Subgroups — C₁₁₃×#13235 #43281255

Order 28928.

510 conjugacy classes of subgroups.

The subgroup lattice has 244 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×D₁₆×C₁₁₃	C₂	1	1	yes	maximal, characteristic
2	C₂×#366×C₁₁₃	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×#357×C₁₁₃	C₂	1	2	yes	maximal, complemented
4	C₂×C₄×C₁₆×C₁₁₃	C₂	1	1	yes	maximal, characteristic, complemented
5	C₁₁₃×#1970	C₂	1	2	yes	maximal, complemented
6	C₂²×D₈×C₁₁₃	C₂²	1	2	yes	—
7	C₂×D₁₆×C₁₁₃	—	2	2	no	—
8	C₂×D₁₆×C₁₁₃	—	2	2	no	—
9	C₂×D₁₆×C₁₁₃	C₄	1	4	yes	complemented
10	C₂²×C₁₆×C₁₁₃	C₂²	1	1	yes	characteristic
11	C₂²×C₁₆×C₁₁₃	C₂²	1	1	yes	characteristic
12	C₂×#107×C₁₁₃	C₂²	1	2	yes	—
13	C₂×#102×C₁₁₃	—	2	2	no	—
14	C₂×C₄×C₈×C₁₁₃	C₂²	1	1	yes	characteristic
15	#390×C₁₁₃	—	2	2	no	—
16	#366×C₁₁₃	—	2	2	no	—
17	#357×C₁₁₃	—	2	4	no	—
18	C₄×C₁₆×C₁₁₃	—	2	2	no	—
19	C₂²×D₄×C₁₁₃	—	2	2	no	—
20	C₂×D₈×C₁₁₃	—	2	2	no	—
21	C₂×D₈×C₁₁₃	—	2	2	no	—
22	C₂×D₈×C₁₁₃	—	2	2	no	—
23	C₂×D₈×C₁₁₃	—	2	2	no	—
24	C₂×D₈×C₁₁₃	—	2	2	no	—
25	C₂×D₈×C₁₁₃	C₂×C₄	1	4	yes	—
26	C₂²×C₈×C₁₁₃	D₄	1	1	yes	characteristic
27	C₂²×C₈×C₁₁₃	C₂³	1	1	yes	characteristic
28	C₂×C₄⋊C₄×C₁₁₃	—	2	2	no	—
29	C₂×C₂²⋊C₄×C₁₁₃	—	4	4	no	—
30	C₂×C₄²×C₁₁₃	D₄	1	1	yes	characteristic
31	D₁₆×C₁₁₃	—	2	8	no	—
32	C₂×C₁₆×C₁₁₃	—	2	1	no	—
33	C₂×C₁₆×C₁₁₃	—	2	1	no	—
34	C₂×C₁₆×C₁₁₃	C₂×C₄	1	2	yes	—
35	C₂×C₁₆×C₁₁₃	D₄	1	2	yes	—
36	C₂×C₁₆×C₁₁₃	D₄	1	2	yes	—
37	C₂×C₁₆×C₁₁₃	D₄	1	2	yes	—
38	#107×C₁₁₃	—	2	2	no	—
39	#107×C₁₁₃	—	2	2	no	—
40	#102×C₁₁₃	—	4	4	no	—
41	C₄×C₈×C₁₁₃	—	2	1	no	—
42	C₄×C₈×C₁₁₃	—	2	1	no	—
43	C₂⁴×C₁₁₃	—	4	2	no	—
44	C₂×D₄×C₁₁₃	—	2	4	no	—
45	C₂×D₄×C₁₁₃	—	4	2	no	—
46	C₂×D₄×C₁₁₃	—	4	2	no	—
47	C₂×D₄×C₁₁₃	—	4	2	no	—
48	C₂×D₄×C₁₁₃	—	4	2	no	—
49	C₂×D₄×C₁₁₃	—	4	2	no	—
50	C₂²×C₄×C₁₁₃	—	4	2	no	—
51	C₂²×C₄×C₁₁₃	D₈	1	2	yes	—
52	C₂²×C₄×C₁₁₃	C₂×D₄	1	1	yes	characteristic
53	D₈×C₁₁₃	—	2	4	no	—
54	D₈×C₁₁₃	—	2	4	no	—
55	D₈×C₁₁₃	—	4	2	no	—
56	D₈×C₁₁₃	—	4	2	no	—
57	C₂×C₈×C₁₁₃	—	2	1	no	—
58	C₂×C₈×C₁₁₃	—	2	1	no	—
59	C₂×C₈×C₁₁₃	D₈	1	2	yes	—
60	C₂×C₈×C₁₁₃	C₂²×C₄	1	1	yes	characteristic
61	C₂×C₈×C₁₁₃	C₂×D₄	1	1	yes	characteristic
62	C₂×C₈×C₁₁₃	C₂×D₄	1	1	yes	characteristic
63	C₂×C₈×C₁₁₃	C₂×D₄	1	1	yes	characteristic
64	C₂×C₈×C₁₁₃	#41	1	1	yes	characteristic
65	C₂×C₈×C₁₁₃	#41	1	1	yes	characteristic
66	C₄⋊C₄×C₁₁₃	—	4	2	no	—
67	C₄⋊C₄×C₁₁₃	—	4	2	no	—
68	C₂²⋊C₄×C₁₁₃	—	8	8	no	—
69	C₄²×C₁₁₃	—	2	1	no	—
70	C₄²×C₁₁₃	—	2	1	no	—
71	C₁₆×C₁₁₃	—	2	2	no	—
72	C₁₆×C₁₁₃	C₂²⋊C₄	1	4	yes	complemented
73	C₂³×C₁₁₃	—	4	4	no	—
74	C₂³×C₁₁₃	—	8	2	no	—
75	C₂³×C₁₁₃	—	8	2	no	—
76	C₂³×C₁₁₃	—	8	4	no	—
77	C₂³×C₁₁₃	—	8	4	no	—
78	C₂³×C₁₁₃	C₂×D₈	1	1	yes	characteristic
79	D₄×C₁₁₃	—	4	4	no	—
80	D₄×C₁₁₃	—	4	4	no	—
81	D₄×C₁₁₃	—	8	2	no	—
82	D₄×C₁₁₃	—	8	2	no	—
83	C₂×C₄×C₁₁₃	—	2	2	no	—
84	C₂×C₄×C₁₁₃	—	2	2	no	—
85	C₂×C₄×C₁₁₃	—	8	2	no	—
86	C₂×C₄×C₁₁₃	—	8	4	no	—
87	C₂×C₄×C₁₁₃	D₁₆	1	4	yes	complemented
88	C₂×C₄×C₁₁₃	C₄×D₄	1	1	yes	characteristic
89	C₂×C₄×C₁₁₃	#121	1	1	yes	characteristic
90	C₂×C₄×C₁₁₃	#127	1	1	yes	characteristic
91	C₂×C₄×C₁₁₃	C₂×D₈	1	1	yes	characteristic
92	C₂×C₄×C₁₁₃	#135	1	1	yes	characteristic
93	C₂×C₄×C₁₁₃	#135	1	1	yes	characteristic
94	C₈×C₁₁₃	—	2	2	no	—
95	C₈×C₁₁₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
96	C₈×C₁₁₃	C₄×D₄	1	1	yes	characteristic
97	C₈×C₁₁₃	#121	1	1	yes	characteristic
98	C₈×C₁₁₃	#124	1	1	yes	characteristic
99	C₂²×C₁₁₃	—	8	8	no	—
100	C₂²×C₁₁₃	—	8	4	no	—
101	C₂²×C₁₁₃	—	16	4	no	—
102	C₂²×C₁₁₃	—	16	2	no	—
103	C₂²×C₁₁₃	—	16	2	no	—
104	C₂²×C₁₁₃	C₄×D₈	1	1	yes	characteristic
105	C₂²×C₁₁₃	#466	1	1	yes	characteristic
106	C₂²×C₁₁₃	#493	1	1	yes	characteristic
107	C₂²×C₁₁₃	C₂×D₁₆	1	2	yes	—
108	C₂²×C₁₁₃	#508	1	2	yes	—
109	C₄×C₁₁₃	—	2	4	no	—
110	C₄×C₁₁₃	—	16	2	no	—
111	C₄×C₁₁₃	#390	1	1	yes	characteristic
112	C₄×C₁₁₃	C₄×D₈	1	1	yes	characteristic
113	C₄×C₁₁₃	#466	1	1	yes	characteristic
114	C₄×C₁₁₃	#495	1	1	yes	characteristic
115	#13235	C₁₁₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
116	C₂×C₁₁₃	—	16	4	no	—
117	C₂×C₁₁₃	#1970	1	1	yes	characteristic
118	C₂×C₁₁₃	C₄×D₁₆	1	2	yes	—
119	C₂×C₁₁₃	#2215	1	2	yes	—
120	C₂×C₁₁₃	#2246	1	1	yes	characteristic
121	C₂×C₁₁₃	#2249	1	1	yes	characteristic
122	C₂²×D₁₆	C₂×C₁₁₃	1	1	yes	characteristic
123	C₂×#366	C₂×C₁₁₃	1	1	yes	characteristic, complemented
124	C₂×#357	C₂×C₁₁₃	1	2	yes	complemented
125	C₂×C₄×C₁₆	C₂×C₁₁₃	1	1	yes	characteristic, complemented
126	#1970	C₂×C₁₁₃	1	2	yes	complemented
127	C₁₁₃	#13235	1	1	yes	characteristic, direct product factor, Sylow
128	C₂²×D₈	C₂²×C₁₁₃	1	2	yes	—
129	C₂×D₁₆	—	2	2	no	—
130	C₂×D₁₆	—	2	2	no	—
131	C₂×D₁₆	C₄×C₁₁₃	1	4	yes	complemented
132	C₂²×C₁₆	C₂²×C₁₁₃	1	1	yes	characteristic
133	C₂²×C₁₆	C₂²×C₁₁₃	1	1	yes	characteristic
134	C₂×#107	C₂²×C₁₁₃	1	2	yes	—
135	C₂×#102	—	2	2	no	—
136	C₂×C₄×C₈	C₂²×C₁₁₃	1	1	yes	characteristic
137	#390	—	2	2	no	—
138	#366	—	2	2	no	—
139	#357	—	2	4	no	—
140	C₄×C₁₆	—	2	2	no	—
141	C₂²×D₄	—	2	2	no	—
142	C₂×D₈	—	2	2	no	—
143	C₂×D₈	—	2	2	no	—
144	C₂×D₈	—	2	2	no	—
145	C₂×D₈	—	2	2	no	—
146	C₂×D₈	—	2	2	no	—
147	C₂×D₈	C₂×C₄×C₁₁₃	1	4	yes	—
148	C₂²×C₈	D₄×C₁₁₃	1	1	yes	characteristic
149	C₂²×C₈	C₂³×C₁₁₃	1	1	yes	characteristic
150	C₂×C₄⋊C₄	—	2	2	no	—
151	C₂×C₂²⋊C₄	—	4	4	no	—
152	C₂×C₄²	D₄×C₁₁₃	1	1	yes	characteristic
153	D₁₆	—	2	8	no	—
154	C₂×C₁₆	—	2	1	no	—
155	C₂×C₁₆	—	2	1	no	—
156	C₂×C₁₆	C₂×C₄×C₁₁₃	1	2	yes	—
157	C₂×C₁₆	D₄×C₁₁₃	1	2	yes	—
158	C₂×C₁₆	D₄×C₁₁₃	1	2	yes	—
159	C₂×C₁₆	D₄×C₁₁₃	1	2	yes	—
160	#107	—	2	2	no	—
161	#107	—	2	2	no	—
162	#102	—	4	4	no	—
163	C₄×C₈	—	2	1	no	—
164	C₄×C₈	—	2	1	no	—
165	C₂⁴	—	4	2	no	—
166	C₂×D₄	—	2	4	no	—
167	C₂×D₄	—	4	2	no	—
168	C₂×D₄	—	4	2	no	—
169	C₂×D₄	—	4	2	no	—
170	C₂×D₄	—	4	2	no	—
171	C₂×D₄	—	4	2	no	—
172	C₂²×C₄	—	4	2	no	—
173	C₂²×C₄	D₈×C₁₁₃	1	2	yes	—
174	C₂²×C₄	C₂×D₄×C₁₁₃	1	1	yes	characteristic
175	D₈	—	2	4	no	—
176	D₈	—	2	4	no	—
177	D₈	—	4	2	no	—
178	D₈	—	4	2	no	—
179	C₂×C₈	—	2	1	no	—
180	C₂×C₈	—	2	1	no	—
181	C₂×C₈	D₈×C₁₁₃	1	2	yes	—
182	C₂×C₈	C₂²×C₄×C₁₁₃	1	1	yes	characteristic
183	C₂×C₈	C₂×D₄×C₁₁₃	1	1	yes	characteristic
184	C₂×C₈	C₂×D₄×C₁₁₃	1	1	yes	characteristic
185	C₂×C₈	C₂×D₄×C₁₁₃	1	1	yes	characteristic
186	C₂×C₈	#41×C₁₁₃	1	1	yes	characteristic
187	C₂×C₈	#41×C₁₁₃	1	1	yes	characteristic
188	C₄⋊C₄	—	4	2	no	—
189	C₄⋊C₄	—	4	2	no	—
190	C₂²⋊C₄	—	8	8	no	—
191	C₄²	—	2	1	no	—
192	C₄²	—	2	1	no	—
193	C₁₆	—	2	2	no	—
194	C₁₆	C₂²⋊C₄×C₁₁₃	1	4	yes	complemented
195	C₂³	—	4	4	no	—
196	C₂³	—	8	2	no	—
197	C₂³	—	8	2	no	—
198	C₂³	—	8	4	no	—
199	C₂³	—	8	4	no	—
200	C₂³	C₂×D₈×C₁₁₃	1	1	yes	characteristic
201	D₄	—	4	4	no	—
202	D₄	—	4	4	no	—
203	D₄	—	8	2	no	—
204	D₄	—	8	2	no	—
205	C₂×C₄	—	2	2	no	—
206	C₂×C₄	—	2	2	no	—
207	C₂×C₄	—	8	2	no	—
208	C₂×C₄	—	8	4	no	—
209	C₂×C₄	D₁₆×C₁₁₃	1	4	yes	complemented
210	C₂×C₄	C₄×D₄×C₁₁₃	1	1	yes	characteristic
211	C₂×C₄	#121×C₁₁₃	1	1	yes	characteristic
212	C₂×C₄	#127×C₁₁₃	1	1	yes	characteristic
213	C₂×C₄	C₂×D₈×C₁₁₃	1	1	yes	characteristic
214	C₂×C₄	#135×C₁₁₃	1	1	yes	characteristic
215	C₂×C₄	#135×C₁₁₃	1	1	yes	characteristic
216	C₈	—	2	2	no	—
217	C₈	C₂×C₂²⋊C₄×C₁₁₃	1	1	yes	characteristic
218	C₈	C₄×D₄×C₁₁₃	1	1	yes	characteristic
219	C₈	#121×C₁₁₃	1	1	yes	characteristic
220	C₈	#124×C₁₁₃	1	1	yes	characteristic
221	C₂²	—	8	8	no	—
222	C₂²	—	8	4	no	—
223	C₂²	—	16	4	no	—
224	C₂²	—	16	2	no	—
225	C₂²	—	16	2	no	—
226	C₂²	C₄×D₈×C₁₁₃	1	1	yes	characteristic
227	C₂²	#466×C₁₁₃	1	1	yes	characteristic
228	C₂²	#493×C₁₁₃	1	1	yes	characteristic
229	C₂²	C₂×D₁₆×C₁₁₃	1	2	yes	—
230	C₂²	#508×C₁₁₃	1	2	yes	—
231	C₄	—	2	4	no	—
232	C₄	—	16	2	no	—
233	C₄	#390×C₁₁₃	1	1	yes	characteristic
234	C₄	C₄×D₈×C₁₁₃	1	1	yes	characteristic
235	C₄	#466×C₁₁₃	1	1	yes	characteristic
236	C₄	#495×C₁₁₃	1	1	yes	characteristic
237	C₂	—	16	4	no	—
238	C₂	C₁₁₃×#1970	1	1	yes	characteristic
239	C₂	C₄×D₁₆×C₁₁₃	1	2	yes	—
240	C₂	C₁₁₃×#2215	1	2	yes	—
241	C₂	C₁₁₃×#2246	1	1	yes	characteristic
242	C₂	C₁₁₃×#2249	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂³×C₁₁₃	C₂×D₈
commutator	C₂×C₈	C₂²×C₄×C₁₁₃
Frattini, cosocle	C₂²×C₈	C₂³×C₁₁₃
prime radical	C₂²×D₁₆×C₁₁₃	C₂
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₁₁₃ ◁ C₂×C₄²×C₁₁₃ ◁ C₂×C₄×C₈×C₁₁₃ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₂×#366×C₁₁₃	C₂	semidirect
3. C₂×#357×C₁₁₃	C₂	semidirect
4. C₂×C₄×C₁₆×C₁₁₃	C₂	semidirect
5. C₁₁₃×#1970	C₂	semidirect
9. C₂×D₁₆×C₁₁₃	C₄	semidirect
72. C₁₆×C₁₁₃	C₂²⋊C₄	semidirect
87. C₂×C₄×C₁₁₃	D₁₆	semidirect
115. #13235	C₁₁₃	direct (N × H)
123. C₂×#366	C₂×C₁₁₃	semidirect
124. C₂×#357	C₂×C₁₁₃	semidirect
125. C₂×C₄×C₁₆	C₂×C₁₁₃	semidirect
126. #1970	C₂×C₁₁₃	semidirect
127. C₁₁₃	#13235	direct (N × H)
131. C₂×D₁₆	C₄×C₁₁₃	semidirect
194. C₁₆	C₂²⋊C₄×C₁₁₃	semidirect
209. C₂×C₄	D₁₆×C₁₁₃	semidirect

Maximal subgroups

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

subgroup	index
C₂²×D₁₆×C₁₁₃	2
C₂×#366×C₁₁₃	2
C₂×#357×C₁₁₃	2
C₂×C₄×C₁₆×C₁₁₃	2
C₁₁₃×#1970	2
#13235	113

Subgroups — C113×#13235 #43281255

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₁₃×#13235 #43281255