Finite Group Databasebeta

Subgroups — #21461978

Order 7808.

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

normal subgroups only

This group has 284 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	#16840772	C₂	1	1	yes	maximal, characteristic, complemented
2	#16840761	C₂	1	2	yes	maximal, complemented
3	#16840743	C₂	1	2	yes	maximal, complemented
4	#16840725	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂×#14519800	C₂	1	1	yes	maximal, characteristic
6	D₆₁⋊M₁₆	C₂²	1	1	yes	characteristic
7	#14519856	C₂²	1	2	yes	—
8	#14519856	C₂²	1	1	yes	characteristic
9	#14519846	—	2	1	no	—
10	#14519845	—	2	1	no	—
11	#14519842	—	2	4	no	—
12	#14519836	—	2	4	no	—
13	#14519831	—	2	1	no	—
14	C₈×#6923	—	2	1	no	—
15	C₂×#11758226	C₂²	1	2	yes	—
16	#14519800	—	2	1	no	—
17	#14519800	—	2	1	no	—
18	#14519800	—	2	1	no	—
19	#14519800	—	2	1	no	—
20	C₂×#11758201	C₄	1	2	yes	complemented
21	C₂×D₄₈₈	C₄	1	1	yes	characteristic, complemented
22	C₂×C₈×D₆₁	C₂²	1	1	yes	characteristic
23	C₂×Q₁₆×C₆₁	C₄	1	1	yes	characteristic, complemented
24	C₂×Q₈×C₆₁	C₂×C₄	1	2	yes	—
25	#11758226	—	2	2	no	—
26	#11758226	—	2	2	no	—
27	#11758226	—	2	2	no	—
28	#11758226	—	2	2	no	—
29	C₂×D₂₄₄	—	2	2	no	—
30	C₂×D₂₄₄	C₂×C₄	1	2	yes	—
31	C₂×C₄×D₆₁	—	2	2	no	—
32	C₂×C₄×D₆₁	C₂³	1	1	yes	characteristic
33	C₂²⋊#6923	—	2	2	no	—
34	C₂²⋊#6923	D₄	1	2	yes	—
35	#11758217	—	2	1	no	—
36	C₄⋊#6923	—	2	2	no	—
37	C₄⋊#6923	—	2	1	no	—
38	C₄×#6923	—	2	1	no	—
39	C₄×#6923	—	2	2	no	—
40	#11758213	D₄	1	2	yes	—
41	D₆₁⋊C₈	—	2	1	no	—
42	Q₁₆×C₆₁	—	2	1	no	—
43	Q₁₆×C₆₁	—	2	1	no	—
44	C₂×C₈×C₆₁	C₂×C₄	1	1	yes	characteristic
45	#11758201	—	2	2	no	—
46	#11758201	—	2	2	no	—
47	C₂×#92418	C₂×C₄	1	1	yes	characteristic
48	D₄₈₈	—	2	1	no	—
49	D₄₈₈	—	2	1	no	—
50	C₈×D₆₁	D₄	1	1	yes	characteristic
51	C₈×D₆₁	D₄	1	1	yes	characteristic
52	C₈×D₆₁	D₄	1	1	yes	characteristic
53	C₈×D₆₁	D₄	1	1	yes	characteristic
54	C₂²×D₆₁	—	2	2	no	—
55	C₂²×D₆₁	C₂×D₄	1	1	yes	characteristic
56	C₂×#6923	—	2	2	no	—
57	C₂×#6923	—	4	2	no	—
58	Q₈×C₆₁	—	2	2	no	—
59	Q₈×C₆₁	—	2	2	no	—
60	C₂×C₄×C₆₁	—	2	2	no	—
61	C₂×C₄×C₆₁	C₂²×C₄	1	1	yes	characteristic
62	C₂×Dic₆₁	#41	1	1	yes	characteristic
63	D₂₄₄	—	2	2	no	—
64	D₂₄₄	—	2	2	no	—
65	D₂₄₄	—	4	2	no	—
66	D₂₄₄	—	4	2	no	—
67	C₄×D₆₁	—	4	2	no	—
68	C₄×D₆₁	—	4	2	no	—
69	C₄×D₆₁	C₂×D₄	1	1	yes	characteristic
70	C₄×D₆₁	C₂×D₄	1	1	yes	characteristic
71	C₄×D₆₁	C₂×D₄	1	1	yes	characteristic
72	C₄×D₆₁	#41	1	1	yes	characteristic
73	#92420	—	2	2	no	—
74	C₈×C₆₁	C₂²⋊C₄	1	1	yes	characteristic
75	C₈×C₆₁	C₂²⋊C₄	1	1	yes	characteristic
76	#92418	C₂²⋊C₄	1	1	yes	characteristic
77	#92418	C₂²⋊C₄	1	1	yes	characteristic
78	C₂²×C₆₁	C₄×D₄	1	1	yes	characteristic
79	C₂×D₆₁	—	4	1	no	—
80	C₂×D₆₁	—	4	2	no	—
81	C₂×D₆₁	—	8	2	no	—
82	C₂×D₆₁	#121	1	1	yes	characteristic
83	C₂×D₆₁	#127	1	1	yes	characteristic
84	#6923	—	4	2	no	—
85	#6923	—	8	2	no	—
86	C₄×C₆₁	—	4	2	no	—
87	C₄×C₆₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
88	C₄×C₆₁	C₄×D₄	1	1	yes	characteristic
89	Dic₆₁	#121	1	1	yes	characteristic
90	Dic₆₁	#124	1	1	yes	characteristic
91	#1976	—	61	1	no	maximal, Sylow
92	C₂×C₆₁	—	2	1	no	—
93	C₂×C₆₁	#390	1	1	yes	characteristic
94	D₆₁	—	2	1	no	—
95	D₆₁	—	2	1	no	—
96	D₆₁	—	8	2	no	—
97	C₂×#135	—	61	1	no	—
98	#427	—	61	1	no	—
99	#421	—	61	2	no	—
100	#410	—	61	2	no	—
101	#405	—	61	1	no	—
102	C₆₁	#1976	1	1	yes	characteristic, complemented, Sylow
103	C₂×#41	—	61	2	no	—
104	#135	—	122	1	no	—
105	#135	—	122	1	no	—
106	#135	—	122	1	no	—
107	#135	—	122	1	no	—
108	C₂×Q₁₆	#6923	1	1	yes	characteristic, complemented
109	C₂×SD₁₆	—	61	2	no	—
110	C₂×D₈	—	61	1	no	—
111	C₂×M₁₆	—	61	1	no	—
112	C₂²×C₈	—	61	1	no	—
113	#117	—	61	1	no	—
114	#117	—	61	2	no	—
115	#107	—	122	1	no	—
116	#106	—	122	1	no	—
117	#104	—	122	4	no	—
118	#99	—	122	4	no	—
119	#97	—	122	1	no	—
120	C₄×C₈	—	122	1	no	—
121	#41	—	122	2	no	—
122	#41	—	122	2	no	—
123	#41	—	122	2	no	—
124	#41	—	122	2	no	—
125	C₂×Q₈	C₂×#6923	1	2	yes	—
126	C₂×D₄	—	61	2	no	—
127	C₂×D₄	—	122	2	no	—
128	C₂²×C₄	—	61	1	no	—
129	C₂²×C₄	—	122	2	no	—
130	Q₁₆	—	2	1	no	—
131	Q₁₆	—	2	1	no	—
132	SD₁₆	—	122	2	no	—
133	SD₁₆	—	122	2	no	—
134	D₈	—	122	1	no	—
135	D₈	—	122	1	no	—
136	M₁₆	—	61	2	no	—
137	M₁₆	—	122	1	no	—
138	C₂×C₈	—	61	1	no	—
139	C₂×C₈	—	61	1	no	—
140	C₂×C₈	—	61	1	no	—
141	C₂×C₈	—	61	1	no	—
142	C₂×C₈	—	61	1	no	—
143	C₂×C₈	—	122	1	no	—
144	C₂×C₈	C₂×#6923	1	1	yes	characteristic
145	C₄⋊C₄	—	122	1	no	—
146	C₄⋊C₄	—	122	2	no	—
147	C₂²⋊C₄	—	61	2	no	—
148	C₂²⋊C₄	—	122	2	no	—
149	C₄²	—	122	1	no	—
150	C₄²	—	122	2	no	—
151	C₂³	—	61	1	no	—
152	C₂³	—	122	2	no	—
153	Q₈	—	2	2	no	—
154	Q₈	—	2	2	no	—
155	D₄	—	122	2	no	—
156	D₄	—	122	2	no	—
157	D₄	—	244	2	no	—
158	D₄	—	244	2	no	—
159	C₂×C₄	—	2	2	no	—
160	C₂×C₄	—	61	1	no	—
161	C₂×C₄	—	61	1	no	—
162	C₂×C₄	—	61	1	no	—
163	C₂×C₄	—	61	1	no	—
164	C₂×C₄	—	61	1	no	—
165	C₂×C₄	—	122	2	no	—
166	C₂×C₄	—	244	2	no	—
167	C₂×C₄	—	244	2	no	—
168	C₂×C₄	—	244	2	no	—
169	C₂×C₄	C₂²×#6923	1	1	yes	characteristic
170	C₈	—	61	1	no	—
171	C₈	—	61	1	no	—
172	C₈	—	122	2	no	—
173	C₈	C₂²⋊#6923	1	1	yes	characteristic
174	C₈	C₂²⋊#6923	1	1	yes	characteristic
175	C₂²	—	61	1	no	—
176	C₂²	—	61	1	no	—
177	C₂²	—	488	2	no	—
178	C₂²	—	244	1	no	—
179	C₂²	—	244	2	no	—
180	C₂²	D₄×#6923	1	1	yes	characteristic
181	C₄	—	4	2	no	—
182	C₄	—	61	1	no	—
183	C₄	—	61	1	no	—
184	C₄	—	488	2	no	—
185	C₄	—	244	2	no	—
186	C₄	C₂×C₂²⋊#6923	1	1	yes	characteristic
187	C₄	D₄×#6923	1	1	yes	characteristic
188	C₂	—	2	1	no	—
189	C₂	—	122	1	no	—
190	C₂	—	122	1	no	—
191	C₂	—	488	2	no	—
192	C₂	#16840714	1	1	yes	characteristic

	subgroup	G/H
center	C₂	#16840714
commutator	C₂×C₄×C₆₁	C₂²×C₄
Frattini	C₂×C₄	C₂²×#6923
Fitting	C₂×Q₁₆×C₆₁	C₄
socle	C₂×C₆₁	#390
hypercenter	C₂×Q₁₆	#6923
hypocenter	C₆₁	#1976
LCS	G ▷ C₂×C₄×C₆₁ ▷ C₂×C₆₁ ▷ C₆₁
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ C₂×Q₁₆

Complements

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

N	H ≅ G/N	type
1. #16840772	C₂	semidirect
2. #16840761	C₂	semidirect
3. #16840743	C₂	semidirect
4. #16840725	C₂	semidirect
20. C₂×#11758201	C₄	semidirect
21. C₂×D₄₈₈	C₄	semidirect
23. C₂×Q₁₆×C₆₁	C₄	semidirect
102. C₆₁	#1976	semidirect
108. C₂×Q₁₆	#6923	semidirect