Subgroups — C₁₁₃×#9850 #43277870

Order 28928.

454 conjugacy classes of subgroups.

The subgroup lattice has 228 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₁₁₃×#3170	C₂	1	1	yes	maximal, characteristic
2	C₁₁₃×#1994	C₂	1	2	yes	maximal
3	C₄×#97×C₁₁₃	C₂	1	1	yes	maximal, characteristic
4	C₁₁₃×#1566	C₂	1	1	yes	maximal, characteristic
5	C₁₁₃×#1493	C₂	1	2	yes	maximal
6	#533×C₁₁₃	C₂²	1	2	yes	—
7	#500×C₁₁₃	C₄	1	1	yes	characteristic, complemented
8	#494×C₁₁₃	C₄	1	1	yes	characteristic, complemented
9	#487×C₁₁₃	C₄	1	2	yes	complemented
10	#467×C₁₁₃	—	2	2	no	—
11	C₄×Q₁₆×C₁₁₃	—	2	2	no	—
12	#423×C₁₁₃	C₂²	1	2	yes	—
13	C₂×#97×C₁₁₃	—	2	2	no	—
14	C₂×C₄×C₈×C₁₁₃	C₂²	1	1	yes	characteristic
15	C₂×C₄×C₈×C₁₁₃	C₂²	1	1	yes	characteristic
16	C₄³×C₁₁₃	C₂²	1	1	yes	characteristic
17	#339×C₁₁₃	—	2	4	no	—
18	#334×C₁₁₃	—	2	2	no	—
19	Q₈⋊C₈×C₁₁₃	—	2	4	no	—
20	C₂×Q₁₆×C₁₁₃	—	2	2	no	—
21	C₂×M₁₆×C₁₁₃	—	2	2	no	—
22	C₂²×C₈×C₁₁₃	—	2	1	no	—
23	C₂²×C₈×C₁₁₃	D₄	1	2	yes	—
24	C₄⋊Q₈×C₁₁₃	C₂×C₄	1	2	yes	—
25	C₄⋊Q₈×C₁₁₃	C₂×C₄	1	2	yes	—
26	#125×C₁₁₃	—	2	2	no	—
27	C₂²⋊Q₈×C₁₁₃	—	2	2	no	—
28	C₂²⋊Q₈×C₁₁₃	—	2	2	no	—
29	C₄×Q₈×C₁₁₃	—	2	2	no	—
30	C₄×Q₈×C₁₁₃	—	2	2	no	—
31	#117×C₁₁₃	—	2	2	no	—
32	C₂×C₄²×C₁₁₃	—	2	1	no	—
33	C₂×C₄²×C₁₁₃	—	2	1	no	—
34	C₂×C₄²×C₁₁₃	D₄	1	2	yes	—
35	C₂×C₄²×C₁₁₃	C₂³	1	1	yes	characteristic
36	#107×C₁₁₃	—	2	2	no	—
37	C₄⋊C₈×C₁₁₃	—	2	2	no	—
38	C₄⋊C₈×C₁₁₃	—	2	2	no	—
39	#103×C₁₁₃	—	2	4	no	—
40	#97×C₁₁₃	—	2	8	no	—
41	C₄×C₈×C₁₁₃	—	2	1	no	—
42	C₄×C₈×C₁₁₃	C₂×C₄	1	1	yes	characteristic
43	C₄×C₈×C₁₁₃	C₂×C₄	1	1	yes	characteristic
44	C₄×C₈×C₁₁₃	D₄	1	2	yes	—
45	C₄×C₈×C₁₁₃	D₄	1	2	yes	—
46	C₂×Q₈×C₁₁₃	—	2	2	no	—
47	C₂×Q₈×C₁₁₃	—	2	2	no	—
48	C₂²×C₄×C₁₁₃	—	2	2	no	—
49	C₂²×C₄×C₁₁₃	C₂×D₄	1	1	yes	characteristic
50	C₂²×C₄×C₁₁₃	C₂×D₄	1	1	yes	characteristic
51	C₂²×C₄×C₁₁₃	#41	1	1	yes	characteristic
52	Q₁₆×C₁₁₃	—	4	4	no	—
53	M₁₆×C₁₁₃	—	4	4	no	—
54	C₂×C₈×C₁₁₃	—	2	4	no	—
55	C₂×C₈×C₁₁₃	—	2	2	no	—
56	C₂×C₈×C₁₁₃	—	2	4	no	—
57	C₂×C₈×C₁₁₃	—	4	2	no	—
58	C₂×C₈×C₁₁₃	C₂²⋊C₄	1	2	yes	—
59	C₂×C₈×C₁₁₃	C₂²⋊C₄	1	2	yes	—
60	C₄⋊C₄×C₁₁₃	—	2	2	no	—
61	C₄⋊C₄×C₁₁₃	—	2	2	no	—
62	C₄⋊C₄×C₁₁₃	—	2	2	no	—
63	C₄⋊C₄×C₁₁₃	—	2	2	no	—
64	C₄⋊C₄×C₁₁₃	—	4	2	no	—
65	C₄⋊C₄×C₁₁₃	—	4	2	no	—
66	C₂²⋊C₄×C₁₁₃	—	4	2	no	—
67	C₄²×C₁₁₃	—	2	4	no	—
68	C₄²×C₁₁₃	—	2	4	no	—
69	C₄²×C₁₁₃	—	4	2	no	—
70	C₄²×C₁₁₃	—	4	2	no	—
71	C₄²×C₁₁₃	C₂²×C₄	1	1	yes	characteristic
72	C₄²×C₁₁₃	C₂×D₄	1	1	yes	characteristic
73	C₄²×C₁₁₃	C₂×D₄	1	1	yes	characteristic
74	C₄²×C₁₁₃	#41	1	1	yes	characteristic
75	C₂³×C₁₁₃	#121	1	1	yes	characteristic
76	Q₈×C₁₁₃	—	4	4	no	—
77	Q₈×C₁₁₃	—	4	4	no	—
78	C₂×C₄×C₁₁₃	—	2	2	no	—
79	C₂×C₄×C₁₁₃	—	2	2	no	—
80	C₂×C₄×C₁₁₃	—	2	4	no	—
81	C₂×C₄×C₁₁₃	—	4	2	no	—
82	C₂×C₄×C₁₁₃	—	4	2	no	—
83	C₂×C₄×C₁₁₃	—	4	2	no	—
84	C₂×C₄×C₁₁₃	—	4	2	no	—
85	C₂×C₄×C₁₁₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
86	C₂×C₄×C₁₁₃	C₄×D₄	1	1	yes	characteristic
87	C₂×C₄×C₁₁₃	C₄×D₄	1	1	yes	characteristic
88	C₂×C₄×C₁₁₃	#121	1	1	yes	characteristic
89	C₂×C₄×C₁₁₃	#124	1	1	yes	characteristic
90	C₂×C₄×C₁₁₃	#127	1	1	yes	characteristic
91	C₂×C₄×C₁₁₃	#137	1	4	yes	—
92	C₈×C₁₁₃	—	2	4	no	—
93	C₈×C₁₁₃	—	2	2	no	—
94	C₈×C₁₁₃	—	8	2	no	—
95	C₈×C₁₁₃	#104	1	4	yes	—
96	C₂²×C₁₁₃	—	2	1	no	—
97	C₂²×C₁₁₃	—	2	1	no	—
98	C₂²×C₁₁₃	#390	1	1	yes	characteristic
99	C₂²×C₁₁₃	#470	1	2	yes	—
100	C₄×C₁₁₃	—	2	1	no	—
101	C₄×C₁₁₃	—	2	1	no	—
102	C₄×C₁₁₃	—	4	4	no	—
103	C₄×C₁₁₃	—	8	2	no	—
104	C₄×C₁₁₃	—	8	2	no	—
105	C₄×C₁₁₃	C₂×#104	1	2	yes	—
106	C₄×C₁₁₃	#441	1	2	yes	—
107	C₄×C₁₁₃	#444	1	2	yes	—
108	C₄×C₁₁₃	#497	1	2	yes	—
109	#9850	C₁₁₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
110	C₂×C₁₁₃	—	2	2	no	—
111	C₂×C₁₁₃	#1962	1	1	yes	characteristic
112	C₂×C₁₁₃	#1971	1	1	yes	characteristic
113	C₂×C₁₁₃	#1975	1	1	yes	characteristic
114	#3170	C₂×C₁₁₃	1	1	yes	characteristic
115	#1994	C₂×C₁₁₃	1	2	yes	—
116	C₄×#97	C₂×C₁₁₃	1	1	yes	characteristic
117	#1566	C₂×C₁₁₃	1	1	yes	characteristic
118	#1493	C₂×C₁₁₃	1	2	yes	—
119	C₁₁₃	#9850	1	1	yes	characteristic, direct product factor, Sylow
120	#533	C₂²×C₁₁₃	1	2	yes	—
121	#500	C₄×C₁₁₃	1	1	yes	characteristic, complemented
122	#494	C₄×C₁₁₃	1	1	yes	characteristic, complemented
123	#487	C₄×C₁₁₃	1	2	yes	complemented
124	#467	—	2	2	no	—
125	C₄×Q₁₆	—	2	2	no	—
126	#423	C₂²×C₁₁₃	1	2	yes	—
127	C₂×#97	—	2	2	no	—
128	C₂×C₄×C₈	C₂²×C₁₁₃	1	1	yes	characteristic
129	C₂×C₄×C₈	C₂²×C₁₁₃	1	1	yes	characteristic
130	C₄³	C₂²×C₁₁₃	1	1	yes	characteristic
131	#339	—	2	4	no	—
132	#334	—	2	2	no	—
133	Q₈⋊C₈	—	2	4	no	—
134	C₂×Q₁₆	—	2	2	no	—
135	C₂×M₁₆	—	2	2	no	—
136	C₂²×C₈	—	2	1	no	—
137	C₂²×C₈	D₄×C₁₁₃	1	2	yes	—
138	C₄⋊Q₈	C₂×C₄×C₁₁₃	1	2	yes	—
139	C₄⋊Q₈	C₂×C₄×C₁₁₃	1	2	yes	—
140	#125	—	2	2	no	—
141	C₂²⋊Q₈	—	2	2	no	—
142	C₂²⋊Q₈	—	2	2	no	—
143	C₄×Q₈	—	2	2	no	—
144	C₄×Q₈	—	2	2	no	—
145	#117	—	2	2	no	—
146	C₂×C₄²	—	2	1	no	—
147	C₂×C₄²	—	2	1	no	—
148	C₂×C₄²	D₄×C₁₁₃	1	2	yes	—
149	C₂×C₄²	C₂³×C₁₁₃	1	1	yes	characteristic
150	#107	—	2	2	no	—
151	C₄⋊C₈	—	2	2	no	—
152	C₄⋊C₈	—	2	2	no	—
153	#103	—	2	4	no	—
154	#97	—	2	8	no	—
155	C₄×C₈	—	2	1	no	—
156	C₄×C₈	C₂×C₄×C₁₁₃	1	1	yes	characteristic
157	C₄×C₈	C₂×C₄×C₁₁₃	1	1	yes	characteristic
158	C₄×C₈	D₄×C₁₁₃	1	2	yes	—
159	C₄×C₈	D₄×C₁₁₃	1	2	yes	—
160	C₂×Q₈	—	2	2	no	—
161	C₂×Q₈	—	2	2	no	—
162	C₂²×C₄	—	2	2	no	—
163	C₂²×C₄	C₂×D₄×C₁₁₃	1	1	yes	characteristic
164	C₂²×C₄	C₂×D₄×C₁₁₃	1	1	yes	characteristic
165	C₂²×C₄	#41×C₁₁₃	1	1	yes	characteristic
166	Q₁₆	—	4	4	no	—
167	M₁₆	—	4	4	no	—
168	C₂×C₈	—	2	4	no	—
169	C₂×C₈	—	2	2	no	—
170	C₂×C₈	—	2	4	no	—
171	C₂×C₈	—	4	2	no	—
172	C₂×C₈	C₂²⋊C₄×C₁₁₃	1	2	yes	—
173	C₂×C₈	C₂²⋊C₄×C₁₁₃	1	2	yes	—
174	C₄⋊C₄	—	2	2	no	—
175	C₄⋊C₄	—	2	2	no	—
176	C₄⋊C₄	—	2	2	no	—
177	C₄⋊C₄	—	2	2	no	—
178	C₄⋊C₄	—	4	2	no	—
179	C₄⋊C₄	—	4	2	no	—
180	C₂²⋊C₄	—	4	2	no	—
181	C₄²	—	2	4	no	—
182	C₄²	—	2	4	no	—
183	C₄²	—	4	2	no	—
184	C₄²	—	4	2	no	—
185	C₄²	C₂²×C₄×C₁₁₃	1	1	yes	characteristic
186	C₄²	C₂×D₄×C₁₁₃	1	1	yes	characteristic
187	C₄²	C₂×D₄×C₁₁₃	1	1	yes	characteristic
188	C₄²	#41×C₁₁₃	1	1	yes	characteristic
189	C₂³	#121×C₁₁₃	1	1	yes	characteristic
190	Q₈	—	4	4	no	—
191	Q₈	—	4	4	no	—
192	C₂×C₄	—	2	2	no	—
193	C₂×C₄	—	2	2	no	—
194	C₂×C₄	—	2	4	no	—
195	C₂×C₄	—	4	2	no	—
196	C₂×C₄	—	4	2	no	—
197	C₂×C₄	—	4	2	no	—
198	C₂×C₄	—	4	2	no	—
199	C₂×C₄	C₂×C₂²⋊C₄×C₁₁₃	1	1	yes	characteristic
200	C₂×C₄	C₄×D₄×C₁₁₃	1	1	yes	characteristic
201	C₂×C₄	C₄×D₄×C₁₁₃	1	1	yes	characteristic
202	C₂×C₄	#121×C₁₁₃	1	1	yes	characteristic
203	C₂×C₄	#124×C₁₁₃	1	1	yes	characteristic
204	C₂×C₄	#127×C₁₁₃	1	1	yes	characteristic
205	C₂×C₄	#137×C₁₁₃	1	4	yes	—
206	C₈	—	2	4	no	—
207	C₈	—	2	2	no	—
208	C₈	—	8	2	no	—
209	C₈	#104×C₁₁₃	1	4	yes	—
210	C₂²	—	2	1	no	—
211	C₂²	—	2	1	no	—
212	C₂²	#390×C₁₁₃	1	1	yes	characteristic
213	C₂²	#470×C₁₁₃	1	2	yes	—
214	C₄	—	2	1	no	—
215	C₄	—	2	1	no	—
216	C₄	—	4	4	no	—
217	C₄	—	8	2	no	—
218	C₄	—	8	2	no	—
219	C₄	C₂×#104×C₁₁₃	1	2	yes	—
220	C₄	#441×C₁₁₃	1	2	yes	—
221	C₄	#444×C₁₁₃	1	2	yes	—
222	C₄	#497×C₁₁₃	1	2	yes	—
223	C₂	—	2	2	no	—
224	C₂	C₁₁₃×#1962	1	1	yes	characteristic
225	C₂	C₁₁₃×#1971	1	1	yes	characteristic
226	C₂	C₁₁₃×#1975	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄×C₁₁₃	#127
commutator	C₄²	C₂²×C₄×C₁₁₃
Frattini, cosocle	C₂×C₄²	C₂³×C₁₁₃
socle	C₂²×C₁₁₃	#390
prime radical	C₂³×C₁₁₃	#121
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
7. #500×C₁₁₃	C₄	semidirect
8. #494×C₁₁₃	C₄	semidirect
9. #487×C₁₁₃	C₄	semidirect
109. #9850	C₁₁₃	direct (N × H)
119. C₁₁₃	#9850	direct (N × H)
121. #500	C₄×C₁₁₃	semidirect
122. #494	C₄×C₁₁₃	semidirect
123. #487	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×#3170	2
C₁₁₃×#1994	2
C₄×#97×C₁₁₃	2
C₁₁₃×#1566	2
C₁₁₃×#1493	2
#9850	113

Subgroups — C113×#9850 #43277870

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₁₃×#9850 #43277870