Finite Group Databasebeta

Subgroups — #21445351

Order 7808.

170 conjugacy classes of subgroups; only the 49 normal subgroups are shown.

normal subgroups only

This group has 170 conjugacy classes of subgroups — too many to draw. Only its 49 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	#16840053	C₂	1	1	yes	maximal, characteristic, complemented
3	#16839954	C₂	1	1	yes	maximal, characteristic
4	#16839839	C₂	1	1	yes	maximal, characteristic, complemented
5	#16839514	C₂	1	1	yes	maximal, characteristic, complemented
6	#16839510	C₂	1	1	yes	maximal, characteristic, complemented
7	C₆₁×#331	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×#11758199	—	2	1	no	—
9	C₂×#11758189	—	2	1	no	—
10	#14519777	C₂²	1	1	yes	characteristic
11	C₄×D₂₄₄	C₂²	1	1	yes	characteristic
12	#14519709	—	2	1	no	—
13	C₄⋊#92418	C₂²	1	1	yes	characteristic
14	#14519699	—	2	1	no	—
15	#14519698	—	2	1	no	—
16	#14519698	—	2	1	no	—
17	#14519698	—	2	1	no	—
18	#14519689	—	2	1	no	—
19	C₄×#92418	C₂²	1	1	yes	characteristic
20	#127×C₆₁	C₂²	1	1	yes	characteristic
21	C₄⋊C₈×C₆₁	C₂²	1	1	yes	characteristic
22	C₄⋊C₈×C₆₁	C₂²	1	1	yes	characteristic
23	C₂×D₄×C₆₁	—	2	1	no	—
24	C₂×D₄×C₆₁	—	2	1	no	—
25	C₂×D₄×C₆₁	—	2	1	no	—
26	C₂×D₂₄₄	D₄	1	1	yes	characteristic
27	C₂×C₄×D₆₁	—	2	1	no	—
28	C₂×Dic₁₂₂	—	2	1	no	—
29	C₂×C₈×C₆₁	—	2	1	no	—
30	C₂×C₈×C₆₁	—	2	1	no	—
31	C₄²×C₆₁	C₂³	1	1	yes	characteristic
32	#11758199	—	4	2	no	—
33	#11758198	—	2	1	no	—
34	C₄⋊Dic₆₁	—	2	1	no	—
35	C₄⋊Dic₆₁	—	2	1	no	—
36	C₄⋊Dic₆₁	D₄	1	1	yes	characteristic
37	C₂×#92418	—	2	1	no	—
38	C₂×#92418	D₄	1	1	yes	characteristic
39	C₂×#92418	D₄	1	1	yes	characteristic
40	#11758189	—	4	2	no	—
41	C₂³×C₆₁	—	4	1	no	—
42	C₂²×D₆₁	—	2	1	no	—
43	D₄×C₆₁	—	4	1	no	—
44	D₄×C₆₁	—	4	1	no	—
45	D₄×C₆₁	—	4	1	no	—
46	D₄×C₆₁	—	4	1	no	—
47	D₄×C₆₁	—	8	1	no	—
48	C₂×C₄×C₆₁	C₂×D₄	1	1	yes	characteristic
49	C₂×C₄×C₆₁	C₂×D₄	1	1	yes	characteristic
50	C₂×C₄×C₆₁	#41	1	1	yes	characteristic
51	C₂×Dic₆₁	—	2	1	no	—
52	C₂×Dic₆₁	—	4	1	no	—
53	D₂₄₄	—	2	1	no	—
54	D₂₄₄	—	2	1	no	—
55	C₄×D₆₁	—	4	1	no	—
56	C₄×D₆₁	—	4	1	no	—
57	Dic₁₂₂	—	8	1	no	—
58	C₈×C₆₁	—	4	1	no	—
59	C₈×C₆₁	—	4	1	no	—
60	#92418	—	2	1	no	—
61	#92418	—	2	1	no	—
62	#92418	—	4	1	no	—
63	C₂²×C₆₁	—	8	1	no	—
64	C₂²×C₆₁	—	8	1	no	—
65	C₂²×C₆₁	—	8	1	no	—
66	C₂²×C₆₁	#121	1	1	yes	characteristic
67	C₂×D₆₁	—	4	1	no	—
68	C₂×D₆₁	—	4	1	no	—
69	C₂×D₆₁	—	4	1	no	—
70	C₄×C₆₁	—	2	1	no	—
71	C₄×C₆₁	#135	1	1	yes	characteristic
72	C₄×C₆₁	AGL(1,ℤ/8)	1	1	yes	characteristic
73	C₄×C₆₁	AGL(1,ℤ/8)	1	1	yes	characteristic
74	C₄×C₆₁	#137	1	1	yes	characteristic
75	Dic₆₁	—	4	1	no	—
76	Dic₆₁	—	8	1	no	—
77	#1697	—	61	1	no	maximal, Sylow
78	C₂×C₆₁	—	16	1	no	—
79	C₂×C₆₁	#463	1	1	yes	characteristic
80	C₂×C₆₁	#468	1	1	yes	characteristic
81	C₂×C₆₁	#472	1	1	yes	characteristic
82	D₆₁	—	8	1	no	—
83	#486	—	61	1	no	—
84	#476	—	61	1	no	—
85	#459	—	61	1	no	—
86	#436	—	61	1	no	—
87	#335	—	61	1	no	—
88	#333	—	61	1	no	—
89	#331	D₆₁	1	1	yes	characteristic, complemented
90	C₆₁	#1697	1	1	yes	characteristic, complemented, Sylow
91	C₂×D₈	—	122	1	no	—
92	C₂×M₁₆	—	122	1	no	—
93	C₄⋊Q₈	—	61	1	no	—
94	#127	C₂×D₆₁	1	1	yes	characteristic
95	C₄×D₄	—	61	1	no	—
96	#106	—	122	1	no	—
97	C₄⋊C₈	—	61	1	no	—
98	C₄⋊C₈	C₂×D₆₁	1	1	yes	characteristic
99	C₄⋊C₈	C₂×D₆₁	1	1	yes	characteristic
100	#102	—	122	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	—	61	1	no	—
111	C₂²×C₄	—	122	1	no	—
112	D₈	—	244	2	no	—
113	M₁₆	—	244	2	no	—
114	C₂×C₈	—	2	1	no	—
115	C₂×C₈	—	2	1	no	—
116	C₂×C₈	—	61	1	no	—
117	C₂×C₈	—	61	1	no	—
118	C₂×C₈	—	122	1	no	—
119	C₄⋊C₄	—	61	1	no	—
120	C₄⋊C₄	—	122	1	no	—
121	C₄⋊C₄	—	122	1	no	—
122	C₂²⋊C₄	—	122	1	no	—
123	C₄²	C₂²×D₆₁	1	1	yes	characteristic
124	C₂³	—	4	1	no	—
125	C₂³	—	122	1	no	—
126	Q₈	—	488	1	no	—
127	D₄	—	4	1	no	—
128	D₄	—	4	1	no	—
129	D₄	—	4	1	no	—
130	D₄	—	4	1	no	—
131	D₄	—	8	1	no	—
132	D₄	—	122	1	no	—
133	D₄	—	122	1	no	—
134	C₂×C₄	—	122	1	no	—
135	C₂×C₄	—	244	1	no	—
136	C₂×C₄	—	244	1	no	—
137	C₂×C₄	—	244	1	no	—
138	C₂×C₄	#11758222	1	1	yes	characteristic
139	C₂×C₄	D₄×D₆₁	1	1	yes	characteristic
140	C₂×C₄	D₄×D₆₁	1	1	yes	characteristic
141	C₈	—	4	1	no	—
142	C₈	—	4	1	no	—
143	C₈	—	122	1	no	—
144	C₈	—	122	1	no	—
145	C₈	—	244	1	no	—
146	C₂²	—	8	1	no	—
147	C₂²	—	8	1	no	—
148	C₂²	—	8	1	no	—
149	C₂²	—	244	1	no	—
150	C₂²	—	244	1	no	—
151	C₂²	—	244	1	no	—
152	C₂²	#14519750	1	1	yes	characteristic
153	C₄	—	2	1	no	—
154	C₄	—	488	1	no	—
155	C₄	—	244	1	no	—
156	C₄	#14519798	1	1	yes	characteristic
157	C₄	#14519801	1	1	yes	characteristic
158	C₄	#14519801	1	1	yes	characteristic
159	C₄	#14519806	1	1	yes	characteristic
160	C₂	—	16	1	no	—
161	C₂	—	488	1	no	—
162	C₂	#16839976	1	1	yes	characteristic
163	C₂	#16840007	1	1	yes	characteristic
164	C₂	#16840031	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	#14519750
commutator	C₄²×C₆₁	C₂³
Frattini	C₄²	C₂²×D₆₁
Fitting	C₆₁×#331	C₂
socle	C₂²×C₆₁	#121
hypercenter	#331	D₆₁
hypocenter	C₆₁	#1697
LCS	G ▷ C₄²×C₆₁ ▷ C₂²×C₆₁ ▷ C₆₁
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #331

Complements

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

N	H ≅ G/N	type
1. #16840108	C₂	semidirect
2. #16840053	C₂	semidirect
4. #16839839	C₂	semidirect
5. #16839514	C₂	semidirect
6. #16839510	C₂	semidirect
7. C₆₁×#331	C₂	semidirect
89. #331	D₆₁	semidirect
90. C₆₁	#1697	semidirect