Subgroups — C₁₁₃×#3512 #28388308

Order 14464.

1032 conjugacy classes of subgroups.

The subgroup lattice has 272 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#553×C₁₁₃	C₂	1	4	yes	maximal, complemented
2	#553×C₁₁₃	C₂	1	8	yes	maximal, complemented
3	#552×C₁₁₃	C₂	1	4	yes	maximal, complemented
4	#541×C₁₁₃	C₂	1	2	yes	maximal, complemented
5	#540×C₁₁₃	C₂	1	2	yes	maximal, complemented
6	#537×C₁₁₃	C₂	1	2	yes	maximal, complemented
7	#532×C₁₁₃	C₂	1	1	yes	maximal, characteristic, complemented
8	#529×C₁₁₃	C₂	1	4	yes	maximal, complemented
9	C₂×#125×C₁₁₃	C₂	1	1	yes	maximal, characteristic, complemented
10	#520×C₁₁₃	C₂	1	2	yes	maximal, complemented
11	C₄×#41×C₁₁₃	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×#41×C₁₁₃	C₂²	1	1	yes	characteristic
13	C₂×#41×C₁₁₃	C₂²	1	2	yes	—
14	C₄⋊Q₈×C₁₁₃	C₂²	1	1	yes	characteristic
15	#127×C₁₁₃	C₂²	1	1	yes	characteristic
16	#126×C₁₁₃	C₂²	1	8	yes	—
17	#126×C₁₁₃	C₂²	1	8	yes	—
18	#125×C₁₁₃	C₂²	1	2	yes	—
19	#125×C₁₁₃	C₂²	1	2	yes	—
20	#125×C₁₁₃	C₂²	1	4	yes	—
21	#125×C₁₁₃	C₂²	1	8	yes	—
22	#124×C₁₁₃	C₂²	1	2	yes	—
23	#123×C₁₁₃	C₂²	1	8	yes	—
24	#123×C₁₁₃	C₂²	1	8	yes	—
25	C₂²⋊Q₈×C₁₁₃	C₂²	1	4	yes	—
26	C₂²⋊Q₈×C₁₁₃	C₂²	1	4	yes	—
27	#121×C₁₁₃	C₂²	1	4	yes	—
28	#121×C₁₁₃	C₂²	1	4	yes	—
29	#121×C₁₁₃	C₂²	1	8	yes	—
30	#121×C₁₁₃	C₂²	1	4	yes	—
31	#121×C₁₁₃	C₂²	1	8	yes	—
32	C₄×Q₈×C₁₁₃	C₂²	1	2	yes	—
33	C₄×Q₈×C₁₁₃	C₂²	1	4	yes	—
34	C₄×D₄×C₁₁₃	C₂²	1	4	yes	—
35	C₄×D₄×C₁₁₃	C₂²	1	2	yes	—
36	C₄×D₄×C₁₁₃	C₂²	1	4	yes	—
37	C₄×D₄×C₁₁₃	C₂²	1	8	yes	—
38	C₄×D₄×C₁₁₃	C₂²	1	8	yes	—
39	C₄×D₄×C₁₁₃	C₂²	1	4	yes	—
40	#117×C₁₁₃	C₂²	1	1	yes	characteristic
41	#117×C₁₁₃	C₂²	1	2	yes	—
42	#117×C₁₁₃	C₂²	1	4	yes	—
43	#117×C₁₁₃	C₂²	1	8	yes	—
44	C₂×C₄⋊C₄×C₁₁₃	C₂²	1	2	yes	—
45	C₂×C₄⋊C₄×C₁₁₃	C₂²	1	4	yes	—
46	C₂×C₄⋊C₄×C₁₁₃	C₂²	1	4	yes	—
47	C₂×C₄²×C₁₁₃	C₂²	1	1	yes	characteristic
48	C₂×C₄²×C₁₁₃	C₂²	1	2	yes	—
49	#41×C₁₁₃	—	2	4	no	—
50	#41×C₁₁₃	—	2	8	no	—
51	C₂×Q₈×C₁₁₃	C₂³	1	1	yes	characteristic
52	C₂×Q₈×C₁₁₃	C₂³	1	2	yes	—
53	C₂×D₄×C₁₁₃	C₂³	1	2	yes	—
54	C₂×D₄×C₁₁₃	C₂³	1	4	yes	—
55	C₂×D₄×C₁₁₃	C₂³	1	1	yes	characteristic
56	C₂×D₄×C₁₁₃	C₂³	1	2	yes	—
57	C₂×D₄×C₁₁₃	C₂³	1	8	yes	—
58	C₂×D₄×C₁₁₃	C₂³	1	4	yes	—
59	C₂²×C₄×C₁₁₃	C₂³	1	2	yes	—
60	C₂²×C₄×C₁₁₃	C₂³	1	4	yes	—
61	C₂²×C₄×C₁₁₃	C₂³	1	2	yes	—
62	C₂²×C₄×C₁₁₃	C₂³	1	4	yes	—
63	C₂²×C₄×C₁₁₃	C₂³	1	4	yes	—
64	C₂²×C₄×C₁₁₃	C₂³	1	8	yes	—
65	C₂²×C₄×C₁₁₃	C₂³	1	1	yes	characteristic
66	C₄⋊C₄×C₁₁₃	C₂³	1	2	yes	—
67	C₄⋊C₄×C₁₁₃	C₂³	1	4	yes	—
68	C₄⋊C₄×C₁₁₃	C₂³	1	8	yes	—
69	C₄⋊C₄×C₁₁₃	C₂³	1	4	yes	—
70	C₄⋊C₄×C₁₁₃	C₂³	1	8	yes	—
71	C₄⋊C₄×C₁₁₃	C₂³	1	8	yes	—
72	C₄⋊C₄×C₁₁₃	C₂³	1	4	yes	—
73	C₄⋊C₄×C₁₁₃	C₂³	1	8	yes	—
74	C₄⋊C₄×C₁₁₃	C₂³	1	4	yes	—
75	C₂²⋊C₄×C₁₁₃	C₂³	1	4	yes	—
76	C₂²⋊C₄×C₁₁₃	C₂³	1	2	yes	—
77	C₂²⋊C₄×C₁₁₃	C₂³	1	4	yes	—
78	C₂²⋊C₄×C₁₁₃	C₂³	1	8	yes	—
79	C₂²⋊C₄×C₁₁₃	C₂³	1	8	yes	—
80	C₂²⋊C₄×C₁₁₃	C₂³	1	4	yes	—
81	C₂²⋊C₄×C₁₁₃	C₂³	1	8	yes	—
82	C₄²×C₁₁₃	C₂³	1	1	yes	characteristic
83	C₄²×C₁₁₃	C₂³	1	2	yes	—
84	C₄²×C₁₁₃	C₂³	1	4	yes	—
85	C₄²×C₁₁₃	C₂³	1	1	yes	characteristic
86	C₄²×C₁₁₃	C₂³	1	2	yes	—
87	C₄²×C₁₁₃	C₂³	1	8	yes	—
88	C₂³×C₁₁₃	C₂⁴	1	4	yes	—
89	C₂³×C₁₁₃	C₂⁴	1	2	yes	—
90	C₂³×C₁₁₃	C₂⁴	1	1	yes	characteristic
91	Q₈×C₁₁₃	—	2	2	no	—
92	Q₈×C₁₁₃	—	2	4	no	—
93	D₄×C₁₁₃	—	2	4	no	—
94	D₄×C₁₁₃	—	2	8	no	—
95	D₄×C₁₁₃	—	2	2	no	—
96	D₄×C₁₁₃	—	2	4	no	—
97	D₄×C₁₁₃	—	4	8	no	—
98	D₄×C₁₁₃	—	4	4	no	—
99	C₂×C₄×C₁₁₃	—	2	8	no	—
100	C₂×C₄×C₁₁₃	—	2	4	no	—
101	C₂×C₄×C₁₁₃	—	4	4	no	—
102	C₂×C₄×C₁₁₃	—	4	8	no	—
103	C₂×C₄×C₁₁₃	—	4	2	no	—
104	C₂×C₄×C₁₁₃	—	4	4	no	—
105	C₂×C₄×C₁₁₃	#41	1	4	yes	—
106	C₂×C₄×C₁₁₃	C₂⁴	1	2	yes	—
107	C₂×C₄×C₁₁₃	C₂⁴	1	4	yes	—
108	C₂×C₄×C₁₁₃	C₂⁴	1	2	yes	—
109	C₂×C₄×C₁₁₃	C₂⁴	1	4	yes	—
110	C₂×C₄×C₁₁₃	C₂⁴	1	2	yes	—
111	C₂×C₄×C₁₁₃	C₂⁴	1	8	yes	—
112	C₂×C₄×C₁₁₃	C₂⁴	1	1	yes	characteristic
113	C₂×C₄×C₁₁₃	C₂⁴	1	1	yes	characteristic
114	C₂²×C₁₁₃	—	2	2	no	—
115	C₂²×C₁₁₃	—	2	4	no	—
116	C₂²×C₁₁₃	—	2	2	no	—
117	C₂²×C₁₁₃	—	2	4	no	—
118	C₂²×C₁₁₃	—	2	4	no	—
119	C₂²×C₁₁₃	—	2	4	no	—
120	C₂²×C₁₁₃	C₂×#41	1	2	yes	—
121	C₂²×C₁₁₃	C₂⁵	1	1	yes	characteristic
122	C₄×C₁₁₃	—	2	4	no	—
123	C₄×C₁₁₃	—	2	2	no	—
124	C₄×C₁₁₃	—	2	2	no	—
125	C₄×C₁₁₃	—	2	4	no	—
126	C₄×C₁₁₃	—	2	8	no	—
127	C₄×C₁₁₃	—	2	2	no	—
128	C₄×C₁₁₃	C₂×#41	1	2	yes	—
129	C₄×C₁₁₃	C₂×#41	1	2	yes	—
130	C₂×C₁₁₃	—	2	2	no	—
131	C₂×C₁₁₃	—	4	4	no	—
132	C₂×C₁₁₃	—	4	2	no	—
133	C₂×C₁₁₃	C₂²×#41	1	1	yes	characteristic
134	C₂×C₁₁₃	#585	1	2	yes	—
135	#3512	C₁₁₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
136	C₁₁₃	#3512	1	1	yes	characteristic, direct product factor, Sylow
137	#553	C₂×C₁₁₃	1	4	yes	complemented
138	#553	C₂×C₁₁₃	1	8	yes	complemented
139	#552	C₂×C₁₁₃	1	4	yes	complemented
140	#541	C₂×C₁₁₃	1	2	yes	complemented
141	#540	C₂×C₁₁₃	1	2	yes	complemented
142	#537	C₂×C₁₁₃	1	2	yes	complemented
143	#532	C₂×C₁₁₃	1	1	yes	characteristic, complemented
144	#529	C₂×C₁₁₃	1	4	yes	complemented
145	C₂×#125	C₂×C₁₁₃	1	1	yes	characteristic, complemented
146	#520	C₂×C₁₁₃	1	2	yes	complemented
147	C₄×#41	C₂×C₁₁₃	1	1	yes	characteristic, complemented
148	C₂×#41	C₂²×C₁₁₃	1	1	yes	characteristic
149	C₂×#41	C₂²×C₁₁₃	1	2	yes	—
150	C₄⋊Q₈	C₂²×C₁₁₃	1	1	yes	characteristic
151	#127	C₂²×C₁₁₃	1	1	yes	characteristic
152	#126	C₂²×C₁₁₃	1	8	yes	—
153	#126	C₂²×C₁₁₃	1	8	yes	—
154	#125	C₂²×C₁₁₃	1	2	yes	—
155	#125	C₂²×C₁₁₃	1	2	yes	—
156	#125	C₂²×C₁₁₃	1	4	yes	—
157	#125	C₂²×C₁₁₃	1	8	yes	—
158	#124	C₂²×C₁₁₃	1	2	yes	—
159	#123	C₂²×C₁₁₃	1	8	yes	—
160	#123	C₂²×C₁₁₃	1	8	yes	—
161	C₂²⋊Q₈	C₂²×C₁₁₃	1	4	yes	—
162	C₂²⋊Q₈	C₂²×C₁₁₃	1	4	yes	—
163	#121	C₂²×C₁₁₃	1	4	yes	—
164	#121	C₂²×C₁₁₃	1	4	yes	—
165	#121	C₂²×C₁₁₃	1	8	yes	—
166	#121	C₂²×C₁₁₃	1	4	yes	—
167	#121	C₂²×C₁₁₃	1	8	yes	—
168	C₄×Q₈	C₂²×C₁₁₃	1	2	yes	—
169	C₄×Q₈	C₂²×C₁₁₃	1	4	yes	—
170	C₄×D₄	C₂²×C₁₁₃	1	4	yes	—
171	C₄×D₄	C₂²×C₁₁₃	1	2	yes	—
172	C₄×D₄	C₂²×C₁₁₃	1	4	yes	—
173	C₄×D₄	C₂²×C₁₁₃	1	8	yes	—
174	C₄×D₄	C₂²×C₁₁₃	1	8	yes	—
175	C₄×D₄	C₂²×C₁₁₃	1	4	yes	—
176	#117	C₂²×C₁₁₃	1	1	yes	characteristic
177	#117	C₂²×C₁₁₃	1	2	yes	—
178	#117	C₂²×C₁₁₃	1	4	yes	—
179	#117	C₂²×C₁₁₃	1	8	yes	—
180	C₂×C₄⋊C₄	C₂²×C₁₁₃	1	2	yes	—
181	C₂×C₄⋊C₄	C₂²×C₁₁₃	1	4	yes	—
182	C₂×C₄⋊C₄	C₂²×C₁₁₃	1	4	yes	—
183	C₂×C₄²	C₂²×C₁₁₃	1	1	yes	characteristic
184	C₂×C₄²	C₂²×C₁₁₃	1	2	yes	—
185	#41	—	2	4	no	—
186	#41	—	2	8	no	—
187	C₂×Q₈	C₂³×C₁₁₃	1	1	yes	characteristic
188	C₂×Q₈	C₂³×C₁₁₃	1	2	yes	—
189	C₂×D₄	C₂³×C₁₁₃	1	2	yes	—
190	C₂×D₄	C₂³×C₁₁₃	1	4	yes	—
191	C₂×D₄	C₂³×C₁₁₃	1	1	yes	characteristic
192	C₂×D₄	C₂³×C₁₁₃	1	2	yes	—
193	C₂×D₄	C₂³×C₁₁₃	1	8	yes	—
194	C₂×D₄	C₂³×C₁₁₃	1	4	yes	—
195	C₂²×C₄	C₂³×C₁₁₃	1	2	yes	—
196	C₂²×C₄	C₂³×C₁₁₃	1	4	yes	—
197	C₂²×C₄	C₂³×C₁₁₃	1	2	yes	—
198	C₂²×C₄	C₂³×C₁₁₃	1	4	yes	—
199	C₂²×C₄	C₂³×C₁₁₃	1	4	yes	—
200	C₂²×C₄	C₂³×C₁₁₃	1	8	yes	—
201	C₂²×C₄	C₂³×C₁₁₃	1	1	yes	characteristic
202	C₄⋊C₄	C₂³×C₁₁₃	1	2	yes	—
203	C₄⋊C₄	C₂³×C₁₁₃	1	4	yes	—
204	C₄⋊C₄	C₂³×C₁₁₃	1	8	yes	—
205	C₄⋊C₄	C₂³×C₁₁₃	1	4	yes	—
206	C₄⋊C₄	C₂³×C₁₁₃	1	8	yes	—
207	C₄⋊C₄	C₂³×C₁₁₃	1	8	yes	—
208	C₄⋊C₄	C₂³×C₁₁₃	1	4	yes	—
209	C₄⋊C₄	C₂³×C₁₁₃	1	8	yes	—
210	C₄⋊C₄	C₂³×C₁₁₃	1	4	yes	—
211	C₂²⋊C₄	C₂³×C₁₁₃	1	4	yes	—
212	C₂²⋊C₄	C₂³×C₁₁₃	1	2	yes	—
213	C₂²⋊C₄	C₂³×C₁₁₃	1	4	yes	—
214	C₂²⋊C₄	C₂³×C₁₁₃	1	8	yes	—
215	C₂²⋊C₄	C₂³×C₁₁₃	1	8	yes	—
216	C₂²⋊C₄	C₂³×C₁₁₃	1	4	yes	—
217	C₂²⋊C₄	C₂³×C₁₁₃	1	8	yes	—
218	C₄²	C₂³×C₁₁₃	1	1	yes	characteristic
219	C₄²	C₂³×C₁₁₃	1	2	yes	—
220	C₄²	C₂³×C₁₁₃	1	4	yes	—
221	C₄²	C₂³×C₁₁₃	1	1	yes	characteristic
222	C₄²	C₂³×C₁₁₃	1	2	yes	—
223	C₄²	C₂³×C₁₁₃	1	8	yes	—
224	C₂³	C₂⁴×C₁₁₃	1	4	yes	—
225	C₂³	C₂⁴×C₁₁₃	1	2	yes	—
226	C₂³	C₂⁴×C₁₁₃	1	1	yes	characteristic
227	Q₈	—	2	2	no	—
228	Q₈	—	2	4	no	—
229	D₄	—	2	4	no	—
230	D₄	—	2	8	no	—
231	D₄	—	2	2	no	—
232	D₄	—	2	4	no	—
233	D₄	—	4	8	no	—
234	D₄	—	4	4	no	—
235	C₂×C₄	—	2	8	no	—
236	C₂×C₄	—	2	4	no	—
237	C₂×C₄	—	4	4	no	—
238	C₂×C₄	—	4	8	no	—
239	C₂×C₄	—	4	2	no	—
240	C₂×C₄	—	4	4	no	—
241	C₂×C₄	#41×C₁₁₃	1	4	yes	—
242	C₂×C₄	C₂⁴×C₁₁₃	1	2	yes	—
243	C₂×C₄	C₂⁴×C₁₁₃	1	4	yes	—
244	C₂×C₄	C₂⁴×C₁₁₃	1	2	yes	—
245	C₂×C₄	C₂⁴×C₁₁₃	1	4	yes	—
246	C₂×C₄	C₂⁴×C₁₁₃	1	2	yes	—
247	C₂×C₄	C₂⁴×C₁₁₃	1	8	yes	—
248	C₂×C₄	C₂⁴×C₁₁₃	1	1	yes	characteristic
249	C₂×C₄	C₂⁴×C₁₁₃	1	1	yes	characteristic
250	C₂²	—	2	2	no	—
251	C₂²	—	2	4	no	—
252	C₂²	—	2	2	no	—
253	C₂²	—	2	4	no	—
254	C₂²	—	2	4	no	—
255	C₂²	—	2	4	no	—
256	C₂²	C₂×#41×C₁₁₃	1	2	yes	—
257	C₂²	C₂⁵×C₁₁₃	1	1	yes	characteristic
258	C₄	—	2	4	no	—
259	C₄	—	2	2	no	—
260	C₄	—	2	2	no	—
261	C₄	—	2	4	no	—
262	C₄	—	2	8	no	—
263	C₄	—	2	2	no	—
264	C₄	C₂×#41×C₁₁₃	1	2	yes	—
265	C₄	C₂×#41×C₁₁₃	1	2	yes	—
266	C₂	—	2	2	no	—
267	C₂	—	4	4	no	—
268	C₂	—	4	2	no	—
269	C₂	C₂²×#41×C₁₁₃	1	1	yes	characteristic
270	C₂	#585×C₁₁₃	1	2	yes	—

	subgroup	G/H
center, socle	C₂²×C₁₁₃	C₂⁵
commutator, Frattini, cosocle	C₂²	C₂⁵×C₁₁₃
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₁₁₃ ◁ G

Complements

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

N	H ≅ G/N	type
1. #553×C₁₁₃	C₂	semidirect
2. #553×C₁₁₃	C₂	semidirect
3. #552×C₁₁₃	C₂	semidirect
4. #541×C₁₁₃	C₂	semidirect
5. #540×C₁₁₃	C₂	semidirect
6. #537×C₁₁₃	C₂	semidirect
7. #532×C₁₁₃	C₂	semidirect
8. #529×C₁₁₃	C₂	semidirect
9. C₂×#125×C₁₁₃	C₂	semidirect
10. #520×C₁₁₃	C₂	semidirect
11. C₄×#41×C₁₁₃	C₂	semidirect
135. #3512	C₁₁₃	direct (N × H)
136. C₁₁₃	#3512	direct (N × H)
137. #553	C₂×C₁₁₃	semidirect
138. #553	C₂×C₁₁₃	semidirect
139. #552	C₂×C₁₁₃	semidirect
140. #541	C₂×C₁₁₃	semidirect
141. #540	C₂×C₁₁₃	semidirect
142. #537	C₂×C₁₁₃	semidirect
143. #532	C₂×C₁₁₃	semidirect
144. #529	C₂×C₁₁₃	semidirect
145. C₂×#125	C₂×C₁₁₃	semidirect
146. #520	C₂×C₁₁₃	semidirect
147. C₄×#41	C₂×C₁₁₃	semidirect

Maximal subgroups

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

subgroup	index
#553×C₁₁₃	2
#552×C₁₁₃	2
#541×C₁₁₃	2
#540×C₁₁₃	2
#537×C₁₁₃	2
#532×C₁₁₃	2
#529×C₁₁₃	2
C₂×#125×C₁₁₃	2
#520×C₁₁₃	2
C₄×#41×C₁₁₃	2
#3512	113

Subgroups — C113×#3512 #28388308

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₁₃×#3512 #28388308