Subgroups — C₁₁₃×#20799 #43288819

Order 28928.

1086 conjugacy classes of subgroups.

The subgroup lattice has 194 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#427×C₁₁₃	C₂	1	4	yes	maximal
2	C₂²×#97×C₁₁₃	C₂	1	1	yes	maximal, characteristic
3	C₁₁₃×#2285	C₂	1	2	yes	maximal
4	C₁₁₃×#1847	C₂	1	8	yes	maximal, complemented
5	C₂³×C₈×C₁₁₃	C₂²	1	2	yes	—
6	C₂×#117×C₁₁₃	C₂²	1	4	yes	—
7	C₂²×C₄²×C₁₁₃	C₂²	1	1	yes	characteristic
8	#427×C₁₁₃	—	2	16	no	—
9	C₂×#107×C₁₁₃	C₂²	1	8	yes	—
10	C₂×#106×C₁₁₃	C₂²	1	8	yes	—
11	C₂×#97×C₁₁₃	C₄	1	8	yes	complemented
12	C₂×#97×C₁₁₃	C₂²	1	4	yes	—
13	#384×C₁₁₃	C₂²	1	4	yes	—
14	#382×C₁₁₃	C₂²	1	4	yes	—
15	#380×C₁₁₃	—	2	4	no	—
16	C₄×C₄⋊C₄×C₁₁₃	—	2	4	no	—
17	C₂³×C₄×C₁₁₃	D₄	1	2	yes	—
18	C₂³×C₄×C₁₁₃	C₂³	1	1	yes	characteristic
19	C₂²×C₈×C₁₁₃	C₂×C₄	1	8	yes	—
20	C₂²×C₈×C₁₁₃	D₄	1	8	yes	—
21	C₂²×C₈×C₁₁₃	Q₈	1	8	yes	—
22	C₂²×C₈×C₁₁₃	C₂³	1	4	yes	—
23	#117×C₁₁₃	—	2	16	no	—
24	C₂×C₄⋊C₄×C₁₁₃	—	2	8	no	—
25	C₂×C₄⋊C₄×C₁₁₃	C₂³	1	8	yes	—
26	C₂×C₂²⋊C₄×C₁₁₃	—	2	4	no	—
27	C₂×C₄²×C₁₁₃	—	2	4	no	—
28	C₂×C₄²×C₁₁₃	C₂×C₄	1	4	yes	—
29	C₂×C₄²×C₁₁₃	D₄	1	2	yes	—
30	C₂×C₄²×C₁₁₃	Q₈	1	4	yes	—
31	C₂×C₄²×C₁₁₃	C₂³	1	2	yes	—
32	#107×C₁₁₃	—	2	16	no	—
33	#106×C₁₁₃	—	2	16	no	—
34	#97×C₁₁₃	C₂×C₄	1	16	yes	—
35	#95×C₁₁₃	—	2	8	no	—
36	C₂⁴×C₁₁₃	C₂×D₄	1	1	yes	characteristic
37	C₂²×C₄×C₁₁₃	—	2	8	no	—
38	C₂²×C₄×C₁₁₃	—	4	4	no	—
39	C₂²×C₄×C₁₁₃	C₄⋊C₄	1	8	yes	—
40	C₂²×C₄×C₁₁₃	C₂²×C₄	1	2	yes	—
41	C₂²×C₄×C₁₁₃	C₂×D₄	1	4	yes	—
42	C₂²×C₄×C₁₁₃	C₂×D₄	1	4	yes	—
43	C₂²×C₄×C₁₁₃	C₂×D₄	1	1	yes	characteristic
44	C₂²×C₄×C₁₁₃	C₂×Q₈	1	4	yes	—
45	C₂²×C₄×C₁₁₃	C₂×Q₈	1	2	yes	—
46	C₂²×C₄×C₁₁₃	C₂⁴	1	1	yes	characteristic
47	C₂×C₈×C₁₁₃	C₄⋊C₄	1	32	yes	complemented
48	C₂×C₈×C₁₁₃	C₂²×C₄	1	8	yes	—
49	C₂×C₈×C₁₁₃	C₂×D₄	1	8	yes	—
50	C₂×C₈×C₁₁₃	C₂×Q₈	1	8	yes	—
51	C₄⋊C₄×C₁₁₃	—	2	16	no	—
52	C₄⋊C₄×C₁₁₃	—	4	16	no	—
53	C₂²⋊C₄×C₁₁₃	—	4	8	no	—
54	C₄²×C₁₁₃	—	4	8	no	—
55	C₄²×C₁₁₃	C₄⋊C₄	1	8	yes	—
56	C₄²×C₁₁₃	C₂²×C₄	1	4	yes	—
57	C₄²×C₁₁₃	C₂×Q₈	1	4	yes	—
58	C₂³×C₁₁₃	—	2	4	no	—
59	C₂³×C₁₁₃	C₂×C₄⋊C₄	1	2	yes	—
60	C₂³×C₁₁₃	#127	1	2	yes	—
61	C₂³×C₁₁₃	C₄⋊Q₈	1	2	yes	—
62	C₂³×C₁₁₃	C₂²×D₄	1	1	yes	characteristic
63	C₂×C₄×C₁₁₃	—	2	8	no	—
64	C₂×C₄×C₁₁₃	—	4	8	no	—
65	C₂×C₄×C₁₁₃	—	4	8	no	—
66	C₂×C₄×C₁₁₃	—	4	16	no	—
67	C₂×C₄×C₁₁₃	C₂×C₄⋊C₄	1	8	yes	—
68	C₂×C₄×C₁₁₃	C₂×C₄⋊C₄	1	8	yes	—
69	C₂×C₄×C₁₁₃	C₂×C₄⋊C₄	1	2	yes	—
70	C₂×C₄×C₁₁₃	#127	1	2	yes	—
71	C₂×C₄×C₁₁₃	C₄⋊Q₈	1	8	yes	—
72	C₂×C₄×C₁₁₃	C₄⋊Q₈	1	2	yes	—
73	C₂×C₄×C₁₁₃	C₂³×C₄	1	1	yes	characteristic
74	C₂×C₄×C₁₁₃	C₂²×D₄	1	2	yes	—
75	C₂×C₄×C₁₁₃	C₂²×Q₈	1	2	yes	—
76	C₂×C₄×C₁₁₃	C₂²×Q₈	1	1	yes	characteristic
77	C₈×C₁₁₃	C₂×C₄⋊C₄	1	16	yes	—
78	C₂²×C₁₁₃	—	2	4	no	—
79	C₂²×C₁₁₃	—	2	4	no	—
80	C₂²×C₁₁₃	—	2	4	no	—
81	C₂²×C₁₁₃	#384	1	4	yes	—
82	C₂²×C₁₁₃	C₂²×C₄⋊C₄	1	1	yes	characteristic
83	C₂²×C₁₁₃	C₂×#127	1	1	yes	characteristic
84	C₂²×C₁₁₃	C₂×C₄⋊Q₈	1	1	yes	characteristic
85	C₂²×C₁₁₃	#575	1	4	yes	—
86	C₄×C₁₁₃	—	2	8	no	—
87	C₄×C₁₁₃	—	8	8	no	—
88	C₄×C₁₁₃	#384	1	4	yes	—
89	C₄×C₁₁₃	C₂²×C₄⋊C₄	1	2	yes	—
90	C₄×C₁₁₃	C₂×C₄⋊Q₈	1	2	yes	—
91	#20799	C₁₁₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
92	C₂×C₁₁₃	—	2	4	no	—
93	C₂×C₁₁₃	C₂×#384	1	1	yes	characteristic
94	C₂×C₁₁₃	#2911	1	2	yes	—
95	C₂×C₁₁₃	#3150	1	2	yes	—
96	C₂×C₁₁₃	#3162	1	2	yes	—
97	C₂×#427	C₂×C₁₁₃	1	4	yes	—
98	C₂²×#97	C₂×C₁₁₃	1	1	yes	characteristic
99	#2285	C₂×C₁₁₃	1	2	yes	—
100	#1847	C₂×C₁₁₃	1	8	yes	complemented
101	C₁₁₃	#20799	1	1	yes	characteristic, direct product factor, Sylow
102	C₂³×C₈	C₂²×C₁₁₃	1	2	yes	—
103	C₂×#117	C₂²×C₁₁₃	1	4	yes	—
104	C₂²×C₄²	C₂²×C₁₁₃	1	1	yes	characteristic
105	#427	—	2	16	no	—
106	C₂×#107	C₂²×C₁₁₃	1	8	yes	—
107	C₂×#106	C₂²×C₁₁₃	1	8	yes	—
108	C₂×#97	C₄×C₁₁₃	1	8	yes	complemented
109	C₂×#97	C₂²×C₁₁₃	1	4	yes	—
110	#384	C₂²×C₁₁₃	1	4	yes	—
111	#382	C₂²×C₁₁₃	1	4	yes	—
112	#380	—	2	4	no	—
113	C₄×C₄⋊C₄	—	2	4	no	—
114	C₂³×C₄	D₄×C₁₁₃	1	2	yes	—
115	C₂³×C₄	C₂³×C₁₁₃	1	1	yes	characteristic
116	C₂²×C₈	C₂×C₄×C₁₁₃	1	8	yes	—
117	C₂²×C₈	D₄×C₁₁₃	1	8	yes	—
118	C₂²×C₈	Q₈×C₁₁₃	1	8	yes	—
119	C₂²×C₈	C₂³×C₁₁₃	1	4	yes	—
120	#117	—	2	16	no	—
121	C₂×C₄⋊C₄	—	2	8	no	—
122	C₂×C₄⋊C₄	C₂³×C₁₁₃	1	8	yes	—
123	C₂×C₂²⋊C₄	—	2	4	no	—
124	C₂×C₄²	—	2	4	no	—
125	C₂×C₄²	C₂×C₄×C₁₁₃	1	4	yes	—
126	C₂×C₄²	D₄×C₁₁₃	1	2	yes	—
127	C₂×C₄²	Q₈×C₁₁₃	1	4	yes	—
128	C₂×C₄²	C₂³×C₁₁₃	1	2	yes	—
129	#107	—	2	16	no	—
130	#106	—	2	16	no	—
131	#97	C₂×C₄×C₁₁₃	1	16	yes	—
132	#95	—	2	8	no	—
133	C₂⁴	C₂×D₄×C₁₁₃	1	1	yes	characteristic
134	C₂²×C₄	—	2	8	no	—
135	C₂²×C₄	—	4	4	no	—
136	C₂²×C₄	C₄⋊C₄×C₁₁₃	1	8	yes	—
137	C₂²×C₄	C₂²×C₄×C₁₁₃	1	2	yes	—
138	C₂²×C₄	C₂×D₄×C₁₁₃	1	4	yes	—
139	C₂²×C₄	C₂×D₄×C₁₁₃	1	1	yes	characteristic
140	C₂²×C₄	C₂×D₄×C₁₁₃	1	4	yes	—
141	C₂²×C₄	C₂×Q₈×C₁₁₃	1	4	yes	—
142	C₂²×C₄	C₂×Q₈×C₁₁₃	1	2	yes	—
143	C₂²×C₄	C₂⁴×C₁₁₃	1	1	yes	characteristic
144	C₂×C₈	C₄⋊C₄×C₁₁₃	1	32	yes	complemented
145	C₂×C₈	C₂²×C₄×C₁₁₃	1	8	yes	—
146	C₂×C₈	C₂×D₄×C₁₁₃	1	8	yes	—
147	C₂×C₈	C₂×Q₈×C₁₁₃	1	8	yes	—
148	C₄⋊C₄	—	2	16	no	—
149	C₄⋊C₄	—	4	16	no	—
150	C₂²⋊C₄	—	4	8	no	—
151	C₄²	—	4	8	no	—
152	C₄²	C₄⋊C₄×C₁₁₃	1	8	yes	—
153	C₄²	C₂²×C₄×C₁₁₃	1	4	yes	—
154	C₄²	C₂×Q₈×C₁₁₃	1	4	yes	—
155	C₂³	—	2	4	no	—
156	C₂³	C₂×C₄⋊C₄×C₁₁₃	1	2	yes	—
157	C₂³	#127×C₁₁₃	1	2	yes	—
158	C₂³	C₄⋊Q₈×C₁₁₃	1	2	yes	—
159	C₂³	C₂²×D₄×C₁₁₃	1	1	yes	characteristic
160	C₂×C₄	—	2	8	no	—
161	C₂×C₄	—	4	8	no	—
162	C₂×C₄	—	4	8	no	—
163	C₂×C₄	—	4	16	no	—
164	C₂×C₄	C₂×C₄⋊C₄×C₁₁₃	1	8	yes	—
165	C₂×C₄	C₂×C₄⋊C₄×C₁₁₃	1	8	yes	—
166	C₂×C₄	C₂×C₄⋊C₄×C₁₁₃	1	2	yes	—
167	C₂×C₄	#127×C₁₁₃	1	2	yes	—
168	C₂×C₄	C₄⋊Q₈×C₁₁₃	1	8	yes	—
169	C₂×C₄	C₄⋊Q₈×C₁₁₃	1	2	yes	—
170	C₂×C₄	C₂³×C₄×C₁₁₃	1	1	yes	characteristic
171	C₂×C₄	C₂²×D₄×C₁₁₃	1	2	yes	—
172	C₂×C₄	C₂²×Q₈×C₁₁₃	1	2	yes	—
173	C₂×C₄	C₂²×Q₈×C₁₁₃	1	1	yes	characteristic
174	C₈	C₂×C₄⋊C₄×C₁₁₃	1	16	yes	—
175	C₂²	—	2	4	no	—
176	C₂²	—	2	4	no	—
177	C₂²	—	2	4	no	—
178	C₂²	#384×C₁₁₃	1	4	yes	—
179	C₂²	C₂²×C₄⋊C₄×C₁₁₃	1	1	yes	characteristic
180	C₂²	C₂×#127×C₁₁₃	1	1	yes	characteristic
181	C₂²	C₂×C₄⋊Q₈×C₁₁₃	1	1	yes	characteristic
182	C₂²	#575×C₁₁₃	1	4	yes	—
183	C₄	—	2	8	no	—
184	C₄	—	8	8	no	—
185	C₄	#384×C₁₁₃	1	4	yes	—
186	C₄	C₂²×C₄⋊C₄×C₁₁₃	1	2	yes	—
187	C₄	C₂×C₄⋊Q₈×C₁₁₃	1	2	yes	—
188	C₂	—	2	4	no	—
189	C₂	C₂×#384×C₁₁₃	1	1	yes	characteristic
190	C₂	C₁₁₃×#2911	1	2	yes	—
191	C₂	C₁₁₃×#3150	1	2	yes	—
192	C₂	C₁₁₃×#3162	1	2	yes	—

	subgroup	G/H
center	C₂²×C₄×C₁₁₃	C₂×D₄
commutator	C₂×C₄	C₂³×C₄×C₁₁₃
Frattini, cosocle	C₂²×C₄	C₂⁴×C₁₁₃
socle	C₂³×C₁₁₃	C₂²×D₄
prime radical	C₂⁴×C₁₁₃	C₂×D₄
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₄×C₁₁₃ ◁ C₂²×C₄²×C₁₁₃ ◁ G

Complements

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

N	H ≅ G/N	type
4. C₁₁₃×#1847	C₂	semidirect
11. C₂×#97×C₁₁₃	C₄	semidirect
47. C₂×C₈×C₁₁₃	C₄⋊C₄	semidirect
91. #20799	C₁₁₃	direct (N × H)
100. #1847	C₂×C₁₁₃	semidirect
101. C₁₁₃	#20799	direct (N × H)
108. C₂×#97	C₄×C₁₁₃	semidirect
144. C₂×C₈	C₄⋊C₄×C₁₁₃	semidirect

Maximal subgroups

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

subgroup	index
C₂×#427×C₁₁₃	2
C₂²×#97×C₁₁₃	2
C₁₁₃×#2285	2
C₁₁₃×#1847	2
#20799	113

Subgroups — C113×#20799 #43288819

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₁₃×#20799 #43288819