Finite Group Databasebeta

Subgroups — #28379440

Order 13952.

142 conjugacy classes of subgroups; only the 44 normal subgroups are shown.

normal subgroups only

This group has 142 conjugacy classes of subgroups — too many to draw. Only its 44 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	#20296072	C₂	1	1	yes	maximal, characteristic
2	#20296066	C₂	1	1	yes	maximal, characteristic, complemented
3	#20295951	C₂	1	1	yes	maximal, characteristic, complemented
4	#20295945	C₂	1	1	yes	maximal, characteristic, complemented
5	#20295943	C₂	1	1	yes	maximal, characteristic
6	#20295917	C₂	1	1	yes	maximal, characteristic, complemented
7	#20295468	C₂	1	1	yes	maximal, characteristic
8	#16812787	—	2	1	no	—
9	Q₈×#89137	—	2	1	no	—
10	#16812774	C₂²	1	1	yes	characteristic
11	#16812769	C₂²	1	1	yes	characteristic
12	#16812767	C₂²	1	1	yes	characteristic
13	#16812764	—	2	1	no	—
14	#16812762	—	2	1	no	—
15	#16812753	C₂²	1	1	yes	characteristic
16	#16812722	C₂²	1	1	yes	characteristic
17	Q₁₆×D₁₀₉	C₂²	1	1	yes	characteristic
18	#16812645	C₄	1	1	yes	characteristic, complemented
19	#16812644	C₄	1	1	yes	characteristic, complemented
20	#16812642	C₄	1	1	yes	characteristic, complemented
21	#16812638	C₂²	1	1	yes	characteristic
22	Q₃₂×C₁₀₉	C₄	1	1	yes	characteristic, complemented
23	#13182039	—	2	1	no	—
24	#13182039	—	4	1	no	—
25	D₁₀₉⋊C₈	—	4	1	no	—
26	D₁₀₉⋊C₈	D₄	1	1	yes	characteristic
27	C₄⋊#89137	—	2	1	no	—
28	C₄⋊#89137	—	4	1	no	—
29	C₄×#89137	—	4	1	no	—
30	C₄×#89137	D₄	1	1	yes	characteristic
31	#13182032	—	2	1	no	—
32	#13182030	—	2	1	no	—
33	Q₈×D₁₀₉	—	2	1	no	—
34	Dic₄₃₆	C₂×C₄	1	1	yes	characteristic
35	#13182019	—	2	1	no	—
36	#13182016	—	2	1	no	—
37	D₈₇₂	C₂×C₄	1	1	yes	characteristic
38	C₈×D₁₀₉	C₂³	1	1	yes	characteristic
39	#13182004	C₂×C₄	1	1	yes	characteristic
40	Q₁₆×C₁₀₉	C₂×C₄	1	1	yes	characteristic
41	Q₁₆×C₁₀₉	C₂×C₄	1	1	yes	characteristic
42	C₁₆×C₁₀₉	C₂×C₄	1	1	yes	characteristic
43	C₂×#89137	—	2	1	no	—
44	C₂×#89137	—	4	1	no	—
45	Q₈×C₁₀₉	—	2	1	no	—
46	Q₈×C₁₀₉	—	2	1	no	—
47	D₄₃₆	—	2	1	no	—
48	D₄₃₆	—	4	1	no	—
49	C₄×D₁₀₉	—	4	1	no	—
50	C₄×D₁₀₉	—	4	1	no	—
51	C₄×D₁₀₉	C₂×D₄	1	1	yes	characteristic
52	Dic₂₁₈	—	2	1	no	—
53	Dic₂₁₈	—	4	1	no	—
54	#11724786	—	2	1	no	—
55	#11724786	—	4	1	no	—
56	C₈×C₁₀₉	C₂²×C₄	1	1	yes	characteristic
57	#11724784	#41	1	1	yes	characteristic
58	C₂×D₁₀₉	—	4	1	no	—
59	C₂×D₁₀₉	AGL(1,ℤ/8)	1	1	yes	characteristic
60	#89137	—	4	1	no	—
61	#89137	—	8	1	no	—
62	C₄×C₁₀₉	—	4	1	no	—
63	C₄×C₁₀₉	—	4	1	no	—
64	C₄×C₁₀₉	C₄×D₄	1	1	yes	characteristic
65	Dic₁₀₉	—	4	1	no	—
66	Dic₁₀₉	AGL(1,ℤ/8)	1	1	yes	characteristic
67	C₂×C₁₀₉	#442	1	1	yes	characteristic
68	D₁₀₉	—	2	1	no	—
69	D₁₀₉	—	8	1	no	—
70	#2180	—	109	1	no	maximal, Sylow
71	C₁₀₉	#2180	1	1	yes	characteristic, complemented, Sylow
72	#510	—	109	1	no	—
73	#444	—	109	1	no	—
74	#441	—	109	1	no	—
75	#365	—	109	1	no	—
76	#362	—	109	1	no	—
77	#360	—	109	1	no	—
78	#347	—	109	1	no	—
79	#135	—	109	1	no	—
80	C₂×Q₁₆	—	109	1	no	—
81	#131	—	218	1	no	—
82	C₄×Q₈	—	218	1	no	—
83	Q₃₂	—	109	1	no	—
84	Q₃₂	#89137	1	1	yes	characteristic, complemented
85	SD₃₂	—	109	1	no	—
86	SD₃₂	—	109	1	no	—
87	M₃₂	—	109	1	no	—
88	M₃₂	—	109	1	no	—
89	#108	—	109	1	no	—
90	#106	—	109	1	no	—
91	#104	—	218	1	no	—
92	#103	—	218	1	no	—
93	#97	—	109	1	no	—
94	#41	—	218	1	no	—
95	C₂×Q₈	—	218	1	no	—
96	Q₁₆	—	109	1	no	—
97	Q₁₆	—	218	1	no	—
98	Q₁₆	C₂×#89137	1	1	yes	characteristic
99	Q₁₆	C₂×#89137	1	1	yes	characteristic
100	SD₁₆	—	218	1	no	—
101	D₈	—	109	1	no	—
102	M₁₆	—	436	1	no	—
103	M₁₆	—	218	1	no	—
104	C₂×C₈	—	109	1	no	—
105	C₂×C₈	—	109	1	no	—
106	C₂×C₈	—	436	1	no	—
107	C₄⋊C₄	—	436	1	no	—
108	C₄⋊C₄	—	218	1	no	—
109	C₄²	—	109	1	no	—
110	C₄²	—	436	1	no	—
111	C₁₆	—	109	1	no	—
112	C₁₆	—	218	1	no	—
113	C₁₆	C₂×#89137	1	1	yes	characteristic
114	Q₈	—	2	1	no	—
115	Q₈	—	2	1	no	—
116	Q₈	—	436	1	no	—
117	Q₈	—	218	1	no	—
118	D₄	—	436	1	no	—
119	D₄	—	218	1	no	—
120	C₂×C₄	—	109	1	no	—
121	C₂×C₄	—	436	1	no	—
122	C₂×C₄	—	436	1	no	—
123	C₂×C₄	—	436	1	no	—
124	C₂×C₄	—	218	1	no	—
125	C₈	—	109	1	no	—
126	C₈	—	436	1	no	—
127	C₈	—	218	1	no	—
128	C₈	C₂²×#89137	1	1	yes	characteristic
129	C₂²	—	109	1	no	—
130	C₂²	—	436	1	no	—
131	C₄	—	4	1	no	—
132	C₄	—	4	1	no	—
133	C₄	—	872	1	no	—
134	C₄	—	109	1	no	—
135	C₄	—	436	1	no	—
136	C₄	—	436	1	no	—
137	C₄	D₄×#89137	1	1	yes	characteristic
138	C₂	—	872	1	no	—
139	C₂	—	218	1	no	—
140	C₂	#20296067	1	1	yes	characteristic

	subgroup	G/H
center	C₂	#20296067
commutator	C₈×C₁₀₉	C₂²×C₄
Frattini	C₈	C₂²×#89137
Fitting	Q₃₂×C₁₀₉	C₄
socle	C₂×C₁₀₉	#442
prime radical	#16812722	C₂²
cosocle	C₈×D₁₀₉	C₂³
hypercenter	Q₃₂	#89137
hypocenter	C₁₀₉	#2180
LCS	G ▷ C₈×C₁₀₉ ▷ C₄×C₁₀₉ ▷ C₂×C₁₀₉ ▷ C₁₀₉
UCS	C₁ ◁ C₂ ◁ C₄ ◁ C₈ ◁ Q₃₂

Complements

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

N	H ≅ G/N	type
2. #20296066	C₂	semidirect
3. #20295951	C₂	semidirect
4. #20295945	C₂	semidirect
6. #20295917	C₂	semidirect
18. #16812645	C₄	semidirect
19. #16812644	C₄	semidirect
20. #16812642	C₄	semidirect
22. Q₃₂×C₁₀₉	C₄	semidirect
71. C₁₀₉	#2180	semidirect
84. Q₃₂	#89137	semidirect

Maximal subgroups

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

subgroup	index
#20296072	2
#20296066	2
#20295951	2
#20295945	2
#20295943	2
#20295917	2
#20295468	2
#2180	109