Subgroups — C₁₁₃×#9701 #43277721

Order 28928.

486 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×#2862	C₂	1	1	yes	maximal, characteristic
2	C₁₁₃×#1842	C₂	1	1	yes	maximal, characteristic
3	C₁₁₃×#1767	C₂	1	1	yes	maximal, characteristic
4	C₂×#333×C₁₁₃	C₂	1	2	yes	maximal
5	C₂×#332×C₁₁₃	C₂	1	2	yes	maximal
6	C₂×C₄⋊Q₈×C₁₁₃	C₂²	1	1	yes	characteristic
7	C₂×C₄⋊Q₈×C₁₁₃	C₂²	1	1	yes	characteristic
8	C₂×C₄⋊C₈×C₁₁₃	—	2	2	no	—
9	C₂×C₄⋊C₈×C₁₁₃	C₂²	1	2	yes	—
10	C₂×C₄⋊C₈×C₁₁₃	C₂²	1	2	yes	—
11	C₂×#97×C₁₁₃	—	2	1	no	—
12	#391×C₁₁₃	—	2	2	no	—
13	#391×C₁₁₃	C₄	1	4	yes	—
14	#382×C₁₁₃	—	2	1	no	—
15	#382×C₁₁₃	—	2	1	no	—
16	C₄³×C₁₁₃	C₂²	1	1	yes	characteristic
17	#336×C₁₁₃	—	2	1	no	—
18	#333×C₁₁₃	—	2	4	no	—
19	#332×C₁₁₃	—	2	4	no	—
20	C₂²×Q₈×C₁₁₃	—	2	1	no	—
21	C₂²×Q₈×C₁₁₃	C₂×C₄	1	2	yes	—
22	C₂²×C₈×C₁₁₃	—	2	2	no	—
23	C₂²×C₈×C₁₁₃	—	2	2	no	—
24	C₄⋊Q₈×C₁₁₃	—	2	2	no	—
25	C₄⋊Q₈×C₁₁₃	—	2	2	no	—
26	C₄⋊Q₈×C₁₁₃	—	2	2	no	—
27	C₄⋊Q₈×C₁₁₃	—	2	2	no	—
28	C₂×C₄⋊C₄×C₁₁₃	—	2	1	no	—
29	C₂×C₄⋊C₄×C₁₁₃	—	2	1	no	—
30	C₂×C₄⋊C₄×C₁₁₃	—	2	1	no	—
31	C₂×C₄⋊C₄×C₁₁₃	C₂×C₄	1	2	yes	—
32	C₂×C₄²×C₁₁₃	C₂×C₄	1	2	yes	—
33	C₂×C₄²×C₁₁₃	D₄	1	1	yes	characteristic
34	C₂×C₄²×C₁₁₃	D₄	1	2	yes	—
35	C₂×C₄²×C₁₁₃	D₄	1	1	yes	characteristic
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	2	no	—
40	C₄⋊C₈×C₁₁₃	—	4	4	no	—
41	C₄⋊C₈×C₁₁₃	D₄	1	4	yes	—
42	#97×C₁₁₃	—	4	2	no	—
43	#95×C₁₁₃	—	2	4	no	—
44	#95×C₁₁₃	—	2	4	no	—
45	C₂×Q₈×C₁₁₃	—	2	4	no	—
46	C₂×Q₈×C₁₁₃	—	4	4	no	—
47	C₂×Q₈×C₁₁₃	—	4	1	no	—
48	C₂×Q₈×C₁₁₃	—	4	1	no	—
49	C₂×Q₈×C₁₁₃	—	4	2	no	—
50	C₂×Q₈×C₁₁₃	—	4	2	no	—
51	C₂²×C₄×C₁₁₃	—	2	2	no	—
52	C₂²×C₄×C₁₁₃	—	2	2	no	—
53	C₂²×C₄×C₁₁₃	C₂²⋊C₄	1	2	yes	—
54	C₂²×C₄×C₁₁₃	C₂²⋊C₄	1	2	yes	—
55	C₂²×C₄×C₁₁₃	C₂²×C₄	1	1	yes	characteristic
56	C₂²×C₄×C₁₁₃	C₂×D₄	1	1	yes	characteristic
57	C₂²×C₄×C₁₁₃	C₂×D₄	1	1	yes	characteristic
58	C₂×C₈×C₁₁₃	—	2	4	no	—
59	C₂×C₈×C₁₁₃	—	4	2	no	—
60	C₂×C₈×C₁₁₃	—	4	2	no	—
61	C₂×C₈×C₁₁₃	—	4	2	no	—
62	C₂×C₈×C₁₁₃	—	4	4	no	—
63	C₄⋊C₄×C₁₁₃	—	2	4	no	—
64	C₄⋊C₄×C₁₁₃	—	4	2	no	—
65	C₄⋊C₄×C₁₁₃	—	4	2	no	—
66	C₄⋊C₄×C₁₁₃	—	4	1	no	—
67	C₄⋊C₄×C₁₁₃	—	4	1	no	—
68	C₄²×C₁₁₃	—	2	2	no	—
69	C₄²×C₁₁₃	—	2	2	no	—
70	C₄²×C₁₁₃	—	2	2	no	—
71	C₄²×C₁₁₃	—	4	2	no	—
72	C₄²×C₁₁₃	SD₁₆	1	2	yes	—
73	C₄²×C₁₁₃	Q₁₆	1	2	yes	—
74	C₄²×C₁₁₃	C₂×D₄	1	2	yes	—
75	C₄²×C₁₁₃	#41	1	2	yes	—
76	C₂³×C₁₁₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
77	Q₈×C₁₁₃	—	8	4	no	—
78	Q₈×C₁₁₃	—	8	2	no	—
79	Q₈×C₁₁₃	—	8	1	no	—
80	Q₈×C₁₁₃	—	8	1	no	—
81	C₂×C₄×C₁₁₃	—	2	2	no	—
82	C₂×C₄×C₁₁₃	—	2	2	no	—
83	C₂×C₄×C₁₁₃	—	2	2	no	—
84	C₂×C₄×C₁₁₃	—	2	2	no	—
85	C₂×C₄×C₁₁₃	—	2	2	no	—
86	C₂×C₄×C₁₁₃	—	2	2	no	—
87	C₂×C₄×C₁₁₃	—	4	2	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	#103	1	4	yes	—
92	C₂×C₄×C₁₁₃	C₄×D₄	1	2	yes	—
93	C₂×C₄×C₁₁₃	#121	1	2	yes	—
94	C₂×C₄×C₁₁₃	#124	1	1	yes	characteristic
95	C₂×C₄×C₁₁₃	#127	1	1	yes	characteristic
96	C₂×C₄×C₁₁₃	C₂×SD₁₆	1	1	yes	characteristic
97	C₂×C₄×C₁₁₃	C₂×Q₁₆	1	1	yes	characteristic
98	C₂×C₄×C₁₁₃	AGL(1,ℤ/8)	1	2	yes	—
99	C₂×C₄×C₁₁₃	#137	1	2	yes	—
100	C₂×C₄×C₁₁₃	#137	1	2	yes	—
101	C₈×C₁₁₃	—	4	4	no	—
102	C₈×C₁₁₃	—	8	2	no	—
103	C₂²×C₁₁₃	#390	1	1	yes	characteristic
104	C₂²×C₁₁₃	#413	1	2	yes	—
105	C₂²×C₁₁₃	C₂×#103	1	1	yes	characteristic
106	C₂²×C₁₁₃	#417	1	2	yes	—
107	C₂²×C₁₁₃	#419	1	1	yes	characteristic
108	C₄×C₁₁₃	—	2	4	no	—
109	C₄×C₁₁₃	—	2	2	no	—
110	C₄×C₁₁₃	—	4	2	no	—
111	C₄×C₁₁₃	—	8	2	no	—
112	C₄×C₁₁₃	—	8	2	no	—
113	C₄×C₁₁₃	#461	1	2	yes	—
114	C₄×C₁₁₃	#462	1	2	yes	—
115	C₄×C₁₁₃	#489	1	2	yes	—
116	C₄×C₁₁₃	#496	1	1	yes	characteristic
117	C₄×C₁₁₃	#497	1	1	yes	characteristic
118	#9701	C₁₁₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
119	C₂×C₁₁₃	#1550	1	2	yes	—
120	C₂×C₁₁₃	#1559	1	2	yes	—
121	C₂×C₁₁₃	#1959	1	1	yes	characteristic
122	C₂×C₁₁₃	#1967	1	1	yes	characteristic
123	C₂×C₁₁₃	#1981	1	1	yes	characteristic
124	#2862	C₂×C₁₁₃	1	1	yes	characteristic
125	#1842	C₂×C₁₁₃	1	1	yes	characteristic
126	#1767	C₂×C₁₁₃	1	1	yes	characteristic
127	C₂×#333	C₂×C₁₁₃	1	2	yes	—
128	C₂×#332	C₂×C₁₁₃	1	2	yes	—
129	C₁₁₃	#9701	1	1	yes	characteristic, direct product factor, Sylow
130	C₂×C₄⋊Q₈	C₂²×C₁₁₃	1	1	yes	characteristic
131	C₂×C₄⋊Q₈	C₂²×C₁₁₃	1	1	yes	characteristic
132	C₂×C₄⋊C₈	—	2	2	no	—
133	C₂×C₄⋊C₈	C₂²×C₁₁₃	1	2	yes	—
134	C₂×C₄⋊C₈	C₂²×C₁₁₃	1	2	yes	—
135	C₂×#97	—	2	1	no	—
136	#391	—	2	2	no	—
137	#391	C₄×C₁₁₃	1	4	yes	—
138	#382	—	2	1	no	—
139	#382	—	2	1	no	—
140	C₄³	C₂²×C₁₁₃	1	1	yes	characteristic
141	#336	—	2	1	no	—
142	#333	—	2	4	no	—
143	#332	—	2	4	no	—
144	C₂²×Q₈	—	2	1	no	—
145	C₂²×Q₈	C₂×C₄×C₁₁₃	1	2	yes	—
146	C₂²×C₈	—	2	2	no	—
147	C₂²×C₈	—	2	2	no	—
148	C₄⋊Q₈	—	2	2	no	—
149	C₄⋊Q₈	—	2	2	no	—
150	C₄⋊Q₈	—	2	2	no	—
151	C₄⋊Q₈	—	2	2	no	—
152	C₂×C₄⋊C₄	—	2	1	no	—
153	C₂×C₄⋊C₄	—	2	1	no	—
154	C₂×C₄⋊C₄	—	2	1	no	—
155	C₂×C₄⋊C₄	C₂×C₄×C₁₁₃	1	2	yes	—
156	C₂×C₄²	C₂×C₄×C₁₁₃	1	2	yes	—
157	C₂×C₄²	D₄×C₁₁₃	1	1	yes	characteristic
158	C₂×C₄²	D₄×C₁₁₃	1	2	yes	—
159	C₂×C₄²	D₄×C₁₁₃	1	1	yes	characteristic
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	2	no	—
164	C₄⋊C₈	—	4	4	no	—
165	C₄⋊C₈	D₄×C₁₁₃	1	4	yes	—
166	#97	—	4	2	no	—
167	#95	—	2	4	no	—
168	#95	—	2	4	no	—
169	C₂×Q₈	—	2	4	no	—
170	C₂×Q₈	—	4	4	no	—
171	C₂×Q₈	—	4	1	no	—
172	C₂×Q₈	—	4	1	no	—
173	C₂×Q₈	—	4	2	no	—
174	C₂×Q₈	—	4	2	no	—
175	C₂²×C₄	—	2	2	no	—
176	C₂²×C₄	—	2	2	no	—
177	C₂²×C₄	C₂²⋊C₄×C₁₁₃	1	2	yes	—
178	C₂²×C₄	C₂²⋊C₄×C₁₁₃	1	2	yes	—
179	C₂²×C₄	C₂²×C₄×C₁₁₃	1	1	yes	characteristic
180	C₂²×C₄	C₂×D₄×C₁₁₃	1	1	yes	characteristic
181	C₂²×C₄	C₂×D₄×C₁₁₃	1	1	yes	characteristic
182	C₂×C₈	—	2	4	no	—
183	C₂×C₈	—	4	2	no	—
184	C₂×C₈	—	4	2	no	—
185	C₂×C₈	—	4	2	no	—
186	C₂×C₈	—	4	4	no	—
187	C₄⋊C₄	—	2	4	no	—
188	C₄⋊C₄	—	4	2	no	—
189	C₄⋊C₄	—	4	2	no	—
190	C₄⋊C₄	—	4	1	no	—
191	C₄⋊C₄	—	4	1	no	—
192	C₄²	—	2	2	no	—
193	C₄²	—	2	2	no	—
194	C₄²	—	2	2	no	—
195	C₄²	—	4	2	no	—
196	C₄²	SD₁₆×C₁₁₃	1	2	yes	—
197	C₄²	Q₁₆×C₁₁₃	1	2	yes	—
198	C₄²	C₂×D₄×C₁₁₃	1	2	yes	—
199	C₄²	#41×C₁₁₃	1	2	yes	—
200	C₂³	C₂×C₂²⋊C₄×C₁₁₃	1	1	yes	characteristic
201	Q₈	—	8	4	no	—
202	Q₈	—	8	2	no	—
203	Q₈	—	8	1	no	—
204	Q₈	—	8	1	no	—
205	C₂×C₄	—	2	2	no	—
206	C₂×C₄	—	2	2	no	—
207	C₂×C₄	—	2	2	no	—
208	C₂×C₄	—	2	2	no	—
209	C₂×C₄	—	2	2	no	—
210	C₂×C₄	—	2	2	no	—
211	C₂×C₄	—	4	2	no	—
212	C₂×C₄	—	4	2	no	—
213	C₂×C₄	—	4	4	no	—
214	C₂×C₄	—	4	4	no	—
215	C₂×C₄	#103×C₁₁₃	1	4	yes	—
216	C₂×C₄	C₄×D₄×C₁₁₃	1	2	yes	—
217	C₂×C₄	#121×C₁₁₃	1	2	yes	—
218	C₂×C₄	#124×C₁₁₃	1	1	yes	characteristic
219	C₂×C₄	#127×C₁₁₃	1	1	yes	characteristic
220	C₂×C₄	C₂×SD₁₆×C₁₁₃	1	1	yes	characteristic
221	C₂×C₄	C₂×Q₁₆×C₁₁₃	1	1	yes	characteristic
222	C₂×C₄	AGL(1,ℤ/8)×C₁₁₃	1	2	yes	—
223	C₂×C₄	#137×C₁₁₃	1	2	yes	—
224	C₂×C₄	#137×C₁₁₃	1	2	yes	—
225	C₈	—	4	4	no	—
226	C₈	—	8	2	no	—
227	C₂²	#390×C₁₁₃	1	1	yes	characteristic
228	C₂²	#413×C₁₁₃	1	2	yes	—
229	C₂²	C₂×#103×C₁₁₃	1	1	yes	characteristic
230	C₂²	#417×C₁₁₃	1	2	yes	—
231	C₂²	#419×C₁₁₃	1	1	yes	characteristic
232	C₄	—	2	4	no	—
233	C₄	—	2	2	no	—
234	C₄	—	4	2	no	—
235	C₄	—	8	2	no	—
236	C₄	—	8	2	no	—
237	C₄	#461×C₁₁₃	1	2	yes	—
238	C₄	#462×C₁₁₃	1	2	yes	—
239	C₄	#489×C₁₁₃	1	2	yes	—
240	C₄	#496×C₁₁₃	1	1	yes	characteristic
241	C₄	#497×C₁₁₃	1	1	yes	characteristic
242	C₂	C₁₁₃×#1550	1	2	yes	—
243	C₂	C₁₁₃×#1559	1	2	yes	—
244	C₂	C₁₁₃×#1959	1	1	yes	characteristic
245	C₂	C₁₁₃×#1967	1	1	yes	characteristic
246	C₂	C₁₁₃×#1981	1	1	yes	characteristic

	subgroup	G/H
center, socle, prime radical	C₂³×C₁₁₃	C₂×C₂²⋊C₄
commutator	C₂²×C₄	C₂²×C₄×C₁₁₃
Frattini, cosocle	C₂×C₄²	C₂³×C₁₁₃
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³×C₁₁₃ ◁ C₄³×C₁₁₃ ◁ G

Complements

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

N	H ≅ G/N	type
118. #9701	C₁₁₃	direct (N × H)
129. C₁₁₃	#9701	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×#2862	2
C₁₁₃×#1842	2
C₁₁₃×#1767	2
C₂×#333×C₁₁₃	2
C₂×#332×C₁₁₃	2
#9701	113

Subgroups — C113×#9701 #43277721

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₁₃×#9701 #43277721