Subgroups — #3508×C₂₂₃ #43252523

Order 28544.

1232 conjugacy classes of subgroups.

The subgroup lattice has 248 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#559×C₂₂₃	C₂	1	4	yes	maximal, complemented
2	#550×C₂₂₃	C₂	1	4	yes	maximal, complemented
3	#546×C₂₂₃	C₂	1	8	yes	maximal, complemented
4	#540×C₂₂₃	C₂	1	4	yes	maximal, complemented
5	#535×C₂₂₃	C₂	1	2	yes	maximal, complemented
6	#533×C₂₂₃	C₂	1	1	yes	maximal, characteristic, complemented
7	#532×C₂₂₃	C₂	1	4	yes	maximal, complemented
8	C₂×#127×C₂₂₃	C₂	1	1	yes	maximal, characteristic, complemented
9	#518×C₂₂₃	C₂	1	2	yes	maximal, complemented
10	C₄×#41×C₂₂₃	C₂	1	1	yes	maximal, characteristic, complemented
11	C₂×#41×C₂₂₃	C₂²	1	1	yes	characteristic
12	C₂×#41×C₂₂₃	C₂²	1	8	yes	complemented
13	C₂²×D₄×C₂₂₃	C₂²	1	2	yes	—
14	C₂²×D₄×C₂₂₃	C₂²	1	4	yes	—
15	C₄⋊Q₈×C₂₂₃	C₂²	1	2	yes	complemented
16	#127×C₂₂₃	C₂²	1	2	yes	—
17	#127×C₂₂₃	C₂²	1	2	yes	—
18	#127×C₂₂₃	C₂²	1	4	yes	—
19	#127×C₂₂₃	C₂²	1	4	yes	complemented
20	#125×C₂₂₃	C₂²	1	2	yes	complemented
21	#124×C₂₂₃	C₂²	1	8	yes	complemented
22	#124×C₂₂₃	C₂²	1	4	yes	complemented
23	#123×C₂₂₃	C₂²	1	16	yes	complemented
24	C₂²⋊Q₈×C₂₂₃	C₂²	1	4	yes	—
25	#121×C₂₂₃	C₂²	1	8	yes	complemented
26	#121×C₂₂₃	C₂²	1	8	yes	complemented
27	#121×C₂₂₃	C₂²	1	16	yes	complemented
28	#120×C₂₂₃	C₂²	1	8	yes	—
29	C₄×Q₈×C₂₂₃	C₂²	1	4	yes	complemented
30	C₄×Q₈×C₂₂₃	C₂²	1	2	yes	complemented
31	C₄×D₄×C₂₂₃	C₂²	1	4	yes	—
32	C₄×D₄×C₂₂₃	C₂²	1	8	yes	complemented
33	C₄×D₄×C₂₂₃	C₂²	1	8	yes	complemented
34	C₄×D₄×C₂₂₃	C₂²	1	2	yes	complemented
35	C₄×D₄×C₂₂₃	C₂²	1	8	yes	complemented
36	#117×C₂₂₃	C₂²	1	2	yes	—
37	#117×C₂₂₃	C₂²	1	1	yes	characteristic
38	#117×C₂₂₃	C₂²	1	2	yes	complemented
39	C₂×C₂²⋊C₄×C₂₂₃	C₂²	1	8	yes	—
40	C₂×C₄²×C₂₂₃	C₂²	1	1	yes	characteristic
41	C₂×C₄²×C₂₂₃	C₂²	1	2	yes	complemented
42	C₂⁴×C₂₂₃	C₂³	1	4	yes	—
43	#41×C₂₂₃	—	2	4	no	—
44	#41×C₂₂₃	—	4	16	no	—
45	C₂×Q₈×C₂₂₃	C₂³	1	1	yes	characteristic
46	C₂×Q₈×C₂₂₃	C₂³	1	4	yes	—
47	C₂×D₄×C₂₂₃	—	2	8	no	—
48	C₂×D₄×C₂₂₃	—	4	8	no	—
49	C₂×D₄×C₂₂₃	C₂³	1	2	yes	—
50	C₂×D₄×C₂₂₃	C₂³	1	4	yes	—
51	C₂×D₄×C₂₂₃	C₂³	1	4	yes	—
52	C₂×D₄×C₂₂₃	C₂³	1	1	yes	characteristic
53	C₂×D₄×C₂₂₃	C₂³	1	8	yes	—
54	C₂×D₄×C₂₂₃	C₂³	1	16	yes	—
55	C₂×D₄×C₂₂₃	C₂³	1	8	yes	—
56	C₂²×C₄×C₂₂₃	C₂³	1	2	yes	—
57	C₂²×C₄×C₂₂₃	C₂³	1	2	yes	—
58	C₂²×C₄×C₂₂₃	C₂³	1	4	yes	—
59	C₂²×C₄×C₂₂₃	C₂³	1	16	yes	—
60	C₂²×C₄×C₂₂₃	C₂³	1	1	yes	characteristic
61	C₄⋊C₄×C₂₂₃	C₂³	1	2	yes	—
62	C₄⋊C₄×C₂₂₃	C₂³	1	4	yes	—
63	C₄⋊C₄×C₂₂₃	C₂³	1	4	yes	—
64	C₄⋊C₄×C₂₂₃	C₂³	1	4	yes	—
65	C₄⋊C₄×C₂₂₃	C₂³	1	8	yes	—
66	C₂²⋊C₄×C₂₂₃	C₂³	1	2	yes	—
67	C₂²⋊C₄×C₂₂₃	C₂³	1	4	yes	—
68	C₂²⋊C₄×C₂₂₃	C₂³	1	16	yes	—
69	C₂²⋊C₄×C₂₂₃	C₂³	1	4	yes	—
70	C₂²⋊C₄×C₂₂₃	C₂³	1	16	yes	—
71	C₄²×C₂₂₃	C₂³	1	1	yes	characteristic
72	C₄²×C₂₂₃	C₂³	1	1	yes	characteristic
73	C₄²×C₂₂₃	C₂³	1	2	yes	—
74	C₄²×C₂₂₃	C₂³	1	2	yes	—
75	C₄²×C₂₂₃	C₂³	1	4	yes	—
76	C₄²×C₂₂₃	C₂³	1	4	yes	—
77	C₂³×C₂₂₃	—	2	8	no	—
78	C₂³×C₂₂₃	—	4	8	no	—
79	C₂³×C₂₂₃	C₂⁴	1	8	yes	—
80	C₂³×C₂₂₃	C₂⁴	1	2	yes	—
81	C₂³×C₂₂₃	C₂⁴	1	1	yes	characteristic
82	Q₈×C₂₂₃	—	2	2	no	—
83	Q₈×C₂₂₃	—	2	8	no	—
84	D₄×C₂₂₃	—	2	4	no	—
85	D₄×C₂₂₃	—	2	8	no	—
86	D₄×C₂₂₃	—	2	8	no	—
87	D₄×C₂₂₃	—	2	2	no	—
88	D₄×C₂₂₃	—	2	16	no	—
89	D₄×C₂₂₃	—	4	8	no	—
90	D₄×C₂₂₃	—	4	16	no	—
91	C₂×C₄×C₂₂₃	—	2	4	no	—
92	C₂×C₄×C₂₂₃	—	2	4	no	—
93	C₂×C₄×C₂₂₃	—	2	8	no	—
94	C₂×C₄×C₂₂₃	—	4	16	no	—
95	C₂×C₄×C₂₂₃	#41	1	4	yes	complemented
96	C₂×C₄×C₂₂₃	C₂⁴	1	2	yes	—
97	C₂×C₄×C₂₂₃	C₂⁴	1	4	yes	—
98	C₂×C₄×C₂₂₃	C₂⁴	1	2	yes	—
99	C₂×C₄×C₂₂₃	C₂⁴	1	8	yes	—
100	C₂×C₄×C₂₂₃	C₂⁴	1	1	yes	characteristic
101	C₂×C₄×C₂₂₃	C₂⁴	1	1	yes	characteristic
102	C₂×C₄×C₂₂₃	C₂⁴	1	2	yes	—
103	C₂²×C₂₂₃	—	2	2	no	—
104	C₂²×C₂₂₃	—	2	2	no	—
105	C₂²×C₂₂₃	—	2	8	no	—
106	C₂²×C₂₂₃	—	2	4	no	—
107	C₂²×C₂₂₃	—	2	16	no	—
108	C₂²×C₂₂₃	—	8	8	no	—
109	C₂²×C₂₂₃	C₂×#41	1	2	yes	—
110	C₂²×C₂₂₃	C₂⁵	1	1	yes	characteristic
111	C₄×C₂₂₃	—	2	2	no	—
112	C₄×C₂₂₃	—	2	4	no	—
113	C₄×C₂₂₃	—	2	2	no	—
114	C₄×C₂₂₃	—	2	8	no	—
115	C₄×C₂₂₃	C₂×#41	1	2	yes	—
116	C₄×C₂₂₃	C₂×#41	1	2	yes	—
117	C₄×C₂₂₃	H(5,𝔽₂)	1	4	yes	—
118	C₂×C₂₂₃	—	2	2	no	—
119	C₂×C₂₂₃	—	4	2	no	—
120	C₂×C₂₂₃	—	4	8	no	—
121	C₂×C₂₂₃	C₂²×#41	1	1	yes	characteristic
122	C₂×C₂₂₃	C₂×H(5,𝔽₂)	1	2	yes	—
123	C₂₂₃	#3508	1	1	yes	characteristic, direct product factor, Sylow
124	#3508	C₂₂₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
125	#559	C₂×C₂₂₃	1	4	yes	complemented
126	#550	C₂×C₂₂₃	1	4	yes	complemented
127	#546	C₂×C₂₂₃	1	8	yes	complemented
128	#540	C₂×C₂₂₃	1	4	yes	complemented
129	#535	C₂×C₂₂₃	1	2	yes	complemented
130	#533	C₂×C₂₂₃	1	1	yes	characteristic, complemented
131	#532	C₂×C₂₂₃	1	4	yes	complemented
132	C₂×#127	C₂×C₂₂₃	1	1	yes	characteristic, complemented
133	#518	C₂×C₂₂₃	1	2	yes	complemented
134	C₄×#41	C₂×C₂₂₃	1	1	yes	characteristic, complemented
135	C₂×#41	C₂²×C₂₂₃	1	1	yes	characteristic
136	C₂×#41	C₂²×C₂₂₃	1	8	yes	complemented
137	C₂²×D₄	C₂²×C₂₂₃	1	2	yes	—
138	C₂²×D₄	C₂²×C₂₂₃	1	4	yes	—
139	C₄⋊Q₈	C₂²×C₂₂₃	1	2	yes	complemented
140	#127	C₂²×C₂₂₃	1	2	yes	—
141	#127	C₂²×C₂₂₃	1	2	yes	—
142	#127	C₂²×C₂₂₃	1	4	yes	—
143	#127	C₂²×C₂₂₃	1	4	yes	complemented
144	#125	C₂²×C₂₂₃	1	2	yes	complemented
145	#124	C₂²×C₂₂₃	1	8	yes	complemented
146	#124	C₂²×C₂₂₃	1	4	yes	complemented
147	#123	C₂²×C₂₂₃	1	16	yes	complemented
148	C₂²⋊Q₈	C₂²×C₂₂₃	1	4	yes	—
149	#121	C₂²×C₂₂₃	1	8	yes	complemented
150	#121	C₂²×C₂₂₃	1	8	yes	complemented
151	#121	C₂²×C₂₂₃	1	16	yes	complemented
152	#120	C₂²×C₂₂₃	1	8	yes	—
153	C₄×Q₈	C₂²×C₂₂₃	1	2	yes	complemented
154	C₄×Q₈	C₂²×C₂₂₃	1	4	yes	complemented
155	C₄×D₄	C₂²×C₂₂₃	1	4	yes	—
156	C₄×D₄	C₂²×C₂₂₃	1	2	yes	complemented
157	C₄×D₄	C₂²×C₂₂₃	1	8	yes	complemented
158	C₄×D₄	C₂²×C₂₂₃	1	8	yes	complemented
159	C₄×D₄	C₂²×C₂₂₃	1	8	yes	complemented
160	#117	C₂²×C₂₂₃	1	1	yes	characteristic
161	#117	C₂²×C₂₂₃	1	2	yes	—
162	#117	C₂²×C₂₂₃	1	2	yes	complemented
163	C₂×C₂²⋊C₄	C₂²×C₂₂₃	1	8	yes	—
164	C₂×C₄²	C₂²×C₂₂₃	1	1	yes	characteristic
165	C₂×C₄²	C₂²×C₂₂₃	1	2	yes	complemented
166	C₂⁴	C₂³×C₂₂₃	1	4	yes	—
167	#41	—	2	4	no	—
168	#41	—	4	16	no	—
169	C₂×Q₈	C₂³×C₂₂₃	1	1	yes	characteristic
170	C₂×Q₈	C₂³×C₂₂₃	1	4	yes	—
171	C₂×D₄	—	2	8	no	—
172	C₂×D₄	—	4	8	no	—
173	C₂×D₄	C₂³×C₂₂₃	1	2	yes	—
174	C₂×D₄	C₂³×C₂₂₃	1	4	yes	—
175	C₂×D₄	C₂³×C₂₂₃	1	4	yes	—
176	C₂×D₄	C₂³×C₂₂₃	1	1	yes	characteristic
177	C₂×D₄	C₂³×C₂₂₃	1	8	yes	—
178	C₂×D₄	C₂³×C₂₂₃	1	16	yes	—
179	C₂×D₄	C₂³×C₂₂₃	1	8	yes	—
180	C₂²×C₄	C₂³×C₂₂₃	1	2	yes	—
181	C₂²×C₄	C₂³×C₂₂₃	1	2	yes	—
182	C₂²×C₄	C₂³×C₂₂₃	1	4	yes	—
183	C₂²×C₄	C₂³×C₂₂₃	1	16	yes	—
184	C₂²×C₄	C₂³×C₂₂₃	1	1	yes	characteristic
185	C₄⋊C₄	C₂³×C₂₂₃	1	2	yes	—
186	C₄⋊C₄	C₂³×C₂₂₃	1	4	yes	—
187	C₄⋊C₄	C₂³×C₂₂₃	1	4	yes	—
188	C₄⋊C₄	C₂³×C₂₂₃	1	4	yes	—
189	C₄⋊C₄	C₂³×C₂₂₃	1	8	yes	—
190	C₂²⋊C₄	C₂³×C₂₂₃	1	2	yes	—
191	C₂²⋊C₄	C₂³×C₂₂₃	1	4	yes	—
192	C₂²⋊C₄	C₂³×C₂₂₃	1	16	yes	—
193	C₂²⋊C₄	C₂³×C₂₂₃	1	4	yes	—
194	C₂²⋊C₄	C₂³×C₂₂₃	1	16	yes	—
195	C₄²	C₂³×C₂₂₃	1	1	yes	characteristic
196	C₄²	C₂³×C₂₂₃	1	1	yes	characteristic
197	C₄²	C₂³×C₂₂₃	1	2	yes	—
198	C₄²	C₂³×C₂₂₃	1	2	yes	—
199	C₄²	C₂³×C₂₂₃	1	4	yes	—
200	C₄²	C₂³×C₂₂₃	1	4	yes	—
201	C₂³	—	2	8	no	—
202	C₂³	—	4	8	no	—
203	C₂³	C₂⁴×C₂₂₃	1	8	yes	—
204	C₂³	C₂⁴×C₂₂₃	1	2	yes	—
205	C₂³	C₂⁴×C₂₂₃	1	1	yes	characteristic
206	Q₈	—	2	2	no	—
207	Q₈	—	2	8	no	—
208	D₄	—	2	4	no	—
209	D₄	—	2	8	no	—
210	D₄	—	2	8	no	—
211	D₄	—	2	2	no	—
212	D₄	—	2	16	no	—
213	D₄	—	4	8	no	—
214	D₄	—	4	16	no	—
215	C₂×C₄	—	2	4	no	—
216	C₂×C₄	—	2	4	no	—
217	C₂×C₄	—	2	8	no	—
218	C₂×C₄	—	4	16	no	—
219	C₂×C₄	#41×C₂₂₃	1	4	yes	complemented
220	C₂×C₄	C₂⁴×C₂₂₃	1	2	yes	—
221	C₂×C₄	C₂⁴×C₂₂₃	1	4	yes	—
222	C₂×C₄	C₂⁴×C₂₂₃	1	2	yes	—
223	C₂×C₄	C₂⁴×C₂₂₃	1	8	yes	—
224	C₂×C₄	C₂⁴×C₂₂₃	1	1	yes	characteristic
225	C₂×C₄	C₂⁴×C₂₂₃	1	1	yes	characteristic
226	C₂×C₄	C₂⁴×C₂₂₃	1	2	yes	—
227	C₂²	—	2	2	no	—
228	C₂²	—	2	2	no	—
229	C₂²	—	2	8	no	—
230	C₂²	—	2	4	no	—
231	C₂²	—	2	16	no	—
232	C₂²	—	8	8	no	—
233	C₂²	C₂×#41×C₂₂₃	1	2	yes	—
234	C₂²	C₂⁵×C₂₂₃	1	1	yes	characteristic
235	C₄	—	2	2	no	—
236	C₄	—	2	4	no	—
237	C₄	—	2	2	no	—
238	C₄	—	2	8	no	—
239	C₄	C₂×#41×C₂₂₃	1	2	yes	—
240	C₄	C₂×#41×C₂₂₃	1	2	yes	—
241	C₄	H(5,𝔽₂)×C₂₂₃	1	4	yes	—
242	C₂	—	2	2	no	—
243	C₂	—	4	8	no	—
244	C₂	—	4	2	no	—
245	C₂	C₂²×#41×C₂₂₃	1	1	yes	characteristic
246	C₂	C₂×H(5,𝔽₂)×C₂₂₃	1	2	yes	—

	subgroup	G/H
center, socle	C₂²×C₂₂₃	C₂⁵
commutator, Frattini, cosocle	C₂²	C₂⁵×C₂₂₃
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₂₂₃ ◁ G

Complements

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

N	H ≅ G/N	type
1. #559×C₂₂₃	C₂	semidirect
2. #550×C₂₂₃	C₂	semidirect
3. #546×C₂₂₃	C₂	semidirect
4. #540×C₂₂₃	C₂	semidirect
5. #535×C₂₂₃	C₂	semidirect
6. #533×C₂₂₃	C₂	semidirect
7. #532×C₂₂₃	C₂	semidirect
8. C₂×#127×C₂₂₃	C₂	semidirect
9. #518×C₂₂₃	C₂	semidirect
10. C₄×#41×C₂₂₃	C₂	semidirect
12. C₂×#41×C₂₂₃	C₂²	semidirect
15. C₄⋊Q₈×C₂₂₃	C₂²	semidirect
19. #127×C₂₂₃	C₂²	semidirect
20. #125×C₂₂₃	C₂²	semidirect
21. #124×C₂₂₃	C₂²	semidirect
22. #124×C₂₂₃	C₂²	semidirect
23. #123×C₂₂₃	C₂²	semidirect
25. #121×C₂₂₃	C₂²	semidirect
26. #121×C₂₂₃	C₂²	semidirect
27. #121×C₂₂₃	C₂²	semidirect
29. C₄×Q₈×C₂₂₃	C₂²	semidirect
30. C₄×Q₈×C₂₂₃	C₂²	semidirect
32. C₄×D₄×C₂₂₃	C₂²	semidirect
33. C₄×D₄×C₂₂₃	C₂²	semidirect
34. C₄×D₄×C₂₂₃	C₂²	semidirect
35. C₄×D₄×C₂₂₃	C₂²	semidirect
38. #117×C₂₂₃	C₂²	semidirect
41. C₂×C₄²×C₂₂₃	C₂²	semidirect
95. C₂×C₄×C₂₂₃	#41	semidirect
123. C₂₂₃	#3508	direct (N × H)
124. #3508	C₂₂₃	direct (N × H)
125. #559	C₂×C₂₂₃	semidirect
126. #550	C₂×C₂₂₃	semidirect
127. #546	C₂×C₂₂₃	semidirect
128. #540	C₂×C₂₂₃	semidirect
129. #535	C₂×C₂₂₃	semidirect
130. #533	C₂×C₂₂₃	semidirect
131. #532	C₂×C₂₂₃	semidirect
132. C₂×#127	C₂×C₂₂₃	semidirect
133. #518	C₂×C₂₂₃	semidirect
134. C₄×#41	C₂×C₂₂₃	semidirect
136. C₂×#41	C₂²×C₂₂₃	semidirect
139. C₄⋊Q₈	C₂²×C₂₂₃	semidirect
143. #127	C₂²×C₂₂₃	semidirect
144. #125	C₂²×C₂₂₃	semidirect
145. #124	C₂²×C₂₂₃	semidirect
146. #124	C₂²×C₂₂₃	semidirect
147. #123	C₂²×C₂₂₃	semidirect
149. #121	C₂²×C₂₂₃	semidirect
150. #121	C₂²×C₂₂₃	semidirect
151. #121	C₂²×C₂₂₃	semidirect
153. C₄×Q₈	C₂²×C₂₂₃	semidirect
154. C₄×Q₈	C₂²×C₂₂₃	semidirect
156. C₄×D₄	C₂²×C₂₂₃	semidirect
157. C₄×D₄	C₂²×C₂₂₃	semidirect
158. C₄×D₄	C₂²×C₂₂₃	semidirect
159. C₄×D₄	C₂²×C₂₂₃	semidirect
162. #117	C₂²×C₂₂₃	semidirect
165. C₂×C₄²	C₂²×C₂₂₃	semidirect
219. C₂×C₄	#41×C₂₂₃	semidirect

Maximal subgroups

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

subgroup	index
#559×C₂₂₃	2
#550×C₂₂₃	2
#546×C₂₂₃	2
#540×C₂₂₃	2
#535×C₂₂₃	2
#533×C₂₂₃	2
#532×C₂₂₃	2
C₂×#127×C₂₂₃	2
#518×C₂₂₃	2
C₄×#41×C₂₂₃	2
#3508	223

Subgroups — #3508×C223 #43252523

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — #3508×C₂₂₃ #43252523