Subgroups — #31909026

Order 18176.

242 conjugacy classes of subgroups.

The subgroup lattice has 170 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#22590096	C₂	1	1	yes	maximal, characteristic
2	C₂×C₄⋊(C₇₁⋊C₁₆)	C₂	1	1	yes	maximal, characteristic
3	#22586410	C₂	1	4	yes	maximal, complemented
4	C₇₁×#1450	C₂	1	1	yes	maximal, characteristic
5	C₂×#14556068	—	2	1	no	—
6	C₂²×C₇₁⋊C₁₆	—	2	1	no	—
7	C₄⋊(C₇₁⋊C₁₆)	—	2	1	no	—
8	C₄⋊(C₇₁⋊C₁₆)	—	2	1	no	—
9	C₄⋊(C₇₁⋊C₁₆)	C₂²	1	2	yes	—
10	C₄⋊(C₇₁⋊C₁₆)	C₂²	1	2	yes	—
11	#431×C₇₁	C₄	1	1	yes	characteristic
12	#431×C₇₁	C₄	1	1	yes	characteristic
13	C₂×C₄×C₈×C₇₁	C₂²	1	1	yes	characteristic
14	#322×C₇₁	C₄	1	2	yes	—
15	C₈²×C₇₁	C₂²	1	2	yes	—
16	#14556068	—	4	2	no	—
17	C₂×C₇₁⋊C₁₆	—	2	2	no	—
18	C₂×C₇₁⋊C₁₆	—	2	2	no	—
19	C₂×C₇₁⋊C₁₆	—	4	2	no	—
20	C₂²×C₈×C₇₁	D₄	1	1	yes	characteristic
21	C₂²×C₈×C₇₁	Q₈	1	1	yes	characteristic
22	C₂×C₄²×C₇₁	C₂×C₄	1	1	yes	characteristic
23	C₄⋊C₈×C₇₁	—	2	1	no	—
24	C₄⋊C₈×C₇₁	C₈	1	2	yes	—
25	C₂²⋊C₈×C₇₁	—	2	1	no	—
26	C₂²⋊C₈×C₇₁	C₈	1	2	yes	—
27	C₄×C₈×C₇₁	C₂×C₄	1	2	yes	—
28	C₄×C₈×C₇₁	C₂×C₄	1	1	yes	characteristic
29	C₄×C₈×C₇₁	C₂×C₄	1	2	yes	—
30	C₄×C₈×C₇₁	D₄	1	1	yes	characteristic
31	C₄×C₈×C₇₁	Q₈	1	1	yes	characteristic
32	C₄×C₈×C₇₁	C₂³	1	1	yes	characteristic
33	C₇₁⋊C₁₆	—	4	2	no	—
34	C₇₁⋊C₁₆	—	4	2	no	—
35	C₂²×C₄×C₇₁	C₄⋊C₄	1	1	yes	characteristic
36	C₂²×C₄×C₇₁	C₂×C₈	1	1	yes	characteristic
37	C₂²×C₄×C₇₁	M₁₆	1	1	yes	characteristic
38	C₂×C₈×C₇₁	—	2	2	no	—
39	C₂×C₈×C₇₁	—	2	2	no	—
40	C₂×C₈×C₇₁	C₄⋊C₄	1	1	yes	characteristic
41	C₂×C₈×C₇₁	C₄⋊C₄	1	1	yes	characteristic
42	C₂×C₈×C₇₁	C₂×C₈	1	4	yes	—
43	C₂×C₈×C₇₁	D₈	1	2	yes	—
44	C₂×C₈×C₇₁	Q₁₆	1	2	yes	—
45	C₂×C₈×C₇₁	C₂×D₄	1	1	yes	characteristic
46	C₂×C₈×C₇₁	C₂×Q₈	1	1	yes	characteristic
47	C₄²×C₇₁	C₄⋊C₄	1	1	yes	characteristic
48	C₄²×C₇₁	C₂×C₈	1	1	yes	characteristic
49	C₄²×C₇₁	M₁₆	1	1	yes	characteristic
50	C₄²×C₇₁	C₂²×C₄	1	1	yes	characteristic
51	C₂³×C₇₁	C₄⋊C₈	1	1	yes	characteristic
52	C₂×C₄×C₇₁	—	2	2	no	—
53	C₂×C₄×C₇₁	—	4	1	no	—
54	C₂×C₄×C₇₁	C₄⋊C₈	1	1	yes	characteristic
55	C₂×C₄×C₇₁	C₄⋊C₈	1	1	yes	characteristic
56	C₂×C₄×C₇₁	C₄⋊C₈	1	1	yes	characteristic
57	C₂×C₄×C₇₁	#107	1	2	yes	—
58	C₂×C₄×C₇₁	#108	1	2	yes	—
59	C₂×C₄×C₇₁	C₂×C₄⋊C₄	1	1	yes	characteristic
60	C₂×C₄×C₇₁	C₂²×C₈	1	1	yes	characteristic
61	C₂×C₄×C₇₁	C₂×M₁₆	1	1	yes	characteristic
62	C₈×C₇₁	—	2	1	no	—
63	C₈×C₇₁	—	2	1	no	—
64	C₈×C₇₁	—	2	4	no	—
65	C₈×C₇₁	—	4	2	no	—
66	C₈×C₇₁	#107	1	2	yes	—
67	C₈×C₇₁	C₂×D₈	1	1	yes	characteristic
68	C₈×C₇₁	C₂×Q₁₆	1	1	yes	characteristic
69	C₂²×C₇₁	—	2	1	no	—
70	C₂²×C₇₁	—	2	1	no	—
71	C₂²×C₇₁	#335	1	2	yes	—
72	C₂²×C₇₁	C₂×C₄⋊C₈	1	1	yes	characteristic
73	C₄×C₇₁	—	2	1	no	—
74	C₄×C₇₁	—	2	1	no	—
75	C₄×C₇₁	—	2	1	no	—
76	C₄×C₇₁	#335	1	2	yes	—
77	C₄×C₇₁	C₂×#107	1	1	yes	characteristic
78	C₄×C₇₁	C₂×#108	1	1	yes	characteristic
79	C₄×C₇₁	#504	1	2	yes	—
80	#12005	—	71	1	no	maximal, Sylow
81	C₂×C₇₁	—	2	2	no	—
82	C₂×C₇₁	C₂×#335	1	1	yes	characteristic
83	C₂×C₇₁	#2151	1	1	yes	characteristic
84	C₂×C₇₁	#2153	1	1	yes	characteristic
85	#2150	—	71	1	no	—
86	C₂×C₄⋊C₁₆	—	71	1	no	—
87	#1450	D₇₁	1	1	yes	characteristic
88	#1370	—	71	4	no	—
89	C₇₁	#12005	1	1	yes	characteristic, complemented, Sylow
90	C₂×M₃₂	—	142	1	no	—
91	C₂²×C₁₆	—	142	1	no	—
92	#431	Dic₇₁	1	1	yes	characteristic
93	#431	Dic₇₁	1	1	yes	characteristic
94	C₂×C₄×C₈	C₂×D₇₁	1	1	yes	characteristic
95	C₄⋊C₁₆	—	71	2	no	—
96	C₄⋊C₁₆	—	71	2	no	—
97	C₄⋊C₁₆	—	142	1	no	—
98	C₄⋊C₁₆	—	142	1	no	—
99	#322	Dic₇₁	1	2	yes	—
100	C₈²	C₂×D₇₁	1	2	yes	—
101	C₂²×C₈	Dic₁₄₂	1	1	yes	characteristic
102	C₂²×C₈	D₂₈₄	1	1	yes	characteristic
103	C₂×C₄²	C₂×Dic₇₁	1	1	yes	characteristic
104	M₃₂	—	284	2	no	—
105	C₂×C₁₆	—	284	2	no	—
106	C₂×C₁₆	—	142	2	no	—
107	C₂×C₁₆	—	142	2	no	—
108	C₄⋊C₈	—	2	1	no	—
109	C₄⋊C₈	C₇₁⋊C₈	1	2	yes	—
110	C₂²⋊C₈	—	2	1	no	—
111	C₂²⋊C₈	C₇₁⋊C₈	1	2	yes	—
112	C₄×C₈	Dic₁₄₂	1	1	yes	characteristic
113	C₄×C₈	D₂₈₄	1	1	yes	characteristic
114	C₄×C₈	C₂×Dic₇₁	1	2	yes	—
115	C₄×C₈	C₂×Dic₇₁	1	1	yes	characteristic
116	C₄×C₈	C₂×Dic₇₁	1	2	yes	—
117	C₄×C₈	C₂²×D₇₁	1	1	yes	characteristic
118	C₂²×C₄	C₄⋊Dic₇₁	1	1	yes	characteristic
119	C₂²×C₄	C₂×C₇₁⋊C₈	1	1	yes	characteristic
120	C₂²×C₄	#11765849	1	1	yes	characteristic
121	C₂×C₈	—	2	2	no	—
122	C₂×C₈	—	2	2	no	—
123	C₂×C₈	C₄⋊Dic₇₁	1	1	yes	characteristic
124	C₂×C₈	C₄⋊Dic₇₁	1	1	yes	characteristic
125	C₂×C₈	C₂×C₇₁⋊C₈	1	4	yes	—
126	C₂×C₈	D₅₆₈	1	2	yes	—
127	C₂×C₈	Dic₂₈₄	1	2	yes	—
128	C₂×C₈	C₂×D₂₈₄	1	1	yes	characteristic
129	C₂×C₈	C₂×Dic₁₄₂	1	1	yes	characteristic
130	C₄²	C₄⋊Dic₇₁	1	1	yes	characteristic
131	C₄²	C₂×C₇₁⋊C₈	1	1	yes	characteristic
132	C₄²	#11765849	1	1	yes	characteristic
133	C₄²	C₂²×Dic₇₁	1	1	yes	characteristic
134	C₁₆	—	284	2	no	—
135	C₁₆	—	284	2	no	—
136	C₂³	C₄⋊(C₇₁⋊C₈)	1	1	yes	characteristic
137	C₂×C₄	—	2	2	no	—
138	C₂×C₄	—	4	1	no	—
139	C₂×C₄	C₄⋊(C₇₁⋊C₈)	1	1	yes	characteristic
140	C₂×C₄	C₄⋊(C₇₁⋊C₈)	1	1	yes	characteristic
141	C₂×C₄	C₄⋊(C₇₁⋊C₈)	1	1	yes	characteristic
142	C₂×C₄	#14556062	1	2	yes	—
143	C₂×C₄	#14556064	1	2	yes	—
144	C₂×C₄	C₂×C₄⋊Dic₇₁	1	1	yes	characteristic
145	C₂×C₄	C₂²×C₇₁⋊C₈	1	1	yes	characteristic
146	C₂×C₄	C₂×#11765849	1	1	yes	characteristic
147	C₈	—	2	4	no	—
148	C₈	—	2	1	no	—
149	C₈	—	2	1	no	—
150	C₈	—	4	2	no	—
151	C₈	#14556062	1	2	yes	—
152	C₈	C₂×D₅₆₈	1	1	yes	characteristic
153	C₈	C₂×Dic₂₈₄	1	1	yes	characteristic
154	C₂²	—	2	1	no	—
155	C₂²	—	2	1	no	—
156	C₂²	#17985752	1	2	yes	—
157	C₂²	C₂×C₄⋊(C₇₁⋊C₈)	1	1	yes	characteristic
158	C₄	—	2	1	no	—
159	C₄	—	2	1	no	—
160	C₄	—	2	1	no	—
161	C₄	#17985752	1	2	yes	—
162	C₄	C₂×#14556062	1	1	yes	characteristic
163	C₄	C₂×#14556064	1	1	yes	characteristic
164	C₄	#17986416	1	2	yes	—
165	C₂	—	2	2	no	—
166	C₂	C₂×#17985752	1	1	yes	characteristic
167	C₂	#22590099	1	1	yes	characteristic
168	C₂	#22590106	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₈	C₂×D₂₈₄
commutator	C₂×C₄×C₇₁	C₂²×C₈
Frattini	C₄×C₈	C₂²×D₇₁
Fitting	C₇₁×#1450	C₂
socle	C₂²×C₇₁	C₂×C₄⋊C₈
prime radical	C₂³×C₇₁	C₄⋊C₈
cosocle	C₄×C₈×C₇₁	C₂³
hypercenter	#1450	D₇₁
hypocenter	C₇₁	#12005
LCS	G ▷ C₂×C₄×C₇₁ ▷ C₂×C₇₁ ▷ C₇₁
UCS	C₁ ◁ C₂×C₈ ◁ C₂×C₄×C₈ ◁ #1450

Complements

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

N	H ≅ G/N	type
3. #22586410	C₂	semidirect
89. C₇₁	#12005	semidirect

Maximal subgroups

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

subgroup	index
#22590096	2
C₂×C₄⋊(C₇₁⋊C₁₆)	2
#22586410	2
C₇₁×#1450	2
#12005	71