Finite Group Databasebeta

Subgroups — #49097214

Order 35968.

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×D₂₂₄₈	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×#22582820	C₂	1	2	yes	maximal, complemented
3	#31823263	C₂	1	2	yes	maximal, complemented
4	C₂²⋊#17985113	C₂	1	1	yes	maximal, characteristic, complemented
5	#466×C₂₈₁	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂²×D₁₁₂₄	—	2	1	no	—
7	C₂×D₂₂₄₈	—	2	4	no	—
8	C₂×D₂₂₄₈	—	2	1	no	—
9	C₂×D₂₂₄₈	C₂²	1	2	yes	—
10	#22582820	—	2	4	no	—
11	C₂×#17985113	C₂²	1	2	yes	complemented
12	C₂×D₈×C₂₈₁	C₂²	1	1	yes	characteristic, complemented
13	C₂²×C₈×C₂₈₁	C₂²	1	1	yes	characteristic
14	#121×C₂₈₁	—	2	1	no	—
15	#107×C₂₈₁	C₂²	1	1	yes	characteristic, complemented
16	#102×C₂₈₁	—	2	1	no	—
17	C₂³×D₂₈₁	—	4	1	no	—
18	C₂×D₁₁₂₄	—	2	4	no	—
19	C₂×D₁₁₂₄	—	2	2	no	—
20	C₂×D₁₁₂₄	—	4	2	no	—
21	C₂×D₁₁₂₄	—	4	1	no	—
22	D₂₂₄₈	—	2	4	no	—
23	D₂₂₄₈	—	2	2	no	—
24	D₂₂₄₈	D₄	1	4	yes	complemented
25	#17985113	—	2	2	no	—
26	C₂×D₄×C₂₈₁	—	2	1	no	—
27	C₂×D₄×C₂₈₁	—	4	1	no	—
28	C₂²×C₄×C₂₈₁	D₄	1	1	yes	characteristic
29	D₈×C₂₈₁	—	2	2	no	—
30	C₂×C₈×C₂₈₁	—	2	2	no	—
31	C₂×C₈×C₂₈₁	D₄	1	1	yes	characteristic
32	C₂×C₈×C₂₈₁	C₂³	1	1	yes	characteristic
33	C₄⋊C₄×C₂₈₁	—	2	1	no	—
34	C₂²⋊C₄×C₂₈₁	—	4	1	no	—
35	C₂²×D₂₈₁	—	4	4	no	—
36	C₂²×D₂₈₁	—	4	2	no	—
37	C₂²×D₂₈₁	—	8	2	no	—
38	C₂²×D₂₈₁	—	8	2	no	—
39	D₁₁₂₄	—	2	4	no	—
40	D₁₁₂₄	—	4	4	no	—
41	D₁₁₂₄	—	4	2	no	—
42	C₂³×C₂₈₁	—	4	1	no	—
43	C₂³×C₂₈₁	D₈	1	1	yes	characteristic
44	D₄×C₂₈₁	—	4	1	no	—
45	D₄×C₂₈₁	—	4	1	no	—
46	D₄×C₂₈₁	—	8	2	no	—
47	C₂×C₄×C₂₈₁	—	2	1	no	—
48	C₂×C₄×C₂₈₁	—	2	1	no	—
49	C₂×C₄×C₂₈₁	—	4	1	no	—
50	C₂×C₄×C₂₈₁	D₈	1	1	yes	characteristic
51	C₂×C₄×C₂₈₁	C₂×D₄	1	1	yes	characteristic
52	C₈×C₂₈₁	—	2	1	no	—
53	C₈×C₂₈₁	C₂×D₄	1	2	yes	—
54	C₂²×C₂₈₁	—	2	1	no	—
55	C₂²×C₂₈₁	—	2	1	no	—
56	C₂²×C₂₈₁	—	8	1	no	—
57	C₂²×C₂₈₁	—	8	1	no	—
58	C₂²×C₂₈₁	—	8	1	no	—
59	C₂²×C₂₈₁	D₁₆	1	2	yes	complemented
60	C₂²×C₂₈₁	C₂×D₈	1	1	yes	characteristic
61	C₂×D₂₈₁	—	4	4	no	—
62	C₂×D₂₈₁	—	8	8	no	—
63	C₂×D₂₈₁	—	8	2	no	—
64	C₂×D₂₈₁	—	8	2	no	—
65	C₄×C₂₈₁	—	2	1	no	—
66	C₄×C₂₈₁	—	8	1	no	—
67	C₄×C₂₈₁	#120	1	1	yes	characteristic
68	C₄×C₂₈₁	AGL(1,ℤ/8)	1	1	yes	characteristic
69	C₂×C₂₈₁	—	2	2	no	—
70	C₂×C₂₈₁	—	16	1	no	—
71	C₂×C₂₈₁	#447	1	1	yes	characteristic
72	C₂×C₂₈₁	C₂×D₁₆	1	1	yes	characteristic
73	C₂×C₂₈₁	#509	1	1	yes	characteristic
74	D₂₈₁	—	8	4	no	—
75	C₂₈₁	#2184	1	1	yes	characteristic, complemented, Sylow
76	#2184	—	281	1	no	maximal, Sylow
77	C₂²×D₈	—	281	1	no	—
78	C₂×D₁₆	—	281	2	no	—
79	#466	D₂₈₁	1	1	yes	characteristic, complemented
80	#357	—	281	2	no	—
81	C₂²⋊C₁₆	—	281	1	no	—
82	C₂²×D₄	—	562	1	no	—
83	C₂×D₈	—	281	2	no	—
84	C₂×D₈	—	562	4	no	—
85	C₂×D₈	—	562	1	no	—
86	C₂×D₈	C₂×D₂₈₁	1	1	yes	characteristic, complemented
87	C₂²×C₈	C₂×D₂₈₁	1	1	yes	characteristic
88	#121	—	2	1	no	—
89	D₁₆	—	562	4	no	—
90	C₂×C₁₆	—	281	2	no	—
91	#107	C₂×D₂₈₁	1	1	yes	characteristic, complemented
92	#102	—	2	1	no	—
93	C₂⁴	—	1124	1	no	—
94	C₂×D₄	—	2	1	no	—
95	C₂×D₄	—	4	1	no	—
96	C₂×D₄	—	562	4	no	—
97	C₂×D₄	—	562	2	no	—
98	C₂×D₄	—	1124	2	no	—
99	C₂×D₄	—	1124	1	no	—
100	C₂²×C₄	C₂²⋊D₂₈₁	1	1	yes	characteristic
101	D₈	—	2	2	no	—
102	D₈	—	281	4	no	—
103	D₈	—	562	4	no	—
104	D₈	—	562	2	no	—
105	C₂×C₈	—	2	2	no	—
106	C₂×C₈	C₂²⋊D₂₈₁	1	1	yes	characteristic
107	C₂×C₈	C₂²×D₂₈₁	1	1	yes	characteristic
108	C₄⋊C₄	—	2	1	no	—
109	C₂²⋊C₄	—	4	1	no	—
110	C₁₆	—	562	2	no	—
111	C₂³	—	4	1	no	—
112	C₂³	—	1124	4	no	—
113	C₂³	—	1124	2	no	—
114	C₂³	—	2248	2	no	—
115	C₂³	—	2248	2	no	—
116	C₂³	#17985123	1	1	yes	characteristic
117	D₄	—	4	1	no	—
118	D₄	—	4	1	no	—
119	D₄	—	8	2	no	—
120	D₄	—	562	4	no	—
121	D₄	—	1124	4	no	—
122	D₄	—	1124	2	no	—
123	C₂×C₄	—	2	1	no	—
124	C₂×C₄	—	2	1	no	—
125	C₂×C₄	—	4	1	no	—
126	C₂×C₄	#17985123	1	1	yes	characteristic
127	C₂×C₄	C₂×C₂²⋊D₂₈₁	1	1	yes	characteristic
128	C₈	—	2	1	no	—
129	C₈	D₄×D₂₈₁	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	—	1124	4	no	—
136	C₂²	—	2248	8	no	—
137	C₂²	—	2248	2	no	—
138	C₂²	—	2248	2	no	—
139	C₂²	#22582820	1	2	yes	complemented
140	C₂²	C₂×#17985123	1	1	yes	characteristic
141	C₄	—	2	1	no	—
142	C₄	—	8	1	no	—
143	C₄	#22582847	1	1	yes	characteristic
144	C₄	#22582904	1	1	yes	characteristic
145	C₂	—	2	2	no	—
146	C₂	—	16	1	no	—
147	C₂	—	2248	4	no	—
148	C₂	#31823596	1	1	yes	characteristic
149	C₂	C₂×#22582820	1	1	yes	characteristic
150	C₂	#31823898	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×#17985123
commutator, cosocle	C₂×C₈×C₂₈₁	C₂³
Frattini	C₂×C₈	C₂²×D₂₈₁
Fitting	#466×C₂₈₁	C₂
socle	C₂²×C₂₈₁	C₂×D₈
hypercenter	#466	D₂₈₁
hypocenter	C₂₈₁	#2184
LCS	G ▷ C₂×C₈×C₂₈₁ ▷ C₄×C₂₈₁ ▷ C₂×C₂₈₁ ▷ C₂₈₁
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×D₂₂₄₈	C₂	semidirect
2. C₂×#22582820	C₂	semidirect
3. #31823263	C₂	semidirect
4. C₂²⋊#17985113	C₂	semidirect
5. #466×C₂₈₁	C₂	semidirect
11. C₂×#17985113	C₂²	semidirect
12. C₂×D₈×C₂₈₁	C₂²	semidirect
15. #107×C₂₈₁	C₂²	semidirect
24. D₂₂₄₈	D₄	semidirect
59. C₂²×C₂₈₁	D₁₆	semidirect
75. C₂₈₁	#2184	semidirect
79. #466	D₂₈₁	semidirect
86. C₂×D₈	C₂×D₂₈₁	semidirect
91. #107	C₂×D₂₈₁	semidirect
139. C₂²	#22582820	semidirect

Maximal subgroups

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

subgroup	index
C₂²×D₂₂₄₈	2
C₂×#22582820	2
#31823263	2
C₂²⋊#17985113	2
#466×C₂₈₁	2
#2184	281