Subgroups — #31886815

Order 18176.

558 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	C₄×#14556039	C₂	1	1	yes	maximal, characteristic
2	C₄²×C₇₁⋊C₈	C₂	1	1	yes	maximal, characteristic
3	C₂×#17985721	C₂	1	2	yes	maximal
4	C₂×#17985720	C₂	1	2	yes	maximal
5	C₇₁×#1744	C₂	1	1	yes	maximal, characteristic
6	C₂×#14556039	C₄	1	4	yes	—
7	C₂×C₄×C₇₁⋊C₈	C₄	1	4	yes	—
8	C₂×C₄×C₇₁⋊C₈	C₂²	1	1	yes	characteristic
9	C₂×C₄×C₇₁⋊C₈	C₂²	1	1	yes	characteristic
10	C₂×C₄×C₇₁⋊C₈	C₂²	1	2	yes	—
11	#17985721	C₄	1	8	yes	complemented
12	#17985720	C₄	1	8	yes	complemented
13	C₂×C₄×C₈×C₇₁	C₂²	1	2	yes	—
14	C₄³×C₇₁	C₂²	1	1	yes	characteristic
15	#336×C₇₁	C₄	1	4	yes	—
16	C₂²×C₇₁⋊C₈	C₂×C₄	1	2	yes	—
17	C₂²×C₇₁⋊C₈	C₂×C₄	1	4	yes	—
18	C₂²×C₇₁⋊C₈	C₂×C₄	1	2	yes	—
19	#14556039	—	2	8	no	—
20	C₄×C₇₁⋊C₈	—	2	8	no	—
21	C₄×C₇₁⋊C₈	C₂×C₄	1	4	yes	—
22	C₄×C₇₁⋊C₈	C₂×C₄	1	4	yes	—
23	C₄×C₇₁⋊C₈	C₂×C₄	1	4	yes	—
24	C₄×C₇₁⋊C₈	C₂×C₄	1	4	yes	—
25	C₂²×C₈×C₇₁	C₂×C₄	1	4	yes	—
26	C₂×C₄²×C₇₁	C₂×C₄	1	2	yes	—
27	C₂×C₄²×C₇₁	C₂×C₄	1	2	yes	—
28	C₂×C₄²×C₇₁	C₂×C₄	1	2	yes	—
29	C₂×C₄²×C₇₁	C₂³	1	1	yes	characteristic
30	C₄×C₈×C₇₁	C₂×C₄	1	4	yes	—
31	C₄×C₈×C₇₁	C₂×C₄	1	4	yes	—
32	C₂×C₇₁⋊C₈	—	2	4	no	—
33	C₂×C₇₁⋊C₈	—	2	8	no	—
34	C₂×C₇₁⋊C₈	C₄²	1	4	yes	—
35	C₂×C₇₁⋊C₈	C₄²	1	8	yes	—
36	C₂×C₇₁⋊C₈	C₄²	1	4	yes	—
37	C₂×C₇₁⋊C₈	M₁₆	1	8	yes	—
38	C₂²×C₄×C₇₁	C₄²	1	4	yes	—
39	C₂²×C₄×C₇₁	C₂²×C₄	1	1	yes	characteristic
40	C₂²×C₄×C₇₁	C₂²×C₄	1	1	yes	characteristic
41	C₂²×C₄×C₇₁	C₂²×C₄	1	1	yes	characteristic
42	C₂×C₈×C₇₁	—	2	8	no	—
43	C₂×C₈×C₇₁	C₄²	1	8	yes	—
44	C₄²×C₇₁	—	2	4	no	—
45	C₄²×C₇₁	—	2	4	no	—
46	C₄²×C₇₁	M₁₆	1	8	yes	—
47	C₄²×C₇₁	C₂²×C₄	1	2	yes	—
48	C₄²×C₇₁	C₂²×C₄	1	2	yes	—
49	C₂³×C₇₁	C₂×C₄²	1	1	yes	characteristic
50	C₂×C₄×C₇₁	—	2	4	no	—
51	C₂×C₄×C₇₁	—	2	4	no	—
52	C₂×C₄×C₇₁	#97	1	8	yes	—
53	C₂×C₄×C₇₁	C₂×C₄²	1	1	yes	characteristic
54	C₂×C₄×C₇₁	C₂×C₄²	1	2	yes	—
55	C₂×C₄×C₇₁	C₂×C₄²	1	1	yes	characteristic
56	C₂×C₄×C₇₁	C₂×C₄²	1	2	yes	—
57	C₂×C₄×C₇₁	C₂×M₁₆	1	4	yes	—
58	C₂×C₄×C₇₁	C₂×M₁₆	1	4	yes	—
59	C₂×C₄×C₇₁	#131	1	4	yes	—
60	C₈×C₇₁	—	2	8	no	—
61	C₇₁⋊C₈	—	2	8	no	—
62	C₇₁⋊C₈	—	2	4	no	—
63	C₇₁⋊C₈	#97	1	8	yes	complemented
64	C₂²×C₇₁	C₄³	1	1	yes	characteristic
65	C₂²×C₇₁	C₂×#97	1	2	yes	—
66	C₂²×C₇₁	C₄×M₁₆	1	2	yes	—
67	C₂²×C₇₁	#405	1	2	yes	—
68	C₄×C₇₁	—	2	8	no	—
69	C₄×C₇₁	C₂×#97	1	2	yes	—
70	C₄×C₇₁	C₄×M₁₆	1	4	yes	—
71	C₄×C₇₁	C₄×M₁₆	1	2	yes	—
72	C₄×C₇₁	#405	1	4	yes	—
73	#7842	—	71	1	no	maximal, Sylow
74	C₂×C₇₁	#1451	1	4	yes	—
75	C₂×C₇₁	C₄×#97	1	1	yes	characteristic
76	C₂×C₇₁	C₄×#97	1	1	yes	characteristic
77	C₂×C₇₁	#1726	1	1	yes	characteristic
78	#1744	D₇₁	1	1	yes	characteristic
79	C₄×#97	—	71	1	no	—
80	C₄²×C₈	—	71	1	no	—
81	C₂×#322	—	71	2	no	—
82	C₂×#322	—	71	2	no	—
83	C₇₁	#7842	1	1	yes	characteristic, complemented, Sylow
84	C₂×#97	—	71	4	no	—
85	C₂×C₄×C₈	—	71	1	no	—
86	C₂×C₄×C₈	—	71	1	no	—
87	C₂×C₄×C₈	—	71	2	no	—
88	C₂×C₄×C₈	—	71	4	no	—
89	C₂×C₄×C₈	C₂×D₇₁	1	2	yes	—
90	C₄³	C₂×D₇₁	1	1	yes	characteristic
91	#336	Dic₇₁	1	4	yes	—
92	#322	—	71	8	no	—
93	#322	—	71	8	no	—
94	C₂²×C₈	—	71	2	no	—
95	C₂²×C₈	—	71	2	no	—
96	C₂²×C₈	—	71	4	no	—
97	C₂²×C₈	C₂×Dic₇₁	1	4	yes	—
98	C₂×C₄²	C₄×D₇₁	1	2	yes	—
99	C₂×C₄²	C₄×D₇₁	1	2	yes	—
100	C₂×C₄²	C₂×Dic₇₁	1	2	yes	—
101	C₂×C₄²	C₂²×D₇₁	1	1	yes	characteristic
102	#97	—	142	8	no	—
103	C₄×C₈	—	71	4	no	—
104	C₄×C₈	—	71	4	no	—
105	C₄×C₈	—	71	4	no	—
106	C₄×C₈	—	71	4	no	—
107	C₄×C₈	—	142	8	no	—
108	C₄×C₈	C₄×D₇₁	1	4	yes	—
109	C₄×C₈	C₄×D₇₁	1	4	yes	—
110	C₂²×C₄	C₄×Dic₇₁	1	4	yes	—
111	C₂²×C₄	C₂×C₄×D₇₁	1	1	yes	characteristic
112	C₂²×C₄	C₂×C₄×D₇₁	1	1	yes	characteristic
113	C₂²×C₄	C₂²×Dic₇₁	1	1	yes	characteristic
114	C₂×C₈	—	2	8	no	—
115	C₂×C₈	—	71	4	no	—
116	C₂×C₈	—	71	8	no	—
117	C₂×C₈	—	71	8	no	—
118	C₂×C₈	—	71	4	no	—
119	C₂×C₈	—	142	4	no	—
120	C₂×C₈	—	142	8	no	—
121	C₂×C₈	C₄×Dic₇₁	1	8	yes	—
122	C₄²	—	2	4	no	—
123	C₄²	—	2	4	no	—
124	C₄²	#11765848	1	8	yes	—
125	C₄²	C₂×C₄×D₇₁	1	2	yes	—
126	C₄²	C₂×C₄×D₇₁	1	2	yes	—
127	C₂³	C₂×C₄×Dic₇₁	1	1	yes	characteristic
128	C₂×C₄	—	2	4	no	—
129	C₂×C₄	—	2	4	no	—
130	C₂×C₄	#14556038	1	8	yes	—
131	C₂×C₄	C₄²×D₇₁	1	1	yes	characteristic
132	C₂×C₄	C₄²×D₇₁	1	2	yes	—
133	C₂×C₄	C₄²×D₇₁	1	1	yes	characteristic
134	C₂×C₄	C₂×C₄×Dic₇₁	1	2	yes	—
135	C₂×C₄	M₁₆×D₇₁	1	4	yes	—
136	C₂×C₄	C₂×#11765848	1	4	yes	—
137	C₂×C₄	#14556135	1	4	yes	—
138	C₈	—	2	8	no	—
139	C₈	—	71	8	no	—
140	C₈	—	142	8	no	—
141	C₈	—	142	4	no	—
142	C₂²	C₄²×Dic₇₁	1	1	yes	characteristic
143	C₂²	C₂×#14556038	1	2	yes	—
144	C₂²	M₁₆×Dic₇₁	1	2	yes	—
145	C₂²	#17985955	1	2	yes	—
146	C₄	—	2	8	no	—
147	C₄	#97×D₇₁	1	2	yes	—
148	C₄	C₄×#11765848	1	4	yes	—
149	C₄	#17985951	1	2	yes	—
150	C₄	#17985955	1	4	yes	—
151	C₂	#22586630	1	4	yes	—
152	C₂	#97×Dic₇₁	1	1	yes	characteristic
153	C₂	C₄×#14556038	1	1	yes	characteristic
154	C₂	#22588027	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂×C₄²	C₂²×D₇₁
commutator	C₂²×C₇₁	C₄³
Fitting	C₇₁×#1744	C₂
socle, prime radical	C₂³×C₇₁	C₂×C₄²
cosocle	C₂×C₄²×C₇₁	C₂³
hypercenter	#1744	D₇₁
hypocenter	C₇₁	#7842
LCS	G ▷ C₂²×C₇₁ ▷ C₇₁
UCS	C₁ ◁ C₂×C₄² ◁ #1744

Complements

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

N	H ≅ G/N	type
11. #17985721	C₄	semidirect
12. #17985720	C₄	semidirect
63. C₇₁⋊C₈	#97	semidirect
83. C₇₁	#7842	semidirect

Maximal subgroups

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

subgroup	index
C₄×#14556039	2
C₄²×C₇₁⋊C₈	2
C₂×#17985721	2
C₂×#17985720	2
C₇₁×#1744	2
#7842	71