Subgroups — C₁₁₃×#7910 #43275930

Order 28928.

710 conjugacy classes of subgroups.

The subgroup lattice has 230 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#432×C₁₁₃	C₂	1	2	yes	maximal
2	C₂×#376×C₁₁₃	C₂	1	1	yes	maximal, characteristic
3	C₁₁₃×#1280	C₂	1	4	yes	maximal
4	C₂²×C₄²×C₁₁₃	C₄	1	2	yes	—
5	C₂²×C₄²×C₁₁₃	C₂²	1	1	yes	characteristic
6	#432×C₁₁₃	—	2	4	no	—
7	#432×C₁₁₃	C₄	1	8	yes	complemented
8	C₂×C₄⋊C₈×C₁₁₃	—	2	2	no	—
9	C₂×C₂²⋊C₈×C₁₁₃	C₂²	1	4	yes	—
10	C₂×#97×C₁₁₃	—	2	2	no	—
11	#376×C₁₁₃	—	2	2	no	—
12	#376×C₁₁₃	—	2	2	no	—
13	C₂×#95×C₁₁₃	C₄	1	2	yes	—
14	C₂×#95×C₁₁₃	C₂²	1	2	yes	—
15	C₂³×C₄×C₁₁₃	C₂×C₄	1	1	yes	characteristic
16	C₂³×C₄×C₁₁₃	C₂×C₄	1	4	yes	—
17	C₂³×C₄×C₁₁₃	C₂×C₄	1	1	yes	characteristic
18	C₂³×C₄×C₁₁₃	C₂³	1	1	yes	characteristic
19	C₂²×C₈×C₁₁₃	—	2	4	no	—
20	C₂×C₄²×C₁₁₃	—	2	2	no	—
21	C₂×C₄²×C₁₁₃	—	2	2	no	—
22	C₂×C₄²×C₁₁₃	—	2	8	no	—
23	C₂×C₄²×C₁₁₃	C₂×C₄	1	4	yes	—
24	C₂×C₄²×C₁₁₃	D₄	1	2	yes	—
25	C₂×C₄²×C₁₁₃	D₄	1	2	yes	—
26	C₂×C₄²×C₁₁₃	D₄	1	2	yes	—
27	C₂×C₄²×C₁₁₃	Q₈	1	2	yes	—
28	C₄⋊C₈×C₁₁₃	—	2	8	no	—
29	C₂²⋊C₈×C₁₁₃	—	2	4	no	—
30	C₂²⋊C₈×C₁₁₃	C₂×C₄	1	8	yes	—
31	#97×C₁₁₃	—	2	8	no	—
32	#95×C₁₁₃	—	2	4	no	—
33	#95×C₁₁₃	—	2	4	no	—
34	C₂⁴×C₁₁₃	C₂²×C₄	1	1	yes	characteristic
35	C₂²×C₄×C₁₁₃	—	2	4	no	—
36	C₂²×C₄×C₁₁₃	—	2	4	no	—
37	C₂²×C₄×C₁₁₃	—	2	4	no	—
38	C₂²×C₄×C₁₁₃	—	4	2	no	—
39	C₂²×C₄×C₁₁₃	—	4	2	no	—
40	C₂²×C₄×C₁₁₃	—	4	4	no	—
41	C₂²×C₄×C₁₁₃	—	4	4	no	—
42	C₂²×C₄×C₁₁₃	C₄²	1	2	yes	—
43	C₂²×C₄×C₁₁₃	C₄²	1	2	yes	—
44	C₂²×C₄×C₁₁₃	C₂²⋊C₄	1	2	yes	—
45	C₂²×C₄×C₁₁₃	C₂²⋊C₄	1	1	yes	characteristic
46	C₂²×C₄×C₁₁₃	C₂²⋊C₄	1	1	yes	characteristic
47	C₂²×C₄×C₁₁₃	C₄⋊C₄	1	2	yes	—
48	C₂²×C₄×C₁₁₃	C₄⋊C₄	1	2	yes	—
49	C₂²×C₄×C₁₁₃	M₁₆	1	8	yes	—
50	C₂²×C₄×C₁₁₃	C₂²×C₄	1	2	yes	—
51	C₂²×C₄×C₁₁₃	C₂×D₄	1	2	yes	—
52	C₂²×C₄×C₁₁₃	C₂×D₄	1	1	yes	characteristic
53	C₂²×C₄×C₁₁₃	C₂×Q₈	1	1	yes	characteristic
54	C₂×C₈×C₁₁₃	—	2	8	no	—
55	C₂×C₈×C₁₁₃	—	4	8	no	—
56	C₄²×C₁₁₃	—	2	8	no	—
57	C₄²×C₁₁₃	—	2	8	no	—
58	C₄²×C₁₁₃	C₂²⋊C₄	1	4	yes	—
59	C₄²×C₁₁₃	C₂²⋊C₄	1	4	yes	—
60	C₄²×C₁₁₃	C₄⋊C₄	1	4	yes	—
61	C₄²×C₁₁₃	C₄⋊C₄	1	4	yes	—
62	C₂³×C₁₁₃	—	2	2	no	—
63	C₂³×C₁₁₃	—	2	2	no	—
64	C₂³×C₁₁₃	C₂×C₄²	1	1	yes	characteristic
65	C₂³×C₁₁₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
66	C₂³×C₁₁₃	C₂×C₄⋊C₄	1	1	yes	characteristic
67	C₂³×C₁₁₃	C₂×M₁₆	1	4	yes	—
68	C₂×C₄×C₁₁₃	—	2	4	no	—
69	C₂×C₄×C₁₁₃	—	2	2	no	—
70	C₂×C₄×C₁₁₃	—	2	2	no	—
71	C₂×C₄×C₁₁₃	—	2	4	no	—
72	C₂×C₄×C₁₁₃	—	2	4	no	—
73	C₂×C₄×C₁₁₃	—	2	4	no	—
74	C₂×C₄×C₁₁₃	—	2	8	no	—
75	C₂×C₄×C₁₁₃	—	4	2	no	—
76	C₂×C₄×C₁₁₃	—	4	2	no	—
77	C₂×C₄×C₁₁₃	—	4	2	no	—
78	C₂×C₄×C₁₁₃	—	4	1	no	—
79	C₂×C₄×C₁₁₃	—	4	1	no	—
80	C₂×C₄×C₁₁₃	—	4	8	no	—
81	C₂×C₄×C₁₁₃	—	4	4	no	—
82	C₂×C₄×C₁₁₃	—	4	4	no	—
83	C₂×C₄×C₁₁₃	—	4	8	no	—
84	C₂×C₄×C₁₁₃	#95	1	2	yes	—
85	C₂×C₄×C₁₁₃	#95	1	2	yes	—
86	C₂×C₄×C₁₁₃	#95	1	4	yes	—
87	C₂×C₄×C₁₁₃	C₂×C₂²⋊C₄	1	2	yes	—
88	C₂×C₄×C₁₁₃	C₂×C₄⋊C₄	1	2	yes	—
89	C₈×C₁₁₃	—	4	8	no	—
90	C₂²×C₁₁₃	—	2	4	no	—
91	C₂²×C₁₁₃	—	2	4	no	—
92	C₂²×C₁₁₃	—	2	1	no	—
93	C₂²×C₁₁₃	—	2	1	no	—
94	C₂²×C₁₁₃	—	2	1	no	—
95	C₂²×C₁₁₃	—	2	1	no	—
96	C₂²×C₁₁₃	C₂×#95	1	1	yes	characteristic
97	C₂²×C₁₁₃	C₄×M₁₆	1	2	yes	—
98	C₂²×C₁₁₃	#407	1	2	yes	—
99	C₂²×C₁₁₃	#410	1	2	yes	—
100	C₂²×C₁₁₃	#413	1	2	yes	—
101	C₂²×C₁₁₃	#423	1	2	yes	—
102	C₄×C₁₁₃	—	2	2	no	—
103	C₄×C₁₁₃	—	2	2	no	—
104	C₄×C₁₁₃	—	2	2	no	—
105	C₄×C₁₁₃	—	2	2	no	—
106	C₄×C₁₁₃	—	2	4	no	—
107	C₄×C₁₁₃	—	4	8	no	—
108	#7910	C₁₁₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
109	C₂×C₁₁₃	—	2	2	no	—
110	C₂×C₁₁₃	—	2	2	no	—
111	C₂×C₁₁₃	#1464	1	4	yes	—
112	C₂×C₁₁₃	#1728	1	1	yes	characteristic
113	C₂×C₁₁₃	#1740	1	1	yes	characteristic
114	C₂×C₁₁₃	#1742	1	1	yes	characteristic
115	C₂×#432	C₂×C₁₁₃	1	2	yes	—
116	C₂×#376	C₂×C₁₁₃	1	1	yes	characteristic
117	#1280	C₂×C₁₁₃	1	4	yes	—
118	C₁₁₃	#7910	1	1	yes	characteristic, direct product factor, Sylow
119	C₂²×C₄²	C₄×C₁₁₃	1	2	yes	—
120	C₂²×C₄²	C₂²×C₁₁₃	1	1	yes	characteristic
121	#432	—	2	4	no	—
122	#432	C₄×C₁₁₃	1	8	yes	complemented
123	C₂×C₄⋊C₈	—	2	2	no	—
124	C₂×C₂²⋊C₈	C₂²×C₁₁₃	1	4	yes	—
125	C₂×#97	—	2	2	no	—
126	#376	—	2	2	no	—
127	#376	—	2	2	no	—
128	C₂×#95	C₄×C₁₁₃	1	2	yes	—
129	C₂×#95	C₂²×C₁₁₃	1	2	yes	—
130	C₂³×C₄	C₂×C₄×C₁₁₃	1	1	yes	characteristic
131	C₂³×C₄	C₂×C₄×C₁₁₃	1	4	yes	—
132	C₂³×C₄	C₂×C₄×C₁₁₃	1	1	yes	characteristic
133	C₂³×C₄	C₂³×C₁₁₃	1	1	yes	characteristic
134	C₂²×C₈	—	2	4	no	—
135	C₂×C₄²	—	2	2	no	—
136	C₂×C₄²	—	2	2	no	—
137	C₂×C₄²	—	2	8	no	—
138	C₂×C₄²	C₂×C₄×C₁₁₃	1	4	yes	—
139	C₂×C₄²	D₄×C₁₁₃	1	2	yes	—
140	C₂×C₄²	D₄×C₁₁₃	1	2	yes	—
141	C₂×C₄²	D₄×C₁₁₃	1	2	yes	—
142	C₂×C₄²	Q₈×C₁₁₃	1	2	yes	—
143	C₄⋊C₈	—	2	8	no	—
144	C₂²⋊C₈	—	2	4	no	—
145	C₂²⋊C₈	C₂×C₄×C₁₁₃	1	8	yes	—
146	#97	—	2	8	no	—
147	#95	—	2	4	no	—
148	#95	—	2	4	no	—
149	C₂⁴	C₂²×C₄×C₁₁₃	1	1	yes	characteristic
150	C₂²×C₄	—	2	4	no	—
151	C₂²×C₄	—	2	4	no	—
152	C₂²×C₄	—	2	4	no	—
153	C₂²×C₄	—	4	2	no	—
154	C₂²×C₄	—	4	2	no	—
155	C₂²×C₄	—	4	4	no	—
156	C₂²×C₄	—	4	4	no	—
157	C₂²×C₄	C₄²×C₁₁₃	1	2	yes	—
158	C₂²×C₄	C₄²×C₁₁₃	1	2	yes	—
159	C₂²×C₄	C₂²⋊C₄×C₁₁₃	1	2	yes	—
160	C₂²×C₄	C₂²⋊C₄×C₁₁₃	1	1	yes	characteristic
161	C₂²×C₄	C₂²⋊C₄×C₁₁₃	1	1	yes	characteristic
162	C₂²×C₄	C₄⋊C₄×C₁₁₃	1	2	yes	—
163	C₂²×C₄	C₄⋊C₄×C₁₁₃	1	2	yes	—
164	C₂²×C₄	M₁₆×C₁₁₃	1	8	yes	—
165	C₂²×C₄	C₂²×C₄×C₁₁₃	1	2	yes	—
166	C₂²×C₄	C₂×D₄×C₁₁₃	1	2	yes	—
167	C₂²×C₄	C₂×D₄×C₁₁₃	1	1	yes	characteristic
168	C₂²×C₄	C₂×Q₈×C₁₁₃	1	1	yes	characteristic
169	C₂×C₈	—	2	8	no	—
170	C₂×C₈	—	4	8	no	—
171	C₄²	—	2	8	no	—
172	C₄²	—	2	8	no	—
173	C₄²	C₂²⋊C₄×C₁₁₃	1	4	yes	—
174	C₄²	C₂²⋊C₄×C₁₁₃	1	4	yes	—
175	C₄²	C₄⋊C₄×C₁₁₃	1	4	yes	—
176	C₄²	C₄⋊C₄×C₁₁₃	1	4	yes	—
177	C₂³	—	2	2	no	—
178	C₂³	—	2	2	no	—
179	C₂³	C₂×C₄²×C₁₁₃	1	1	yes	characteristic
180	C₂³	C₂×C₂²⋊C₄×C₁₁₃	1	1	yes	characteristic
181	C₂³	C₂×C₄⋊C₄×C₁₁₃	1	1	yes	characteristic
182	C₂³	C₂×M₁₆×C₁₁₃	1	4	yes	—
183	C₂×C₄	—	2	4	no	—
184	C₂×C₄	—	2	2	no	—
185	C₂×C₄	—	2	2	no	—
186	C₂×C₄	—	2	4	no	—
187	C₂×C₄	—	2	4	no	—
188	C₂×C₄	—	2	4	no	—
189	C₂×C₄	—	2	8	no	—
190	C₂×C₄	—	4	2	no	—
191	C₂×C₄	—	4	2	no	—
192	C₂×C₄	—	4	2	no	—
193	C₂×C₄	—	4	1	no	—
194	C₂×C₄	—	4	1	no	—
195	C₂×C₄	—	4	8	no	—
196	C₂×C₄	—	4	4	no	—
197	C₂×C₄	—	4	4	no	—
198	C₂×C₄	—	4	8	no	—
199	C₂×C₄	#95×C₁₁₃	1	2	yes	—
200	C₂×C₄	#95×C₁₁₃	1	2	yes	—
201	C₂×C₄	#95×C₁₁₃	1	4	yes	—
202	C₂×C₄	C₂×C₂²⋊C₄×C₁₁₃	1	2	yes	—
203	C₂×C₄	C₂×C₄⋊C₄×C₁₁₃	1	2	yes	—
204	C₈	—	4	8	no	—
205	C₂²	—	2	4	no	—
206	C₂²	—	2	4	no	—
207	C₂²	—	2	1	no	—
208	C₂²	—	2	1	no	—
209	C₂²	—	2	1	no	—
210	C₂²	—	2	1	no	—
211	C₂²	C₂×#95×C₁₁₃	1	1	yes	characteristic
212	C₂²	C₄×M₁₆×C₁₁₃	1	2	yes	—
213	C₂²	#407×C₁₁₃	1	2	yes	—
214	C₂²	#410×C₁₁₃	1	2	yes	—
215	C₂²	#413×C₁₁₃	1	2	yes	—
216	C₂²	#423×C₁₁₃	1	2	yes	—
217	C₄	—	2	2	no	—
218	C₄	—	2	2	no	—
219	C₄	—	2	2	no	—
220	C₄	—	2	2	no	—
221	C₄	—	2	4	no	—
222	C₄	—	4	8	no	—
223	C₂	—	2	2	no	—
224	C₂	—	2	2	no	—
225	C₂	C₁₁₃×#1464	1	4	yes	—
226	C₂	C₁₁₃×#1728	1	1	yes	characteristic
227	C₂	C₁₁₃×#1740	1	1	yes	characteristic
228	C₂	C₁₁₃×#1742	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂³×C₁₁₃	C₂×C₂²⋊C₄
commutator	C₂³	C₂×C₄²×C₁₁₃
Frattini, cosocle	C₂³×C₄	C₂³×C₁₁₃
prime radical	C₂⁴×C₁₁₃	C₂²×C₄
LCS	G ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₁₁₃ ◁ C₂²×C₄²×C₁₁₃ ◁ G

Complements

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

N	H ≅ G/N	type
7. #432×C₁₁₃	C₄	semidirect
108. #7910	C₁₁₃	direct (N × H)
118. C₁₁₃	#7910	direct (N × H)
122. #432	C₄×C₁₁₃	semidirect

Maximal subgroups

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

subgroup	index
C₂×#432×C₁₁₃	2
C₂×#376×C₁₁₃	2
C₁₁₃×#1280	2
#7910	113

Subgroups — C113×#7910 #43275930

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₁₃×#7910 #43275930