Finite Group Databasebeta

Subgroups — #21462201

Order 7808.

164 conjugacy classes of subgroups; only the 66 normal subgroups are shown.

normal subgroups only

This group has 164 conjugacy classes of subgroups — too many to draw. Only its 66 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	C₂×#14519852	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂²×#11758187	C₂	1	1	yes	maximal, characteristic, complemented
3	#16840683	C₂	1	1	yes	maximal, characteristic, complemented
4	#16840657	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂²⋊#11758187	C₂	1	1	yes	maximal, characteristic, complemented
6	#16840651	C₂	1	1	yes	maximal, characteristic, complemented
7	#16839825	C₂	1	1	yes	maximal, characteristic
8	#14519852	—	2	2	no	—
9	C₂×#11758187	—	2	2	no	—
10	C₂×#11758187	C₂²	1	1	yes	characteristic
11	C₂×#11758187	C₂²	1	1	yes	characteristic
12	C₂×#11758187	C₂²	1	1	yes	characteristic
13	C₂×#11758187	C₂²	1	1	yes	characteristic
14	C₂²×#92418	C₂²	1	1	yes	characteristic
15	C₂×C₈×D₆₁	C₂²	1	1	yes	characteristic
16	C₄×C₂²⋊D₆₁	C₄	1	1	yes	characteristic
17	#14519706	C₄	1	1	yes	characteristic
18	#14519689	C₄	1	1	yes	characteristic
19	C₈×Dic₆₁	C₂²	1	1	yes	characteristic
20	C₂²⋊C₈×C₆₁	C₄	1	1	yes	characteristic
21	C₂²×C₄×C₆₁	C₂×C₄	1	1	yes	characteristic
22	C₂×C₂²⋊D₆₁	C₈	1	1	yes	characteristic
23	C₂×C₄×D₆₁	C₂×C₄	1	1	yes	characteristic
24	C₂×C₈×C₆₁	C₂×C₄	1	1	yes	characteristic
25	C₂×C₈×C₆₁	C₂×C₄	1	1	yes	characteristic
26	C₂²⋊Dic₆₁	C₈	1	1	yes	characteristic
27	#11758198	C₈	1	1	yes	characteristic
28	#11758196	C₈	1	1	yes	characteristic
29	C₄×Dic₆₁	C₂×C₄	1	1	yes	characteristic
30	C₂×#92418	—	2	1	no	—
31	C₂×#92418	—	2	1	no	—
32	C₂×#92418	C₂×C₄	1	1	yes	characteristic
33	C₂×#92418	C₂³	1	1	yes	characteristic
34	C₈×D₆₁	—	2	1	no	—
35	C₈×D₆₁	—	2	1	no	—
36	#11758187	—	2	1	no	—
37	#11758187	—	2	1	no	—
38	#11758187	—	2	1	no	—
39	#11758187	D₄	1	2	yes	complemented
40	C₂³×C₆₁	C₂×C₈	1	1	yes	characteristic
41	C₂²×D₆₁	C₂×C₈	1	1	yes	characteristic
42	C₂×C₄×C₆₁	—	2	1	no	—
43	C₂×C₄×C₆₁	—	2	1	no	—
44	C₂×C₄×C₆₁	C₂×C₈	1	1	yes	characteristic
45	C₂×C₄×C₆₁	C₂²×C₄	1	1	yes	characteristic
46	C₂²⋊D₆₁	—	2	2	no	—
47	C₂×Dic₆₁	C₂×C₈	1	1	yes	characteristic
48	C₂×Dic₆₁	C₂×C₈	1	1	yes	characteristic
49	C₂×Dic₆₁	C₂×C₈	1	1	yes	characteristic
50	C₄×D₆₁	—	2	1	no	—
51	C₄×D₆₁	—	2	1	no	—
52	C₈×C₆₁	—	2	1	no	—
53	C₈×C₆₁	—	2	1	no	—
54	#92418	—	2	1	no	—
55	#92418	C₂×D₄	1	1	yes	characteristic
56	#92418	#41	1	1	yes	characteristic
57	C₂²×C₆₁	—	2	1	no	—
58	C₂²×C₆₁	—	2	1	no	—
59	C₂²×C₆₁	M₃₂	1	2	yes	complemented
60	C₂²×C₆₁	C₂²×C₈	1	1	yes	characteristic
61	C₂×D₆₁	—	2	1	no	—
62	C₂×D₆₁	—	2	1	no	—
63	C₂×D₆₁	—	2	1	no	—
64	C₄×C₆₁	—	2	1	no	—
65	C₄×C₆₁	C₄×D₄	1	1	yes	characteristic
66	C₄×C₆₁	#131	1	1	yes	characteristic
67	Dic₆₁	—	2	1	no	—
68	Dic₆₁	—	2	1	no	—
69	Dic₆₁	—	2	1	no	—
70	#2168	—	61	1	no	maximal, Sylow
71	C₂×C₆₁	—	2	2	no	—
72	C₂×C₆₁	C₈×D₄	1	1	yes	characteristic
73	C₂×C₆₁	C₂×M₃₂	1	1	yes	characteristic
74	C₂×C₆₁	#504	1	1	yes	characteristic
75	D₆₁	—	4	1	no	—
76	C₂×M₃₂	—	61	1	no	—
77	C₂²×C₁₆	—	61	1	no	—
78	C₈×D₄	—	61	1	no	—
79	C₄⋊C₁₆	—	61	1	no	—
80	C₂²⋊C₁₆	—	61	1	no	—
81	C₂²⋊C₁₆	—	61	1	no	—
82	#346	—	61	1	no	—
83	C₆₁	#2168	1	1	yes	characteristic, complemented, Sylow
84	C₂²×C₈	—	61	1	no	—
85	C₂²×C₈	—	61	1	no	—
86	C₄×D₄	—	61	1	no	—
87	M₃₂	—	122	2	no	—
88	C₂×C₁₆	—	61	1	no	—
89	C₂×C₁₆	—	61	1	no	—
90	C₂×C₁₆	—	61	1	no	—
91	C₂×C₁₆	—	61	1	no	—
92	C₂×C₁₆	—	122	2	no	—
93	C₄⋊C₈	—	61	1	no	—
94	C₂²⋊C₈	—	61	1	no	—
95	C₂²⋊C₈	#6923	1	1	yes	characteristic
96	C₄×C₈	—	61	1	no	—
97	C₂×D₄	—	61	1	no	—
98	C₂²×C₄	—	61	1	no	—
99	C₂²×C₄	C₂×#6923	1	1	yes	characteristic
100	C₂×C₈	—	61	1	no	—
101	C₂×C₈	—	61	1	no	—
102	C₂×C₈	—	122	1	no	—
103	C₂×C₈	—	122	1	no	—
104	C₂×C₈	—	122	1	no	—
105	C₂×C₈	—	122	1	no	—
106	C₂×C₈	C₂×#6923	1	1	yes	characteristic
107	C₂×C₈	C₂×#6923	1	1	yes	characteristic
108	C₄⋊C₄	—	61	1	no	—
109	C₂²⋊C₄	—	61	1	no	—
110	C₂²⋊C₄	—	61	1	no	—
111	C₄²	—	61	1	no	—
112	C₁₆	—	61	2	no	—
113	C₁₆	—	122	1	no	—
114	C₁₆	—	122	1	no	—
115	C₁₆	—	122	1	no	—
116	C₂³	—	61	1	no	—
117	C₂³	D₆₁⋊C₈	1	1	yes	characteristic
118	D₄	—	122	2	no	—
119	C₂×C₄	—	2	1	no	—
120	C₂×C₄	—	2	1	no	—
121	C₂×C₄	—	61	1	no	—
122	C₂×C₄	—	61	1	no	—
123	C₂×C₄	—	61	1	no	—
124	C₂×C₄	—	122	1	no	—
125	C₂×C₄	—	122	1	no	—
126	C₂×C₄	D₆₁⋊C₈	1	1	yes	characteristic
127	C₂×C₄	C₂²×#6923	1	1	yes	characteristic
128	C₈	—	2	1	no	—
129	C₈	—	2	1	no	—
130	C₈	—	61	1	no	—
131	C₈	—	61	1	no	—
132	C₈	—	122	1	no	—
133	C₂²	—	2	1	no	—
134	C₂²	—	2	1	no	—
135	C₂²	—	122	1	no	—
136	C₂²	—	122	1	no	—
137	C₂²	—	122	1	no	—
138	C₂²	#14519852	1	2	yes	complemented
139	C₂²	C₂×D₆₁⋊C₈	1	1	yes	characteristic
140	C₄	—	2	1	no	—
141	C₄	—	122	1	no	—
142	C₄	—	122	1	no	—
143	C₄	—	122	1	no	—
144	C₄	D₄×#6923	1	1	yes	characteristic
145	C₄	#14519864	1	1	yes	characteristic
146	C₂	—	2	2	no	—
147	C₂	—	244	1	no	—
148	C₂	#16840789	1	1	yes	characteristic
149	C₂	C₂×#14519852	1	1	yes	characteristic
150	C₂	#16840821	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂×C₄	C₂²×#6923
commutator, socle	C₂²×C₆₁	C₂²×C₈
Fitting	C₂²⋊C₈×C₆₁	C₄
hypercenter	C₂²⋊C₈	#6923
hypocenter	C₆₁	#2168
LCS	G ▷ C₂²×C₆₁ ▷ C₆₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂²⋊C₈

Complements

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

N	H ≅ G/N	type
1. C₂×#14519852	C₂	semidirect
2. C₂²×#11758187	C₂	semidirect
3. #16840683	C₂	semidirect
4. #16840657	C₂	semidirect
5. C₂²⋊#11758187	C₂	semidirect
6. #16840651	C₂	semidirect
39. #11758187	D₄	semidirect
59. C₂²×C₆₁	M₃₂	semidirect
83. C₆₁	#2168	semidirect
138. C₂²	#14519852	semidirect