Subgroups — C₁₁₃×#17352 #43285372

Order 28928.

1162 conjugacy classes of subgroups.

The subgroup lattice has 230 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×#2982	C₂	1	2	yes	maximal, complemented
3	C₁₁₃×#1461	C₂	1	6	yes	maximal, complemented
4	C₁₁₃×#1458	C₂	1	6	yes	maximal
5	C₂³×Q₈×C₁₁₃	C₄	1	1	yes	characteristic
6	#519×C₁₁₃	—	2	8	no	—
7	C₂×C₄×Q₈×C₁₁₃	C₄	1	3	yes	—
8	C₂×C₄×Q₈×C₁₁₃	C₄	1	1	yes	characteristic
9	C₂×C₄×Q₈×C₁₁₃	C₂²	1	3	yes	—
10	C₂×C₄×Q₈×C₁₁₃	C₂²	1	1	yes	characteristic
11	C₂×#117×C₁₁₃	C₄	1	3	yes	—
12	C₂×#117×C₁₁₃	C₂²	1	3	yes	—
13	C₈×Q₈×C₁₁₃	—	2	8	no	—
14	#431×C₁₁₃	C₂²	1	12	yes	—
15	#324×C₁₁₃	C₂²	1	12	yes	—
16	#323×C₁₁₃	C₂²	1	4	yes	—
17	C₂²×Q₈×C₁₁₃	—	2	4	no	—
18	C₂²×Q₈×C₁₁₃	—	2	4	no	—
19	C₂²×Q₈×C₁₁₃	C₈	1	8	yes	complemented
20	C₂²×Q₈×C₁₁₃	C₂×C₄	1	3	yes	—
21	C₂²×Q₈×C₁₁₃	C₂×C₄	1	1	yes	characteristic
22	C₂³×C₄×C₁₁₃	C₂×C₄	1	3	yes	—
23	C₄×Q₈×C₁₁₃	—	2	4	no	—
24	C₄×Q₈×C₁₁₃	—	2	12	no	—
25	C₄×Q₈×C₁₁₃	—	2	12	no	—
26	C₄×Q₈×C₁₁₃	D₄	1	8	yes	—
27	#117×C₁₁₃	—	2	12	no	—
28	#117×C₁₁₃	—	2	12	no	—
29	C₂×C₄⋊C₄×C₁₁₃	C₂×C₄	1	3	yes	—
30	C₂×C₄⋊C₄×C₁₁₃	C₂×C₄	1	3	yes	—
31	C₂×C₄⋊C₄×C₁₁₃	C₂×C₄	1	3	yes	—
32	C₂×C₄⋊C₄×C₁₁₃	C₂³	1	3	yes	—
33	C₂×C₂²⋊C₄×C₁₁₃	C₂×C₄	1	3	yes	—
34	C₂×C₂²⋊C₄×C₁₁₃	C₂³	1	1	yes	characteristic
35	C₂×C₄²×C₁₁₃	C₂×C₄	1	6	yes	—
36	C₂×C₄²×C₁₁₃	C₂×C₄	1	3	yes	—
37	C₂×C₄²×C₁₁₃	C₂³	1	3	yes	—
38	C₄⋊C₈×C₁₁₃	—	2	12	no	—
39	C₂²⋊C₈×C₁₁₃	C₂³	1	8	yes	—
40	C₄×C₈×C₁₁₃	—	2	12	no	—
41	C₂⁴×C₁₁₃	C₂²×C₄	1	1	yes	characteristic
42	C₂×Q₈×C₁₁₃	—	2	32	no	—
43	C₂×Q₈×C₁₁₃	—	2	6	no	—
44	C₂×Q₈×C₁₁₃	—	2	6	no	—
45	C₂×Q₈×C₁₁₃	C₂²⋊C₄	1	4	yes	—
46	C₂×Q₈×C₁₁₃	C₂×C₈	1	12	yes	—
47	C₂×Q₈×C₁₁₃	C₂×C₈	1	4	yes	—
48	C₂×Q₈×C₁₁₃	M₁₆	1	4	yes	—
49	C₂²×C₄×C₁₁₃	—	2	6	no	—
50	C₂²×C₄×C₁₁₃	—	2	6	no	—
51	C₂²×C₄×C₁₁₃	C₂×C₈	1	12	yes	—
52	C₂²×C₄×C₁₁₃	C₂²×C₄	1	3	yes	—
53	C₂²×C₄×C₁₁₃	C₂²×C₄	1	3	yes	—
54	C₂²×C₄×C₁₁₃	C₂²×C₄	1	1	yes	characteristic
55	C₂²×C₄×C₁₁₃	C₂²×C₄	1	3	yes	—
56	C₂²×C₄×C₁₁₃	C₂²×C₄	1	3	yes	—
57	C₂²×C₄×C₁₁₃	C₂⁴	1	1	yes	characteristic
58	C₂×C₈×C₁₁₃	—	2	8	no	—
59	C₄⋊C₄×C₁₁₃	—	2	6	no	—
60	C₄⋊C₄×C₁₁₃	—	2	6	no	—
61	C₄⋊C₄×C₁₁₃	—	2	6	no	—
62	C₄⋊C₄×C₁₁₃	C₂²⋊C₄	1	12	yes	—
63	C₂²⋊C₄×C₁₁₃	—	2	6	no	—
64	C₂²⋊C₄×C₁₁₃	—	2	2	no	—
65	C₄²×C₁₁₃	—	2	12	no	—
66	C₄²×C₁₁₃	—	2	6	no	—
67	C₄²×C₁₁₃	C₂×D₄	1	12	yes	—
68	C₂³×C₁₁₃	—	2	1	no	—
69	C₂³×C₁₁₃	—	2	1	no	—
70	C₂³×C₁₁₃	—	4	1	no	—
71	C₂³×C₁₁₃	—	4	1	no	—
72	C₂³×C₁₁₃	C₂²×C₈	1	2	yes	—
73	C₂³×C₁₁₃	C₂³×C₄	1	1	yes	characteristic
74	Q₈×C₁₁₃	—	2	24	no	—
75	Q₈×C₁₁₃	C₂²⋊C₈	1	16	yes	—
76	C₂×C₄×C₁₁₃	—	2	3	no	—
77	C₂×C₄×C₁₁₃	—	2	3	no	—
78	C₂×C₄×C₁₁₃	—	2	24	no	—
79	C₂×C₄×C₁₁₃	—	2	3	no	—
80	C₂×C₄×C₁₁₃	—	2	1	no	—
81	C₂×C₄×C₁₁₃	—	4	3	no	—
82	C₂×C₄×C₁₁₃	—	4	1	no	—
83	C₂×C₄×C₁₁₃	—	4	3	no	—
84	C₂×C₄×C₁₁₃	—	4	1	no	—
85	C₂×C₄×C₁₁₃	C₂×C₂²⋊C₄	1	6	yes	—
86	C₂×C₄×C₁₁₃	C₂×C₂²⋊C₄	1	6	yes	—
87	C₂×C₄×C₁₁₃	C₂²×C₈	1	6	yes	—
88	C₂×C₄×C₁₁₃	C₂²×C₈	1	6	yes	—
89	C₂×C₄×C₁₁₃	C₂×M₁₆	1	6	yes	—
90	C₂×C₄×C₁₁₃	C₂²×D₄	1	2	yes	—
91	C₈×C₁₁₃	—	4	8	no	—
92	C₂²×C₁₁₃	—	2	1	no	—
93	C₂²×C₁₁₃	—	2	1	no	—
94	C₂²×C₁₁₃	—	2	2	no	—
95	C₂²×C₁₁₃	—	4	2	no	—
96	C₂²×C₁₁₃	—	4	2	no	—
97	C₂²×C₁₁₃	—	4	1	no	—
98	C₂²×C₁₁₃	—	4	1	no	—
99	C₂²×C₁₁₃	C₂²×C₂²⋊C₄	1	1	yes	characteristic
100	C₂²×C₁₁₃	C₂³×C₈	1	1	yes	characteristic
101	C₂²×C₁₁₃	C₂²×M₁₆	1	1	yes	characteristic
102	C₄×C₁₁₃	—	2	6	no	—
103	C₄×C₁₁₃	—	2	2	no	—
104	C₄×C₁₁₃	—	2	6	no	—
105	C₄×C₁₁₃	—	4	3	no	—
106	C₄×C₁₁₃	—	4	1	no	—
107	C₄×C₁₁₃	C₂×C₂²⋊C₈	1	12	yes	—
108	#17352	C₁₁₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
109	C₂×C₁₁₃	—	2	1	no	—
110	C₂×C₁₁₃	—	2	1	no	—
111	C₂×C₁₁₃	—	4	2	no	—
112	C₂×C₁₁₃	C₂²×C₂²⋊C₈	1	1	yes	characteristic
113	C₂×C₁₁₃	#2884	1	1	yes	characteristic
114	C₂×C₁₁₃	#2889	1	1	yes	characteristic
115	C₂×#519	C₂×C₁₁₃	1	1	yes	characteristic
116	#2982	C₂×C₁₁₃	1	2	yes	complemented
117	#1461	C₂×C₁₁₃	1	6	yes	complemented
118	#1458	C₂×C₁₁₃	1	6	yes	—
119	C₁₁₃	#17352	1	1	yes	characteristic, direct product factor, Sylow
120	C₂³×Q₈	C₄×C₁₁₃	1	1	yes	characteristic
121	#519	—	2	8	no	—
122	C₂×C₄×Q₈	C₄×C₁₁₃	1	3	yes	—
123	C₂×C₄×Q₈	C₄×C₁₁₃	1	1	yes	characteristic
124	C₂×C₄×Q₈	C₂²×C₁₁₃	1	3	yes	—
125	C₂×C₄×Q₈	C₂²×C₁₁₃	1	1	yes	characteristic
126	C₂×#117	C₄×C₁₁₃	1	3	yes	—
127	C₂×#117	C₂²×C₁₁₃	1	3	yes	—
128	C₈×Q₈	—	2	8	no	—
129	#431	C₂²×C₁₁₃	1	12	yes	—
130	#324	C₂²×C₁₁₃	1	12	yes	—
131	#323	C₂²×C₁₁₃	1	4	yes	—
132	C₂²×Q₈	—	2	4	no	—
133	C₂²×Q₈	—	2	4	no	—
134	C₂²×Q₈	C₈×C₁₁₃	1	8	yes	complemented
135	C₂²×Q₈	C₂×C₄×C₁₁₃	1	3	yes	—
136	C₂²×Q₈	C₂×C₄×C₁₁₃	1	1	yes	characteristic
137	C₂³×C₄	C₂×C₄×C₁₁₃	1	3	yes	—
138	C₄×Q₈	—	2	4	no	—
139	C₄×Q₈	—	2	12	no	—
140	C₄×Q₈	—	2	12	no	—
141	C₄×Q₈	D₄×C₁₁₃	1	8	yes	—
142	#117	—	2	12	no	—
143	#117	—	2	12	no	—
144	C₂×C₄⋊C₄	C₂×C₄×C₁₁₃	1	3	yes	—
145	C₂×C₄⋊C₄	C₂×C₄×C₁₁₃	1	3	yes	—
146	C₂×C₄⋊C₄	C₂×C₄×C₁₁₃	1	3	yes	—
147	C₂×C₄⋊C₄	C₂³×C₁₁₃	1	3	yes	—
148	C₂×C₂²⋊C₄	C₂×C₄×C₁₁₃	1	3	yes	—
149	C₂×C₂²⋊C₄	C₂³×C₁₁₃	1	1	yes	characteristic
150	C₂×C₄²	C₂×C₄×C₁₁₃	1	6	yes	—
151	C₂×C₄²	C₂×C₄×C₁₁₃	1	3	yes	—
152	C₂×C₄²	C₂³×C₁₁₃	1	3	yes	—
153	C₄⋊C₈	—	2	12	no	—
154	C₂²⋊C₈	C₂³×C₁₁₃	1	8	yes	—
155	C₄×C₈	—	2	12	no	—
156	C₂⁴	C₂²×C₄×C₁₁₃	1	1	yes	characteristic
157	C₂×Q₈	—	2	32	no	—
158	C₂×Q₈	—	2	6	no	—
159	C₂×Q₈	—	2	6	no	—
160	C₂×Q₈	C₂²⋊C₄×C₁₁₃	1	4	yes	—
161	C₂×Q₈	C₂×C₈×C₁₁₃	1	12	yes	—
162	C₂×Q₈	C₂×C₈×C₁₁₃	1	4	yes	—
163	C₂×Q₈	M₁₆×C₁₁₃	1	4	yes	—
164	C₂²×C₄	—	2	6	no	—
165	C₂²×C₄	—	2	6	no	—
166	C₂²×C₄	C₂×C₈×C₁₁₃	1	12	yes	—
167	C₂²×C₄	C₂²×C₄×C₁₁₃	1	3	yes	—
168	C₂²×C₄	C₂²×C₄×C₁₁₃	1	3	yes	—
169	C₂²×C₄	C₂²×C₄×C₁₁₃	1	1	yes	characteristic
170	C₂²×C₄	C₂²×C₄×C₁₁₃	1	3	yes	—
171	C₂²×C₄	C₂²×C₄×C₁₁₃	1	3	yes	—
172	C₂²×C₄	C₂⁴×C₁₁₃	1	1	yes	characteristic
173	C₂×C₈	—	2	8	no	—
174	C₄⋊C₄	—	2	6	no	—
175	C₄⋊C₄	—	2	6	no	—
176	C₄⋊C₄	—	2	6	no	—
177	C₄⋊C₄	C₂²⋊C₄×C₁₁₃	1	12	yes	—
178	C₂²⋊C₄	—	2	6	no	—
179	C₂²⋊C₄	—	2	2	no	—
180	C₄²	—	2	12	no	—
181	C₄²	—	2	6	no	—
182	C₄²	C₂×D₄×C₁₁₃	1	12	yes	—
183	C₂³	—	2	1	no	—
184	C₂³	—	2	1	no	—
185	C₂³	—	4	1	no	—
186	C₂³	—	4	1	no	—
187	C₂³	C₂²×C₈×C₁₁₃	1	2	yes	—
188	C₂³	C₂³×C₄×C₁₁₃	1	1	yes	characteristic
189	Q₈	—	2	24	no	—
190	Q₈	C₂²⋊C₈×C₁₁₃	1	16	yes	—
191	C₂×C₄	—	2	3	no	—
192	C₂×C₄	—	2	3	no	—
193	C₂×C₄	—	2	24	no	—
194	C₂×C₄	—	2	3	no	—
195	C₂×C₄	—	2	1	no	—
196	C₂×C₄	—	4	3	no	—
197	C₂×C₄	—	4	1	no	—
198	C₂×C₄	—	4	3	no	—
199	C₂×C₄	—	4	1	no	—
200	C₂×C₄	C₂×C₂²⋊C₄×C₁₁₃	1	6	yes	—
201	C₂×C₄	C₂×C₂²⋊C₄×C₁₁₃	1	6	yes	—
202	C₂×C₄	C₂²×C₈×C₁₁₃	1	6	yes	—
203	C₂×C₄	C₂²×C₈×C₁₁₃	1	6	yes	—
204	C₂×C₄	C₂×M₁₆×C₁₁₃	1	6	yes	—
205	C₂×C₄	C₂²×D₄×C₁₁₃	1	2	yes	—
206	C₈	—	4	8	no	—
207	C₂²	—	2	1	no	—
208	C₂²	—	2	1	no	—
209	C₂²	—	2	2	no	—
210	C₂²	—	4	2	no	—
211	C₂²	—	4	2	no	—
212	C₂²	—	4	1	no	—
213	C₂²	—	4	1	no	—
214	C₂²	C₂²×C₂²⋊C₄×C₁₁₃	1	1	yes	characteristic
215	C₂²	C₂³×C₈×C₁₁₃	1	1	yes	characteristic
216	C₂²	C₂²×M₁₆×C₁₁₃	1	1	yes	characteristic
217	C₄	—	2	6	no	—
218	C₄	—	2	2	no	—
219	C₄	—	2	6	no	—
220	C₄	—	4	3	no	—
221	C₄	—	4	1	no	—
222	C₄	C₂×C₂²⋊C₈×C₁₁₃	1	12	yes	—
223	C₂	—	2	1	no	—
224	C₂	—	2	1	no	—
225	C₂	—	4	2	no	—
226	C₂	C₂²×C₂²⋊C₈×C₁₁₃	1	1	yes	characteristic
227	C₂	C₁₁₃×#2884	1	1	yes	characteristic
228	C₂	C₁₁₃×#2889	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₁₁₃	C₂²×C₂²⋊C₄
commutator	C₂²	C₂³×C₈×C₁₁₃
Frattini, cosocle	C₂²×C₄	C₂⁴×C₁₁₃
prime radical	C₂⁴×C₁₁₃	C₂²×C₄
LCS	G ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₁₁₃ ◁ C₂×C₄×Q₈×C₁₁₃ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₁₁₃×#2982	C₂	semidirect
3. C₁₁₃×#1461	C₂	semidirect
19. C₂²×Q₈×C₁₁₃	C₈	semidirect
108. #17352	C₁₁₃	direct (N × H)
116. #2982	C₂×C₁₁₃	semidirect
117. #1461	C₂×C₁₁₃	semidirect
119. C₁₁₃	#17352	direct (N × H)
134. C₂²×Q₈	C₈×C₁₁₃	semidirect

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×#2982	2
C₁₁₃×#1461	2
C₁₁₃×#1458	2
#17352	113

Subgroups — C113×#17352 #43285372

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₁₃×#17352 #43285372