Subgroups — C₁₁₃×#22260 #43290280

Order 28928.

1106 conjugacy classes of subgroups.

The subgroup lattice has 274 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₈×D₄×C₁₁₃	C₂	1	2	yes	maximal, complemented
2	C₄×#127×C₁₁₃	C₂	1	1	yes	maximal, characteristic
3	C₁₁₃×#1954	C₂	1	2	yes	maximal, complemented
4	C₁₁₃×#1869	C₂	1	8	yes	maximal, complemented
5	C₁₁₃×#1831	C₂	1	1	yes	maximal, characteristic, complemented
6	C₁₁₃×#1744	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂³×C₈×C₁₁₃	C₂²	1	4	yes	—
8	C₂×#127×C₁₁₃	C₄	1	1	yes	characteristic
9	C₂×C₄×D₄×C₁₁₃	C₂²	1	2	yes	—
10	C₂×C₄×D₄×C₁₁₃	C₂²	1	4	yes	—
11	C₈×D₄×C₁₁₃	—	2	8	no	—
12	C₂×C₄⋊C₈×C₁₁₃	C₂²	1	2	yes	—
13	C₂×C₂²⋊C₈×C₁₁₃	C₂²	1	4	yes	—
14	C₂×C₂²⋊C₈×C₁₁₃	C₂²	1	8	yes	—
15	C₂×C₄×C₈×C₁₁₃	C₂²	1	1	yes	characteristic
16	C₂×C₄×C₈×C₁₁₃	C₂²	1	1	yes	characteristic
17	#390×C₁₁₃	C₄	1	2	yes	—
18	#390×C₁₁₃	C₄	1	4	yes	—
19	#384×C₁₁₃	C₄	1	1	yes	characteristic
20	C₄³×C₁₁₃	C₂²	1	1	yes	characteristic
21	#336×C₁₁₃	C₂²	1	4	yes	—
22	#336×C₁₁₃	C₂²	1	4	yes	—
23	C₂²×D₄×C₁₁₃	C₂×C₄	1	2	yes	—
24	C₂²×D₄×C₁₁₃	C₂×C₄	1	4	yes	—
25	C₂³×C₄×C₁₁₃	C₂³	1	4	yes	—
26	C₂²×C₈×C₁₁₃	—	2	8	no	—
27	C₂²×C₈×C₁₁₃	—	4	8	no	—
28	C₂²×C₈×C₁₁₃	C₂³	1	4	yes	—
29	C₂²×C₈×C₁₁₃	C₂³	1	2	yes	—
30	C₂²×C₈×C₁₁₃	C₂³	1	2	yes	—
31	#127×C₁₁₃	C₈	1	8	yes	complemented
32	C₄×D₄×C₁₁₃	—	2	4	no	—
33	C₄×D₄×C₁₁₃	—	2	4	no	—
34	C₄×D₄×C₁₁₃	—	4	8	no	—
35	C₂×C₄⋊C₄×C₁₁₃	C₂×C₄	1	2	yes	—
36	C₂×C₄⋊C₄×C₁₁₃	C₂×C₄	1	4	yes	—
37	C₂×C₂²⋊C₄×C₁₁₃	C₂×C₄	1	4	yes	—
38	C₂×C₂²⋊C₄×C₁₁₃	C₂×C₄	1	8	yes	—
39	C₂×C₄²×C₁₁₃	C₂×C₄	1	1	yes	characteristic
40	C₂×C₄²×C₁₁₃	C₂×C₄	1	2	yes	—
41	C₂×C₄²×C₁₁₃	C₂×C₄	1	1	yes	characteristic
42	C₂×C₄²×C₁₁₃	C₂³	1	2	yes	—
43	C₂×C₄²×C₁₁₃	C₂³	1	1	yes	characteristic
44	C₄⋊C₈×C₁₁₃	—	2	4	no	—
45	C₂²⋊C₈×C₁₁₃	—	2	8	no	—
46	C₂²⋊C₈×C₁₁₃	—	4	8	no	—
47	C₄×C₈×C₁₁₃	—	2	2	no	—
48	C₄×C₈×C₁₁₃	D₄	1	4	yes	complemented
49	C₂⁴×C₁₁₃	C₂²×C₄	1	4	yes	—
50	C₂×D₄×C₁₁₃	—	2	4	no	—
51	C₂×D₄×C₁₁₃	—	2	4	no	—
52	C₂×D₄×C₁₁₃	—	4	4	no	—
53	C₂×D₄×C₁₁₃	—	4	4	no	—
54	C₂×D₄×C₁₁₃	C₂×C₈	1	8	yes	complemented
55	C₂×D₄×C₁₁₃	C₂×C₈	1	16	yes	complemented
56	C₂²×C₄×C₁₁₃	—	2	4	no	—
57	C₂²×C₄×C₁₁₃	—	2	4	no	—
58	C₂²×C₄×C₁₁₃	—	2	8	no	—
59	C₂²×C₄×C₁₁₃	—	2	8	no	—
60	C₂²×C₄×C₁₁₃	C₂²×C₄	1	2	yes	—
61	C₂²×C₄×C₁₁₃	C₂²×C₄	1	4	yes	—
62	C₂²×C₄×C₁₁₃	C₂²×C₄	1	1	yes	characteristic
63	C₂²×C₄×C₁₁₃	C₂²×C₄	1	2	yes	—
64	C₂²×C₄×C₁₁₃	C₂²×C₄	1	1	yes	characteristic
65	C₂²×C₄×C₁₁₃	C₂⁴	1	1	yes	characteristic
66	C₂×C₈×C₁₁₃	—	2	4	no	—
67	C₂×C₈×C₁₁₃	—	2	4	no	—
68	C₂×C₈×C₁₁₃	—	2	2	no	—
69	C₂×C₈×C₁₁₃	—	2	4	no	—
70	C₂×C₈×C₁₁₃	—	2	8	no	—
71	C₂×C₈×C₁₁₃	—	4	16	no	—
72	C₂×C₈×C₁₁₃	C₂×D₄	1	4	yes	—
73	C₄⋊C₄×C₁₁₃	—	2	4	no	—
74	C₄⋊C₄×C₁₁₃	—	4	4	no	—
75	C₂²⋊C₄×C₁₁₃	—	2	8	no	—
76	C₂²⋊C₄×C₁₁₃	—	4	8	no	—
77	C₄²×C₁₁₃	—	2	2	no	—
78	C₄²×C₁₁₃	—	2	4	no	—
79	C₄²×C₁₁₃	—	4	2	no	—
80	C₄²×C₁₁₃	C₂×C₈	1	4	yes	complemented
81	C₄²×C₁₁₃	C₂×D₄	1	2	yes	—
82	C₄²×C₁₁₃	#41	1	2	yes	—
83	C₂³×C₁₁₃	—	2	4	no	—
84	C₂³×C₁₁₃	—	2	8	no	—
85	C₂³×C₁₁₃	—	2	4	no	—
86	C₂³×C₁₁₃	—	2	8	no	—
87	C₂³×C₁₁₃	C₂²×C₈	1	8	yes	—
88	C₂³×C₁₁₃	C₂³×C₄	1	1	yes	characteristic
89	D₄×C₁₁₃	—	2	16	no	—
90	D₄×C₁₁₃	—	4	16	no	—
91	C₂×C₄×C₁₁₃	—	2	2	no	—
92	C₂×C₄×C₁₁₃	—	2	2	no	—
93	C₂×C₄×C₁₁₃	—	2	2	no	—
94	C₂×C₄×C₁₁₃	—	2	4	no	—
95	C₂×C₄×C₁₁₃	—	2	2	no	—
96	C₂×C₄×C₁₁₃	—	2	8	no	—
97	C₂×C₄×C₁₁₃	—	2	2	no	—
98	C₂×C₄×C₁₁₃	—	2	2	no	—
99	C₂×C₄×C₁₁₃	—	4	4	no	—
100	C₂×C₄×C₁₁₃	—	4	4	no	—
101	C₂×C₄×C₁₁₃	—	4	8	no	—
102	C₂×C₄×C₁₁₃	C₄×D₄	1	2	yes	—
103	C₂×C₄×C₁₁₃	C₄×D₄	1	2	yes	—
104	C₂×C₄×C₁₁₃	C₂²×C₈	1	4	yes	—
105	C₂×C₄×C₁₁₃	C₂²×C₈	1	2	yes	—
106	C₂×C₄×C₁₁₃	#131	1	2	yes	—
107	C₂×C₄×C₁₁₃	C₂²×D₄	1	1	yes	characteristic
108	C₂×C₄×C₁₁₃	C₂×#41	1	1	yes	characteristic
109	C₂×C₄×C₁₁₃	H(5,𝔽₂)	1	4	yes	—
110	C₈×C₁₁₃	—	2	4	no	—
111	C₈×C₁₁₃	—	4	4	no	—
112	C₈×C₁₁₃	—	4	2	no	—
113	C₂²×C₁₁₃	—	2	8	no	—
114	C₂²×C₁₁₃	—	2	16	no	—
115	C₂²×C₁₁₃	—	4	4	no	—
116	C₂²×C₁₁₃	—	4	4	no	—
117	C₂²×C₁₁₃	—	4	8	no	—
118	C₂²×C₁₁₃	C₂×C₄×D₄	1	1	yes	characteristic
119	C₂²×C₁₁₃	#518	1	2	yes	—
120	C₂²×C₁₁₃	C₂³×C₈	1	1	yes	characteristic
121	C₂²×C₁₁₃	C₂×#131	1	1	yes	characteristic
122	C₂²×C₁₁₃	#568	1	2	yes	—
123	C₄×C₁₁₃	—	2	2	no	—
124	C₄×C₁₁₃	—	2	4	no	—
125	C₄×C₁₁₃	—	4	4	no	—
126	C₄×C₁₁₃	—	4	2	no	—
127	C₄×C₁₁₃	C₈×D₄	1	4	yes	—
128	C₄×C₁₁₃	#535	1	2	yes	—
129	C₄×C₁₁₃	#540	1	2	yes	—
130	#22260	C₁₁₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
131	C₂×C₁₁₃	—	4	8	no	—
132	C₂×C₁₁₃	#2328	1	1	yes	characteristic
133	C₂×C₁₁₃	C₂×C₈×D₄	1	1	yes	characteristic
134	C₂×C₁₁₃	#2932	1	1	yes	characteristic
135	C₂×C₁₁₃	#2972	1	2	yes	—
136	C₂×C₁₁₃	#2981	1	2	yes	—
137	C₂×C₈×D₄	C₂×C₁₁₃	1	2	yes	complemented
138	C₄×#127	C₂×C₁₁₃	1	1	yes	characteristic
139	#1954	C₂×C₁₁₃	1	2	yes	complemented
140	#1869	C₂×C₁₁₃	1	8	yes	complemented
141	#1831	C₂×C₁₁₃	1	1	yes	characteristic, complemented
142	#1744	C₂×C₁₁₃	1	1	yes	characteristic, complemented
143	C₁₁₃	#22260	1	1	yes	characteristic, direct product factor, Sylow
144	C₂³×C₈	C₂²×C₁₁₃	1	4	yes	—
145	C₂×#127	C₄×C₁₁₃	1	1	yes	characteristic
146	C₂×C₄×D₄	C₂²×C₁₁₃	1	2	yes	—
147	C₂×C₄×D₄	C₂²×C₁₁₃	1	4	yes	—
148	C₈×D₄	—	2	8	no	—
149	C₂×C₄⋊C₈	C₂²×C₁₁₃	1	2	yes	—
150	C₂×C₂²⋊C₈	C₂²×C₁₁₃	1	4	yes	—
151	C₂×C₂²⋊C₈	C₂²×C₁₁₃	1	8	yes	—
152	C₂×C₄×C₈	C₂²×C₁₁₃	1	1	yes	characteristic
153	C₂×C₄×C₈	C₂²×C₁₁₃	1	1	yes	characteristic
154	#390	C₄×C₁₁₃	1	2	yes	—
155	#390	C₄×C₁₁₃	1	4	yes	—
156	#384	C₄×C₁₁₃	1	1	yes	characteristic
157	C₄³	C₂²×C₁₁₃	1	1	yes	characteristic
158	#336	C₂²×C₁₁₃	1	4	yes	—
159	#336	C₂²×C₁₁₃	1	4	yes	—
160	C₂²×D₄	C₂×C₄×C₁₁₃	1	2	yes	—
161	C₂²×D₄	C₂×C₄×C₁₁₃	1	4	yes	—
162	C₂³×C₄	C₂³×C₁₁₃	1	4	yes	—
163	C₂²×C₈	—	2	8	no	—
164	C₂²×C₈	—	4	8	no	—
165	C₂²×C₈	C₂³×C₁₁₃	1	4	yes	—
166	C₂²×C₈	C₂³×C₁₁₃	1	2	yes	—
167	C₂²×C₈	C₂³×C₁₁₃	1	2	yes	—
168	#127	C₈×C₁₁₃	1	8	yes	complemented
169	C₄×D₄	—	2	4	no	—
170	C₄×D₄	—	2	4	no	—
171	C₄×D₄	—	4	8	no	—
172	C₂×C₄⋊C₄	C₂×C₄×C₁₁₃	1	2	yes	—
173	C₂×C₄⋊C₄	C₂×C₄×C₁₁₃	1	4	yes	—
174	C₂×C₂²⋊C₄	C₂×C₄×C₁₁₃	1	4	yes	—
175	C₂×C₂²⋊C₄	C₂×C₄×C₁₁₃	1	8	yes	—
176	C₂×C₄²	C₂×C₄×C₁₁₃	1	1	yes	characteristic
177	C₂×C₄²	C₂×C₄×C₁₁₃	1	2	yes	—
178	C₂×C₄²	C₂×C₄×C₁₁₃	1	1	yes	characteristic
179	C₂×C₄²	C₂³×C₁₁₃	1	2	yes	—
180	C₂×C₄²	C₂³×C₁₁₃	1	1	yes	characteristic
181	C₄⋊C₈	—	2	4	no	—
182	C₂²⋊C₈	—	2	8	no	—
183	C₂²⋊C₈	—	4	8	no	—
184	C₄×C₈	—	2	2	no	—
185	C₄×C₈	D₄×C₁₁₃	1	4	yes	complemented
186	C₂⁴	C₂²×C₄×C₁₁₃	1	4	yes	—
187	C₂×D₄	—	2	4	no	—
188	C₂×D₄	—	2	4	no	—
189	C₂×D₄	—	4	4	no	—
190	C₂×D₄	—	4	4	no	—
191	C₂×D₄	C₂×C₈×C₁₁₃	1	8	yes	complemented
192	C₂×D₄	C₂×C₈×C₁₁₃	1	16	yes	complemented
193	C₂²×C₄	—	2	4	no	—
194	C₂²×C₄	—	2	4	no	—
195	C₂²×C₄	—	2	8	no	—
196	C₂²×C₄	—	2	8	no	—
197	C₂²×C₄	C₂²×C₄×C₁₁₃	1	2	yes	—
198	C₂²×C₄	C₂²×C₄×C₁₁₃	1	4	yes	—
199	C₂²×C₄	C₂²×C₄×C₁₁₃	1	1	yes	characteristic
200	C₂²×C₄	C₂²×C₄×C₁₁₃	1	2	yes	—
201	C₂²×C₄	C₂²×C₄×C₁₁₃	1	1	yes	characteristic
202	C₂²×C₄	C₂⁴×C₁₁₃	1	1	yes	characteristic
203	C₂×C₈	—	2	4	no	—
204	C₂×C₈	—	2	4	no	—
205	C₂×C₈	—	2	2	no	—
206	C₂×C₈	—	2	4	no	—
207	C₂×C₈	—	2	8	no	—
208	C₂×C₈	—	4	16	no	—
209	C₂×C₈	C₂×D₄×C₁₁₃	1	4	yes	—
210	C₄⋊C₄	—	2	4	no	—
211	C₄⋊C₄	—	4	4	no	—
212	C₂²⋊C₄	—	2	8	no	—
213	C₂²⋊C₄	—	4	8	no	—
214	C₄²	—	2	2	no	—
215	C₄²	—	2	4	no	—
216	C₄²	—	4	2	no	—
217	C₄²	C₂×C₈×C₁₁₃	1	4	yes	complemented
218	C₄²	C₂×D₄×C₁₁₃	1	2	yes	—
219	C₄²	#41×C₁₁₃	1	2	yes	—
220	C₂³	—	2	4	no	—
221	C₂³	—	2	8	no	—
222	C₂³	—	2	4	no	—
223	C₂³	—	2	8	no	—
224	C₂³	C₂²×C₈×C₁₁₃	1	8	yes	—
225	C₂³	C₂³×C₄×C₁₁₃	1	1	yes	characteristic
226	D₄	—	2	16	no	—
227	D₄	—	4	16	no	—
228	C₂×C₄	—	2	2	no	—
229	C₂×C₄	—	2	2	no	—
230	C₂×C₄	—	2	2	no	—
231	C₂×C₄	—	2	4	no	—
232	C₂×C₄	—	2	2	no	—
233	C₂×C₄	—	2	8	no	—
234	C₂×C₄	—	2	2	no	—
235	C₂×C₄	—	2	2	no	—
236	C₂×C₄	—	4	4	no	—
237	C₂×C₄	—	4	4	no	—
238	C₂×C₄	—	4	8	no	—
239	C₂×C₄	C₄×D₄×C₁₁₃	1	2	yes	—
240	C₂×C₄	C₄×D₄×C₁₁₃	1	2	yes	—
241	C₂×C₄	C₂²×C₈×C₁₁₃	1	4	yes	—
242	C₂×C₄	C₂²×C₈×C₁₁₃	1	2	yes	—
243	C₂×C₄	#131×C₁₁₃	1	2	yes	—
244	C₂×C₄	C₂²×D₄×C₁₁₃	1	1	yes	characteristic
245	C₂×C₄	C₂×#41×C₁₁₃	1	1	yes	characteristic
246	C₂×C₄	H(5,𝔽₂)×C₁₁₃	1	4	yes	—
247	C₈	—	2	4	no	—
248	C₈	—	4	4	no	—
249	C₈	—	4	2	no	—
250	C₂²	—	2	8	no	—
251	C₂²	—	2	16	no	—
252	C₂²	—	4	4	no	—
253	C₂²	—	4	4	no	—
254	C₂²	—	4	8	no	—
255	C₂²	C₂×C₄×D₄×C₁₁₃	1	1	yes	characteristic
256	C₂²	#518×C₁₁₃	1	2	yes	—
257	C₂²	C₂³×C₈×C₁₁₃	1	1	yes	characteristic
258	C₂²	C₂×#131×C₁₁₃	1	1	yes	characteristic
259	C₂²	#568×C₁₁₃	1	2	yes	—
260	C₄	—	2	2	no	—
261	C₄	—	2	4	no	—
262	C₄	—	4	4	no	—
263	C₄	—	4	2	no	—
264	C₄	C₈×D₄×C₁₁₃	1	4	yes	—
265	C₄	#535×C₁₁₃	1	2	yes	—
266	C₄	#540×C₁₁₃	1	2	yes	—
267	C₂	—	4	8	no	—
268	C₂	C₁₁₃×#2328	1	1	yes	characteristic
269	C₂	C₂×C₈×D₄×C₁₁₃	1	1	yes	characteristic
270	C₂	C₁₁₃×#2932	1	1	yes	characteristic
271	C₂	C₁₁₃×#2972	1	2	yes	—
272	C₂	C₁₁₃×#2981	1	2	yes	—

	subgroup	G/H
center	C₂²×C₄×C₁₁₃	C₂⁴
commutator	C₂²	C₂³×C₈×C₁₁₃
Frattini, cosocle	C₂²×C₄	C₂⁴×C₁₁₃
socle	C₂³×C₁₁₃	C₂³×C₄
prime radical	C₂×#127×C₁₁₃	C₄
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₄×C₁₁₃ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₂×C₈×D₄×C₁₁₃	C₂	semidirect
3. C₁₁₃×#1954	C₂	semidirect
4. C₁₁₃×#1869	C₂	semidirect
5. C₁₁₃×#1831	C₂	semidirect
6. C₁₁₃×#1744	C₂	semidirect
31. #127×C₁₁₃	C₈	semidirect
48. C₄×C₈×C₁₁₃	D₄	semidirect
54. C₂×D₄×C₁₁₃	C₂×C₈	semidirect
55. C₂×D₄×C₁₁₃	C₂×C₈	semidirect
80. C₄²×C₁₁₃	C₂×C₈	semidirect
130. #22260	C₁₁₃	direct (N × H)
137. C₂×C₈×D₄	C₂×C₁₁₃	semidirect
139. #1954	C₂×C₁₁₃	semidirect
140. #1869	C₂×C₁₁₃	semidirect
141. #1831	C₂×C₁₁₃	semidirect
142. #1744	C₂×C₁₁₃	semidirect
143. C₁₁₃	#22260	direct (N × H)
168. #127	C₈×C₁₁₃	semidirect
185. C₄×C₈	D₄×C₁₁₃	semidirect
191. C₂×D₄	C₂×C₈×C₁₁₃	semidirect
192. C₂×D₄	C₂×C₈×C₁₁₃	semidirect
217. C₄²	C₂×C₈×C₁₁₃	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₈×D₄×C₁₁₃	2
C₄×#127×C₁₁₃	2
C₁₁₃×#1954	2
C₁₁₃×#1869	2
C₁₁₃×#1831	2
C₁₁₃×#1744	2
#22260	113

Subgroups — C113×#22260 #43290280

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₁₃×#22260 #43290280