Subgroups — C₁₁₃×#8302 #43276322

Order 28928.

1482 conjugacy classes of subgroups.

The subgroup lattice has 310 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₄×D₄×C₁₁₃	C₂	1	1	yes	maximal, characteristic
2	C₁₁₃×#1817	C₂	1	2	yes	maximal, complemented
3	C₂×#323×C₁₁₃	C₂	1	4	yes	maximal, complemented
4	C₂³×D₄×C₁₁₃	C₄	1	1	yes	characteristic
5	C₂⁴×C₄×C₁₁₃	C₄	1	2	yes	—
6	C₂×C₄×D₄×C₁₁₃	—	2	8	no	—
7	C₂×C₄×D₄×C₁₁₃	—	2	4	no	—
8	C₂²×C₄⋊C₄×C₁₁₃	C₄	1	1	yes	characteristic
9	C₂²×C₂²⋊C₄×C₁₁₃	C₂²	1	2	yes	—
10	C₂²×C₄²×C₁₁₃	C₂²	1	1	yes	characteristic
11	C₂×C₂²⋊C₈×C₁₁₃	C₂²	1	4	yes	—
12	#336×C₁₁₃	—	2	4	no	—
13	#323×C₁₁₃	—	2	8	no	—
14	C₂⁵×C₁₁₃	C₂×C₄	1	2	yes	—
15	C₂²×D₄×C₁₁₃	—	2	4	no	—
16	C₂²×D₄×C₁₁₃	—	2	4	no	—
17	C₂²×D₄×C₁₁₃	—	2	2	no	—
18	C₂²×D₄×C₁₁₃	—	2	2	no	—
19	C₂²×D₄×C₁₁₃	C₈	1	4	yes	complemented
20	C₂³×C₄×C₁₁₃	—	2	4	no	—
21	C₂³×C₄×C₁₁₃	—	2	4	no	—
22	C₂³×C₄×C₁₁₃	—	2	4	no	—
23	C₂³×C₄×C₁₁₃	—	4	8	no	—
24	C₂³×C₄×C₁₁₃	C₂×C₄	1	1	yes	characteristic
25	C₂³×C₄×C₁₁₃	C₂×C₄	1	1	yes	characteristic
26	C₂³×C₄×C₁₁₃	C₂×C₄	1	2	yes	—
27	C₂³×C₄×C₁₁₃	C₂³	1	1	yes	characteristic
28	C₂²×C₈×C₁₁₃	—	2	4	no	—
29	C₄×D₄×C₁₁₃	—	2	32	no	—
30	C₂×C₄⋊C₄×C₁₁₃	—	2	4	no	—
31	C₂×C₄⋊C₄×C₁₁₃	—	2	2	no	—
32	C₂×C₂²⋊C₄×C₁₁₃	—	2	4	no	—
33	C₂×C₂²⋊C₄×C₁₁₃	—	2	4	no	—
34	C₂×C₂²⋊C₄×C₁₁₃	D₄	1	8	yes	—
35	C₂×C₄²×C₁₁₃	—	2	4	no	—
36	C₂×C₄²×C₁₁₃	D₄	1	4	yes	—
37	C₂²⋊C₈×C₁₁₃	—	2	8	no	—
38	C₂⁴×C₁₁₃	—	2	2	no	—
39	C₂⁴×C₁₁₃	—	2	2	no	—
40	C₂⁴×C₁₁₃	—	2	2	no	—
41	C₂⁴×C₁₁₃	—	2	2	no	—
42	C₂⁴×C₁₁₃	—	4	4	no	—
43	C₂⁴×C₁₁₃	—	4	4	no	—
44	C₂⁴×C₁₁₃	C₂²⋊C₄	1	4	yes	—
45	C₂⁴×C₁₁₃	C₂×C₈	1	4	yes	—
46	C₂⁴×C₁₁₃	M₁₆	1	4	yes	—
47	C₂⁴×C₁₁₃	C₂²×C₄	1	1	yes	characteristic
48	C₂×D₄×C₁₁₃	—	2	8	no	—
49	C₂×D₄×C₁₁₃	—	2	8	no	—
50	C₂×D₄×C₁₁₃	—	2	16	no	—
51	C₂×D₄×C₁₁₃	—	2	16	no	—
52	C₂×D₄×C₁₁₃	—	2	8	no	—
53	C₂²×C₄×C₁₁₃	—	2	2	no	—
54	C₂²×C₄×C₁₁₃	—	2	2	no	—
55	C₂²×C₄×C₁₁₃	—	2	2	no	—
56	C₂²×C₄×C₁₁₃	—	2	4	no	—
57	C₂²×C₄×C₁₁₃	—	2	16	no	—
58	C₂²×C₄×C₁₁₃	—	2	2	no	—
59	C₂²×C₄×C₁₁₃	—	2	2	no	—
60	C₂²×C₄×C₁₁₃	—	4	2	no	—
61	C₂²×C₄×C₁₁₃	—	4	16	no	—
62	C₂²×C₄×C₁₁₃	—	4	16	no	—
63	C₂²×C₄×C₁₁₃	—	4	8	no	—
64	C₂²×C₄×C₁₁₃	—	4	8	no	—
65	C₂²×C₄×C₁₁₃	—	4	4	no	—
66	C₂²×C₄×C₁₁₃	C₂²⋊C₄	1	2	yes	—
67	C₂²×C₄×C₁₁₃	C₂²⋊C₄	1	2	yes	—
68	C₂²×C₄×C₁₁₃	C₂²⋊C₄	1	4	yes	—
69	C₂²×C₄×C₁₁₃	C₂×C₈	1	2	yes	—
70	C₂²×C₄×C₁₁₃	M₁₆	1	2	yes	—
71	C₂²×C₄×C₁₁₃	C₂×D₄	1	2	yes	—
72	C₂²×C₄×C₁₁₃	C₂×D₄	1	4	yes	—
73	C₂×C₈×C₁₁₃	—	4	8	no	—
74	C₂×C₈×C₁₁₃	—	4	4	no	—
75	C₄⋊C₄×C₁₁₃	—	2	8	no	—
76	C₂²⋊C₄×C₁₁₃	—	2	16	no	—
77	C₄²×C₁₁₃	—	2	8	no	—
78	C₂³×C₁₁₃	—	2	2	no	—
79	C₂³×C₁₁₃	—	2	2	no	—
80	C₂³×C₁₁₃	—	2	4	no	—
81	C₂³×C₁₁₃	—	2	4	no	—
82	C₂³×C₁₁₃	—	2	4	no	—
83	C₂³×C₁₁₃	—	2	4	no	—
84	C₂³×C₁₁₃	—	2	4	no	—
85	C₂³×C₁₁₃	—	2	4	no	—
86	C₂³×C₁₁₃	—	4	4	no	—
87	C₂³×C₁₁₃	—	4	4	no	—
88	C₂³×C₁₁₃	—	4	8	no	—
89	C₂³×C₁₁₃	—	4	8	no	—
90	C₂³×C₁₁₃	—	4	4	no	—
91	C₂³×C₁₁₃	—	4	8	no	—
92	C₂³×C₁₁₃	—	4	8	no	—
93	C₂³×C₁₁₃	—	4	4	no	—
94	C₂³×C₁₁₃	—	4	4	no	—
95	C₂³×C₁₁₃	—	4	4	no	—
96	C₂³×C₁₁₃	C₂²⋊C₈	1	8	yes	—
97	C₂³×C₁₁₃	C₂×C₂²⋊C₄	1	2	yes	—
98	C₂³×C₁₁₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
99	C₂³×C₁₁₃	C₂²×C₈	1	1	yes	characteristic
100	C₂³×C₁₁₃	C₂×M₁₆	1	2	yes	—
101	C₂³×C₁₁₃	C₂×M₁₆	1	1	yes	characteristic
102	D₄×C₁₁₃	—	2	32	no	—
103	C₂×C₄×C₁₁₃	—	2	8	no	—
104	C₂×C₄×C₁₁₃	—	2	2	no	—
105	C₂×C₄×C₁₁₃	—	2	2	no	—
106	C₂×C₄×C₁₁₃	—	2	4	no	—
107	C₂×C₄×C₁₁₃	—	2	2	no	—
108	C₂×C₄×C₁₁₃	—	2	4	no	—
109	C₂×C₄×C₁₁₃	—	2	4	no	—
110	C₂×C₄×C₁₁₃	—	2	2	no	—
111	C₂×C₄×C₁₁₃	—	2	4	no	—
112	C₂×C₄×C₁₁₃	—	4	4	no	—
113	C₂×C₄×C₁₁₃	—	4	32	no	—
114	C₂×C₄×C₁₁₃	—	4	4	no	—
115	C₂×C₄×C₁₁₃	—	4	4	no	—
116	C₂×C₄×C₁₁₃	—	4	4	no	—
117	C₂×C₄×C₁₁₃	C₂²⋊C₈	1	4	yes	—
118	C₂×C₄×C₁₁₃	#100	1	4	yes	—
119	C₂×C₄×C₁₁₃	#120	1	4	yes	—
120	C₈×C₁₁₃	—	8	4	no	—
121	C₂²×C₁₁₃	—	2	2	no	—
122	C₂²×C₁₁₃	—	2	2	no	—
123	C₂²×C₁₁₃	—	2	4	no	—
124	C₂²×C₁₁₃	—	2	2	no	—
125	C₂²×C₁₁₃	—	2	2	no	—
126	C₂²×C₁₁₃	—	2	8	no	—
127	C₂²×C₁₁₃	—	4	8	no	—
128	C₂²×C₁₁₃	—	4	8	no	—
129	C₂²×C₁₁₃	—	4	8	no	—
130	C₂²×C₁₁₃	—	4	8	no	—
131	C₂²×C₁₁₃	—	4	8	no	—
132	C₂²×C₁₁₃	—	4	8	no	—
133	C₂²×C₁₁₃	—	4	4	no	—
134	C₂²×C₁₁₃	—	4	4	no	—
135	C₂²×C₁₁₃	#379	1	1	yes	characteristic
136	C₂²×C₁₁₃	C₂×C₂²⋊C₈	1	2	yes	—
137	C₂²×C₁₁₃	C₂×C₂²⋊C₈	1	1	yes	characteristic
138	C₂²×C₁₁₃	#407	1	2	yes	—
139	C₂²×C₁₁₃	#407	1	1	yes	characteristic
140	C₂²×C₁₁₃	#410	1	2	yes	—
141	C₂²×C₁₁₃	C₂×#100	1	2	yes	—
142	C₄×C₁₁₃	—	2	4	no	—
143	C₄×C₁₁₃	—	2	4	no	—
144	C₄×C₁₁₃	—	2	4	no	—
145	C₄×C₁₁₃	—	4	4	no	—
146	#8302	C₁₁₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
147	C₂×C₁₁₃	—	2	2	no	—
148	C₂×C₁₁₃	—	2	2	no	—
149	C₂×C₁₁₃	—	4	8	no	—
150	C₂×C₁₁₃	#1460	1	2	yes	—
151	C₂×C₁₁₃	#1466	1	2	yes	—
152	C₂×C₁₁₃	#1779	1	1	yes	characteristic
153	C₂×C₁₁₃	#1782	1	1	yes	characteristic
154	C₂×C₁₁₃	#1783	1	1	yes	characteristic
155	C₂²×C₄×D₄	C₂×C₁₁₃	1	1	yes	characteristic
156	#1817	C₂×C₁₁₃	1	2	yes	complemented
157	C₂×#323	C₂×C₁₁₃	1	4	yes	complemented
158	C₁₁₃	#8302	1	1	yes	characteristic, direct product factor, Sylow
159	C₂³×D₄	C₄×C₁₁₃	1	1	yes	characteristic
160	C₂⁴×C₄	C₄×C₁₁₃	1	2	yes	—
161	C₂×C₄×D₄	—	2	8	no	—
162	C₂×C₄×D₄	—	2	4	no	—
163	C₂²×C₄⋊C₄	C₄×C₁₁₃	1	1	yes	characteristic
164	C₂²×C₂²⋊C₄	C₂²×C₁₁₃	1	2	yes	—
165	C₂²×C₄²	C₂²×C₁₁₃	1	1	yes	characteristic
166	C₂×C₂²⋊C₈	C₂²×C₁₁₃	1	4	yes	—
167	#336	—	2	4	no	—
168	#323	—	2	8	no	—
169	C₂⁵	C₂×C₄×C₁₁₃	1	2	yes	—
170	C₂²×D₄	—	2	4	no	—
171	C₂²×D₄	—	2	4	no	—
172	C₂²×D₄	—	2	2	no	—
173	C₂²×D₄	—	2	2	no	—
174	C₂²×D₄	C₈×C₁₁₃	1	4	yes	complemented
175	C₂³×C₄	—	2	4	no	—
176	C₂³×C₄	—	2	4	no	—
177	C₂³×C₄	—	2	4	no	—
178	C₂³×C₄	—	4	8	no	—
179	C₂³×C₄	C₂×C₄×C₁₁₃	1	1	yes	characteristic
180	C₂³×C₄	C₂×C₄×C₁₁₃	1	1	yes	characteristic
181	C₂³×C₄	C₂×C₄×C₁₁₃	1	2	yes	—
182	C₂³×C₄	C₂³×C₁₁₃	1	1	yes	characteristic
183	C₂²×C₈	—	2	4	no	—
184	C₄×D₄	—	2	32	no	—
185	C₂×C₄⋊C₄	—	2	4	no	—
186	C₂×C₄⋊C₄	—	2	2	no	—
187	C₂×C₂²⋊C₄	—	2	4	no	—
188	C₂×C₂²⋊C₄	—	2	4	no	—
189	C₂×C₂²⋊C₄	D₄×C₁₁₃	1	8	yes	—
190	C₂×C₄²	—	2	4	no	—
191	C₂×C₄²	D₄×C₁₁₃	1	4	yes	—
192	C₂²⋊C₈	—	2	8	no	—
193	C₂⁴	—	2	2	no	—
194	C₂⁴	—	2	2	no	—
195	C₂⁴	—	2	2	no	—
196	C₂⁴	—	2	2	no	—
197	C₂⁴	—	4	4	no	—
198	C₂⁴	—	4	4	no	—
199	C₂⁴	C₂²⋊C₄×C₁₁₃	1	4	yes	—
200	C₂⁴	C₂×C₈×C₁₁₃	1	4	yes	—
201	C₂⁴	M₁₆×C₁₁₃	1	4	yes	—
202	C₂⁴	C₂²×C₄×C₁₁₃	1	1	yes	characteristic
203	C₂×D₄	—	2	8	no	—
204	C₂×D₄	—	2	8	no	—
205	C₂×D₄	—	2	16	no	—
206	C₂×D₄	—	2	16	no	—
207	C₂×D₄	—	2	8	no	—
208	C₂²×C₄	—	2	2	no	—
209	C₂²×C₄	—	2	2	no	—
210	C₂²×C₄	—	2	2	no	—
211	C₂²×C₄	—	2	4	no	—
212	C₂²×C₄	—	2	16	no	—
213	C₂²×C₄	—	2	2	no	—
214	C₂²×C₄	—	2	2	no	—
215	C₂²×C₄	—	4	2	no	—
216	C₂²×C₄	—	4	16	no	—
217	C₂²×C₄	—	4	16	no	—
218	C₂²×C₄	—	4	8	no	—
219	C₂²×C₄	—	4	8	no	—
220	C₂²×C₄	—	4	4	no	—
221	C₂²×C₄	C₂²⋊C₄×C₁₁₃	1	2	yes	—
222	C₂²×C₄	C₂²⋊C₄×C₁₁₃	1	2	yes	—
223	C₂²×C₄	C₂²⋊C₄×C₁₁₃	1	4	yes	—
224	C₂²×C₄	C₂×C₈×C₁₁₃	1	2	yes	—
225	C₂²×C₄	M₁₆×C₁₁₃	1	2	yes	—
226	C₂²×C₄	C₂×D₄×C₁₁₃	1	2	yes	—
227	C₂²×C₄	C₂×D₄×C₁₁₃	1	4	yes	—
228	C₂×C₈	—	4	8	no	—
229	C₂×C₈	—	4	4	no	—
230	C₄⋊C₄	—	2	8	no	—
231	C₂²⋊C₄	—	2	16	no	—
232	C₄²	—	2	8	no	—
233	C₂³	—	2	2	no	—
234	C₂³	—	2	2	no	—
235	C₂³	—	2	4	no	—
236	C₂³	—	2	4	no	—
237	C₂³	—	2	4	no	—
238	C₂³	—	2	4	no	—
239	C₂³	—	2	4	no	—
240	C₂³	—	2	4	no	—
241	C₂³	—	4	4	no	—
242	C₂³	—	4	4	no	—
243	C₂³	—	4	8	no	—
244	C₂³	—	4	8	no	—
245	C₂³	—	4	4	no	—
246	C₂³	—	4	4	no	—
247	C₂³	—	4	8	no	—
248	C₂³	—	4	8	no	—
249	C₂³	—	4	4	no	—
250	C₂³	—	4	4	no	—
251	C₂³	C₂²⋊C₈×C₁₁₃	1	8	yes	—
252	C₂³	C₂×C₂²⋊C₄×C₁₁₃	1	2	yes	—
253	C₂³	C₂×C₂²⋊C₄×C₁₁₃	1	1	yes	characteristic
254	C₂³	C₂²×C₈×C₁₁₃	1	1	yes	characteristic
255	C₂³	C₂×M₁₆×C₁₁₃	1	2	yes	—
256	C₂³	C₂×M₁₆×C₁₁₃	1	1	yes	characteristic
257	D₄	—	2	32	no	—
258	C₂×C₄	—	2	8	no	—
259	C₂×C₄	—	2	2	no	—
260	C₂×C₄	—	2	2	no	—
261	C₂×C₄	—	2	4	no	—
262	C₂×C₄	—	2	2	no	—
263	C₂×C₄	—	2	4	no	—
264	C₂×C₄	—	2	4	no	—
265	C₂×C₄	—	2	2	no	—
266	C₂×C₄	—	2	4	no	—
267	C₂×C₄	—	4	4	no	—
268	C₂×C₄	—	4	32	no	—
269	C₂×C₄	—	4	4	no	—
270	C₂×C₄	—	4	4	no	—
271	C₂×C₄	—	4	4	no	—
272	C₂×C₄	C₂²⋊C₈×C₁₁₃	1	4	yes	—
273	C₂×C₄	#100×C₁₁₃	1	4	yes	—
274	C₂×C₄	#120×C₁₁₃	1	4	yes	—
275	C₈	—	8	4	no	—
276	C₂²	—	2	2	no	—
277	C₂²	—	2	2	no	—
278	C₂²	—	2	2	no	—
279	C₂²	—	2	2	no	—
280	C₂²	—	2	4	no	—
281	C₂²	—	2	8	no	—
282	C₂²	—	4	8	no	—
283	C₂²	—	4	8	no	—
284	C₂²	—	4	8	no	—
285	C₂²	—	4	8	no	—
286	C₂²	—	4	4	no	—
287	C₂²	—	4	4	no	—
288	C₂²	—	4	8	no	—
289	C₂²	—	4	8	no	—
290	C₂²	#379×C₁₁₃	1	1	yes	characteristic
291	C₂²	C₂×C₂²⋊C₈×C₁₁₃	1	2	yes	—
292	C₂²	C₂×C₂²⋊C₈×C₁₁₃	1	1	yes	characteristic
293	C₂²	#407×C₁₁₃	1	2	yes	—
294	C₂²	#407×C₁₁₃	1	1	yes	characteristic
295	C₂²	#410×C₁₁₃	1	2	yes	—
296	C₂²	C₂×#100×C₁₁₃	1	2	yes	—
297	C₄	—	2	4	no	—
298	C₄	—	2	4	no	—
299	C₄	—	2	4	no	—
300	C₄	—	4	4	no	—
301	C₂	—	2	2	no	—
302	C₂	—	2	2	no	—
303	C₂	—	4	8	no	—
304	C₂	C₁₁₃×#1460	1	2	yes	—
305	C₂	C₁₁₃×#1466	1	2	yes	—
306	C₂	C₁₁₃×#1779	1	1	yes	characteristic
307	C₂	C₁₁₃×#1782	1	1	yes	characteristic
308	C₂	C₁₁₃×#1783	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂³×C₁₁₃	C₂×C₂²⋊C₄
commutator	C₂³	C₂²×C₈×C₁₁₃
Frattini, cosocle	C₂³×C₄	C₂³×C₁₁₃
prime radical	C₂³×D₄×C₁₁₃	C₄
LCS	G ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₁₁₃ ◁ C₂²×C₄²×C₁₁₃ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₁₁₃×#1817	C₂	semidirect
3. C₂×#323×C₁₁₃	C₂	semidirect
19. C₂²×D₄×C₁₁₃	C₈	semidirect
146. #8302	C₁₁₃	direct (N × H)
156. #1817	C₂×C₁₁₃	semidirect
157. C₂×#323	C₂×C₁₁₃	semidirect
158. C₁₁₃	#8302	direct (N × H)
174. C₂²×D₄	C₈×C₁₁₃	semidirect

Maximal subgroups

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

subgroup	index
C₂²×C₄×D₄×C₁₁₃	2
C₁₁₃×#1817	2
C₂×#323×C₁₁₃	2
#8302	113

Subgroups — C113×#8302 #43276322

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₁₃×#8302 #43276322