Subgroups — C₁₁₃×#9719 #43277739

Order 28928.

506 conjugacy classes of subgroups.

The subgroup lattice has 292 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₁₁₃×#3169	C₂	1	1	yes	maximal, characteristic
2	C₁₁₃×#1980	C₂	1	2	yes	maximal, complemented
3	C₄×#107×C₁₁₃	C₂	1	1	yes	maximal, characteristic, complemented
4	C₁₁₃×#1502	C₂	1	2	yes	maximal, complemented
5	C₈×M₁₆×C₁₁₃	C₂	1	1	yes	maximal, characteristic, complemented
6	#532×C₁₁₃	C₂²	1	2	yes	—
7	#500×C₁₁₃	C₄	1	1	yes	characteristic, complemented
8	#493×C₁₁₃	C₄	1	1	yes	characteristic, complemented
9	#486×C₁₁₃	C₄	1	2	yes	complemented
10	#466×C₁₁₃	—	2	1	no	—
11	#466×C₁₁₃	—	2	1	no	—
12	C₄×D₈×C₁₁₃	—	2	1	no	—
13	C₄×D₈×C₁₁₃	—	2	1	no	—
14	#431×C₁₁₃	C₂²	1	1	yes	characteristic
15	C₂×#107×C₁₁₃	—	2	1	no	—
16	C₂×#104×C₁₁₃	—	2	4	no	—
17	C₄×M₁₆×C₁₁₃	C₂²	1	1	yes	characteristic
18	C₂×C₄×C₈×C₁₁₃	C₂²	1	1	yes	characteristic
19	C₄×C₄⋊C₄×C₁₁₃	C₂²	1	2	yes	—
20	#340×C₁₁₃	—	2	1	no	—
21	#327×C₁₁₃	—	2	4	no	—
22	#322×C₁₁₃	—	2	1	no	—
23	C₈²×C₁₁₃	—	2	1	no	—
24	C₂×#41×C₁₁₃	—	2	2	no	—
25	C₂×D₈×C₁₁₃	—	2	1	no	—
26	C₂×D₈×C₁₁₃	—	2	1	no	—
27	C₂×M₁₆×C₁₁₃	D₄	1	2	yes	—
28	C₂²×C₈×C₁₁₃	D₄	1	1	yes	characteristic
29	C₂²×C₈×C₁₁₃	D₄	1	1	yes	characteristic
30	C₄⋊Q₈×C₁₁₃	C₂×C₄	1	2	yes	—
31	#127×C₁₁₃	C₂×C₄	1	2	yes	—
32	#124×C₁₁₃	—	2	2	no	—
33	#121×C₁₁₃	—	2	2	no	—
34	#121×C₁₁₃	—	2	2	no	—
35	C₄×D₄×C₁₁₃	—	2	2	no	—
36	C₄×D₄×C₁₁₃	—	2	2	no	—
37	C₂×C₄⋊C₄×C₁₁₃	—	2	2	no	—
38	C₂×C₄²×C₁₁₃	—	2	2	no	—
39	C₂×C₄²×C₁₁₃	C₂³	1	1	yes	characteristic
40	#107×C₁₁₃	—	2	4	no	—
41	#107×C₁₁₃	—	2	1	no	—
42	#107×C₁₁₃	—	2	1	no	—
43	C₄⋊C₈×C₁₁₃	—	2	1	no	—
44	#104×C₁₁₃	—	4	8	no	—
45	#102×C₁₁₃	—	2	2	no	—
46	#102×C₁₁₃	—	2	2	no	—
47	C₂²⋊C₈×C₁₁₃	D₄	1	2	yes	—
48	#97×C₁₁₃	—	2	1	no	—
49	C₄×C₈×C₁₁₃	—	2	1	no	—
50	C₄×C₈×C₁₁₃	—	2	1	no	—
51	C₄×C₈×C₁₁₃	C₂×C₄	1	1	yes	characteristic
52	C₄×C₈×C₁₁₃	C₂×C₄	1	1	yes	characteristic
53	C₄×C₈×C₁₁₃	D₄	1	1	yes	characteristic
54	C₄×C₈×C₁₁₃	D₄	1	1	yes	characteristic
55	#95×C₁₁₃	—	2	2	no	—
56	#41×C₁₁₃	—	4	4	no	—
57	#41×C₁₁₃	—	4	4	no	—
58	C₂×Q₈×C₁₁₃	—	2	2	no	—
59	C₂×D₄×C₁₁₃	—	2	2	no	—
60	C₂×D₄×C₁₁₃	—	2	2	no	—
61	C₂×D₄×C₁₁₃	—	2	2	no	—
62	C₂×D₄×C₁₁₃	—	4	2	no	—
63	C₂²×C₄×C₁₁₃	—	4	2	no	—
64	C₂²×C₄×C₁₁₃	—	4	2	no	—
65	C₂²×C₄×C₁₁₃	C₂×D₄	1	1	yes	characteristic
66	C₂²×C₄×C₁₁₃	C₂×D₄	1	1	yes	characteristic
67	C₂²×C₄×C₁₁₃	C₂×D₄	1	1	yes	characteristic
68	D₈×C₁₁₃	—	4	2	no	—
69	D₈×C₁₁₃	—	4	2	no	—
70	M₁₆×C₁₁₃	—	2	2	no	—
71	M₁₆×C₁₁₃	D₈	1	4	yes	complemented
72	C₂×C₈×C₁₁₃	—	2	2	no	—
73	C₂×C₈×C₁₁₃	—	2	2	no	—
74	C₂×C₈×C₁₁₃	—	2	2	no	—
75	C₂×C₈×C₁₁₃	—	4	1	no	—
76	C₂×C₈×C₁₁₃	C₂²⋊C₄	1	1	yes	characteristic
77	C₂×C₈×C₁₁₃	C₂²⋊C₄	1	1	yes	characteristic
78	C₂×C₈×C₁₁₃	C₂²⋊C₄	1	1	yes	characteristic
79	C₂×C₈×C₁₁₃	C₂²⋊C₄	1	1	yes	characteristic
80	C₄⋊C₄×C₁₁₃	—	2	4	no	—
81	C₄⋊C₄×C₁₁₃	—	2	2	no	—
82	C₄⋊C₄×C₁₁₃	—	2	2	no	—
83	C₄⋊C₄×C₁₁₃	—	4	2	no	—
84	C₂²⋊C₄×C₁₁₃	—	4	2	no	—
85	C₂²⋊C₄×C₁₁₃	—	4	2	no	—
86	C₄²×C₁₁₃	—	4	4	no	—
87	C₄²×C₁₁₃	C₂²×C₄	1	1	yes	characteristic
88	C₄²×C₁₁₃	C₂×D₄	1	1	yes	characteristic
89	C₄²×C₁₁₃	#41	1	1	yes	characteristic
90	C₄²×C₁₁₃	#41	1	1	yes	characteristic
91	C₂³×C₁₁₃	—	4	2	no	—
92	C₂³×C₁₁₃	#127	1	1	yes	characteristic
93	Q₈×C₁₁₃	—	4	4	no	—
94	D₄×C₁₁₃	—	4	2	no	—
95	D₄×C₁₁₃	—	4	2	no	—
96	D₄×C₁₁₃	—	4	4	no	—
97	D₄×C₁₁₃	—	8	2	no	—
98	D₄×C₁₁₃	—	8	2	no	—
99	D₄×C₁₁₃	—	8	2	no	—
100	C₂×C₄×C₁₁₃	—	2	1	no	—
101	C₂×C₄×C₁₁₃	—	2	1	no	—
102	C₂×C₄×C₁₁₃	—	4	1	no	—
103	C₂×C₄×C₁₁₃	—	4	4	no	—
104	C₂×C₄×C₁₁₃	—	4	2	no	—
105	C₂×C₄×C₁₁₃	—	8	2	no	—
106	C₂×C₄×C₁₁₃	—	8	2	no	—
107	C₂×C₄×C₁₁₃	—	8	4	no	—
108	C₂×C₄×C₁₁₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
109	C₂×C₄×C₁₁₃	C₄×D₄	1	1	yes	characteristic
110	C₂×C₄×C₁₁₃	C₄×D₄	1	1	yes	characteristic
111	C₂×C₄×C₁₁₃	#121	1	1	yes	characteristic
112	C₂×C₄×C₁₁₃	#121	1	1	yes	characteristic
113	C₂×C₄×C₁₁₃	#124	1	1	yes	characteristic
114	C₂×C₄×C₁₁₃	C₂×D₈	1	2	yes	—
115	C₂×C₄×C₁₁₃	#135	1	2	yes	—
116	C₈×C₁₁₃	—	2	1	no	—
117	C₈×C₁₁₃	—	2	1	no	—
118	C₈×C₁₁₃	—	2	1	no	—
119	C₈×C₁₁₃	—	2	4	no	—
120	C₈×C₁₁₃	—	4	2	no	—
121	C₈×C₁₁₃	#104	1	2	yes	complemented
122	C₂²×C₁₁₃	—	2	1	no	—
123	C₂²×C₁₁₃	—	2	1	no	—
124	C₂²×C₁₁₃	—	8	2	no	—
125	C₂²×C₁₁₃	—	8	2	no	—
126	C₂²×C₁₁₃	—	8	2	no	—
127	C₂²×C₁₁₃	#390	1	1	yes	characteristic
128	C₂²×C₁₁₃	#493	1	1	yes	characteristic
129	C₂²×C₁₁₃	#495	1	1	yes	characteristic
130	C₄×C₁₁₃	—	2	1	no	—
131	C₄×C₁₁₃	—	2	1	no	—
132	C₄×C₁₁₃	—	2	1	no	—
133	C₄×C₁₁₃	—	8	4	no	—
134	C₄×C₁₁₃	—	8	2	no	—
135	C₄×C₁₁₃	C₂×#104	1	1	yes	characteristic
136	C₄×C₁₁₃	#421	1	1	yes	characteristic
137	C₄×C₁₁₃	C₄×D₈	1	2	yes	—
138	C₄×C₁₁₃	#466	1	2	yes	—
139	#9719	C₁₁₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
140	C₂×C₁₁₃	—	2	1	no	—
141	C₂×C₁₁₃	—	2	1	no	—
142	C₂×C₁₁₃	—	16	2	no	—
143	C₂×C₁₁₃	#1970	1	1	yes	characteristic
144	C₂×C₁₁₃	#1976	1	1	yes	characteristic
145	C₂×C₁₁₃	#1980	1	1	yes	characteristic
146	#3169	C₂×C₁₁₃	1	1	yes	characteristic
147	#1980	C₂×C₁₁₃	1	2	yes	complemented
148	C₄×#107	C₂×C₁₁₃	1	1	yes	characteristic, complemented
149	#1502	C₂×C₁₁₃	1	2	yes	complemented
150	C₈×M₁₆	C₂×C₁₁₃	1	1	yes	characteristic, complemented
151	C₁₁₃	#9719	1	1	yes	characteristic, direct product factor, Sylow
152	#532	C₂²×C₁₁₃	1	2	yes	—
153	#500	C₄×C₁₁₃	1	1	yes	characteristic, complemented
154	#493	C₄×C₁₁₃	1	1	yes	characteristic, complemented
155	#486	C₄×C₁₁₃	1	2	yes	complemented
156	#466	—	2	1	no	—
157	#466	—	2	1	no	—
158	C₄×D₈	—	2	1	no	—
159	C₄×D₈	—	2	1	no	—
160	#431	C₂²×C₁₁₃	1	1	yes	characteristic
161	C₂×#107	—	2	1	no	—
162	C₂×#104	—	2	4	no	—
163	C₄×M₁₆	C₂²×C₁₁₃	1	1	yes	characteristic
164	C₂×C₄×C₈	C₂²×C₁₁₃	1	1	yes	characteristic
165	C₄×C₄⋊C₄	C₂²×C₁₁₃	1	2	yes	—
166	#340	—	2	1	no	—
167	#327	—	2	4	no	—
168	#322	—	2	1	no	—
169	C₈²	—	2	1	no	—
170	C₂×#41	—	2	2	no	—
171	C₂×D₈	—	2	1	no	—
172	C₂×D₈	—	2	1	no	—
173	C₂×M₁₆	D₄×C₁₁₃	1	2	yes	—
174	C₂²×C₈	D₄×C₁₁₃	1	1	yes	characteristic
175	C₂²×C₈	D₄×C₁₁₃	1	1	yes	characteristic
176	C₄⋊Q₈	C₂×C₄×C₁₁₃	1	2	yes	—
177	#127	C₂×C₄×C₁₁₃	1	2	yes	—
178	#124	—	2	2	no	—
179	#121	—	2	2	no	—
180	#121	—	2	2	no	—
181	C₄×D₄	—	2	2	no	—
182	C₄×D₄	—	2	2	no	—
183	C₂×C₄⋊C₄	—	2	2	no	—
184	C₂×C₄²	—	2	2	no	—
185	C₂×C₄²	C₂³×C₁₁₃	1	1	yes	characteristic
186	#107	—	2	4	no	—
187	#107	—	2	1	no	—
188	#107	—	2	1	no	—
189	C₄⋊C₈	—	2	1	no	—
190	#104	—	4	8	no	—
191	#102	—	2	2	no	—
192	#102	—	2	2	no	—
193	C₂²⋊C₈	D₄×C₁₁₃	1	2	yes	—
194	#97	—	2	1	no	—
195	C₄×C₈	—	2	1	no	—
196	C₄×C₈	—	2	1	no	—
197	C₄×C₈	C₂×C₄×C₁₁₃	1	1	yes	characteristic
198	C₄×C₈	C₂×C₄×C₁₁₃	1	1	yes	characteristic
199	C₄×C₈	D₄×C₁₁₃	1	1	yes	characteristic
200	C₄×C₈	D₄×C₁₁₃	1	1	yes	characteristic
201	#95	—	2	2	no	—
202	#41	—	4	4	no	—
203	#41	—	4	4	no	—
204	C₂×Q₈	—	2	2	no	—
205	C₂×D₄	—	2	2	no	—
206	C₂×D₄	—	2	2	no	—
207	C₂×D₄	—	2	2	no	—
208	C₂×D₄	—	4	2	no	—
209	C₂²×C₄	—	4	2	no	—
210	C₂²×C₄	—	4	2	no	—
211	C₂²×C₄	C₂×D₄×C₁₁₃	1	1	yes	characteristic
212	C₂²×C₄	C₂×D₄×C₁₁₃	1	1	yes	characteristic
213	C₂²×C₄	C₂×D₄×C₁₁₃	1	1	yes	characteristic
214	D₈	—	4	2	no	—
215	D₈	—	4	2	no	—
216	M₁₆	—	2	2	no	—
217	M₁₆	D₈×C₁₁₃	1	4	yes	complemented
218	C₂×C₈	—	2	2	no	—
219	C₂×C₈	—	2	2	no	—
220	C₂×C₈	—	2	2	no	—
221	C₂×C₈	—	4	1	no	—
222	C₂×C₈	C₂²⋊C₄×C₁₁₃	1	1	yes	characteristic
223	C₂×C₈	C₂²⋊C₄×C₁₁₃	1	1	yes	characteristic
224	C₂×C₈	C₂²⋊C₄×C₁₁₃	1	1	yes	characteristic
225	C₂×C₈	C₂²⋊C₄×C₁₁₃	1	1	yes	characteristic
226	C₄⋊C₄	—	2	4	no	—
227	C₄⋊C₄	—	2	2	no	—
228	C₄⋊C₄	—	2	2	no	—
229	C₄⋊C₄	—	4	2	no	—
230	C₂²⋊C₄	—	4	2	no	—
231	C₂²⋊C₄	—	4	2	no	—
232	C₄²	—	4	4	no	—
233	C₄²	C₂²×C₄×C₁₁₃	1	1	yes	characteristic
234	C₄²	C₂×D₄×C₁₁₃	1	1	yes	characteristic
235	C₄²	#41×C₁₁₃	1	1	yes	characteristic
236	C₄²	#41×C₁₁₃	1	1	yes	characteristic
237	C₂³	—	4	2	no	—
238	C₂³	#127×C₁₁₃	1	1	yes	characteristic
239	Q₈	—	4	4	no	—
240	D₄	—	4	2	no	—
241	D₄	—	4	2	no	—
242	D₄	—	4	4	no	—
243	D₄	—	8	2	no	—
244	D₄	—	8	2	no	—
245	D₄	—	8	2	no	—
246	C₂×C₄	—	2	1	no	—
247	C₂×C₄	—	2	1	no	—
248	C₂×C₄	—	4	1	no	—
249	C₂×C₄	—	4	4	no	—
250	C₂×C₄	—	4	2	no	—
251	C₂×C₄	—	8	2	no	—
252	C₂×C₄	—	8	2	no	—
253	C₂×C₄	—	8	4	no	—
254	C₂×C₄	C₂×C₂²⋊C₄×C₁₁₃	1	1	yes	characteristic
255	C₂×C₄	C₄×D₄×C₁₁₃	1	1	yes	characteristic
256	C₂×C₄	C₄×D₄×C₁₁₃	1	1	yes	characteristic
257	C₂×C₄	#121×C₁₁₃	1	1	yes	characteristic
258	C₂×C₄	#121×C₁₁₃	1	1	yes	characteristic
259	C₂×C₄	#124×C₁₁₃	1	1	yes	characteristic
260	C₂×C₄	C₂×D₈×C₁₁₃	1	2	yes	—
261	C₂×C₄	#135×C₁₁₃	1	2	yes	—
262	C₈	—	2	1	no	—
263	C₈	—	2	1	no	—
264	C₈	—	2	1	no	—
265	C₈	—	2	4	no	—
266	C₈	—	4	2	no	—
267	C₈	#104×C₁₁₃	1	2	yes	complemented
268	C₂²	—	2	1	no	—
269	C₂²	—	2	1	no	—
270	C₂²	—	8	2	no	—
271	C₂²	—	8	2	no	—
272	C₂²	—	8	2	no	—
273	C₂²	#390×C₁₁₃	1	1	yes	characteristic
274	C₂²	#493×C₁₁₃	1	1	yes	characteristic
275	C₂²	#495×C₁₁₃	1	1	yes	characteristic
276	C₄	—	2	1	no	—
277	C₄	—	2	1	no	—
278	C₄	—	2	1	no	—
279	C₄	—	8	4	no	—
280	C₄	—	8	2	no	—
281	C₄	C₂×#104×C₁₁₃	1	1	yes	characteristic
282	C₄	#421×C₁₁₃	1	1	yes	characteristic
283	C₄	C₄×D₈×C₁₁₃	1	2	yes	—
284	C₄	#466×C₁₁₃	1	2	yes	—
285	C₂	—	2	1	no	—
286	C₂	—	2	1	no	—
287	C₂	—	16	2	no	—
288	C₂	C₁₁₃×#1970	1	1	yes	characteristic
289	C₂	C₁₁₃×#1976	1	1	yes	characteristic
290	C₂	C₁₁₃×#1980	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄×C₁₁₃	#121
commutator	C₄²	C₂²×C₄×C₁₁₃
Frattini, cosocle	C₂×C₄²	C₂³×C₁₁₃
socle	C₂²×C₁₁₃	#390
prime radical	C₁₁₃×#3169	C₂
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₁₁₃ ◁ C₂×C₄²×C₁₁₃ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₁₁₃×#1980	C₂	semidirect
3. C₄×#107×C₁₁₃	C₂	semidirect
4. C₁₁₃×#1502	C₂	semidirect
5. C₈×M₁₆×C₁₁₃	C₂	semidirect
7. #500×C₁₁₃	C₄	semidirect
8. #493×C₁₁₃	C₄	semidirect
9. #486×C₁₁₃	C₄	semidirect
71. M₁₆×C₁₁₃	D₈	semidirect
121. C₈×C₁₁₃	#104	semidirect
139. #9719	C₁₁₃	direct (N × H)
147. #1980	C₂×C₁₁₃	semidirect
148. C₄×#107	C₂×C₁₁₃	semidirect
149. #1502	C₂×C₁₁₃	semidirect
150. C₈×M₁₆	C₂×C₁₁₃	semidirect
151. C₁₁₃	#9719	direct (N × H)
153. #500	C₄×C₁₁₃	semidirect
154. #493	C₄×C₁₁₃	semidirect
155. #486	C₄×C₁₁₃	semidirect
217. M₁₆	D₈×C₁₁₃	semidirect
267. C₈	#104×C₁₁₃	semidirect

Maximal subgroups

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

subgroup	index
C₁₁₃×#3169	2
C₁₁₃×#1980	2
C₄×#107×C₁₁₃	2
C₁₁₃×#1502	2
C₈×M₁₆×C₁₁₃	2
#9719	113

Subgroups — C113×#9719 #43277739

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₁₃×#9719 #43277739