Subgroups — C₁₁₃×#17667 #43285687

Order 28928.

962 conjugacy classes of subgroups.

The subgroup lattice has 290 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×#125×C₁₁₃	C₂	1	1	yes	maximal, characteristic
2	C₁₁₃×#2920	C₂	1	2	yes	maximal
3	C₁₁₃×#1534	C₂	1	4	yes	maximal, complemented
4	C₁₁₃×#1528	C₂	1	8	yes	maximal, complemented
5	C₂×#125×C₁₁₃	—	2	2	no	—
6	C₂×#125×C₁₁₃	—	2	2	no	—
7	C₂×#125×C₁₁₃	—	2	2	no	—
8	C₂×#125×C₁₁₃	C₄	1	4	yes	—
9	C₂×#125×C₁₁₃	C₄	1	2	yes	—
10	C₂×#125×C₁₁₃	C₂²	1	4	yes	—
11	C₂×#125×C₁₁₃	C₂²	1	2	yes	—
12	C₂²×C₄⋊C₄×C₁₁₃	—	2	2	no	—
13	C₂²×C₄⋊C₄×C₁₁₃	C₄	1	2	yes	—
14	C₂²×C₄²×C₁₁₃	C₂²	1	1	yes	characteristic
15	#432×C₁₁₃	C₂²	1	4	yes	—
16	#432×C₁₁₃	C₂²	1	4	yes	—
17	#431×C₁₁₃	—	2	2	no	—
18	#431×C₁₁₃	—	2	2	no	—
19	#423×C₁₁₃	—	2	2	no	—
20	#407×C₁₁₃	—	2	2	no	—
21	C₄×M₁₆×C₁₁₃	C₂²	1	4	yes	—
22	#330×C₁₁₃	C₂²	1	16	yes	complemented
23	C₂³×C₄×C₁₁₃	—	2	2	no	—
24	C₂³×C₄×C₁₁₃	C₂×C₄	1	1	yes	characteristic
25	C₂³×C₄×C₁₁₃	D₄	1	2	yes	—
26	C₂×M₁₆×C₁₁₃	—	2	4	no	—
27	#125×C₁₁₃	—	2	2	no	—
28	#125×C₁₁₃	—	2	2	no	—
29	#125×C₁₁₃	—	2	4	no	—
30	#125×C₁₁₃	—	2	4	no	—
31	#125×C₁₁₃	—	2	4	no	—
32	#125×C₁₁₃	—	2	4	no	—
33	#125×C₁₁₃	—	2	4	no	—
34	#125×C₁₁₃	C₂×C₄	1	8	yes	—
35	#125×C₁₁₃	C₂×C₄	1	4	yes	—
36	#125×C₁₁₃	C₂³	1	4	yes	—
37	C₂×C₄⋊C₄×C₁₁₃	—	2	4	no	—
38	C₂×C₄⋊C₄×C₁₁₃	—	2	4	no	—
39	C₂×C₄⋊C₄×C₁₁₃	—	2	4	no	—
40	C₂×C₄⋊C₄×C₁₁₃	—	2	2	no	—
41	C₂×C₄⋊C₄×C₁₁₃	—	2	2	no	—
42	C₂×C₄⋊C₄×C₁₁₃	—	2	2	no	—
43	C₂×C₄⋊C₄×C₁₁₃	—	2	4	no	—
44	C₂×C₄⋊C₄×C₁₁₃	—	2	4	no	—
45	C₂×C₄⋊C₄×C₁₁₃	—	2	4	no	—
46	C₂×C₄⋊C₄×C₁₁₃	C₂×C₄	1	4	yes	—
47	C₂×C₄⋊C₄×C₁₁₃	C₂×C₄	1	4	yes	—
48	C₂×C₄⋊C₄×C₁₁₃	C₂×C₄	1	4	yes	—
49	C₂×C₄²×C₁₁₃	C₂×C₄	1	2	yes	—
50	C₂×C₄²×C₁₁₃	C₂×C₄	1	1	yes	characteristic
51	C₂×C₄²×C₁₁₃	D₄	1	2	yes	—
52	C₂×C₄²×C₁₁₃	D₄	1	2	yes	—
53	C₂×C₄²×C₁₁₃	D₄	1	2	yes	—
54	C₂×C₄²×C₁₁₃	C₂³	1	2	yes	—
55	C₂×C₄²×C₁₁₃	C₂³	1	1	yes	characteristic
56	C₄⋊C₈×C₁₁₃	—	2	4	no	—
57	C₄⋊C₈×C₁₁₃	—	2	4	no	—
58	C₂²⋊C₈×C₁₁₃	—	2	4	no	—
59	C₂²⋊C₈×C₁₁₃	—	2	4	no	—
60	#97×C₁₁₃	C₂³	1	8	yes	—
61	C₄×C₈×C₁₁₃	—	2	4	no	—
62	C₂⁴×C₁₁₃	C₂²⋊C₄	1	1	yes	characteristic
63	C₂²×C₄×C₁₁₃	—	2	4	no	—
64	C₂²×C₄×C₁₁₃	—	2	4	no	—
65	C₂²×C₄×C₁₁₃	—	2	4	no	—
66	C₂²×C₄×C₁₁₃	—	2	4	no	—
67	C₂²×C₄×C₁₁₃	—	4	4	no	—
68	C₂²×C₄×C₁₁₃	—	4	8	no	—
69	C₂²×C₄×C₁₁₃	C₂²⋊C₄	1	2	yes	—
70	C₂²×C₄×C₁₁₃	C₂²⋊C₄	1	2	yes	—
71	C₂²×C₄×C₁₁₃	C₂²⋊C₄	1	2	yes	—
72	C₂²×C₄×C₁₁₃	C₂²⋊C₄	1	2	yes	—
73	C₂²×C₄×C₁₁₃	C₂²⋊C₄	1	1	yes	characteristic
74	C₂²×C₄×C₁₁₃	C₂²×C₄	1	1	yes	characteristic
75	C₂²×C₄×C₁₁₃	C₂²×C₄	1	2	yes	—
76	C₂²×C₄×C₁₁₃	C₂×D₄	1	2	yes	—
77	C₂²×C₄×C₁₁₃	C₂×D₄	1	2	yes	—
78	C₂²×C₄×C₁₁₃	C₂×D₄	1	2	yes	—
79	M₁₆×C₁₁₃	—	4	8	no	—
80	C₂×C₈×C₁₁₃	—	2	8	no	—
81	C₄⋊C₄×C₁₁₃	—	2	8	no	—
82	C₄⋊C₄×C₁₁₃	—	2	8	no	—
83	C₄⋊C₄×C₁₁₃	—	2	8	no	—
84	C₄⋊C₄×C₁₁₃	—	2	4	no	—
85	C₄⋊C₄×C₁₁₃	—	2	4	no	—
86	C₄⋊C₄×C₁₁₃	—	2	4	no	—
87	C₄⋊C₄×C₁₁₃	—	2	8	no	—
88	C₄⋊C₄×C₁₁₃	C₂²×C₄	1	8	yes	—
89	C₄²×C₁₁₃	C₂²⋊C₄	1	1	yes	characteristic
90	C₄²×C₁₁₃	C₂²⋊C₄	1	1	yes	characteristic
91	C₄²×C₁₁₃	C₂²⋊C₄	1	2	yes	—
92	C₄²×C₁₁₃	C₂²⋊C₄	1	2	yes	—
93	C₄²×C₁₁₃	C₂²×C₄	1	2	yes	—
94	C₄²×C₁₁₃	C₂²×C₄	1	1	yes	characteristic
95	C₄²×C₁₁₃	C₂×D₄	1	2	yes	—
96	C₄²×C₁₁₃	C₂×D₄	1	2	yes	—
97	C₄²×C₁₁₃	C₂×D₄	1	2	yes	—
98	C₄²×C₁₁₃	C₂⁴	1	1	yes	characteristic
99	C₂³×C₁₁₃	—	2	2	no	—
100	C₂³×C₁₁₃	—	2	4	no	—
101	C₂³×C₁₁₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
102	C₂³×C₁₁₃	C₂×C₂²⋊C₄	1	2	yes	—
103	C₂×C₄×C₁₁₃	—	2	2	no	—
104	C₂×C₄×C₁₁₃	—	2	4	no	—
105	C₂×C₄×C₁₁₃	—	2	4	no	—
106	C₂×C₄×C₁₁₃	—	2	4	no	—
107	C₂×C₄×C₁₁₃	—	2	4	no	—
108	C₂×C₄×C₁₁₃	—	2	8	no	—
109	C₂×C₄×C₁₁₃	—	4	2	no	—
110	C₂×C₄×C₁₁₃	—	4	2	no	—
111	C₂×C₄×C₁₁₃	—	4	2	no	—
112	C₂×C₄×C₁₁₃	—	4	2	no	—
113	C₂×C₄×C₁₁₃	—	4	8	no	—
114	C₂×C₄×C₁₁₃	—	4	8	no	—
115	C₂×C₄×C₁₁₃	—	4	8	no	—
116	C₂×C₄×C₁₁₃	#101	1	4	yes	—
117	C₂×C₄×C₁₁₃	C₂×C₂²⋊C₄	1	2	yes	—
118	C₂×C₄×C₁₁₃	C₂×C₂²⋊C₄	1	2	yes	—
119	C₂×C₄×C₁₁₃	C₂×C₂²⋊C₄	1	2	yes	—
120	C₂×C₄×C₁₁₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
121	C₂×C₄×C₁₁₃	C₂×C₂²⋊C₄	1	2	yes	—
122	C₂×C₄×C₁₁₃	C₂³×C₄	1	1	yes	characteristic
123	C₂×C₄×C₁₁₃	C₂²×D₄	1	2	yes	—
124	C₈×C₁₁₃	—	4	8	no	—
125	C₂²×C₁₁₃	—	2	8	no	—
126	C₂²×C₁₁₃	—	2	2	no	—
127	C₂²×C₁₁₃	—	2	2	no	—
128	C₂²×C₁₁₃	—	2	2	no	—
129	C₂²×C₁₁₃	C₂×#101	1	2	yes	—
130	C₂²×C₁₁₃	#421	1	4	yes	—
131	C₂²×C₁₁₃	C₂²×C₂²⋊C₄	1	1	yes	characteristic
132	C₄×C₁₁₃	—	2	1	no	—
133	C₄×C₁₁₃	—	2	1	no	—
134	C₄×C₁₁₃	—	2	2	no	—
135	C₄×C₁₁₃	—	2	2	no	—
136	C₄×C₁₁₃	—	2	2	no	—
137	C₄×C₁₁₃	—	2	2	no	—
138	C₄×C₁₁₃	—	4	8	no	—
139	C₄×C₁₁₃	C₂×#101	1	4	yes	—
140	#17667	C₁₁₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
141	C₂×C₁₁₃	—	2	2	no	—
142	C₂×C₁₁₃	—	2	4	no	—
143	C₂×C₁₁₃	C₂²×#101	1	1	yes	characteristic
144	C₂×C₁₁₃	C₂×#421	1	2	yes	—
145	C₂²×#125	C₂×C₁₁₃	1	1	yes	characteristic
146	#2920	C₂×C₁₁₃	1	2	yes	—
147	#1534	C₂×C₁₁₃	1	4	yes	complemented
148	#1528	C₂×C₁₁₃	1	8	yes	complemented
149	C₁₁₃	#17667	1	1	yes	characteristic, direct product factor, Sylow
150	C₂×#125	—	2	2	no	—
151	C₂×#125	—	2	2	no	—
152	C₂×#125	—	2	2	no	—
153	C₂×#125	C₄×C₁₁₃	1	4	yes	—
154	C₂×#125	C₄×C₁₁₃	1	2	yes	—
155	C₂×#125	C₂²×C₁₁₃	1	4	yes	—
156	C₂×#125	C₂²×C₁₁₃	1	2	yes	—
157	C₂²×C₄⋊C₄	—	2	2	no	—
158	C₂²×C₄⋊C₄	C₄×C₁₁₃	1	2	yes	—
159	C₂²×C₄²	C₂²×C₁₁₃	1	1	yes	characteristic
160	#432	C₂²×C₁₁₃	1	4	yes	—
161	#432	C₂²×C₁₁₃	1	4	yes	—
162	#431	—	2	2	no	—
163	#431	—	2	2	no	—
164	#423	—	2	2	no	—
165	#407	—	2	2	no	—
166	C₄×M₁₆	C₂²×C₁₁₃	1	4	yes	—
167	#330	C₂²×C₁₁₃	1	16	yes	complemented
168	C₂³×C₄	—	2	2	no	—
169	C₂³×C₄	C₂×C₄×C₁₁₃	1	1	yes	characteristic
170	C₂³×C₄	D₄×C₁₁₃	1	2	yes	—
171	C₂×M₁₆	—	2	4	no	—
172	#125	—	2	2	no	—
173	#125	—	2	2	no	—
174	#125	—	2	4	no	—
175	#125	—	2	4	no	—
176	#125	—	2	4	no	—
177	#125	—	2	4	no	—
178	#125	—	2	4	no	—
179	#125	C₂×C₄×C₁₁₃	1	8	yes	—
180	#125	C₂×C₄×C₁₁₃	1	4	yes	—
181	#125	C₂³×C₁₁₃	1	4	yes	—
182	C₂×C₄⋊C₄	—	2	4	no	—
183	C₂×C₄⋊C₄	—	2	4	no	—
184	C₂×C₄⋊C₄	—	2	4	no	—
185	C₂×C₄⋊C₄	—	2	2	no	—
186	C₂×C₄⋊C₄	—	2	2	no	—
187	C₂×C₄⋊C₄	—	2	2	no	—
188	C₂×C₄⋊C₄	—	2	4	no	—
189	C₂×C₄⋊C₄	—	2	4	no	—
190	C₂×C₄⋊C₄	—	2	4	no	—
191	C₂×C₄⋊C₄	C₂×C₄×C₁₁₃	1	4	yes	—
192	C₂×C₄⋊C₄	C₂×C₄×C₁₁₃	1	4	yes	—
193	C₂×C₄⋊C₄	C₂×C₄×C₁₁₃	1	4	yes	—
194	C₂×C₄²	C₂×C₄×C₁₁₃	1	2	yes	—
195	C₂×C₄²	C₂×C₄×C₁₁₃	1	1	yes	characteristic
196	C₂×C₄²	D₄×C₁₁₃	1	2	yes	—
197	C₂×C₄²	D₄×C₁₁₃	1	2	yes	—
198	C₂×C₄²	D₄×C₁₁₃	1	2	yes	—
199	C₂×C₄²	C₂³×C₁₁₃	1	2	yes	—
200	C₂×C₄²	C₂³×C₁₁₃	1	1	yes	characteristic
201	C₄⋊C₈	—	2	4	no	—
202	C₄⋊C₈	—	2	4	no	—
203	C₂²⋊C₈	—	2	4	no	—
204	C₂²⋊C₈	—	2	4	no	—
205	#97	C₂³×C₁₁₃	1	8	yes	—
206	C₄×C₈	—	2	4	no	—
207	C₂⁴	C₂²⋊C₄×C₁₁₃	1	1	yes	characteristic
208	C₂²×C₄	—	2	4	no	—
209	C₂²×C₄	—	2	4	no	—
210	C₂²×C₄	—	2	4	no	—
211	C₂²×C₄	—	2	4	no	—
212	C₂²×C₄	—	4	4	no	—
213	C₂²×C₄	—	4	8	no	—
214	C₂²×C₄	C₂²⋊C₄×C₁₁₃	1	2	yes	—
215	C₂²×C₄	C₂²⋊C₄×C₁₁₃	1	2	yes	—
216	C₂²×C₄	C₂²⋊C₄×C₁₁₃	1	2	yes	—
217	C₂²×C₄	C₂²⋊C₄×C₁₁₃	1	2	yes	—
218	C₂²×C₄	C₂²⋊C₄×C₁₁₃	1	1	yes	characteristic
219	C₂²×C₄	C₂²×C₄×C₁₁₃	1	1	yes	characteristic
220	C₂²×C₄	C₂²×C₄×C₁₁₃	1	2	yes	—
221	C₂²×C₄	C₂×D₄×C₁₁₃	1	2	yes	—
222	C₂²×C₄	C₂×D₄×C₁₁₃	1	2	yes	—
223	C₂²×C₄	C₂×D₄×C₁₁₃	1	2	yes	—
224	M₁₆	—	4	8	no	—
225	C₂×C₈	—	2	8	no	—
226	C₄⋊C₄	—	2	8	no	—
227	C₄⋊C₄	—	2	8	no	—
228	C₄⋊C₄	—	2	8	no	—
229	C₄⋊C₄	—	2	4	no	—
230	C₄⋊C₄	—	2	4	no	—
231	C₄⋊C₄	—	2	4	no	—
232	C₄⋊C₄	—	2	8	no	—
233	C₄⋊C₄	C₂²×C₄×C₁₁₃	1	8	yes	—
234	C₄²	C₂²⋊C₄×C₁₁₃	1	1	yes	characteristic
235	C₄²	C₂²⋊C₄×C₁₁₃	1	1	yes	characteristic
236	C₄²	C₂²⋊C₄×C₁₁₃	1	2	yes	—
237	C₄²	C₂²⋊C₄×C₁₁₃	1	2	yes	—
238	C₄²	C₂²×C₄×C₁₁₃	1	2	yes	—
239	C₄²	C₂²×C₄×C₁₁₃	1	1	yes	characteristic
240	C₄²	C₂×D₄×C₁₁₃	1	2	yes	—
241	C₄²	C₂×D₄×C₁₁₃	1	2	yes	—
242	C₄²	C₂×D₄×C₁₁₃	1	2	yes	—
243	C₄²	C₂⁴×C₁₁₃	1	1	yes	characteristic
244	C₂³	—	2	2	no	—
245	C₂³	—	2	4	no	—
246	C₂³	C₂×C₂²⋊C₄×C₁₁₃	1	1	yes	characteristic
247	C₂³	C₂×C₂²⋊C₄×C₁₁₃	1	2	yes	—
248	C₂×C₄	—	2	2	no	—
249	C₂×C₄	—	2	4	no	—
250	C₂×C₄	—	2	4	no	—
251	C₂×C₄	—	2	4	no	—
252	C₂×C₄	—	2	4	no	—
253	C₂×C₄	—	2	8	no	—
254	C₂×C₄	—	4	2	no	—
255	C₂×C₄	—	4	2	no	—
256	C₂×C₄	—	4	2	no	—
257	C₂×C₄	—	4	2	no	—
258	C₂×C₄	—	4	8	no	—
259	C₂×C₄	—	4	8	no	—
260	C₂×C₄	—	4	8	no	—
261	C₂×C₄	#101×C₁₁₃	1	4	yes	—
262	C₂×C₄	C₂×C₂²⋊C₄×C₁₁₃	1	2	yes	—
263	C₂×C₄	C₂×C₂²⋊C₄×C₁₁₃	1	2	yes	—
264	C₂×C₄	C₂×C₂²⋊C₄×C₁₁₃	1	2	yes	—
265	C₂×C₄	C₂×C₂²⋊C₄×C₁₁₃	1	1	yes	characteristic
266	C₂×C₄	C₂×C₂²⋊C₄×C₁₁₃	1	2	yes	—
267	C₂×C₄	C₂³×C₄×C₁₁₃	1	1	yes	characteristic
268	C₂×C₄	C₂²×D₄×C₁₁₃	1	2	yes	—
269	C₈	—	4	8	no	—
270	C₂²	—	2	8	no	—
271	C₂²	—	2	2	no	—
272	C₂²	—	2	2	no	—
273	C₂²	—	2	2	no	—
274	C₂²	C₂×#101×C₁₁₃	1	2	yes	—
275	C₂²	#421×C₁₁₃	1	4	yes	—
276	C₂²	C₂²×C₂²⋊C₄×C₁₁₃	1	1	yes	characteristic
277	C₄	—	2	1	no	—
278	C₄	—	2	1	no	—
279	C₄	—	2	2	no	—
280	C₄	—	2	2	no	—
281	C₄	—	2	2	no	—
282	C₄	—	2	2	no	—
283	C₄	—	4	8	no	—
284	C₄	C₂×#101×C₁₁₃	1	4	yes	—
285	C₂	—	2	2	no	—
286	C₂	—	2	4	no	—
287	C₂	C₂²×#101×C₁₁₃	1	1	yes	characteristic
288	C₂	C₂×#421×C₁₁₃	1	2	yes	—

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

Complements

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

N	H ≅ G/N	type
3. C₁₁₃×#1534	C₂	semidirect
4. C₁₁₃×#1528	C₂	semidirect
22. #330×C₁₁₃	C₂²	semidirect
140. #17667	C₁₁₃	direct (N × H)
147. #1534	C₂×C₁₁₃	semidirect
148. #1528	C₂×C₁₁₃	semidirect
149. C₁₁₃	#17667	direct (N × H)
167. #330	C₂²×C₁₁₃	semidirect

Maximal subgroups

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

subgroup	index
C₂²×#125×C₁₁₃	2
C₁₁₃×#2920	2
C₁₁₃×#1534	2
C₁₁₃×#1528	2
#17667	113

Subgroups — C113×#17667 #43285687

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₁₃×#17667 #43285687