Finite Group Databasebeta

Subgroups — #20261269

Order 6208.

234 conjugacy classes of subgroups; only the 103 normal subgroups are shown.

normal subgroups only

This group has 234 conjugacy classes of subgroups — too many to draw. Only its 103 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	#15690804	C₂	1	2	yes	maximal, complemented
2	#15690802	C₂	1	1	yes	maximal, characteristic, complemented
3	#15690798	C₂	1	1	yes	maximal, characteristic, complemented
4	#15690793	C₂	1	2	yes	maximal, complemented
5	#15690789	C₂	1	1	yes	maximal, characteristic, complemented
6	#15690784	C₂	1	1	yes	maximal, characteristic
7	C₄×Dic₁₉₄	C₂	1	1	yes	maximal, characteristic, complemented
8	C₄×D₃₈₈	C₂	1	1	yes	maximal, characteristic, complemented
9	C₄×C₂²⋊D₉₇	C₂	1	2	yes	maximal, complemented
10	#15690767	C₂	1	2	yes	maximal, complemented
11	C₂×C₄²×C₉₇	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×Dic₁₉₄	C₂²	1	1	yes	characteristic
13	C₂×D₃₈₈	C₂²	1	1	yes	characteristic
14	C₂×C₂²⋊D₉₇	C₂²	1	2	yes	—
15	C₂×C₄×D₉₇	C₂²	1	2	yes	—
16	C₄⋊Dic₉₇	C₂²	1	1	yes	characteristic
17	C₄⋊Dic₉₇	C₂²	1	1	yes	characteristic
18	#13095490	C₂²	1	2	yes	—
19	#13095490	C₂²	1	2	yes	—
20	#13095490	C₂²	1	2	yes	—
21	#13095490	C₂²	1	2	yes	—
22	C₂²⋊Dic₉₇	C₂²	1	2	yes	—
23	#13095488	C₂²	1	2	yes	—
24	#13095488	C₂²	1	2	yes	—
25	#13095488	C₂²	1	2	yes	—
26	#13095488	C₂²	1	2	yes	—
27	C₄×Dic₉₇	C₂²	1	2	yes	—
28	C₂²×C₄×C₉₇	C₂²	1	1	yes	characteristic
29	C₂²×C₄×C₉₇	C₂²	1	1	yes	characteristic
30	C₂²×C₄×C₉₇	C₂²	1	1	yes	characteristic
31	C₄²×C₉₇	C₂²	1	1	yes	characteristic
32	C₄²×C₉₇	C₂²	1	1	yes	characteristic
33	C₄²×C₉₇	C₂²	1	1	yes	characteristic
34	C₄²×C₉₇	C₂²	1	1	yes	characteristic
35	C₂³×C₉₇	C₂³	1	1	yes	characteristic
36	C₂²×D₉₇	C₂³	1	2	yes	—
37	C₂×C₄×C₉₇	—	2	2	no	—
38	C₂×C₄×C₉₇	—	2	2	no	—
39	C₂×C₄×C₉₇	—	2	2	no	—
40	C₂×C₄×C₉₇	C₂³	1	1	yes	characteristic
41	C₂×C₄×C₉₇	C₂³	1	1	yes	characteristic
42	C₂×C₄×C₉₇	C₂³	1	1	yes	characteristic
43	C₂×C₄×C₉₇	C₂³	1	1	yes	characteristic
44	C₂×C₄×C₉₇	C₂³	1	1	yes	characteristic
45	C₂×C₄×C₉₇	C₂³	1	1	yes	characteristic
46	C₂²⋊D₉₇	—	2	4	no	—
47	C₂×Dic₉₇	C₂³	1	2	yes	—
48	C₂×Dic₉₇	C₂³	1	2	yes	—
49	C₂×Dic₉₇	C₂³	1	2	yes	—
50	D₃₈₈	—	2	2	no	—
51	C₄×D₉₇	—	2	4	no	—
52	Dic₁₉₄	—	2	2	no	—
53	C₂²×C₉₇	—	2	1	no	—
54	C₂²×C₉₇	—	2	1	no	—
55	C₂²×C₉₇	#41	1	2	yes	—
56	C₂²×C₉₇	C₂⁴	1	1	yes	characteristic
57	C₂×D₉₇	—	2	2	no	—
58	C₂×D₉₇	—	2	2	no	—
59	C₂×D₉₇	—	2	2	no	—
60	C₄×C₉₇	—	2	1	no	—
61	C₄×C₉₇	—	2	1	no	—
62	C₄×C₉₇	—	2	1	no	—
63	C₄×C₉₇	—	2	1	no	—
64	C₄×C₉₇	#41	1	2	yes	—
65	C₄×C₉₇	#41	1	2	yes	—
66	Dic₉₇	—	2	2	no	—
67	Dic₉₇	—	2	2	no	—
68	Dic₉₇	—	2	2	no	—
69	C₂×C₉₇	—	2	2	no	—
70	C₂×C₉₇	C₂×#41	1	1	yes	characteristic
71	C₂×C₉₇	C₂×#41	1	1	yes	characteristic
72	C₂×C₉₇	C₂×#41	1	1	yes	characteristic
73	D₉₇	—	4	2	no	—
74	C₉₇	#529	1	1	yes	characteristic, complemented, Sylow
75	#529	—	97	1	no	maximal, Sylow
76	#126	—	97	2	no	—
77	#125	—	97	1	no	—
78	#124	—	97	1	no	—
79	#123	—	97	2	no	—
80	C₂²⋊Q₈	—	97	1	no	—
81	#121	—	97	1	no	—
82	C₄×Q₈	—	97	1	no	—
83	C₄×D₄	—	97	2	no	—
84	C₄×D₄	—	97	1	no	—
85	#117	—	97	2	no	—
86	C₂×C₄²	D₉₇	1	1	yes	characteristic, complemented
87	C₂×Q₈	—	97	1	no	—
88	C₂×D₄	—	97	2	no	—
89	C₂×D₄	—	97	1	no	—
90	C₂²×C₄	—	97	2	no	—
91	C₂²×C₄	C₂×D₉₇	1	1	yes	characteristic
92	C₂²×C₄	C₂×D₉₇	1	1	yes	characteristic
93	C₂²×C₄	C₂×D₉₇	1	1	yes	characteristic
94	C₄⋊C₄	—	97	2	no	—
95	C₄⋊C₄	—	97	2	no	—
96	C₄⋊C₄	—	97	2	no	—
97	C₄⋊C₄	—	97	2	no	—
98	C₄⋊C₄	—	97	1	no	—
99	C₄⋊C₄	—	97	1	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	—	97	2	no	—
105	C₄²	—	97	2	no	—
106	C₄²	C₂×D₉₇	1	1	yes	characteristic
107	C₄²	C₂×D₉₇	1	1	yes	characteristic
108	C₄²	C₂×D₉₇	1	1	yes	characteristic
109	C₄²	C₂×D₉₇	1	1	yes	characteristic
110	C₂³	—	97	2	no	—
111	C₂³	C₂²×D₉₇	1	1	yes	characteristic
112	Q₈	—	194	2	no	—
113	D₄	—	194	4	no	—
114	D₄	—	194	2	no	—
115	C₂×C₄	—	2	2	no	—
116	C₂×C₄	—	2	2	no	—
117	C₂×C₄	—	2	2	no	—
118	C₂×C₄	—	97	2	no	—
119	C₂×C₄	—	97	2	no	—
120	C₂×C₄	—	97	2	no	—
121	C₂×C₄	—	194	4	no	—
122	C₂×C₄	C₂²×D₉₇	1	1	yes	characteristic
123	C₂×C₄	C₂²×D₉₇	1	1	yes	characteristic
124	C₂×C₄	C₂²×D₉₇	1	1	yes	characteristic
125	C₂×C₄	C₂²×D₉₇	1	1	yes	characteristic
126	C₂×C₄	C₂²×D₉₇	1	1	yes	characteristic
127	C₂×C₄	C₂²×D₉₇	1	1	yes	characteristic
128	C₂²	—	2	1	no	—
129	C₂²	—	2	1	no	—
130	C₂²	—	194	2	no	—
131	C₂²	—	194	2	no	—
132	C₂²	—	194	2	no	—
133	C₂²	#13095511	1	2	yes	—
134	C₂²	C₂³×D₉₇	1	1	yes	characteristic
135	C₄	—	2	1	no	—
136	C₄	—	2	1	no	—
137	C₄	—	2	1	no	—
138	C₄	—	2	1	no	—
139	C₄	—	194	2	no	—
140	C₄	—	194	2	no	—
141	C₄	—	194	2	no	—
142	C₄	#13095511	1	2	yes	—
143	C₄	#13095511	1	2	yes	—
144	C₂	—	2	2	no	—
145	C₂	—	388	2	no	—
146	C₂	C₂×#13095511	1	1	yes	characteristic
147	C₂	C₂×#13095511	1	1	yes	characteristic
148	C₂	C₂×#13095511	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₂²×D₉₇
commutator, socle	C₂²×C₉₇	C₂⁴
Frattini	C₂²	C₂³×D₉₇
Fitting	C₂×C₄²×C₉₇	C₂
hypercenter	C₂×C₄²	D₉₇
hypocenter	C₉₇	#529
LCS	G ▷ C₂²×C₉₇ ▷ C₉₇
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄²

Complements

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

N	H ≅ G/N	type
1. #15690804	C₂	semidirect
2. #15690802	C₂	semidirect
3. #15690798	C₂	semidirect
4. #15690793	C₂	semidirect
5. #15690789	C₂	semidirect
7. C₄×Dic₁₉₄	C₂	semidirect
8. C₄×D₃₈₈	C₂	semidirect
9. C₄×C₂²⋊D₉₇	C₂	semidirect
10. #15690767	C₂	semidirect
11. C₂×C₄²×C₉₇	C₂	semidirect
74. C₉₇	#529	semidirect
86. C₂×C₄²	D₉₇	semidirect