Subgroups — C₁₁₃×#9756 #43277776

Order 28928.

502 conjugacy classes of subgroups.

The subgroup lattice has 266 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₁₁₃×#3147	C₂	1	1	yes	maximal, characteristic
2	C₁₁₃×#1846	C₂	1	1	yes	maximal, characteristic, complemented
3	C₁₁₃×#1517	C₂	1	4	yes	maximal, complemented
4	C₁₁₃×#1450	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂×#135×C₁₁₃	—	2	1	no	—
6	C₂×#135×C₁₁₃	—	2	1	no	—
7	#532×C₁₁₃	C₂²	1	2	yes	—
8	#495×C₁₁₃	—	2	1	no	—
9	#495×C₁₁₃	—	2	1	no	—
10	#494×C₁₁₃	C₄	1	1	yes	characteristic, complemented
11	#493×C₁₁₃	C₄	1	1	yes	characteristic, complemented
12	#492×C₁₁₃	C₄	1	2	yes	complemented
13	#431×C₁₁₃	C₂²	1	2	yes	—
14	C₂×#107×C₁₁₃	—	2	1	no	—
15	C₂×#106×C₁₁₃	—	2	1	no	—
16	C₂×C₄×C₈×C₁₁₃	C₂²	1	1	yes	characteristic
17	#382×C₁₁₃	C₂²	1	2	yes	—
18	#327×C₁₁₃	—	2	8	no	—
19	#322×C₁₁₃	—	2	1	no	—
20	C₈²×C₁₁₃	—	2	1	no	—
21	C₂×#41×C₁₁₃	—	2	2	no	—
22	#135×C₁₁₃	—	4	2	no	—
23	#135×C₁₁₃	—	4	2	no	—
24	#135×C₁₁₃	—	4	2	no	—
25	#135×C₁₁₃	—	4	2	no	—
26	C₂×Q₁₆×C₁₁₃	—	2	1	no	—
27	C₂×Q₁₆×C₁₁₃	—	2	1	no	—
28	C₂×SD₁₆×C₁₁₃	—	2	2	no	—
29	C₂×SD₁₆×C₁₁₃	—	2	2	no	—
30	C₂×D₈×C₁₁₃	—	2	1	no	—
31	C₂×D₈×C₁₁₃	—	2	1	no	—
32	C₂²×C₈×C₁₁₃	D₄	1	1	yes	characteristic
33	C₂²×C₈×C₁₁₃	D₄	1	1	yes	characteristic
34	C₄⋊Q₈×C₁₁₃	C₂×C₄	1	2	yes	—
35	#127×C₁₁₃	C₂×C₄	1	2	yes	—
36	#124×C₁₁₃	—	2	2	no	—
37	#121×C₁₁₃	—	2	4	no	—
38	C₄×D₄×C₁₁₃	—	2	4	no	—
39	C₂×C₄⋊C₄×C₁₁₃	—	2	2	no	—
40	C₂×C₄²×C₁₁₃	C₂³	1	1	yes	characteristic
41	#107×C₁₁₃	—	4	2	no	—
42	#106×C₁₁₃	—	4	2	no	—
43	C₄⋊C₈×C₁₁₃	—	2	2	no	—
44	C₂²⋊C₈×C₁₁₃	D₄	1	4	yes	—
45	C₄×C₈×C₁₁₃	—	2	2	no	—
46	C₄×C₈×C₁₁₃	C₂×C₄	1	1	yes	characteristic
47	C₄×C₈×C₁₁₃	C₂×C₄	1	1	yes	characteristic
48	C₄×C₈×C₁₁₃	D₄	1	1	yes	characteristic
49	C₄×C₈×C₁₁₃	D₄	1	1	yes	characteristic
50	#95×C₁₁₃	—	2	4	no	—
51	#41×C₁₁₃	—	4	2	no	—
52	#41×C₁₁₃	—	4	2	no	—
53	#41×C₁₁₃	—	4	2	no	—
54	#41×C₁₁₃	—	4	2	no	—
55	C₂×Q₈×C₁₁₃	—	2	2	no	—
56	C₂×D₄×C₁₁₃	—	2	2	no	—
57	C₂×D₄×C₁₁₃	—	2	4	no	—
58	C₂×D₄×C₁₁₃	—	4	2	no	—
59	C₂²×C₄×C₁₁₃	—	4	2	no	—
60	C₂²×C₄×C₁₁₃	—	4	2	no	—
61	C₂²×C₄×C₁₁₃	C₂×D₄	1	2	yes	—
62	C₂²×C₄×C₁₁₃	C₂×D₄	1	1	yes	characteristic
63	Q₁₆×C₁₁₃	—	4	2	no	—
64	Q₁₆×C₁₁₃	—	4	2	no	—
65	SD₁₆×C₁₁₃	—	4	4	no	—
66	SD₁₆×C₁₁₃	—	4	4	no	—
67	D₈×C₁₁₃	—	4	2	no	—
68	D₈×C₁₁₃	—	4	2	no	—
69	C₂×C₈×C₁₁₃	—	2	4	no	—
70	C₂×C₈×C₁₁₃	C₂²⋊C₄	1	1	yes	characteristic
71	C₂×C₈×C₁₁₃	C₂²⋊C₄	1	1	yes	characteristic
72	C₂×C₈×C₁₁₃	C₂²⋊C₄	1	1	yes	characteristic
73	C₂×C₈×C₁₁₃	C₂²⋊C₄	1	1	yes	characteristic
74	C₂×C₈×C₁₁₃	D₈	1	2	yes	—
75	C₂×C₈×C₁₁₃	D₈	1	2	yes	—
76	C₂×C₈×C₁₁₃	SD₁₆	1	2	yes	—
77	C₂×C₈×C₁₁₃	SD₁₆	1	2	yes	—
78	C₄⋊C₄×C₁₁₃	—	2	4	no	—
79	C₄⋊C₄×C₁₁₃	—	4	2	no	—
80	C₄⋊C₄×C₁₁₃	—	4	2	no	—
81	C₂²⋊C₄×C₁₁₃	—	4	4	no	—
82	C₄²×C₁₁₃	C₂²×C₄	1	1	yes	characteristic
83	C₄²×C₁₁₃	C₂×D₄	1	1	yes	characteristic
84	C₄²×C₁₁₃	#41	1	2	yes	—
85	C₂³×C₁₁₃	—	4	2	no	—
86	C₂³×C₁₁₃	#127	1	1	yes	characteristic
87	Q₈×C₁₁₃	—	4	2	no	—
88	Q₈×C₁₁₃	—	4	2	no	—
89	D₄×C₁₁₃	—	4	2	no	—
90	D₄×C₁₁₃	—	4	2	no	—
91	D₄×C₁₁₃	—	8	4	no	—
92	D₄×C₁₁₃	—	8	2	no	—
93	D₄×C₁₁₃	—	8	2	no	—
94	C₂×C₄×C₁₁₃	—	4	2	no	—
95	C₂×C₄×C₁₁₃	—	4	2	no	—
96	C₂×C₄×C₁₁₃	—	8	2	no	—
97	C₂×C₄×C₁₁₃	—	8	2	no	—
98	C₂×C₄×C₁₁₃	—	8	4	no	—
99	C₂×C₄×C₁₁₃	—	8	2	no	—
100	C₂×C₄×C₁₁₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
101	C₂×C₄×C₁₁₃	C₄×D₄	1	2	yes	—
102	C₂×C₄×C₁₁₃	#121	1	2	yes	—
103	C₂×C₄×C₁₁₃	#124	1	1	yes	characteristic
104	C₂×C₄×C₁₁₃	C₂×D₈	1	1	yes	characteristic
105	C₂×C₄×C₁₁₃	C₂×D₈	1	1	yes	characteristic
106	C₂×C₄×C₁₁₃	C₂×SD₁₆	1	1	yes	characteristic
107	C₂×C₄×C₁₁₃	C₂×SD₁₆	1	1	yes	characteristic
108	C₈×C₁₁₃	—	4	4	no	—
109	C₈×C₁₁₃	#102	1	2	yes	—
110	C₈×C₁₁₃	#102	1	2	yes	—
111	C₈×C₁₁₃	#102	1	2	yes	—
112	C₈×C₁₁₃	#102	1	2	yes	—
113	C₂²×C₁₁₃	—	2	2	no	—
114	C₂²×C₁₁₃	—	8	2	no	—
115	C₂²×C₁₁₃	—	8	4	no	—
116	C₂²×C₁₁₃	#390	1	1	yes	characteristic
117	C₂²×C₁₁₃	#492	1	1	yes	characteristic
118	C₂²×C₁₁₃	#493	1	1	yes	characteristic
119	C₄×C₁₁₃	—	2	2	no	—
120	C₄×C₁₁₃	—	2	2	no	—
121	C₄×C₁₁₃	—	8	2	no	—
122	C₄×C₁₁₃	—	16	2	no	—
123	C₄×C₁₁₃	C₂×#102	1	1	yes	characteristic
124	C₄×C₁₁₃	C₂×#102	1	1	yes	characteristic
125	C₄×C₁₁₃	#486	1	1	yes	characteristic
126	C₄×C₁₁₃	#486	1	1	yes	characteristic
127	#9756	C₁₁₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
128	C₂×C₁₁₃	—	2	1	no	—
129	C₂×C₁₁₃	—	2	1	no	—
130	C₂×C₁₁₃	—	16	2	no	—
131	C₂×C₁₁₃	#1957	1	1	yes	characteristic
132	C₂×C₁₁₃	#1976	1	2	yes	—
133	#3147	C₂×C₁₁₃	1	1	yes	characteristic
134	#1846	C₂×C₁₁₃	1	1	yes	characteristic, complemented
135	#1517	C₂×C₁₁₃	1	4	yes	complemented
136	#1450	C₂×C₁₁₃	1	1	yes	characteristic, complemented
137	C₁₁₃	#9756	1	1	yes	characteristic, direct product factor, Sylow
138	C₂×#135	—	2	1	no	—
139	C₂×#135	—	2	1	no	—
140	#532	C₂²×C₁₁₃	1	2	yes	—
141	#495	—	2	1	no	—
142	#495	—	2	1	no	—
143	#494	C₄×C₁₁₃	1	1	yes	characteristic, complemented
144	#493	C₄×C₁₁₃	1	1	yes	characteristic, complemented
145	#492	C₄×C₁₁₃	1	2	yes	complemented
146	#431	C₂²×C₁₁₃	1	2	yes	—
147	C₂×#107	—	2	1	no	—
148	C₂×#106	—	2	1	no	—
149	C₂×C₄×C₈	C₂²×C₁₁₃	1	1	yes	characteristic
150	#382	C₂²×C₁₁₃	1	2	yes	—
151	#327	—	2	8	no	—
152	#322	—	2	1	no	—
153	C₈²	—	2	1	no	—
154	C₂×#41	—	2	2	no	—
155	#135	—	4	2	no	—
156	#135	—	4	2	no	—
157	#135	—	4	2	no	—
158	#135	—	4	2	no	—
159	C₂×Q₁₆	—	2	1	no	—
160	C₂×Q₁₆	—	2	1	no	—
161	C₂×SD₁₆	—	2	2	no	—
162	C₂×SD₁₆	—	2	2	no	—
163	C₂×D₈	—	2	1	no	—
164	C₂×D₈	—	2	1	no	—
165	C₂²×C₈	D₄×C₁₁₃	1	1	yes	characteristic
166	C₂²×C₈	D₄×C₁₁₃	1	1	yes	characteristic
167	C₄⋊Q₈	C₂×C₄×C₁₁₃	1	2	yes	—
168	#127	C₂×C₄×C₁₁₃	1	2	yes	—
169	#124	—	2	2	no	—
170	#121	—	2	4	no	—
171	C₄×D₄	—	2	4	no	—
172	C₂×C₄⋊C₄	—	2	2	no	—
173	C₂×C₄²	C₂³×C₁₁₃	1	1	yes	characteristic
174	#107	—	4	2	no	—
175	#106	—	4	2	no	—
176	C₄⋊C₈	—	2	2	no	—
177	C₂²⋊C₈	D₄×C₁₁₃	1	4	yes	—
178	C₄×C₈	—	2	2	no	—
179	C₄×C₈	C₂×C₄×C₁₁₃	1	1	yes	characteristic
180	C₄×C₈	C₂×C₄×C₁₁₃	1	1	yes	characteristic
181	C₄×C₈	D₄×C₁₁₃	1	1	yes	characteristic
182	C₄×C₈	D₄×C₁₁₃	1	1	yes	characteristic
183	#95	—	2	4	no	—
184	#41	—	4	2	no	—
185	#41	—	4	2	no	—
186	#41	—	4	2	no	—
187	#41	—	4	2	no	—
188	C₂×Q₈	—	2	2	no	—
189	C₂×D₄	—	2	2	no	—
190	C₂×D₄	—	2	4	no	—
191	C₂×D₄	—	4	2	no	—
192	C₂²×C₄	—	4	2	no	—
193	C₂²×C₄	—	4	2	no	—
194	C₂²×C₄	C₂×D₄×C₁₁₃	1	2	yes	—
195	C₂²×C₄	C₂×D₄×C₁₁₃	1	1	yes	characteristic
196	Q₁₆	—	4	2	no	—
197	Q₁₆	—	4	2	no	—
198	SD₁₆	—	4	4	no	—
199	SD₁₆	—	4	4	no	—
200	D₈	—	4	2	no	—
201	D₈	—	4	2	no	—
202	C₂×C₈	—	2	4	no	—
203	C₂×C₈	C₂²⋊C₄×C₁₁₃	1	1	yes	characteristic
204	C₂×C₈	C₂²⋊C₄×C₁₁₃	1	1	yes	characteristic
205	C₂×C₈	C₂²⋊C₄×C₁₁₃	1	1	yes	characteristic
206	C₂×C₈	C₂²⋊C₄×C₁₁₃	1	1	yes	characteristic
207	C₂×C₈	D₈×C₁₁₃	1	2	yes	—
208	C₂×C₈	D₈×C₁₁₃	1	2	yes	—
209	C₂×C₈	SD₁₆×C₁₁₃	1	2	yes	—
210	C₂×C₈	SD₁₆×C₁₁₃	1	2	yes	—
211	C₄⋊C₄	—	2	4	no	—
212	C₄⋊C₄	—	4	2	no	—
213	C₄⋊C₄	—	4	2	no	—
214	C₂²⋊C₄	—	4	4	no	—
215	C₄²	C₂²×C₄×C₁₁₃	1	1	yes	characteristic
216	C₄²	C₂×D₄×C₁₁₃	1	1	yes	characteristic
217	C₄²	#41×C₁₁₃	1	2	yes	—
218	C₂³	—	4	2	no	—
219	C₂³	#127×C₁₁₃	1	1	yes	characteristic
220	Q₈	—	4	2	no	—
221	Q₈	—	4	2	no	—
222	D₄	—	4	2	no	—
223	D₄	—	4	2	no	—
224	D₄	—	8	4	no	—
225	D₄	—	8	2	no	—
226	D₄	—	8	2	no	—
227	C₂×C₄	—	4	2	no	—
228	C₂×C₄	—	4	2	no	—
229	C₂×C₄	—	8	2	no	—
230	C₂×C₄	—	8	2	no	—
231	C₂×C₄	—	8	4	no	—
232	C₂×C₄	—	8	2	no	—
233	C₂×C₄	C₂×C₂²⋊C₄×C₁₁₃	1	1	yes	characteristic
234	C₂×C₄	C₄×D₄×C₁₁₃	1	2	yes	—
235	C₂×C₄	#121×C₁₁₃	1	2	yes	—
236	C₂×C₄	#124×C₁₁₃	1	1	yes	characteristic
237	C₂×C₄	C₂×D₈×C₁₁₃	1	1	yes	characteristic
238	C₂×C₄	C₂×D₈×C₁₁₃	1	1	yes	characteristic
239	C₂×C₄	C₂×SD₁₆×C₁₁₃	1	1	yes	characteristic
240	C₂×C₄	C₂×SD₁₆×C₁₁₃	1	1	yes	characteristic
241	C₈	—	4	4	no	—
242	C₈	#102×C₁₁₃	1	2	yes	—
243	C₈	#102×C₁₁₃	1	2	yes	—
244	C₈	#102×C₁₁₃	1	2	yes	—
245	C₈	#102×C₁₁₃	1	2	yes	—
246	C₂²	—	2	2	no	—
247	C₂²	—	8	2	no	—
248	C₂²	—	8	4	no	—
249	C₂²	#390×C₁₁₃	1	1	yes	characteristic
250	C₂²	#492×C₁₁₃	1	1	yes	characteristic
251	C₂²	#493×C₁₁₃	1	1	yes	characteristic
252	C₄	—	2	2	no	—
253	C₄	—	2	2	no	—
254	C₄	—	8	2	no	—
255	C₄	—	16	2	no	—
256	C₄	C₂×#102×C₁₁₃	1	1	yes	characteristic
257	C₄	C₂×#102×C₁₁₃	1	1	yes	characteristic
258	C₄	#486×C₁₁₃	1	1	yes	characteristic
259	C₄	#486×C₁₁₃	1	1	yes	characteristic
260	C₂	—	2	1	no	—
261	C₂	—	2	1	no	—
262	C₂	—	16	2	no	—
263	C₂	C₁₁₃×#1957	1	1	yes	characteristic
264	C₂	C₁₁₃×#1976	1	2	yes	—

	subgroup	G/H
center, socle	C₂²×C₁₁₃	#390
commutator	C₄²	C₂²×C₄×C₁₁₃
Frattini, cosocle	C₂×C₄²	C₂³×C₁₁₃
prime radical	C₁₁₃×#3147	C₂
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₁₁₃ ◁ C₂×C₄²×C₁₁₃ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₁₁₃×#1846	C₂	semidirect
3. C₁₁₃×#1517	C₂	semidirect
4. C₁₁₃×#1450	C₂	semidirect
10. #494×C₁₁₃	C₄	semidirect
11. #493×C₁₁₃	C₄	semidirect
12. #492×C₁₁₃	C₄	semidirect
127. #9756	C₁₁₃	direct (N × H)
134. #1846	C₂×C₁₁₃	semidirect
135. #1517	C₂×C₁₁₃	semidirect
136. #1450	C₂×C₁₁₃	semidirect
137. C₁₁₃	#9756	direct (N × H)
143. #494	C₄×C₁₁₃	semidirect
144. #493	C₄×C₁₁₃	semidirect
145. #492	C₄×C₁₁₃	semidirect

Maximal subgroups

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

subgroup	index
C₁₁₃×#3147	2
C₁₁₃×#1846	2
C₁₁₃×#1517	2
C₁₁₃×#1450	2
#9756	113

Subgroups — C113×#9756 #43277776

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₁₃×#9756 #43277776