Subgroups — #3449×C₂₂₃ #43252464

Order 28544.

1556 conjugacy classes of subgroups.

The subgroup lattice has 224 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×#41×C₂₂₃	C₂	1	2	yes	maximal, complemented
2	#537×C₂₂₃	C₂	1	4	yes	maximal, complemented
3	#536×C₂₂₃	C₂	1	4	yes	maximal, complemented
4	#535×C₂₂₃	C₂	1	4	yes	maximal, complemented
5	#534×C₂₂₃	C₂	1	4	yes	maximal, complemented
6	#532×C₂₂₃	C₂	1	4	yes	maximal, complemented
7	#525×C₂₂₃	C₂	1	8	yes	maximal, complemented
8	C₂×#117×C₂₂₃	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂×#41×C₂₂₃	—	2	16	no	—
10	C₂×#41×C₂₂₃	C₂²	1	4	yes	—
11	C₂×#41×C₂₂₃	C₂²	1	4	yes	—
12	C₂×#41×C₂₂₃	C₂²	1	4	yes	complemented
13	C₂×#41×C₂₂₃	C₂²	1	4	yes	complemented
14	C₂²×Q₈×C₂₂₃	C₂²	1	2	yes	complemented
15	C₂²×D₄×C₂₂₃	C₂²	1	4	yes	—
16	C₂²×D₄×C₂₂₃	C₂²	1	2	yes	complemented
17	C₂³×C₄×C₂₂₃	C₂²	1	1	yes	characteristic
18	C₂³×C₄×C₂₂₃	C₂²	1	4	yes	complemented
19	C₄⋊Q₈×C₂₂₃	C₂²	1	4	yes	complemented
20	#127×C₂₂₃	C₂²	1	4	yes	complemented
21	#124×C₂₂₃	C₂²	1	8	yes	complemented
22	#123×C₂₂₃	C₂²	1	16	yes	complemented
23	C₂²⋊Q₈×C₂₂₃	C₂²	1	16	yes	complemented
24	#121×C₂₂₃	C₂²	1	16	yes	complemented
25	#121×C₂₂₃	C₂²	1	16	yes	complemented
26	#120×C₂₂₃	C₂²	1	16	yes	complemented
27	C₄×D₄×C₂₂₃	C₂²	1	16	yes	complemented
28	#117×C₂₂₃	C₂²	1	4	yes	complemented
29	#117×C₂₂₃	C₂²	1	4	yes	complemented
30	C₂×C₄⋊C₄×C₂₂₃	C₂²	1	2	yes	complemented
31	C₂×C₂²⋊C₄×C₂₂₃	C₂²	1	2	yes	—
32	C₂×C₄²×C₂₂₃	C₂²	1	2	yes	complemented
33	C₂⁴×C₂₂₃	C₂³	1	4	yes	—
34	C₂⁴×C₂₂₃	C₂³	1	1	yes	characteristic
35	#41×C₂₂₃	—	2	16	no	—
36	#41×C₂₂₃	—	2	16	no	—
37	#41×C₂₂₃	—	2	16	no	—
38	#41×C₂₂₃	—	2	16	no	—
39	C₂×Q₈×C₂₂₃	—	2	8	no	—
40	C₂×Q₈×C₂₂₃	C₂³	1	4	yes	—
41	C₂×Q₈×C₂₂₃	C₂³	1	4	yes	—
42	C₂×D₄×C₂₂₃	—	2	16	no	—
43	C₂×D₄×C₂₂₃	—	2	8	no	—
44	C₂×D₄×C₂₂₃	C₂³	1	8	yes	—
45	C₂×D₄×C₂₂₃	C₂³	1	8	yes	—
46	C₂×D₄×C₂₂₃	C₂³	1	4	yes	—
47	C₂×D₄×C₂₂₃	C₂³	1	4	yes	—
48	C₂×D₄×C₂₂₃	C₂³	1	16	yes	—
49	C₂²×C₄×C₂₂₃	—	2	16	no	—
50	C₂²×C₄×C₂₂₃	D₄	1	8	yes	complemented
51	C₂²×C₄×C₂₂₃	C₂³	1	8	yes	—
52	C₂²×C₄×C₂₂₃	C₂³	1	4	yes	—
53	C₂²×C₄×C₂₂₃	C₂³	1	8	yes	—
54	C₂²×C₄×C₂₂₃	C₂³	1	4	yes	—
55	C₂²×C₄×C₂₂₃	C₂³	1	1	yes	characteristic
56	C₂²×C₄×C₂₂₃	C₂³	1	2	yes	—
57	C₂²×C₄×C₂₂₃	C₂³	1	1	yes	characteristic
58	C₂²×C₄×C₂₂₃	C₂³	1	2	yes	—
59	C₄⋊C₄×C₂₂₃	C₂³	1	8	yes	—
60	C₄⋊C₄×C₂₂₃	C₂³	1	16	yes	—
61	C₂²⋊C₄×C₂₂₃	C₂³	1	8	yes	—
62	C₂²⋊C₄×C₂₂₃	C₂³	1	32	yes	—
63	C₄²×C₂₂₃	C₂³	1	4	yes	—
64	C₄²×C₂₂₃	C₂³	1	4	yes	—
65	C₂³×C₂₂₃	—	2	8	no	—
66	C₂³×C₂₂₃	—	4	2	no	—
67	C₂³×C₂₂₃	—	4	8	no	—
68	C₂³×C₂₂₃	C₂×D₄	1	4	yes	—
69	C₂³×C₂₂₃	C₂⁴	1	8	yes	—
70	C₂³×C₂₂₃	C₂⁴	1	2	yes	—
71	C₂³×C₂₂₃	C₂⁴	1	1	yes	characteristic
72	Q₈×C₂₂₃	—	2	8	no	—
73	Q₈×C₂₂₃	—	2	8	no	—
74	D₄×C₂₂₃	—	2	16	no	—
75	D₄×C₂₂₃	—	2	16	no	—
76	D₄×C₂₂₃	—	2	8	no	—
77	D₄×C₂₂₃	—	2	8	no	—
78	D₄×C₂₂₃	—	4	16	no	—
79	C₂×C₄×C₂₂₃	—	2	8	no	—
80	C₂×C₄×C₂₂₃	—	2	16	no	—
81	C₂×C₄×C₂₂₃	—	2	16	no	—
82	C₂×C₄×C₂₂₃	—	4	4	no	—
83	C₂×C₄×C₂₂₃	C₂×D₄	1	8	yes	—
84	C₂×C₄×C₂₂₃	C₂×D₄	1	8	yes	—
85	C₂×C₄×C₂₂₃	C₂×D₄	1	4	yes	—
86	C₂×C₄×C₂₂₃	C₂×D₄	1	4	yes	—
87	C₂×C₄×C₂₂₃	C₂⁴	1	8	yes	—
88	C₂×C₄×C₂₂₃	C₂⁴	1	8	yes	—
89	C₂×C₄×C₂₂₃	C₂⁴	1	1	yes	characteristic
90	C₂×C₄×C₂₂₃	C₂⁴	1	2	yes	—
91	C₂×C₄×C₂₂₃	C₂⁴	1	1	yes	characteristic
92	C₂²×C₂₂₃	—	2	2	no	—
93	C₂²×C₂₂₃	—	2	4	no	—
94	C₂²×C₂₂₃	—	2	8	no	—
95	C₂²×C₂₂₃	—	2	16	no	—
96	C₂²×C₂₂₃	—	4	4	no	—
97	C₂²×C₂₂₃	—	4	16	no	—
98	C₂²×C₂₂₃	C₂²×D₄	1	4	yes	—
99	C₂²×C₂₂₃	C₂²×D₄	1	2	yes	—
100	C₂²×C₂₂₃	C₂⁵	1	1	yes	characteristic
101	C₄×C₂₂₃	—	2	8	no	—
102	C₄×C₂₂₃	—	2	8	no	—
103	C₄×C₂₂₃	C₂²×D₄	1	2	yes	—
104	C₄×C₂₂₃	C₂²×D₄	1	4	yes	—
105	C₄×C₂₂₃	C₂²×D₄	1	2	yes	—
106	C₂×C₂₂₃	—	2	4	no	—
107	C₂×C₂₂₃	—	2	2	no	—
108	C₂×C₂₂₃	—	4	8	no	—
109	C₂×C₂₂₃	C₂³×D₄	1	1	yes	characteristic
110	C₂×C₂₂₃	#585	1	2	yes	—
111	C₂₂₃	#3449	1	1	yes	characteristic, direct product factor, Sylow
112	#3449	C₂₂₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
113	C₂²×#41	C₂×C₂₂₃	1	2	yes	complemented
114	#537	C₂×C₂₂₃	1	4	yes	complemented
115	#536	C₂×C₂₂₃	1	4	yes	complemented
116	#535	C₂×C₂₂₃	1	4	yes	complemented
117	#534	C₂×C₂₂₃	1	4	yes	complemented
118	#532	C₂×C₂₂₃	1	4	yes	complemented
119	#525	C₂×C₂₂₃	1	8	yes	complemented
120	C₂×#117	C₂×C₂₂₃	1	1	yes	characteristic, complemented
121	C₂×#41	—	2	16	no	—
122	C₂×#41	C₂²×C₂₂₃	1	4	yes	—
123	C₂×#41	C₂²×C₂₂₃	1	4	yes	—
124	C₂×#41	C₂²×C₂₂₃	1	4	yes	complemented
125	C₂×#41	C₂²×C₂₂₃	1	4	yes	complemented
126	C₂²×Q₈	C₂²×C₂₂₃	1	2	yes	complemented
127	C₂²×D₄	C₂²×C₂₂₃	1	4	yes	—
128	C₂²×D₄	C₂²×C₂₂₃	1	2	yes	complemented
129	C₂³×C₄	C₂²×C₂₂₃	1	1	yes	characteristic
130	C₂³×C₄	C₂²×C₂₂₃	1	4	yes	complemented
131	C₄⋊Q₈	C₂²×C₂₂₃	1	4	yes	complemented
132	#127	C₂²×C₂₂₃	1	4	yes	complemented
133	#124	C₂²×C₂₂₃	1	8	yes	complemented
134	#123	C₂²×C₂₂₃	1	16	yes	complemented
135	C₂²⋊Q₈	C₂²×C₂₂₃	1	16	yes	complemented
136	#121	C₂²×C₂₂₃	1	16	yes	complemented
137	#121	C₂²×C₂₂₃	1	16	yes	complemented
138	#120	C₂²×C₂₂₃	1	16	yes	complemented
139	C₄×D₄	C₂²×C₂₂₃	1	16	yes	complemented
140	#117	C₂²×C₂₂₃	1	4	yes	complemented
141	#117	C₂²×C₂₂₃	1	4	yes	complemented
142	C₂×C₄⋊C₄	C₂²×C₂₂₃	1	2	yes	complemented
143	C₂×C₂²⋊C₄	C₂²×C₂₂₃	1	2	yes	—
144	C₂×C₄²	C₂²×C₂₂₃	1	2	yes	complemented
145	C₂⁴	C₂³×C₂₂₃	1	4	yes	—
146	C₂⁴	C₂³×C₂₂₃	1	1	yes	characteristic
147	#41	—	2	16	no	—
148	#41	—	2	16	no	—
149	#41	—	2	16	no	—
150	#41	—	2	16	no	—
151	C₂×Q₈	—	2	8	no	—
152	C₂×Q₈	C₂³×C₂₂₃	1	4	yes	—
153	C₂×Q₈	C₂³×C₂₂₃	1	4	yes	—
154	C₂×D₄	—	2	16	no	—
155	C₂×D₄	—	2	8	no	—
156	C₂×D₄	C₂³×C₂₂₃	1	8	yes	—
157	C₂×D₄	C₂³×C₂₂₃	1	8	yes	—
158	C₂×D₄	C₂³×C₂₂₃	1	4	yes	—
159	C₂×D₄	C₂³×C₂₂₃	1	4	yes	—
160	C₂×D₄	C₂³×C₂₂₃	1	16	yes	—
161	C₂²×C₄	—	2	16	no	—
162	C₂²×C₄	D₄×C₂₂₃	1	8	yes	complemented
163	C₂²×C₄	C₂³×C₂₂₃	1	8	yes	—
164	C₂²×C₄	C₂³×C₂₂₃	1	8	yes	—
165	C₂²×C₄	C₂³×C₂₂₃	1	4	yes	—
166	C₂²×C₄	C₂³×C₂₂₃	1	4	yes	—
167	C₂²×C₄	C₂³×C₂₂₃	1	1	yes	characteristic
168	C₂²×C₄	C₂³×C₂₂₃	1	1	yes	characteristic
169	C₂²×C₄	C₂³×C₂₂₃	1	2	yes	—
170	C₂²×C₄	C₂³×C₂₂₃	1	2	yes	—
171	C₄⋊C₄	C₂³×C₂₂₃	1	8	yes	—
172	C₄⋊C₄	C₂³×C₂₂₃	1	16	yes	—
173	C₂²⋊C₄	C₂³×C₂₂₃	1	8	yes	—
174	C₂²⋊C₄	C₂³×C₂₂₃	1	32	yes	—
175	C₄²	C₂³×C₂₂₃	1	4	yes	—
176	C₄²	C₂³×C₂₂₃	1	4	yes	—
177	C₂³	—	2	8	no	—
178	C₂³	—	4	2	no	—
179	C₂³	—	4	8	no	—
180	C₂³	C₂×D₄×C₂₂₃	1	4	yes	—
181	C₂³	C₂⁴×C₂₂₃	1	8	yes	—
182	C₂³	C₂⁴×C₂₂₃	1	2	yes	—
183	C₂³	C₂⁴×C₂₂₃	1	1	yes	characteristic
184	Q₈	—	2	8	no	—
185	Q₈	—	2	8	no	—
186	D₄	—	2	16	no	—
187	D₄	—	2	16	no	—
188	D₄	—	2	8	no	—
189	D₄	—	2	8	no	—
190	D₄	—	4	16	no	—
191	C₂×C₄	—	2	8	no	—
192	C₂×C₄	—	2	16	no	—
193	C₂×C₄	—	2	16	no	—
194	C₂×C₄	—	4	4	no	—
195	C₂×C₄	C₂×D₄×C₂₂₃	1	8	yes	—
196	C₂×C₄	C₂×D₄×C₂₂₃	1	8	yes	—
197	C₂×C₄	C₂×D₄×C₂₂₃	1	4	yes	—
198	C₂×C₄	C₂×D₄×C₂₂₃	1	4	yes	—
199	C₂×C₄	C₂⁴×C₂₂₃	1	8	yes	—
200	C₂×C₄	C₂⁴×C₂₂₃	1	8	yes	—
201	C₂×C₄	C₂⁴×C₂₂₃	1	1	yes	characteristic
202	C₂×C₄	C₂⁴×C₂₂₃	1	1	yes	characteristic
203	C₂×C₄	C₂⁴×C₂₂₃	1	2	yes	—
204	C₂²	—	2	2	no	—
205	C₂²	—	2	4	no	—
206	C₂²	—	2	8	no	—
207	C₂²	—	2	16	no	—
208	C₂²	—	4	4	no	—
209	C₂²	—	4	16	no	—
210	C₂²	C₂²×D₄×C₂₂₃	1	4	yes	—
211	C₂²	C₂²×D₄×C₂₂₃	1	2	yes	—
212	C₂²	C₂⁵×C₂₂₃	1	1	yes	characteristic
213	C₄	—	2	8	no	—
214	C₄	—	2	8	no	—
215	C₄	C₂²×D₄×C₂₂₃	1	2	yes	—
216	C₄	C₂²×D₄×C₂₂₃	1	4	yes	—
217	C₄	C₂²×D₄×C₂₂₃	1	2	yes	—
218	C₂	—	2	4	no	—
219	C₂	—	2	2	no	—
220	C₂	—	4	8	no	—
221	C₂	C₂³×D₄×C₂₂₃	1	1	yes	characteristic
222	C₂	#585×C₂₂₃	1	2	yes	—

	subgroup	G/H
center	C₂×C₄×C₂₂₃	C₂⁴
commutator, Frattini, cosocle	C₂²	C₂⁵×C₂₂₃
socle	C₂²×C₂₂₃	C₂⁵
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
1. C₂²×#41×C₂₂₃	C₂	semidirect
2. #537×C₂₂₃	C₂	semidirect
3. #536×C₂₂₃	C₂	semidirect
4. #535×C₂₂₃	C₂	semidirect
5. #534×C₂₂₃	C₂	semidirect
6. #532×C₂₂₃	C₂	semidirect
7. #525×C₂₂₃	C₂	semidirect
8. C₂×#117×C₂₂₃	C₂	semidirect
12. C₂×#41×C₂₂₃	C₂²	semidirect
13. C₂×#41×C₂₂₃	C₂²	semidirect
14. C₂²×Q₈×C₂₂₃	C₂²	semidirect
16. C₂²×D₄×C₂₂₃	C₂²	semidirect
18. C₂³×C₄×C₂₂₃	C₂²	semidirect
19. C₄⋊Q₈×C₂₂₃	C₂²	semidirect
20. #127×C₂₂₃	C₂²	semidirect
21. #124×C₂₂₃	C₂²	semidirect
22. #123×C₂₂₃	C₂²	semidirect
23. C₂²⋊Q₈×C₂₂₃	C₂²	semidirect
24. #121×C₂₂₃	C₂²	semidirect
25. #121×C₂₂₃	C₂²	semidirect
26. #120×C₂₂₃	C₂²	semidirect
27. C₄×D₄×C₂₂₃	C₂²	semidirect
28. #117×C₂₂₃	C₂²	semidirect
29. #117×C₂₂₃	C₂²	semidirect
30. C₂×C₄⋊C₄×C₂₂₃	C₂²	semidirect
32. C₂×C₄²×C₂₂₃	C₂²	semidirect
50. C₂²×C₄×C₂₂₃	D₄	semidirect
111. C₂₂₃	#3449	direct (N × H)
112. #3449	C₂₂₃	direct (N × H)
113. C₂²×#41	C₂×C₂₂₃	semidirect
114. #537	C₂×C₂₂₃	semidirect
115. #536	C₂×C₂₂₃	semidirect
116. #535	C₂×C₂₂₃	semidirect
117. #534	C₂×C₂₂₃	semidirect
118. #532	C₂×C₂₂₃	semidirect
119. #525	C₂×C₂₂₃	semidirect
120. C₂×#117	C₂×C₂₂₃	semidirect
124. C₂×#41	C₂²×C₂₂₃	semidirect
125. C₂×#41	C₂²×C₂₂₃	semidirect
126. C₂²×Q₈	C₂²×C₂₂₃	semidirect
128. C₂²×D₄	C₂²×C₂₂₃	semidirect
130. C₂³×C₄	C₂²×C₂₂₃	semidirect
131. C₄⋊Q₈	C₂²×C₂₂₃	semidirect
132. #127	C₂²×C₂₂₃	semidirect
133. #124	C₂²×C₂₂₃	semidirect
134. #123	C₂²×C₂₂₃	semidirect
135. C₂²⋊Q₈	C₂²×C₂₂₃	semidirect
136. #121	C₂²×C₂₂₃	semidirect
137. #121	C₂²×C₂₂₃	semidirect
138. #120	C₂²×C₂₂₃	semidirect
139. C₄×D₄	C₂²×C₂₂₃	semidirect
140. #117	C₂²×C₂₂₃	semidirect
141. #117	C₂²×C₂₂₃	semidirect
142. C₂×C₄⋊C₄	C₂²×C₂₂₃	semidirect
144. C₂×C₄²	C₂²×C₂₂₃	semidirect
162. C₂²×C₄	D₄×C₂₂₃	semidirect

Maximal subgroups

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

subgroup	index
C₂²×#41×C₂₂₃	2
#537×C₂₂₃	2
#536×C₂₂₃	2
#535×C₂₂₃	2
#534×C₂₂₃	2
#532×C₂₂₃	2
#525×C₂₂₃	2
C₂×#117×C₂₂₃	2
#3449	223

Subgroups — #3449×C223 #43252464

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — #3449×C₂₂₃ #43252464