Subgroups — C₂×#418×C₂₂₃ #43251908

Order 28544.

1008 conjugacy classes of subgroups.

The subgroup lattice has 196 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂³×D₄×C₂₂₃	C₂	1	1	yes	maximal, characteristic
2	C₂²×M₁₆×C₂₂₃	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×#117×C₂₂₃	C₂	1	1	yes	maximal, characteristic, complemented
4	#418×C₂₂₃	C₂	1	8	yes	maximal, direct product factor
5	C₂×#102×C₂₂₃	C₂	1	4	yes	maximal, complemented
6	C₂⁵×C₂₂₃	—	2	1	no	—
7	C₂²×D₄×C₂₂₃	—	2	2	no	—
8	C₂²×D₄×C₂₂₃	—	2	1	no	—
9	C₂²×D₄×C₂₂₃	—	2	4	no	—
10	C₂²×D₄×C₂₂₃	C₄	1	8	yes	complemented
11	C₂²×D₄×C₂₂₃	C₂²	1	2	yes	—
12	C₂²×D₄×C₂₂₃	C₂²	1	4	yes	—
13	C₂³×C₄×C₂₂₃	C₂²	1	1	yes	characteristic
14	C₂×M₁₆×C₂₂₃	—	2	4	no	—
15	C₂×M₁₆×C₂₂₃	C₂²	1	4	yes	complemented
16	C₂²×C₈×C₂₂₃	C₂²	1	2	yes	complemented
17	#117×C₂₂₃	—	2	2	no	—
18	#117×C₂₂₃	C₂²	1	4	yes	complemented
19	C₂×C₄⋊C₄×C₂₂₃	C₂²	1	2	yes	complemented
20	C₂×C₂²⋊C₄×C₂₂₃	—	2	1	no	—
21	C₂×C₄²×C₂₂₃	—	2	1	no	—
22	#102×C₂₂₃	C₂²	1	16	yes	complemented
23	C₂⁴×C₂₂₃	—	2	8	no	—
24	C₂⁴×C₂₂₃	—	2	4	no	—
25	C₂⁴×C₂₂₃	—	2	2	no	—
26	C₂⁴×C₂₂₃	—	4	8	no	—
27	C₂⁴×C₂₂₃	D₄	1	1	yes	characteristic
28	C₂×D₄×C₂₂₃	—	2	16	no	—
29	C₂×D₄×C₂₂₃	—	2	8	no	—
30	C₂×D₄×C₂₂₃	—	2	4	no	—
31	C₂×D₄×C₂₂₃	—	2	2	no	—
32	C₂×D₄×C₂₂₃	—	2	4	no	—
33	C₂×D₄×C₂₂₃	—	2	8	no	—
34	C₂×D₄×C₂₂₃	C₂×C₄	1	8	yes	—
35	C₂×D₄×C₂₂₃	C₂×C₄	1	16	yes	complemented
36	C₂×D₄×C₂₂₃	C₂³	1	4	yes	—
37	C₂²×C₄×C₂₂₃	—	2	2	no	—
38	C₂²×C₄×C₂₂₃	C₂×C₄	1	4	yes	—
39	C₂²×C₄×C₂₂₃	D₄	1	2	yes	—
40	C₂²×C₄×C₂₂₃	D₄	1	1	yes	characteristic
41	C₂²×C₄×C₂₂₃	D₄	1	4	yes	—
42	C₂²×C₄×C₂₂₃	C₂³	1	2	yes	—
43	C₂²×C₄×C₂₂₃	C₂³	1	1	yes	characteristic
44	M₁₆×C₂₂₃	—	2	8	no	—
45	C₂×C₈×C₂₂₃	—	2	4	no	—
46	C₂×C₈×C₂₂₃	C₂³	1	4	yes	complemented
47	C₄⋊C₄×C₂₂₃	—	2	2	no	—
48	C₄⋊C₄×C₂₂₃	C₂³	1	4	yes	complemented
49	C₂²⋊C₄×C₂₂₃	—	2	4	no	—
50	C₄²×C₂₂₃	—	2	4	no	—
51	C₂³×C₂₂₃	—	2	4	no	—
52	C₂³×C₂₂₃	—	2	8	no	—
53	C₂³×C₂₂₃	—	2	16	no	—
54	C₂³×C₂₂₃	—	2	4	no	—
55	C₂³×C₂₂₃	—	4	16	no	—
56	C₂³×C₂₂₃	—	4	32	no	—
57	C₂³×C₂₂₃	—	4	8	no	—
58	C₂³×C₂₂₃	C₂²⋊C₄	1	4	yes	—
59	C₂³×C₂₂₃	C₂×D₄	1	2	yes	—
60	C₂³×C₂₂₃	C₂×D₄	1	1	yes	characteristic
61	D₄×C₂₂₃	—	2	8	no	—
62	D₄×C₂₂₃	—	2	16	no	—
63	D₄×C₂₂₃	—	2	4	no	—
64	D₄×C₂₂₃	C₂²×C₄	1	8	yes	—
65	C₂×C₄×C₂₂₃	—	2	4	no	—
66	C₂×C₄×C₂₂₃	—	4	4	no	—
67	C₂×C₄×C₂₂₃	C₂²⋊C₄	1	8	yes	—
68	C₂×C₄×C₂₂₃	C₂²⋊C₄	1	4	yes	—
69	C₂×C₄×C₂₂₃	C₂²×C₄	1	2	yes	—
70	C₂×C₄×C₂₂₃	C₂²×C₄	1	4	yes	—
71	C₂×C₄×C₂₂₃	C₂×D₄	1	2	yes	—
72	C₂×C₄×C₂₂₃	C₂×D₄	1	1	yes	characteristic
73	C₂×C₄×C₂₂₃	C₂×D₄	1	2	yes	—
74	C₂×C₄×C₂₂₃	C₂×D₄	1	4	yes	—
75	C₂×C₄×C₂₂₃	C₂⁴	1	1	yes	characteristic
76	C₈×C₂₂₃	—	2	4	no	—
77	C₂²×C₂₂₃	—	2	4	no	—
78	C₂²×C₂₂₃	—	2	8	no	—
79	C₂²×C₂₂₃	—	2	8	no	—
80	C₂²×C₂₂₃	—	4	32	no	—
81	C₂²×C₂₂₃	—	4	8	no	—
82	C₂²×C₂₂₃	—	4	16	no	—
83	C₂²×C₂₂₃	C₂×C₂²⋊C₄	1	4	yes	—
84	C₂²×C₂₂₃	C₂×C₂²⋊C₄	1	2	yes	—
85	C₂²×C₂₂₃	AGL(1,ℤ/8)	1	4	yes	—
86	C₂²×C₂₂₃	C₂²×D₄	1	1	yes	characteristic
87	C₄×C₂₂₃	—	4	4	no	—
88	C₄×C₂₂₃	C₂×C₂²⋊C₄	1	2	yes	—
89	C₄×C₂₂₃	C₂×C₂²⋊C₄	1	4	yes	—
90	C₄×C₂₂₃	C₂³×C₄	1	1	yes	characteristic
91	C₄×C₂₂₃	C₂²×D₄	1	1	yes	characteristic
92	C₂×C₂₂₃	—	2	4	no	—
93	C₂×C₂₂₃	—	4	8	no	—
94	C₂×C₂₂₃	#418	1	4	yes	direct product factor
95	C₂×C₂₂₃	C₂²×C₂²⋊C₄	1	1	yes	characteristic
96	C₂×C₂₂₃	C₂×AGL(1,ℤ/8)	1	2	yes	—
97	C₂₂₃	C₂×#418	1	1	yes	characteristic, direct product factor, Sylow
98	C₂×#418	C₂₂₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
99	C₂³×D₄	C₂×C₂₂₃	1	1	yes	characteristic
100	C₂²×M₁₆	C₂×C₂₂₃	1	1	yes	characteristic, complemented
101	C₂×#117	C₂×C₂₂₃	1	1	yes	characteristic, complemented
102	#418	C₂×C₂₂₃	1	8	yes	direct product factor
103	C₂×#102	C₂×C₂₂₃	1	4	yes	complemented
104	C₂⁵	—	2	1	no	—
105	C₂²×D₄	—	2	1	no	—
106	C₂²×D₄	—	2	2	no	—
107	C₂²×D₄	—	2	4	no	—
108	C₂²×D₄	C₄×C₂₂₃	1	8	yes	complemented
109	C₂²×D₄	C₂²×C₂₂₃	1	2	yes	—
110	C₂²×D₄	C₂²×C₂₂₃	1	4	yes	—
111	C₂³×C₄	C₂²×C₂₂₃	1	1	yes	characteristic
112	C₂×M₁₆	—	2	4	no	—
113	C₂×M₁₆	C₂²×C₂₂₃	1	4	yes	complemented
114	C₂²×C₈	C₂²×C₂₂₃	1	2	yes	complemented
115	#117	—	2	2	no	—
116	#117	C₂²×C₂₂₃	1	4	yes	complemented
117	C₂×C₄⋊C₄	C₂²×C₂₂₃	1	2	yes	complemented
118	C₂×C₂²⋊C₄	—	2	1	no	—
119	C₂×C₄²	—	2	1	no	—
120	#102	C₂²×C₂₂₃	1	16	yes	complemented
121	C₂⁴	—	2	8	no	—
122	C₂⁴	—	2	2	no	—
123	C₂⁴	—	2	4	no	—
124	C₂⁴	—	4	8	no	—
125	C₂⁴	D₄×C₂₂₃	1	1	yes	characteristic
126	C₂×D₄	—	2	16	no	—
127	C₂×D₄	—	2	8	no	—
128	C₂×D₄	—	2	4	no	—
129	C₂×D₄	—	2	2	no	—
130	C₂×D₄	—	2	4	no	—
131	C₂×D₄	—	2	8	no	—
132	C₂×D₄	C₂×C₄×C₂₂₃	1	8	yes	—
133	C₂×D₄	C₂×C₄×C₂₂₃	1	16	yes	complemented
134	C₂×D₄	C₂³×C₂₂₃	1	4	yes	—
135	C₂²×C₄	—	2	2	no	—
136	C₂²×C₄	C₂×C₄×C₂₂₃	1	4	yes	—
137	C₂²×C₄	D₄×C₂₂₃	1	1	yes	characteristic
138	C₂²×C₄	D₄×C₂₂₃	1	2	yes	—
139	C₂²×C₄	D₄×C₂₂₃	1	4	yes	—
140	C₂²×C₄	C₂³×C₂₂₃	1	1	yes	characteristic
141	C₂²×C₄	C₂³×C₂₂₃	1	2	yes	—
142	M₁₆	—	2	8	no	—
143	C₂×C₈	—	2	4	no	—
144	C₂×C₈	C₂³×C₂₂₃	1	4	yes	complemented
145	C₄⋊C₄	—	2	2	no	—
146	C₄⋊C₄	C₂³×C₂₂₃	1	4	yes	complemented
147	C₂²⋊C₄	—	2	4	no	—
148	C₄²	—	2	4	no	—
149	C₂³	—	2	4	no	—
150	C₂³	—	2	8	no	—
151	C₂³	—	2	16	no	—
152	C₂³	—	2	4	no	—
153	C₂³	—	4	16	no	—
154	C₂³	—	4	32	no	—
155	C₂³	—	4	8	no	—
156	C₂³	C₂²⋊C₄×C₂₂₃	1	4	yes	—
157	C₂³	C₂×D₄×C₂₂₃	1	2	yes	—
158	C₂³	C₂×D₄×C₂₂₃	1	1	yes	characteristic
159	D₄	—	2	8	no	—
160	D₄	—	2	16	no	—
161	D₄	—	2	4	no	—
162	D₄	C₂²×C₄×C₂₂₃	1	8	yes	—
163	C₂×C₄	—	2	4	no	—
164	C₂×C₄	—	4	4	no	—
165	C₂×C₄	C₂²⋊C₄×C₂₂₃	1	8	yes	—
166	C₂×C₄	C₂²⋊C₄×C₂₂₃	1	4	yes	—
167	C₂×C₄	C₂²×C₄×C₂₂₃	1	2	yes	—
168	C₂×C₄	C₂²×C₄×C₂₂₃	1	4	yes	—
169	C₂×C₄	C₂×D₄×C₂₂₃	1	2	yes	—
170	C₂×C₄	C₂×D₄×C₂₂₃	1	1	yes	characteristic
171	C₂×C₄	C₂×D₄×C₂₂₃	1	2	yes	—
172	C₂×C₄	C₂×D₄×C₂₂₃	1	4	yes	—
173	C₂×C₄	C₂⁴×C₂₂₃	1	1	yes	characteristic
174	C₈	—	2	4	no	—
175	C₂²	—	2	4	no	—
176	C₂²	—	2	8	no	—
177	C₂²	—	2	8	no	—
178	C₂²	—	4	32	no	—
179	C₂²	—	4	8	no	—
180	C₂²	—	4	16	no	—
181	C₂²	C₂×C₂²⋊C₄×C₂₂₃	1	4	yes	—
182	C₂²	C₂×C₂²⋊C₄×C₂₂₃	1	2	yes	—
183	C₂²	AGL(1,ℤ/8)×C₂₂₃	1	4	yes	—
184	C₂²	C₂²×D₄×C₂₂₃	1	1	yes	characteristic
185	C₄	—	4	4	no	—
186	C₄	C₂×C₂²⋊C₄×C₂₂₃	1	2	yes	—
187	C₄	C₂×C₂²⋊C₄×C₂₂₃	1	4	yes	—
188	C₄	C₂³×C₄×C₂₂₃	1	1	yes	characteristic
189	C₄	C₂²×D₄×C₂₂₃	1	1	yes	characteristic
190	C₂	—	2	4	no	—
191	C₂	—	4	8	no	—
192	C₂	#418×C₂₂₃	1	4	yes	direct product factor
193	C₂	C₂²×C₂²⋊C₄×C₂₂₃	1	1	yes	characteristic
194	C₂	C₂×AGL(1,ℤ/8)×C₂₂₃	1	2	yes	—

	subgroup	G/H
center, socle	C₂³×C₂₂₃	C₂×D₄
commutator	C₄	C₂³×C₄×C₂₂₃
Frattini, cosocle	C₂×C₄	C₂⁴×C₂₂₃
prime radical	C₂³×D₄×C₂₂₃	C₂
LCS	G ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₂₂₃ ◁ C₂³×C₄×C₂₂₃ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₂²×M₁₆×C₂₂₃	C₂	semidirect
3. C₂×#117×C₂₂₃	C₂	semidirect
4. #418×C₂₂₃	C₂	direct (N × H)
5. C₂×#102×C₂₂₃	C₂	semidirect
10. C₂²×D₄×C₂₂₃	C₄	semidirect
15. C₂×M₁₆×C₂₂₃	C₂²	semidirect
16. C₂²×C₈×C₂₂₃	C₂²	semidirect
18. #117×C₂₂₃	C₂²	semidirect
19. C₂×C₄⋊C₄×C₂₂₃	C₂²	semidirect
22. #102×C₂₂₃	C₂²	semidirect
35. C₂×D₄×C₂₂₃	C₂×C₄	semidirect
46. C₂×C₈×C₂₂₃	C₂³	semidirect
48. C₄⋊C₄×C₂₂₃	C₂³	semidirect
94. C₂×C₂₂₃	#418	direct (N × H)
97. C₂₂₃	C₂×#418	direct (N × H)
98. C₂×#418	C₂₂₃	direct (N × H)
100. C₂²×M₁₆	C₂×C₂₂₃	semidirect
101. C₂×#117	C₂×C₂₂₃	semidirect
102. #418	C₂×C₂₂₃	direct (N × H)
103. C₂×#102	C₂×C₂₂₃	semidirect
108. C₂²×D₄	C₄×C₂₂₃	semidirect
113. C₂×M₁₆	C₂²×C₂₂₃	semidirect
114. C₂²×C₈	C₂²×C₂₂₃	semidirect
116. #117	C₂²×C₂₂₃	semidirect
117. C₂×C₄⋊C₄	C₂²×C₂₂₃	semidirect
120. #102	C₂²×C₂₂₃	semidirect
133. C₂×D₄	C₂×C₄×C₂₂₃	semidirect
144. C₂×C₈	C₂³×C₂₂₃	semidirect
146. C₄⋊C₄	C₂³×C₂₂₃	semidirect
192. C₂	#418×C₂₂₃	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂³×D₄×C₂₂₃	2
C₂²×M₁₆×C₂₂₃	2
C₂×#117×C₂₂₃	2
#418×C₂₂₃	2
C₂×#102×C₂₂₃	2
C₂×#418	223

Subgroups — C2×#418×C223 #43251908

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#418×C₂₂₃ #43251908