Finite Group Databasebeta

Subgroups — #28406889

Order 14464.

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

normal subgroups only

This group has 220 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	#20299654	C₂	1	1	yes	maximal, characteristic, complemented
2	D₁₁₃⋊#108	C₂	1	1	yes	maximal, characteristic, complemented
3	#20299512	C₂	1	4	yes	maximal, complemented
4	C₂×#16813641	C₂	1	1	yes	maximal, characteristic
5	D₁₁₃⋊M₁₆	—	2	1	no	—
6	#16813697	—	2	1	no	—
7	#16813696	—	2	1	no	—
8	#16813695	C₂²	1	2	yes	complemented
9	#16813693	C₂²	1	2	yes	complemented
10	#16813692	—	2	1	no	—
11	C₂×#14268646	—	2	1	no	—
12	#16813641	—	2	1	no	—
13	#16813641	—	2	1	no	—
14	#16813641	C₄	1	2	yes	—
15	#16813641	C₂²	1	2	yes	—
16	C₂×Dic₄₅₂	C₄	1	1	yes	characteristic
17	C₂×#14268635	—	2	1	no	—
18	C₂×C₈×D₁₁₃	C₂²	1	1	yes	characteristic
19	C₂×D₈×C₁₁₃	C₄	1	1	yes	characteristic
20	#14268657	—	2	2	no	—
21	#14268656	—	4	1	no	—
22	#14268655	—	4	1	no	—
23	#14268650	—	2	2	no	—
24	#14268646	—	2	2	no	—
25	#14268646	—	2	2	no	—
26	#14268646	—	4	1	no	—
27	#14268646	—	4	1	no	—
28	C₂×Dic₂₂₆	—	2	1	no	—
29	C₂×C₂²⋊D₁₁₃	—	4	1	no	—
30	C₂²×Dic₁₁₃	—	4	1	no	—
31	C₂×C₄×D₁₁₃	D₄	1	1	yes	characteristic
32	Dic₄₅₂	—	2	1	no	—
33	Dic₄₅₂	C₂×C₄	1	2	yes	—
34	#14268635	—	2	2	no	—
35	#14268635	—	2	2	no	—
36	C₂×#11744751	D₄	1	1	yes	characteristic
37	C₈×D₁₁₃	C₂×C₄	1	1	yes	characteristic
38	C₈×D₁₁₃	D₄	1	1	yes	characteristic
39	C₈×D₁₁₃	D₄	1	1	yes	characteristic
40	C₈×D₁₁₃	C₂³	1	1	yes	characteristic
41	C₂×D₄×C₁₁₃	—	2	1	no	—
42	D₈×C₁₁₃	—	2	1	no	—
43	D₈×C₁₁₃	C₂×C₄	1	2	yes	—
44	C₂×C₈×C₁₁₃	C₂×C₄	1	1	yes	characteristic
45	C₂³×C₁₁₃	—	4	1	no	—
46	C₂²×D₁₁₃	SD₁₆	1	1	yes	characteristic
47	D₄×C₁₁₃	—	2	2	no	—
48	D₄×C₁₁₃	—	4	1	no	—
49	C₂×C₄×C₁₁₃	C₂²⋊C₄	1	1	yes	characteristic
50	C₂²⋊D₁₁₃	—	4	2	no	—
51	C₂²⋊D₁₁₃	—	4	2	no	—
52	C₂×Dic₁₁₃	—	4	1	no	—
53	C₂×Dic₁₁₃	—	4	2	no	—
54	C₂×Dic₁₁₃	—	4	2	no	—
55	C₂×Dic₁₁₃	D₈	1	1	yes	characteristic
56	C₄×D₁₁₃	C₂²⋊C₄	1	1	yes	characteristic
57	C₄×D₁₁₃	D₈	1	1	yes	characteristic
58	C₄×D₁₁₃	SD₁₆	1	1	yes	characteristic
59	C₄×D₁₁₃	C₂×D₄	1	1	yes	characteristic
60	Dic₂₂₆	—	2	2	no	—
61	Dic₂₂₆	—	4	1	no	—
62	#11744753	—	4	2	no	—
63	C₈×C₁₁₃	C₂²⋊C₄	1	1	yes	characteristic
64	C₈×C₁₁₃	C₂²×C₄	1	1	yes	characteristic
65	#11744751	C₂²⋊C₄	1	1	yes	characteristic
66	#11744751	C₂×D₄	1	1	yes	characteristic
67	C₂²×C₁₁₃	—	4	2	no	—
68	C₂²×C₁₁₃	—	8	2	no	—
69	C₂²×C₁₁₃	#102	1	1	yes	characteristic
70	C₂×D₁₁₃	—	4	1	no	—
71	C₂×D₁₁₃	#102	1	1	yes	characteristic
72	C₂×D₁₁₃	C₂×SD₁₆	1	1	yes	characteristic
73	C₄×C₁₁₃	#102	1	1	yes	characteristic
74	C₄×C₁₁₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
75	Dic₁₁₃	—	4	2	no	—
76	Dic₁₁₃	#102	1	1	yes	characteristic
77	Dic₁₁₃	C₂×D₈	1	1	yes	characteristic
78	C₂×C₁₁₃	—	2	1	no	—
79	C₂×C₁₁₃	—	8	2	no	—
80	C₂×C₁₁₃	C₂×#102	1	1	yes	characteristic
81	D₁₁₃	—	2	1	no	—
82	D₁₁₃	—	2	1	no	—
83	#2143	—	113	1	no	maximal, Sylow
84	C₁₁₃	#2143	1	1	yes	characteristic, complemented, Sylow
85	C₂×#135	—	113	1	no	—
86	C₂×M₃₂	—	113	1	no	—
87	C₂×#108	—	113	1	no	—
88	#359	—	113	4	no	—
89	C₂×#41	—	226	1	no	—
90	#135	—	113	2	no	—
91	#135	—	113	2	no	—
92	#135	—	226	1	no	—
93	#135	—	226	1	no	—
94	C₂×Q₁₆	—	113	1	no	—
95	C₂×SD₁₆	—	226	1	no	—
96	C₂×D₈	#90673	1	1	yes	characteristic
97	C₂×M₁₆	—	226	1	no	—
98	C₂²×C₈	—	113	1	no	—
99	M₃₂	—	226	1	no	—
100	M₃₂	—	226	1	no	—
101	C₂×C₁₆	—	113	2	no	—
102	#108	—	113	2	no	—
103	#108	—	226	1	no	—
104	#41	—	452	1	no	—
105	#41	—	452	1	no	—
106	#41	—	226	2	no	—
107	#41	—	226	2	no	—
108	C₂×Q₈	—	226	1	no	—
109	C₂×D₄	—	2	1	no	—
110	C₂×D₄	—	452	1	no	—
111	C₂²×C₄	—	113	1	no	—
112	C₂²×C₄	—	452	1	no	—
113	Q₁₆	—	113	2	no	—
114	Q₁₆	—	226	1	no	—
115	SD₁₆	—	226	2	no	—
116	SD₁₆	—	226	2	no	—
117	D₈	—	2	1	no	—
118	D₈	C₂×#90673	1	2	yes	—
119	M₁₆	—	452	1	no	—
120	M₁₆	—	226	2	no	—
121	C₂×C₈	—	113	1	no	—
122	C₂×C₈	—	113	1	no	—
123	C₂×C₈	—	113	1	no	—
124	C₂×C₈	—	113	1	no	—
125	C₂×C₈	—	113	1	no	—
126	C₂×C₈	—	452	1	no	—
127	C₂×C₈	C₂×#90673	1	1	yes	characteristic
128	C₁₆	—	226	2	no	—
129	C₂³	—	4	1	no	—
130	C₂³	—	113	1	no	—
131	Q₈	—	452	1	no	—
132	Q₈	—	226	2	no	—
133	D₄	—	2	2	no	—
134	D₄	—	4	1	no	—
135	D₄	—	452	2	no	—
136	D₄	—	452	2	no	—
137	C₂×C₄	—	113	1	no	—
138	C₂×C₄	—	113	1	no	—
139	C₂×C₄	—	113	1	no	—
140	C₂×C₄	—	113	1	no	—
141	C₂×C₄	—	113	1	no	—
142	C₂×C₄	—	452	2	no	—
143	C₂×C₄	—	452	2	no	—
144	C₂×C₄	—	452	1	no	—
145	C₂×C₄	C₂²⋊#90673	1	1	yes	characteristic
146	C₈	—	113	1	no	—
147	C₈	—	113	1	no	—
148	C₈	—	452	2	no	—
149	C₈	C₂²⋊#90673	1	1	yes	characteristic
150	C₈	C₂²×#90673	1	1	yes	characteristic
151	C₂²	—	4	2	no	—
152	C₂²	—	8	2	no	—
153	C₂²	—	113	1	no	—
154	C₂²	—	113	1	no	—
155	C₂²	—	452	1	no	—
156	C₂²	#16813684	1	1	yes	characteristic
157	C₄	—	113	1	no	—
158	C₄	—	113	1	no	—
159	C₄	—	452	2	no	—
160	C₄	#16813684	1	1	yes	characteristic
161	C₄	C₂×C₂²⋊#90673	1	1	yes	characteristic
162	C₂	—	2	1	no	—
163	C₂	—	8	2	no	—
164	C₂	—	226	1	no	—
165	C₂	—	226	1	no	—
166	C₂	C₂×#16813684	1	1	yes	characteristic

	subgroup	G/H
center	C₂	C₂×#16813684
commutator	C₈×C₁₁₃	C₂²×C₄
Frattini	C₈	C₂²×#90673
Fitting	C₂×D₈×C₁₁₃	C₄
socle	C₂×C₁₁₃	C₂×#102
prime radical	C₂×#16813641	C₂
cosocle	C₈×D₁₁₃	C₂³
hypercenter	C₂×D₈	#90673
hypocenter	C₁₁₃	#2143
LCS	G ▷ C₈×C₁₁₃ ▷ C₄×C₁₁₃ ▷ C₂×C₁₁₃ ▷ C₁₁₃
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ C₂×C₈ ◁ C₂×D₈

Complements

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

N	H ≅ G/N	type
1. #20299654	C₂	semidirect
2. D₁₁₃⋊#108	C₂	semidirect
3. #20299512	C₂	semidirect
8. #16813695	C₂²	semidirect
9. #16813693	C₂²	semidirect
84. C₁₁₃	#2143	semidirect

Maximal subgroups

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

subgroup	index
#20299654	2
D₁₁₃⋊#108	2
#20299512	2
C₂×#16813641	2
#2143	113