Finite Group Databasebeta

Subgroups — #21447799

Order 7808.

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

normal subgroups only

This group has 206 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
2	#16840205	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×Dic₄₈₈	C₂	1	1	yes	maximal, characteristic, complemented
4	#16840037	C₂	1	1	yes	maximal, characteristic, complemented
5	#16839579	C₂	1	1	yes	maximal, characteristic, complemented
6	#16839570	C₂	1	1	yes	maximal, characteristic, complemented
7	C₆₁×#350	C₂	1	1	yes	maximal, characteristic, complemented
8	#14519824	—	2	1	no	—
9	#14519810	—	2	1	no	—
10	#14519810	—	2	2	no	—
11	#14519807	—	2	1	no	—
12	#14519799	—	2	1	no	—
13	#14519799	C₂²	1	1	yes	characteristic
14	C₂×Dic₂₄₄	—	2	1	no	—
15	C₂×Dic₂₄₄	C₂²	1	1	yes	characteristic
16	C₂×Dic₂₄₄	C₂²	1	1	yes	characteristic
17	Dic₄₈₈	—	2	1	no	—
18	Dic₄₈₈	—	2	2	no	—
19	#14519722	—	2	1	no	—
20	#14519713	C₂²	1	1	yes	characteristic
21	#14519695	—	2	1	no	—
22	#131×C₆₁	C₂²	1	1	yes	characteristic
23	M₃₂×C₆₁	C₂²	1	1	yes	characteristic
24	C₂×C₁₆×C₆₁	C₂²	1	1	yes	characteristic
25	#41×C₆₁	D₄	1	1	yes	characteristic
26	Q₈×D₆₁	—	4	1	no	—
27	#11758224	—	4	2	no	—
28	#11758224	—	4	1	no	—
29	#11758222	—	2	2	no	—
30	#11758222	—	2	1	no	—
31	C₂×Dic₁₂₂	—	2	2	no	—
32	C₂×Dic₁₂₂	—	2	1	no	—
33	C₂×Dic₁₂₂	—	2	1	no	—
34	M₁₆×C₆₁	—	2	1	no	—
35	M₁₆×C₆₁	D₄	1	1	yes	characteristic
36	C₂×C₈×C₆₁	—	2	1	no	—
37	C₂×C₈×C₆₁	C₂³	1	1	yes	characteristic
38	#11758202	—	4	1	no	—
39	#11758200	—	4	1	no	—
40	#11758194	—	2	1	no	—
41	Dic₂₄₄	—	2	2	no	—
42	Dic₂₄₄	—	2	1	no	—
43	Dic₂₄₄	—	2	1	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	C₁₆×C₆₁	—	2	1	no	—
51	C₁₆×C₆₁	—	2	1	no	—
52	Q₈×C₆₁	D₈	1	1	yes	characteristic
53	D₄×C₆₁	—	2	1	no	—
54	D₄×C₆₁	D₈	1	1	yes	characteristic
55	C₂×C₄×C₆₁	—	2	1	no	—
56	C₂×C₄×C₆₁	C₂×D₄	1	1	yes	characteristic
57	C₂²⋊D₆₁	—	4	2	no	—
58	C₂²⋊D₆₁	—	4	1	no	—
59	C₂×Dic₆₁	—	4	2	no	—
60	C₂×Dic₆₁	—	4	1	no	—
61	C₂×Dic₆₁	—	4	1	no	—
62	D₂₄₄	—	2	1	no	—
63	C₄×D₆₁	—	4	2	no	—
64	C₄×D₆₁	—	4	1	no	—
65	Dic₁₂₂	—	2	2	no	—
66	Dic₁₂₂	—	2	1	no	—
67	Dic₁₂₂	—	4	2	no	—
68	Dic₁₂₂	—	4	1	no	—
69	Dic₁₂₂	—	4	1	no	—
70	Dic₁₂₂	—	4	1	no	—
71	C₈×C₆₁	—	2	1	no	—
72	C₈×C₆₁	C₂×D₄	1	1	yes	characteristic
73	C₈×C₆₁	C₂×D₄	1	1	yes	characteristic
74	#92418	—	4	1	no	—
75	C₂²×C₆₁	—	2	1	no	—
76	C₂²×C₆₁	AGL(1,ℤ/8)	1	1	yes	characteristic
77	C₂×D₆₁	—	4	1	no	—
78	C₄×C₆₁	—	2	1	no	—
79	C₄×C₆₁	#120	1	1	yes	characteristic
80	C₄×C₆₁	C₂×D₈	1	1	yes	characteristic
81	Dic₆₁	—	4	1	no	—
82	Dic₆₁	—	4	2	no	—
83	Dic₆₁	—	8	1	no	—
84	#2195	—	61	1	no	maximal, Sylow
85	C₂×C₆₁	—	2	1	no	—
86	C₂×C₆₁	—	4	1	no	—
87	C₂×C₆₁	#447	1	1	yes	characteristic
88	D₆₁	—	8	1	no	—
89	#578	—	61	1	no	—
90	#510	—	61	1	no	—
91	C₂×Q₃₂	—	61	1	no	—
92	#473	—	61	1	no	—
93	#362	—	61	1	no	—
94	#359	—	61	1	no	—
95	#350	D₆₁	1	1	yes	characteristic, complemented
96	C₆₁	#2195	1	1	yes	characteristic, complemented, Sylow
97	2₋¹⁺⁴	—	122	1	no	—
98	#137	—	122	2	no	—
99	#137	—	122	1	no	—
100	#137	—	122	1	no	—
101	#135	—	61	1	no	—
102	#135	—	122	1	no	—
103	C₂×Q₁₆	—	61	1	no	—
104	C₂×Q₁₆	—	61	1	no	—
105	C₂×Q₁₆	—	122	1	no	—
106	#131	C₂×D₆₁	1	1	yes	characteristic
107	Q₃₂	—	122	2	no	—
108	Q₃₂	—	122	1	no	—
109	SD₃₂	—	122	1	no	—
110	M₃₂	C₂×D₆₁	1	1	yes	characteristic
111	C₂×C₁₆	C₂×D₆₁	1	1	yes	characteristic
112	#108	—	61	1	no	—
113	#101	—	122	1	no	—
114	#41	—	122	2	no	—
115	#41	—	122	1	no	—
116	#41	—	244	2	no	—
117	#41	—	244	1	no	—
118	#41	D₂₄₄	1	1	yes	characteristic
119	C₂×Q₈	—	122	2	no	—
120	C₂×Q₈	—	122	1	no	—
121	C₂×Q₈	—	122	1	no	—
122	C₂×Q₈	—	244	1	no	—
123	Q₁₆	—	61	2	no	—
124	Q₁₆	—	61	1	no	—
125	Q₁₆	—	122	2	no	—
126	Q₁₆	—	122	1	no	—
127	Q₁₆	—	122	1	no	—
128	Q₁₆	—	122	1	no	—
129	Q₁₆	—	244	1	no	—
130	SD₁₆	—	122	2	no	—
131	SD₁₆	—	122	1	no	—
132	SD₁₆	—	244	1	no	—
133	D₈	—	61	1	no	—
134	M₁₆	—	2	1	no	—
135	M₁₆	—	122	1	no	—
136	M₁₆	D₂₄₄	1	1	yes	characteristic
137	C₂×C₈	—	2	1	no	—
138	C₂×C₈	C₂²×D₆₁	1	1	yes	characteristic
139	C₁₆	—	2	1	no	—
140	C₁₆	—	2	1	no	—
141	Q₈	—	122	2	no	—
142	Q₈	—	122	1	no	—
143	Q₈	—	244	2	no	—
144	Q₈	—	244	1	no	—
145	Q₈	—	244	1	no	—
146	Q₈	—	244	1	no	—
147	Q₈	D₄₈₈	1	1	yes	characteristic
148	D₄	—	2	1	no	—
149	D₄	—	122	1	no	—
150	D₄	—	244	2	no	—
151	D₄	—	244	1	no	—
152	D₄	D₄₈₈	1	1	yes	characteristic
153	C₂×C₄	—	2	1	no	—
154	C₂×C₄	—	244	2	no	—
155	C₂×C₄	—	244	1	no	—
156	C₂×C₄	—	244	2	no	—
157	C₂×C₄	—	244	1	no	—
158	C₂×C₄	—	244	1	no	—
159	C₂×C₄	C₂×D₂₄₄	1	1	yes	characteristic
160	C₈	—	2	1	no	—
161	C₈	—	244	1	no	—
162	C₈	D₄×D₆₁	1	1	yes	characteristic
163	C₈	D₄×D₆₁	1	1	yes	characteristic
164	C₂²	—	2	1	no	—
165	C₂²	—	244	1	no	—
166	C₂²	#14519805	1	1	yes	characteristic
167	C₄	—	2	1	no	—
168	C₄	—	488	1	no	—
169	C₄	—	244	2	no	—
170	C₄	—	244	1	no	—
171	C₄	#14519747	1	1	yes	characteristic
172	C₄	C₂×D₄₈₈	1	1	yes	characteristic
173	C₂	—	2	1	no	—
174	C₂	—	4	1	no	—
175	C₂	—	488	1	no	—
176	C₂	#16839898	1	1	yes	characteristic

	subgroup	G/H
center	C₂	#16839898
commutator	C₂×C₈×C₆₁	C₂³
Frattini	C₂×C₈	C₂²×D₆₁
Fitting	C₆₁×#350	C₂
socle	C₂×C₆₁	#447
hypercenter	#350	D₆₁
hypocenter	C₆₁	#2195
LCS	G ▷ C₂×C₈×C₆₁ ▷ C₄×C₆₁ ▷ C₂×C₆₁ ▷ C₆₁
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ #131 ◁ #350

Complements

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

N	H ≅ G/N	type
2. #16840205	C₂	semidirect
3. C₂×Dic₄₈₈	C₂	semidirect
4. #16840037	C₂	semidirect
5. #16839579	C₂	semidirect
6. #16839570	C₂	semidirect
7. C₆₁×#350	C₂	semidirect
95. #350	D₆₁	semidirect
96. C₆₁	#2195	semidirect