Finite Group Databasebeta

Subgroups — #28361633

Order 13952.

212 conjugacy classes of subgroups; only the 73 normal subgroups are shown.

normal subgroups only

This group has 212 conjugacy classes of subgroups — too many to draw. Only its 73 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	C₂×#16812691	C₂	1	1	yes	maximal, characteristic
2	#20295083	C₂	1	1	yes	maximal, characteristic
3	#20294767	C₂	1	4	yes	maximal, complemented
4	C₄×M₁₆×C₁₀₉	C₂	1	1	yes	maximal, characteristic
5	#16812691	—	2	1	no	—
6	#16812691	—	2	1	no	—
7	#16812691	C₄	1	2	yes	—
8	#16812691	C₂²	1	2	yes	—
9	C₂×C₄⋊Dic₁₀₉	C₄	1	1	yes	characteristic
10	C₂×C₄⋊Dic₁₀₉	C₄	1	1	yes	characteristic
11	C₂×#13182008	—	2	1	no	—
12	C₂×#13182008	—	2	1	no	—
13	C₄⋊#11724784	—	2	1	no	—
14	C₂²⋊#11724784	—	2	1	no	—
15	#16812610	C₂²	1	2	yes	—
16	C₂×M₁₆×C₁₀₉	—	2	1	no	—
17	C₂×C₄²×C₁₀₉	C₂²	1	1	yes	characteristic
18	#97×C₁₀₉	C₂²	1	2	yes	—
19	C₄×C₈×C₁₀₉	—	2	1	no	—
20	C₂²×Dic₁₀₉	—	2	1	no	—
21	C₂²×Dic₁₀₉	—	2	1	no	—
22	C₂×#11724784	—	2	2	no	—
23	C₄⋊Dic₁₀₉	—	2	1	no	—
24	C₄⋊Dic₁₀₉	—	2	1	no	—
25	C₄⋊Dic₁₀₉	C₂×C₄	1	2	yes	—
26	C₄⋊Dic₁₀₉	C₂×C₄	1	2	yes	—
27	#13182008	—	2	2	no	—
28	#13182008	—	2	2	no	—
29	#13182008	—	2	2	no	—
30	#13182008	—	2	2	no	—
31	C₂²×C₄×C₁₀₉	C₂×C₄	1	1	yes	characteristic
32	C₂²×C₄×C₁₀₉	D₄	1	1	yes	characteristic
33	C₂²×C₄×C₁₀₉	D₄	1	1	yes	characteristic
34	M₁₆×C₁₀₉	—	2	4	no	—
35	C₂×C₈×C₁₀₉	—	2	2	no	—
36	C₄²×C₁₀₉	C₂×C₄	1	1	yes	characteristic
37	C₄²×C₁₀₉	D₄	1	1	yes	characteristic
38	C₄²×C₁₀₉	D₄	1	1	yes	characteristic
39	C₄²×C₁₀₉	C₂³	1	1	yes	characteristic
40	C₂³×C₁₀₉	C₂²⋊C₄	1	1	yes	characteristic
41	C₂×C₄×C₁₀₉	—	2	2	no	—
42	C₂×C₄×C₁₀₉	—	2	2	no	—
43	C₂×C₄×C₁₀₉	—	4	1	no	—
44	C₂×C₄×C₁₀₉	C₂²⋊C₄	1	1	yes	characteristic
45	C₂×C₄×C₁₀₉	C₂²⋊C₄	1	1	yes	characteristic
46	C₂×C₄×C₁₀₉	C₂²⋊C₄	1	1	yes	characteristic
47	C₂×C₄×C₁₀₉	C₂²×C₄	1	1	yes	characteristic
48	C₂×C₄×C₁₀₉	C₂×D₄	1	1	yes	characteristic
49	C₂×C₄×C₁₀₉	C₂×D₄	1	1	yes	characteristic
50	C₂×Dic₁₀₉	—	2	2	no	—
51	C₂×Dic₁₀₉	—	2	2	no	—
52	C₂×Dic₁₀₉	—	4	2	no	—
53	C₂×Dic₁₀₉	—	4	2	no	—
54	C₈×C₁₀₉	—	2	4	no	—
55	#11724784	—	4	2	no	—
56	C₂²×C₁₀₉	—	2	1	no	—
57	C₂²×C₁₀₉	—	2	1	no	—
58	C₂²×C₁₀₉	#104	1	2	yes	—
59	C₂²×C₁₀₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
60	C₄×C₁₀₉	—	2	1	no	—
61	C₄×C₁₀₉	—	2	1	no	—
62	C₄×C₁₀₉	—	2	1	no	—
63	C₄×C₁₀₉	—	2	1	no	—
64	C₄×C₁₀₉	—	2	1	no	—
65	C₄×C₁₀₉	#101	1	2	yes	—
66	Dic₁₀₉	—	4	2	no	—
67	Dic₁₀₉	—	4	2	no	—
68	C₂×C₁₀₉	—	2	2	no	—
69	C₂×C₁₀₉	C₂×#101	1	1	yes	characteristic
70	C₂×C₁₀₉	C₂×#104	1	1	yes	characteristic
71	C₂×C₁₀₉	#421	1	1	yes	characteristic
72	#1528	—	109	1	no	maximal, Sylow
73	C₁₀₉	#1528	1	1	yes	characteristic, complemented, Sylow
74	C₂×#125	—	109	1	no	—
75	#432	—	109	1	no	—
76	C₄×M₁₆	D₁₀₉	1	1	yes	characteristic
77	#330	—	109	4	no	—
78	C₂×M₁₆	—	2	1	no	—
79	#125	—	109	2	no	—
80	#125	—	109	2	no	—
81	#125	—	218	1	no	—
82	#125	—	218	1	no	—
83	C₂×C₄⋊C₄	—	109	1	no	—
84	C₂×C₄⋊C₄	—	109	1	no	—
85	C₂×C₄⋊C₄	—	218	1	no	—
86	C₂×C₄⋊C₄	—	218	1	no	—
87	C₂×C₄²	C₂×D₁₀₉	1	1	yes	characteristic
88	C₄⋊C₈	—	218	1	no	—
89	C₂²⋊C₈	—	218	1	no	—
90	#97	—	109	2	no	—
91	#97	C₂×D₁₀₉	1	2	yes	—
92	C₄×C₈	—	2	1	no	—
93	C₂²×C₄	—	218	1	no	—
94	C₂²×C₄	—	218	1	no	—
95	C₂²×C₄	C₄×D₁₀₉	1	1	yes	characteristic
96	C₂²×C₄	D₄₃₆	1	1	yes	characteristic
97	C₂²×C₄	C₂²⋊D₁₀₉	1	1	yes	characteristic
98	M₁₆	—	2	4	no	—
99	C₂×C₈	—	2	2	no	—
100	C₂×C₈	—	218	2	no	—
101	C₄⋊C₄	—	109	2	no	—
102	C₄⋊C₄	—	109	2	no	—
103	C₄⋊C₄	—	218	2	no	—
104	C₄⋊C₄	—	218	2	no	—
105	C₄⋊C₄	—	218	1	no	—
106	C₄⋊C₄	—	218	1	no	—
107	C₄⋊C₄	—	218	2	no	—
108	C₄⋊C₄	—	218	2	no	—
109	C₄²	C₄×D₁₀₉	1	1	yes	characteristic
110	C₄²	D₄₃₆	1	1	yes	characteristic
111	C₄²	C₂²⋊D₁₀₉	1	1	yes	characteristic
112	C₄²	C₂²×D₁₀₉	1	1	yes	characteristic
113	C₂³	#13182006	1	1	yes	characteristic
114	C₂×C₄	—	2	2	no	—
115	C₂×C₄	—	2	2	no	—
116	C₂×C₄	—	4	1	no	—
117	C₂×C₄	—	436	2	no	—
118	C₂×C₄	—	436	2	no	—
119	C₂×C₄	—	218	2	no	—
120	C₂×C₄	—	218	2	no	—
121	C₂×C₄	#13182006	1	1	yes	characteristic
122	C₂×C₄	#13182006	1	1	yes	characteristic
123	C₂×C₄	#13182006	1	1	yes	characteristic
124	C₂×C₄	C₂×C₄×D₁₀₉	1	1	yes	characteristic
125	C₂×C₄	C₂×C₂²⋊D₁₀₉	1	1	yes	characteristic
126	C₂×C₄	C₂×D₄₃₆	1	1	yes	characteristic
127	C₈	—	2	4	no	—
128	C₈	—	436	2	no	—
129	C₂²	—	2	1	no	—
130	C₂²	—	2	1	no	—
131	C₂²	#16812627	1	2	yes	—
132	C₂²	C₂×#13182006	1	1	yes	characteristic
133	C₄	—	2	1	no	—
134	C₄	—	2	1	no	—
135	C₄	—	2	1	no	—
136	C₄	—	2	1	no	—
137	C₄	—	2	1	no	—
138	C₄	—	436	2	no	—
139	C₄	—	436	2	no	—
140	C₄	#16812617	1	2	yes	—
141	C₂	—	2	2	no	—
142	C₂	C₂×#16812617	1	1	yes	characteristic
143	C₂	C₂×#16812627	1	1	yes	characteristic
144	C₂	#20295042	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×#13182006
commutator	C₂×C₄×C₁₀₉	C₂²×C₄
Frattini	C₄²	C₂²×D₁₀₉
Fitting	C₄×M₁₆×C₁₀₉	C₂
socle	C₂²×C₁₀₉	C₂×C₂²⋊C₄
prime radical	C₂³×C₁₀₉	C₂²⋊C₄
cosocle	C₄²×C₁₀₉	C₂³
hypercenter	C₄×M₁₆	D₁₀₉
hypocenter	C₁₀₉	#1528
LCS	G ▷ C₂×C₄×C₁₀₉ ▷ C₂²×C₁₀₉ ▷ C₁₀₉
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₄×M₁₆

Complements

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

N	H ≅ G/N	type
3. #20294767	C₂	semidirect
73. C₁₀₉	#1528	semidirect

Maximal subgroups

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

subgroup	index
C₂×#16812691	2
#20295083	2
#20294767	2
C₄×M₁₆×C₁₀₉	2
#1528	109