Finite Group Databasebeta

Subgroups — #28379215

Order 13952.

260 conjugacy classes of subgroups; only the 78 normal subgroups are shown.

normal subgroups only

This group has 260 conjugacy classes of subgroups — too many to draw. Only its 78 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	#20295985	C₂	1	1	yes	maximal, characteristic
2	#20295972	C₂	1	1	yes	maximal, characteristic
3	#20295899	C₂	1	4	yes	maximal
4	#125×D₁₀₉	C₂	1	1	yes	maximal, characteristic
5	C₂×D₁₀₉⋊C₈	C₂²	1	2	yes	—
6	C₂×C₄⋊#89137	—	2	2	no	—
7	C₂×C₄⋊#89137	C₂²	1	2	yes	—
8	#16812754	—	2	2	no	—
9	#16812691	C₄	1	1	yes	characteristic, complemented
10	#16812690	C₄	1	2	yes	complemented
11	#16812689	—	2	2	no	—
12	C₄⋊C₄×D₁₀₉	—	2	2	no	—
13	C₄⋊C₄×D₁₀₉	C₂²	1	2	yes	—
14	C₄²×D₁₀₉	C₂²	1	1	yes	characteristic
15	#125×C₁₀₉	C₄	1	1	yes	characteristic, complemented
16	C₂²×#89137	—	2	2	no	—
17	D₁₀₉⋊C₈	—	2	2	no	—
18	D₁₀₉⋊C₈	Q₈	1	4	yes	—
19	C₂×#11724786	—	2	2	no	—
20	C₄⋊#89137	—	2	4	no	—
21	C₄⋊#89137	—	4	4	no	—
22	C₂×C₄×D₁₀₉	—	2	2	no	—
23	C₂×C₄×D₁₀₉	D₄	1	2	yes	—
24	C₂×C₄×D₁₀₉	C₂³	1	1	yes	characteristic
25	C₄⋊Dic₁₀₉	—	2	2	no	—
26	C₄⋊Dic₁₀₉	C₂×C₄	1	2	yes	—
27	#13182008	—	2	2	no	—
28	#13182008	—	2	2	no	—
29	#13182008	—	2	2	no	—
30	C₄×Dic₁₀₉	C₂×C₄	1	1	yes	characteristic
31	C₄×Dic₁₀₉	D₄	1	2	yes	—
32	C₄⋊C₄×C₁₀₉	—	2	2	no	—
33	C₄⋊C₄×C₁₀₉	C₂×C₄	1	2	yes	—
34	C₄²×C₁₀₉	C₂×C₄	1	1	yes	characteristic
35	C₂²×D₁₀₉	C₂×D₄	1	1	yes	characteristic
36	C₂×#89137	—	4	2	no	—
37	C₂×#89137	—	4	4	no	—
38	C₂×C₄×C₁₀₉	—	2	2	no	—
39	C₂×C₄×C₁₀₉	C₂²⋊C₄	1	2	yes	—
40	C₂×C₄×C₁₀₉	C₂²×C₄	1	1	yes	characteristic
41	C₂×Dic₁₀₉	—	2	2	no	—
42	C₂×Dic₁₀₉	C₂²⋊C₄	1	2	yes	—
43	C₂×Dic₁₀₉	C₂×D₄	1	1	yes	characteristic
44	C₄×D₁₀₉	—	2	2	no	—
45	C₄×D₁₀₉	—	2	2	no	—
46	C₄×D₁₀₉	—	4	2	no	—
47	C₄×D₁₀₉	—	4	2	no	—
48	C₄×D₁₀₉	C₂×Q₈	1	2	yes	—
49	C₄×D₁₀₉	#41	1	2	yes	—
50	#11724786	—	2	4	no	—
51	C₂²×C₁₀₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
52	C₂×D₁₀₉	#124	1	1	yes	characteristic
53	C₂×D₁₀₉	C₄⋊Q₈	1	1	yes	characteristic
54	C₂×D₁₀₉	AGL(1,ℤ/8)	1	2	yes	—
55	C₂×D₁₀₉	#137	1	2	yes	—
56	#89137	—	8	2	no	—
57	C₄×C₁₀₉	—	2	2	no	—
58	C₄×C₁₀₉	—	4	2	no	—
59	C₄×C₁₀₉	C₄×Q₈	1	2	yes	—
60	Dic₁₀₉	—	2	2	no	—
61	Dic₁₀₉	—	4	2	no	—
62	Dic₁₀₉	C₂²⋊Q₈	1	2	yes	—
63	C₂×C₁₀₉	#391	1	1	yes	characteristic
64	C₂×C₁₀₉	#417	1	2	yes	—
65	D₁₀₉	#490	1	1	yes	characteristic
66	D₁₀₉	#491	1	1	yes	characteristic
67	D₁₀₉	#501	1	2	yes	—
68	#1992	—	109	1	no	maximal, Sylow
69	C₁₀₉	#1992	1	1	yes	characteristic, complemented, Sylow
70	C₂×#125	—	109	1	no	—
71	C₂×#97	—	109	1	no	—
72	#384	—	109	1	no	—
73	#340	—	109	4	no	—
74	C₂²×C₈	—	109	2	no	—
75	#125	—	109	2	no	—
76	#125	—	109	1	no	—
77	#125	—	218	2	no	—
78	#125	#89137	1	1	yes	characteristic, complemented
79	C₂×C₄⋊C₄	—	109	2	no	—
80	C₂×C₄⋊C₄	—	109	2	no	—
81	C₂×C₄⋊C₄	—	218	2	no	—
82	C₂×C₄⋊C₄	—	218	2	no	—
83	C₂×C₄²	—	109	1	no	—
84	#97	—	218	2	no	—
85	C₂²×C₄	—	109	1	no	—
86	C₂²×C₄	—	109	2	no	—
87	C₂²×C₄	—	218	2	no	—
88	C₂²×C₄	—	218	2	no	—
89	C₂×C₈	—	109	4	no	—
90	C₂×C₈	—	218	2	no	—
91	C₂×C₈	—	218	2	no	—
92	C₄⋊C₄	—	2	2	no	—
93	C₄⋊C₄	—	109	2	no	—
94	C₄⋊C₄	—	436	4	no	—
95	C₄⋊C₄	—	218	2	no	—
96	C₄⋊C₄	—	218	2	no	—
97	C₄⋊C₄	—	218	2	no	—
98	C₄⋊C₄	—	218	4	no	—
99	C₄⋊C₄	—	218	2	no	—
100	C₄⋊C₄	C₂×#89137	1	2	yes	—
101	C₄²	—	109	2	no	—
102	C₄²	—	109	1	no	—
103	C₄²	C₂×#89137	1	1	yes	characteristic
104	C₂³	—	109	1	no	—
105	C₂×C₄	—	2	2	no	—
106	C₂×C₄	—	109	1	no	—
107	C₂×C₄	—	109	2	no	—
108	C₂×C₄	—	109	2	no	—
109	C₂×C₄	—	109	2	no	—
110	C₂×C₄	—	436	2	no	—
111	C₂×C₄	—	436	2	no	—
112	C₂×C₄	—	436	4	no	—
113	C₂×C₄	—	436	2	no	—
114	C₂×C₄	—	218	2	no	—
115	C₂×C₄	—	218	2	no	—
116	C₂×C₄	—	218	2	no	—
117	C₂×C₄	C₂²⋊#89137	1	2	yes	—
118	C₂×C₄	C₂²×#89137	1	1	yes	characteristic
119	C₈	—	218	4	no	—
120	C₂²	—	109	1	no	—
121	C₂²	—	109	1	no	—
122	C₂²	—	109	2	no	—
123	C₂²	—	109	2	no	—
124	C₂²	C₂×C₂²⋊#89137	1	1	yes	characteristic
125	C₄	—	2	2	no	—
126	C₄	—	4	2	no	—
127	C₄	—	872	2	no	—
128	C₄	—	109	2	no	—
129	C₄	—	436	2	no	—
130	C₄	—	218	2	no	—
131	C₄	Q₈×#89137	1	2	yes	—
132	C₂	—	109	1	no	—
133	C₂	—	109	1	no	—
134	C₂	—	109	2	no	—
135	C₂	(C₂×#89137)⋊Q₈	1	1	yes	characteristic
136	C₂	#20296017	1	2	yes	—

	subgroup	G/H
center	C₂²	C₂×C₂²⋊#89137
commutator	C₂×C₄×C₁₀₉	C₂²×C₄
Frattini	C₂×C₄	C₂²×#89137
Fitting	#125×C₁₀₉	C₄
socle	C₂²×C₁₀₉	C₂×C₂²⋊C₄
prime radical	C₂²×D₁₀₉	C₂×D₄
cosocle	C₂×C₄×D₁₀₉	C₂³
hypercenter	#125	#89137
hypocenter	C₁₀₉	#1992
LCS	G ▷ C₂×C₄×C₁₀₉ ▷ C₂×C₁₀₉ ▷ C₁₀₉
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #125

Complements

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

N	H ≅ G/N	type
9. #16812691	C₄	semidirect
10. #16812690	C₄	semidirect
15. #125×C₁₀₉	C₄	semidirect
69. C₁₀₉	#1992	semidirect
78. #125	#89137	semidirect

Maximal subgroups

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

subgroup	index
#20295985	2
#20295972	2
#20295899	2
#125×D₁₀₉	2
#1992	109