Finite Group Databasebeta

Subgroups — #28379166

Order 13952.

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

normal subgroups only

This group has 360 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	#20296030	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×#16812763	C₂	1	4	yes	maximal, complemented
3	C₄²×#89137	C₂	1	1	yes	maximal, characteristic, complemented
4	#20295574	C₂	1	1	yes	maximal, characteristic
5	C₂×#13182039	C₂²	1	2	yes	—
6	C₂×C₄×#89137	—	2	2	no	—
7	C₂×C₄×#89137	C₂²	1	2	yes	—
8	C₄⋊#11724786	—	2	2	no	—
9	#16812763	—	2	8	no	—
10	C₂×#13182028	C₂²	1	2	yes	—
11	#16812699	C₄	1	1	yes	characteristic, complemented
12	#16812686	C₄	1	2	yes	complemented
13	#16812670	—	2	2	no	—
14	D₄×Dic₁₀₉	—	2	2	no	—
15	C₄²×D₁₀₉	C₂²	1	1	yes	characteristic
16	#127×C₁₀₉	C₄	1	1	yes	characteristic, complemented
17	C₂²×#89137	—	2	2	no	—
18	#13182039	—	2	4	no	—
19	C₂×#11724786	—	2	2	no	—
20	C₄×#89137	—	2	8	no	—
21	C₄×#89137	—	2	2	no	—
22	C₄×#89137	D₄	1	4	yes	complemented
23	#13182028	—	2	4	no	—
24	#13182028	—	2	4	no	—
25	C₂×Dic₂₁₈	C₂×C₄	1	2	yes	—
26	C₂×C₂²⋊D₁₀₉	—	2	2	no	—
27	C₂²×Dic₁₀₉	—	2	2	no	—
28	C₂×C₄×D₁₀₉	D₄	1	2	yes	—
29	C₂×C₄×D₁₀₉	C₂³	1	1	yes	characteristic
30	C₄⋊Dic₁₀₉	—	2	2	no	—
31	C₂²⋊Dic₁₀₉	—	2	4	no	—
32	C₄×Dic₁₀₉	C₂×C₄	1	1	yes	characteristic
33	C₄×Dic₁₀₉	D₄	1	2	yes	—
34	C₂×D₄×C₁₀₉	—	2	2	no	—
35	C₂×D₄×C₁₀₉	C₂×C₄	1	2	yes	—
36	C₄²×C₁₀₉	C₂×C₄	1	1	yes	characteristic
37	C₂³×C₁₀₉	—	2	2	no	—
38	C₂²×D₁₀₉	C₂×D₄	1	1	yes	characteristic
39	C₂×#89137	—	2	8	no	—
40	C₂×#89137	—	2	4	no	—
41	D₄×C₁₀₉	—	2	4	no	—
42	D₄×C₁₀₉	—	4	4	no	—
43	C₂×C₄×C₁₀₉	C₂²⋊C₄	1	2	yes	—
44	C₂×C₄×C₁₀₉	C₂²×C₄	1	1	yes	characteristic
45	C₂²⋊D₁₀₉	—	4	4	no	—
46	C₂×Dic₁₀₉	—	2	2	no	—
47	C₂×Dic₁₀₉	—	4	2	no	—
48	C₂×Dic₁₀₉	—	4	2	no	—
49	C₂×Dic₁₀₉	C₂²⋊C₄	1	2	yes	—
50	C₂×Dic₁₀₉	C₂×D₄	1	1	yes	characteristic
51	C₄×D₁₀₉	—	2	4	no	—
52	C₄×D₁₀₉	C₂×D₄	1	2	yes	—
53	C₄×D₁₀₉	#41	1	2	yes	—
54	Dic₂₁₈	—	2	4	no	—
55	#11724786	—	4	2	no	—
56	C₂²×C₁₀₉	—	4	2	no	—
57	C₂²×C₁₀₉	—	4	4	no	—
58	C₂²×C₁₀₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
59	C₂×D₁₀₉	—	2	2	no	—
60	C₂×D₁₀₉	#121	1	2	yes	—
61	#89137	—	2	8	no	—
62	C₄×C₁₀₉	#104	1	4	yes	complemented
63	C₄×C₁₀₉	C₄×D₄	1	2	yes	—
64	Dic₁₀₉	—	2	2	no	—
65	Dic₁₀₉	—	4	2	no	—
66	Dic₁₀₉	#124	1	1	yes	characteristic
67	Dic₁₀₉	#127	1	1	yes	characteristic
68	C₂×C₁₀₉	—	8	2	no	—
69	C₂×C₁₀₉	#390	1	1	yes	characteristic
70	C₂×C₁₀₉	C₂×#104	1	2	yes	—
71	D₁₀₉	—	2	2	no	—
72	#1962	—	109	1	no	maximal, Sylow
73	C₁₀₉	#1962	1	1	yes	characteristic, complemented, Sylow
74	#532	—	109	1	no	—
75	#423	—	109	1	no	—
76	C₂×#104	—	109	4	no	—
77	C₄³	—	109	1	no	—
78	C₂×#41	—	109	2	no	—
79	C₂×M₁₆	—	109	2	no	—
80	C₄⋊Q₈	—	109	1	no	—
81	#127	#89137	1	1	yes	characteristic, complemented
82	#124	—	109	2	no	—
83	#121	—	218	2	no	—
84	C₄×D₄	—	218	2	no	—
85	C₂×C₄²	—	109	1	no	—
86	C₂×C₄²	—	109	2	no	—
87	C₂×C₄²	—	218	2	no	—
88	C₄⋊C₈	—	218	2	no	—
89	#104	—	218	8	no	—
90	#41	—	218	4	no	—
91	#41	—	218	4	no	—
92	C₂×Q₈	—	109	2	no	—
93	C₂×D₄	—	2	2	no	—
94	C₂×D₄	—	218	2	no	—
95	C₂×D₄	C₂×#89137	1	2	yes	—
96	C₂²×C₄	—	109	1	no	—
97	C₂²×C₄	—	109	2	no	—
98	C₂²×C₄	—	218	2	no	—
99	C₂²×C₄	—	218	2	no	—
100	M₁₆	—	218	4	no	—
101	C₂×C₈	—	218	2	no	—
102	C₄⋊C₄	—	218	2	no	—
103	C₂²⋊C₄	—	218	4	no	—
104	C₄²	—	109	2	no	—
105	C₄²	—	109	1	no	—
106	C₄²	—	109	4	no	—
107	C₄²	—	218	2	no	—
108	C₄²	—	218	8	no	—
109	C₄²	C₂×#89137	1	1	yes	characteristic
110	C₂³	—	2	2	no	—
111	C₂³	—	109	1	no	—
112	Q₈	—	218	4	no	—
113	D₄	—	2	4	no	—
114	D₄	—	4	4	no	—
115	D₄	—	436	4	no	—
116	C₂×C₄	—	109	1	no	—
117	C₂×C₄	—	109	2	no	—
118	C₂×C₄	—	109	2	no	—
119	C₂×C₄	—	109	2	no	—
120	C₂×C₄	—	436	2	no	—
121	C₂×C₄	—	436	2	no	—
122	C₂×C₄	—	218	2	no	—
123	C₂×C₄	—	218	4	no	—
124	C₂×C₄	—	218	8	no	—
125	C₂×C₄	—	218	4	no	—
126	C₂×C₄	C₂²⋊#89137	1	2	yes	—
127	C₂×C₄	C₂²×#89137	1	1	yes	characteristic
128	C₈	—	436	2	no	—
129	C₂²	—	4	2	no	—
130	C₂²	—	4	4	no	—
131	C₂²	—	109	2	no	—
132	C₂²	—	218	2	no	—
133	C₂²	C₂×C₂²⋊#89137	1	1	yes	characteristic
134	C₄	—	109	1	no	—
135	C₄	—	109	1	no	—
136	C₄	—	436	2	no	—
137	C₄	—	218	8	no	—
138	C₄	—	218	2	no	—
139	C₄	#16812763	1	4	yes	complemented
140	C₄	D₄×#89137	1	2	yes	—
141	C₂	—	8	2	no	—
142	C₂	—	218	2	no	—
143	C₂	#20295978	1	1	yes	characteristic
144	C₂	C₂×#16812763	1	2	yes	—

	subgroup	G/H
center	C₂²	C₂×C₂²⋊#89137
commutator	C₂×C₄×C₁₀₉	C₂²×C₄
Frattini	C₂×C₄	C₂²×#89137
Fitting	#127×C₁₀₉	C₄
socle	C₂²×C₁₀₉	C₂×C₂²⋊C₄
prime radical	#20295574	C₂
cosocle	C₂×C₄×D₁₀₉	C₂³
hypercenter	#127	#89137
hypocenter	C₁₀₉	#1962
LCS	G ▷ C₂×C₄×C₁₀₉ ▷ C₂×C₁₀₉ ▷ C₁₀₉
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #127

Complements

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

N	H ≅ G/N	type
1. #20296030	C₂	semidirect
2. C₂×#16812763	C₂	semidirect
3. C₄²×#89137	C₂	semidirect
11. #16812699	C₄	semidirect
12. #16812686	C₄	semidirect
16. #127×C₁₀₉	C₄	semidirect
22. C₄×#89137	D₄	semidirect
62. C₄×C₁₀₉	#104	semidirect
73. C₁₀₉	#1962	semidirect
81. #127	#89137	semidirect
139. C₄	#16812763	semidirect

Maximal subgroups

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

subgroup	index
#20296030	2
C₂×#16812763	2
C₄²×#89137	2
#20295574	2
#1962	109