Subgroups — #3523×C₂₂₃ #43252538

Order 28544.

992 conjugacy classes of subgroups.

The subgroup lattice has 250 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#557×C₂₂₃	C₂	1	1	yes	maximal, characteristic, complemented
2	#557×C₂₂₃	C₂	1	3	yes	maximal, complemented
3	#556×C₂₂₃	C₂	1	3	yes	maximal, complemented
4	#552×C₂₂₃	C₂	1	6	yes	maximal, complemented
5	D₄×Q₈×C₂₂₃	C₂	1	2	yes	maximal, complemented
6	D₄×Q₈×C₂₂₃	C₂	1	1	yes	maximal, characteristic, complemented
7	#541×C₂₂₃	C₂	1	6	yes	maximal, complemented
8	#533×C₂₂₃	C₂	1	3	yes	maximal, complemented
9	#529×C₂₂₃	C₂	1	3	yes	maximal
10	#519×C₂₂₃	C₂	1	2	yes	maximal, complemented
11	C₂×C₄×Q₈×C₂₂₃	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂²×Q₈×C₂₂₃	C₂²	1	1	yes	characteristic
13	C₂²×Q₈×C₂₂₃	C₂²	1	2	yes	—
14	C₄⋊Q₈×C₂₂₃	C₂²	1	3	yes	—
15	C₄⋊Q₈×C₂₂₃	C₂²	1	3	yes	—
16	C₄⋊Q₈×C₂₂₃	C₂²	1	6	yes	—
17	#126×C₂₂₃	C₂²	1	6	yes	—
18	#126×C₂₂₃	C₂²	1	12	yes	—
19	#125×C₂₂₃	C₂²	1	3	yes	—
20	#125×C₂₂₃	C₂²	1	6	yes	—
21	#125×C₂₂₃	C₂²	1	12	yes	—
22	#125×C₂₂₃	C₂²	1	6	yes	—
23	#124×C₂₂₃	C₂²	1	3	yes	—
24	#123×C₂₂₃	C₂²	1	6	yes	—
25	C₂²⋊Q₈×C₂₂₃	C₂²	1	3	yes	—
26	C₂²⋊Q₈×C₂₂₃	C₂²	1	6	yes	—
27	C₂²⋊Q₈×C₂₂₃	C₂²	1	6	yes	—
28	C₂²⋊Q₈×C₂₂₃	C₂²	1	6	yes	—
29	#121×C₂₂₃	C₂²	1	1	yes	characteristic
30	C₄×Q₈×C₂₂₃	C₂²	1	3	yes	—
31	C₄×Q₈×C₂₂₃	C₂²	1	6	yes	—
32	C₄×Q₈×C₂₂₃	C₂²	1	6	yes	—
33	C₄×Q₈×C₂₂₃	C₂²	1	3	yes	—
34	C₄×Q₈×C₂₂₃	C₂²	1	1	yes	characteristic
35	C₄×Q₈×C₂₂₃	C₂²	1	2	yes	—
36	C₄×Q₈×C₂₂₃	C₂²	1	6	yes	—
37	C₄×Q₈×C₂₂₃	C₂²	1	3	yes	—
38	C₄×Q₈×C₂₂₃	C₂²	1	1	yes	characteristic
39	C₄×D₄×C₂₂₃	C₂²	1	6	yes	—
40	C₄×D₄×C₂₂₃	C₂²	1	3	yes	—
41	#117×C₂₂₃	C₂²	1	6	yes	—
42	#117×C₂₂₃	C₂²	1	6	yes	—
43	#117×C₂₂₃	C₂²	1	6	yes	—
44	C₂×C₄⋊C₄×C₂₂₃	C₂²	1	3	yes	—
45	C₂×C₄²×C₂₂₃	C₂²	1	3	yes	—
46	C₂×Q₈×C₂₂₃	—	2	4	no	—
47	C₂×Q₈×C₂₂₃	—	2	8	no	—
48	C₂×Q₈×C₂₂₃	C₂³	1	3	yes	—
49	C₂×Q₈×C₂₂₃	C₂³	1	6	yes	—
50	C₂×Q₈×C₂₂₃	C₂³	1	3	yes	—
51	C₂×Q₈×C₂₂₃	C₂³	1	1	yes	characteristic
52	C₂×D₄×C₂₂₃	C₂³	1	2	yes	—
53	C₂×D₄×C₂₂₃	C₂³	1	1	yes	characteristic
54	C₂²×C₄×C₂₂₃	C₂³	1	1	yes	characteristic
55	C₂²×C₄×C₂₂₃	C₂³	1	3	yes	—
56	C₂²×C₄×C₂₂₃	C₂³	1	3	yes	—
57	C₂²×C₄×C₂₂₃	C₂³	1	6	yes	—
58	C₄⋊C₄×C₂₂₃	C₂³	1	3	yes	—
59	C₄⋊C₄×C₂₂₃	C₂³	1	3	yes	—
60	C₄⋊C₄×C₂₂₃	C₂³	1	6	yes	—
61	C₄⋊C₄×C₂₂₃	C₂³	1	6	yes	—
62	C₄⋊C₄×C₂₂₃	C₂³	1	6	yes	—
63	C₄⋊C₄×C₂₂₃	C₂³	1	6	yes	—
64	C₄⋊C₄×C₂₂₃	C₂³	1	6	yes	—
65	C₄⋊C₄×C₂₂₃	C₂³	1	6	yes	—
66	C₄⋊C₄×C₂₂₃	C₂³	1	3	yes	—
67	C₄⋊C₄×C₂₂₃	C₂³	1	6	yes	—
68	C₄⋊C₄×C₂₂₃	C₂³	1	1	yes	characteristic
69	C₄⋊C₄×C₂₂₃	C₂³	1	12	yes	—
70	C₄⋊C₄×C₂₂₃	C₂³	1	6	yes	—
71	C₂²⋊C₄×C₂₂₃	C₂³	1	2	yes	—
72	C₂²⋊C₄×C₂₂₃	C₂³	1	6	yes	—
73	C₂²⋊C₄×C₂₂₃	C₂³	1	6	yes	—
74	C₂²⋊C₄×C₂₂₃	C₂³	1	6	yes	—
75	C₂²⋊C₄×C₂₂₃	C₂³	1	6	yes	—
76	C₄²×C₂₂₃	C₂³	1	3	yes	—
77	C₄²×C₂₂₃	C₂³	1	3	yes	—
78	C₄²×C₂₂₃	C₂³	1	3	yes	—
79	C₄²×C₂₂₃	C₂³	1	3	yes	—
80	C₄²×C₂₂₃	C₂³	1	12	yes	—
81	C₄²×C₂₂₃	C₂³	1	6	yes	—
82	C₂³×C₂₂₃	C₂⁴	1	2	yes	—
83	C₂³×C₂₂₃	C₂⁴	1	1	yes	characteristic
84	Q₈×C₂₂₃	—	2	6	no	—
85	Q₈×C₂₂₃	—	2	12	no	—
86	Q₈×C₂₂₃	—	2	6	no	—
87	Q₈×C₂₂₃	#41	1	4	yes	—
88	D₄×C₂₂₃	—	2	4	no	—
89	D₄×C₂₂₃	—	2	2	no	—
90	C₂×C₄×C₂₂₃	—	2	6	no	—
91	C₂×C₄×C₂₂₃	—	2	6	no	—
92	C₂×C₄×C₂₂₃	—	2	12	no	—
93	C₂×C₄×C₂₂₃	—	2	2	no	—
94	C₂×C₄×C₂₂₃	C₂⁴	1	1	yes	characteristic
95	C₂×C₄×C₂₂₃	C₂⁴	1	3	yes	—
96	C₂×C₄×C₂₂₃	C₂⁴	1	6	yes	—
97	C₂×C₄×C₂₂₃	C₂⁴	1	2	yes	—
98	C₂×C₄×C₂₂₃	C₂⁴	1	3	yes	—
99	C₂×C₄×C₂₂₃	C₂⁴	1	3	yes	—
100	C₂×C₄×C₂₂₃	C₂⁴	1	6	yes	—
101	C₂×C₄×C₂₂₃	C₂⁴	1	3	yes	—
102	C₂×C₄×C₂₂₃	C₂⁴	1	1	yes	characteristic
103	C₂²×C₂₂₃	—	2	1	no	—
104	C₂²×C₂₂₃	—	2	1	no	—
105	C₂²×C₂₂₃	—	2	2	no	—
106	C₂²×C₂₂₃	—	2	2	no	—
107	C₂²×C₂₂₃	—	2	2	no	—
108	C₂²×C₂₂₃	2₋¹⁺⁴	1	2	yes	—
109	C₂²×C₂₂₃	C₂⁵	1	1	yes	characteristic
110	C₄×C₂₂₃	—	2	3	no	—
111	C₄×C₂₂₃	—	2	6	no	—
112	C₄×C₂₂₃	—	2	3	no	—
113	C₄×C₂₂₃	—	2	3	no	—
114	C₄×C₂₂₃	—	2	1	no	—
115	C₄×C₂₂₃	—	2	2	no	—
116	C₄×C₂₂₃	—	2	6	no	—
117	C₄×C₂₂₃	C₂×#41	1	6	yes	—
118	C₄×C₂₂₃	2₋¹⁺⁴	1	2	yes	—
119	C₂×C₂₂₃	—	2	2	no	—
120	C₂×C₂₂₃	—	4	2	no	—
121	C₂×C₂₂₃	C₂²×#41	1	1	yes	characteristic
122	C₂×C₂₂₃	C₂×2₋¹⁺⁴	1	1	yes	characteristic
123	C₂×C₂₂₃	C₂×2₋¹⁺⁴	1	1	yes	characteristic
124	C₂₂₃	#3523	1	1	yes	characteristic, direct product factor, Sylow
125	#3523	C₂₂₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
126	#557	C₂×C₂₂₃	1	1	yes	characteristic, complemented
127	#557	C₂×C₂₂₃	1	3	yes	complemented
128	#556	C₂×C₂₂₃	1	3	yes	complemented
129	#552	C₂×C₂₂₃	1	6	yes	complemented
130	D₄×Q₈	C₂×C₂₂₃	1	2	yes	complemented
131	D₄×Q₈	C₂×C₂₂₃	1	1	yes	characteristic, complemented
132	#541	C₂×C₂₂₃	1	6	yes	complemented
133	#533	C₂×C₂₂₃	1	3	yes	complemented
134	#529	C₂×C₂₂₃	1	3	yes	—
135	#519	C₂×C₂₂₃	1	2	yes	complemented
136	C₂×C₄×Q₈	C₂×C₂₂₃	1	1	yes	characteristic, complemented
137	C₂²×Q₈	C₂²×C₂₂₃	1	1	yes	characteristic
138	C₂²×Q₈	C₂²×C₂₂₃	1	2	yes	—
139	C₄⋊Q₈	C₂²×C₂₂₃	1	3	yes	—
140	C₄⋊Q₈	C₂²×C₂₂₃	1	3	yes	—
141	C₄⋊Q₈	C₂²×C₂₂₃	1	6	yes	—
142	#126	C₂²×C₂₂₃	1	6	yes	—
143	#126	C₂²×C₂₂₃	1	12	yes	—
144	#125	C₂²×C₂₂₃	1	3	yes	—
145	#125	C₂²×C₂₂₃	1	6	yes	—
146	#125	C₂²×C₂₂₃	1	12	yes	—
147	#125	C₂²×C₂₂₃	1	6	yes	—
148	#124	C₂²×C₂₂₃	1	3	yes	—
149	#123	C₂²×C₂₂₃	1	6	yes	—
150	C₂²⋊Q₈	C₂²×C₂₂₃	1	3	yes	—
151	C₂²⋊Q₈	C₂²×C₂₂₃	1	6	yes	—
152	C₂²⋊Q₈	C₂²×C₂₂₃	1	6	yes	—
153	C₂²⋊Q₈	C₂²×C₂₂₃	1	6	yes	—
154	#121	C₂²×C₂₂₃	1	1	yes	characteristic
155	C₄×Q₈	C₂²×C₂₂₃	1	1	yes	characteristic
156	C₄×Q₈	C₂²×C₂₂₃	1	3	yes	—
157	C₄×Q₈	C₂²×C₂₂₃	1	1	yes	characteristic
158	C₄×Q₈	C₂²×C₂₂₃	1	3	yes	—
159	C₄×Q₈	C₂²×C₂₂₃	1	6	yes	—
160	C₄×Q₈	C₂²×C₂₂₃	1	2	yes	—
161	C₄×Q₈	C₂²×C₂₂₃	1	6	yes	—
162	C₄×Q₈	C₂²×C₂₂₃	1	3	yes	—
163	C₄×Q₈	C₂²×C₂₂₃	1	6	yes	—
164	C₄×D₄	C₂²×C₂₂₃	1	6	yes	—
165	C₄×D₄	C₂²×C₂₂₃	1	3	yes	—
166	#117	C₂²×C₂₂₃	1	6	yes	—
167	#117	C₂²×C₂₂₃	1	6	yes	—
168	#117	C₂²×C₂₂₃	1	6	yes	—
169	C₂×C₄⋊C₄	C₂²×C₂₂₃	1	3	yes	—
170	C₂×C₄²	C₂²×C₂₂₃	1	3	yes	—
171	C₂×Q₈	—	2	4	no	—
172	C₂×Q₈	—	2	8	no	—
173	C₂×Q₈	C₂³×C₂₂₃	1	3	yes	—
174	C₂×Q₈	C₂³×C₂₂₃	1	3	yes	—
175	C₂×Q₈	C₂³×C₂₂₃	1	6	yes	—
176	C₂×Q₈	C₂³×C₂₂₃	1	1	yes	characteristic
177	C₂×D₄	C₂³×C₂₂₃	1	2	yes	—
178	C₂×D₄	C₂³×C₂₂₃	1	1	yes	characteristic
179	C₂²×C₄	C₂³×C₂₂₃	1	1	yes	characteristic
180	C₂²×C₄	C₂³×C₂₂₃	1	3	yes	—
181	C₂²×C₄	C₂³×C₂₂₃	1	3	yes	—
182	C₂²×C₄	C₂³×C₂₂₃	1	6	yes	—
183	C₄⋊C₄	C₂³×C₂₂₃	1	3	yes	—
184	C₄⋊C₄	C₂³×C₂₂₃	1	3	yes	—
185	C₄⋊C₄	C₂³×C₂₂₃	1	6	yes	—
186	C₄⋊C₄	C₂³×C₂₂₃	1	6	yes	—
187	C₄⋊C₄	C₂³×C₂₂₃	1	6	yes	—
188	C₄⋊C₄	C₂³×C₂₂₃	1	6	yes	—
189	C₄⋊C₄	C₂³×C₂₂₃	1	6	yes	—
190	C₄⋊C₄	C₂³×C₂₂₃	1	6	yes	—
191	C₄⋊C₄	C₂³×C₂₂₃	1	3	yes	—
192	C₄⋊C₄	C₂³×C₂₂₃	1	6	yes	—
193	C₄⋊C₄	C₂³×C₂₂₃	1	1	yes	characteristic
194	C₄⋊C₄	C₂³×C₂₂₃	1	12	yes	—
195	C₄⋊C₄	C₂³×C₂₂₃	1	6	yes	—
196	C₂²⋊C₄	C₂³×C₂₂₃	1	2	yes	—
197	C₂²⋊C₄	C₂³×C₂₂₃	1	6	yes	—
198	C₂²⋊C₄	C₂³×C₂₂₃	1	6	yes	—
199	C₂²⋊C₄	C₂³×C₂₂₃	1	6	yes	—
200	C₂²⋊C₄	C₂³×C₂₂₃	1	6	yes	—
201	C₄²	C₂³×C₂₂₃	1	3	yes	—
202	C₄²	C₂³×C₂₂₃	1	3	yes	—
203	C₄²	C₂³×C₂₂₃	1	3	yes	—
204	C₄²	C₂³×C₂₂₃	1	3	yes	—
205	C₄²	C₂³×C₂₂₃	1	12	yes	—
206	C₄²	C₂³×C₂₂₃	1	6	yes	—
207	C₂³	C₂⁴×C₂₂₃	1	2	yes	—
208	C₂³	C₂⁴×C₂₂₃	1	1	yes	characteristic
209	Q₈	—	2	6	no	—
210	Q₈	—	2	12	no	—
211	Q₈	—	2	6	no	—
212	Q₈	#41×C₂₂₃	1	4	yes	—
213	D₄	—	2	4	no	—
214	D₄	—	2	2	no	—
215	C₂×C₄	—	2	6	no	—
216	C₂×C₄	—	2	6	no	—
217	C₂×C₄	—	2	12	no	—
218	C₂×C₄	—	2	2	no	—
219	C₂×C₄	C₂⁴×C₂₂₃	1	1	yes	characteristic
220	C₂×C₄	C₂⁴×C₂₂₃	1	3	yes	—
221	C₂×C₄	C₂⁴×C₂₂₃	1	6	yes	—
222	C₂×C₄	C₂⁴×C₂₂₃	1	2	yes	—
223	C₂×C₄	C₂⁴×C₂₂₃	1	3	yes	—
224	C₂×C₄	C₂⁴×C₂₂₃	1	3	yes	—
225	C₂×C₄	C₂⁴×C₂₂₃	1	6	yes	—
226	C₂×C₄	C₂⁴×C₂₂₃	1	3	yes	—
227	C₂×C₄	C₂⁴×C₂₂₃	1	1	yes	characteristic
228	C₂²	—	2	1	no	—
229	C₂²	—	2	1	no	—
230	C₂²	—	2	2	no	—
231	C₂²	—	2	2	no	—
232	C₂²	—	2	2	no	—
233	C₂²	2₋¹⁺⁴×C₂₂₃	1	2	yes	—
234	C₂²	C₂⁵×C₂₂₃	1	1	yes	characteristic
235	C₄	—	2	3	no	—
236	C₄	—	2	6	no	—
237	C₄	—	2	3	no	—
238	C₄	—	2	3	no	—
239	C₄	—	2	1	no	—
240	C₄	—	2	2	no	—
241	C₄	—	2	6	no	—
242	C₄	C₂×#41×C₂₂₃	1	6	yes	—
243	C₄	2₋¹⁺⁴×C₂₂₃	1	2	yes	—
244	C₂	—	2	2	no	—
245	C₂	—	4	2	no	—
246	C₂	C₂²×#41×C₂₂₃	1	1	yes	characteristic
247	C₂	C₂×2₋¹⁺⁴×C₂₂₃	1	1	yes	characteristic
248	C₂	C₂×2₋¹⁺⁴×C₂₂₃	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₂₂₃	C₂⁵
commutator, Frattini, cosocle	C₂²	C₂⁵×C₂₂₃
prime radical	#121×C₂₂₃	C₂²
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. #557×C₂₂₃	C₂	semidirect
2. #557×C₂₂₃	C₂	semidirect
3. #556×C₂₂₃	C₂	semidirect
4. #552×C₂₂₃	C₂	semidirect
5. D₄×Q₈×C₂₂₃	C₂	semidirect
6. D₄×Q₈×C₂₂₃	C₂	semidirect
7. #541×C₂₂₃	C₂	semidirect
8. #533×C₂₂₃	C₂	semidirect
10. #519×C₂₂₃	C₂	semidirect
11. C₂×C₄×Q₈×C₂₂₃	C₂	semidirect
124. C₂₂₃	#3523	direct (N × H)
125. #3523	C₂₂₃	direct (N × H)
126. #557	C₂×C₂₂₃	semidirect
127. #557	C₂×C₂₂₃	semidirect
128. #556	C₂×C₂₂₃	semidirect
129. #552	C₂×C₂₂₃	semidirect
130. D₄×Q₈	C₂×C₂₂₃	semidirect
131. D₄×Q₈	C₂×C₂₂₃	semidirect
132. #541	C₂×C₂₂₃	semidirect
133. #533	C₂×C₂₂₃	semidirect
135. #519	C₂×C₂₂₃	semidirect
136. C₂×C₄×Q₈	C₂×C₂₂₃	semidirect

Maximal subgroups

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

subgroup	index
#557×C₂₂₃	2
#556×C₂₂₃	2
#552×C₂₂₃	2
D₄×Q₈×C₂₂₃	2
#541×C₂₂₃	2
#533×C₂₂₃	2
#529×C₂₂₃	2
#519×C₂₂₃	2
C₂×C₄×Q₈×C₂₂₃	2
#3523	223

Subgroups — #3523×C223 #43252538

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — #3523×C₂₂₃ #43252538