Finite Group Databasebeta

Subgroups — #20261325

Order 6208.

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	#15690805	C₂	1	2	yes	maximal, complemented
2	#15690799	C₂	1	2	yes	maximal, complemented
3	#15690795	C₂	1	2	yes	maximal, complemented
4	#15690785	C₂	1	2	yes	maximal, complemented
5	#15690784	C₂	1	1	yes	maximal, characteristic, complemented
6	#15690780	C₂	1	2	yes	maximal, complemented
7	#15690770	C₂	1	2	yes	maximal, complemented
8	C₂×#13095488	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂²⋊Q₈×C₉₇	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂³×D₉₇	C₂²	1	1	yes	characteristic
11	C₂×D₃₈₈	C₂²	1	2	yes	—
12	C₂×D₃₈₈	C₂²	1	2	yes	—
13	C₂×D₃₈₈	C₂²	1	1	yes	characteristic, complemented
14	C₂×C₂²⋊D₉₇	C₂²	1	2	yes	—
15	C₂²×Dic₉₇	C₂²	1	1	yes	characteristic
16	C₂×C₄×D₉₇	C₂²	1	2	yes	complemented
17	C₄⋊Dic₉₇	C₂²	1	1	yes	characteristic, complemented
18	#13095490	C₂²	1	2	yes	complemented
19	#13095488	C₂²	1	2	yes	—
20	#13095488	C₂²	1	2	yes	—
21	#13095488	C₂²	1	2	yes	—
22	#13095488	C₂²	1	2	yes	—
23	#13095488	C₂²	1	2	yes	complemented
24	#13095488	C₂²	1	2	yes	complemented
25	C₄×Dic₉₇	C₂²	1	2	yes	complemented
26	C₂×Q₈×C₉₇	C₂²	1	1	yes	characteristic, complemented
27	C₂²×C₄×C₉₇	C₂²	1	1	yes	characteristic
28	C₄⋊C₄×C₉₇	C₂²	1	2	yes	complemented
29	C₄⋊C₄×C₉₇	C₂²	1	1	yes	characteristic, complemented
30	C₂²⋊C₄×C₉₇	C₂²	1	2	yes	—
31	C₂³×C₉₇	C₂³	1	1	yes	characteristic
32	C₂²×D₉₇	—	2	2	no	—
33	C₂²×D₉₇	—	4	1	no	—
34	C₂²×D₉₇	—	4	1	no	—
35	C₂²×D₉₇	C₂³	1	2	yes	—
36	C₂²×D₉₇	C₂³	1	2	yes	—
37	Q₈×C₉₇	—	4	1	no	—
38	C₂×C₄×C₉₇	—	4	1	no	—
39	C₂×C₄×C₉₇	C₂³	1	2	yes	—
40	C₂×C₄×C₉₇	C₂³	1	2	yes	—
41	C₂×C₄×C₉₇	C₂³	1	1	yes	characteristic
42	C₂×C₄×C₉₇	C₂³	1	1	yes	characteristic
43	C₂²⋊D₉₇	—	2	4	no	—
44	C₂×Dic₉₇	—	4	1	no	—
45	C₂×Dic₉₇	C₂³	1	2	yes	—
46	C₂×Dic₉₇	C₂³	1	2	yes	—
47	D₃₈₈	—	4	2	no	—
48	D₃₈₈	—	4	2	no	—
49	D₃₈₈	—	4	1	no	—
50	C₄×D₉₇	—	4	2	no	—
51	C₂²×C₉₇	—	2	1	no	—
52	C₂²×C₉₇	—	2	1	no	—
53	C₂²×C₉₇	#41	1	2	yes	—
54	C₂²×C₉₇	C₂⁴	1	1	yes	characteristic
55	C₂×D₉₇	—	2	2	no	—
56	C₂×D₉₇	—	2	2	no	—
57	C₂×D₉₇	—	2	2	no	—
58	C₂×D₉₇	—	2	2	no	—
59	C₂×D₉₇	—	2	2	no	—
60	C₂×D₉₇	—	2	2	no	—
61	C₂×D₉₇	—	4	4	no	—
62	C₄×C₉₇	—	2	2	no	—
63	C₄×C₉₇	—	2	1	no	—
64	C₄×C₉₇	—	2	2	no	—
65	C₄×C₉₇	—	2	1	no	—
66	Dic₉₇	—	2	2	no	—
67	Dic₉₇	—	2	2	no	—
68	C₂×C₉₇	—	2	2	no	—
69	C₂×C₉₇	C₂×#41	1	1	yes	characteristic
70	C₂×C₉₇	H(5,𝔽₂)	1	1	yes	characteristic
71	C₂×C₉₇	H(5,𝔽₂)	1	1	yes	characteristic
72	D₉₇	—	4	2	no	—
73	D₉₇	—	4	2	no	—
74	C₉₇	#538	1	1	yes	characteristic, complemented, Sylow
75	#538	—	97	1	no	maximal, Sylow
76	#126	—	97	2	no	—
77	#124	—	97	2	no	—
78	#123	—	97	2	no	—
79	C₂²⋊Q₈	D₉₇	1	1	yes	characteristic, complemented
80	#121	—	97	1	no	—
81	#121	—	97	2	no	—
82	#120	—	97	2	no	—
83	C₄×D₄	—	97	2	no	—
84	C₂×C₂²⋊C₄	—	97	1	no	—
85	C₂⁴	—	97	1	no	—
86	C₂×Q₈	C₂×D₉₇	1	1	yes	characteristic, complemented
87	C₂×D₄	—	97	2	no	—
88	C₂×D₄	—	97	2	no	—
89	C₂×D₄	—	97	2	no	—
90	C₂×D₄	—	97	1	no	—
91	C₂²×C₄	—	97	1	no	—
92	C₂²×C₄	—	97	2	no	—
93	C₂²×C₄	C₂×D₉₇	1	1	yes	characteristic
94	C₄⋊C₄	—	97	2	no	—
95	C₄⋊C₄	—	97	1	no	—
96	C₄⋊C₄	C₂×D₉₇	1	2	yes	complemented
97	C₄⋊C₄	C₂×D₉₇	1	1	yes	characteristic, complemented
98	C₂²⋊C₄	—	97	2	no	—
99	C₂²⋊C₄	—	97	2	no	—
100	C₂²⋊C₄	—	97	2	no	—
101	C₂²⋊C₄	—	97	2	no	—
102	C₂²⋊C₄	—	97	2	no	—
103	C₂²⋊C₄	—	97	2	no	—
104	C₂²⋊C₄	C₂×D₉₇	1	2	yes	—
105	C₄²	—	97	2	no	—
106	C₂³	—	97	2	no	—
107	C₂³	—	97	2	no	—
108	C₂³	—	388	1	no	—
109	C₂³	—	388	1	no	—
110	C₂³	—	194	2	no	—
111	C₂³	C₂²×D₉₇	1	1	yes	characteristic
112	Q₈	—	4	1	no	—
113	D₄	—	388	1	no	—
114	D₄	—	388	2	no	—
115	D₄	—	388	2	no	—
116	D₄	—	194	4	no	—
117	C₂×C₄	—	4	1	no	—
118	C₂×C₄	—	97	2	no	—
119	C₂×C₄	—	97	2	no	—
120	C₂×C₄	—	388	2	no	—
121	C₂×C₄	—	388	1	no	—
122	C₂×C₄	C₂²×D₉₇	1	2	yes	—
123	C₂×C₄	C₂²×D₉₇	1	1	yes	characteristic
124	C₂×C₄	C₂²×D₉₇	1	1	yes	characteristic
125	C₂×C₄	C₂²×D₉₇	1	2	yes	—
126	C₂²	—	2	1	no	—
127	C₂²	—	2	1	no	—
128	C₂²	—	388	4	no	—
129	C₂²	—	194	2	no	—
130	C₂²	—	194	2	no	—
131	C₂²	—	194	2	no	—
132	C₂²	—	194	2	no	—
133	C₂²	—	194	2	no	—
134	C₂²	—	194	2	no	—
135	C₂²	#13095512	1	2	yes	—
136	C₂²	C₂³×D₉₇	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	—	194	2	no	—
142	C₄	—	194	2	no	—
143	C₂	—	2	2	no	—
144	C₂	—	388	2	no	—
145	C₂	—	388	2	no	—
146	C₂	C₂×#13095512	1	1	yes	characteristic
147	C₂	#15690855	1	1	yes	characteristic
148	C₂	#15690856	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂²	C₂³×D₉₇
commutator, socle	C₂²×C₉₇	C₂⁴
Fitting	C₂²⋊Q₈×C₉₇	C₂
hypercenter	C₂²⋊Q₈	D₉₇
hypocenter	C₉₇	#538
LCS	G ▷ C₂²×C₉₇ ▷ C₉₇
UCS	C₁ ◁ C₂² ◁ C₂²⋊Q₈

Complements

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

N	H ≅ G/N	type
1. #15690805	C₂	semidirect
2. #15690799	C₂	semidirect
3. #15690795	C₂	semidirect
4. #15690785	C₂	semidirect
5. #15690784	C₂	semidirect
6. #15690780	C₂	semidirect
7. #15690770	C₂	semidirect
8. C₂×#13095488	C₂	semidirect
9. C₂²⋊Q₈×C₉₇	C₂	semidirect
13. C₂×D₃₈₈	C₂²	semidirect
16. C₂×C₄×D₉₇	C₂²	semidirect
17. C₄⋊Dic₉₇	C₂²	semidirect
18. #13095490	C₂²	semidirect
23. #13095488	C₂²	semidirect
24. #13095488	C₂²	semidirect
25. C₄×Dic₉₇	C₂²	semidirect
26. C₂×Q₈×C₉₇	C₂²	semidirect
28. C₄⋊C₄×C₉₇	C₂²	semidirect
29. C₄⋊C₄×C₉₇	C₂²	semidirect
74. C₉₇	#538	semidirect
79. C₂²⋊Q₈	D₉₇	semidirect
86. C₂×Q₈	C₂×D₉₇	semidirect
96. C₄⋊C₄	C₂×D₉₇	semidirect
97. C₄⋊C₄	C₂×D₉₇	semidirect