Subgroups — C₁₁₃×#23553 #43291573

Order 28928.

1142 conjugacy classes of subgroups.

The subgroup lattice has 272 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×#125×C₁₁₃	C₂	1	1	yes	maximal, characteristic
2	C₂×#428×C₁₁₃	C₂	1	2	yes	maximal
3	C₁₁₃×#2296	C₂	1	1	yes	maximal, characteristic
4	C₁₁₃×#2059	C₂	1	8	yes	maximal, complemented
5	C₁₁₃×#1736	C₂	1	2	yes	maximal
6	C₁₁₃×#1728	C₂	1	1	yes	maximal, characteristic
7	C₂²×M₁₆×C₁₁₃	C₂²	1	2	yes	—
8	C₂×#125×C₁₁₃	—	2	8	no	—
9	C₂×#125×C₁₁₃	—	2	2	no	—
10	C₂×#125×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×C₄⋊C₄×C₁₁₃	—	2	2	no	—
14	C₂²×C₄⋊C₄×C₁₁₃	C₂²	1	2	yes	—
15	C₂²×C₄²×C₁₁₃	C₂²	1	1	yes	characteristic
16	#428×C₁₁₃	—	2	8	no	—
17	C₂×#107×C₁₁₃	C₂²	1	4	yes	—
18	C₂×#106×C₁₁₃	C₂²	1	4	yes	—
19	#389×C₁₁₃	C₂²	1	4	yes	—
20	#385×C₁₁₃	—	2	2	no	—
21	C₄×C₄⋊C₄×C₁₁₃	—	2	2	no	—
22	#340×C₁₁₃	C₂²	1	8	yes	—
23	#336×C₁₁₃	C₂²	1	4	yes	—
24	C₂³×C₄×C₁₁₃	—	2	2	no	—
25	C₂³×C₄×C₁₁₃	D₄	1	2	yes	—
26	C₂³×C₄×C₁₁₃	C₂³	1	1	yes	characteristic
27	C₂×M₁₆×C₁₁₃	—	2	4	no	—
28	C₂×M₁₆×C₁₁₃	—	2	4	no	—
29	C₂×M₁₆×C₁₁₃	Q₈	1	8	yes	—
30	C₂²×C₈×C₁₁₃	C₂³	1	4	yes	—
31	#125×C₁₁₃	—	2	16	no	—
32	#125×C₁₁₃	—	2	16	no	—
33	#117×C₁₁₃	—	2	2	no	—
34	#117×C₁₁₃	—	2	4	no	—
35	#117×C₁₁₃	Q₈	1	4	yes	—
36	C₂×C₄⋊C₄×C₁₁₃	—	2	16	no	—
37	C₂×C₄⋊C₄×C₁₁₃	—	2	8	no	—
38	C₂×C₄⋊C₄×C₁₁₃	—	2	2	no	—
39	C₂×C₄⋊C₄×C₁₁₃	—	2	4	no	—
40	C₂×C₄⋊C₄×C₁₁₃	—	2	4	no	—
41	C₂×C₄⋊C₄×C₁₁₃	D₄	1	8	yes	—
42	C₂×C₄⋊C₄×C₁₁₃	C₂³	1	2	yes	—
43	C₂×C₄⋊C₄×C₁₁₃	C₂³	1	4	yes	—
44	C₂×C₄⋊C₄×C₁₁₃	C₂³	1	2	yes	—
45	C₂×C₂²⋊C₄×C₁₁₃	—	2	1	no	—
46	C₂×C₂²⋊C₄×C₁₁₃	—	2	1	no	—
47	C₂×C₄²×C₁₁₃	—	2	4	no	—
48	C₂×C₄²×C₁₁₃	—	2	1	no	—
49	C₂×C₄²×C₁₁₃	—	2	1	no	—
50	C₂×C₄²×C₁₁₃	D₄	1	2	yes	—
51	C₂×C₄²×C₁₁₃	C₂³	1	2	yes	—
52	#107×C₁₁₃	—	2	8	no	—
53	#106×C₁₁₃	—	2	8	no	—
54	#95×C₁₁₃	—	2	4	no	—
55	C₂⁴×C₁₁₃	C₂×D₄	1	1	yes	characteristic
56	C₂²×C₄×C₁₁₃	—	2	4	no	—
57	C₂²×C₄×C₁₁₃	—	2	8	no	—
58	C₂²×C₄×C₁₁₃	—	2	2	no	—
59	C₂²×C₄×C₁₁₃	—	2	4	no	—
60	C₂²×C₄×C₁₁₃	—	2	2	no	—
61	C₂²×C₄×C₁₁₃	—	4	4	no	—
62	C₂²×C₄×C₁₁₃	—	4	8	no	—
63	C₂²×C₄×C₁₁₃	C₂×D₄	1	4	yes	—
64	C₂²×C₄×C₁₁₃	C₂×D₄	1	1	yes	characteristic
65	C₂²×C₄×C₁₁₃	C₂×D₄	1	4	yes	—
66	C₂²×C₄×C₁₁₃	C₂×Q₈	1	4	yes	—
67	C₂²×C₄×C₁₁₃	C₂×Q₈	1	2	yes	—
68	C₂²×C₄×C₁₁₃	#41	1	2	yes	—
69	C₂²×C₄×C₁₁₃	C₂⁴	1	1	yes	characteristic
70	M₁₆×C₁₁₃	—	2	16	no	—
71	C₂×C₈×C₁₁₃	—	2	4	no	—
72	C₂×C₈×C₁₁₃	—	2	4	no	—
73	C₂×C₈×C₁₁₃	C₂×Q₈	1	8	yes	—
74	C₄⋊C₄×C₁₁₃	—	2	32	no	—
75	C₄⋊C₄×C₁₁₃	—	2	8	no	—
76	C₄⋊C₄×C₁₁₃	—	2	4	no	—
77	C₄⋊C₄×C₁₁₃	—	2	2	no	—
78	C₄⋊C₄×C₁₁₃	—	2	2	no	—
79	C₄⋊C₄×C₁₁₃	—	2	2	no	—
80	C₄⋊C₄×C₁₁₃	—	4	8	no	—
81	C₄⋊C₄×C₁₁₃	C₂×D₄	1	8	yes	—
82	C₄⋊C₄×C₁₁₃	C₂×Q₈	1	4	yes	—
83	C₂²⋊C₄×C₁₁₃	—	2	4	no	—
84	C₂²⋊C₄×C₁₁₃	—	4	2	no	—
85	C₄²×C₁₁₃	—	2	4	no	—
86	C₄²×C₁₁₃	—	2	4	no	—
87	C₄²×C₁₁₃	—	2	4	no	—
88	C₄²×C₁₁₃	—	4	2	no	—
89	C₂³×C₁₁₃	—	2	4	no	—
90	C₂³×C₁₁₃	#120	1	2	yes	—
91	C₂³×C₁₁₃	C₂²⋊Q₈	1	2	yes	—
92	C₂³×C₁₁₃	C₄⋊Q₈	1	2	yes	—
93	C₂³×C₁₁₃	C₂²×D₄	1	1	yes	characteristic
94	C₂×C₄×C₁₁₃	—	2	4	no	—
95	C₂×C₄×C₁₁₃	—	2	8	no	—
96	C₂×C₄×C₁₁₃	—	2	4	no	—
97	C₂×C₄×C₁₁₃	—	2	4	no	—
98	C₂×C₄×C₁₁₃	—	2	4	no	—
99	C₂×C₄×C₁₁₃	—	4	8	no	—
100	C₂×C₄×C₁₁₃	—	4	16	no	—
101	C₂×C₄×C₁₁₃	—	4	4	no	—
102	C₂×C₄×C₁₁₃	—	4	8	no	—
103	C₂×C₄×C₁₁₃	—	4	2	no	—
104	C₂×C₄×C₁₁₃	—	4	4	no	—
105	C₂×C₄×C₁₁₃	#120	1	2	yes	—
106	C₂×C₄×C₁₁₃	C₂²⋊Q₈	1	8	yes	—
107	C₂×C₄×C₁₁₃	C₂²⋊Q₈	1	2	yes	—
108	C₂×C₄×C₁₁₃	C₄⋊Q₈	1	8	yes	—
109	C₂×C₄×C₁₁₃	C₄⋊Q₈	1	2	yes	—
110	C₂×C₄×C₁₁₃	C₂²×D₄	1	2	yes	—
111	C₂×C₄×C₁₁₃	C₂²×Q₈	1	2	yes	—
112	C₂×C₄×C₁₁₃	C₂²×Q₈	1	1	yes	characteristic
113	C₂×C₄×C₁₁₃	C₂×#41	1	1	yes	characteristic
114	C₈×C₁₁₃	—	2	8	no	—
115	C₂²×C₁₁₃	—	2	4	no	—
116	C₂²×C₁₁₃	—	2	4	no	—
117	C₂²×C₁₁₃	—	2	4	no	—
118	C₂²×C₁₁₃	#395	1	4	yes	—
119	C₂²×C₁₁₃	C₂×#120	1	1	yes	characteristic
120	C₂²×C₁₁₃	C₂×C₂²⋊Q₈	1	1	yes	characteristic
121	C₂²×C₁₁₃	C₂×C₄⋊Q₈	1	1	yes	characteristic
122	C₂²×C₁₁₃	#575	1	4	yes	—
123	C₄×C₁₁₃	—	4	4	no	—
124	C₄×C₁₁₃	—	4	4	no	—
125	C₄×C₁₁₃	—	4	8	no	—
126	C₄×C₁₁₃	—	8	2	no	—
127	C₄×C₁₁₃	#395	1	4	yes	—
128	C₄×C₁₁₃	C₂×C₂²⋊Q₈	1	2	yes	—
129	C₄×C₁₁₃	C₂×C₄⋊Q₈	1	2	yes	—
130	#23553	C₁₁₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
131	C₂×C₁₁₃	—	2	4	no	—
132	C₂×C₁₁₃	C₂×#395	1	1	yes	characteristic
133	C₂×C₁₁₃	#3005	1	2	yes	—
134	C₂×C₁₁₃	#3080	1	2	yes	—
135	C₂×C₁₁₃	#3166	1	2	yes	—
136	C₂²×#125	C₂×C₁₁₃	1	1	yes	characteristic
137	C₂×#428	C₂×C₁₁₃	1	2	yes	—
138	#2296	C₂×C₁₁₃	1	1	yes	characteristic
139	#2059	C₂×C₁₁₃	1	8	yes	complemented
140	#1736	C₂×C₁₁₃	1	2	yes	—
141	#1728	C₂×C₁₁₃	1	1	yes	characteristic
142	C₁₁₃	#23553	1	1	yes	characteristic, direct product factor, Sylow
143	C₂²×M₁₆	C₂²×C₁₁₃	1	2	yes	—
144	C₂×#125	—	2	8	no	—
145	C₂×#125	—	2	2	no	—
146	C₂×#125	C₂²×C₁₁₃	1	4	yes	—
147	C₂×#117	C₂²×C₁₁₃	1	1	yes	characteristic
148	C₂×#117	C₂²×C₁₁₃	1	1	yes	characteristic
149	C₂²×C₄⋊C₄	—	2	2	no	—
150	C₂²×C₄⋊C₄	C₂²×C₁₁₃	1	2	yes	—
151	C₂²×C₄²	C₂²×C₁₁₃	1	1	yes	characteristic
152	#428	—	2	8	no	—
153	C₂×#107	C₂²×C₁₁₃	1	4	yes	—
154	C₂×#106	C₂²×C₁₁₃	1	4	yes	—
155	#389	C₂²×C₁₁₃	1	4	yes	—
156	#385	—	2	2	no	—
157	C₄×C₄⋊C₄	—	2	2	no	—
158	#340	C₂²×C₁₁₃	1	8	yes	—
159	#336	C₂²×C₁₁₃	1	4	yes	—
160	C₂³×C₄	—	2	2	no	—
161	C₂³×C₄	D₄×C₁₁₃	1	2	yes	—
162	C₂³×C₄	C₂³×C₁₁₃	1	1	yes	characteristic
163	C₂×M₁₆	—	2	4	no	—
164	C₂×M₁₆	—	2	4	no	—
165	C₂×M₁₆	Q₈×C₁₁₃	1	8	yes	—
166	C₂²×C₈	C₂³×C₁₁₃	1	4	yes	—
167	#125	—	2	16	no	—
168	#125	—	2	16	no	—
169	#117	—	2	2	no	—
170	#117	—	2	4	no	—
171	#117	Q₈×C₁₁₃	1	4	yes	—
172	C₂×C₄⋊C₄	—	2	16	no	—
173	C₂×C₄⋊C₄	—	2	8	no	—
174	C₂×C₄⋊C₄	—	2	2	no	—
175	C₂×C₄⋊C₄	—	2	4	no	—
176	C₂×C₄⋊C₄	—	2	4	no	—
177	C₂×C₄⋊C₄	D₄×C₁₁₃	1	8	yes	—
178	C₂×C₄⋊C₄	C₂³×C₁₁₃	1	2	yes	—
179	C₂×C₄⋊C₄	C₂³×C₁₁₃	1	4	yes	—
180	C₂×C₄⋊C₄	C₂³×C₁₁₃	1	2	yes	—
181	C₂×C₂²⋊C₄	—	2	1	no	—
182	C₂×C₂²⋊C₄	—	2	1	no	—
183	C₂×C₄²	—	2	1	no	—
184	C₂×C₄²	—	2	4	no	—
185	C₂×C₄²	—	2	1	no	—
186	C₂×C₄²	D₄×C₁₁₃	1	2	yes	—
187	C₂×C₄²	C₂³×C₁₁₃	1	2	yes	—
188	#107	—	2	8	no	—
189	#106	—	2	8	no	—
190	#95	—	2	4	no	—
191	C₂⁴	C₂×D₄×C₁₁₃	1	1	yes	characteristic
192	C₂²×C₄	—	2	4	no	—
193	C₂²×C₄	—	2	8	no	—
194	C₂²×C₄	—	2	2	no	—
195	C₂²×C₄	—	2	4	no	—
196	C₂²×C₄	—	2	2	no	—
197	C₂²×C₄	—	4	4	no	—
198	C₂²×C₄	—	4	8	no	—
199	C₂²×C₄	C₂×D₄×C₁₁₃	1	4	yes	—
200	C₂²×C₄	C₂×D₄×C₁₁₃	1	1	yes	characteristic
201	C₂²×C₄	C₂×D₄×C₁₁₃	1	4	yes	—
202	C₂²×C₄	C₂×Q₈×C₁₁₃	1	4	yes	—
203	C₂²×C₄	C₂×Q₈×C₁₁₃	1	2	yes	—
204	C₂²×C₄	#41×C₁₁₃	1	2	yes	—
205	C₂²×C₄	C₂⁴×C₁₁₃	1	1	yes	characteristic
206	M₁₆	—	2	16	no	—
207	C₂×C₈	—	2	4	no	—
208	C₂×C₈	—	2	4	no	—
209	C₂×C₈	C₂×Q₈×C₁₁₃	1	8	yes	—
210	C₄⋊C₄	—	2	32	no	—
211	C₄⋊C₄	—	2	8	no	—
212	C₄⋊C₄	—	2	4	no	—
213	C₄⋊C₄	—	2	2	no	—
214	C₄⋊C₄	—	2	2	no	—
215	C₄⋊C₄	—	2	2	no	—
216	C₄⋊C₄	—	4	8	no	—
217	C₄⋊C₄	C₂×D₄×C₁₁₃	1	8	yes	—
218	C₄⋊C₄	C₂×Q₈×C₁₁₃	1	4	yes	—
219	C₂²⋊C₄	—	2	4	no	—
220	C₂²⋊C₄	—	4	2	no	—
221	C₄²	—	2	4	no	—
222	C₄²	—	2	4	no	—
223	C₄²	—	2	4	no	—
224	C₄²	—	4	2	no	—
225	C₂³	—	2	4	no	—
226	C₂³	#120×C₁₁₃	1	2	yes	—
227	C₂³	C₂²⋊Q₈×C₁₁₃	1	2	yes	—
228	C₂³	C₄⋊Q₈×C₁₁₃	1	2	yes	—
229	C₂³	C₂²×D₄×C₁₁₃	1	1	yes	characteristic
230	C₂×C₄	—	2	4	no	—
231	C₂×C₄	—	2	8	no	—
232	C₂×C₄	—	2	4	no	—
233	C₂×C₄	—	2	4	no	—
234	C₂×C₄	—	2	4	no	—
235	C₂×C₄	—	4	8	no	—
236	C₂×C₄	—	4	16	no	—
237	C₂×C₄	—	4	4	no	—
238	C₂×C₄	—	4	8	no	—
239	C₂×C₄	—	4	2	no	—
240	C₂×C₄	—	4	4	no	—
241	C₂×C₄	#120×C₁₁₃	1	2	yes	—
242	C₂×C₄	C₂²⋊Q₈×C₁₁₃	1	8	yes	—
243	C₂×C₄	C₂²⋊Q₈×C₁₁₃	1	2	yes	—
244	C₂×C₄	C₄⋊Q₈×C₁₁₃	1	8	yes	—
245	C₂×C₄	C₄⋊Q₈×C₁₁₃	1	2	yes	—
246	C₂×C₄	C₂²×D₄×C₁₁₃	1	2	yes	—
247	C₂×C₄	C₂²×Q₈×C₁₁₃	1	2	yes	—
248	C₂×C₄	C₂²×Q₈×C₁₁₃	1	1	yes	characteristic
249	C₂×C₄	C₂×#41×C₁₁₃	1	1	yes	characteristic
250	C₈	—	2	8	no	—
251	C₂²	—	2	4	no	—
252	C₂²	—	2	4	no	—
253	C₂²	—	2	4	no	—
254	C₂²	#395×C₁₁₃	1	4	yes	—
255	C₂²	C₂×#120×C₁₁₃	1	1	yes	characteristic
256	C₂²	C₂×C₂²⋊Q₈×C₁₁₃	1	1	yes	characteristic
257	C₂²	C₂×C₄⋊Q₈×C₁₁₃	1	1	yes	characteristic
258	C₂²	#575×C₁₁₃	1	4	yes	—
259	C₄	—	4	4	no	—
260	C₄	—	4	4	no	—
261	C₄	—	4	8	no	—
262	C₄	—	8	2	no	—
263	C₄	#395×C₁₁₃	1	4	yes	—
264	C₄	C₂×C₂²⋊Q₈×C₁₁₃	1	2	yes	—
265	C₄	C₂×C₄⋊Q₈×C₁₁₃	1	2	yes	—
266	C₂	—	2	4	no	—
267	C₂	C₂×#395×C₁₁₃	1	1	yes	characteristic
268	C₂	C₁₁₃×#3005	1	2	yes	—
269	C₂	C₁₁₃×#3080	1	2	yes	—
270	C₂	C₁₁₃×#3166	1	2	yes	—

	subgroup	G/H
center, socle	C₂³×C₁₁₃	C₂²×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁴×C₁₁₃
prime radical	C₂⁴×C₁₁₃	C₂×D₄
LCS	G ▷ C₂²×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
4. C₁₁₃×#2059	C₂	semidirect
130. #23553	C₁₁₃	direct (N × H)
139. #2059	C₂×C₁₁₃	semidirect
142. C₁₁₃	#23553	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂²×#125×C₁₁₃	2
C₂×#428×C₁₁₃	2
C₁₁₃×#2296	2
C₁₁₃×#2059	2
C₁₁₃×#1736	2
C₁₁₃×#1728	2
#23553	113

Subgroups — C113×#23553 #43291573

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₁₃×#23553 #43291573