Finite Group Databasebeta

Subgroups — #31986781

Order 18176.

736 conjugacy classes of subgroups; only the 179 normal subgroups are shown.

normal subgroups only

This group has 736 conjugacy classes of subgroups — too many to draw. Only its 179 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#22599085	C₂	1	3	yes	maximal, complemented
2	#22598715	C₂	1	3	yes	maximal
3	#22588000	C₂	1	6	yes	maximal, complemented
4	#22587952	C₂	1	2	yes	maximal, complemented
5	C₇₁×#1455	C₂	1	1	yes	maximal, characteristic, complemented
6	#17986556	C₂²	1	3	yes	—
7	C₂×#14556125	C₂²	1	1	yes	characteristic
8	#17986397	C₂²	1	3	yes	complemented
9	#17986379	C₂²	1	3	yes	complemented
10	#17986364	—	2	6	no	—
11	#17986350	C₂²	1	6	yes	complemented
12	#17986346	C₂²	1	6	yes	—
13	#17986346	C₂²	1	6	yes	—
14	#17986257	—	2	6	no	—
15	#17986107	—	2	6	no	—
16	#17986005	—	2	6	no	—
17	C₄×M₁₆×C₇₁	C₂²	1	3	yes	—
18	#322×C₇₁	C₂²	1	3	yes	complemented
19	C₈²×C₇₁	C₂²	1	1	yes	characteristic, complemented
20	C₂²×D₂₈₄	—	2	3	no	—
21	C₂×Dic₂₈₄	—	2	6	no	—
22	#14556128	C₂³	1	6	yes	—
23	#14556125	—	2	3	no	—
24	#14556125	C₂³	1	2	yes	—
25	#14556120	—	2	3	no	—
26	#14556107	—	2	6	no	—
27	C₄×Dic₁₄₂	—	2	6	no	—
28	#14556087	—	2	3	no	—
29	#14556060	—	2	6	no	—
30	#14556053	—	2	12	no	—
31	#14556050	—	2	12	no	—
32	#14556050	—	2	12	no	—
33	C₂×M₁₆×C₇₁	D₄	1	6	yes	—
34	C₂×C₄²×C₇₁	C₂³	1	1	yes	characteristic
35	#97×C₇₁	D₄	1	6	yes	—
36	C₄×C₈×C₇₁	C₂³	1	3	yes	—
37	C₄×C₈×C₇₁	C₂³	1	3	yes	—
38	C₂³×D₇₁	—	4	1	no	—
39	C₂×Dic₁₄₂	—	2	6	no	—
40	C₂×D₂₈₄	—	2	6	no	—
41	C₂×D₂₈₄	—	4	12	no	—
42	C₂×D₂₈₄	—	4	3	no	—
43	Dic₂₈₄	—	4	12	no	—
44	C₄⋊Dic₇₁	—	2	6	no	—
45	C₄⋊Dic₇₁	—	2	6	no	—
46	Dic₇₁⋊C₄	—	4	6	no	—
47	C₂²⋊Dic₇₁	—	4	3	no	—
48	C₄×Dic₇₁	—	4	3	no	—
49	C₂²×C₄×C₇₁	C₂×D₄	1	3	yes	—
50	M₁₆×C₇₁	—	4	6	no	—
51	C₂×C₈×C₇₁	C₂×D₄	1	6	yes	—
52	C₂×C₈×C₇₁	C₂×D₄	1	6	yes	—
53	C₄²×C₇₁	C₂×D₄	1	3	yes	—
54	C₄²×C₇₁	C₂⁴	1	1	yes	characteristic
55	C₂³×C₇₁	#127	1	1	yes	characteristic
56	C₂²×D₇₁	—	4	2	no	—
57	C₂²×D₇₁	—	8	6	no	—
58	C₂×C₄×C₇₁	—	2	6	no	—
59	C₂×C₄×C₇₁	#127	1	3	yes	—
60	C₂×C₄×C₇₁	C₂²×D₄	1	3	yes	—
61	C₂×Dic₇₁	—	4	6	no	—
62	D₂₈₄	—	4	12	no	—
63	D₂₈₄	—	8	6	no	—
64	Dic₁₄₂	—	4	12	no	—
65	C₈×C₇₁	—	2	6	no	—
66	C₈×C₇₁	—	2	6	no	—
67	C₂²×C₇₁	—	2	3	no	—
68	C₂²×C₇₁	C₂×#127	1	1	yes	characteristic
69	C₂×D₇₁	—	8	6	no	—
70	C₂×D₇₁	—	16	4	no	—
71	C₄×C₇₁	—	2	3	no	—
72	C₄×C₇₁	#575	1	6	yes	—
73	Dic₇₁	—	8	6	no	—
74	#19753	—	71	1	no	maximal, Sylow
75	C₂×C₇₁	—	4	1	no	—
76	C₂×C₇₁	#3153	1	3	yes	—
77	D₇₁	—	16	2	no	—
78	#3185	—	71	3	no	—
79	#3138	—	71	3	no	—
80	#1720	—	71	6	no	—
81	#1711	—	71	2	no	—
82	#1455	D₇₁	1	1	yes	characteristic, complemented
83	C₇₁	#19753	1	1	yes	characteristic, complemented, Sylow
84	#533	—	71	3	no	—
85	C₂×#127	—	71	1	no	—
86	#498	—	71	3	no	—
87	#494	—	71	3	no	—
88	#490	—	142	6	no	—
89	#487	—	71	6	no	—
90	#486	—	71	6	no	—
91	#486	—	71	6	no	—
92	#470	—	142	6	no	—
93	#441	—	142	6	no	—
94	#418	—	142	6	no	—
95	C₄×M₁₆	C₂×D₇₁	1	3	yes	—
96	#322	C₂×D₇₁	1	3	yes	complemented
97	C₈²	C₂×D₇₁	1	1	yes	characteristic, complemented
98	C₂²×D₄	—	142	3	no	—
99	C₂×Q₁₆	—	142	6	no	—
100	C₂×M₁₆	D₂₈₄	1	6	yes	—
101	C₄⋊Q₈	—	71	6	no	—
102	#127	—	71	2	no	—
103	#127	—	142	3	no	—
104	#125	—	142	3	no	—
105	C₂²⋊Q₈	—	142	6	no	—
106	C₄×Q₈	—	142	6	no	—
107	#117	—	142	3	no	—
108	C₂×C₄²	C₂²×D₇₁	1	1	yes	characteristic
109	#106	—	142	6	no	—
110	#103	—	142	12	no	—
111	#102	—	142	12	no	—
112	#102	—	142	12	no	—
113	#97	D₂₈₄	1	6	yes	—
114	C₄×C₈	C₂²×D₇₁	1	3	yes	—
115	C₄×C₈	C₂²×D₇₁	1	3	yes	—
116	C₂⁴	—	284	1	no	—
117	C₂×Q₈	—	142	6	no	—
118	C₂×D₄	—	284	12	no	—
119	C₂×D₄	—	284	3	no	—
120	C₂×D₄	—	142	6	no	—
121	C₂²×C₄	C₂×D₂₈₄	1	3	yes	—
122	Q₁₆	—	284	12	no	—
123	M₁₆	—	4	6	no	—
124	C₂×C₈	C₂×D₂₈₄	1	6	yes	—
125	C₂×C₈	C₂×D₂₈₄	1	6	yes	—
126	C₄⋊C₄	—	284	6	no	—
127	C₄⋊C₄	—	142	6	no	—
128	C₄⋊C₄	—	142	6	no	—
129	C₂²⋊C₄	—	284	3	no	—
130	C₄²	—	284	3	no	—
131	C₄²	C₂×D₂₈₄	1	3	yes	—
132	C₄²	C₂³×D₇₁	1	1	yes	characteristic
133	C₂³	—	284	2	no	—
134	C₂³	—	568	6	no	—
135	C₂³	#14556125	1	1	yes	characteristic
136	Q₈	—	284	12	no	—
137	D₄	—	284	12	no	—
138	D₄	—	568	6	no	—
139	C₂×C₄	—	2	6	no	—
140	C₂×C₄	—	284	6	no	—
141	C₂×C₄	#14556125	1	3	yes	—
142	C₂×C₄	C₂²×D₂₈₄	1	3	yes	—
143	C₈	—	2	6	no	—
144	C₈	—	2	6	no	—
145	C₂²	—	2	3	no	—
146	C₂²	—	568	6	no	—
147	C₂²	—	1136	4	no	—
148	C₂²	C₂×#14556125	1	1	yes	characteristic
149	C₄	—	2	3	no	—
150	C₄	—	568	6	no	—
151	C₄	#17986799	1	6	yes	—
152	C₂	—	4	1	no	—
153	C₂	—	1136	2	no	—
154	C₂	#22598806	1	3	yes	—

	subgroup	G/H
center	C₂²	C₂×#14556125
commutator, cosocle	C₄²×C₇₁	C₂⁴
Frattini	C₄²	C₂³×D₇₁
Fitting	C₇₁×#1455	C₂
socle	C₂²×C₇₁	C₂×#127
prime radical	C₂×#14556125	C₂²
hypercenter	#1455	D₇₁
hypocenter	C₇₁	#19753
LCS	G ▷ C₄²×C₇₁ ▷ C₂²×C₇₁ ▷ C₇₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #1455

Complements

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

N	H ≅ G/N	type
1. #22599085	C₂	semidirect
3. #22588000	C₂	semidirect
4. #22587952	C₂	semidirect
5. C₇₁×#1455	C₂	semidirect
8. #17986397	C₂²	semidirect
9. #17986379	C₂²	semidirect
11. #17986350	C₂²	semidirect
18. #322×C₇₁	C₂²	semidirect
19. C₈²×C₇₁	C₂²	semidirect
82. #1455	D₇₁	semidirect
83. C₇₁	#19753	semidirect
96. #322	C₂×D₇₁	semidirect
97. C₈²	C₂×D₇₁	semidirect

Maximal subgroups

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

subgroup	index
#22599085	2
#22598715	2
#22588000	2
#22587952	2
C₇₁×#1455	2
#19753	71