Finite Group Databasebeta

Subgroups — #28378583

Order 13952.

180 conjugacy classes of subgroups; only the 92 normal subgroups are shown.

normal subgroups only

This group has 180 conjugacy classes of subgroups — too many to draw. Only its 92 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	#20296042	C₂	1	1	yes	maximal, characteristic
2	C₂×C₄⋊#11724786	C₂	1	1	yes	maximal, characteristic
3	#20295889	C₂	1	2	yes	maximal, complemented
4	#20295887	C₂	1	2	yes	maximal, complemented
5	M₁₆×Dic₁₀₉	C₂	1	1	yes	maximal, characteristic
6	C₂²×#11724786	—	2	1	no	—
7	C₄⋊#11724786	—	2	1	no	—
8	C₄⋊#11724786	C₂²	1	2	yes	—
9	C₄⋊#11724786	C₂²	1	1	yes	characteristic
10	C₄⋊#11724786	C₂²	1	1	yes	characteristic
11	C₂²⋊#11724786	—	2	1	no	—
12	C₄×#11724786	—	2	1	no	—
13	C₂×#13182013	C₄	1	1	yes	characteristic
14	C₂×C₄×Dic₁₀₉	C₂²	1	1	yes	characteristic
15	#16812609	C₄	1	2	yes	—
16	C₈×Dic₁₀₉	C₂²	1	2	yes	—
17	C₂×M₁₆×C₁₀₉	C₄	1	1	yes	characteristic
18	C₂×#11724786	—	2	1	no	—
19	C₂×#11724786	—	2	1	no	—
20	C₂×#11724786	—	2	2	no	—
21	C₂×#11724786	—	4	2	no	—
22	C₂²×Dic₁₀₉	D₄	1	1	yes	characteristic
23	C₂²×Dic₁₀₉	Q₈	1	1	yes	characteristic
24	#13182013	—	2	2	no	—
25	C₂×#11724784	C₂×C₄	1	2	yes	—
26	C₄×Dic₁₀₉	C₂×C₄	1	1	yes	characteristic
27	C₄×Dic₁₀₉	D₄	1	1	yes	characteristic
28	C₄×Dic₁₀₉	Q₈	1	1	yes	characteristic
29	C₄×Dic₁₀₉	C₂³	1	1	yes	characteristic
30	C₂²×C₄×C₁₀₉	C₂×C₄	1	1	yes	characteristic
31	M₁₆×C₁₀₉	C₈	1	4	yes	complemented
32	C₂×C₈×C₁₀₉	C₂×C₄	1	2	yes	—
33	C₂³×C₁₀₉	C₄⋊C₄	1	1	yes	characteristic
34	C₂×C₄×C₁₀₉	C₄⋊C₄	1	1	yes	characteristic
35	C₂×C₄×C₁₀₉	C₂×C₈	1	2	yes	—
36	C₂×C₄×C₁₀₉	M₁₆	1	2	yes	—
37	C₂×C₄×C₁₀₉	C₂²×C₄	1	1	yes	characteristic
38	C₂×Dic₁₀₉	—	2	1	no	—
39	C₂×Dic₁₀₉	—	2	1	no	—
40	C₂×Dic₁₀₉	—	4	1	no	—
41	C₂×Dic₁₀₉	C₄⋊C₄	1	1	yes	characteristic
42	C₂×Dic₁₀₉	C₄⋊C₄	1	1	yes	characteristic
43	C₂×Dic₁₀₉	C₂×D₄	1	1	yes	characteristic
44	C₂×Dic₁₀₉	C₂×Q₈	1	1	yes	characteristic
45	#11724786	—	4	1	no	—
46	#11724786	—	4	2	no	—
47	#11724786	—	4	1	no	—
48	C₈×C₁₀₉	C₂×C₈	1	4	yes	complemented
49	#11724784	—	2	2	no	—
50	C₂²×C₁₀₉	—	2	1	no	—
51	C₂²×C₁₀₉	—	2	1	no	—
52	C₂²×C₁₀₉	C₄⋊C₈	1	2	yes	—
53	C₂²×C₁₀₉	C₂×C₄⋊C₄	1	1	yes	characteristic
54	C₄×C₁₀₉	C₄⋊C₈	1	2	yes	—
55	C₄×C₁₀₉	C₂²×C₈	1	1	yes	characteristic
56	C₄×C₁₀₉	C₂×M₁₆	1	1	yes	characteristic
57	Dic₁₀₉	—	2	1	no	—
58	Dic₁₀₉	—	2	1	no	—
59	Dic₁₀₉	—	2	1	no	—
60	Dic₁₀₉	AGL(1,ℤ/8)	1	1	yes	characteristic
61	Dic₁₀₉	#137	1	1	yes	characteristic
62	C₂×C₁₀₉	—	2	2	no	—
63	C₂×C₁₀₉	C₂×C₄⋊C₈	1	1	yes	characteristic
64	C₂×C₁₀₉	#428	1	1	yes	characteristic
65	C₂×C₁₀₉	#430	1	1	yes	characteristic
66	#1565	—	109	1	no	maximal, Sylow
67	C₁₀₉	#1565	1	1	yes	characteristic, complemented, Sylow
68	#431	—	109	1	no	—
69	C₂×C₄⋊C₈	—	109	1	no	—
70	C₄×M₁₆	—	109	1	no	—
71	#335	—	109	2	no	—
72	#334	—	109	2	no	—
73	C₂×M₁₆	—	109	1	no	—
74	C₂×M₁₆	#89137	1	1	yes	characteristic
75	C₂²×C₈	—	218	1	no	—
76	C₂×C₄²	—	109	1	no	—
77	C₄⋊C₈	—	109	1	no	—
78	C₄⋊C₈	—	109	2	no	—
79	C₄⋊C₈	—	109	1	no	—
80	C₄⋊C₈	—	218	1	no	—
81	C₂²⋊C₈	—	218	1	no	—
82	#97	—	109	2	no	—
83	C₄×C₈	—	109	2	no	—
84	C₄×C₈	—	218	1	no	—
85	C₂²×C₄	—	109	1	no	—
86	C₂²×C₄	—	109	1	no	—
87	C₂²×C₄	C₂×#89137	1	1	yes	characteristic
88	M₁₆	—	218	2	no	—
89	M₁₆	#11724786	1	4	yes	complemented
90	C₂×C₈	—	109	2	no	—
91	C₂×C₈	—	436	2	no	—
92	C₂×C₈	—	218	1	no	—
93	C₂×C₈	—	218	1	no	—
94	C₂×C₈	—	218	2	no	—
95	C₂×C₈	C₂×#89137	1	2	yes	—
96	C₄²	—	109	1	no	—
97	C₄²	—	109	1	no	—
98	C₄²	—	109	1	no	—
99	C₄²	—	109	1	no	—
100	C₂³	C₄⋊#89137	1	1	yes	characteristic
101	C₂×C₄	—	109	1	no	—
102	C₂×C₄	—	109	1	no	—
103	C₂×C₄	—	109	1	no	—
104	C₂×C₄	—	109	1	no	—
105	C₂×C₄	—	436	1	no	—
106	C₂×C₄	—	218	1	no	—
107	C₂×C₄	—	218	1	no	—
108	C₂×C₄	C₄⋊#89137	1	1	yes	characteristic
109	C₂×C₄	C₂×#11724786	1	2	yes	—
110	C₂×C₄	#13182039	1	2	yes	—
111	C₂×C₄	C₂²×#89137	1	1	yes	characteristic
112	C₈	—	436	2	no	—
113	C₈	—	436	1	no	—
114	C₈	—	436	1	no	—
115	C₈	—	218	2	no	—
116	C₈	C₂×#11724786	1	4	yes	complemented
117	C₂²	—	2	1	no	—
118	C₂²	—	2	1	no	—
119	C₂²	C₄⋊#11724786	1	2	yes	—
120	C₂²	C₂×C₄⋊#89137	1	1	yes	characteristic
121	C₄	—	109	1	no	—
122	C₄	—	109	1	no	—
123	C₄	—	218	1	no	—
124	C₄	—	218	1	no	—
125	C₄	—	218	1	no	—
126	C₄	C₄⋊#11724786	1	2	yes	—
127	C₄	C₂²×#11724786	1	1	yes	characteristic
128	C₄	C₂×#13182039	1	1	yes	characteristic
129	C₂	—	2	2	no	—
130	C₂	C₂×C₄⋊#11724786	1	1	yes	characteristic
131	C₂	#20296037	1	1	yes	characteristic
132	C₂	#20296041	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×C₄⋊#89137
commutator	C₄×C₁₀₉	C₂²×C₈
Frattini	C₂×C₄	C₂²×#89137
Fitting	C₂×M₁₆×C₁₀₉	C₄
socle	C₂²×C₁₀₉	C₂×C₄⋊C₄
prime radical	C₂³×C₁₀₉	C₄⋊C₄
cosocle	C₄×Dic₁₀₉	C₂³
hypercenter	C₂×M₁₆	#89137
hypocenter	C₁₀₉	#1565
LCS	G ▷ C₄×C₁₀₉ ▷ C₂×C₁₀₉ ▷ C₁₀₉
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×M₁₆

Complements

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

N	H ≅ G/N	type
3. #20295889	C₂	semidirect
4. #20295887	C₂	semidirect
31. M₁₆×C₁₀₉	C₈	semidirect
48. C₈×C₁₀₉	C₂×C₈	semidirect
67. C₁₀₉	#1565	semidirect
89. M₁₆	#11724786	semidirect
116. C₈	C₂×#11724786	semidirect

Maximal subgroups

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

subgroup	index
#20296042	2
C₂×C₄⋊#11724786	2
#20295889	2
#20295887	2
M₁₆×Dic₁₀₉	2
#1565	109