Subgroups — #31912457

Order 18176.

358 conjugacy classes of subgroups.

The subgroup lattice has 142 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#17985775	C₂	1	1	yes	maximal, characteristic
2	C₂×C₄×C₇₁⋊C₁₆	C₂	1	1	yes	maximal, characteristic
3	#22586418	C₂	1	4	yes	maximal
4	C₇₁×#1726	C₂	1	1	yes	maximal, characteristic
5	C₂²×C₇₁⋊C₁₆	C₂²	1	2	yes	—
6	C₂²×C₇₁⋊C₁₆	C₂²	1	2	yes	—
7	#17985775	C₄	1	4	yes	complemented
8	C₄×C₇₁⋊C₁₆	C₄	1	4	yes	complemented
9	C₂×#97×C₇₁	C₄	1	2	yes	—
10	C₂×#97×C₇₁	C₄	1	1	yes	characteristic
11	C₂×C₄×C₈×C₇₁	C₂²	1	2	yes	—
12	C₂×C₄×C₈×C₇₁	C₂²	1	1	yes	characteristic
13	#376×C₇₁	C₄	1	1	yes	characteristic
14	C₂×C₇₁⋊C₁₆	—	2	4	no	—
15	C₂×C₇₁⋊C₁₆	—	2	4	no	—
16	C₂×C₇₁⋊C₁₆	C₂×C₄	1	4	yes	—
17	C₂×C₇₁⋊C₁₆	C₂×C₄	1	4	yes	—
18	C₂²×C₈×C₇₁	C₂×C₄	1	2	yes	—
19	C₂²×C₈×C₇₁	C₂×C₄	1	1	yes	characteristic
20	C₂²×C₈×C₇₁	C₂³	1	1	yes	characteristic
21	C₂×C₄²×C₇₁	C₂×C₄	1	2	yes	—
22	C₂×C₄²×C₇₁	C₂×C₄	1	1	yes	characteristic
23	#97×C₇₁	—	2	4	no	—
24	#97×C₇₁	C₈	1	4	yes	—
25	C₄×C₈×C₇₁	—	2	4	no	—
26	C₄×C₈×C₇₁	C₂×C₄	1	2	yes	—
27	C₄×C₈×C₇₁	C₂×C₄	1	2	yes	—
28	#95×C₇₁	C₈	1	4	yes	—
29	C₇₁⋊C₁₆	—	2	4	no	—
30	C₇₁⋊C₁₆	—	2	4	no	—
31	C₂²×C₄×C₇₁	C₂×C₈	1	4	yes	—
32	C₂²×C₄×C₇₁	C₂×C₈	1	2	yes	—
33	C₂²×C₄×C₇₁	C₂²×C₄	1	1	yes	characteristic
34	C₂×C₈×C₇₁	—	2	2	no	—
35	C₂×C₈×C₇₁	—	2	2	no	—
36	C₂×C₈×C₇₁	C₄²	1	2	yes	—
37	C₂×C₈×C₇₁	C₂×C₈	1	4	yes	—
38	C₂×C₈×C₇₁	M₁₆	1	4	yes	—
39	C₂×C₈×C₇₁	C₂²×C₄	1	1	yes	characteristic
40	C₂×C₈×C₇₁	C₂²×C₄	1	1	yes	characteristic
41	C₂×C₈×C₇₁	#41	1	4	yes	—
42	C₄²×C₇₁	—	2	2	no	—
43	C₄²×C₇₁	—	2	2	no	—
44	C₄²×C₇₁	C₄²	1	2	yes	—
45	C₄²×C₇₁	C₂×C₈	1	2	yes	—
46	C₂³×C₇₁	C₂²×C₈	1	1	yes	characteristic
47	C₂×C₄×C₇₁	—	2	4	no	—
48	C₂×C₄×C₇₁	—	2	4	no	—
49	C₂×C₄×C₇₁	—	2	8	no	—
50	C₂×C₄×C₇₁	C₄×C₈	1	4	yes	—
51	C₂×C₄×C₇₁	C₂×C₄²	1	1	yes	characteristic
52	C₂×C₄×C₇₁	#117	1	2	yes	—
53	C₂×C₄×C₇₁	C₂²×C₈	1	1	yes	characteristic
54	C₂×C₄×C₇₁	C₂×M₁₆	1	2	yes	—
55	C₈×C₇₁	—	2	4	no	—
56	C₈×C₇₁	#97	1	4	yes	—
57	C₈×C₇₁	#117	1	2	yes	—
58	C₈×C₇₁	#117	1	2	yes	—
59	C₂²×C₇₁	C₂×C₄×C₈	1	1	yes	characteristic
60	C₂²×C₇₁	#431	1	2	yes	—
61	C₂²×C₇₁	#504	1	4	yes	—
62	C₄×C₇₁	—	2	4	no	—
63	C₄×C₇₁	—	4	4	no	—
64	C₄×C₇₁	#376	1	1	yes	characteristic
65	C₄×C₇₁	C₂×#97	1	1	yes	characteristic
66	C₄×C₇₁	#431	1	2	yes	—
67	#12592	—	71	1	no	maximal, Sylow
68	C₂×C₇₁	#1744	1	1	yes	characteristic
69	C₂×C₇₁	#2108	1	2	yes	—
70	C₂×C₇₁	#2164	1	4	yes	—
71	C₂×#346	—	71	1	no	—
72	C₂×C₄×C₁₆	—	71	1	no	—
73	#1726	D₇₁	1	1	yes	characteristic
74	#1374	—	71	4	no	—
75	C₇₁	#12592	1	1	yes	characteristic, complemented, Sylow
76	C₂²×C₁₆	—	71	2	no	—
77	C₂²×C₁₆	—	71	2	no	—
78	C₂×#97	Dic₇₁	1	2	yes	—
79	C₂×#97	Dic₇₁	1	1	yes	characteristic
80	C₂×C₄×C₈	C₂×D₇₁	1	2	yes	—
81	C₂×C₄×C₈	C₂×D₇₁	1	1	yes	characteristic
82	#376	Dic₇₁	1	1	yes	characteristic
83	#346	—	71	4	no	—
84	C₄×C₁₆	—	71	4	no	—
85	C₂²×C₈	C₂×Dic₇₁	1	2	yes	—
86	C₂²×C₈	C₂×Dic₇₁	1	1	yes	characteristic
87	C₂²×C₈	C₂²×D₇₁	1	1	yes	characteristic
88	C₂×C₄²	C₂×Dic₇₁	1	2	yes	—
89	C₂×C₄²	C₂×Dic₇₁	1	1	yes	characteristic
90	C₂×C₁₆	—	71	4	no	—
91	C₂×C₁₆	—	71	4	no	—
92	C₂×C₁₆	—	142	4	no	—
93	C₂×C₁₆	—	142	4	no	—
94	#97	—	2	4	no	—
95	#97	C₇₁⋊C₈	1	4	yes	—
96	C₄×C₈	—	2	4	no	—
97	C₄×C₈	C₄×D₇₁	1	2	yes	—
98	C₄×C₈	C₄×D₇₁	1	2	yes	—
99	#95	C₇₁⋊C₈	1	4	yes	—
100	C₂²×C₄	C₂×C₇₁⋊C₈	1	4	yes	—
101	C₂²×C₄	C₂×C₇₁⋊C₈	1	2	yes	—
102	C₂²×C₄	C₂²×Dic₇₁	1	1	yes	characteristic
103	C₂×C₈	—	2	2	no	—
104	C₂×C₈	—	2	2	no	—
105	C₂×C₈	C₄×Dic₇₁	1	2	yes	—
106	C₂×C₈	C₂×C₇₁⋊C₈	1	4	yes	—
107	C₂×C₈	#11765849	1	4	yes	—
108	C₂×C₈	C₂×C₄×D₇₁	1	1	yes	characteristic
109	C₂×C₈	C₂×C₄×D₇₁	1	1	yes	characteristic
110	C₂×C₈	#11765865	1	4	yes	—
111	C₄²	—	2	2	no	—
112	C₄²	—	2	2	no	—
113	C₄²	C₄×Dic₇₁	1	2	yes	—
114	C₄²	C₂×C₇₁⋊C₈	1	2	yes	—
115	C₁₆	—	142	4	no	—
116	C₁₆	—	142	4	no	—
117	C₂³	C₂²×C₇₁⋊C₈	1	1	yes	characteristic
118	C₂×C₄	—	2	4	no	—
119	C₂×C₄	—	2	4	no	—
120	C₂×C₄	—	2	8	no	—
121	C₂×C₄	C₄×C₇₁⋊C₈	1	4	yes	—
122	C₂×C₄	C₂×C₄×Dic₇₁	1	1	yes	characteristic
123	C₂×C₄	#14556087	1	2	yes	—
124	C₂×C₄	C₂²×C₇₁⋊C₈	1	1	yes	characteristic
125	C₂×C₄	C₂×#11765849	1	2	yes	—
126	C₈	—	2	4	no	—
127	C₈	#14556039	1	4	yes	—
128	C₈	#14556085	1	2	yes	—
129	C₈	#14556085	1	2	yes	—
130	C₂²	C₂×C₄×C₇₁⋊C₈	1	1	yes	characteristic
131	C₂²	#17986052	1	2	yes	—
132	C₂²	#17986416	1	4	yes	—
133	C₄	—	2	4	no	—
134	C₄	—	4	4	no	—
135	C₄	#17985845	1	1	yes	characteristic
136	C₄	C₂×#14556039	1	1	yes	characteristic
137	C₄	#17986052	1	2	yes	—
138	C₂	#22588086	1	1	yes	characteristic
139	C₂	#22589937	1	2	yes	—
140	C₂	#22590147	1	4	yes	—

	subgroup	G/H
center, Frattini	C₂²×C₈	C₂²×D₇₁
commutator	C₂²×C₇₁	C₂×C₄×C₈
Fitting	C₇₁×#1726	C₂
socle, prime radical	C₂³×C₇₁	C₂²×C₈
cosocle	C₂²×C₈×C₇₁	C₂³
hypercenter	#1726	D₇₁
hypocenter	C₇₁	#12592
LCS	G ▷ C₂²×C₇₁ ▷ C₇₁
UCS	C₁ ◁ C₂²×C₈ ◁ #1726

Complements

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

N	H ≅ G/N	type
7. #17985775	C₄	semidirect
8. C₄×C₇₁⋊C₁₆	C₄	semidirect
75. C₇₁	#12592	semidirect

Maximal subgroups

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

subgroup	index
C₂×#17985775	2
C₂×C₄×C₇₁⋊C₁₆	2
#22586418	2
C₇₁×#1726	2
#12592	71