Finite Group Databasebeta

Subgroups — #21447765

Order 7808.

226 conjugacy classes of subgroups; only the 53 normal subgroups are shown.

normal subgroups only

This group has 226 conjugacy classes of subgroups — too many to draw. Only its 53 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	#16840197	C₂	1	2	yes	maximal, complemented
2	#16840015	C₂	1	1	yes	maximal, characteristic, complemented
3	#16839572	C₂	1	2	yes	maximal, complemented
4	#16839544	C₂	1	1	yes	maximal, characteristic, complemented
5	C₆₁×#575	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×D₄₈₈	C₂²	1	1	yes	characteristic, complemented
7	#14519751	—	2	1	no	—
8	#14519720	—	2	2	no	—
9	#14519718	—	2	2	no	—
10	#14519717	C₂²	1	2	yes	complemented
11	#14519709	C₂²	1	1	yes	characteristic, complemented
12	#14519699	—	2	1	no	—
13	C₂×#41×C₆₁	—	2	1	no	—
14	#137×C₆₁	—	2	2	no	—
15	AGL(1,ℤ/8)×C₆₁	—	2	2	no	—
16	#135×C₆₁	—	2	1	no	—
17	#135×C₆₁	C₂²	1	2	yes	—
18	C₂×M₁₆×C₆₁	C₂²	1	1	yes	characteristic
19	#41×C₆₁	—	2	2	no	—
20	#41×C₆₁	—	2	2	no	—
21	#41×C₆₁	—	4	1	no	—
22	#41×C₆₁	—	4	1	no	—
23	C₂×Q₈×C₆₁	—	2	1	no	—
24	C₂×D₄×C₆₁	—	2	1	no	—
25	C₂×D₄×C₆₁	—	4	1	no	—
26	C₂²×C₄×C₆₁	—	4	1	no	—
27	C₂²×C₄×C₆₁	D₄	1	1	yes	characteristic
28	C₂×C₂²⋊D₆₁	—	4	1	no	—
29	C₂×D₂₄₄	—	2	1	no	—
30	Q₁₆×C₆₁	—	2	1	no	—
31	Q₁₆×C₆₁	D₄	1	2	yes	complemented
32	SD₁₆×C₆₁	—	2	2	no	—
33	SD₁₆×C₆₁	—	2	2	no	—
34	D₈×C₆₁	—	2	1	no	—
35	D₈×C₆₁	D₄	1	2	yes	complemented
36	M₁₆×C₆₁	—	2	2	no	—
37	C₂×C₈×C₆₁	D₄	1	1	yes	characteristic
38	C₂×C₈×C₆₁	C₂³	1	1	yes	characteristic
39	#11758198	—	4	1	no	—
40	C₄⋊Dic₆₁	—	2	1	no	—
41	D₄₈₈	—	2	2	no	—
42	#11758185	—	2	2	no	—
43	C₂³×C₆₁	—	4	1	no	—
44	C₂³×C₆₁	D₈	1	1	yes	characteristic
45	C₂²×D₆₁	—	4	1	no	—
46	Q₈×C₆₁	—	2	2	no	—
47	Q₈×C₆₁	—	4	1	no	—
48	D₄×C₆₁	—	2	2	no	—
49	D₄×C₆₁	—	4	1	no	—
50	D₄×C₆₁	—	4	2	no	—
51	D₄×C₆₁	—	4	2	no	—
52	C₂×C₄×C₆₁	—	2	1	no	—
53	C₂×C₄×C₆₁	—	2	1	no	—
54	C₂×C₄×C₆₁	—	4	2	no	—
55	C₂×C₄×C₆₁	—	4	1	no	—
56	C₂×C₄×C₆₁	—	4	2	no	—
57	C₂×C₄×C₆₁	D₈	1	1	yes	characteristic
58	C₂×C₄×C₆₁	C₂×D₄	1	1	yes	characteristic
59	C₂²⋊D₆₁	—	8	2	no	—
60	C₂×Dic₆₁	—	4	1	no	—
61	D₂₄₄	—	4	1	no	—
62	D₂₄₄	—	4	1	no	—
63	C₈×C₆₁	—	2	1	no	—
64	C₈×C₆₁	C₂×D₄	1	2	yes	—
65	C₂²×C₆₁	—	2	1	no	—
66	C₂²×C₆₁	—	4	1	no	—
67	C₂²×C₆₁	—	4	2	no	—
68	C₂²×C₆₁	—	8	2	no	—
69	C₂²×C₆₁	C₂×D₈	1	1	yes	characteristic
70	C₂×D₆₁	—	8	1	no	—
71	C₂×D₆₁	—	8	1	no	—
72	C₂×D₆₁	—	8	1	no	—
73	C₄×C₆₁	—	2	1	no	—
74	C₄×C₆₁	—	4	2	no	—
75	C₄×C₆₁	#120	1	1	yes	characteristic
76	C₄×C₆₁	AGL(1,ℤ/8)	1	1	yes	characteristic
77	Dic₆₁	—	8	1	no	—
78	#2190	—	61	1	no	maximal, Sylow
79	C₂×C₆₁	—	2	1	no	—
80	C₂×C₆₁	—	4	1	no	—
81	C₂×C₆₁	—	8	2	no	—
82	C₂×C₆₁	#447	1	1	yes	characteristic
83	D₆₁	—	16	1	no	—
84	#575	D₆₁	1	1	yes	characteristic, complemented
85	#509	—	61	2	no	—
86	#469	—	61	1	no	—
87	#360	—	61	2	no	—
88	#349	—	61	1	no	—
89	C₆₁	#2190	1	1	yes	characteristic, complemented, Sylow
90	C₂×#41	—	2	1	no	—
91	#137	—	2	2	no	—
92	AGL(1,ℤ/8)	—	2	2	no	—
93	#135	—	2	1	no	—
94	#135	C₂×D₆₁	1	2	yes	—
95	C₂×D₈	—	61	1	no	—
96	C₂×M₁₆	C₂×D₆₁	1	1	yes	characteristic
97	#121	—	122	1	no	—
98	SD₃₂	—	122	2	no	—
99	D₁₆	—	122	2	no	—
100	M₃₂	—	61	2	no	—
101	#106	—	61	1	no	—
102	#102	—	122	1	no	—
103	#41	—	2	2	no	—
104	#41	—	2	2	no	—
105	#41	—	4	1	no	—
106	#41	—	4	1	no	—
107	C₂×Q₈	—	2	1	no	—
108	C₂×D₄	—	2	1	no	—
109	C₂×D₄	—	4	1	no	—
110	C₂×D₄	—	122	1	no	—
111	C₂×D₄	—	244	1	no	—
112	C₂²×C₄	—	4	1	no	—
113	C₂²×C₄	C₂²⋊D₆₁	1	1	yes	characteristic
114	Q₁₆	—	2	1	no	—
115	Q₁₆	C₂²⋊D₆₁	1	2	yes	complemented
116	SD₁₆	—	2	2	no	—
117	SD₁₆	—	2	2	no	—
118	D₈	—	2	1	no	—
119	D₈	—	122	2	no	—
120	D₈	C₂²⋊D₆₁	1	2	yes	complemented
121	M₁₆	—	2	2	no	—
122	C₂×C₈	C₂²⋊D₆₁	1	1	yes	characteristic
123	C₂×C₈	C₂²×D₆₁	1	1	yes	characteristic
124	C₄⋊C₄	—	122	1	no	—
125	C₂²⋊C₄	—	244	1	no	—
126	C₁₆	—	122	2	no	—
127	C₂³	—	4	1	no	—
128	C₂³	—	244	1	no	—
129	C₂³	#11758199	1	1	yes	characteristic
130	Q₈	—	2	2	no	—
131	Q₈	—	4	1	no	—
132	D₄	—	2	2	no	—
133	D₄	—	4	1	no	—
134	D₄	—	4	2	no	—
135	D₄	—	4	2	no	—
136	D₄	—	488	2	no	—
137	D₄	—	244	1	no	—
138	D₄	—	244	1	no	—
139	C₂×C₄	—	2	1	no	—
140	C₂×C₄	—	2	1	no	—
141	C₂×C₄	—	4	2	no	—
142	C₂×C₄	—	4	1	no	—
143	C₂×C₄	—	4	2	no	—
144	C₂×C₄	—	244	1	no	—
145	C₂×C₄	#11758199	1	1	yes	characteristic
146	C₂×C₄	C₂×C₂²⋊D₆₁	1	1	yes	characteristic
147	C₈	—	2	1	no	—
148	C₈	C₂×C₂²⋊D₆₁	1	2	yes	—
149	C₂²	—	2	1	no	—
150	C₂²	—	4	1	no	—
151	C₂²	—	4	2	no	—
152	C₂²	—	8	2	no	—
153	C₂²	—	488	1	no	—
154	C₂²	—	488	1	no	—
155	C₂²	—	488	1	no	—
156	C₂²	C₂×#11758199	1	1	yes	characteristic
157	C₄	—	2	1	no	—
158	C₄	—	4	2	no	—
159	C₄	—	488	1	no	—
160	C₄	#14519746	1	1	yes	characteristic
161	C₄	#14519803	1	1	yes	characteristic
162	C₂	—	2	1	no	—
163	C₂	—	4	1	no	—
164	C₂	—	8	2	no	—
165	C₂	—	976	1	no	—
166	C₂	#16839896	1	1	yes	characteristic

	subgroup	G/H
center	C₂	#16839896
commutator	C₂×C₈×C₆₁	C₂³
Frattini	C₂×C₈	C₂²×D₆₁
Fitting	C₆₁×#575	C₂
socle	C₂×C₆₁	#447
hypercenter	#575	D₆₁
hypocenter	C₆₁	#2190
LCS	G ▷ C₂×C₈×C₆₁ ▷ C₄×C₆₁ ▷ C₂×C₆₁ ▷ C₆₁
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ C₂×M₁₆ ◁ #575

Complements

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

N	H ≅ G/N	type
1. #16840197	C₂	semidirect
2. #16840015	C₂	semidirect
3. #16839572	C₂	semidirect
4. #16839544	C₂	semidirect
5. C₆₁×#575	C₂	semidirect
6. C₂×D₄₈₈	C₂²	semidirect
10. #14519717	C₂²	semidirect
11. #14519709	C₂²	semidirect
31. Q₁₆×C₆₁	D₄	semidirect
35. D₈×C₆₁	D₄	semidirect
84. #575	D₆₁	semidirect
89. C₆₁	#2190	semidirect
115. Q₁₆	C₂²⋊D₆₁	semidirect
120. D₈	C₂²⋊D₆₁	semidirect