Finite Group Databasebeta

Subgroups — #28362465

Order 13952.

150 conjugacy classes of subgroups; only the 51 normal subgroups are shown.

normal subgroups only

This group has 150 conjugacy classes of subgroups — too many to draw. Only its 51 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	#20295378	C₂	1	1	yes	maximal, characteristic
2	#20295312	C₂	1	1	yes	maximal, characteristic
3	#20295233	C₂	1	1	yes	maximal, characteristic
4	#20295150	C₂	1	1	yes	maximal, characteristic
5	#20294776	C₂	1	1	yes	maximal, characteristic
6	#20294773	C₂	1	1	yes	maximal, characteristic
7	#332×C₁₀₉	C₂	1	1	yes	maximal, characteristic
8	C₂×#13182019	—	2	1	no	—
9	#16812699	C₂²	1	1	yes	characteristic
10	C₄×Dic₂₁₈	C₂²	1	1	yes	characteristic
11	#16812631	—	2	1	no	—
12	C₄⋊#11724784	C₂²	1	1	yes	characteristic
13	#16812628	—	2	1	no	—
14	#16812624	—	2	1	no	—
15	#16812623	—	2	1	no	—
16	#16812622	—	2	1	no	—
17	#16812622	—	2	1	no	—
18	C₄×#11724784	C₂²	1	1	yes	characteristic
19	C₈×Dic₁₀₉	—	2	1	no	—
20	C₄⋊Q₈×C₁₀₉	C₂²	1	1	yes	characteristic
21	C₄⋊C₈×C₁₀₉	C₂²	1	1	yes	characteristic
22	C₄⋊C₈×C₁₀₉	C₂²	1	1	yes	characteristic
23	C₂×Dic₂₁₈	—	2	1	no	—
24	C₂×Dic₂₁₈	D₄	1	1	yes	characteristic
25	#13182019	—	4	2	no	—
26	C₂×#11724784	—	2	1	no	—
27	C₂×#11724784	D₄	1	1	yes	characteristic
28	C₂×#11724784	D₄	1	1	yes	characteristic
29	C₄⋊Dic₁₀₉	—	2	1	no	—
30	C₄⋊Dic₁₀₉	—	2	1	no	—
31	C₄⋊Dic₁₀₉	D₄	1	1	yes	characteristic
32	#13182008	—	2	1	no	—
33	C₄×Dic₁₀₉	—	2	1	no	—
34	C₂×Q₈×C₁₀₉	—	2	1	no	—
35	C₂×C₈×C₁₀₉	—	2	1	no	—
36	C₂×C₈×C₁₀₉	—	2	1	no	—
37	C₄⋊C₄×C₁₀₉	—	2	1	no	—
38	C₄⋊C₄×C₁₀₉	—	2	1	no	—
39	C₄²×C₁₀₉	C₂³	1	1	yes	characteristic
40	Q₈×C₁₀₉	—	4	1	no	—
41	Q₈×C₁₀₉	—	4	1	no	—
42	C₂×C₄×C₁₀₉	—	4	1	no	—
43	C₂×C₄×C₁₀₉	C₂×D₄	1	1	yes	characteristic
44	C₂×C₄×C₁₀₉	C₂×D₄	1	1	yes	characteristic
45	C₂×C₄×C₁₀₉	#41	1	1	yes	characteristic
46	C₂×Dic₁₀₉	—	2	1	no	—
47	C₂×Dic₁₀₉	—	2	1	no	—
48	C₂×Dic₁₀₉	—	4	1	no	—
49	Dic₂₁₈	—	2	1	no	—
50	Dic₂₁₈	—	4	1	no	—
51	Dic₂₁₈	—	4	1	no	—
52	Dic₂₁₈	SD₁₆	1	2	yes	—
53	C₈×C₁₀₉	—	4	1	no	—
54	C₈×C₁₀₉	—	4	1	no	—
55	#11724784	—	2	1	no	—
56	#11724784	—	2	1	no	—
57	#11724784	—	4	1	no	—
58	C₂²×C₁₀₉	#121	1	1	yes	characteristic
59	C₄×C₁₀₉	—	2	1	no	—
60	C₄×C₁₀₉	—	8	1	no	—
61	C₄×C₁₀₉	C₂×SD₁₆	1	1	yes	characteristic
62	C₄×C₁₀₉	#135	1	1	yes	characteristic
63	C₄×C₁₀₉	#135	1	1	yes	characteristic
64	C₄×C₁₀₉	#137	1	1	yes	characteristic
65	Dic₁₀₉	—	2	2	no	—
66	Dic₁₀₉	—	4	1	no	—
67	Dic₁₀₉	—	8	1	no	—
68	C₂×C₁₀₉	#464	1	1	yes	characteristic
69	C₂×C₁₀₉	#465	1	1	yes	characteristic
70	C₂×C₁₀₉	#473	1	1	yes	characteristic
71	#1680	—	109	1	no	maximal, Sylow
72	C₁₀₉	#1680	1	1	yes	characteristic, complemented, Sylow
73	#487	—	109	1	no	—
74	#475	—	109	1	no	—
75	#462	—	109	1	no	—
76	C₈×Q₈	—	109	1	no	—
77	#334	—	109	1	no	—
78	#332	—	109	1	no	—
79	#332	D₁₀₉	1	1	yes	characteristic
80	C₂×Q₁₆	—	218	1	no	—
81	C₄⋊Q₈	—	109	1	no	—
82	C₄⋊Q₈	C₂×D₁₀₉	1	1	yes	characteristic
83	C₄×Q₈	—	109	1	no	—
84	#106	—	218	1	no	—
85	C₄⋊C₈	—	109	1	no	—
86	C₄⋊C₈	—	218	1	no	—
87	C₄⋊C₈	C₂×D₁₀₉	1	1	yes	characteristic
88	C₄⋊C₈	C₂×D₁₀₉	1	1	yes	characteristic
89	#103	—	218	1	no	—
90	#103	—	218	1	no	—
91	#103	—	218	1	no	—
92	#103	—	218	1	no	—
93	C₄×C₈	—	109	1	no	—
94	C₄×C₈	—	218	1	no	—
95	C₂×Q₈	—	2	1	no	—
96	C₂×Q₈	—	109	1	no	—
97	C₂×Q₈	—	218	1	no	—
98	Q₁₆	—	436	2	no	—
99	C₂×C₈	—	2	1	no	—
100	C₂×C₈	—	2	1	no	—
101	C₂×C₈	—	109	1	no	—
102	C₂×C₈	—	109	1	no	—
103	C₂×C₈	—	218	1	no	—
104	C₄⋊C₄	—	2	1	no	—
105	C₄⋊C₄	—	2	1	no	—
106	C₄⋊C₄	—	109	1	no	—
107	C₄⋊C₄	—	218	1	no	—
108	C₄⋊C₄	—	218	1	no	—
109	C₄⋊C₄	—	218	1	no	—
110	C₄²	—	218	1	no	—
111	C₄²	C₂²×D₁₀₉	1	1	yes	characteristic
112	Q₈	—	4	1	no	—
113	Q₈	—	4	1	no	—
114	Q₈	—	109	2	no	—
115	Q₈	—	436	1	no	—
116	Q₈	—	436	1	no	—
117	Q₈	—	218	1	no	—
118	C₂×C₄	—	4	1	no	—
119	C₂×C₄	—	436	1	no	—
120	C₂×C₄	—	218	1	no	—
121	C₂×C₄	—	218	1	no	—
122	C₂×C₄	D₄×D₁₀₉	1	1	yes	characteristic
123	C₂×C₄	D₄×D₁₀₉	1	1	yes	characteristic
124	C₂×C₄	#13182029	1	1	yes	characteristic
125	C₈	—	4	1	no	—
126	C₈	—	4	1	no	—
127	C₈	—	436	1	no	—
128	C₈	—	218	1	no	—
129	C₈	—	218	1	no	—
130	C₂²	#16812672	1	1	yes	characteristic
131	C₄	—	2	1	no	—
132	C₄	—	8	1	no	—
133	C₄	—	872	1	no	—
134	C₄	—	436	1	no	—
135	C₄	—	218	2	no	—
136	C₄	SD₁₆×D₁₀₉	1	1	yes	characteristic
137	C₄	#16812718	1	1	yes	characteristic
138	C₄	#16812720	1	1	yes	characteristic
139	C₄	#16812731	1	1	yes	characteristic
140	C₂	#20295251	1	1	yes	characteristic
141	C₂	#20295254	1	1	yes	characteristic
142	C₂	#20295300	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	#16812672
commutator, cosocle	C₄²×C₁₀₉	C₂³
Frattini	C₄²	C₂²×D₁₀₉
Fitting	#332×C₁₀₉	C₂
socle, prime radical	C₂²×C₁₀₉	#121
hypercenter	#332	D₁₀₉
hypocenter	C₁₀₉	#1680
LCS	G ▷ C₄²×C₁₀₉ ▷ C₂²×C₁₀₉ ▷ C₁₀₉
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #332

Complements

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

N	H ≅ G/N	type
72. C₁₀₉	#1680	semidirect

Maximal subgroups

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

subgroup	index
#20295378	2
#20295312	2
#20295233	2
#20295150	2
#20294776	2
#20294773	2
#332×C₁₀₉	2
#1680	109