Subgroups — C₁₁₃×#8731 #43276751

Order 28928.

522 conjugacy classes of subgroups.

The subgroup lattice has 274 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄×#117×C₁₁₃	C₂	1	1	yes	maximal, characteristic
2	C₁₁₃×#1458	C₂	1	2	yes	maximal
3	C₁₁₃×#1276	C₂	1	2	yes	maximal, complemented
4	C₁₁₃×#1274	C₂	1	2	yes	maximal, complemented
5	C₂×#117×C₁₁₃	C₄	1	1	yes	characteristic
6	C₂×#117×C₁₁₃	C₂²	1	1	yes	characteristic
7	C₂²×C₄²×C₁₁₃	C₄	1	1	yes	characteristic
8	#431×C₁₁₃	C₂²	1	4	yes	—
9	C₄×C₄⋊C₄×C₁₁₃	—	2	2	no	—
10	C₄×C₂²⋊C₄×C₁₁₃	—	2	2	no	—
11	#376×C₁₁₃	C₄	1	1	yes	characteristic
12	#376×C₁₁₃	C₂²	1	1	yes	characteristic
13	C₄³×C₁₁₃	C₄	1	1	yes	characteristic
14	C₄³×C₁₁₃	C₂²	1	1	yes	characteristic
15	#324×C₁₁₃	—	2	2	no	—
16	#323×C₁₁₃	—	2	2	no	—
17	C₂³×C₄×C₁₁₃	—	2	1	no	—
18	C₂³×C₄×C₁₁₃	C₂×C₄	1	1	yes	characteristic
19	#117×C₁₁₃	—	2	2	no	—
20	#117×C₁₁₃	—	2	2	no	—
21	#117×C₁₁₃	—	2	2	no	—
22	#117×C₁₁₃	C₈	1	4	yes	complemented
23	C₂×C₄⋊C₄×C₁₁₃	—	2	1	no	—
24	C₂×C₄⋊C₄×C₁₁₃	D₄	1	2	yes	—
25	C₂×C₂²⋊C₄×C₁₁₃	—	2	1	no	—
26	C₂×C₂²⋊C₄×C₁₁₃	D₄	1	2	yes	—
27	C₂×C₄²×C₁₁₃	—	2	1	no	—
28	C₂×C₄²×C₁₁₃	—	2	2	no	—
29	C₂×C₄²×C₁₁₃	—	2	2	no	—
30	C₂×C₄²×C₁₁₃	—	2	2	no	—
31	C₂×C₄²×C₁₁₃	—	2	2	no	—
32	C₂×C₄²×C₁₁₃	C₂×C₄	1	1	yes	characteristic
33	C₂×C₄²×C₁₁₃	C₂×C₄	1	1	yes	characteristic
34	C₂×C₄²×C₁₁₃	C₂×C₄	1	1	yes	characteristic
35	C₂×C₄²×C₁₁₃	C₂×C₄	1	1	yes	characteristic
36	C₂×C₄²×C₁₁₃	C₂×C₄	1	1	yes	characteristic
37	C₂×C₄²×C₁₁₃	C₂³	1	1	yes	characteristic
38	C₄⋊C₈×C₁₁₃	—	2	4	no	—
39	C₂²⋊C₈×C₁₁₃	—	2	4	no	—
40	C₄×C₈×C₁₁₃	—	2	4	no	—
41	#95×C₁₁₃	—	2	2	no	—
42	#95×C₁₁₃	—	2	2	no	—
43	C₂⁴×C₁₁₃	C₂²⋊C₄	1	1	yes	characteristic
44	C₂²×C₄×C₁₁₃	—	2	1	no	—
45	C₂²×C₄×C₁₁₃	—	2	1	no	—
46	C₂²×C₄×C₁₁₃	—	2	1	no	—
47	C₂²×C₄×C₁₁₃	—	2	1	no	—
48	C₂²×C₄×C₁₁₃	—	2	1	no	—
49	C₂²×C₄×C₁₁₃	—	2	4	no	—
50	C₂²×C₄×C₁₁₃	—	2	1	no	—
51	C₂²×C₄×C₁₁₃	—	2	1	no	—
52	C₂²×C₄×C₁₁₃	—	2	1	no	—
53	C₂²×C₄×C₁₁₃	—	4	4	no	—
54	C₂²×C₄×C₁₁₃	C₂²⋊C₄	1	1	yes	characteristic
55	C₂²×C₄×C₁₁₃	C₂²⋊C₄	1	2	yes	—
56	C₂²×C₄×C₁₁₃	C₂×C₈	1	2	yes	—
57	C₂²×C₄×C₁₁₃	M₁₆	1	2	yes	—
58	C₂²×C₄×C₁₁₃	C₂²×C₄	1	1	yes	characteristic
59	C₂²×C₄×C₁₁₃	C₂×D₄	1	2	yes	—
60	C₂×C₈×C₁₁₃	—	4	4	no	—
61	C₄⋊C₄×C₁₁₃	—	2	4	no	—
62	C₄⋊C₄×C₁₁₃	—	2	4	no	—
63	C₂²⋊C₄×C₁₁₃	—	2	4	no	—
64	C₂²⋊C₄×C₁₁₃	—	2	4	no	—
65	C₄²×C₁₁₃	—	2	4	no	—
66	C₄²×C₁₁₃	—	2	1	no	—
67	C₄²×C₁₁₃	—	2	1	no	—
68	C₄²×C₁₁₃	—	2	1	no	—
69	C₄²×C₁₁₃	—	2	1	no	—
70	C₄²×C₁₁₃	—	2	1	no	—
71	C₄²×C₁₁₃	—	2	1	no	—
72	C₄²×C₁₁₃	—	4	4	no	—
73	C₄²×C₁₁₃	—	4	4	no	—
74	C₄²×C₁₁₃	C₂×C₈	1	2	yes	—
75	C₄²×C₁₁₃	C₂×C₈	1	2	yes	—
76	C₄²×C₁₁₃	M₁₆	1	2	yes	—
77	C₄²×C₁₁₃	M₁₆	1	2	yes	—
78	C₄²×C₁₁₃	#41	1	2	yes	—
79	C₄²×C₁₁₃	#41	1	2	yes	—
80	C₂³×C₁₁₃	—	2	1	no	—
81	C₂³×C₁₁₃	—	2	1	no	—
82	C₂³×C₁₁₃	—	4	2	no	—
83	C₂³×C₁₁₃	C₂²⋊C₈	1	2	yes	—
84	C₂³×C₁₁₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
85	C₂×C₄×C₁₁₃	—	2	4	no	—
86	C₂×C₄×C₁₁₃	—	2	1	no	—
87	C₂×C₄×C₁₁₃	—	2	1	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	2	no	—
93	C₂×C₄×C₁₁₃	—	2	2	no	—
94	C₂×C₄×C₁₁₃	—	2	2	no	—
95	C₂×C₄×C₁₁₃	—	2	2	no	—
96	C₂×C₄×C₁₁₃	—	2	2	no	—
97	C₂×C₄×C₁₁₃	—	4	2	no	—
98	C₂×C₄×C₁₁₃	—	4	8	no	—
99	C₂×C₄×C₁₁₃	—	4	2	no	—
100	C₂×C₄×C₁₁₃	—	4	2	no	—
101	C₂×C₄×C₁₁₃	—	4	2	no	—
102	C₂×C₄×C₁₁₃	—	4	2	no	—
103	C₂×C₄×C₁₁₃	C₂²⋊C₈	1	2	yes	—
104	C₂×C₄×C₁₁₃	#117	1	1	yes	characteristic
105	C₂×C₄×C₁₁₃	#117	1	1	yes	characteristic
106	C₂×C₄×C₁₁₃	#123	1	4	yes	—
107	C₂×C₄×C₁₁₃	C₂²×C₈	1	1	yes	characteristic
108	C₂×C₄×C₁₁₃	C₂×M₁₆	1	1	yes	characteristic
109	C₂×C₄×C₁₁₃	C₂×M₁₆	1	1	yes	characteristic
110	C₂×C₄×C₁₁₃	C₂×M₁₆	1	1	yes	characteristic
111	C₈×C₁₁₃	—	8	4	no	—
112	C₂²×C₁₁₃	—	2	2	no	—
113	C₂²×C₁₁₃	—	2	2	no	—
114	C₂²×C₁₁₃	—	4	2	no	—
115	C₂²×C₁₁₃	—	4	2	no	—
116	C₂²×C₁₁₃	—	4	2	no	—
117	C₂²×C₁₁₃	#381	1	1	yes	characteristic
118	C₂²×C₁₁₃	C₂×C₂²⋊C₈	1	1	yes	characteristic
119	C₂²×C₁₁₃	#407	1	1	yes	characteristic
120	C₄×C₁₁₃	—	2	4	no	—
121	C₄×C₁₁₃	—	2	2	no	—
122	C₄×C₁₁₃	—	2	4	no	—
123	C₄×C₁₁₃	—	4	2	no	—
124	C₄×C₁₁₃	—	4	2	no	—
125	C₄×C₁₁₃	—	4	2	no	—
126	C₄×C₁₁₃	—	4	2	no	—
127	C₄×C₁₁₃	#431	1	1	yes	characteristic
128	C₄×C₁₁₃	#431	1	1	yes	characteristic
129	C₄×C₁₁₃	#432	1	1	yes	characteristic
130	C₄×C₁₁₃	#432	1	1	yes	characteristic
131	#8731	C₁₁₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
132	C₂×C₁₁₃	—	2	1	no	—
133	C₂×C₁₁₃	—	2	1	no	—
134	C₂×C₁₁₃	—	4	2	no	—
135	C₂×C₁₁₃	#1817	1	1	yes	characteristic
136	C₂×C₁₁₃	#1827	1	2	yes	—
137	C₄×#117	C₂×C₁₁₃	1	1	yes	characteristic
138	#1458	C₂×C₁₁₃	1	2	yes	—
139	#1276	C₂×C₁₁₃	1	2	yes	complemented
140	#1274	C₂×C₁₁₃	1	2	yes	complemented
141	C₁₁₃	#8731	1	1	yes	characteristic, direct product factor, Sylow
142	C₂×#117	C₄×C₁₁₃	1	1	yes	characteristic
143	C₂×#117	C₂²×C₁₁₃	1	1	yes	characteristic
144	C₂²×C₄²	C₄×C₁₁₃	1	1	yes	characteristic
145	#431	C₂²×C₁₁₃	1	4	yes	—
146	C₄×C₄⋊C₄	—	2	2	no	—
147	C₄×C₂²⋊C₄	—	2	2	no	—
148	#376	C₄×C₁₁₃	1	1	yes	characteristic
149	#376	C₂²×C₁₁₃	1	1	yes	characteristic
150	C₄³	C₄×C₁₁₃	1	1	yes	characteristic
151	C₄³	C₂²×C₁₁₃	1	1	yes	characteristic
152	#324	—	2	2	no	—
153	#323	—	2	2	no	—
154	C₂³×C₄	—	2	1	no	—
155	C₂³×C₄	C₂×C₄×C₁₁₃	1	1	yes	characteristic
156	#117	—	2	2	no	—
157	#117	—	2	2	no	—
158	#117	—	2	2	no	—
159	#117	C₈×C₁₁₃	1	4	yes	complemented
160	C₂×C₄⋊C₄	—	2	1	no	—
161	C₂×C₄⋊C₄	D₄×C₁₁₃	1	2	yes	—
162	C₂×C₂²⋊C₄	—	2	1	no	—
163	C₂×C₂²⋊C₄	D₄×C₁₁₃	1	2	yes	—
164	C₂×C₄²	—	2	1	no	—
165	C₂×C₄²	—	2	2	no	—
166	C₂×C₄²	—	2	2	no	—
167	C₂×C₄²	—	2	2	no	—
168	C₂×C₄²	—	2	2	no	—
169	C₂×C₄²	C₂×C₄×C₁₁₃	1	1	yes	characteristic
170	C₂×C₄²	C₂×C₄×C₁₁₃	1	1	yes	characteristic
171	C₂×C₄²	C₂×C₄×C₁₁₃	1	1	yes	characteristic
172	C₂×C₄²	C₂×C₄×C₁₁₃	1	1	yes	characteristic
173	C₂×C₄²	C₂×C₄×C₁₁₃	1	1	yes	characteristic
174	C₂×C₄²	C₂³×C₁₁₃	1	1	yes	characteristic
175	C₄⋊C₈	—	2	4	no	—
176	C₂²⋊C₈	—	2	4	no	—
177	C₄×C₈	—	2	4	no	—
178	#95	—	2	2	no	—
179	#95	—	2	2	no	—
180	C₂⁴	C₂²⋊C₄×C₁₁₃	1	1	yes	characteristic
181	C₂²×C₄	—	2	1	no	—
182	C₂²×C₄	—	2	1	no	—
183	C₂²×C₄	—	2	1	no	—
184	C₂²×C₄	—	2	1	no	—
185	C₂²×C₄	—	2	1	no	—
186	C₂²×C₄	—	2	4	no	—
187	C₂²×C₄	—	2	1	no	—
188	C₂²×C₄	—	2	1	no	—
189	C₂²×C₄	—	2	1	no	—
190	C₂²×C₄	—	4	4	no	—
191	C₂²×C₄	C₂²⋊C₄×C₁₁₃	1	1	yes	characteristic
192	C₂²×C₄	C₂²⋊C₄×C₁₁₃	1	2	yes	—
193	C₂²×C₄	C₂×C₈×C₁₁₃	1	2	yes	—
194	C₂²×C₄	M₁₆×C₁₁₃	1	2	yes	—
195	C₂²×C₄	C₂²×C₄×C₁₁₃	1	1	yes	characteristic
196	C₂²×C₄	C₂×D₄×C₁₁₃	1	2	yes	—
197	C₂×C₈	—	4	4	no	—
198	C₄⋊C₄	—	2	4	no	—
199	C₄⋊C₄	—	2	4	no	—
200	C₂²⋊C₄	—	2	4	no	—
201	C₂²⋊C₄	—	2	4	no	—
202	C₄²	—	2	4	no	—
203	C₄²	—	2	1	no	—
204	C₄²	—	2	1	no	—
205	C₄²	—	2	1	no	—
206	C₄²	—	2	1	no	—
207	C₄²	—	2	1	no	—
208	C₄²	—	2	1	no	—
209	C₄²	—	4	4	no	—
210	C₄²	—	4	4	no	—
211	C₄²	C₂×C₈×C₁₁₃	1	2	yes	—
212	C₄²	C₂×C₈×C₁₁₃	1	2	yes	—
213	C₄²	M₁₆×C₁₁₃	1	2	yes	—
214	C₄²	M₁₆×C₁₁₃	1	2	yes	—
215	C₄²	#41×C₁₁₃	1	2	yes	—
216	C₄²	#41×C₁₁₃	1	2	yes	—
217	C₂³	—	2	1	no	—
218	C₂³	—	2	1	no	—
219	C₂³	—	4	2	no	—
220	C₂³	C₂²⋊C₈×C₁₁₃	1	2	yes	—
221	C₂³	C₂×C₂²⋊C₄×C₁₁₃	1	1	yes	characteristic
222	C₂×C₄	—	2	4	no	—
223	C₂×C₄	—	2	1	no	—
224	C₂×C₄	—	2	1	no	—
225	C₂×C₄	—	2	2	no	—
226	C₂×C₄	—	2	2	no	—
227	C₂×C₄	—	2	2	no	—
228	C₂×C₄	—	2	2	no	—
229	C₂×C₄	—	2	2	no	—
230	C₂×C₄	—	2	2	no	—
231	C₂×C₄	—	2	2	no	—
232	C₂×C₄	—	2	2	no	—
233	C₂×C₄	—	2	2	no	—
234	C₂×C₄	—	4	2	no	—
235	C₂×C₄	—	4	8	no	—
236	C₂×C₄	—	4	2	no	—
237	C₂×C₄	—	4	2	no	—
238	C₂×C₄	—	4	2	no	—
239	C₂×C₄	—	4	2	no	—
240	C₂×C₄	C₂²⋊C₈×C₁₁₃	1	2	yes	—
241	C₂×C₄	#117×C₁₁₃	1	1	yes	characteristic
242	C₂×C₄	#117×C₁₁₃	1	1	yes	characteristic
243	C₂×C₄	#123×C₁₁₃	1	4	yes	—
244	C₂×C₄	C₂²×C₈×C₁₁₃	1	1	yes	characteristic
245	C₂×C₄	C₂×M₁₆×C₁₁₃	1	1	yes	characteristic
246	C₂×C₄	C₂×M₁₆×C₁₁₃	1	1	yes	characteristic
247	C₂×C₄	C₂×M₁₆×C₁₁₃	1	1	yes	characteristic
248	C₈	—	8	4	no	—
249	C₂²	—	2	2	no	—
250	C₂²	—	2	2	no	—
251	C₂²	—	4	2	no	—
252	C₂²	—	4	2	no	—
253	C₂²	—	4	2	no	—
254	C₂²	#381×C₁₁₃	1	1	yes	characteristic
255	C₂²	C₂×C₂²⋊C₈×C₁₁₃	1	1	yes	characteristic
256	C₂²	#407×C₁₁₃	1	1	yes	characteristic
257	C₄	—	2	4	no	—
258	C₄	—	2	2	no	—
259	C₄	—	2	4	no	—
260	C₄	—	4	2	no	—
261	C₄	—	4	2	no	—
262	C₄	—	4	2	no	—
263	C₄	—	4	2	no	—
264	C₄	#431×C₁₁₃	1	1	yes	characteristic
265	C₄	#431×C₁₁₃	1	1	yes	characteristic
266	C₄	#432×C₁₁₃	1	1	yes	characteristic
267	C₄	#432×C₁₁₃	1	1	yes	characteristic
268	C₂	—	2	1	no	—
269	C₂	—	2	1	no	—
270	C₂	—	4	2	no	—
271	C₂	C₁₁₃×#1817	1	1	yes	characteristic
272	C₂	C₁₁₃×#1827	1	2	yes	—

	subgroup	G/H
center	C₂×C₄×C₁₁₃	#117
commutator	C₂×C₄	C₂²×C₈×C₁₁₃
Frattini, cosocle	C₂×C₄²	C₂³×C₁₁₃
socle	C₂²×C₁₁₃	#381
prime radical	C₂⁴×C₁₁₃	C₂²⋊C₄
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₁₁₃ ◁ #376×C₁₁₃ ◁ G

Complements

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

N	H ≅ G/N	type
3. C₁₁₃×#1276	C₂	semidirect
4. C₁₁₃×#1274	C₂	semidirect
22. #117×C₁₁₃	C₈	semidirect
131. #8731	C₁₁₃	direct (N × H)
139. #1276	C₂×C₁₁₃	semidirect
140. #1274	C₂×C₁₁₃	semidirect
141. C₁₁₃	#8731	direct (N × H)
159. #117	C₈×C₁₁₃	semidirect

Maximal subgroups

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

subgroup	index
C₄×#117×C₁₁₃	2
C₁₁₃×#1458	2
C₁₁₃×#1276	2
C₁₁₃×#1274	2
#8731	113

Subgroups — C113×#8731 #43276751

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₁₃×#8731 #43276751