Subgroups — #3234×C₂₂₃ #43252249

Order 28544.

368 conjugacy classes of subgroups.

The subgroup lattice has 214 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#532×C₂₂₃	C₂	1	1	yes	maximal, characteristic
2	C₂×#125×C₂₂₃	C₂	1	1	yes	maximal, characteristic, complemented
3	#501×C₂₂₃	C₂	1	2	yes	maximal, complemented
4	#490×C₂₂₃	C₂	1	2	yes	maximal, complemented
5	#481×C₂₂₃	C₂	1	2	yes	maximal, complemented
6	#480×C₂₂₃	C₂	1	2	yes	maximal, complemented
7	#478×C₂₂₃	C₂	1	4	yes	maximal, complemented
8	#432×C₂₂₃	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂×#41×C₂₂₃	—	2	1	no	—
10	C₄⋊Q₈×C₂₂₃	C₂²	1	1	yes	characteristic
11	#127×C₂₂₃	C₂²	1	1	yes	characteristic
12	#125×C₂₂₃	—	2	1	no	—
13	#125×C₂₂₃	—	2	1	no	—
14	#125×C₂₂₃	C₂²	1	2	yes	—
15	#125×C₂₂₃	C₂²	1	2	yes	—
16	#124×C₂₂₃	—	2	1	no	—
17	#121×C₂₂₃	—	2	1	no	—
18	#121×C₂₂₃	C₂²	1	2	yes	—
19	C₄×D₄×C₂₂₃	—	2	1	no	—
20	C₄×D₄×C₂₂₃	C₂²	1	2	yes	—
21	C₂×C₄⋊C₄×C₂₂₃	—	2	1	no	—
22	C₂×C₄⋊C₄×C₂₂₃	—	2	1	no	—
23	C₂×C₄⋊C₄×C₂₂₃	C₂²	1	1	yes	characteristic
24	C₂×C₄⋊C₄×C₂₂₃	C₂²	1	1	yes	characteristic
25	C₂×C₄²×C₂₂₃	C₂²	1	1	yes	characteristic
26	#107×C₂₂₃	C₂²	1	4	yes	—
27	#106×C₂₂₃	C₂²	1	4	yes	—
28	C₄⋊C₈×C₂₂₃	C₂²	1	2	yes	—
29	#102×C₂₂₃	C₂²	1	4	yes	—
30	#102×C₂₂₃	C₂²	1	4	yes	—
31	C₂²⋊C₈×C₂₂₃	C₂²	1	2	yes	—
32	#97×C₂₂₃	C₂²	1	2	yes	—
33	#41×C₂₂₃	—	4	4	no	—
34	C₂×Q₈×C₂₂₃	—	2	1	no	—
35	C₂×D₄×C₂₂₃	—	2	2	no	—
36	C₂×D₄×C₂₂₃	—	2	1	no	—
37	C₂×D₄×C₂₂₃	—	2	1	no	—
38	C₂×D₄×C₂₂₃	C₂³	1	2	yes	—
39	C₂²×C₄×C₂₂₃	—	2	1	no	—
40	C₂²×C₄×C₂₂₃	—	2	1	no	—
41	C₂²×C₄×C₂₂₃	—	2	2	no	—
42	C₂²×C₄×C₂₂₃	D₄	1	1	yes	characteristic
43	C₂²×C₄×C₂₂₃	D₄	1	1	yes	characteristic
44	C₂²×C₄×C₂₂₃	C₂³	1	1	yes	characteristic
45	C₂×C₈×C₂₂₃	C₂³	1	4	yes	—
46	C₄⋊C₄×C₂₂₃	—	2	2	no	—
47	C₄⋊C₄×C₂₂₃	—	2	2	no	—
48	C₄⋊C₄×C₂₂₃	—	2	2	no	—
49	C₄⋊C₄×C₂₂₃	—	2	2	no	—
50	C₄⋊C₄×C₂₂₃	—	2	1	no	—
51	C₄⋊C₄×C₂₂₃	—	2	1	no	—
52	C₄⋊C₄×C₂₂₃	—	2	1	no	—
53	C₄⋊C₄×C₂₂₃	C₂³	1	2	yes	—
54	C₄⋊C₄×C₂₂₃	C₂³	1	2	yes	—
55	C₄⋊C₄×C₂₂₃	C₂³	1	2	yes	—
56	C₂²⋊C₄×C₂₂₃	—	2	2	no	—
57	C₂²⋊C₄×C₂₂₃	—	2	2	no	—
58	C₄²×C₂₂₃	D₄	1	1	yes	characteristic
59	C₄²×C₂₂₃	D₄	1	1	yes	characteristic
60	C₄²×C₂₂₃	C₂³	1	1	yes	characteristic
61	C₄²×C₂₂₃	C₂³	1	1	yes	characteristic
62	C₂³×C₂₂₃	—	2	2	no	—
63	C₂³×C₂₂₃	C₂×D₄	1	1	yes	characteristic
64	Q₈×C₂₂₃	—	4	2	no	—
65	D₄×C₂₂₃	—	2	4	no	—
66	D₄×C₂₂₃	—	4	2	no	—
67	D₄×C₂₂₃	—	4	2	no	—
68	D₄×C₂₂₃	—	4	4	no	—
69	C₂×C₄×C₂₂₃	—	2	2	no	—
70	C₂×C₄×C₂₂₃	—	2	2	no	—
71	C₂×C₄×C₂₂₃	—	2	2	no	—
72	C₂×C₄×C₂₂₃	—	2	2	no	—
73	C₂×C₄×C₂₂₃	—	2	2	no	—
74	C₂×C₄×C₂₂₃	—	4	1	no	—
75	C₂×C₄×C₂₂₃	—	4	4	no	—
76	C₂×C₄×C₂₂₃	—	4	2	no	—
77	C₂×C₄×C₂₂₃	—	4	2	no	—
78	C₂×C₄×C₂₂₃	C₂×D₄	1	1	yes	characteristic
79	C₂×C₄×C₂₂₃	C₂×D₄	1	1	yes	characteristic
80	C₂×C₄×C₂₂₃	C₂×D₄	1	1	yes	characteristic
81	C₂×C₄×C₂₂₃	C₂×D₄	1	1	yes	characteristic
82	C₂×C₄×C₂₂₃	C₂×D₄	1	1	yes	characteristic
83	C₂×C₄×C₂₂₃	C₂⁴	1	1	yes	characteristic
84	C₈×C₂₂₃	—	2	4	no	—
85	C₂²×C₂₂₃	—	2	1	no	—
86	C₂²×C₂₂₃	—	2	1	no	—
87	C₂²×C₂₂₃	—	4	2	no	—
88	C₂²×C₂₂₃	—	4	2	no	—
89	C₂²×C₂₂₃	—	4	2	no	—
90	C₂²×C₂₂₃	AGL(1,ℤ/8)	1	2	yes	—
91	C₂²×C₂₂₃	C₂²×D₄	1	1	yes	characteristic
92	C₄×C₂₂₃	—	2	1	no	—
93	C₄×C₂₂₃	—	2	1	no	—
94	C₄×C₂₂₃	—	2	1	no	—
95	C₄×C₂₂₃	—	2	1	no	—
96	C₄×C₂₂₃	—	2	1	no	—
97	C₄×C₂₂₃	—	4	2	no	—
98	C₄×C₂₂₃	—	4	2	no	—
99	C₄×C₂₂₃	—	4	2	no	—
100	C₄×C₂₂₃	2₋¹⁺⁴	1	2	yes	—
101	C₂×C₂₂₃	—	2	2	no	—
102	C₂×C₂₂₃	—	8	2	no	—
103	C₂×C₂₂₃	#536	1	1	yes	characteristic
104	C₂×C₂₂₃	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
105	C₂×C₂₂₃	#575	1	1	yes	characteristic
106	C₂₂₃	#3234	1	1	yes	characteristic, direct product factor, Sylow
107	#3234	C₂₂₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
108	#532	C₂×C₂₂₃	1	1	yes	characteristic
109	C₂×#125	C₂×C₂₂₃	1	1	yes	characteristic, complemented
110	#501	C₂×C₂₂₃	1	2	yes	complemented
111	#490	C₂×C₂₂₃	1	2	yes	complemented
112	#481	C₂×C₂₂₃	1	2	yes	complemented
113	#480	C₂×C₂₂₃	1	2	yes	complemented
114	#478	C₂×C₂₂₃	1	4	yes	complemented
115	#432	C₂×C₂₂₃	1	1	yes	characteristic, complemented
116	C₂×#41	—	2	1	no	—
117	C₄⋊Q₈	C₂²×C₂₂₃	1	1	yes	characteristic
118	#127	C₂²×C₂₂₃	1	1	yes	characteristic
119	#125	—	2	1	no	—
120	#125	—	2	1	no	—
121	#125	C₂²×C₂₂₃	1	2	yes	—
122	#125	C₂²×C₂₂₃	1	2	yes	—
123	#124	—	2	1	no	—
124	#121	—	2	1	no	—
125	#121	C₂²×C₂₂₃	1	2	yes	—
126	C₄×D₄	—	2	1	no	—
127	C₄×D₄	C₂²×C₂₂₃	1	2	yes	—
128	C₂×C₄⋊C₄	—	2	1	no	—
129	C₂×C₄⋊C₄	—	2	1	no	—
130	C₂×C₄⋊C₄	C₂²×C₂₂₃	1	1	yes	characteristic
131	C₂×C₄⋊C₄	C₂²×C₂₂₃	1	1	yes	characteristic
132	C₂×C₄²	C₂²×C₂₂₃	1	1	yes	characteristic
133	#107	C₂²×C₂₂₃	1	4	yes	—
134	#106	C₂²×C₂₂₃	1	4	yes	—
135	C₄⋊C₈	C₂²×C₂₂₃	1	2	yes	—
136	#102	C₂²×C₂₂₃	1	4	yes	—
137	#102	C₂²×C₂₂₃	1	4	yes	—
138	C₂²⋊C₈	C₂²×C₂₂₃	1	2	yes	—
139	#97	C₂²×C₂₂₃	1	2	yes	—
140	#41	—	4	4	no	—
141	C₂×Q₈	—	2	1	no	—
142	C₂×D₄	—	2	2	no	—
143	C₂×D₄	—	2	1	no	—
144	C₂×D₄	—	2	1	no	—
145	C₂×D₄	C₂³×C₂₂₃	1	2	yes	—
146	C₂²×C₄	—	2	1	no	—
147	C₂²×C₄	—	2	1	no	—
148	C₂²×C₄	—	2	2	no	—
149	C₂²×C₄	D₄×C₂₂₃	1	1	yes	characteristic
150	C₂²×C₄	D₄×C₂₂₃	1	1	yes	characteristic
151	C₂²×C₄	C₂³×C₂₂₃	1	1	yes	characteristic
152	C₂×C₈	C₂³×C₂₂₃	1	4	yes	—
153	C₄⋊C₄	—	2	2	no	—
154	C₄⋊C₄	—	2	2	no	—
155	C₄⋊C₄	—	2	2	no	—
156	C₄⋊C₄	—	2	2	no	—
157	C₄⋊C₄	—	2	1	no	—
158	C₄⋊C₄	—	2	1	no	—
159	C₄⋊C₄	—	2	1	no	—
160	C₄⋊C₄	C₂³×C₂₂₃	1	2	yes	—
161	C₄⋊C₄	C₂³×C₂₂₃	1	2	yes	—
162	C₄⋊C₄	C₂³×C₂₂₃	1	2	yes	—
163	C₂²⋊C₄	—	2	2	no	—
164	C₂²⋊C₄	—	2	2	no	—
165	C₄²	D₄×C₂₂₃	1	1	yes	characteristic
166	C₄²	D₄×C₂₂₃	1	1	yes	characteristic
167	C₄²	C₂³×C₂₂₃	1	1	yes	characteristic
168	C₄²	C₂³×C₂₂₃	1	1	yes	characteristic
169	C₂³	—	2	2	no	—
170	C₂³	C₂×D₄×C₂₂₃	1	1	yes	characteristic
171	Q₈	—	4	2	no	—
172	D₄	—	2	4	no	—
173	D₄	—	4	2	no	—
174	D₄	—	4	2	no	—
175	D₄	—	4	4	no	—
176	C₂×C₄	—	2	2	no	—
177	C₂×C₄	—	2	2	no	—
178	C₂×C₄	—	2	2	no	—
179	C₂×C₄	—	2	2	no	—
180	C₂×C₄	—	2	2	no	—
181	C₂×C₄	—	4	1	no	—
182	C₂×C₄	—	4	4	no	—
183	C₂×C₄	—	4	2	no	—
184	C₂×C₄	—	4	2	no	—
185	C₂×C₄	C₂×D₄×C₂₂₃	1	1	yes	characteristic
186	C₂×C₄	C₂×D₄×C₂₂₃	1	1	yes	characteristic
187	C₂×C₄	C₂×D₄×C₂₂₃	1	1	yes	characteristic
188	C₂×C₄	C₂×D₄×C₂₂₃	1	1	yes	characteristic
189	C₂×C₄	C₂×D₄×C₂₂₃	1	1	yes	characteristic
190	C₂×C₄	C₂⁴×C₂₂₃	1	1	yes	characteristic
191	C₈	—	2	4	no	—
192	C₂²	—	2	1	no	—
193	C₂²	—	2	1	no	—
194	C₂²	—	4	2	no	—
195	C₂²	—	4	2	no	—
196	C₂²	—	4	2	no	—
197	C₂²	AGL(1,ℤ/8)×C₂₂₃	1	2	yes	—
198	C₂²	C₂²×D₄×C₂₂₃	1	1	yes	characteristic
199	C₄	—	2	1	no	—
200	C₄	—	2	1	no	—
201	C₄	—	2	1	no	—
202	C₄	—	2	1	no	—
203	C₄	—	2	1	no	—
204	C₄	—	4	2	no	—
205	C₄	—	4	2	no	—
206	C₄	—	4	2	no	—
207	C₄	2₋¹⁺⁴×C₂₂₃	1	2	yes	—
208	C₂	—	2	2	no	—
209	C₂	—	8	2	no	—
210	C₂	#536×C₂₂₃	1	1	yes	characteristic
211	C₂	C₂×AGL(1,ℤ/8)×C₂₂₃	1	1	yes	characteristic
212	C₂	#575×C₂₂₃	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₂₂₃	C₂²×D₄
commutator, Frattini, cosocle	C₂×C₄	C₂⁴×C₂₂₃
prime radical	#532×C₂₂₃	C₂
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₂₂₃ ◁ C₂×C₄²×C₂₂₃ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₂×#125×C₂₂₃	C₂	semidirect
3. #501×C₂₂₃	C₂	semidirect
4. #490×C₂₂₃	C₂	semidirect
5. #481×C₂₂₃	C₂	semidirect
6. #480×C₂₂₃	C₂	semidirect
7. #478×C₂₂₃	C₂	semidirect
8. #432×C₂₂₃	C₂	semidirect
106. C₂₂₃	#3234	direct (N × H)
107. #3234	C₂₂₃	direct (N × H)
109. C₂×#125	C₂×C₂₂₃	semidirect
110. #501	C₂×C₂₂₃	semidirect
111. #490	C₂×C₂₂₃	semidirect
112. #481	C₂×C₂₂₃	semidirect
113. #480	C₂×C₂₂₃	semidirect
114. #478	C₂×C₂₂₃	semidirect
115. #432	C₂×C₂₂₃	semidirect

Maximal subgroups

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

subgroup	index
#532×C₂₂₃	2
C₂×#125×C₂₂₃	2
#501×C₂₂₃	2
#490×C₂₂₃	2
#481×C₂₂₃	2
#480×C₂₂₃	2
#478×C₂₂₃	2
#432×C₂₂₃	2
#3234	223

Subgroups — #3234×C223 #43252249

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — #3234×C₂₂₃ #43252249