Subgroups — C₁₁₃×#8277 #43276297

Order 28928.

394 conjugacy classes of subgroups.

The subgroup lattice has 188 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₁₁₃×#1767	C₂	1	1	yes	maximal, characteristic
2	C₄×C₄⋊C₈×C₁₁₃	C₂	1	1	yes	maximal, characteristic
3	C₁₁₃×#1744	C₂	1	1	yes	maximal, characteristic
4	C₂×#334×C₁₁₃	C₂	1	2	yes	maximal
5	C₁₁₃×#1277	C₂	1	2	yes	maximal
6	C₂×C₄⋊C₈×C₁₁₃	C₂²	1	2	yes	—
7	C₂×C₄⋊C₈×C₁₁₃	C₂²	1	2	yes	—
8	C₂×#97×C₁₁₃	—	2	1	no	—
9	C₂×C₄×C₈×C₁₁₃	—	2	1	no	—
10	C₂×C₄×C₈×C₁₁₃	C₂²	1	1	yes	characteristic
11	C₂×C₄×C₈×C₁₁₃	C₂²	1	1	yes	characteristic
12	C₄³×C₁₁₃	C₂²	1	1	yes	characteristic
13	#336×C₁₁₃	—	2	1	no	—
14	#336×C₁₁₃	—	2	1	no	—
15	#336×C₁₁₃	C₄	1	4	yes	—
16	#334×C₁₁₃	C₄	1	8	yes	complemented
17	C₂²×C₈×C₁₁₃	—	2	2	no	—
18	C₂²×C₈×C₁₁₃	—	2	2	no	—
19	C₂²×C₈×C₁₁₃	C₂×C₄	1	2	yes	—
20	C₂²×C₈×C₁₁₃	C₂×C₄	1	2	yes	—
21	C₂×C₄²×C₁₁₃	C₂×C₄	1	2	yes	—
22	C₂×C₄²×C₁₁₃	D₄	1	1	yes	characteristic
23	C₂×C₄²×C₁₁₃	D₄	1	1	yes	characteristic
24	C₂×C₄²×C₁₁₃	Q₈	1	2	yes	—
25	C₂×C₄²×C₁₁₃	C₂³	1	1	yes	characteristic
26	C₄⋊C₈×C₁₁₃	—	2	2	no	—
27	C₄⋊C₈×C₁₁₃	—	2	2	no	—
28	C₄⋊C₈×C₁₁₃	C₂×C₄	1	4	yes	—
29	C₄⋊C₈×C₁₁₃	C₂×C₄	1	4	yes	—
30	#97×C₁₁₃	—	4	2	no	—
31	C₄×C₈×C₁₁₃	—	2	2	no	—
32	C₄×C₈×C₁₁₃	—	4	2	no	—
33	C₄×C₈×C₁₁₃	C₂×C₄	1	4	yes	—
34	C₂²×C₄×C₁₁₃	C₄⋊C₄	1	2	yes	—
35	C₂²×C₄×C₁₁₃	C₄⋊C₄	1	2	yes	—
36	C₂²×C₄×C₁₁₃	C₂²×C₄	1	1	yes	characteristic
37	C₂²×C₄×C₁₁₃	C₂×D₄	1	1	yes	characteristic
38	C₂²×C₄×C₁₁₃	C₂×Q₈	1	1	yes	characteristic
39	C₂×C₈×C₁₁₃	—	2	4	no	—
40	C₂×C₈×C₁₁₃	—	2	2	no	—
41	C₂×C₈×C₁₁₃	—	2	4	no	—
42	C₂×C₈×C₁₁₃	—	2	4	no	—
43	C₂×C₈×C₁₁₃	—	2	4	no	—
44	C₂×C₈×C₁₁₃	—	4	2	no	—
45	C₂×C₈×C₁₁₃	—	4	2	no	—
46	C₂×C₈×C₁₁₃	—	4	2	no	—
47	C₂×C₈×C₁₁₃	—	4	2	no	—
48	C₂×C₈×C₁₁₃	C₄²	1	4	yes	—
49	C₄²×C₁₁₃	—	2	4	no	—
50	C₄²×C₁₁₃	—	2	2	no	—
51	C₄²×C₁₁₃	—	4	2	no	—
52	C₄²×C₁₁₃	SD₁₆	1	2	yes	—
53	C₄²×C₁₁₃	SD₁₆	1	2	yes	—
54	C₄²×C₁₁₃	C₂²×C₄	1	2	yes	—
55	C₄²×C₁₁₃	#41	1	2	yes	—
56	C₂³×C₁₁₃	C₂×C₄⋊C₄	1	1	yes	characteristic
57	C₂×C₄×C₁₁₃	—	2	2	no	—
58	C₂×C₄×C₁₁₃	—	2	2	no	—
59	C₂×C₄×C₁₁₃	—	2	2	no	—
60	C₂×C₄×C₁₁₃	—	2	2	no	—
61	C₂×C₄×C₁₁₃	—	2	2	no	—
62	C₂×C₄×C₁₁₃	—	2	2	no	—
63	C₂×C₄×C₁₁₃	#106	1	4	yes	—
64	C₂×C₄×C₁₁₃	C₂×C₄²	1	1	yes	characteristic
65	C₂×C₄×C₁₁₃	#117	1	1	yes	characteristic
66	C₂×C₄×C₁₁₃	C₄×D₄	1	2	yes	—
67	C₂×C₄×C₁₁₃	C₄×Q₈	1	2	yes	—
68	C₂×C₄×C₁₁₃	#131	1	4	yes	—
69	C₂×C₄×C₁₁₃	C₂×SD₁₆	1	1	yes	characteristic
70	C₂×C₄×C₁₁₃	C₂×SD₁₆	1	1	yes	characteristic
71	C₂×C₄×C₁₁₃	#135	1	2	yes	—
72	C₈×C₁₁₃	—	2	4	no	—
73	C₈×C₁₁₃	—	4	2	no	—
74	C₈×C₁₁₃	—	4	4	no	—
75	C₈×C₁₁₃	—	4	4	no	—
76	C₂²×C₁₁₃	C₄×C₄⋊C₄	1	1	yes	characteristic
77	C₂²×C₁₁₃	#424	1	2	yes	—
78	C₂²×C₁₁₃	C₂×#106	1	1	yes	characteristic
79	C₂²×C₁₁₃	#427	1	1	yes	characteristic
80	C₂²×C₁₁₃	#430	1	2	yes	—
81	C₄×C₁₁₃	—	2	4	no	—
82	C₄×C₁₁₃	—	2	2	no	—
83	C₄×C₁₁₃	—	4	2	no	—
84	C₄×C₁₁₃	#405	1	2	yes	—
85	C₄×C₁₁₃	#433	1	2	yes	—
86	C₄×C₁₁₃	C₄×SD₁₆	1	2	yes	—
87	C₄×C₁₁₃	C₄×SD₁₆	1	2	yes	—
88	#8277	C₁₁₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
89	C₂×C₁₁₃	#1570	1	2	yes	—
90	C₂×C₁₁₃	#1572	1	2	yes	—
91	C₂×C₁₁₃	#1768	1	1	yes	characteristic
92	C₂×C₁₁₃	C₄×#106	1	1	yes	characteristic
93	C₂×C₁₁₃	#1778	1	1	yes	characteristic
94	#1767	C₂×C₁₁₃	1	1	yes	characteristic
95	C₄×C₄⋊C₈	C₂×C₁₁₃	1	1	yes	characteristic
96	#1744	C₂×C₁₁₃	1	1	yes	characteristic
97	C₂×#334	C₂×C₁₁₃	1	2	yes	—
98	#1277	C₂×C₁₁₃	1	2	yes	—
99	C₁₁₃	#8277	1	1	yes	characteristic, direct product factor, Sylow
100	C₂×C₄⋊C₈	C₂²×C₁₁₃	1	2	yes	—
101	C₂×C₄⋊C₈	C₂²×C₁₁₃	1	2	yes	—
102	C₂×#97	—	2	1	no	—
103	C₂×C₄×C₈	—	2	1	no	—
104	C₂×C₄×C₈	C₂²×C₁₁₃	1	1	yes	characteristic
105	C₂×C₄×C₈	C₂²×C₁₁₃	1	1	yes	characteristic
106	C₄³	C₂²×C₁₁₃	1	1	yes	characteristic
107	#336	—	2	1	no	—
108	#336	—	2	1	no	—
109	#336	C₄×C₁₁₃	1	4	yes	—
110	#334	C₄×C₁₁₃	1	8	yes	complemented
111	C₂²×C₈	—	2	2	no	—
112	C₂²×C₈	—	2	2	no	—
113	C₂²×C₈	C₂×C₄×C₁₁₃	1	2	yes	—
114	C₂²×C₈	C₂×C₄×C₁₁₃	1	2	yes	—
115	C₂×C₄²	C₂×C₄×C₁₁₃	1	2	yes	—
116	C₂×C₄²	D₄×C₁₁₃	1	1	yes	characteristic
117	C₂×C₄²	D₄×C₁₁₃	1	1	yes	characteristic
118	C₂×C₄²	Q₈×C₁₁₃	1	2	yes	—
119	C₂×C₄²	C₂³×C₁₁₃	1	1	yes	characteristic
120	C₄⋊C₈	—	2	2	no	—
121	C₄⋊C₈	—	2	2	no	—
122	C₄⋊C₈	C₂×C₄×C₁₁₃	1	4	yes	—
123	C₄⋊C₈	C₂×C₄×C₁₁₃	1	4	yes	—
124	#97	—	4	2	no	—
125	C₄×C₈	—	2	2	no	—
126	C₄×C₈	—	4	2	no	—
127	C₄×C₈	C₂×C₄×C₁₁₃	1	4	yes	—
128	C₂²×C₄	C₄⋊C₄×C₁₁₃	1	2	yes	—
129	C₂²×C₄	C₄⋊C₄×C₁₁₃	1	2	yes	—
130	C₂²×C₄	C₂²×C₄×C₁₁₃	1	1	yes	characteristic
131	C₂²×C₄	C₂×D₄×C₁₁₃	1	1	yes	characteristic
132	C₂²×C₄	C₂×Q₈×C₁₁₃	1	1	yes	characteristic
133	C₂×C₈	—	2	4	no	—
134	C₂×C₈	—	2	2	no	—
135	C₂×C₈	—	2	4	no	—
136	C₂×C₈	—	2	4	no	—
137	C₂×C₈	—	2	4	no	—
138	C₂×C₈	—	4	2	no	—
139	C₂×C₈	—	4	2	no	—
140	C₂×C₈	—	4	2	no	—
141	C₂×C₈	—	4	2	no	—
142	C₂×C₈	C₄²×C₁₁₃	1	4	yes	—
143	C₄²	—	2	4	no	—
144	C₄²	—	2	2	no	—
145	C₄²	—	4	2	no	—
146	C₄²	SD₁₆×C₁₁₃	1	2	yes	—
147	C₄²	SD₁₆×C₁₁₃	1	2	yes	—
148	C₄²	C₂²×C₄×C₁₁₃	1	2	yes	—
149	C₄²	#41×C₁₁₃	1	2	yes	—
150	C₂³	C₂×C₄⋊C₄×C₁₁₃	1	1	yes	characteristic
151	C₂×C₄	—	2	2	no	—
152	C₂×C₄	—	2	2	no	—
153	C₂×C₄	—	2	2	no	—
154	C₂×C₄	—	2	2	no	—
155	C₂×C₄	—	2	2	no	—
156	C₂×C₄	—	2	2	no	—
157	C₂×C₄	#106×C₁₁₃	1	4	yes	—
158	C₂×C₄	C₂×C₄²×C₁₁₃	1	1	yes	characteristic
159	C₂×C₄	#117×C₁₁₃	1	1	yes	characteristic
160	C₂×C₄	C₄×D₄×C₁₁₃	1	2	yes	—
161	C₂×C₄	C₄×Q₈×C₁₁₃	1	2	yes	—
162	C₂×C₄	#131×C₁₁₃	1	4	yes	—
163	C₂×C₄	C₂×SD₁₆×C₁₁₃	1	1	yes	characteristic
164	C₂×C₄	C₂×SD₁₆×C₁₁₃	1	1	yes	characteristic
165	C₂×C₄	#135×C₁₁₃	1	2	yes	—
166	C₈	—	2	4	no	—
167	C₈	—	4	2	no	—
168	C₈	—	4	4	no	—
169	C₈	—	4	4	no	—
170	C₂²	C₄×C₄⋊C₄×C₁₁₃	1	1	yes	characteristic
171	C₂²	#424×C₁₁₃	1	2	yes	—
172	C₂²	C₂×#106×C₁₁₃	1	1	yes	characteristic
173	C₂²	#427×C₁₁₃	1	1	yes	characteristic
174	C₂²	#430×C₁₁₃	1	2	yes	—
175	C₄	—	2	2	no	—
176	C₄	—	2	4	no	—
177	C₄	—	4	2	no	—
178	C₄	#405×C₁₁₃	1	2	yes	—
179	C₄	#433×C₁₁₃	1	2	yes	—
180	C₄	C₄×SD₁₆×C₁₁₃	1	2	yes	—
181	C₄	C₄×SD₁₆×C₁₁₃	1	2	yes	—
182	C₂	C₁₁₃×#1570	1	2	yes	—
183	C₂	C₁₁₃×#1572	1	2	yes	—
184	C₂	C₁₁₃×#1768	1	1	yes	characteristic
185	C₂	C₄×#106×C₁₁₃	1	1	yes	characteristic
186	C₂	C₁₁₃×#1778	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄×C₁₁₃	C₂×D₄
commutator	C₂×C₄	C₂×C₄²×C₁₁₃
Frattini, cosocle	C₂×C₄²	C₂³×C₁₁₃
socle, prime radical	C₂³×C₁₁₃	C₂×C₄⋊C₄
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₄×C₁₁₃ ◁ C₄³×C₁₁₃ ◁ G

Complements

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

N	H ≅ G/N	type
16. #334×C₁₁₃	C₄	semidirect
88. #8277	C₁₁₃	direct (N × H)
99. C₁₁₃	#8277	direct (N × H)
110. #334	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×#1767	2
C₄×C₄⋊C₈×C₁₁₃	2
C₁₁₃×#1744	2
C₂×#334×C₁₁₃	2
C₁₁₃×#1277	2
#8277	113

Subgroups — C113×#8277 #43276297

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₁₃×#8277 #43276297