Subgroups — C₁₁₃×#20762 #43288782

Order 28928.

910 conjugacy classes of subgroups.

The subgroup lattice has 168 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×#97×C₁₁₃	C₂	1	1	yes	maximal, characteristic
2	C₁₁₃×#2285	C₂	1	2	yes	maximal
3	C₁₁₃×#1836	C₂	1	8	yes	maximal, complemented
4	C₁₁₃×#1735	C₂	1	4	yes	maximal
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	C₂×#97×C₁₁₃	C₄	1	8	yes	complemented
9	C₂×#97×C₁₁₃	C₂²	1	4	yes	—
10	#384×C₁₁₃	C₂²	1	4	yes	—
11	#382×C₁₁₃	C₂²	1	4	yes	—
12	#380×C₁₁₃	—	2	4	no	—
13	C₄×C₄⋊C₄×C₁₁₃	—	2	4	no	—
14	#340×C₁₁₃	C₂²	1	16	yes	—
15	C₂³×C₄×C₁₁₃	D₄	1	2	yes	—
16	C₂³×C₄×C₁₁₃	C₂³	1	1	yes	characteristic
17	C₂²×C₈×C₁₁₃	—	2	8	no	—
18	C₂²×C₈×C₁₁₃	C₂×C₄	1	8	yes	—
19	C₂²×C₈×C₁₁₃	C₂³	1	4	yes	—
20	#117×C₁₁₃	—	2	16	no	—
21	C₂×C₄⋊C₄×C₁₁₃	—	2	8	no	—
22	C₂×C₄⋊C₄×C₁₁₃	C₂³	1	8	yes	—
23	C₂×C₂²⋊C₄×C₁₁₃	—	2	4	no	—
24	C₂×C₄²×C₁₁₃	—	2	4	no	—
25	C₂×C₄²×C₁₁₃	C₂×C₄	1	4	yes	—
26	C₂×C₄²×C₁₁₃	D₄	1	2	yes	—
27	C₂×C₄²×C₁₁₃	Q₈	1	4	yes	—
28	C₂×C₄²×C₁₁₃	C₂³	1	2	yes	—
29	#97×C₁₁₃	C₂×C₄	1	16	yes	—
30	#95×C₁₁₃	—	2	8	no	—
31	C₂⁴×C₁₁₃	C₂×D₄	1	1	yes	characteristic
32	C₂²×C₄×C₁₁₃	—	2	8	no	—
33	C₂²×C₄×C₁₁₃	—	4	4	no	—
34	C₂²×C₄×C₁₁₃	C₄⋊C₄	1	8	yes	—
35	C₂²×C₄×C₁₁₃	C₂²×C₄	1	2	yes	—
36	C₂²×C₄×C₁₁₃	C₂×D₄	1	4	yes	—
37	C₂²×C₄×C₁₁₃	C₂×D₄	1	1	yes	characteristic
38	C₂²×C₄×C₁₁₃	C₂×Q₈	1	2	yes	—
39	C₂²×C₄×C₁₁₃	#41	1	8	yes	—
40	C₂²×C₄×C₁₁₃	C₂⁴	1	1	yes	characteristic
41	C₂×C₈×C₁₁₃	—	2	16	no	—
42	C₂×C₈×C₁₁₃	—	2	8	no	—
43	C₂×C₈×C₁₁₃	C₂²×C₄	1	8	yes	—
44	C₄⋊C₄×C₁₁₃	—	2	16	no	—
45	C₄⋊C₄×C₁₁₃	—	4	16	no	—
46	C₂²⋊C₄×C₁₁₃	—	4	8	no	—
47	C₄²×C₁₁₃	—	4	8	no	—
48	C₄²×C₁₁₃	C₄⋊C₄	1	8	yes	—
49	C₄²×C₁₁₃	C₂²×C₄	1	4	yes	—
50	C₄²×C₁₁₃	C₂×Q₈	1	4	yes	—
51	C₂³×C₁₁₃	—	2	4	no	—
52	C₂³×C₁₁₃	C₂×C₄⋊C₄	1	2	yes	—
53	C₂³×C₁₁₃	#124	1	4	yes	—
54	C₂³×C₁₁₃	C₂²×D₄	1	1	yes	characteristic
55	C₂×C₄×C₁₁₃	—	2	8	no	—
56	C₂×C₄×C₁₁₃	—	4	8	no	—
57	C₂×C₄×C₁₁₃	—	4	8	no	—
58	C₂×C₄×C₁₁₃	—	4	16	no	—
59	C₂×C₄×C₁₁₃	C₂×C₄⋊C₄	1	8	yes	—
60	C₂×C₄×C₁₁₃	C₂×C₄⋊C₄	1	2	yes	—
61	C₂×C₄×C₁₁₃	#117	1	8	yes	—
62	C₂×C₄×C₁₁₃	#124	1	4	yes	—
63	C₂×C₄×C₁₁₃	#125	1	8	yes	—
64	C₂×C₄×C₁₁₃	C₂³×C₄	1	1	yes	characteristic
65	C₂×C₄×C₁₁₃	C₂²×Q₈	1	1	yes	characteristic
66	C₂×C₄×C₁₁₃	C₂×#41	1	4	yes	—
67	C₈×C₁₁₃	—	2	8	no	—
68	C₂²×C₁₁₃	—	2	8	no	—
69	C₂²×C₁₁₃	—	2	4	no	—
70	C₂²×C₁₁₃	#382	1	4	yes	—
71	C₂²×C₁₁₃	C₂²×C₄⋊C₄	1	1	yes	characteristic
72	C₂²×C₁₁₃	C₂×#124	1	2	yes	—
73	C₂²×C₁₁₃	#575	1	4	yes	—
74	C₄×C₁₁₃	—	2	8	no	—
75	C₄×C₁₁₃	—	8	8	no	—
76	C₄×C₁₁₃	#382	1	4	yes	—
77	C₄×C₁₁₃	C₂×#117	1	2	yes	—
78	C₄×C₁₁₃	C₂×#125	1	2	yes	—
79	#20762	C₁₁₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
80	C₂×C₁₁₃	—	2	4	no	—
81	C₂×C₁₁₃	C₂×#382	1	1	yes	characteristic
82	C₂×C₁₁₃	#2911	1	2	yes	—
83	C₂×C₁₁₃	#3135	1	4	yes	—
84	C₂²×#97	C₂×C₁₁₃	1	1	yes	characteristic
85	#2285	C₂×C₁₁₃	1	2	yes	—
86	#1836	C₂×C₁₁₃	1	8	yes	complemented
87	#1735	C₂×C₁₁₃	1	4	yes	—
88	C₁₁₃	#20762	1	1	yes	characteristic, direct product factor, Sylow
89	C₂³×C₈	C₂²×C₁₁₃	1	2	yes	—
90	C₂×#117	C₂²×C₁₁₃	1	4	yes	—
91	C₂²×C₄²	C₂²×C₁₁₃	1	1	yes	characteristic
92	C₂×#97	C₄×C₁₁₃	1	8	yes	complemented
93	C₂×#97	C₂²×C₁₁₃	1	4	yes	—
94	#384	C₂²×C₁₁₃	1	4	yes	—
95	#382	C₂²×C₁₁₃	1	4	yes	—
96	#380	—	2	4	no	—
97	C₄×C₄⋊C₄	—	2	4	no	—
98	#340	C₂²×C₁₁₃	1	16	yes	—
99	C₂³×C₄	D₄×C₁₁₃	1	2	yes	—
100	C₂³×C₄	C₂³×C₁₁₃	1	1	yes	characteristic
101	C₂²×C₈	—	2	8	no	—
102	C₂²×C₈	C₂×C₄×C₁₁₃	1	8	yes	—
103	C₂²×C₈	C₂³×C₁₁₃	1	4	yes	—
104	#117	—	2	16	no	—
105	C₂×C₄⋊C₄	—	2	8	no	—
106	C₂×C₄⋊C₄	C₂³×C₁₁₃	1	8	yes	—
107	C₂×C₂²⋊C₄	—	2	4	no	—
108	C₂×C₄²	—	2	4	no	—
109	C₂×C₄²	C₂×C₄×C₁₁₃	1	4	yes	—
110	C₂×C₄²	D₄×C₁₁₃	1	2	yes	—
111	C₂×C₄²	Q₈×C₁₁₃	1	4	yes	—
112	C₂×C₄²	C₂³×C₁₁₃	1	2	yes	—
113	#97	C₂×C₄×C₁₁₃	1	16	yes	—
114	#95	—	2	8	no	—
115	C₂⁴	C₂×D₄×C₁₁₃	1	1	yes	characteristic
116	C₂²×C₄	—	2	8	no	—
117	C₂²×C₄	—	4	4	no	—
118	C₂²×C₄	C₄⋊C₄×C₁₁₃	1	8	yes	—
119	C₂²×C₄	C₂²×C₄×C₁₁₃	1	2	yes	—
120	C₂²×C₄	C₂×D₄×C₁₁₃	1	4	yes	—
121	C₂²×C₄	C₂×D₄×C₁₁₃	1	1	yes	characteristic
122	C₂²×C₄	C₂×Q₈×C₁₁₃	1	2	yes	—
123	C₂²×C₄	#41×C₁₁₃	1	8	yes	—
124	C₂²×C₄	C₂⁴×C₁₁₃	1	1	yes	characteristic
125	C₂×C₈	—	2	16	no	—
126	C₂×C₈	—	2	8	no	—
127	C₂×C₈	C₂²×C₄×C₁₁₃	1	8	yes	—
128	C₄⋊C₄	—	2	16	no	—
129	C₄⋊C₄	—	4	16	no	—
130	C₂²⋊C₄	—	4	8	no	—
131	C₄²	—	4	8	no	—
132	C₄²	C₄⋊C₄×C₁₁₃	1	8	yes	—
133	C₄²	C₂²×C₄×C₁₁₃	1	4	yes	—
134	C₄²	C₂×Q₈×C₁₁₃	1	4	yes	—
135	C₂³	—	2	4	no	—
136	C₂³	C₂×C₄⋊C₄×C₁₁₃	1	2	yes	—
137	C₂³	#124×C₁₁₃	1	4	yes	—
138	C₂³	C₂²×D₄×C₁₁₃	1	1	yes	characteristic
139	C₂×C₄	—	2	8	no	—
140	C₂×C₄	—	4	8	no	—
141	C₂×C₄	—	4	8	no	—
142	C₂×C₄	—	4	16	no	—
143	C₂×C₄	C₂×C₄⋊C₄×C₁₁₃	1	8	yes	—
144	C₂×C₄	C₂×C₄⋊C₄×C₁₁₃	1	2	yes	—
145	C₂×C₄	#117×C₁₁₃	1	8	yes	—
146	C₂×C₄	#124×C₁₁₃	1	4	yes	—
147	C₂×C₄	#125×C₁₁₃	1	8	yes	—
148	C₂×C₄	C₂³×C₄×C₁₁₃	1	1	yes	characteristic
149	C₂×C₄	C₂²×Q₈×C₁₁₃	1	1	yes	characteristic
150	C₂×C₄	C₂×#41×C₁₁₃	1	4	yes	—
151	C₈	—	2	8	no	—
152	C₂²	—	2	8	no	—
153	C₂²	—	2	4	no	—
154	C₂²	#382×C₁₁₃	1	4	yes	—
155	C₂²	C₂²×C₄⋊C₄×C₁₁₃	1	1	yes	characteristic
156	C₂²	C₂×#124×C₁₁₃	1	2	yes	—
157	C₂²	#575×C₁₁₃	1	4	yes	—
158	C₄	—	2	8	no	—
159	C₄	—	8	8	no	—
160	C₄	#382×C₁₁₃	1	4	yes	—
161	C₄	C₂×#117×C₁₁₃	1	2	yes	—
162	C₄	C₂×#125×C₁₁₃	1	2	yes	—
163	C₂	—	2	4	no	—
164	C₂	C₂×#382×C₁₁₃	1	1	yes	characteristic
165	C₂	C₁₁₃×#2911	1	2	yes	—
166	C₂	C₁₁₃×#3135	1	4	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
3. C₁₁₃×#1836	C₂	semidirect
8. C₂×#97×C₁₁₃	C₄	semidirect
79. #20762	C₁₁₃	direct (N × H)
86. #1836	C₂×C₁₁₃	semidirect
88. C₁₁₃	#20762	direct (N × H)
92. C₂×#97	C₄×C₁₁₃	semidirect

Maximal subgroups

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

subgroup	index
C₂²×#97×C₁₁₃	2
C₁₁₃×#2285	2
C₁₁₃×#1836	2
C₁₁₃×#1735	2
#20762	113

Subgroups — C113×#20762 #43288782

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₁₃×#20762 #43288782