Subgroups — C₁₁₃×#21882 #43289902

Order 28928.

1082 conjugacy classes of subgroups.

The subgroup lattice has 314 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#540×C₁₁₃	C₂	1	1	yes	maximal, characteristic
2	C₁₁₃×#2921	C₂	1	1	yes	maximal, characteristic, complemented
3	C₁₁₃×#2905	C₂	1	1	yes	maximal, characteristic, complemented
4	C₁₁₃×#1998	C₂	1	2	yes	maximal, complemented
5	C₁₁₃×#1965	C₂	1	2	yes	maximal, complemented
6	C₁₁₃×#1900	C₂	1	4	yes	maximal, complemented
7	C₁₁₃×#1800	C₂	1	4	yes	maximal, complemented
8	C₂²×M₁₆×C₁₁₃	C₂²	1	2	yes	—
9	#540×C₁₁₃	—	2	4	no	—
10	#540×C₁₁₃	—	2	4	no	—
11	C₂×#127×C₁₁₃	C₂²	1	1	yes	characteristic
12	C₂×#125×C₁₁₃	C₂²	1	1	yes	characteristic
13	C₂×#123×C₁₁₃	C₄	1	4	yes	—
14	C₂×#121×C₁₁₃	C₄	1	4	yes	—
15	C₂×#121×C₁₁₃	C₂²	1	2	yes	—
16	C₂×C₄×D₄×C₁₁₃	C₂²	1	2	yes	—
17	C₂×#117×C₁₁₃	C₂²	1	1	yes	characteristic
18	#433×C₁₁₃	—	2	4	no	—
19	#424×C₁₁₃	—	2	4	no	—
20	#423×C₁₁₃	C₂²	1	2	yes	—
21	#423×C₁₁₃	C₂²	1	2	yes	—
22	#423×C₁₁₃	C₂²	1	2	yes	—
23	C₂×#100×C₁₁₃	C₂²	1	8	yes	—
24	#407×C₁₁₃	C₂²	1	2	yes	—
25	C₄×M₁₆×C₁₁₃	C₂²	1	2	yes	—
26	#343×C₁₁₃	C₂²	1	8	yes	—
27	C₂²×D₄×C₁₁₃	C₂×C₄	1	4	yes	—
28	C₂²×D₄×C₁₁₃	C₂×C₄	1	4	yes	—
29	C₂²×D₄×C₁₁₃	C₂³	1	2	yes	—
30	C₂³×C₄×C₁₁₃	C₂×C₄	1	4	yes	—
31	C₂³×C₄×C₁₁₃	C₂³	1	1	yes	characteristic
32	C₂×M₁₆×C₁₁₃	—	2	2	no	—
33	C₂×M₁₆×C₁₁₃	—	4	4	no	—
34	C₂×M₁₆×C₁₁₃	C₂³	1	4	yes	—
35	C₂×M₁₆×C₁₁₃	C₂³	1	4	yes	—
36	C₂²×C₈×C₁₁₃	—	2	2	no	—
37	#127×C₁₁₃	—	2	2	no	—
38	#127×C₁₁₃	—	2	2	no	—
39	#125×C₁₁₃	—	2	2	no	—
40	#125×C₁₁₃	—	2	2	no	—
41	#123×C₁₁₃	—	2	16	no	—
42	#121×C₁₁₃	—	2	4	no	—
43	#121×C₁₁₃	—	2	4	no	—
44	#121×C₁₁₃	—	2	16	no	—
45	C₄×D₄×C₁₁₃	—	2	4	no	—
46	C₄×D₄×C₁₁₃	—	2	4	no	—
47	#117×C₁₁₃	D₄	1	4	yes	—
48	#117×C₁₁₃	D₄	1	4	yes	—
49	C₂×C₄⋊C₄×C₁₁₃	C₂×C₄	1	4	yes	—
50	C₂×C₄⋊C₄×C₁₁₃	C₂×C₄	1	2	yes	—
51	C₂×C₄⋊C₄×C₁₁₃	C₂³	1	2	yes	—
52	C₂×C₂²⋊C₄×C₁₁₃	C₂×C₄	1	4	yes	—
53	C₂×C₂²⋊C₄×C₁₁₃	C₂×C₄	1	4	yes	—
54	C₂×C₂²⋊C₄×C₁₁₃	C₂×C₄	1	2	yes	—
55	C₂×C₄²×C₁₁₃	C₂³	1	1	yes	characteristic
56	C₂×C₄²×C₁₁₃	C₂³	1	1	yes	characteristic
57	C₄⋊C₈×C₁₁₃	—	2	4	no	—
58	C₄⋊C₈×C₁₁₃	—	2	4	no	—
59	C₄⋊C₈×C₁₁₃	—	2	4	no	—
60	#100×C₁₁₃	—	4	8	no	—
61	C₂²⋊C₈×C₁₁₃	—	2	4	no	—
62	#97×C₁₁₃	—	2	2	no	—
63	C₄×C₈×C₁₁₃	—	2	2	no	—
64	C₂⁴×C₁₁₃	C₂²×C₄	1	4	yes	—
65	C₂⁴×C₁₁₃	C₂²×C₄	1	1	yes	characteristic
66	C₂×D₄×C₁₁₃	—	2	8	no	—
67	C₂×D₄×C₁₁₃	—	2	8	no	—
68	C₂×D₄×C₁₁₃	—	2	2	no	—
69	C₂×D₄×C₁₁₃	—	2	2	no	—
70	C₂×D₄×C₁₁₃	—	2	4	no	—
71	C₂×D₄×C₁₁₃	—	2	4	no	—
72	C₂×D₄×C₁₁₃	—	8	4	no	—
73	C₂×D₄×C₁₁₃	—	8	4	no	—
74	C₂²×C₄×C₁₁₃	—	2	2	no	—
75	C₂²×C₄×C₁₁₃	—	2	2	no	—
76	C₂²×C₄×C₁₁₃	—	2	8	no	—
77	C₂²×C₄×C₁₁₃	—	8	4	no	—
78	C₂²×C₄×C₁₁₃	C₂²×C₄	1	1	yes	characteristic
79	C₂²×C₄×C₁₁₃	C₂²×C₄	1	4	yes	—
80	C₂²×C₄×C₁₁₃	C₂²×C₄	1	2	yes	—
81	C₂²×C₄×C₁₁₃	C₂²×C₄	1	2	yes	—
82	C₂²×C₄×C₁₁₃	C₂×D₄	1	2	yes	—
83	C₂²×C₄×C₁₁₃	C₂×D₄	1	2	yes	—
84	C₂²×C₄×C₁₁₃	C₂⁴	1	1	yes	characteristic
85	M₁₆×C₁₁₃	—	2	4	no	—
86	M₁₆×C₁₁₃	—	2	8	no	—
87	M₁₆×C₁₁₃	—	2	4	no	—
88	M₁₆×C₁₁₃	—	4	4	no	—
89	C₂×C₈×C₁₁₃	—	2	4	no	—
90	C₂×C₈×C₁₁₃	—	2	4	no	—
91	C₂×C₈×C₁₁₃	—	4	4	no	—
92	C₄⋊C₄×C₁₁₃	—	2	8	no	—
93	C₄⋊C₄×C₁₁₃	—	2	2	no	—
94	C₄⋊C₄×C₁₁₃	—	2	2	no	—
95	C₄⋊C₄×C₁₁₃	C₂²⋊C₄	1	8	yes	—
96	C₂²⋊C₄×C₁₁₃	—	2	8	no	—
97	C₂²⋊C₄×C₁₁₃	—	2	8	no	—
98	C₂²⋊C₄×C₁₁₃	C₂²⋊C₄	1	8	yes	—
99	C₄²×C₁₁₃	C₂×D₄	1	2	yes	—
100	C₄²×C₁₁₃	C₂×D₄	1	2	yes	—
101	C₄²×C₁₁₃	C₂×D₄	1	2	yes	—
102	C₄²×C₁₁₃	C₂×D₄	1	2	yes	—
103	C₂³×C₁₁₃	—	2	2	no	—
104	C₂³×C₁₁₃	—	2	4	no	—
105	C₂³×C₁₁₃	—	2	8	no	—
106	C₂³×C₁₁₃	—	4	2	no	—
107	C₂³×C₁₁₃	—	4	4	no	—
108	C₂³×C₁₁₃	—	4	4	no	—
109	C₂³×C₁₁₃	C₂×C₂²⋊C₄	1	2	yes	—
110	C₂³×C₁₁₃	C₂³×C₄	1	1	yes	characteristic
111	D₄×C₁₁₃	—	4	2	no	—
112	D₄×C₁₁₃	—	4	2	no	—
113	D₄×C₁₁₃	—	4	4	no	—
114	D₄×C₁₁₃	—	8	8	no	—
115	D₄×C₁₁₃	—	8	8	no	—
116	C₂×C₄×C₁₁₃	—	2	2	no	—
117	C₂×C₄×C₁₁₃	—	2	2	no	—
118	C₂×C₄×C₁₁₃	—	2	2	no	—
119	C₂×C₄×C₁₁₃	—	2	4	no	—
120	C₂×C₄×C₁₁₃	—	2	4	no	—
121	C₂×C₄×C₁₁₃	—	4	2	no	—
122	C₂×C₄×C₁₁₃	—	4	2	no	—
123	C₂×C₄×C₁₁₃	—	4	4	no	—
124	C₂×C₄×C₁₁₃	—	8	8	no	—
125	C₂×C₄×C₁₁₃	C₂×C₂²⋊C₄	1	4	yes	—
126	C₂×C₄×C₁₁₃	C₂×C₂²⋊C₄	1	4	yes	—
127	C₂×C₄×C₁₁₃	C₂×C₂²⋊C₄	1	2	yes	—
128	C₂×C₄×C₁₁₃	C₂²×D₄	1	1	yes	characteristic
129	C₂×C₄×C₁₁₃	C₂²×D₄	1	1	yes	characteristic
130	C₂×C₄×C₁₁₃	H(5,𝔽₂)	1	2	yes	—
131	C₂×C₄×C₁₁₃	2₋¹⁺⁴	1	2	yes	—
132	C₈×C₁₁₃	—	4	4	no	—
133	C₈×C₁₁₃	—	4	4	no	—
134	C₂²×C₁₁₃	—	2	2	no	—
135	C₂²×C₁₁₃	—	2	2	no	—
136	C₂²×C₁₁₃	—	2	4	no	—
137	C₂²×C₁₁₃	—	4	4	no	—
138	C₂²×C₁₁₃	—	4	4	no	—
139	C₂²×C₁₁₃	—	4	4	no	—
140	C₂²×C₁₁₃	—	8	2	no	—
141	C₂²×C₁₁₃	—	8	8	no	—
142	C₂²×C₁₁₃	C₂²×C₂²⋊C₄	1	1	yes	characteristic
143	C₂²×C₁₁₃	#520	1	2	yes	—
144	C₄×C₁₁₃	—	2	2	no	—
145	C₄×C₁₁₃	—	2	4	no	—
146	C₄×C₁₁₃	—	2	4	no	—
147	C₄×C₁₁₃	—	4	4	no	—
148	C₄×C₁₁₃	#535	1	1	yes	characteristic
149	C₄×C₁₁₃	#536	1	1	yes	characteristic
150	C₄×C₁₁₃	#537	1	2	yes	—
151	#21882	C₁₁₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
152	C₂×C₁₁₃	—	2	2	no	—
153	C₂×C₁₁₃	—	4	2	no	—
154	C₂×C₁₁₃	—	8	4	no	—
155	C₂×C₁₁₃	#2311	1	1	yes	characteristic
156	C₂×C₁₁₃	#2888	1	2	yes	—
157	C₂×#540	C₂×C₁₁₃	1	1	yes	characteristic
158	#2921	C₂×C₁₁₃	1	1	yes	characteristic, complemented
159	#2905	C₂×C₁₁₃	1	1	yes	characteristic, complemented
160	#1998	C₂×C₁₁₃	1	2	yes	complemented
161	#1965	C₂×C₁₁₃	1	2	yes	complemented
162	#1900	C₂×C₁₁₃	1	4	yes	complemented
163	#1800	C₂×C₁₁₃	1	4	yes	complemented
164	C₁₁₃	#21882	1	1	yes	characteristic, direct product factor, Sylow
165	C₂²×M₁₆	C₂²×C₁₁₃	1	2	yes	—
166	#540	—	2	4	no	—
167	#540	—	2	4	no	—
168	C₂×#127	C₂²×C₁₁₃	1	1	yes	characteristic
169	C₂×#125	C₂²×C₁₁₃	1	1	yes	characteristic
170	C₂×#123	C₄×C₁₁₃	1	4	yes	—
171	C₂×#121	C₄×C₁₁₃	1	4	yes	—
172	C₂×#121	C₂²×C₁₁₃	1	2	yes	—
173	C₂×C₄×D₄	C₂²×C₁₁₃	1	2	yes	—
174	C₂×#117	C₂²×C₁₁₃	1	1	yes	characteristic
175	#433	—	2	4	no	—
176	#424	—	2	4	no	—
177	#423	C₂²×C₁₁₃	1	2	yes	—
178	#423	C₂²×C₁₁₃	1	2	yes	—
179	#423	C₂²×C₁₁₃	1	2	yes	—
180	C₂×#100	C₂²×C₁₁₃	1	8	yes	—
181	#407	C₂²×C₁₁₃	1	2	yes	—
182	C₄×M₁₆	C₂²×C₁₁₃	1	2	yes	—
183	#343	C₂²×C₁₁₃	1	8	yes	—
184	C₂²×D₄	C₂×C₄×C₁₁₃	1	4	yes	—
185	C₂²×D₄	C₂×C₄×C₁₁₃	1	4	yes	—
186	C₂²×D₄	C₂³×C₁₁₃	1	2	yes	—
187	C₂³×C₄	C₂×C₄×C₁₁₃	1	4	yes	—
188	C₂³×C₄	C₂³×C₁₁₃	1	1	yes	characteristic
189	C₂×M₁₆	—	2	2	no	—
190	C₂×M₁₆	—	4	4	no	—
191	C₂×M₁₆	C₂³×C₁₁₃	1	4	yes	—
192	C₂×M₁₆	C₂³×C₁₁₃	1	4	yes	—
193	C₂²×C₈	—	2	2	no	—
194	#127	—	2	2	no	—
195	#127	—	2	2	no	—
196	#125	—	2	2	no	—
197	#125	—	2	2	no	—
198	#123	—	2	16	no	—
199	#121	—	2	4	no	—
200	#121	—	2	4	no	—
201	#121	—	2	16	no	—
202	C₄×D₄	—	2	4	no	—
203	C₄×D₄	—	2	4	no	—
204	#117	D₄×C₁₁₃	1	4	yes	—
205	#117	D₄×C₁₁₃	1	4	yes	—
206	C₂×C₄⋊C₄	C₂×C₄×C₁₁₃	1	4	yes	—
207	C₂×C₄⋊C₄	C₂×C₄×C₁₁₃	1	2	yes	—
208	C₂×C₄⋊C₄	C₂³×C₁₁₃	1	2	yes	—
209	C₂×C₂²⋊C₄	C₂×C₄×C₁₁₃	1	4	yes	—
210	C₂×C₂²⋊C₄	C₂×C₄×C₁₁₃	1	4	yes	—
211	C₂×C₂²⋊C₄	C₂×C₄×C₁₁₃	1	2	yes	—
212	C₂×C₄²	C₂³×C₁₁₃	1	1	yes	characteristic
213	C₂×C₄²	C₂³×C₁₁₃	1	1	yes	characteristic
214	C₄⋊C₈	—	2	4	no	—
215	C₄⋊C₈	—	2	4	no	—
216	C₄⋊C₈	—	2	4	no	—
217	#100	—	4	8	no	—
218	C₂²⋊C₈	—	2	4	no	—
219	#97	—	2	2	no	—
220	C₄×C₈	—	2	2	no	—
221	C₂⁴	C₂²×C₄×C₁₁₃	1	4	yes	—
222	C₂⁴	C₂²×C₄×C₁₁₃	1	1	yes	characteristic
223	C₂×D₄	—	2	8	no	—
224	C₂×D₄	—	2	8	no	—
225	C₂×D₄	—	2	2	no	—
226	C₂×D₄	—	2	2	no	—
227	C₂×D₄	—	2	4	no	—
228	C₂×D₄	—	2	4	no	—
229	C₂×D₄	—	8	4	no	—
230	C₂×D₄	—	8	4	no	—
231	C₂²×C₄	—	2	2	no	—
232	C₂²×C₄	—	2	2	no	—
233	C₂²×C₄	—	2	8	no	—
234	C₂²×C₄	—	8	4	no	—
235	C₂²×C₄	C₂²×C₄×C₁₁₃	1	1	yes	characteristic
236	C₂²×C₄	C₂²×C₄×C₁₁₃	1	4	yes	—
237	C₂²×C₄	C₂²×C₄×C₁₁₃	1	2	yes	—
238	C₂²×C₄	C₂²×C₄×C₁₁₃	1	2	yes	—
239	C₂²×C₄	C₂×D₄×C₁₁₃	1	2	yes	—
240	C₂²×C₄	C₂×D₄×C₁₁₃	1	2	yes	—
241	C₂²×C₄	C₂⁴×C₁₁₃	1	1	yes	characteristic
242	M₁₆	—	2	4	no	—
243	M₁₆	—	2	8	no	—
244	M₁₆	—	2	4	no	—
245	M₁₆	—	4	4	no	—
246	C₂×C₈	—	2	4	no	—
247	C₂×C₈	—	2	4	no	—
248	C₂×C₈	—	4	4	no	—
249	C₄⋊C₄	—	2	8	no	—
250	C₄⋊C₄	—	2	2	no	—
251	C₄⋊C₄	—	2	2	no	—
252	C₄⋊C₄	C₂²⋊C₄×C₁₁₃	1	8	yes	—
253	C₂²⋊C₄	—	2	8	no	—
254	C₂²⋊C₄	—	2	8	no	—
255	C₂²⋊C₄	C₂²⋊C₄×C₁₁₃	1	8	yes	—
256	C₄²	C₂×D₄×C₁₁₃	1	2	yes	—
257	C₄²	C₂×D₄×C₁₁₃	1	2	yes	—
258	C₄²	C₂×D₄×C₁₁₃	1	2	yes	—
259	C₄²	C₂×D₄×C₁₁₃	1	2	yes	—
260	C₂³	—	2	2	no	—
261	C₂³	—	2	4	no	—
262	C₂³	—	2	8	no	—
263	C₂³	—	4	2	no	—
264	C₂³	—	4	4	no	—
265	C₂³	—	4	4	no	—
266	C₂³	C₂×C₂²⋊C₄×C₁₁₃	1	2	yes	—
267	C₂³	C₂³×C₄×C₁₁₃	1	1	yes	characteristic
268	D₄	—	4	2	no	—
269	D₄	—	4	2	no	—
270	D₄	—	4	4	no	—
271	D₄	—	8	8	no	—
272	D₄	—	8	8	no	—
273	C₂×C₄	—	2	2	no	—
274	C₂×C₄	—	2	2	no	—
275	C₂×C₄	—	2	2	no	—
276	C₂×C₄	—	2	4	no	—
277	C₂×C₄	—	2	4	no	—
278	C₂×C₄	—	4	2	no	—
279	C₂×C₄	—	4	2	no	—
280	C₂×C₄	—	4	4	no	—
281	C₂×C₄	—	8	8	no	—
282	C₂×C₄	C₂×C₂²⋊C₄×C₁₁₃	1	4	yes	—
283	C₂×C₄	C₂×C₂²⋊C₄×C₁₁₃	1	4	yes	—
284	C₂×C₄	C₂×C₂²⋊C₄×C₁₁₃	1	2	yes	—
285	C₂×C₄	C₂²×D₄×C₁₁₃	1	1	yes	characteristic
286	C₂×C₄	C₂²×D₄×C₁₁₃	1	1	yes	characteristic
287	C₂×C₄	H(5,𝔽₂)×C₁₁₃	1	2	yes	—
288	C₂×C₄	2₋¹⁺⁴×C₁₁₃	1	2	yes	—
289	C₈	—	4	4	no	—
290	C₈	—	4	4	no	—
291	C₂²	—	2	2	no	—
292	C₂²	—	2	2	no	—
293	C₂²	—	2	4	no	—
294	C₂²	—	4	4	no	—
295	C₂²	—	4	4	no	—
296	C₂²	—	4	4	no	—
297	C₂²	—	8	2	no	—
298	C₂²	—	8	8	no	—
299	C₂²	C₂²×C₂²⋊C₄×C₁₁₃	1	1	yes	characteristic
300	C₂²	#520×C₁₁₃	1	2	yes	—
301	C₄	—	2	2	no	—
302	C₄	—	2	4	no	—
303	C₄	—	2	4	no	—
304	C₄	—	4	4	no	—
305	C₄	#535×C₁₁₃	1	1	yes	characteristic
306	C₄	#536×C₁₁₃	1	1	yes	characteristic
307	C₄	#537×C₁₁₃	1	2	yes	—
308	C₂	—	2	2	no	—
309	C₂	—	4	2	no	—
310	C₂	—	8	4	no	—
311	C₂	C₁₁₃×#2311	1	1	yes	characteristic
312	C₂	C₁₁₃×#2888	1	2	yes	—

	subgroup	G/H
center, socle	C₂²×C₁₁₃	C₂²×C₂²⋊C₄
commutator	C₂³	C₂³×C₄×C₁₁₃
Frattini, cosocle	C₂²×C₄	C₂⁴×C₁₁₃
prime radical	C₂×#540×C₁₁₃	C₂
LCS	G ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₁₁₃ ◁ C₂×#117×C₁₁₃ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₁₁₃×#2921	C₂	semidirect
3. C₁₁₃×#2905	C₂	semidirect
4. C₁₁₃×#1998	C₂	semidirect
5. C₁₁₃×#1965	C₂	semidirect
6. C₁₁₃×#1900	C₂	semidirect
7. C₁₁₃×#1800	C₂	semidirect
151. #21882	C₁₁₃	direct (N × H)
158. #2921	C₂×C₁₁₃	semidirect
159. #2905	C₂×C₁₁₃	semidirect
160. #1998	C₂×C₁₁₃	semidirect
161. #1965	C₂×C₁₁₃	semidirect
162. #1900	C₂×C₁₁₃	semidirect
163. #1800	C₂×C₁₁₃	semidirect
164. C₁₁₃	#21882	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂×#540×C₁₁₃	2
C₁₁₃×#2921	2
C₁₁₃×#2905	2
C₁₁₃×#1998	2
C₁₁₃×#1965	2
C₁₁₃×#1900	2
C₁₁₃×#1800	2
#21882	113

Subgroups — C113×#21882 #43289902

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₁₃×#21882 #43289902