Finite Group Databasebeta

Subgroups — #28362446

Order 13952.

160 conjugacy classes of subgroups; only the 55 normal subgroups are shown.

normal subgroups only

This group has 160 conjugacy classes of subgroups — too many to draw. Only its 55 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	#20295372	C₂	1	1	yes	maximal, characteristic
2	#20295321	C₂	1	1	yes	maximal, characteristic, complemented
3	Q₈×#11724784	C₂	1	1	yes	maximal, characteristic, complemented
4	#20294776	C₂	1	1	yes	maximal, characteristic, complemented
5	#20294774	C₂	1	1	yes	maximal, characteristic, complemented
6	#20294770	C₂	1	1	yes	maximal, characteristic
7	#460×C₁₀₉	C₂	1	1	yes	maximal, characteristic
8	#16812699	C₂²	1	1	yes	characteristic
9	#16812632	—	2	1	no	—
10	C₄⋊#11724784	—	2	1	no	—
11	C₄⋊#11724784	C₂²	1	1	yes	characteristic
12	C₄⋊#11724784	C₂²	1	1	yes	characteristic
13	#16812623	—	2	1	no	—
14	#16812620	—	2	1	no	—
15	#16812620	—	2	1	no	—
16	C₄×#11724784	—	2	1	no	—
17	C₄×#11724784	C₂²	1	1	yes	characteristic
18	C₂×SD₁₆×C₁₀₉	—	2	1	no	—
19	#127×C₁₀₉	C₂²	1	1	yes	characteristic
20	C₄×Q₈×C₁₀₉	C₂²	1	1	yes	characteristic
21	C₄⋊C₈×C₁₀₉	C₂²	1	1	yes	characteristic
22	#102×C₁₀₉	—	2	1	no	—
23	C₂×Dic₂₁₈	—	2	1	no	—
24	C₂×#11724784	—	2	1	no	—
25	C₂×#11724784	—	2	1	no	—
26	C₂×#11724784	D₄	1	1	yes	characteristic
27	C₂×#11724784	D₄	1	1	yes	characteristic
28	C₄⋊Dic₁₀₉	—	2	1	no	—
29	C₄⋊Dic₁₀₉	—	2	1	no	—
30	C₂×Q₈×C₁₀₉	D₄	1	1	yes	characteristic
31	C₂×D₄×C₁₀₉	—	2	1	no	—
32	C₂×D₄×C₁₀₉	—	2	1	no	—
33	C₂×D₄×C₁₀₉	—	2	1	no	—
34	SD₁₆×C₁₀₉	—	4	2	no	—
35	C₂×C₈×C₁₀₉	—	2	1	no	—
36	C₄⋊C₄×C₁₀₉	—	2	1	no	—
37	C₄⋊C₄×C₁₀₉	D₄	1	1	yes	characteristic
38	C₄²×C₁₀₉	—	2	1	no	—
39	C₄²×C₁₀₉	C₂³	1	1	yes	characteristic
40	C₂³×C₁₀₉	—	4	1	no	—
41	Q₈×C₁₀₉	—	2	1	no	—
42	Q₈×C₁₀₉	SD₁₆	1	2	yes	—
43	D₄×C₁₀₉	—	4	1	no	—
44	D₄×C₁₀₉	—	4	1	no	—
45	D₄×C₁₀₉	—	4	1	no	—
46	D₄×C₁₀₉	—	4	1	no	—
47	D₄×C₁₀₉	—	8	1	no	—
48	C₂×C₄×C₁₀₉	—	2	1	no	—
49	C₂×C₄×C₁₀₉	—	2	1	no	—
50	C₂×C₄×C₁₀₉	C₂×D₄	1	1	yes	characteristic
51	C₂×C₄×C₁₀₉	C₂×D₄	1	1	yes	characteristic
52	C₂×C₄×C₁₀₉	#41	1	1	yes	characteristic
53	C₂×Dic₁₀₉	—	4	1	no	—
54	Dic₂₁₈	—	8	1	no	—
55	C₈×C₁₀₉	—	4	1	no	—
56	#11724784	—	2	1	no	—
57	#11724784	—	2	1	no	—
58	#11724784	—	4	1	no	—
59	#11724784	—	4	1	no	—
60	C₂²×C₁₀₉	—	8	1	no	—
61	C₂²×C₁₀₉	—	8	1	no	—
62	C₂²×C₁₀₉	—	8	1	no	—
63	C₂²×C₁₀₉	#121	1	1	yes	characteristic
64	C₄×C₁₀₉	—	2	2	no	—
65	C₄×C₁₀₉	—	2	1	no	—
66	C₄×C₁₀₉	—	4	1	no	—
67	C₄×C₁₀₉	C₂×SD₁₆	1	1	yes	characteristic
68	C₄×C₁₀₉	#135	1	1	yes	characteristic
69	C₄×C₁₀₉	#135	1	1	yes	characteristic
70	C₄×C₁₀₉	#137	1	1	yes	characteristic
71	Dic₁₀₉	—	8	1	no	—
72	C₂×C₁₀₉	—	16	1	no	—
73	C₂×C₁₀₉	#464	1	1	yes	characteristic
74	C₂×C₁₀₉	#465	1	1	yes	characteristic
75	C₂×C₁₀₉	#472	1	1	yes	characteristic
76	#1678	—	109	1	no	maximal, Sylow
77	C₁₀₉	#1678	1	1	yes	characteristic, complemented, Sylow
78	#486	—	109	1	no	—
79	#477	—	109	1	no	—
80	#460	D₁₀₉	1	1	yes	characteristic
81	C₈×Q₈	—	109	1	no	—
82	#334	—	109	1	no	—
83	#333	—	109	1	no	—
84	#331	—	109	1	no	—
85	C₂×SD₁₆	—	2	1	no	—
86	C₄⋊Q₈	—	109	1	no	—
87	#127	C₂×D₁₀₉	1	1	yes	characteristic
88	C₄×Q₈	C₂×D₁₀₉	1	1	yes	characteristic
89	#107	—	218	1	no	—
90	C₄⋊C₈	—	109	1	no	—
91	C₄⋊C₈	—	109	1	no	—
92	C₄⋊C₈	—	218	1	no	—
93	C₄⋊C₈	C₂×D₁₀₉	1	1	yes	characteristic
94	#103	—	218	1	no	—
95	#102	—	2	1	no	—
96	#102	—	218	1	no	—
97	#102	—	218	1	no	—
98	C₄×C₈	—	109	1	no	—
99	C₄×C₈	—	218	1	no	—
100	C₂×Q₈	—	218	1	no	—
101	C₂×Q₈	C₂²⋊D₁₀₉	1	1	yes	characteristic
102	C₂×D₄	—	2	1	no	—
103	C₂×D₄	—	2	1	no	—
104	C₂×D₄	—	2	1	no	—
105	SD₁₆	—	4	2	no	—
106	C₂×C₈	—	2	1	no	—
107	C₂×C₈	—	109	1	no	—
108	C₂×C₈	—	109	1	no	—
109	C₂×C₈	—	218	1	no	—
110	C₂×C₈	—	218	1	no	—
111	C₄⋊C₄	—	2	1	no	—
112	C₄⋊C₄	—	218	1	no	—
113	C₄⋊C₄	—	218	1	no	—
114	C₄⋊C₄	C₂²⋊D₁₀₉	1	1	yes	characteristic
115	C₄²	—	2	1	no	—
116	C₄²	C₂²×D₁₀₉	1	1	yes	characteristic
117	C₂³	—	4	1	no	—
118	Q₈	—	2	1	no	—
119	Q₈	—	872	1	no	—
120	Q₈	#13182017	1	2	yes	—
121	D₄	—	4	1	no	—
122	D₄	—	4	1	no	—
123	D₄	—	4	1	no	—
124	D₄	—	4	1	no	—
125	D₄	—	8	1	no	—
126	C₂×C₄	—	2	1	no	—
127	C₂×C₄	—	2	1	no	—
128	C₂×C₄	—	436	1	no	—
129	C₂×C₄	D₄×D₁₀₉	1	1	yes	characteristic
130	C₂×C₄	C₂×C₂²⋊D₁₀₉	1	1	yes	characteristic
131	C₂×C₄	#13182028	1	1	yes	characteristic
132	C₈	—	4	1	no	—
133	C₈	—	436	1	no	—
134	C₈	—	436	1	no	—
135	C₈	—	218	1	no	—
136	C₈	—	218	1	no	—
137	C₂²	—	8	1	no	—
138	C₂²	—	8	1	no	—
139	C₂²	—	8	1	no	—
140	C₂²	#16812671	1	1	yes	characteristic
141	C₄	—	2	2	no	—
142	C₄	—	2	1	no	—
143	C₄	—	4	1	no	—
144	C₄	—	872	1	no	—
145	C₄	C₂×#13182017	1	1	yes	characteristic
146	C₄	#16812719	1	1	yes	characteristic
147	C₄	#16812720	1	1	yes	characteristic
148	C₄	#16812728	1	1	yes	characteristic
149	C₂	—	16	1	no	—
150	C₂	#20295246	1	1	yes	characteristic
151	C₂	#20295253	1	1	yes	characteristic
152	C₂	#20295294	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	#16812671
commutator, cosocle	C₄²×C₁₀₉	C₂³
Frattini	C₄²	C₂²×D₁₀₉
Fitting	#460×C₁₀₉	C₂
socle	C₂²×C₁₀₉	#121
prime radical	#127×C₁₀₉	C₂²
hypercenter	#460	D₁₀₉
hypocenter	C₁₀₉	#1678
LCS	G ▷ C₄²×C₁₀₉ ▷ C₂²×C₁₀₉ ▷ C₁₀₉
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #460

Complements

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

N	H ≅ G/N	type
2. #20295321	C₂	semidirect
3. Q₈×#11724784	C₂	semidirect
4. #20294776	C₂	semidirect
5. #20294774	C₂	semidirect
77. C₁₀₉	#1678	semidirect

Maximal subgroups

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

subgroup	index
#20295372	2
#20295321	2
Q₈×#11724784	2
#20294776	2
#20294774	2
#20294770	2
#460×C₁₀₉	2
#1678	109