Subgroups — C₁₁₃×#20587 #43288607

Order 28928.

3634 conjugacy classes of subgroups.

The subgroup lattice has 278 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×2₋¹⁺⁴×C₁₁₃	C₂	1	1	yes	maximal, characteristic
2	C₂×#410×C₁₁₃	C₂	1	4	yes	maximal, complemented
3	C₁₁₃×#2310	C₂	1	2	yes	maximal, complemented
4	C₁₁₃×#1782	C₂	1	8	yes	maximal, complemented
5	C₂×2₋¹⁺⁴×C₁₁₃	—	2	16	no	—
6	C₂×2₋¹⁺⁴×C₁₁₃	—	2	8	no	—
7	C₂²×#41×C₁₁₃	C₄	1	4	yes	complemented
8	C₂²×#41×C₁₁₃	C₂²	1	4	yes	—
9	C₂²×#41×C₁₁₃	C₂²	1	2	yes	—
10	C₂³×Q₈×C₁₁₃	C₄	1	4	yes	complemented
11	C₂³×Q₈×C₁₁₃	C₂²	1	1	yes	characteristic
12	C₂×#117×C₁₁₃	C₂²	1	4	yes	—
13	#410×C₁₁₃	—	2	16	no	—
14	C₂×#99×C₁₁₃	C₂²	1	16	yes	—
15	#391×C₁₁₃	—	2	8	no	—
16	#381×C₁₁₃	C₂²	1	8	yes	—
17	2₋¹⁺⁴×C₁₁₃	—	2	64	no	—
18	2₋¹⁺⁴×C₁₁₃	—	2	64	no	—
19	C₂×#41×C₁₁₃	—	2	16	no	—
20	C₂×#41×C₁₁₃	—	2	16	no	—
21	C₂×#41×C₁₁₃	—	2	32	no	—
22	C₂×#41×C₁₁₃	—	2	16	no	—
23	C₂×#41×C₁₁₃	—	2	16	no	—
24	C₂×#41×C₁₁₃	—	2	8	no	—
25	C₂×#41×C₁₁₃	—	2	8	no	—
26	C₂×#41×C₁₁₃	D₄	1	16	yes	—
27	C₂²×Q₈×C₁₁₃	—	2	16	no	—
28	C₂²×Q₈×C₁₁₃	—	2	16	no	—
29	C₂²×Q₈×C₁₁₃	—	2	16	no	—
30	C₂²×Q₈×C₁₁₃	—	2	8	no	—
31	C₂²×Q₈×C₁₁₃	C₂×C₄	1	4	yes	—
32	C₂²×Q₈×C₁₁₃	C₂×C₄	1	2	yes	—
33	C₂²×Q₈×C₁₁₃	C₂×C₄	1	4	yes	—
34	C₂²×Q₈×C₁₁₃	D₄	1	8	yes	—
35	C₂²×D₄×C₁₁₃	C₂×C₄	1	4	yes	—
36	C₂²×D₄×C₁₁₃	C₂×C₄	1	2	yes	—
37	C₂²×D₄×C₁₁₃	C₂³	1	4	yes	—
38	C₂³×C₄×C₁₁₃	C₂×C₄	1	8	yes	—
39	C₂³×C₄×C₁₁₃	C₂×C₄	1	4	yes	—
40	C₂³×C₄×C₁₁₃	C₂³	1	2	yes	—
41	C₂³×C₄×C₁₁₃	C₂³	1	1	yes	characteristic
42	#117×C₁₁₃	—	2	16	no	—
43	C₂×C₄⋊C₄×C₁₁₃	—	2	4	no	—
44	C₂×C₂²⋊C₄×C₁₁₃	C₂³	1	8	yes	—
45	C₂×C₄²×C₁₁₃	—	2	4	no	—
46	#99×C₁₁₃	—	2	32	no	—
47	#95×C₁₁₃	—	2	16	no	—
48	C₂⁴×C₁₁₃	C₂²×C₄	1	4	yes	—
49	C₂⁴×C₁₁₃	C₂⁴	1	1	yes	characteristic
50	#41×C₁₁₃	—	2	128	no	—
51	#41×C₁₁₃	—	2	128	no	—
52	#41×C₁₁₃	—	2	32	no	—
53	#41×C₁₁₃	—	2	32	no	—
54	C₂×Q₈×C₁₁₃	—	2	128	no	—
55	C₂×Q₈×C₁₁₃	—	2	16	no	—
56	C₂×Q₈×C₁₁₃	—	2	8	no	—
57	C₂×Q₈×C₁₁₃	—	2	4	no	—
58	C₂×Q₈×C₁₁₃	—	2	8	no	—
59	C₂×Q₈×C₁₁₃	—	2	8	no	—
60	C₂×Q₈×C₁₁₃	—	2	4	no	—
61	C₂×Q₈×C₁₁₃	—	2	16	no	—
62	C₂×Q₈×C₁₁₃	—	2	16	no	—
63	C₂×Q₈×C₁₁₃	C₂²⋊C₄	1	8	yes	—
64	C₂×Q₈×C₁₁₃	C₂²⋊C₄	1	16	yes	—
65	C₂×D₄×C₁₁₃	—	2	16	no	—
66	C₂×D₄×C₁₁₃	—	2	8	no	—
67	C₂×D₄×C₁₁₃	—	2	8	no	—
68	C₂×D₄×C₁₁₃	—	2	8	no	—
69	C₂×D₄×C₁₁₃	—	2	4	no	—
70	C₂×D₄×C₁₁₃	—	2	4	no	—
71	C₂×D₄×C₁₁₃	C₂²⋊C₄	1	8	yes	—
72	C₂×D₄×C₁₁₃	C₂×D₄	1	16	yes	—
73	C₂²×C₄×C₁₁₃	—	2	16	no	—
74	C₂²×C₄×C₁₁₃	—	2	8	no	—
75	C₂²×C₄×C₁₁₃	—	2	16	no	—
76	C₂²×C₄×C₁₁₃	—	2	16	no	—
77	C₂²×C₄×C₁₁₃	—	2	8	no	—
78	C₂²×C₄×C₁₁₃	—	2	8	no	—
79	C₂²×C₄×C₁₁₃	—	2	8	no	—
80	C₂²×C₄×C₁₁₃	C₂²⋊C₄	1	16	yes	—
81	C₂²×C₄×C₁₁₃	C₂²×C₄	1	4	yes	—
82	C₂²×C₄×C₁₁₃	C₂²×C₄	1	4	yes	—
83	C₂²×C₄×C₁₁₃	C₂²×C₄	1	2	yes	—
84	C₂²×C₄×C₁₁₃	C₂×D₄	1	8	yes	—
85	C₂²×C₄×C₁₁₃	C₂×D₄	1	4	yes	—
86	C₂²×C₄×C₁₁₃	C₂×D₄	1	8	yes	—
87	C₄⋊C₄×C₁₁₃	—	4	8	no	—
88	C₂²⋊C₄×C₁₁₃	—	2	16	no	—
89	C₄²×C₁₁₃	—	4	8	no	—
90	C₂³×C₁₁₃	—	2	4	no	—
91	C₂³×C₁₁₃	—	2	4	no	—
92	C₂³×C₁₁₃	—	4	2	no	—
93	C₂³×C₁₁₃	—	4	8	no	—
94	C₂³×C₁₁₃	C₂×C₂²⋊C₄	1	8	yes	—
95	C₂³×C₁₁₃	C₂³×C₄	1	1	yes	characteristic
96	C₂³×C₁₁₃	C₂²×D₄	1	2	yes	—
97	C₂³×C₁₁₃	C₂²×D₄	1	4	yes	—
98	Q₈×C₁₁₃	—	2	32	no	—
99	Q₈×C₁₁₃	—	2	16	no	—
100	Q₈×C₁₁₃	—	2	32	no	—
101	D₄×C₁₁₃	—	2	32	no	—
102	D₄×C₁₁₃	—	2	32	no	—
103	D₄×C₁₁₃	—	2	16	no	—
104	C₂×C₄×C₁₁₃	—	2	64	no	—
105	C₂×C₄×C₁₁₃	—	2	4	no	—
106	C₂×C₄×C₁₁₃	—	2	8	no	—
107	C₂×C₄×C₁₁₃	—	2	4	no	—
108	C₂×C₄×C₁₁₃	—	2	16	no	—
109	C₂×C₄×C₁₁₃	—	2	32	no	—
110	C₂×C₄×C₁₁₃	—	4	16	no	—
111	C₂×C₄×C₁₁₃	—	4	8	no	—
112	C₂×C₄×C₁₁₃	C₂×C₂²⋊C₄	1	8	yes	—
113	C₂×C₄×C₁₁₃	C₂×C₂²⋊C₄	1	8	yes	—
114	C₂×C₄×C₁₁₃	C₂×C₂²⋊C₄	1	4	yes	—
115	C₂×C₄×C₁₁₃	C₂×C₂²⋊C₄	1	4	yes	—
116	C₂×C₄×C₁₁₃	C₂×C₂²⋊C₄	1	4	yes	—
117	C₂×C₄×C₁₁₃	#120	1	8	yes	—
118	C₂×C₄×C₁₁₃	#120	1	8	yes	—
119	C₂²×C₁₁₃	—	2	4	no	—
120	C₂²×C₁₁₃	—	2	2	no	—
121	C₂²×C₁₁₃	—	2	8	no	—
122	C₂²×C₁₁₃	—	4	4	no	—
123	C₂²×C₁₁₃	—	4	8	no	—
124	C₂²×C₁₁₃	—	4	8	no	—
125	C₂²×C₁₁₃	—	4	8	no	—
126	C₂²×C₁₁₃	C₂²×C₂²⋊C₄	1	2	yes	—
127	C₂²×C₁₁₃	C₂²×C₂²⋊C₄	1	1	yes	characteristic
128	C₂²×C₁₁₃	C₂×#120	1	4	yes	—
129	C₄×C₁₁₃	—	2	8	no	—
130	C₄×C₁₁₃	—	2	8	no	—
131	C₄×C₁₁₃	—	8	8	no	—
132	C₄×C₁₁₃	#379	1	8	yes	—
133	#20587	C₁₁₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
134	C₂×C₁₁₃	—	2	4	no	—
135	C₂×C₁₁₃	—	2	2	no	—
136	C₂×C₁₁₃	—	4	8	no	—
137	C₂×C₁₁₃	C₂×#379	1	1	yes	characteristic
138	C₂×C₁₁₃	#2884	1	2	yes	—
139	C₂²×2₋¹⁺⁴	C₂×C₁₁₃	1	1	yes	characteristic
140	C₂×#410	C₂×C₁₁₃	1	4	yes	complemented
141	#2310	C₂×C₁₁₃	1	2	yes	complemented
142	#1782	C₂×C₁₁₃	1	8	yes	complemented
143	C₁₁₃	#20587	1	1	yes	characteristic, direct product factor, Sylow
144	C₂×2₋¹⁺⁴	—	2	16	no	—
145	C₂×2₋¹⁺⁴	—	2	8	no	—
146	C₂²×#41	C₄×C₁₁₃	1	4	yes	complemented
147	C₂²×#41	C₂²×C₁₁₃	1	4	yes	—
148	C₂²×#41	C₂²×C₁₁₃	1	2	yes	—
149	C₂³×Q₈	C₄×C₁₁₃	1	4	yes	complemented
150	C₂³×Q₈	C₂²×C₁₁₃	1	1	yes	characteristic
151	C₂×#117	C₂²×C₁₁₃	1	4	yes	—
152	#410	—	2	16	no	—
153	C₂×#99	C₂²×C₁₁₃	1	16	yes	—
154	#391	—	2	8	no	—
155	#381	C₂²×C₁₁₃	1	8	yes	—
156	2₋¹⁺⁴	—	2	64	no	—
157	2₋¹⁺⁴	—	2	64	no	—
158	C₂×#41	—	2	16	no	—
159	C₂×#41	—	2	16	no	—
160	C₂×#41	—	2	32	no	—
161	C₂×#41	—	2	16	no	—
162	C₂×#41	—	2	16	no	—
163	C₂×#41	—	2	8	no	—
164	C₂×#41	—	2	8	no	—
165	C₂×#41	D₄×C₁₁₃	1	16	yes	—
166	C₂²×Q₈	—	2	16	no	—
167	C₂²×Q₈	—	2	16	no	—
168	C₂²×Q₈	—	2	16	no	—
169	C₂²×Q₈	—	2	8	no	—
170	C₂²×Q₈	C₂×C₄×C₁₁₃	1	4	yes	—
171	C₂²×Q₈	C₂×C₄×C₁₁₃	1	2	yes	—
172	C₂²×Q₈	C₂×C₄×C₁₁₃	1	4	yes	—
173	C₂²×Q₈	D₄×C₁₁₃	1	8	yes	—
174	C₂²×D₄	C₂×C₄×C₁₁₃	1	4	yes	—
175	C₂²×D₄	C₂×C₄×C₁₁₃	1	2	yes	—
176	C₂²×D₄	C₂³×C₁₁₃	1	4	yes	—
177	C₂³×C₄	C₂×C₄×C₁₁₃	1	8	yes	—
178	C₂³×C₄	C₂×C₄×C₁₁₃	1	4	yes	—
179	C₂³×C₄	C₂³×C₁₁₃	1	2	yes	—
180	C₂³×C₄	C₂³×C₁₁₃	1	1	yes	characteristic
181	#117	—	2	16	no	—
182	C₂×C₄⋊C₄	—	2	4	no	—
183	C₂×C₂²⋊C₄	C₂³×C₁₁₃	1	8	yes	—
184	C₂×C₄²	—	2	4	no	—
185	#99	—	2	32	no	—
186	#95	—	2	16	no	—
187	C₂⁴	C₂²×C₄×C₁₁₃	1	4	yes	—
188	C₂⁴	C₂⁴×C₁₁₃	1	1	yes	characteristic
189	#41	—	2	128	no	—
190	#41	—	2	128	no	—
191	#41	—	2	32	no	—
192	#41	—	2	32	no	—
193	C₂×Q₈	—	2	128	no	—
194	C₂×Q₈	—	2	16	no	—
195	C₂×Q₈	—	2	8	no	—
196	C₂×Q₈	—	2	4	no	—
197	C₂×Q₈	—	2	8	no	—
198	C₂×Q₈	—	2	4	no	—
199	C₂×Q₈	—	2	8	no	—
200	C₂×Q₈	—	2	16	no	—
201	C₂×Q₈	—	2	16	no	—
202	C₂×Q₈	C₂²⋊C₄×C₁₁₃	1	8	yes	—
203	C₂×Q₈	C₂²⋊C₄×C₁₁₃	1	16	yes	—
204	C₂×D₄	—	2	16	no	—
205	C₂×D₄	—	2	8	no	—
206	C₂×D₄	—	2	8	no	—
207	C₂×D₄	—	2	8	no	—
208	C₂×D₄	—	2	4	no	—
209	C₂×D₄	—	2	4	no	—
210	C₂×D₄	C₂²⋊C₄×C₁₁₃	1	8	yes	—
211	C₂×D₄	C₂×D₄×C₁₁₃	1	16	yes	—
212	C₂²×C₄	—	2	16	no	—
213	C₂²×C₄	—	2	8	no	—
214	C₂²×C₄	—	2	16	no	—
215	C₂²×C₄	—	2	16	no	—
216	C₂²×C₄	—	2	8	no	—
217	C₂²×C₄	—	2	8	no	—
218	C₂²×C₄	—	2	8	no	—
219	C₂²×C₄	C₂²⋊C₄×C₁₁₃	1	16	yes	—
220	C₂²×C₄	C₂²×C₄×C₁₁₃	1	4	yes	—
221	C₂²×C₄	C₂²×C₄×C₁₁₃	1	4	yes	—
222	C₂²×C₄	C₂²×C₄×C₁₁₃	1	2	yes	—
223	C₂²×C₄	C₂×D₄×C₁₁₃	1	8	yes	—
224	C₂²×C₄	C₂×D₄×C₁₁₃	1	4	yes	—
225	C₂²×C₄	C₂×D₄×C₁₁₃	1	8	yes	—
226	C₄⋊C₄	—	4	8	no	—
227	C₂²⋊C₄	—	2	16	no	—
228	C₄²	—	4	8	no	—
229	C₂³	—	2	4	no	—
230	C₂³	—	2	4	no	—
231	C₂³	—	4	2	no	—
232	C₂³	—	4	8	no	—
233	C₂³	C₂×C₂²⋊C₄×C₁₁₃	1	8	yes	—
234	C₂³	C₂³×C₄×C₁₁₃	1	1	yes	characteristic
235	C₂³	C₂²×D₄×C₁₁₃	1	2	yes	—
236	C₂³	C₂²×D₄×C₁₁₃	1	4	yes	—
237	Q₈	—	2	32	no	—
238	Q₈	—	2	16	no	—
239	Q₈	—	2	32	no	—
240	D₄	—	2	32	no	—
241	D₄	—	2	32	no	—
242	D₄	—	2	16	no	—
243	C₂×C₄	—	2	64	no	—
244	C₂×C₄	—	2	4	no	—
245	C₂×C₄	—	2	8	no	—
246	C₂×C₄	—	2	4	no	—
247	C₂×C₄	—	2	16	no	—
248	C₂×C₄	—	2	32	no	—
249	C₂×C₄	—	4	16	no	—
250	C₂×C₄	—	4	8	no	—
251	C₂×C₄	C₂×C₂²⋊C₄×C₁₁₃	1	8	yes	—
252	C₂×C₄	C₂×C₂²⋊C₄×C₁₁₃	1	8	yes	—
253	C₂×C₄	C₂×C₂²⋊C₄×C₁₁₃	1	4	yes	—
254	C₂×C₄	C₂×C₂²⋊C₄×C₁₁₃	1	4	yes	—
255	C₂×C₄	C₂×C₂²⋊C₄×C₁₁₃	1	4	yes	—
256	C₂×C₄	#120×C₁₁₃	1	8	yes	—
257	C₂×C₄	#120×C₁₁₃	1	8	yes	—
258	C₂²	—	2	4	no	—
259	C₂²	—	2	2	no	—
260	C₂²	—	2	8	no	—
261	C₂²	—	4	4	no	—
262	C₂²	—	4	8	no	—
263	C₂²	—	4	8	no	—
264	C₂²	—	4	8	no	—
265	C₂²	C₂²×C₂²⋊C₄×C₁₁₃	1	2	yes	—
266	C₂²	C₂²×C₂²⋊C₄×C₁₁₃	1	1	yes	characteristic
267	C₂²	C₂×#120×C₁₁₃	1	4	yes	—
268	C₄	—	2	8	no	—
269	C₄	—	2	8	no	—
270	C₄	—	8	8	no	—
271	C₄	#379×C₁₁₃	1	8	yes	—
272	C₂	—	2	4	no	—
273	C₂	—	2	2	no	—
274	C₂	—	4	8	no	—
275	C₂	C₂×#379×C₁₁₃	1	1	yes	characteristic
276	C₂	C₁₁₃×#2884	1	2	yes	—

	subgroup	G/H
center, socle	C₂²×C₁₁₃	C₂²×C₂²⋊C₄
commutator	C₂³	C₂³×C₄×C₁₁₃
Frattini, cosocle	C₂⁴	C₂⁴×C₁₁₃
prime radical	C₂²×2₋¹⁺⁴×C₁₁₃	C₂
LCS	G ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₁₁₃ ◁ C₂³×Q₈×C₁₁₃ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₂×#410×C₁₁₃	C₂	semidirect
3. C₁₁₃×#2310	C₂	semidirect
4. C₁₁₃×#1782	C₂	semidirect
7. C₂²×#41×C₁₁₃	C₄	semidirect
10. C₂³×Q₈×C₁₁₃	C₄	semidirect
133. #20587	C₁₁₃	direct (N × H)
140. C₂×#410	C₂×C₁₁₃	semidirect
141. #2310	C₂×C₁₁₃	semidirect
142. #1782	C₂×C₁₁₃	semidirect
143. C₁₁₃	#20587	direct (N × H)
146. C₂²×#41	C₄×C₁₁₃	semidirect
149. C₂³×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
C₂²×2₋¹⁺⁴×C₁₁₃	2
C₂×#410×C₁₁₃	2
C₁₁₃×#2310	2
C₁₁₃×#1782	2
#20587	113

Subgroups — C113×#20587 #43288607

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₁₃×#20587 #43288607