Subgroups — #28407519

Order 14464.

354 conjugacy classes of subgroups.

The subgroup lattice has 244 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20299649	C₂	1	1	yes	maximal, characteristic, complemented
2	Q₈×#11744753	C₂	1	1	yes	maximal, characteristic, complemented
3	#20299632	C₂	1	1	yes	maximal, characteristic, complemented
4	#20299631	C₂	1	2	yes	maximal, complemented
5	#20299630	C₂	1	2	yes	maximal, complemented
6	#20299626	C₂	1	1	yes	maximal, characteristic, complemented
7	#20299623	C₂	1	2	yes	maximal, complemented
8	#20299614	C₂	1	1	yes	maximal, characteristic, complemented
9	#20299605	C₂	1	2	yes	maximal, complemented
10	#20299601	C₂	1	1	yes	maximal, characteristic, complemented
11	#20299134	C₂	1	1	yes	maximal, characteristic
12	C₂×#14268657	C₂²	1	2	yes	—
13	C₂×#14268656	C₂²	1	2	yes	—
14	C₂×#14268655	C₂²	1	2	yes	—
15	#16813689	C₂²	1	2	yes	—
16	#16813689	C₂²	1	2	yes	—
17	C₄⋊#11744753	C₂²	1	1	yes	characteristic
18	C₄⋊#11744753	C₂²	1	1	yes	characteristic
19	C₄⋊#11744753	C₂²	1	1	yes	characteristic
20	C₄⋊#11744753	C₂²	1	2	yes	—
21	C₄⋊#11744753	C₂²	1	1	yes	characteristic
22	#16813679	C₂²	1	2	yes	—
23	#16813679	C₂²	1	2	yes	—
24	C₂²⋊#11744753	C₂²	1	1	yes	characteristic
25	C₂²⋊#11744753	C₂²	1	1	yes	characteristic
26	#16813677	C₂²	1	2	yes	—
27	C₄×#11744753	C₂²	1	1	yes	characteristic
28	C₄×#11744753	C₂²	1	1	yes	characteristic
29	C₄×#11744753	C₂²	1	2	yes	—
30	#16813617	C₄	1	2	yes	—
31	#16813613	C₄	1	1	yes	characteristic
32	#16813611	C₄	1	1	yes	characteristic
33	#16813606	C₄	1	2	yes	—
34	#16813596	C₄	1	1	yes	characteristic
35	Q₈×Dic₁₁₃	C₂²	1	1	yes	characteristic
36	#16813584	C₂²	1	1	yes	characteristic
37	#16813582	C₂²	1	2	yes	—
38	#16813580	C₂²	1	2	yes	—
39	C₂×C₄×Dic₁₁₃	C₂²	1	1	yes	characteristic
40	C₂²⋊Q₈×C₁₁₃	C₄	1	1	yes	characteristic
41	#14268657	—	4	2	no	—
42	#14268656	—	4	2	no	—
43	#14268655	—	4	2	no	—
44	C₂×#11744753	C₂³	1	1	yes	characteristic
45	C₂×#11744753	C₂³	1	1	yes	characteristic
46	C₂×#11744753	C₂³	1	1	yes	characteristic
47	C₂×#11744753	C₂³	1	1	yes	characteristic
48	C₂×#11744753	C₂³	1	2	yes	—
49	C₂×#11744753	C₂³	1	2	yes	—
50	C₂×D₄₅₂	C₂×C₄	1	1	yes	characteristic
51	C₂×C₂²⋊D₁₁₃	C₂×C₄	1	2	yes	—
52	C₂²×Dic₁₁₃	C₂×C₄	1	1	yes	characteristic
53	C₂²×Dic₁₁₃	C₂³	1	1	yes	characteristic
54	C₂×C₄×D₁₁₃	C₂³	1	2	yes	—
55	C₄⋊Dic₁₁₃	C₂×C₄	1	2	yes	—
56	C₄⋊Dic₁₁₃	C₂×C₄	1	1	yes	characteristic
57	C₄⋊Dic₁₁₃	C₂×C₄	1	2	yes	—
58	#14268626	C₂×C₄	1	2	yes	—
59	#14268624	C₂×C₄	1	2	yes	—
60	#14268624	C₂×C₄	1	2	yes	—
61	#14268624	C₂×C₄	1	2	yes	—
62	#14268624	C₂×C₄	1	2	yes	—
63	C₄×Dic₁₁₃	C₂×C₄	1	1	yes	characteristic
64	C₄×Dic₁₁₃	C₂×C₄	1	1	yes	characteristic
65	C₄×Dic₁₁₃	C₂³	1	1	yes	characteristic
66	C₄×Dic₁₁₃	C₂³	1	1	yes	characteristic
67	C₄×Dic₁₁₃	C₂³	1	2	yes	—
68	C₂×Q₈×C₁₁₃	C₂×C₄	1	1	yes	characteristic
69	C₂²×C₄×C₁₁₃	C₂×C₄	1	1	yes	characteristic
70	C₄⋊C₄×C₁₁₃	C₂×C₄	1	2	yes	—
71	C₄⋊C₄×C₁₁₃	C₂×C₄	1	1	yes	characteristic
72	C₂²⋊C₄×C₁₁₃	C₂×C₄	1	2	yes	—
73	C₂³×C₁₁₃	C₂²×C₄	1	1	yes	characteristic
74	C₂²×D₁₁₃	C₂²×C₄	1	2	yes	—
75	Q₈×C₁₁₃	—	2	2	no	—
76	C₂×C₄×C₁₁₃	—	2	2	no	—
77	C₂×C₄×C₁₁₃	C₂²×C₄	1	2	yes	—
78	C₂×C₄×C₁₁₃	C₂²×C₄	1	2	yes	—
79	C₂×C₄×C₁₁₃	C₂²×C₄	1	1	yes	characteristic
80	C₂×C₄×C₁₁₃	C₂²×C₄	1	1	yes	characteristic
81	C₂²⋊D₁₁₃	—	4	2	no	—
82	C₂×Dic₁₁₃	—	2	1	no	—
83	C₂×Dic₁₁₃	—	2	1	no	—
84	C₂×Dic₁₁₃	—	4	1	no	—
85	C₂×Dic₁₁₃	C₂²×C₄	1	1	yes	characteristic
86	C₂×Dic₁₁₃	C₂²×C₄	1	1	yes	characteristic
87	C₂×Dic₁₁₃	C₂²×C₄	1	1	yes	characteristic
88	C₂×Dic₁₁₃	C₂²×C₄	1	2	yes	—
89	C₂×Dic₁₁₃	C₂⁴	1	1	yes	characteristic
90	D₄₅₂	—	2	2	no	—
91	C₄×D₁₁₃	—	2	2	no	—
92	C₄×D₁₁₃	—	2	2	no	—
93	#11744753	—	2	1	no	—
94	#11744753	—	2	2	no	—
95	#11744753	—	2	2	no	—
96	#11744753	—	2	1	no	—
97	#11744753	—	2	1	no	—
98	#11744753	—	2	1	no	—
99	C₂²×C₁₁₃	—	2	1	no	—
100	C₂²×C₁₁₃	—	2	1	no	—
101	C₂²×C₁₁₃	—	2	1	no	—
102	C₂²×C₁₁₃	C₂³×C₄	1	1	yes	characteristic
103	C₂×D₁₁₃	—	2	2	no	—
104	C₂×D₁₁₃	—	2	2	no	—
105	C₂×D₁₁₃	—	2	2	no	—
106	C₄×C₁₁₃	—	2	2	no	—
107	C₄×C₁₁₃	—	2	1	no	—
108	C₄×C₁₁₃	—	2	2	no	—
109	C₄×C₁₁₃	#131	1	2	yes	—
110	Dic₁₁₃	—	2	1	no	—
111	Dic₁₁₃	—	2	1	no	—
112	Dic₁₁₃	—	2	1	no	—
113	Dic₁₁₃	—	2	2	no	—
114	Dic₁₁₃	H(5,𝔽₂)	1	1	yes	characteristic
115	Dic₁₁₃	2₋¹⁺⁴	1	1	yes	characteristic
116	C₂×C₁₁₃	—	4	1	no	—
117	C₂×C₁₁₃	#520	1	1	yes	characteristic
118	C₂×C₁₁₃	C₂×#131	1	1	yes	characteristic
119	C₂×C₁₁₃	#568	1	1	yes	characteristic
120	D₁₁₃	—	4	2	no	—
121	#2993	—	113	1	no	maximal, Sylow
122	C₁₁₃	#2993	1	1	yes	characteristic, complemented, Sylow
123	#529	—	113	1	no	—
124	#446	—	113	1	no	—
125	C₈×Q₈	—	113	1	no	—
126	#436	—	113	2	no	—
127	#436	—	113	1	no	—
128	#435	—	113	2	no	—
129	C₈×D₄	—	113	1	no	—
130	#433	—	113	2	no	—
131	#431	—	113	1	no	—
132	#424	—	113	2	no	—
133	#423	—	113	1	no	—
134	C₂×M₁₆	—	113	2	no	—
135	C₂×M₁₆	—	113	2	no	—
136	C₂²×C₈	—	113	2	no	—
137	#126	—	113	2	no	—
138	#125	—	113	1	no	—
139	#124	—	113	1	no	—
140	#123	—	113	2	no	—
141	C₂²⋊Q₈	#90673	1	1	yes	characteristic
142	#121	—	113	1	no	—
143	C₄×Q₈	—	113	1	no	—
144	C₄×D₄	—	113	1	no	—
145	C₄×D₄	—	113	2	no	—
146	#117	—	113	2	no	—
147	C₂×C₄²	—	113	1	no	—
148	C₄⋊C₈	—	113	1	no	—
149	C₄⋊C₈	—	113	1	no	—
150	C₄⋊C₈	—	113	1	no	—
151	C₄⋊C₈	—	113	2	no	—
152	C₄⋊C₈	—	113	1	no	—
153	C₄⋊C₈	—	113	2	no	—
154	C₄⋊C₈	—	113	2	no	—
155	C₂²⋊C₈	—	113	1	no	—
156	C₂²⋊C₈	—	113	1	no	—
157	C₂²⋊C₈	—	113	2	no	—
158	C₂²⋊C₈	—	113	2	no	—
159	#97	—	113	2	no	—
160	C₄×C₈	—	113	1	no	—
161	C₄×C₈	—	113	1	no	—
162	C₄×C₈	—	113	2	no	—
163	C₂×Q₈	C₂×#90673	1	1	yes	characteristic
164	C₂×D₄	—	113	2	no	—
165	C₂×D₄	—	113	1	no	—
166	C₂²×C₄	—	113	1	no	—
167	C₂²×C₄	—	113	1	no	—
168	C₂²×C₄	—	113	2	no	—
169	C₂²×C₄	C₂×#90673	1	1	yes	characteristic
170	M₁₆	—	452	2	no	—
171	M₁₆	—	452	2	no	—
172	C₂×C₈	—	113	1	no	—
173	C₂×C₈	—	113	1	no	—
174	C₂×C₈	—	113	1	no	—
175	C₂×C₈	—	113	1	no	—
176	C₂×C₈	—	113	2	no	—
177	C₂×C₈	—	113	2	no	—
178	C₂×C₈	—	452	2	no	—
179	C₄⋊C₄	—	113	2	no	—
180	C₄⋊C₄	—	113	2	no	—
181	C₄⋊C₄	—	113	1	no	—
182	C₄⋊C₄	—	113	2	no	—
183	C₄⋊C₄	C₂×#90673	1	2	yes	—
184	C₄⋊C₄	C₂×#90673	1	1	yes	characteristic
185	C₂²⋊C₄	—	113	2	no	—
186	C₂²⋊C₄	—	113	2	no	—
187	C₂²⋊C₄	—	113	2	no	—
188	C₂²⋊C₄	—	113	2	no	—
189	C₂²⋊C₄	C₂×#90673	1	2	yes	—
190	C₄²	—	113	1	no	—
191	C₄²	—	113	2	no	—
192	C₄²	—	113	1	no	—
193	C₄²	—	113	1	no	—
194	C₄²	—	113	1	no	—
195	C₂³	—	113	2	no	—
196	C₂³	C₂²×#90673	1	1	yes	characteristic
197	Q₈	—	2	2	no	—
198	D₄	—	452	2	no	—
199	D₄	—	226	2	no	—
200	C₂×C₄	—	2	2	no	—
201	C₂×C₄	—	113	1	no	—
202	C₂×C₄	—	113	1	no	—
203	C₂×C₄	—	113	1	no	—
204	C₂×C₄	—	113	1	no	—
205	C₂×C₄	—	113	2	no	—
206	C₂×C₄	—	452	1	no	—
207	C₂×C₄	—	226	2	no	—
208	C₂×C₄	—	226	2	no	—
209	C₂×C₄	—	226	1	no	—
210	C₂×C₄	—	226	1	no	—
211	C₂×C₄	C₂²×#90673	1	1	yes	characteristic
212	C₂×C₄	C₂²×#90673	1	2	yes	—
213	C₂×C₄	C₂²×#90673	1	2	yes	—
214	C₂×C₄	C₂²×#90673	1	1	yes	characteristic
215	C₈	—	226	1	no	—
216	C₈	—	226	2	no	—
217	C₈	—	226	2	no	—
218	C₈	—	226	1	no	—
219	C₈	—	226	1	no	—
220	C₈	—	226	1	no	—
221	C₂²	—	2	1	no	—
222	C₂²	—	2	1	no	—
223	C₂²	—	2	1	no	—
224	C₂²	—	226	2	no	—
225	C₂²	—	226	2	no	—
226	C₂²	—	226	2	no	—
227	C₂²	C₂³×#90673	1	1	yes	characteristic
228	C₄	—	2	2	no	—
229	C₄	—	2	1	no	—
230	C₄	—	2	2	no	—
231	C₄	—	113	1	no	—
232	C₄	—	113	1	no	—
233	C₄	—	226	1	no	—
234	C₄	—	226	1	no	—
235	C₄	—	226	1	no	—
236	C₄	—	226	2	no	—
237	C₄	#16813709	1	2	yes	—
238	C₂	—	4	1	no	—
239	C₂	—	452	2	no	—
240	C₂	#20299667	1	1	yes	characteristic
241	C₂	C₂×#16813709	1	1	yes	characteristic
242	C₂	#20299678	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂²	C₂³×#90673
commutator, socle	C₂²×C₁₁₃	C₂³×C₄
Fitting	C₂²⋊Q₈×C₁₁₃	C₄
prime radical	#16813596	C₄
cosocle	C₂×Dic₁₁₃	C₂⁴
hypercenter	C₂²⋊Q₈	#90673
hypocenter	C₁₁₃	#2993
LCS	G ▷ C₂²×C₁₁₃ ▷ C₁₁₃
UCS	C₁ ◁ C₂² ◁ C₂²⋊Q₈

Complements

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

N	H ≅ G/N	type
1. #20299649	C₂	semidirect
2. Q₈×#11744753	C₂	semidirect
3. #20299632	C₂	semidirect
4. #20299631	C₂	semidirect
5. #20299630	C₂	semidirect
6. #20299626	C₂	semidirect
7. #20299623	C₂	semidirect
8. #20299614	C₂	semidirect
9. #20299605	C₂	semidirect
10. #20299601	C₂	semidirect
122. C₁₁₃	#2993	semidirect

Maximal subgroups

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

subgroup	index
#20299649	2
Q₈×#11744753	2
#20299632	2
#20299631	2
#20299630	2
#20299626	2
#20299623	2
#20299614	2
#20299605	2
#20299601	2
#20299134	2
#2993	113