Finite Group Databasebeta

Subgroups — #27164033

Order 12736.

154 conjugacy classes of subgroups; only the 51 normal subgroups are shown.

normal subgroups only

This group has 154 conjugacy classes of subgroups — too many to draw. Only its 51 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×D₁₉₉	C₂	1	1	yes	maximal, characteristic
2	M₁₆×D₁₉₉	C₂	1	1	yes	maximal, characteristic, complemented
3	#20263912	C₂	1	1	yes	maximal, characteristic, complemented
4	#20263883	C₂	1	1	yes	maximal, characteristic, complemented
5	#20263882	C₂	1	1	yes	maximal, characteristic, complemented
6	#20263881	C₂	1	1	yes	maximal, characteristic, complemented
7	#104×C₁₉₉	C₂	1	1	yes	maximal, characteristic, complemented
8	#15691887	C₄	1	1	yes	characteristic, complemented
9	#15691886	—	2	1	no	—
10	#15691886	C₂²	1	1	yes	characteristic
11	#15691885	—	2	1	no	—
12	#15691885	C₄	1	1	yes	characteristic, complemented
13	Q₈×D₁₉₉	C₄	1	1	yes	characteristic, complemented
14	D₄×D₁₉₉	—	2	1	no	—
15	D₄×D₁₉₉	C₄	1	1	yes	characteristic, complemented
16	C₂×C₄×D₁₉₉	—	2	1	no	—
17	C₂×C₄×D₁₉₉	C₂²	1	1	yes	characteristic
18	#15691870	C₂²	1	1	yes	characteristic, complemented
19	#15691869	—	2	1	no	—
20	C₈×D₁₉₉	—	2	1	no	—
21	Dic₁₉₉⋊C₄	—	2	1	no	—
22	(C₂×D₁₉₉)⋊C₄	—	2	1	no	—
23	C₄×Dic₁₉₉	C₂²	1	1	yes	characteristic, complemented
24	#41×C₁₉₉	C₂²	1	1	yes	characteristic
25	M₁₆×C₁₉₉	C₂²	1	1	yes	characteristic, complemented
26	C₄²×C₁₉₉	C₂²	1	1	yes	characteristic, complemented
27	C₂²×D₁₉₉	—	2	1	no	—
28	C₂²×D₁₉₉	D₄	1	1	yes	characteristic
29	Dic₃₉₈	—	2	1	no	—
30	Dic₃₉₈	C₂×C₄	1	1	yes	characteristic
31	D₇₉₆	—	2	1	no	—
32	D₇₉₆	C₂×C₄	1	1	yes	characteristic
33	C₂²⋊D₁₉₉	—	2	1	no	—
34	C₂²⋊D₁₉₉	—	2	1	no	—
35	C₂²⋊D₁₉₉	—	2	1	no	—
36	C₂×Dic₁₉₉	—	2	1	no	—
37	C₂×Dic₁₉₉	—	2	1	no	—
38	C₂×Dic₁₉₉	D₄	1	1	yes	characteristic
39	C₄×D₁₉₉	—	2	1	no	—
40	C₄×D₁₉₉	—	2	1	no	—
41	C₄×D₁₉₉	—	2	1	no	—
42	C₄×D₁₉₉	C₂×C₄	1	1	yes	characteristic
43	C₄×D₁₉₉	C₂×C₄	1	1	yes	characteristic
44	C₄×D₁₉₉	D₄	1	1	yes	characteristic
45	C₄×D₁₉₉	D₄	1	1	yes	characteristic
46	C₁₉₉⋊C₈	—	2	1	no	—
47	Q₈×C₁₉₉	C₂×C₄	1	1	yes	characteristic
48	D₄×C₁₉₉	—	2	1	no	—
49	D₄×C₁₉₉	C₂×C₄	1	1	yes	characteristic
50	C₂×C₄×C₁₉₉	—	2	1	no	—
51	C₂×C₄×C₁₉₉	—	2	1	no	—
52	C₂×C₄×C₁₉₉	C₂³	1	1	yes	characteristic
53	C₈×C₁₉₉	—	2	1	no	—
54	C₂²×C₁₉₉	—	2	1	no	—
55	C₂²×C₁₉₉	C₂×D₄	1	1	yes	characteristic
56	C₂×D₁₉₉	—	2	1	no	—
57	C₂×D₁₉₉	—	4	1	no	—
58	C₂×D₁₉₉	—	4	2	no	—
59	C₂×D₁₉₉	C₂²⋊C₄	1	1	yes	characteristic
60	C₂×D₁₉₉	C₂²⋊C₄	1	1	yes	characteristic
61	C₄×C₁₉₉	—	2	1	no	—
62	C₄×C₁₉₉	—	4	1	no	—
63	C₄×C₁₉₉	C₂²×C₄	1	1	yes	characteristic
64	C₄×C₁₉₉	C₂×D₄	1	1	yes	characteristic
65	Dic₁₉₉	—	2	1	no	—
66	Dic₁₉₉	—	4	1	no	—
67	Dic₁₉₉	C₂²⋊C₄	1	1	yes	characteristic
68	Dic₁₉₉	C₂²⋊C₄	1	1	yes	characteristic
69	C₂×C₁₉₉	—	2	1	no	—
70	C₂×C₁₉₉	—	4	1	no	—
71	C₂×C₁₉₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
72	D₁₉₉	—	2	1	no	—
73	D₁₉₉	—	2	1	no	—
74	D₁₉₉	—	4	1	no	—
75	C₁₉₉	#421	1	1	yes	characteristic, complemented, Sylow
76	#421	—	199	1	no	maximal, Sylow
77	C₂×#41	—	199	1	no	—
78	C₂×M₁₆	—	199	1	no	—
79	#117	—	199	1	no	—
80	#104	—	199	1	no	—
81	#104	—	199	1	no	—
82	#104	—	199	1	no	—
83	#104	D₁₉₉	1	1	yes	characteristic, complemented
84	#41	—	398	1	no	—
85	#41	—	398	1	no	—
86	#41	—	199	1	no	—
87	#41	—	199	1	no	—
88	#41	—	199	1	no	—
89	#41	C₂×D₁₉₉	1	1	yes	characteristic
90	C₂×Q₈	—	199	1	no	—
91	C₂×D₄	—	398	1	no	—
92	C₂×D₄	—	199	1	no	—
93	C₂²×C₄	—	398	1	no	—
94	C₂²×C₄	—	199	1	no	—
95	M₁₆	—	398	1	no	—
96	M₁₆	—	199	1	no	—
97	M₁₆	C₂×D₁₉₉	1	1	yes	characteristic, complemented
98	C₂×C₈	—	398	1	no	—
99	C₄⋊C₄	—	398	1	no	—
100	C₂²⋊C₄	—	398	1	no	—
101	C₄²	—	199	1	no	—
102	C₄²	C₂×D₁₉₉	1	1	yes	characteristic, complemented
103	C₂³	—	398	1	no	—
104	C₂³	—	199	1	no	—
105	Q₈	—	398	1	no	—
106	Q₈	—	199	1	no	—
107	Q₈	C₄×D₁₉₉	1	1	yes	characteristic
108	D₄	—	2	1	no	—
109	D₄	—	398	1	no	—
110	D₄	—	398	1	no	—
111	D₄	—	398	1	no	—
112	D₄	—	398	1	no	—
113	D₄	—	199	1	no	—
114	D₄	C₄×D₁₉₉	1	1	yes	characteristic
115	C₂×C₄	—	2	1	no	—
116	C₂×C₄	—	2	1	no	—
117	C₂×C₄	—	398	1	no	—
118	C₂×C₄	—	398	1	no	—
119	C₂×C₄	—	398	1	no	—
120	C₂×C₄	—	398	1	no	—
121	C₂×C₄	—	398	1	no	—
122	C₂×C₄	—	199	1	no	—
123	C₂×C₄	—	199	1	no	—
124	C₂×C₄	—	199	1	no	—
125	C₂×C₄	—	199	1	no	—
126	C₂×C₄	—	199	1	no	—
127	C₂×C₄	C₂²×D₁₉₉	1	1	yes	characteristic
128	C₈	—	2	1	no	—
129	C₈	—	398	1	no	—
130	C₂²	—	2	1	no	—
131	C₂²	—	796	1	no	—
132	C₂²	—	796	2	no	—
133	C₂²	—	398	1	no	—
134	C₂²	—	199	1	no	—
135	C₂²	—	199	1	no	—
136	C₂²	D₄×D₁₉₉	1	1	yes	characteristic
137	C₄	—	2	1	no	—
138	C₄	—	4	1	no	—
139	C₄	—	796	1	no	—
140	C₄	—	398	1	no	—
141	C₄	—	199	1	no	—
142	C₄	—	199	1	no	—
143	C₄	C₂×C₄×D₁₉₉	1	1	yes	characteristic
144	C₄	D₄×D₁₉₉	1	1	yes	characteristic
145	C₂	—	2	1	no	—
146	C₂	—	4	1	no	—
147	C₂	—	796	1	no	—
148	C₂	—	398	1	no	—
149	C₂	—	398	1	no	—
150	C₂	C₂²⋊C₄×D₁₉₉	1	1	yes	characteristic

	subgroup	G/H
center	C₄	D₄×D₁₉₉
commutator	C₄×C₁₉₉	C₂²×C₄
Frattini	C₂×C₄	C₂²×D₁₉₉
Fitting	#104×C₁₉₉	C₂
socle	C₂×C₁₉₉	C₂×C₂²⋊C₄
prime radical	#41×D₁₉₉	C₂
cosocle	C₂×C₄×C₁₉₉	C₂³
hypercenter	#104	D₁₉₉
hypocenter	C₁₉₉	#421
LCS	G ▷ C₄×C₁₉₉ ▷ C₂×C₁₉₉ ▷ C₁₉₉
UCS	C₁ ◁ C₄ ◁ C₂×C₄ ◁ #104

Complements

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

N	H ≅ G/N	type
2. M₁₆×D₁₉₉	C₂	semidirect
3. #20263912	C₂	semidirect
4. #20263883	C₂	semidirect
5. #20263882	C₂	semidirect
6. #20263881	C₂	semidirect
7. #104×C₁₉₉	C₂	semidirect
8. #15691887	C₄	semidirect
12. #15691885	C₄	semidirect
13. Q₈×D₁₉₉	C₄	semidirect
15. D₄×D₁₉₉	C₄	semidirect
18. #15691870	C₂²	semidirect
23. C₄×Dic₁₉₉	C₂²	semidirect
25. M₁₆×C₁₉₉	C₂²	semidirect
26. C₄²×C₁₉₉	C₂²	semidirect
75. C₁₉₉	#421	semidirect
83. #104	D₁₉₉	semidirect
97. M₁₆	C₂×D₁₉₉	semidirect
102. C₄²	C₂×D₁₉₉	semidirect

Maximal subgroups

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

subgroup	index
#41×D₁₉₉	2
M₁₆×D₁₉₉	2
#20263912	2
#20263883	2
#20263882	2
#20263881	2
#104×C₁₉₉	2
#421	199