Subgroups — C₂×C₁₁₃×#1991 #43289337

Order 28928.

1246 conjugacy classes of subgroups.

The subgroup lattice has 240 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×#125×C₁₁₃	C₂	1	1	yes	maximal, characteristic
2	C₂²×C₄×C₈×C₁₁₃	C₂	1	1	yes	maximal, characteristic
3	C₂×#382×C₁₁₃	C₂	1	1	yes	maximal, characteristic
4	C₁₁₃×#1991	C₂	1	8	yes	maximal, direct product factor
5	C₂×#340×C₁₁₃	C₂	1	4	yes	maximal
6	C₂³×C₈×C₁₁₃	C₂²	1	2	yes	—
7	C₂×#125×C₁₁₃	—	2	2	no	—
8	C₂×#125×C₁₁₃	—	2	4	no	—
9	C₂×#125×C₁₁₃	C₄	1	8	yes	complemented
10	C₂×#125×C₁₁₃	C₂²	1	4	yes	—
11	C₂²×C₄⋊C₄×C₁₁₃	—	2	2	no	—
12	C₂²×C₄⋊C₄×C₁₁₃	C₂²	1	2	yes	—
13	C₂²×C₄⋊C₄×C₁₁₃	C₂²	1	2	yes	—
14	C₂²×C₄²×C₁₁₃	C₂²	1	1	yes	characteristic
15	C₂×C₄×C₈×C₁₁₃	—	2	4	no	—
16	C₂×C₄×C₈×C₁₁₃	C₂²	1	4	yes	—
17	#382×C₁₁₃	—	2	2	no	—
18	#382×C₁₁₃	C₂²	1	4	yes	—
19	C₂×#95×C₁₁₃	—	2	2	no	—
20	#340×C₁₁₃	C₂²	1	16	yes	—
21	C₂³×C₄×C₁₁₃	—	2	2	no	—
22	C₂³×C₄×C₁₁₃	—	2	2	no	—
23	C₂³×C₄×C₁₁₃	D₄	1	2	yes	—
24	C₂³×C₄×C₁₁₃	C₂³	1	1	yes	characteristic
25	C₂²×C₈×C₁₁₃	—	2	4	no	—
26	C₂²×C₈×C₁₁₃	—	2	4	no	—
27	C₂²×C₈×C₁₁₃	Q₈	1	8	yes	—
28	C₂²×C₈×C₁₁₃	C₂³	1	4	yes	—
29	#125×C₁₁₃	—	2	16	no	—
30	#125×C₁₁₃	—	2	8	no	—
31	#125×C₁₁₃	C₂×C₄	1	16	yes	complemented
32	C₂×C₄⋊C₄×C₁₁₃	—	2	16	no	—
33	C₂×C₄⋊C₄×C₁₁₃	—	2	8	no	—
34	C₂×C₄⋊C₄×C₁₁₃	—	2	2	no	—
35	C₂×C₄⋊C₄×C₁₁₃	—	2	4	no	—
36	C₂×C₄⋊C₄×C₁₁₃	—	2	8	no	—
37	C₂×C₄⋊C₄×C₁₁₃	—	2	2	no	—
38	C₂×C₄⋊C₄×C₁₁₃	C₂×C₄	1	8	yes	—
39	C₂×C₄⋊C₄×C₁₁₃	C₂³	1	4	yes	—
40	C₂×C₄⋊C₄×C₁₁₃	C₂³	1	4	yes	—
41	C₂×C₄²×C₁₁₃	C₂×C₄	1	4	yes	—
42	C₂×C₄²×C₁₁₃	D₄	1	2	yes	—
43	C₂×C₄²×C₁₁₃	D₄	1	4	yes	—
44	C₂×C₄²×C₁₁₃	C₂³	1	2	yes	—
45	C₄×C₈×C₁₁₃	—	2	8	no	—
46	#95×C₁₁₃	—	2	8	no	—
47	C₂⁴×C₁₁₃	C₂×D₄	1	1	yes	characteristic
48	C₂²×C₄×C₁₁₃	—	2	8	no	—
49	C₂²×C₄×C₁₁₃	—	2	8	no	—
50	C₂²×C₄×C₁₁₃	—	2	4	no	—
51	C₂²×C₄×C₁₁₃	—	2	4	no	—
52	C₂²×C₄×C₁₁₃	—	4	8	no	—
53	C₂²×C₄×C₁₁₃	—	4	4	no	—
54	C₂²×C₄×C₁₁₃	—	4	8	no	—
55	C₂²×C₄×C₁₁₃	C₂²⋊C₄	1	8	yes	—
56	C₂²×C₄×C₁₁₃	C₂²×C₄	1	2	yes	—
57	C₂²×C₄×C₁₁₃	C₂×D₄	1	4	yes	—
58	C₂²×C₄×C₁₁₃	C₂×D₄	1	1	yes	characteristic
59	C₂²×C₄×C₁₁₃	C₂×D₄	1	2	yes	—
60	C₂²×C₄×C₁₁₃	C₂×Q₈	1	4	yes	—
61	C₂²×C₄×C₁₁₃	#41	1	4	yes	—
62	C₂²×C₄×C₁₁₃	C₂⁴	1	1	yes	characteristic
63	C₂×C₈×C₁₁₃	—	2	16	no	—
64	C₂×C₈×C₁₁₃	—	2	4	no	—
65	C₂×C₈×C₁₁₃	—	2	4	no	—
66	C₂×C₈×C₁₁₃	C₂×Q₈	1	8	yes	—
67	C₄⋊C₄×C₁₁₃	—	2	32	no	—
68	C₄⋊C₄×C₁₁₃	—	2	8	no	—
69	C₄⋊C₄×C₁₁₃	—	2	4	no	—
70	C₄⋊C₄×C₁₁₃	—	2	8	no	—
71	C₄⋊C₄×C₁₁₃	—	4	4	no	—
72	C₄⋊C₄×C₁₁₃	C₂²×C₄	1	8	yes	—
73	C₄²×C₁₁₃	C₂²⋊C₄	1	8	yes	—
74	C₄²×C₁₁₃	C₂²×C₄	1	4	yes	—
75	C₄²×C₁₁₃	C₂×D₄	1	4	yes	—
76	C₂³×C₁₁₃	C₂×C₂²⋊C₄	1	2	yes	—
77	C₂³×C₁₁₃	#124	1	2	yes	—
78	C₂³×C₁₁₃	C₄⋊Q₈	1	2	yes	—
79	C₂³×C₁₁₃	#135	1	8	yes	—
80	C₂³×C₁₁₃	C₂²×D₄	1	1	yes	characteristic
81	C₂×C₄×C₁₁₃	—	2	8	no	—
82	C₂×C₄×C₁₁₃	—	2	16	no	—
83	C₂×C₄×C₁₁₃	—	2	8	no	—
84	C₂×C₄×C₁₁₃	—	4	8	no	—
85	C₂×C₄×C₁₁₃	—	4	16	no	—
86	C₂×C₄×C₁₁₃	—	4	4	no	—
87	C₂×C₄×C₁₁₃	—	4	8	no	—
88	C₂×C₄×C₁₁₃	—	8	8	no	—
89	C₂×C₄×C₁₁₃	C₂×C₂²⋊C₄	1	8	yes	—
90	C₂×C₄×C₁₁₃	C₂×C₂²⋊C₄	1	2	yes	—
91	C₂×C₄×C₁₁₃	C₄×Q₈	1	8	yes	—
92	C₂×C₄×C₁₁₃	C₂²⋊Q₈	1	8	yes	—
93	C₂×C₄×C₁₁₃	#124	1	2	yes	—
94	C₂×C₄×C₁₁₃	C₄⋊Q₈	1	2	yes	—
95	C₂×C₄×C₁₁₃	C₂³×C₄	1	1	yes	characteristic
96	C₂×C₄×C₁₁₃	C₂²×D₄	1	1	yes	characteristic
97	C₂×C₄×C₁₁₃	C₂²×Q₈	1	2	yes	—
98	C₂×C₄×C₁₁₃	C₂×#41	1	2	yes	—
99	C₈×C₁₁₃	—	2	8	no	—
100	C₂²×C₁₁₃	#391	1	4	yes	—
101	C₂²×C₁₁₃	#416	1	8	yes	—
102	C₂²×C₁₁₃	#488	1	8	yes	—
103	C₂²×C₁₁₃	#499	1	8	yes	—
104	C₂²×C₁₁₃	C₂²×C₂²⋊C₄	1	1	yes	characteristic
105	C₂²×C₁₁₃	C₂×#124	1	1	yes	characteristic
106	C₂²×C₁₁₃	C₂×C₄⋊Q₈	1	1	yes	characteristic
107	C₂²×C₁₁₃	C₂×#135	1	4	yes	—
108	C₄×C₁₁₃	—	2	8	no	—
109	C₄×C₁₁₃	—	4	8	no	—
110	C₄×C₁₁₃	—	8	4	no	—
111	C₄×C₁₁₃	#391	1	4	yes	—
112	C₄×C₁₁₃	C₂×C₄×Q₈	1	2	yes	—
113	C₄×C₁₁₃	C₂×C₂²⋊Q₈	1	2	yes	—
114	C₂×#1991	C₁₁₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
115	C₂×C₁₁₃	#1991	1	8	yes	direct product factor
116	C₂×C₁₁₃	C₂×#391	1	1	yes	characteristic
117	C₂×C₁₁₃	C₂×#416	1	2	yes	—
118	C₂×C₁₁₃	C₂×#488	1	2	yes	—
119	C₂×C₁₁₃	C₂×#499	1	2	yes	—
120	C₂²×#125	C₂×C₁₁₃	1	1	yes	characteristic
121	C₂²×C₄×C₈	C₂×C₁₁₃	1	1	yes	characteristic
122	C₂×#382	C₂×C₁₁₃	1	1	yes	characteristic
123	#1991	C₂×C₁₁₃	1	8	yes	direct product factor
124	C₂×#340	C₂×C₁₁₃	1	4	yes	—
125	C₁₁₃	C₂×#1991	1	1	yes	characteristic, direct product factor, Sylow
126	C₂³×C₈	C₂²×C₁₁₃	1	2	yes	—
127	C₂×#125	—	2	2	no	—
128	C₂×#125	—	2	4	no	—
129	C₂×#125	C₄×C₁₁₃	1	8	yes	complemented
130	C₂×#125	C₂²×C₁₁₃	1	4	yes	—
131	C₂²×C₄⋊C₄	—	2	2	no	—
132	C₂²×C₄⋊C₄	C₂²×C₁₁₃	1	2	yes	—
133	C₂²×C₄⋊C₄	C₂²×C₁₁₃	1	2	yes	—
134	C₂²×C₄²	C₂²×C₁₁₃	1	1	yes	characteristic
135	C₂×C₄×C₈	—	2	4	no	—
136	C₂×C₄×C₈	C₂²×C₁₁₃	1	4	yes	—
137	#382	—	2	2	no	—
138	#382	C₂²×C₁₁₃	1	4	yes	—
139	C₂×#95	—	2	2	no	—
140	#340	C₂²×C₁₁₃	1	16	yes	—
141	C₂³×C₄	—	2	2	no	—
142	C₂³×C₄	—	2	2	no	—
143	C₂³×C₄	D₄×C₁₁₃	1	2	yes	—
144	C₂³×C₄	C₂³×C₁₁₃	1	1	yes	characteristic
145	C₂²×C₈	—	2	4	no	—
146	C₂²×C₈	—	2	4	no	—
147	C₂²×C₈	Q₈×C₁₁₃	1	8	yes	—
148	C₂²×C₈	C₂³×C₁₁₃	1	4	yes	—
149	#125	—	2	16	no	—
150	#125	—	2	8	no	—
151	#125	C₂×C₄×C₁₁₃	1	16	yes	complemented
152	C₂×C₄⋊C₄	—	2	16	no	—
153	C₂×C₄⋊C₄	—	2	8	no	—
154	C₂×C₄⋊C₄	—	2	2	no	—
155	C₂×C₄⋊C₄	—	2	4	no	—
156	C₂×C₄⋊C₄	—	2	8	no	—
157	C₂×C₄⋊C₄	—	2	2	no	—
158	C₂×C₄⋊C₄	C₂×C₄×C₁₁₃	1	8	yes	—
159	C₂×C₄⋊C₄	C₂³×C₁₁₃	1	4	yes	—
160	C₂×C₄⋊C₄	C₂³×C₁₁₃	1	4	yes	—
161	C₂×C₄²	C₂×C₄×C₁₁₃	1	4	yes	—
162	C₂×C₄²	D₄×C₁₁₃	1	2	yes	—
163	C₂×C₄²	D₄×C₁₁₃	1	4	yes	—
164	C₂×C₄²	C₂³×C₁₁₃	1	2	yes	—
165	C₄×C₈	—	2	8	no	—
166	#95	—	2	8	no	—
167	C₂⁴	C₂×D₄×C₁₁₃	1	1	yes	characteristic
168	C₂²×C₄	—	2	8	no	—
169	C₂²×C₄	—	2	8	no	—
170	C₂²×C₄	—	2	4	no	—
171	C₂²×C₄	—	2	4	no	—
172	C₂²×C₄	—	4	8	no	—
173	C₂²×C₄	—	4	4	no	—
174	C₂²×C₄	—	4	8	no	—
175	C₂²×C₄	C₂²⋊C₄×C₁₁₃	1	8	yes	—
176	C₂²×C₄	C₂²×C₄×C₁₁₃	1	2	yes	—
177	C₂²×C₄	C₂×D₄×C₁₁₃	1	4	yes	—
178	C₂²×C₄	C₂×D₄×C₁₁₃	1	1	yes	characteristic
179	C₂²×C₄	C₂×D₄×C₁₁₃	1	2	yes	—
180	C₂²×C₄	C₂×Q₈×C₁₁₃	1	4	yes	—
181	C₂²×C₄	#41×C₁₁₃	1	4	yes	—
182	C₂²×C₄	C₂⁴×C₁₁₃	1	1	yes	characteristic
183	C₂×C₈	—	2	16	no	—
184	C₂×C₈	—	2	4	no	—
185	C₂×C₈	—	2	4	no	—
186	C₂×C₈	C₂×Q₈×C₁₁₃	1	8	yes	—
187	C₄⋊C₄	—	2	32	no	—
188	C₄⋊C₄	—	2	8	no	—
189	C₄⋊C₄	—	2	4	no	—
190	C₄⋊C₄	—	2	8	no	—
191	C₄⋊C₄	—	4	4	no	—
192	C₄⋊C₄	C₂²×C₄×C₁₁₃	1	8	yes	—
193	C₄²	C₂²⋊C₄×C₁₁₃	1	8	yes	—
194	C₄²	C₂²×C₄×C₁₁₃	1	4	yes	—
195	C₄²	C₂×D₄×C₁₁₃	1	4	yes	—
196	C₂³	C₂×C₂²⋊C₄×C₁₁₃	1	2	yes	—
197	C₂³	#124×C₁₁₃	1	2	yes	—
198	C₂³	C₄⋊Q₈×C₁₁₃	1	2	yes	—
199	C₂³	#135×C₁₁₃	1	8	yes	—
200	C₂³	C₂²×D₄×C₁₁₃	1	1	yes	characteristic
201	C₂×C₄	—	2	8	no	—
202	C₂×C₄	—	2	16	no	—
203	C₂×C₄	—	2	8	no	—
204	C₂×C₄	—	4	8	no	—
205	C₂×C₄	—	4	16	no	—
206	C₂×C₄	—	4	4	no	—
207	C₂×C₄	—	4	8	no	—
208	C₂×C₄	—	8	8	no	—
209	C₂×C₄	C₂×C₂²⋊C₄×C₁₁₃	1	8	yes	—
210	C₂×C₄	C₂×C₂²⋊C₄×C₁₁₃	1	2	yes	—
211	C₂×C₄	C₄×Q₈×C₁₁₃	1	8	yes	—
212	C₂×C₄	C₂²⋊Q₈×C₁₁₃	1	8	yes	—
213	C₂×C₄	#124×C₁₁₃	1	2	yes	—
214	C₂×C₄	C₄⋊Q₈×C₁₁₃	1	2	yes	—
215	C₂×C₄	C₂³×C₄×C₁₁₃	1	1	yes	characteristic
216	C₂×C₄	C₂²×D₄×C₁₁₃	1	1	yes	characteristic
217	C₂×C₄	C₂²×Q₈×C₁₁₃	1	2	yes	—
218	C₂×C₄	C₂×#41×C₁₁₃	1	2	yes	—
219	C₈	—	2	8	no	—
220	C₂²	#391×C₁₁₃	1	4	yes	—
221	C₂²	#416×C₁₁₃	1	8	yes	—
222	C₂²	#488×C₁₁₃	1	8	yes	—
223	C₂²	#499×C₁₁₃	1	8	yes	—
224	C₂²	C₂²×C₂²⋊C₄×C₁₁₃	1	1	yes	characteristic
225	C₂²	C₂×#124×C₁₁₃	1	1	yes	characteristic
226	C₂²	C₂×C₄⋊Q₈×C₁₁₃	1	1	yes	characteristic
227	C₂²	C₂×#135×C₁₁₃	1	4	yes	—
228	C₄	—	2	8	no	—
229	C₄	—	4	8	no	—
230	C₄	—	8	4	no	—
231	C₄	#391×C₁₁₃	1	4	yes	—
232	C₄	C₂×C₄×Q₈×C₁₁₃	1	2	yes	—
233	C₄	C₂×C₂²⋊Q₈×C₁₁₃	1	2	yes	—
234	C₂	C₁₁₃×#1991	1	8	yes	direct product factor
235	C₂	C₂×#391×C₁₁₃	1	1	yes	characteristic
236	C₂	C₂×#416×C₁₁₃	1	2	yes	—
237	C₂	C₂×#488×C₁₁₃	1	2	yes	—
238	C₂	C₂×#499×C₁₁₃	1	2	yes	—

	subgroup	G/H
center, socle, prime radical	C₂⁴×C₁₁₃	C₂×D₄
commutator	C₂×C₄	C₂³×C₄×C₁₁₃
Frattini, cosocle	C₂²×C₄	C₂⁴×C₁₁₃
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂⁴×C₁₁₃ ◁ C₂²×C₄²×C₁₁₃ ◁ G

Complements

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

N	H ≅ G/N	type
4. C₁₁₃×#1991	C₂	direct (N × H)
9. C₂×#125×C₁₁₃	C₄	semidirect
31. #125×C₁₁₃	C₂×C₄	semidirect
114. C₂×#1991	C₁₁₃	direct (N × H)
115. C₂×C₁₁₃	#1991	direct (N × H)
123. #1991	C₂×C₁₁₃	direct (N × H)
125. C₁₁₃	C₂×#1991	direct (N × H)
129. C₂×#125	C₄×C₁₁₃	semidirect
151. #125	C₂×C₄×C₁₁₃	semidirect
234. C₂	C₁₁₃×#1991	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂²×#125×C₁₁₃	2
C₂²×C₄×C₈×C₁₁₃	2
C₂×#382×C₁₁₃	2
C₁₁₃×#1991	2
C₂×#340×C₁₁₃	2
C₂×#1991	113

Subgroups — C2×C113×#1991 #43289337

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₁₁₃×#1991 #43289337