Finite Group Databasebeta

Subgroups — #21445226

Order 7808.

176 conjugacy classes of subgroups; only the 55 normal subgroups are shown.

normal subgroups only

This group has 176 conjugacy classes of subgroups — too many to draw. Only its 55 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	#16840108	C₂	1	1	yes	maximal, characteristic, complemented
2	#16840052	C₂	1	1	yes	maximal, characteristic, complemented
3	D₄×#92418	C₂	1	1	yes	maximal, characteristic, complemented
4	#16839512	C₂	1	1	yes	maximal, characteristic, complemented
5	#16839510	C₂	1	1	yes	maximal, characteristic, complemented
6	#16839506	C₂	1	1	yes	maximal, characteristic, complemented
7	C₆₁×#459	C₂	1	1	yes	maximal, characteristic
8	C₂²×#92418	—	2	1	no	—
9	#14519777	C₂²	1	1	yes	characteristic
10	#14519708	—	2	1	no	—
11	C₄⋊#92418	C₂²	1	1	yes	characteristic
12	C₄⋊#92418	C₂²	1	1	yes	characteristic
13	#14519698	—	2	1	no	—
14	#14519698	—	2	1	no	—
15	#14519698	—	2	1	no	—
16	C₂²⋊#92418	—	2	1	no	—
17	C₄×#92418	C₂²	1	1	yes	characteristic
18	C₂×D₈×C₆₁	—	2	1	no	—
19	#127×C₆₁	C₂²	1	1	yes	characteristic
20	C₄×D₄×C₆₁	C₂²	1	1	yes	characteristic
21	C₄⋊C₈×C₆₁	C₂²	1	1	yes	characteristic
22	#102×C₆₁	—	2	1	no	—
23	C₂×D₄×C₆₁	—	2	1	no	—
24	C₂×D₄×C₆₁	—	2	1	no	—
25	C₂×D₄×C₆₁	—	2	1	no	—
26	C₂×D₄×C₆₁	D₄	1	1	yes	characteristic
27	C₂²×C₄×C₆₁	—	2	1	no	—
28	C₂×Dic₁₂₂	—	2	1	no	—
29	D₈×C₆₁	—	4	2	no	—
30	C₂×C₈×C₆₁	—	2	1	no	—
31	C₄⋊C₄×C₆₁	D₄	1	1	yes	characteristic
32	C₂²⋊C₄×C₆₁	—	2	1	no	—
33	C₄²×C₆₁	C₂³	1	1	yes	characteristic
34	C₄⋊Dic₆₁	—	2	1	no	—
35	C₄⋊Dic₆₁	—	2	1	no	—
36	C₂×#92418	—	2	1	no	—
37	C₂×#92418	—	2	1	no	—
38	C₂×#92418	—	4	2	no	—
39	C₂×#92418	D₄	1	1	yes	characteristic
40	C₂×#92418	D₄	1	1	yes	characteristic
41	C₂³×C₆₁	—	2	1	no	—
42	C₂³×C₆₁	—	4	1	no	—
43	D₄×C₆₁	—	2	1	no	—
44	D₄×C₆₁	—	4	1	no	—
45	D₄×C₆₁	—	4	1	no	—
46	D₄×C₆₁	—	4	1	no	—
47	D₄×C₆₁	—	4	1	no	—
48	D₄×C₆₁	—	8	1	no	—
49	D₄×C₆₁	SD₁₆	1	2	yes	—
50	C₂×C₄×C₆₁	—	2	1	no	—
51	C₂×C₄×C₆₁	—	4	1	no	—
52	C₂×C₄×C₆₁	—	4	1	no	—
53	C₂×C₄×C₆₁	C₂×D₄	1	1	yes	characteristic
54	C₂×C₄×C₆₁	C₂×D₄	1	1	yes	characteristic
55	C₂×C₄×C₆₁	#41	1	1	yes	characteristic
56	C₂×Dic₆₁	—	4	1	no	—
57	Dic₁₂₂	—	8	1	no	—
58	C₈×C₆₁	—	4	1	no	—
59	#92418	—	2	1	no	—
60	#92418	—	2	1	no	—
61	#92418	—	4	1	no	—
62	#92418	—	4	1	no	—
63	C₂²×C₆₁	—	2	2	no	—
64	C₂²×C₆₁	—	4	1	no	—
65	C₂²×C₆₁	—	4	1	no	—
66	C₂²×C₆₁	—	8	1	no	—
67	C₂²×C₆₁	—	8	1	no	—
68	C₂²×C₆₁	—	8	1	no	—
69	C₂²×C₆₁	#121	1	1	yes	characteristic
70	C₄×C₆₁	—	2	1	no	—
71	C₄×C₆₁	—	4	1	no	—
72	C₄×C₆₁	C₂×SD₁₆	1	1	yes	characteristic
73	C₄×C₆₁	#135	1	1	yes	characteristic
74	C₄×C₆₁	#135	1	1	yes	characteristic
75	C₄×C₆₁	AGL(1,ℤ/8)	1	1	yes	characteristic
76	Dic₆₁	—	8	1	no	—
77	#1677	—	61	1	no	maximal, Sylow
78	C₂×C₆₁	—	4	2	no	—
79	C₂×C₆₁	—	16	1	no	—
80	C₂×C₆₁	#463	1	1	yes	characteristic
81	C₂×C₆₁	#465	1	1	yes	characteristic
82	C₂×C₆₁	#472	1	1	yes	characteristic
83	#486	—	61	1	no	—
84	#476	—	61	1	no	—
85	#459	D₆₁	1	1	yes	characteristic
86	C₈×D₄	—	61	1	no	—
87	#334	—	61	1	no	—
88	#333	—	61	1	no	—
89	#331	—	61	1	no	—
90	C₆₁	#1677	1	1	yes	characteristic, complemented, Sylow
91	C₂×D₈	—	2	1	no	—
92	C₂²×C₈	—	122	1	no	—
93	C₄⋊Q₈	—	61	1	no	—
94	#127	C₂×D₆₁	1	1	yes	characteristic
95	C₄×D₄	C₂×D₆₁	1	1	yes	characteristic
96	#106	—	122	1	no	—
97	C₄⋊C₈	—	61	1	no	—
98	C₄⋊C₈	—	61	1	no	—
99	C₄⋊C₈	C₂×D₆₁	1	1	yes	characteristic
100	#102	—	2	1	no	—
101	#102	—	122	1	no	—
102	#102	—	122	1	no	—
103	#102	—	122	1	no	—
104	C₂²⋊C₈	—	122	1	no	—
105	C₄×C₈	—	61	1	no	—
106	C₂×Q₈	—	122	1	no	—
107	C₂×D₄	—	2	1	no	—
108	C₂×D₄	—	2	1	no	—
109	C₂×D₄	—	2	1	no	—
110	C₂×D₄	C₂²⋊D₆₁	1	1	yes	characteristic
111	C₂²×C₄	—	2	1	no	—
112	D₈	—	4	2	no	—
113	C₂×C₈	—	2	1	no	—
114	C₂×C₈	—	61	1	no	—
115	C₂×C₈	—	61	1	no	—
116	C₂×C₈	—	122	1	no	—
117	C₂×C₈	—	122	1	no	—
118	C₂×C₈	—	244	2	no	—
119	C₄⋊C₄	—	122	1	no	—
120	C₄⋊C₄	—	122	1	no	—
121	C₄⋊C₄	C₂²⋊D₆₁	1	1	yes	characteristic
122	C₂²⋊C₄	—	2	1	no	—
123	C₄²	C₂²×D₆₁	1	1	yes	characteristic
124	C₂³	—	2	1	no	—
125	C₂³	—	4	1	no	—
126	Q₈	—	488	1	no	—
127	D₄	—	2	1	no	—
128	D₄	—	4	1	no	—
129	D₄	—	4	1	no	—
130	D₄	—	4	1	no	—
131	D₄	—	4	1	no	—
132	D₄	—	8	1	no	—
133	D₄	#11758200	1	2	yes	—
134	C₂×C₄	—	2	1	no	—
135	C₂×C₄	—	4	1	no	—
136	C₂×C₄	—	4	1	no	—
137	C₂×C₄	—	244	1	no	—
138	C₂×C₄	D₄×D₆₁	1	1	yes	characteristic
139	C₂×C₄	#11758224	1	1	yes	characteristic
140	C₂×C₄	C₂×C₂²⋊D₆₁	1	1	yes	characteristic
141	C₈	—	4	1	no	—
142	C₈	—	122	1	no	—
143	C₈	—	122	1	no	—
144	C₈	—	244	1	no	—
145	C₈	—	244	1	no	—
146	C₂²	—	2	2	no	—
147	C₂²	—	4	1	no	—
148	C₂²	—	4	1	no	—
149	C₂²	—	8	1	no	—
150	C₂²	—	8	1	no	—
151	C₂²	—	8	1	no	—
152	C₂²	#14519749	1	1	yes	characteristic
153	C₄	—	2	1	no	—
154	C₄	—	4	1	no	—
155	C₄	—	488	1	no	—
156	C₄	C₂×#11758200	1	1	yes	characteristic
157	C₄	#14519796	1	1	yes	characteristic
158	C₄	#14519797	1	1	yes	characteristic
159	C₄	#14519801	1	1	yes	characteristic
160	C₂	—	4	2	no	—
161	C₂	—	16	1	no	—
162	C₂	#16839975	1	1	yes	characteristic
163	C₂	#16839989	1	1	yes	characteristic
164	C₂	#16840030	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	#14519749
commutator	C₄²×C₆₁	C₂³
Frattini	C₄²	C₂²×D₆₁
Fitting	C₆₁×#459	C₂
socle	C₂²×C₆₁	#121
hypercenter	#459	D₆₁
hypocenter	C₆₁	#1677
LCS	G ▷ C₄²×C₆₁ ▷ C₂²×C₆₁ ▷ C₆₁
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #459

Complements

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

N	H ≅ G/N	type
1. #16840108	C₂	semidirect
2. #16840052	C₂	semidirect
3. D₄×#92418	C₂	semidirect
4. #16839512	C₂	semidirect
5. #16839510	C₂	semidirect
6. #16839506	C₂	semidirect
90. C₆₁	#1677	semidirect