Subgroups — C₁₁₃×#8689 #43276709

Order 28928.

346 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×#2960	C₂	1	1	yes	maximal, characteristic
2	C₁₁₃×#1837	C₂	1	1	yes	maximal, characteristic
3	C₂×#335×C₁₁₃	C₂	1	1	yes	maximal, characteristic
4	C₁₁₃×#1550	C₂	1	2	yes	maximal
5	C₁₁₃×#1292	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	—
9	C₈×Q₈×C₁₁₃	—	2	1	no	—
10	C₈×Q₈×C₁₁₃	C₄	1	2	yes	—
11	#424×C₁₁₃	—	2	1	no	—
12	#423×C₁₁₃	C₂²	1	2	yes	—
13	#423×C₁₁₃	C₂²	1	1	yes	characteristic
14	C₂×C₄⋊C₈×C₁₁₃	C₂²	1	2	yes	—
15	#405×C₁₁₃	—	2	1	no	—
16	C₂×C₄×C₈×C₁₁₃	C₂²	1	1	yes	characteristic
17	#341×C₁₁₃	—	2	1	no	—
18	#341×C₁₁₃	—	2	1	no	—
19	#335×C₁₁₃	—	2	1	no	—
20	#335×C₁₁₃	—	2	1	no	—
21	#333×C₁₁₃	—	2	2	no	—
22	#332×C₁₁₃	—	2	2	no	—
23	C₂×M₁₆×C₁₁₃	—	2	1	no	—
24	C₂×M₁₆×C₁₁₃	—	2	2	no	—
25	C₂²×C₈×C₁₁₃	—	2	2	no	—
26	C₂²×C₈×C₁₁₃	D₄	1	1	yes	characteristic
27	C₂²×C₈×C₁₁₃	D₄	1	1	yes	characteristic
28	C₄⋊Q₈×C₁₁₃	D₄	1	1	yes	characteristic
29	C₄⋊Q₈×C₁₁₃	D₄	1	1	yes	characteristic
30	#125×C₁₁₃	—	2	1	no	—
31	C₂²⋊Q₈×C₁₁₃	—	2	1	no	—
32	C₂²⋊Q₈×C₁₁₃	Q₈	1	2	yes	—
33	C₄×Q₈×C₁₁₃	—	2	1	no	—
34	C₄×Q₈×C₁₁₃	C₂×C₄	1	2	yes	—
35	#117×C₁₁₃	—	2	1	no	—
36	C₂×C₄²×C₁₁₃	C₂³	1	1	yes	characteristic
37	C₄⋊C₈×C₁₁₃	—	2	2	no	—
38	C₄⋊C₈×C₁₁₃	—	2	2	no	—
39	C₄⋊C₈×C₁₁₃	—	2	1	no	—
40	C₄⋊C₈×C₁₁₃	—	2	2	no	—
41	C₄⋊C₈×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	2	yes	—
45	#97×C₁₁₃	—	2	2	no	—
46	C₄×C₈×C₁₁₃	—	2	2	no	—
47	C₄×C₈×C₁₁₃	C₂×C₄	1	1	yes	characteristic
48	C₄×C₈×C₁₁₃	C₂×C₄	1	1	yes	characteristic
49	C₄×C₈×C₁₁₃	D₄	1	1	yes	characteristic
50	C₄×C₈×C₁₁₃	D₄	1	1	yes	characteristic
51	C₂×Q₈×C₁₁₃	—	2	1	no	—
52	C₂×Q₈×C₁₁₃	C₄⋊C₄	1	2	yes	—
53	C₂²×C₄×C₁₁₃	C₂×D₄	1	1	yes	characteristic
54	C₂²×C₄×C₁₁₃	C₂×Q₈	1	1	yes	characteristic
55	C₂²×C₄×C₁₁₃	#41	1	1	yes	characteristic
56	M₁₆×C₁₁₃	—	4	2	no	—
57	M₁₆×C₁₁₃	—	4	2	no	—
58	M₁₆×C₁₁₃	—	4	2	no	—
59	C₂×C₈×C₁₁₃	—	2	2	no	—
60	C₂×C₈×C₁₁₃	—	2	2	no	—
61	C₂×C₈×C₁₁₃	—	2	2	no	—
62	C₂×C₈×C₁₁₃	—	4	2	no	—
63	C₂×C₈×C₁₁₃	—	4	2	no	—
64	C₂×C₈×C₁₁₃	—	8	2	no	—
65	C₂×C₈×C₁₁₃	C₂²⋊C₄	1	1	yes	characteristic
66	C₂×C₈×C₁₁₃	C₂²⋊C₄	1	1	yes	characteristic
67	C₂×C₈×C₁₁₃	C₂²⋊C₄	1	1	yes	characteristic
68	C₂×C₈×C₁₁₃	C₂²⋊C₄	1	1	yes	characteristic
69	C₄⋊C₄×C₁₁₃	—	2	2	no	—
70	C₄⋊C₄×C₁₁₃	—	2	1	no	—
71	C₄⋊C₄×C₁₁₃	—	2	1	no	—
72	C₄⋊C₄×C₁₁₃	—	2	1	no	—
73	C₄⋊C₄×C₁₁₃	—	2	2	no	—
74	C₄⋊C₄×C₁₁₃	C₄⋊C₄	1	2	yes	—
75	C₂²⋊C₄×C₁₁₃	—	2	2	no	—
76	C₄²×C₁₁₃	—	2	2	no	—
77	C₄²×C₁₁₃	C₂²×C₄	1	1	yes	characteristic
78	C₄²×C₁₁₃	C₂×D₄	1	1	yes	characteristic
79	C₄²×C₁₁₃	C₂×D₄	1	1	yes	characteristic
80	C₄²×C₁₁₃	#41	1	1	yes	characteristic
81	C₂³×C₁₁₃	C₂²⋊Q₈	1	1	yes	characteristic
82	Q₈×C₁₁₃	—	4	2	no	—
83	Q₈×C₁₁₃	—	4	1	no	—
84	Q₈×C₁₁₃	—	4	1	no	—
85	C₂×C₄×C₁₁₃	—	2	1	no	—
86	C₂×C₄×C₁₁₃	—	2	1	no	—
87	C₂×C₄×C₁₁₃	—	2	2	no	—
88	C₂×C₄×C₁₁₃	—	2	2	no	—
89	C₂×C₄×C₁₁₃	—	4	1	no	—
90	C₂×C₄×C₁₁₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
91	C₂×C₄×C₁₁₃	C₂×C₄⋊C₄	1	1	yes	characteristic
92	C₂×C₄×C₁₁₃	#117	1	1	yes	characteristic
93	C₂×C₄×C₁₁₃	#121	1	1	yes	characteristic
94	C₂×C₄×C₁₁₃	#121	1	1	yes	characteristic
95	C₂×C₄×C₁₁₃	C₂²⋊Q₈	1	1	yes	characteristic
96	C₂×C₄×C₁₁₃	#135	1	2	yes	—
97	C₂×C₄×C₁₁₃	#137	1	2	yes	—
98	C₈×C₁₁₃	—	2	1	no	—
99	C₈×C₁₁₃	—	2	1	no	—
100	C₈×C₁₁₃	—	2	1	no	—
101	C₈×C₁₁₃	—	2	1	no	—
102	C₈×C₁₁₃	—	4	2	no	—
103	C₈×C₁₁₃	—	8	2	no	—
104	C₈×C₁₁₃	—	8	2	no	—
105	C₂²×C₁₁₃	—	2	1	no	—
106	C₂²×C₁₁₃	—	2	1	no	—
107	C₂²×C₁₁₃	#380	1	1	yes	characteristic
108	C₂²×C₁₁₃	#479	1	2	yes	—
109	C₄×C₁₁₃	—	2	1	no	—
110	C₄×C₁₁₃	—	2	1	no	—
111	C₄×C₁₁₃	—	2	1	no	—
112	C₄×C₁₁₃	—	4	2	no	—
113	C₄×C₁₁₃	—	4	2	no	—
114	C₄×C₁₁₃	#416	1	1	yes	characteristic
115	C₄×C₁₁₃	#419	1	1	yes	characteristic
116	C₄×C₁₁₃	#464	1	2	yes	—
117	C₄×C₁₁₃	#472	1	1	yes	characteristic
118	C₄×C₁₁₃	#473	1	1	yes	characteristic
119	#8689	C₁₁₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
120	C₂×C₁₁₃	—	2	2	no	—
121	C₂×C₁₁₃	#1805	1	1	yes	characteristic
122	C₂×C₁₁₃	#1813	1	1	yes	characteristic
123	C₂×C₁₁₃	#1816	1	1	yes	characteristic
124	#2960	C₂×C₁₁₃	1	1	yes	characteristic
125	#1837	C₂×C₁₁₃	1	1	yes	characteristic
126	C₂×#335	C₂×C₁₁₃	1	1	yes	characteristic
127	#1550	C₂×C₁₁₃	1	2	yes	—
128	#1292	C₂×C₁₁₃	1	2	yes	—
129	C₁₁₃	#8689	1	1	yes	characteristic, direct product factor, Sylow
130	#533	C₂²×C₁₁₃	1	1	yes	characteristic
131	#446	—	2	1	no	—
132	#446	C₄×C₁₁₃	1	2	yes	—
133	C₈×Q₈	—	2	1	no	—
134	C₈×Q₈	C₄×C₁₁₃	1	2	yes	—
135	#424	—	2	1	no	—
136	#423	C₂²×C₁₁₃	1	2	yes	—
137	#423	C₂²×C₁₁₃	1	1	yes	characteristic
138	C₂×C₄⋊C₈	C₂²×C₁₁₃	1	2	yes	—
139	#405	—	2	1	no	—
140	C₂×C₄×C₈	C₂²×C₁₁₃	1	1	yes	characteristic
141	#341	—	2	1	no	—
142	#341	—	2	1	no	—
143	#335	—	2	1	no	—
144	#335	—	2	1	no	—
145	#333	—	2	2	no	—
146	#332	—	2	2	no	—
147	C₂×M₁₆	—	2	1	no	—
148	C₂×M₁₆	—	2	2	no	—
149	C₂²×C₈	—	2	2	no	—
150	C₂²×C₈	D₄×C₁₁₃	1	1	yes	characteristic
151	C₂²×C₈	D₄×C₁₁₃	1	1	yes	characteristic
152	C₄⋊Q₈	D₄×C₁₁₃	1	1	yes	characteristic
153	C₄⋊Q₈	D₄×C₁₁₃	1	1	yes	characteristic
154	#125	—	2	1	no	—
155	C₂²⋊Q₈	—	2	1	no	—
156	C₂²⋊Q₈	Q₈×C₁₁₃	1	2	yes	—
157	C₄×Q₈	—	2	1	no	—
158	C₄×Q₈	C₂×C₄×C₁₁₃	1	2	yes	—
159	#117	—	2	1	no	—
160	C₂×C₄²	C₂³×C₁₁₃	1	1	yes	characteristic
161	C₄⋊C₈	—	2	2	no	—
162	C₄⋊C₈	—	2	2	no	—
163	C₄⋊C₈	—	2	1	no	—
164	C₄⋊C₈	—	2	2	no	—
165	C₄⋊C₈	—	2	2	no	—
166	C₄⋊C₈	—	2	2	no	—
167	C₄⋊C₈	—	2	2	no	—
168	C₄⋊C₈	C₂×C₄×C₁₁₃	1	2	yes	—
169	#97	—	2	2	no	—
170	C₄×C₈	—	2	2	no	—
171	C₄×C₈	C₂×C₄×C₁₁₃	1	1	yes	characteristic
172	C₄×C₈	C₂×C₄×C₁₁₃	1	1	yes	characteristic
173	C₄×C₈	D₄×C₁₁₃	1	1	yes	characteristic
174	C₄×C₈	D₄×C₁₁₃	1	1	yes	characteristic
175	C₂×Q₈	—	2	1	no	—
176	C₂×Q₈	C₄⋊C₄×C₁₁₃	1	2	yes	—
177	C₂²×C₄	C₂×D₄×C₁₁₃	1	1	yes	characteristic
178	C₂²×C₄	C₂×Q₈×C₁₁₃	1	1	yes	characteristic
179	C₂²×C₄	#41×C₁₁₃	1	1	yes	characteristic
180	M₁₆	—	4	2	no	—
181	M₁₆	—	4	2	no	—
182	M₁₆	—	4	2	no	—
183	C₂×C₈	—	2	2	no	—
184	C₂×C₈	—	2	2	no	—
185	C₂×C₈	—	2	2	no	—
186	C₂×C₈	—	4	2	no	—
187	C₂×C₈	—	4	2	no	—
188	C₂×C₈	—	8	2	no	—
189	C₂×C₈	C₂²⋊C₄×C₁₁₃	1	1	yes	characteristic
190	C₂×C₈	C₂²⋊C₄×C₁₁₃	1	1	yes	characteristic
191	C₂×C₈	C₂²⋊C₄×C₁₁₃	1	1	yes	characteristic
192	C₂×C₈	C₂²⋊C₄×C₁₁₃	1	1	yes	characteristic
193	C₄⋊C₄	—	2	2	no	—
194	C₄⋊C₄	—	2	1	no	—
195	C₄⋊C₄	—	2	1	no	—
196	C₄⋊C₄	—	2	1	no	—
197	C₄⋊C₄	—	2	2	no	—
198	C₄⋊C₄	C₄⋊C₄×C₁₁₃	1	2	yes	—
199	C₂²⋊C₄	—	2	2	no	—
200	C₄²	—	2	2	no	—
201	C₄²	C₂²×C₄×C₁₁₃	1	1	yes	characteristic
202	C₄²	C₂×D₄×C₁₁₃	1	1	yes	characteristic
203	C₄²	C₂×D₄×C₁₁₃	1	1	yes	characteristic
204	C₄²	#41×C₁₁₃	1	1	yes	characteristic
205	C₂³	C₂²⋊Q₈×C₁₁₃	1	1	yes	characteristic
206	Q₈	—	4	2	no	—
207	Q₈	—	4	1	no	—
208	Q₈	—	4	1	no	—
209	C₂×C₄	—	2	1	no	—
210	C₂×C₄	—	2	1	no	—
211	C₂×C₄	—	2	2	no	—
212	C₂×C₄	—	2	2	no	—
213	C₂×C₄	—	4	1	no	—
214	C₂×C₄	C₂×C₂²⋊C₄×C₁₁₃	1	1	yes	characteristic
215	C₂×C₄	C₂×C₄⋊C₄×C₁₁₃	1	1	yes	characteristic
216	C₂×C₄	#117×C₁₁₃	1	1	yes	characteristic
217	C₂×C₄	#121×C₁₁₃	1	1	yes	characteristic
218	C₂×C₄	#121×C₁₁₃	1	1	yes	characteristic
219	C₂×C₄	C₂²⋊Q₈×C₁₁₃	1	1	yes	characteristic
220	C₂×C₄	#135×C₁₁₃	1	2	yes	—
221	C₂×C₄	#137×C₁₁₃	1	2	yes	—
222	C₈	—	2	1	no	—
223	C₈	—	2	1	no	—
224	C₈	—	2	1	no	—
225	C₈	—	2	1	no	—
226	C₈	—	4	2	no	—
227	C₈	—	8	2	no	—
228	C₈	—	8	2	no	—
229	C₂²	—	2	1	no	—
230	C₂²	—	2	1	no	—
231	C₂²	#380×C₁₁₃	1	1	yes	characteristic
232	C₂²	#479×C₁₁₃	1	2	yes	—
233	C₄	—	2	1	no	—
234	C₄	—	2	1	no	—
235	C₄	—	2	1	no	—
236	C₄	—	4	2	no	—
237	C₄	—	4	2	no	—
238	C₄	#416×C₁₁₃	1	1	yes	characteristic
239	C₄	#419×C₁₁₃	1	1	yes	characteristic
240	C₄	#464×C₁₁₃	1	2	yes	—
241	C₄	#472×C₁₁₃	1	1	yes	characteristic
242	C₄	#473×C₁₁₃	1	1	yes	characteristic
243	C₂	—	2	2	no	—
244	C₂	C₁₁₃×#1805	1	1	yes	characteristic
245	C₂	C₁₁₃×#1813	1	1	yes	characteristic
246	C₂	C₁₁₃×#1816	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	C₂²⋊Q₈
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
119. #8689	C₁₁₃	direct (N × H)
129. C₁₁₃	#8689	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×#2960	2
C₁₁₃×#1837	2
C₂×#335×C₁₁₃	2
C₁₁₃×#1550	2
C₁₁₃×#1292	2
#8689	113

Subgroups — C113×#8689 #43276709

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₁₃×#8689 #43276709