Finite Group Databasebeta

Subgroups — #21461383

Order 7808.

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

normal subgroups only

This group has 166 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	#16840783	C₂	1	1	yes	maximal, characteristic
2	#16840765	C₂	1	1	yes	maximal, characteristic
3	#16840626	C₂	1	1	yes	maximal, characteristic, complemented
4	#16840625	C₂	1	1	yes	maximal, characteristic, complemented
5	#16840624	C₂	1	1	yes	maximal, characteristic, complemented
6	#16840623	C₂	1	1	yes	maximal, characteristic, complemented
7	M₁₆×Dic₆₁	C₂	1	1	yes	maximal, characteristic
8	C₂×#11758217	—	2	1	no	—
9	#14519844	—	2	1	no	—
10	C₄⋊#92420	C₂²	1	1	yes	characteristic
11	C₄⋊#92420	C₂²	1	1	yes	characteristic
12	C₄⋊#92420	C₂²	1	1	yes	characteristic
13	C₄⋊#92420	C₂²	1	1	yes	characteristic
14	C₂²⋊#92420	—	2	1	no	—
15	#14519832	—	2	1	no	—
16	C₂×#11758194	C₄	1	1	yes	characteristic
17	C₂×C₄×Dic₆₁	C₂²	1	1	yes	characteristic
18	#14519687	C₄	1	1	yes	characteristic
19	#14519687	C₄	1	1	yes	characteristic
20	C₈×Dic₆₁	C₂²	1	1	yes	characteristic
21	C₈×Dic₆₁	C₂²	1	1	yes	characteristic
22	C₂×M₁₆×C₆₁	C₄	1	1	yes	characteristic
23	C₂²×C₄×C₆₁	C₂×C₄	1	1	yes	characteristic
24	C₂²×Dic₆₁	D₄	1	1	yes	characteristic
25	C₂²×Dic₆₁	Q₈	1	1	yes	characteristic
26	#11758217	—	4	2	no	—
27	C₂×#92420	—	2	1	no	—
28	C₂×#92420	—	2	1	no	—
29	C₂×#92420	—	2	1	no	—
30	C₂×#92420	—	2	1	no	—
31	M₁₆×C₆₁	—	2	2	no	—
32	C₂×C₈×C₆₁	C₂×C₄	1	1	yes	characteristic
33	C₂×C₈×C₆₁	C₂×C₄	1	1	yes	characteristic
34	C₄×Dic₆₁	C₂×C₄	1	1	yes	characteristic
35	C₄×Dic₆₁	D₄	1	1	yes	characteristic
36	C₄×Dic₆₁	Q₈	1	1	yes	characteristic
37	C₄×Dic₆₁	C₂³	1	1	yes	characteristic
38	#11758194	—	2	2	no	—
39	C₂×#92418	C₂×C₄	1	1	yes	characteristic
40	C₂×#92418	C₂×C₄	1	1	yes	characteristic
41	C₂³×C₆₁	C₄⋊C₄	1	1	yes	characteristic
42	C₂×C₄×C₆₁	—	2	1	no	—
43	C₂×C₄×C₆₁	—	2	1	no	—
44	C₂×C₄×C₆₁	C₄⋊C₄	1	1	yes	characteristic
45	C₂×C₄×C₆₁	C₂²×C₄	1	1	yes	characteristic
46	C₂×Dic₆₁	—	2	1	no	—
47	C₂×Dic₆₁	—	2	1	no	—
48	C₂×Dic₆₁	—	4	1	no	—
49	C₂×Dic₆₁	C₄⋊C₄	1	1	yes	characteristic
50	C₂×Dic₆₁	C₄⋊C₄	1	1	yes	characteristic
51	C₂×Dic₆₁	C₂×D₄	1	1	yes	characteristic
52	C₂×Dic₆₁	C₂×Q₈	1	1	yes	characteristic
53	#92420	—	4	1	no	—
54	#92420	—	4	1	no	—
55	#92420	—	4	1	no	—
56	#92420	—	4	1	no	—
57	C₈×C₆₁	—	2	1	no	—
58	C₈×C₆₁	M₁₆	1	2	yes	complemented
59	#92418	—	2	1	no	—
60	#92418	M₁₆	1	2	yes	complemented
61	C₂²×C₆₁	—	2	1	no	—
62	C₂²×C₆₁	—	2	1	no	—
63	C₂²×C₆₁	—	2	1	no	—
64	C₂²×C₆₁	C₂×C₄⋊C₄	1	1	yes	characteristic
65	C₄×C₆₁	—	2	1	no	—
66	C₄×C₆₁	C₂×M₁₆	1	1	yes	characteristic
67	C₄×C₆₁	C₂×M₁₆	1	1	yes	characteristic
68	Dic₆₁	—	2	1	no	—
69	Dic₆₁	—	2	1	no	—
70	Dic₆₁	—	2	1	no	—
71	Dic₆₁	AGL(1,ℤ/8)	1	1	yes	characteristic
72	Dic₆₁	#137	1	1	yes	characteristic
73	#1569	—	61	1	no	maximal, Sylow
74	C₂×C₆₁	—	4	1	no	—
75	C₂×C₆₁	#423	1	1	yes	characteristic
76	C₂×C₆₁	#428	1	1	yes	characteristic
77	C₂×C₆₁	#430	1	1	yes	characteristic
78	#432	—	61	1	no	—
79	#423	—	61	1	no	—
80	C₄×M₁₆	—	61	1	no	—
81	#335	—	61	1	no	—
82	#335	—	61	1	no	—
83	#334	—	61	1	no	—
84	#334	—	61	1	no	—
85	C₆₁	#1569	1	1	yes	characteristic, complemented, Sylow
86	C₂×M₁₆	—	61	1	no	—
87	C₂×M₁₆	—	122	1	no	—
88	C₂×M₁₆	#6923	1	1	yes	characteristic
89	C₂×C₄²	—	61	1	no	—
90	C₄⋊C₈	—	61	1	no	—
91	C₄⋊C₈	—	61	1	no	—
92	C₄⋊C₈	—	61	1	no	—
93	C₄⋊C₈	—	61	1	no	—
94	C₄⋊C₈	—	122	1	no	—
95	C₂²⋊C₈	—	122	1	no	—
96	#97	—	61	1	no	—
97	#97	—	61	1	no	—
98	#97	—	122	1	no	—
99	C₄×C₈	—	61	1	no	—
100	C₄×C₈	—	61	1	no	—
101	C₂²×C₄	—	61	1	no	—
102	C₂²×C₄	—	61	1	no	—
103	C₂²×C₄	C₂×#6923	1	1	yes	characteristic
104	M₁₆	—	2	2	no	—
105	M₁₆	—	122	2	no	—
106	M₁₆	—	244	2	no	—
107	C₂×C₈	—	61	1	no	—
108	C₂×C₈	—	61	1	no	—
109	C₂×C₈	—	122	1	no	—
110	C₂×C₈	—	122	1	no	—
111	C₂×C₈	—	122	1	no	—
112	C₂×C₈	—	122	1	no	—
113	C₂×C₈	C₂×#6923	1	1	yes	characteristic
114	C₂×C₈	C₂×#6923	1	1	yes	characteristic
115	C₄²	—	61	1	no	—
116	C₄²	—	61	1	no	—
117	C₄²	—	61	1	no	—
118	C₄²	—	61	1	no	—
119	C₂³	C₄⋊#6923	1	1	yes	characteristic
120	C₂×C₄	—	2	1	no	—
121	C₂×C₄	—	2	1	no	—
122	C₂×C₄	—	61	1	no	—
123	C₂×C₄	—	61	1	no	—
124	C₂×C₄	—	61	1	no	—
125	C₂×C₄	—	61	1	no	—
126	C₂×C₄	—	122	1	no	—
127	C₂×C₄	—	122	1	no	—
128	C₂×C₄	—	244	1	no	—
129	C₂×C₄	C₄⋊#6923	1	1	yes	characteristic
130	C₂×C₄	C₂²×#6923	1	1	yes	characteristic
131	C₈	—	2	1	no	—
132	C₈	—	61	2	no	—
133	C₈	—	122	1	no	—
134	C₈	—	244	1	no	—
135	C₈	—	244	1	no	—
136	C₈	—	244	1	no	—
137	C₈	—	244	1	no	—
138	C₈	#11758217	1	2	yes	complemented
139	C₂²	—	2	1	no	—
140	C₂²	—	2	1	no	—
141	C₂²	—	2	1	no	—
142	C₂²	C₂×C₄⋊#6923	1	1	yes	characteristic
143	C₄	—	2	1	no	—
144	C₄	—	61	1	no	—
145	C₄	—	61	1	no	—
146	C₄	—	122	1	no	—
147	C₄	—	122	1	no	—
148	C₄	—	122	1	no	—
149	C₄	C₂×#11758217	1	1	yes	characteristic
150	C₄	D₆₁⋊M₁₆	1	1	yes	characteristic
151	C₂	—	4	1	no	—
152	C₂	#16840765	1	1	yes	characteristic
153	C₂	#16840773	1	1	yes	characteristic
154	C₂	#16840777	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×C₄⋊#6923
commutator	C₂×C₄×C₆₁	C₂²×C₄
Frattini	C₂×C₄	C₂²×#6923
Fitting	C₂×M₁₆×C₆₁	C₄
socle	C₂²×C₆₁	C₂×C₄⋊C₄
hypercenter	C₂×M₁₆	#6923
hypocenter	C₆₁	#1569
LCS	G ▷ C₂×C₄×C₆₁ ▷ C₂×C₆₁ ▷ C₆₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×M₁₆

Complements

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

N	H ≅ G/N	type
3. #16840626	C₂	semidirect
4. #16840625	C₂	semidirect
5. #16840624	C₂	semidirect
6. #16840623	C₂	semidirect
58. C₈×C₆₁	M₁₆	semidirect
60. #92418	M₁₆	semidirect
85. C₆₁	#1569	semidirect
138. C₈	#11758217	semidirect