Subgroups — C₁₁₃×#13961 #43281981

Order 28928.

1622 conjugacy classes of subgroups.

The subgroup lattice has 328 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₄×Q₈×C₁₁₃	C₂	1	1	yes	maximal, characteristic
2	C₂×#383×C₁₁₃	C₂	1	1	yes	maximal, characteristic
3	C₂×#376×C₁₁₃	C₂	1	1	yes	maximal, characteristic
4	C₁₁₃×#1446	C₂	1	2	yes	maximal
5	C₁₁₃×#1445	C₂	1	2	yes	maximal
6	C₁₁₃×#1436	C₂	1	4	yes	maximal
7	C₁₁₃×#1434	C₂	1	4	yes	maximal
8	C₂³×Q₈×C₁₁₃	C₂²	1	1	yes	characteristic
9	C₂×C₄×Q₈×C₁₁₃	—	2	4	no	—
10	C₂×C₄×Q₈×C₁₁₃	—	2	4	no	—
11	C₂×C₄×Q₈×C₁₁₃	C₄	1	8	yes	complemented
12	C₂²×C₄⋊C₄×C₁₁₃	C₂²	1	2	yes	—
13	C₂²×C₄⋊C₄×C₁₁₃	C₂²	1	1	yes	characteristic
14	C₂²×C₄²×C₁₁₃	C₂²	1	2	yes	—
15	C₂²×C₄²×C₁₁₃	C₂²	1	2	yes	—
16	C₂²×C₄²×C₁₁₃	C₂²	1	1	yes	characteristic
17	#383×C₁₁₃	—	2	4	no	—
18	#376×C₁₁₃	—	2	4	no	—
19	C₂×#95×C₁₁₃	C₂²	1	2	yes	—
20	C₂×#95×C₁₁₃	C₂²	1	2	yes	—
21	C₂×#95×C₁₁₃	C₂²	1	2	yes	—
22	C₂×#95×C₁₁₃	C₂²	1	2	yes	—
23	C₂×#95×C₁₁₃	C₂²	1	8	yes	—
24	C₂×#95×C₁₁₃	C₂²	1	8	yes	—
25	C₂×#95×C₁₁₃	C₂²	1	2	yes	—
26	C₂²×Q₈×C₁₁₃	—	2	2	no	—
27	C₂²×Q₈×C₁₁₃	—	2	2	no	—
28	C₂²×Q₈×C₁₁₃	—	2	8	no	—
29	C₂²×Q₈×C₁₁₃	C₂×C₄	1	4	yes	—
30	C₂³×C₄×C₁₁₃	C₂³	1	4	yes	—
31	C₂³×C₄×C₁₁₃	C₂³	1	2	yes	—
32	C₂³×C₄×C₁₁₃	C₂³	1	4	yes	—
33	C₂³×C₄×C₁₁₃	C₂³	1	1	yes	characteristic
34	C₂³×C₄×C₁₁₃	C₂³	1	1	yes	characteristic
35	C₂³×C₄×C₁₁₃	C₂³	1	1	yes	characteristic
36	C₂³×C₄×C₁₁₃	C₂³	1	2	yes	—
37	C₄×Q₈×C₁₁₃	—	2	32	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	C₂×C₄	1	4	yes	—
43	C₂×C₄⋊C₄×C₁₁₃	C₂×C₄	1	8	yes	—
44	C₂×C₄²×C₁₁₃	—	2	4	no	—
45	C₂×C₄²×C₁₁₃	—	2	4	no	—
46	C₂×C₄²×C₁₁₃	—	2	4	no	—
47	C₂×C₄²×C₁₁₃	—	2	8	no	—
48	C₂×C₄²×C₁₁₃	C₂×C₄	1	8	yes	—
49	C₂×C₄²×C₁₁₃	C₂×C₄	1	4	yes	—
50	C₂×C₄²×C₁₁₃	D₄	1	4	yes	—
51	C₂×C₄²×C₁₁₃	D₄	1	4	yes	—
52	#95×C₁₁₃	—	2	4	no	—
53	#95×C₁₁₃	—	2	4	no	—
54	#95×C₁₁₃	—	2	4	no	—
55	#95×C₁₁₃	—	2	4	no	—
56	#95×C₁₁₃	—	2	4	no	—
57	#95×C₁₁₃	—	2	16	no	—
58	#95×C₁₁₃	—	2	16	no	—
59	C₂⁴×C₁₁₃	C₂⁴	1	1	yes	characteristic
60	C₂×Q₈×C₁₁₃	—	2	16	no	—
61	C₂×Q₈×C₁₁₃	—	2	16	no	—
62	C₂×Q₈×C₁₁₃	—	2	8	no	—
63	C₂×Q₈×C₁₁₃	—	2	16	no	—
64	C₂²×C₄×C₁₁₃	—	2	2	no	—
65	C₂²×C₄×C₁₁₃	—	2	2	no	—
66	C₂²×C₄×C₁₁₃	—	2	4	no	—
67	C₂²×C₄×C₁₁₃	—	2	4	no	—
68	C₂²×C₄×C₁₁₃	—	2	4	no	—
69	C₂²×C₄×C₁₁₃	—	2	4	no	—
70	C₂²×C₄×C₁₁₃	—	2	4	no	—
71	C₂²×C₄×C₁₁₃	—	2	2	no	—
72	C₂²×C₄×C₁₁₃	—	2	2	no	—
73	C₂²×C₄×C₁₁₃	—	2	8	no	—
74	C₂²×C₄×C₁₁₃	—	2	8	no	—
75	C₂²×C₄×C₁₁₃	—	2	4	no	—
76	C₂²×C₄×C₁₁₃	—	2	8	no	—
77	C₂²×C₄×C₁₁₃	—	2	4	no	—
78	C₂²×C₄×C₁₁₃	—	2	8	no	—
79	C₂²×C₄×C₁₁₃	—	2	8	no	—
80	C₂²×C₄×C₁₁₃	—	2	4	no	—
81	C₂²×C₄×C₁₁₃	—	2	4	no	—
82	C₂²×C₄×C₁₁₃	C₂²×C₄	1	4	yes	—
83	C₂²×C₄×C₁₁₃	C₂²×C₄	1	2	yes	—
84	C₂²×C₄×C₁₁₃	C₂²×C₄	1	2	yes	—
85	C₂²×C₄×C₁₁₃	C₂²×C₄	1	2	yes	—
86	C₂²×C₄×C₁₁₃	C₂²×C₄	1	4	yes	—
87	C₂²×C₄×C₁₁₃	C₂×D₄	1	2	yes	—
88	C₂²×C₄×C₁₁₃	C₂×D₄	1	2	yes	—
89	C₂²×C₄×C₁₁₃	C₂×D₄	1	2	yes	—
90	C₂²×C₄×C₁₁₃	C₂×D₄	1	2	yes	—
91	C₂²×C₄×C₁₁₃	C₂×D₄	1	2	yes	—
92	C₂²×C₄×C₁₁₃	C₂×D₄	1	2	yes	—
93	C₂²×C₄×C₁₁₃	#41	1	8	yes	—
94	C₂²×C₄×C₁₁₃	#41	1	8	yes	—
95	C₄⋊C₄×C₁₁₃	—	2	16	no	—
96	C₄⋊C₄×C₁₁₃	—	2	8	no	—
97	C₄²×C₁₁₃	—	2	16	no	—
98	C₄²×C₁₁₃	—	2	16	no	—
99	C₄²×C₁₁₃	C₂²⋊C₄	1	16	yes	—
100	C₂³×C₁₁₃	C₂³×C₄	1	1	yes	characteristic
101	C₂³×C₁₁₃	C₂²×D₄	1	1	yes	characteristic
102	C₂³×C₁₁₃	C₂²×D₄	1	1	yes	characteristic
103	C₂³×C₁₁₃	C₂×#41	1	4	yes	—
104	C₂³×C₁₁₃	C₂×#41	1	4	yes	—
105	C₂³×C₁₁₃	2₋¹⁺⁴	1	4	yes	—
106	Q₈×C₁₁₃	—	2	32	no	—
107	C₂×C₄×C₁₁₃	—	2	4	no	—
108	C₂×C₄×C₁₁₃	—	2	4	no	—
109	C₂×C₄×C₁₁₃	—	2	4	no	—
110	C₂×C₄×C₁₁₃	—	2	4	no	—
111	C₂×C₄×C₁₁₃	—	2	4	no	—
112	C₂×C₄×C₁₁₃	—	2	4	no	—
113	C₂×C₄×C₁₁₃	—	2	4	no	—
114	C₂×C₄×C₁₁₃	—	2	4	no	—
115	C₂×C₄×C₁₁₃	—	2	8	no	—
116	C₂×C₄×C₁₁₃	—	2	8	no	—
117	C₂×C₄×C₁₁₃	—	2	4	no	—
118	C₂×C₄×C₁₁₃	—	2	4	no	—
119	C₂×C₄×C₁₁₃	—	2	4	no	—
120	C₂×C₄×C₁₁₃	—	2	4	no	—
121	C₂×C₄×C₁₁₃	—	2	8	no	—
122	C₂×C₄×C₁₁₃	—	2	8	no	—
123	C₂×C₄×C₁₁₃	—	2	16	no	—
124	C₂×C₄×C₁₁₃	—	4	4	no	—
125	C₂×C₄×C₁₁₃	—	4	4	no	—
126	C₂×C₄×C₁₁₃	—	4	4	no	—
127	C₂×C₄×C₁₁₃	—	4	4	no	—
128	C₂×C₄×C₁₁₃	—	4	8	no	—
129	C₂×C₄×C₁₁₃	—	4	4	no	—
130	C₂×C₄×C₁₁₃	—	4	4	no	—
131	C₂×C₄×C₁₁₃	—	4	4	no	—
132	C₂×C₄×C₁₁₃	—	4	8	no	—
133	C₂×C₄×C₁₁₃	—	4	8	no	—
134	C₂×C₄×C₁₁₃	C₂×C₂²⋊C₄	1	4	yes	—
135	C₂×C₄×C₁₁₃	C₂×C₂²⋊C₄	1	4	yes	—
136	C₂×C₄×C₁₁₃	C₂×C₂²⋊C₄	1	4	yes	—
137	C₂×C₄×C₁₁₃	#117	1	8	yes	—
138	C₂²×C₁₁₃	C₂²×C₂²⋊C₄	1	1	yes	characteristic
139	C₂²×C₁₁₃	C₂×#117	1	2	yes	—
140	C₂²×C₁₁₃	C₄×#41	1	2	yes	—
141	C₂²×C₁₁₃	#519	1	2	yes	—
142	C₂²×C₁₁₃	#525	1	2	yes	—
143	C₂²×C₁₁₃	#525	1	2	yes	—
144	C₂²×C₁₁₃	#525	1	2	yes	—
145	C₂²×C₁₁₃	#532	1	2	yes	—
146	C₂²×C₁₁₃	#536	1	2	yes	—
147	C₂²×C₁₁₃	#536	1	2	yes	—
148	C₂²×C₁₁₃	#552	1	8	yes	—
149	C₂²×C₁₁₃	#556	1	8	yes	—
150	C₄×C₁₁₃	—	2	4	no	—
151	C₄×C₁₁₃	—	2	4	no	—
152	C₄×C₁₁₃	—	2	8	no	—
153	C₄×C₁₁₃	—	2	4	no	—
154	C₄×C₁₁₃	—	4	8	no	—
155	C₄×C₁₁₃	—	4	4	no	—
156	C₄×C₁₁₃	—	8	4	no	—
157	#13961	C₁₁₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
158	C₂×C₁₁₃	#2280	1	1	yes	characteristic
159	C₂×C₁₁₃	#2297	1	1	yes	characteristic
160	C₂×C₁₁₃	#2310	1	1	yes	characteristic
161	C₂×C₁₁₃	#2356	1	4	yes	—
162	C₂×C₁₁₃	#2599	1	4	yes	—
163	C₂×C₁₁₃	#2604	1	4	yes	—
164	C₂²×C₄×Q₈	C₂×C₁₁₃	1	1	yes	characteristic
165	C₂×#383	C₂×C₁₁₃	1	1	yes	characteristic
166	C₂×#376	C₂×C₁₁₃	1	1	yes	characteristic
167	#1446	C₂×C₁₁₃	1	2	yes	—
168	#1445	C₂×C₁₁₃	1	2	yes	—
169	#1436	C₂×C₁₁₃	1	4	yes	—
170	#1434	C₂×C₁₁₃	1	4	yes	—
171	C₁₁₃	#13961	1	1	yes	characteristic, direct product factor, Sylow
172	C₂³×Q₈	C₂²×C₁₁₃	1	1	yes	characteristic
173	C₂×C₄×Q₈	—	2	4	no	—
174	C₂×C₄×Q₈	—	2	4	no	—
175	C₂×C₄×Q₈	C₄×C₁₁₃	1	8	yes	complemented
176	C₂²×C₄⋊C₄	C₂²×C₁₁₃	1	2	yes	—
177	C₂²×C₄⋊C₄	C₂²×C₁₁₃	1	1	yes	characteristic
178	C₂²×C₄²	C₂²×C₁₁₃	1	2	yes	—
179	C₂²×C₄²	C₂²×C₁₁₃	1	2	yes	—
180	C₂²×C₄²	C₂²×C₁₁₃	1	1	yes	characteristic
181	#383	—	2	4	no	—
182	#376	—	2	4	no	—
183	C₂×#95	C₂²×C₁₁₃	1	2	yes	—
184	C₂×#95	C₂²×C₁₁₃	1	2	yes	—
185	C₂×#95	C₂²×C₁₁₃	1	8	yes	—
186	C₂×#95	C₂²×C₁₁₃	1	8	yes	—
187	C₂×#95	C₂²×C₁₁₃	1	2	yes	—
188	C₂×#95	C₂²×C₁₁₃	1	2	yes	—
189	C₂×#95	C₂²×C₁₁₃	1	2	yes	—
190	C₂²×Q₈	—	2	2	no	—
191	C₂²×Q₈	—	2	2	no	—
192	C₂²×Q₈	—	2	8	no	—
193	C₂²×Q₈	C₂×C₄×C₁₁₃	1	4	yes	—
194	C₂³×C₄	C₂³×C₁₁₃	1	4	yes	—
195	C₂³×C₄	C₂³×C₁₁₃	1	2	yes	—
196	C₂³×C₄	C₂³×C₁₁₃	1	4	yes	—
197	C₂³×C₄	C₂³×C₁₁₃	1	1	yes	characteristic
198	C₂³×C₄	C₂³×C₁₁₃	1	1	yes	characteristic
199	C₂³×C₄	C₂³×C₁₁₃	1	1	yes	characteristic
200	C₂³×C₄	C₂³×C₁₁₃	1	2	yes	—
201	C₄×Q₈	—	2	32	no	—
202	C₂×C₄⋊C₄	—	2	4	no	—
203	C₂×C₄⋊C₄	—	2	4	no	—
204	C₂×C₄⋊C₄	—	2	2	no	—
205	C₂×C₄⋊C₄	—	2	2	no	—
206	C₂×C₄⋊C₄	C₂×C₄×C₁₁₃	1	4	yes	—
207	C₂×C₄⋊C₄	C₂×C₄×C₁₁₃	1	8	yes	—
208	C₂×C₄²	—	2	4	no	—
209	C₂×C₄²	—	2	4	no	—
210	C₂×C₄²	—	2	4	no	—
211	C₂×C₄²	—	2	8	no	—
212	C₂×C₄²	C₂×C₄×C₁₁₃	1	8	yes	—
213	C₂×C₄²	C₂×C₄×C₁₁₃	1	4	yes	—
214	C₂×C₄²	D₄×C₁₁₃	1	4	yes	—
215	C₂×C₄²	D₄×C₁₁₃	1	4	yes	—
216	#95	—	2	4	no	—
217	#95	—	2	4	no	—
218	#95	—	2	4	no	—
219	#95	—	2	4	no	—
220	#95	—	2	4	no	—
221	#95	—	2	16	no	—
222	#95	—	2	16	no	—
223	C₂⁴	C₂⁴×C₁₁₃	1	1	yes	characteristic
224	C₂×Q₈	—	2	16	no	—
225	C₂×Q₈	—	2	16	no	—
226	C₂×Q₈	—	2	8	no	—
227	C₂×Q₈	—	2	16	no	—
228	C₂²×C₄	—	2	2	no	—
229	C₂²×C₄	—	2	2	no	—
230	C₂²×C₄	—	2	4	no	—
231	C₂²×C₄	—	2	4	no	—
232	C₂²×C₄	—	2	4	no	—
233	C₂²×C₄	—	2	4	no	—
234	C₂²×C₄	—	2	4	no	—
235	C₂²×C₄	—	2	2	no	—
236	C₂²×C₄	—	2	2	no	—
237	C₂²×C₄	—	2	8	no	—
238	C₂²×C₄	—	2	8	no	—
239	C₂²×C₄	—	2	4	no	—
240	C₂²×C₄	—	2	8	no	—
241	C₂²×C₄	—	2	4	no	—
242	C₂²×C₄	—	2	8	no	—
243	C₂²×C₄	—	2	8	no	—
244	C₂²×C₄	—	2	4	no	—
245	C₂²×C₄	—	2	4	no	—
246	C₂²×C₄	C₂²×C₄×C₁₁₃	1	4	yes	—
247	C₂²×C₄	C₂²×C₄×C₁₁₃	1	2	yes	—
248	C₂²×C₄	C₂²×C₄×C₁₁₃	1	2	yes	—
249	C₂²×C₄	C₂²×C₄×C₁₁₃	1	2	yes	—
250	C₂²×C₄	C₂²×C₄×C₁₁₃	1	4	yes	—
251	C₂²×C₄	C₂×D₄×C₁₁₃	1	2	yes	—
252	C₂²×C₄	C₂×D₄×C₁₁₃	1	2	yes	—
253	C₂²×C₄	C₂×D₄×C₁₁₃	1	2	yes	—
254	C₂²×C₄	C₂×D₄×C₁₁₃	1	2	yes	—
255	C₂²×C₄	C₂×D₄×C₁₁₃	1	2	yes	—
256	C₂²×C₄	C₂×D₄×C₁₁₃	1	2	yes	—
257	C₂²×C₄	#41×C₁₁₃	1	8	yes	—
258	C₂²×C₄	#41×C₁₁₃	1	8	yes	—
259	C₄⋊C₄	—	2	16	no	—
260	C₄⋊C₄	—	2	8	no	—
261	C₄²	—	2	16	no	—
262	C₄²	—	2	16	no	—
263	C₄²	C₂²⋊C₄×C₁₁₃	1	16	yes	—
264	C₂³	C₂³×C₄×C₁₁₃	1	1	yes	characteristic
265	C₂³	C₂²×D₄×C₁₁₃	1	1	yes	characteristic
266	C₂³	C₂²×D₄×C₁₁₃	1	1	yes	characteristic
267	C₂³	C₂×#41×C₁₁₃	1	4	yes	—
268	C₂³	C₂×#41×C₁₁₃	1	4	yes	—
269	C₂³	2₋¹⁺⁴×C₁₁₃	1	4	yes	—
270	Q₈	—	2	32	no	—
271	C₂×C₄	—	2	4	no	—
272	C₂×C₄	—	2	4	no	—
273	C₂×C₄	—	2	4	no	—
274	C₂×C₄	—	2	4	no	—
275	C₂×C₄	—	2	4	no	—
276	C₂×C₄	—	2	4	no	—
277	C₂×C₄	—	2	4	no	—
278	C₂×C₄	—	2	4	no	—
279	C₂×C₄	—	2	8	no	—
280	C₂×C₄	—	2	8	no	—
281	C₂×C₄	—	2	4	no	—
282	C₂×C₄	—	2	4	no	—
283	C₂×C₄	—	2	4	no	—
284	C₂×C₄	—	2	4	no	—
285	C₂×C₄	—	2	8	no	—
286	C₂×C₄	—	2	8	no	—
287	C₂×C₄	—	2	16	no	—
288	C₂×C₄	—	4	4	no	—
289	C₂×C₄	—	4	4	no	—
290	C₂×C₄	—	4	4	no	—
291	C₂×C₄	—	4	4	no	—
292	C₂×C₄	—	4	8	no	—
293	C₂×C₄	—	4	4	no	—
294	C₂×C₄	—	4	4	no	—
295	C₂×C₄	—	4	4	no	—
296	C₂×C₄	—	4	8	no	—
297	C₂×C₄	—	4	8	no	—
298	C₂×C₄	C₂×C₂²⋊C₄×C₁₁₃	1	4	yes	—
299	C₂×C₄	C₂×C₂²⋊C₄×C₁₁₃	1	4	yes	—
300	C₂×C₄	C₂×C₂²⋊C₄×C₁₁₃	1	4	yes	—
301	C₂×C₄	#117×C₁₁₃	1	8	yes	—
302	C₂²	C₂²×C₂²⋊C₄×C₁₁₃	1	1	yes	characteristic
303	C₂²	C₂×#117×C₁₁₃	1	2	yes	—
304	C₂²	C₄×#41×C₁₁₃	1	2	yes	—
305	C₂²	#519×C₁₁₃	1	2	yes	—
306	C₂²	#525×C₁₁₃	1	2	yes	—
307	C₂²	#525×C₁₁₃	1	2	yes	—
308	C₂²	#525×C₁₁₃	1	2	yes	—
309	C₂²	#532×C₁₁₃	1	2	yes	—
310	C₂²	#536×C₁₁₃	1	2	yes	—
311	C₂²	#536×C₁₁₃	1	2	yes	—
312	C₂²	#552×C₁₁₃	1	8	yes	—
313	C₂²	#556×C₁₁₃	1	8	yes	—
314	C₄	—	2	4	no	—
315	C₄	—	2	8	no	—
316	C₄	—	2	4	no	—
317	C₄	—	2	4	no	—
318	C₄	—	4	4	no	—
319	C₄	—	4	8	no	—
320	C₄	—	8	4	no	—
321	C₂	C₁₁₃×#2280	1	1	yes	characteristic
322	C₂	C₁₁₃×#2297	1	1	yes	characteristic
323	C₂	C₁₁₃×#2310	1	1	yes	characteristic
324	C₂	C₁₁₃×#2356	1	4	yes	—
325	C₂	C₁₁₃×#2599	1	4	yes	—
326	C₂	C₁₁₃×#2604	1	4	yes	—

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

Complements

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

N	H ≅ G/N	type
11. C₂×C₄×Q₈×C₁₁₃	C₄	semidirect
157. #13961	C₁₁₃	direct (N × H)
171. C₁₁₃	#13961	direct (N × H)
175. C₂×C₄×Q₈	C₄×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×Q₈×C₁₁₃	2
C₂×#383×C₁₁₃	2
C₂×#376×C₁₁₃	2
C₁₁₃×#1446	2
C₁₁₃×#1445	2
C₁₁₃×#1436	2
C₁₁₃×#1434	2
#13961	113

Subgroups — C113×#13961 #43281981

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₁₃×#13961 #43281981