Subgroups — #31912342

Order 18176.

554 conjugacy classes of subgroups.

The subgroup lattice has 174 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×#14556064	C₂	1	2	yes	maximal
2	#22586434	C₂	1	4	yes	maximal, complemented
3	C₂²×M₃₂×C₇₁	C₂	1	1	yes	maximal, characteristic
4	C₂²×#11765849	—	2	2	no	—
5	C₂×#14556064	—	2	2	no	—
6	C₂×#14556064	—	2	4	no	—
7	C₂×#14556064	C₄	1	8	yes	—
8	C₂×#14556064	C₂²	1	4	yes	—
9	C₂³×C₈×C₇₁	C₂²	1	1	yes	characteristic
10	C₂×M₃₂×C₇₁	—	2	4	no	—
11	C₂×M₃₂×C₇₁	C₄	1	4	yes	—
12	C₂²×C₁₆×C₇₁	C₂²	1	2	yes	—
13	C₂×#11765849	—	2	8	no	—
14	C₂×#11765849	—	2	4	no	—
15	C₂×#11765849	—	4	2	no	—
16	C₂×#11765849	—	4	4	no	—
17	C₂²×C₇₁⋊C₈	—	4	2	no	—
18	#14556064	—	2	8	no	—
19	#14556064	—	2	4	no	—
20	#14556064	C₂×C₄	1	8	yes	—
21	C₂³×C₄×C₇₁	D₄	1	1	yes	characteristic
22	C₂²×C₈×C₇₁	C₂×C₄	1	2	yes	—
23	C₂²×C₈×C₇₁	C₂×C₄	1	4	yes	—
24	C₂²×C₈×C₇₁	D₄	1	1	yes	characteristic
25	C₂²×C₈×C₇₁	D₄	1	4	yes	—
26	C₂²×C₈×C₇₁	Q₈	1	2	yes	—
27	C₂²×C₈×C₇₁	C₂³	1	1	yes	characteristic
28	M₃₂×C₇₁	—	2	8	no	—
29	C₂×C₁₆×C₇₁	—	2	4	no	—
30	C₂×C₁₆×C₇₁	C₂×C₄	1	4	yes	—
31	#11765849	—	2	8	no	—
32	#11765849	—	4	8	no	—
33	#11765849	—	4	4	no	—
34	C₂×C₇₁⋊C₈	—	4	8	no	—
35	C₂×C₇₁⋊C₈	—	4	4	no	—
36	C₂⁴×C₇₁	Q₁₆	1	1	yes	characteristic
37	C₂²×C₄×C₇₁	C₂²⋊C₄	1	4	yes	—
38	C₂²×C₄×C₇₁	C₄⋊C₄	1	2	yes	—
39	C₂²×C₄×C₇₁	D₈	1	2	yes	—
40	C₂²×C₄×C₇₁	SD₁₆	1	4	yes	—
41	C₂²×C₄×C₇₁	Q₁₆	1	1	yes	characteristic
42	C₂²×C₄×C₇₁	C₂×D₄	1	1	yes	characteristic
43	C₂×C₈×C₇₁	C₄²	1	4	yes	—
44	C₂×C₈×C₇₁	C₂²⋊C₄	1	4	yes	—
45	C₂×C₈×C₇₁	C₂²⋊C₄	1	4	yes	—
46	C₂×C₈×C₇₁	C₄⋊C₄	1	8	yes	—
47	C₂×C₈×C₇₁	C₄⋊C₄	1	2	yes	—
48	C₂×C₈×C₇₁	C₂²×C₄	1	2	yes	—
49	C₂×C₈×C₇₁	C₂²×C₄	1	1	yes	characteristic
50	C₂×C₈×C₇₁	C₂×D₄	1	2	yes	—
51	C₂×C₈×C₇₁	C₂×Q₈	1	1	yes	characteristic
52	C₁₆×C₇₁	—	2	4	no	—
53	C₂³×C₇₁	—	4	2	no	—
54	C₂³×C₇₁	#103	1	4	yes	—
55	C₂³×C₇₁	#107	1	2	yes	—
56	C₂³×C₇₁	C₂×Q₁₆	1	1	yes	characteristic
57	C₂×C₄×C₇₁	#95	1	4	yes	—
58	C₂×C₄×C₇₁	#102	1	4	yes	—
59	C₂×C₄×C₇₁	#102	1	4	yes	—
60	C₂×C₄×C₇₁	#103	1	4	yes	—
61	C₂×C₄×C₇₁	#106	1	4	yes	—
62	C₂×C₄×C₇₁	#107	1	2	yes	—
63	C₂×C₄×C₇₁	C₂×C₂²⋊C₄	1	2	yes	—
64	C₂×C₄×C₇₁	C₂×C₄⋊C₄	1	1	yes	characteristic
65	C₂×C₄×C₇₁	C₂×D₈	1	1	yes	characteristic
66	C₂×C₄×C₇₁	C₂×SD₁₆	1	2	yes	—
67	C₈×C₇₁	#95	1	4	yes	—
68	C₈×C₇₁	C₂×C₄²	1	1	yes	characteristic
69	C₈×C₇₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
70	C₈×C₇₁	C₂×C₄⋊C₄	1	2	yes	—
71	C₇₁⋊C₈	—	4	8	no	—
72	C₂²×C₇₁	—	2	4	no	—
73	C₂²×C₇₁	—	2	2	no	—
74	C₂²×C₇₁	—	4	4	no	—
75	C₂²×C₇₁	#340	1	4	yes	—
76	C₂²×C₇₁	C₂×#103	1	2	yes	—
77	C₂²×C₇₁	C₂×#107	1	1	yes	characteristic
78	C₄×C₇₁	#340	1	4	yes	—
79	C₄×C₇₁	C₂×#95	1	1	yes	characteristic
80	C₄×C₇₁	C₂×#102	1	2	yes	—
81	C₄×C₇₁	C₂×#106	1	1	yes	characteristic
82	#12559	—	71	1	no	maximal, Sylow
83	C₂×C₇₁	—	2	4	no	—
84	C₂×C₇₁	—	2	2	no	—
85	C₂×C₇₁	C₂×#340	1	1	yes	characteristic
86	C₂×C₇₁	#2161	1	2	yes	—
87	C₂²×M₃₂	D₇₁	1	1	yes	characteristic
88	C₂²×#108	—	71	2	no	—
89	#1381	—	71	4	no	—
90	C₇₁	#12559	1	1	yes	characteristic, complemented, Sylow
91	C₂²×M₁₆	—	142	2	no	—
92	C₂³×C₈	C₂×D₇₁	1	1	yes	characteristic
93	C₂×M₃₂	—	2	4	no	—
94	C₂×M₃₂	Dic₇₁	1	4	yes	—
95	C₂²×C₁₆	C₂×D₇₁	1	2	yes	—
96	C₂×#108	—	71	8	no	—
97	C₂×#108	—	71	4	no	—
98	C₂×#108	—	142	2	no	—
99	C₂×#108	—	142	4	no	—
100	C₂³×C₄	D₂₈₄	1	1	yes	characteristic
101	C₂×M₁₆	—	284	2	no	—
102	C₂×M₁₆	—	284	4	no	—
103	C₂×M₁₆	—	142	8	no	—
104	C₂×M₁₆	—	142	4	no	—
105	C₂²×C₈	—	284	2	no	—
106	C₂²×C₈	Dic₁₄₂	1	2	yes	—
107	C₂²×C₈	C₄×D₇₁	1	4	yes	—
108	C₂²×C₈	D₂₈₄	1	1	yes	characteristic
109	C₂²×C₈	C₂×Dic₇₁	1	2	yes	—
110	C₂²×C₈	C₂²⋊D₇₁	1	4	yes	—
111	C₂²×C₈	C₂²×D₇₁	1	1	yes	characteristic
112	M₃₂	—	2	8	no	—
113	C₂×C₁₆	—	2	4	no	—
114	C₂×C₁₆	C₂×Dic₇₁	1	4	yes	—
115	#108	—	71	8	no	—
116	#108	—	142	8	no	—
117	#108	—	142	4	no	—
118	C₂⁴	Dic₂₈₄	1	1	yes	characteristic
119	C₂²×C₄	(C₂×D₇₁)⋊C₄	1	4	yes	—
120	C₂²×C₄	C₄⋊Dic₇₁	1	2	yes	—
121	C₂²×C₄	D₅₆₈	1	2	yes	—
122	C₂²×C₄	#11765854	1	4	yes	—
123	C₂²×C₄	Dic₂₈₄	1	1	yes	characteristic
124	C₂²×C₄	C₂×D₂₈₄	1	1	yes	characteristic
125	M₁₆	—	284	8	no	—
126	M₁₆	—	284	4	no	—
127	M₁₆	—	142	8	no	—
128	C₂×C₈	—	284	8	no	—
129	C₂×C₈	—	284	4	no	—
130	C₂×C₈	C₄×Dic₇₁	1	4	yes	—
131	C₂×C₈	(C₂×D₇₁)⋊C₄	1	4	yes	—
132	C₂×C₈	C₂²⋊Dic₇₁	1	4	yes	—
133	C₂×C₈	Dic₇₁⋊C₄	1	8	yes	—
134	C₂×C₈	C₄⋊Dic₇₁	1	2	yes	—
135	C₂×C₈	C₂×C₄×D₇₁	1	2	yes	—
136	C₂×C₈	C₂²×Dic₇₁	1	1	yes	characteristic
137	C₂×C₈	C₂×C₂²⋊D₇₁	1	2	yes	—
138	C₂×C₈	C₂×Dic₁₄₂	1	1	yes	characteristic
139	C₁₆	—	2	4	no	—
140	C₂³	—	4	2	no	—
141	C₂³	#14556053	1	4	yes	—
142	C₂³	#14556062	1	2	yes	—
143	C₂³	C₂×Dic₂₈₄	1	1	yes	characteristic
144	C₂×C₄	C₇₁⋊#95	1	4	yes	—
145	C₂×C₄	#14556050	1	4	yes	—
146	C₂×C₄	#14556050	1	4	yes	—
147	C₂×C₄	#14556053	1	4	yes	—
148	C₂×C₄	#14556060	1	4	yes	—
149	C₂×C₄	#14556062	1	2	yes	—
150	C₂×C₄	C₂×(C₂×D₇₁)⋊C₄	1	2	yes	—
151	C₂×C₄	C₂×C₄⋊Dic₇₁	1	1	yes	characteristic
152	C₂×C₄	C₂×D₅₆₈	1	1	yes	characteristic
153	C₂×C₄	C₂×#11765854	1	2	yes	—
154	C₈	—	284	8	no	—
155	C₈	C₇₁⋊#95	1	4	yes	—
156	C₈	C₂×C₄×Dic₇₁	1	1	yes	characteristic
157	C₈	C₂×C₂²⋊Dic₇₁	1	1	yes	characteristic
158	C₈	C₂×Dic₇₁⋊C₄	1	2	yes	—
159	C₂²	—	2	4	no	—
160	C₂²	—	2	2	no	—
161	C₂²	—	4	4	no	—
162	C₂²	#17985762	1	4	yes	—
163	C₂²	C₂×#14556053	1	2	yes	—
164	C₂²	C₂×#14556062	1	1	yes	characteristic
165	C₄	#17985762	1	4	yes	—
166	C₄	C₂×C₇₁⋊#95	1	1	yes	characteristic
167	C₄	C₂×#14556050	1	2	yes	—
168	C₄	C₂×#14556060	1	1	yes	characteristic
169	C₂	—	2	2	no	—
170	C₂	—	2	4	no	—
171	C₂	C₂×#17985762	1	1	yes	characteristic
172	C₂	#22590135	1	2	yes	—

	subgroup	G/H
center	C₂×C₄	C₂×D₅₆₈
commutator	C₈×C₇₁	C₂×C₄²
Frattini	C₂²×C₈	C₂²×D₇₁
Fitting	C₂²×M₃₂×C₇₁	C₂
socle	C₂²×C₇₁	C₂×#107
prime radical	C₂⁴×C₇₁	Q₁₆
cosocle	C₂²×C₈×C₇₁	C₂³
hypercenter	C₂²×M₃₂	D₇₁
hypocenter	C₇₁	#12559
LCS	G ▷ C₈×C₇₁ ▷ C₄×C₇₁ ▷ C₂×C₇₁ ▷ C₇₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₄ ◁ C₂³×C₈ ◁ C₂²×M₃₂

Complements

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

N	H ≅ G/N	type
2. #22586434	C₂	semidirect
90. C₇₁	#12559	semidirect

Maximal subgroups

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

subgroup	index
C₂²×#14556064	2
#22586434	2
C₂²×M₃₂×C₇₁	2
#12559	71