Subgroups — C₁₁₃×#2693 #28387489

Order 14464.

568 conjugacy classes of subgroups.

The subgroup lattice has 260 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#123×C₁₁₃	C₂	1	2	yes	maximal, complemented
2	#400×C₁₁₃	C₂	1	2	yes	maximal, complemented
3	#397×C₁₁₃	C₂	1	1	yes	maximal, characteristic, complemented
4	#396×C₁₁₃	C₂	1	2	yes	maximal, complemented
5	#392×C₁₁₃	C₂	1	1	yes	maximal, characteristic
6	#386×C₁₁₃	C₂	1	2	yes	maximal, complemented
7	#386×C₁₁₃	C₂	1	2	yes	maximal, complemented
8	#385×C₁₁₃	C₂	1	1	yes	maximal, characteristic, complemented
9	#380×C₁₁₃	C₂	1	2	yes	maximal, complemented
10	C₂²×D₄×C₁₁₃	C₂²	1	2	yes	—
11	C₂²×D₄×C₁₁₃	C₂²	1	1	yes	characteristic
12	C₂³×C₄×C₁₁₃	C₂²	1	2	yes	—
13	C₂³×C₄×C₁₁₃	C₂²	1	1	yes	characteristic
14	#123×C₁₁₃	—	2	8	no	—
15	C₂×C₄⋊C₄×C₁₁₃	C₂²	1	2	yes	—
16	C₂×C₄⋊C₄×C₁₁₃	C₂²	1	2	yes	—
17	C₂×C₄⋊C₄×C₁₁₃	C₂²	1	2	yes	—
18	C₂×C₄⋊C₄×C₁₁₃	C₂²	1	1	yes	characteristic
19	C₂×C₂²⋊C₄×C₁₁₃	C₂²	1	2	yes	—
20	C₂×C₂²⋊C₄×C₁₁₃	C₂²	1	2	yes	—
21	C₂×C₂²⋊C₄×C₁₁₃	C₂²	1	2	yes	—
22	C₂×C₂²⋊C₄×C₁₁₃	C₂²	1	2	yes	—
23	C₂×C₂²⋊C₄×C₁₁₃	C₂²	1	1	yes	characteristic
24	C₂×C₂²⋊C₄×C₁₁₃	C₂²	1	2	yes	—
25	C₂×C₂²⋊C₄×C₁₁₃	C₂²	1	1	yes	characteristic
26	#95×C₁₁₃	C₂²	1	1	yes	characteristic
27	#95×C₁₁₃	C₂²	1	2	yes	—
28	#95×C₁₁₃	C₂²	1	2	yes	—
29	#95×C₁₁₃	C₂²	1	2	yes	—
30	#95×C₁₁₃	C₂²	1	2	yes	—
31	#95×C₁₁₃	C₂²	1	1	yes	characteristic
32	C₂⁴×C₁₁₃	C₂³	1	1	yes	characteristic
33	C₂⁴×C₁₁₃	C₂³	1	2	yes	—
34	C₂×D₄×C₁₁₃	—	2	4	no	—
35	C₂×D₄×C₁₁₃	—	2	4	no	—
36	C₂×D₄×C₁₁₃	—	2	4	no	—
37	C₂×D₄×C₁₁₃	—	4	1	no	—
38	C₂×D₄×C₁₁₃	—	4	2	no	—
39	C₂²×C₄×C₁₁₃	—	2	4	no	—
40	C₂²×C₄×C₁₁₃	—	2	4	no	—
41	C₂²×C₄×C₁₁₃	—	4	2	no	—
42	C₂²×C₄×C₁₁₃	—	4	1	no	—
43	C₂²×C₄×C₁₁₃	—	4	2	no	—
44	C₂²×C₄×C₁₁₃	C₂³	1	1	yes	characteristic
45	C₂²×C₄×C₁₁₃	C₂³	1	2	yes	—
46	C₂²×C₄×C₁₁₃	C₂³	1	2	yes	—
47	C₂²×C₄×C₁₁₃	C₂³	1	2	yes	—
48	C₂²×C₄×C₁₁₃	C₂³	1	2	yes	—
49	C₂²×C₄×C₁₁₃	C₂³	1	1	yes	characteristic
50	C₂²×C₄×C₁₁₃	C₂³	1	2	yes	—
51	C₄⋊C₄×C₁₁₃	—	2	4	no	—
52	C₄⋊C₄×C₁₁₃	—	2	4	no	—
53	C₄⋊C₄×C₁₁₃	—	4	2	no	—
54	C₄⋊C₄×C₁₁₃	—	4	1	no	—
55	C₂²⋊C₄×C₁₁₃	—	2	4	no	—
56	C₂²⋊C₄×C₁₁₃	—	2	4	no	—
57	C₂²⋊C₄×C₁₁₃	—	4	2	no	—
58	C₂²⋊C₄×C₁₁₃	—	4	2	no	—
59	C₂²⋊C₄×C₁₁₃	—	4	1	no	—
60	C₂²⋊C₄×C₁₁₃	—	4	2	no	—
61	C₂²⋊C₄×C₁₁₃	D₄	1	4	yes	complemented
62	C₂³×C₁₁₃	—	2	2	no	—
63	C₂³×C₁₁₃	—	2	2	no	—
64	C₂³×C₁₁₃	—	2	2	no	—
65	C₂³×C₁₁₃	—	2	2	no	—
66	C₂³×C₁₁₃	—	2	1	no	—
67	C₂³×C₁₁₃	—	2	2	no	—
68	C₂³×C₁₁₃	—	2	2	no	—
69	C₂³×C₁₁₃	—	2	2	no	—
70	C₂³×C₁₁₃	—	2	1	no	—
71	C₂³×C₁₁₃	—	2	2	no	—
72	C₂³×C₁₁₃	C₂×D₄	1	2	yes	—
73	C₂³×C₁₁₃	#41	1	4	yes	—
74	C₂³×C₁₁₃	C₂⁴	1	1	yes	characteristic
75	D₄×C₁₁₃	—	4	8	no	—
76	D₄×C₁₁₃	—	4	4	no	—
77	C₂×C₄×C₁₁₃	—	2	2	no	—
78	C₂×C₄×C₁₁₃	—	2	2	no	—
79	C₂×C₄×C₁₁₃	—	2	2	no	—
80	C₂×C₄×C₁₁₃	—	2	2	no	—
81	C₂×C₄×C₁₁₃	—	2	2	no	—
82	C₂×C₄×C₁₁₃	—	2	2	no	—
83	C₂×C₄×C₁₁₃	—	2	2	no	—
84	C₂×C₄×C₁₁₃	—	2	2	no	—
85	C₂×C₄×C₁₁₃	—	2	2	no	—
86	C₂×C₄×C₁₁₃	—	2	1	no	—
87	C₂×C₄×C₁₁₃	—	2	2	no	—
88	C₂×C₄×C₁₁₃	—	2	2	no	—
89	C₂×C₄×C₁₁₃	—	2	2	no	—
90	C₂×C₄×C₁₁₃	—	2	2	no	—
91	C₂×C₄×C₁₁₃	—	2	2	no	—
92	C₂×C₄×C₁₁₃	—	2	1	no	—
93	C₂×C₄×C₁₁₃	—	2	2	no	—
94	C₂×C₄×C₁₁₃	—	2	2	no	—
95	C₂×C₄×C₁₁₃	—	4	8	no	—
96	C₂×C₄×C₁₁₃	—	4	4	no	—
97	C₂×C₄×C₁₁₃	C₂×D₄	1	4	yes	—
98	C₂²×C₁₁₃	—	2	2	no	—
99	C₂²×C₁₁₃	—	2	4	no	—
100	C₂²×C₁₁₃	—	2	4	no	—
101	C₂²×C₁₁₃	—	2	4	no	—
102	C₂²×C₁₁₃	—	2	4	no	—
103	C₂²×C₁₁₃	—	4	2	no	—
104	C₂²×C₁₁₃	—	4	2	no	—
105	C₂²×C₁₁₃	—	4	2	no	—
106	C₂²×C₁₁₃	—	4	2	no	—
107	C₂²×C₁₁₃	—	4	1	no	—
108	C₂²×C₁₁₃	—	4	1	no	—
109	C₂²×C₁₁₃	—	4	2	no	—
110	C₂²×C₁₁₃	C₂²×D₄	1	1	yes	characteristic
111	C₂²×C₁₁₃	C₂×#41	1	2	yes	—
112	C₂²×C₁₁₃	H(5,𝔽₂)	1	1	yes	characteristic
113	C₂²×C₁₁₃	H(5,𝔽₂)	1	2	yes	—
114	C₂²×C₁₁₃	2₋¹⁺⁴	1	1	yes	characteristic
115	C₄×C₁₁₃	—	2	4	no	—
116	C₄×C₁₁₃	—	4	2	no	—
117	C₄×C₁₁₃	—	4	2	no	—
118	C₄×C₁₁₃	—	4	2	no	—
119	C₄×C₁₁₃	—	4	1	no	—
120	C₄×C₁₁₃	—	4	1	no	—
121	C₄×C₁₁₃	—	4	2	no	—
122	C₂×C₁₁₃	—	4	2	no	—
123	C₂×C₁₁₃	—	4	4	no	—
124	C₂×C₁₁₃	#537	1	1	yes	characteristic
125	C₂×C₁₁₃	#539	1	2	yes	—
126	C₂×C₁₁₃	#546	1	2	yes	—
127	C₂×C₁₁₃	#551	1	1	yes	characteristic
128	C₂×C₁₁₃	#560	1	1	yes	characteristic
129	#2693	C₁₁₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
130	C₁₁₃	#2693	1	1	yes	characteristic, direct product factor, Sylow
131	C₂×#123	C₂×C₁₁₃	1	2	yes	complemented
132	#400	C₂×C₁₁₃	1	2	yes	complemented
133	#397	C₂×C₁₁₃	1	1	yes	characteristic, complemented
134	#396	C₂×C₁₁₃	1	2	yes	complemented
135	#392	C₂×C₁₁₃	1	1	yes	characteristic
136	#386	C₂×C₁₁₃	1	2	yes	complemented
137	#386	C₂×C₁₁₃	1	2	yes	complemented
138	#385	C₂×C₁₁₃	1	1	yes	characteristic, complemented
139	#380	C₂×C₁₁₃	1	2	yes	complemented
140	C₂²×D₄	C₂²×C₁₁₃	1	2	yes	—
141	C₂²×D₄	C₂²×C₁₁₃	1	1	yes	characteristic
142	C₂³×C₄	C₂²×C₁₁₃	1	2	yes	—
143	C₂³×C₄	C₂²×C₁₁₃	1	1	yes	characteristic
144	#123	—	2	8	no	—
145	C₂×C₄⋊C₄	C₂²×C₁₁₃	1	2	yes	—
146	C₂×C₄⋊C₄	C₂²×C₁₁₃	1	2	yes	—
147	C₂×C₄⋊C₄	C₂²×C₁₁₃	1	2	yes	—
148	C₂×C₄⋊C₄	C₂²×C₁₁₃	1	1	yes	characteristic
149	C₂×C₂²⋊C₄	C₂²×C₁₁₃	1	2	yes	—
150	C₂×C₂²⋊C₄	C₂²×C₁₁₃	1	2	yes	—
151	C₂×C₂²⋊C₄	C₂²×C₁₁₃	1	2	yes	—
152	C₂×C₂²⋊C₄	C₂²×C₁₁₃	1	2	yes	—
153	C₂×C₂²⋊C₄	C₂²×C₁₁₃	1	1	yes	characteristic
154	C₂×C₂²⋊C₄	C₂²×C₁₁₃	1	2	yes	—
155	C₂×C₂²⋊C₄	C₂²×C₁₁₃	1	1	yes	characteristic
156	#95	C₂²×C₁₁₃	1	1	yes	characteristic
157	#95	C₂²×C₁₁₃	1	2	yes	—
158	#95	C₂²×C₁₁₃	1	2	yes	—
159	#95	C₂²×C₁₁₃	1	2	yes	—
160	#95	C₂²×C₁₁₃	1	2	yes	—
161	#95	C₂²×C₁₁₃	1	1	yes	characteristic
162	C₂⁴	C₂³×C₁₁₃	1	1	yes	characteristic
163	C₂⁴	C₂³×C₁₁₃	1	2	yes	—
164	C₂×D₄	—	2	4	no	—
165	C₂×D₄	—	2	4	no	—
166	C₂×D₄	—	2	4	no	—
167	C₂×D₄	—	4	1	no	—
168	C₂×D₄	—	4	2	no	—
169	C₂²×C₄	—	2	4	no	—
170	C₂²×C₄	—	2	4	no	—
171	C₂²×C₄	—	4	2	no	—
172	C₂²×C₄	—	4	1	no	—
173	C₂²×C₄	—	4	2	no	—
174	C₂²×C₄	C₂³×C₁₁₃	1	1	yes	characteristic
175	C₂²×C₄	C₂³×C₁₁₃	1	2	yes	—
176	C₂²×C₄	C₂³×C₁₁₃	1	2	yes	—
177	C₂²×C₄	C₂³×C₁₁₃	1	2	yes	—
178	C₂²×C₄	C₂³×C₁₁₃	1	2	yes	—
179	C₂²×C₄	C₂³×C₁₁₃	1	1	yes	characteristic
180	C₂²×C₄	C₂³×C₁₁₃	1	2	yes	—
181	C₄⋊C₄	—	2	4	no	—
182	C₄⋊C₄	—	2	4	no	—
183	C₄⋊C₄	—	4	2	no	—
184	C₄⋊C₄	—	4	1	no	—
185	C₂²⋊C₄	—	2	4	no	—
186	C₂²⋊C₄	—	2	4	no	—
187	C₂²⋊C₄	—	4	2	no	—
188	C₂²⋊C₄	—	4	2	no	—
189	C₂²⋊C₄	—	4	1	no	—
190	C₂²⋊C₄	—	4	2	no	—
191	C₂²⋊C₄	D₄×C₁₁₃	1	4	yes	complemented
192	C₂³	—	2	2	no	—
193	C₂³	—	2	2	no	—
194	C₂³	—	2	2	no	—
195	C₂³	—	2	2	no	—
196	C₂³	—	2	1	no	—
197	C₂³	—	2	2	no	—
198	C₂³	—	2	2	no	—
199	C₂³	—	2	2	no	—
200	C₂³	—	2	1	no	—
201	C₂³	—	2	2	no	—
202	C₂³	C₂×D₄×C₁₁₃	1	2	yes	—
203	C₂³	#41×C₁₁₃	1	4	yes	—
204	C₂³	C₂⁴×C₁₁₃	1	1	yes	characteristic
205	D₄	—	4	8	no	—
206	D₄	—	4	4	no	—
207	C₂×C₄	—	2	2	no	—
208	C₂×C₄	—	2	2	no	—
209	C₂×C₄	—	2	2	no	—
210	C₂×C₄	—	2	2	no	—
211	C₂×C₄	—	2	2	no	—
212	C₂×C₄	—	2	2	no	—
213	C₂×C₄	—	2	2	no	—
214	C₂×C₄	—	2	2	no	—
215	C₂×C₄	—	2	2	no	—
216	C₂×C₄	—	2	1	no	—
217	C₂×C₄	—	2	2	no	—
218	C₂×C₄	—	2	2	no	—
219	C₂×C₄	—	2	2	no	—
220	C₂×C₄	—	2	2	no	—
221	C₂×C₄	—	2	2	no	—
222	C₂×C₄	—	2	1	no	—
223	C₂×C₄	—	2	2	no	—
224	C₂×C₄	—	2	2	no	—
225	C₂×C₄	—	4	8	no	—
226	C₂×C₄	—	4	4	no	—
227	C₂×C₄	C₂×D₄×C₁₁₃	1	4	yes	—
228	C₂²	—	2	2	no	—
229	C₂²	—	2	4	no	—
230	C₂²	—	2	4	no	—
231	C₂²	—	2	4	no	—
232	C₂²	—	2	4	no	—
233	C₂²	—	4	2	no	—
234	C₂²	—	4	2	no	—
235	C₂²	—	4	2	no	—
236	C₂²	—	4	2	no	—
237	C₂²	—	4	1	no	—
238	C₂²	—	4	1	no	—
239	C₂²	—	4	2	no	—
240	C₂²	C₂²×D₄×C₁₁₃	1	1	yes	characteristic
241	C₂²	C₂×#41×C₁₁₃	1	2	yes	—
242	C₂²	H(5,𝔽₂)×C₁₁₃	1	1	yes	characteristic
243	C₂²	H(5,𝔽₂)×C₁₁₃	1	2	yes	—
244	C₂²	2₋¹⁺⁴×C₁₁₃	1	1	yes	characteristic
245	C₄	—	2	4	no	—
246	C₄	—	4	2	no	—
247	C₄	—	4	2	no	—
248	C₄	—	4	2	no	—
249	C₄	—	4	1	no	—
250	C₄	—	4	2	no	—
251	C₄	—	4	1	no	—
252	C₂	—	4	2	no	—
253	C₂	—	4	4	no	—
254	C₂	#537×C₁₁₃	1	1	yes	characteristic
255	C₂	#539×C₁₁₃	1	2	yes	—
256	C₂	#546×C₁₁₃	1	2	yes	—
257	C₂	#551×C₁₁₃	1	1	yes	characteristic
258	C₂	#560×C₁₁₃	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂³×C₁₁₃	C₂⁴
commutator, Frattini, cosocle	C₂³	C₂⁴×C₁₁₃
prime radical	#392×C₁₁₃	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³×C₁₁₃ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₂×#123×C₁₁₃	C₂	semidirect
2. #400×C₁₁₃	C₂	semidirect
3. #397×C₁₁₃	C₂	semidirect
4. #396×C₁₁₃	C₂	semidirect
6. #386×C₁₁₃	C₂	semidirect
7. #386×C₁₁₃	C₂	semidirect
8. #385×C₁₁₃	C₂	semidirect
9. #380×C₁₁₃	C₂	semidirect
61. C₂²⋊C₄×C₁₁₃	D₄	semidirect
129. #2693	C₁₁₃	direct (N × H)
130. C₁₁₃	#2693	direct (N × H)
131. C₂×#123	C₂×C₁₁₃	semidirect
132. #400	C₂×C₁₁₃	semidirect
133. #397	C₂×C₁₁₃	semidirect
134. #396	C₂×C₁₁₃	semidirect
136. #386	C₂×C₁₁₃	semidirect
137. #386	C₂×C₁₁₃	semidirect
138. #385	C₂×C₁₁₃	semidirect
139. #380	C₂×C₁₁₃	semidirect
191. C₂²⋊C₄	D₄×C₁₁₃	semidirect

Maximal subgroups

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

subgroup	index
C₂×#123×C₁₁₃	2
#400×C₁₁₃	2
#397×C₁₁₃	2
#396×C₁₁₃	2
#392×C₁₁₃	2
#386×C₁₁₃	2
#385×C₁₁₃	2
#380×C₁₁₃	2
#2693	113

Subgroups — C113×#2693 #28387489

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₁₃×#2693 #28387489