Subgroups — #3522×C₂₂₃ #43252537

Order 28544.

1104 conjugacy classes of subgroups.

The subgroup lattice has 258 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	3	yes	maximal, complemented
2	Q₈²×C₂₂₃	C₂	1	1	yes	maximal, characteristic, complemented
3	#554×C₂₂₃	C₂	1	3	yes	maximal, complemented
4	#550×C₂₂₃	C₂	1	1	yes	maximal, characteristic, complemented
5	#548×C₂₂₃	C₂	1	2	yes	maximal, complemented
6	#547×C₂₂₃	C₂	1	6	yes	maximal, complemented
7	#542×C₂₂₃	C₂	1	6	yes	maximal, complemented
8	#532×C₂₂₃	C₂	1	3	yes	maximal, complemented
9	#529×C₂₂₃	C₂	1	3	yes	maximal, complemented
10	#520×C₂₂₃	C₂	1	2	yes	maximal, complemented
11	C₂×C₄×Q₈×C₂₂₃	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×#41×C₂₂₃	C₂²	1	2	yes	—
13	C₂×#41×C₂₂₃	C₂²	1	6	yes	—
14	C₂²×Q₈×C₂₂₃	C₂²	1	1	yes	characteristic
15	C₄⋊Q₈×C₂₂₃	C₂²	1	3	yes	—
16	C₄⋊Q₈×C₂₂₃	C₂²	1	6	yes	—
17	#127×C₂₂₃	C₂²	1	3	yes	—
18	#126×C₂₂₃	C₂²	1	6	yes	—
19	#125×C₂₂₃	C₂²	1	3	yes	—
20	#124×C₂₂₃	C₂²	1	3	yes	—
21	#124×C₂₂₃	C₂²	1	6	yes	—
22	#124×C₂₂₃	C₂²	1	6	yes	—
23	#123×C₂₂₃	C₂²	1	6	yes	—
24	#123×C₂₂₃	C₂²	1	12	yes	—
25	C₂²⋊Q₈×C₂₂₃	C₂²	1	1	yes	characteristic
26	C₂²⋊Q₈×C₂₂₃	C₂²	1	12	yes	—
27	C₂²⋊Q₈×C₂₂₃	C₂²	1	6	yes	—
28	#121×C₂₂₃	C₂²	1	3	yes	—
29	#121×C₂₂₃	C₂²	1	6	yes	—
30	#121×C₂₂₃	C₂²	1	6	yes	—
31	C₄×Q₈×C₂₂₃	C₂²	1	1	yes	characteristic
32	C₄×Q₈×C₂₂₃	C₂²	1	2	yes	—
33	C₄×Q₈×C₂₂₃	C₂²	1	3	yes	—
34	C₄×Q₈×C₂₂₃	C₂²	1	3	yes	—
35	C₄×Q₈×C₂₂₃	C₂²	1	1	yes	characteristic
36	C₄×Q₈×C₂₂₃	C₂²	1	3	yes	—
37	C₄×D₄×C₂₂₃	C₂²	1	6	yes	—
38	C₄×D₄×C₂₂₃	C₂²	1	6	yes	—
39	C₄×D₄×C₂₂₃	C₂²	1	6	yes	—
40	C₄×D₄×C₂₂₃	C₂²	1	3	yes	—
41	C₄×D₄×C₂₂₃	C₂²	1	6	yes	—
42	#117×C₂₂₃	C₂²	1	6	yes	—
43	C₂×C₄⋊C₄×C₂₂₃	C₂²	1	6	yes	—
44	C₂×C₄⋊C₄×C₂₂₃	C₂²	1	3	yes	—
45	C₂×C₄²×C₂₂₃	C₂²	1	3	yes	—
46	#41×C₂₂₃	—	2	8	no	—
47	#41×C₂₂₃	—	4	12	no	—
48	C₂×Q₈×C₂₂₃	—	2	4	no	—
49	C₂×Q₈×C₂₂₃	C₂³	1	1	yes	characteristic
50	C₂×Q₈×C₂₂₃	C₂³	1	3	yes	—
51	C₂×Q₈×C₂₂₃	C₂³	1	6	yes	—
52	C₂×Q₈×C₂₂₃	C₂³	1	1	yes	characteristic
53	C₂×D₄×C₂₂₃	C₂³	1	3	yes	—
54	C₂×D₄×C₂₂₃	C₂³	1	6	yes	—
55	C₂×D₄×C₂₂₃	C₂³	1	6	yes	—
56	C₂×D₄×C₂₂₃	C₂³	1	6	yes	—
57	C₂²×C₄×C₂₂₃	C₂³	1	1	yes	characteristic
58	C₂²×C₄×C₂₂₃	C₂³	1	3	yes	—
59	C₂²×C₄×C₂₂₃	C₂³	1	3	yes	—
60	C₂²×C₄×C₂₂₃	C₂³	1	6	yes	—
61	C₂²×C₄×C₂₂₃	C₂³	1	12	yes	—
62	C₄⋊C₄×C₂₂₃	C₂³	1	3	yes	—
63	C₄⋊C₄×C₂₂₃	C₂³	1	6	yes	—
64	C₄⋊C₄×C₂₂₃	C₂³	1	6	yes	—
65	C₄⋊C₄×C₂₂₃	C₂³	1	3	yes	—
66	C₄⋊C₄×C₂₂₃	C₂³	1	6	yes	—
67	C₄⋊C₄×C₂₂₃	C₂³	1	2	yes	—
68	C₄⋊C₄×C₂₂₃	C₂³	1	1	yes	characteristic
69	C₄⋊C₄×C₂₂₃	C₂³	1	3	yes	—
70	C₄⋊C₄×C₂₂₃	C₂³	1	6	yes	—
71	C₄⋊C₄×C₂₂₃	C₂³	1	6	yes	—
72	C₂²⋊C₄×C₂₂₃	C₂³	1	6	yes	—
73	C₂²⋊C₄×C₂₂₃	C₂³	1	6	yes	—
74	C₂²⋊C₄×C₂₂₃	C₂³	1	2	yes	—
75	C₂²⋊C₄×C₂₂₃	C₂³	1	6	yes	—
76	C₂²⋊C₄×C₂₂₃	C₂³	1	12	yes	—
77	C₂²⋊C₄×C₂₂₃	C₂³	1	6	yes	—
78	C₄²×C₂₂₃	C₂³	1	3	yes	—
79	C₄²×C₂₂₃	C₂³	1	3	yes	—
80	C₄²×C₂₂₃	C₂³	1	3	yes	—
81	C₄²×C₂₂₃	C₂³	1	3	yes	—
82	C₄²×C₂₂₃	C₂³	1	6	yes	—
83	C₂³×C₂₂₃	C₂⁴	1	6	yes	—
84	C₂³×C₂₂₃	C₂⁴	1	1	yes	characteristic
85	Q₈×C₂₂₃	—	2	6	no	—
86	Q₈×C₂₂₃	—	2	2	no	—
87	Q₈×C₂₂₃	—	4	6	no	—
88	Q₈×C₂₂₃	#41	1	4	yes	complemented
89	D₄×C₂₂₃	—	2	12	no	—
90	D₄×C₂₂₃	—	2	12	no	—
91	D₄×C₂₂₃	—	2	6	no	—
92	D₄×C₂₂₃	—	4	6	no	—
93	C₂×C₄×C₂₂₃	—	2	6	no	—
94	C₂×C₄×C₂₂₃	—	2	6	no	—
95	C₂×C₄×C₂₂₃	—	2	12	no	—
96	C₂×C₄×C₂₂₃	—	2	2	no	—
97	C₂×C₄×C₂₂₃	—	4	12	no	—
98	C₂×C₄×C₂₂₃	C₂⁴	1	1	yes	characteristic
99	C₂×C₄×C₂₂₃	C₂⁴	1	2	yes	—
100	C₂×C₄×C₂₂₃	C₂⁴	1	3	yes	—
101	C₂×C₄×C₂₂₃	C₂⁴	1	3	yes	—
102	C₂×C₄×C₂₂₃	C₂⁴	1	3	yes	—
103	C₂×C₄×C₂₂₃	C₂⁴	1	6	yes	—
104	C₂×C₄×C₂₂₃	C₂⁴	1	2	yes	—
105	C₂×C₄×C₂₂₃	C₂⁴	1	3	yes	—
106	C₂×C₄×C₂₂₃	C₂⁴	1	1	yes	characteristic
107	C₂²×C₂₂₃	—	2	1	no	—
108	C₂²×C₂₂₃	—	2	1	no	—
109	C₂²×C₂₂₃	—	2	6	no	—
110	C₂²×C₂₂₃	—	2	6	no	—
111	C₂²×C₂₂₃	—	2	6	no	—
112	C₂²×C₂₂₃	H(5,𝔽₂)	1	2	yes	—
113	C₂²×C₂₂₃	C₂⁵	1	1	yes	characteristic
114	C₄×C₂₂₃	—	2	2	no	—
115	C₄×C₂₂₃	—	2	3	no	—
116	C₄×C₂₂₃	—	2	3	no	—
117	C₄×C₂₂₃	—	2	3	no	—
118	C₄×C₂₂₃	—	2	6	no	—
119	C₄×C₂₂₃	—	2	1	no	—
120	C₄×C₂₂₃	—	2	2	no	—
121	C₄×C₂₂₃	C₂×#41	1	6	yes	—
122	C₄×C₂₂₃	2₋¹⁺⁴	1	2	yes	—
123	C₂×C₂₂₃	—	2	2	no	—
124	C₂×C₂₂₃	—	4	6	no	—
125	C₂×C₂₂₃	C₂²×#41	1	1	yes	characteristic
126	C₂×C₂₂₃	C₂×H(5,𝔽₂)	1	1	yes	characteristic
127	C₂×C₂₂₃	C₂×2₋¹⁺⁴	1	1	yes	characteristic
128	C₂₂₃	#3522	1	1	yes	characteristic, direct product factor, Sylow
129	#3522	C₂₂₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
130	#559	C₂×C₂₂₃	1	3	yes	complemented
131	Q₈²	C₂×C₂₂₃	1	1	yes	characteristic, complemented
132	#554	C₂×C₂₂₃	1	3	yes	complemented
133	#550	C₂×C₂₂₃	1	1	yes	characteristic, complemented
134	#548	C₂×C₂₂₃	1	2	yes	complemented
135	#547	C₂×C₂₂₃	1	6	yes	complemented
136	#542	C₂×C₂₂₃	1	6	yes	complemented
137	#532	C₂×C₂₂₃	1	3	yes	complemented
138	#529	C₂×C₂₂₃	1	3	yes	complemented
139	#520	C₂×C₂₂₃	1	2	yes	complemented
140	C₂×C₄×Q₈	C₂×C₂₂₃	1	1	yes	characteristic, complemented
141	C₂×#41	C₂²×C₂₂₃	1	2	yes	—
142	C₂×#41	C₂²×C₂₂₃	1	6	yes	—
143	C₂²×Q₈	C₂²×C₂₂₃	1	1	yes	characteristic
144	C₄⋊Q₈	C₂²×C₂₂₃	1	3	yes	—
145	C₄⋊Q₈	C₂²×C₂₂₃	1	6	yes	—
146	#127	C₂²×C₂₂₃	1	3	yes	—
147	#126	C₂²×C₂₂₃	1	6	yes	—
148	#125	C₂²×C₂₂₃	1	3	yes	—
149	#124	C₂²×C₂₂₃	1	3	yes	—
150	#124	C₂²×C₂₂₃	1	6	yes	—
151	#124	C₂²×C₂₂₃	1	6	yes	—
152	#123	C₂²×C₂₂₃	1	6	yes	—
153	#123	C₂²×C₂₂₃	1	12	yes	—
154	C₂²⋊Q₈	C₂²×C₂₂₃	1	1	yes	characteristic
155	C₂²⋊Q₈	C₂²×C₂₂₃	1	12	yes	—
156	C₂²⋊Q₈	C₂²×C₂₂₃	1	6	yes	—
157	#121	C₂²×C₂₂₃	1	3	yes	—
158	#121	C₂²×C₂₂₃	1	6	yes	—
159	#121	C₂²×C₂₂₃	1	6	yes	—
160	C₄×Q₈	C₂²×C₂₂₃	1	1	yes	characteristic
161	C₄×Q₈	C₂²×C₂₂₃	1	3	yes	—
162	C₄×Q₈	C₂²×C₂₂₃	1	1	yes	characteristic
163	C₄×Q₈	C₂²×C₂₂₃	1	3	yes	—
164	C₄×Q₈	C₂²×C₂₂₃	1	2	yes	—
165	C₄×Q₈	C₂²×C₂₂₃	1	3	yes	—
166	C₄×D₄	C₂²×C₂₂₃	1	6	yes	—
167	C₄×D₄	C₂²×C₂₂₃	1	6	yes	—
168	C₄×D₄	C₂²×C₂₂₃	1	6	yes	—
169	C₄×D₄	C₂²×C₂₂₃	1	3	yes	—
170	C₄×D₄	C₂²×C₂₂₃	1	6	yes	—
171	#117	C₂²×C₂₂₃	1	6	yes	—
172	C₂×C₄⋊C₄	C₂²×C₂₂₃	1	3	yes	—
173	C₂×C₄⋊C₄	C₂²×C₂₂₃	1	6	yes	—
174	C₂×C₄²	C₂²×C₂₂₃	1	3	yes	—
175	#41	—	2	8	no	—
176	#41	—	4	12	no	—
177	C₂×Q₈	—	2	4	no	—
178	C₂×Q₈	C₂³×C₂₂₃	1	1	yes	characteristic
179	C₂×Q₈	C₂³×C₂₂₃	1	3	yes	—
180	C₂×Q₈	C₂³×C₂₂₃	1	6	yes	—
181	C₂×Q₈	C₂³×C₂₂₃	1	1	yes	characteristic
182	C₂×D₄	C₂³×C₂₂₃	1	3	yes	—
183	C₂×D₄	C₂³×C₂₂₃	1	6	yes	—
184	C₂×D₄	C₂³×C₂₂₃	1	6	yes	—
185	C₂×D₄	C₂³×C₂₂₃	1	6	yes	—
186	C₂²×C₄	C₂³×C₂₂₃	1	1	yes	characteristic
187	C₂²×C₄	C₂³×C₂₂₃	1	3	yes	—
188	C₂²×C₄	C₂³×C₂₂₃	1	3	yes	—
189	C₂²×C₄	C₂³×C₂₂₃	1	6	yes	—
190	C₂²×C₄	C₂³×C₂₂₃	1	12	yes	—
191	C₄⋊C₄	C₂³×C₂₂₃	1	3	yes	—
192	C₄⋊C₄	C₂³×C₂₂₃	1	6	yes	—
193	C₄⋊C₄	C₂³×C₂₂₃	1	6	yes	—
194	C₄⋊C₄	C₂³×C₂₂₃	1	3	yes	—
195	C₄⋊C₄	C₂³×C₂₂₃	1	6	yes	—
196	C₄⋊C₄	C₂³×C₂₂₃	1	2	yes	—
197	C₄⋊C₄	C₂³×C₂₂₃	1	1	yes	characteristic
198	C₄⋊C₄	C₂³×C₂₂₃	1	3	yes	—
199	C₄⋊C₄	C₂³×C₂₂₃	1	6	yes	—
200	C₄⋊C₄	C₂³×C₂₂₃	1	6	yes	—
201	C₂²⋊C₄	C₂³×C₂₂₃	1	6	yes	—
202	C₂²⋊C₄	C₂³×C₂₂₃	1	6	yes	—
203	C₂²⋊C₄	C₂³×C₂₂₃	1	2	yes	—
204	C₂²⋊C₄	C₂³×C₂₂₃	1	6	yes	—
205	C₂²⋊C₄	C₂³×C₂₂₃	1	12	yes	—
206	C₂²⋊C₄	C₂³×C₂₂₃	1	6	yes	—
207	C₄²	C₂³×C₂₂₃	1	3	yes	—
208	C₄²	C₂³×C₂₂₃	1	3	yes	—
209	C₄²	C₂³×C₂₂₃	1	3	yes	—
210	C₄²	C₂³×C₂₂₃	1	3	yes	—
211	C₄²	C₂³×C₂₂₃	1	6	yes	—
212	C₂³	C₂⁴×C₂₂₃	1	6	yes	—
213	C₂³	C₂⁴×C₂₂₃	1	1	yes	characteristic
214	Q₈	—	2	6	no	—
215	Q₈	—	2	2	no	—
216	Q₈	—	4	6	no	—
217	Q₈	#41×C₂₂₃	1	4	yes	complemented
218	D₄	—	2	12	no	—
219	D₄	—	2	12	no	—
220	D₄	—	2	6	no	—
221	D₄	—	4	6	no	—
222	C₂×C₄	—	2	6	no	—
223	C₂×C₄	—	2	6	no	—
224	C₂×C₄	—	2	12	no	—
225	C₂×C₄	—	2	2	no	—
226	C₂×C₄	—	4	12	no	—
227	C₂×C₄	C₂⁴×C₂₂₃	1	1	yes	characteristic
228	C₂×C₄	C₂⁴×C₂₂₃	1	2	yes	—
229	C₂×C₄	C₂⁴×C₂₂₃	1	3	yes	—
230	C₂×C₄	C₂⁴×C₂₂₃	1	3	yes	—
231	C₂×C₄	C₂⁴×C₂₂₃	1	3	yes	—
232	C₂×C₄	C₂⁴×C₂₂₃	1	6	yes	—
233	C₂×C₄	C₂⁴×C₂₂₃	1	2	yes	—
234	C₂×C₄	C₂⁴×C₂₂₃	1	3	yes	—
235	C₂×C₄	C₂⁴×C₂₂₃	1	1	yes	characteristic
236	C₂²	—	2	1	no	—
237	C₂²	—	2	1	no	—
238	C₂²	—	2	6	no	—
239	C₂²	—	2	6	no	—
240	C₂²	—	2	6	no	—
241	C₂²	H(5,𝔽₂)×C₂₂₃	1	2	yes	—
242	C₂²	C₂⁵×C₂₂₃	1	1	yes	characteristic
243	C₄	—	2	2	no	—
244	C₄	—	2	3	no	—
245	C₄	—	2	3	no	—
246	C₄	—	2	3	no	—
247	C₄	—	2	6	no	—
248	C₄	—	2	1	no	—
249	C₄	—	2	2	no	—
250	C₄	C₂×#41×C₂₂₃	1	6	yes	—
251	C₄	2₋¹⁺⁴×C₂₂₃	1	2	yes	—
252	C₂	—	2	2	no	—
253	C₂	—	4	6	no	—
254	C₂	C₂²×#41×C₂₂₃	1	1	yes	characteristic
255	C₂	C₂×H(5,𝔽₂)×C₂₂₃	1	1	yes	characteristic
256	C₂	C₂×2₋¹⁺⁴×C₂₂₃	1	1	yes	characteristic

	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. Q₈²×C₂₂₃	C₂	semidirect
3. #554×C₂₂₃	C₂	semidirect
4. #550×C₂₂₃	C₂	semidirect
5. #548×C₂₂₃	C₂	semidirect
6. #547×C₂₂₃	C₂	semidirect
7. #542×C₂₂₃	C₂	semidirect
8. #532×C₂₂₃	C₂	semidirect
9. #529×C₂₂₃	C₂	semidirect
10. #520×C₂₂₃	C₂	semidirect
11. C₂×C₄×Q₈×C₂₂₃	C₂	semidirect
88. Q₈×C₂₂₃	#41	semidirect
128. C₂₂₃	#3522	direct (N × H)
129. #3522	C₂₂₃	direct (N × H)
130. #559	C₂×C₂₂₃	semidirect
131. Q₈²	C₂×C₂₂₃	semidirect
132. #554	C₂×C₂₂₃	semidirect
133. #550	C₂×C₂₂₃	semidirect
134. #548	C₂×C₂₂₃	semidirect
135. #547	C₂×C₂₂₃	semidirect
136. #542	C₂×C₂₂₃	semidirect
137. #532	C₂×C₂₂₃	semidirect
138. #529	C₂×C₂₂₃	semidirect
139. #520	C₂×C₂₂₃	semidirect
140. C₂×C₄×Q₈	C₂×C₂₂₃	semidirect
217. Q₈	#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
Q₈²×C₂₂₃	2
#554×C₂₂₃	2
#550×C₂₂₃	2
#548×C₂₂₃	2
#547×C₂₂₃	2
#542×C₂₂₃	2
#532×C₂₂₃	2
#529×C₂₂₃	2
#520×C₂₂₃	2
C₂×C₄×Q₈×C₂₂₃	2
#3522	223

Subgroups — #3522×C223 #43252537

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — #3522×C₂₂₃ #43252537