Subgroups — C₁₁₃×#3325 #28388121

Order 14464.

460 conjugacy classes of subgroups.

The subgroup lattice has 282 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#555×C₁₁₃	C₂	1	1	yes	maximal, characteristic, complemented
2	D₄²×C₁₁₃	C₂	1	1	yes	maximal, characteristic, complemented
3	#486×C₁₁₃	C₂	1	1	yes	maximal, characteristic, complemented
4	#482×C₁₁₃	C₂	1	2	yes	maximal, complemented
5	#476×C₁₁₃	C₂	1	1	yes	maximal, characteristic, complemented
6	#469×C₁₁₃	C₂	1	2	yes	maximal, complemented
7	#459×C₁₁₃	C₂	1	1	yes	maximal, characteristic, complemented
8	#447×C₁₁₃	C₂	1	2	yes	maximal, complemented
9	C₄×D₈×C₁₁₃	C₂	1	1	yes	maximal, characteristic, complemented
10	#436×C₁₁₃	C₂	1	1	yes	maximal, characteristic, complemented
11	#418×C₁₁₃	C₂	1	2	yes	maximal, complemented
12	C₂²×D₄×C₁₁₃	—	2	1	no	—
13	C₂²×D₄×C₁₁₃	C₂²	1	2	yes	—
14	C₂×D₈×C₁₁₃	C₂²	1	1	yes	characteristic
15	C₂×D₈×C₁₁₃	C₂²	1	2	yes	complemented
16	C₂×M₁₆×C₁₁₃	C₂²	1	2	yes	complemented
17	C₄⋊Q₈×C₁₁₃	C₂²	1	1	yes	characteristic, complemented
18	#127×C₁₁₃	C₂²	1	1	yes	characteristic
19	#124×C₁₁₃	—	2	2	no	—
20	#121×C₁₁₃	—	2	1	no	—
21	#121×C₁₁₃	—	2	1	no	—
22	#121×C₁₁₃	C₂²	1	2	yes	—
23	#121×C₁₁₃	C₂²	1	2	yes	complemented
24	#120×C₁₁₃	—	2	2	no	—
25	C₄×D₄×C₁₁₃	C₂²	1	1	yes	characteristic
26	C₄×D₄×C₁₁₃	C₂²	1	1	yes	characteristic
27	C₄×D₄×C₁₁₃	C₂²	1	1	yes	characteristic, complemented
28	#117×C₁₁₃	—	2	1	no	—
29	#117×C₁₁₃	C₂²	1	2	yes	complemented
30	#107×C₁₁₃	C₂²	1	1	yes	characteristic, complemented
31	#106×C₁₁₃	C₂²	1	2	yes	complemented
32	C₄⋊C₈×C₁₁₃	C₂²	1	1	yes	characteristic, complemented
33	#102×C₁₁₃	C₂²	1	1	yes	characteristic
34	#102×C₁₁₃	C₂²	1	2	yes	—
35	#102×C₁₁₃	C₂²	1	2	yes	complemented
36	#102×C₁₁₃	C₂²	1	2	yes	complemented
37	#102×C₁₁₃	C₂²	1	1	yes	characteristic, complemented
38	#102×C₁₁₃	C₂²	1	2	yes	complemented
39	C₂²⋊C₈×C₁₁₃	C₂²	1	2	yes	complemented
40	C₄×C₈×C₁₁₃	C₂²	1	1	yes	characteristic, complemented
41	C₂⁴×C₁₁₃	—	2	2	no	—
42	C₂×Q₈×C₁₁₃	—	2	1	no	—
43	C₂×D₄×C₁₁₃	—	2	4	no	—
44	C₂×D₄×C₁₁₃	—	2	2	no	—
45	C₂×D₄×C₁₁₃	—	2	2	no	—
46	C₂×D₄×C₁₁₃	—	2	2	no	—
47	C₂×D₄×C₁₁₃	—	2	1	no	—
48	C₂×D₄×C₁₁₃	—	2	1	no	—
49	C₂×D₄×C₁₁₃	—	4	2	no	—
50	C₂×D₄×C₁₁₃	—	4	4	no	—
51	C₂×D₄×C₁₁₃	D₄	1	1	yes	characteristic
52	C₂×D₄×C₁₁₃	C₂³	1	2	yes	—
53	C₂×D₄×C₁₁₃	C₂³	1	1	yes	characteristic
54	C₂×D₄×C₁₁₃	C₂³	1	1	yes	characteristic
55	C₂²×C₄×C₁₁₃	—	2	1	no	—
56	C₂²×C₄×C₁₁₃	—	2	1	no	—
57	C₂²×C₄×C₁₁₃	C₂³	1	2	yes	—
58	D₈×C₁₁₃	—	2	2	no	—
59	D₈×C₁₁₃	—	4	2	no	—
60	M₁₆×C₁₁₃	—	4	2	no	—
61	C₂×C₈×C₁₁₃	C₂³	1	1	yes	characteristic
62	C₂×C₈×C₁₁₃	C₂³	1	1	yes	characteristic
63	C₂×C₈×C₁₁₃	C₂³	1	2	yes	—
64	C₄⋊C₄×C₁₁₃	—	2	1	no	—
65	C₄⋊C₄×C₁₁₃	D₄	1	1	yes	characteristic
66	C₄⋊C₄×C₁₁₃	C₂³	1	2	yes	—
67	C₄⋊C₄×C₁₁₃	C₂³	1	1	yes	characteristic
68	C₄⋊C₄×C₁₁₃	C₂³	1	1	yes	characteristic
69	C₂²⋊C₄×C₁₁₃	—	2	2	no	—
70	C₂²⋊C₄×C₁₁₃	—	2	2	no	—
71	C₂²⋊C₄×C₁₁₃	—	2	1	no	—
72	C₂²⋊C₄×C₁₁₃	—	2	1	no	—
73	C₂²⋊C₄×C₁₁₃	—	2	2	no	—
74	C₂²⋊C₄×C₁₁₃	D₄	1	2	yes	—
75	C₄²×C₁₁₃	—	2	2	no	—
76	C₄²×C₁₁₃	C₂³	1	1	yes	characteristic
77	C₂³×C₁₁₃	—	2	1	no	—
78	C₂³×C₁₁₃	—	2	1	no	—
79	C₂³×C₁₁₃	—	2	2	no	—
80	C₂³×C₁₁₃	—	4	4	no	—
81	C₂³×C₁₁₃	—	4	4	no	—
82	C₂³×C₁₁₃	—	8	2	no	—
83	C₂³×C₁₁₃	C₂×D₄	1	2	yes	—
84	Q₈×C₁₁₃	—	8	1	no	—
85	D₄×C₁₁₃	—	2	2	no	—
86	D₄×C₁₁₃	—	2	1	no	—
87	D₄×C₁₁₃	—	2	2	no	—
88	D₄×C₁₁₃	—	2	1	no	—
89	D₄×C₁₁₃	—	2	1	no	—
90	D₄×C₁₁₃	—	2	2	no	—
91	D₄×C₁₁₃	—	4	4	no	—
92	D₄×C₁₁₃	—	4	4	no	—
93	D₄×C₁₁₃	—	4	2	no	—
94	D₄×C₁₁₃	—	8	1	no	—
95	D₄×C₁₁₃	—	8	2	no	—
96	D₄×C₁₁₃	#41	1	2	yes	—
97	C₂×C₄×C₁₁₃	—	2	2	no	—
98	C₂×C₄×C₁₁₃	—	2	2	no	—
99	C₂×C₄×C₁₁₃	—	2	1	no	—
100	C₂×C₄×C₁₁₃	—	2	2	no	—
101	C₂×C₄×C₁₁₃	—	2	1	no	—
102	C₂×C₄×C₁₁₃	—	4	2	no	—
103	C₂×C₄×C₁₁₃	—	4	2	no	—
104	C₂×C₄×C₁₁₃	C₂×D₄	1	2	yes	—
105	C₂×C₄×C₁₁₃	C₂×D₄	1	1	yes	characteristic
106	C₂×C₄×C₁₁₃	C₂×D₄	1	1	yes	characteristic
107	C₂×C₄×C₁₁₃	C₂⁴	1	1	yes	characteristic
108	C₈×C₁₁₃	—	2	1	no	—
109	C₈×C₁₁₃	—	2	1	no	—
110	C₈×C₁₁₃	—	2	2	no	—
111	C₂²×C₁₁₃	—	2	2	no	—
112	C₂²×C₁₁₃	—	2	2	no	—
113	C₂²×C₁₁₃	—	2	2	no	—
114	C₂²×C₁₁₃	—	2	2	no	—
115	C₂²×C₁₁₃	—	4	1	no	—
116	C₂²×C₁₁₃	—	4	1	no	—
117	C₂²×C₁₁₃	—	4	2	no	—
118	C₂²×C₁₁₃	—	4	1	no	—
119	C₂²×C₁₁₃	—	4	1	no	—
120	C₂²×C₁₁₃	—	4	2	no	—
121	C₂²×C₁₁₃	—	4	1	no	—
122	C₂²×C₁₁₃	—	4	2	no	—
123	C₂²×C₁₁₃	—	8	8	no	—
124	C₂²×C₁₁₃	C₂²×D₄	1	1	yes	characteristic
125	C₄×C₁₁₃	—	2	2	no	—
126	C₄×C₁₁₃	—	2	1	no	—
127	C₄×C₁₁₃	—	4	1	no	—
128	C₄×C₁₁₃	—	4	1	no	—
129	C₄×C₁₁₃	—	4	2	no	—
130	C₄×C₁₁₃	AGL(1,ℤ/8)	1	2	yes	—
131	C₄×C₁₁₃	C₂×#41	1	1	yes	characteristic
132	C₄×C₁₁₃	H(5,𝔽₂)	1	1	yes	characteristic
133	C₂×C₁₁₃	—	4	2	no	—
134	C₂×C₁₁₃	—	4	2	no	—
135	C₂×C₁₁₃	—	8	1	no	—
136	C₂×C₁₁₃	—	8	2	no	—
137	C₂×C₁₁₃	#546	1	1	yes	characteristic
138	C₂×C₁₁₃	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
139	C₂×C₁₁₃	#576	1	1	yes	characteristic
140	#3325	C₁₁₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
141	C₁₁₃	#3325	1	1	yes	characteristic, direct product factor, Sylow
142	#555	C₂×C₁₁₃	1	1	yes	characteristic, complemented
143	D₄²	C₂×C₁₁₃	1	1	yes	characteristic, complemented
144	#486	C₂×C₁₁₃	1	1	yes	characteristic, complemented
145	#482	C₂×C₁₁₃	1	2	yes	complemented
146	#476	C₂×C₁₁₃	1	1	yes	characteristic, complemented
147	#469	C₂×C₁₁₃	1	2	yes	complemented
148	#459	C₂×C₁₁₃	1	1	yes	characteristic, complemented
149	#447	C₂×C₁₁₃	1	2	yes	complemented
150	C₄×D₈	C₂×C₁₁₃	1	1	yes	characteristic, complemented
151	#436	C₂×C₁₁₃	1	1	yes	characteristic, complemented
152	#418	C₂×C₁₁₃	1	2	yes	complemented
153	C₂²×D₄	—	2	1	no	—
154	C₂²×D₄	C₂²×C₁₁₃	1	2	yes	—
155	C₂×D₈	C₂²×C₁₁₃	1	1	yes	characteristic
156	C₂×D₈	C₂²×C₁₁₃	1	2	yes	complemented
157	C₂×M₁₆	C₂²×C₁₁₃	1	2	yes	complemented
158	C₄⋊Q₈	C₂²×C₁₁₃	1	1	yes	characteristic, complemented
159	#127	C₂²×C₁₁₃	1	1	yes	characteristic
160	#124	—	2	2	no	—
161	#121	—	2	1	no	—
162	#121	—	2	1	no	—
163	#121	C₂²×C₁₁₃	1	2	yes	—
164	#121	C₂²×C₁₁₃	1	2	yes	complemented
165	#120	—	2	2	no	—
166	C₄×D₄	C₂²×C₁₁₃	1	1	yes	characteristic
167	C₄×D₄	C₂²×C₁₁₃	1	1	yes	characteristic
168	C₄×D₄	C₂²×C₁₁₃	1	1	yes	characteristic, complemented
169	#117	—	2	1	no	—
170	#117	C₂²×C₁₁₃	1	2	yes	complemented
171	#107	C₂²×C₁₁₃	1	1	yes	characteristic, complemented
172	#106	C₂²×C₁₁₃	1	2	yes	complemented
173	C₄⋊C₈	C₂²×C₁₁₃	1	1	yes	characteristic, complemented
174	#102	C₂²×C₁₁₃	1	1	yes	characteristic
175	#102	C₂²×C₁₁₃	1	2	yes	—
176	#102	C₂²×C₁₁₃	1	2	yes	complemented
177	#102	C₂²×C₁₁₃	1	2	yes	complemented
178	#102	C₂²×C₁₁₃	1	1	yes	characteristic, complemented
179	#102	C₂²×C₁₁₃	1	2	yes	complemented
180	C₂²⋊C₈	C₂²×C₁₁₃	1	2	yes	complemented
181	C₄×C₈	C₂²×C₁₁₃	1	1	yes	characteristic, complemented
182	C₂⁴	—	2	2	no	—
183	C₂×Q₈	—	2	1	no	—
184	C₂×D₄	—	2	4	no	—
185	C₂×D₄	—	2	2	no	—
186	C₂×D₄	—	2	2	no	—
187	C₂×D₄	—	2	2	no	—
188	C₂×D₄	—	2	1	no	—
189	C₂×D₄	—	2	1	no	—
190	C₂×D₄	—	4	2	no	—
191	C₂×D₄	—	4	4	no	—
192	C₂×D₄	D₄×C₁₁₃	1	1	yes	characteristic
193	C₂×D₄	C₂³×C₁₁₃	1	2	yes	—
194	C₂×D₄	C₂³×C₁₁₃	1	1	yes	characteristic
195	C₂×D₄	C₂³×C₁₁₃	1	1	yes	characteristic
196	C₂²×C₄	—	2	1	no	—
197	C₂²×C₄	—	2	1	no	—
198	C₂²×C₄	C₂³×C₁₁₃	1	2	yes	—
199	D₈	—	2	2	no	—
200	D₈	—	4	2	no	—
201	M₁₆	—	4	2	no	—
202	C₂×C₈	C₂³×C₁₁₃	1	1	yes	characteristic
203	C₂×C₈	C₂³×C₁₁₃	1	1	yes	characteristic
204	C₂×C₈	C₂³×C₁₁₃	1	2	yes	—
205	C₄⋊C₄	—	2	1	no	—
206	C₄⋊C₄	D₄×C₁₁₃	1	1	yes	characteristic
207	C₄⋊C₄	C₂³×C₁₁₃	1	2	yes	—
208	C₄⋊C₄	C₂³×C₁₁₃	1	1	yes	characteristic
209	C₄⋊C₄	C₂³×C₁₁₃	1	1	yes	characteristic
210	C₂²⋊C₄	—	2	2	no	—
211	C₂²⋊C₄	—	2	2	no	—
212	C₂²⋊C₄	—	2	1	no	—
213	C₂²⋊C₄	—	2	1	no	—
214	C₂²⋊C₄	—	2	2	no	—
215	C₂²⋊C₄	D₄×C₁₁₃	1	2	yes	—
216	C₄²	—	2	2	no	—
217	C₄²	C₂³×C₁₁₃	1	1	yes	characteristic
218	C₂³	—	2	1	no	—
219	C₂³	—	2	1	no	—
220	C₂³	—	2	2	no	—
221	C₂³	—	4	4	no	—
222	C₂³	—	4	4	no	—
223	C₂³	—	8	2	no	—
224	C₂³	C₂×D₄×C₁₁₃	1	2	yes	—
225	Q₈	—	8	1	no	—
226	D₄	—	2	2	no	—
227	D₄	—	2	1	no	—
228	D₄	—	2	2	no	—
229	D₄	—	2	1	no	—
230	D₄	—	2	1	no	—
231	D₄	—	2	2	no	—
232	D₄	—	4	4	no	—
233	D₄	—	4	4	no	—
234	D₄	—	4	2	no	—
235	D₄	—	8	1	no	—
236	D₄	—	8	2	no	—
237	D₄	#41×C₁₁₃	1	2	yes	—
238	C₂×C₄	—	2	2	no	—
239	C₂×C₄	—	2	2	no	—
240	C₂×C₄	—	2	1	no	—
241	C₂×C₄	—	2	2	no	—
242	C₂×C₄	—	2	1	no	—
243	C₂×C₄	—	4	2	no	—
244	C₂×C₄	—	4	2	no	—
245	C₂×C₄	C₂×D₄×C₁₁₃	1	2	yes	—
246	C₂×C₄	C₂×D₄×C₁₁₃	1	1	yes	characteristic
247	C₂×C₄	C₂×D₄×C₁₁₃	1	1	yes	characteristic
248	C₂×C₄	C₂⁴×C₁₁₃	1	1	yes	characteristic
249	C₈	—	2	1	no	—
250	C₈	—	2	1	no	—
251	C₈	—	2	2	no	—
252	C₂²	—	2	2	no	—
253	C₂²	—	2	2	no	—
254	C₂²	—	2	2	no	—
255	C₂²	—	2	2	no	—
256	C₂²	—	4	1	no	—
257	C₂²	—	4	1	no	—
258	C₂²	—	4	2	no	—
259	C₂²	—	4	1	no	—
260	C₂²	—	4	1	no	—
261	C₂²	—	4	2	no	—
262	C₂²	—	4	1	no	—
263	C₂²	—	4	2	no	—
264	C₂²	—	8	8	no	—
265	C₂²	C₂²×D₄×C₁₁₃	1	1	yes	characteristic
266	C₄	—	2	2	no	—
267	C₄	—	2	1	no	—
268	C₄	—	4	1	no	—
269	C₄	—	4	1	no	—
270	C₄	—	4	2	no	—
271	C₄	AGL(1,ℤ/8)×C₁₁₃	1	2	yes	—
272	C₄	C₂×#41×C₁₁₃	1	1	yes	characteristic
273	C₄	H(5,𝔽₂)×C₁₁₃	1	1	yes	characteristic
274	C₂	—	4	2	no	—
275	C₂	—	4	2	no	—
276	C₂	—	8	2	no	—
277	C₂	—	8	1	no	—
278	C₂	#546×C₁₁₃	1	1	yes	characteristic
279	C₂	C₂×AGL(1,ℤ/8)×C₁₁₃	1	1	yes	characteristic
280	C₂	#576×C₁₁₃	1	1	yes	characteristic

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

Complements

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

N	H ≅ G/N	type
1. #555×C₁₁₃	C₂	semidirect
2. D₄²×C₁₁₃	C₂	semidirect
3. #486×C₁₁₃	C₂	semidirect
4. #482×C₁₁₃	C₂	semidirect
5. #476×C₁₁₃	C₂	semidirect
6. #469×C₁₁₃	C₂	semidirect
7. #459×C₁₁₃	C₂	semidirect
8. #447×C₁₁₃	C₂	semidirect
9. C₄×D₈×C₁₁₃	C₂	semidirect
10. #436×C₁₁₃	C₂	semidirect
11. #418×C₁₁₃	C₂	semidirect
15. C₂×D₈×C₁₁₃	C₂²	semidirect
16. C₂×M₁₆×C₁₁₃	C₂²	semidirect
17. C₄⋊Q₈×C₁₁₃	C₂²	semidirect
23. #121×C₁₁₃	C₂²	semidirect
27. C₄×D₄×C₁₁₃	C₂²	semidirect
29. #117×C₁₁₃	C₂²	semidirect
30. #107×C₁₁₃	C₂²	semidirect
31. #106×C₁₁₃	C₂²	semidirect
32. C₄⋊C₈×C₁₁₃	C₂²	semidirect
35. #102×C₁₁₃	C₂²	semidirect
36. #102×C₁₁₃	C₂²	semidirect
37. #102×C₁₁₃	C₂²	semidirect
38. #102×C₁₁₃	C₂²	semidirect
39. C₂²⋊C₈×C₁₁₃	C₂²	semidirect
40. C₄×C₈×C₁₁₃	C₂²	semidirect
140. #3325	C₁₁₃	direct (N × H)
141. C₁₁₃	#3325	direct (N × H)
142. #555	C₂×C₁₁₃	semidirect
143. D₄²	C₂×C₁₁₃	semidirect
144. #486	C₂×C₁₁₃	semidirect
145. #482	C₂×C₁₁₃	semidirect
146. #476	C₂×C₁₁₃	semidirect
147. #469	C₂×C₁₁₃	semidirect
148. #459	C₂×C₁₁₃	semidirect
149. #447	C₂×C₁₁₃	semidirect
150. C₄×D₈	C₂×C₁₁₃	semidirect
151. #436	C₂×C₁₁₃	semidirect
152. #418	C₂×C₁₁₃	semidirect
156. C₂×D₈	C₂²×C₁₁₃	semidirect
157. C₂×M₁₆	C₂²×C₁₁₃	semidirect
158. C₄⋊Q₈	C₂²×C₁₁₃	semidirect
164. #121	C₂²×C₁₁₃	semidirect
168. C₄×D₄	C₂²×C₁₁₃	semidirect
170. #117	C₂²×C₁₁₃	semidirect
171. #107	C₂²×C₁₁₃	semidirect
172. #106	C₂²×C₁₁₃	semidirect
173. C₄⋊C₈	C₂²×C₁₁₃	semidirect
176. #102	C₂²×C₁₁₃	semidirect
177. #102	C₂²×C₁₁₃	semidirect
178. #102	C₂²×C₁₁₃	semidirect
179. #102	C₂²×C₁₁₃	semidirect
180. C₂²⋊C₈	C₂²×C₁₁₃	semidirect
181. C₄×C₈	C₂²×C₁₁₃	semidirect

Maximal subgroups

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

subgroup	index
#555×C₁₁₃	2
D₄²×C₁₁₃	2
#486×C₁₁₃	2
#482×C₁₁₃	2
#476×C₁₁₃	2
#469×C₁₁₃	2
#459×C₁₁₃	2
#447×C₁₁₃	2
C₄×D₈×C₁₁₃	2
#436×C₁₁₃	2
#418×C₁₁₃	2
#3325	113

Subgroups — C113×#3325 #28388121

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₁₃×#3325 #28388121