Subgroups — C₁₁₃×#23746 #43291766

Order 28928.

814 conjugacy classes of subgroups.

The subgroup lattice has 248 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₁₁₃×#2296	C₂	1	1	yes	maximal, characteristic
2	C₂×#382×C₁₁₃	C₂	1	1	yes	maximal, characteristic
3	C₁₁₃×#2101	C₂	1	8	yes	maximal, complemented
4	C₁₁₃×#1736	C₂	1	2	yes	maximal
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×#117×C₁₁₃	C₂²	1	1	yes	characteristic
9	C₂×#117×C₁₁₃	C₂²	1	1	yes	characteristic
10	C₂²×C₄⋊C₄×C₁₁₃	C₂²	1	2	yes	—
11	C₂²×C₄²×C₁₁₃	C₂²	1	1	yes	characteristic
12	#389×C₁₁₃	C₂²	1	4	yes	—
13	#385×C₁₁₃	—	2	2	no	—
14	#382×C₁₁₃	—	2	2	no	—
15	#382×C₁₁₃	C₂²	1	4	yes	—
16	C₄×C₄⋊C₄×C₁₁₃	—	2	2	no	—
17	C₂×#95×C₁₁₃	—	2	2	no	—
18	#340×C₁₁₃	C₂²	1	8	yes	—
19	#340×C₁₁₃	C₂²	1	8	yes	—
20	#336×C₁₁₃	C₂²	1	4	yes	—
21	C₂³×C₄×C₁₁₃	—	2	2	no	—
22	C₂³×C₄×C₁₁₃	D₄	1	2	yes	—
23	C₂³×C₄×C₁₁₃	C₂³	1	1	yes	characteristic
24	C₂×M₁₆×C₁₁₃	—	2	4	no	—
25	C₂×M₁₆×C₁₁₃	—	2	4	no	—
26	C₂×M₁₆×C₁₁₃	—	2	4	no	—
27	C₂²×C₈×C₁₁₃	C₂³	1	4	yes	—
28	#117×C₁₁₃	—	2	2	no	—
29	#117×C₁₁₃	—	2	4	no	—
30	#117×C₁₁₃	Q₈	1	4	yes	—
31	C₂×C₄⋊C₄×C₁₁₃	—	2	8	no	—
32	C₂×C₄⋊C₄×C₁₁₃	—	2	2	no	—
33	C₂×C₄⋊C₄×C₁₁₃	—	2	4	no	—
34	C₂×C₄⋊C₄×C₁₁₃	C₂³	1	4	yes	—
35	C₂×C₄⋊C₄×C₁₁₃	C₂³	1	2	yes	—
36	C₂×C₄⋊C₄×C₁₁₃	C₂³	1	2	yes	—
37	C₂×C₂²⋊C₄×C₁₁₃	—	2	1	no	—
38	C₂×C₂²⋊C₄×C₁₁₃	—	2	1	no	—
39	C₂×C₄²×C₁₁₃	—	2	1	no	—
40	C₂×C₄²×C₁₁₃	—	2	4	no	—
41	C₂×C₄²×C₁₁₃	—	2	1	no	—
42	C₂×C₄²×C₁₁₃	D₄	1	2	yes	—
43	C₂×C₄²×C₁₁₃	C₂³	1	2	yes	—
44	#95×C₁₁₃	—	2	8	no	—
45	#95×C₁₁₃	—	2	4	no	—
46	C₂⁴×C₁₁₃	C₂×D₄	1	1	yes	characteristic
47	C₂²×C₄×C₁₁₃	—	2	4	no	—
48	C₂²×C₄×C₁₁₃	—	2	2	no	—
49	C₂²×C₄×C₁₁₃	—	2	4	no	—
50	C₂²×C₄×C₁₁₃	—	2	2	no	—
51	C₂²×C₄×C₁₁₃	—	4	4	no	—
52	C₂²×C₄×C₁₁₃	—	4	8	no	—
53	C₂²×C₄×C₁₁₃	—	4	4	no	—
54	C₂²×C₄×C₁₁₃	C₂×D₄	1	4	yes	—
55	C₂²×C₄×C₁₁₃	C₂×D₄	1	1	yes	characteristic
56	C₂²×C₄×C₁₁₃	C₂×Q₈	1	2	yes	—
57	C₂²×C₄×C₁₁₃	#41	1	2	yes	—
58	C₂²×C₄×C₁₁₃	#41	1	4	yes	—
59	C₂²×C₄×C₁₁₃	#41	1	4	yes	—
60	C₂²×C₄×C₁₁₃	C₂⁴	1	1	yes	characteristic
61	M₁₆×C₁₁₃	—	4	8	no	—
62	C₂×C₈×C₁₁₃	—	2	4	no	—
63	C₂×C₈×C₁₁₃	—	2	4	no	—
64	C₂×C₈×C₁₁₃	—	2	4	no	—
65	C₄⋊C₄×C₁₁₃	—	2	2	no	—
66	C₄⋊C₄×C₁₁₃	—	2	8	no	—
67	C₄⋊C₄×C₁₁₃	—	2	4	no	—
68	C₄⋊C₄×C₁₁₃	—	4	8	no	—
69	C₄⋊C₄×C₁₁₃	—	4	4	no	—
70	C₄⋊C₄×C₁₁₃	C₂×Q₈	1	4	yes	—
71	C₂²⋊C₄×C₁₁₃	—	2	4	no	—
72	C₂²⋊C₄×C₁₁₃	—	4	2	no	—
73	C₄²×C₁₁₃	—	2	4	no	—
74	C₄²×C₁₁₃	—	2	4	no	—
75	C₄²×C₁₁₃	—	2	4	no	—
76	C₄²×C₁₁₃	—	4	2	no	—
77	C₂³×C₁₁₃	—	2	4	no	—
78	C₂³×C₁₁₃	C₂²⋊Q₈	1	2	yes	—
79	C₂³×C₁₁₃	#123	1	2	yes	—
80	C₂³×C₁₁₃	#124	1	2	yes	—
81	C₂³×C₁₁₃	C₂²×D₄	1	1	yes	characteristic
82	C₂×C₄×C₁₁₃	—	2	4	no	—
83	C₂×C₄×C₁₁₃	—	2	4	no	—
84	C₂×C₄×C₁₁₃	—	2	4	no	—
85	C₂×C₄×C₁₁₃	—	2	4	no	—
86	C₂×C₄×C₁₁₃	—	4	8	no	—
87	C₂×C₄×C₁₁₃	—	4	4	no	—
88	C₂×C₄×C₁₁₃	—	4	2	no	—
89	C₂×C₄×C₁₁₃	—	4	4	no	—
90	C₂×C₄×C₁₁₃	—	4	4	no	—
91	C₂×C₄×C₁₁₃	—	4	8	no	—
92	C₂×C₄×C₁₁₃	—	8	8	no	—
93	C₂×C₄×C₁₁₃	C₂²⋊Q₈	1	2	yes	—
94	C₂×C₄×C₁₁₃	#123	1	2	yes	—
95	C₂×C₄×C₁₁₃	#124	1	2	yes	—
96	C₂×C₄×C₁₁₃	#125	1	8	yes	—
97	C₂×C₄×C₁₁₃	#126	1	8	yes	—
98	C₂×C₄×C₁₁₃	C₂²×Q₈	1	1	yes	characteristic
99	C₂×C₄×C₁₁₃	C₂×#41	1	2	yes	—
100	C₂×C₄×C₁₁₃	C₂×#41	1	2	yes	—
101	C₂×C₄×C₁₁₃	C₂×#41	1	1	yes	characteristic
102	C₈×C₁₁₃	—	4	4	no	—
103	C₂²×C₁₁₃	—	2	4	no	—
104	C₂²×C₁₁₃	—	2	4	no	—
105	C₂²×C₁₁₃	—	2	4	no	—
106	C₂²×C₁₁₃	#400	1	4	yes	—
107	C₂²×C₁₁₃	C₂×C₂²⋊Q₈	1	1	yes	characteristic
108	C₂²×C₁₁₃	C₂×#123	1	1	yes	characteristic
109	C₂²×C₁₁₃	C₂×#124	1	1	yes	characteristic
110	C₂²×C₁₁₃	#575	1	4	yes	—
111	C₄×C₁₁₃	—	4	4	no	—
112	C₄×C₁₁₃	—	4	4	no	—
113	C₄×C₁₁₃	—	8	2	no	—
114	C₄×C₁₁₃	—	8	4	no	—
115	C₄×C₁₁₃	#400	1	4	yes	—
116	C₄×C₁₁₃	C₂×#125	1	2	yes	—
117	C₄×C₁₁₃	C₂×#126	1	2	yes	—
118	#23746	C₁₁₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
119	C₂×C₁₁₃	—	2	4	no	—
120	C₂×C₁₁₃	C₂×#400	1	1	yes	characteristic
121	C₂×C₁₁₃	#3080	1	2	yes	—
122	C₂×C₁₁₃	#3093	1	2	yes	—
123	C₂×C₁₁₃	#3140	1	2	yes	—
124	#2296	C₂×C₁₁₃	1	1	yes	characteristic
125	C₂×#382	C₂×C₁₁₃	1	1	yes	characteristic
126	#2101	C₂×C₁₁₃	1	8	yes	complemented
127	#1736	C₂×C₁₁₃	1	2	yes	—
128	#1736	C₂×C₁₁₃	1	2	yes	—
129	#1728	C₂×C₁₁₃	1	1	yes	characteristic
130	C₁₁₃	#23746	1	1	yes	characteristic, direct product factor, Sylow
131	C₂²×M₁₆	C₂²×C₁₁₃	1	2	yes	—
132	C₂×#117	C₂²×C₁₁₃	1	1	yes	characteristic
133	C₂×#117	C₂²×C₁₁₃	1	1	yes	characteristic
134	C₂²×C₄⋊C₄	C₂²×C₁₁₃	1	2	yes	—
135	C₂²×C₄²	C₂²×C₁₁₃	1	1	yes	characteristic
136	#389	C₂²×C₁₁₃	1	4	yes	—
137	#385	—	2	2	no	—
138	#382	—	2	2	no	—
139	#382	C₂²×C₁₁₃	1	4	yes	—
140	C₄×C₄⋊C₄	—	2	2	no	—
141	C₂×#95	—	2	2	no	—
142	#340	C₂²×C₁₁₃	1	8	yes	—
143	#340	C₂²×C₁₁₃	1	8	yes	—
144	#336	C₂²×C₁₁₃	1	4	yes	—
145	C₂³×C₄	—	2	2	no	—
146	C₂³×C₄	D₄×C₁₁₃	1	2	yes	—
147	C₂³×C₄	C₂³×C₁₁₃	1	1	yes	characteristic
148	C₂×M₁₆	—	2	4	no	—
149	C₂×M₁₆	—	2	4	no	—
150	C₂×M₁₆	—	2	4	no	—
151	C₂²×C₈	C₂³×C₁₁₃	1	4	yes	—
152	#117	—	2	2	no	—
153	#117	—	2	4	no	—
154	#117	Q₈×C₁₁₃	1	4	yes	—
155	C₂×C₄⋊C₄	—	2	8	no	—
156	C₂×C₄⋊C₄	—	2	2	no	—
157	C₂×C₄⋊C₄	—	2	4	no	—
158	C₂×C₄⋊C₄	C₂³×C₁₁₃	1	4	yes	—
159	C₂×C₄⋊C₄	C₂³×C₁₁₃	1	2	yes	—
160	C₂×C₄⋊C₄	C₂³×C₁₁₃	1	2	yes	—
161	C₂×C₂²⋊C₄	—	2	1	no	—
162	C₂×C₂²⋊C₄	—	2	1	no	—
163	C₂×C₄²	—	2	1	no	—
164	C₂×C₄²	—	2	4	no	—
165	C₂×C₄²	—	2	1	no	—
166	C₂×C₄²	D₄×C₁₁₃	1	2	yes	—
167	C₂×C₄²	C₂³×C₁₁₃	1	2	yes	—
168	#95	—	2	8	no	—
169	#95	—	2	4	no	—
170	C₂⁴	C₂×D₄×C₁₁₃	1	1	yes	characteristic
171	C₂²×C₄	—	2	4	no	—
172	C₂²×C₄	—	2	2	no	—
173	C₂²×C₄	—	2	4	no	—
174	C₂²×C₄	—	2	2	no	—
175	C₂²×C₄	—	4	4	no	—
176	C₂²×C₄	—	4	8	no	—
177	C₂²×C₄	—	4	4	no	—
178	C₂²×C₄	C₂×D₄×C₁₁₃	1	4	yes	—
179	C₂²×C₄	C₂×D₄×C₁₁₃	1	1	yes	characteristic
180	C₂²×C₄	C₂×Q₈×C₁₁₃	1	2	yes	—
181	C₂²×C₄	#41×C₁₁₃	1	2	yes	—
182	C₂²×C₄	#41×C₁₁₃	1	4	yes	—
183	C₂²×C₄	#41×C₁₁₃	1	4	yes	—
184	C₂²×C₄	C₂⁴×C₁₁₃	1	1	yes	characteristic
185	M₁₆	—	4	8	no	—
186	C₂×C₈	—	2	4	no	—
187	C₂×C₈	—	2	4	no	—
188	C₂×C₈	—	2	4	no	—
189	C₄⋊C₄	—	2	2	no	—
190	C₄⋊C₄	—	2	8	no	—
191	C₄⋊C₄	—	2	4	no	—
192	C₄⋊C₄	—	4	8	no	—
193	C₄⋊C₄	—	4	4	no	—
194	C₄⋊C₄	C₂×Q₈×C₁₁₃	1	4	yes	—
195	C₂²⋊C₄	—	2	4	no	—
196	C₂²⋊C₄	—	4	2	no	—
197	C₄²	—	2	4	no	—
198	C₄²	—	2	4	no	—
199	C₄²	—	2	4	no	—
200	C₄²	—	4	2	no	—
201	C₂³	—	2	4	no	—
202	C₂³	C₂²⋊Q₈×C₁₁₃	1	2	yes	—
203	C₂³	#123×C₁₁₃	1	2	yes	—
204	C₂³	#124×C₁₁₃	1	2	yes	—
205	C₂³	C₂²×D₄×C₁₁₃	1	1	yes	characteristic
206	C₂×C₄	—	2	4	no	—
207	C₂×C₄	—	2	4	no	—
208	C₂×C₄	—	2	4	no	—
209	C₂×C₄	—	2	4	no	—
210	C₂×C₄	—	4	8	no	—
211	C₂×C₄	—	4	4	no	—
212	C₂×C₄	—	4	2	no	—
213	C₂×C₄	—	4	4	no	—
214	C₂×C₄	—	4	4	no	—
215	C₂×C₄	—	4	8	no	—
216	C₂×C₄	—	8	8	no	—
217	C₂×C₄	C₂²⋊Q₈×C₁₁₃	1	2	yes	—
218	C₂×C₄	#123×C₁₁₃	1	2	yes	—
219	C₂×C₄	#124×C₁₁₃	1	2	yes	—
220	C₂×C₄	#125×C₁₁₃	1	8	yes	—
221	C₂×C₄	#126×C₁₁₃	1	8	yes	—
222	C₂×C₄	C₂²×Q₈×C₁₁₃	1	1	yes	characteristic
223	C₂×C₄	C₂×#41×C₁₁₃	1	2	yes	—
224	C₂×C₄	C₂×#41×C₁₁₃	1	2	yes	—
225	C₂×C₄	C₂×#41×C₁₁₃	1	1	yes	characteristic
226	C₈	—	4	4	no	—
227	C₂²	—	2	4	no	—
228	C₂²	—	2	4	no	—
229	C₂²	—	2	4	no	—
230	C₂²	#400×C₁₁₃	1	4	yes	—
231	C₂²	C₂×C₂²⋊Q₈×C₁₁₃	1	1	yes	characteristic
232	C₂²	C₂×#123×C₁₁₃	1	1	yes	characteristic
233	C₂²	C₂×#124×C₁₁₃	1	1	yes	characteristic
234	C₂²	#575×C₁₁₃	1	4	yes	—
235	C₄	—	4	4	no	—
236	C₄	—	4	4	no	—
237	C₄	—	8	2	no	—
238	C₄	—	8	4	no	—
239	C₄	#400×C₁₁₃	1	4	yes	—
240	C₄	C₂×#125×C₁₁₃	1	2	yes	—
241	C₄	C₂×#126×C₁₁₃	1	2	yes	—
242	C₂	—	2	4	no	—
243	C₂	C₂×#400×C₁₁₃	1	1	yes	characteristic
244	C₂	C₁₁₃×#3080	1	2	yes	—
245	C₂	C₁₁₃×#3093	1	2	yes	—
246	C₂	C₁₁₃×#3140	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
3. C₁₁₃×#2101	C₂	semidirect
118. #23746	C₁₁₃	direct (N × H)
126. #2101	C₂×C₁₁₃	semidirect
130. C₁₁₃	#23746	direct (N × H)

Maximal subgroups

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

subgroup	index
C₁₁₃×#2296	2
C₂×#382×C₁₁₃	2
C₁₁₃×#2101	2
C₁₁₃×#1736	2
C₁₁₃×#1728	2
#23746	113

Subgroups — C113×#23746 #43291766

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₁₃×#23746 #43291766