Subgroups — C₁₁₃×#11223 #43279243

Order 28928.

706 conjugacy classes of subgroups.

The subgroup lattice has 318 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₁₁₃×#2905	C₂	1	1	yes	maximal, characteristic
2	C₁₁₃×#2296	C₂	1	1	yes	maximal, characteristic
3	C₂×#344×C₁₁₃	C₂	1	2	yes	maximal
4	C₁₁₃×#1742	C₂	1	1	yes	maximal, characteristic
5	C₁₁₃×#1728	C₂	1	2	yes	maximal
6	C₂²×M₁₆×C₁₁₃	C₂²	1	2	yes	—
7	C₂²×M₁₆×C₁₁₃	C₂²	1	2	yes	—
8	C₂×#117×C₁₁₃	C₂²	1	1	yes	characteristic
9	C₂×#117×C₁₁₃	C₂²	1	1	yes	characteristic
10	C₂²×C₄²×C₁₁₃	C₂²	1	1	yes	characteristic
11	#424×C₁₁₃	—	2	4	no	—
12	#423×C₁₁₃	—	2	2	no	—
13	#389×C₁₁₃	—	2	2	no	—
14	#385×C₁₁₃	C₄	1	4	yes	—
15	C₄×C₄⋊C₄×C₁₁₃	—	2	2	no	—
16	#344×C₁₁₃	—	2	4	no	—
17	#343×C₁₁₃	—	2	2	no	—
18	#336×C₁₁₃	—	2	4	no	—
19	C₂³×C₄×C₁₁₃	C₂×C₄	1	2	yes	—
20	C₂³×C₄×C₁₁₃	C₂³	1	1	yes	characteristic
21	C₂×M₁₆×C₁₁₃	—	2	2	no	—
22	C₂×M₁₆×C₁₁₃	—	2	2	no	—
23	C₂×M₁₆×C₁₁₃	—	2	2	no	—
24	C₂×M₁₆×C₁₁₃	—	2	2	no	—
25	C₂×M₁₆×C₁₁₃	—	2	2	no	—
26	C₂×M₁₆×C₁₁₃	—	2	2	no	—
27	C₂×M₁₆×C₁₁₃	—	2	2	no	—
28	C₂×M₁₆×C₁₁₃	—	2	2	no	—
29	C₂×M₁₆×C₁₁₃	—	2	2	no	—
30	C₂×M₁₆×C₁₁₃	—	2	2	no	—
31	C₂×M₁₆×C₁₁₃	—	2	2	no	—
32	C₂×M₁₆×C₁₁₃	Q₈	1	4	yes	—
33	C₂²×C₈×C₁₁₃	—	2	2	no	—
34	C₂²×C₈×C₁₁₃	D₄	1	4	yes	—
35	#117×C₁₁₃	—	2	2	no	—
36	#117×C₁₁₃	—	2	1	no	—
37	#117×C₁₁₃	—	2	1	no	—
38	#117×C₁₁₃	—	2	2	no	—
39	#117×C₁₁₃	D₄	1	2	yes	—
40	#117×C₁₁₃	Q₈	1	2	yes	—
41	C₂×C₄⋊C₄×C₁₁₃	—	2	1	no	—
42	C₂×C₄⋊C₄×C₁₁₃	—	2	4	no	—
43	C₂×C₄⋊C₄×C₁₁₃	D₄	1	2	yes	—
44	C₂×C₂²⋊C₄×C₁₁₃	C₂×C₄	1	2	yes	—
45	C₂×C₂²⋊C₄×C₁₁₃	C₂×C₄	1	2	yes	—
46	C₂×C₄²×C₁₁₃	—	2	1	no	—
47	C₂×C₄²×C₁₁₃	—	2	2	no	—
48	C₂×C₄²×C₁₁₃	—	2	1	no	—
49	C₂×C₄²×C₁₁₃	—	2	1	no	—
50	C₂×C₄²×C₁₁₃	—	2	2	no	—
51	C₂×C₄²×C₁₁₃	D₄	1	2	yes	—
52	C₄⋊C₈×C₁₁₃	—	2	8	no	—
53	#95×C₁₁₃	—	2	4	no	—
54	C₂⁴×C₁₁₃	C₂²×C₄	1	1	yes	characteristic
55	C₂²×C₄×C₁₁₃	—	2	2	no	—
56	C₂²×C₄×C₁₁₃	—	2	2	no	—
57	C₂²×C₄×C₁₁₃	—	2	4	no	—
58	C₂²×C₄×C₁₁₃	—	4	4	no	—
59	C₂²×C₄×C₁₁₃	C₂²⋊C₄	1	4	yes	—
60	C₂²×C₄×C₁₁₃	C₂×D₄	1	2	yes	—
61	C₂²×C₄×C₁₁₃	C₂×D₄	1	1	yes	characteristic
62	C₂²×C₄×C₁₁₃	C₂×D₄	1	1	yes	characteristic
63	C₂²×C₄×C₁₁₃	C₂×D₄	1	1	yes	characteristic
64	C₂²×C₄×C₁₁₃	C₂×Q₈	1	2	yes	—
65	C₂²×C₄×C₁₁₃	C₂×Q₈	1	1	yes	characteristic
66	C₂²×C₄×C₁₁₃	#41	1	1	yes	characteristic
67	C₂²×C₄×C₁₁₃	#41	1	1	yes	characteristic
68	C₂²×C₄×C₁₁₃	#41	1	2	yes	—
69	C₂²×C₄×C₁₁₃	#41	1	2	yes	—
70	M₁₆×C₁₁₃	—	2	4	no	—
71	M₁₆×C₁₁₃	—	2	4	no	—
72	M₁₆×C₁₁₃	—	4	2	no	—
73	M₁₆×C₁₁₃	—	4	2	no	—
74	M₁₆×C₁₁₃	—	4	2	no	—
75	M₁₆×C₁₁₃	—	4	2	no	—
76	M₁₆×C₁₁₃	—	4	2	no	—
77	M₁₆×C₁₁₃	—	4	2	no	—
78	C₂×C₈×C₁₁₃	—	2	4	no	—
79	C₂×C₈×C₁₁₃	—	2	4	no	—
80	C₂×C₈×C₁₁₃	—	2	4	no	—
81	C₂×C₈×C₁₁₃	—	2	4	no	—
82	C₂×C₈×C₁₁₃	—	4	2	no	—
83	C₂×C₈×C₁₁₃	—	4	2	no	—
84	C₄⋊C₄×C₁₁₃	—	2	2	no	—
85	C₄⋊C₄×C₁₁₃	—	2	4	no	—
86	C₄⋊C₄×C₁₁₃	—	2	2	no	—
87	C₄⋊C₄×C₁₁₃	—	4	8	no	—
88	C₂²⋊C₄×C₁₁₃	—	2	2	no	—
89	C₂²⋊C₄×C₁₁₃	—	2	4	no	—
90	C₂²⋊C₄×C₁₁₃	C₄⋊C₄	1	4	yes	—
91	C₄²×C₁₁₃	—	2	2	no	—
92	C₄²×C₁₁₃	—	2	2	no	—
93	C₄²×C₁₁₃	—	2	2	no	—
94	C₄²×C₁₁₃	—	2	2	no	—
95	C₄²×C₁₁₃	—	4	1	no	—
96	C₄²×C₁₁₃	—	4	2	no	—
97	C₄²×C₁₁₃	—	4	1	no	—
98	C₂³×C₁₁₃	—	4	2	no	—
99	C₂³×C₁₁₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
100	C₂³×C₁₁₃	C₂×C₄⋊C₄	1	1	yes	characteristic
101	C₂³×C₁₁₃	#117	1	1	yes	characteristic
102	C₂³×C₁₁₃	C₄×D₄	1	2	yes	—
103	C₂³×C₁₁₃	C₄×Q₈	1	2	yes	—
104	C₂×C₄×C₁₁₃	—	2	4	no	—
105	C₂×C₄×C₁₁₃	—	2	8	no	—
106	C₂×C₄×C₁₁₃	—	2	4	no	—
107	C₂×C₄×C₁₁₃	—	2	4	no	—
108	C₂×C₄×C₁₁₃	—	4	4	no	—
109	C₂×C₄×C₁₁₃	—	4	2	no	—
110	C₂×C₄×C₁₁₃	—	4	4	no	—
111	C₂×C₄×C₁₁₃	—	4	2	no	—
112	C₂×C₄×C₁₁₃	—	4	4	no	—
113	C₂×C₄×C₁₁₃	#120	1	1	yes	characteristic
114	C₂×C₄×C₁₁₃	#121	1	2	yes	—
115	C₂×C₄×C₁₁₃	#121	1	1	yes	characteristic
116	C₂×C₄×C₁₁₃	#121	1	1	yes	characteristic
117	C₂×C₄×C₁₁₃	#121	1	2	yes	—
118	C₂×C₄×C₁₁₃	C₂²⋊Q₈	1	2	yes	—
119	C₂×C₄×C₁₁₃	C₂²⋊Q₈	1	1	yes	characteristic
120	C₂×C₄×C₁₁₃	C₂²⋊Q₈	1	2	yes	—
121	C₂×C₄×C₁₁₃	C₂²⋊Q₈	1	2	yes	—
122	C₂×C₄×C₁₁₃	C₂²⋊Q₈	1	1	yes	characteristic
123	C₂×C₄×C₁₁₃	#123	1	2	yes	—
124	C₂×C₄×C₁₁₃	#123	1	1	yes	characteristic
125	C₂×C₄×C₁₁₃	#124	1	1	yes	characteristic
126	C₂×C₄×C₁₁₃	#125	1	2	yes	—
127	C₂×C₄×C₁₁₃	#125	1	2	yes	—
128	C₂×C₄×C₁₁₃	#126	1	2	yes	—
129	C₂×C₄×C₁₁₃	C₄⋊Q₈	1	2	yes	—
130	C₂×C₄×C₁₁₃	C₄⋊Q₈	1	1	yes	characteristic
131	C₈×C₁₁₃	—	4	4	no	—
132	C₈×C₁₁₃	—	4	4	no	—
133	C₂²×C₁₁₃	—	2	2	no	—
134	C₂²×C₁₁₃	—	2	4	no	—
135	C₂²×C₁₁₃	—	4	1	no	—
136	C₂²×C₁₁₃	—	4	1	no	—
137	C₂²×C₁₁₃	—	4	2	no	—
138	C₂²×C₁₁₃	#380	1	1	yes	characteristic
139	C₂²×C₁₁₃	#386	1	1	yes	characteristic
140	C₂²×C₁₁₃	#387	1	2	yes	—
141	C₂²×C₁₁₃	#389	1	2	yes	—
142	C₂²×C₁₁₃	#391	1	1	yes	characteristic
143	C₄×C₁₁₃	—	4	4	no	—
144	C₄×C₁₁₃	—	4	4	no	—
145	C₄×C₁₁₃	—	4	4	no	—
146	C₄×C₁₁₃	#394	1	1	yes	characteristic
147	C₄×C₁₁₃	#395	1	1	yes	characteristic
148	C₄×C₁₁₃	#396	1	2	yes	—
149	C₄×C₁₁₃	#398	1	1	yes	characteristic
150	C₄×C₁₁₃	#398	1	2	yes	—
151	C₄×C₁₁₃	#400	1	1	yes	characteristic
152	#11223	C₁₁₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
153	C₂×C₁₁₃	—	2	1	no	—
154	C₂×C₁₁₃	—	2	2	no	—
155	C₂×C₁₁₃	—	2	2	no	—
156	C₂×C₁₁₃	—	2	1	no	—
157	C₂×C₁₁₃	#1441	1	1	yes	characteristic
158	C₂×C₁₁₃	#1891	1	2	yes	—
159	#2905	C₂×C₁₁₃	1	1	yes	characteristic
160	#2296	C₂×C₁₁₃	1	1	yes	characteristic
161	C₂×#344	C₂×C₁₁₃	1	2	yes	—
162	#1742	C₂×C₁₁₃	1	1	yes	characteristic
163	#1728	C₂×C₁₁₃	1	2	yes	—
164	C₁₁₃	#11223	1	1	yes	characteristic, direct product factor, Sylow
165	C₂²×M₁₆	C₂²×C₁₁₃	1	2	yes	—
166	C₂²×M₁₆	C₂²×C₁₁₃	1	2	yes	—
167	C₂×#117	C₂²×C₁₁₃	1	1	yes	characteristic
168	C₂×#117	C₂²×C₁₁₃	1	1	yes	characteristic
169	C₂²×C₄²	C₂²×C₁₁₃	1	1	yes	characteristic
170	#424	—	2	4	no	—
171	#423	—	2	2	no	—
172	#389	—	2	2	no	—
173	#385	C₄×C₁₁₃	1	4	yes	—
174	C₄×C₄⋊C₄	—	2	2	no	—
175	#344	—	2	4	no	—
176	#343	—	2	2	no	—
177	#336	—	2	4	no	—
178	C₂³×C₄	C₂×C₄×C₁₁₃	1	2	yes	—
179	C₂³×C₄	C₂³×C₁₁₃	1	1	yes	characteristic
180	C₂×M₁₆	—	2	2	no	—
181	C₂×M₁₆	—	2	2	no	—
182	C₂×M₁₆	—	2	2	no	—
183	C₂×M₁₆	—	2	2	no	—
184	C₂×M₁₆	—	2	2	no	—
185	C₂×M₁₆	—	2	2	no	—
186	C₂×M₁₆	—	2	2	no	—
187	C₂×M₁₆	—	2	2	no	—
188	C₂×M₁₆	—	2	2	no	—
189	C₂×M₁₆	—	2	2	no	—
190	C₂×M₁₆	—	2	2	no	—
191	C₂×M₁₆	Q₈×C₁₁₃	1	4	yes	—
192	C₂²×C₈	—	2	2	no	—
193	C₂²×C₈	D₄×C₁₁₃	1	4	yes	—
194	#117	—	2	2	no	—
195	#117	—	2	1	no	—
196	#117	—	2	1	no	—
197	#117	—	2	2	no	—
198	#117	D₄×C₁₁₃	1	2	yes	—
199	#117	Q₈×C₁₁₃	1	2	yes	—
200	C₂×C₄⋊C₄	—	2	1	no	—
201	C₂×C₄⋊C₄	—	2	4	no	—
202	C₂×C₄⋊C₄	D₄×C₁₁₃	1	2	yes	—
203	C₂×C₂²⋊C₄	C₂×C₄×C₁₁₃	1	2	yes	—
204	C₂×C₂²⋊C₄	C₂×C₄×C₁₁₃	1	2	yes	—
205	C₂×C₄²	—	2	1	no	—
206	C₂×C₄²	—	2	2	no	—
207	C₂×C₄²	—	2	1	no	—
208	C₂×C₄²	—	2	1	no	—
209	C₂×C₄²	—	2	2	no	—
210	C₂×C₄²	D₄×C₁₁₃	1	2	yes	—
211	C₄⋊C₈	—	2	8	no	—
212	#95	—	2	4	no	—
213	C₂⁴	C₂²×C₄×C₁₁₃	1	1	yes	characteristic
214	C₂²×C₄	—	2	2	no	—
215	C₂²×C₄	—	2	2	no	—
216	C₂²×C₄	—	2	4	no	—
217	C₂²×C₄	—	4	4	no	—
218	C₂²×C₄	C₂²⋊C₄×C₁₁₃	1	4	yes	—
219	C₂²×C₄	C₂×D₄×C₁₁₃	1	2	yes	—
220	C₂²×C₄	C₂×D₄×C₁₁₃	1	1	yes	characteristic
221	C₂²×C₄	C₂×D₄×C₁₁₃	1	1	yes	characteristic
222	C₂²×C₄	C₂×D₄×C₁₁₃	1	1	yes	characteristic
223	C₂²×C₄	C₂×Q₈×C₁₁₃	1	2	yes	—
224	C₂²×C₄	C₂×Q₈×C₁₁₃	1	1	yes	characteristic
225	C₂²×C₄	#41×C₁₁₃	1	1	yes	characteristic
226	C₂²×C₄	#41×C₁₁₃	1	1	yes	characteristic
227	C₂²×C₄	#41×C₁₁₃	1	2	yes	—
228	C₂²×C₄	#41×C₁₁₃	1	2	yes	—
229	M₁₆	—	2	4	no	—
230	M₁₆	—	2	4	no	—
231	M₁₆	—	4	2	no	—
232	M₁₆	—	4	2	no	—
233	M₁₆	—	4	2	no	—
234	M₁₆	—	4	2	no	—
235	M₁₆	—	4	2	no	—
236	M₁₆	—	4	2	no	—
237	C₂×C₈	—	2	4	no	—
238	C₂×C₈	—	2	4	no	—
239	C₂×C₈	—	2	4	no	—
240	C₂×C₈	—	2	4	no	—
241	C₂×C₈	—	4	2	no	—
242	C₂×C₈	—	4	2	no	—
243	C₄⋊C₄	—	2	2	no	—
244	C₄⋊C₄	—	2	4	no	—
245	C₄⋊C₄	—	2	2	no	—
246	C₄⋊C₄	—	4	8	no	—
247	C₂²⋊C₄	—	2	2	no	—
248	C₂²⋊C₄	—	2	4	no	—
249	C₂²⋊C₄	C₄⋊C₄×C₁₁₃	1	4	yes	—
250	C₄²	—	2	2	no	—
251	C₄²	—	2	2	no	—
252	C₄²	—	2	2	no	—
253	C₄²	—	2	2	no	—
254	C₄²	—	4	1	no	—
255	C₄²	—	4	2	no	—
256	C₄²	—	4	1	no	—
257	C₂³	—	4	2	no	—
258	C₂³	C₂×C₂²⋊C₄×C₁₁₃	1	1	yes	characteristic
259	C₂³	C₂×C₄⋊C₄×C₁₁₃	1	1	yes	characteristic
260	C₂³	#117×C₁₁₃	1	1	yes	characteristic
261	C₂³	C₄×D₄×C₁₁₃	1	2	yes	—
262	C₂³	C₄×Q₈×C₁₁₃	1	2	yes	—
263	C₂×C₄	—	2	4	no	—
264	C₂×C₄	—	2	8	no	—
265	C₂×C₄	—	2	4	no	—
266	C₂×C₄	—	2	4	no	—
267	C₂×C₄	—	4	4	no	—
268	C₂×C₄	—	4	2	no	—
269	C₂×C₄	—	4	4	no	—
270	C₂×C₄	—	4	2	no	—
271	C₂×C₄	—	4	4	no	—
272	C₂×C₄	#120×C₁₁₃	1	1	yes	characteristic
273	C₂×C₄	#121×C₁₁₃	1	2	yes	—
274	C₂×C₄	#121×C₁₁₃	1	1	yes	characteristic
275	C₂×C₄	#121×C₁₁₃	1	1	yes	characteristic
276	C₂×C₄	#121×C₁₁₃	1	2	yes	—
277	C₂×C₄	C₂²⋊Q₈×C₁₁₃	1	2	yes	—
278	C₂×C₄	C₂²⋊Q₈×C₁₁₃	1	1	yes	characteristic
279	C₂×C₄	C₂²⋊Q₈×C₁₁₃	1	2	yes	—
280	C₂×C₄	C₂²⋊Q₈×C₁₁₃	1	2	yes	—
281	C₂×C₄	C₂²⋊Q₈×C₁₁₃	1	1	yes	characteristic
282	C₂×C₄	#123×C₁₁₃	1	2	yes	—
283	C₂×C₄	#123×C₁₁₃	1	1	yes	characteristic
284	C₂×C₄	#124×C₁₁₃	1	1	yes	characteristic
285	C₂×C₄	#125×C₁₁₃	1	2	yes	—
286	C₂×C₄	#125×C₁₁₃	1	2	yes	—
287	C₂×C₄	#126×C₁₁₃	1	2	yes	—
288	C₂×C₄	C₄⋊Q₈×C₁₁₃	1	2	yes	—
289	C₂×C₄	C₄⋊Q₈×C₁₁₃	1	1	yes	characteristic
290	C₈	—	4	4	no	—
291	C₈	—	4	4	no	—
292	C₂²	—	2	2	no	—
293	C₂²	—	2	4	no	—
294	C₂²	—	4	1	no	—
295	C₂²	—	4	1	no	—
296	C₂²	—	4	2	no	—
297	C₂²	#380×C₁₁₃	1	1	yes	characteristic
298	C₂²	#386×C₁₁₃	1	1	yes	characteristic
299	C₂²	#387×C₁₁₃	1	2	yes	—
300	C₂²	#389×C₁₁₃	1	2	yes	—
301	C₂²	#391×C₁₁₃	1	1	yes	characteristic
302	C₄	—	4	4	no	—
303	C₄	—	4	4	no	—
304	C₄	—	4	4	no	—
305	C₄	#394×C₁₁₃	1	1	yes	characteristic
306	C₄	#395×C₁₁₃	1	1	yes	characteristic
307	C₄	#396×C₁₁₃	1	2	yes	—
308	C₄	#398×C₁₁₃	1	1	yes	characteristic
309	C₄	#398×C₁₁₃	1	2	yes	—
310	C₄	#400×C₁₁₃	1	1	yes	characteristic
311	C₂	—	2	1	no	—
312	C₂	—	2	2	no	—
313	C₂	—	2	2	no	—
314	C₂	—	2	1	no	—
315	C₂	C₁₁₃×#1441	1	1	yes	characteristic
316	C₂	C₁₁₃×#1891	1	2	yes	—

	subgroup	G/H
center	C₂×C₄×C₁₁₃	#120
commutator	C₂⁴	C₂²×C₄×C₁₁₃
Frattini, cosocle	C₂³×C₄	C₂³×C₁₁₃
socle	C₂²×C₁₁₃	#386
prime radical	C₂⁴×C₁₁₃	C₂²×C₄
LCS	G ▷ C₂⁴ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₁₁₃ ◁ C₂³×C₄×C₁₁₃ ◁ G

Complements

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

N	H ≅ G/N	type
152. #11223	C₁₁₃	direct (N × H)
164. C₁₁₃	#11223	direct (N × H)

Maximal subgroups

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

subgroup	index
C₁₁₃×#2905	2
C₁₁₃×#2296	2
C₂×#344×C₁₁₃	2
C₁₁₃×#1742	2
C₁₁₃×#1728	2
#11223	113

Subgroups — C113×#11223 #43279243

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₁₃×#11223 #43279243