Subgroups — C₁₁₃×#20638 #43288658

Order 28928.

1530 conjugacy classes of subgroups.

The subgroup lattice has 280 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#518×C₁₁₃	C₂	1	1	yes	maximal, characteristic
2	C₁₁₃×#2905	C₂	1	2	yes	maximal, complemented
3	C₂²×#100×C₁₁₃	C₂	1	2	yes	maximal
4	C₁₁₃×#1800	C₂	1	8	yes	maximal, complemented
5	C₁₁₃×#1742	C₂	1	2	yes	maximal, complemented
6	C₂³×D₄×C₁₁₃	C₂²	1	1	yes	characteristic
7	C₂²×M₁₆×C₁₁₃	C₂²	1	4	yes	—
8	#518×C₁₁₃	—	2	8	no	—
9	C₂×C₄×D₄×C₁₁₃	C₄	1	4	yes	—
10	C₂×C₄×D₄×C₁₁₃	C₂²	1	4	yes	—
11	C₂×#117×C₁₁₃	C₂²	1	1	yes	characteristic
12	C₂×#117×C₁₁₃	C₂²	1	1	yes	characteristic
13	C₂²×C₂²⋊C₄×C₁₁₃	C₄	1	4	yes	—
14	#424×C₁₁₃	—	2	8	no	—
15	#423×C₁₁₃	C₂²	1	8	yes	complemented
16	C₂×#100×C₁₁₃	—	2	8	no	—
17	C₂×#100×C₁₁₃	C₂²	1	8	yes	—
18	#343×C₁₁₃	C₂²	1	8	yes	complemented
19	C₂⁵×C₁₁₃	C₂×C₄	1	2	yes	—
20	C₂²×D₄×C₁₁₃	—	2	4	no	—
21	C₂²×D₄×C₁₁₃	—	2	4	no	—
22	C₂²×D₄×C₁₁₃	C₂×C₄	1	2	yes	—
23	C₂²×D₄×C₁₁₃	D₄	1	4	yes	—
24	C₂²×D₄×C₁₁₃	Q₈	1	4	yes	—
25	C₂²×D₄×C₁₁₃	C₂³	1	2	yes	—
26	C₂³×C₄×C₁₁₃	C₂×C₄	1	8	yes	—
27	C₂³×C₄×C₁₁₃	C₂³	1	1	yes	characteristic
28	C₂×M₁₆×C₁₁₃	—	2	4	no	—
29	C₂×M₁₆×C₁₁₃	—	2	8	no	—
30	C₂×M₁₆×C₁₁₃	—	2	8	no	—
31	C₂×M₁₆×C₁₁₃	C₂³	1	8	yes	—
32	C₂²×C₈×C₁₁₃	—	2	4	no	—
33	C₄×D₄×C₁₁₃	—	2	16	no	—
34	C₄×D₄×C₁₁₃	—	2	16	no	—
35	#117×C₁₁₃	—	2	4	no	—
36	#117×C₁₁₃	D₄	1	8	yes	—
37	C₂×C₄⋊C₄×C₁₁₃	C₂×C₄	1	2	yes	—
38	C₂×C₄⋊C₄×C₁₁₃	C₂³	1	2	yes	—
39	C₂×C₂²⋊C₄×C₁₁₃	—	2	16	no	—
40	C₂×C₂²⋊C₄×C₁₁₃	C₂×C₄	1	2	yes	—
41	C₂×C₂²⋊C₄×C₁₁₃	C₂×C₄	1	8	yes	—
42	C₂×C₂²⋊C₄×C₁₁₃	C₂×C₄	1	2	yes	—
43	C₂×C₄²×C₁₁₃	C₂×C₄	1	2	yes	—
44	C₂×C₄²×C₁₁₃	C₂³	1	2	yes	—
45	C₄⋊C₈×C₁₁₃	—	2	16	no	—
46	#100×C₁₁₃	—	2	16	no	—
47	C₂⁴×C₁₁₃	—	2	4	no	—
48	C₂⁴×C₁₁₃	—	2	4	no	—
49	C₂⁴×C₁₁₃	—	8	4	no	—
50	C₂⁴×C₁₁₃	C₄⋊C₄	1	8	yes	—
51	C₂⁴×C₁₁₃	C₂²×C₄	1	4	yes	—
52	C₂⁴×C₁₁₃	C₂²×C₄	1	1	yes	characteristic
53	C₂×D₄×C₁₁₃	—	2	4	no	—
54	C₂×D₄×C₁₁₃	—	2	4	no	—
55	C₂×D₄×C₁₁₃	—	2	16	no	—
56	C₂×D₄×C₁₁₃	—	2	16	no	—
57	C₂×D₄×C₁₁₃	—	2	4	no	—
58	C₂×D₄×C₁₁₃	—	2	4	no	—
59	C₂×D₄×C₁₁₃	C₄⋊C₄	1	8	yes	—
60	C₂×D₄×C₁₁₃	C₂×D₄	1	4	yes	—
61	C₂×D₄×C₁₁₃	C₂×Q₈	1	4	yes	—
62	C₂²×C₄×C₁₁₃	—	2	16	no	—
63	C₂²×C₄×C₁₁₃	—	8	8	no	—
64	C₂²×C₄×C₁₁₃	C₂²×C₄	1	4	yes	—
65	C₂²×C₄×C₁₁₃	C₂²×C₄	1	4	yes	—
66	C₂²×C₄×C₁₁₃	C₂²×C₄	1	1	yes	characteristic
67	C₂²×C₄×C₁₁₃	C₂×D₄	1	4	yes	—
68	C₂²×C₄×C₁₁₃	C₂×D₄	1	2	yes	—
69	C₂²×C₄×C₁₁₃	C₂×Q₈	1	2	yes	—
70	C₂²×C₄×C₁₁₃	#41	1	4	yes	—
71	C₂²×C₄×C₁₁₃	C₂⁴	1	1	yes	characteristic
72	M₁₆×C₁₁₃	—	2	8	no	—
73	M₁₆×C₁₁₃	—	2	8	no	—
74	M₁₆×C₁₁₃	—	4	8	no	—
75	C₂×C₈×C₁₁₃	—	2	8	no	—
76	C₂×C₈×C₁₁₃	—	4	8	no	—
77	C₄⋊C₄×C₁₁₃	—	2	4	no	—
78	C₄⋊C₄×C₁₁₃	C₂²⋊C₄	1	8	yes	—
79	C₂²⋊C₄×C₁₁₃	—	2	4	no	—
80	C₂²⋊C₄×C₁₁₃	—	2	16	no	—
81	C₂²⋊C₄×C₁₁₃	C₂²⋊C₄	1	8	yes	—
82	C₄²×C₁₁₃	—	2	4	no	—
83	C₄²×C₁₁₃	C₂×D₄	1	8	yes	—
84	C₂³×C₁₁₃	—	2	4	no	—
85	C₂³×C₁₁₃	—	2	4	no	—
86	C₂³×C₁₁₃	—	2	16	no	—
87	C₂³×C₁₁₃	—	4	2	no	—
88	C₂³×C₁₁₃	—	4	8	no	—
89	C₂³×C₁₁₃	—	8	8	no	—
90	C₂³×C₁₁₃	—	8	8	no	—
91	C₂³×C₁₁₃	—	8	8	no	—
92	C₂³×C₁₁₃	C₂×C₂²⋊C₄	1	2	yes	—
93	C₂³×C₁₁₃	C₂×C₄⋊C₄	1	8	yes	—
94	C₂³×C₁₁₃	C₂×C₄⋊C₄	1	2	yes	—
95	C₂³×C₁₁₃	#117	1	2	yes	—
96	C₂³×C₁₁₃	C₂³×C₄	1	1	yes	characteristic
97	D₄×C₁₁₃	—	2	8	no	—
98	D₄×C₁₁₃	—	2	8	no	—
99	D₄×C₁₁₃	—	2	16	no	—
100	C₂×C₄×C₁₁₃	—	2	4	no	—
101	C₂×C₄×C₁₁₃	—	4	4	no	—
102	C₂×C₄×C₁₁₃	—	4	4	no	—
103	C₂×C₄×C₁₁₃	—	8	16	no	—
104	C₂×C₄×C₁₁₃	C₂×C₂²⋊C₄	1	8	yes	—
105	C₂×C₄×C₁₁₃	C₂×C₂²⋊C₄	1	2	yes	—
106	C₂×C₄×C₁₁₃	C₂×C₄⋊C₄	1	2	yes	—
107	C₂×C₄×C₁₁₃	#117	1	2	yes	—
108	C₂×C₄×C₁₁₃	#121	1	8	yes	—
109	C₂×C₄×C₁₁₃	C₂²⋊Q₈	1	8	yes	—
110	C₂×C₄×C₁₁₃	C₂²×D₄	1	1	yes	characteristic
111	C₂×C₄×C₁₁₃	C₂²×D₄	1	2	yes	—
112	C₂×C₄×C₁₁₃	C₂²×Q₈	1	1	yes	characteristic
113	C₂×C₄×C₁₁₃	C₂×#41	1	2	yes	—
114	C₈×C₁₁₃	—	4	8	no	—
115	C₂²×C₁₁₃	—	2	2	no	—
116	C₂²×C₁₁₃	—	2	4	no	—
117	C₂²×C₁₁₃	—	2	8	no	—
118	C₂²×C₁₁₃	—	4	2	no	—
119	C₂²×C₁₁₃	—	4	2	no	—
120	C₂²×C₁₁₃	—	4	8	no	—
121	C₂²×C₁₁₃	—	4	8	no	—
122	C₂²×C₁₁₃	—	4	8	no	—
123	C₂²×C₁₁₃	—	8	16	no	—
124	C₂²×C₁₁₃	#380	1	4	yes	—
125	C₂²×C₁₁₃	C₂²×C₂²⋊C₄	1	1	yes	characteristic
126	C₂²×C₁₁₃	C₂²×C₄⋊C₄	1	1	yes	characteristic
127	C₂²×C₁₁₃	C₂×#117	1	1	yes	characteristic
128	C₄×C₁₁₃	—	2	8	no	—
129	C₄×C₁₁₃	—	4	4	no	—
130	C₄×C₁₁₃	#380	1	4	yes	—
131	C₄×C₁₁₃	C₂×#121	1	2	yes	—
132	C₄×C₁₁₃	C₂×C₂²⋊Q₈	1	2	yes	—
133	#20638	C₁₁₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
134	C₂×C₁₁₃	—	2	1	no	—
135	C₂×C₁₁₃	—	2	1	no	—
136	C₂×C₁₁₃	—	2	4	no	—
137	C₂×C₁₁₃	—	4	8	no	—
138	C₂×C₁₁₃	C₂×#380	1	1	yes	characteristic
139	C₂×C₁₁₃	#2888	1	2	yes	—
140	C₂×#518	C₂×C₁₁₃	1	1	yes	characteristic
141	#2905	C₂×C₁₁₃	1	2	yes	complemented
142	C₂²×#100	C₂×C₁₁₃	1	2	yes	—
143	#1800	C₂×C₁₁₃	1	8	yes	complemented
144	#1742	C₂×C₁₁₃	1	2	yes	complemented
145	C₁₁₃	#20638	1	1	yes	characteristic, direct product factor, Sylow
146	C₂³×D₄	C₂²×C₁₁₃	1	1	yes	characteristic
147	C₂²×M₁₆	C₂²×C₁₁₃	1	4	yes	—
148	#518	—	2	8	no	—
149	C₂×C₄×D₄	C₄×C₁₁₃	1	4	yes	—
150	C₂×C₄×D₄	C₂²×C₁₁₃	1	4	yes	—
151	C₂×#117	C₂²×C₁₁₃	1	1	yes	characteristic
152	C₂×#117	C₂²×C₁₁₃	1	1	yes	characteristic
153	C₂²×C₂²⋊C₄	C₄×C₁₁₃	1	4	yes	—
154	#424	—	2	8	no	—
155	#423	C₂²×C₁₁₃	1	8	yes	complemented
156	C₂×#100	—	2	8	no	—
157	C₂×#100	C₂²×C₁₁₃	1	8	yes	—
158	#343	C₂²×C₁₁₃	1	8	yes	complemented
159	C₂⁵	C₂×C₄×C₁₁₃	1	2	yes	—
160	C₂²×D₄	—	2	4	no	—
161	C₂²×D₄	—	2	4	no	—
162	C₂²×D₄	C₂×C₄×C₁₁₃	1	2	yes	—
163	C₂²×D₄	D₄×C₁₁₃	1	4	yes	—
164	C₂²×D₄	Q₈×C₁₁₃	1	4	yes	—
165	C₂²×D₄	C₂³×C₁₁₃	1	2	yes	—
166	C₂³×C₄	C₂×C₄×C₁₁₃	1	8	yes	—
167	C₂³×C₄	C₂³×C₁₁₃	1	1	yes	characteristic
168	C₂×M₁₆	—	2	4	no	—
169	C₂×M₁₆	—	2	8	no	—
170	C₂×M₁₆	—	2	8	no	—
171	C₂×M₁₆	C₂³×C₁₁₃	1	8	yes	—
172	C₂²×C₈	—	2	4	no	—
173	C₄×D₄	—	2	16	no	—
174	C₄×D₄	—	2	16	no	—
175	#117	—	2	4	no	—
176	#117	D₄×C₁₁₃	1	8	yes	—
177	C₂×C₄⋊C₄	C₂×C₄×C₁₁₃	1	2	yes	—
178	C₂×C₄⋊C₄	C₂³×C₁₁₃	1	2	yes	—
179	C₂×C₂²⋊C₄	—	2	16	no	—
180	C₂×C₂²⋊C₄	C₂×C₄×C₁₁₃	1	2	yes	—
181	C₂×C₂²⋊C₄	C₂×C₄×C₁₁₃	1	8	yes	—
182	C₂×C₂²⋊C₄	C₂×C₄×C₁₁₃	1	2	yes	—
183	C₂×C₄²	C₂×C₄×C₁₁₃	1	2	yes	—
184	C₂×C₄²	C₂³×C₁₁₃	1	2	yes	—
185	C₄⋊C₈	—	2	16	no	—
186	#100	—	2	16	no	—
187	C₂⁴	—	2	4	no	—
188	C₂⁴	—	2	4	no	—
189	C₂⁴	—	8	4	no	—
190	C₂⁴	C₄⋊C₄×C₁₁₃	1	8	yes	—
191	C₂⁴	C₂²×C₄×C₁₁₃	1	4	yes	—
192	C₂⁴	C₂²×C₄×C₁₁₃	1	1	yes	characteristic
193	C₂×D₄	—	2	4	no	—
194	C₂×D₄	—	2	4	no	—
195	C₂×D₄	—	2	16	no	—
196	C₂×D₄	—	2	16	no	—
197	C₂×D₄	—	2	4	no	—
198	C₂×D₄	—	2	4	no	—
199	C₂×D₄	C₄⋊C₄×C₁₁₃	1	8	yes	—
200	C₂×D₄	C₂×D₄×C₁₁₃	1	4	yes	—
201	C₂×D₄	C₂×Q₈×C₁₁₃	1	4	yes	—
202	C₂²×C₄	—	2	16	no	—
203	C₂²×C₄	—	8	8	no	—
204	C₂²×C₄	C₂²×C₄×C₁₁₃	1	4	yes	—
205	C₂²×C₄	C₂²×C₄×C₁₁₃	1	4	yes	—
206	C₂²×C₄	C₂²×C₄×C₁₁₃	1	1	yes	characteristic
207	C₂²×C₄	C₂×D₄×C₁₁₃	1	4	yes	—
208	C₂²×C₄	C₂×D₄×C₁₁₃	1	2	yes	—
209	C₂²×C₄	C₂×Q₈×C₁₁₃	1	2	yes	—
210	C₂²×C₄	#41×C₁₁₃	1	4	yes	—
211	C₂²×C₄	C₂⁴×C₁₁₃	1	1	yes	characteristic
212	M₁₆	—	2	8	no	—
213	M₁₆	—	2	8	no	—
214	M₁₆	—	4	8	no	—
215	C₂×C₈	—	2	8	no	—
216	C₂×C₈	—	4	8	no	—
217	C₄⋊C₄	—	2	4	no	—
218	C₄⋊C₄	C₂²⋊C₄×C₁₁₃	1	8	yes	—
219	C₂²⋊C₄	—	2	4	no	—
220	C₂²⋊C₄	—	2	16	no	—
221	C₂²⋊C₄	C₂²⋊C₄×C₁₁₃	1	8	yes	—
222	C₄²	—	2	4	no	—
223	C₄²	C₂×D₄×C₁₁₃	1	8	yes	—
224	C₂³	—	2	4	no	—
225	C₂³	—	2	4	no	—
226	C₂³	—	2	16	no	—
227	C₂³	—	4	2	no	—
228	C₂³	—	4	8	no	—
229	C₂³	—	8	8	no	—
230	C₂³	—	8	8	no	—
231	C₂³	—	8	8	no	—
232	C₂³	C₂×C₂²⋊C₄×C₁₁₃	1	2	yes	—
233	C₂³	C₂×C₄⋊C₄×C₁₁₃	1	8	yes	—
234	C₂³	C₂×C₄⋊C₄×C₁₁₃	1	2	yes	—
235	C₂³	#117×C₁₁₃	1	2	yes	—
236	C₂³	C₂³×C₄×C₁₁₃	1	1	yes	characteristic
237	D₄	—	2	8	no	—
238	D₄	—	2	8	no	—
239	D₄	—	2	16	no	—
240	C₂×C₄	—	2	4	no	—
241	C₂×C₄	—	4	4	no	—
242	C₂×C₄	—	4	4	no	—
243	C₂×C₄	—	8	16	no	—
244	C₂×C₄	C₂×C₂²⋊C₄×C₁₁₃	1	8	yes	—
245	C₂×C₄	C₂×C₂²⋊C₄×C₁₁₃	1	2	yes	—
246	C₂×C₄	C₂×C₄⋊C₄×C₁₁₃	1	2	yes	—
247	C₂×C₄	#117×C₁₁₃	1	2	yes	—
248	C₂×C₄	#121×C₁₁₃	1	8	yes	—
249	C₂×C₄	C₂²⋊Q₈×C₁₁₃	1	8	yes	—
250	C₂×C₄	C₂²×D₄×C₁₁₃	1	1	yes	characteristic
251	C₂×C₄	C₂²×D₄×C₁₁₃	1	2	yes	—
252	C₂×C₄	C₂²×Q₈×C₁₁₃	1	1	yes	characteristic
253	C₂×C₄	C₂×#41×C₁₁₃	1	2	yes	—
254	C₈	—	4	8	no	—
255	C₂²	—	2	2	no	—
256	C₂²	—	2	4	no	—
257	C₂²	—	2	8	no	—
258	C₂²	—	4	2	no	—
259	C₂²	—	4	2	no	—
260	C₂²	—	4	8	no	—
261	C₂²	—	4	8	no	—
262	C₂²	—	4	8	no	—
263	C₂²	—	8	16	no	—
264	C₂²	#380×C₁₁₃	1	4	yes	—
265	C₂²	C₂²×C₂²⋊C₄×C₁₁₃	1	1	yes	characteristic
266	C₂²	C₂²×C₄⋊C₄×C₁₁₃	1	1	yes	characteristic
267	C₂²	C₂×#117×C₁₁₃	1	1	yes	characteristic
268	C₄	—	2	8	no	—
269	C₄	—	4	4	no	—
270	C₄	#380×C₁₁₃	1	4	yes	—
271	C₄	C₂×#121×C₁₁₃	1	2	yes	—
272	C₄	C₂×C₂²⋊Q₈×C₁₁₃	1	2	yes	—
273	C₂	—	2	1	no	—
274	C₂	—	2	1	no	—
275	C₂	—	2	4	no	—
276	C₂	—	4	8	no	—
277	C₂	C₂×#380×C₁₁₃	1	1	yes	characteristic
278	C₂	C₁₁₃×#2888	1	2	yes	—

	subgroup	G/H
center, socle	C₂²×C₁₁₃	C₂²×C₂²⋊C₄
commutator	C₂³	C₂³×C₄×C₁₁₃
Frattini, cosocle	C₂²×C₄	C₂⁴×C₁₁₃
prime radical	C₂³×D₄×C₁₁₃	C₂²
LCS	G ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₁₁₃ ◁ C₂×#117×C₁₁₃ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₁₁₃×#2905	C₂	semidirect
4. C₁₁₃×#1800	C₂	semidirect
5. C₁₁₃×#1742	C₂	semidirect
15. #423×C₁₁₃	C₂²	semidirect
18. #343×C₁₁₃	C₂²	semidirect
133. #20638	C₁₁₃	direct (N × H)
141. #2905	C₂×C₁₁₃	semidirect
143. #1800	C₂×C₁₁₃	semidirect
144. #1742	C₂×C₁₁₃	semidirect
145. C₁₁₃	#20638	direct (N × H)
155. #423	C₂²×C₁₁₃	semidirect
158. #343	C₂²×C₁₁₃	semidirect

Maximal subgroups

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

subgroup	index
C₂×#518×C₁₁₃	2
C₁₁₃×#2905	2
C₂²×#100×C₁₁₃	2
C₁₁₃×#1800	2
C₁₁₃×#1742	2
#20638	113

Subgroups — C113×#20638 #43288658

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₁₃×#20638 #43288658