Subgroups — #28407328

Order 14464.

492 conjugacy classes of subgroups.

The subgroup lattice has 256 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#41×#90673	C₂	1	2	yes	maximal, complemented
2	#20299639	C₂	1	2	yes	maximal, complemented
3	#20299638	C₂	1	2	yes	maximal, complemented
4	#20299637	C₂	1	4	yes	maximal, complemented
5	#20299608	C₂	1	1	yes	maximal, characteristic, complemented
6	#20299588	C₂	1	2	yes	maximal, complemented
7	C₂×#16813676	C₂	1	1	yes	maximal, characteristic, complemented
8	#135×D₁₁₃	C₂	1	1	yes	maximal, characteristic
9	C₂×#14268655	C₂²	1	1	yes	characteristic
10	Q₈×#90673	C₂²	1	4	yes	—
11	D₄×#90673	—	2	4	no	—
12	D₄×#90673	C₂²	1	4	yes	—
13	#16813701	—	2	2	no	—
14	#16813701	C₂²	1	2	yes	—
15	C₂×C₄×#90673	—	2	2	no	—
16	C₂×C₄×#90673	C₂²	1	1	yes	characteristic
17	#16813691	C₂²	1	2	yes	—
18	#16813690	C₂²	1	2	yes	—
19	#16813685	C₂²	1	4	yes	—
20	#16813684	C₂²	1	4	yes	—
21	#16813676	C₂²	1	2	yes	—
22	#16813676	C₂²	1	2	yes	—
23	#41×D₁₁₃	C₂²	1	2	yes	—
24	#16813645	C₄	1	1	yes	characteristic, complemented
25	#16813644	C₄	1	1	yes	characteristic, complemented
26	#16813643	C₄	1	2	yes	complemented
27	#16813642	C₄	1	2	yes	complemented
28	#16813641	C₄	1	1	yes	characteristic, complemented
29	Q₁₆×D₁₁₃	C₂²	1	1	yes	characteristic
30	SD₁₆×D₁₁₃	C₂²	1	2	yes	—
31	D₈×D₁₁₃	C₂²	1	1	yes	characteristic
32	C₂×C₈×D₁₁₃	C₂²	1	1	yes	characteristic
33	#135×C₁₁₃	C₄	1	1	yes	characteristic, complemented
34	C₂²×#90673	—	2	4	no	—
35	C₂²×#90673	D₄	1	2	yes	—
36	#14268655	—	2	1	no	—
37	#14268655	C₂³	1	2	yes	—
38	C₂×#11744753	—	2	1	no	—
39	C₄⋊#90673	—	2	4	no	—
40	C₄⋊#90673	C₂³	1	4	yes	—
41	C₂²⋊#90673	—	2	2	no	—
42	C₂²⋊#90673	—	2	4	no	—
43	C₄×#90673	—	2	4	no	—
44	C₄×#90673	—	2	2	no	—
45	C₄×#90673	D₄	1	2	yes	—
46	C₄×#90673	C₂³	1	2	yes	—
47	#14268648	C₂×C₄	1	2	yes	complemented
48	#14268647	—	2	2	no	—
49	#14268647	C₂×C₄	1	2	yes	complemented
50	#14268646	—	2	2	no	—
51	#14268646	C₂×C₄	1	2	yes	complemented
52	Q₈×D₁₁₃	C₂³	1	2	yes	—
53	D₄×D₁₁₃	—	2	2	no	—
54	D₄×D₁₁₃	C₂³	1	2	yes	—
55	C₂×C₄×D₁₁₃	—	2	2	no	—
56	C₂×C₄×D₁₁₃	C₂³	1	1	yes	characteristic
57	Dic₄₅₂	C₂×C₄	1	1	yes	characteristic, complemented
58	#14268637	C₂×C₄	1	2	yes	complemented
59	#14268636	C₂×C₄	1	2	yes	complemented
60	#14268635	C₂×C₄	1	2	yes	complemented
61	#14268634	C₂×C₄	1	2	yes	complemented
62	D₉₀₄	C₂×C₄	1	1	yes	characteristic, complemented
63	#14268632	C₂×C₄	1	2	yes	complemented
64	C₂×#11744751	C₂×C₄	1	1	yes	characteristic, complemented
65	C₈×D₁₁₃	C₂×C₄	1	1	yes	characteristic, complemented
66	C₈×D₁₁₃	C₂×C₄	1	1	yes	characteristic, complemented
67	C₈×D₁₁₃	C₂³	1	1	yes	characteristic
68	C₈×D₁₁₃	C₂³	1	1	yes	characteristic
69	#41×C₁₁₃	C₂×C₄	1	2	yes	complemented
70	Q₁₆×C₁₁₃	C₂×C₄	1	1	yes	characteristic, complemented
71	SD₁₆×C₁₁₃	C₂×C₄	1	2	yes	complemented
72	D₈×C₁₁₃	C₂×C₄	1	1	yes	characteristic, complemented
73	C₂×C₈×C₁₁₃	C₂×C₄	1	1	yes	characteristic, complemented
74	C₂²×D₁₁₃	—	2	2	no	—
75	C₂²×D₁₁₃	C₂×D₄	1	1	yes	characteristic
76	C₂×#90673	—	2	4	no	—
77	C₂×#90673	—	4	2	no	—
78	C₂×#90673	—	4	8	no	—
79	C₂×#90673	C₂×D₄	1	4	yes	—
80	Q₈×C₁₁₃	C₂²×C₄	1	2	yes	—
81	D₄×C₁₁₃	—	2	2	no	—
82	D₄×C₁₁₃	C₂²×C₄	1	2	yes	—
83	C₂×C₄×C₁₁₃	—	2	2	no	—
84	C₂×C₄×C₁₁₃	C₂²×C₄	1	1	yes	characteristic
85	C₂²⋊D₁₁₃	—	2	2	no	—
86	C₂²⋊D₁₁₃	—	2	2	no	—
87	C₂²⋊D₁₁₃	—	2	2	no	—
88	C₂×Dic₁₁₃	—	2	2	no	—
89	C₂×Dic₁₁₃	#41	1	1	yes	characteristic
90	D₄₅₂	—	2	2	no	—
91	D₄₅₂	C₂²×C₄	1	2	yes	—
92	C₄×D₁₁₃	—	2	2	no	—
93	C₄×D₁₁₃	—	2	2	no	—
94	C₄×D₁₁₃	—	2	2	no	—
95	C₄×D₁₁₃	C₂²×C₄	1	1	yes	characteristic
96	C₄×D₁₁₃	C₂×D₄	1	1	yes	characteristic
97	C₄×D₁₁₃	#41	1	1	yes	characteristic
98	C₄×D₁₁₃	C₂⁴	1	1	yes	characteristic
99	Dic₂₂₆	—	2	2	no	—
100	Dic₂₂₆	C₂²×C₄	1	2	yes	—
101	#11744753	—	2	2	no	—
102	C₈×C₁₁₃	C₂²×C₄	1	1	yes	characteristic
103	C₈×C₁₁₃	C₂²×C₄	1	1	yes	characteristic
104	#11744751	C₂²×C₄	1	1	yes	characteristic
105	#11744751	C₂²×C₄	1	1	yes	characteristic
106	C₂²×C₁₁₃	—	2	2	no	—
107	C₂²×C₁₁₃	C₄×D₄	1	1	yes	characteristic
108	C₂×D₁₁₃	—	2	2	no	—
109	C₂×D₁₁₃	—	2	2	no	—
110	C₂×D₁₁₃	—	4	4	no	—
111	C₂×D₁₁₃	C₄×D₄	1	1	yes	characteristic
112	C₂×D₁₁₃	C₂²×D₄	1	1	yes	characteristic
113	#90673	—	2	4	no	—
114	#90673	—	4	4	no	—
115	C₄×C₁₁₃	—	2	2	no	—
116	C₄×C₁₁₃	C₄×D₄	1	1	yes	characteristic
117	C₄×C₁₁₃	C₂³×C₄	1	1	yes	characteristic
118	Dic₁₁₃	—	2	2	no	—
119	Dic₁₁₃	C₄×D₄	1	1	yes	characteristic
120	Dic₁₁₃	C₂×#41	1	1	yes	characteristic
121	C₂×C₁₁₃	—	2	1	no	—
122	C₂×C₁₁₃	—	4	2	no	—
123	C₂×C₁₁₃	C₂×C₄×D₄	1	1	yes	characteristic
124	D₁₁₃	—	2	1	no	—
125	D₁₁₃	—	4	2	no	—
126	D₁₁₃	#575	1	2	yes	—
127	#2943	—	113	1	no	maximal, Sylow
128	C₁₁₃	#2943	1	1	yes	characteristic, complemented, Sylow
129	C₂×#135	—	113	1	no	—
130	C₄×#41	—	113	2	no	—
131	#442	—	113	2	no	—
132	#441	—	113	2	no	—
133	#440	—	113	4	no	—
134	#427	—	113	1	no	—
135	#416	—	113	2	no	—
136	C₂×#97	—	113	1	no	—
137	C₂×#41	—	113	2	no	—
138	#135	—	113	2	no	—
139	#135	—	113	1	no	—
140	#135	—	113	2	no	—
141	#135	—	113	1	no	—
142	#135	—	113	1	no	—
143	#135	#90673	1	1	yes	characteristic, complemented
144	C₂×Q₁₆	—	113	1	no	—
145	C₂×SD₁₆	—	113	2	no	—
146	C₂×D₈	—	113	1	no	—
147	C₂²×C₈	—	113	1	no	—
148	C₂²×C₈	—	113	1	no	—
149	C₄×Q₈	—	113	4	no	—
150	C₄×D₄	—	113	4	no	—
151	C₄×D₄	—	226	4	no	—
152	#117	—	113	2	no	—
153	#117	—	226	2	no	—
154	C₂×C₄²	—	113	1	no	—
155	C₂×C₄²	—	226	2	no	—
156	#107	—	113	2	no	—
157	#106	—	113	2	no	—
158	#103	—	113	4	no	—
159	#102	—	113	4	no	—
160	#97	—	113	2	no	—
161	#97	—	113	2	no	—
162	#41	—	113	2	no	—
163	#41	—	113	2	no	—
164	#41	—	113	2	no	—
165	#41	—	226	2	no	—
166	#41	—	226	2	no	—
167	#41	C₂×#90673	1	2	yes	complemented
168	C₂×Q₈	—	113	2	no	—
169	C₂×D₄	—	113	2	no	—
170	C₂×D₄	—	226	2	no	—
171	C₂²×C₄	—	113	1	no	—
172	C₂²×C₄	—	113	2	no	—
173	C₂²×C₄	—	226	2	no	—
174	C₂²×C₄	—	226	4	no	—
175	Q₁₆	—	113	2	no	—
176	Q₁₆	—	113	1	no	—
177	Q₁₆	C₂×#90673	1	1	yes	characteristic, complemented
178	SD₁₆	—	113	2	no	—
179	SD₁₆	—	113	2	no	—
180	SD₁₆	—	113	2	no	—
181	SD₁₆	C₂×#90673	1	2	yes	complemented
182	D₈	—	113	2	no	—
183	D₈	—	113	1	no	—
184	D₈	C₂×#90673	1	1	yes	characteristic, complemented
185	C₂×C₈	—	113	1	no	—
186	C₂×C₈	—	113	1	no	—
187	C₂×C₈	—	113	1	no	—
188	C₂×C₈	—	113	1	no	—
189	C₂×C₈	—	113	1	no	—
190	C₂×C₈	—	113	2	no	—
191	C₂×C₈	—	226	1	no	—
192	C₂×C₈	—	226	1	no	—
193	C₂×C₈	C₂×#90673	1	1	yes	characteristic, complemented
194	C₄⋊C₄	—	113	4	no	—
195	C₄⋊C₄	—	226	4	no	—
196	C₂²⋊C₄	—	226	4	no	—
197	C₂²⋊C₄	—	226	2	no	—
198	C₄²	—	113	2	no	—
199	C₄²	—	113	2	no	—
200	C₄²	—	226	2	no	—
201	C₄²	—	226	4	no	—
202	C₂³	—	113	1	no	—
203	C₂³	—	226	2	no	—
204	Q₈	—	113	2	no	—
205	Q₈	—	226	2	no	—
206	Q₈	C₂²×#90673	1	2	yes	—
207	D₄	—	2	2	no	—
208	D₄	—	113	2	no	—
209	D₄	—	226	2	no	—
210	D₄	—	226	2	no	—
211	D₄	—	226	2	no	—
212	D₄	—	226	2	no	—
213	D₄	C₂²×#90673	1	2	yes	—
214	C₂×C₄	—	2	2	no	—
215	C₂×C₄	—	113	1	no	—
216	C₂×C₄	—	113	1	no	—
217	C₂×C₄	—	113	1	no	—
218	C₂×C₄	—	113	1	no	—
219	C₂×C₄	—	113	1	no	—
220	C₂×C₄	—	113	4	no	—
221	C₂×C₄	—	452	8	no	—
222	C₂×C₄	—	452	2	no	—
223	C₂×C₄	—	226	2	no	—
224	C₂×C₄	—	226	2	no	—
225	C₂×C₄	—	226	2	no	—
226	C₂×C₄	—	226	4	no	—
227	C₂×C₄	—	226	2	no	—
228	C₂×C₄	C₂²×#90673	1	1	yes	characteristic
229	C₈	—	113	1	no	—
230	C₈	—	113	1	no	—
231	C₈	—	226	2	no	—
232	C₈	C₂²×#90673	1	1	yes	characteristic
233	C₈	C₂²×#90673	1	1	yes	characteristic
234	C₂²	—	2	2	no	—
235	C₂²	—	113	1	no	—
236	C₂²	—	113	1	no	—
237	C₂²	—	452	4	no	—
238	C₂²	—	226	2	no	—
239	C₂²	—	226	2	no	—
240	C₂²	D₄×#90673	1	1	yes	characteristic
241	C₄	—	2	2	no	—
242	C₄	—	113	1	no	—
243	C₄	—	113	1	no	—
244	C₄	—	452	4	no	—
245	C₄	—	226	4	no	—
246	C₄	—	226	2	no	—
247	C₄	D₄×#90673	1	1	yes	characteristic
248	C₄	C₂³×#90673	1	1	yes	characteristic
249	C₂	—	2	1	no	—
250	C₂	—	4	2	no	—
251	C₂	—	113	2	no	—
252	C₂	—	452	2	no	—
253	C₂	—	226	1	no	—
254	C₂	C₂×D₄×#90673	1	1	yes	characteristic

	subgroup	G/H
center	C₄	D₄×#90673
commutator	C₄×C₁₁₃	C₂³×C₄
Frattini	C₄	C₂³×#90673
Fitting	#135×C₁₁₃	C₄
socle	C₂×C₁₁₃	C₂×C₄×D₄
prime radical	#135×D₁₁₃	C₂
cosocle	C₄×D₁₁₃	C₂⁴
hypercenter	#135	#90673
hypocenter	C₁₁₃	#2943
LCS	G ▷ C₄×C₁₁₃ ▷ C₂×C₁₁₃ ▷ C₁₁₃
UCS	C₁ ◁ C₄ ◁ C₂×C₄ ◁ #135

Complements

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

N	H ≅ G/N	type
1. #41×#90673	C₂	semidirect
2. #20299639	C₂	semidirect
3. #20299638	C₂	semidirect
4. #20299637	C₂	semidirect
5. #20299608	C₂	semidirect
6. #20299588	C₂	semidirect
7. C₂×#16813676	C₂	semidirect
24. #16813645	C₄	semidirect
25. #16813644	C₄	semidirect
26. #16813643	C₄	semidirect
27. #16813642	C₄	semidirect
28. #16813641	C₄	semidirect
33. #135×C₁₁₃	C₄	semidirect
47. #14268648	C₂×C₄	semidirect
49. #14268647	C₂×C₄	semidirect
51. #14268646	C₂×C₄	semidirect
57. Dic₄₅₂	C₂×C₄	semidirect
58. #14268637	C₂×C₄	semidirect
59. #14268636	C₂×C₄	semidirect
60. #14268635	C₂×C₄	semidirect
61. #14268634	C₂×C₄	semidirect
62. D₉₀₄	C₂×C₄	semidirect
63. #14268632	C₂×C₄	semidirect
64. C₂×#11744751	C₂×C₄	semidirect
65. C₈×D₁₁₃	C₂×C₄	semidirect
66. C₈×D₁₁₃	C₂×C₄	semidirect
69. #41×C₁₁₃	C₂×C₄	semidirect
70. Q₁₆×C₁₁₃	C₂×C₄	semidirect
71. SD₁₆×C₁₁₃	C₂×C₄	semidirect
72. D₈×C₁₁₃	C₂×C₄	semidirect
73. C₂×C₈×C₁₁₃	C₂×C₄	semidirect
128. C₁₁₃	#2943	semidirect
143. #135	#90673	semidirect
167. #41	C₂×#90673	semidirect
177. Q₁₆	C₂×#90673	semidirect
181. SD₁₆	C₂×#90673	semidirect
184. D₈	C₂×#90673	semidirect
193. C₂×C₈	C₂×#90673	semidirect

Maximal subgroups

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

subgroup	index
#41×#90673	2
#20299639	2
#20299638	2
#20299637	2
#20299608	2
#20299588	2
C₂×#16813676	2
#135×D₁₁₃	2
#2943	113