Finite Group Databasebeta

Subgroups — #21462727

Order 7808.

322 conjugacy classes of subgroups; only the 148 normal subgroups are shown.

normal subgroups only

This group has 322 conjugacy classes of subgroups — too many to draw. Only its 148 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	#16840813	C₂	1	4	yes	maximal, complemented
2	#16840812	C₂	1	2	yes	maximal, complemented
3	#16840811	C₂	1	2	yes	maximal, complemented
4	#16840787	C₂	1	4	yes	maximal
5	#16840784	C₂	1	1	yes	maximal, characteristic
6	#16840780	C₂	1	1	yes	maximal, characteristic
7	#16840317	C₂	1	1	yes	maximal, characteristic
8	#14519844	C₂²	1	2	yes	—
9	#14519844	C₂²	1	4	yes	—
10	C₄⋊#92420	C₂²	1	2	yes	—
11	C₄⋊#92420	C₂²	1	4	yes	—
12	#14519834	C₂²	1	2	yes	—
13	#14519834	C₂²	1	2	yes	—
14	#14519832	C₂²	1	2	yes	—
15	#14519832	C₂²	1	4	yes	—
16	C₄×#92420	C₂²	1	1	yes	characteristic
17	C₄×#92420	C₂²	1	1	yes	characteristic
18	C₄×#92420	C₂²	1	4	yes	—
19	#14519777	C₄	1	1	yes	characteristic
20	#14519768	C₄	1	2	yes	—
21	#14519757	C₄	1	2	yes	—
22	#14519753	C₄	1	2	yes	—
23	Q₈×Dic₆₁	C₂²	1	2	yes	—
24	#14519742	C₂²	1	2	yes	—
25	#14519735	C₂²	1	2	yes	—
26	C₄²×D₆₁	C₂²	1	1	yes	characteristic
27	C₄⋊Q₈×C₆₁	C₄	1	1	yes	characteristic
28	C₂×Q₈×C₆₁	C₂×C₄	1	2	yes	—
29	C₂×C₄×D₆₁	C₂×C₄	1	1	yes	characteristic
30	C₂×C₄×D₆₁	C₂×C₄	1	1	yes	characteristic
31	C₂×C₄×D₆₁	C₂³	1	1	yes	characteristic
32	C₂×Dic₁₂₂	C₂×C₄	1	2	yes	—
33	C₂×#92420	C₂³	1	2	yes	—
34	C₂×#92420	C₂³	1	2	yes	—
35	C₂×#92420	C₂³	1	4	yes	—
36	C₄⋊C₄×C₆₁	C₂×C₄	1	2	yes	—
37	C₄⋊C₄×C₆₁	C₂×C₄	1	2	yes	—
38	C₄²×C₆₁	C₂×C₄	1	1	yes	characteristic
39	#11758198	C₂×C₄	1	4	yes	—
40	C₄⋊Dic₆₁	C₂×C₄	1	2	yes	—
41	C₄⋊Dic₆₁	C₂×C₄	1	2	yes	—
42	C₄⋊Dic₆₁	C₂×C₄	1	4	yes	—
43	#11758196	C₂×C₄	1	4	yes	—
44	C₄×Dic₆₁	C₂×C₄	1	1	yes	characteristic
45	C₄×Dic₆₁	C₂³	1	1	yes	characteristic
46	C₄×Dic₆₁	C₂³	1	4	yes	—
47	C₄×Dic₆₁	C₂³	1	1	yes	characteristic
48	C₂²×D₆₁	C₂²×C₄	1	1	yes	characteristic
49	Q₈×C₆₁	—	4	2	no	—
50	C₂×C₄×C₆₁	C₂²×C₄	1	1	yes	characteristic
51	C₂×C₄×C₆₁	C₂²×C₄	1	4	yes	—
52	C₂×C₄×C₆₁	C₂²×C₄	1	1	yes	characteristic
53	C₂×C₄×C₆₁	C₂²×C₄	1	1	yes	characteristic
54	C₂×Dic₆₁	C₂²×C₄	1	1	yes	characteristic
55	C₂×Dic₆₁	C₂²×C₄	1	1	yes	characteristic
56	C₂×Dic₆₁	C₂²×C₄	1	4	yes	—
57	C₂×Dic₆₁	C₂⁴	1	1	yes	characteristic
58	C₄×D₆₁	—	2	2	no	—
59	C₄×D₆₁	—	2	2	no	—
60	C₄×D₆₁	—	4	1	no	—
61	Dic₁₂₂	—	2	4	no	—
62	#92420	—	2	2	no	—
63	#92420	—	2	2	no	—
64	#92420	—	2	4	no	—
65	C₂²×C₆₁	C₂³×C₄	1	1	yes	characteristic
66	C₂×D₆₁	—	2	1	no	—
67	C₂×D₆₁	—	2	1	no	—
68	C₂×D₆₁	—	2	1	no	—
69	C₄×C₆₁	—	2	1	no	—
70	C₄×C₆₁	—	2	1	no	—
71	C₄×C₆₁	—	2	4	no	—
72	C₄×C₆₁	#131	1	2	yes	—
73	Dic₆₁	—	2	4	no	—
74	Dic₆₁	—	2	1	no	—
75	Dic₆₁	—	2	1	no	—
76	Dic₆₁	2₋¹⁺⁴	1	2	yes	—
77	#2985	—	61	1	no	maximal, Sylow
78	C₂×C₆₁	#519	1	1	yes	characteristic
79	C₂×C₆₁	C₂×#131	1	1	yes	characteristic
80	C₂×C₆₁	#568	1	1	yes	characteristic
81	D₆₁	—	4	1	no	—
82	#533	—	61	1	no	—
83	#446	—	61	2	no	—
84	#446	—	61	4	no	—
85	C₈×Q₈	—	61	2	no	—
86	#433	—	61	4	no	—
87	#432	—	61	1	no	—
88	#431	—	61	1	no	—
89	C₆₁	#2985	1	1	yes	characteristic, complemented, Sylow
90	C₄⋊Q₈	—	61	1	no	—
91	C₄⋊Q₈	#6923	1	1	yes	characteristic
92	#125	—	61	2	no	—
93	C₂²⋊Q₈	—	61	2	no	—
94	C₂²⋊Q₈	—	61	2	no	—
95	C₄×Q₈	—	61	2	no	—
96	C₄×Q₈	—	61	2	no	—
97	#117	—	61	2	no	—
98	C₂×C₄²	—	61	1	no	—
99	C₄⋊C₈	—	61	2	no	—
100	C₄⋊C₈	—	61	4	no	—
101	C₄⋊C₈	—	61	2	no	—
102	C₄⋊C₈	—	61	4	no	—
103	C₂²⋊C₈	—	61	2	no	—
104	C₂²⋊C₈	—	61	2	no	—
105	#97	—	61	2	no	—
106	#97	—	61	4	no	—
107	C₄×C₈	—	61	1	no	—
108	C₄×C₈	—	61	1	no	—
109	C₄×C₈	—	61	4	no	—
110	C₂×Q₈	—	61	2	no	—
111	C₂×Q₈	C₂×#6923	1	2	yes	—
112	C₂²×C₄	—	61	1	no	—
113	C₂²×C₄	—	61	1	no	—
114	C₂²×C₄	—	61	1	no	—
115	C₂×C₈	—	61	2	no	—
116	C₂×C₈	—	61	2	no	—
117	C₂×C₈	—	61	4	no	—
118	C₄⋊C₄	—	61	4	no	—
119	C₄⋊C₄	—	61	2	no	—
120	C₄⋊C₄	—	61	4	no	—
121	C₄⋊C₄	—	61	2	no	—
122	C₄⋊C₄	C₂×#6923	1	2	yes	—
123	C₄⋊C₄	C₂×#6923	1	2	yes	—
124	C₂²⋊C₄	—	61	4	no	—
125	C₄²	—	61	1	no	—
126	C₄²	—	61	4	no	—
127	C₄²	—	61	1	no	—
128	C₄²	—	61	1	no	—
129	C₄²	C₂×#6923	1	1	yes	characteristic
130	C₂³	—	61	1	no	—
131	Q₈	—	4	2	no	—
132	Q₈	—	122	4	no	—
133	C₂×C₄	—	61	1	no	—
134	C₂×C₄	—	61	1	no	—
135	C₂×C₄	—	61	1	no	—
136	C₂×C₄	—	61	4	no	—
137	C₂×C₄	—	122	2	no	—
138	C₂×C₄	—	122	2	no	—
139	C₂×C₄	—	244	1	no	—
140	C₂×C₄	C₂²×#6923	1	1	yes	characteristic
141	C₂×C₄	C₂²×#6923	1	1	yes	characteristic
142	C₂×C₄	C₂²×#6923	1	4	yes	—
143	C₂×C₄	C₂²×#6923	1	1	yes	characteristic
144	C₈	—	122	2	no	—
145	C₈	—	122	2	no	—
146	C₈	—	122	4	no	—
147	C₂²	—	122	1	no	—
148	C₂²	—	122	1	no	—
149	C₂²	—	122	1	no	—
150	C₂²	C₂³×#6923	1	1	yes	characteristic
151	C₄	—	2	1	no	—
152	C₄	—	2	1	no	—
153	C₄	—	2	4	no	—
154	C₄	—	61	2	no	—
155	C₄	—	122	1	no	—
156	C₄	—	122	4	no	—
157	C₄	—	122	1	no	—
158	C₄	#14519865	1	2	yes	—
159	C₂	—	244	1	no	—
160	C₂	#16840830	1	1	yes	characteristic
161	C₂	C₂×#14519865	1	1	yes	characteristic
162	C₂	#16840840	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂²	C₂³×#6923
commutator, socle	C₂²×C₆₁	C₂³×C₄
Fitting	C₄⋊Q₈×C₆₁	C₄
hypercenter	C₄⋊Q₈	#6923
hypocenter	C₆₁	#2985
LCS	G ▷ C₂²×C₆₁ ▷ C₆₁
UCS	C₁ ◁ C₂² ◁ C₄⋊Q₈

Complements

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

N	H ≅ G/N	type
1. #16840813	C₂	semidirect
2. #16840812	C₂	semidirect
3. #16840811	C₂	semidirect
89. C₆₁	#2985	semidirect