C₂×C₈	—	142	4	no	—
142	C₂×C₈	—	142	2	no	—
143	C₂×C₈	—	142	2	no	—
144	C₂×C₈	C₈×D₇₁	1	4	yes	—
145	C₄²	—	71	4	no	—
146	C₄²	—	71	4	no	—
147	C₄²	—	142	4	no	—
148	C₄²	—	142	4	no	—
149	C₄²	#11765864	1	2	yes	—
150	C₄²	#11765864	1	1	yes	characteristic
151	C₄²	#11765866	1	1	yes	characteristic
152	C₂³	C₄²×D₇₁	1	1	yes	characteristic
153	C₂×C₄	—	71	2	no	—
154	C₂×C₄	—	71	2	no	—
155	C₂×C₄	—	71	4	no	—
156	C₂×C₄	—	142	2	no	—
157	C₂×C₄	—	142	2	no	—
158	C₂×C₄	—	142	2	no	—
159	C₂×C₄	—	142	2	no	—
160	C₂×C₄	#14556084	1	1	yes	characteristic
161	C₂×C₄	#14556084	1	2	yes	—
162	C₂×C₄	#14556086	1	1	yes	characteristic
163	C₂×C₄	#14556086	1	2	yes	—
164	C₂×C₄	C₂×C₈×D₇₁	1	2	yes	—
165	C₂×C₄	M₁₆×D₇₁	1	2	yes	—
166	C₂×C₄	#14556134	1	4	yes	—
167	C₂×C₄	#14556136	1	4	yes	—
168	C₈	—	2	4	no	—
169	C₈	—	4	4	no	—
170	C₈	—	284	4	no	—
171	C₈	—	142	4	no	—
172	C₂²	#17985844	1	1	yes	characteristic
173	C₂²	C₄×C₈×D₇₁	1	1	yes	characteristic
174	C₂²	#97×D₇₁	1	1	yes	characteristic
175	C₂²	#17985953	1	2	yes	—
176	C₂²	#17985956	1	2	yes	—
177	C₄	—	142	2	no	—
178	C₄	—	142	4	no	—
179	C₄	—	142	2	no	—
180	C₄	#17986049	1	2	yes	—
181	C₄	#17986051	1	2	yes	—
182	C₄	#17986057	1	4	yes	—
183	C₄	#17986059	1	4	yes	—
184	C₂	#22586622	1	1	yes	characteristic
185	C₂	#22586625	1	1	yes	characteristic
186	C₂	#22586631	1	2	yes	—
187	C₂	#22588084	1	1	yes	characteristic
188	C₂	#22588090	1	2	yes	—

	subgroup	G/H
center, Frattini	C₂×C₄²	C₂²×D₇₁
commutator	C₂²×C₇₁	C₂×C₄×C₈
Fitting	C₇₁×#1273	C₂
socle, prime radical	C₂³×C₇₁	C₂×C₄²
cosocle	C₂×C₄²×C₇₁	C₂³
hypercenter	#1273	D₇₁
hypocenter	C₇₁	#8133
LCS	G ▷ C₂²×C₇₁ ▷ C₇₁
UCS	C₁ ◁ C₂×C₄² ◁ #1273

Complements

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

N	H ≅ G/N	type
26. C₈×Dic₇₁	C₈	semidirect
100. C₇₁	#8133	semidirect

Maximal subgroups

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

subgroup	index
#22588085	2
C₄×C₈×Dic₇₁	2
C₇₁⋊#1273	2
C₇₁×#1273	2
#8133	71