Finite Group Databasebeta

Subgroups — #44506848

Order 30848.

144 conjugacy classes of subgroups; only the 53 normal subgroups are shown.

normal subgroups only

This group has 144 conjugacy classes of subgroups — too many to draw. Only its 53 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	#29508120	C₂	1	1	yes	maximal, characteristic
2	#29508013	C₂	1	1	yes	maximal, characteristic
3	#29508009	C₂	1	1	yes	maximal, characteristic
4	#29507841	C₂	1	1	yes	maximal, characteristic
5	#29507469	C₂	1	1	yes	maximal, characteristic
6	#29507465	C₂	1	1	yes	maximal, characteristic
7	#333×C₂₄₁	C₂	1	1	yes	maximal, characteristic
8	#21440642	C₂²	1	1	yes	characteristic
9	C₄×Dic₄₈₂	C₂²	1	1	yes	characteristic
10	#21440576	—	2	1	no	—
11	#21440575	—	2	1	no	—
12	#21440575	—	2	1	no	—
13	#21440575	—	2	1	no	—
14	C₄⋊#14515349	C₂²	1	1	yes	characteristic
15	#21440571	—	2	1	no	—
16	#21440567	—	2	1	no	—
17	#21440565	—	2	1	no	—
18	C₄×#14515349	C₂²	1	1	yes	characteristic
19	C₈×Dic₂₄₁	—	2	1	no	—
20	C₄⋊Q₈×C₂₄₁	C₂²	1	1	yes	characteristic
21	C₄⋊C₈×C₂₄₁	C₂²	1	1	yes	characteristic
22	C₄⋊C₈×C₂₄₁	C₂²	1	1	yes	characteristic
23	C₂×Dic₄₈₂	—	2	1	no	—
24	C₂×Dic₄₈₂	D₄	1	1	yes	characteristic
25	C₂×#14515349	—	2	1	no	—
26	C₂×#14515349	D₄	1	1	yes	characteristic
27	C₂×#14515349	D₄	1	1	yes	characteristic
28	C₄⋊Dic₂₄₁	—	2	1	no	—
29	C₄⋊Dic₂₄₁	—	2	1	no	—
30	C₄⋊Dic₂₄₁	D₄	1	1	yes	characteristic
31	#16838749	—	2	1	no	—
32	C₄×Dic₂₄₁	—	2	1	no	—
33	C₂×Q₈×C₂₄₁	—	2	1	no	—
34	C₂×C₈×C₂₄₁	—	2	1	no	—
35	C₂×C₈×C₂₄₁	—	2	1	no	—
36	C₄⋊C₄×C₂₄₁	—	2	1	no	—
37	C₄⋊C₄×C₂₄₁	—	2	1	no	—
38	C₄²×C₂₄₁	C₂³	1	1	yes	characteristic
39	Dic₄₈₂	—	2	1	no	—
40	Dic₄₈₂	—	8	1	no	—
41	Dic₄₈₂	Q₁₆	1	2	yes	—
42	C₂×Dic₂₄₁	—	2	1	no	—
43	C₂×Dic₂₄₁	—	2	1	no	—
44	C₂×Dic₂₄₁	—	4	1	no	—
45	#14515349	—	2	1	no	—
46	#14515349	—	4	1	no	—
47	#14515349	Q₁₆	1	2	yes	—
48	Q₈×C₂₄₁	—	8	1	no	—
49	C₂×C₄×C₂₄₁	—	4	1	no	—
50	C₂×C₄×C₂₄₁	C₂×D₄	1	1	yes	characteristic
51	C₂×C₄×C₂₄₁	C₂×D₄	1	1	yes	characteristic
52	C₂×C₄×C₂₄₁	#41	1	1	yes	characteristic
53	C₈×C₂₄₁	—	4	1	no	—
54	C₈×C₂₄₁	—	4	1	no	—
55	C₂²×C₂₄₁	#121	1	1	yes	characteristic
56	C₄×C₂₄₁	—	2	1	no	—
57	C₄×C₂₄₁	—	8	1	no	—
58	C₄×C₂₄₁	C₂×Q₁₆	1	1	yes	characteristic
59	C₄×C₂₄₁	C₂×Q₁₆	1	1	yes	characteristic
60	C₄×C₂₄₁	#135	1	1	yes	characteristic
61	C₄×C₂₄₁	#137	1	1	yes	characteristic
62	Dic₂₄₁	—	2	2	no	—
63	Dic₂₄₁	—	4	1	no	—
64	Dic₂₄₁	—	8	1	no	—
65	C₂×C₂₄₁	#462	1	1	yes	characteristic
66	C₂×C₂₄₁	#467	1	1	yes	characteristic
67	C₂×C₂₄₁	#472	1	1	yes	characteristic
68	C₂₄₁	#1676	1	1	yes	characteristic, complemented, Sylow
69	#1676	—	241	1	no	maximal, Sylow
70	#500	—	241	1	no	—
71	#477	—	241	1	no	—
72	#477	—	241	1	no	—
73	C₈×Q₈	—	241	1	no	—
74	#335	—	241	1	no	—
75	#333	—	241	1	no	—
76	#333	D₂₄₁	1	1	yes	characteristic
77	C₄⋊Q₈	—	241	1	no	—
78	C₄⋊Q₈	C₂×D₂₄₁	1	1	yes	characteristic
79	C₄×Q₈	—	241	1	no	—
80	#107	—	482	1	no	—
81	#107	—	482	1	no	—
82	#107	—	482	1	no	—
83	#107	—	482	1	no	—
84	C₄⋊C₈	—	482	1	no	—
85	C₄⋊C₈	—	241	1	no	—
86	C₄⋊C₈	C₂×D₂₄₁	1	1	yes	characteristic
87	C₄⋊C₈	C₂×D₂₄₁	1	1	yes	characteristic
88	#103	—	482	1	no	—
89	#103	—	482	1	no	—
90	C₄×C₈	—	482	1	no	—
91	C₄×C₈	—	241	1	no	—
92	C₂×Q₈	—	2	1	no	—
93	C₂×Q₈	—	482	1	no	—
94	C₂×Q₈	—	241	1	no	—
95	C₂×C₈	—	2	1	no	—
96	C₂×C₈	—	2	1	no	—
97	C₂×C₈	—	482	1	no	—
98	C₂×C₈	—	241	1	no	—
99	C₂×C₈	—	241	1	no	—
100	C₄⋊C₄	—	2	1	no	—
101	C₄⋊C₄	—	2	1	no	—
102	C₄⋊C₄	—	482	1	no	—
103	C₄⋊C₄	—	482	1	no	—
104	C₄⋊C₄	—	482	1	no	—
105	C₄⋊C₄	—	241	1	no	—
106	C₄²	—	482	1	no	—
107	C₄²	C₂²×D₂₄₁	1	1	yes	characteristic
108	Q₈	—	8	1	no	—
109	Q₈	—	1928	1	no	—
110	Q₈	—	482	1	no	—
111	Q₈	—	241	2	no	—
112	C₂×C₄	—	4	1	no	—
113	C₂×C₄	—	964	1	no	—
114	C₂×C₄	—	482	1	no	—
115	C₂×C₄	—	482	1	no	—
116	C₂×C₄	D₄×D₂₄₁	1	1	yes	characteristic
117	C₂×C₄	D₄×D₂₄₁	1	1	yes	characteristic
118	C₂×C₄	#16838770	1	1	yes	characteristic
119	C₈	—	4	1	no	—
120	C₈	—	4	1	no	—
121	C₈	—	964	1	no	—
122	C₈	—	482	1	no	—
123	C₈	—	241	2	no	—
124	C₂²	#21440615	1	1	yes	characteristic
125	C₄	—	2	1	no	—
126	C₄	—	8	1	no	—
127	C₄	—	1928	1	no	—
128	C₄	—	964	1	no	—
129	C₄	—	482	2	no	—
130	C₄	Q₁₆×D₂₄₁	1	1	yes	characteristic
131	C₄	Q₁₆×D₂₄₁	1	1	yes	characteristic
132	C₄	#21440663	1	1	yes	characteristic
133	C₄	#21440671	1	1	yes	characteristic
134	C₂	#29507926	1	1	yes	characteristic
135	C₂	#29507957	1	1	yes	characteristic
136	C₂	#29507986	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	#21440615
commutator, cosocle	C₄²×C₂₄₁	C₂³
Frattini	C₄²	C₂²×D₂₄₁
Fitting	#333×C₂₄₁	C₂
socle, prime radical	C₂²×C₂₄₁	#121
hypercenter	#333	D₂₄₁
hypocenter	C₂₄₁	#1676
LCS	G ▷ C₄²×C₂₄₁ ▷ C₂²×C₂₄₁ ▷ C₂₄₁
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #333

Complements

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

N	H ≅ G/N	type
68. C₂₄₁	#1676	semidirect

Maximal subgroups

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

subgroup	index
#29508120	2
#29508013	2
#29508009	2
#29507841	2
#29507469	2
#29507465	2
#333×C₂₄₁	2
#1676	241