Finite Group Databasebeta

Subgroups — #16840352

Order 3904.

250 conjugacy classes of subgroups; only the 95 normal subgroups are shown.

normal subgroups only

This group has 250 conjugacy classes of subgroups — too many to draw. Only its 95 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	#14519772	C₂	1	2	yes	maximal, complemented
2	#14519766	C₂	1	2	yes	maximal, complemented
3	#14519762	C₂	1	2	yes	maximal, complemented
4	#14519752	C₂	1	2	yes	maximal, complemented
5	#14519751	C₂	1	1	yes	maximal, characteristic, complemented
6	#14519747	C₂	1	2	yes	maximal, complemented
7	#14519737	C₂	1	2	yes	maximal, complemented
8	C₂×#11758198	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂²⋊Q₈×C₆₁	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂³×D₆₁	C₂²	1	1	yes	characteristic
11	C₂×Q₈×C₆₁	C₂²	1	1	yes	characteristic, complemented
12	C₂²×C₄×C₆₁	C₂²	1	1	yes	characteristic
13	C₂×C₂²⋊D₆₁	C₂²	1	2	yes	—
14	C₂²×Dic₆₁	C₂²	1	1	yes	characteristic
15	C₂×D₂₄₄	C₂²	1	2	yes	—
16	C₂×D₂₄₄	C₂²	1	2	yes	—
17	C₂×D₂₄₄	C₂²	1	1	yes	characteristic, complemented
18	C₂×C₄×D₆₁	C₂²	1	2	yes	complemented
19	C₄⋊C₄×C₆₁	C₂²	1	2	yes	complemented
20	C₄⋊C₄×C₆₁	C₂²	1	1	yes	characteristic, complemented
21	C₂²⋊C₄×C₆₁	C₂²	1	2	yes	—
22	#11758198	C₂²	1	2	yes	—
23	#11758198	C₂²	1	2	yes	—
24	#11758198	C₂²	1	2	yes	—
25	#11758198	C₂²	1	2	yes	—
26	#11758198	C₂²	1	2	yes	complemented
27	#11758198	C₂²	1	2	yes	complemented
28	C₄⋊Dic₆₁	C₂²	1	1	yes	characteristic, complemented
29	#11758196	C₂²	1	2	yes	complemented
30	C₄×Dic₆₁	C₂²	1	2	yes	complemented
31	C₂³×C₆₁	C₂³	1	1	yes	characteristic
32	C₂²×D₆₁	—	2	2	no	—
33	C₂²×D₆₁	—	4	1	no	—
34	C₂²×D₆₁	—	4	1	no	—
35	C₂²×D₆₁	C₂³	1	2	yes	—
36	C₂²×D₆₁	C₂³	1	2	yes	—
37	Q₈×C₆₁	—	4	1	no	—
38	C₂×C₄×C₆₁	—	4	1	no	—
39	C₂×C₄×C₆₁	C₂³	1	2	yes	—
40	C₂×C₄×C₆₁	C₂³	1	2	yes	—
41	C₂×C₄×C₆₁	C₂³	1	1	yes	characteristic
42	C₂×C₄×C₆₁	C₂³	1	1	yes	characteristic
43	C₂²⋊D₆₁	—	2	4	no	—
44	C₂×Dic₆₁	—	4	1	no	—
45	C₂×Dic₆₁	C₂³	1	2	yes	—
46	C₂×Dic₆₁	C₂³	1	2	yes	—
47	D₂₄₄	—	4	2	no	—
48	D₂₄₄	—	4	2	no	—
49	D₂₄₄	—	4	1	no	—
50	C₄×D₆₁	—	4	2	no	—
51	C₂²×C₆₁	—	2	1	no	—
52	C₂²×C₆₁	—	2	1	no	—
53	C₂²×C₆₁	#41	1	2	yes	—
54	C₂²×C₆₁	C₂⁴	1	1	yes	characteristic
55	C₂×D₆₁	—	2	2	no	—
56	C₂×D₆₁	—	2	2	no	—
57	C₂×D₆₁	—	2	2	no	—
58	C₂×D₆₁	—	2	2	no	—
59	C₂×D₆₁	—	2	2	no	—
60	C₂×D₆₁	—	2	2	no	—
61	C₂×D₆₁	—	4	4	no	—
62	C₄×C₆₁	—	2	2	no	—
63	C₄×C₆₁	—	2	1	no	—
64	C₄×C₆₁	—	2	2	no	—
65	C₄×C₆₁	—	2	1	no	—
66	Dic₆₁	—	2	2	no	—
67	Dic₆₁	—	2	2	no	—
68	C₂×C₆₁	—	2	2	no	—
69	C₂×C₆₁	C₂×#41	1	1	yes	characteristic
70	C₂×C₆₁	H(5,𝔽₂)	1	1	yes	characteristic
71	C₂×C₆₁	H(5,𝔽₂)	1	1	yes	characteristic
72	D₆₁	—	4	2	no	—
73	D₆₁	—	4	2	no	—
74	#538	—	61	1	no	maximal, Sylow
75	C₆₁	#538	1	1	yes	characteristic, complemented, Sylow
76	#126	—	61	2	no	—
77	#124	—	61	2	no	—
78	#123	—	61	2	no	—
79	C₂²⋊Q₈	D₆₁	1	1	yes	characteristic, complemented
80	#121	—	61	1	no	—
81	#121	—	61	2	no	—
82	#120	—	61	2	no	—
83	C₄×D₄	—	61	2	no	—
84	C₂×C₂²⋊C₄	—	61	1	no	—
85	C₂⁴	—	61	1	no	—
86	C₂×Q₈	C₂×D₆₁	1	1	yes	characteristic, complemented
87	C₂×D₄	—	61	2	no	—
88	C₂×D₄	—	61	2	no	—
89	C₂×D₄	—	61	2	no	—
90	C₂×D₄	—	61	1	no	—
91	C₂²×C₄	—	61	1	no	—
92	C₂²×C₄	—	61	2	no	—
93	C₂²×C₄	C₂×D₆₁	1	1	yes	characteristic
94	C₄⋊C₄	—	61	2	no	—
95	C₄⋊C₄	—	61	1	no	—
96	C₄⋊C₄	C₂×D₆₁	1	2	yes	complemented
97	C₄⋊C₄	C₂×D₆₁	1	1	yes	characteristic, complemented
98	C₂²⋊C₄	—	61	2	no	—
99	C₂²⋊C₄	—	61	2	no	—
100	C₂²⋊C₄	—	61	2	no	—
101	C₂²⋊C₄	—	61	2	no	—
102	C₂²⋊C₄	—	61	2	no	—
103	C₂²⋊C₄	—	61	2	no	—
104	C₂²⋊C₄	C₂×D₆₁	1	2	yes	—
105	C₄²	—	61	2	no	—
106	C₂³	—	61	2	no	—
107	C₂³	—	61	2	no	—
108	C₂³	—	122	2	no	—
109	C₂³	—	244	1	no	—
110	C₂³	—	244	1	no	—
111	C₂³	C₂²×D₆₁	1	1	yes	characteristic
112	Q₈	—	4	1	no	—
113	D₄	—	122	4	no	—
114	D₄	—	244	1	no	—
115	D₄	—	244	2	no	—
116	D₄	—	244	2	no	—
117	C₂×C₄	—	4	1	no	—
118	C₂×C₄	—	61	2	no	—
119	C₂×C₄	—	61	2	no	—
120	C₂×C₄	—	244	2	no	—
121	C₂×C₄	—	244	1	no	—
122	C₂×C₄	C₂²×D₆₁	1	2	yes	—
123	C₂×C₄	C₂²×D₆₁	1	1	yes	characteristic
124	C₂×C₄	C₂²×D₆₁	1	1	yes	characteristic
125	C₂×C₄	C₂²×D₆₁	1	2	yes	—
126	C₂²	—	2	1	no	—
127	C₂²	—	2	1	no	—
128	C₂²	—	122	2	no	—
129	C₂²	—	122	2	no	—
130	C₂²	—	122	2	no	—
131	C₂²	—	122	2	no	—
132	C₂²	—	122	2	no	—
133	C₂²	—	122	2	no	—
134	C₂²	—	244	4	no	—
135	C₂²	#11758226	1	2	yes	—
136	C₂²	C₂³×D₆₁	1	1	yes	characteristic
137	C₄	—	2	1	no	—
138	C₄	—	2	1	no	—
139	C₄	—	2	2	no	—
140	C₄	—	2	2	no	—
141	C₄	—	122	2	no	—
142	C₄	—	122	2	no	—
143	C₂	—	2	2	no	—
144	C₂	—	244	2	no	—
145	C₂	—	244	2	no	—
146	C₂	C₂×#11758226	1	1	yes	characteristic
147	C₂	#14519822	1	1	yes	characteristic
148	C₂	#14519823	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂²	C₂³×D₆₁
commutator, socle, cosocle	C₂²×C₆₁	C₂⁴
Fitting	C₂²⋊Q₈×C₆₁	C₂
hypercenter	C₂²⋊Q₈	D₆₁
hypocenter	C₆₁	#538
LCS	G ▷ C₂²×C₆₁ ▷ C₆₁
UCS	C₁ ◁ C₂² ◁ C₂²⋊Q₈

Complements

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

N	H ≅ G/N	type
1. #14519772	C₂	semidirect
2. #14519766	C₂	semidirect
3. #14519762	C₂	semidirect
4. #14519752	C₂	semidirect
5. #14519751	C₂	semidirect
6. #14519747	C₂	semidirect
7. #14519737	C₂	semidirect
8. C₂×#11758198	C₂	semidirect
9. C₂²⋊Q₈×C₆₁	C₂	semidirect
11. C₂×Q₈×C₆₁	C₂²	semidirect
17. C₂×D₂₄₄	C₂²	semidirect
18. C₂×C₄×D₆₁	C₂²	semidirect
19. C₄⋊C₄×C₆₁	C₂²	semidirect
20. C₄⋊C₄×C₆₁	C₂²	semidirect
26. #11758198	C₂²	semidirect
27. #11758198	C₂²	semidirect
28. C₄⋊Dic₆₁	C₂²	semidirect
29. #11758196	C₂²	semidirect
30. C₄×Dic₆₁	C₂²	semidirect
75. C₆₁	#538	semidirect
79. C₂²⋊Q₈	D₆₁	semidirect
86. C₂×Q₈	C₂×D₆₁	semidirect
96. C₄⋊C₄	C₂×D₆₁	semidirect
97. C₄⋊C₄	C₂×D₆₁	semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
#14519772	2
#14519766	2
#14519762	2
#14519752	2
#14519751	2
#14519747	2
#14519737	2
C₂×#11758198	2
C₂²⋊Q₈×C₆₁	2
#538	61