Subgroups — #28407023

Order 14464.

656 conjugacy classes of subgroups.

The subgroup lattice has 222 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#16813701	C₂	1	2	yes	maximal
2	#20299544	C₂	1	2	yes	maximal, complemented
3	#20299542	C₂	1	2	yes	maximal, complemented
4	#20299540	C₂	1	4	yes	maximal
5	C₄×C₄⋊#90673	C₂	1	4	yes	maximal, complemented
6	C₂×C₄²×D₁₁₃	C₂	1	1	yes	maximal, characteristic
7	#16813701	—	2	8	no	—
8	C₂×C₄⋊#90673	C₂²	1	8	yes	—
9	C₂×C₄⋊#90673	C₂²	1	4	yes	—
10	C₂×C₂²⋊#90673	C₂²	1	4	yes	—
11	C₂×C₄×#90673	C₂²	1	4	yes	—
12	(C₂×#90673)⋊C₄	C₂²	1	8	yes	—
13	C₂²×C₄×D₁₁₃	C₂²	1	2	yes	—
14	C₂²×C₄×D₁₁₃	C₂²	1	1	yes	characteristic
15	C₂×C₄×Dic₁₁₃	C₄	1	2	yes	complemented
16	C₂×C₄×Dic₁₁₃	C₄	1	1	yes	characteristic, complemented
17	C₄²×D₁₁₃	C₄	1	1	yes	characteristic, complemented
18	C₄²×D₁₁₃	C₄	1	1	yes	characteristic, complemented
19	C₄²×D₁₁₃	C₄	1	2	yes	complemented
20	C₄²×D₁₁₃	C₂²	1	2	yes	—
21	C₄²×D₁₁₃	C₂²	1	1	yes	characteristic
22	C₄²×D₁₁₃	C₂²	1	1	yes	characteristic
23	C₂×C₄²×C₁₁₃	C₄	1	1	yes	characteristic, complemented
24	C₂²×#90673	C₂³	1	8	yes	—
25	C₄⋊#90673	—	2	8	no	—
26	C₄⋊#90673	—	2	16	no	—
27	C₂²⋊#90673	—	2	8	no	—
28	C₄×#90673	—	2	8	no	—
29	C₂³×D₁₁₃	C₂³	1	1	yes	characteristic
30	C₂²×Dic₁₁₃	C₂×C₄	1	2	yes	—
31	C₂²×Dic₁₁₃	C₂×C₄	1	1	yes	characteristic
32	C₂×C₄×D₁₁₃	—	2	8	no	—
33	C₂×C₄×D₁₁₃	C₂×C₄	1	2	yes	—
34	C₂×C₄×D₁₁₃	C₂×C₄	1	2	yes	—
35	C₂×C₄×D₁₁₃	C₂×C₄	1	1	yes	characteristic
36	C₂×C₄×D₁₁₃	C₂×C₄	1	1	yes	characteristic
37	C₂×C₄×D₁₁₃	D₄	1	4	yes	—
38	C₂×C₄×D₁₁₃	Q₈	1	4	yes	—
39	C₂×C₄×D₁₁₃	C₂³	1	2	yes	—
40	C₂×C₄×D₁₁₃	C₂³	1	1	yes	characteristic
41	C₂×C₄×D₁₁₃	C₂³	1	1	yes	characteristic
42	C₂×C₄×D₁₁₃	C₂³	1	2	yes	—
43	C₄×Dic₁₁₃	C₂×C₄	1	1	yes	characteristic
44	C₄×Dic₁₁₃	C₂×C₄	1	2	yes	—
45	C₄×Dic₁₁₃	C₂×C₄	1	2	yes	—
46	C₄×Dic₁₁₃	C₂×C₄	1	2	yes	—
47	C₄×Dic₁₁₃	C₂×C₄	1	2	yes	—
48	C₄×Dic₁₁₃	C₂×C₄	1	1	yes	characteristic
49	C₄×Dic₁₁₃	C₂×C₄	1	2	yes	—
50	C₂²×C₄×C₁₁₃	C₂×C₄	1	2	yes	—
51	C₂²×C₄×C₁₁₃	C₂×C₄	1	1	yes	characteristic
52	C₄²×C₁₁₃	C₂×C₄	1	1	yes	characteristic
53	C₄²×C₁₁₃	C₂×C₄	1	1	yes	characteristic
54	C₄²×C₁₁₃	C₂×C₄	1	2	yes	—
55	C₂³×C₁₁₃	C₂²×C₄	1	1	yes	characteristic
56	C₂²×D₁₁₃	—	2	4	no	—
57	C₂²×D₁₁₃	C₂²×C₄	1	1	yes	characteristic
58	C₂²×D₁₁₃	C₂×D₄	1	2	yes	—
59	C₂²×D₁₁₃	C₂×Q₈	1	2	yes	—
60	C₂²×D₁₁₃	C₂⁴	1	1	yes	characteristic
61	C₂×#90673	—	2	8	no	—
62	C₂×#90673	—	2	16	no	—
63	C₂×C₄×C₁₁₃	—	2	4	no	—
64	C₂×C₄×C₁₁₃	C₄⋊C₄	1	4	yes	complemented
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	2	yes	—
68	C₂×C₄×C₁₁₃	C₂²×C₄	1	1	yes	characteristic
69	C₂×Dic₁₁₃	—	2	4	no	—
70	C₂×Dic₁₁₃	C₄⋊C₄	1	4	yes	complemented
71	C₂×Dic₁₁₃	C₂²×C₄	1	2	yes	—
72	C₂×Dic₁₁₃	C₂²×C₄	1	1	yes	characteristic
73	C₂×Dic₁₁₃	C₂²×C₄	1	1	yes	characteristic
74	C₂×Dic₁₁₃	C₂²×C₄	1	2	yes	—
75	C₄×D₁₁₃	—	2	4	no	—
76	C₄×D₁₁₃	—	2	4	no	—
77	C₄×D₁₁₃	—	2	4	no	—
78	C₄×D₁₁₃	C₄⋊C₄	1	4	yes	complemented
79	C₄×D₁₁₃	C₄⋊C₄	1	4	yes	complemented
80	C₄×D₁₁₃	C₂×D₄	1	2	yes	—
81	C₄×D₁₁₃	C₂×D₄	1	2	yes	—
82	C₄×D₁₁₃	C₂×Q₈	1	2	yes	—
83	C₄×D₁₁₃	C₂×Q₈	1	2	yes	—
84	C₄×D₁₁₃	#41	1	8	yes	—
85	C₂²×C₁₁₃	—	2	2	no	—
86	C₂²×C₁₁₃	C₂×C₄⋊C₄	1	2	yes	—
87	C₂²×C₁₁₃	C₂³×C₄	1	1	yes	characteristic
88	C₂×D₁₁₃	—	2	2	no	—
89	C₂×D₁₁₃	—	2	4	no	—
90	C₂×D₁₁₃	—	2	4	no	—
91	C₂×D₁₁₃	C₂×C₄⋊C₄	1	2	yes	—
92	C₂×D₁₁₃	C₂²×D₄	1	1	yes	characteristic
93	C₂×D₁₁₃	C₂²×Q₈	1	1	yes	characteristic
94	C₂×D₁₁₃	C₂×#41	1	4	yes	—
95	#90673	—	4	8	no	—
96	C₄×C₁₁₃	—	2	2	no	—
97	C₄×C₁₁₃	C₂×C₄⋊C₄	1	2	yes	—
98	C₄×C₁₁₃	C₂×C₄⋊C₄	1	2	yes	—
99	C₄×C₁₁₃	#117	1	4	yes	complemented
100	Dic₁₁₃	—	2	2	no	—
101	Dic₁₁₃	C₂×C₄⋊C₄	1	2	yes	—
102	Dic₁₁₃	C₂×C₄⋊C₄	1	2	yes	—
103	Dic₁₁₃	#117	1	4	yes	complemented
104	C₂×C₁₁₃	—	2	2	no	—
105	C₂×C₁₁₃	C₂²×C₄⋊C₄	1	1	yes	characteristic
106	C₂×C₁₁₃	C₂×#117	1	2	yes	—
107	D₁₁₃	—	2	2	no	—
108	D₁₁₃	#532	1	2	yes	—
109	D₁₁₃	#533	1	2	yes	—
110	#2285	—	113	1	no	maximal, Sylow
111	C₁₁₃	#2285	1	1	yes	characteristic, complemented, Sylow
112	C₂×#117	—	113	2	no	—
113	C₂²×C₄²	—	113	1	no	—
114	#384	—	113	2	no	—
115	#382	—	113	2	no	—
116	#380	—	113	4	no	—
117	C₄×C₄⋊C₄	—	113	4	no	—
118	C₂³×C₄	—	113	1	no	—
119	C₂³×C₄	—	113	2	no	—
120	#117	—	226	8	no	—
121	C₂×C₄⋊C₄	—	113	4	no	—
122	C₂×C₄⋊C₄	—	113	8	no	—
123	C₂×C₂²⋊C₄	—	113	4	no	—
124	C₂×C₄²	—	113	2	no	—
125	C₂×C₄²	—	113	1	no	—
126	C₂×C₄²	—	113	1	no	—
127	C₂×C₄²	—	113	1	no	—
128	C₂×C₄²	—	113	2	no	—
129	C₂×C₄²	—	113	2	no	—
130	C₂×C₄²	—	113	1	no	—
131	C₂×C₄²	—	113	1	no	—
132	C₂×C₄²	—	113	4	no	—
133	C₂×C₄²	#90673	1	1	yes	characteristic, complemented
134	#95	—	113	8	no	—
135	C₂⁴	—	113	1	no	—
136	C₂²×C₄	—	113	1	no	—
137	C₂²×C₄	—	113	4	no	—
138	C₂²×C₄	—	113	4	no	—
139	C₂²×C₄	—	113	1	no	—
140	C₂²×C₄	—	113	2	no	—
141	C₂²×C₄	—	113	1	no	—
142	C₂²×C₄	—	113	1	no	—
143	C₂²×C₄	—	113	1	no	—
144	C₂²×C₄	—	113	2	no	—
145	C₂²×C₄	—	113	2	no	—
146	C₂²×C₄	—	113	2	no	—
147	C₂²×C₄	—	113	2	no	—
148	C₂²×C₄	—	113	8	no	—
149	C₂²×C₄	—	226	8	no	—
150	C₂²×C₄	C₂×#90673	1	2	yes	—
151	C₂²×C₄	C₂×#90673	1	1	yes	characteristic
152	C₄⋊C₄	—	226	8	no	—
153	C₄⋊C₄	—	226	16	no	—
154	C₂²⋊C₄	—	226	8	no	—
155	C₄²	—	113	2	no	—
156	C₄²	—	113	1	no	—
157	C₄²	—	113	2	no	—
158	C₄²	—	113	2	no	—
159	C₄²	—	113	2	no	—
160	C₄²	—	113	1	no	—
161	C₄²	—	113	2	no	—
162	C₄²	—	226	8	no	—
163	C₄²	C₂×#90673	1	1	yes	characteristic
164	C₄²	C₂×#90673	1	1	yes	characteristic
165	C₄²	C₂×#90673	1	2	yes	—
166	C₂³	—	113	1	no	—
167	C₂³	—	113	2	no	—
168	C₂³	—	113	2	no	—
169	C₂³	—	113	1	no	—
170	C₂³	—	226	4	no	—
171	C₂³	C₂²×#90673	1	1	yes	characteristic
172	C₂×C₄	—	2	4	no	—
173	C₂×C₄	—	113	4	no	—
174	C₂×C₄	—	113	1	no	—
175	C₂×C₄	—	113	1	no	—
176	C₂×C₄	—	113	2	no	—
177	C₂×C₄	—	113	2	no	—
178	C₂×C₄	—	113	2	no	—
179	C₂×C₄	—	113	4	no	—
180	C₂×C₄	—	113	4	no	—
181	C₂×C₄	—	113	2	no	—
182	C₂×C₄	—	113	2	no	—
183	C₂×C₄	—	113	8	no	—
184	C₂×C₄	—	113	2	no	—
185	C₂×C₄	—	226	4	no	—
186	C₂×C₄	—	226	4	no	—
187	C₂×C₄	—	226	4	no	—
188	C₂×C₄	—	226	8	no	—
189	C₂×C₄	—	226	16	no	—
190	C₂×C₄	—	226	4	no	—
191	C₂×C₄	C₄⋊#90673	1	4	yes	complemented
192	C₂×C₄	C₂²×#90673	1	2	yes	—
193	C₂×C₄	C₂²×#90673	1	1	yes	characteristic
194	C₂×C₄	C₂²×#90673	1	2	yes	—
195	C₂×C₄	C₂²×#90673	1	1	yes	characteristic
196	C₂²	—	2	2	no	—
197	C₂²	—	113	1	no	—
198	C₂²	—	113	1	no	—
199	C₂²	—	113	4	no	—
200	C₂²	—	113	2	no	—
201	C₂²	—	226	4	no	—
202	C₂²	—	226	4	no	—
203	C₂²	—	226	2	no	—
204	C₂²	C₂×C₄⋊#90673	1	2	yes	—
205	C₂²	C₂³×#90673	1	1	yes	characteristic
206	C₄	—	2	2	no	—
207	C₄	—	113	2	no	—
208	C₄	—	113	2	no	—
209	C₄	—	113	4	no	—
210	C₄	—	452	8	no	—
211	C₄	—	226	2	no	—
212	C₄	C₂×C₄⋊#90673	1	2	yes	—
213	C₄	C₂×C₄⋊#90673	1	2	yes	—
214	C₄	#16813701	1	4	yes	complemented
215	C₂	—	2	2	no	—
216	C₂	—	113	2	no	—
217	C₂	—	113	2	no	—
218	C₂	—	226	2	no	—
219	C₂	C₂²×C₄⋊#90673	1	1	yes	characteristic
220	C₂	C₂×#16813701	1	2	yes	—

	subgroup	G/H
center	C₂×C₄	C₂²×#90673
commutator, socle	C₂²×C₁₁₃	C₂³×C₄
Frattini	C₂²	C₂³×#90673
Fitting	C₂×C₄²×C₁₁₃	C₄
prime radical	C₂³×D₁₁₃	C₂³
cosocle	C₂²×D₁₁₃	C₂⁴
hypercenter	C₂×C₄²	#90673
hypocenter	C₁₁₃	#2285
LCS	G ▷ C₂²×C₁₁₃ ▷ C₁₁₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄²

Complements

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

N	H ≅ G/N	type
2. #20299544	C₂	semidirect
3. #20299542	C₂	semidirect
5. C₄×C₄⋊#90673	C₂	semidirect
15. C₂×C₄×Dic₁₁₃	C₄	semidirect
16. C₂×C₄×Dic₁₁₃	C₄	semidirect
17. C₄²×D₁₁₃	C₄	semidirect
18. C₄²×D₁₁₃	C₄	semidirect
19. C₄²×D₁₁₃	C₄	semidirect
23. C₂×C₄²×C₁₁₃	C₄	semidirect
64. C₂×C₄×C₁₁₃	C₄⋊C₄	semidirect
70. C₂×Dic₁₁₃	C₄⋊C₄	semidirect
78. C₄×D₁₁₃	C₄⋊C₄	semidirect
79. C₄×D₁₁₃	C₄⋊C₄	semidirect
99. C₄×C₁₁₃	#117	semidirect
103. Dic₁₁₃	#117	semidirect
111. C₁₁₃	#2285	semidirect
133. C₂×C₄²	#90673	semidirect
191. C₂×C₄	C₄⋊#90673	semidirect
214. C₄	#16813701	semidirect

Maximal subgroups

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

subgroup	index
C₂×#16813701	2
#20299544	2
#20299542	2
#20299540	2
C₄×C₄⋊#90673	2
C₂×C₄²×D₁₁₃	2
#2285	113