Subgroups — #28404458

Order 14464.

992 conjugacy classes of subgroups.

The subgroup lattice has 228 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20299323	C₂	1	4	yes	maximal, complemented
2	#20299322	C₂	1	4	yes	maximal, complemented
3	#20299283	C₂	1	8	yes	maximal, complemented
4	#20299209	C₂	1	4	yes	maximal, complemented
5	#20299172	C₂	1	2	yes	maximal
6	#20299155	C₂	1	1	yes	maximal, characteristic, complemented
7	#20299153	C₂	1	4	yes	maximal, complemented
8	#20299076	C₂	1	2	yes	maximal, complemented
9	C₄×#14268647	C₂	1	1	yes	maximal, characteristic
10	C₂×C₄⋊Q₈×C₁₁₃	C₂	1	1	yes	maximal, characteristic, complemented
11	C₂×#14268647	C₂²	1	1	yes	characteristic
12	#16813622	C₂²	1	2	yes	—
13	#16813620	C₂²	1	4	yes	—
14	#16813617	C₂²	1	16	yes	—
15	#16813614	C₂²	1	2	yes	—
16	#16813613	C₂²	1	8	yes	—
17	#16813612	C₂²	1	16	yes	—
18	#16813611	C₂²	1	4	yes	—
19	#16813605	C₂²	1	8	yes	—
20	#16813602	C₂²	1	8	yes	—
21	#16813601	C₂²	1	4	yes	—
22	#16813598	C₂²	1	8	yes	—
23	Q₈×Dic₁₁₃	C₂²	1	8	yes	—
24	Q₈×Dic₁₁₃	C₂²	1	8	yes	—
25	C₄×Dic₂₂₆	C₂²	1	4	yes	—
26	C₄×Dic₂₂₆	C₂²	1	2	yes	—
27	#16813587	C₂²	1	8	yes	—
28	C₄×D₄₅₂	C₂²	1	2	yes	—
29	C₄×C₂²⋊D₁₁₃	C₂²	1	4	yes	—
30	#16813581	C₂²	1	8	yes	—
31	#16813581	C₂²	1	1	yes	characteristic
32	#16813581	C₂²	1	2	yes	—
33	#16813580	C₂²	1	8	yes	—
34	#16813579	C₂²	1	2	yes	—
35	C₄²×D₁₁₃	C₂²	1	2	yes	—
36	C₂²×Q₈×C₁₁₃	C₂²	1	2	yes	—
37	C₄⋊Q₈×C₁₁₃	C₂²	1	4	yes	—
38	C₄⋊Q₈×C₁₁₃	C₂²	1	2	yes	—
39	C₄⋊Q₈×C₁₁₃	C₂²	1	2	yes	—
40	C₂×C₄⋊C₄×C₁₁₃	C₂²	1	4	yes	—
41	C₂×C₄²×C₁₁₃	C₂²	1	1	yes	characteristic
42	#14268647	—	2	4	no	—
43	C₂×Dic₂₂₆	C₂³	1	4	yes	—
44	C₂×Dic₂₂₆	C₂³	1	1	yes	characteristic
45	C₂×D₄₅₂	C₂³	1	1	yes	characteristic
46	C₂×C₂²⋊D₁₁₃	C₂³	1	2	yes	—
47	C₂×C₄×D₁₁₃	C₂³	1	2	yes	—
48	C₂×C₄×D₁₁₃	C₂³	1	4	yes	—
49	C₄⋊Dic₁₁₃	C₂³	1	4	yes	—
50	C₄⋊Dic₁₁₃	C₂³	1	2	yes	—
51	C₄⋊Dic₁₁₃	C₂³	1	4	yes	—
52	C₄⋊Dic₁₁₃	C₂³	1	16	yes	—
53	#14268626	C₂³	1	4	yes	—
54	#14268626	C₂³	1	16	yes	—
55	#14268626	C₂³	1	8	yes	—
56	C₂²⋊Dic₁₁₃	C₂³	1	2	yes	—
57	C₂²⋊Dic₁₁₃	C₂³	1	4	yes	—
58	#14268624	C₂³	1	4	yes	—
59	#14268624	C₂³	1	16	yes	—
60	C₄×Dic₁₁₃	C₂³	1	2	yes	—
61	C₄×Dic₁₁₃	C₂³	1	4	yes	—
62	C₄×Dic₁₁₃	C₂³	1	16	yes	—
63	C₄×Dic₁₁₃	C₂³	1	4	yes	—
64	C₂×Q₈×C₁₁₃	—	2	8	no	—
65	C₂×Q₈×C₁₁₃	C₂³	1	4	yes	—
66	C₂×Q₈×C₁₁₃	C₂³	1	4	yes	—
67	C₂²×C₄×C₁₁₃	C₂³	1	4	yes	—
68	C₂²×C₄×C₁₁₃	C₂³	1	2	yes	—
69	C₂²×C₄×C₁₁₃	C₂³	1	1	yes	characteristic
70	C₄⋊C₄×C₁₁₃	C₂³	1	8	yes	—
71	C₄⋊C₄×C₁₁₃	C₂³	1	8	yes	—
72	C₄²×C₁₁₃	C₂³	1	1	yes	characteristic
73	C₄²×C₁₁₃	C₂³	1	1	yes	characteristic
74	C₄²×C₁₁₃	C₂³	1	2	yes	—
75	C₂³×C₁₁₃	C₂⁴	1	1	yes	characteristic
76	C₂²×D₁₁₃	C₂⁴	1	2	yes	—
77	Q₈×C₁₁₃	—	2	8	no	—
78	Q₈×C₁₁₃	—	2	8	no	—
79	C₂×C₄×C₁₁₃	—	2	8	no	—
80	C₂×C₄×C₁₁₃	—	2	4	no	—
81	C₂×C₄×C₁₁₃	#41	1	4	yes	—
82	C₂×C₄×C₁₁₃	C₂⁴	1	2	yes	—
83	C₂×C₄×C₁₁₃	C₂⁴	1	8	yes	—
84	C₂×C₄×C₁₁₃	C₂⁴	1	1	yes	characteristic
85	C₂×C₄×C₁₁₃	C₂⁴	1	1	yes	characteristic
86	C₂×C₄×C₁₁₃	C₂⁴	1	2	yes	—
87	C₂²⋊D₁₁₃	—	2	4	no	—
88	C₂×Dic₁₁₃	C₂⁴	1	4	yes	—
89	C₂×Dic₁₁₃	C₂⁴	1	8	yes	—
90	C₂×Dic₁₁₃	C₂⁴	1	2	yes	—
91	D₄₅₂	—	2	2	no	—
92	C₄×D₁₁₃	—	2	4	no	—
93	C₄×D₁₁₃	—	2	8	no	—
94	Dic₂₂₆	—	2	2	no	—
95	Dic₂₂₆	—	2	8	no	—
96	C₂²×C₁₁₃	—	2	2	no	—
97	C₂²×C₁₁₃	C₂×#41	1	2	yes	—
98	C₂²×C₁₁₃	C₂⁵	1	1	yes	characteristic
99	C₂×D₁₁₃	—	2	2	no	—
100	C₂×D₁₁₃	—	2	4	no	—
101	C₄×C₁₁₃	—	2	8	no	—
102	C₄×C₁₁₃	—	2	2	no	—
103	C₄×C₁₁₃	C₂×#41	1	2	yes	—
104	C₄×C₁₁₃	C₂×#41	1	2	yes	—
105	C₄×C₁₁₃	2₋¹⁺⁴	1	4	yes	—
106	Dic₁₁₃	—	2	2	no	—
107	Dic₁₁₃	—	2	8	no	—
108	Dic₁₁₃	—	2	4	no	—
109	C₂×C₁₁₃	—	2	2	no	—
110	C₂×C₁₁₃	C₂²×#41	1	1	yes	characteristic
111	C₂×C₁₁₃	C₂×2₋¹⁺⁴	1	2	yes	—
112	D₁₁₃	—	4	2	no	—
113	#3509	—	113	1	no	maximal, Sylow
114	C₁₁₃	#3509	1	1	yes	characteristic, complemented, Sylow
115	#557	—	113	4	no	—
116	#556	—	113	4	no	—
117	#552	—	113	8	no	—
118	#541	—	113	4	no	—
119	#536	—	113	2	no	—
120	#533	—	113	4	no	—
121	#533	—	113	1	no	—
122	C₂×C₄⋊Q₈	D₁₁₃	1	1	yes	characteristic, complemented
123	#519	—	113	2	no	—
124	C₄×#41	—	113	1	no	—
125	C₂×#41	—	113	1	no	—
126	C₂²×Q₈	C₂×D₁₁₃	1	2	yes	—
127	C₄⋊Q₈	—	113	4	no	—
128	C₄⋊Q₈	—	113	2	no	—
129	C₄⋊Q₈	C₂×D₁₁₃	1	2	yes	—
130	C₄⋊Q₈	C₂×D₁₁₃	1	2	yes	—
131	C₄⋊Q₈	C₂×D₁₁₃	1	4	yes	—
132	#126	—	113	16	no	—
133	#125	—	113	8	no	—
134	#125	—	113	16	no	—
135	#125	—	113	2	no	—
136	#124	—	113	4	no	—
137	#123	—	113	8	no	—
138	C₂²⋊Q₈	—	113	8	no	—
139	C₂²⋊Q₈	—	113	4	no	—
140	C₂²⋊Q₈	—	113	8	no	—
141	C₄×Q₈	—	113	8	no	—
142	C₄×Q₈	—	113	8	no	—
143	C₄×Q₈	—	113	8	no	—
144	C₄×Q₈	—	113	2	no	—
145	C₄×Q₈	—	113	4	no	—
146	C₄×D₄	—	113	4	no	—
147	C₄×D₄	—	113	2	no	—
148	#117	—	113	1	no	—
149	#117	—	113	2	no	—
150	#117	—	113	8	no	—
151	#117	—	113	8	no	—
152	#117	—	113	2	no	—
153	C₂×C₄⋊C₄	C₂×D₁₁₃	1	4	yes	—
154	C₂×C₄²	—	113	2	no	—
155	C₂×C₄²	C₂×D₁₁₃	1	1	yes	characteristic
156	#41	—	226	4	no	—
157	C₂×Q₈	—	2	8	no	—
158	C₂×Q₈	—	113	4	no	—
159	C₂×Q₈	—	113	1	no	—
160	C₂×Q₈	C₂²×D₁₁₃	1	4	yes	—
161	C₂×Q₈	C₂²×D₁₁₃	1	4	yes	—
162	C₂×D₄	—	113	2	no	—
163	C₂×D₄	—	113	1	no	—
164	C₂²×C₄	—	113	4	no	—
165	C₂²×C₄	—	113	2	no	—
166	C₂²×C₄	C₂²×D₁₁₃	1	2	yes	—
167	C₂²×C₄	C₂²×D₁₁₃	1	4	yes	—
168	C₂²×C₄	C₂²×D₁₁₃	1	1	yes	characteristic
169	C₄⋊C₄	—	113	4	no	—
170	C₄⋊C₄	—	113	2	no	—
171	C₄⋊C₄	—	113	4	no	—
172	C₄⋊C₄	—	113	4	no	—
173	C₄⋊C₄	—	113	16	no	—
174	C₄⋊C₄	—	113	16	no	—
175	C₄⋊C₄	—	113	8	no	—
176	C₄⋊C₄	C₂²×D₁₁₃	1	8	yes	—
177	C₄⋊C₄	C₂²×D₁₁₃	1	8	yes	—
178	C₂²⋊C₄	—	113	2	no	—
179	C₂²⋊C₄	—	113	4	no	—
180	C₂²⋊C₄	—	113	4	no	—
181	C₂²⋊C₄	—	113	16	no	—
182	C₄²	—	113	2	no	—
183	C₄²	—	113	4	no	—
184	C₄²	—	113	16	no	—
185	C₄²	—	113	4	no	—
186	C₄²	C₂²×D₁₁₃	1	1	yes	characteristic
187	C₄²	C₂²×D₁₁₃	1	1	yes	characteristic
188	C₄²	C₂²×D₁₁₃	1	2	yes	—
189	C₂³	—	113	2	no	—
190	C₂³	C₂³×D₁₁₃	1	1	yes	characteristic
191	Q₈	—	2	8	no	—
192	Q₈	—	2	8	no	—
193	Q₈	—	226	2	no	—
194	Q₈	—	226	8	no	—
195	D₄	—	226	4	no	—
196	D₄	—	226	2	no	—
197	C₂×C₄	—	2	8	no	—
198	C₂×C₄	—	2	4	no	—
199	C₂×C₄	—	113	4	no	—
200	C₂×C₄	—	113	8	no	—
201	C₂×C₄	—	113	2	no	—
202	C₂×C₄	—	226	4	no	—
203	C₂×C₄	—	226	8	no	—
204	C₂×C₄	#14268646	1	4	yes	—
205	C₂×C₄	C₂³×D₁₁₃	1	8	yes	—
206	C₂×C₄	C₂³×D₁₁₃	1	2	yes	—
207	C₂×C₄	C₂³×D₁₁₃	1	1	yes	characteristic
208	C₂×C₄	C₂³×D₁₁₃	1	1	yes	characteristic
209	C₂×C₄	C₂³×D₁₁₃	1	2	yes	—
210	C₂²	—	2	2	no	—
211	C₂²	—	226	2	no	—
212	C₂²	—	226	4	no	—
213	C₂²	C₂×#14268646	1	2	yes	—
214	C₂²	C₂⁴×D₁₁₃	1	1	yes	characteristic
215	C₄	—	2	8	no	—
216	C₄	—	2	2	no	—
217	C₄	—	226	2	no	—
218	C₄	—	226	4	no	—
219	C₄	—	226	8	no	—
220	C₄	C₂×#14268646	1	2	yes	—
221	C₄	C₂×#14268646	1	2	yes	—
222	C₄	#16813670	1	4	yes	—
223	C₂	—	2	2	no	—
224	C₂	—	452	2	no	—
225	C₂	C₂²×#14268646	1	1	yes	characteristic
226	C₂	C₂×#16813670	1	2	yes	—

	subgroup	G/H
center, Frattini	C₂²	C₂⁴×D₁₁₃
commutator, socle, cosocle	C₂²×C₁₁₃	C₂⁵
Fitting	C₂×C₄⋊Q₈×C₁₁₃	C₂
prime radical	C₂×#14268647	C₂²
hypercenter	C₂×C₄⋊Q₈	D₁₁₃
hypocenter	C₁₁₃	#3509
LCS	G ▷ C₂²×C₁₁₃ ▷ C₁₁₃
UCS	C₁ ◁ C₂² ◁ C₂×C₄⋊Q₈

Complements

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

N	H ≅ G/N	type
1. #20299323	C₂	semidirect
2. #20299322	C₂	semidirect
3. #20299283	C₂	semidirect
4. #20299209	C₂	semidirect
6. #20299155	C₂	semidirect
7. #20299153	C₂	semidirect
8. #20299076	C₂	semidirect
10. C₂×C₄⋊Q₈×C₁₁₃	C₂	semidirect
114. C₁₁₃	#3509	semidirect
122. C₂×C₄⋊Q₈	D₁₁₃	semidirect

Maximal subgroups

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

subgroup	index
#20299323	2
#20299322	2
#20299283	2
#20299209	2
#20299172	2
#20299155	2
#20299153	2
#20299076	2
C₄×#14268647	2
C₂×C₄⋊Q₈×C₁₁₃	2
#3509	113