Finite Group Databasebeta

Subgroups — #28365211

Order 13952.

150 conjugacy classes of subgroups; only the 47 normal subgroups are shown.

normal subgroups only

This group has 150 conjugacy classes of subgroups — too many to draw. Only its 47 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	#20295259	C₂	1	1	yes	maximal, characteristic
2	#107×D₁₀₉	C₂	1	1	yes	maximal, characteristic, complemented
3	#20294852	C₂	1	1	yes	maximal, characteristic, complemented
4	#20294851	C₂	1	1	yes	maximal, characteristic, complemented
5	#20294827	C₂	1	1	yes	maximal, characteristic, complemented
6	#20294805	C₂	1	1	yes	maximal, characteristic, complemented
7	#357×C₁₀₉	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×C₈×D₁₀₉	C₂²	1	1	yes	characteristic
9	#16812670	—	2	1	no	—
10	C₄⋊C₄×D₁₀₉	—	2	1	no	—
11	C₂×#13182004	C₂²	1	1	yes	characteristic
12	#16812633	C₂²	1	1	yes	characteristic
13	#16812633	C₂²	1	1	yes	characteristic
14	#16812632	—	2	1	no	—
15	#16812620	—	2	1	no	—
16	C₂×D₈×C₁₀₉	C₂²	1	1	yes	characteristic
17	C₂×C₁₆×C₁₀₉	C₂²	1	1	yes	characteristic
18	#107×C₁₀₉	C₂²	1	1	yes	characteristic
19	C₂×C₂²⋊D₁₀₉	—	4	1	no	—
20	C₂×C₄×D₁₀₉	—	4	1	no	—
21	C₂×C₄×D₁₀₉	D₄	1	1	yes	characteristic
22	C₂×#11724784	D₄	1	1	yes	characteristic
23	C₈×D₁₀₉	—	2	1	no	—
24	C₈×D₁₀₉	—	2	1	no	—
25	C₄⋊Dic₁₀₉	—	2	1	no	—
26	C₄⋊Dic₁₀₉	—	2	1	no	—
27	#13182008	—	4	1	no	—
28	C₂²⋊Dic₁₀₉	—	4	1	no	—
29	#13182004	—	2	1	no	—
30	C₂×D₄×C₁₀₉	—	2	1	no	—
31	D₈×C₁₀₉	—	2	1	no	—
32	D₈×C₁₀₉	—	2	1	no	—
33	C₂×C₈×C₁₀₉	C₂³	1	1	yes	characteristic
34	C₄⋊C₄×C₁₀₉	—	2	1	no	—
35	C₁₆×C₁₀₉	—	2	1	no	—
36	C₂³×C₁₀₉	—	4	1	no	—
37	C₂²×D₁₀₉	D₈	1	1	yes	characteristic
38	D₄×C₁₀₉	—	4	1	no	—
39	D₄×C₁₀₉	—	4	1	no	—
40	C₂×C₄×C₁₀₉	—	4	1	no	—
41	C₂×C₄×C₁₀₉	C₂×D₄	1	1	yes	characteristic
42	C₂²⋊D₁₀₉	—	8	2	no	—
43	C₂×Dic₁₀₉	—	4	1	no	—
44	C₂×Dic₁₀₉	—	4	1	no	—
45	C₂×Dic₁₀₉	D₈	1	1	yes	characteristic
46	C₄×D₁₀₉	—	2	1	no	—
47	C₄×D₁₀₉	—	2	1	no	—
48	C₄×D₁₀₉	—	8	2	no	—
49	C₈×C₁₀₉	#41	1	1	yes	characteristic
50	C₈×C₁₀₉	#41	1	1	yes	characteristic
51	#11724784	—	2	1	no	—
52	C₂²×C₁₀₉	—	8	1	no	—
53	C₂²×C₁₀₉	—	8	1	no	—
54	C₂²×C₁₀₉	—	8	1	no	—
55	C₂²×C₁₀₉	C₂×D₈	1	1	yes	characteristic
56	C₂×D₁₀₉	—	2	1	no	—
57	C₂×D₁₀₉	—	2	1	no	—
58	C₂×D₁₀₉	D₁₆	1	2	yes	—
59	C₄×C₁₀₉	—	8	1	no	—
60	C₄×C₁₀₉	#123	1	1	yes	characteristic
61	C₄×C₁₀₉	#137	1	1	yes	characteristic
62	Dic₁₀₉	—	2	1	no	—
63	Dic₁₀₉	—	8	1	no	—
64	Dic₁₀₉	—	8	1	no	—
65	C₂×C₁₀₉	—	16	1	no	—
66	C₂×C₁₀₉	#480	1	1	yes	characteristic
67	C₂×C₁₀₉	C₂×D₁₆	1	1	yes	characteristic
68	C₂×C₁₀₉	#510	1	1	yes	characteristic
69	D₁₀₉	—	2	2	no	—
70	#2232	—	109	1	no	maximal, Sylow
71	C₁₀₉	#2232	1	1	yes	characteristic, complemented, Sylow
72	#466	—	109	1	no	—
73	C₂×#107	—	109	1	no	—
74	#366	—	109	1	no	—
75	#366	—	109	1	no	—
76	#357	—	109	1	no	—
77	#357	D₁₀₉	1	1	yes	characteristic, complemented
78	C₂²⋊C₁₆	—	109	1	no	—
79	C₂×D₈	C₂×D₁₀₉	1	1	yes	characteristic
80	C₂²×C₈	—	109	1	no	—
81	#121	—	218	1	no	—
82	C₂×C₄⋊C₄	—	218	1	no	—
83	C₂×C₁₆	—	109	1	no	—
84	C₂×C₁₆	C₂×D₁₀₉	1	1	yes	characteristic
85	#107	—	109	1	no	—
86	#107	—	109	1	no	—
87	#107	—	218	1	no	—
88	#107	C₂×D₁₀₉	1	1	yes	characteristic
89	#102	—	218	1	no	—
90	C₂×D₄	—	2	1	no	—
91	C₂×D₄	—	436	1	no	—
92	C₂²×C₄	—	109	1	no	—
93	C₂²×C₄	—	436	1	no	—
94	D₈	—	2	1	no	—
95	D₈	—	2	1	no	—
96	C₂×C₈	—	109	1	no	—
97	C₂×C₈	—	218	1	no	—
98	C₂×C₈	—	218	1	no	—
99	C₂×C₈	C₂²×D₁₀₉	1	1	yes	characteristic
100	C₄⋊C₄	—	2	1	no	—
101	C₄⋊C₄	—	436	1	no	—
102	C₄⋊C₄	—	218	1	no	—
103	C₄⋊C₄	—	218	1	no	—
104	C₂²⋊C₄	—	436	1	no	—
105	C₁₆	—	2	1	no	—
106	C₁₆	—	218	1	no	—
107	C₂³	—	4	1	no	—
108	C₂³	—	109	1	no	—
109	D₄	—	4	1	no	—
110	D₄	—	4	1	no	—
111	D₄	—	872	2	no	—
112	C₂×C₄	—	4	1	no	—
113	C₂×C₄	—	872	2	no	—
114	C₂×C₄	—	109	1	no	—
115	C₂×C₄	—	436	1	no	—
116	C₂×C₄	—	436	1	no	—
117	C₂×C₄	—	218	1	no	—
118	C₂×C₄	—	218	1	no	—
119	C₂×C₄	D₄×D₁₀₉	1	1	yes	characteristic
120	C₈	—	218	1	no	—
121	C₈	#13182028	1	1	yes	characteristic
122	C₈	#13182029	1	1	yes	characteristic
123	C₂²	—	8	1	no	—
124	C₂²	—	8	1	no	—
125	C₂²	—	8	1	no	—
126	C₂²	—	109	2	no	—
127	C₂²	—	218	1	no	—
128	C₂²	—	218	1	no	—
129	C₂²	D₈×D₁₀₉	1	1	yes	characteristic
130	C₄	—	8	1	no	—
131	C₄	—	872	1	no	—
132	C₄	—	872	1	no	—
133	C₄	—	218	1	no	—
134	C₄	#16812681	1	1	yes	characteristic
135	C₄	#16812728	1	1	yes	characteristic
136	C₂	—	16	1	no	—
137	C₂	—	218	2	no	—
138	C₂	#20295338	1	1	yes	characteristic
139	C₂	D₁₆×D₁₀₉	1	1	yes	characteristic
140	C₂	#20295465	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	D₈×D₁₀₉
commutator, cosocle	C₂×C₈×C₁₀₉	C₂³
Frattini	C₂×C₈	C₂²×D₁₀₉
Fitting	#357×C₁₀₉	C₂
socle	C₂²×C₁₀₉	C₂×D₈
prime radical	#20295259	C₂
hypercenter	#357	D₁₀₉
hypocenter	C₁₀₉	#2232
LCS	G ▷ C₂×C₈×C₁₀₉ ▷ C₄×C₁₀₉ ▷ C₂×C₁₀₉ ▷ C₁₀₉
UCS	C₁ ◁ C₂² ◁ C₂×C₄ ◁ C₂×C₈ ◁ #357

Complements

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

N	H ≅ G/N	type
2. #107×D₁₀₉	C₂	semidirect
3. #20294852	C₂	semidirect
4. #20294851	C₂	semidirect
5. #20294827	C₂	semidirect
6. #20294805	C₂	semidirect
7. #357×C₁₀₉	C₂	semidirect
71. C₁₀₉	#2232	semidirect
77. #357	D₁₀₉	semidirect

Maximal subgroups

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

subgroup	index
#20295259	2
#107×D₁₀₉	2
#20294852	2
#20294851	2
#20294827	2
#20294805	2
#357×C₁₀₉	2
#2232	109