Subgroups — C₁₁₃×#20729 #43288749

Order 28928.

1382 conjugacy classes of subgroups.

The subgroup lattice has 268 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×#2921	C₂	1	2	yes	maximal, complemented
3	C₁₁₃×#1822	C₂	1	2	yes	maximal, complemented
4	C₁₁₃×#1800	C₂	1	4	yes	maximal, complemented
5	C₁₁₃×#1783	C₂	1	2	yes	maximal
6	C₄×#100×C₁₁₃	C₂	1	4	yes	maximal, complemented
7	C₂³×D₄×C₁₁₃	C₂²	1	1	yes	characteristic
8	#518×C₁₁₃	—	2	8	no	—
9	C₂×C₄×D₄×C₁₁₃	C₄	1	4	yes	—
10	C₂×C₄×D₄×C₁₁₃	C₂²	1	2	yes	—
11	C₂×C₄×D₄×C₁₁₃	C₂²	1	2	yes	—
12	C₂×#117×C₁₁₃	C₂²	1	1	yes	characteristic
13	C₂×#117×C₁₁₃	C₂²	1	1	yes	characteristic
14	C₂²×C₂²⋊C₄×C₁₁₃	C₄	1	4	yes	—
15	#433×C₁₁₃	—	2	8	no	—
16	#423×C₁₁₃	C₂²	1	4	yes	complemented
17	C₂×#100×C₁₁₃	C₂²	1	8	yes	—
18	#407×C₁₁₃	C₂²	1	4	yes	—
19	C₄×M₁₆×C₁₁₃	C₂²	1	4	yes	complemented
20	#343×C₁₁₃	C₂²	1	8	yes	complemented
21	C₂⁵×C₁₁₃	C₂×C₄	1	2	yes	—
22	C₂²×D₄×C₁₁₃	—	2	8	no	—
23	C₂²×D₄×C₁₁₃	—	2	4	no	—
24	C₂²×D₄×C₁₁₃	C₂×C₄	1	2	yes	—
25	C₂²×D₄×C₁₁₃	C₂³	1	2	yes	—
26	C₂³×C₄×C₁₁₃	C₂×C₄	1	4	yes	—
27	C₂³×C₄×C₁₁₃	C₂×C₄	1	4	yes	—
28	C₂³×C₄×C₁₁₃	C₂³	1	1	yes	characteristic
29	C₂×M₁₆×C₁₁₃	C₂³	1	8	yes	—
30	C₄×D₄×C₁₁₃	—	2	16	no	—
31	C₄×D₄×C₁₁₃	—	2	8	no	—
32	C₄×D₄×C₁₁₃	—	2	8	no	—
33	#117×C₁₁₃	—	2	4	no	—
34	#117×C₁₁₃	D₄	1	8	yes	—
35	C₂×C₄⋊C₄×C₁₁₃	C₂×C₄	1	2	yes	—
36	C₂×C₄⋊C₄×C₁₁₃	C₂³	1	2	yes	—
37	C₂×C₂²⋊C₄×C₁₁₃	—	2	16	no	—
38	C₂×C₂²⋊C₄×C₁₁₃	C₂×C₄	1	2	yes	—
39	C₂×C₂²⋊C₄×C₁₁₃	C₂×C₄	1	4	yes	—
40	C₂×C₂²⋊C₄×C₁₁₃	C₂×C₄	1	4	yes	—
41	C₂×C₂²⋊C₄×C₁₁₃	C₂×C₄	1	2	yes	—
42	C₂×C₄²×C₁₁₃	C₂×C₄	1	2	yes	—
43	C₂×C₄²×C₁₁₃	C₂³	1	1	yes	characteristic
44	C₂×C₄²×C₁₁₃	C₂³	1	1	yes	characteristic
45	C₄⋊C₈×C₁₁₃	—	2	8	no	—
46	#100×C₁₁₃	—	2	16	no	—
47	C₂²⋊C₈×C₁₁₃	—	2	8	no	—
48	#97×C₁₁₃	—	2	4	no	—
49	C₄×C₈×C₁₁₃	—	2	4	no	—
50	C₂⁴×C₁₁₃	—	2	4	no	—
51	C₂⁴×C₁₁₃	—	2	8	no	—
52	C₂⁴×C₁₁₃	—	8	4	no	—
53	C₂⁴×C₁₁₃	C₂²×C₄	1	1	yes	characteristic
54	C₂⁴×C₁₁₃	C₂²×C₄	1	4	yes	—
55	C₂×D₄×C₁₁₃	—	2	32	no	—
56	C₂×D₄×C₁₁₃	—	2	8	no	—
57	C₂×D₄×C₁₁₃	—	2	4	no	—
58	C₂×D₄×C₁₁₃	—	2	4	no	—
59	C₂×D₄×C₁₁₃	—	2	4	no	—
60	C₂×D₄×C₁₁₃	#41	1	8	yes	—
61	C₂²×C₄×C₁₁₃	—	2	4	no	—
62	C₂²×C₄×C₁₁₃	—	2	8	no	—
63	C₂²×C₄×C₁₁₃	—	2	8	no	—
64	C₂²×C₄×C₁₁₃	—	8	4	no	—
65	C₂²×C₄×C₁₁₃	—	8	4	no	—
66	C₂²×C₄×C₁₁₃	C₂²×C₄	1	1	yes	characteristic
67	C₂²×C₄×C₁₁₃	C₂²×C₄	1	4	yes	—
68	C₂²×C₄×C₁₁₃	C₂²×C₄	1	2	yes	—
69	C₂²×C₄×C₁₁₃	C₂²×C₄	1	2	yes	—
70	C₂²×C₄×C₁₁₃	C₂×D₄	1	4	yes	—
71	C₂²×C₄×C₁₁₃	C₂⁴	1	1	yes	characteristic
72	M₁₆×C₁₁₃	—	2	16	no	—
73	C₂×C₈×C₁₁₃	—	2	8	no	—
74	C₄⋊C₄×C₁₁₃	—	2	4	no	—
75	C₄⋊C₄×C₁₁₃	C₂²⋊C₄	1	8	yes	—
76	C₂²⋊C₄×C₁₁₃	—	2	4	no	—
77	C₂²⋊C₄×C₁₁₃	—	2	8	no	—
78	C₂²⋊C₄×C₁₁₃	—	2	8	no	—
79	C₂²⋊C₄×C₁₁₃	C₂²⋊C₄	1	8	yes	—
80	C₄²×C₁₁₃	—	2	4	no	—
81	C₄²×C₁₁₃	C₂×D₄	1	4	yes	—
82	C₄²×C₁₁₃	C₂×D₄	1	4	yes	—
83	C₂³×C₁₁₃	—	2	16	no	—
84	C₂³×C₁₁₃	—	2	4	no	—
85	C₂³×C₁₁₃	—	2	2	no	—
86	C₂³×C₁₁₃	—	2	4	no	—
87	C₂³×C₁₁₃	—	4	2	no	—
88	C₂³×C₁₁₃	—	4	4	no	—
89	C₂³×C₁₁₃	—	4	4	no	—
90	C₂³×C₁₁₃	—	8	8	no	—
91	C₂³×C₁₁₃	—	8	16	no	—
92	C₂³×C₁₁₃	C₂×C₂²⋊C₄	1	2	yes	—
93	C₂³×C₁₁₃	#117	1	8	yes	—
94	C₂³×C₁₁₃	C₂³×C₄	1	1	yes	characteristic
95	D₄×C₁₁₃	—	2	16	no	—
96	D₄×C₁₁₃	—	2	16	no	—
97	C₂×C₄×C₁₁₃	—	2	2	no	—
98	C₂×C₄×C₁₁₃	—	2	8	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	4	no	—
103	C₂×C₄×C₁₁₃	—	8	8	no	—
104	C₂×C₄×C₁₁₃	—	8	8	no	—
105	C₂×C₄×C₁₁₃	C₂×C₂²⋊C₄	1	2	yes	—
106	C₂×C₄×C₁₁₃	C₂×C₂²⋊C₄	1	4	yes	—
107	C₂×C₄×C₁₁₃	C₂×C₂²⋊C₄	1	4	yes	—
108	C₂×C₄×C₁₁₃	C₂²×D₄	1	2	yes	—
109	C₂×C₄×C₁₁₃	C₂×#41	1	4	yes	—
110	C₈×C₁₁₃	—	4	8	no	—
111	C₂²×C₁₁₃	—	2	2	no	—
112	C₂²×C₁₁₃	—	2	4	no	—
113	C₂²×C₁₁₃	—	2	2	no	—
114	C₂²×C₁₁₃	—	2	8	no	—
115	C₂²×C₁₁₃	—	4	8	no	—
116	C₂²×C₁₁₃	—	4	8	no	—
117	C₂²×C₁₁₃	—	4	4	no	—
118	C₂²×C₁₁₃	—	4	4	no	—
119	C₂²×C₁₁₃	—	4	4	no	—
120	C₂²×C₁₁₃	—	8	16	no	—
121	C₂²×C₁₁₃	C₂²×C₂²⋊C₄	1	1	yes	characteristic
122	C₂²×C₁₁₃	C₂×#117	1	2	yes	—
123	C₄×C₁₁₃	—	2	4	no	—
124	C₄×C₁₁₃	—	2	4	no	—
125	C₄×C₁₁₃	—	2	2	no	—
126	C₄×C₁₁₃	—	4	4	no	—
127	C₄×C₁₁₃	#525	1	4	yes	—
128	#20729	C₁₁₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
129	C₂×C₁₁₃	—	2	2	no	—
130	C₂×C₁₁₃	—	4	2	no	—
131	C₂×C₁₁₃	—	4	8	no	—
132	C₂×C₁₁₃	#2280	1	1	yes	characteristic
133	C₂×C₁₁₃	#2888	1	2	yes	—
134	C₂×#518	C₂×C₁₁₃	1	1	yes	characteristic
135	#2921	C₂×C₁₁₃	1	2	yes	complemented
136	#1822	C₂×C₁₁₃	1	2	yes	complemented
137	#1800	C₂×C₁₁₃	1	4	yes	complemented
138	#1783	C₂×C₁₁₃	1	2	yes	—
139	C₄×#100	C₂×C₁₁₃	1	4	yes	complemented
140	C₁₁₃	#20729	1	1	yes	characteristic, direct product factor, Sylow
141	C₂³×D₄	C₂²×C₁₁₃	1	1	yes	characteristic
142	#518	—	2	8	no	—
143	C₂×C₄×D₄	C₄×C₁₁₃	1	4	yes	—
144	C₂×C₄×D₄	C₂²×C₁₁₃	1	2	yes	—
145	C₂×C₄×D₄	C₂²×C₁₁₃	1	2	yes	—
146	C₂×#117	C₂²×C₁₁₃	1	1	yes	characteristic
147	C₂×#117	C₂²×C₁₁₃	1	1	yes	characteristic
148	C₂²×C₂²⋊C₄	C₄×C₁₁₃	1	4	yes	—
149	#433	—	2	8	no	—
150	#423	C₂²×C₁₁₃	1	4	yes	complemented
151	C₂×#100	C₂²×C₁₁₃	1	8	yes	—
152	#407	C₂²×C₁₁₃	1	4	yes	—
153	C₄×M₁₆	C₂²×C₁₁₃	1	4	yes	complemented
154	#343	C₂²×C₁₁₃	1	8	yes	complemented
155	C₂⁵	C₂×C₄×C₁₁₃	1	2	yes	—
156	C₂²×D₄	—	2	8	no	—
157	C₂²×D₄	—	2	4	no	—
158	C₂²×D₄	C₂×C₄×C₁₁₃	1	2	yes	—
159	C₂²×D₄	C₂³×C₁₁₃	1	2	yes	—
160	C₂³×C₄	C₂×C₄×C₁₁₃	1	4	yes	—
161	C₂³×C₄	C₂×C₄×C₁₁₃	1	4	yes	—
162	C₂³×C₄	C₂³×C₁₁₃	1	1	yes	characteristic
163	C₂×M₁₆	C₂³×C₁₁₃	1	8	yes	—
164	C₄×D₄	—	2	16	no	—
165	C₄×D₄	—	2	8	no	—
166	C₄×D₄	—	2	8	no	—
167	#117	—	2	4	no	—
168	#117	D₄×C₁₁₃	1	8	yes	—
169	C₂×C₄⋊C₄	C₂×C₄×C₁₁₃	1	2	yes	—
170	C₂×C₄⋊C₄	C₂³×C₁₁₃	1	2	yes	—
171	C₂×C₂²⋊C₄	—	2	16	no	—
172	C₂×C₂²⋊C₄	C₂×C₄×C₁₁₃	1	2	yes	—
173	C₂×C₂²⋊C₄	C₂×C₄×C₁₁₃	1	4	yes	—
174	C₂×C₂²⋊C₄	C₂×C₄×C₁₁₃	1	4	yes	—
175	C₂×C₂²⋊C₄	C₂×C₄×C₁₁₃	1	2	yes	—
176	C₂×C₄²	C₂×C₄×C₁₁₃	1	2	yes	—
177	C₂×C₄²	C₂³×C₁₁₃	1	1	yes	characteristic
178	C₂×C₄²	C₂³×C₁₁₃	1	1	yes	characteristic
179	C₄⋊C₈	—	2	8	no	—
180	#100	—	2	16	no	—
181	C₂²⋊C₈	—	2	8	no	—
182	#97	—	2	4	no	—
183	C₄×C₈	—	2	4	no	—
184	C₂⁴	—	2	4	no	—
185	C₂⁴	—	2	8	no	—
186	C₂⁴	—	8	4	no	—
187	C₂⁴	C₂²×C₄×C₁₁₃	1	1	yes	characteristic
188	C₂⁴	C₂²×C₄×C₁₁₃	1	4	yes	—
189	C₂×D₄	—	2	32	no	—
190	C₂×D₄	—	2	8	no	—
191	C₂×D₄	—	2	4	no	—
192	C₂×D₄	—	2	4	no	—
193	C₂×D₄	—	2	4	no	—
194	C₂×D₄	#41×C₁₁₃	1	8	yes	—
195	C₂²×C₄	—	2	4	no	—
196	C₂²×C₄	—	2	8	no	—
197	C₂²×C₄	—	2	8	no	—
198	C₂²×C₄	—	8	4	no	—
199	C₂²×C₄	—	8	4	no	—
200	C₂²×C₄	C₂²×C₄×C₁₁₃	1	1	yes	characteristic
201	C₂²×C₄	C₂²×C₄×C₁₁₃	1	4	yes	—
202	C₂²×C₄	C₂²×C₄×C₁₁₃	1	2	yes	—
203	C₂²×C₄	C₂²×C₄×C₁₁₃	1	2	yes	—
204	C₂²×C₄	C₂×D₄×C₁₁₃	1	4	yes	—
205	C₂²×C₄	C₂⁴×C₁₁₃	1	1	yes	characteristic
206	M₁₆	—	2	16	no	—
207	C₂×C₈	—	2	8	no	—
208	C₄⋊C₄	—	2	4	no	—
209	C₄⋊C₄	C₂²⋊C₄×C₁₁₃	1	8	yes	—
210	C₂²⋊C₄	—	2	4	no	—
211	C₂²⋊C₄	—	2	8	no	—
212	C₂²⋊C₄	—	2	8	no	—
213	C₂²⋊C₄	C₂²⋊C₄×C₁₁₃	1	8	yes	—
214	C₄²	—	2	4	no	—
215	C₄²	C₂×D₄×C₁₁₃	1	4	yes	—
216	C₄²	C₂×D₄×C₁₁₃	1	4	yes	—
217	C₂³	—	2	16	no	—
218	C₂³	—	2	4	no	—
219	C₂³	—	2	2	no	—
220	C₂³	—	2	4	no	—
221	C₂³	—	4	2	no	—
222	C₂³	—	4	4	no	—
223	C₂³	—	4	4	no	—
224	C₂³	—	8	8	no	—
225	C₂³	—	8	16	no	—
226	C₂³	C₂×C₂²⋊C₄×C₁₁₃	1	2	yes	—
227	C₂³	#117×C₁₁₃	1	8	yes	—
228	C₂³	C₂³×C₄×C₁₁₃	1	1	yes	characteristic
229	D₄	—	2	16	no	—
230	D₄	—	2	16	no	—
231	C₂×C₄	—	2	2	no	—
232	C₂×C₄	—	2	8	no	—
233	C₂×C₄	—	2	4	no	—
234	C₂×C₄	—	4	2	no	—
235	C₂×C₄	—	4	2	no	—
236	C₂×C₄	—	4	4	no	—
237	C₂×C₄	—	8	8	no	—
238	C₂×C₄	—	8	8	no	—
239	C₂×C₄	C₂×C₂²⋊C₄×C₁₁₃	1	2	yes	—
240	C₂×C₄	C₂×C₂²⋊C₄×C₁₁₃	1	4	yes	—
241	C₂×C₄	C₂×C₂²⋊C₄×C₁₁₃	1	4	yes	—
242	C₂×C₄	C₂²×D₄×C₁₁₃	1	2	yes	—
243	C₂×C₄	C₂×#41×C₁₁₃	1	4	yes	—
244	C₈	—	4	8	no	—
245	C₂²	—	2	2	no	—
246	C₂²	—	2	2	no	—
247	C₂²	—	2	8	no	—
248	C₂²	—	2	4	no	—
249	C₂²	—	4	8	no	—
250	C₂²	—	4	4	no	—
251	C₂²	—	4	8	no	—
252	C₂²	—	4	4	no	—
253	C₂²	—	4	4	no	—
254	C₂²	—	8	16	no	—
255	C₂²	C₂²×C₂²⋊C₄×C₁₁₃	1	1	yes	characteristic
256	C₂²	C₂×#117×C₁₁₃	1	2	yes	—
257	C₄	—	2	2	no	—
258	C₄	—	2	4	no	—
259	C₄	—	2	4	no	—
260	C₄	—	4	4	no	—
261	C₄	#525×C₁₁₃	1	4	yes	—
262	C₂	—	2	2	no	—
263	C₂	—	4	2	no	—
264	C₂	—	4	8	no	—
265	C₂	C₁₁₃×#2280	1	1	yes	characteristic
266	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×#2921	C₂	semidirect
3. C₁₁₃×#1822	C₂	semidirect
4. C₁₁₃×#1800	C₂	semidirect
6. C₄×#100×C₁₁₃	C₂	semidirect
16. #423×C₁₁₃	C₂²	semidirect
19. C₄×M₁₆×C₁₁₃	C₂²	semidirect
20. #343×C₁₁₃	C₂²	semidirect
128. #20729	C₁₁₃	direct (N × H)
135. #2921	C₂×C₁₁₃	semidirect
136. #1822	C₂×C₁₁₃	semidirect
137. #1800	C₂×C₁₁₃	semidirect
139. C₄×#100	C₂×C₁₁₃	semidirect
140. C₁₁₃	#20729	direct (N × H)
150. #423	C₂²×C₁₁₃	semidirect
153. C₄×M₁₆	C₂²×C₁₁₃	semidirect
154. #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×#2921	2
C₁₁₃×#1822	2
C₁₁₃×#1800	2
C₁₁₃×#1783	2
C₄×#100×C₁₁₃	2
#20729	113

Subgroups — C113×#20729 #43288749

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₁₃×#20729 #43288749