Finite Group Databasebeta

Subgroups — #28379434

Order 13952.

144 conjugacy classes of subgroups; only the 46 normal subgroups are shown.

normal subgroups only

This group has 144 conjugacy classes of subgroups — too many to draw. Only its 46 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	#20296070	C₂	1	1	yes	maximal, characteristic, complemented
2	#20296068	C₂	1	1	yes	maximal, characteristic, complemented
3	#20295954	C₂	1	1	yes	maximal, characteristic, complemented
4	#20295941	C₂	1	1	yes	maximal, characteristic, complemented
5	#20295940	C₂	1	1	yes	maximal, characteristic, complemented
6	#20295913	C₂	1	1	yes	maximal, characteristic, complemented
7	#20295457	C₂	1	1	yes	maximal, characteristic
8	#16812787	—	2	1	no	—
9	#16812786	—	2	1	no	—
10	D₁₀₉⋊C₁₆	C₂²	1	1	yes	characteristic
11	#16812770	C₂²	1	1	yes	characteristic
12	#16812770	C₂²	1	1	yes	characteristic
13	#16812764	—	2	1	no	—
14	#16812763	—	2	1	no	—
15	C₈×#89137	C₂²	1	1	yes	characteristic
16	#16812722	C₂²	1	1	yes	characteristic
17	#16812718	C₂²	1	1	yes	characteristic
18	#16812645	C₄	1	1	yes	characteristic, complemented
19	#16812644	C₄	1	1	yes	characteristic, complemented
20	#16812640	C₄	1	1	yes	characteristic, complemented
21	C₁₆×D₁₀₉	C₂²	1	1	yes	characteristic
22	SD₃₂×C₁₀₉	C₄	1	1	yes	characteristic, complemented
23	#13182039	—	2	1	no	—
24	#13182039	—	2	1	no	—
25	#13182039	—	4	1	no	—
26	#13182038	—	4	1	no	—
27	D₁₀₉⋊C₈	—	4	1	no	—
28	D₁₀₉⋊C₈	D₄	1	1	yes	characteristic
29	C₂×#11724786	—	4	1	no	—
30	C₄×#89137	D₄	1	1	yes	characteristic
31	#13182032	—	2	1	no	—
32	#13182030	—	2	1	no	—
33	#13182028	—	2	1	no	—
34	Dic₄₃₆	C₂×C₄	1	1	yes	characteristic
35	#13182017	—	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	D₈×C₁₀₉	C₂×C₄	1	1	yes	characteristic
42	C₁₆×C₁₀₉	C₂×C₄	1	1	yes	characteristic
43	C₂×#89137	—	2	1	no	—
44	Q₈×C₁₀₉	—	2	1	no	—
45	D₄×C₁₀₉	—	2	1	no	—
46	C₂²⋊D₁₀₉	—	4	1	no	—
47	C₂×Dic₁₀₉	—	4	1	no	—
48	D₄₃₆	—	2	1	no	—
49	D₄₃₆	—	4	1	no	—
50	C₄×D₁₀₉	—	4	1	no	—
51	C₄×D₁₀₉	C₂×D₄	1	1	yes	characteristic
52	Dic₂₁₈	—	2	1	no	—
53	#11724786	—	4	1	no	—
54	#11724786	—	4	1	no	—
55	#11724786	D₈	1	1	yes	characteristic
56	#11724786	D₈	1	1	yes	characteristic
57	C₈×C₁₀₉	C₂²×C₄	1	1	yes	characteristic
58	#11724784	#41	1	1	yes	characteristic
59	C₂²×C₁₀₉	—	4	1	no	—
60	C₂×D₁₀₉	—	4	1	no	—
61	C₂×D₁₀₉	#135	1	1	yes	characteristic
62	#89137	—	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	C₂×D₈	1	1	yes	characteristic
67	C₂×C₁₀₉	—	8	1	no	—
68	C₂×C₁₀₉	C₄×D₈	1	1	yes	characteristic
69	D₁₀₉	—	2	1	no	—
70	D₁₀₉	—	8	1	no	—
71	#2179	—	109	1	no	maximal, Sylow
72	C₁₀₉	#2179	1	1	yes	characteristic, complemented, Sylow
73	#508	—	109	1	no	—
74	#443	—	109	1	no	—
75	#443	—	109	1	no	—
76	#368	—	109	1	no	—
77	#359	—	109	1	no	—
78	#359	—	109	1	no	—
79	#346	—	109	1	no	—
80	#135	—	109	1	no	—
81	#135	—	109	1	no	—
82	#131	—	218	1	no	—
83	#131	—	218	1	no	—
84	Q₃₂	—	109	1	no	—
85	SD₃₂	—	109	1	no	—
86	SD₃₂	#89137	1	1	yes	characteristic, complemented
87	D₁₆	—	109	1	no	—
88	C₂×C₁₆	—	109	1	no	—
89	C₂×C₁₆	—	109	1	no	—
90	#108	—	109	1	no	—
91	#108	—	109	1	no	—
92	#104	—	218	1	no	—
93	#104	—	218	1	no	—
94	C₄×C₈	—	109	1	no	—
95	#41	—	218	1	no	—
96	#41	—	218	1	no	—
97	Q₁₆	—	109	1	no	—
98	Q₁₆	C₂×#89137	1	1	yes	characteristic
99	SD₁₆	—	218	1	no	—
100	SD₁₆	—	218	1	no	—
101	D₈	—	109	1	no	—
102	D₈	C₂×#89137	1	1	yes	characteristic
103	M₁₆	—	436	1	no	—
104	M₁₆	—	436	1	no	—
105	M₁₆	—	218	1	no	—
106	M₁₆	—	218	1	no	—
107	C₂×C₈	—	109	1	no	—
108	C₂×C₈	—	109	1	no	—
109	C₂×C₈	—	436	1	no	—
110	C₂×C₈	—	436	1	no	—
111	C₄²	—	109	1	no	—
112	C₁₆	—	109	1	no	—
113	C₁₆	—	218	1	no	—
114	C₁₆	C₂×#89137	1	1	yes	characteristic
115	Q₈	—	2	1	no	—
116	Q₈	—	218	1	no	—
117	D₄	—	2	1	no	—
118	D₄	—	436	1	no	—
119	D₄	—	436	1	no	—
120	D₄	—	218	1	no	—
121	C₂×C₄	—	109	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	—	109	1	no	—
127	C₈	—	109	1	no	—
128	C₈	—	436	1	no	—
129	C₈	—	436	1	no	—
130	C₈	C₂²×#89137	1	1	yes	characteristic
131	C₂²	—	4	1	no	—
132	C₂²	—	109	1	no	—
133	C₂²	—	436	1	no	—
134	C₄	—	4	1	no	—
135	C₄	—	109	1	no	—
136	C₄	—	436	1	no	—
137	C₄	—	436	1	no	—
138	C₄	D₄×#89137	1	1	yes	characteristic
139	C₂	—	8	1	no	—
140	C₂	—	872	1	no	—
141	C₂	—	218	1	no	—
142	C₂	D₈×#89137	1	1	yes	characteristic

	subgroup	G/H
center	C₂	D₈×#89137
commutator	C₈×C₁₀₉	C₂²×C₄
Frattini	C₈	C₂²×#89137
Fitting	SD₃₂×C₁₀₉	C₄
socle	C₂×C₁₀₉	C₄×D₈
prime radical	#20295457	C₂
cosocle	C₈×D₁₀₉	C₂³
hypercenter	SD₃₂	#89137
hypocenter	C₁₀₉	#2179
LCS	G ▷ C₈×C₁₀₉ ▷ C₄×C₁₀₉ ▷ C₂×C₁₀₉ ▷ C₁₀₉
UCS	C₁ ◁ C₂ ◁ C₄ ◁ C₈ ◁ SD₃₂

Complements

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

N	H ≅ G/N	type
1. #20296070	C₂	semidirect
2. #20296068	C₂	semidirect
3. #20295954	C₂	semidirect
4. #20295941	C₂	semidirect
5. #20295940	C₂	semidirect
6. #20295913	C₂	semidirect
18. #16812645	C₄	semidirect
19. #16812644	C₄	semidirect
20. #16812640	C₄	semidirect
22. SD₃₂×C₁₀₉	C₄	semidirect
72. C₁₀₉	#2179	semidirect
86. SD₃₂	#89137	semidirect

Maximal subgroups

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

subgroup	index
#20296070	2
#20296068	2
#20295954	2
#20295941	2
#20295940	2
#20295913	2
#20295457	2
#2179	109