Finite Group Databasebeta

Subgroups — #28379423

Order 13952.

144 conjugacy classes of subgroups; only the 58 normal subgroups are shown.

normal subgroups only

This group has 144 conjugacy classes of subgroups — too many to draw. Only its 58 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	#20296085	C₂	1	1	yes	maximal, characteristic, complemented
2	#20296084	C₂	1	1	yes	maximal, characteristic, complemented
3	#20295949	C₂	1	1	yes	maximal, characteristic, complemented
4	#20295925	C₂	1	1	yes	maximal, characteristic, complemented
5	#20295924	C₂	1	1	yes	maximal, characteristic, complemented
6	#20295914	C₂	1	1	yes	maximal, characteristic, complemented
7	#20295140	C₂	1	1	yes	maximal, characteristic
8	#16812774	—	2	1	no	—
9	#16812773	—	2	1	no	—
10	#16812773	C₂²	1	1	yes	characteristic
11	#16812773	C₂²	1	1	yes	characteristic
12	D₁₀₉⋊C₁₆	—	2	1	no	—
13	C₂×#13182032	—	2	1	no	—
14	C₂×#13182032	C₂²	1	1	yes	characteristic
15	C₂×#13182032	C₂²	1	1	yes	characteristic
16	#16812719	C₄	1	1	yes	characteristic
17	#16812707	C₂²	1	1	yes	characteristic
18	#16812706	C₂²	1	1	yes	characteristic
19	#16812634	C₄	1	1	yes	characteristic
20	#16812627	C₄	1	1	yes	characteristic
21	C₄×#11724784	C₂²	1	1	yes	characteristic
22	#104×C₁₀₉	C₄	1	1	yes	characteristic
23	#13182032	—	2	1	no	—
24	#13182032	—	2	1	no	—
25	#13182032	—	2	1	no	—
26	#13182032	D₄	1	1	yes	characteristic
27	#13182032	D₄	1	1	yes	characteristic
28	#13182029	C₂×C₄	1	1	yes	characteristic
29	#13182019	C₈	1	1	yes	characteristic
30	#13182017	C₈	1	1	yes	characteristic
31	#13182016	C₈	1	1	yes	characteristic
32	#13182014	C₈	1	1	yes	characteristic
33	#13182013	—	2	1	no	—
34	#13182013	C₂×C₄	1	1	yes	characteristic
35	#13182012	—	2	1	no	—
36	C₂×#11724784	—	2	1	no	—
37	C₂×#11724784	C₂×C₄	1	1	yes	characteristic
38	C₂×#11724784	C₂³	1	1	yes	characteristic
39	C₈×D₁₀₉	—	2	1	no	—
40	#41×C₁₀₉	C₂×C₄	1	1	yes	characteristic
41	M₁₆×C₁₀₉	C₂×C₄	1	1	yes	characteristic
42	C₄²×C₁₀₉	C₂×C₄	1	1	yes	characteristic
43	Q₈×C₁₀₉	C₂×C₈	1	1	yes	characteristic
44	D₄×C₁₀₉	—	2	1	no	—
45	D₄×C₁₀₉	C₂×C₈	1	1	yes	characteristic
46	C₂×C₄×C₁₀₉	—	2	1	no	—
47	C₂×C₄×C₁₀₉	—	2	1	no	—
48	C₂×C₄×C₁₀₉	C₂²×C₄	1	1	yes	characteristic
49	C₂²⋊D₁₀₉	—	2	1	no	—
50	D₄₃₆	C₂×C₈	1	1	yes	characteristic
51	C₄×D₁₀₉	—	2	1	no	—
52	Dic₂₁₈	C₂×C₈	1	1	yes	characteristic
53	C₈×C₁₀₉	—	2	1	no	—
54	#11724784	—	2	1	no	—
55	#11724784	C₂×C₈	1	1	yes	characteristic
56	#11724784	C₂×C₈	1	1	yes	characteristic
57	#11724784	C₂×D₄	1	1	yes	characteristic
58	#11724784	#41	1	1	yes	characteristic
59	C₂²×C₁₀₉	—	2	1	no	—
60	C₂²×C₁₀₉	#131	1	1	yes	characteristic
61	C₂×D₁₀₉	—	2	1	no	—
62	C₄×C₁₀₉	—	2	1	no	—
63	C₄×C₁₀₉	—	4	1	no	—
64	C₄×C₁₀₉	C₄×D₄	1	1	yes	characteristic
65	C₄×C₁₀₉	C₂²×C₈	1	1	yes	characteristic
66	Dic₁₀₉	—	2	1	no	—
67	C₂×C₁₀₉	—	2	1	no	—
68	C₂×C₁₀₉	—	4	1	no	—
69	C₂×C₁₀₉	C₈×D₄	1	1	yes	characteristic
70	D₁₀₉	—	4	1	no	—
71	#2171	—	109	1	no	maximal, Sylow
72	C₁₀₉	#2171	1	1	yes	characteristic, complemented, Sylow
73	#504	—	109	1	no	—
74	#504	—	109	1	no	—
75	#443	—	109	1	no	—
76	#364	—	109	1	no	—
77	#350	—	109	1	no	—
78	#350	—	109	1	no	—
79	#346	—	109	1	no	—
80	#135	—	109	1	no	—
81	#131	—	109	1	no	—
82	#131	—	109	1	no	—
83	M₃₂	—	109	1	no	—
84	M₃₂	—	109	1	no	—
85	M₃₂	—	218	1	no	—
86	M₃₂	—	218	1	no	—
87	C₂×C₁₆	—	109	1	no	—
88	C₂×C₁₆	—	109	1	no	—
89	C₂×C₁₆	—	218	1	no	—
90	C₂×C₁₆	—	218	1	no	—
91	#108	—	109	1	no	—
92	#104	—	109	1	no	—
93	#104	#89137	1	1	yes	characteristic
94	C₄×C₈	—	109	1	no	—
95	#41	—	109	1	no	—
96	#41	C₂×#89137	1	1	yes	characteristic
97	Q₁₆	—	109	1	no	—
98	SD₁₆	—	109	1	no	—
99	SD₁₆	—	109	1	no	—
100	D₈	—	109	1	no	—
101	M₁₆	—	109	1	no	—
102	M₁₆	—	218	1	no	—
103	M₁₆	—	218	1	no	—
104	M₁₆	C₂×#89137	1	1	yes	characteristic
105	C₂×C₈	—	109	1	no	—
106	C₂×C₈	—	109	1	no	—
107	C₂×C₈	—	218	1	no	—
108	C₂×C₈	—	218	1	no	—
109	C₄²	C₂×#89137	1	1	yes	characteristic
110	C₁₆	—	109	1	no	—
111	C₁₆	—	109	1	no	—
112	C₁₆	—	218	1	no	—
113	C₁₆	—	218	1	no	—
114	C₁₆	—	218	1	no	—
115	Q₈	—	109	1	no	—
116	Q₈	D₁₀₉⋊C₈	1	1	yes	characteristic
117	D₄	—	2	1	no	—
118	D₄	—	109	1	no	—
119	D₄	—	218	1	no	—
120	D₄	D₁₀₉⋊C₈	1	1	yes	characteristic
121	C₂×C₄	—	2	1	no	—
122	C₂×C₄	—	2	1	no	—
123	C₂×C₄	—	218	1	no	—
124	C₂×C₄	C₂²×#89137	1	1	yes	characteristic
125	C₈	—	2	1	no	—
126	C₈	—	109	1	no	—
127	C₈	—	109	1	no	—
128	C₈	—	109	1	no	—
129	C₈	—	109	1	no	—
130	C₈	—	218	1	no	—
131	C₂²	—	2	1	no	—
132	C₂²	—	218	1	no	—
133	C₂²	#16812786	1	1	yes	characteristic
134	C₄	—	2	1	no	—
135	C₄	—	4	1	no	—
136	C₄	—	218	1	no	—
137	C₄	D₄×#89137	1	1	yes	characteristic
138	C₄	C₂×D₁₀₉⋊C₈	1	1	yes	characteristic
139	C₂	—	2	1	no	—
140	C₂	—	4	1	no	—
141	C₂	—	436	1	no	—
142	C₂	#20296053	1	1	yes	characteristic

	subgroup	G/H
center	C₄	D₄×#89137
commutator	C₄×C₁₀₉	C₂²×C₈
Frattini	C₂×C₄	C₂²×#89137
Fitting	#104×C₁₀₉	C₄
socle	C₂×C₁₀₉	C₈×D₄
prime radical	#16812719	C₄
cosocle	C₂×#11724784	C₂³
hypercenter	#104	#89137
hypocenter	C₁₀₉	#2171
LCS	G ▷ C₄×C₁₀₉ ▷ C₂×C₁₀₉ ▷ C₁₀₉
UCS	C₁ ◁ C₄ ◁ C₂×C₄ ◁ #104

Complements

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

N	H ≅ G/N	type
1. #20296085	C₂	semidirect
2. #20296084	C₂	semidirect
3. #20295949	C₂	semidirect
4. #20295925	C₂	semidirect
5. #20295924	C₂	semidirect
6. #20295914	C₂	semidirect
72. C₁₀₉	#2171	semidirect

Maximal subgroups

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

subgroup	index
#20296085	2
#20296084	2
#20295949	2
#20295925	2
#20295924	2
#20295914	2
#20295140	2
#2171	109