Subgroups — #31963043

Order 18176.

890 conjugacy classes of subgroups.

The subgroup lattice has 156 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#41×C₇₁⋊C₈	C₂	1	2	yes	maximal
2	#22586623	C₂	1	3	yes	maximal, complemented
3	M₁₆×C₇₁⋊C₈	C₂	1	6	yes	maximal, complemented
4	C₂×#17985720	C₂	1	3	yes	maximal, complemented
5	C₄×#131×C₇₁	C₂	1	1	yes	maximal, characteristic
6	Q₈×C₇₁⋊C₈	C₄	1	4	yes	—
7	D₄×C₇₁⋊C₈	C₄	1	12	yes	—
8	#17986052	C₂²	1	6	yes	—
9	C₂×C₄×C₇₁⋊C₈	C₂²	1	6	yes	—
10	#17985720	C₂²	1	1	yes	characteristic
11	#17985720	C₂²	1	6	yes	—
12	#17985720	C₂²	1	3	yes	—
13	C₈×C₇₁⋊C₈	C₂²	1	6	yes	—
14	C₂×#131×C₇₁	C₄	1	2	yes	—
15	C₄×#41×C₇₁	C₂²	1	1	yes	characteristic
16	#405×C₇₁	C₄	1	6	yes	—
17	C₄×M₁₆×C₇₁	C₂²	1	3	yes	—
18	C₂×C₄×C₈×C₇₁	C₂²	1	3	yes	—
19	C₂²×C₇₁⋊C₈	C₂×C₄	1	12	yes	—
20	C₄⋊(C₇₁⋊C₈)	C₂×C₄	1	12	yes	—
21	C₂²⋊(C₇₁⋊C₈)	C₂×C₄	1	12	yes	—
22	C₄×C₇₁⋊C₈	C₂×C₄	1	12	yes	—
23	C₄×C₇₁⋊C₈	C₂³	1	6	yes	—
24	C₄×C₇₁⋊C₈	C₂³	1	2	yes	—
25	C₂×#41×C₇₁	C₂×C₄	1	1	yes	characteristic
26	#131×C₇₁	C₈	1	16	yes	complemented
27	C₂×M₁₆×C₇₁	C₂×C₄	1	6	yes	—
28	C₂²×C₈×C₇₁	C₂×C₄	1	6	yes	—
29	C₄×Q₈×C₇₁	C₂×C₄	1	2	yes	—
30	C₄×D₄×C₇₁	C₂×C₄	1	6	yes	—
31	#117×C₇₁	C₂×C₄	1	3	yes	—
32	C₂×C₄²×C₇₁	C₂³	1	3	yes	—
33	#97×C₇₁	C₂×C₄	1	6	yes	—
34	C₄×C₈×C₇₁	C₂×C₄	1	6	yes	—
35	C₄×C₈×C₇₁	C₂³	1	3	yes	—
36	C₄×C₈×C₇₁	C₂³	1	1	yes	characteristic
37	C₂×C₇₁⋊C₈	—	4	12	no	—
38	C₂×C₇₁⋊C₈	C₂²×C₄	1	4	yes	—
39	C₂×C₇₁⋊C₈	C₂²×C₄	1	12	yes	—
40	#41×C₇₁	C₂×C₈	1	8	yes	—
41	C₂×Q₈×C₇₁	C₄²	1	1	yes	characteristic
42	C₂×D₄×C₇₁	C₄²	1	3	yes	—
43	C₂²×C₄×C₇₁	C₂²×C₄	1	6	yes	—
44	C₂²×C₄×C₇₁	C₂²×C₄	1	3	yes	—
45	M₁₆×C₇₁	C₂×C₈	1	24	yes	complemented
46	C₂×C₈×C₇₁	C₂×C₈	1	24	yes	complemented
47	C₂×C₈×C₇₁	C₂²×C₄	1	2	yes	—
48	C₂×C₈×C₇₁	C₂²×C₄	1	6	yes	—
49	C₄⋊C₄×C₇₁	C₄²	1	6	yes	—
50	C₂²⋊C₄×C₇₁	C₄²	1	6	yes	—
51	C₄²×C₇₁	C₂²×C₄	1	6	yes	—
52	C₄²×C₇₁	C₂²×C₄	1	3	yes	—
53	C₄²×C₇₁	C₂⁴	1	1	yes	characteristic
54	C₂³×C₇₁	C₂×C₄²	1	3	yes	—
55	Q₈×C₇₁	C₄×C₈	1	4	yes	—
56	D₄×C₇₁	C₄×C₈	1	12	yes	—
57	C₂×C₄×C₇₁	—	2	12	no	—
58	C₂×C₄×C₇₁	C₂×C₄²	1	6	yes	—
59	C₂×C₄×C₇₁	C₂×C₄²	1	3	yes	—
60	C₂×C₄×C₇₁	C₂²×C₈	1	12	yes	—
61	C₂×C₄×C₇₁	C₂³×C₄	1	2	yes	—
62	C₂×C₄×C₇₁	C₂³×C₄	1	1	yes	characteristic
63	C₈×C₇₁	C₂²×C₈	1	4	yes	—
64	C₈×C₇₁	C₂²×C₈	1	12	yes	—
65	C₇₁⋊C₈	—	2	4	no	—
66	C₇₁⋊C₈	—	2	12	no	—
67	C₂²×C₇₁	—	2	6	no	—
68	C₂²×C₇₁	C₂×C₄×C₈	1	6	yes	—
69	C₂²×C₇₁	C₂²×C₄²	1	1	yes	characteristic
70	C₄×C₇₁	—	2	6	no	—
71	C₄×C₇₁	C₂×C₄×C₈	1	6	yes	—
72	C₄×C₇₁	C₂³×C₈	1	2	yes	—
73	C₄×C₇₁	#568	1	4	yes	—
74	#17316	—	71	1	no	maximal, Sylow
75	C₂×C₇₁	—	2	6	no	—
76	C₂×C₇₁	C₂²×C₄×C₈	1	1	yes	characteristic
77	C₂×C₇₁	#2875	1	2	yes	—
78	C₈×#41	—	71	2	no	—
79	C₄×#131	D₇₁	1	1	yes	characteristic
80	#1450	—	71	3	no	—
81	C₈×M₁₆	—	71	6	no	—
82	C₂×#322	—	71	3	no	—
83	C₇₁	#17316	1	1	yes	characteristic, complemented, Sylow
84	C₂×#131	Dic₇₁	1	2	yes	—
85	C₄×#41	C₂×D₇₁	1	1	yes	characteristic
86	C₈×Q₈	—	71	4	no	—
87	C₈×D₄	—	71	12	no	—
88	#431	—	71	6	no	—
89	#405	Dic₇₁	1	6	yes	—
90	C₄×M₁₆	C₂×D₇₁	1	3	yes	—
91	C₂×C₄×C₈	—	71	6	no	—
92	C₂×C₄×C₈	C₂×D₇₁	1	3	yes	—
93	#322	—	71	1	no	—
94	#322	—	71	6	no	—
95	#322	—	71	3	no	—
96	C₈²	—	71	6	no	—
97	C₂×#41	C₂×Dic₇₁	1	1	yes	characteristic
98	#131	C₇₁⋊C₈	1	16	yes	complemented
99	C₂×M₁₆	C₂×Dic₇₁	1	6	yes	—
100	C₂²×C₈	—	71	12	no	—
101	C₂²×C₈	C₂×Dic₇₁	1	6	yes	—
102	C₄×Q₈	C₄×D₇₁	1	2	yes	—
103	C₄×D₄	C₄×D₇₁	1	6	yes	—
104	#117	C₂×Dic₇₁	1	3	yes	—
105	C₂×C₄²	C₂²×D₇₁	1	3	yes	—
106	C₄⋊C₈	—	71	12	no	—
107	C₂²⋊C₈	—	71	12	no	—
108	#97	C₂×Dic₇₁	1	6	yes	—
109	C₄×C₈	—	71	12	no	—
110	C₄×C₈	—	71	6	no	—
111	C₄×C₈	—	71	2	no	—
112	C₄×C₈	C₂×Dic₇₁	1	6	yes	—
113	C₄×C₈	C₂²×D₇₁	1	3	yes	—
114	C₄×C₈	C₂²×D₇₁	1	1	yes	characteristic
115	#41	C₂×C₇₁⋊C₈	1	8	yes	—
116	C₂×Q₈	C₄×Dic₇₁	1	1	yes	characteristic
117	C₂×D₄	C₄×Dic₇₁	1	3	yes	—
118	C₂²×C₄	C₂×C₄×D₇₁	1	6	yes	—
119	C₂²×C₄	C₂²×Dic₇₁	1	3	yes	—
120	M₁₆	C₂×C₇₁⋊C₈	1	24	yes	complemented
121	C₂×C₈	—	284	12	no	—
122	C₂×C₈	—	71	4	no	—
123	C₂×C₈	—	71	12	no	—
124	C₂×C₈	C₂×C₇₁⋊C₈	1	24	yes	complemented
125	C₂×C₈	C₂²×Dic₇₁	1	2	yes	—
126	C₂×C₈	C₂²×Dic₇₁	1	6	yes	—
127	C₄⋊C₄	C₄×Dic₇₁	1	6	yes	—
128	C₂²⋊C₄	C₄×Dic₇₁	1	6	yes	—
129	C₄²	C₂×C₄×D₇₁	1	6	yes	—
130	C₄²	C₂²×Dic₇₁	1	3	yes	—
131	C₄²	C₂³×D₇₁	1	1	yes	characteristic
132	C₂³	C₂×C₄×Dic₇₁	1	3	yes	—
133	Q₈	C₄×C₇₁⋊C₈	1	4	yes	—
134	D₄	C₄×C₇₁⋊C₈	1	12	yes	—
135	C₂×C₄	—	2	12	no	—
136	C₂×C₄	C₂×C₄×Dic₇₁	1	6	yes	—
137	C₂×C₄	C₂×C₄×Dic₇₁	1	3	yes	—
138	C₂×C₄	C₂²×C₇₁⋊C₈	1	12	yes	—
139	C₂×C₄	C₂²×C₄×D₇₁	1	2	yes	—
140	C₂×C₄	C₂³×Dic₇₁	1	1	yes	characteristic
141	C₈	—	142	4	no	—
142	C₈	—	142	12	no	—
143	C₈	C₂²×C₇₁⋊C₈	1	4	yes	—
144	C₈	C₂²×C₇₁⋊C₈	1	12	yes	—
145	C₂²	—	2	6	no	—
146	C₂²	C₂×C₄×C₇₁⋊C₈	1	6	yes	—
147	C₂²	C₂²×C₄×Dic₇₁	1	1	yes	characteristic
148	C₄	—	2	6	no	—
149	C₄	C₂×C₄×C₇₁⋊C₈	1	6	yes	—
150	C₄	C₂³×C₇₁⋊C₈	1	2	yes	—
151	C₄	#17986764	1	4	yes	—
152	C₂	—	2	6	no	—
153	C₂	C₂²×C₄×C₇₁⋊C₈	1	1	yes	characteristic
154	C₂	#22596766	1	2	yes	—

	subgroup	G/H
center, Frattini	C₄²	C₂³×D₇₁
commutator	C₂×C₇₁	C₂²×C₄×C₈
Fitting	C₄×#131×C₇₁	C₂
socle	C₂²×C₇₁	C₂²×C₄²
prime radical	C₂×#41×C₇₁	C₂×C₄
cosocle	C₄²×C₇₁	C₂⁴
hypercenter	C₄×#131	D₇₁
hypocenter	C₇₁	#17316
LCS	G ▷ C₂×C₇₁ ▷ C₇₁
UCS	C₁ ◁ C₄² ◁ C₄×#131

Complements

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

N	H ≅ G/N	type
2. #22586623	C₂	semidirect
3. M₁₆×C₇₁⋊C₈	C₂	semidirect
4. C₂×#17985720	C₂	semidirect
26. #131×C₇₁	C₈	semidirect
45. M₁₆×C₇₁	C₂×C₈	semidirect
46. C₂×C₈×C₇₁	C₂×C₈	semidirect
83. C₇₁	#17316	semidirect
98. #131	C₇₁⋊C₈	semidirect
120. M₁₆	C₂×C₇₁⋊C₈	semidirect
124. C₂×C₈	C₂×C₇₁⋊C₈	semidirect

Maximal subgroups

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

subgroup	index
#41×C₇₁⋊C₈	2
#22586623	2
M₁₆×C₇₁⋊C₈	2
C₂×#17985720	2
C₄×#131×C₇₁	2
#17316	71