Subgroups — C₁₁₃×#2392 #28387188

Order 14464.

696 conjugacy classes of subgroups.

The subgroup lattice has 298 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₄×D₄×C₁₁₃	C₂	1	2	yes	maximal, complemented
2	C₂×C₄×D₄×C₁₁₃	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×#117×C₁₁₃	C₂	1	1	yes	maximal, characteristic, complemented
4	#398×C₁₁₃	C₂	1	1	yes	maximal, characteristic, complemented
5	#397×C₁₁₃	C₂	1	1	yes	maximal, characteristic, complemented
6	#396×C₁₁₃	C₂	1	2	yes	maximal, complemented
7	#394×C₁₁₃	C₂	1	2	yes	maximal, complemented
8	#394×C₁₁₃	C₂	1	1	yes	maximal, characteristic, complemented
9	#392×C₁₁₃	C₂	1	1	yes	maximal, characteristic
10	C₄×C₄⋊C₄×C₁₁₃	C₂	1	1	yes	maximal, characteristic, complemented
11	C₄×C₂²⋊C₄×C₁₁₃	C₂	1	2	yes	maximal, complemented
12	C₂²×D₄×C₁₁₃	C₂²	1	1	yes	characteristic
13	C₂²×D₄×C₁₁₃	C₂²	1	2	yes	—
14	C₂³×C₄×C₁₁₃	C₂²	1	2	yes	—
15	C₂³×C₄×C₁₁₃	C₂²	1	1	yes	characteristic
16	C₄×D₄×C₁₁₃	—	2	8	no	—
17	C₄×D₄×C₁₁₃	—	2	4	no	—
18	#117×C₁₁₃	—	2	4	no	—
19	C₂×C₄⋊C₄×C₁₁₃	C₂²	1	1	yes	characteristic
20	C₂×C₄⋊C₄×C₁₁₃	C₂²	1	1	yes	characteristic
21	C₂×C₄⋊C₄×C₁₁₃	C₂²	1	2	yes	—
22	C₂×C₄⋊C₄×C₁₁₃	C₂²	1	1	yes	characteristic
23	C₂×C₄⋊C₄×C₁₁₃	C₂²	1	2	yes	—
24	C₂×C₂²⋊C₄×C₁₁₃	C₂²	1	1	yes	characteristic
25	C₂×C₂²⋊C₄×C₁₁₃	C₂²	1	1	yes	characteristic
26	C₂×C₂²⋊C₄×C₁₁₃	C₂²	1	2	yes	—
27	C₂×C₂²⋊C₄×C₁₁₃	C₂²	1	2	yes	—
28	C₂×C₂²⋊C₄×C₁₁₃	C₂²	1	2	yes	—
29	C₂×C₂²⋊C₄×C₁₁₃	C₂²	1	2	yes	—
30	C₂×C₂²⋊C₄×C₁₁₃	C₂²	1	2	yes	—
31	C₂×C₄²×C₁₁₃	C₂²	1	1	yes	characteristic
32	C₂×C₄²×C₁₁₃	C₂²	1	2	yes	—
33	C₂×C₄²×C₁₁₃	C₂²	1	1	yes	characteristic
34	#95×C₁₁₃	C₂²	1	2	yes	—
35	#95×C₁₁₃	C₂²	1	2	yes	—
36	#95×C₁₁₃	C₂²	1	1	yes	characteristic
37	#95×C₁₁₃	C₂²	1	1	yes	characteristic
38	C₂⁴×C₁₁₃	C₂³	1	1	yes	characteristic
39	C₂⁴×C₁₁₃	C₂³	1	2	yes	—
40	C₂×D₄×C₁₁₃	—	2	4	no	—
41	C₂×D₄×C₁₁₃	—	2	4	no	—
42	C₂×D₄×C₁₁₃	—	2	4	no	—
43	C₂×D₄×C₁₁₃	—	2	2	no	—
44	C₂×D₄×C₁₁₃	—	2	4	no	—
45	C₂²×C₄×C₁₁₃	—	2	2	no	—
46	C₂²×C₄×C₁₁₃	—	2	2	no	—
47	C₂²×C₄×C₁₁₃	—	2	4	no	—
48	C₂²×C₄×C₁₁₃	—	2	4	no	—
49	C₂²×C₄×C₁₁₃	—	2	4	no	—
50	C₂²×C₄×C₁₁₃	D₄	1	4	yes	complemented
51	C₂²×C₄×C₁₁₃	C₂³	1	2	yes	—
52	C₂²×C₄×C₁₁₃	C₂³	1	1	yes	characteristic
53	C₂²×C₄×C₁₁₃	C₂³	1	1	yes	characteristic
54	C₂²×C₄×C₁₁₃	C₂³	1	2	yes	—
55	C₂²×C₄×C₁₁₃	C₂³	1	1	yes	characteristic
56	C₂²×C₄×C₁₁₃	C₂³	1	1	yes	characteristic
57	C₂²×C₄×C₁₁₃	C₂³	1	1	yes	characteristic
58	C₂²×C₄×C₁₁₃	C₂³	1	2	yes	—
59	C₂²×C₄×C₁₁₃	C₂³	1	1	yes	characteristic
60	C₄⋊C₄×C₁₁₃	—	2	2	no	—
61	C₄⋊C₄×C₁₁₃	—	2	2	no	—
62	C₄⋊C₄×C₁₁₃	—	2	4	no	—
63	C₄⋊C₄×C₁₁₃	—	2	2	no	—
64	C₄⋊C₄×C₁₁₃	—	2	4	no	—
65	C₂²⋊C₄×C₁₁₃	—	2	4	no	—
66	C₂²⋊C₄×C₁₁₃	—	2	4	no	—
67	C₂²⋊C₄×C₁₁₃	—	2	2	no	—
68	C₂²⋊C₄×C₁₁₃	—	2	2	no	—
69	C₂²⋊C₄×C₁₁₃	—	2	4	no	—
70	C₂²⋊C₄×C₁₁₃	—	2	4	no	—
71	C₂²⋊C₄×C₁₁₃	—	2	4	no	—
72	C₄²×C₁₁₃	—	2	2	no	—
73	C₄²×C₁₁₃	—	2	4	no	—
74	C₄²×C₁₁₃	D₄	1	4	yes	complemented
75	C₂³×C₁₁₃	—	2	1	no	—
76	C₂³×C₁₁₃	—	2	2	no	—
77	C₂³×C₁₁₃	—	2	2	no	—
78	C₂³×C₁₁₃	—	2	2	no	—
79	C₂³×C₁₁₃	—	2	1	no	—
80	C₂³×C₁₁₃	—	2	2	no	—
81	C₂³×C₁₁₃	—	2	2	no	—
82	C₂³×C₁₁₃	—	2	2	no	—
83	C₂³×C₁₁₃	—	2	2	no	—
84	C₂³×C₁₁₃	—	2	2	no	—
85	C₂³×C₁₁₃	C₂×D₄	1	2	yes	—
86	C₂³×C₁₁₃	#41	1	4	yes	—
87	C₂³×C₁₁₃	C₂⁴	1	1	yes	characteristic
88	D₄×C₁₁₃	—	2	8	no	—
89	D₄×C₁₁₃	—	2	16	no	—
90	C₂×C₄×C₁₁₃	—	2	2	no	—
91	C₂×C₄×C₁₁₃	—	2	1	no	—
92	C₂×C₄×C₁₁₃	—	2	2	no	—
93	C₂×C₄×C₁₁₃	—	2	1	no	—
94	C₂×C₄×C₁₁₃	—	2	2	no	—
95	C₂×C₄×C₁₁₃	—	2	2	no	—
96	C₂×C₄×C₁₁₃	—	2	8	no	—
97	C₂×C₄×C₁₁₃	—	2	2	no	—
98	C₂×C₄×C₁₁₃	—	2	1	no	—
99	C₂×C₄×C₁₁₃	—	2	1	no	—
100	C₂×C₄×C₁₁₃	—	2	2	no	—
101	C₂×C₄×C₁₁₃	—	2	2	no	—
102	C₂×C₄×C₁₁₃	—	2	2	no	—
103	C₂×C₄×C₁₁₃	—	2	2	no	—
104	C₂×C₄×C₁₁₃	—	2	2	no	—
105	C₂×C₄×C₁₁₃	—	2	2	no	—
106	C₂×C₄×C₁₁₃	—	2	2	no	—
107	C₂×C₄×C₁₁₃	—	4	8	no	—
108	C₂×C₄×C₁₁₃	C₂×D₄	1	2	yes	—
109	C₂×C₄×C₁₁₃	C₂×D₄	1	2	yes	—
110	C₂×C₄×C₁₁₃	C₂×D₄	1	2	yes	—
111	C₂×C₄×C₁₁₃	C₂×D₄	1	2	yes	—
112	C₂×C₄×C₁₁₃	C₂×D₄	1	2	yes	—
113	C₂×C₄×C₁₁₃	#41	1	4	yes	—
114	C₂×C₄×C₁₁₃	#41	1	2	yes	—
115	C₂²×C₁₁₃	—	2	2	no	—
116	C₂²×C₁₁₃	—	2	4	no	—
117	C₂²×C₁₁₃	—	2	4	no	—
118	C₂²×C₁₁₃	—	2	4	no	—
119	C₂²×C₁₁₃	—	2	4	no	—
120	C₂²×C₁₁₃	—	4	2	no	—
121	C₂²×C₁₁₃	—	4	1	no	—
122	C₂²×C₁₁₃	—	4	2	no	—
123	C₂²×C₁₁₃	—	4	2	no	—
124	C₂²×C₁₁₃	—	4	2	no	—
125	C₂²×C₁₁₃	—	4	1	no	—
126	C₂²×C₁₁₃	—	4	2	no	—
127	C₂²×C₁₁₃	C₂²×D₄	1	1	yes	characteristic
128	C₂²×C₁₁₃	C₂²×D₄	1	1	yes	characteristic
129	C₂²×C₁₁₃	C₂×#41	1	1	yes	characteristic
130	C₂²×C₁₁₃	C₂×#41	1	2	yes	—
131	C₂²×C₁₁₃	C₂×#41	1	2	yes	—
132	C₄×C₁₁₃	—	2	2	no	—
133	C₄×C₁₁₃	—	2	2	no	—
134	C₄×C₁₁₃	—	2	4	no	—
135	C₄×C₁₁₃	—	2	2	no	—
136	C₄×C₁₁₃	—	4	2	no	—
137	C₄×C₁₁₃	—	4	2	no	—
138	C₄×C₁₁₃	—	4	1	no	—
139	C₄×C₁₁₃	—	4	1	no	—
140	C₄×C₁₁₃	#121	1	4	yes	—
141	C₂×C₁₁₃	—	4	2	no	—
142	C₂×C₁₁₃	—	4	4	no	—
143	C₂×C₁₁₃	C₂×#121	1	1	yes	characteristic
144	C₂×C₁₁₃	#525	1	1	yes	characteristic
145	C₂×C₁₁₃	#525	1	2	yes	—
146	C₂×C₁₁₃	#529	1	2	yes	—
147	C₂×C₁₁₃	#532	1	1	yes	characteristic
148	#2392	C₁₁₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
149	C₁₁₃	#2392	1	1	yes	characteristic, direct product factor, Sylow
150	C₂×C₄×D₄	C₂×C₁₁₃	1	2	yes	complemented
151	C₂×C₄×D₄	C₂×C₁₁₃	1	1	yes	characteristic, complemented
152	C₂×#117	C₂×C₁₁₃	1	1	yes	characteristic, complemented
153	#398	C₂×C₁₁₃	1	1	yes	characteristic, complemented
154	#397	C₂×C₁₁₃	1	1	yes	characteristic, complemented
155	#396	C₂×C₁₁₃	1	2	yes	complemented
156	#394	C₂×C₁₁₃	1	2	yes	complemented
157	#394	C₂×C₁₁₃	1	1	yes	characteristic, complemented
158	#392	C₂×C₁₁₃	1	1	yes	characteristic
159	C₄×C₄⋊C₄	C₂×C₁₁₃	1	1	yes	characteristic, complemented
160	C₄×C₂²⋊C₄	C₂×C₁₁₃	1	2	yes	complemented
161	C₂²×D₄	C₂²×C₁₁₃	1	1	yes	characteristic
162	C₂²×D₄	C₂²×C₁₁₃	1	2	yes	—
163	C₂³×C₄	C₂²×C₁₁₃	1	2	yes	—
164	C₂³×C₄	C₂²×C₁₁₃	1	1	yes	characteristic
165	C₄×D₄	—	2	8	no	—
166	C₄×D₄	—	2	4	no	—
167	#117	—	2	4	no	—
168	C₂×C₄⋊C₄	C₂²×C₁₁₃	1	1	yes	characteristic
169	C₂×C₄⋊C₄	C₂²×C₁₁₃	1	1	yes	characteristic
170	C₂×C₄⋊C₄	C₂²×C₁₁₃	1	2	yes	—
171	C₂×C₄⋊C₄	C₂²×C₁₁₃	1	1	yes	characteristic
172	C₂×C₄⋊C₄	C₂²×C₁₁₃	1	2	yes	—
173	C₂×C₂²⋊C₄	C₂²×C₁₁₃	1	1	yes	characteristic
174	C₂×C₂²⋊C₄	C₂²×C₁₁₃	1	1	yes	characteristic
175	C₂×C₂²⋊C₄	C₂²×C₁₁₃	1	2	yes	—
176	C₂×C₂²⋊C₄	C₂²×C₁₁₃	1	2	yes	—
177	C₂×C₂²⋊C₄	C₂²×C₁₁₃	1	2	yes	—
178	C₂×C₂²⋊C₄	C₂²×C₁₁₃	1	2	yes	—
179	C₂×C₂²⋊C₄	C₂²×C₁₁₃	1	2	yes	—
180	C₂×C₄²	C₂²×C₁₁₃	1	1	yes	characteristic
181	C₂×C₄²	C₂²×C₁₁₃	1	2	yes	—
182	C₂×C₄²	C₂²×C₁₁₃	1	1	yes	characteristic
183	#95	C₂²×C₁₁₃	1	2	yes	—
184	#95	C₂²×C₁₁₃	1	2	yes	—
185	#95	C₂²×C₁₁₃	1	1	yes	characteristic
186	#95	C₂²×C₁₁₃	1	1	yes	characteristic
187	C₂⁴	C₂³×C₁₁₃	1	1	yes	characteristic
188	C₂⁴	C₂³×C₁₁₃	1	2	yes	—
189	C₂×D₄	—	2	4	no	—
190	C₂×D₄	—	2	4	no	—
191	C₂×D₄	—	2	4	no	—
192	C₂×D₄	—	2	2	no	—
193	C₂×D₄	—	2	4	no	—
194	C₂²×C₄	—	2	2	no	—
195	C₂²×C₄	—	2	2	no	—
196	C₂²×C₄	—	2	4	no	—
197	C₂²×C₄	—	2	4	no	—
198	C₂²×C₄	—	2	4	no	—
199	C₂²×C₄	D₄×C₁₁₃	1	4	yes	complemented
200	C₂²×C₄	C₂³×C₁₁₃	1	2	yes	—
201	C₂²×C₄	C₂³×C₁₁₃	1	1	yes	characteristic
202	C₂²×C₄	C₂³×C₁₁₃	1	1	yes	characteristic
203	C₂²×C₄	C₂³×C₁₁₃	1	2	yes	—
204	C₂²×C₄	C₂³×C₁₁₃	1	1	yes	characteristic
205	C₂²×C₄	C₂³×C₁₁₃	1	1	yes	characteristic
206	C₂²×C₄	C₂³×C₁₁₃	1	1	yes	characteristic
207	C₂²×C₄	C₂³×C₁₁₃	1	2	yes	—
208	C₂²×C₄	C₂³×C₁₁₃	1	1	yes	characteristic
209	C₄⋊C₄	—	2	2	no	—
210	C₄⋊C₄	—	2	2	no	—
211	C₄⋊C₄	—	2	4	no	—
212	C₄⋊C₄	—	2	2	no	—
213	C₄⋊C₄	—	2	4	no	—
214	C₂²⋊C₄	—	2	4	no	—
215	C₂²⋊C₄	—	2	4	no	—
216	C₂²⋊C₄	—	2	2	no	—
217	C₂²⋊C₄	—	2	2	no	—
218	C₂²⋊C₄	—	2	4	no	—
219	C₂²⋊C₄	—	2	4	no	—
220	C₂²⋊C₄	—	2	4	no	—
221	C₄²	—	2	2	no	—
222	C₄²	—	2	4	no	—
223	C₄²	D₄×C₁₁₃	1	4	yes	complemented
224	C₂³	—	2	1	no	—
225	C₂³	—	2	2	no	—
226	C₂³	—	2	2	no	—
227	C₂³	—	2	2	no	—
228	C₂³	—	2	1	no	—
229	C₂³	—	2	2	no	—
230	C₂³	—	2	2	no	—
231	C₂³	—	2	2	no	—
232	C₂³	—	2	2	no	—
233	C₂³	—	2	2	no	—
234	C₂³	C₂×D₄×C₁₁₃	1	2	yes	—
235	C₂³	#41×C₁₁₃	1	4	yes	—
236	C₂³	C₂⁴×C₁₁₃	1	1	yes	characteristic
237	D₄	—	2	8	no	—
238	D₄	—	2	16	no	—
239	C₂×C₄	—	2	2	no	—
240	C₂×C₄	—	2	1	no	—
241	C₂×C₄	—	2	2	no	—
242	C₂×C₄	—	2	1	no	—
243	C₂×C₄	—	2	2	no	—
244	C₂×C₄	—	2	2	no	—
245	C₂×C₄	—	2	8	no	—
246	C₂×C₄	—	2	2	no	—
247	C₂×C₄	—	2	1	no	—
248	C₂×C₄	—	2	1	no	—
249	C₂×C₄	—	2	2	no	—
250	C₂×C₄	—	2	2	no	—
251	C₂×C₄	—	2	2	no	—
252	C₂×C₄	—	2	2	no	—
253	C₂×C₄	—	2	2	no	—
254	C₂×C₄	—	2	2	no	—
255	C₂×C₄	—	2	2	no	—
256	C₂×C₄	—	4	8	no	—
257	C₂×C₄	C₂×D₄×C₁₁₃	1	2	yes	—
258	C₂×C₄	C₂×D₄×C₁₁₃	1	2	yes	—
259	C₂×C₄	C₂×D₄×C₁₁₃	1	2	yes	—
260	C₂×C₄	C₂×D₄×C₁₁₃	1	2	yes	—
261	C₂×C₄	C₂×D₄×C₁₁₃	1	2	yes	—
262	C₂×C₄	#41×C₁₁₃	1	4	yes	—
263	C₂×C₄	#41×C₁₁₃	1	2	yes	—
264	C₂²	—	2	2	no	—
265	C₂²	—	2	4	no	—
266	C₂²	—	2	4	no	—
267	C₂²	—	2	4	no	—
268	C₂²	—	2	4	no	—
269	C₂²	—	4	2	no	—
270	C₂²	—	4	1	no	—
271	C₂²	—	4	2	no	—
272	C₂²	—	4	2	no	—
273	C₂²	—	4	2	no	—
274	C₂²	—	4	1	no	—
275	C₂²	—	4	2	no	—
276	C₂²	C₂²×D₄×C₁₁₃	1	1	yes	characteristic
277	C₂²	C₂²×D₄×C₁₁₃	1	1	yes	characteristic
278	C₂²	C₂×#41×C₁₁₃	1	1	yes	characteristic
279	C₂²	C₂×#41×C₁₁₃	1	2	yes	—
280	C₂²	C₂×#41×C₁₁₃	1	2	yes	—
281	C₄	—	2	2	no	—
282	C₄	—	2	2	no	—
283	C₄	—	2	4	no	—
284	C₄	—	2	2	no	—
285	C₄	—	4	2	no	—
286	C₄	—	4	2	no	—
287	C₄	—	4	1	no	—
288	C₄	—	4	1	no	—
289	C₄	#121×C₁₁₃	1	4	yes	—
290	C₂	—	4	2	no	—
291	C₂	—	4	4	no	—
292	C₂	C₂×#121×C₁₁₃	1	1	yes	characteristic
293	C₂	#525×C₁₁₃	1	1	yes	characteristic
294	C₂	#525×C₁₁₃	1	2	yes	—
295	C₂	#529×C₁₁₃	1	2	yes	—
296	C₂	#532×C₁₁₃	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄×C₁₁₃	C₂³
commutator, Frattini, cosocle	C₂³	C₂⁴×C₁₁₃
socle	C₂³×C₁₁₃	C₂⁴
prime radical	#392×C₁₁₃	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂²×C₄×C₁₁₃ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₂×C₄×D₄×C₁₁₃	C₂	semidirect
2. C₂×C₄×D₄×C₁₁₃	C₂	semidirect
3. C₂×#117×C₁₁₃	C₂	semidirect
4. #398×C₁₁₃	C₂	semidirect
5. #397×C₁₁₃	C₂	semidirect
6. #396×C₁₁₃	C₂	semidirect
7. #394×C₁₁₃	C₂	semidirect
8. #394×C₁₁₃	C₂	semidirect
10. C₄×C₄⋊C₄×C₁₁₃	C₂	semidirect
11. C₄×C₂²⋊C₄×C₁₁₃	C₂	semidirect
50. C₂²×C₄×C₁₁₃	D₄	semidirect
74. C₄²×C₁₁₃	D₄	semidirect
148. #2392	C₁₁₃	direct (N × H)
149. C₁₁₃	#2392	direct (N × H)
150. C₂×C₄×D₄	C₂×C₁₁₃	semidirect
151. C₂×C₄×D₄	C₂×C₁₁₃	semidirect
152. C₂×#117	C₂×C₁₁₃	semidirect
153. #398	C₂×C₁₁₃	semidirect
154. #397	C₂×C₁₁₃	semidirect
155. #396	C₂×C₁₁₃	semidirect
156. #394	C₂×C₁₁₃	semidirect
157. #394	C₂×C₁₁₃	semidirect
159. C₄×C₄⋊C₄	C₂×C₁₁₃	semidirect
160. C₄×C₂²⋊C₄	C₂×C₁₁₃	semidirect
199. C₂²×C₄	D₄×C₁₁₃	semidirect
223. C₄²	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×C₄×D₄×C₁₁₃	2
C₂×#117×C₁₁₃	2
#398×C₁₁₃	2
#397×C₁₁₃	2
#396×C₁₁₃	2
#394×C₁₁₃	2
#392×C₁₁₃	2
C₄×C₄⋊C₄×C₁₁₃	2
C₄×C₂²⋊C₄×C₁₁₃	2
#2392	113

Subgroups — C113×#2392 #28387188

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₁₃×#2392 #28387188