Subgroups — C₁₁₃×#23360 #43291380

Order 28928.

688 conjugacy classes of subgroups.

The subgroup lattice has 256 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₁₁₃×#3152	C₂	1	1	yes	maximal, characteristic
2	C₁₁₃×#1905	C₂	1	4	yes	maximal, complemented
3	C₁₁₃×#1902	C₂	1	4	yes	maximal, complemented
4	C₁₁₃×#1886	C₂	1	2	yes	maximal, complemented
5	C₁₁₃×#1884	C₂	1	2	yes	maximal
6	C₁₁₃×#1841	C₂	1	2	yes	maximal, complemented
7	C₂×#137×C₁₁₃	C₂²	1	4	yes	—
8	#568×C₁₁₃	C₂²	1	2	yes	—
9	#532×C₁₁₃	C₂²	1	1	yes	characteristic
10	C₂×C₄⋊Q₈×C₁₁₃	C₂²	1	1	yes	characteristic
11	#519×C₁₁₃	C₂²	1	2	yes	—
12	#497×C₁₁₃	C₄	1	4	yes	complemented
13	#494×C₁₁₃	C₄	1	2	yes	complemented
14	#492×C₁₁₃	C₄	1	2	yes	complemented
15	#421×C₁₁₃	C₂²	1	4	yes	—
16	#419×C₁₁₃	C₂²	1	4	yes	—
17	C₂×#101×C₁₁₃	C₂²	1	2	yes	—
18	#410×C₁₁₃	C₂²	1	2	yes	—
19	C₄×M₁₆×C₁₁₃	C₂²	1	1	yes	characteristic
20	#344×C₁₁₃	C₂²	1	4	yes	—
21	#339×C₁₁₃	C₂²	1	4	yes	—
22	#338×C₁₁₃	C₂²	1	2	yes	—
23	H(3,ℤ/4)×C₁₁₃	C₂²	1	2	yes	—
24	C₂×#41×C₁₁₃	C₂³	1	2	yes	—
25	C₂²×Q₈×C₁₁₃	C₂³	1	2	yes	—
26	#137×C₁₁₃	—	4	4	no	—
27	#137×C₁₁₃	—	4	4	no	—
28	C₂×Q₁₆×C₁₁₃	C₂×C₄	1	8	yes	—
29	C₂×SD₁₆×C₁₁₃	C₂×C₄	1	8	yes	—
30	#131×C₁₁₃	—	4	4	no	—
31	C₂×M₁₆×C₁₁₃	—	4	2	no	—
32	C₂×M₁₆×C₁₁₃	C₂³	1	4	yes	—
33	C₂×M₁₆×C₁₁₃	C₂³	1	2	yes	—
34	C₄⋊Q₈×C₁₁₃	—	4	1	no	—
35	C₄⋊Q₈×C₁₁₃	C₂×C₄	1	2	yes	—
36	C₄⋊Q₈×C₁₁₃	C₂×C₄	1	2	yes	—
37	C₄⋊Q₈×C₁₁₃	C₂×C₄	1	1	yes	characteristic
38	#127×C₁₁₃	C₂×C₄	1	1	yes	characteristic
39	#124×C₁₁₃	C₂×C₄	1	2	yes	—
40	#121×C₁₁₃	—	4	1	no	—
41	C₄×Q₈×C₁₁₃	—	4	4	no	—
42	C₄×D₄×C₁₁₃	—	4	1	no	—
43	#117×C₁₁₃	—	4	2	no	—
44	#117×C₁₁₃	C₂³	1	4	yes	—
45	C₂×C₄⋊C₄×C₁₁₃	—	4	1	no	—
46	C₂×C₄²×C₁₁₃	C₂³	1	1	yes	characteristic
47	#104×C₁₁₃	—	4	4	no	—
48	#103×C₁₁₃	—	4	4	no	—
49	#101×C₁₁₃	—	4	2	no	—
50	#99×C₁₁₃	—	4	2	no	—
51	#97×C₁₁₃	C₂×C₄	1	2	yes	—
52	C₄×C₈×C₁₁₃	C₂×C₄	1	2	yes	—
53	#41×C₁₁₃	—	2	4	no	—
54	#41×C₁₁₃	—	2	4	no	—
55	C₂×Q₈×C₁₁₃	—	2	2	no	—
56	C₂×Q₈×C₁₁₃	—	2	4	no	—
57	C₂×Q₈×C₁₁₃	—	2	4	no	—
58	C₂×Q₈×C₁₁₃	C₂²×C₄	1	2	yes	—
59	C₂×Q₈×C₁₁₃	C₂²×C₄	1	4	yes	—
60	C₂×D₄×C₁₁₃	—	2	2	no	—
61	C₂×D₄×C₁₁₃	—	4	1	no	—
62	C₂×D₄×C₁₁₃	C₂²×C₄	1	2	yes	—
63	C₂²×C₄×C₁₁₃	—	2	1	no	—
64	C₂²×C₄×C₁₁₃	—	2	2	no	—
65	C₂²×C₄×C₁₁₃	—	2	2	no	—
66	C₂²×C₄×C₁₁₃	C₂⁴	1	1	yes	characteristic
67	Q₁₆×C₁₁₃	—	4	8	no	—
68	SD₁₆×C₁₁₃	—	4	8	no	—
69	M₁₆×C₁₁₃	—	2	4	no	—
70	M₁₆×C₁₁₃	—	2	4	no	—
71	M₁₆×C₁₁₃	—	2	4	no	—
72	M₁₆×C₁₁₃	—	4	4	no	—
73	C₂×C₈×C₁₁₃	—	2	4	no	—
74	C₂×C₈×C₁₁₃	—	4	4	no	—
75	C₂×C₈×C₁₁₃	C₂²×C₄	1	4	yes	—
76	C₄⋊C₄×C₁₁₃	—	2	4	no	—
77	C₄⋊C₄×C₁₁₃	—	4	2	no	—
78	C₄⋊C₄×C₁₁₃	—	4	2	no	—
79	C₄⋊C₄×C₁₁₃	—	4	1	no	—
80	C₄⋊C₄×C₁₁₃	—	4	4	no	—
81	C₂²⋊C₄×C₁₁₃	—	2	4	no	—
82	C₂²⋊C₄×C₁₁₃	—	4	2	no	—
83	C₄²×C₁₁₃	—	2	1	no	—
84	C₄²×C₁₁₃	—	2	4	no	—
85	C₄²×C₁₁₃	—	4	4	no	—
86	C₄²×C₁₁₃	C₂²×C₄	1	1	yes	characteristic
87	C₄²×C₁₁₃	C₂²×C₄	1	1	yes	characteristic
88	C₂³×C₁₁₃	—	2	2	no	—
89	C₂³×C₁₁₃	H(5,𝔽₂)	1	1	yes	characteristic
90	Q₈×C₁₁₃	—	2	8	no	—
91	Q₈×C₁₁₃	—	2	4	no	—
92	Q₈×C₁₁₃	—	4	4	no	—
93	D₄×C₁₁₃	—	2	4	no	—
94	D₄×C₁₁₃	—	4	4	no	—
95	D₄×C₁₁₃	—	8	2	no	—
96	C₂×C₄×C₁₁₃	—	2	1	no	—
97	C₂×C₄×C₁₁₃	—	2	1	no	—
98	C₂×C₄×C₁₁₃	—	2	2	no	—
99	C₂×C₄×C₁₁₃	—	2	4	no	—
100	C₂×C₄×C₁₁₃	—	4	2	no	—
101	C₂×C₄×C₁₁₃	—	4	2	no	—
102	C₂×C₄×C₁₁₃	—	4	2	no	—
103	C₂×C₄×C₁₁₃	—	4	2	no	—
104	C₂×C₄×C₁₁₃	—	4	2	no	—
105	C₂×C₄×C₁₁₃	—	4	4	no	—
106	C₂×C₄×C₁₁₃	C₂³×C₄	1	1	yes	characteristic
107	C₂×C₄×C₁₁₃	H(5,𝔽₂)	1	4	yes	—
108	C₂×C₄×C₁₁₃	H(5,𝔽₂)	1	1	yes	characteristic
109	C₈×C₁₁₃	—	2	4	no	—
110	C₈×C₁₁₃	—	4	4	no	—
111	C₂²×C₁₁₃	—	4	1	no	—
112	C₂²×C₁₁₃	—	4	2	no	—
113	C₂²×C₁₁₃	—	8	2	no	—
114	C₂²×C₁₁₃	#518	1	1	yes	characteristic
115	C₂²×C₁₁₃	#560	1	2	yes	—
116	C₄×C₁₁₃	—	2	2	no	—
117	C₄×C₁₁₃	—	4	1	no	—
118	C₄×C₁₁₃	—	4	2	no	—
119	C₄×C₁₁₃	—	4	4	no	—
120	C₄×C₁₁₃	—	8	4	no	—
121	C₄×C₁₁₃	#518	1	2	yes	—
122	C₄×C₁₁₃	#560	1	2	yes	—
123	#23360	C₁₁₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
124	C₂×C₁₁₃	—	2	1	no	—
125	C₂×C₁₁₃	—	2	2	no	—
126	C₂×C₁₁₃	—	8	2	no	—
127	C₂×C₁₁₃	#2380	1	1	yes	characteristic
128	#3152	C₂×C₁₁₃	1	1	yes	characteristic
129	#1905	C₂×C₁₁₃	1	4	yes	complemented
130	#1902	C₂×C₁₁₃	1	4	yes	complemented
131	#1886	C₂×C₁₁₃	1	2	yes	complemented
132	#1884	C₂×C₁₁₃	1	2	yes	—
133	#1841	C₂×C₁₁₃	1	2	yes	complemented
134	C₁₁₃	#23360	1	1	yes	characteristic, direct product factor, Sylow
135	C₂×#137	C₂²×C₁₁₃	1	4	yes	—
136	#568	C₂²×C₁₁₃	1	2	yes	—
137	#532	C₂²×C₁₁₃	1	1	yes	characteristic
138	C₂×C₄⋊Q₈	C₂²×C₁₁₃	1	1	yes	characteristic
139	#519	C₂²×C₁₁₃	1	2	yes	—
140	#497	C₄×C₁₁₃	1	4	yes	complemented
141	#494	C₄×C₁₁₃	1	2	yes	complemented
142	#492	C₄×C₁₁₃	1	2	yes	complemented
143	#421	C₂²×C₁₁₃	1	4	yes	—
144	#419	C₂²×C₁₁₃	1	4	yes	—
145	C₂×#101	C₂²×C₁₁₃	1	2	yes	—
146	#410	C₂²×C₁₁₃	1	2	yes	—
147	C₄×M₁₆	C₂²×C₁₁₃	1	1	yes	characteristic
148	#344	C₂²×C₁₁₃	1	4	yes	—
149	#339	C₂²×C₁₁₃	1	4	yes	—
150	#338	C₂²×C₁₁₃	1	2	yes	—
151	H(3,ℤ/4)	C₂²×C₁₁₃	1	2	yes	—
152	C₂×#41	C₂³×C₁₁₃	1	2	yes	—
153	C₂²×Q₈	C₂³×C₁₁₃	1	2	yes	—
154	#137	—	4	4	no	—
155	#137	—	4	4	no	—
156	C₂×Q₁₆	C₂×C₄×C₁₁₃	1	8	yes	—
157	C₂×SD₁₆	C₂×C₄×C₁₁₃	1	8	yes	—
158	#131	—	4	4	no	—
159	C₂×M₁₆	—	4	2	no	—
160	C₂×M₁₆	C₂³×C₁₁₃	1	4	yes	—
161	C₂×M₁₆	C₂³×C₁₁₃	1	2	yes	—
162	C₄⋊Q₈	—	4	1	no	—
163	C₄⋊Q₈	C₂×C₄×C₁₁₃	1	2	yes	—
164	C₄⋊Q₈	C₂×C₄×C₁₁₃	1	2	yes	—
165	C₄⋊Q₈	C₂×C₄×C₁₁₃	1	1	yes	characteristic
166	#127	C₂×C₄×C₁₁₃	1	1	yes	characteristic
167	#124	C₂×C₄×C₁₁₃	1	2	yes	—
168	#121	—	4	1	no	—
169	C₄×Q₈	—	4	4	no	—
170	C₄×D₄	—	4	1	no	—
171	#117	—	4	2	no	—
172	#117	C₂³×C₁₁₃	1	4	yes	—
173	C₂×C₄⋊C₄	—	4	1	no	—
174	C₂×C₄²	C₂³×C₁₁₃	1	1	yes	characteristic
175	#104	—	4	4	no	—
176	#103	—	4	4	no	—
177	#101	—	4	2	no	—
178	#99	—	4	2	no	—
179	#97	C₂×C₄×C₁₁₃	1	2	yes	—
180	C₄×C₈	C₂×C₄×C₁₁₃	1	2	yes	—
181	#41	—	2	4	no	—
182	#41	—	2	4	no	—
183	C₂×Q₈	—	2	2	no	—
184	C₂×Q₈	—	2	4	no	—
185	C₂×Q₈	—	2	4	no	—
186	C₂×Q₈	C₂²×C₄×C₁₁₃	1	2	yes	—
187	C₂×Q₈	C₂²×C₄×C₁₁₃	1	4	yes	—
188	C₂×D₄	—	2	2	no	—
189	C₂×D₄	—	4	1	no	—
190	C₂×D₄	C₂²×C₄×C₁₁₃	1	2	yes	—
191	C₂²×C₄	—	2	1	no	—
192	C₂²×C₄	—	2	2	no	—
193	C₂²×C₄	—	2	2	no	—
194	C₂²×C₄	C₂⁴×C₁₁₃	1	1	yes	characteristic
195	Q₁₆	—	4	8	no	—
196	SD₁₆	—	4	8	no	—
197	M₁₆	—	2	4	no	—
198	M₁₆	—	2	4	no	—
199	M₁₆	—	2	4	no	—
200	M₁₆	—	4	4	no	—
201	C₂×C₈	—	2	4	no	—
202	C₂×C₈	—	4	4	no	—
203	C₂×C₈	C₂²×C₄×C₁₁₃	1	4	yes	—
204	C₄⋊C₄	—	2	4	no	—
205	C₄⋊C₄	—	4	2	no	—
206	C₄⋊C₄	—	4	2	no	—
207	C₄⋊C₄	—	4	1	no	—
208	C₄⋊C₄	—	4	4	no	—
209	C₂²⋊C₄	—	2	4	no	—
210	C₂²⋊C₄	—	4	2	no	—
211	C₄²	—	2	1	no	—
212	C₄²	—	2	4	no	—
213	C₄²	—	4	4	no	—
214	C₄²	C₂²×C₄×C₁₁₃	1	1	yes	characteristic
215	C₄²	C₂²×C₄×C₁₁₃	1	1	yes	characteristic
216	C₂³	—	2	2	no	—
217	C₂³	H(5,𝔽₂)×C₁₁₃	1	1	yes	characteristic
218	Q₈	—	2	8	no	—
219	Q₈	—	2	4	no	—
220	Q₈	—	4	4	no	—
221	D₄	—	2	4	no	—
222	D₄	—	4	4	no	—
223	D₄	—	8	2	no	—
224	C₂×C₄	—	2	1	no	—
225	C₂×C₄	—	2	1	no	—
226	C₂×C₄	—	2	2	no	—
227	C₂×C₄	—	2	4	no	—
228	C₂×C₄	—	4	2	no	—
229	C₂×C₄	—	4	2	no	—
230	C₂×C₄	—	4	2	no	—
231	C₂×C₄	—	4	2	no	—
232	C₂×C₄	—	4	2	no	—
233	C₂×C₄	—	4	4	no	—
234	C₂×C₄	C₂³×C₄×C₁₁₃	1	1	yes	characteristic
235	C₂×C₄	H(5,𝔽₂)×C₁₁₃	1	4	yes	—
236	C₂×C₄	H(5,𝔽₂)×C₁₁₃	1	1	yes	characteristic
237	C₈	—	2	4	no	—
238	C₈	—	4	4	no	—
239	C₂²	—	4	1	no	—
240	C₂²	—	4	2	no	—
241	C₂²	—	8	2	no	—
242	C₂²	#518×C₁₁₃	1	1	yes	characteristic
243	C₂²	#560×C₁₁₃	1	2	yes	—
244	C₄	—	2	2	no	—
245	C₄	—	4	1	no	—
246	C₄	—	4	2	no	—
247	C₄	—	4	4	no	—
248	C₄	—	8	4	no	—
249	C₄	#518×C₁₁₃	1	2	yes	—
250	C₄	#560×C₁₁₃	1	2	yes	—
251	C₂	—	2	1	no	—
252	C₂	—	2	2	no	—
253	C₂	—	8	2	no	—
254	C₂	C₁₁₃×#2380	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂×C₁₁₃	#2380
commutator	C₂×C₄	C₂³×C₄×C₁₁₃
Frattini, cosocle	C₂²×C₄	C₂⁴×C₁₁₃
prime radical	#532×C₁₁₃	C₂²
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₁₁₃ ◁ C₂²×C₄×C₁₁₃ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₁₁₃×#1905	C₂	semidirect
3. C₁₁₃×#1902	C₂	semidirect
4. C₁₁₃×#1886	C₂	semidirect
6. C₁₁₃×#1841	C₂	semidirect
12. #497×C₁₁₃	C₄	semidirect
13. #494×C₁₁₃	C₄	semidirect
14. #492×C₁₁₃	C₄	semidirect
123. #23360	C₁₁₃	direct (N × H)
129. #1905	C₂×C₁₁₃	semidirect
130. #1902	C₂×C₁₁₃	semidirect
131. #1886	C₂×C₁₁₃	semidirect
133. #1841	C₂×C₁₁₃	semidirect
134. C₁₁₃	#23360	direct (N × H)
140. #497	C₄×C₁₁₃	semidirect
141. #494	C₄×C₁₁₃	semidirect
142. #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×#3152	2
C₁₁₃×#1905	2
C₁₁₃×#1902	2
C₁₁₃×#1886	2
C₁₁₃×#1884	2
C₁₁₃×#1841	2
#23360	113

Subgroups — C113×#23360 #43291380

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₁₃×#23360 #43291380