Subgroups — C₁₁₃×#20801 #43288821

Order 28928.

1102 conjugacy classes of subgroups.

The subgroup lattice has 258 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#428×C₁₁₃	C₂	1	4	yes	maximal
2	C₂×C₄×M₁₆×C₁₁₃	C₂	1	1	yes	maximal, characteristic
3	C₁₁₃×#2285	C₂	1	1	yes	maximal, characteristic
4	C₂×#382×C₁₁₃	C₂	1	1	yes	maximal, characteristic
5	C₁₁₃×#1847	C₂	1	4	yes	maximal, complemented
6	C₁₁₃×#1846	C₂	1	4	yes	maximal, complemented
7	C₂²×M₁₆×C₁₁₃	C₂²	1	2	yes	—
8	C₂×#117×C₁₁₃	C₂²	1	2	yes	—
9	C₂²×C₄⋊C₄×C₁₁₃	C₂²	1	2	yes	—
10	C₂²×C₄²×C₁₁₃	C₂²	1	1	yes	characteristic
11	#428×C₁₁₃	—	2	16	no	—
12	C₂×#107×C₁₁₃	C₂²	1	8	yes	—
13	C₂×#106×C₁₁₃	C₂²	1	8	yes	—
14	C₄×M₁₆×C₁₁₃	C₄	1	8	yes	complemented
15	C₂×#97×C₁₁₃	C₂²	1	2	yes	—
16	C₂×C₄×C₈×C₁₁₃	C₂²	1	2	yes	—
17	#384×C₁₁₃	C₂²	1	2	yes	—
18	#382×C₁₁₃	—	2	2	no	—
19	#382×C₁₁₃	C₂²	1	2	yes	—
20	#382×C₁₁₃	C₂²	1	2	yes	—
21	#382×C₁₁₃	C₂²	1	2	yes	—
22	#380×C₁₁₃	—	2	2	no	—
23	C₄×C₄⋊C₄×C₁₁₃	—	2	2	no	—
24	C₂×#95×C₁₁₃	—	2	2	no	—
25	C₂³×C₄×C₁₁₃	—	2	2	no	—
26	C₂³×C₄×C₁₁₃	D₄	1	2	yes	—
27	C₂³×C₄×C₁₁₃	C₂³	1	1	yes	characteristic
28	C₂×M₁₆×C₁₁₃	C₂×C₄	1	8	yes	—
29	C₂×M₁₆×C₁₁₃	D₄	1	8	yes	—
30	C₂×M₁₆×C₁₁₃	Q₈	1	8	yes	—
31	C₂²×C₈×C₁₁₃	C₂³	1	4	yes	—
32	#117×C₁₁₃	—	2	8	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	C₂³	1	4	yes	—
37	C₂×C₄⋊C₄×C₁₁₃	C₂³	1	4	yes	—
38	C₂×C₂²⋊C₄×C₁₁₃	—	2	2	no	—
39	C₂×C₄²×C₁₁₃	—	2	2	no	—
40	C₂×C₄²×C₁₁₃	C₂×C₄	1	4	yes	—
41	C₂×C₄²×C₁₁₃	D₄	1	2	yes	—
42	C₂×C₄²×C₁₁₃	Q₈	1	4	yes	—
43	C₂×C₄²×C₁₁₃	C₂³	1	1	yes	characteristic
44	C₂×C₄²×C₁₁₃	C₂³	1	1	yes	characteristic
45	#107×C₁₁₃	—	2	16	no	—
46	#106×C₁₁₃	—	2	16	no	—
47	#97×C₁₁₃	C₂×C₄	1	8	yes	complemented
48	C₄×C₈×C₁₁₃	C₂×C₄	1	8	yes	complemented
49	#95×C₁₁₃	—	2	4	no	—
50	#95×C₁₁₃	—	2	4	no	—
51	#95×C₁₁₃	—	2	4	no	—
52	C₂⁴×C₁₁₃	C₂×D₄	1	1	yes	characteristic
53	C₂²×C₄×C₁₁₃	—	2	4	no	—
54	C₂²×C₄×C₁₁₃	—	2	4	no	—
55	C₂²×C₄×C₁₁₃	—	4	4	no	—
56	C₂²×C₄×C₁₁₃	—	4	8	no	—
57	C₂²×C₄×C₁₁₃	—	4	4	no	—
58	C₂²×C₄×C₁₁₃	C₄⋊C₄	1	8	yes	—
59	C₂²×C₄×C₁₁₃	C₂²×C₄	1	2	yes	—
60	C₂²×C₄×C₁₁₃	C₂×D₄	1	2	yes	—
61	C₂²×C₄×C₁₁₃	C₂×D₄	1	2	yes	—
62	C₂²×C₄×C₁₁₃	C₂×D₄	1	1	yes	characteristic
63	C₂²×C₄×C₁₁₃	C₂×D₄	1	4	yes	—
64	C₂²×C₄×C₁₁₃	C₂×Q₈	1	4	yes	—
65	C₂²×C₄×C₁₁₃	C₂×Q₈	1	2	yes	—
66	C₂²×C₄×C₁₁₃	C₂⁴	1	1	yes	characteristic
67	M₁₆×C₁₁₃	C₄⋊C₄	1	32	yes	complemented
68	C₂×C₈×C₁₁₃	C₂²×C₄	1	8	yes	—
69	C₂×C₈×C₁₁₃	C₂×D₄	1	8	yes	—
70	C₂×C₈×C₁₁₃	C₂×Q₈	1	8	yes	—
71	C₄⋊C₄×C₁₁₃	—	2	4	no	—
72	C₄⋊C₄×C₁₁₃	—	2	8	no	—
73	C₄⋊C₄×C₁₁₃	—	2	4	no	—
74	C₄⋊C₄×C₁₁₃	—	4	8	no	—
75	C₄⋊C₄×C₁₁₃	—	4	4	no	—
76	C₂²⋊C₄×C₁₁₃	—	4	4	no	—
77	C₄²×C₁₁₃	—	4	4	no	—
78	C₄²×C₁₁₃	C₄⋊C₄	1	8	yes	—
79	C₄²×C₁₁₃	C₂²×C₄	1	2	yes	—
80	C₄²×C₁₁₃	C₂²×C₄	1	2	yes	—
81	C₄²×C₁₁₃	C₂×Q₈	1	2	yes	—
82	C₄²×C₁₁₃	C₂×Q₈	1	2	yes	—
83	C₂³×C₁₁₃	—	2	4	no	—
84	C₂³×C₁₁₃	C₂×C₄⋊C₄	1	2	yes	—
85	C₂³×C₁₁₃	#127	1	2	yes	—
86	C₂³×C₁₁₃	C₄⋊Q₈	1	2	yes	—
87	C₂³×C₁₁₃	C₂²×D₄	1	1	yes	characteristic
88	C₂×C₄×C₁₁₃	—	2	4	no	—
89	C₂×C₄×C₁₁₃	—	2	4	no	—
90	C₂×C₄×C₁₁₃	—	4	8	no	—
91	C₂×C₄×C₁₁₃	—	4	4	no	—
92	C₂×C₄×C₁₁₃	—	4	4	no	—
93	C₂×C₄×C₁₁₃	—	4	8	no	—
94	C₂×C₄×C₁₁₃	—	4	8	no	—
95	C₂×C₄×C₁₁₃	—	8	8	no	—
96	C₂×C₄×C₁₁₃	C₂×C₄⋊C₄	1	4	yes	—
97	C₂×C₄×C₁₁₃	C₂×C₄⋊C₄	1	4	yes	—
98	C₂×C₄×C₁₁₃	C₂×C₄⋊C₄	1	8	yes	—
99	C₂×C₄×C₁₁₃	C₂×C₄⋊C₄	1	2	yes	—
100	C₂×C₄×C₁₁₃	#127	1	2	yes	—
101	C₂×C₄×C₁₁₃	C₄⋊Q₈	1	8	yes	—
102	C₂×C₄×C₁₁₃	C₄⋊Q₈	1	2	yes	—
103	C₂×C₄×C₁₁₃	C₂³×C₄	1	1	yes	characteristic
104	C₂×C₄×C₁₁₃	C₂²×D₄	1	2	yes	—
105	C₂×C₄×C₁₁₃	C₂²×Q₈	1	2	yes	—
106	C₂×C₄×C₁₁₃	C₂²×Q₈	1	1	yes	characteristic
107	C₈×C₁₁₃	C₂×C₄⋊C₄	1	16	yes	—
108	C₂²×C₁₁₃	—	2	4	no	—
109	C₂²×C₁₁₃	—	2	4	no	—
110	C₂²×C₁₁₃	—	2	4	no	—
111	C₂²×C₁₁₃	#384	1	4	yes	—
112	C₂²×C₁₁₃	C₂²×C₄⋊C₄	1	1	yes	characteristic
113	C₂²×C₁₁₃	C₂×#127	1	1	yes	characteristic
114	C₂²×C₁₁₃	C₂×C₄⋊Q₈	1	1	yes	characteristic
115	C₂²×C₁₁₃	#575	1	4	yes	—
116	C₄×C₁₁₃	—	2	4	no	—
117	C₄×C₁₁₃	—	2	4	no	—
118	C₄×C₁₁₃	—	8	4	no	—
119	C₄×C₁₁₃	—	8	4	no	—
120	C₄×C₁₁₃	#384	1	4	yes	—
121	C₄×C₁₁₃	C₂²×C₄⋊C₄	1	2	yes	—
122	C₄×C₁₁₃	C₂×C₄⋊Q₈	1	2	yes	—
123	#20801	C₁₁₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
124	C₂×C₁₁₃	—	2	4	no	—
125	C₂×C₁₁₃	C₂×#384	1	1	yes	characteristic
126	C₂×C₁₁₃	#2911	1	2	yes	—
127	C₂×C₁₁₃	#3153	1	2	yes	—
128	C₂×C₁₁₃	#3166	1	2	yes	—
129	C₂×#428	C₂×C₁₁₃	1	4	yes	—
130	C₂×C₄×M₁₆	C₂×C₁₁₃	1	1	yes	characteristic
131	#2285	C₂×C₁₁₃	1	1	yes	characteristic
132	C₂×#382	C₂×C₁₁₃	1	1	yes	characteristic
133	#1847	C₂×C₁₁₃	1	4	yes	complemented
134	#1846	C₂×C₁₁₃	1	4	yes	complemented
135	C₁₁₃	#20801	1	1	yes	characteristic, direct product factor, Sylow
136	C₂²×M₁₆	C₂²×C₁₁₃	1	2	yes	—
137	C₂×#117	C₂²×C₁₁₃	1	2	yes	—
138	C₂²×C₄⋊C₄	C₂²×C₁₁₃	1	2	yes	—
139	C₂²×C₄²	C₂²×C₁₁₃	1	1	yes	characteristic
140	#428	—	2	16	no	—
141	C₂×#107	C₂²×C₁₁₃	1	8	yes	—
142	C₂×#106	C₂²×C₁₁₃	1	8	yes	—
143	C₄×M₁₆	C₄×C₁₁₃	1	8	yes	complemented
144	C₂×#97	C₂²×C₁₁₃	1	2	yes	—
145	C₂×C₄×C₈	C₂²×C₁₁₃	1	2	yes	—
146	#384	C₂²×C₁₁₃	1	2	yes	—
147	#382	—	2	2	no	—
148	#382	C₂²×C₁₁₃	1	2	yes	—
149	#382	C₂²×C₁₁₃	1	2	yes	—
150	#382	C₂²×C₁₁₃	1	2	yes	—
151	#380	—	2	2	no	—
152	C₄×C₄⋊C₄	—	2	2	no	—
153	C₂×#95	—	2	2	no	—
154	C₂³×C₄	—	2	2	no	—
155	C₂³×C₄	D₄×C₁₁₃	1	2	yes	—
156	C₂³×C₄	C₂³×C₁₁₃	1	1	yes	characteristic
157	C₂×M₁₆	C₂×C₄×C₁₁₃	1	8	yes	—
158	C₂×M₁₆	D₄×C₁₁₃	1	8	yes	—
159	C₂×M₁₆	Q₈×C₁₁₃	1	8	yes	—
160	C₂²×C₈	C₂³×C₁₁₃	1	4	yes	—
161	#117	—	2	8	no	—
162	C₂×C₄⋊C₄	—	2	8	no	—
163	C₂×C₄⋊C₄	—	2	2	no	—
164	C₂×C₄⋊C₄	—	2	4	no	—
165	C₂×C₄⋊C₄	C₂³×C₁₁₃	1	4	yes	—
166	C₂×C₄⋊C₄	C₂³×C₁₁₃	1	4	yes	—
167	C₂×C₂²⋊C₄	—	2	2	no	—
168	C₂×C₄²	—	2	2	no	—
169	C₂×C₄²	C₂×C₄×C₁₁₃	1	4	yes	—
170	C₂×C₄²	D₄×C₁₁₃	1	2	yes	—
171	C₂×C₄²	Q₈×C₁₁₃	1	4	yes	—
172	C₂×C₄²	C₂³×C₁₁₃	1	1	yes	characteristic
173	C₂×C₄²	C₂³×C₁₁₃	1	1	yes	characteristic
174	#107	—	2	16	no	—
175	#106	—	2	16	no	—
176	#97	C₂×C₄×C₁₁₃	1	8	yes	complemented
177	C₄×C₈	C₂×C₄×C₁₁₃	1	8	yes	complemented
178	#95	—	2	4	no	—
179	#95	—	2	4	no	—
180	#95	—	2	4	no	—
181	C₂⁴	C₂×D₄×C₁₁₃	1	1	yes	characteristic
182	C₂²×C₄	—	2	4	no	—
183	C₂²×C₄	—	2	4	no	—
184	C₂²×C₄	—	4	4	no	—
185	C₂²×C₄	—	4	8	no	—
186	C₂²×C₄	—	4	4	no	—
187	C₂²×C₄	C₄⋊C₄×C₁₁₃	1	8	yes	—
188	C₂²×C₄	C₂²×C₄×C₁₁₃	1	2	yes	—
189	C₂²×C₄	C₂×D₄×C₁₁₃	1	2	yes	—
190	C₂²×C₄	C₂×D₄×C₁₁₃	1	2	yes	—
191	C₂²×C₄	C₂×D₄×C₁₁₃	1	1	yes	characteristic
192	C₂²×C₄	C₂×D₄×C₁₁₃	1	4	yes	—
193	C₂²×C₄	C₂×Q₈×C₁₁₃	1	4	yes	—
194	C₂²×C₄	C₂×Q₈×C₁₁₃	1	2	yes	—
195	C₂²×C₄	C₂⁴×C₁₁₃	1	1	yes	characteristic
196	M₁₆	C₄⋊C₄×C₁₁₃	1	32	yes	complemented
197	C₂×C₈	C₂²×C₄×C₁₁₃	1	8	yes	—
198	C₂×C₈	C₂×D₄×C₁₁₃	1	8	yes	—
199	C₂×C₈	C₂×Q₈×C₁₁₃	1	8	yes	—
200	C₄⋊C₄	—	2	4	no	—
201	C₄⋊C₄	—	2	8	no	—
202	C₄⋊C₄	—	2	4	no	—
203	C₄⋊C₄	—	4	8	no	—
204	C₄⋊C₄	—	4	4	no	—
205	C₂²⋊C₄	—	4	4	no	—
206	C₄²	—	4	4	no	—
207	C₄²	C₄⋊C₄×C₁₁₃	1	8	yes	—
208	C₄²	C₂²×C₄×C₁₁₃	1	2	yes	—
209	C₄²	C₂²×C₄×C₁₁₃	1	2	yes	—
210	C₄²	C₂×Q₈×C₁₁₃	1	2	yes	—
211	C₄²	C₂×Q₈×C₁₁₃	1	2	yes	—
212	C₂³	—	2	4	no	—
213	C₂³	C₂×C₄⋊C₄×C₁₁₃	1	2	yes	—
214	C₂³	#127×C₁₁₃	1	2	yes	—
215	C₂³	C₄⋊Q₈×C₁₁₃	1	2	yes	—
216	C₂³	C₂²×D₄×C₁₁₃	1	1	yes	characteristic
217	C₂×C₄	—	2	4	no	—
218	C₂×C₄	—	2	4	no	—
219	C₂×C₄	—	4	8	no	—
220	C₂×C₄	—	4	4	no	—
221	C₂×C₄	—	4	4	no	—
222	C₂×C₄	—	4	8	no	—
223	C₂×C₄	—	4	8	no	—
224	C₂×C₄	—	8	8	no	—
225	C₂×C₄	C₂×C₄⋊C₄×C₁₁₃	1	4	yes	—
226	C₂×C₄	C₂×C₄⋊C₄×C₁₁₃	1	4	yes	—
227	C₂×C₄	C₂×C₄⋊C₄×C₁₁₃	1	8	yes	—
228	C₂×C₄	C₂×C₄⋊C₄×C₁₁₃	1	2	yes	—
229	C₂×C₄	#127×C₁₁₃	1	2	yes	—
230	C₂×C₄	C₄⋊Q₈×C₁₁₃	1	8	yes	—
231	C₂×C₄	C₄⋊Q₈×C₁₁₃	1	2	yes	—
232	C₂×C₄	C₂³×C₄×C₁₁₃	1	1	yes	characteristic
233	C₂×C₄	C₂²×D₄×C₁₁₃	1	2	yes	—
234	C₂×C₄	C₂²×Q₈×C₁₁₃	1	2	yes	—
235	C₂×C₄	C₂²×Q₈×C₁₁₃	1	1	yes	characteristic
236	C₈	C₂×C₄⋊C₄×C₁₁₃	1	16	yes	—
237	C₂²	—	2	4	no	—
238	C₂²	—	2	4	no	—
239	C₂²	—	2	4	no	—
240	C₂²	#384×C₁₁₃	1	4	yes	—
241	C₂²	C₂²×C₄⋊C₄×C₁₁₃	1	1	yes	characteristic
242	C₂²	C₂×#127×C₁₁₃	1	1	yes	characteristic
243	C₂²	C₂×C₄⋊Q₈×C₁₁₃	1	1	yes	characteristic
244	C₂²	#575×C₁₁₃	1	4	yes	—
245	C₄	—	2	4	no	—
246	C₄	—	2	4	no	—
247	C₄	—	8	4	no	—
248	C₄	—	8	4	no	—
249	C₄	#384×C₁₁₃	1	4	yes	—
250	C₄	C₂²×C₄⋊C₄×C₁₁₃	1	2	yes	—
251	C₄	C₂×C₄⋊Q₈×C₁₁₃	1	2	yes	—
252	C₂	—	2	4	no	—
253	C₂	C₂×#384×C₁₁₃	1	1	yes	characteristic
254	C₂	C₁₁₃×#2911	1	2	yes	—
255	C₂	C₁₁₃×#3153	1	2	yes	—
256	C₂	C₁₁₃×#3166	1	2	yes	—

	subgroup	G/H
center, socle	C₂³×C₁₁₃	C₂²×D₄
commutator	C₂×C₄	C₂³×C₄×C₁₁₃
Frattini, cosocle	C₂²×C₄	C₂⁴×C₁₁₃
prime radical	C₂⁴×C₁₁₃	C₂×D₄
LCS	G ▷ C₂×C₄ ▷ 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
5. C₁₁₃×#1847	C₂	semidirect
6. C₁₁₃×#1846	C₂	semidirect
14. C₄×M₁₆×C₁₁₃	C₄	semidirect
47. #97×C₁₁₃	C₂×C₄	semidirect
48. C₄×C₈×C₁₁₃	C₂×C₄	semidirect
67. M₁₆×C₁₁₃	C₄⋊C₄	semidirect
123. #20801	C₁₁₃	direct (N × H)
133. #1847	C₂×C₁₁₃	semidirect
134. #1846	C₂×C₁₁₃	semidirect
135. C₁₁₃	#20801	direct (N × H)
143. C₄×M₁₆	C₄×C₁₁₃	semidirect
176. #97	C₂×C₄×C₁₁₃	semidirect
177. C₄×C₈	C₂×C₄×C₁₁₃	semidirect
196. M₁₆	C₄⋊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×#428×C₁₁₃	2
C₂×C₄×M₁₆×C₁₁₃	2
C₁₁₃×#2285	2
C₂×#382×C₁₁₃	2
C₁₁₃×#1847	2
C₁₁₃×#1846	2
#20801	113

Subgroups — C113×#20801 #43288821

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₁₃×#20801 #43288821