Subgroups — C₁₁₃×#20639 #43288659

Order 28928.

1314 conjugacy classes of subgroups.

The subgroup lattice has 256 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#519×C₁₁₃	C₂	1	1	yes	maximal, characteristic
2	C₁₁₃×#2905	C₂	1	2	yes	maximal
3	C₂²×#101×C₁₁₃	C₂	1	2	yes	maximal
4	C₁₁₃×#1801	C₂	1	8	yes	maximal, complemented
5	C₁₁₃×#1742	C₂	1	2	yes	maximal
6	C₂³×Q₈×C₁₁₃	C₂²	1	1	yes	characteristic
7	C₂²×M₁₆×C₁₁₃	C₂²	1	4	yes	—
8	#519×C₁₁₃	—	2	8	no	—
9	C₂×C₄×Q₈×C₁₁₃	C₄	1	4	yes	—
10	C₂×C₄×Q₈×C₁₁₃	C₂²	1	4	yes	—
11	C₂×#117×C₁₁₃	C₄	1	4	yes	—
12	C₂×#117×C₁₁₃	C₂²	1	1	yes	characteristic
13	C₂×#117×C₁₁₃	C₂²	1	1	yes	characteristic
14	#424×C₁₁₃	—	2	8	no	—
15	#423×C₁₁₃	C₂²	1	8	yes	—
16	C₂×#101×C₁₁₃	—	2	8	no	—
17	C₂×#101×C₁₁₃	C₂²	1	8	yes	—
18	#343×C₁₁₃	C₂²	1	8	yes	—
19	C₂²×Q₈×C₁₁₃	—	2	4	no	—
20	C₂²×Q₈×C₁₁₃	—	2	4	no	—
21	C₂²×Q₈×C₁₁₃	C₂×C₄	1	2	yes	—
22	C₂²×Q₈×C₁₁₃	D₄	1	4	yes	—
23	C₂²×Q₈×C₁₁₃	Q₈	1	4	yes	—
24	C₂²×Q₈×C₁₁₃	C₂³	1	2	yes	—
25	C₂³×C₄×C₁₁₃	C₂×C₄	1	2	yes	—
26	C₂³×C₄×C₁₁₃	C₂³	1	1	yes	characteristic
27	C₂×M₁₆×C₁₁₃	—	2	4	no	—
28	C₂×M₁₆×C₁₁₃	—	2	8	no	—
29	C₂×M₁₆×C₁₁₃	—	2	8	no	—
30	C₂×M₁₆×C₁₁₃	C₂³	1	8	yes	—
31	C₂²×C₈×C₁₁₃	—	2	4	no	—
32	C₄×Q₈×C₁₁₃	—	2	16	no	—
33	C₄×Q₈×C₁₁₃	—	2	16	no	—
34	#117×C₁₁₃	—	2	4	no	—
35	#117×C₁₁₃	—	2	16	no	—
36	#117×C₁₁₃	D₄	1	8	yes	—
37	C₂×C₄⋊C₄×C₁₁₃	C₂×C₄	1	8	yes	—
38	C₂×C₄⋊C₄×C₁₁₃	C₂×C₄	1	2	yes	—
39	C₂×C₄⋊C₄×C₁₁₃	C₂³	1	2	yes	—
40	C₂×C₂²⋊C₄×C₁₁₃	C₂×C₄	1	2	yes	—
41	C₂×C₂²⋊C₄×C₁₁₃	C₂×C₄	1	2	yes	—
42	C₂×C₄²×C₁₁₃	C₂×C₄	1	8	yes	—
43	C₂×C₄²×C₁₁₃	C₂×C₄	1	2	yes	—
44	C₂×C₄²×C₁₁₃	C₂³	1	2	yes	—
45	C₄⋊C₈×C₁₁₃	—	2	16	no	—
46	#101×C₁₁₃	—	2	16	no	—
47	C₂⁴×C₁₁₃	C₂²×C₄	1	1	yes	characteristic
48	C₂×Q₈×C₁₁₃	—	2	4	no	—
49	C₂×Q₈×C₁₁₃	—	2	4	no	—
50	C₂×Q₈×C₁₁₃	—	2	16	no	—
51	C₂×Q₈×C₁₁₃	—	2	16	no	—
52	C₂×Q₈×C₁₁₃	—	2	4	no	—
53	C₂×Q₈×C₁₁₃	—	2	4	no	—
54	C₂×Q₈×C₁₁₃	C₄⋊C₄	1	8	yes	—
55	C₂×Q₈×C₁₁₃	C₂×D₄	1	4	yes	—
56	C₂×Q₈×C₁₁₃	C₂×Q₈	1	4	yes	—
57	C₂²×C₄×C₁₁₃	—	2	4	no	—
58	C₂²×C₄×C₁₁₃	—	2	4	no	—
59	C₂²×C₄×C₁₁₃	C₄⋊C₄	1	8	yes	—
60	C₂²×C₄×C₁₁₃	C₂²×C₄	1	4	yes	—
61	C₂²×C₄×C₁₁₃	C₂²×C₄	1	4	yes	—
62	C₂²×C₄×C₁₁₃	C₂²×C₄	1	4	yes	—
63	C₂²×C₄×C₁₁₃	C₂²×C₄	1	1	yes	characteristic
64	C₂²×C₄×C₁₁₃	C₂×D₄	1	4	yes	—
65	C₂²×C₄×C₁₁₃	C₂×D₄	1	2	yes	—
66	C₂²×C₄×C₁₁₃	C₂×Q₈	1	2	yes	—
67	C₂²×C₄×C₁₁₃	#41	1	4	yes	—
68	C₂²×C₄×C₁₁₃	C₂⁴	1	1	yes	characteristic
69	M₁₆×C₁₁₃	—	2	8	no	—
70	M₁₆×C₁₁₃	—	2	8	no	—
71	M₁₆×C₁₁₃	—	4	8	no	—
72	C₂×C₈×C₁₁₃	—	2	8	no	—
73	C₂×C₈×C₁₁₃	—	4	8	no	—
74	C₄⋊C₄×C₁₁₃	—	2	16	no	—
75	C₄⋊C₄×C₁₁₃	—	2	4	no	—
76	C₄⋊C₄×C₁₁₃	C₂²⋊C₄	1	8	yes	—
77	C₂²⋊C₄×C₁₁₃	—	2	4	no	—
78	C₂²⋊C₄×C₁₁₃	C₂²⋊C₄	1	8	yes	—
79	C₄²×C₁₁₃	—	2	16	no	—
80	C₄²×C₁₁₃	—	2	4	no	—
81	C₄²×C₁₁₃	C₂×D₄	1	8	yes	—
82	C₂³×C₁₁₃	—	4	2	no	—
83	C₂³×C₁₁₃	C₂×C₂²⋊C₄	1	2	yes	—
84	C₂³×C₁₁₃	C₂×C₄⋊C₄	1	2	yes	—
85	C₂³×C₁₁₃	#117	1	2	yes	—
86	C₂³×C₁₁₃	C₂³×C₄	1	1	yes	characteristic
87	Q₈×C₁₁₃	—	2	8	no	—
88	Q₈×C₁₁₃	—	2	8	no	—
89	Q₈×C₁₁₃	—	2	16	no	—
90	C₂×C₄×C₁₁₃	—	2	4	no	—
91	C₂×C₄×C₁₁₃	—	2	4	no	—
92	C₂×C₄×C₁₁₃	—	2	16	no	—
93	C₂×C₄×C₁₁₃	—	2	4	no	—
94	C₂×C₄×C₁₁₃	—	4	4	no	—
95	C₂×C₄×C₁₁₃	—	4	4	no	—
96	C₂×C₄×C₁₁₃	C₂×C₂²⋊C₄	1	8	yes	—
97	C₂×C₄×C₁₁₃	C₂×C₂²⋊C₄	1	2	yes	—
98	C₂×C₄×C₁₁₃	C₂×C₄⋊C₄	1	8	yes	—
99	C₂×C₄×C₁₁₃	C₂×C₄⋊C₄	1	2	yes	—
100	C₂×C₄×C₁₁₃	#117	1	2	yes	—
101	C₂×C₄×C₁₁₃	#121	1	8	yes	—
102	C₂×C₄×C₁₁₃	C₂²⋊Q₈	1	8	yes	—
103	C₂×C₄×C₁₁₃	C₂²×D₄	1	1	yes	characteristic
104	C₂×C₄×C₁₁₃	C₂²×D₄	1	2	yes	—
105	C₂×C₄×C₁₁₃	C₂²×Q₈	1	1	yes	characteristic
106	C₂×C₄×C₁₁₃	C₂×#41	1	2	yes	—
107	C₈×C₁₁₃	—	4	8	no	—
108	C₂²×C₁₁₃	—	2	2	no	—
109	C₂²×C₁₁₃	—	2	4	no	—
110	C₂²×C₁₁₃	—	4	2	no	—
111	C₂²×C₁₁₃	—	4	2	no	—
112	C₂²×C₁₁₃	#380	1	4	yes	—
113	C₂²×C₁₁₃	C₂²×C₂²⋊C₄	1	1	yes	characteristic
114	C₂²×C₁₁₃	C₂²×C₄⋊C₄	1	1	yes	characteristic
115	C₂²×C₁₁₃	C₂×#117	1	1	yes	characteristic
116	C₄×C₁₁₃	—	2	8	no	—
117	C₄×C₁₁₃	—	2	8	no	—
118	C₄×C₁₁₃	—	4	4	no	—
119	C₄×C₁₁₃	#380	1	4	yes	—
120	C₄×C₁₁₃	C₂×#121	1	2	yes	—
121	C₄×C₁₁₃	C₂×C₂²⋊Q₈	1	2	yes	—
122	#20639	C₁₁₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
123	C₂×C₁₁₃	—	2	1	no	—
124	C₂×C₁₁₃	—	2	1	no	—
125	C₂×C₁₁₃	—	2	4	no	—
126	C₂×C₁₁₃	C₂×#380	1	1	yes	characteristic
127	C₂×C₁₁₃	#2889	1	2	yes	—
128	C₂×#519	C₂×C₁₁₃	1	1	yes	characteristic
129	#2905	C₂×C₁₁₃	1	2	yes	—
130	C₂²×#101	C₂×C₁₁₃	1	2	yes	—
131	#1801	C₂×C₁₁₃	1	8	yes	complemented
132	#1742	C₂×C₁₁₃	1	2	yes	—
133	C₁₁₃	#20639	1	1	yes	characteristic, direct product factor, Sylow
134	C₂³×Q₈	C₂²×C₁₁₃	1	1	yes	characteristic
135	C₂²×M₁₆	C₂²×C₁₁₃	1	4	yes	—
136	#519	—	2	8	no	—
137	C₂×C₄×Q₈	C₄×C₁₁₃	1	4	yes	—
138	C₂×C₄×Q₈	C₂²×C₁₁₃	1	4	yes	—
139	C₂×#117	C₄×C₁₁₃	1	4	yes	—
140	C₂×#117	C₂²×C₁₁₃	1	1	yes	characteristic
141	C₂×#117	C₂²×C₁₁₃	1	1	yes	characteristic
142	#424	—	2	8	no	—
143	#423	C₂²×C₁₁₃	1	8	yes	—
144	C₂×#101	—	2	8	no	—
145	C₂×#101	C₂²×C₁₁₃	1	8	yes	—
146	#343	C₂²×C₁₁₃	1	8	yes	—
147	C₂²×Q₈	—	2	4	no	—
148	C₂²×Q₈	—	2	4	no	—
149	C₂²×Q₈	C₂×C₄×C₁₁₃	1	2	yes	—
150	C₂²×Q₈	D₄×C₁₁₃	1	4	yes	—
151	C₂²×Q₈	Q₈×C₁₁₃	1	4	yes	—
152	C₂²×Q₈	C₂³×C₁₁₃	1	2	yes	—
153	C₂³×C₄	C₂×C₄×C₁₁₃	1	2	yes	—
154	C₂³×C₄	C₂³×C₁₁₃	1	1	yes	characteristic
155	C₂×M₁₆	—	2	4	no	—
156	C₂×M₁₆	—	2	8	no	—
157	C₂×M₁₆	—	2	8	no	—
158	C₂×M₁₆	C₂³×C₁₁₃	1	8	yes	—
159	C₂²×C₈	—	2	4	no	—
160	C₄×Q₈	—	2	16	no	—
161	C₄×Q₈	—	2	16	no	—
162	#117	—	2	4	no	—
163	#117	—	2	16	no	—
164	#117	D₄×C₁₁₃	1	8	yes	—
165	C₂×C₄⋊C₄	C₂×C₄×C₁₁₃	1	8	yes	—
166	C₂×C₄⋊C₄	C₂×C₄×C₁₁₃	1	2	yes	—
167	C₂×C₄⋊C₄	C₂³×C₁₁₃	1	2	yes	—
168	C₂×C₂²⋊C₄	C₂×C₄×C₁₁₃	1	2	yes	—
169	C₂×C₂²⋊C₄	C₂×C₄×C₁₁₃	1	2	yes	—
170	C₂×C₄²	C₂×C₄×C₁₁₃	1	8	yes	—
171	C₂×C₄²	C₂×C₄×C₁₁₃	1	2	yes	—
172	C₂×C₄²	C₂³×C₁₁₃	1	2	yes	—
173	C₄⋊C₈	—	2	16	no	—
174	#101	—	2	16	no	—
175	C₂⁴	C₂²×C₄×C₁₁₃	1	1	yes	characteristic
176	C₂×Q₈	—	2	4	no	—
177	C₂×Q₈	—	2	4	no	—
178	C₂×Q₈	—	2	16	no	—
179	C₂×Q₈	—	2	16	no	—
180	C₂×Q₈	—	2	4	no	—
181	C₂×Q₈	—	2	4	no	—
182	C₂×Q₈	C₄⋊C₄×C₁₁₃	1	8	yes	—
183	C₂×Q₈	C₂×D₄×C₁₁₃	1	4	yes	—
184	C₂×Q₈	C₂×Q₈×C₁₁₃	1	4	yes	—
185	C₂²×C₄	—	2	4	no	—
186	C₂²×C₄	—	2	4	no	—
187	C₂²×C₄	C₄⋊C₄×C₁₁₃	1	8	yes	—
188	C₂²×C₄	C₂²×C₄×C₁₁₃	1	4	yes	—
189	C₂²×C₄	C₂²×C₄×C₁₁₃	1	4	yes	—
190	C₂²×C₄	C₂²×C₄×C₁₁₃	1	4	yes	—
191	C₂²×C₄	C₂²×C₄×C₁₁₃	1	1	yes	characteristic
192	C₂²×C₄	C₂×D₄×C₁₁₃	1	4	yes	—
193	C₂²×C₄	C₂×D₄×C₁₁₃	1	2	yes	—
194	C₂²×C₄	C₂×Q₈×C₁₁₃	1	2	yes	—
195	C₂²×C₄	#41×C₁₁₃	1	4	yes	—
196	C₂²×C₄	C₂⁴×C₁₁₃	1	1	yes	characteristic
197	M₁₆	—	2	8	no	—
198	M₁₆	—	2	8	no	—
199	M₁₆	—	4	8	no	—
200	C₂×C₈	—	2	8	no	—
201	C₂×C₈	—	4	8	no	—
202	C₄⋊C₄	—	2	16	no	—
203	C₄⋊C₄	—	2	4	no	—
204	C₄⋊C₄	C₂²⋊C₄×C₁₁₃	1	8	yes	—
205	C₂²⋊C₄	—	2	4	no	—
206	C₂²⋊C₄	C₂²⋊C₄×C₁₁₃	1	8	yes	—
207	C₄²	—	2	16	no	—
208	C₄²	—	2	4	no	—
209	C₄²	C₂×D₄×C₁₁₃	1	8	yes	—
210	C₂³	—	4	2	no	—
211	C₂³	C₂×C₂²⋊C₄×C₁₁₃	1	2	yes	—
212	C₂³	C₂×C₄⋊C₄×C₁₁₃	1	2	yes	—
213	C₂³	#117×C₁₁₃	1	2	yes	—
214	C₂³	C₂³×C₄×C₁₁₃	1	1	yes	characteristic
215	Q₈	—	2	8	no	—
216	Q₈	—	2	8	no	—
217	Q₈	—	2	16	no	—
218	C₂×C₄	—	2	4	no	—
219	C₂×C₄	—	2	4	no	—
220	C₂×C₄	—	2	16	no	—
221	C₂×C₄	—	2	4	no	—
222	C₂×C₄	—	4	4	no	—
223	C₂×C₄	—	4	4	no	—
224	C₂×C₄	C₂×C₂²⋊C₄×C₁₁₃	1	8	yes	—
225	C₂×C₄	C₂×C₂²⋊C₄×C₁₁₃	1	2	yes	—
226	C₂×C₄	C₂×C₄⋊C₄×C₁₁₃	1	8	yes	—
227	C₂×C₄	C₂×C₄⋊C₄×C₁₁₃	1	2	yes	—
228	C₂×C₄	#117×C₁₁₃	1	2	yes	—
229	C₂×C₄	#121×C₁₁₃	1	8	yes	—
230	C₂×C₄	C₂²⋊Q₈×C₁₁₃	1	8	yes	—
231	C₂×C₄	C₂²×D₄×C₁₁₃	1	1	yes	characteristic
232	C₂×C₄	C₂²×D₄×C₁₁₃	1	2	yes	—
233	C₂×C₄	C₂²×Q₈×C₁₁₃	1	1	yes	characteristic
234	C₂×C₄	C₂×#41×C₁₁₃	1	2	yes	—
235	C₈	—	4	8	no	—
236	C₂²	—	2	2	no	—
237	C₂²	—	2	4	no	—
238	C₂²	—	4	2	no	—
239	C₂²	—	4	2	no	—
240	C₂²	#380×C₁₁₃	1	4	yes	—
241	C₂²	C₂²×C₂²⋊C₄×C₁₁₃	1	1	yes	characteristic
242	C₂²	C₂²×C₄⋊C₄×C₁₁₃	1	1	yes	characteristic
243	C₂²	C₂×#117×C₁₁₃	1	1	yes	characteristic
244	C₄	—	2	8	no	—
245	C₄	—	2	8	no	—
246	C₄	—	4	4	no	—
247	C₄	#380×C₁₁₃	1	4	yes	—
248	C₄	C₂×#121×C₁₁₃	1	2	yes	—
249	C₄	C₂×C₂²⋊Q₈×C₁₁₃	1	2	yes	—
250	C₂	—	2	1	no	—
251	C₂	—	2	1	no	—
252	C₂	—	2	4	no	—
253	C₂	C₂×#380×C₁₁₃	1	1	yes	characteristic
254	C₂	C₁₁₃×#2889	1	2	yes	—

	subgroup	G/H
center, socle	C₂²×C₁₁₃	C₂²×C₂²⋊C₄
commutator	C₂³	C₂³×C₄×C₁₁₃
Frattini, cosocle	C₂²×C₄	C₂⁴×C₁₁₃
prime radical	C₂⁴×C₁₁₃	C₂²×C₄
LCS	G ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₁₁₃ ◁ C₂×#117×C₁₁₃ ◁ G

Complements

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

N	H ≅ G/N	type
4. C₁₁₃×#1801	C₂	semidirect
122. #20639	C₁₁₃	direct (N × H)
131. #1801	C₂×C₁₁₃	semidirect
133. C₁₁₃	#20639	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂×#519×C₁₁₃	2
C₁₁₃×#2905	2
C₂²×#101×C₁₁₃	2
C₁₁₃×#1801	2
C₁₁₃×#1742	2
#20639	113

Subgroups — C113×#20639 #43288659

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₁₃×#20639 #43288659