Subgroups — C₁₁₃×#22276 #43290296

Order 28928.

1050 conjugacy classes of subgroups.

The subgroup lattice has 206 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#493×C₁₁₃	C₂	1	1	yes	maximal, characteristic
2	C₁₁₃×#2328	C₂	1	2	yes	maximal, complemented
3	C₁₁₃×#1961	C₂	1	2	yes	maximal, complemented
4	C₁₁₃×#1879	C₂	1	8	yes	maximal, complemented
5	C₁₁₃×#1835	C₂	1	1	yes	maximal, characteristic, complemented
6	C₁₁₃×#1746	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂²×D₈×C₁₁₃	C₂²	1	4	yes	—
8	C₂×#127×C₁₁₃	C₂²	1	2	yes	—
9	C₂×C₄×D₄×C₁₁₃	C₂²	1	4	yes	—
10	#493×C₁₁₃	C₄	1	8	yes	complemented
11	C₂×#102×C₁₁₃	C₂²	1	8	yes	—
12	C₂×#97×C₁₁₃	C₂²	1	1	yes	characteristic
13	C₂×C₄×C₈×C₁₁₃	C₂²	1	1	yes	characteristic
14	#390×C₁₁₃	C₂²	1	4	yes	—
15	#386×C₁₁₃	—	2	8	no	—
16	#382×C₁₁₃	C₂²	1	2	yes	—
17	#376×C₁₁₃	C₂²	1	1	yes	characteristic
18	#340×C₁₁₃	C₂²	1	4	yes	—
19	#336×C₁₁₃	C₂²	1	4	yes	—
20	C₂²×D₄×C₁₁₃	—	2	4	no	—
21	C₂²×D₄×C₁₁₃	C₂³	1	4	yes	—
22	C₂³×C₄×C₁₁₃	—	2	4	no	—
23	C₂×D₈×C₁₁₃	—	4	8	no	—
24	C₂×D₈×C₁₁₃	C₂×C₄	1	16	yes	complemented
25	C₂²×C₈×C₁₁₃	C₂³	1	2	yes	—
26	C₂²×C₈×C₁₁₃	C₂³	1	2	yes	—
27	#127×C₁₁₃	—	2	4	no	—
28	#127×C₁₁₃	C₂×C₄	1	8	yes	complemented
29	C₄×D₄×C₁₁₃	—	4	8	no	—
30	C₂×C₄⋊C₄×C₁₁₃	C₂³	1	4	yes	—
31	C₂×C₂²⋊C₄×C₁₁₃	—	2	4	no	—
32	C₂×C₂²⋊C₄×C₁₁₃	—	2	8	no	—
33	C₂×C₄²×C₁₁₃	C₂³	1	2	yes	—
34	C₂×C₄²×C₁₁₃	C₂³	1	1	yes	characteristic
35	#102×C₁₁₃	—	4	8	no	—
36	#97×C₁₁₃	—	2	2	no	—
37	C₄×C₈×C₁₁₃	C₂×C₄	1	4	yes	complemented
38	#95×C₁₁₃	—	2	4	no	—
39	#95×C₁₁₃	D₄	1	4	yes	—
40	C₂⁴×C₁₁₃	—	2	4	no	—
41	C₂×D₄×C₁₁₃	—	2	16	no	—
42	C₂×D₄×C₁₁₃	—	2	4	no	—
43	C₂×D₄×C₁₁₃	—	2	8	no	—
44	C₂×D₄×C₁₁₃	—	2	8	no	—
45	C₂×D₄×C₁₁₃	—	8	8	no	—
46	C₂×D₄×C₁₁₃	C₂²×C₄	1	8	yes	—
47	C₂²×C₄×C₁₁₃	—	2	4	no	—
48	C₂²×C₄×C₁₁₃	—	4	8	no	—
49	C₂²×C₄×C₁₁₃	—	8	8	no	—
50	C₂²×C₄×C₁₁₃	C₂×D₄	1	4	yes	—
51	C₂²×C₄×C₁₁₃	C₂×D₄	1	2	yes	—
52	C₂²×C₄×C₁₁₃	C₂⁴	1	1	yes	characteristic
53	D₈×C₁₁₃	—	4	16	no	—
54	C₂×C₈×C₁₁₃	—	2	4	no	—
55	C₂×C₈×C₁₁₃	—	2	2	no	—
56	C₂×C₈×C₁₁₃	—	2	4	no	—
57	C₂×C₈×C₁₁₃	C₂²×C₄	1	4	yes	—
58	C₄⋊C₄×C₁₁₃	—	2	8	no	—
59	C₂²⋊C₄×C₁₁₃	—	4	8	no	—
60	C₂²⋊C₄×C₁₁₃	—	8	8	no	—
61	C₄²×C₁₁₃	—	2	4	no	—
62	C₄²×C₁₁₃	C₂²×C₄	1	2	yes	—
63	C₄²×C₁₁₃	#41	1	2	yes	—
64	C₂³×C₁₁₃	—	2	8	no	—
65	C₂³×C₁₁₃	—	4	4	no	—
66	C₂³×C₁₁₃	—	4	4	no	—
67	C₂³×C₁₁₃	—	4	8	no	—
68	C₂³×C₁₁₃	—	4	8	no	—
69	C₂³×C₁₁₃	C₂²×D₄	1	1	yes	characteristic
70	D₄×C₁₁₃	—	2	16	no	—
71	D₄×C₁₁₃	—	4	8	no	—
72	D₄×C₁₁₃	—	8	16	no	—
73	C₂×C₄×C₁₁₃	—	2	4	no	—
74	C₂×C₄×C₁₁₃	—	2	4	no	—
75	C₂×C₄×C₁₁₃	—	2	8	no	—
76	C₂×C₄×C₁₁₃	—	4	4	no	—
77	C₂×C₄×C₁₁₃	—	4	8	no	—
78	C₂×C₄×C₁₁₃	—	8	16	no	—
79	C₂×C₄×C₁₁₃	C₄×D₄	1	4	yes	—
80	C₂×C₄×C₁₁₃	C₂³×C₄	1	1	yes	characteristic
81	C₂×C₄×C₁₁₃	C₂×#41	1	1	yes	characteristic
82	C₂×C₄×C₁₁₃	H(5,𝔽₂)	1	4	yes	—
83	C₈×C₁₁₃	—	2	4	no	—
84	C₈×C₁₁₃	—	4	2	no	—
85	C₂²×C₁₁₃	—	4	8	no	—
86	C₂²×C₁₁₃	—	4	16	no	—
87	C₂²×C₁₁₃	—	8	4	no	—
88	C₂²×C₁₁₃	—	8	4	no	—
89	C₂²×C₁₁₃	—	8	8	no	—
90	C₂²×C₁₁₃	C₂×C₄×D₄	1	1	yes	characteristic
91	C₂²×C₁₁₃	#535	1	2	yes	—
92	C₂²×C₁₁₃	#576	1	4	yes	—
93	C₄×C₁₁₃	—	2	4	no	—
94	C₄×C₁₁₃	—	4	4	no	—
95	C₄×C₁₁₃	—	8	4	no	—
96	C₄×C₁₁₃	#518	1	2	yes	—
97	C₄×C₁₁₃	#540	1	2	yes	—
98	#22276	C₁₁₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
99	C₂×C₁₁₃	—	8	8	no	—
100	C₂×C₁₁₃	#2328	1	1	yes	characteristic
101	C₂×C₁₁₃	#2948	1	2	yes	—
102	C₂×C₁₁₃	#3220	1	4	yes	—
103	C₂×#493	C₂×C₁₁₃	1	1	yes	characteristic
104	#2328	C₂×C₁₁₃	1	2	yes	complemented
105	#1961	C₂×C₁₁₃	1	2	yes	complemented
106	#1879	C₂×C₁₁₃	1	8	yes	complemented
107	#1835	C₂×C₁₁₃	1	1	yes	characteristic, complemented
108	#1746	C₂×C₁₁₃	1	1	yes	characteristic, complemented
109	C₁₁₃	#22276	1	1	yes	characteristic, direct product factor, Sylow
110	C₂²×D₈	C₂²×C₁₁₃	1	4	yes	—
111	C₂×#127	C₂²×C₁₁₃	1	2	yes	—
112	C₂×C₄×D₄	C₂²×C₁₁₃	1	4	yes	—
113	#493	C₄×C₁₁₃	1	8	yes	complemented
114	C₂×#102	C₂²×C₁₁₃	1	8	yes	—
115	C₂×#97	C₂²×C₁₁₃	1	1	yes	characteristic
116	C₂×C₄×C₈	C₂²×C₁₁₃	1	1	yes	characteristic
117	#390	C₂²×C₁₁₃	1	4	yes	—
118	#386	—	2	8	no	—
119	#382	C₂²×C₁₁₃	1	2	yes	—
120	#376	C₂²×C₁₁₃	1	1	yes	characteristic
121	#340	C₂²×C₁₁₃	1	4	yes	—
122	#336	C₂²×C₁₁₃	1	4	yes	—
123	C₂²×D₄	—	2	4	no	—
124	C₂²×D₄	C₂³×C₁₁₃	1	4	yes	—
125	C₂³×C₄	—	2	4	no	—
126	C₂×D₈	—	4	8	no	—
127	C₂×D₈	C₂×C₄×C₁₁₃	1	16	yes	complemented
128	C₂²×C₈	C₂³×C₁₁₃	1	2	yes	—
129	C₂²×C₈	C₂³×C₁₁₃	1	2	yes	—
130	#127	—	2	4	no	—
131	#127	C₂×C₄×C₁₁₃	1	8	yes	complemented
132	C₄×D₄	—	4	8	no	—
133	C₂×C₄⋊C₄	C₂³×C₁₁₃	1	4	yes	—
134	C₂×C₂²⋊C₄	—	2	4	no	—
135	C₂×C₂²⋊C₄	—	2	8	no	—
136	C₂×C₄²	C₂³×C₁₁₃	1	2	yes	—
137	C₂×C₄²	C₂³×C₁₁₃	1	1	yes	characteristic
138	#102	—	4	8	no	—
139	#97	—	2	2	no	—
140	C₄×C₈	C₂×C₄×C₁₁₃	1	4	yes	complemented
141	#95	—	2	4	no	—
142	#95	D₄×C₁₁₃	1	4	yes	—
143	C₂⁴	—	2	4	no	—
144	C₂×D₄	—	2	16	no	—
145	C₂×D₄	—	2	4	no	—
146	C₂×D₄	—	2	8	no	—
147	C₂×D₄	—	2	8	no	—
148	C₂×D₄	—	8	8	no	—
149	C₂×D₄	C₂²×C₄×C₁₁₃	1	8	yes	—
150	C₂²×C₄	—	2	4	no	—
151	C₂²×C₄	—	4	8	no	—
152	C₂²×C₄	—	8	8	no	—
153	C₂²×C₄	C₂×D₄×C₁₁₃	1	4	yes	—
154	C₂²×C₄	C₂×D₄×C₁₁₃	1	2	yes	—
155	C₂²×C₄	C₂⁴×C₁₁₃	1	1	yes	characteristic
156	D₈	—	4	16	no	—
157	C₂×C₈	—	2	4	no	—
158	C₂×C₈	—	2	2	no	—
159	C₂×C₈	—	2	4	no	—
160	C₂×C₈	C₂²×C₄×C₁₁₃	1	4	yes	—
161	C₄⋊C₄	—	2	8	no	—
162	C₂²⋊C₄	—	4	8	no	—
163	C₂²⋊C₄	—	8	8	no	—
164	C₄²	—	2	4	no	—
165	C₄²	C₂²×C₄×C₁₁₃	1	2	yes	—
166	C₄²	#41×C₁₁₃	1	2	yes	—
167	C₂³	—	2	8	no	—
168	C₂³	—	4	4	no	—
169	C₂³	—	4	4	no	—
170	C₂³	—	4	8	no	—
171	C₂³	—	4	8	no	—
172	C₂³	C₂²×D₄×C₁₁₃	1	1	yes	characteristic
173	D₄	—	2	16	no	—
174	D₄	—	4	8	no	—
175	D₄	—	8	16	no	—
176	C₂×C₄	—	2	4	no	—
177	C₂×C₄	—	2	4	no	—
178	C₂×C₄	—	2	8	no	—
179	C₂×C₄	—	4	4	no	—
180	C₂×C₄	—	4	8	no	—
181	C₂×C₄	—	8	16	no	—
182	C₂×C₄	C₄×D₄×C₁₁₃	1	4	yes	—
183	C₂×C₄	C₂³×C₄×C₁₁₃	1	1	yes	characteristic
184	C₂×C₄	C₂×#41×C₁₁₃	1	1	yes	characteristic
185	C₂×C₄	H(5,𝔽₂)×C₁₁₃	1	4	yes	—
186	C₈	—	2	4	no	—
187	C₈	—	4	2	no	—
188	C₂²	—	4	8	no	—
189	C₂²	—	4	16	no	—
190	C₂²	—	8	4	no	—
191	C₂²	—	8	4	no	—
192	C₂²	—	8	8	no	—
193	C₂²	C₂×C₄×D₄×C₁₁₃	1	1	yes	characteristic
194	C₂²	#535×C₁₁₃	1	2	yes	—
195	C₂²	#576×C₁₁₃	1	4	yes	—
196	C₄	—	2	4	no	—
197	C₄	—	4	4	no	—
198	C₄	—	8	4	no	—
199	C₄	#518×C₁₁₃	1	2	yes	—
200	C₄	#540×C₁₁₃	1	2	yes	—
201	C₂	—	8	8	no	—
202	C₂	C₁₁₃×#2328	1	1	yes	characteristic
203	C₂	C₁₁₃×#2948	1	2	yes	—
204	C₂	C₁₁₃×#3220	1	4	yes	—

	subgroup	G/H
center, socle	C₂³×C₁₁₃	C₂²×D₄
commutator	C₂×C₄	C₂³×C₄×C₁₁₃
Frattini, cosocle	C₂²×C₄	C₂⁴×C₁₁₃
prime radical	C₂×#493×C₁₁₃	C₂
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₁₁₃ ◁ #376×C₁₁₃ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₁₁₃×#2328	C₂	semidirect
3. C₁₁₃×#1961	C₂	semidirect
4. C₁₁₃×#1879	C₂	semidirect
5. C₁₁₃×#1835	C₂	semidirect
6. C₁₁₃×#1746	C₂	semidirect
10. #493×C₁₁₃	C₄	semidirect
24. C₂×D₈×C₁₁₃	C₂×C₄	semidirect
28. #127×C₁₁₃	C₂×C₄	semidirect
37. C₄×C₈×C₁₁₃	C₂×C₄	semidirect
98. #22276	C₁₁₃	direct (N × H)
104. #2328	C₂×C₁₁₃	semidirect
105. #1961	C₂×C₁₁₃	semidirect
106. #1879	C₂×C₁₁₃	semidirect
107. #1835	C₂×C₁₁₃	semidirect
108. #1746	C₂×C₁₁₃	semidirect
109. C₁₁₃	#22276	direct (N × H)
113. #493	C₄×C₁₁₃	semidirect
127. C₂×D₈	C₂×C₄×C₁₁₃	semidirect
131. #127	C₂×C₄×C₁₁₃	semidirect
140. C₄×C₈	C₂×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×#493×C₁₁₃	2
C₁₁₃×#2328	2
C₁₁₃×#1961	2
C₁₁₃×#1879	2
C₁₁₃×#1835	2
C₁₁₃×#1746	2
#22276	113

Subgroups — C113×#22276 #43290296

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₁₃×#22276 #43290296