Finite Group Databasebeta

Subgroups — #21445484

Order 7808.

242 conjugacy classes of subgroups; only the 61 normal subgroups are shown.

normal subgroups only

This group has 242 conjugacy classes of subgroups — too many to draw. Only its 61 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	#16840138	C₂	1	1	yes	maximal, characteristic, complemented
2	#16840138	C₂	1	1	yes	maximal, characteristic, complemented
3	#16840134	C₂	1	1	yes	maximal, characteristic, complemented
4	#16840109	C₂	1	1	yes	maximal, characteristic, complemented
5	#16840108	C₂	1	1	yes	maximal, characteristic, complemented
6	#16839484	C₂	1	1	yes	maximal, characteristic, complemented
7	C₆₁×#493	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×#11758200	—	2	2	no	—
9	C₂×#11758199	—	2	2	no	—
10	C₂×#11758199	—	2	2	no	—
11	#14519777	C₂²	1	1	yes	characteristic
12	#14519774	C₂²	1	1	yes	characteristic
13	#14519699	—	2	2	no	—
14	#14519698	—	2	2	no	—
15	C₄×#92418	C₂²	1	1	yes	characteristic
16	C₄×#92418	C₂²	1	1	yes	characteristic
17	C₂×D₈×C₆₁	—	2	2	no	—
18	#127×C₆₁	C₂²	1	1	yes	characteristic
19	#127×C₆₁	C₂²	1	1	yes	characteristic
20	C₄×C₈×C₆₁	C₂²	1	1	yes	characteristic
21	C₂×D₄×C₆₁	—	2	1	no	—
22	C₂×D₄×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	—	2	1	no	—
27	C₂×D₂₄₄	—	2	1	no	—
28	C₂×D₂₄₄	—	2	1	no	—
29	C₂×D₂₄₄	—	2	1	no	—
30	C₂×Dic₁₂₂	—	2	1	no	—
31	D₈×C₆₁	—	4	4	no	—
32	C₂×C₈×C₆₁	D₄	1	2	yes	—
33	C₄²×C₆₁	C₂³	1	1	yes	characteristic
34	#11758200	—	4	4	no	—
35	#11758199	—	4	4	no	—
36	#11758199	—	4	4	no	—
37	C₄⋊Dic₆₁	—	2	1	no	—
38	C₄⋊Dic₆₁	—	2	1	no	—
39	C₂×#92418	D₄	1	2	yes	—
40	C₂×#92418	D₄	1	2	yes	—
41	C₂³×C₆₁	—	4	1	no	—
42	C₂³×C₆₁	—	4	1	no	—
43	C₂²×D₆₁	—	4	1	no	—
44	D₄×C₆₁	—	4	2	no	—
45	D₄×C₆₁	—	4	2	no	—
46	D₄×C₆₁	—	4	2	no	—
47	D₄×C₆₁	—	4	2	no	—
48	D₄×C₆₁	—	4	2	no	—
49	D₄×C₆₁	—	4	2	no	—
50	C₂×C₄×C₆₁	C₂×D₄	1	1	yes	characteristic
51	C₂×C₄×C₆₁	C₂×D₄	1	1	yes	characteristic
52	C₂×C₄×C₆₁	C₂×D₄	1	1	yes	characteristic
53	C₂×Dic₆₁	—	4	1	no	—
54	D₂₄₄	—	4	2	no	—
55	D₂₄₄	—	4	2	no	—
56	D₂₄₄	—	4	2	no	—
57	Dic₁₂₂	—	4	2	no	—
58	C₈×C₆₁	—	2	2	no	—
59	#92418	—	2	2	no	—
60	#92418	D₈	1	4	yes	complemented
61	C₂²×C₆₁	—	8	1	no	—
62	C₂²×C₆₁	—	8	1	no	—
63	C₂²×C₆₁	—	8	1	no	—
64	C₂²×C₆₁	—	8	1	no	—
65	C₂²×C₆₁	—	8	1	no	—
66	C₂²×C₆₁	—	8	1	no	—
67	C₂²×C₆₁	#127	1	1	yes	characteristic
68	C₂×D₆₁	—	8	1	no	—
69	C₂×D₆₁	—	8	1	no	—
70	C₂×D₆₁	—	8	1	no	—
71	C₄×C₆₁	C₂×D₈	1	2	yes	—
72	C₄×C₆₁	AGL(1,ℤ/8)	1	2	yes	—
73	C₄×C₆₁	AGL(1,ℤ/8)	1	2	yes	—
74	Dic₆₁	—	8	1	no	—
75	#1721	—	61	1	no	maximal, Sylow
76	C₂×C₆₁	—	16	1	no	—
77	C₂×C₆₁	—	16	1	no	—
78	C₂×C₆₁	#493	1	1	yes	characteristic
79	C₂×C₆₁	#496	1	1	yes	characteristic
80	C₂×C₆₁	#496	1	1	yes	characteristic
81	D₆₁	—	16	1	no	—
82	#493	—	61	1	no	—
83	#493	—	61	1	no	—
84	#493	D₆₁	1	1	yes	characteristic, complemented
85	#492	—	61	1	no	—
86	#486	—	61	1	no	—
87	#486	—	61	1	no	—
88	#322	—	61	1	no	—
89	C₆₁	#1721	1	1	yes	characteristic, complemented, Sylow
90	C₂×SD₁₆	—	122	2	no	—
91	C₂×D₈	—	2	2	no	—
92	C₂×D₈	—	122	2	no	—
93	C₂×D₈	—	122	2	no	—
94	C₄⋊Q₈	—	61	1	no	—
95	#127	—	61	1	no	—
96	#127	C₂×D₆₁	1	1	yes	characteristic
97	#127	C₂×D₆₁	1	1	yes	characteristic
98	#102	—	122	2	no	—
99	#102	—	122	2	no	—
100	C₄×C₈	—	61	1	no	—
101	C₄×C₈	—	61	1	no	—
102	C₄×C₈	C₂×D₆₁	1	1	yes	characteristic
103	C₂×Q₈	—	122	1	no	—
104	C₂×D₄	—	2	1	no	—
105	C₂×D₄	—	2	1	no	—
106	C₂×D₄	—	2	1	no	—
107	C₂×D₄	—	2	1	no	—
108	C₂×D₄	—	2	1	no	—
109	C₂×D₄	—	2	1	no	—
110	C₂×D₄	—	122	1	no	—
111	C₂×D₄	—	122	1	no	—
112	C₂×D₄	—	122	1	no	—
113	SD₁₆	—	244	4	no	—
114	D₈	—	4	4	no	—
115	D₈	—	244	4	no	—
116	D₈	—	244	4	no	—
117	C₂×C₈	—	61	2	no	—
118	C₂×C₈	—	61	2	no	—
119	C₂×C₈	C₂²⋊D₆₁	1	2	yes	—
120	C₄⋊C₄	—	122	1	no	—
121	C₄⋊C₄	—	122	1	no	—
122	C₄²	C₂²×D₆₁	1	1	yes	characteristic
123	C₂³	—	4	1	no	—
124	C₂³	—	4	1	no	—
125	C₂³	—	244	1	no	—
126	Q₈	—	244	2	no	—
127	D₄	—	4	2	no	—
128	D₄	—	4	2	no	—
129	D₄	—	4	2	no	—
130	D₄	—	4	2	no	—
131	D₄	—	4	2	no	—
132	D₄	—	4	2	no	—
133	D₄	—	244	2	no	—
134	D₄	—	244	2	no	—
135	D₄	—	244	2	no	—
136	C₂×C₄	—	244	1	no	—
137	C₂×C₄	D₄×D₆₁	1	1	yes	characteristic
138	C₂×C₄	D₄×D₆₁	1	1	yes	characteristic
139	C₂×C₄	C₂×C₂²⋊D₆₁	1	1	yes	characteristic
140	C₈	—	2	2	no	—
141	C₈	—	61	4	no	—
142	C₈	—	122	2	no	—
143	C₂²	—	8	1	no	—
144	C₂²	—	8	1	no	—
145	C₂²	—	8	1	no	—
146	C₂²	—	8	1	no	—
147	C₂²	—	8	1	no	—
148	C₂²	—	8	1	no	—
149	C₂²	—	488	1	no	—
150	C₂²	—	488	1	no	—
151	C₂²	—	488	1	no	—
152	C₂²	#14519773	1	1	yes	characteristic
153	C₄	—	488	1	no	—
154	C₄	D₈×D₆₁	1	2	yes	—
155	C₄	#14519801	1	2	yes	—
156	C₄	#14519803	1	2	yes	—
157	C₂	—	16	1	no	—
158	C₂	—	16	1	no	—
159	C₂	—	976	1	no	—
160	C₂	#16840137	1	1	yes	characteristic
161	C₂	#16840149	1	1	yes	characteristic
162	C₂	#16840150	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	#14519773
commutator	C₄²×C₆₁	C₂³
Frattini	C₄²	C₂²×D₆₁
Fitting	C₆₁×#493	C₂
socle	C₂²×C₆₁	#127
hypercenter	#493	D₆₁
hypocenter	C₆₁	#1721
LCS	G ▷ C₄²×C₆₁ ▷ C₂²×C₆₁ ▷ C₆₁
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #493

Complements

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

N	H ≅ G/N	type
1. #16840138	C₂	semidirect
2. #16840138	C₂	semidirect
3. #16840134	C₂	semidirect
4. #16840109	C₂	semidirect
5. #16840108	C₂	semidirect
6. #16839484	C₂	semidirect
7. C₆₁×#493	C₂	semidirect
60. #92418	D₈	semidirect
84. #493	D₆₁	semidirect
89. C₆₁	#1721	semidirect