Subgroups — #28407375

Order 14464.

388 conjugacy classes of subgroups.

The subgroup lattice has 208 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20299650	C₂	1	4	yes	maximal, complemented
2	#20299648	C₂	1	2	yes	maximal, complemented
3	#20299647	C₂	1	2	yes	maximal, complemented
4	#20299601	C₂	1	1	yes	maximal, characteristic
5	#20299601	C₂	1	2	yes	maximal
6	C₄×#14268656	C₂	1	1	yes	maximal, characteristic
7	C₄×#14268656	C₂	1	2	yes	maximal
8	C₂×#16813587	C₂	1	1	yes	maximal, characteristic
9	C₂×#14268656	C₂²	1	1	yes	characteristic
10	C₂×#14268656	C₂²	1	2	yes	—
11	C₂×#14268656	C₂²	1	1	yes	characteristic
12	#16813689	C₂²	1	4	yes	—
13	#16813689	C₂²	1	2	yes	—
14	C₄⋊#11744753	C₂²	1	2	yes	—
15	C₄⋊#11744753	C₂²	1	4	yes	—
16	#16813677	C₂²	1	4	yes	—
17	#16813677	C₂²	1	2	yes	—
18	C₄×#11744753	C₂²	1	2	yes	—
19	C₄×#11744753	C₂²	1	4	yes	—
20	C₂²×Dic₂₂₆	C₄	1	1	yes	characteristic
21	#16813587	C₄	1	2	yes	—
22	#16813587	C₄	1	1	yes	characteristic
23	#16813587	C₄	1	1	yes	characteristic
24	#16813587	C₂²	1	1	yes	characteristic
25	#16813587	C₂²	1	1	yes	characteristic
26	#16813587	C₂²	1	2	yes	—
27	C₂×#14268626	C₄	1	2	yes	—
28	C₂×C₄×Dic₁₁₃	C₂²	1	1	yes	characteristic
29	C₂×C₄×Dic₁₁₃	C₂²	1	2	yes	—
30	C₂×C₄⋊C₄×C₁₁₃	C₄	1	1	yes	characteristic
31	#14268656	—	2	2	no	—
32	#14268656	—	2	4	no	—
33	#14268656	Q₈	1	4	yes	complemented
34	C₂×#11744753	C₂³	1	2	yes	—
35	C₂×#11744753	C₂³	1	4	yes	—
36	C₂×#11744753	C₂³	1	2	yes	—
37	C₂×Dic₂₂₆	—	2	4	no	—
38	C₂×Dic₂₂₆	C₂×C₄	1	1	yes	characteristic
39	C₂×Dic₂₂₆	C₂×C₄	1	2	yes	—
40	C₂×Dic₂₂₆	C₂×C₄	1	1	yes	characteristic
41	C₂²×Dic₁₁₃	C₂×C₄	1	2	yes	—
42	C₂²×Dic₁₁₃	C₂×C₄	1	1	yes	characteristic
43	C₂²×Dic₁₁₃	C₂³	1	1	yes	characteristic
44	#14268626	C₂×C₄	1	2	yes	—
45	#14268626	C₂×C₄	1	2	yes	—
46	#14268626	C₂×C₄	1	2	yes	—
47	#14268626	C₂×C₄	1	2	yes	—
48	C₄×Dic₁₁₃	C₂×C₄	1	1	yes	characteristic
49	C₄×Dic₁₁₃	C₂×C₄	1	1	yes	characteristic
50	C₄×Dic₁₁₃	C₂×C₄	1	2	yes	—
51	C₄×Dic₁₁₃	C₂×C₄	1	2	yes	—
52	C₄×Dic₁₁₃	C₂³	1	1	yes	characteristic
53	C₄×Dic₁₁₃	C₂³	1	2	yes	—
54	C₄×Dic₁₁₃	C₂³	1	2	yes	—
55	C₄×Dic₁₁₃	C₂³	1	1	yes	characteristic
56	C₂²×C₄×C₁₁₃	C₂×C₄	1	1	yes	characteristic
57	C₂²×C₄×C₁₁₃	C₂×C₄	1	2	yes	—
58	C₄⋊C₄×C₁₁₃	C₂×C₄	1	2	yes	—
59	C₄⋊C₄×C₁₁₃	C₂×C₄	1	1	yes	characteristic
60	C₄⋊C₄×C₁₁₃	C₂×C₄	1	1	yes	characteristic
61	C₂³×C₁₁₃	C₂²×C₄	1	1	yes	characteristic
62	C₂×C₄×C₁₁₃	—	2	2	no	—
63	C₂×C₄×C₁₁₃	—	4	2	no	—
64	C₂×C₄×C₁₁₃	C₂²×C₄	1	2	yes	—
65	C₂×C₄×C₁₁₃	C₂²×C₄	1	2	yes	—
66	C₂×C₄×C₁₁₃	C₂²×C₄	1	1	yes	characteristic
67	C₂×C₄×C₁₁₃	C₂²×C₄	1	1	yes	characteristic
68	C₂×Dic₁₁₃	—	2	4	no	—
69	C₂×Dic₁₁₃	—	4	1	no	—
70	C₂×Dic₁₁₃	C₂²×C₄	1	1	yes	characteristic
71	C₂×Dic₁₁₃	C₂²×C₄	1	1	yes	characteristic
72	C₂×Dic₁₁₃	C₂²×C₄	1	2	yes	—
73	C₂×Dic₁₁₃	C₂²×C₄	1	1	yes	characteristic
74	C₂×Dic₁₁₃	C₂²×C₄	1	2	yes	—
75	C₂×Dic₁₁₃	C₂×Q₈	1	2	yes	—
76	C₂×Dic₁₁₃	#41	1	2	yes	—
77	C₂×Dic₁₁₃	C₂⁴	1	1	yes	characteristic
78	Dic₂₂₆	—	2	2	no	—
79	Dic₂₂₆	—	2	4	no	—
80	Dic₂₂₆	M₁₆	1	4	yes	complemented
81	#11744753	—	2	2	no	—
82	#11744753	—	2	4	no	—
83	#11744753	C₂×Q₈	1	4	yes	complemented
84	C₂²×C₁₁₃	—	2	1	no	—
85	C₂²×C₁₁₃	—	2	1	no	—
86	C₂²×C₁₁₃	C₄×Q₈	1	2	yes	—
87	C₂²×C₁₁₃	C₂³×C₄	1	1	yes	characteristic
88	C₄×C₁₁₃	—	2	1	no	—
89	C₄×C₁₁₃	—	2	2	no	—
90	C₄×C₁₁₃	—	2	2	no	—
91	C₄×C₁₁₃	C₂×M₁₆	1	2	yes	—
92	Dic₁₁₃	—	2	2	no	—
93	Dic₁₁₃	—	2	1	no	—
94	Dic₁₁₃	—	2	1	no	—
95	Dic₁₁₃	C₄×Q₈	1	2	yes	—
96	Dic₁₁₃	C₂×M₁₆	1	4	yes	—
97	Dic₁₁₃	C₂²×Q₈	1	1	yes	characteristic
98	Dic₁₁₃	C₂×#41	1	1	yes	characteristic
99	C₂×C₁₁₃	—	2	2	no	—
100	C₂×C₁₁₃	C₂×C₄×Q₈	1	1	yes	characteristic
101	C₂×C₁₁₃	C₂²×M₁₆	1	1	yes	characteristic
102	C₂×C₁₁₃	#568	1	1	yes	characteristic
103	Q₈×M₁₆	—	113	1	no	maximal, Sylow
104	C₁₁₃	Q₈×M₁₆	1	1	yes	characteristic, complemented, Sylow
105	C₂×C₄×Q₈	—	113	1	no	—
106	#446	—	113	2	no	—
107	#446	—	113	4	no	—
108	C₈×Q₈	—	113	2	no	—
109	#423	—	113	1	no	—
110	#423	—	113	2	no	—
111	C₄×M₁₆	—	113	1	no	—
112	C₄×M₁₆	—	113	2	no	—
113	C₂²×Q₈	—	113	1	no	—
114	C₂×M₁₆	—	113	1	no	—
115	C₂×M₁₆	—	113	2	no	—
116	C₂×M₁₆	—	113	1	no	—
117	C₄×Q₈	—	113	1	no	—
118	C₄×Q₈	—	113	2	no	—
119	C₄×Q₈	—	113	1	no	—
120	C₄×Q₈	—	113	1	no	—
121	C₄×Q₈	—	113	1	no	—
122	C₄×Q₈	—	113	2	no	—
123	C₂×C₄⋊C₄	—	113	2	no	—
124	C₂×C₄⋊C₄	#90673	1	1	yes	characteristic
125	C₂×C₄²	—	113	1	no	—
126	C₂×C₄²	—	113	2	no	—
127	C₄⋊C₈	—	113	2	no	—
128	C₄⋊C₈	—	113	4	no	—
129	C₄⋊C₈	—	113	4	no	—
130	C₄⋊C₈	—	113	2	no	—
131	#97	—	113	4	no	—
132	#97	—	113	2	no	—
133	C₄×C₈	—	113	2	no	—
134	C₄×C₈	—	113	4	no	—
135	C₂×Q₈	—	113	1	no	—
136	C₂×Q₈	—	113	2	no	—
137	C₂×Q₈	—	113	1	no	—
138	C₂×Q₈	—	226	4	no	—
139	C₂²×C₄	—	113	1	no	—
140	C₂²×C₄	—	113	1	no	—
141	C₂²×C₄	—	113	2	no	—
142	C₂²×C₄	C₂×#90673	1	1	yes	characteristic
143	C₂²×C₄	C₂×#90673	1	2	yes	—
144	M₁₆	—	113	4	no	—
145	M₁₆	—	226	4	no	—
146	M₁₆	—	226	2	no	—
147	C₂×C₈	—	113	2	no	—
148	C₂×C₈	—	113	4	no	—
149	C₂×C₈	—	113	2	no	—
150	C₄⋊C₄	—	113	2	no	—
151	C₄⋊C₄	—	113	2	no	—
152	C₄⋊C₄	—	113	2	no	—
153	C₄⋊C₄	—	113	2	no	—
154	C₄⋊C₄	C₂×#90673	1	2	yes	—
155	C₄⋊C₄	C₂×#90673	1	1	yes	characteristic
156	C₄⋊C₄	C₂×#90673	1	1	yes	characteristic
157	C₄²	—	113	2	no	—
158	C₄²	—	113	1	no	—
159	C₄²	—	113	1	no	—
160	C₄²	—	113	2	no	—
161	C₄²	—	113	2	no	—
162	C₄²	—	113	1	no	—
163	C₄²	—	113	1	no	—
164	C₄²	—	113	2	no	—
165	C₂³	C₂²×#90673	1	1	yes	characteristic
166	Q₈	—	113	4	no	—
167	Q₈	—	226	4	no	—
168	Q₈	—	226	2	no	—
169	C₂×C₄	—	2	2	no	—
170	C₂×C₄	—	4	2	no	—
171	C₂×C₄	—	113	2	no	—
172	C₂×C₄	—	113	2	no	—
173	C₂×C₄	—	113	1	no	—
174	C₂×C₄	—	113	1	no	—
175	C₂×C₄	—	113	2	no	—
176	C₂×C₄	—	113	1	no	—
177	C₂×C₄	—	113	1	no	—
178	C₂×C₄	—	113	2	no	—
179	C₂×C₄	—	452	1	no	—
180	C₂×C₄	—	226	4	no	—
181	C₂×C₄	C₂²×#90673	1	2	yes	—
182	C₂×C₄	C₂²×#90673	1	2	yes	—
183	C₂×C₄	C₂²×#90673	1	1	yes	characteristic
184	C₂×C₄	C₂²×#90673	1	1	yes	characteristic
185	C₈	—	113	4	no	—
186	C₈	—	226	4	no	—
187	C₈	—	226	2	no	—
188	C₂²	—	2	1	no	—
189	C₂²	—	2	1	no	—
190	C₂²	Q₈×#90673	1	2	yes	—
191	C₂²	C₂³×#90673	1	1	yes	characteristic
192	C₄	—	2	1	no	—
193	C₄	—	2	2	no	—
194	C₄	—	2	2	no	—
195	C₄	—	113	1	no	—
196	C₄	—	113	1	no	—
197	C₄	—	113	2	no	—
198	C₄	—	113	4	no	—
199	C₄	—	226	2	no	—
200	C₄	—	226	1	no	—
201	C₄	—	226	1	no	—
202	C₄	D₁₁₃⋊M₁₆	1	2	yes	—
203	C₂	—	2	2	no	—
204	C₂	C₂×Q₈×#90673	1	1	yes	characteristic
205	C₂	C₂×D₁₁₃⋊M₁₆	1	1	yes	characteristic
206	C₂	#20299676	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂²	C₂³×#90673
commutator, socle	C₂²×C₁₁₃	C₂³×C₄
Fitting	C₂×C₄⋊C₄×C₁₁₃	C₄
prime radical	C₂³×C₁₁₃	C₂²×C₄
cosocle	C₂×Dic₁₁₃	C₂⁴
hypercenter	C₂×C₄⋊C₄	#90673
hypocenter	C₁₁₃	Q₈×M₁₆
LCS	G ▷ C₂²×C₁₁₃ ▷ C₁₁₃
UCS	C₁ ◁ C₂² ◁ C₂×C₄⋊C₄

Complements

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

N	H ≅ G/N	type
1. #20299650	C₂	semidirect
2. #20299648	C₂	semidirect
3. #20299647	C₂	semidirect
33. #14268656	Q₈	semidirect
80. Dic₂₂₆	M₁₆	semidirect
83. #11744753	C₂×Q₈	semidirect
104. C₁₁₃	Q₈×M₁₆	semidirect

Maximal subgroups

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

subgroup	index
#20299650	2
#20299648	2
#20299647	2
#20299601	2
C₄×#14268656	2
C₂×#16813587	2
Q₈×M₁₆	113