Subgroups — #31884656

Order 18176.

226 conjugacy classes of subgroups.

The subgroup lattice has 192 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#22588512	C₂	1	1	yes	maximal, characteristic
2	C₂×#17985750	C₂	1	1	yes	maximal, characteristic
3	C₇₁×#2150	C₂	1	1	yes	maximal, characteristic
4	C₂×#14556064	—	2	1	no	—
5	C₂×#14556064	C₄	1	2	yes	—
6	#17986023	—	2	1	no	—
7	C₂×C₄⋊(C₇₁⋊C₈)	—	2	1	no	—
8	#17985750	—	2	1	no	—
9	#17985750	C₄	1	2	yes	—
10	C₂×M₃₂×C₇₁	—	2	1	no	—
11	C₂×C₄×C₈×C₇₁	C₂²	1	1	yes	characteristic
12	C₄⋊C₁₆×C₇₁	—	2	1	no	—
13	C₄⋊C₁₆×C₇₁	C₄	1	2	yes	—
14	C₂×#11765849	—	2	2	no	—
15	C₂²×C₇₁⋊C₈	—	4	1	no	—
16	#14556064	—	4	2	no	—
17	#14556064	—	4	1	no	—
18	#14556064	—	4	1	no	—
19	C₄⋊(C₇₁⋊C₈)	—	2	2	no	—
20	C₄⋊(C₇₁⋊C₈)	—	4	1	no	—
21	C₄⋊(C₇₁⋊C₈)	—	4	1	no	—
22	C₄⋊(C₇₁⋊C₈)	—	4	1	no	—
23	C₂²×C₈×C₇₁	C₂×C₄	1	1	yes	characteristic
24	C₂²×C₈×C₇₁	D₄	1	1	yes	characteristic
25	C₂×C₄²×C₇₁	D₄	1	1	yes	characteristic
26	M₃₂×C₇₁	—	4	1	no	—
27	M₃₂×C₇₁	—	4	1	no	—
28	C₂×C₁₆×C₇₁	—	2	2	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	#11765849	—	4	2	no	—
34	#11765849	—	4	2	no	—
35	C₂×C₇₁⋊C₈	—	4	2	no	—
36	C₂×C₇₁⋊C₈	—	4	2	no	—
37	C₂×C₇₁⋊C₈	—	8	2	no	—
38	C₂²×C₄×C₇₁	C₂²⋊C₄	1	1	yes	characteristic
39	C₂²×C₄×C₇₁	SD₁₆	1	1	yes	characteristic
40	C₂²×C₄×C₇₁	Q₁₆	1	1	yes	characteristic
41	C₂×C₈×C₇₁	—	2	1	no	—
42	C₂×C₈×C₇₁	—	2	1	no	—
43	C₂×C₈×C₇₁	C₄²	1	1	yes	characteristic
44	C₂×C₈×C₇₁	C₂²⋊C₄	1	1	yes	characteristic
45	C₂×C₈×C₇₁	C₄⋊C₄	1	1	yes	characteristic
46	C₂×C₈×C₇₁	C₄⋊C₄	1	1	yes	characteristic
47	C₂×C₈×C₇₁	D₈	1	1	yes	characteristic
48	C₂×C₈×C₇₁	SD₁₆	1	1	yes	characteristic
49	C₂×C₈×C₇₁	SD₁₆	1	1	yes	characteristic
50	C₂×C₈×C₇₁	Q₁₆	1	1	yes	characteristic
51	C₄²×C₇₁	C₂²⋊C₄	1	1	yes	characteristic
52	C₄²×C₇₁	C₄⋊C₄	1	1	yes	characteristic
53	C₄²×C₇₁	D₈	1	1	yes	characteristic
54	C₄²×C₇₁	SD₁₆	1	1	yes	characteristic
55	C₁₆×C₇₁	—	4	2	no	—
56	C₂³×C₇₁	#103	1	1	yes	characteristic
57	C₂×C₄×C₇₁	—	2	2	no	—
58	C₂×C₄×C₇₁	—	4	1	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	#102	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	C₈×C₇₁	—	2	1	no	—
70	C₈×C₇₁	—	2	1	no	—
71	C₈×C₇₁	#102	1	1	yes	characteristic
72	C₈×C₇₁	#103	1	1	yes	characteristic
73	C₈×C₇₁	#106	1	1	yes	characteristic
74	C₈×C₇₁	#107	1	1	yes	characteristic
75	C₇₁⋊C₈	—	8	2	no	—
76	C₇₁⋊C₈	—	8	2	no	—
77	C₂²×C₇₁	—	2	1	no	—
78	C₂²×C₇₁	—	2	1	no	—
79	C₂²×C₇₁	#332	1	1	yes	characteristic
80	C₂²×C₇₁	#333	1	1	yes	characteristic
81	C₂²×C₇₁	#340	1	1	yes	characteristic
82	C₄×C₇₁	—	2	1	no	—
83	C₄×C₇₁	—	2	1	no	—
84	C₄×C₇₁	—	2	1	no	—
85	C₄×C₇₁	#331	1	1	yes	characteristic
86	C₄×C₇₁	#332	1	1	yes	characteristic
87	C₄×C₇₁	#340	1	1	yes	characteristic
88	C₄×C₇₁	#343	1	1	yes	characteristic
89	C₄×C₇₁	#365	1	2	yes	—
90	#7280	—	71	1	no	maximal, Sylow
91	C₂×C₇₁	—	2	1	no	—
92	C₂×C₇₁	—	2	1	no	—
93	C₂×C₇₁	#1296	1	1	yes	characteristic
94	C₂×C₇₁	#1383	1	1	yes	characteristic
95	C₂×C₇₁	#1388	1	1	yes	characteristic
96	#2150	D₇₁	1	1	yes	characteristic
97	#1848	—	71	1	no	—
98	C₂×#334	—	71	1	no	—
99	C₇₁	#7280	1	1	yes	characteristic, complemented, Sylow
100	C₂×M₃₂	—	2	1	no	—
101	C₂×#108	—	71	2	no	—
102	C₂×#108	—	142	1	no	—
103	#423	—	142	1	no	—
104	C₂×C₄⋊C₈	—	142	1	no	—
105	C₂×C₄×C₈	C₂×D₇₁	1	1	yes	characteristic
106	C₄⋊C₁₆	—	2	1	no	—
107	C₄⋊C₁₆	Dic₇₁	1	2	yes	—
108	#334	—	71	2	no	—
109	#334	—	142	1	no	—
110	C₂×M₁₆	—	142	2	no	—
111	C₂²×C₈	—	284	1	no	—
112	C₂²×C₈	C₄×D₇₁	1	1	yes	characteristic
113	C₂²×C₈	C₂²⋊D₇₁	1	1	yes	characteristic
114	C₂×C₄²	D₂₈₄	1	1	yes	characteristic
115	M₃₂	—	4	1	no	—
116	M₃₂	—	4	1	no	—
117	C₂×C₁₆	—	2	2	no	—
118	#108	—	284	2	no	—
119	#108	—	284	1	no	—
120	#108	—	284	1	no	—
121	C₄⋊C₈	—	284	1	no	—
122	C₄⋊C₈	—	284	1	no	—
123	C₄⋊C₈	—	284	1	no	—
124	C₄⋊C₈	—	142	2	no	—
125	C₄×C₈	Dic₁₄₂	1	1	yes	characteristic
126	C₄×C₈	C₄×D₇₁	1	1	yes	characteristic
127	C₄×C₈	C₂×Dic₇₁	1	1	yes	characteristic
128	C₄×C₈	C₂²⋊D₇₁	1	1	yes	characteristic
129	C₂²×C₄	(C₂×D₇₁)⋊C₄	1	1	yes	characteristic
130	C₂²×C₄	#11765854	1	1	yes	characteristic
131	C₂²×C₄	Dic₂₈₄	1	1	yes	characteristic
132	M₁₆	—	284	2	no	—
133	M₁₆	—	284	2	no	—
134	C₂×C₈	—	2	1	no	—
135	C₂×C₈	—	2	1	no	—
136	C₂×C₈	—	284	2	no	—
137	C₂×C₈	—	284	2	no	—
138	C₂×C₈	—	568	2	no	—
139	C₂×C₈	C₄×Dic₇₁	1	1	yes	characteristic
140	C₂×C₈	C₂²⋊Dic₇₁	1	1	yes	characteristic
141	C₂×C₈	Dic₇₁⋊C₄	1	1	yes	characteristic
142	C₂×C₈	Dic₇₁⋊C₄	1	1	yes	characteristic
143	C₂×C₈	#11765850	1	1	yes	characteristic
144	C₂×C₈	#11765852	1	1	yes	characteristic
145	C₂×C₈	#11765853	1	1	yes	characteristic
146	C₂×C₈	#11765855	1	1	yes	characteristic
147	C₄²	(C₂×D₇₁)⋊C₄	1	1	yes	characteristic
148	C₄²	C₄⋊Dic₇₁	1	1	yes	characteristic
149	C₄²	D₅₆₈	1	1	yes	characteristic
150	C₄²	#11765854	1	1	yes	characteristic
151	C₁₆	—	4	2	no	—
152	C₂³	#14556053	1	1	yes	characteristic
153	C₂×C₄	—	2	2	no	—
154	C₂×C₄	—	4	1	no	—
155	C₂×C₄	C₇₁⋊#95	1	1	yes	characteristic
156	C₂×C₄	#14556044	1	1	yes	characteristic
157	C₂×C₄	#14556046	1	1	yes	characteristic
158	C₂×C₄	#14556048	1	1	yes	characteristic
159	C₂×C₄	#14556050	1	1	yes	characteristic
160	C₂×C₄	#14556050	1	1	yes	characteristic
161	C₂×C₄	#14556051	1	1	yes	characteristic
162	C₂×C₄	#14556053	1	1	yes	characteristic
163	C₂×C₄	#14556060	1	1	yes	characteristic
164	C₂×C₄	#14556062	1	1	yes	characteristic
165	C₈	—	2	1	no	—
166	C₈	—	2	1	no	—
167	C₈	—	568	2	no	—
168	C₈	—	568	2	no	—
169	C₈	#14556049	1	1	yes	characteristic
170	C₈	#14556052	1	1	yes	characteristic
171	C₈	#14556059	1	1	yes	characteristic
172	C₈	#14556061	1	1	yes	characteristic
173	C₂²	—	2	1	no	—
174	C₂²	—	2	1	no	—
175	C₂²	#17985746	1	1	yes	characteristic
176	C₂²	#17985747	1	1	yes	characteristic
177	C₂²	#17985762	1	1	yes	characteristic
178	C₄	—	2	1	no	—
179	C₄	—	2	1	no	—
180	C₄	—	2	1	no	—
181	C₄	#17985742	1	1	yes	characteristic
182	C₄	#17985744	1	1	yes	characteristic
183	C₄	#17985761	1	1	yes	characteristic
184	C₄	#17985767	1	1	yes	characteristic
185	C₄	#17985822	1	2	yes	—
186	C₂	—	2	1	no	—
187	C₂	—	2	1	no	—
188	C₂	#22586224	1	1	yes	characteristic
189	C₂	#22586439	1	1	yes	characteristic
190	C₂	#22586452	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	#17985762
commutator	C₂×C₈×C₇₁	C₄²
Frattini	C₂×C₄×C₈	C₂×D₇₁
Fitting	C₇₁×#2150	C₂
socle	C₂²×C₇₁	#340
prime radical	C₂³×C₇₁	#103
cosocle	C₂×C₄×C₈×C₇₁	C₂²
hypercenter	#2150	D₇₁
hypocenter	C₇₁	#7280
LCS	G ▷ C₂×C₈×C₇₁ ▷ C₂×C₄×C₇₁ ▷ C₂×C₇₁ ▷ C₇₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ #2150

Complements

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

N	H ≅ G/N	type
99. C₇₁	#7280	semidirect

Maximal subgroups

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

subgroup	index
#22588512	2
C₂×#17985750	2
C₇₁×#2150	2
#7280	71