Finite Group Databasebeta

Subgroups — #49097507

Order 35968.

140 conjugacy classes of subgroups; only the 47 normal subgroups are shown.

normal subgroups only

This group has 140 conjugacy classes of subgroups — too many to draw. Only its 47 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	#31823705	C₂	1	1	yes	maximal, characteristic
2	#107×D₂₈₁	C₂	1	1	yes	maximal, characteristic
3	#31823288	C₂	1	1	yes	maximal, characteristic, complemented
4	#31823268	C₂	1	1	yes	maximal, characteristic, complemented
5	#31823266	C₂	1	1	yes	maximal, characteristic, complemented
6	#31823242	C₂	1	1	yes	maximal, characteristic
7	#366×C₂₈₁	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×Dic₁₁₂₄	C₂²	1	1	yes	characteristic
9	C₂×C₈×D₂₈₁	C₂²	1	1	yes	characteristic
10	#22582859	—	2	1	no	—
11	C₄⋊C₄×D₂₈₁	—	2	1	no	—
12	C₂×#17985113	C₂²	1	1	yes	characteristic
13	#22582813	C₂²	1	1	yes	characteristic
14	#22582812	—	2	1	no	—
15	#22582802	—	2	1	no	—
16	C₂×C₁₆×C₂₈₁	C₂²	1	1	yes	characteristic
17	#107×C₂₈₁	C₂²	1	1	yes	characteristic
18	#107×C₂₈₁	C₂²	1	1	yes	characteristic
19	C₂×Dic₅₆₂	—	2	1	no	—
20	C₂×C₄×D₂₈₁	—	4	1	no	—
21	C₂×C₄×D₂₈₁	D₄	1	1	yes	characteristic
22	Dic₁₁₂₄	—	2	1	no	—
23	Dic₁₁₂₄	—	2	1	no	—
24	C₂×#14555886	D₄	1	1	yes	characteristic
25	C₈×D₂₈₁	—	2	1	no	—
26	C₈×D₂₈₁	—	2	1	no	—
27	C₄⋊Dic₂₈₁	—	2	1	no	—
28	C₄⋊Dic₂₈₁	—	4	1	no	—
29	#17985117	—	4	1	no	—
30	#17985115	—	4	1	no	—
31	#17985113	—	2	1	no	—
32	C₂×C₈×C₂₈₁	C₂³	1	1	yes	characteristic
33	C₄⋊C₄×C₂₈₁	—	2	1	no	—
34	C₄⋊C₄×C₂₈₁	—	2	1	no	—
35	C₁₆×C₂₈₁	—	2	1	no	—
36	C₂²×D₂₈₁	D₈	1	1	yes	characteristic
37	Dic₅₆₂	—	4	1	no	—
38	Dic₅₆₂	—	4	1	no	—
39	C₂×Dic₂₈₁	—	4	1	no	—
40	C₂×Dic₂₈₁	—	4	1	no	—
41	C₂×Dic₂₈₁	D₈	1	1	yes	characteristic
42	C₄×D₂₈₁	—	2	1	no	—
43	C₄×D₂₈₁	—	2	1	no	—
44	C₄×D₂₈₁	—	8	2	no	—
45	#14555886	—	2	1	no	—
46	C₂×C₄×C₂₈₁	—	4	1	no	—
47	C₂×C₄×C₂₈₁	—	4	1	no	—
48	C₂×C₄×C₂₈₁	C₂×D₄	1	1	yes	characteristic
49	C₈×C₂₈₁	#41	1	1	yes	characteristic
50	C₈×C₂₈₁	#41	1	1	yes	characteristic
51	C₂²×C₂₈₁	C₂×D₈	1	1	yes	characteristic
52	C₂×D₂₈₁	—	2	1	no	—
53	C₂×D₂₈₁	—	2	1	no	—
54	C₂×D₂₈₁	Q₃₂	1	2	yes	—
55	C₄×C₂₈₁	—	8	1	no	—
56	C₄×C₂₈₁	—	8	1	no	—
57	C₄×C₂₈₁	#123	1	1	yes	characteristic
58	C₄×C₂₈₁	#137	1	1	yes	characteristic
59	Dic₂₈₁	—	2	1	no	—
60	Dic₂₈₁	—	8	1	no	—
61	Dic₂₈₁	—	8	1	no	—
62	C₂×C₂₈₁	#480	1	1	yes	characteristic
63	C₂×C₂₈₁	C₂×Q₃₂	1	1	yes	characteristic
64	C₂×C₂₈₁	#509	1	1	yes	characteristic
65	D₂₈₁	—	2	2	no	—
66	C₂₈₁	#2236	1	1	yes	characteristic, complemented, Sylow
67	#2236	—	281	1	no	maximal, Sylow
68	#467	—	281	1	no	—
69	C₂×#107	—	281	1	no	—
70	#366	—	281	1	no	—
71	#366	D₂₈₁	1	1	yes	characteristic, complemented
72	#358	—	281	1	no	—
73	#358	—	281	1	no	—
74	C₂²⋊C₁₆	—	281	1	no	—
75	C₂×Q₁₆	—	281	1	no	—
76	C₂²×C₈	—	281	1	no	—
77	C₂²⋊Q₈	—	562	1	no	—
78	C₂×C₄⋊C₄	—	562	1	no	—
79	C₂×C₁₆	—	281	1	no	—
80	C₂×C₁₆	C₂×D₂₈₁	1	1	yes	characteristic
81	#107	—	281	1	no	—
82	#107	—	562	1	no	—
83	#107	C₂×D₂₈₁	1	1	yes	characteristic
84	#107	C₂×D₂₈₁	1	1	yes	characteristic
85	#103	—	562	1	no	—
86	C₂×Q₈	—	562	1	no	—
87	C₂²×C₄	—	281	1	no	—
88	C₂²×C₄	—	1124	1	no	—
89	Q₁₆	—	562	1	no	—
90	Q₁₆	—	562	1	no	—
91	C₂×C₈	—	281	1	no	—
92	C₂×C₈	—	562	1	no	—
93	C₂×C₈	—	562	1	no	—
94	C₂×C₈	C₂²×D₂₈₁	1	1	yes	characteristic
95	C₄⋊C₄	—	2	1	no	—
96	C₄⋊C₄	—	2	1	no	—
97	C₄⋊C₄	—	562	1	no	—
98	C₄⋊C₄	—	1124	1	no	—
99	C₄⋊C₄	—	1124	1	no	—
100	C₂²⋊C₄	—	1124	1	no	—
101	C₁₆	—	2	1	no	—
102	C₁₆	—	562	1	no	—
103	C₂³	—	281	1	no	—
104	Q₈	—	1124	1	no	—
105	Q₈	—	1124	1	no	—
106	C₂×C₄	—	4	1	no	—
107	C₂×C₄	—	4	1	no	—
108	C₂×C₄	—	281	1	no	—
109	C₂×C₄	—	562	1	no	—
110	C₂×C₄	—	562	1	no	—
111	C₂×C₄	—	1124	1	no	—
112	C₂×C₄	—	1124	1	no	—
113	C₂×C₄	—	2248	2	no	—
114	C₂×C₄	D₄×D₂₈₁	1	1	yes	characteristic
115	C₈	—	562	1	no	—
116	C₈	#17985138	1	1	yes	characteristic
117	C₈	#17985139	1	1	yes	characteristic
118	C₂²	—	281	2	no	—
119	C₂²	—	562	1	no	—
120	C₂²	—	562	1	no	—
121	C₂²	D₈×D₂₈₁	1	1	yes	characteristic
122	C₄	—	8	1	no	—
123	C₄	—	8	1	no	—
124	C₄	—	562	1	no	—
125	C₄	—	2248	1	no	—
126	C₄	—	2248	1	no	—
127	C₄	#22582864	1	1	yes	characteristic
128	C₄	#22582911	1	1	yes	characteristic
129	C₂	—	562	2	no	—
130	C₂	#31823778	1	1	yes	characteristic
131	C₂	Q₃₂×D₂₈₁	1	1	yes	characteristic
132	C₂	#31823897	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	D₈×D₂₈₁
commutator, cosocle	C₂×C₈×C₂₈₁	C₂³
Frattini	C₂×C₈	C₂²×D₂₈₁
Fitting	#366×C₂₈₁	C₂
socle	C₂²×C₂₈₁	C₂×D₈
prime radical	C₂²×D₂₈₁	D₈
hypercenter	#366	D₂₈₁
hypocenter	C₂₈₁	#2236
LCS	G ▷ C₂×C₈×C₂₈₁ ▷ C₄×C₂₈₁ ▷ C₂×C₂₈₁ ▷ C₂₈₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄ ◁ C₂×C₈ ◁ #366

Complements

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

N	H ≅ G/N	type
3. #31823288	C₂	semidirect
4. #31823268	C₂	semidirect
5. #31823266	C₂	semidirect
7. #366×C₂₈₁	C₂	semidirect
66. C₂₈₁	#2236	semidirect
71. #366	D₂₈₁	semidirect

Maximal subgroups

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

subgroup	index
#31823705	2
#107×D₂₈₁	2
#31823288	2
#31823268	2
#31823266	2
#31823242	2
#366×C₂₈₁	2
#2236	281