Finite Group Databasebeta

Subgroups — #38770716

Order 25472.

160 conjugacy classes of subgroups; only the 61 normal subgroups are shown.

normal subgroups only

This group has 160 conjugacy classes of subgroups — too many to draw. Only its 61 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	C₂×#20263900	C₂	1	1	yes	maximal, characteristic
2	C₄×Dic₇₉₆	C₂	1	1	yes	maximal, characteristic, complemented
3	C₄×D₁₅₉₂	C₂	1	1	yes	maximal, characteristic
4	#27163844	C₂	1	1	yes	maximal, characteristic, complemented
5	#27163823	C₂	1	1	yes	maximal, characteristic, complemented
6	#27163820	C₂	1	1	yes	maximal, characteristic
7	#346×C₁₉₉	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×Dic₇₉₆	C₂²	1	1	yes	characteristic
9	C₂×D₁₅₉₂	C₂²	1	1	yes	characteristic
10	C₄×Dic₃₉₈	—	2	1	no	—
11	C₄×D₇₉₆	—	2	1	no	—
12	#20263900	C₄	1	4	yes	complemented
13	#20263889	C₂²	1	1	yes	characteristic
14	#20263889	C₂²	1	1	yes	characteristic
15	#20263880	—	2	1	no	—
16	#20263877	—	2	1	no	—
17	C₂×C₁₆×C₁₉₉	C₂²	1	1	yes	characteristic
18	C₂×C₁₆×C₁₉₉	C₂²	1	1	yes	characteristic
19	C₄×C₈×C₁₉₉	C₂²	1	1	yes	characteristic
20	C₂×Dic₃₉₈	—	2	1	no	—
21	C₂×D₇₉₆	—	2	1	no	—
22	C₂×C₄×D₁₉₉	—	4	1	no	—
23	Dic₇₉₆	—	2	1	no	—
24	Dic₇₉₆	C₂×C₄	1	2	yes	complemented
25	D₁₅₉₂	—	2	1	no	—
26	D₁₅₉₂	C₂×C₄	1	2	yes	—
27	C₄⋊Dic₁₉₉	—	2	1	no	—
28	C₄⋊Dic₁₉₉	—	2	1	no	—
29	Dic₁₉₉⋊C₄	—	4	1	no	—
30	(C₂×D₁₉₉)⋊C₄	—	4	1	no	—
31	C₄×Dic₁₉₉	—	4	1	no	—
32	C₂×C₈×C₁₉₉	D₄	1	1	yes	characteristic
33	C₂×C₈×C₁₉₉	C₂³	1	1	yes	characteristic
34	C₄²×C₁₉₉	D₄	1	1	yes	characteristic
35	C₁₆×C₁₉₉	—	2	1	no	—
36	C₁₆×C₁₉₉	C₂×C₄	1	2	yes	complemented
37	C₂²×D₁₉₉	—	4	1	no	—
38	Dic₃₉₈	—	2	2	no	—
39	Dic₃₉₈	—	4	1	no	—
40	D₇₉₆	—	2	2	no	—
41	D₇₉₆	—	4	1	no	—
42	C₂×Dic₁₉₉	—	4	1	no	—
43	C₂×Dic₁₉₉	—	4	1	no	—
44	C₂×Dic₁₉₉	—	4	1	no	—
45	C₄×D₁₉₉	—	8	2	no	—
46	C₂×C₄×C₁₉₉	D₈	1	1	yes	characteristic
47	C₂×C₄×C₁₉₉	D₈	1	1	yes	characteristic
48	C₂×C₄×C₁₉₉	C₂×D₄	1	1	yes	characteristic
49	C₈×C₁₉₉	—	2	1	no	—
50	C₈×C₁₉₉	C₂²×C₄	1	1	yes	characteristic
51	C₈×C₁₉₉	#41	1	1	yes	characteristic
52	C₂²×C₁₉₉	C₂×D₈	1	1	yes	characteristic
53	C₂×D₁₉₉	—	4	2	no	—
54	C₂×D₁₉₉	—	8	1	no	—
55	C₂×D₁₉₉	—	8	1	no	—
56	C₄×C₁₉₉	—	2	1	no	—
57	C₄×C₁₉₉	—	2	1	no	—
58	C₄×C₁₉₉	C₄×D₄	1	1	yes	characteristic
59	C₄×C₁₉₉	#135	1	1	yes	characteristic
60	Dic₁₉₉	—	4	2	no	—
61	Dic₁₉₉	—	8	1	no	—
62	Dic₁₉₉	—	8	1	no	—
63	C₂×C₁₉₉	C₄×D₈	1	1	yes	characteristic
64	C₂×C₁₉₉	#509	1	1	yes	characteristic
65	C₂×C₁₉₉	#510	1	1	yes	characteristic
66	D₁₉₉	—	8	2	no	—
67	C₁₉₉	#2175	1	1	yes	characteristic, complemented, Sylow
68	#2175	—	199	1	no	maximal, Sylow
69	C₂×SD₃₂	—	199	1	no	—
70	C₄×Q₁₆	—	199	1	no	—
71	C₄×D₈	—	199	1	no	—
72	#366	—	199	1	no	—
73	#358	—	199	1	no	—
74	#357	—	199	1	no	—
75	#346	D₁₉₉	1	1	yes	characteristic, complemented
76	C₂×Q₁₆	—	199	1	no	—
77	C₂×D₈	—	199	1	no	—
78	C₄×Q₈	—	398	1	no	—
79	C₄×D₄	—	398	1	no	—
80	SD₃₂	—	199	4	no	—
81	C₂×C₁₆	C₂×D₁₉₉	1	1	yes	characteristic
82	C₂×C₁₆	C₂×D₁₉₉	1	1	yes	characteristic
83	#107	—	199	1	no	—
84	#107	—	199	1	no	—
85	#103	—	398	1	no	—
86	#102	—	398	1	no	—
87	C₄×C₈	C₂×D₁₉₉	1	1	yes	characteristic
88	C₂×Q₈	—	398	1	no	—
89	C₂×D₄	—	398	1	no	—
90	C₂²×C₄	—	796	1	no	—
91	Q₁₆	—	398	1	no	—
92	Q₁₆	—	199	2	no	—
93	D₈	—	398	1	no	—
94	D₈	—	199	2	no	—
95	C₂×C₈	D₇₉₆	1	1	yes	characteristic
96	C₂×C₈	C₂²×D₁₉₉	1	1	yes	characteristic
97	C₄⋊C₄	—	796	1	no	—
98	C₄⋊C₄	—	398	1	no	—
99	C₄⋊C₄	—	398	1	no	—
100	C₂²⋊C₄	—	796	1	no	—
101	C₄²	—	796	1	no	—
102	C₄²	D₇₉₆	1	1	yes	characteristic
103	C₁₆	—	2	1	no	—
104	C₁₆	C₄×D₁₉₉	1	2	yes	complemented
105	C₂³	—	796	1	no	—
106	Q₈	—	796	1	no	—
107	Q₈	—	398	2	no	—
108	D₄	—	796	1	no	—
109	D₄	—	398	2	no	—
110	C₂×C₄	—	796	1	no	—
111	C₂×C₄	—	796	1	no	—
112	C₂×C₄	—	796	1	no	—
113	C₂×C₄	—	1592	2	no	—
114	C₂×C₄	D₁₅₉₂	1	1	yes	characteristic
115	C₂×C₄	D₁₅₉₂	1	1	yes	characteristic
116	C₂×C₄	C₂×D₇₉₆	1	1	yes	characteristic
117	C₈	—	2	1	no	—
118	C₈	C₂×C₄×D₁₉₉	1	1	yes	characteristic
119	C₈	#15691886	1	1	yes	characteristic
120	C₂²	—	796	2	no	—
121	C₂²	—	1592	1	no	—
122	C₂²	—	1592	1	no	—
123	C₂²	C₂×D₁₅₉₂	1	1	yes	characteristic
124	C₄	—	2	1	no	—
125	C₄	—	2	1	no	—
126	C₄	—	796	2	no	—
127	C₄	—	1592	1	no	—
128	C₄	—	1592	1	no	—
129	C₄	C₄×D₇₉₆	1	1	yes	characteristic
130	C₄	#20263977	1	1	yes	characteristic
131	C₂	—	1592	2	no	—
132	C₂	C₄×D₁₅₉₂	1	1	yes	characteristic
133	C₂	#27164456	1	1	yes	characteristic
134	C₂	#27164461	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×D₁₅₉₂
commutator	C₈×C₁₉₉	C₂²×C₄
Frattini	C₂×C₈	C₂²×D₁₉₉
Fitting	#346×C₁₉₉	C₂
socle	C₂²×C₁₉₉	C₂×D₈
prime radical	C₂×D₁₅₉₂	C₂²
cosocle	C₂×C₈×C₁₉₉	C₂³
hypercenter	#346	D₁₉₉
hypocenter	C₁₉₉	#2175
LCS	G ▷ C₈×C₁₉₉ ▷ C₄×C₁₉₉ ▷ C₂×C₁₉₉ ▷ C₁₉₉
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₄×C₈ ◁ #346

Complements

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

N	H ≅ G/N	type
2. C₄×Dic₇₉₆	C₂	semidirect
4. #27163844	C₂	semidirect
5. #27163823	C₂	semidirect
7. #346×C₁₉₉	C₂	semidirect
12. #20263900	C₄	semidirect
24. Dic₇₉₆	C₂×C₄	semidirect
36. C₁₆×C₁₉₉	C₂×C₄	semidirect
67. C₁₉₉	#2175	semidirect
75. #346	D₁₉₉	semidirect
104. C₁₆	C₄×D₁₉₉	semidirect

Maximal subgroups

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

subgroup	index
C₂×#20263900	2
C₄×Dic₇₉₆	2
C₄×D₁₅₉₂	2
#27163844	2
#27163823	2
#27163820	2
#346×C₁₉₉	2
#2175	199