Finite Group Databasebeta

Subgroups — #28379152

Order 13952.

184 conjugacy classes of subgroups; only the 66 normal subgroups are shown.

normal subgroups only

This group has 184 conjugacy classes of subgroups — too many to draw. Only its 66 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	#20296041	C₂	1	2	yes	maximal
2	C₂×#16812770	C₂	1	1	yes	maximal, characteristic
3	#20295989	C₂	1	1	yes	maximal, characteristic
4	#20295909	C₂	1	2	yes	maximal
5	#108×D₁₀₉	C₂	1	1	yes	maximal, characteristic
6	C₂×#13182039	C₂²	1	2	yes	—
7	D₁₀₉⋊M₁₆	C₂²	1	1	yes	characteristic
8	#16812778	C₂²	1	1	yes	characteristic
9	#16812770	—	2	2	no	—
10	#16812770	—	2	1	no	—
11	#16812770	—	2	1	no	—
12	#16812769	—	2	2	no	—
13	#16812753	—	2	1	no	—
14	C₈×#89137	—	2	1	no	—
15	M₁₆×D₁₀₉	C₂²	1	2	yes	—
16	C₂×C₈×D₁₀₉	C₂²	1	1	yes	characteristic
17	#16812635	C₄	1	1	yes	characteristic, complemented
18	#16812634	C₄	1	2	yes	complemented
19	#108×C₁₀₉	C₄	1	1	yes	characteristic, complemented
20	#13182039	—	2	2	no	—
21	#13182039	—	2	2	no	—
22	#13182039	D₄	1	1	yes	characteristic
23	#13182039	D₄	1	1	yes	characteristic
24	#13182038	—	2	1	no	—
25	D₁₀₉⋊C₈	—	2	1	no	—
26	C₂×#11724786	—	2	2	no	—
27	C₄⋊#89137	—	2	1	no	—
28	C₂²⋊#89137	Q₈	1	2	yes	—
29	C₄×#89137	—	2	1	no	—
30	C₂×C₄×D₁₀₉	C₂³	1	1	yes	characteristic
31	#13182013	C₂×C₄	1	2	yes	—
32	#13182012	—	2	2	no	—
33	C₂×#11724784	C₂×C₄	1	1	yes	characteristic
34	C₈×D₁₀₉	—	2	2	no	—
35	C₈×D₁₀₉	D₄	1	1	yes	characteristic
36	C₈×D₁₀₉	D₄	1	1	yes	characteristic
37	C₈×D₁₀₉	Q₈	1	1	yes	characteristic
38	C₈×D₁₀₉	Q₈	1	1	yes	characteristic
39	M₁₆×C₁₀₉	C₂×C₄	1	2	yes	—
40	C₂×C₈×C₁₀₉	C₂×C₄	1	1	yes	characteristic
41	C₂²×D₁₀₉	C₂×Q₈	1	1	yes	characteristic
42	C₂×#89137	—	2	2	no	—
43	C₂×C₄×C₁₀₉	C₂²×C₄	1	1	yes	characteristic
44	C₂×Dic₁₀₉	#41	1	1	yes	characteristic
45	C₄×D₁₀₉	C₂×D₄	1	1	yes	characteristic
46	C₄×D₁₀₉	C₂×D₄	1	1	yes	characteristic
47	C₄×D₁₀₉	C₂×Q₈	1	1	yes	characteristic
48	C₄×D₁₀₉	#41	1	1	yes	characteristic
49	#11724786	—	2	1	no	—
50	#11724786	—	2	1	no	—
51	#11724786	—	4	2	no	—
52	C₈×C₁₀₉	—	2	2	no	—
53	C₈×C₁₀₉	C₄⋊C₄	1	1	yes	characteristic
54	C₈×C₁₀₉	C₄⋊C₄	1	1	yes	characteristic
55	#11724784	—	2	2	no	—
56	#11724784	C₄⋊C₄	1	1	yes	characteristic
57	#11724784	C₄⋊C₄	1	1	yes	characteristic
58	C₂²×C₁₀₉	C₄×Q₈	1	1	yes	characteristic
59	C₂×D₁₀₉	—	4	1	no	—
60	C₂×D₁₀₉	C₂²⋊Q₈	1	1	yes	characteristic
61	C₂×D₁₀₉	C₄⋊Q₈	1	1	yes	characteristic
62	#89137	—	4	2	no	—
63	C₄×C₁₀₉	C₂×C₄⋊C₄	1	1	yes	characteristic
64	C₄×C₁₀₉	C₄×D₄	1	1	yes	characteristic
65	Dic₁₀₉	#121	1	1	yes	characteristic
66	Dic₁₀₉	#125	1	1	yes	characteristic
67	C₂×C₁₀₉	—	2	1	no	—
68	C₂×C₁₀₉	#389	1	1	yes	characteristic
69	D₁₀₉	—	2	1	no	—
70	D₁₀₉	—	2	1	no	—
71	#1953	—	109	1	no	maximal, Sylow
72	C₁₀₉	#1953	1	1	yes	characteristic, complemented, Sylow
73	#430	—	109	2	no	—
74	C₂×#108	—	109	1	no	—
75	C₂×#108	—	109	1	no	—
76	#405	—	109	1	no	—
77	#344	—	109	2	no	—
78	C₂×M₁₆	—	109	1	no	—
79	C₂×M₁₆	—	109	2	no	—
80	C₂×M₁₆	—	109	2	no	—
81	C₂²×C₈	—	109	1	no	—
82	#117	—	109	1	no	—
83	#108	—	109	2	no	—
84	#108	—	109	1	no	—
85	#108	—	218	1	no	—
86	#108	—	218	1	no	—
87	#108	—	218	2	no	—
88	#108	—	218	2	no	—
89	#108	#89137	1	1	yes	characteristic, complemented
90	#97	—	218	1	no	—
91	C₄×C₈	—	218	1	no	—
92	C₂²×C₄	—	109	1	no	—
93	M₁₆	—	109	1	no	—
94	M₁₆	—	109	1	no	—
95	M₁₆	—	109	2	no	—
96	M₁₆	—	218	2	no	—
97	M₁₆	—	218	2	no	—
98	M₁₆	—	218	2	no	—
99	M₁₆	—	218	1	no	—
100	M₁₆	C₂×#89137	1	2	yes	—
101	C₂×C₈	—	109	1	no	—
102	C₂×C₈	—	109	1	no	—
103	C₂×C₈	—	109	1	no	—
104	C₂×C₈	—	109	1	no	—
105	C₂×C₈	—	109	1	no	—
106	C₂×C₈	—	218	2	no	—
107	C₂×C₈	—	218	2	no	—
108	C₂×C₈	—	218	1	no	—
109	C₂×C₈	C₂×#89137	1	1	yes	characteristic
110	C₄⋊C₄	—	218	1	no	—
111	C₂²⋊C₄	—	109	2	no	—
112	C₄²	—	218	1	no	—
113	C₂³	—	109	1	no	—
114	C₂×C₄	—	109	1	no	—
115	C₂×C₄	—	109	1	no	—
116	C₂×C₄	—	109	1	no	—
117	C₂×C₄	—	109	1	no	—
118	C₂×C₄	—	109	1	no	—
119	C₂×C₄	—	218	2	no	—
120	C₂×C₄	C₂²×#89137	1	1	yes	characteristic
121	C₈	—	2	2	no	—
122	C₈	—	109	1	no	—
123	C₈	—	109	1	no	—
124	C₈	—	436	2	no	—
125	C₈	—	218	1	no	—
126	C₈	—	218	1	no	—
127	C₈	—	218	2	no	—
128	C₈	C₄⋊#89137	1	1	yes	characteristic
129	C₈	C₄⋊#89137	1	1	yes	characteristic
130	C₂²	—	109	1	no	—
131	C₂²	—	109	1	no	—
132	C₂²	—	436	1	no	—
133	C₂²	Q₈×#89137	1	1	yes	characteristic
134	C₄	—	109	1	no	—
135	C₄	—	109	1	no	—
136	C₄	—	436	2	no	—
137	C₄	C₂×C₄⋊#89137	1	1	yes	characteristic
138	C₄	D₄×#89137	1	1	yes	characteristic
139	C₂	—	2	1	no	—
140	C₂	—	218	1	no	—
141	C₂	—	218	1	no	—
142	C₂	#20295977	1	1	yes	characteristic

	subgroup	G/H
center	C₂	#20295977
commutator	C₂×C₄×C₁₀₉	C₂²×C₄
Frattini	C₂×C₄	C₂²×#89137
Fitting	#108×C₁₀₉	C₄
socle	C₂×C₁₀₉	#389
prime radical	C₂²×D₁₀₉	C₂×Q₈
cosocle	C₂×C₄×D₁₀₉	C₂³
hypercenter	#108	#89137
hypocenter	C₁₀₉	#1953
LCS	G ▷ C₂×C₄×C₁₀₉ ▷ C₂×C₁₀₉ ▷ C₁₀₉
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ #108

Complements

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

N	H ≅ G/N	type
17. #16812635	C₄	semidirect
18. #16812634	C₄	semidirect
19. #108×C₁₀₉	C₄	semidirect
72. C₁₀₉	#1953	semidirect
89. #108	#89137	semidirect

Maximal subgroups

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

subgroup	index
#20296041	2
C₂×#16812770	2
#20295989	2
#20295909	2
#108×D₁₀₉	2
#1953	109