Subgroups — C₂²×AGL(1,ℤ/8)×C₂₃₃ #44449610

Order 29824.

1720 conjugacy classes of subgroups.

The subgroup lattice has 190 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×#41×C₂₃₃	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂³×D₄×C₂₃₃	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×AGL(1,ℤ/8)×C₂₃₃	C₂	1	24	yes	maximal, direct product factor
4	C₂²×SD₁₆×C₂₃₃	C₂	1	2	yes	maximal, complemented
5	C₂²×D₈×C₂₃₃	C₂	1	2	yes	maximal, complemented
6	C₂²×M₁₆×C₂₃₃	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂⁵×C₂₃₃	—	2	1	no	—
8	C₂×#41×C₂₃₃	—	2	6	no	—
9	C₂×#41×C₂₃₃	C₂²	1	12	yes	complemented
10	C₂²×Q₈×C₂₃₃	C₂²	1	1	yes	characteristic, complemented
11	C₂²×D₄×C₂₃₃	—	2	1	no	—
12	C₂²×D₄×C₂₃₃	—	2	6	no	—
13	C₂²×D₄×C₂₃₃	—	2	1	no	—
14	C₂²×D₄×C₂₃₃	C₂²	1	2	yes	—
15	C₂²×D₄×C₂₃₃	C₂²	1	12	yes	complemented
16	C₂²×D₄×C₂₃₃	C₂²	1	1	yes	characteristic, complemented
17	C₂³×C₄×C₂₃₃	—	2	1	no	—
18	C₂³×C₄×C₂₃₃	C₂²	1	1	yes	characteristic
19	AGL(1,ℤ/8)×C₂₃₃	C₂²	1	64	yes	direct product factor
20	C₂×SD₁₆×C₂₃₃	C₂²	1	24	yes	complemented
21	C₂×D₈×C₂₃₃	C₂²	1	24	yes	complemented
22	C₂×M₁₆×C₂₃₃	C₂²	1	12	yes	complemented
23	C₂²×C₈×C₂₃₃	C₂²	1	2	yes	complemented
24	C₂⁴×C₂₃₃	—	2	12	no	—
25	C₂⁴×C₂₃₃	—	2	1	no	—
26	C₂⁴×C₂₃₃	—	2	2	no	—
27	C₂⁴×C₂₃₃	—	4	8	no	—
28	C₂⁴×C₂₃₃	D₄	1	1	yes	characteristic
29	#41×C₂₃₃	—	2	24	no	—
30	#41×C₂₃₃	C₂³	1	16	yes	complemented
31	C₂×Q₈×C₂₃₃	—	2	3	no	—
32	C₂×Q₈×C₂₃₃	C₂³	1	6	yes	complemented
33	C₂×D₄×C₂₃₃	—	2	12	no	—
34	C₂×D₄×C₂₃₃	—	2	6	no	—
35	C₂×D₄×C₂₃₃	—	2	12	no	—
36	C₂×D₄×C₂₃₃	—	2	24	no	—
37	C₂×D₄×C₂₃₃	—	2	3	no	—
38	C₂×D₄×C₂₃₃	C₂³	1	12	yes	—
39	C₂×D₄×C₂₃₃	C₂³	1	16	yes	complemented
40	C₂×D₄×C₂₃₃	C₂³	1	6	yes	complemented
41	C₂²×C₄×C₂₃₃	—	2	1	no	—
42	C₂²×C₄×C₂₃₃	—	2	12	no	—
43	C₂²×C₄×C₂₃₃	D₄	1	6	yes	—
44	C₂²×C₄×C₂₃₃	D₄	1	1	yes	characteristic
45	C₂²×C₄×C₂₃₃	C₂³	1	6	yes	—
46	C₂²×C₄×C₂₃₃	C₂³	1	1	yes	characteristic
47	SD₁₆×C₂₃₃	C₂³	1	32	yes	complemented
48	D₈×C₂₃₃	C₂³	1	32	yes	complemented
49	M₁₆×C₂₃₃	C₂³	1	16	yes	complemented
50	C₂×C₈×C₂₃₃	C₂³	1	12	yes	complemented
51	C₂³×C₂₃₃	—	2	4	no	—
52	C₂³×C₂₃₃	—	2	6	no	—
53	C₂³×C₂₃₃	—	2	12	no	—
54	C₂³×C₂₃₃	—	2	16	no	—
55	C₂³×C₂₃₃	—	4	48	no	—
56	C₂³×C₂₃₃	—	4	4	no	—
57	C₂³×C₂₃₃	—	4	8	no	—
58	C₂³×C₂₃₃	C₂×D₄	1	6	yes	—
59	C₂³×C₂₃₃	C₂×D₄	1	1	yes	characteristic
60	Q₈×C₂₃₃	—	2	6	no	—
61	Q₈×C₂₃₃	C₂⁴	1	4	yes	complemented
62	D₄×C₂₃₃	—	2	16	no	—
63	D₄×C₂₃₃	—	2	12	no	—
64	D₄×C₂₃₃	—	2	16	no	—
65	D₄×C₂₃₃	—	2	6	no	—
66	D₄×C₂₃₃	C₂⁴	1	8	yes	—
67	D₄×C₂₃₃	C₂⁴	1	4	yes	complemented
68	C₂×C₄×C₂₃₃	—	2	6	no	—
69	C₂×C₄×C₂₃₃	—	2	16	no	—
70	C₂×C₄×C₂₃₃	C₂×D₄	1	6	yes	—
71	C₂×C₄×C₂₃₃	C₂×D₄	1	3	yes	—
72	C₂×C₄×C₂₃₃	C₂×D₄	1	12	yes	—
73	C₂×C₄×C₂₃₃	C₂⁴	1	3	yes	—
74	C₂×C₄×C₂₃₃	C₂⁴	1	4	yes	—
75	C₈×C₂₃₃	C₂⁴	1	8	yes	complemented
76	C₂²×C₂₃₃	—	2	12	no	—
77	C₂²×C₂₃₃	—	2	4	no	—
78	C₂²×C₂₃₃	—	2	8	no	—
79	C₂²×C₂₃₃	—	4	32	no	—
80	C₂²×C₂₃₃	—	4	12	no	—
81	C₂²×C₂₃₃	—	4	24	no	—
82	C₂²×C₂₃₃	AGL(1,ℤ/8)	1	4	yes	direct product factor
83	C₂²×C₂₃₃	C₂²×D₄	1	4	yes	—
84	C₂²×C₂₃₃	C₂²×D₄	1	3	yes	—
85	C₄×C₂₃₃	—	2	4	no	—
86	C₄×C₂₃₃	C₂²×D₄	1	4	yes	—
87	C₄×C₂₃₃	C₂²×D₄	1	3	yes	—
88	C₄×C₂₃₃	C₂⁵	1	1	yes	characteristic
89	C₂×C₂₃₃	—	2	4	no	—
90	C₂×C₂₃₃	—	4	4	no	—
91	C₂×C₂₃₃	—	4	8	no	—
92	C₂×C₂₃₃	C₂×AGL(1,ℤ/8)	1	6	yes	direct product factor
93	C₂×C₂₃₃	C₂³×D₄	1	1	yes	characteristic
94	C₂₃₃	C₂²×AGL(1,ℤ/8)	1	1	yes	characteristic, direct product factor, Sylow
95	C₂²×AGL(1,ℤ/8)	C₂₃₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
96	C₂²×#41	C₂×C₂₃₃	1	1	yes	characteristic, complemented
97	C₂³×D₄	C₂×C₂₃₃	1	1	yes	characteristic, complemented
98	C₂×AGL(1,ℤ/8)	C₂×C₂₃₃	1	24	yes	direct product factor
99	C₂²×SD₁₆	C₂×C₂₃₃	1	2	yes	complemented
100	C₂²×D₈	C₂×C₂₃₃	1	2	yes	complemented
101	C₂²×M₁₆	C₂×C₂₃₃	1	1	yes	characteristic, complemented
102	C₂⁵	—	2	1	no	—
103	C₂×#41	—	2	6	no	—
104	C₂×#41	C₂²×C₂₃₃	1	12	yes	complemented
105	C₂²×Q₈	C₂²×C₂₃₃	1	1	yes	characteristic, complemented
106	C₂²×D₄	—	2	1	no	—
107	C₂²×D₄	—	2	1	no	—
108	C₂²×D₄	—	2	6	no	—
109	C₂²×D₄	C₂²×C₂₃₃	1	2	yes	—
110	C₂²×D₄	C₂²×C₂₃₃	1	12	yes	complemented
111	C₂²×D₄	C₂²×C₂₃₃	1	1	yes	characteristic, complemented
112	C₂³×C₄	—	2	1	no	—
113	C₂³×C₄	C₂²×C₂₃₃	1	1	yes	characteristic
114	AGL(1,ℤ/8)	C₂²×C₂₃₃	1	64	yes	direct product factor
115	C₂×SD₁₆	C₂²×C₂₃₃	1	24	yes	complemented
116	C₂×D₈	C₂²×C₂₃₃	1	24	yes	complemented
117	C₂×M₁₆	C₂²×C₂₃₃	1	12	yes	complemented
118	C₂²×C₈	C₂²×C₂₃₃	1	2	yes	complemented
119	C₂⁴	—	2	1	no	—
120	C₂⁴	—	2	2	no	—
121	C₂⁴	—	2	12	no	—
122	C₂⁴	—	4	8	no	—
123	C₂⁴	D₄×C₂₃₃	1	1	yes	characteristic
124	#41	—	2	24	no	—
125	#41	C₂³×C₂₃₃	1	16	yes	complemented
126	C₂×Q₈	—	2	3	no	—
127	C₂×Q₈	C₂³×C₂₃₃	1	6	yes	complemented
128	C₂×D₄	—	2	12	no	—
129	C₂×D₄	—	2	6	no	—
130	C₂×D₄	—	2	12	no	—
131	C₂×D₄	—	2	24	no	—
132	C₂×D₄	—	2	3	no	—
133	C₂×D₄	C₂³×C₂₃₃	1	12	yes	—
134	C₂×D₄	C₂³×C₂₃₃	1	16	yes	complemented
135	C₂×D₄	C₂³×C₂₃₃	1	6	yes	complemented
136	C₂²×C₄	—	2	1	no	—
137	C₂²×C₄	—	2	12	no	—
138	C₂²×C₄	D₄×C₂₃₃	1	1	yes	characteristic
139	C₂²×C₄	D₄×C₂₃₃	1	6	yes	—
140	C₂²×C₄	C₂³×C₂₃₃	1	1	yes	characteristic
141	C₂²×C₄	C₂³×C₂₃₃	1	6	yes	—
142	SD₁₆	C₂³×C₂₃₃	1	32	yes	complemented
143	D₈	C₂³×C₂₃₃	1	32	yes	complemented
144	M₁₆	C₂³×C₂₃₃	1	16	yes	complemented
145	C₂×C₈	C₂³×C₂₃₃	1	12	yes	complemented
146	C₂³	—	2	4	no	—
147	C₂³	—	2	6	no	—
148	C₂³	—	2	12	no	—
149	C₂³	—	2	16	no	—
150	C₂³	—	4	48	no	—
151	C₂³	—	4	8	no	—
152	C₂³	—	4	4	no	—
153	C₂³	C₂×D₄×C₂₃₃	1	6	yes	—
154	C₂³	C₂×D₄×C₂₃₃	1	1	yes	characteristic
155	Q₈	—	2	6	no	—
156	Q₈	C₂⁴×C₂₃₃	1	4	yes	complemented
157	D₄	—	2	16	no	—
158	D₄	—	2	12	no	—
159	D₄	—	2	16	no	—
160	D₄	—	2	6	no	—
161	D₄	C₂⁴×C₂₃₃	1	8	yes	—
162	D₄	C₂⁴×C₂₃₃	1	4	yes	complemented
163	C₂×C₄	—	2	6	no	—
164	C₂×C₄	—	2	16	no	—
165	C₂×C₄	C₂×D₄×C₂₃₃	1	6	yes	—
166	C₂×C₄	C₂×D₄×C₂₃₃	1	3	yes	—
167	C₂×C₄	C₂×D₄×C₂₃₃	1	12	yes	—
168	C₂×C₄	C₂⁴×C₂₃₃	1	3	yes	—
169	C₂×C₄	C₂⁴×C₂₃₃	1	4	yes	—
170	C₈	C₂⁴×C₂₃₃	1	8	yes	complemented
171	C₂²	—	2	12	no	—
172	C₂²	—	2	4	no	—
173	C₂²	—	2	8	no	—
174	C₂²	—	4	32	no	—
175	C₂²	—	4	24	no	—
176	C₂²	—	4	12	no	—
177	C₂²	AGL(1,ℤ/8)×C₂₃₃	1	4	yes	direct product factor
178	C₂²	C₂²×D₄×C₂₃₃	1	4	yes	—
179	C₂²	C₂²×D₄×C₂₃₃	1	3	yes	—
180	C₄	—	2	4	no	—
181	C₄	C₂²×D₄×C₂₃₃	1	4	yes	—
182	C₄	C₂²×D₄×C₂₃₃	1	3	yes	—
183	C₄	C₂⁵×C₂₃₃	1	1	yes	characteristic
184	C₂	—	2	4	no	—
185	C₂	—	4	8	no	—
186	C₂	—	4	4	no	—
187	C₂	C₂×AGL(1,ℤ/8)×C₂₃₃	1	6	yes	direct product factor
188	C₂	C₂³×D₄×C₂₃₃	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂³×C₂₃₃	C₂×D₄
commutator, Frattini, cosocle	C₄	C₂⁵×C₂₃₃
LCS	G ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₂₃₃ ◁ C₂³×C₄×C₂₃₃ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₂²×#41×C₂₃₃	C₂	semidirect
2. C₂³×D₄×C₂₃₃	C₂	semidirect
3. C₂×AGL(1,ℤ/8)×C₂₃₃	C₂	direct (N × H)
4. C₂²×SD₁₆×C₂₃₃	C₂	semidirect
5. C₂²×D₈×C₂₃₃	C₂	semidirect
6. C₂²×M₁₆×C₂₃₃	C₂	semidirect
9. C₂×#41×C₂₃₃	C₂²	semidirect
10. C₂²×Q₈×C₂₃₃	C₂²	semidirect
15. C₂²×D₄×C₂₃₃	C₂²	semidirect
16. C₂²×D₄×C₂₃₃	C₂²	semidirect
19. AGL(1,ℤ/8)×C₂₃₃	C₂²	direct (N × H)
20. C₂×SD₁₆×C₂₃₃	C₂²	semidirect
21. C₂×D₈×C₂₃₃	C₂²	semidirect
22. C₂×M₁₆×C₂₃₃	C₂²	semidirect
23. C₂²×C₈×C₂₃₃	C₂²	semidirect
30. #41×C₂₃₃	C₂³	semidirect
32. C₂×Q₈×C₂₃₃	C₂³	semidirect
39. C₂×D₄×C₂₃₃	C₂³	semidirect
40. C₂×D₄×C₂₃₃	C₂³	semidirect
47. SD₁₆×C₂₃₃	C₂³	semidirect
48. D₈×C₂₃₃	C₂³	semidirect
49. M₁₆×C₂₃₃	C₂³	semidirect
50. C₂×C₈×C₂₃₃	C₂³	semidirect
61. Q₈×C₂₃₃	C₂⁴	semidirect
67. D₄×C₂₃₃	C₂⁴	semidirect
75. C₈×C₂₃₃	C₂⁴	semidirect
82. C₂²×C₂₃₃	AGL(1,ℤ/8)	direct (N × H)
92. C₂×C₂₃₃	C₂×AGL(1,ℤ/8)	direct (N × H)
94. C₂₃₃	C₂²×AGL(1,ℤ/8)	direct (N × H)
95. C₂²×AGL(1,ℤ/8)	C₂₃₃	direct (N × H)
96. C₂²×#41	C₂×C₂₃₃	semidirect
97. C₂³×D₄	C₂×C₂₃₃	semidirect
98. C₂×AGL(1,ℤ/8)	C₂×C₂₃₃	direct (N × H)
99. C₂²×SD₁₆	C₂×C₂₃₃	semidirect
100. C₂²×D₈	C₂×C₂₃₃	semidirect
101. C₂²×M₁₆	C₂×C₂₃₃	semidirect
104. C₂×#41	C₂²×C₂₃₃	semidirect
105. C₂²×Q₈	C₂²×C₂₃₃	semidirect
110. C₂²×D₄	C₂²×C₂₃₃	semidirect
111. C₂²×D₄	C₂²×C₂₃₃	semidirect
114. AGL(1,ℤ/8)	C₂²×C₂₃₃	direct (N × H)
115. C₂×SD₁₆	C₂²×C₂₃₃	semidirect
116. C₂×D₈	C₂²×C₂₃₃	semidirect
117. C₂×M₁₆	C₂²×C₂₃₃	semidirect
118. C₂²×C₈	C₂²×C₂₃₃	semidirect
125. #41	C₂³×C₂₃₃	semidirect
127. C₂×Q₈	C₂³×C₂₃₃	semidirect
134. C₂×D₄	C₂³×C₂₃₃	semidirect
135. C₂×D₄	C₂³×C₂₃₃	semidirect
142. SD₁₆	C₂³×C₂₃₃	semidirect
143. D₈	C₂³×C₂₃₃	semidirect
144. M₁₆	C₂³×C₂₃₃	semidirect
145. C₂×C₈	C₂³×C₂₃₃	semidirect
156. Q₈	C₂⁴×C₂₃₃	semidirect
162. D₄	C₂⁴×C₂₃₃	semidirect
170. C₈	C₂⁴×C₂₃₃	semidirect
177. C₂²	AGL(1,ℤ/8)×C₂₃₃	direct (N × H)
187. C₂	C₂×AGL(1,ℤ/8)×C₂₃₃	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂²×#41×C₂₃₃	2
C₂³×D₄×C₂₃₃	2
C₂×AGL(1,ℤ/8)×C₂₃₃	2
C₂²×SD₁₆×C₂₃₃	2
C₂²×D₈×C₂₃₃	2
C₂²×M₁₆×C₂₃₃	2
C₂²×AGL(1,ℤ/8)	233

Subgroups — C22×AGL(1,ℤ/8)×C233 #44449610

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂²×AGL(1,ℤ/8)×C₂₃₃ #44449610