Subgroups — C₁₁₃×#20336 #43288356

Order 28928.

2162 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	C₂²×#131×C₁₁₃	C₂	1	2	yes	maximal
2	C₂×C₄×#41×C₁₁₃	C₂	1	1	yes	maximal, characteristic
3	C₁₁₃×#1729	C₂	1	6	yes	maximal, complemented
4	C₁₁₃×#1728	C₂	1	3	yes	maximal, complemented
5	C₂×#336×C₁₁₃	C₂	1	3	yes	maximal, complemented
6	C₂²×#41×C₁₁₃	C₂²	1	1	yes	characteristic
7	C₂×#131×C₁₁₃	—	2	8	no	—
8	C₂×#131×C₁₁₃	—	2	8	no	—
9	C₂×#131×C₁₁₃	C₄	1	16	yes	complemented
10	C₂²×M₁₆×C₁₁₃	C₂²	1	6	yes	—
11	C₂³×C₈×C₁₁₃	C₂²	1	6	yes	—
12	C₄×#41×C₁₁₃	—	2	8	no	—
13	C₂×C₄×Q₈×C₁₁₃	C₄	1	2	yes	—
14	C₂×C₄×D₄×C₁₁₃	C₄	1	6	yes	—
15	C₂×#117×C₁₁₃	C₂²	1	3	yes	—
16	C₂²×C₄²×C₁₁₃	C₂²	1	3	yes	—
17	#336×C₁₁₃	C₂²	1	1	yes	characteristic
18	#336×C₁₁₃	C₂²	1	3	yes	—
19	#336×C₁₁₃	C₂²	1	6	yes	—
20	#336×C₁₁₃	C₂²	1	6	yes	—
21	C₂×#41×C₁₁₃	C₂×C₄	1	8	yes	—
22	C₂×#41×C₁₁₃	D₄	1	8	yes	—
23	C₂×#41×C₁₁₃	D₄	1	4	yes	—
24	C₂×#41×C₁₁₃	Q₈	1	4	yes	—
25	C₂²×Q₈×C₁₁₃	C₂×C₄	1	1	yes	characteristic
26	C₂²×D₄×C₁₁₃	C₂×C₄	1	3	yes	—
27	C₂³×C₄×C₁₁₃	C₂×C₄	1	6	yes	—
28	C₂³×C₄×C₁₁₃	C₂³	1	3	yes	—
29	#131×C₁₁₃	—	2	64	no	—
30	C₂×M₁₆×C₁₁₃	—	2	12	no	—
31	C₂×M₁₆×C₁₁₃	—	2	12	no	—
32	C₂×M₁₆×C₁₁₃	C₂×C₄	1	24	yes	complemented
33	C₂²×C₈×C₁₁₃	—	2	12	no	—
34	C₂²×C₈×C₁₁₃	—	2	12	no	—
35	C₂²×C₈×C₁₁₃	C₂×C₄	1	24	yes	complemented
36	C₂²×C₈×C₁₁₃	C₂³	1	2	yes	—
37	C₂²×C₈×C₁₁₃	C₂³	1	6	yes	—
38	C₄×Q₈×C₁₁₃	—	2	8	no	—
39	C₄×D₄×C₁₁₃	—	2	24	no	—
40	#117×C₁₁₃	—	2	12	no	—
41	C₂×C₄⋊C₄×C₁₁₃	C₂×C₄	1	6	yes	—
42	C₂×C₂²⋊C₄×C₁₁₃	C₂×C₄	1	6	yes	—
43	C₂×C₄²×C₁₁₃	—	2	12	no	—
44	C₂×C₄²×C₁₁₃	C₂×C₄	1	6	yes	—
45	C₂×C₄²×C₁₁₃	C₂³	1	1	yes	characteristic
46	C₂×C₄²×C₁₁₃	C₂³	1	3	yes	—
47	C₂⁴×C₁₁₃	C₂²×C₄	1	3	yes	—
48	#41×C₁₁₃	C₂²⋊C₄	1	32	yes	—
49	#41×C₁₁₃	C₄⋊C₄	1	32	yes	—
50	C₂×Q₈×C₁₁₃	C₄²	1	4	yes	—
51	C₂×Q₈×C₁₁₃	C₂²⋊C₄	1	4	yes	—
52	C₂×Q₈×C₁₁₃	C₄⋊C₄	1	4	yes	—
53	C₂×D₄×C₁₁₃	C₄²	1	12	yes	—
54	C₂×D₄×C₁₁₃	C₂²⋊C₄	1	12	yes	—
55	C₂×D₄×C₁₁₃	C₄⋊C₄	1	12	yes	—
56	C₂²×C₄×C₁₁₃	—	2	12	no	—
57	C₂²×C₄×C₁₁₃	—	2	12	no	—
58	C₂²×C₄×C₁₁₃	—	4	6	no	—
59	C₂²×C₄×C₁₁₃	C₂²×C₄	1	6	yes	—
60	C₂²×C₄×C₁₁₃	C₂²×C₄	1	2	yes	—
61	C₂²×C₄×C₁₁₃	C₂²×C₄	1	3	yes	—
62	C₂²×C₄×C₁₁₃	C₂²×C₄	1	12	yes	—
63	C₂²×C₄×C₁₁₃	C₂×D₄	1	6	yes	—
64	C₂²×C₄×C₁₁₃	C₂×D₄	1	12	yes	—
65	C₂²×C₄×C₁₁₃	C₂×Q₈	1	6	yes	—
66	C₂²×C₄×C₁₁₃	C₂⁴	1	1	yes	characteristic
67	M₁₆×C₁₁₃	—	2	48	no	—
68	C₂×C₈×C₁₁₃	—	2	48	no	—
69	C₂×C₈×C₁₁₃	—	2	2	no	—
70	C₂×C₈×C₁₁₃	—	2	2	no	—
71	C₂×C₈×C₁₁₃	—	2	6	no	—
72	C₂×C₈×C₁₁₃	—	2	6	no	—
73	C₂×C₈×C₁₁₃	C₂²×C₄	1	4	yes	—
74	C₂×C₈×C₁₁₃	C₂²×C₄	1	12	yes	—
75	C₄⋊C₄×C₁₁₃	—	2	12	no	—
76	C₂²⋊C₄×C₁₁₃	—	2	12	no	—
77	C₄²×C₁₁₃	—	2	12	no	—
78	C₄²×C₁₁₃	—	2	2	no	—
79	C₄²×C₁₁₃	—	2	6	no	—
80	C₂³×C₁₁₃	—	2	6	no	—
81	C₂³×C₁₁₃	—	2	6	no	—
82	C₂³×C₁₁₃	C₂×C₄²	1	6	yes	—
83	C₂³×C₁₁₃	C₂×C₂²⋊C₄	1	6	yes	—
84	C₂³×C₁₁₃	C₂×C₄⋊C₄	1	6	yes	—
85	C₂³×C₁₁₃	C₂³×C₄	1	1	yes	characteristic
86	Q₈×C₁₁₃	#95	1	16	yes	—
87	D₄×C₁₁₃	#95	1	48	yes	—
88	C₂×C₄×C₁₁₃	—	2	2	no	—
89	C₂×C₄×C₁₁₃	—	2	2	no	—
90	C₂×C₄×C₁₁₃	—	2	6	no	—
91	C₂×C₄×C₁₁₃	—	2	6	no	—
92	C₂×C₄×C₁₁₃	—	2	6	no	—
93	C₂×C₄×C₁₁₃	—	4	24	no	—
94	C₂×C₄×C₁₁₃	C₂×C₄²	1	6	yes	—
95	C₂×C₄×C₁₁₃	C₂×C₂²⋊C₄	1	24	yes	—
96	C₂×C₄×C₁₁₃	C₂×C₂²⋊C₄	1	6	yes	—
97	C₂×C₄×C₁₁₃	C₂×C₄⋊C₄	1	24	yes	—
98	C₂×C₄×C₁₁₃	C₂×C₄⋊C₄	1	6	yes	—
99	C₂×C₄×C₁₁₃	#131	1	4	yes	complemented
100	C₂×C₄×C₁₁₃	C₂³×C₄	1	2	yes	—
101	C₂×C₄×C₁₁₃	C₂²×D₄	1	1	yes	characteristic
102	C₂×C₄×C₁₁₃	C₂²×D₄	1	2	yes	—
103	C₂×C₄×C₁₁₃	C₂²×Q₈	1	1	yes	characteristic
104	C₈×C₁₁₃	—	2	4	no	—
105	C₈×C₁₁₃	—	2	12	no	—
106	C₂²×C₁₁₃	—	2	6	no	—
107	C₂²×C₁₁₃	—	2	6	no	—
108	C₂²×C₁₁₃	—	2	12	no	—
109	C₂²×C₁₁₃	—	2	12	no	—
110	C₂²×C₁₁₃	C₂×#95	1	12	yes	—
111	C₂²×C₁₁₃	C₂²×C₄²	1	1	yes	characteristic
112	C₂²×C₁₁₃	C₂²×C₂²⋊C₄	1	1	yes	characteristic
113	C₂²×C₁₁₃	C₂²×C₄⋊C₄	1	1	yes	characteristic
114	C₂²×C₁₁₃	C₂×#131	1	2	yes	—
115	C₂²×C₁₁₃	#568	1	2	yes	—
116	C₄×C₁₁₃	—	2	4	no	—
117	C₄×C₁₁₃	—	4	6	no	—
118	C₄×C₁₁₃	C₂×#95	1	12	yes	—
119	C₄×C₁₁₃	C₂²×C₂²⋊C₄	1	2	yes	—
120	C₄×C₁₁₃	C₂²×C₄⋊C₄	1	2	yes	—
121	#20336	C₁₁₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
122	C₂×C₁₁₃	—	2	12	no	—
123	C₂×C₁₁₃	C₂²×#95	1	1	yes	characteristic
124	C₂×C₁₁₃	C₄×#131	1	1	yes	characteristic
125	C₂×C₁₁₃	#2875	1	1	yes	characteristic
126	C₂×C₁₁₃	#2880	1	2	yes	—
127	C₂×C₁₁₃	#2906	1	2	yes	—
128	C₂²×#131	C₂×C₁₁₃	1	2	yes	—
129	C₂×C₄×#41	C₂×C₁₁₃	1	1	yes	characteristic
130	#1729	C₂×C₁₁₃	1	6	yes	complemented
131	#1728	C₂×C₁₁₃	1	3	yes	complemented
132	C₂×#336	C₂×C₁₁₃	1	3	yes	complemented
133	C₁₁₃	#20336	1	1	yes	characteristic, direct product factor, Sylow
134	C₂²×#41	C₂²×C₁₁₃	1	1	yes	characteristic
135	C₂×#131	—	2	8	no	—
136	C₂×#131	—	2	8	no	—
137	C₂×#131	C₄×C₁₁₃	1	16	yes	complemented
138	C₂²×M₁₆	C₂²×C₁₁₃	1	6	yes	—
139	C₂³×C₈	C₂²×C₁₁₃	1	6	yes	—
140	C₄×#41	—	2	8	no	—
141	C₂×C₄×Q₈	C₄×C₁₁₃	1	2	yes	—
142	C₂×C₄×D₄	C₄×C₁₁₃	1	6	yes	—
143	C₂×#117	C₂²×C₁₁₃	1	3	yes	—
144	C₂²×C₄²	C₂²×C₁₁₃	1	3	yes	—
145	#336	C₂²×C₁₁₃	1	1	yes	characteristic
146	#336	C₂²×C₁₁₃	1	3	yes	—
147	#336	C₂²×C₁₁₃	1	6	yes	—
148	#336	C₂²×C₁₁₃	1	6	yes	—
149	C₂×#41	C₂×C₄×C₁₁₃	1	8	yes	—
150	C₂×#41	D₄×C₁₁₃	1	8	yes	—
151	C₂×#41	D₄×C₁₁₃	1	4	yes	—
152	C₂×#41	Q₈×C₁₁₃	1	4	yes	—
153	C₂²×Q₈	C₂×C₄×C₁₁₃	1	1	yes	characteristic
154	C₂²×D₄	C₂×C₄×C₁₁₃	1	3	yes	—
155	C₂³×C₄	C₂×C₄×C₁₁₃	1	6	yes	—
156	C₂³×C₄	C₂³×C₁₁₃	1	3	yes	—
157	#131	—	2	64	no	—
158	C₂×M₁₆	—	2	12	no	—
159	C₂×M₁₆	—	2	12	no	—
160	C₂×M₁₆	C₂×C₄×C₁₁₃	1	24	yes	complemented
161	C₂²×C₈	—	2	12	no	—
162	C₂²×C₈	—	2	12	no	—
163	C₂²×C₈	C₂×C₄×C₁₁₃	1	24	yes	complemented
164	C₂²×C₈	C₂³×C₁₁₃	1	2	yes	—
165	C₂²×C₈	C₂³×C₁₁₃	1	6	yes	—
166	C₄×Q₈	—	2	8	no	—
167	C₄×D₄	—	2	24	no	—
168	#117	—	2	12	no	—
169	C₂×C₄⋊C₄	C₂×C₄×C₁₁₃	1	6	yes	—
170	C₂×C₂²⋊C₄	C₂×C₄×C₁₁₃	1	6	yes	—
171	C₂×C₄²	—	2	12	no	—
172	C₂×C₄²	C₂×C₄×C₁₁₃	1	6	yes	—
173	C₂×C₄²	C₂³×C₁₁₃	1	3	yes	—
174	C₂×C₄²	C₂³×C₁₁₃	1	1	yes	characteristic
175	C₂⁴	C₂²×C₄×C₁₁₃	1	3	yes	—
176	#41	C₂²⋊C₄×C₁₁₃	1	32	yes	—
177	#41	C₄⋊C₄×C₁₁₃	1	32	yes	—
178	C₂×Q₈	C₄²×C₁₁₃	1	4	yes	—
179	C₂×Q₈	C₂²⋊C₄×C₁₁₃	1	4	yes	—
180	C₂×Q₈	C₄⋊C₄×C₁₁₃	1	4	yes	—
181	C₂×D₄	C₄²×C₁₁₃	1	12	yes	—
182	C₂×D₄	C₂²⋊C₄×C₁₁₃	1	12	yes	—
183	C₂×D₄	C₄⋊C₄×C₁₁₃	1	12	yes	—
184	C₂²×C₄	—	2	12	no	—
185	C₂²×C₄	—	2	12	no	—
186	C₂²×C₄	—	4	6	no	—
187	C₂²×C₄	C₂²×C₄×C₁₁₃	1	6	yes	—
188	C₂²×C₄	C₂²×C₄×C₁₁₃	1	2	yes	—
189	C₂²×C₄	C₂²×C₄×C₁₁₃	1	3	yes	—
190	C₂²×C₄	C₂²×C₄×C₁₁₃	1	12	yes	—
191	C₂²×C₄	C₂×D₄×C₁₁₃	1	6	yes	—
192	C₂²×C₄	C₂×D₄×C₁₁₃	1	12	yes	—
193	C₂²×C₄	C₂×Q₈×C₁₁₃	1	6	yes	—
194	C₂²×C₄	C₂⁴×C₁₁₃	1	1	yes	characteristic
195	M₁₆	—	2	48	no	—
196	C₂×C₈	—	2	48	no	—
197	C₂×C₈	—	2	2	no	—
198	C₂×C₈	—	2	2	no	—
199	C₂×C₈	—	2	6	no	—
200	C₂×C₈	—	2	6	no	—
201	C₂×C₈	C₂²×C₄×C₁₁₃	1	4	yes	—
202	C₂×C₈	C₂²×C₄×C₁₁₃	1	12	yes	—
203	C₄⋊C₄	—	2	12	no	—
204	C₂²⋊C₄	—	2	12	no	—
205	C₄²	—	2	12	no	—
206	C₄²	—	2	2	no	—
207	C₄²	—	2	6	no	—
208	C₂³	—	2	6	no	—
209	C₂³	—	2	6	no	—
210	C₂³	C₂×C₄²×C₁₁₃	1	6	yes	—
211	C₂³	C₂×C₂²⋊C₄×C₁₁₃	1	6	yes	—
212	C₂³	C₂×C₄⋊C₄×C₁₁₃	1	6	yes	—
213	C₂³	C₂³×C₄×C₁₁₃	1	1	yes	characteristic
214	Q₈	#95×C₁₁₃	1	16	yes	—
215	D₄	#95×C₁₁₃	1	48	yes	—
216	C₂×C₄	—	2	2	no	—
217	C₂×C₄	—	2	2	no	—
218	C₂×C₄	—	2	6	no	—
219	C₂×C₄	—	2	6	no	—
220	C₂×C₄	—	2	6	no	—
221	C₂×C₄	—	4	24	no	—
222	C₂×C₄	C₂×C₄²×C₁₁₃	1	6	yes	—
223	C₂×C₄	C₂×C₂²⋊C₄×C₁₁₃	1	24	yes	—
224	C₂×C₄	C₂×C₂²⋊C₄×C₁₁₃	1	6	yes	—
225	C₂×C₄	C₂×C₄⋊C₄×C₁₁₃	1	24	yes	—
226	C₂×C₄	C₂×C₄⋊C₄×C₁₁₃	1	6	yes	—
227	C₂×C₄	#131×C₁₁₃	1	4	yes	complemented
228	C₂×C₄	C₂³×C₄×C₁₁₃	1	2	yes	—
229	C₂×C₄	C₂²×D₄×C₁₁₃	1	1	yes	characteristic
230	C₂×C₄	C₂²×D₄×C₁₁₃	1	2	yes	—
231	C₂×C₄	C₂²×Q₈×C₁₁₃	1	1	yes	characteristic
232	C₈	—	2	4	no	—
233	C₈	—	2	12	no	—
234	C₂²	—	2	6	no	—
235	C₂²	—	2	6	no	—
236	C₂²	—	2	12	no	—
237	C₂²	—	2	12	no	—
238	C₂²	C₂×#95×C₁₁₃	1	12	yes	—
239	C₂²	C₂²×C₄²×C₁₁₃	1	1	yes	characteristic
240	C₂²	C₂²×C₂²⋊C₄×C₁₁₃	1	1	yes	characteristic
241	C₂²	C₂²×C₄⋊C₄×C₁₁₃	1	1	yes	characteristic
242	C₂²	C₂×#131×C₁₁₃	1	2	yes	—
243	C₂²	#568×C₁₁₃	1	2	yes	—
244	C₄	—	2	4	no	—
245	C₄	—	4	6	no	—
246	C₄	C₂×#95×C₁₁₃	1	12	yes	—
247	C₄	C₂²×C₂²⋊C₄×C₁₁₃	1	2	yes	—
248	C₄	C₂²×C₄⋊C₄×C₁₁₃	1	2	yes	—
249	C₂	—	2	12	no	—
250	C₂	C₂²×#95×C₁₁₃	1	1	yes	characteristic
251	C₂	C₄×#131×C₁₁₃	1	1	yes	characteristic
252	C₂	C₁₁₃×#2875	1	1	yes	characteristic
253	C₂	C₁₁₃×#2880	1	2	yes	—
254	C₂	C₁₁₃×#2906	1	2	yes	—

	subgroup	G/H
center	C₂²×C₄×C₁₁₃	C₂⁴
commutator	C₂²	C₂²×C₄²×C₁₁₃
Frattini, cosocle	C₂²×C₄	C₂⁴×C₁₁₃
socle	C₂³×C₁₁₃	C₂³×C₄
prime radical	C₂²×#41×C₁₁₃	C₂²
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₄×C₁₁₃ ◁ G

Complements

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

N	H ≅ G/N	type
3. C₁₁₃×#1729	C₂	semidirect
4. C₁₁₃×#1728	C₂	semidirect
5. C₂×#336×C₁₁₃	C₂	semidirect
9. C₂×#131×C₁₁₃	C₄	semidirect
32. C₂×M₁₆×C₁₁₃	C₂×C₄	semidirect
35. C₂²×C₈×C₁₁₃	C₂×C₄	semidirect
99. C₂×C₄×C₁₁₃	#131	semidirect
121. #20336	C₁₁₃	direct (N × H)
130. #1729	C₂×C₁₁₃	semidirect
131. #1728	C₂×C₁₁₃	semidirect
132. C₂×#336	C₂×C₁₁₃	semidirect
133. C₁₁₃	#20336	direct (N × H)
137. C₂×#131	C₄×C₁₁₃	semidirect
160. C₂×M₁₆	C₂×C₄×C₁₁₃	semidirect
163. C₂²×C₈	C₂×C₄×C₁₁₃	semidirect
227. C₂×C₄	#131×C₁₁₃	semidirect

Maximal subgroups

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

subgroup	index
C₂²×#131×C₁₁₃	2
C₂×C₄×#41×C₁₁₃	2
C₁₁₃×#1729	2
C₁₁₃×#1728	2
C₂×#336×C₁₁₃	2
#20336	113

Subgroups — C113×#20336 #43288356

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₁₃×#20336 #43288356