Subgroups — C₁₁₃×#8681 #43276701

Order 28928.

350 conjugacy classes of subgroups.

The subgroup lattice has 244 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₁₁₃×#2960	C₂	1	1	yes	maximal, characteristic
2	C₁₁₃×#1835	C₂	1	1	yes	maximal, characteristic
3	C₁₁₃×#1564	C₂	1	1	yes	maximal, characteristic
4	C₁₁₃×#1518	C₂	1	2	yes	maximal
5	C₁₁₃×#1286	C₂	1	2	yes	maximal
6	#533×C₁₁₃	C₂²	1	1	yes	characteristic
7	#446×C₁₁₃	—	2	1	no	—
8	#446×C₁₁₃	C₄	1	2	yes	complemented
9	C₈×Q₈×C₁₁₃	—	2	1	no	—
10	C₈×Q₈×C₁₁₃	C₄	1	2	yes	complemented
11	#431×C₁₁₃	C₂²	1	2	yes	—
12	#424×C₁₁₃	—	2	1	no	—
13	#423×C₁₁₃	C₂²	1	1	yes	characteristic
14	#405×C₁₁₃	—	2	1	no	—
15	C₂×C₄×C₈×C₁₁₃	C₂²	1	1	yes	characteristic
16	#382×C₁₁₃	C₂²	1	2	yes	—
17	#340×C₁₁₃	—	2	1	no	—
18	#340×C₁₁₃	—	2	1	no	—
19	#335×C₁₁₃	—	2	1	no	—
20	#334×C₁₁₃	—	2	1	no	—
21	#328×C₁₁₃	—	2	4	no	—
22	C₂×M₁₆×C₁₁₃	—	2	1	no	—
23	C₂²×C₈×C₁₁₃	D₄	1	1	yes	characteristic
24	C₂²×C₈×C₁₁₃	D₄	1	1	yes	characteristic
25	C₄⋊Q₈×C₁₁₃	Q₈	1	1	yes	characteristic
26	C₄⋊Q₈×C₁₁₃	Q₈	1	1	yes	characteristic
27	#125×C₁₁₃	—	2	1	no	—
28	C₂²⋊Q₈×C₁₁₃	—	2	1	no	—
29	C₂²⋊Q₈×C₁₁₃	D₄	1	2	yes	—
30	C₄×Q₈×C₁₁₃	—	2	1	no	—
31	C₄×Q₈×C₁₁₃	C₂×C₄	1	2	yes	—
32	#117×C₁₁₃	—	2	1	no	—
33	C₂×C₄⋊C₄×C₁₁₃	—	2	2	no	—
34	C₂×C₄²×C₁₁₃	C₂³	1	1	yes	characteristic
35	C₄⋊C₈×C₁₁₃	—	2	2	no	—
36	C₄⋊C₈×C₁₁₃	—	2	2	no	—
37	C₄⋊C₈×C₁₁₃	—	2	1	no	—
38	C₄⋊C₈×C₁₁₃	—	2	2	no	—
39	C₄⋊C₈×C₁₁₃	C₂×C₄	1	2	yes	—
40	C₂²⋊C₈×C₁₁₃	—	2	2	no	—
41	#97×C₁₁₃	—	2	2	no	—
42	C₄×C₈×C₁₁₃	—	2	2	no	—
43	C₄×C₈×C₁₁₃	—	2	2	no	—
44	C₄×C₈×C₁₁₃	C₂×C₄	1	1	yes	characteristic
45	C₄×C₈×C₁₁₃	C₂×C₄	1	1	yes	characteristic
46	C₄×C₈×C₁₁₃	D₄	1	1	yes	characteristic
47	C₄×C₈×C₁₁₃	D₄	1	1	yes	characteristic
48	#95×C₁₁₃	—	2	2	no	—
49	#95×C₁₁₃	—	2	2	no	—
50	C₂×Q₈×C₁₁₃	—	2	1	no	—
51	C₂×Q₈×C₁₁₃	C₄⋊C₄	1	2	yes	—
52	C₂²×C₄×C₁₁₃	—	4	2	no	—
53	C₂²×C₄×C₁₁₃	C₂×D₄	1	1	yes	characteristic
54	C₂²×C₄×C₁₁₃	C₂×D₄	1	1	yes	characteristic
55	C₂²×C₄×C₁₁₃	#41	1	1	yes	characteristic
56	M₁₆×C₁₁₃	—	4	2	no	—
57	C₂×C₈×C₁₁₃	—	2	2	no	—
58	C₂×C₈×C₁₁₃	—	2	2	no	—
59	C₂×C₈×C₁₁₃	—	2	2	no	—
60	C₂×C₈×C₁₁₃	—	4	2	no	—
61	C₂×C₈×C₁₁₃	C₂²⋊C₄	1	1	yes	characteristic
62	C₂×C₈×C₁₁₃	C₂²⋊C₄	1	1	yes	characteristic
63	C₂×C₈×C₁₁₃	C₂²⋊C₄	1	1	yes	characteristic
64	C₂×C₈×C₁₁₃	C₂²⋊C₄	1	1	yes	characteristic
65	C₄⋊C₄×C₁₁₃	—	2	2	no	—
66	C₄⋊C₄×C₁₁₃	—	2	2	no	—
67	C₄⋊C₄×C₁₁₃	—	2	1	no	—
68	C₄⋊C₄×C₁₁₃	—	2	1	no	—
69	C₄⋊C₄×C₁₁₃	—	2	1	no	—
70	C₄⋊C₄×C₁₁₃	—	4	2	no	—
71	C₄⋊C₄×C₁₁₃	—	4	2	no	—
72	C₄⋊C₄×C₁₁₃	C₄⋊C₄	1	2	yes	—
73	C₂²⋊C₄×C₁₁₃	—	2	2	no	—
74	C₄²×C₁₁₃	—	2	2	no	—
75	C₄²×C₁₁₃	C₂²×C₄	1	1	yes	characteristic
76	C₄²×C₁₁₃	C₂×D₄	1	1	yes	characteristic
77	C₄²×C₁₁₃	C₂×Q₈	1	1	yes	characteristic
78	C₄²×C₁₁₃	#41	1	1	yes	characteristic
79	C₂³×C₁₁₃	#121	1	1	yes	characteristic
80	Q₈×C₁₁₃	—	4	2	no	—
81	Q₈×C₁₁₃	—	4	2	no	—
82	C₂×C₄×C₁₁₃	—	2	2	no	—
83	C₂×C₄×C₁₁₃	—	2	2	no	—
84	C₂×C₄×C₁₁₃	—	4	1	no	—
85	C₂×C₄×C₁₁₃	—	4	1	no	—
86	C₂×C₄×C₁₁₃	—	8	2	no	—
87	C₂×C₄×C₁₁₃	—	8	2	no	—
88	C₂×C₄×C₁₁₃	—	8	2	no	—
89	C₂×C₄×C₁₁₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
90	C₂×C₄×C₁₁₃	C₂×C₄⋊C₄	1	1	yes	characteristic
91	C₂×C₄×C₁₁₃	#117	1	1	yes	characteristic
92	C₂×C₄×C₁₁₃	#121	1	1	yes	characteristic
93	C₂×C₄×C₁₁₃	C₂²⋊Q₈	1	1	yes	characteristic
94	C₂×C₄×C₁₁₃	C₂²⋊Q₈	1	1	yes	characteristic
95	C₂×C₄×C₁₁₃	#135	1	2	yes	—
96	C₂×C₄×C₁₁₃	#137	1	2	yes	—
97	C₈×C₁₁₃	—	2	1	no	—
98	C₈×C₁₁₃	—	2	1	no	—
99	C₈×C₁₁₃	—	2	1	no	—
100	C₈×C₁₁₃	—	2	1	no	—
101	C₈×C₁₁₃	—	4	2	no	—
102	C₈×C₁₁₃	—	8	2	no	—
103	C₂²×C₁₁₃	—	2	1	no	—
104	C₂²×C₁₁₃	—	2	1	no	—
105	C₂²×C₁₁₃	#380	1	1	yes	characteristic
106	C₂²×C₁₁₃	#464	1	2	yes	—
107	C₄×C₁₁₃	—	2	1	no	—
108	C₄×C₁₁₃	—	2	1	no	—
109	C₄×C₁₁₃	—	2	1	no	—
110	C₄×C₁₁₃	—	2	1	no	—
111	C₄×C₁₁₃	—	4	2	no	—
112	C₄×C₁₁₃	—	4	2	no	—
113	C₄×C₁₁₃	—	16	2	no	—
114	C₄×C₁₁₃	#416	1	1	yes	characteristic
115	C₄×C₁₁₃	#419	1	1	yes	characteristic
116	C₄×C₁₁₃	#479	1	2	yes	—
117	#8681	C₁₁₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
118	C₂×C₁₁₃	—	2	2	no	—
119	C₂×C₁₁₃	#1805	1	1	yes	characteristic
120	C₂×C₁₁₃	#1812	1	1	yes	characteristic
121	C₂×C₁₁₃	#1814	1	1	yes	characteristic
122	#2960	C₂×C₁₁₃	1	1	yes	characteristic
123	#1835	C₂×C₁₁₃	1	1	yes	characteristic
124	#1564	C₂×C₁₁₃	1	1	yes	characteristic
125	#1518	C₂×C₁₁₃	1	2	yes	—
126	#1286	C₂×C₁₁₃	1	2	yes	—
127	C₁₁₃	#8681	1	1	yes	characteristic, direct product factor, Sylow
128	#533	C₂²×C₁₁₃	1	1	yes	characteristic
129	#446	—	2	1	no	—
130	#446	C₄×C₁₁₃	1	2	yes	complemented
131	C₈×Q₈	—	2	1	no	—
132	C₈×Q₈	C₄×C₁₁₃	1	2	yes	complemented
133	#431	C₂²×C₁₁₃	1	2	yes	—
134	#424	—	2	1	no	—
135	#423	C₂²×C₁₁₃	1	1	yes	characteristic
136	#405	—	2	1	no	—
137	C₂×C₄×C₈	C₂²×C₁₁₃	1	1	yes	characteristic
138	#382	C₂²×C₁₁₃	1	2	yes	—
139	#340	—	2	1	no	—
140	#340	—	2	1	no	—
141	#335	—	2	1	no	—
142	#334	—	2	1	no	—
143	#328	—	2	4	no	—
144	C₂×M₁₆	—	2	1	no	—
145	C₂²×C₈	D₄×C₁₁₃	1	1	yes	characteristic
146	C₂²×C₈	D₄×C₁₁₃	1	1	yes	characteristic
147	C₄⋊Q₈	Q₈×C₁₁₃	1	1	yes	characteristic
148	C₄⋊Q₈	Q₈×C₁₁₃	1	1	yes	characteristic
149	#125	—	2	1	no	—
150	C₂²⋊Q₈	—	2	1	no	—
151	C₂²⋊Q₈	D₄×C₁₁₃	1	2	yes	—
152	C₄×Q₈	—	2	1	no	—
153	C₄×Q₈	C₂×C₄×C₁₁₃	1	2	yes	—
154	#117	—	2	1	no	—
155	C₂×C₄⋊C₄	—	2	2	no	—
156	C₂×C₄²	C₂³×C₁₁₃	1	1	yes	characteristic
157	C₄⋊C₈	—	2	2	no	—
158	C₄⋊C₈	—	2	2	no	—
159	C₄⋊C₈	—	2	1	no	—
160	C₄⋊C₈	—	2	2	no	—
161	C₄⋊C₈	C₂×C₄×C₁₁₃	1	2	yes	—
162	C₂²⋊C₈	—	2	2	no	—
163	#97	—	2	2	no	—
164	C₄×C₈	—	2	2	no	—
165	C₄×C₈	—	2	2	no	—
166	C₄×C₈	C₂×C₄×C₁₁₃	1	1	yes	characteristic
167	C₄×C₈	C₂×C₄×C₁₁₃	1	1	yes	characteristic
168	C₄×C₈	D₄×C₁₁₃	1	1	yes	characteristic
169	C₄×C₈	D₄×C₁₁₃	1	1	yes	characteristic
170	#95	—	2	2	no	—
171	#95	—	2	2	no	—
172	C₂×Q₈	—	2	1	no	—
173	C₂×Q₈	C₄⋊C₄×C₁₁₃	1	2	yes	—
174	C₂²×C₄	—	4	2	no	—
175	C₂²×C₄	C₂×D₄×C₁₁₃	1	1	yes	characteristic
176	C₂²×C₄	C₂×D₄×C₁₁₃	1	1	yes	characteristic
177	C₂²×C₄	#41×C₁₁₃	1	1	yes	characteristic
178	M₁₆	—	4	2	no	—
179	C₂×C₈	—	2	2	no	—
180	C₂×C₈	—	2	2	no	—
181	C₂×C₈	—	2	2	no	—
182	C₂×C₈	—	4	2	no	—
183	C₂×C₈	C₂²⋊C₄×C₁₁₃	1	1	yes	characteristic
184	C₂×C₈	C₂²⋊C₄×C₁₁₃	1	1	yes	characteristic
185	C₂×C₈	C₂²⋊C₄×C₁₁₃	1	1	yes	characteristic
186	C₂×C₈	C₂²⋊C₄×C₁₁₃	1	1	yes	characteristic
187	C₄⋊C₄	—	2	2	no	—
188	C₄⋊C₄	—	2	2	no	—
189	C₄⋊C₄	—	2	1	no	—
190	C₄⋊C₄	—	2	1	no	—
191	C₄⋊C₄	—	2	1	no	—
192	C₄⋊C₄	—	4	2	no	—
193	C₄⋊C₄	—	4	2	no	—
194	C₄⋊C₄	C₄⋊C₄×C₁₁₃	1	2	yes	—
195	C₂²⋊C₄	—	2	2	no	—
196	C₄²	—	2	2	no	—
197	C₄²	C₂²×C₄×C₁₁₃	1	1	yes	characteristic
198	C₄²	C₂×D₄×C₁₁₃	1	1	yes	characteristic
199	C₄²	C₂×Q₈×C₁₁₃	1	1	yes	characteristic
200	C₄²	#41×C₁₁₃	1	1	yes	characteristic
201	C₂³	#121×C₁₁₃	1	1	yes	characteristic
202	Q₈	—	4	2	no	—
203	Q₈	—	4	2	no	—
204	C₂×C₄	—	2	2	no	—
205	C₂×C₄	—	2	2	no	—
206	C₂×C₄	—	4	1	no	—
207	C₂×C₄	—	4	1	no	—
208	C₂×C₄	—	8	2	no	—
209	C₂×C₄	—	8	2	no	—
210	C₂×C₄	—	8	2	no	—
211	C₂×C₄	C₂×C₂²⋊C₄×C₁₁₃	1	1	yes	characteristic
212	C₂×C₄	C₂×C₄⋊C₄×C₁₁₃	1	1	yes	characteristic
213	C₂×C₄	#117×C₁₁₃	1	1	yes	characteristic
214	C₂×C₄	#121×C₁₁₃	1	1	yes	characteristic
215	C₂×C₄	C₂²⋊Q₈×C₁₁₃	1	1	yes	characteristic
216	C₂×C₄	C₂²⋊Q₈×C₁₁₃	1	1	yes	characteristic
217	C₂×C₄	#135×C₁₁₃	1	2	yes	—
218	C₂×C₄	#137×C₁₁₃	1	2	yes	—
219	C₈	—	2	1	no	—
220	C₈	—	2	1	no	—
221	C₈	—	2	1	no	—
222	C₈	—	2	1	no	—
223	C₈	—	4	2	no	—
224	C₈	—	8	2	no	—
225	C₂²	—	2	1	no	—
226	C₂²	—	2	1	no	—
227	C₂²	#380×C₁₁₃	1	1	yes	characteristic
228	C₂²	#464×C₁₁₃	1	2	yes	—
229	C₄	—	2	1	no	—
230	C₄	—	2	1	no	—
231	C₄	—	2	1	no	—
232	C₄	—	2	1	no	—
233	C₄	—	4	2	no	—
234	C₄	—	4	2	no	—
235	C₄	—	16	2	no	—
236	C₄	#416×C₁₁₃	1	1	yes	characteristic
237	C₄	#419×C₁₁₃	1	1	yes	characteristic
238	C₄	#479×C₁₁₃	1	2	yes	—
239	C₂	—	2	2	no	—
240	C₂	C₁₁₃×#1805	1	1	yes	characteristic
241	C₂	C₁₁₃×#1812	1	1	yes	characteristic
242	C₂	C₁₁₃×#1814	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₁₁₃	#380
commutator	C₄²	C₂²×C₄×C₁₁₃
Frattini, cosocle	C₂×C₄²	C₂³×C₁₁₃
prime radical	C₂³×C₁₁₃	#121
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₁₁₃ ◁ C₂×C₄²×C₁₁₃ ◁ G

Complements

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

N	H ≅ G/N	type
8. #446×C₁₁₃	C₄	semidirect
10. C₈×Q₈×C₁₁₃	C₄	semidirect
117. #8681	C₁₁₃	direct (N × H)
127. C₁₁₃	#8681	direct (N × H)
130. #446	C₄×C₁₁₃	semidirect
132. C₈×Q₈	C₄×C₁₁₃	semidirect

Maximal subgroups

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

subgroup	index
C₁₁₃×#2960	2
C₁₁₃×#1835	2
C₁₁₃×#1564	2
C₁₁₃×#1518	2
C₁₁₃×#1286	2
#8681	113

Subgroups — C113×#8681 #43276701

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₁₃×#8681 #43276701