Finite Group Databasebeta

Subgroups — #21447781

Order 7808.

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

normal subgroups only

This group has 210 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	#16840579	C₂	1	1	yes	maximal, characteristic, complemented
2	#16840198	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×#14519721	C₂	1	1	yes	maximal, characteristic, complemented
4	#16839577	C₂	1	1	yes	maximal, characteristic, complemented
5	#16839568	C₂	1	1	yes	maximal, characteristic, complemented
6	#16839546	C₂	1	1	yes	maximal, characteristic, complemented
7	C₆₁×#472	C₂	1	1	yes	maximal, characteristic, complemented
8	#14519824	—	2	1	no	—
9	#14519810	—	2	1	no	—
10	#14519810	—	2	2	no	—
11	#14519799	—	2	1	no	—
12	#14519799	C₂²	1	1	yes	characteristic
13	C₂×Dic₂₄₄	—	2	1	no	—
14	C₂×Dic₂₄₄	C₂²	1	1	yes	characteristic
15	#14519721	—	2	1	no	—
16	#14519721	—	2	2	no	—
17	#14519718	—	2	1	no	—
18	#14519717	C₂²	1	1	yes	characteristic
19	C₂×#11758185	C₂²	1	1	yes	characteristic
20	AGL(1,ℤ/8)×C₆₁	—	2	1	no	—
21	C₂×D₈×C₆₁	C₂²	1	1	yes	characteristic
22	#131×C₆₁	C₂²	1	1	yes	characteristic
23	#108×C₆₁	C₂²	1	1	yes	characteristic
24	#100×C₆₁	—	2	1	no	—
25	#41×C₆₁	D₄	1	1	yes	characteristic
26	C₂×D₄×C₆₁	—	2	1	no	—
27	Q₈×D₆₁	—	4	1	no	—
28	#11758224	—	4	2	no	—
29	#11758224	—	4	1	no	—
30	#11758222	—	2	2	no	—
31	#11758222	—	2	1	no	—
32	C₂×Dic₁₂₂	—	2	2	no	—
33	C₂×Dic₁₂₂	—	2	1	no	—
34	SD₁₆×C₆₁	—	4	1	no	—
35	D₈×C₆₁	—	2	1	no	—
36	D₈×C₆₁	—	2	1	no	—
37	D₈×C₆₁	—	4	1	no	—
38	M₁₆×C₆₁	—	2	1	no	—
39	M₁₆×C₆₁	—	2	1	no	—
40	M₁₆×C₆₁	D₄	1	1	yes	characteristic
41	C₂×C₈×C₆₁	—	2	1	no	—
42	C₂×C₈×C₆₁	C₂³	1	1	yes	characteristic
43	Dic₂₄₄	—	2	2	no	—
44	Dic₂₄₄	—	2	1	no	—
45	Dic₂₄₄	D₄	1	2	yes	—
46	Dic₂₄₄	D₄	1	1	yes	characteristic
47	D₄₈₈	D₄	1	1	yes	characteristic
48	#11758190	—	2	2	no	—
49	#11758190	—	2	1	no	—
50	#11758185	—	2	1	no	—
51	#11758185	—	2	1	no	—
52	C₂³×C₆₁	—	4	1	no	—
53	Q₈×C₆₁	D₈	1	1	yes	characteristic
54	D₄×C₆₁	—	2	1	no	—
55	D₄×C₆₁	—	4	1	no	—
56	D₄×C₆₁	—	4	1	no	—
57	D₄×C₆₁	D₈	1	1	yes	characteristic
58	C₂×C₄×C₆₁	—	2	1	no	—
59	C₂×C₄×C₆₁	C₂×D₄	1	1	yes	characteristic
60	C₂²⋊D₆₁	—	4	2	no	—
61	C₂²⋊D₆₁	—	4	1	no	—
62	C₂×Dic₆₁	—	4	2	no	—
63	C₂×Dic₆₁	—	4	1	no	—
64	D₂₄₄	—	2	1	no	—
65	C₄×D₆₁	—	4	2	no	—
66	C₄×D₆₁	—	4	1	no	—
67	Dic₁₂₂	—	2	2	no	—
68	Dic₁₂₂	—	2	1	no	—
69	Dic₁₂₂	—	4	2	no	—
70	Dic₁₂₂	—	4	1	no	—
71	C₈×C₆₁	—	2	1	no	—
72	C₈×C₆₁	—	4	1	no	—
73	C₈×C₆₁	C₂×D₄	1	1	yes	characteristic
74	C₈×C₆₁	C₂×D₄	1	1	yes	characteristic
75	C₂²×C₆₁	—	2	1	no	—
76	C₂²×C₆₁	—	8	1	no	—
77	C₂²×C₆₁	—	16	1	no	—
78	C₂²×C₆₁	AGL(1,ℤ/8)	1	1	yes	characteristic
79	C₂×D₆₁	—	4	1	no	—
80	C₄×C₆₁	—	2	1	no	—
81	C₄×C₆₁	#120	1	1	yes	characteristic
82	C₄×C₆₁	C₂×D₈	1	1	yes	characteristic
83	Dic₆₁	—	4	1	no	—
84	Dic₆₁	—	4	2	no	—
85	#2193	—	61	1	no	maximal, Sylow
86	C₂×C₆₁	—	2	1	no	—
87	C₂×C₆₁	—	4	1	no	—
88	C₂×C₆₁	—	16	1	no	—
89	C₂×C₆₁	#447	1	1	yes	characteristic
90	D₆₁	—	8	1	no	—
91	#578	—	61	1	no	—
92	#509	—	61	1	no	—
93	C₂×SD₃₂	—	61	1	no	—
94	#472	D₆₁	1	1	yes	characteristic, complemented
95	#362	—	61	1	no	—
96	#359	—	61	1	no	—
97	#350	—	61	1	no	—
98	C₆₁	#2193	1	1	yes	characteristic, complemented, Sylow
99	2₋¹⁺⁴	—	122	1	no	—
100	#137	—	122	2	no	—
101	#137	—	122	1	no	—
102	AGL(1,ℤ/8)	—	2	1	no	—
103	#135	—	61	1	no	—
104	#135	—	122	1	no	—
105	C₂×Q₁₆	—	61	1	no	—
106	C₂×Q₁₆	—	122	1	no	—
107	C₂×D₈	C₂×D₆₁	1	1	yes	characteristic
108	#131	C₂×D₆₁	1	1	yes	characteristic
109	SD₃₂	—	122	2	no	—
110	SD₃₂	—	122	1	no	—
111	D₁₆	—	122	1	no	—
112	M₃₂	—	61	1	no	—
113	C₂×C₁₆	—	61	1	no	—
114	#108	C₂×D₆₁	1	1	yes	characteristic
115	#100	—	2	1	no	—
116	#41	—	122	2	no	—
117	#41	—	122	1	no	—
118	#41	—	244	2	no	—
119	#41	—	244	1	no	—
120	#41	C₂²⋊D₆₁	1	1	yes	characteristic
121	C₂×Q₈	—	122	2	no	—
122	C₂×Q₈	—	122	1	no	—
123	C₂×Q₈	—	244	1	no	—
124	C₂×D₄	—	2	1	no	—
125	Q₁₆	—	61	2	no	—
126	Q₁₆	—	61	1	no	—
127	Q₁₆	—	122	2	no	—
128	Q₁₆	—	122	1	no	—
129	SD₁₆	—	4	1	no	—
130	SD₁₆	—	122	2	no	—
131	SD₁₆	—	122	1	no	—
132	D₈	—	2	1	no	—
133	D₈	—	2	1	no	—
134	D₈	—	4	1	no	—
135	D₈	—	61	1	no	—
136	M₁₆	—	2	1	no	—
137	M₁₆	—	2	1	no	—
138	M₁₆	C₂²⋊D₆₁	1	1	yes	characteristic
139	C₂×C₈	—	2	1	no	—
140	C₂×C₈	C₂²×D₆₁	1	1	yes	characteristic
141	C₁₆	—	122	1	no	—
142	C₁₆	—	122	1	no	—
143	C₂³	—	4	1	no	—
144	Q₈	—	122	2	no	—
145	Q₈	—	122	1	no	—
146	Q₈	—	244	2	no	—
147	Q₈	—	244	1	no	—
148	Q₈	#11758199	1	1	yes	characteristic
149	D₄	—	2	1	no	—
150	D₄	—	4	1	no	—
151	D₄	—	4	1	no	—
152	D₄	—	122	1	no	—
153	D₄	—	244	2	no	—
154	D₄	—	244	1	no	—
155	D₄	#11758199	1	1	yes	characteristic
156	C₂×C₄	—	2	1	no	—
157	C₂×C₄	—	244	2	no	—
158	C₂×C₄	—	244	2	no	—
159	C₂×C₄	—	244	1	no	—
160	C₂×C₄	—	244	1	no	—
161	C₂×C₄	C₂×C₂²⋊D₆₁	1	1	yes	characteristic
162	C₈	—	2	1	no	—
163	C₈	—	4	1	no	—
164	C₈	D₄×D₆₁	1	1	yes	characteristic
165	C₈	D₄×D₆₁	1	1	yes	characteristic
166	C₂²	—	2	1	no	—
167	C₂²	—	8	1	no	—
168	C₂²	—	16	1	no	—
169	C₂²	—	244	1	no	—
170	C₂²	#14519802	1	1	yes	characteristic
171	C₄	—	2	1	no	—
172	C₄	—	244	2	no	—
173	C₄	—	244	1	no	—
174	C₄	#14519745	1	1	yes	characteristic
175	C₄	C₂×#11758199	1	1	yes	characteristic
176	C₂	—	2	1	no	—
177	C₂	—	4	1	no	—
178	C₂	—	16	1	no	—
179	C₂	—	488	1	no	—
180	C₂	#16839895	1	1	yes	characteristic

	subgroup	G/H
center	C₂	#16839895
commutator	C₂×C₈×C₆₁	C₂³
Frattini	C₂×C₈	C₂²×D₆₁
Fitting	C₆₁×#472	C₂
socle	C₂×C₆₁	#447
hypercenter	#472	D₆₁
hypocenter	C₆₁	#2193
LCS	G ▷ C₂×C₈×C₆₁ ▷ C₄×C₆₁ ▷ C₂×C₆₁ ▷ C₆₁
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ #131 ◁ #472

Complements

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

N	H ≅ G/N	type
1. #16840579	C₂	semidirect
2. #16840198	C₂	semidirect
3. C₂×#14519721	C₂	semidirect
4. #16839577	C₂	semidirect
5. #16839568	C₂	semidirect
6. #16839546	C₂	semidirect
7. C₆₁×#472	C₂	semidirect
94. #472	D₆₁	semidirect
98. C₆₁	#2193	semidirect