Subgroups — C₁₁₃×#21276 #43289296

Order 28928.

822 conjugacy classes of subgroups.

The subgroup lattice has 290 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₄×Q₁₆×C₁₁₃	C₂	1	1	yes	maximal, characteristic
2	C₁₁₃×#2295	C₂	1	2	yes	maximal
3	C₁₁₃×#1969	C₂	1	2	yes	maximal
4	C₁₁₃×#1942	C₂	1	2	yes	maximal
5	C₁₁₃×#1933	C₂	1	4	yes	maximal
6	C₄×#107×C₁₁₃	C₂	1	1	yes	maximal, characteristic
7	C₄×#103×C₁₁₃	C₂	1	2	yes	maximal
8	C₄²×C₈×C₁₁₃	C₂	1	1	yes	maximal, characteristic
9	C₂²×Q₁₆×C₁₁₃	C₂²	1	1	yes	characteristic
10	C₂×C₄×Q₈×C₁₁₃	C₂²	1	2	yes	—
11	C₄×Q₁₆×C₁₁₃	C₄	1	8	yes	complemented
12	C₂×#107×C₁₁₃	C₂²	1	2	yes	—
13	C₂×#107×C₁₁₃	C₂²	1	1	yes	characteristic
14	C₂×#103×C₁₁₃	C₂²	1	4	yes	—
15	C₂×#103×C₁₁₃	C₂²	1	2	yes	—
16	C₂×C₄×C₈×C₁₁₃	C₂²	1	2	yes	—
17	C₂×C₄×C₈×C₁₁₃	C₂²	1	2	yes	—
18	C₂×C₄×C₈×C₁₁₃	C₂²	1	1	yes	characteristic
19	C₂×C₄×C₈×C₁₁₃	C₂²	1	1	yes	characteristic
20	#391×C₁₁₃	C₂²	1	4	yes	—
21	#389×C₁₁₃	C₂²	1	4	yes	—
22	#387×C₁₁₃	—	2	4	no	—
23	C₄×C₄⋊C₄×C₁₁₃	—	2	2	no	—
24	C₄×C₄⋊C₄×C₁₁₃	C₂²	1	2	yes	—
25	#376×C₁₁₃	—	2	2	no	—
26	C₄³×C₁₁₃	C₂²	1	1	yes	characteristic
27	#340×C₁₁₃	C₂²	1	2	yes	—
28	#340×C₁₁₃	C₂²	1	4	yes	—
29	C₂²×Q₈×C₁₁₃	C₂³	1	2	yes	—
30	C₂×Q₁₆×C₁₁₃	—	2	4	no	—
31	C₂×Q₁₆×C₁₁₃	C₂×C₄	1	4	yes	—
32	C₂²×C₈×C₁₁₃	C₂³	1	1	yes	characteristic
33	C₂²×C₈×C₁₁₃	C₂³	1	2	yes	—
34	C₂²×C₈×C₁₁₃	C₂³	1	1	yes	characteristic
35	C₄×Q₈×C₁₁₃	—	2	4	no	—
36	C₄×Q₈×C₁₁₃	C₂×C₄	1	8	yes	—
37	C₂×C₄⋊C₄×C₁₁₃	—	2	2	no	—
38	C₂×C₄⋊C₄×C₁₁₃	—	2	4	no	—
39	C₂×C₄⋊C₄×C₁₁₃	C₂³	1	4	yes	—
40	C₂×C₄⋊C₄×C₁₁₃	C₂³	1	2	yes	—
41	C₂×C₄²×C₁₁₃	—	2	2	no	—
42	C₂×C₄²×C₁₁₃	—	2	2	no	—
43	C₂×C₄²×C₁₁₃	D₄	1	2	yes	—
44	C₂×C₄²×C₁₁₃	D₄	1	1	yes	characteristic
45	C₂×C₄²×C₁₁₃	D₄	1	1	yes	characteristic
46	C₂×C₄²×C₁₁₃	C₂³	1	2	yes	—
47	C₂×C₄²×C₁₁₃	C₂³	1	1	yes	characteristic
48	#107×C₁₁₃	—	2	4	no	—
49	#107×C₁₁₃	C₂×C₄	1	4	yes	—
50	#103×C₁₁₃	—	2	8	no	—
51	#103×C₁₁₃	C₂×C₄	1	8	yes	—
52	C₄×C₈×C₁₁₃	—	2	4	no	—
53	C₄×C₈×C₁₁₃	—	2	4	no	—
54	C₄×C₈×C₁₁₃	—	2	2	no	—
55	C₄×C₈×C₁₁₃	C₂×C₄	1	4	yes	—
56	#95×C₁₁₃	—	2	4	no	—
57	#95×C₁₁₃	—	2	4	no	—
58	#95×C₁₁₃	—	2	4	no	—
59	#95×C₁₁₃	—	2	2	no	—
60	C₂×Q₈×C₁₁₃	—	2	2	no	—
61	C₂×Q₈×C₁₁₃	—	2	4	no	—
62	C₂×Q₈×C₁₁₃	—	2	4	no	—
63	C₂×Q₈×C₁₁₃	C₂²×C₄	1	4	yes	—
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	C₂×D₄	1	1	yes	characteristic
68	C₂²×C₄×C₁₁₃	C₂×D₄	1	2	yes	—
69	C₂²×C₄×C₁₁₃	C₂×D₄	1	2	yes	—
70	C₂²×C₄×C₁₁₃	C₂×D₄	1	1	yes	characteristic
71	C₂²×C₄×C₁₁₃	C₂⁴	1	1	yes	characteristic
72	Q₁₆×C₁₁₃	—	2	8	no	—
73	C₂×C₈×C₁₁₃	—	2	2	no	—
74	C₂×C₈×C₁₁₃	—	2	2	no	—
75	C₂×C₈×C₁₁₃	—	2	2	no	—
76	C₂×C₈×C₁₁₃	C₂²×C₄	1	2	yes	—
77	C₂×C₈×C₁₁₃	C₂²×C₄	1	2	yes	—
78	C₂×C₈×C₁₁₃	#41	1	4	yes	—
79	C₂×C₈×C₁₁₃	#41	1	4	yes	—
80	C₄⋊C₄×C₁₁₃	—	2	8	no	—
81	C₄⋊C₄×C₁₁₃	—	2	2	no	—
82	C₄⋊C₄×C₁₁₃	—	2	4	no	—
83	C₄⋊C₄×C₁₁₃	—	2	4	no	—
84	C₄⋊C₄×C₁₁₃	—	4	8	no	—
85	C₄⋊C₄×C₁₁₃	C₂²×C₄	1	4	yes	—
86	C₄²×C₁₁₃	—	2	8	no	—
87	C₄²×C₁₁₃	—	2	4	no	—
88	C₄²×C₁₁₃	—	2	2	no	—
89	C₄²×C₁₁₃	—	2	2	no	—
90	C₄²×C₁₁₃	—	2	2	no	—
91	C₄²×C₁₁₃	—	4	4	no	—
92	C₄²×C₁₁₃	Q₁₆	1	4	yes	complemented
93	C₄²×C₁₁₃	C₂²×C₄	1	2	yes	—
94	C₄²×C₁₁₃	#41	1	2	yes	—
95	C₂³×C₁₁₃	C₂²×D₄	1	1	yes	characteristic
96	Q₈×C₁₁₃	—	2	4	no	—
97	Q₈×C₁₁₃	—	2	4	no	—
98	Q₈×C₁₁₃	—	4	4	no	—
99	C₂×C₄×C₁₁₃	—	2	2	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	4	no	—
105	C₂×C₄×C₁₁₃	—	2	4	no	—
106	C₂×C₄×C₁₁₃	—	4	4	no	—
107	C₂×C₄×C₁₁₃	—	4	4	no	—
108	C₂×C₄×C₁₁₃	—	4	4	no	—
109	C₂×C₄×C₁₁₃	—	4	4	no	—
110	C₂×C₄×C₁₁₃	—	4	4	no	—
111	C₂×C₄×C₁₁₃	C₄×D₄	1	2	yes	—
112	C₂×C₄×C₁₁₃	C₄×D₄	1	2	yes	—
113	C₂×C₄×C₁₁₃	C₂×Q₁₆	1	4	yes	—
114	C₂×C₄×C₁₁₃	C₂×Q₁₆	1	2	yes	—
115	C₂×C₄×C₁₁₃	#135	1	4	yes	—
116	C₂×C₄×C₁₁₃	#135	1	2	yes	—
117	C₂×C₄×C₁₁₃	C₂³×C₄	1	1	yes	characteristic
118	C₂×C₄×C₁₁₃	C₂×#41	1	2	yes	—
119	C₂×C₄×C₁₁₃	C₂×#41	1	2	yes	—
120	C₂×C₄×C₁₁₃	C₂×#41	1	1	yes	characteristic
121	C₈×C₁₁₃	—	2	2	no	—
122	C₈×C₁₁₃	—	2	4	no	—
123	C₈×C₁₁₃	#117	1	4	yes	—
124	C₂²×C₁₁₃	C₂×C₄×D₄	1	1	yes	characteristic
125	C₂²×C₁₁₃	#532	1	2	yes	—
126	C₂²×C₁₁₃	C₂²×Q₁₆	1	1	yes	characteristic
127	C₂²×C₁₁₃	C₂×#135	1	1	yes	characteristic
128	C₂²×C₁₁₃	C₂×#135	1	2	yes	—
129	C₄×C₁₁₃	—	2	4	no	—
130	C₄×C₁₁₃	—	2	4	no	—
131	C₄×C₁₁₃	—	2	2	no	—
132	C₄×C₁₁₃	—	4	4	no	—
133	C₄×C₁₁₃	—	4	4	no	—
134	C₄×C₁₁₃	—	8	4	no	—
135	C₄×C₁₁₃	C₄×Q₁₆	1	4	yes	—
136	C₄×C₁₁₃	C₂×#117	1	1	yes	characteristic
137	C₄×C₁₁₃	C₄×#41	1	1	yes	characteristic
138	C₄×C₁₁₃	#529	1	2	yes	—
139	#21276	C₁₁₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
140	C₂×C₁₁₃	#2293	1	1	yes	characteristic
141	C₂×C₁₁₃	C₂×C₄×Q₁₆	1	1	yes	characteristic
142	C₂×C₁₁₃	C₄×#135	1	1	yes	characteristic
143	C₂×C₁₁₃	#3168	1	2	yes	—
144	C₂×C₁₁₃	#3170	1	2	yes	—
145	C₂×C₄×Q₁₆	C₂×C₁₁₃	1	1	yes	characteristic
146	#2295	C₂×C₁₁₃	1	2	yes	—
147	#1969	C₂×C₁₁₃	1	2	yes	—
148	#1942	C₂×C₁₁₃	1	2	yes	—
149	#1933	C₂×C₁₁₃	1	4	yes	—
150	C₄×#107	C₂×C₁₁₃	1	1	yes	characteristic
151	C₄×#103	C₂×C₁₁₃	1	2	yes	—
152	C₄²×C₈	C₂×C₁₁₃	1	1	yes	characteristic
153	C₁₁₃	#21276	1	1	yes	characteristic, direct product factor, Sylow
154	C₂²×Q₁₆	C₂²×C₁₁₃	1	1	yes	characteristic
155	C₂×C₄×Q₈	C₂²×C₁₁₃	1	2	yes	—
156	C₄×Q₁₆	C₄×C₁₁₃	1	8	yes	complemented
157	C₂×#107	C₂²×C₁₁₃	1	2	yes	—
158	C₂×#107	C₂²×C₁₁₃	1	1	yes	characteristic
159	C₂×#103	C₂²×C₁₁₃	1	4	yes	—
160	C₂×#103	C₂²×C₁₁₃	1	2	yes	—
161	C₂×C₄×C₈	C₂²×C₁₁₃	1	2	yes	—
162	C₂×C₄×C₈	C₂²×C₁₁₃	1	2	yes	—
163	C₂×C₄×C₈	C₂²×C₁₁₃	1	1	yes	characteristic
164	C₂×C₄×C₈	C₂²×C₁₁₃	1	1	yes	characteristic
165	#391	C₂²×C₁₁₃	1	4	yes	—
166	#389	C₂²×C₁₁₃	1	4	yes	—
167	#387	—	2	4	no	—
168	C₄×C₄⋊C₄	—	2	2	no	—
169	C₄×C₄⋊C₄	C₂²×C₁₁₃	1	2	yes	—
170	#376	—	2	2	no	—
171	C₄³	C₂²×C₁₁₃	1	1	yes	characteristic
172	#340	C₂²×C₁₁₃	1	2	yes	—
173	#340	C₂²×C₁₁₃	1	4	yes	—
174	C₂²×Q₈	C₂³×C₁₁₃	1	2	yes	—
175	C₂×Q₁₆	—	2	4	no	—
176	C₂×Q₁₆	C₂×C₄×C₁₁₃	1	4	yes	—
177	C₂²×C₈	C₂³×C₁₁₃	1	1	yes	characteristic
178	C₂²×C₈	C₂³×C₁₁₃	1	2	yes	—
179	C₂²×C₈	C₂³×C₁₁₃	1	1	yes	characteristic
180	C₄×Q₈	—	2	4	no	—
181	C₄×Q₈	C₂×C₄×C₁₁₃	1	8	yes	—
182	C₂×C₄⋊C₄	—	2	2	no	—
183	C₂×C₄⋊C₄	—	2	4	no	—
184	C₂×C₄⋊C₄	C₂³×C₁₁₃	1	4	yes	—
185	C₂×C₄⋊C₄	C₂³×C₁₁₃	1	2	yes	—
186	C₂×C₄²	—	2	2	no	—
187	C₂×C₄²	—	2	2	no	—
188	C₂×C₄²	D₄×C₁₁₃	1	1	yes	characteristic
189	C₂×C₄²	D₄×C₁₁₃	1	2	yes	—
190	C₂×C₄²	D₄×C₁₁₃	1	1	yes	characteristic
191	C₂×C₄²	C₂³×C₁₁₃	1	2	yes	—
192	C₂×C₄²	C₂³×C₁₁₃	1	1	yes	characteristic
193	#107	—	2	4	no	—
194	#107	C₂×C₄×C₁₁₃	1	4	yes	—
195	#103	—	2	8	no	—
196	#103	C₂×C₄×C₁₁₃	1	8	yes	—
197	C₄×C₈	—	2	4	no	—
198	C₄×C₈	—	2	4	no	—
199	C₄×C₈	—	2	2	no	—
200	C₄×C₈	C₂×C₄×C₁₁₃	1	4	yes	—
201	#95	—	2	4	no	—
202	#95	—	2	4	no	—
203	#95	—	2	4	no	—
204	#95	—	2	2	no	—
205	C₂×Q₈	—	2	2	no	—
206	C₂×Q₈	—	2	4	no	—
207	C₂×Q₈	—	2	4	no	—
208	C₂×Q₈	C₂²×C₄×C₁₁₃	1	4	yes	—
209	C₂²×C₄	—	2	2	no	—
210	C₂²×C₄	—	2	2	no	—
211	C₂²×C₄	—	2	4	no	—
212	C₂²×C₄	C₂×D₄×C₁₁₃	1	1	yes	characteristic
213	C₂²×C₄	C₂×D₄×C₁₁₃	1	2	yes	—
214	C₂²×C₄	C₂×D₄×C₁₁₃	1	2	yes	—
215	C₂²×C₄	C₂×D₄×C₁₁₃	1	1	yes	characteristic
216	C₂²×C₄	C₂⁴×C₁₁₃	1	1	yes	characteristic
217	Q₁₆	—	2	8	no	—
218	C₂×C₈	—	2	2	no	—
219	C₂×C₈	—	2	2	no	—
220	C₂×C₈	—	2	2	no	—
221	C₂×C₈	C₂²×C₄×C₁₁₃	1	2	yes	—
222	C₂×C₈	C₂²×C₄×C₁₁₃	1	2	yes	—
223	C₂×C₈	#41×C₁₁₃	1	4	yes	—
224	C₂×C₈	#41×C₁₁₃	1	4	yes	—
225	C₄⋊C₄	—	2	8	no	—
226	C₄⋊C₄	—	2	2	no	—
227	C₄⋊C₄	—	2	4	no	—
228	C₄⋊C₄	—	2	4	no	—
229	C₄⋊C₄	—	4	8	no	—
230	C₄⋊C₄	C₂²×C₄×C₁₁₃	1	4	yes	—
231	C₄²	—	2	8	no	—
232	C₄²	—	2	4	no	—
233	C₄²	—	2	2	no	—
234	C₄²	—	2	2	no	—
235	C₄²	—	2	2	no	—
236	C₄²	—	4	4	no	—
237	C₄²	Q₁₆×C₁₁₃	1	4	yes	complemented
238	C₄²	C₂²×C₄×C₁₁₃	1	2	yes	—
239	C₄²	#41×C₁₁₃	1	2	yes	—
240	C₂³	C₂²×D₄×C₁₁₃	1	1	yes	characteristic
241	Q₈	—	2	4	no	—
242	Q₈	—	2	4	no	—
243	Q₈	—	4	4	no	—
244	C₂×C₄	—	2	2	no	—
245	C₂×C₄	—	2	2	no	—
246	C₂×C₄	—	2	2	no	—
247	C₂×C₄	—	2	2	no	—
248	C₂×C₄	—	2	2	no	—
249	C₂×C₄	—	2	4	no	—
250	C₂×C₄	—	2	4	no	—
251	C₂×C₄	—	4	4	no	—
252	C₂×C₄	—	4	4	no	—
253	C₂×C₄	—	4	4	no	—
254	C₂×C₄	—	4	4	no	—
255	C₂×C₄	—	4	4	no	—
256	C₂×C₄	C₄×D₄×C₁₁₃	1	2	yes	—
257	C₂×C₄	C₄×D₄×C₁₁₃	1	2	yes	—
258	C₂×C₄	C₂×Q₁₆×C₁₁₃	1	4	yes	—
259	C₂×C₄	C₂×Q₁₆×C₁₁₃	1	2	yes	—
260	C₂×C₄	#135×C₁₁₃	1	4	yes	—
261	C₂×C₄	#135×C₁₁₃	1	2	yes	—
262	C₂×C₄	C₂³×C₄×C₁₁₃	1	1	yes	characteristic
263	C₂×C₄	C₂×#41×C₁₁₃	1	2	yes	—
264	C₂×C₄	C₂×#41×C₁₁₃	1	2	yes	—
265	C₂×C₄	C₂×#41×C₁₁₃	1	1	yes	characteristic
266	C₈	—	2	2	no	—
267	C₈	—	2	4	no	—
268	C₈	#117×C₁₁₃	1	4	yes	—
269	C₂²	C₂×C₄×D₄×C₁₁₃	1	1	yes	characteristic
270	C₂²	#532×C₁₁₃	1	2	yes	—
271	C₂²	C₂²×Q₁₆×C₁₁₃	1	1	yes	characteristic
272	C₂²	C₂×#135×C₁₁₃	1	1	yes	characteristic
273	C₂²	C₂×#135×C₁₁₃	1	2	yes	—
274	C₄	—	2	4	no	—
275	C₄	—	2	4	no	—
276	C₄	—	2	2	no	—
277	C₄	—	4	4	no	—
278	C₄	—	4	4	no	—
279	C₄	—	8	4	no	—
280	C₄	C₄×Q₁₆×C₁₁₃	1	4	yes	—
281	C₄	C₂×#117×C₁₁₃	1	1	yes	characteristic
282	C₄	C₄×#41×C₁₁₃	1	1	yes	characteristic
283	C₄	#529×C₁₁₃	1	2	yes	—
284	C₂	C₁₁₃×#2293	1	1	yes	characteristic
285	C₂	C₂×C₄×Q₁₆×C₁₁₃	1	1	yes	characteristic
286	C₂	C₄×#135×C₁₁₃	1	1	yes	characteristic
287	C₂	C₁₁₃×#3168	1	2	yes	—
288	C₂	C₁₁₃×#3170	1	2	yes	—

	subgroup	G/H
center	C₂²×C₄×C₁₁₃	C₂×D₄
commutator	C₂×C₄	C₂³×C₄×C₁₁₃
Frattini, cosocle	C₂²×C₄	C₂⁴×C₁₁₃
socle, prime radical	C₂³×C₁₁₃	C₂²×D₄
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₄×C₁₁₃ ◁ C₄³×C₁₁₃ ◁ G

Complements

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

N	H ≅ G/N	type
11. C₄×Q₁₆×C₁₁₃	C₄	semidirect
92. C₄²×C₁₁₃	Q₁₆	semidirect
139. #21276	C₁₁₃	direct (N × H)
153. C₁₁₃	#21276	direct (N × H)
156. C₄×Q₁₆	C₄×C₁₁₃	semidirect
237. C₄²	Q₁₆×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×Q₁₆×C₁₁₃	2
C₁₁₃×#2295	2
C₁₁₃×#1969	2
C₁₁₃×#1942	2
C₁₁₃×#1933	2
C₄×#107×C₁₁₃	2
C₄×#103×C₁₁₃	2
C₄²×C₈×C₁₁₃	2
#21276	113

Subgroups — C113×#21276 #43289296

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₁₃×#21276 #43289296