Finite Group Databasebeta

Subgroups — #44509356

Order 30848.

266 conjugacy classes of subgroups; only the 57 normal subgroups are shown.

normal subgroups only

This group has 266 conjugacy classes of subgroups — too many to draw. Only its 57 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	C₂²×D₁₉₂₈	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×#21440583	C₂	1	2	yes	maximal, complemented
3	#29507517	C₂	1	2	yes	maximal, complemented
4	C₂²⋊#16838745	C₂	1	1	yes	maximal, characteristic, complemented
5	#466×C₂₄₁	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂²×D₉₆₄	—	2	1	no	—
7	C₂×D₁₉₂₈	—	2	4	no	—
8	C₂×D₁₉₂₈	—	2	1	no	—
9	C₂×D₁₉₂₈	C₂²	1	2	yes	—
10	#21440583	—	2	4	no	—
11	C₂×#16838745	C₂²	1	2	yes	complemented
12	C₂×D₈×C₂₄₁	C₂²	1	1	yes	characteristic, complemented
13	C₂²×C₈×C₂₄₁	C₂²	1	1	yes	characteristic
14	#121×C₂₄₁	—	2	1	no	—
15	#107×C₂₄₁	C₂²	1	1	yes	characteristic, complemented
16	#102×C₂₄₁	—	2	1	no	—
17	C₂³×D₂₄₁	—	4	1	no	—
18	C₂×D₉₆₄	—	2	4	no	—
19	C₂×D₉₆₄	—	2	2	no	—
20	C₂×D₉₆₄	—	4	2	no	—
21	C₂×D₉₆₄	—	4	1	no	—
22	D₁₉₂₈	—	2	4	no	—
23	D₁₉₂₈	—	2	2	no	—
24	D₁₉₂₈	D₄	1	4	yes	complemented
25	#16838745	—	2	2	no	—
26	C₂×D₄×C₂₄₁	—	2	1	no	—
27	C₂×D₄×C₂₄₁	—	4	1	no	—
28	C₂²×C₄×C₂₄₁	D₄	1	1	yes	characteristic
29	D₈×C₂₄₁	—	2	2	no	—
30	C₂×C₈×C₂₄₁	—	2	2	no	—
31	C₂×C₈×C₂₄₁	D₄	1	1	yes	characteristic
32	C₂×C₈×C₂₄₁	C₂³	1	1	yes	characteristic
33	C₄⋊C₄×C₂₄₁	—	2	1	no	—
34	C₂²⋊C₄×C₂₄₁	—	4	1	no	—
35	C₂²×D₂₄₁	—	4	4	no	—
36	C₂²×D₂₄₁	—	4	2	no	—
37	C₂²×D₂₄₁	—	8	2	no	—
38	C₂²×D₂₄₁	—	8	2	no	—
39	D₉₆₄	—	2	4	no	—
40	D₉₆₄	—	4	4	no	—
41	D₉₆₄	—	4	2	no	—
42	C₂³×C₂₄₁	—	4	1	no	—
43	C₂³×C₂₄₁	D₈	1	1	yes	characteristic
44	D₄×C₂₄₁	—	4	1	no	—
45	D₄×C₂₄₁	—	4	1	no	—
46	D₄×C₂₄₁	—	8	2	no	—
47	C₂×C₄×C₂₄₁	—	2	1	no	—
48	C₂×C₄×C₂₄₁	—	2	1	no	—
49	C₂×C₄×C₂₄₁	—	4	1	no	—
50	C₂×C₄×C₂₄₁	D₈	1	1	yes	characteristic
51	C₂×C₄×C₂₄₁	C₂×D₄	1	1	yes	characteristic
52	C₈×C₂₄₁	—	2	1	no	—
53	C₈×C₂₄₁	C₂×D₄	1	2	yes	—
54	C₂²×C₂₄₁	—	2	1	no	—
55	C₂²×C₂₄₁	—	2	1	no	—
56	C₂²×C₂₄₁	—	8	1	no	—
57	C₂²×C₂₄₁	—	8	1	no	—
58	C₂²×C₂₄₁	—	8	1	no	—
59	C₂²×C₂₄₁	D₁₆	1	2	yes	complemented
60	C₂²×C₂₄₁	C₂×D₈	1	1	yes	characteristic
61	C₂×D₂₄₁	—	4	4	no	—
62	C₂×D₂₄₁	—	8	8	no	—
63	C₂×D₂₄₁	—	8	2	no	—
64	C₂×D₂₄₁	—	8	2	no	—
65	C₄×C₂₄₁	—	2	1	no	—
66	C₄×C₂₄₁	—	8	1	no	—
67	C₄×C₂₄₁	#120	1	1	yes	characteristic
68	C₄×C₂₄₁	AGL(1,ℤ/8)	1	1	yes	characteristic
69	C₂×C₂₄₁	—	2	2	no	—
70	C₂×C₂₄₁	—	16	1	no	—
71	C₂×C₂₄₁	#447	1	1	yes	characteristic
72	C₂×C₂₄₁	C₂×D₁₆	1	1	yes	characteristic
73	C₂×C₂₄₁	#509	1	1	yes	characteristic
74	D₂₄₁	—	8	4	no	—
75	C₂₄₁	#2184	1	1	yes	characteristic, complemented, Sylow
76	#2184	—	241	1	no	maximal, Sylow
77	C₂²×D₈	—	241	1	no	—
78	C₂×D₁₆	—	241	2	no	—
79	#466	D₂₄₁	1	1	yes	characteristic, complemented
80	#357	—	241	2	no	—
81	C₂²⋊C₁₆	—	241	1	no	—
82	C₂²×D₄	—	482	1	no	—
83	C₂×D₈	—	482	4	no	—
84	C₂×D₈	—	482	1	no	—
85	C₂×D₈	—	241	2	no	—
86	C₂×D₈	C₂×D₂₄₁	1	1	yes	characteristic, complemented
87	C₂²×C₈	C₂×D₂₄₁	1	1	yes	characteristic
88	#121	—	2	1	no	—
89	D₁₆	—	482	4	no	—
90	C₂×C₁₆	—	241	2	no	—
91	#107	C₂×D₂₄₁	1	1	yes	characteristic, complemented
92	#102	—	2	1	no	—
93	C₂⁴	—	964	1	no	—
94	C₂×D₄	—	2	1	no	—
95	C₂×D₄	—	4	1	no	—
96	C₂×D₄	—	964	2	no	—
97	C₂×D₄	—	964	1	no	—
98	C₂×D₄	—	482	4	no	—
99	C₂×D₄	—	482	2	no	—
100	C₂²×C₄	C₂²⋊D₂₄₁	1	1	yes	characteristic
101	D₈	—	2	2	no	—
102	D₈	—	482	4	no	—
103	D₈	—	482	2	no	—
104	D₈	—	241	4	no	—
105	C₂×C₈	—	2	2	no	—
106	C₂×C₈	C₂²⋊D₂₄₁	1	1	yes	characteristic
107	C₂×C₈	C₂²×D₂₄₁	1	1	yes	characteristic
108	C₄⋊C₄	—	2	1	no	—
109	C₂²⋊C₄	—	4	1	no	—
110	C₁₆	—	482	2	no	—
111	C₂³	—	4	1	no	—
112	C₂³	—	1928	2	no	—
113	C₂³	—	1928	2	no	—
114	C₂³	—	964	4	no	—
115	C₂³	—	964	2	no	—
116	C₂³	#16838755	1	1	yes	characteristic
117	D₄	—	4	1	no	—
118	D₄	—	4	1	no	—
119	D₄	—	8	2	no	—
120	D₄	—	964	4	no	—
121	D₄	—	964	2	no	—
122	D₄	—	482	4	no	—
123	C₂×C₄	—	2	1	no	—
124	C₂×C₄	—	2	1	no	—
125	C₂×C₄	—	4	1	no	—
126	C₂×C₄	#16838755	1	1	yes	characteristic
127	C₂×C₄	C₂×C₂²⋊D₂₄₁	1	1	yes	characteristic
128	C₈	—	2	1	no	—
129	C₈	D₄×D₂₄₁	1	2	yes	—
130	C₂²	—	2	1	no	—
131	C₂²	—	2	1	no	—
132	C₂²	—	8	1	no	—
133	C₂²	—	8	1	no	—
134	C₂²	—	8	1	no	—
135	C₂²	—	1928	8	no	—
136	C₂²	—	1928	2	no	—
137	C₂²	—	1928	2	no	—
138	C₂²	—	964	4	no	—
139	C₂²	#21440583	1	2	yes	complemented
140	C₂²	C₂×#16838755	1	1	yes	characteristic
141	C₄	—	2	1	no	—
142	C₄	—	8	1	no	—
143	C₄	#21440610	1	1	yes	characteristic
144	C₄	#21440667	1	1	yes	characteristic
145	C₂	—	2	2	no	—
146	C₂	—	16	1	no	—
147	C₂	—	1928	4	no	—
148	C₂	#29507850	1	1	yes	characteristic
149	C₂	C₂×#21440583	1	1	yes	characteristic
150	C₂	#29508152	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×#16838755
commutator, cosocle	C₂×C₈×C₂₄₁	C₂³
Frattini	C₂×C₈	C₂²×D₂₄₁
Fitting	#466×C₂₄₁	C₂
socle	C₂²×C₂₄₁	C₂×D₈
hypercenter	#466	D₂₄₁
hypocenter	C₂₄₁	#2184
LCS	G ▷ C₂×C₈×C₂₄₁ ▷ C₄×C₂₄₁ ▷ C₂×C₂₄₁ ▷ C₂₄₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂²×C₈ ◁ #466

Complements

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

N	H ≅ G/N	type
1. C₂²×D₁₉₂₈	C₂	semidirect
2. C₂×#21440583	C₂	semidirect
3. #29507517	C₂	semidirect
4. C₂²⋊#16838745	C₂	semidirect
5. #466×C₂₄₁	C₂	semidirect
11. C₂×#16838745	C₂²	semidirect
12. C₂×D₈×C₂₄₁	C₂²	semidirect
15. #107×C₂₄₁	C₂²	semidirect
24. D₁₉₂₈	D₄	semidirect
59. C₂²×C₂₄₁	D₁₆	semidirect
75. C₂₄₁	#2184	semidirect
79. #466	D₂₄₁	semidirect
86. C₂×D₈	C₂×D₂₄₁	semidirect
91. #107	C₂×D₂₄₁	semidirect
139. C₂²	#21440583	semidirect

Maximal subgroups

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

subgroup	index
C₂²×D₁₉₂₈	2
C₂×#21440583	2
#29507517	2
C₂²⋊#16838745	2
#466×C₂₄₁	2
#2184	241