Subgroups — #31884642

Order 18176.

246 conjugacy classes of subgroups.

The subgroup lattice has 198 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	#22588504	C₂	1	1	yes	maximal, characteristic
3	C₂×#334×C₇₁	C₂	1	1	yes	maximal, characteristic
4	C₂²×C₇₁⋊C₁₆	—	2	1	no	—
5	C₂×#14556062	C₄	1	2	yes	—
6	C₂×#14556060	—	2	1	no	—
7	#17985865	—	2	1	no	—
8	C₄⋊(C₇₁⋊C₁₆)	—	2	1	no	—
9	C₄⋊(C₇₁⋊C₁₆)	C₄	1	2	yes	complemented
10	C₂×C₄⋊C₈×C₇₁	—	2	1	no	—
11	C₂×C₄×C₈×C₇₁	C₂²	1	1	yes	characteristic
12	#334×C₇₁	—	2	1	no	—
13	#334×C₇₁	C₄	1	2	yes	complemented
14	C₂×C₄⋊Dic₇₁	—	2	2	no	—
15	C₂×C₇₁⋊C₁₆	—	2	2	no	—
16	C₂×C₇₁⋊C₁₆	—	4	1	no	—
17	C₂×C₇₁⋊C₁₆	—	4	1	no	—
18	#14556062	—	4	2	no	—
19	#14556060	—	4	1	no	—
20	#14556060	—	4	1	no	—
21	C₇₁⋊#95	—	4	1	no	—
22	C₇₁⋊#95	—	4	1	no	—
23	C₂²×C₈×C₇₁	—	4	1	no	—
24	C₂²×C₈×C₇₁	C₂×C₄	1	1	yes	characteristic
25	C₂²×C₈×C₇₁	D₄	1	1	yes	characteristic
26	C₂×C₄²×C₇₁	D₄	1	1	yes	characteristic
27	C₄⋊C₈×C₇₁	—	2	2	no	—
28	C₄⋊C₈×C₇₁	—	4	1	no	—
29	C₄×C₈×C₇₁	C₂×C₄	1	1	yes	characteristic
30	C₄×C₈×C₇₁	C₂×C₄	1	1	yes	characteristic
31	C₄×C₈×C₇₁	D₄	1	1	yes	characteristic
32	C₄×C₈×C₇₁	Q₈	1	1	yes	characteristic
33	C₂²×Dic₇₁	—	4	2	no	—
34	C₄⋊Dic₇₁	—	4	2	no	—
35	C₄⋊Dic₇₁	—	4	2	no	—
36	C₇₁⋊C₁₆	—	4	2	no	—
37	C₂²×C₄×C₇₁	C₂²⋊C₄	1	1	yes	characteristic
38	C₂²×C₄×C₇₁	D₈	1	1	yes	characteristic
39	C₂²×C₄×C₇₁	SD₁₆	1	1	yes	characteristic
40	C₂×C₈×C₇₁	—	2	1	no	—
41	C₂×C₈×C₇₁	—	2	1	no	—
42	C₂×C₈×C₇₁	—	4	2	no	—
43	C₂×C₈×C₇₁	—	8	2	no	—
44	C₂×C₈×C₇₁	C₄²	1	1	yes	characteristic
45	C₂×C₈×C₇₁	C₂²⋊C₄	1	1	yes	characteristic
46	C₂×C₈×C₇₁	C₄⋊C₄	1	1	yes	characteristic
47	C₂×C₈×C₇₁	C₄⋊C₄	1	1	yes	characteristic
48	C₂×C₈×C₇₁	D₈	1	1	yes	characteristic
49	C₂×C₈×C₇₁	SD₁₆	1	1	yes	characteristic
50	C₂×C₈×C₇₁	SD₁₆	1	1	yes	characteristic
51	C₂×C₈×C₇₁	Q₁₆	1	1	yes	characteristic
52	C₄²×C₇₁	C₂²⋊C₄	1	1	yes	characteristic
53	C₄²×C₇₁	C₄⋊C₄	1	1	yes	characteristic
54	C₄²×C₇₁	SD₁₆	1	1	yes	characteristic
55	C₄²×C₇₁	Q₁₆	1	1	yes	characteristic
56	C₂³×C₇₁	#102	1	1	yes	characteristic
57	C₂×C₄×C₇₁	—	2	2	no	—
58	C₂×C₄×C₇₁	—	2	2	no	—
59	C₂×C₄×C₇₁	#95	1	1	yes	characteristic
60	C₂×C₄×C₇₁	#100	1	1	yes	characteristic
61	C₂×C₄×C₇₁	#101	1	1	yes	characteristic
62	C₂×C₄×C₇₁	#102	1	1	yes	characteristic
63	C₂×C₄×C₇₁	#102	1	1	yes	characteristic
64	C₂×C₄×C₇₁	#103	1	1	yes	characteristic
65	C₂×C₄×C₇₁	#103	1	1	yes	characteristic
66	C₂×C₄×C₇₁	#103	1	1	yes	characteristic
67	C₂×C₄×C₇₁	#106	1	1	yes	characteristic
68	C₂×C₄×C₇₁	#107	1	1	yes	characteristic
69	C₂×Dic₇₁	—	8	2	no	—
70	C₂×Dic₇₁	—	8	2	no	—
71	C₂×Dic₇₁	—	8	2	no	—
72	C₈×C₇₁	—	2	1	no	—
73	C₈×C₇₁	—	2	1	no	—
74	C₈×C₇₁	—	8	2	no	—
75	C₈×C₇₁	#102	1	1	yes	characteristic
76	C₈×C₇₁	#103	1	1	yes	characteristic
77	C₈×C₇₁	#106	1	1	yes	characteristic
78	C₈×C₇₁	#107	1	1	yes	characteristic
79	C₂²×C₇₁	#331	1	1	yes	characteristic
80	C₂²×C₇₁	#332	1	1	yes	characteristic
81	C₂²×C₇₁	#340	1	1	yes	characteristic
82	C₂²×C₇₁	#359	1	2	yes	—
83	C₂²×C₇₁	#360	1	2	yes	—
84	C₄×C₇₁	—	2	1	no	—
85	C₄×C₇₁	—	2	1	no	—
86	C₄×C₇₁	—	2	1	no	—
87	C₄×C₇₁	—	2	1	no	—
88	C₄×C₇₁	#332	1	1	yes	characteristic
89	C₄×C₇₁	#333	1	1	yes	characteristic
90	C₄×C₇₁	#340	1	1	yes	characteristic
91	C₄×C₇₁	#343	1	1	yes	characteristic
92	Dic₇₁	—	16	2	no	—
93	#7275	—	71	1	no	maximal, Sylow
94	C₂×C₇₁	#1296	1	1	yes	characteristic
95	C₂×C₇₁	#1360	1	2	yes	—
96	C₂×C₇₁	#1363	1	2	yes	—
97	C₂×C₇₁	#1382	1	1	yes	characteristic
98	C₂×C₇₁	#1386	1	1	yes	characteristic
99	C₂×C₄⋊C₁₆	—	71	1	no	—
100	#1846	—	71	1	no	—
101	C₂×#334	D₇₁	1	1	yes	characteristic
102	C₇₁	#7275	1	1	yes	characteristic, complemented, Sylow
103	C₂²×C₁₆	—	142	1	no	—
104	C₂×#107	—	71	2	no	—
105	C₂×#106	—	142	1	no	—
106	C₂×C₄⋊C₈	—	2	1	no	—
107	C₂×C₄×C₈	C₂×D₇₁	1	1	yes	characteristic
108	#382	—	142	1	no	—
109	C₄⋊C₁₆	—	71	2	no	—
110	C₄⋊C₁₆	—	142	1	no	—
111	#334	—	2	1	no	—
112	#334	Dic₇₁	1	2	yes	complemented
113	C₂²×C₈	—	4	1	no	—
114	C₂²×C₈	C₄×D₇₁	1	1	yes	characteristic
115	C₂²×C₈	D₂₈₄	1	1	yes	characteristic
116	C₂×C₄⋊C₄	—	142	2	no	—
117	C₂×C₄²	C₂²⋊D₇₁	1	1	yes	characteristic
118	C₂×C₁₆	—	284	1	no	—
119	C₂×C₁₆	—	284	1	no	—
120	C₂×C₁₆	—	142	2	no	—
121	#107	—	284	2	no	—
122	#106	—	284	1	no	—
123	#106	—	284	1	no	—
124	C₄⋊C₈	—	2	2	no	—
125	C₄⋊C₈	—	4	1	no	—
126	C₄×C₈	Dic₁₄₂	1	1	yes	characteristic
127	C₄×C₈	C₄×D₇₁	1	1	yes	characteristic
128	C₄×C₈	C₂×Dic₇₁	1	1	yes	characteristic
129	C₄×C₈	C₂²⋊D₇₁	1	1	yes	characteristic
130	#95	—	284	1	no	—
131	#95	—	284	1	no	—
132	C₂²×C₄	—	284	2	no	—
133	C₂²×C₄	(C₂×D₇₁)⋊C₄	1	1	yes	characteristic
134	C₂²×C₄	#11765850	1	1	yes	characteristic
135	C₂²×C₄	#11765852	1	1	yes	characteristic
136	C₂×C₈	—	2	1	no	—
137	C₂×C₈	—	2	1	no	—
138	C₂×C₈	—	4	2	no	—
139	C₂×C₈	—	8	2	no	—
140	C₂×C₈	C₄×Dic₇₁	1	1	yes	characteristic
141	C₂×C₈	(C₂×D₇₁)⋊C₄	1	1	yes	characteristic
142	C₂×C₈	Dic₇₁⋊C₄	1	1	yes	characteristic
143	C₂×C₈	C₄⋊Dic₇₁	1	1	yes	characteristic
144	C₂×C₈	D₅₆₈	1	1	yes	characteristic
145	C₂×C₈	#11765854	1	1	yes	characteristic
146	C₂×C₈	#11765854	1	1	yes	characteristic
147	C₂×C₈	Dic₂₈₄	1	1	yes	characteristic
148	C₄⋊C₄	—	284	2	no	—
149	C₄⋊C₄	—	284	2	no	—
150	C₄²	C₂²⋊Dic₇₁	1	1	yes	characteristic
151	C₄²	Dic₇₁⋊C₄	1	1	yes	characteristic
152	C₄²	#11765853	1	1	yes	characteristic
153	C₄²	#11765855	1	1	yes	characteristic
154	C₁₆	—	284	2	no	—
155	C₂³	#14556048	1	1	yes	characteristic
156	C₂×C₄	—	2	2	no	—
157	C₂×C₄	—	2	2	no	—
158	C₂×C₄	—	568	2	no	—
159	C₂×C₄	—	568	2	no	—
160	C₂×C₄	—	568	2	no	—
161	C₂×C₄	C₇₁⋊#95	1	1	yes	characteristic
162	C₂×C₄	#14556044	1	1	yes	characteristic
163	C₂×C₄	#14556046	1	1	yes	characteristic
164	C₂×C₄	#14556049	1	1	yes	characteristic
165	C₂×C₄	#14556050	1	1	yes	characteristic
166	C₂×C₄	#14556051	1	1	yes	characteristic
167	C₂×C₄	#14556052	1	1	yes	characteristic
168	C₂×C₄	#14556053	1	1	yes	characteristic
169	C₂×C₄	#14556059	1	1	yes	characteristic
170	C₂×C₄	#14556061	1	1	yes	characteristic
171	C₈	—	2	1	no	—
172	C₈	—	2	1	no	—
173	C₈	—	8	2	no	—
174	C₈	#14556050	1	1	yes	characteristic
175	C₈	#14556053	1	1	yes	characteristic
176	C₈	#14556060	1	1	yes	characteristic
177	C₈	#14556062	1	1	yes	characteristic
178	C₂²	#17985742	1	1	yes	characteristic
179	C₂²	#17985746	1	1	yes	characteristic
180	C₂²	#17985761	1	1	yes	characteristic
181	C₂²	#17985805	1	2	yes	—
182	C₂²	#17985808	1	2	yes	—
183	C₄	—	2	1	no	—
184	C₄	—	2	1	no	—
185	C₄	—	2	1	no	—
186	C₄	—	2	1	no	—
187	C₄	—	1136	2	no	—
188	C₄	#17985744	1	1	yes	characteristic
189	C₄	#17985747	1	1	yes	characteristic
190	C₄	#17985762	1	1	yes	characteristic
191	C₄	#17985767	1	1	yes	characteristic
192	C₂	#22586224	1	1	yes	characteristic
193	C₂	#22586382	1	2	yes	—
194	C₂	#22586393	1	2	yes	—
195	C₂	#22586437	1	1	yes	characteristic
196	C₂	#22586447	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	#14556048
commutator	C₂×C₈×C₇₁	C₄²
Frattini	C₂×C₄×C₈	C₂×D₇₁
Fitting	C₂×#334×C₇₁	C₂
socle, prime radical	C₂³×C₇₁	#102
cosocle	C₂×C₄×C₈×C₇₁	C₂²
hypercenter	C₂×#334	D₇₁
hypocenter	C₇₁	#7275
LCS	G ▷ C₂×C₈×C₇₁ ▷ C₂×C₄×C₇₁ ▷ C₂×C₇₁ ▷ C₇₁
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ C₂×#334

Complements

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

N	H ≅ G/N	type
9. C₄⋊(C₇₁⋊C₁₆)	C₄	semidirect
13. #334×C₇₁	C₄	semidirect
102. C₇₁	#7275	semidirect
112. #334	Dic₇₁	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
#22588504	2
C₂×#334×C₇₁	2
#7275	71