Subgroups — #2989×C₂₂₃ #43252004

Order 28544.

388 conjugacy classes of subgroups.

The subgroup lattice has 216 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄×#41×C₂₂₃	C₂	1	1	yes	maximal, characteristic
2	#446×C₂₂₃	C₂	1	2	yes	maximal, complemented
3	#436×C₂₂₃	C₂	1	2	yes	maximal, complemented
4	#435×C₂₂₃	C₂	1	4	yes	maximal, complemented
5	#432×C₂₂₃	C₂	1	2	yes	maximal, complemented
6	#431×C₂₂₃	C₂	1	1	yes	maximal, characteristic, complemented
7	#423×C₂₂₃	C₂	1	2	yes	maximal, complemented
8	C₂×C₄⋊C₈×C₂₂₃	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂×#41×C₂₂₃	C₄	1	1	yes	characteristic
10	C₂×M₁₆×C₂₂₃	C₂²	1	4	yes	—
11	C₂²×C₈×C₂₂₃	C₂²	1	2	yes	—
12	C₄×Q₈×C₂₂₃	C₄	1	1	yes	characteristic
13	C₄×Q₈×C₂₂₃	C₂²	1	1	yes	characteristic
14	C₄×D₄×C₂₂₃	C₄	1	2	yes	—
15	C₄×D₄×C₂₂₃	C₄	1	1	yes	characteristic
16	C₄×D₄×C₂₂₃	C₂²	1	2	yes	—
17	C₄×D₄×C₂₂₃	C₂²	1	1	yes	characteristic
18	#117×C₂₂₃	C₄	1	2	yes	—
19	#117×C₂₂₃	C₄	1	1	yes	characteristic
20	C₂×C₄²×C₂₂₃	C₂²	1	1	yes	characteristic
21	C₂×C₄²×C₂₂₃	C₂²	1	2	yes	—
22	C₄⋊C₈×C₂₂₃	C₂²	1	2	yes	—
23	C₄⋊C₈×C₂₂₃	C₂²	1	2	yes	—
24	C₄⋊C₈×C₂₂₃	C₂²	1	1	yes	characteristic
25	C₄⋊C₈×C₂₂₃	C₂²	1	4	yes	—
26	C₄⋊C₈×C₂₂₃	C₂²	1	1	yes	characteristic
27	C₂²⋊C₈×C₂₂₃	C₂²	1	2	yes	—
28	C₂²⋊C₈×C₂₂₃	C₂²	1	4	yes	—
29	#97×C₂₂₃	C₂²	1	4	yes	—
30	C₄×C₈×C₂₂₃	C₂²	1	2	yes	—
31	#41×C₂₂₃	—	2	4	no	—
32	C₂×Q₈×C₂₂₃	C₂×C₄	1	1	yes	characteristic
33	C₂×D₄×C₂₂₃	C₂×C₄	1	2	yes	—
34	C₂×D₄×C₂₂₃	C₂×C₄	1	1	yes	characteristic
35	C₂²×C₄×C₂₂₃	C₂×C₄	1	2	yes	—
36	C₂²×C₄×C₂₂₃	C₂×C₄	1	2	yes	—
37	C₂²×C₄×C₂₂₃	C₂×C₄	1	1	yes	characteristic
38	C₂²×C₄×C₂₂₃	C₂×C₄	1	1	yes	characteristic
39	C₂²×C₄×C₂₂₃	C₂³	1	1	yes	characteristic
40	C₂²×C₄×C₂₂₃	C₂³	1	2	yes	—
41	M₁₆×C₂₂₃	—	4	4	no	—
42	C₂×C₈×C₂₂₃	—	4	2	no	—
43	C₂×C₈×C₂₂₃	C₂³	1	2	yes	—
44	C₂×C₈×C₂₂₃	C₂³	1	2	yes	—
45	C₂×C₈×C₂₂₃	C₂³	1	4	yes	—
46	C₄⋊C₄×C₂₂₃	C₂×C₄	1	2	yes	—
47	C₄⋊C₄×C₂₂₃	C₂×C₄	1	1	yes	characteristic
48	C₄⋊C₄×C₂₂₃	C₂×C₄	1	2	yes	—
49	C₄⋊C₄×C₂₂₃	C₂×C₄	1	1	yes	characteristic
50	C₂²⋊C₄×C₂₂₃	C₂×C₄	1	2	yes	—
51	C₂²⋊C₄×C₂₂₃	C₂×C₄	1	2	yes	—
52	C₂²⋊C₄×C₂₂₃	C₂×C₄	1	2	yes	—
53	C₄²×C₂₂₃	C₂×C₄	1	1	yes	characteristic
54	C₄²×C₂₂₃	C₂×C₄	1	2	yes	—
55	C₄²×C₂₂₃	C₂×C₄	1	2	yes	—
56	C₄²×C₂₂₃	C₂×C₄	1	1	yes	characteristic
57	C₄²×C₂₂₃	C₂³	1	1	yes	characteristic
58	C₄²×C₂₂₃	C₂³	1	2	yes	—
59	C₄²×C₂₂₃	C₂³	1	1	yes	characteristic
60	C₂³×C₂₂₃	C₂²×C₄	1	2	yes	—
61	C₂³×C₂₂₃	C₂²×C₄	1	1	yes	characteristic
62	Q₈×C₂₂₃	—	2	2	no	—
63	D₄×C₂₂₃	—	2	4	no	—
64	D₄×C₂₂₃	—	2	2	no	—
65	C₂×C₄×C₂₂₃	—	2	4	no	—
66	C₂×C₄×C₂₂₃	—	2	1	no	—
67	C₂×C₄×C₂₂₃	—	2	2	no	—
68	C₂×C₄×C₂₂₃	—	2	1	no	—
69	C₂×C₄×C₂₂₃	—	2	2	no	—
70	C₂×C₄×C₂₂₃	—	4	1	no	—
71	C₂×C₄×C₂₂₃	—	4	2	no	—
72	C₂×C₄×C₂₂₃	M₁₆	1	4	yes	complemented
73	C₂×C₄×C₂₂₃	C₂²×C₄	1	1	yes	characteristic
74	C₂×C₄×C₂₂₃	C₂²×C₄	1	1	yes	characteristic
75	C₂×C₄×C₂₂₃	C₂²×C₄	1	2	yes	—
76	C₂×C₄×C₂₂₃	C₂²×C₄	1	1	yes	characteristic
77	C₂×C₄×C₂₂₃	C₂²×C₄	1	2	yes	—
78	C₂×C₄×C₂₂₃	C₂²×C₄	1	2	yes	—
79	C₂×C₄×C₂₂₃	C₂²×C₄	1	1	yes	characteristic
80	C₂×C₄×C₂₂₃	C₂²×C₄	1	1	yes	characteristic
81	C₂×C₄×C₂₂₃	C₂⁴	1	1	yes	characteristic
82	C₈×C₂₂₃	—	2	4	no	—
83	C₈×C₂₂₃	—	2	2	no	—
84	C₈×C₂₂₃	—	2	2	no	—
85	C₂²×C₂₂₃	—	2	1	no	—
86	C₂²×C₂₂₃	—	2	1	no	—
87	C₂²×C₂₂₃	—	2	2	no	—
88	C₂²×C₂₂₃	—	2	2	no	—
89	C₂²×C₂₂₃	—	2	2	no	—
90	C₂²×C₂₂₃	C₂×M₁₆	1	2	yes	—
91	C₂²×C₂₂₃	C₂³×C₄	1	1	yes	characteristic
92	C₄×C₂₂₃	—	2	1	no	—
93	C₄×C₂₂₃	—	2	2	no	—
94	C₄×C₂₂₃	—	2	2	no	—
95	C₄×C₂₂₃	—	2	1	no	—
96	C₄×C₂₂₃	—	2	1	no	—
97	C₄×C₂₂₃	—	2	2	no	—
98	C₄×C₂₂₃	C₂×M₁₆	1	2	yes	—
99	C₄×C₂₂₃	C₂×M₁₆	1	2	yes	—
100	C₄×C₂₂₃	H(5,𝔽₂)	1	1	yes	characteristic
101	C₄×C₂₂₃	2₋¹⁺⁴	1	1	yes	characteristic
102	C₂×C₂₂₃	—	2	2	no	—
103	C₂×C₂₂₃	—	4	2	no	—
104	C₂×C₂₂₃	#520	1	1	yes	characteristic
105	C₂×C₂₂₃	C₂²×M₁₆	1	1	yes	characteristic
106	C₂×C₂₂₃	#568	1	1	yes	characteristic
107	C₂₂₃	#2989	1	1	yes	characteristic, direct product factor, Sylow
108	#2989	C₂₂₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
109	C₄×#41	C₂×C₂₂₃	1	1	yes	characteristic
110	#446	C₂×C₂₂₃	1	2	yes	complemented
111	#436	C₂×C₂₂₃	1	2	yes	complemented
112	#435	C₂×C₂₂₃	1	4	yes	complemented
113	#432	C₂×C₂₂₃	1	2	yes	complemented
114	#431	C₂×C₂₂₃	1	1	yes	characteristic, complemented
115	#423	C₂×C₂₂₃	1	2	yes	complemented
116	C₂×C₄⋊C₈	C₂×C₂₂₃	1	1	yes	characteristic, complemented
117	C₂×#41	C₄×C₂₂₃	1	1	yes	characteristic
118	C₂×M₁₆	C₂²×C₂₂₃	1	4	yes	—
119	C₂²×C₈	C₂²×C₂₂₃	1	2	yes	—
120	C₄×Q₈	C₄×C₂₂₃	1	1	yes	characteristic
121	C₄×Q₈	C₂²×C₂₂₃	1	1	yes	characteristic
122	C₄×D₄	C₄×C₂₂₃	1	2	yes	—
123	C₄×D₄	C₄×C₂₂₃	1	1	yes	characteristic
124	C₄×D₄	C₂²×C₂₂₃	1	2	yes	—
125	C₄×D₄	C₂²×C₂₂₃	1	1	yes	characteristic
126	#117	C₄×C₂₂₃	1	2	yes	—
127	#117	C₄×C₂₂₃	1	1	yes	characteristic
128	C₂×C₄²	C₂²×C₂₂₃	1	1	yes	characteristic
129	C₂×C₄²	C₂²×C₂₂₃	1	2	yes	—
130	C₄⋊C₈	C₂²×C₂₂₃	1	2	yes	—
131	C₄⋊C₈	C₂²×C₂₂₃	1	2	yes	—
132	C₄⋊C₈	C₂²×C₂₂₃	1	1	yes	characteristic
133	C₄⋊C₈	C₂²×C₂₂₃	1	4	yes	—
134	C₄⋊C₈	C₂²×C₂₂₃	1	1	yes	characteristic
135	C₂²⋊C₈	C₂²×C₂₂₃	1	2	yes	—
136	C₂²⋊C₈	C₂²×C₂₂₃	1	4	yes	—
137	#97	C₂²×C₂₂₃	1	4	yes	—
138	C₄×C₈	C₂²×C₂₂₃	1	2	yes	—
139	#41	—	2	4	no	—
140	C₂×Q₈	C₂×C₄×C₂₂₃	1	1	yes	characteristic
141	C₂×D₄	C₂×C₄×C₂₂₃	1	2	yes	—
142	C₂×D₄	C₂×C₄×C₂₂₃	1	1	yes	characteristic
143	C₂²×C₄	C₂×C₄×C₂₂₃	1	2	yes	—
144	C₂²×C₄	C₂×C₄×C₂₂₃	1	2	yes	—
145	C₂²×C₄	C₂×C₄×C₂₂₃	1	1	yes	characteristic
146	C₂²×C₄	C₂×C₄×C₂₂₃	1	1	yes	characteristic
147	C₂²×C₄	C₂³×C₂₂₃	1	1	yes	characteristic
148	C₂²×C₄	C₂³×C₂₂₃	1	2	yes	—
149	M₁₆	—	4	4	no	—
150	C₂×C₈	—	4	2	no	—
151	C₂×C₈	C₂³×C₂₂₃	1	2	yes	—
152	C₂×C₈	C₂³×C₂₂₃	1	2	yes	—
153	C₂×C₈	C₂³×C₂₂₃	1	4	yes	—
154	C₄⋊C₄	C₂×C₄×C₂₂₃	1	2	yes	—
155	C₄⋊C₄	C₂×C₄×C₂₂₃	1	1	yes	characteristic
156	C₄⋊C₄	C₂×C₄×C₂₂₃	1	2	yes	—
157	C₄⋊C₄	C₂×C₄×C₂₂₃	1	1	yes	characteristic
158	C₂²⋊C₄	C₂×C₄×C₂₂₃	1	2	yes	—
159	C₂²⋊C₄	C₂×C₄×C₂₂₃	1	2	yes	—
160	C₂²⋊C₄	C₂×C₄×C₂₂₃	1	2	yes	—
161	C₄²	C₂×C₄×C₂₂₃	1	1	yes	characteristic
162	C₄²	C₂×C₄×C₂₂₃	1	2	yes	—
163	C₄²	C₂×C₄×C₂₂₃	1	2	yes	—
164	C₄²	C₂×C₄×C₂₂₃	1	1	yes	characteristic
165	C₄²	C₂³×C₂₂₃	1	1	yes	characteristic
166	C₄²	C₂³×C₂₂₃	1	2	yes	—
167	C₄²	C₂³×C₂₂₃	1	1	yes	characteristic
168	C₂³	C₂²×C₄×C₂₂₃	1	2	yes	—
169	C₂³	C₂²×C₄×C₂₂₃	1	1	yes	characteristic
170	Q₈	—	2	2	no	—
171	D₄	—	2	4	no	—
172	D₄	—	2	2	no	—
173	C₂×C₄	—	2	4	no	—
174	C₂×C₄	—	2	1	no	—
175	C₂×C₄	—	2	2	no	—
176	C₂×C₄	—	2	1	no	—
177	C₂×C₄	—	2	2	no	—
178	C₂×C₄	—	4	1	no	—
179	C₂×C₄	—	4	2	no	—
180	C₂×C₄	M₁₆×C₂₂₃	1	4	yes	complemented
181	C₂×C₄	C₂²×C₄×C₂₂₃	1	1	yes	characteristic
182	C₂×C₄	C₂²×C₄×C₂₂₃	1	1	yes	characteristic
183	C₂×C₄	C₂²×C₄×C₂₂₃	1	2	yes	—
184	C₂×C₄	C₂²×C₄×C₂₂₃	1	1	yes	characteristic
185	C₂×C₄	C₂²×C₄×C₂₂₃	1	2	yes	—
186	C₂×C₄	C₂²×C₄×C₂₂₃	1	2	yes	—
187	C₂×C₄	C₂²×C₄×C₂₂₃	1	1	yes	characteristic
188	C₂×C₄	C₂²×C₄×C₂₂₃	1	1	yes	characteristic
189	C₂×C₄	C₂⁴×C₂₂₃	1	1	yes	characteristic
190	C₈	—	2	4	no	—
191	C₈	—	2	2	no	—
192	C₈	—	2	2	no	—
193	C₂²	—	2	1	no	—
194	C₂²	—	2	1	no	—
195	C₂²	—	2	2	no	—
196	C₂²	—	2	2	no	—
197	C₂²	—	2	2	no	—
198	C₂²	C₂×M₁₆×C₂₂₃	1	2	yes	—
199	C₂²	C₂³×C₄×C₂₂₃	1	1	yes	characteristic
200	C₄	—	2	1	no	—
201	C₄	—	2	2	no	—
202	C₄	—	2	2	no	—
203	C₄	—	2	1	no	—
204	C₄	—	2	1	no	—
205	C₄	—	2	2	no	—
206	C₄	C₂×M₁₆×C₂₂₃	1	2	yes	—
207	C₄	C₂×M₁₆×C₂₂₃	1	2	yes	—
208	C₄	H(5,𝔽₂)×C₂₂₃	1	1	yes	characteristic
209	C₄	2₋¹⁺⁴×C₂₂₃	1	1	yes	characteristic
210	C₂	—	2	2	no	—
211	C₂	—	4	2	no	—
212	C₂	#520×C₂₂₃	1	1	yes	characteristic
213	C₂	C₂²×M₁₆×C₂₂₃	1	1	yes	characteristic
214	C₂	#568×C₂₂₃	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄×C₂₂₃	C₂⁴
commutator	C₂²	C₂³×C₄×C₂₂₃
Frattini, cosocle	C₂×C₄	C₂⁴×C₂₂₃
socle	C₂²×C₂₂₃	C₂³×C₄
prime radical	C₂×#41×C₂₂₃	C₄
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₂₂₃ ◁ G

Complements

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

N	H ≅ G/N	type
2. #446×C₂₂₃	C₂	semidirect
3. #436×C₂₂₃	C₂	semidirect
4. #435×C₂₂₃	C₂	semidirect
5. #432×C₂₂₃	C₂	semidirect
6. #431×C₂₂₃	C₂	semidirect
7. #423×C₂₂₃	C₂	semidirect
8. C₂×C₄⋊C₈×C₂₂₃	C₂	semidirect
72. C₂×C₄×C₂₂₃	M₁₆	semidirect
107. C₂₂₃	#2989	direct (N × H)
108. #2989	C₂₂₃	direct (N × H)
110. #446	C₂×C₂₂₃	semidirect
111. #436	C₂×C₂₂₃	semidirect
112. #435	C₂×C₂₂₃	semidirect
113. #432	C₂×C₂₂₃	semidirect
114. #431	C₂×C₂₂₃	semidirect
115. #423	C₂×C₂₂₃	semidirect
116. C₂×C₄⋊C₈	C₂×C₂₂₃	semidirect
180. C₂×C₄	M₁₆×C₂₂₃	semidirect

Maximal subgroups

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

subgroup	index
C₄×#41×C₂₂₃	2
#446×C₂₂₃	2
#436×C₂₂₃	2
#435×C₂₂₃	2
#432×C₂₂₃	2
#431×C₂₂₃	2
#423×C₂₂₃	2
C₂×C₄⋊C₈×C₂₂₃	2
#2989	223

Subgroups — #2989×C223 #43252004

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — #2989×C₂₂₃ #43252004