Subgroups — C₁₁₃×#9721 #43277741

Order 28928.

498 conjugacy classes of subgroups.

The subgroup lattice has 284 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×#106×C₁₁₃	C₂	1	1	yes	maximal, characteristic, complemented
4	C₁₁₃×#1502	C₂	1	2	yes	maximal, complemented
5	C₁₁₃×#1451	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×#106×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	#322×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	1	no	—
41	#107×C₁₁₃	—	2	1	no	—
42	#106×C₁₁₃	—	2	4	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	4	no	—
71	C₂×C₈×C₁₁₃	—	2	2	no	—
72	C₂×C₈×C₁₁₃	—	2	2	no	—
73	C₂×C₈×C₁₁₃	—	2	2	no	—
74	C₂×C₈×C₁₁₃	—	4	1	no	—
75	C₂×C₈×C₁₁₃	C₂²⋊C₄	1	1	yes	characteristic
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	C₄⋊C₄×C₁₁₃	—	2	4	no	—
80	C₄⋊C₄×C₁₁₃	—	2	2	no	—
81	C₄⋊C₄×C₁₁₃	—	2	2	no	—
82	C₄⋊C₄×C₁₁₃	—	4	2	no	—
83	C₂²⋊C₄×C₁₁₃	—	4	2	no	—
84	C₂²⋊C₄×C₁₁₃	—	4	2	no	—
85	C₄²×C₁₁₃	—	4	4	no	—
86	C₄²×C₁₁₃	C₂²×C₄	1	1	yes	characteristic
87	C₄²×C₁₁₃	C₂×D₄	1	1	yes	characteristic
88	C₄²×C₁₁₃	#41	1	1	yes	characteristic
89	C₄²×C₁₁₃	#41	1	1	yes	characteristic
90	C₂³×C₁₁₃	—	4	2	no	—
91	C₂³×C₁₁₃	#127	1	1	yes	characteristic
92	Q₈×C₁₁₃	—	4	4	no	—
93	D₄×C₁₁₃	—	4	2	no	—
94	D₄×C₁₁₃	—	4	2	no	—
95	D₄×C₁₁₃	—	4	4	no	—
96	D₄×C₁₁₃	—	8	2	no	—
97	D₄×C₁₁₃	—	8	2	no	—
98	D₄×C₁₁₃	—	8	2	no	—
99	C₂×C₄×C₁₁₃	—	2	1	no	—
100	C₂×C₄×C₁₁₃	—	2	1	no	—
101	C₂×C₄×C₁₁₃	—	4	1	no	—
102	C₂×C₄×C₁₁₃	—	4	4	no	—
103	C₂×C₄×C₁₁₃	—	4	2	no	—
104	C₂×C₄×C₁₁₃	—	8	2	no	—
105	C₂×C₄×C₁₁₃	—	8	2	no	—
106	C₂×C₄×C₁₁₃	—	8	4	no	—
107	C₂×C₄×C₁₁₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
108	C₂×C₄×C₁₁₃	C₄×D₄	1	1	yes	characteristic
109	C₂×C₄×C₁₁₃	C₄×D₄	1	1	yes	characteristic
110	C₂×C₄×C₁₁₃	#121	1	1	yes	characteristic
111	C₂×C₄×C₁₁₃	#121	1	1	yes	characteristic
112	C₂×C₄×C₁₁₃	#124	1	1	yes	characteristic
113	C₂×C₄×C₁₁₃	AGL(1,ℤ/8)	1	4	yes	—
114	C₈×C₁₁₃	—	2	1	no	—
115	C₈×C₁₁₃	—	2	1	no	—
116	C₈×C₁₁₃	—	2	1	no	—
117	C₈×C₁₁₃	—	4	2	no	—
118	C₈×C₁₁₃	—	4	2	no	—
119	C₈×C₁₁₃	#104	1	2	yes	complemented
120	C₂²×C₁₁₃	—	2	1	no	—
121	C₂²×C₁₁₃	—	2	1	no	—
122	C₂²×C₁₁₃	—	8	2	no	—
123	C₂²×C₁₁₃	—	8	2	no	—
124	C₂²×C₁₁₃	—	8	2	no	—
125	C₂²×C₁₁₃	#390	1	1	yes	characteristic
126	C₂²×C₁₁₃	#496	1	2	yes	—
127	C₄×C₁₁₃	—	2	1	no	—
128	C₄×C₁₁₃	—	2	1	no	—
129	C₄×C₁₁₃	—	2	1	no	—
130	C₄×C₁₁₃	—	8	4	no	—
131	C₄×C₁₁₃	—	8	2	no	—
132	C₄×C₁₁₃	C₂×#104	1	1	yes	characteristic
133	C₄×C₁₁₃	#421	1	1	yes	characteristic
134	C₄×C₁₁₃	#442	1	2	yes	—
135	C₄×C₁₁₃	#469	1	2	yes	—
136	#9721	C₁₁₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
137	C₂×C₁₁₃	—	2	2	no	—
138	C₂×C₁₁₃	—	16	2	no	—
139	C₂×C₁₁₃	#1972	1	1	yes	characteristic
140	C₂×C₁₁₃	#1976	1	1	yes	characteristic
141	C₂×C₁₁₃	#1980	1	1	yes	characteristic
142	#3169	C₂×C₁₁₃	1	1	yes	characteristic
143	#1980	C₂×C₁₁₃	1	2	yes	complemented
144	C₄×#106	C₂×C₁₁₃	1	1	yes	characteristic, complemented
145	#1502	C₂×C₁₁₃	1	2	yes	complemented
146	#1451	C₂×C₁₁₃	1	1	yes	characteristic, complemented
147	C₁₁₃	#9721	1	1	yes	characteristic, direct product factor, Sylow
148	#532	C₂²×C₁₁₃	1	2	yes	—
149	#500	C₄×C₁₁₃	1	1	yes	characteristic, complemented
150	#493	C₄×C₁₁₃	1	1	yes	characteristic, complemented
151	#486	C₄×C₁₁₃	1	2	yes	complemented
152	#466	—	2	1	no	—
153	#466	—	2	1	no	—
154	C₄×D₈	—	2	1	no	—
155	C₄×D₈	—	2	1	no	—
156	#431	C₂²×C₁₁₃	1	1	yes	characteristic
157	C₂×#106	—	2	1	no	—
158	C₂×#104	—	2	4	no	—
159	C₄×M₁₆	C₂²×C₁₁₃	1	1	yes	characteristic
160	C₂×C₄×C₈	C₂²×C₁₁₃	1	1	yes	characteristic
161	C₄×C₄⋊C₄	C₂²×C₁₁₃	1	2	yes	—
162	#340	—	2	1	no	—
163	#327	—	2	4	no	—
164	#322	—	2	1	no	—
165	#322	—	2	1	no	—
166	C₂×#41	—	2	2	no	—
167	C₂×D₈	—	2	1	no	—
168	C₂×D₈	—	2	1	no	—
169	C₂×M₁₆	D₄×C₁₁₃	1	2	yes	—
170	C₂²×C₈	D₄×C₁₁₃	1	1	yes	characteristic
171	C₂²×C₈	D₄×C₁₁₃	1	1	yes	characteristic
172	C₄⋊Q₈	C₂×C₄×C₁₁₃	1	2	yes	—
173	#127	C₂×C₄×C₁₁₃	1	2	yes	—
174	#124	—	2	2	no	—
175	#121	—	2	2	no	—
176	#121	—	2	2	no	—
177	C₄×D₄	—	2	2	no	—
178	C₄×D₄	—	2	2	no	—
179	C₂×C₄⋊C₄	—	2	2	no	—
180	C₂×C₄²	—	2	2	no	—
181	C₂×C₄²	C₂³×C₁₁₃	1	1	yes	characteristic
182	#107	—	2	1	no	—
183	#107	—	2	1	no	—
184	#106	—	2	4	no	—
185	C₄⋊C₈	—	2	1	no	—
186	#104	—	4	8	no	—
187	#102	—	2	2	no	—
188	#102	—	2	2	no	—
189	C₂²⋊C₈	D₄×C₁₁₃	1	2	yes	—
190	#97	—	2	1	no	—
191	C₄×C₈	—	2	1	no	—
192	C₄×C₈	—	2	1	no	—
193	C₄×C₈	C₂×C₄×C₁₁₃	1	1	yes	characteristic
194	C₄×C₈	C₂×C₄×C₁₁₃	1	1	yes	characteristic
195	C₄×C₈	D₄×C₁₁₃	1	1	yes	characteristic
196	C₄×C₈	D₄×C₁₁₃	1	1	yes	characteristic
197	#95	—	2	2	no	—
198	#41	—	4	4	no	—
199	#41	—	4	4	no	—
200	C₂×Q₈	—	2	2	no	—
201	C₂×D₄	—	2	2	no	—
202	C₂×D₄	—	2	2	no	—
203	C₂×D₄	—	2	2	no	—
204	C₂×D₄	—	4	2	no	—
205	C₂²×C₄	—	4	2	no	—
206	C₂²×C₄	—	4	2	no	—
207	C₂²×C₄	C₂×D₄×C₁₁₃	1	1	yes	characteristic
208	C₂²×C₄	C₂×D₄×C₁₁₃	1	1	yes	characteristic
209	C₂²×C₄	C₂×D₄×C₁₁₃	1	1	yes	characteristic
210	D₈	—	4	2	no	—
211	D₈	—	4	2	no	—
212	M₁₆	—	2	4	no	—
213	C₂×C₈	—	2	2	no	—
214	C₂×C₈	—	2	2	no	—
215	C₂×C₈	—	2	2	no	—
216	C₂×C₈	—	4	1	no	—
217	C₂×C₈	C₂²⋊C₄×C₁₁₃	1	1	yes	characteristic
218	C₂×C₈	C₂²⋊C₄×C₁₁₃	1	1	yes	characteristic
219	C₂×C₈	C₂²⋊C₄×C₁₁₃	1	1	yes	characteristic
220	C₂×C₈	C₂²⋊C₄×C₁₁₃	1	1	yes	characteristic
221	C₄⋊C₄	—	2	4	no	—
222	C₄⋊C₄	—	2	2	no	—
223	C₄⋊C₄	—	2	2	no	—
224	C₄⋊C₄	—	4	2	no	—
225	C₂²⋊C₄	—	4	2	no	—
226	C₂²⋊C₄	—	4	2	no	—
227	C₄²	—	4	4	no	—
228	C₄²	C₂²×C₄×C₁₁₃	1	1	yes	characteristic
229	C₄²	C₂×D₄×C₁₁₃	1	1	yes	characteristic
230	C₄²	#41×C₁₁₃	1	1	yes	characteristic
231	C₄²	#41×C₁₁₃	1	1	yes	characteristic
232	C₂³	—	4	2	no	—
233	C₂³	#127×C₁₁₃	1	1	yes	characteristic
234	Q₈	—	4	4	no	—
235	D₄	—	4	2	no	—
236	D₄	—	4	2	no	—
237	D₄	—	4	4	no	—
238	D₄	—	8	2	no	—
239	D₄	—	8	2	no	—
240	D₄	—	8	2	no	—
241	C₂×C₄	—	2	1	no	—
242	C₂×C₄	—	2	1	no	—
243	C₂×C₄	—	4	1	no	—
244	C₂×C₄	—	4	4	no	—
245	C₂×C₄	—	4	2	no	—
246	C₂×C₄	—	8	2	no	—
247	C₂×C₄	—	8	2	no	—
248	C₂×C₄	—	8	4	no	—
249	C₂×C₄	C₂×C₂²⋊C₄×C₁₁₃	1	1	yes	characteristic
250	C₂×C₄	C₄×D₄×C₁₁₃	1	1	yes	characteristic
251	C₂×C₄	C₄×D₄×C₁₁₃	1	1	yes	characteristic
252	C₂×C₄	#121×C₁₁₃	1	1	yes	characteristic
253	C₂×C₄	#121×C₁₁₃	1	1	yes	characteristic
254	C₂×C₄	#124×C₁₁₃	1	1	yes	characteristic
255	C₂×C₄	AGL(1,ℤ/8)×C₁₁₃	1	4	yes	—
256	C₈	—	2	1	no	—
257	C₈	—	2	1	no	—
258	C₈	—	2	1	no	—
259	C₈	—	4	2	no	—
260	C₈	—	4	2	no	—
261	C₈	#104×C₁₁₃	1	2	yes	complemented
262	C₂²	—	2	1	no	—
263	C₂²	—	2	1	no	—
264	C₂²	—	8	2	no	—
265	C₂²	—	8	2	no	—
266	C₂²	—	8	2	no	—
267	C₂²	#390×C₁₁₃	1	1	yes	characteristic
268	C₂²	#496×C₁₁₃	1	2	yes	—
269	C₄	—	2	1	no	—
270	C₄	—	2	1	no	—
271	C₄	—	2	1	no	—
272	C₄	—	8	4	no	—
273	C₄	—	8	2	no	—
274	C₄	C₂×#104×C₁₁₃	1	1	yes	characteristic
275	C₄	#421×C₁₁₃	1	1	yes	characteristic
276	C₄	#442×C₁₁₃	1	2	yes	—
277	C₄	#469×C₁₁₃	1	2	yes	—
278	C₂	—	2	2	no	—
279	C₂	—	16	2	no	—
280	C₂	C₁₁₃×#1972	1	1	yes	characteristic
281	C₂	C₁₁₃×#1976	1	1	yes	characteristic
282	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×#106×C₁₁₃	C₂	semidirect
4. C₁₁₃×#1502	C₂	semidirect
5. C₁₁₃×#1451	C₂	semidirect
7. #500×C₁₁₃	C₄	semidirect
8. #493×C₁₁₃	C₄	semidirect
9. #486×C₁₁₃	C₄	semidirect
119. C₈×C₁₁₃	#104	semidirect
136. #9721	C₁₁₃	direct (N × H)
143. #1980	C₂×C₁₁₃	semidirect
144. C₄×#106	C₂×C₁₁₃	semidirect
145. #1502	C₂×C₁₁₃	semidirect
146. #1451	C₂×C₁₁₃	semidirect
147. C₁₁₃	#9721	direct (N × H)
149. #500	C₄×C₁₁₃	semidirect
150. #493	C₄×C₁₁₃	semidirect
151. #486	C₄×C₁₁₃	semidirect
261. 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×#106×C₁₁₃	2
C₁₁₃×#1502	2
C₁₁₃×#1451	2
#9721	113

Subgroups — C113×#9721 #43277741

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₁₃×#9721 #43277741