Finite Group Databasebeta

Subgroups — #20261532

Order 6208.

262 conjugacy classes of subgroups; only the 99 normal subgroups are shown.

normal subgroups only

This group has 262 conjugacy classes of subgroups — too many to draw. Only its 99 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	#41×D₉₇	C₂	1	2	yes	maximal, complemented
2	#15690842	C₂	1	2	yes	maximal, complemented
3	#15690839	C₂	1	2	yes	maximal, complemented
4	#15690837	C₂	1	2	yes	maximal, complemented
5	#15690834	C₂	1	2	yes	maximal, complemented
6	#15690832	C₂	1	1	yes	maximal, characteristic, complemented
7	#15690830	C₂	1	2	yes	maximal, complemented
8	C₂×#13095494	C₂	1	1	yes	maximal, characteristic, complemented
9	#135×C₉₇	C₂	1	1	yes	maximal, characteristic, complemented
10	#13095512	C₂²	1	2	yes	—
11	#13095511	—	2	2	no	—
12	#13095511	C₂²	1	2	yes	complemented
13	#13095510	—	2	2	no	—
14	#13095510	C₂²	1	2	yes	—
15	Q₈×D₉₇	C₂²	1	2	yes	complemented
16	D₄×D₉₇	—	2	2	no	—
17	D₄×D₉₇	C₂²	1	2	yes	complemented
18	C₂×C₄×D₉₇	—	2	2	no	—
19	C₂×C₄×D₉₇	C₂²	1	1	yes	characteristic
20	Dic₃₈₈	C₂²	1	1	yes	characteristic, complemented
21	#13095501	C₂²	1	2	yes	complemented
22	#13095500	C₂²	1	2	yes	complemented
23	#13095499	C₂²	1	2	yes	complemented
24	#13095498	C₂²	1	2	yes	complemented
25	D₇₇₆	C₂²	1	1	yes	characteristic, complemented
26	#13095496	C₂²	1	2	yes	complemented
27	#13095494	C₂²	1	2	yes	complemented
28	#13095494	C₂²	1	2	yes	complemented
29	C₂×#11715895	C₂²	1	1	yes	characteristic, complemented
30	#41×C₉₇	C₂²	1	2	yes	complemented
31	Q₁₆×C₉₇	C₂²	1	1	yes	characteristic, complemented
32	SD₁₆×C₉₇	C₂²	1	2	yes	complemented
33	D₈×C₉₇	C₂²	1	1	yes	characteristic, complemented
34	C₂×C₈×C₉₇	C₂²	1	1	yes	characteristic, complemented
35	C₂²×D₉₇	—	2	2	no	—
36	C₂²×D₉₇	D₄	1	1	yes	characteristic
37	Q₈×C₉₇	C₂³	1	2	yes	—
38	D₄×C₉₇	—	2	2	no	—
39	D₄×C₉₇	C₂³	1	2	yes	—
40	C₂×C₄×C₉₇	—	2	2	no	—
41	C₂×C₄×C₉₇	C₂³	1	1	yes	characteristic
42	C₂²⋊D₉₇	—	2	2	no	—
43	C₂²⋊D₉₇	—	2	2	no	—
44	C₂²⋊D₉₇	—	2	2	no	—
45	C₂×Dic₉₇	—	2	2	no	—
46	C₂×Dic₉₇	D₄	1	1	yes	characteristic
47	D₃₈₈	—	2	2	no	—
48	D₃₈₈	C₂³	1	2	yes	—
49	C₄×D₉₇	—	2	2	no	—
50	C₄×D₉₇	—	2	2	no	—
51	C₄×D₉₇	—	2	2	no	—
52	C₄×D₉₇	D₄	1	2	yes	—
53	C₄×D₉₇	C₂³	1	2	yes	—
54	Dic₁₉₄	—	2	2	no	—
55	Dic₁₉₄	C₂³	1	2	yes	—
56	C₈×C₉₇	C₂³	1	1	yes	characteristic
57	C₈×C₉₇	C₂³	1	1	yes	characteristic
58	#11715895	C₂³	1	2	yes	—
59	C₂²×C₉₇	—	2	2	no	—
60	C₂²×C₉₇	C₂×D₄	1	1	yes	characteristic
61	C₂×D₉₇	—	2	2	no	—
62	C₂×D₉₇	—	4	1	no	—
63	C₂×D₉₇	—	4	4	no	—
64	C₂×D₉₇	C₂×D₄	1	2	yes	—
65	C₄×C₉₇	—	2	2	no	—
66	C₄×C₉₇	C₂×D₄	1	1	yes	characteristic
67	C₄×C₉₇	C₂⁴	1	1	yes	characteristic
68	Dic₉₇	—	2	2	no	—
69	Dic₉₇	C₂×D₄	1	2	yes	—
70	C₂×C₉₇	—	2	1	no	—
71	C₂×C₉₇	—	4	2	no	—
72	C₂×C₉₇	C₂²×D₄	1	1	yes	characteristic
73	D₉₇	—	2	2	no	—
74	D₉₇	—	4	2	no	—
75	C₉₇	#575	1	1	yes	characteristic, complemented, Sylow
76	#575	—	97	1	no	maximal, Sylow
77	C₂×#41	—	97	2	no	—
78	#137	—	97	2	no	—
79	#137	—	97	2	no	—
80	AGL(1,ℤ/8)	—	97	2	no	—
81	AGL(1,ℤ/8)	—	97	2	no	—
82	#135	—	97	2	no	—
83	#135	—	97	1	no	—
84	#135	D₉₇	1	1	yes	characteristic, complemented
85	C₂×M₁₆	—	97	1	no	—
86	#41	—	97	2	no	—
87	#41	—	97	2	no	—
88	#41	—	97	2	no	—
89	#41	—	194	2	no	—
90	#41	—	194	2	no	—
91	#41	C₂×D₉₇	1	2	yes	complemented
92	C₂×Q₈	—	97	2	no	—
93	C₂×D₄	—	97	2	no	—
94	C₂×D₄	—	194	2	no	—
95	C₂²×C₄	—	97	1	no	—
96	C₂²×C₄	—	194	2	no	—
97	Q₁₆	—	97	2	no	—
98	Q₁₆	—	97	1	no	—
99	Q₁₆	C₂×D₉₇	1	1	yes	characteristic, complemented
100	SD₁₆	—	97	2	no	—
101	SD₁₆	—	97	2	no	—
102	SD₁₆	—	97	2	no	—
103	SD₁₆	C₂×D₉₇	1	2	yes	complemented
104	D₈	—	97	2	no	—
105	D₈	—	97	1	no	—
106	D₈	C₂×D₉₇	1	1	yes	characteristic, complemented
107	M₁₆	—	97	2	no	—
108	M₁₆	—	97	2	no	—
109	C₂×C₈	—	97	1	no	—
110	C₂×C₈	C₂×D₉₇	1	1	yes	characteristic, complemented
111	C₂³	—	97	1	no	—
112	C₂³	—	194	2	no	—
113	Q₈	—	97	2	no	—
114	Q₈	—	194	2	no	—
115	Q₈	C₂²×D₉₇	1	2	yes	—
116	D₄	—	2	2	no	—
117	D₄	—	97	2	no	—
118	D₄	—	194	2	no	—
119	D₄	—	194	2	no	—
120	D₄	—	194	2	no	—
121	D₄	—	194	2	no	—
122	D₄	C₂²×D₉₇	1	2	yes	—
123	C₂×C₄	—	2	2	no	—
124	C₂×C₄	—	97	1	no	—
125	C₂×C₄	—	97	2	no	—
126	C₂×C₄	—	97	2	no	—
127	C₂×C₄	—	194	2	no	—
128	C₂×C₄	—	194	2	no	—
129	C₂×C₄	—	194	2	no	—
130	C₂×C₄	—	194	2	no	—
131	C₂×C₄	C₂²×D₉₇	1	1	yes	characteristic
132	C₈	—	97	2	no	—
133	C₈	C₂²×D₉₇	1	1	yes	characteristic
134	C₈	C₂²×D₉₇	1	1	yes	characteristic
135	C₂²	—	2	2	no	—
136	C₂²	—	97	2	no	—
137	C₂²	—	388	1	no	—
138	C₂²	—	388	4	no	—
139	C₂²	—	194	2	no	—
140	C₂²	D₄×D₉₇	1	1	yes	characteristic
141	C₄	—	2	2	no	—
142	C₄	—	97	2	no	—
143	C₄	—	194	2	no	—
144	C₄	D₄×D₉₇	1	1	yes	characteristic
145	C₄	C₂³×D₉₇	1	1	yes	characteristic
146	C₂	—	2	1	no	—
147	C₂	—	4	2	no	—
148	C₂	—	388	2	no	—
149	C₂	—	194	2	no	—
150	C₂	C₂×D₄×D₉₇	1	1	yes	characteristic

	subgroup	G/H
center	C₄	D₄×D₉₇
commutator	C₄×C₉₇	C₂⁴
Frattini	C₄	C₂³×D₉₇
Fitting	#135×C₉₇	C₂
socle	C₂×C₉₇	C₂²×D₄
hypercenter	#135	D₉₇
hypocenter	C₉₇	#575
LCS	G ▷ C₄×C₉₇ ▷ C₂×C₉₇ ▷ C₉₇
UCS	C₁ ◁ C₄ ◁ C₂×C₄ ◁ #135

Complements

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

N	H ≅ G/N	type
1. #41×D₉₇	C₂	semidirect
2. #15690842	C₂	semidirect
3. #15690839	C₂	semidirect
4. #15690837	C₂	semidirect
5. #15690834	C₂	semidirect
6. #15690832	C₂	semidirect
7. #15690830	C₂	semidirect
8. C₂×#13095494	C₂	semidirect
9. #135×C₉₇	C₂	semidirect
12. #13095511	C₂²	semidirect
15. Q₈×D₉₇	C₂²	semidirect
17. D₄×D₉₇	C₂²	semidirect
20. Dic₃₈₈	C₂²	semidirect
21. #13095501	C₂²	semidirect
22. #13095500	C₂²	semidirect
23. #13095499	C₂²	semidirect
24. #13095498	C₂²	semidirect
25. D₇₇₆	C₂²	semidirect
26. #13095496	C₂²	semidirect
27. #13095494	C₂²	semidirect
28. #13095494	C₂²	semidirect
29. C₂×#11715895	C₂²	semidirect
30. #41×C₉₇	C₂²	semidirect
31. Q₁₆×C₉₇	C₂²	semidirect
32. SD₁₆×C₉₇	C₂²	semidirect
33. D₈×C₉₇	C₂²	semidirect
34. C₂×C₈×C₉₇	C₂²	semidirect
75. C₉₇	#575	semidirect
84. #135	D₉₇	semidirect
91. #41	C₂×D₉₇	semidirect
99. Q₁₆	C₂×D₉₇	semidirect
103. SD₁₆	C₂×D₉₇	semidirect
106. D₈	C₂×D₉₇	semidirect
110. C₂×C₈	C₂×D₉₇	semidirect