Subgroups — #38768452

Order 25472.

190 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#27164391	C₂	1	1	yes	maximal, characteristic
2	#27164368	C₂	1	2	yes	maximal, complemented
3	#27164366	C₂	1	2	yes	maximal
4	C₈×C₁₉₉⋊C₈	C₂	1	1	yes	maximal, characteristic, complemented
5	#498×C₁₉₉	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×#15691875	—	2	2	no	—
7	#20263955	C₂²	1	1	yes	characteristic
8	#20263952	C₂²	1	1	yes	characteristic
9	#20263879	—	2	4	no	—
10	#20263875	—	2	4	no	—
11	C₄×C₁₉₉⋊C₈	C₂²	1	2	yes	—
12	C₄⋊Q₈×C₁₉₉	C₂²	1	2	yes	—
13	#106×C₁₉₉	—	2	2	no	—
14	C₄×C₈×C₁₉₉	C₂²	1	1	yes	characteristic
15	C₂×Dic₃₉₈	—	2	1	no	—
16	C₂×D₇₉₆	—	2	1	no	—
17	C₂×D₇₉₆	—	2	2	no	—
18	#15691875	—	4	4	no	—
19	C₂×C₁₉₉⋊C₈	D₄	1	4	yes	—
20	C₄⋊Dic₁₉₉	—	2	2	no	—
21	C₂×Q₈×C₁₉₉	—	2	2	no	—
22	C₂×C₈×C₁₉₉	D₄	1	2	yes	—
23	C₄⋊C₄×C₁₉₉	—	2	2	no	—
24	C₄⋊C₄×C₁₉₉	—	2	2	no	—
25	C₄²×C₁₉₉	C₂³	1	1	yes	characteristic
26	C₂²×D₁₉₉	—	4	1	no	—
27	Dic₃₉₈	—	4	2	no	—
28	D₇₉₆	—	4	2	no	—
29	D₇₉₆	—	4	4	no	—
30	C₂×Dic₁₉₉	—	4	1	no	—
31	C₁₉₉⋊C₈	—	2	4	no	—
32	Q₈×C₁₉₉	—	4	4	no	—
33	C₂×C₄×C₁₉₉	—	4	2	no	—
34	C₂×C₄×C₁₉₉	C₂×D₄	1	2	yes	—
35	C₂×C₄×C₁₉₉	C₂×D₄	1	1	yes	characteristic
36	C₈×C₁₉₉	SD₁₆	1	4	yes	—
37	C₂²×C₁₉₉	#127	1	1	yes	characteristic
38	C₂×D₁₉₉	—	8	1	no	—
39	C₂×D₁₉₉	—	8	2	no	—
40	C₄×C₁₉₉	—	8	2	no	—
41	C₄×C₁₉₉	C₂×SD₁₆	1	2	yes	—
42	C₄×C₁₉₉	#135	1	4	yes	—
43	Dic₁₉₉	—	8	1	no	—
44	C₂×C₁₉₉	#492	1	1	yes	characteristic
45	C₂×C₁₉₉	#495	1	2	yes	—
46	D₁₉₉	—	16	1	no	—
47	C₁₉₉	#1708	1	1	yes	characteristic, complemented, Sylow
48	#1708	—	199	1	no	maximal, Sylow
49	#498	D₁₉₉	1	1	yes	characteristic, complemented
50	#492	—	199	1	no	—
51	#487	—	199	2	no	—
52	#486	—	199	2	no	—
53	C₈²	—	199	1	no	—
54	C₂×SD₁₆	—	398	2	no	—
55	C₄⋊Q₈	—	199	1	no	—
56	C₄⋊Q₈	C₂×D₁₉₉	1	2	yes	—
57	#127	—	199	1	no	—
58	#106	—	2	2	no	—
59	#103	—	398	4	no	—
60	#102	—	398	4	no	—
61	C₄×C₈	—	199	2	no	—
62	C₄×C₈	C₂×D₁₉₉	1	1	yes	characteristic
63	C₂×Q₈	—	2	2	no	—
64	C₂×Q₈	—	398	1	no	—
65	C₂×D₄	—	398	2	no	—
66	C₂×D₄	—	398	1	no	—
67	SD₁₆	—	796	4	no	—
68	C₂×C₈	—	199	4	no	—
69	C₂×C₈	C₂²⋊D₁₉₉	1	2	yes	—
70	C₄⋊C₄	—	2	2	no	—
71	C₄⋊C₄	—	2	2	no	—
72	C₄⋊C₄	—	398	2	no	—
73	C₄²	C₂²×D₁₉₉	1	1	yes	characteristic
74	C₂³	—	796	1	no	—
75	Q₈	—	4	4	no	—
76	Q₈	—	796	2	no	—
77	D₄	—	796	2	no	—
78	D₄	—	796	4	no	—
79	C₂×C₄	—	4	2	no	—
80	C₂×C₄	—	796	1	no	—
81	C₂×C₄	D₄×D₁₉₉	1	2	yes	—
82	C₂×C₄	C₂×C₂²⋊D₁₉₉	1	1	yes	characteristic
83	C₈	—	398	4	no	—
84	C₈	#15691873	1	4	yes	—
85	C₂²	—	1592	1	no	—
86	C₂²	—	1592	2	no	—
87	C₂²	#20263951	1	1	yes	characteristic
88	C₄	—	8	2	no	—
89	C₄	—	1592	1	no	—
90	C₄	C₂×#15691873	1	2	yes	—
91	C₄	#20263976	1	4	yes	—
92	C₂	—	3184	1	no	—
93	C₂	#27164392	1	1	yes	characteristic
94	C₂	#27164399	1	2	yes	—

	subgroup	G/H
center	C₂²	#20263951
commutator, cosocle	C₄²×C₁₉₉	C₂³
Frattini	C₄²	C₂²×D₁₉₉
Fitting	#498×C₁₉₉	C₂
socle	C₂²×C₁₉₉	#127
prime radical	#20263952	C₂²
hypercenter	#498	D₁₉₉
hypocenter	C₁₉₉	#1708
LCS	G ▷ C₄²×C₁₉₉ ▷ C₂²×C₁₉₉ ▷ C₁₉₉
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #498

Complements

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

N	H ≅ G/N	type
2. #27164368	C₂	semidirect
4. C₈×C₁₉₉⋊C₈	C₂	semidirect
5. #498×C₁₉₉	C₂	semidirect
47. C₁₉₉	#1708	semidirect
49. #498	D₁₉₉	semidirect

Maximal subgroups

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

subgroup	index
#27164391	2
#27164368	2
#27164366	2
C₈×C₁₉₉⋊C₈	2
#498×C₁₉₉	2
#1708	199