Finite Group Databasebeta

Subgroups — #28379354

Order 13952.

224 conjugacy classes of subgroups; only the 62 normal subgroups are shown.

normal subgroups only

This group has 224 conjugacy classes of subgroups — too many to draw. Only its 62 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	D₁₀₉⋊M₃₂	C₂	1	1	yes	maximal, characteristic, complemented
2	#20296036	C₂	1	1	yes	maximal, characteristic, complemented
3	#20295943	C₂	1	4	yes	maximal, complemented
4	C₂×#16812722	C₂	1	1	yes	maximal, characteristic
5	#16812778	—	2	1	no	—
6	#16812774	C₂²	1	2	yes	complemented
7	#16812773	—	2	1	no	—
8	D₁₀₉⋊C₁₆	—	2	1	no	—
9	#16812768	—	2	1	no	—
10	#16812767	C₂²	1	2	yes	complemented
11	C₂×#13182030	—	2	1	no	—
12	#16812722	—	2	1	no	—
13	#16812722	—	2	1	no	—
14	#16812722	C₄	1	2	yes	complemented
15	#16812722	C₂²	1	2	yes	—
16	C₂×#13182016	—	2	1	no	—
17	C₂×D₈₇₂	C₄	1	1	yes	characteristic, complemented
18	C₂×C₈×D₁₀₉	C₂²	1	1	yes	characteristic
19	C₂×Q₁₆×C₁₀₉	C₄	1	1	yes	characteristic, complemented
20	C₄⋊#89137	—	2	2	no	—
21	C₂²⋊#89137	—	4	1	no	—
22	C₄×#89137	—	4	1	no	—
23	#13182032	—	2	2	no	—
24	#13182030	—	2	2	no	—
25	#13182030	—	2	2	no	—
26	#13182030	—	4	1	no	—
27	#13182030	—	4	1	no	—
28	C₂×D₄₃₆	—	2	1	no	—
29	C₂×D₄₃₆	—	4	1	no	—
30	C₂×C₄×D₁₀₉	—	4	1	no	—
31	C₂×C₄×D₁₀₉	D₄	1	1	yes	characteristic
32	#13182016	—	2	2	no	—
33	#13182016	—	2	2	no	—
34	D₈₇₂	—	2	1	no	—
35	D₈₇₂	C₂×C₄	1	2	yes	—
36	C₂×#11724784	D₄	1	1	yes	characteristic
37	C₈×D₁₀₉	C₂×C₄	1	1	yes	characteristic
38	C₈×D₁₀₉	D₄	1	1	yes	characteristic
39	C₈×D₁₀₉	D₄	1	1	yes	characteristic
40	C₈×D₁₀₉	C₂³	1	1	yes	characteristic
41	C₂×Q₈×C₁₀₉	—	2	1	no	—
42	Q₁₆×C₁₀₉	—	2	1	no	—
43	Q₁₆×C₁₀₉	C₂×C₄	1	2	yes	—
44	C₂×C₈×C₁₀₉	C₂×C₄	1	1	yes	characteristic
45	C₂²×D₁₀₉	—	4	1	no	—
46	C₂²×D₁₀₉	D₈	1	1	yes	characteristic
47	C₂×#89137	—	4	2	no	—
48	Q₈×C₁₀₉	—	2	2	no	—
49	Q₈×C₁₀₉	—	4	1	no	—
50	C₂×C₄×C₁₀₉	—	4	1	no	—
51	C₂×C₄×C₁₀₉	C₂²⋊C₄	1	1	yes	characteristic
52	C₂×Dic₁₀₉	SD₁₆	1	1	yes	characteristic
53	D₄₃₆	—	2	2	no	—
54	D₄₃₆	—	4	2	no	—
55	D₄₃₆	—	4	2	no	—
56	D₄₃₆	—	4	1	no	—
57	C₄×D₁₀₉	—	4	2	no	—
58	C₄×D₁₀₉	—	4	2	no	—
59	C₄×D₁₀₉	C₂²⋊C₄	1	1	yes	characteristic
60	C₄×D₁₀₉	D₈	1	1	yes	characteristic
61	C₄×D₁₀₉	SD₁₆	1	1	yes	characteristic
62	C₄×D₁₀₉	C₂×D₄	1	1	yes	characteristic
63	C₈×C₁₀₉	C₂²⋊C₄	1	1	yes	characteristic
64	C₈×C₁₀₉	C₂²×C₄	1	1	yes	characteristic
65	#11724784	C₂²⋊C₄	1	1	yes	characteristic
66	#11724784	C₂×D₄	1	1	yes	characteristic
67	C₂²×C₁₀₉	#102	1	1	yes	characteristic
68	C₂×D₁₀₉	—	4	1	no	—
69	C₂×D₁₀₉	—	4	2	no	—
70	C₂×D₁₀₉	—	8	2	no	—
71	C₂×D₁₀₉	#102	1	1	yes	characteristic
72	C₂×D₁₀₉	C₂×D₈	1	1	yes	characteristic
73	#89137	—	8	2	no	—
74	C₄×C₁₀₉	—	4	2	no	—
75	C₄×C₁₀₉	#102	1	1	yes	characteristic
76	C₄×C₁₀₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
77	Dic₁₀₉	#102	1	1	yes	characteristic
78	Dic₁₀₉	C₂×SD₁₆	1	1	yes	characteristic
79	C₂×C₁₀₉	—	2	1	no	—
80	C₂×C₁₀₉	C₂×#102	1	1	yes	characteristic
81	D₁₀₉	—	2	1	no	—
82	D₁₀₉	—	2	1	no	—
83	D₁₀₉	—	8	2	no	—
84	#2145	—	109	1	no	maximal, Sylow
85	C₁₀₉	#2145	1	1	yes	characteristic, complemented, Sylow
86	C₂×#135	—	109	1	no	—
87	C₂×M₃₂	—	109	1	no	—
88	#427	—	109	1	no	—
89	#360	—	109	4	no	—
90	C₂×#41	—	218	1	no	—
91	#135	—	109	2	no	—
92	#135	—	109	2	no	—
93	#135	—	218	1	no	—
94	#135	—	218	1	no	—
95	C₂×Q₁₆	#89137	1	1	yes	characteristic, complemented
96	C₂×SD₁₆	—	218	1	no	—
97	C₂×D₈	—	109	1	no	—
98	C₂²×C₈	—	109	1	no	—
99	#117	—	218	1	no	—
100	M₃₂	—	109	2	no	—
101	M₃₂	—	218	1	no	—
102	C₂×C₁₆	—	218	1	no	—
103	#107	—	218	1	no	—
104	#106	—	109	2	no	—
105	#41	—	436	1	no	—
106	#41	—	436	1	no	—
107	#41	—	218	2	no	—
108	#41	—	218	2	no	—
109	C₂×Q₈	—	2	1	no	—
110	C₂×D₄	—	436	1	no	—
111	C₂×D₄	—	218	1	no	—
112	C₂²×C₄	—	109	1	no	—
113	C₂²×C₄	—	436	1	no	—
114	Q₁₆	—	2	1	no	—
115	Q₁₆	C₂×#89137	1	2	yes	—
116	SD₁₆	—	218	2	no	—
117	SD₁₆	—	218	2	no	—
118	D₈	—	109	2	no	—
119	D₈	—	218	1	no	—
120	C₂×C₈	—	109	1	no	—
121	C₂×C₈	—	109	1	no	—
122	C₂×C₈	—	109	1	no	—
123	C₂×C₈	—	109	1	no	—
124	C₂×C₈	—	109	1	no	—
125	C₂×C₈	C₂×#89137	1	1	yes	characteristic
126	C₄⋊C₄	—	218	2	no	—
127	C₂²⋊C₄	—	436	1	no	—
128	C₄²	—	436	1	no	—
129	C₁₆	—	218	2	no	—
130	C₂³	—	109	1	no	—
131	C₂³	—	436	1	no	—
132	Q₈	—	2	2	no	—
133	Q₈	—	4	1	no	—
134	D₄	—	436	2	no	—
135	D₄	—	436	2	no	—
136	D₄	—	436	1	no	—
137	D₄	—	218	2	no	—
138	C₂×C₄	—	4	1	no	—
139	C₂×C₄	—	109	1	no	—
140	C₂×C₄	—	109	1	no	—
141	C₂×C₄	—	109	1	no	—
142	C₂×C₄	—	109	1	no	—
143	C₂×C₄	—	109	1	no	—
144	C₂×C₄	—	436	2	no	—
145	C₂×C₄	—	436	2	no	—
146	C₂×C₄	—	436	2	no	—
147	C₂×C₄	C₂²⋊#89137	1	1	yes	characteristic
148	C₈	—	109	1	no	—
149	C₈	—	109	1	no	—
150	C₈	C₂²⋊#89137	1	1	yes	characteristic
151	C₈	C₂²×#89137	1	1	yes	characteristic
152	C₂²	—	872	2	no	—
153	C₂²	—	109	1	no	—
154	C₂²	—	109	1	no	—
155	C₂²	—	436	1	no	—
156	C₂²	—	436	2	no	—
157	C₂²	#16812761	1	1	yes	characteristic
158	C₄	—	4	2	no	—
159	C₄	—	872	2	no	—
160	C₄	—	109	1	no	—
161	C₄	—	109	1	no	—
162	C₄	#16812761	1	1	yes	characteristic
163	C₄	C₂×C₂²⋊#89137	1	1	yes	characteristic
164	C₂	—	2	1	no	—
165	C₂	—	872	2	no	—
166	C₂	—	218	1	no	—
167	C₂	—	218	1	no	—
168	C₂	C₂×#16812761	1	1	yes	characteristic

	subgroup	G/H
center	C₂	C₂×#16812761
commutator	C₈×C₁₀₉	C₂²×C₄
Frattini	C₈	C₂²×#89137
Fitting	C₂×Q₁₆×C₁₀₉	C₄
socle	C₂×C₁₀₉	C₂×#102
prime radical	C₂×#16812722	C₂
cosocle	C₈×D₁₀₉	C₂³
hypercenter	C₂×Q₁₆	#89137
hypocenter	C₁₀₉	#2145
LCS	G ▷ C₈×C₁₀₉ ▷ C₄×C₁₀₉ ▷ C₂×C₁₀₉ ▷ C₁₀₉
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ C₂×C₈ ◁ C₂×Q₁₆

Complements

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

N	H ≅ G/N	type
1. D₁₀₉⋊M₃₂	C₂	semidirect
2. #20296036	C₂	semidirect
3. #20295943	C₂	semidirect
6. #16812774	C₂²	semidirect
10. #16812767	C₂²	semidirect
14. #16812722	C₄	semidirect
17. C₂×D₈₇₂	C₄	semidirect
19. C₂×Q₁₆×C₁₀₉	C₄	semidirect
85. C₁₀₉	#2145	semidirect
95. C₂×Q₁₆	#89137	semidirect

Maximal subgroups

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

subgroup	index
D₁₀₉⋊M₃₂	2
#20296036	2
#20295943	2
C₂×#16812722	2
#2145	109