Subgroups — #3482×C₂₂₃ #43252497

Order 28544.

1094 conjugacy classes of subgroups.

The subgroup lattice has 256 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	#556×C₂₂₃	C₂	1	3	yes	maximal, complemented
3	#552×C₂₂₃	C₂	1	6	yes	maximal, complemented
4	D₄×Q₈×C₂₂₃	C₂	1	1	yes	maximal, characteristic, complemented
5	#548×C₂₂₃	C₂	1	2	yes	maximal, complemented
6	#547×C₂₂₃	C₂	1	3	yes	maximal, complemented
7	#533×C₂₂₃	C₂	1	3	yes	maximal, complemented
8	#529×C₂₂₃	C₂	1	6	yes	maximal, complemented
9	C₄×#41×C₂₂₃	C₂	1	3	yes	maximal, complemented
10	C₄×#41×C₂₂₃	C₂	1	1	yes	maximal, characteristic, complemented
11	C₂×C₄×Q₈×C₂₂₃	C₂	1	2	yes	maximal, complemented
12	C₂×#41×C₂₂₃	C₂²	1	3	yes	—
13	C₂×#41×C₂₂₃	C₂²	1	1	yes	characteristic
14	C₂²×Q₈×C₂₂₃	C₂²	1	2	yes	—
15	C₄⋊Q₈×C₂₂₃	C₂²	1	3	yes	—
16	C₄⋊Q₈×C₂₂₃	C₂²	1	3	yes	—
17	#126×C₂₂₃	C₂²	1	12	yes	—
18	#125×C₂₂₃	C₂²	1	6	yes	—
19	#125×C₂₂₃	C₂²	1	6	yes	—
20	#124×C₂₂₃	C₂²	1	6	yes	—
21	#123×C₂₂₃	C₂²	1	12	yes	—
22	C₂²⋊Q₈×C₂₂₃	C₂²	1	6	yes	—
23	C₂²⋊Q₈×C₂₂₃	C₂²	1	6	yes	—
24	C₂²⋊Q₈×C₂₂₃	C₂²	1	6	yes	—
25	#121×C₂₂₃	C₂²	1	6	yes	—
26	C₄×Q₈×C₂₂₃	C₂²	1	6	yes	—
27	C₄×Q₈×C₂₂₃	C₂²	1	3	yes	—
28	C₄×Q₈×C₂₂₃	C₂²	1	2	yes	—
29	C₄×Q₈×C₂₂₃	C₂²	1	1	yes	characteristic
30	C₄×Q₈×C₂₂₃	C₂²	1	6	yes	—
31	C₄×Q₈×C₂₂₃	C₂²	1	3	yes	—
32	C₄×Q₈×C₂₂₃	C₂²	1	1	yes	characteristic
33	C₄×D₄×C₂₂₃	C₂²	1	3	yes	—
34	C₄×D₄×C₂₂₃	C₂²	1	6	yes	—
35	C₄×D₄×C₂₂₃	C₂²	1	3	yes	—
36	C₄×D₄×C₂₂₃	C₂²	1	3	yes	—
37	C₄×D₄×C₂₂₃	C₂²	1	6	yes	—
38	C₄×D₄×C₂₂₃	C₂²	1	1	yes	characteristic
39	#117×C₂₂₃	C₂²	1	6	yes	—
40	#117×C₂₂₃	C₂²	1	3	yes	—
41	#117×C₂₂₃	C₂²	1	6	yes	—
42	#117×C₂₂₃	C₂²	1	3	yes	—
43	C₂×C₄⋊C₄×C₂₂₃	C₂²	1	6	yes	—
44	C₂×C₄²×C₂₂₃	C₂²	1	6	yes	—
45	C₂×C₄²×C₂₂₃	C₂²	1	3	yes	—
46	#41×C₂₂₃	—	2	4	no	—
47	#41×C₂₂₃	—	2	12	no	—
48	C₂×Q₈×C₂₂₃	—	2	8	no	—
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	6	yes	—
53	C₂×D₄×C₂₂₃	C₂³	1	3	yes	—
54	C₂×D₄×C₂₂₃	C₂³	1	1	yes	characteristic
55	C₂²×C₄×C₂₂₃	C₂³	1	6	yes	—
56	C₂²×C₄×C₂₂₃	C₂³	1	6	yes	—
57	C₂²×C₄×C₂₂₃	C₂³	1	3	yes	—
58	C₂²×C₄×C₂₂₃	C₂³	1	3	yes	—
59	C₂²×C₄×C₂₂₃	C₂³	1	2	yes	—
60	C₂²×C₄×C₂₂₃	C₂³	1	3	yes	—
61	C₄⋊C₄×C₂₂₃	C₂³	1	3	yes	—
62	C₄⋊C₄×C₂₂₃	C₂³	1	6	yes	—
63	C₄⋊C₄×C₂₂₃	C₂³	1	12	yes	—
64	C₄⋊C₄×C₂₂₃	C₂³	1	1	yes	characteristic
65	C₄⋊C₄×C₂₂₃	C₂³	1	6	yes	—
66	C₄⋊C₄×C₂₂₃	C₂³	1	3	yes	—
67	C₄⋊C₄×C₂₂₃	C₂³	1	3	yes	—
68	C₄⋊C₄×C₂₂₃	C₂³	1	6	yes	—
69	C₄⋊C₄×C₂₂₃	C₂³	1	6	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	12	yes	—
76	C₄²×C₂₂₃	C₂³	1	3	yes	—
77	C₄²×C₂₂₃	C₂³	1	6	yes	—
78	C₄²×C₂₂₃	C₂³	1	3	yes	—
79	C₄²×C₂₂₃	C₂³	1	6	yes	—
80	C₄²×C₂₂₃	C₂³	1	3	yes	—
81	C₄²×C₂₂₃	C₂³	1	1	yes	characteristic
82	C₄²×C₂₂₃	C₂³	1	6	yes	—
83	C₂³×C₂₂₃	C₂⁴	1	3	yes	—
84	C₂³×C₂₂₃	C₂⁴	1	2	yes	—
85	Q₈×C₂₂₃	—	2	12	no	—
86	Q₈×C₂₂₃	—	2	6	no	—
87	Q₈×C₂₂₃	#41	1	4	yes	complemented
88	D₄×C₂₂₃	—	2	12	no	—
89	D₄×C₂₂₃	—	2	6	no	—
90	D₄×C₂₂₃	#41	1	4	yes	complemented
91	C₂×C₄×C₂₂₃	—	2	12	no	—
92	C₂×C₄×C₂₂₃	—	2	12	no	—
93	C₂×C₄×C₂₂₃	—	2	6	no	—
94	C₂×C₄×C₂₂₃	—	2	6	no	—
95	C₂×C₄×C₂₂₃	—	2	4	no	—
96	C₂×C₄×C₂₂₃	—	2	6	no	—
97	C₂×C₄×C₂₂₃	C₂⁴	1	2	yes	—
98	C₂×C₄×C₂₂₃	C₂⁴	1	1	yes	characteristic
99	C₂×C₄×C₂₂₃	C₂⁴	1	3	yes	—
100	C₂×C₄×C₂₂₃	C₂⁴	1	3	yes	—
101	C₂×C₄×C₂₂₃	C₂⁴	1	6	yes	—
102	C₂×C₄×C₂₂₃	C₂⁴	1	6	yes	—
103	C₂×C₄×C₂₂₃	C₂⁴	1	3	yes	—
104	C₂×C₄×C₂₂₃	C₂⁴	1	1	yes	characteristic
105	C₂×C₄×C₂₂₃	C₂⁴	1	1	yes	characteristic
106	C₂²×C₂₂₃	—	2	2	no	—
107	C₂²×C₂₂₃	—	2	2	no	—
108	C₂²×C₂₂₃	—	2	3	no	—
109	C₂²×C₂₂₃	—	2	3	no	—
110	C₂²×C₂₂₃	—	2	3	no	—
111	C₂²×C₂₂₃	C₂×#41	1	4	yes	—
112	C₂²×C₂₂₃	C₂⁵	1	1	yes	characteristic
113	C₄×C₂₂₃	—	2	3	no	—
114	C₄×C₂₂₃	—	2	1	no	—
115	C₄×C₂₂₃	—	2	6	no	—
116	C₄×C₂₂₃	—	2	6	no	—
117	C₄×C₂₂₃	—	2	2	no	—
118	C₄×C₂₂₃	—	2	3	no	—
119	C₄×C₂₂₃	C₂×#41	1	6	yes	—
120	C₄×C₂₂₃	C₂×#41	1	2	yes	—
121	C₄×C₂₂₃	2₋¹⁺⁴	1	2	yes	—
122	C₂×C₂₂₃	—	2	4	no	—
123	C₂×C₂₂₃	—	4	3	no	—
124	C₂×C₂₂₃	C₂²×#41	1	1	yes	characteristic
125	C₂×C₂₂₃	C₂²×#41	1	1	yes	characteristic
126	C₂×C₂₂₃	C₂×2₋¹⁺⁴	1	1	yes	characteristic
127	C₂₂₃	#3482	1	1	yes	characteristic, direct product factor, Sylow
128	#3482	C₂₂₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
129	#557	C₂×C₂₂₃	1	1	yes	characteristic, complemented
130	#556	C₂×C₂₂₃	1	3	yes	complemented
131	#552	C₂×C₂₂₃	1	6	yes	complemented
132	D₄×Q₈	C₂×C₂₂₃	1	1	yes	characteristic, complemented
133	#548	C₂×C₂₂₃	1	2	yes	complemented
134	#547	C₂×C₂₂₃	1	3	yes	complemented
135	#533	C₂×C₂₂₃	1	3	yes	complemented
136	#529	C₂×C₂₂₃	1	6	yes	complemented
137	C₄×#41	C₂×C₂₂₃	1	3	yes	complemented
138	C₄×#41	C₂×C₂₂₃	1	1	yes	characteristic, complemented
139	C₂×C₄×Q₈	C₂×C₂₂₃	1	2	yes	complemented
140	C₂×#41	C₂²×C₂₂₃	1	3	yes	—
141	C₂×#41	C₂²×C₂₂₃	1	1	yes	characteristic
142	C₂²×Q₈	C₂²×C₂₂₃	1	2	yes	—
143	C₄⋊Q₈	C₂²×C₂₂₃	1	3	yes	—
144	C₄⋊Q₈	C₂²×C₂₂₃	1	3	yes	—
145	#126	C₂²×C₂₂₃	1	12	yes	—
146	#125	C₂²×C₂₂₃	1	6	yes	—
147	#125	C₂²×C₂₂₃	1	6	yes	—
148	#124	C₂²×C₂₂₃	1	6	yes	—
149	#123	C₂²×C₂₂₃	1	12	yes	—
150	C₂²⋊Q₈	C₂²×C₂₂₃	1	6	yes	—
151	C₂²⋊Q₈	C₂²×C₂₂₃	1	6	yes	—
152	C₂²⋊Q₈	C₂²×C₂₂₃	1	6	yes	—
153	#121	C₂²×C₂₂₃	1	6	yes	—
154	C₄×Q₈	C₂²×C₂₂₃	1	1	yes	characteristic
155	C₄×Q₈	C₂²×C₂₂₃	1	6	yes	—
156	C₄×Q₈	C₂²×C₂₂₃	1	2	yes	—
157	C₄×Q₈	C₂²×C₂₂₃	1	6	yes	—
158	C₄×Q₈	C₂²×C₂₂₃	1	3	yes	—
159	C₄×Q₈	C₂²×C₂₂₃	1	1	yes	characteristic
160	C₄×Q₈	C₂²×C₂₂₃	1	3	yes	—
161	C₄×D₄	C₂²×C₂₂₃	1	1	yes	characteristic
162	C₄×D₄	C₂²×C₂₂₃	1	3	yes	—
163	C₄×D₄	C₂²×C₂₂₃	1	6	yes	—
164	C₄×D₄	C₂²×C₂₂₃	1	3	yes	—
165	C₄×D₄	C₂²×C₂₂₃	1	3	yes	—
166	C₄×D₄	C₂²×C₂₂₃	1	6	yes	—
167	#117	C₂²×C₂₂₃	1	3	yes	—
168	#117	C₂²×C₂₂₃	1	6	yes	—
169	#117	C₂²×C₂₂₃	1	3	yes	—
170	#117	C₂²×C₂₂₃	1	6	yes	—
171	C₂×C₄⋊C₄	C₂²×C₂₂₃	1	6	yes	—
172	C₂×C₄²	C₂²×C₂₂₃	1	6	yes	—
173	C₂×C₄²	C₂²×C₂₂₃	1	3	yes	—
174	#41	—	2	4	no	—
175	#41	—	2	12	no	—
176	C₂×Q₈	—	2	8	no	—
177	C₂×Q₈	C₂³×C₂₂₃	1	6	yes	—
178	C₂×Q₈	C₂³×C₂₂₃	1	3	yes	—
179	C₂×Q₈	C₂³×C₂₂₃	1	1	yes	characteristic
180	C₂×D₄	C₂³×C₂₂₃	1	6	yes	—
181	C₂×D₄	C₂³×C₂₂₃	1	3	yes	—
182	C₂×D₄	C₂³×C₂₂₃	1	1	yes	characteristic
183	C₂²×C₄	C₂³×C₂₂₃	1	2	yes	—
184	C₂²×C₄	C₂³×C₂₂₃	1	6	yes	—
185	C₂²×C₄	C₂³×C₂₂₃	1	6	yes	—
186	C₂²×C₄	C₂³×C₂₂₃	1	3	yes	—
187	C₂²×C₄	C₂³×C₂₂₃	1	3	yes	—
188	C₂²×C₄	C₂³×C₂₂₃	1	3	yes	—
189	C₄⋊C₄	C₂³×C₂₂₃	1	3	yes	—
190	C₄⋊C₄	C₂³×C₂₂₃	1	6	yes	—
191	C₄⋊C₄	C₂³×C₂₂₃	1	12	yes	—
192	C₄⋊C₄	C₂³×C₂₂₃	1	1	yes	characteristic
193	C₄⋊C₄	C₂³×C₂₂₃	1	6	yes	—
194	C₄⋊C₄	C₂³×C₂₂₃	1	3	yes	—
195	C₄⋊C₄	C₂³×C₂₂₃	1	3	yes	—
196	C₄⋊C₄	C₂³×C₂₂₃	1	6	yes	—
197	C₄⋊C₄	C₂³×C₂₂₃	1	6	yes	—
198	C₄⋊C₄	C₂³×C₂₂₃	1	6	yes	—
199	C₂²⋊C₄	C₂³×C₂₂₃	1	2	yes	—
200	C₂²⋊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	12	yes	—
204	C₄²	C₂³×C₂₂₃	1	3	yes	—
205	C₄²	C₂³×C₂₂₃	1	1	yes	characteristic
206	C₄²	C₂³×C₂₂₃	1	3	yes	—
207	C₄²	C₂³×C₂₂₃	1	6	yes	—
208	C₄²	C₂³×C₂₂₃	1	6	yes	—
209	C₄²	C₂³×C₂₂₃	1	3	yes	—
210	C₄²	C₂³×C₂₂₃	1	6	yes	—
211	C₂³	C₂⁴×C₂₂₃	1	3	yes	—
212	C₂³	C₂⁴×C₂₂₃	1	2	yes	—
213	Q₈	—	2	12	no	—
214	Q₈	—	2	6	no	—
215	Q₈	#41×C₂₂₃	1	4	yes	complemented
216	D₄	—	2	12	no	—
217	D₄	—	2	6	no	—
218	D₄	#41×C₂₂₃	1	4	yes	complemented
219	C₂×C₄	—	2	12	no	—
220	C₂×C₄	—	2	12	no	—
221	C₂×C₄	—	2	6	no	—
222	C₂×C₄	—	2	6	no	—
223	C₂×C₄	—	2	4	no	—
224	C₂×C₄	—	2	6	no	—
225	C₂×C₄	C₂⁴×C₂₂₃	1	2	yes	—
226	C₂×C₄	C₂⁴×C₂₂₃	1	1	yes	characteristic
227	C₂×C₄	C₂⁴×C₂₂₃	1	3	yes	—
228	C₂×C₄	C₂⁴×C₂₂₃	1	3	yes	—
229	C₂×C₄	C₂⁴×C₂₂₃	1	6	yes	—
230	C₂×C₄	C₂⁴×C₂₂₃	1	6	yes	—
231	C₂×C₄	C₂⁴×C₂₂₃	1	3	yes	—
232	C₂×C₄	C₂⁴×C₂₂₃	1	1	yes	characteristic
233	C₂×C₄	C₂⁴×C₂₂₃	1	1	yes	characteristic
234	C₂²	—	2	2	no	—
235	C₂²	—	2	2	no	—
236	C₂²	—	2	3	no	—
237	C₂²	—	2	3	no	—
238	C₂²	—	2	3	no	—
239	C₂²	C₂×#41×C₂₂₃	1	4	yes	—
240	C₂²	C₂⁵×C₂₂₃	1	1	yes	characteristic
241	C₄	—	2	3	no	—
242	C₄	—	2	1	no	—
243	C₄	—	2	6	no	—
244	C₄	—	2	6	no	—
245	C₄	—	2	2	no	—
246	C₄	—	2	3	no	—
247	C₄	C₂×#41×C₂₂₃	1	6	yes	—
248	C₄	C₂×#41×C₂₂₃	1	2	yes	—
249	C₄	2₋¹⁺⁴×C₂₂₃	1	2	yes	—
250	C₂	—	2	4	no	—
251	C₂	—	4	3	no	—
252	C₂	C₂²×#41×C₂₂₃	1	1	yes	characteristic
253	C₂	C₂²×#41×C₂₂₃	1	1	yes	characteristic
254	C₂	C₂×2₋¹⁺⁴×C₂₂₃	1	1	yes	characteristic

	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. #557×C₂₂₃	C₂	semidirect
2. #556×C₂₂₃	C₂	semidirect
3. #552×C₂₂₃	C₂	semidirect
4. D₄×Q₈×C₂₂₃	C₂	semidirect
5. #548×C₂₂₃	C₂	semidirect
6. #547×C₂₂₃	C₂	semidirect
7. #533×C₂₂₃	C₂	semidirect
8. #529×C₂₂₃	C₂	semidirect
9. C₄×#41×C₂₂₃	C₂	semidirect
10. C₄×#41×C₂₂₃	C₂	semidirect
11. C₂×C₄×Q₈×C₂₂₃	C₂	semidirect
87. Q₈×C₂₂₃	#41	semidirect
90. D₄×C₂₂₃	#41	semidirect
127. C₂₂₃	#3482	direct (N × H)
128. #3482	C₂₂₃	direct (N × H)
129. #557	C₂×C₂₂₃	semidirect
130. #556	C₂×C₂₂₃	semidirect
131. #552	C₂×C₂₂₃	semidirect
132. D₄×Q₈	C₂×C₂₂₃	semidirect
133. #548	C₂×C₂₂₃	semidirect
134. #547	C₂×C₂₂₃	semidirect
135. #533	C₂×C₂₂₃	semidirect
136. #529	C₂×C₂₂₃	semidirect
137. C₄×#41	C₂×C₂₂₃	semidirect
138. C₄×#41	C₂×C₂₂₃	semidirect
139. C₂×C₄×Q₈	C₂×C₂₂₃	semidirect
215. Q₈	#41×C₂₂₃	semidirect
218. D₄	#41×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
#548×C₂₂₃	2
#547×C₂₂₃	2
#533×C₂₂₃	2
#529×C₂₂₃	2
C₄×#41×C₂₂₃	2
C₂×C₄×Q₈×C₂₂₃	2
#3482	223

Subgroups — #3482×C223 #43252497

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — #3482×C₂₂₃ #43252497