Subgroups — C₁₁₃×#9063 #43277083

Order 28928.

654 conjugacy classes of subgroups.

The subgroup lattice has 300 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#492×C₁₁₃	C₂	1	1	yes	maximal, characteristic
2	C₁₁₃×#2369	C₂	1	1	yes	maximal, characteristic, complemented
3	C₁₁₃×#1982	C₂	1	2	yes	maximal
4	C₂×#333×C₁₁₃	C₂	1	1	yes	maximal, characteristic, complemented
5	C₁₁₃×#1279	C₂	1	2	yes	maximal, complemented
6	C₂²×SD₁₆×C₁₁₃	—	2	2	no	—
7	C₂×C₄⋊Q₈×C₁₁₃	C₂²	1	1	yes	characteristic
8	C₂×#127×C₁₁₃	C₂²	1	1	yes	characteristic
9	#492×C₁₁₃	—	2	2	no	—
10	#492×C₁₁₃	C₄	1	4	yes	complemented
11	C₂×C₄⋊C₈×C₁₁₃	C₂²	1	2	yes	—
12	C₂×#102×C₁₁₃	—	2	2	no	—
13	C₂×#102×C₁₁₃	—	2	2	no	—
14	C₂×C₄×C₈×C₁₁₃	C₂²	1	1	yes	characteristic
15	#391×C₁₁₃	—	2	2	no	—
16	#389×C₁₁₃	—	2	2	no	—
17	C₄×C₄⋊C₄×C₁₁₃	—	2	2	no	—
18	C₄×C₄⋊C₄×C₁₁₃	C₂²	1	2	yes	—
19	#333×C₁₁₃	—	2	2	no	—
20	C₂²×Q₈×C₁₁₃	D₄	1	2	yes	—
21	C₂²×D₄×C₁₁₃	—	2	1	no	—
22	C₂²×D₄×C₁₁₃	—	2	2	no	—
23	C₂×SD₁₆×C₁₁₃	—	2	8	no	—
24	C₂×SD₁₆×C₁₁₃	—	4	4	no	—
25	C₂×SD₁₆×C₁₁₃	—	4	4	no	—
26	C₂²×C₈×C₁₁₃	—	2	1	no	—
27	C₂²×C₈×C₁₁₃	—	2	2	no	—
28	C₄⋊Q₈×C₁₁₃	—	2	2	no	—
29	C₄⋊Q₈×C₁₁₃	C₂×C₄	1	2	yes	—
30	C₄⋊Q₈×C₁₁₃	D₄	1	2	yes	—
31	#127×C₁₁₃	—	2	1	no	—
32	#127×C₁₁₃	—	2	2	no	—
33	#127×C₁₁₃	C₂×C₄	1	2	yes	—
34	C₂×C₄⋊C₄×C₁₁₃	—	2	2	no	—
35	C₂×C₄⋊C₄×C₁₁₃	—	2	2	no	—
36	C₂×C₄⋊C₄×C₁₁₃	D₄	1	2	yes	—
37	C₂×C₄⋊C₄×C₁₁₃	D₄	1	2	yes	—
38	C₂×C₄²×C₁₁₃	—	2	2	no	—
39	C₂×C₄²×C₁₁₃	—	2	2	no	—
40	C₂×C₄²×C₁₁₃	C₂³	1	1	yes	characteristic
41	C₄⋊C₈×C₁₁₃	—	2	2	no	—
42	C₄⋊C₈×C₁₁₃	—	2	2	no	—
43	#102×C₁₁₃	—	4	4	no	—
44	#102×C₁₁₃	—	4	4	no	—
45	C₄×C₈×C₁₁₃	—	2	1	no	—
46	C₄×C₈×C₁₁₃	C₂×C₄	1	2	yes	—
47	#95×C₁₁₃	—	2	4	no	—
48	#95×C₁₁₃	—	2	2	no	—
49	C₂⁴×C₁₁₃	—	4	1	no	—
50	C₂×Q₈×C₁₁₃	—	2	2	no	—
51	C₂×Q₈×C₁₁₃	—	4	2	no	—
52	C₂×Q₈×C₁₁₃	—	4	2	no	—
53	C₂×Q₈×C₁₁₃	C₂²⋊C₄	1	4	yes	—
54	C₂×D₄×C₁₁₃	—	2	2	no	—
55	C₂×D₄×C₁₁₃	—	2	4	no	—
56	C₂×D₄×C₁₁₃	—	4	2	no	—
57	C₂×D₄×C₁₁₃	—	4	2	no	—
58	C₂×D₄×C₁₁₃	—	4	2	no	—
59	C₂×D₄×C₁₁₃	—	4	1	no	—
60	C₂×D₄×C₁₁₃	—	4	2	no	—
61	C₂×D₄×C₁₁₃	—	4	2	no	—
62	C₂×D₄×C₁₁₃	—	4	2	no	—
63	C₂×D₄×C₁₁₃	—	4	2	no	—
64	C₂²×C₄×C₁₁₃	—	2	2	no	—
65	C₂²×C₄×C₁₁₃	—	2	2	no	—
66	C₂²×C₄×C₁₁₃	—	2	2	no	—
67	C₂²×C₄×C₁₁₃	C₂×D₄	1	1	yes	characteristic
68	C₂²×C₄×C₁₁₃	C₂×D₄	1	2	yes	—
69	SD₁₆×C₁₁₃	—	4	16	no	—
70	C₂×C₈×C₁₁₃	—	2	2	no	—
71	C₂×C₈×C₁₁₃	—	2	2	no	—
72	C₂×C₈×C₁₁₃	—	2	2	no	—
73	C₂×C₈×C₁₁₃	—	4	2	no	—
74	C₂×C₈×C₁₁₃	—	4	2	no	—
75	C₂×C₈×C₁₁₃	—	4	2	no	—
76	C₄⋊C₄×C₁₁₃	—	2	4	no	—
77	C₄⋊C₄×C₁₁₃	—	2	4	no	—
78	C₄⋊C₄×C₁₁₃	—	2	2	no	—
79	C₄⋊C₄×C₁₁₃	—	2	2	no	—
80	C₄⋊C₄×C₁₁₃	—	2	2	no	—
81	C₄⋊C₄×C₁₁₃	—	8	2	no	—
82	C₄⋊C₄×C₁₁₃	SD₁₆	1	4	yes	complemented
83	C₄²×C₁₁₃	—	4	2	no	—
84	C₄²×C₁₁₃	—	4	2	no	—
85	C₄²×C₁₁₃	—	8	2	no	—
86	C₄²×C₁₁₃	C₂²×C₄	1	1	yes	characteristic
87	C₄²×C₁₁₃	C₂×D₄	1	1	yes	characteristic
88	C₄²×C₁₁₃	#41	1	2	yes	—
89	C₂³×C₁₁₃	—	4	2	no	—
90	C₂³×C₁₁₃	—	8	2	no	—
91	C₂³×C₁₁₃	—	8	2	no	—
92	C₂³×C₁₁₃	—	8	1	no	—
93	C₂³×C₁₁₃	—	8	1	no	—
94	C₂³×C₁₁₃	#120	1	1	yes	characteristic
95	Q₈×C₁₁₃	—	4	4	no	—
96	Q₈×C₁₁₃	—	8	2	no	—
97	D₄×C₁₁₃	—	4	4	no	—
98	D₄×C₁₁₃	—	4	4	no	—
99	D₄×C₁₁₃	—	4	4	no	—
100	D₄×C₁₁₃	—	8	2	no	—
101	D₄×C₁₁₃	—	8	2	no	—
102	D₄×C₁₁₃	—	8	2	no	—
103	C₂×C₄×C₁₁₃	—	2	1	no	—
104	C₂×C₄×C₁₁₃	—	2	1	no	—
105	C₂×C₄×C₁₁₃	—	2	4	no	—
106	C₂×C₄×C₁₁₃	—	2	4	no	—
107	C₂×C₄×C₁₁₃	—	4	2	no	—
108	C₂×C₄×C₁₁₃	—	4	2	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	2	no	—
114	C₂×C₄×C₁₁₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
115	C₂×C₄×C₁₁₃	C₄×D₄	1	2	yes	—
116	C₂×C₄×C₁₁₃	#121	1	2	yes	—
117	C₂×C₄×C₁₁₃	#123	1	1	yes	characteristic
118	C₂×C₄×C₁₁₃	C₂×SD₁₆	1	2	yes	—
119	C₂×C₄×C₁₁₃	#135	1	2	yes	—
120	C₂×C₄×C₁₁₃	AGL(1,ℤ/8)	1	2	yes	—
121	C₂×C₄×C₁₁₃	#137	1	2	yes	—
122	C₈×C₁₁₃	—	2	4	no	—
123	C₈×C₁₁₃	—	8	2	no	—
124	C₂²×C₁₁₃	—	8	4	no	—
125	C₂²×C₁₁₃	—	8	2	no	—
126	C₂²×C₁₁₃	—	16	2	no	—
127	C₂²×C₁₁₃	—	16	1	no	—
128	C₂²×C₁₁₃	—	16	1	no	—
129	C₂²×C₁₁₃	#386	1	1	yes	characteristic
130	C₂²×C₁₁₃	#448	1	2	yes	—
131	C₂²×C₁₁₃	#449	1	2	yes	—
132	C₂²×C₁₁₃	#453	1	1	yes	characteristic
133	C₂²×C₁₁₃	#453	1	1	yes	characteristic
134	C₄×C₁₁₃	—	2	1	no	—
135	C₄×C₁₁₃	—	2	1	no	—
136	C₄×C₁₁₃	—	4	4	no	—
137	C₄×C₁₁₃	—	4	4	no	—
138	C₄×C₁₁₃	—	8	2	no	—
139	C₄×C₁₁₃	C₄×SD₁₆	1	2	yes	—
140	C₄×C₁₁₃	#440	1	2	yes	—
141	C₄×C₁₁₃	#460	1	2	yes	—
142	C₄×C₁₁₃	#464	1	2	yes	—
143	#9063	C₁₁₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
144	C₂×C₁₁₃	—	16	2	no	—
145	C₂×C₁₁₃	#1622	1	2	yes	—
146	C₂×C₁₁₃	#1623	1	1	yes	characteristic
147	C₂×C₁₁₃	#1624	1	1	yes	characteristic
148	C₂×C₁₁₃	#1880	1	2	yes	—
149	C₂×C₁₁₃	#1887	1	1	yes	characteristic
150	C₂×#492	C₂×C₁₁₃	1	1	yes	characteristic
151	#2369	C₂×C₁₁₃	1	1	yes	characteristic, complemented
152	#1982	C₂×C₁₁₃	1	2	yes	—
153	C₂×#333	C₂×C₁₁₃	1	1	yes	characteristic, complemented
154	#1279	C₂×C₁₁₃	1	2	yes	complemented
155	C₁₁₃	#9063	1	1	yes	characteristic, direct product factor, Sylow
156	C₂²×SD₁₆	—	2	2	no	—
157	C₂×C₄⋊Q₈	C₂²×C₁₁₃	1	1	yes	characteristic
158	C₂×#127	C₂²×C₁₁₃	1	1	yes	characteristic
159	#492	—	2	2	no	—
160	#492	C₄×C₁₁₃	1	4	yes	complemented
161	C₂×C₄⋊C₈	C₂²×C₁₁₃	1	2	yes	—
162	C₂×#102	—	2	2	no	—
163	C₂×#102	—	2	2	no	—
164	C₂×C₄×C₈	C₂²×C₁₁₃	1	1	yes	characteristic
165	#391	—	2	2	no	—
166	#389	—	2	2	no	—
167	C₄×C₄⋊C₄	—	2	2	no	—
168	C₄×C₄⋊C₄	C₂²×C₁₁₃	1	2	yes	—
169	#333	—	2	2	no	—
170	C₂²×Q₈	D₄×C₁₁₃	1	2	yes	—
171	C₂²×D₄	—	2	1	no	—
172	C₂²×D₄	—	2	2	no	—
173	C₂×SD₁₆	—	2	8	no	—
174	C₂×SD₁₆	—	4	4	no	—
175	C₂×SD₁₆	—	4	4	no	—
176	C₂²×C₈	—	2	1	no	—
177	C₂²×C₈	—	2	2	no	—
178	C₄⋊Q₈	—	2	2	no	—
179	C₄⋊Q₈	C₂×C₄×C₁₁₃	1	2	yes	—
180	C₄⋊Q₈	D₄×C₁₁₃	1	2	yes	—
181	#127	—	2	1	no	—
182	#127	—	2	2	no	—
183	#127	C₂×C₄×C₁₁₃	1	2	yes	—
184	C₂×C₄⋊C₄	—	2	2	no	—
185	C₂×C₄⋊C₄	—	2	2	no	—
186	C₂×C₄⋊C₄	D₄×C₁₁₃	1	2	yes	—
187	C₂×C₄⋊C₄	D₄×C₁₁₃	1	2	yes	—
188	C₂×C₄²	—	2	2	no	—
189	C₂×C₄²	—	2	2	no	—
190	C₂×C₄²	C₂³×C₁₁₃	1	1	yes	characteristic
191	C₄⋊C₈	—	2	2	no	—
192	C₄⋊C₈	—	2	2	no	—
193	#102	—	4	4	no	—
194	#102	—	4	4	no	—
195	C₄×C₈	—	2	1	no	—
196	C₄×C₈	C₂×C₄×C₁₁₃	1	2	yes	—
197	#95	—	2	4	no	—
198	#95	—	2	2	no	—
199	C₂⁴	—	4	1	no	—
200	C₂×Q₈	—	2	2	no	—
201	C₂×Q₈	—	4	2	no	—
202	C₂×Q₈	—	4	2	no	—
203	C₂×Q₈	C₂²⋊C₄×C₁₁₃	1	4	yes	—
204	C₂×D₄	—	2	2	no	—
205	C₂×D₄	—	2	4	no	—
206	C₂×D₄	—	4	2	no	—
207	C₂×D₄	—	4	2	no	—
208	C₂×D₄	—	4	2	no	—
209	C₂×D₄	—	4	1	no	—
210	C₂×D₄	—	4	2	no	—
211	C₂×D₄	—	4	2	no	—
212	C₂×D₄	—	4	2	no	—
213	C₂×D₄	—	4	2	no	—
214	C₂²×C₄	—	2	2	no	—
215	C₂²×C₄	—	2	2	no	—
216	C₂²×C₄	—	2	2	no	—
217	C₂²×C₄	C₂×D₄×C₁₁₃	1	1	yes	characteristic
218	C₂²×C₄	C₂×D₄×C₁₁₃	1	2	yes	—
219	SD₁₆	—	4	16	no	—
220	C₂×C₈	—	2	2	no	—
221	C₂×C₈	—	2	2	no	—
222	C₂×C₈	—	2	2	no	—
223	C₂×C₈	—	4	2	no	—
224	C₂×C₈	—	4	2	no	—
225	C₂×C₈	—	4	2	no	—
226	C₄⋊C₄	—	2	4	no	—
227	C₄⋊C₄	—	2	4	no	—
228	C₄⋊C₄	—	2	2	no	—
229	C₄⋊C₄	—	2	2	no	—
230	C₄⋊C₄	—	2	2	no	—
231	C₄⋊C₄	—	8	2	no	—
232	C₄⋊C₄	SD₁₆×C₁₁₃	1	4	yes	complemented
233	C₄²	—	4	2	no	—
234	C₄²	—	4	2	no	—
235	C₄²	—	8	2	no	—
236	C₄²	C₂²×C₄×C₁₁₃	1	1	yes	characteristic
237	C₄²	C₂×D₄×C₁₁₃	1	1	yes	characteristic
238	C₄²	#41×C₁₁₃	1	2	yes	—
239	C₂³	—	4	2	no	—
240	C₂³	—	8	2	no	—
241	C₂³	—	8	2	no	—
242	C₂³	—	8	1	no	—
243	C₂³	—	8	1	no	—
244	C₂³	#120×C₁₁₃	1	1	yes	characteristic
245	Q₈	—	4	4	no	—
246	Q₈	—	8	2	no	—
247	D₄	—	4	4	no	—
248	D₄	—	4	4	no	—
249	D₄	—	4	4	no	—
250	D₄	—	8	2	no	—
251	D₄	—	8	2	no	—
252	D₄	—	8	2	no	—
253	C₂×C₄	—	2	1	no	—
254	C₂×C₄	—	2	1	no	—
255	C₂×C₄	—	2	4	no	—
256	C₂×C₄	—	2	4	no	—
257	C₂×C₄	—	4	2	no	—
258	C₂×C₄	—	4	2	no	—
259	C₂×C₄	—	4	2	no	—
260	C₂×C₄	—	4	2	no	—
261	C₂×C₄	—	4	2	no	—
262	C₂×C₄	—	4	2	no	—
263	C₂×C₄	—	4	2	no	—
264	C₂×C₄	C₂×C₂²⋊C₄×C₁₁₃	1	1	yes	characteristic
265	C₂×C₄	C₄×D₄×C₁₁₃	1	2	yes	—
266	C₂×C₄	#121×C₁₁₃	1	2	yes	—
267	C₂×C₄	#123×C₁₁₃	1	1	yes	characteristic
268	C₂×C₄	C₂×SD₁₆×C₁₁₃	1	2	yes	—
269	C₂×C₄	#135×C₁₁₃	1	2	yes	—
270	C₂×C₄	AGL(1,ℤ/8)×C₁₁₃	1	2	yes	—
271	C₂×C₄	#137×C₁₁₃	1	2	yes	—
272	C₈	—	2	4	no	—
273	C₈	—	8	2	no	—
274	C₂²	—	8	4	no	—
275	C₂²	—	8	2	no	—
276	C₂²	—	16	2	no	—
277	C₂²	—	16	1	no	—
278	C₂²	—	16	1	no	—
279	C₂²	#386×C₁₁₃	1	1	yes	characteristic
280	C₂²	#448×C₁₁₃	1	2	yes	—
281	C₂²	#449×C₁₁₃	1	2	yes	—
282	C₂²	#453×C₁₁₃	1	1	yes	characteristic
283	C₂²	#453×C₁₁₃	1	1	yes	characteristic
284	C₄	—	2	1	no	—
285	C₄	—	2	1	no	—
286	C₄	—	4	4	no	—
287	C₄	—	4	4	no	—
288	C₄	—	8	2	no	—
289	C₄	C₄×SD₁₆×C₁₁₃	1	2	yes	—
290	C₄	#440×C₁₁₃	1	2	yes	—
291	C₄	#460×C₁₁₃	1	2	yes	—
292	C₄	#464×C₁₁₃	1	2	yes	—
293	C₂	—	16	2	no	—
294	C₂	C₁₁₃×#1622	1	2	yes	—
295	C₂	C₁₁₃×#1623	1	1	yes	characteristic
296	C₂	C₁₁₃×#1624	1	1	yes	characteristic
297	C₂	C₁₁₃×#1880	1	2	yes	—
298	C₂	C₁₁₃×#1887	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂³×C₁₁₃	#120
commutator	C₄²	C₂²×C₄×C₁₁₃
Frattini, cosocle	C₂×C₄²	C₂³×C₁₁₃
prime radical	C₂×#127×C₁₁₃	C₂²
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³×C₁₁₃ ◁ C₂×C₄²×C₁₁₃ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₁₁₃×#2369	C₂	semidirect
4. C₂×#333×C₁₁₃	C₂	semidirect
5. C₁₁₃×#1279	C₂	semidirect
10. #492×C₁₁₃	C₄	semidirect
82. C₄⋊C₄×C₁₁₃	SD₁₆	semidirect
143. #9063	C₁₁₃	direct (N × H)
151. #2369	C₂×C₁₁₃	semidirect
153. C₂×#333	C₂×C₁₁₃	semidirect
154. #1279	C₂×C₁₁₃	semidirect
155. C₁₁₃	#9063	direct (N × H)
160. #492	C₄×C₁₁₃	semidirect
232. C₄⋊C₄	SD₁₆×C₁₁₃	semidirect

Maximal subgroups

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

subgroup	index
C₂×#492×C₁₁₃	2
C₁₁₃×#2369	2
C₁₁₃×#1982	2
C₂×#333×C₁₁₃	2
C₁₁₃×#1279	2
#9063	113

Subgroups — C113×#9063 #43277083

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₁₃×#9063 #43277083