Subgroups — D₄×C₂²⋊C₄×C₂₂₃ #43251353

Order 28544.

1252 conjugacy classes of subgroups.

The subgroup lattice has 272 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂³×D₄×C₂₂₃	C₂	1	1	yes	maximal, characteristic
2	C₂×C₄×D₄×C₂₂₃	C₂	1	2	yes	maximal, complemented
3	C₂²×C₂²⋊C₄×C₂₂₃	C₂	1	2	yes	maximal, complemented
4	#390×C₂₂₃	C₂	1	2	yes	maximal, complemented
5	#386×C₂₂₃	C₂	1	4	yes	maximal, complemented
6	#380×C₂₂₃	C₂	1	1	yes	maximal, characteristic, complemented
7	#379×C₂₂₃	C₂	1	2	yes	maximal, complemented
8	C₄×C₂²⋊C₄×C₂₂₃	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂⁵×C₂₂₃	C₂²	1	2	yes	—
10	C₂²×D₄×C₂₂₃	—	2	8	no	—
11	C₂²×D₄×C₂₂₃	C₄	1	8	yes	complemented
12	C₂²×D₄×C₂₂₃	C₂²	1	2	yes	—
13	C₂²×D₄×C₂₂₃	C₂²	1	1	yes	characteristic
14	C₂²×D₄×C₂₂₃	C₂²	1	1	yes	characteristic
15	C₂³×C₄×C₂₂₃	C₂²	1	1	yes	characteristic
16	C₂³×C₄×C₂₂₃	C₂²	1	4	yes	complemented
17	C₄×D₄×C₂₂₃	—	2	8	no	—
18	C₂×C₄⋊C₄×C₂₂₃	C₂²	1	2	yes	complemented
19	C₂×C₂²⋊C₄×C₂₂₃	—	2	8	no	—
20	C₂×C₂²⋊C₄×C₂₂₃	C₂²	1	1	yes	characteristic
21	C₂×C₂²⋊C₄×C₂₂₃	C₂²	1	2	yes	—
22	C₂×C₂²⋊C₄×C₂₂₃	C₂²	1	1	yes	characteristic
23	C₂×C₂²⋊C₄×C₂₂₃	C₂²	1	4	yes	complemented
24	C₂×C₂²⋊C₄×C₂₂₃	C₂²	1	4	yes	complemented
25	C₂×C₂²⋊C₄×C₂₂₃	C₂²	1	4	yes	complemented
26	C₂×C₄²×C₂₂₃	C₂²	1	2	yes	complemented
27	#95×C₂₂₃	C₂²	1	2	yes	complemented
28	#95×C₂₂₃	C₂²	1	2	yes	complemented
29	C₂⁴×C₂₂₃	—	2	4	no	—
30	C₂⁴×C₂₂₃	—	2	8	no	—
31	C₂⁴×C₂₂₃	—	2	4	no	—
32	C₂⁴×C₂₂₃	—	4	4	no	—
33	C₂⁴×C₂₂₃	C₂×C₄	1	8	yes	complemented
34	C₂⁴×C₂₂₃	C₂³	1	2	yes	—
35	C₂⁴×C₂₂₃	C₂³	1	2	yes	—
36	C₂⁴×C₂₂₃	C₂³	1	1	yes	characteristic
37	C₂×D₄×C₂₂₃	—	2	8	no	—
38	C₂×D₄×C₂₂₃	—	2	32	no	—
39	C₂×D₄×C₂₂₃	—	2	4	no	—
40	C₂×D₄×C₂₂₃	C₂×C₄	1	4	yes	—
41	C₂×D₄×C₂₂₃	C₂×C₄	1	8	yes	complemented
42	C₂×D₄×C₂₂₃	C₂×C₄	1	4	yes	complemented
43	C₂×D₄×C₂₂₃	D₄	1	8	yes	—
44	C₂²×C₄×C₂₂₃	—	2	4	no	—
45	C₂²×C₄×C₂₂₃	—	2	8	no	—
46	C₂²×C₄×C₂₂₃	—	2	8	no	—
47	C₂²×C₄×C₂₂₃	—	4	4	no	—
48	C₂²×C₄×C₂₂₃	C₂×C₄	1	4	yes	complemented
49	C₂²×C₄×C₂₂₃	C₂³	1	2	yes	—
50	C₂²×C₄×C₂₂₃	C₂³	1	4	yes	—
51	C₂²×C₄×C₂₂₃	C₂³	1	1	yes	characteristic
52	C₂²×C₄×C₂₂₃	C₂³	1	2	yes	—
53	C₂²×C₄×C₂₂₃	C₂³	1	1	yes	characteristic
54	C₄⋊C₄×C₂₂₃	—	2	4	no	—
55	C₂²⋊C₄×C₂₂₃	—	2	4	no	—
56	C₂²⋊C₄×C₂₂₃	—	2	8	no	—
57	C₂²⋊C₄×C₂₂₃	—	2	2	no	—
58	C₂²⋊C₄×C₂₂₃	—	2	8	no	—
59	C₂²⋊C₄×C₂₂₃	—	4	4	no	—
60	C₂²⋊C₄×C₂₂₃	D₄	1	4	yes	direct product factor
61	C₄²×C₂₂₃	—	2	4	no	—
62	C₂³×C₂₂₃	—	2	2	no	—
63	C₂³×C₂₂₃	—	2	2	no	—
64	C₂³×C₂₂₃	—	2	2	no	—
65	C₂³×C₂₂₃	—	2	4	no	—
66	C₂³×C₂₂₃	—	2	2	no	—
67	C₂³×C₂₂₃	—	2	16	no	—
68	C₂³×C₂₂₃	—	2	16	no	—
69	C₂³×C₂₂₃	—	2	2	no	—
70	C₂³×C₂₂₃	—	2	4	no	—
71	C₂³×C₂₂₃	—	2	2	no	—
72	C₂³×C₂₂₃	—	2	4	no	—
73	C₂³×C₂₂₃	—	4	8	no	—
74	C₂³×C₂₂₃	—	4	16	no	—
75	C₂³×C₂₂₃	—	4	16	no	—
76	C₂³×C₂₂₃	C₂²×C₄	1	4	yes	—
77	C₂³×C₂₂₃	C₂²×C₄	1	4	yes	—
78	C₂³×C₂₂₃	C₂²×C₄	1	2	yes	—
79	C₂³×C₂₂₃	C₂×D₄	1	8	yes	—
80	C₂³×C₂₂₃	C₂×D₄	1	2	yes	—
81	C₂³×C₂₂₃	#41	1	2	yes	—
82	C₂³×C₂₂₃	C₂⁴	1	1	yes	characteristic
83	D₄×C₂₂₃	—	2	16	no	—
84	D₄×C₂₂₃	—	2	8	no	—
85	D₄×C₂₂₃	C₂²⋊C₄	1	16	yes	direct product factor
86	C₂×C₄×C₂₂₃	—	2	2	no	—
87	C₂×C₄×C₂₂₃	—	2	2	no	—
88	C₂×C₄×C₂₂₃	—	2	2	no	—
89	C₂×C₄×C₂₂₃	—	2	8	no	—
90	C₂×C₄×C₂₂₃	—	2	2	no	—
91	C₂×C₄×C₂₂₃	—	2	4	no	—
92	C₂×C₄×C₂₂₃	—	2	4	no	—
93	C₂×C₄×C₂₂₃	—	2	2	no	—
94	C₂×C₄×C₂₂₃	—	2	2	no	—
95	C₂×C₄×C₂₂₃	—	2	4	no	—
96	C₂×C₄×C₂₂₃	—	4	16	no	—
97	C₂×C₄×C₂₂₃	C₂²×C₄	1	2	yes	—
98	C₂×C₄×C₂₂₃	C₂²×C₄	1	2	yes	—
99	C₂×C₄×C₂₂₃	C₂×D₄	1	4	yes	—
100	C₂×C₄×C₂₂₃	C₂×D₄	1	4	yes	complemented
101	C₂²×C₂₂₃	—	2	2	no	—
102	C₂²×C₂₂₃	—	2	4	no	—
103	C₂²×C₂₂₃	—	2	2	no	—
104	C₂²×C₂₂₃	—	2	4	no	—
105	C₂²×C₂₂₃	—	2	8	no	—
106	C₂²×C₂₂₃	—	2	4	no	—
107	C₂²×C₂₂₃	—	2	4	no	—
108	C₂²×C₂₂₃	—	2	8	no	—
109	C₂²×C₂₂₃	—	2	4	no	—
110	C₂²×C₂₂₃	—	2	4	no	—
111	C₂²×C₂₂₃	—	2	4	no	—
112	C₂²×C₂₂₃	—	4	32	no	—
113	C₂²×C₂₂₃	—	4	4	no	—
114	C₂²×C₂₂₃	—	4	4	no	—
115	C₂²×C₂₂₃	C₂×C₂²⋊C₄	1	8	yes	complemented
116	C₂²×C₂₂₃	C₄×D₄	1	4	yes	complemented
117	C₂²×C₂₂₃	C₂³×C₄	1	1	yes	characteristic
118	C₂²×C₂₂₃	C₂²×D₄	1	2	yes	—
119	C₂²×C₂₂₃	C₂²×D₄	1	1	yes	characteristic
120	C₂²×C₂₂₃	C₂×#41	1	1	yes	characteristic
121	C₂²×C₂₂₃	H(5,𝔽₂)	1	2	yes	—
122	C₄×C₂₂₃	—	2	2	no	—
123	C₄×C₂₂₃	—	2	4	no	—
124	C₄×C₂₂₃	—	4	4	no	—
125	C₄×C₂₂₃	—	4	2	no	—
126	C₄×C₂₂₃	C₂×C₂²⋊C₄	1	4	yes	complemented
127	C₂×C₂₂₃	—	2	4	no	—
128	C₂×C₂₂₃	—	2	8	no	—
129	C₂×C₂₂₃	—	4	4	no	—
130	C₂×C₂₂₃	C₂²×C₂²⋊C₄	1	1	yes	characteristic
131	C₂×C₂₂₃	C₂×C₄×D₄	1	1	yes	characteristic
132	C₂×C₂₂₃	#518	1	1	yes	characteristic
133	C₂×C₂₂₃	D₄²	1	2	yes	—
134	C₂×C₂₂₃	#546	1	2	yes	—
135	C₂₂₃	D₄×C₂²⋊C₄	1	1	yes	characteristic, direct product factor, Sylow
136	D₄×C₂²⋊C₄	C₂₂₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
137	C₂³×D₄	C₂×C₂₂₃	1	1	yes	characteristic
138	C₂×C₄×D₄	C₂×C₂₂₃	1	2	yes	complemented
139	C₂²×C₂²⋊C₄	C₂×C₂₂₃	1	2	yes	complemented
140	#390	C₂×C₂₂₃	1	2	yes	complemented
141	#386	C₂×C₂₂₃	1	4	yes	complemented
142	#380	C₂×C₂₂₃	1	1	yes	characteristic, complemented
143	#379	C₂×C₂₂₃	1	2	yes	complemented
144	C₄×C₂²⋊C₄	C₂×C₂₂₃	1	1	yes	characteristic, complemented
145	C₂⁵	C₂²×C₂₂₃	1	2	yes	—
146	C₂²×D₄	—	2	8	no	—
147	C₂²×D₄	C₄×C₂₂₃	1	8	yes	complemented
148	C₂²×D₄	C₂²×C₂₂₃	1	2	yes	—
149	C₂²×D₄	C₂²×C₂₂₃	1	1	yes	characteristic
150	C₂²×D₄	C₂²×C₂₂₃	1	1	yes	characteristic
151	C₂³×C₄	C₂²×C₂₂₃	1	1	yes	characteristic
152	C₂³×C₄	C₂²×C₂₂₃	1	4	yes	complemented
153	C₄×D₄	—	2	8	no	—
154	C₂×C₄⋊C₄	C₂²×C₂₂₃	1	2	yes	complemented
155	C₂×C₂²⋊C₄	—	2	8	no	—
156	C₂×C₂²⋊C₄	C₂²×C₂₂₃	1	2	yes	—
157	C₂×C₂²⋊C₄	C₂²×C₂₂₃	1	1	yes	characteristic
158	C₂×C₂²⋊C₄	C₂²×C₂₂₃	1	1	yes	characteristic
159	C₂×C₂²⋊C₄	C₂²×C₂₂₃	1	4	yes	complemented
160	C₂×C₂²⋊C₄	C₂²×C₂₂₃	1	4	yes	complemented
161	C₂×C₂²⋊C₄	C₂²×C₂₂₃	1	4	yes	complemented
162	C₂×C₄²	C₂²×C₂₂₃	1	2	yes	complemented
163	#95	C₂²×C₂₂₃	1	2	yes	complemented
164	#95	C₂²×C₂₂₃	1	2	yes	complemented
165	C₂⁴	—	2	4	no	—
166	C₂⁴	—	2	8	no	—
167	C₂⁴	—	2	4	no	—
168	C₂⁴	—	4	4	no	—
169	C₂⁴	C₂×C₄×C₂₂₃	1	8	yes	complemented
170	C₂⁴	C₂³×C₂₂₃	1	2	yes	—
171	C₂⁴	C₂³×C₂₂₃	1	2	yes	—
172	C₂⁴	C₂³×C₂₂₃	1	1	yes	characteristic
173	C₂×D₄	—	2	8	no	—
174	C₂×D₄	—	2	32	no	—
175	C₂×D₄	—	2	4	no	—
176	C₂×D₄	C₂×C₄×C₂₂₃	1	4	yes	—
177	C₂×D₄	C₂×C₄×C₂₂₃	1	8	yes	complemented
178	C₂×D₄	C₂×C₄×C₂₂₃	1	4	yes	complemented
179	C₂×D₄	D₄×C₂₂₃	1	8	yes	—
180	C₂²×C₄	—	2	4	no	—
181	C₂²×C₄	—	2	8	no	—
182	C₂²×C₄	—	2	8	no	—
183	C₂²×C₄	—	4	4	no	—
184	C₂²×C₄	C₂×C₄×C₂₂₃	1	4	yes	complemented
185	C₂²×C₄	C₂³×C₂₂₃	1	2	yes	—
186	C₂²×C₄	C₂³×C₂₂₃	1	4	yes	—
187	C₂²×C₄	C₂³×C₂₂₃	1	1	yes	characteristic
188	C₂²×C₄	C₂³×C₂₂₃	1	2	yes	—
189	C₂²×C₄	C₂³×C₂₂₃	1	1	yes	characteristic
190	C₄⋊C₄	—	2	4	no	—
191	C₂²⋊C₄	—	2	4	no	—
192	C₂²⋊C₄	—	2	8	no	—
193	C₂²⋊C₄	—	2	2	no	—
194	C₂²⋊C₄	—	2	8	no	—
195	C₂²⋊C₄	—	4	4	no	—
196	C₂²⋊C₄	D₄×C₂₂₃	1	4	yes	direct product factor
197	C₄²	—	2	4	no	—
198	C₂³	—	2	2	no	—
199	C₂³	—	2	2	no	—
200	C₂³	—	2	2	no	—
201	C₂³	—	2	4	no	—
202	C₂³	—	2	2	no	—
203	C₂³	—	2	16	no	—
204	C₂³	—	2	16	no	—
205	C₂³	—	2	2	no	—
206	C₂³	—	2	4	no	—
207	C₂³	—	2	2	no	—
208	C₂³	—	2	4	no	—
209	C₂³	—	4	8	no	—
210	C₂³	—	4	16	no	—
211	C₂³	—	4	16	no	—
212	C₂³	C₂²×C₄×C₂₂₃	1	4	yes	—
213	C₂³	C₂²×C₄×C₂₂₃	1	4	yes	—
214	C₂³	C₂²×C₄×C₂₂₃	1	2	yes	—
215	C₂³	C₂×D₄×C₂₂₃	1	8	yes	—
216	C₂³	C₂×D₄×C₂₂₃	1	2	yes	—
217	C₂³	#41×C₂₂₃	1	2	yes	—
218	C₂³	C₂⁴×C₂₂₃	1	1	yes	characteristic
219	D₄	—	2	16	no	—
220	D₄	—	2	8	no	—
221	D₄	C₂²⋊C₄×C₂₂₃	1	16	yes	direct product factor
222	C₂×C₄	—	2	2	no	—
223	C₂×C₄	—	2	2	no	—
224	C₂×C₄	—	2	2	no	—
225	C₂×C₄	—	2	8	no	—
226	C₂×C₄	—	2	2	no	—
227	C₂×C₄	—	2	4	no	—
228	C₂×C₄	—	2	4	no	—
229	C₂×C₄	—	2	2	no	—
230	C₂×C₄	—	2	2	no	—
231	C₂×C₄	—	2	4	no	—
232	C₂×C₄	—	4	16	no	—
233	C₂×C₄	C₂²×C₄×C₂₂₃	1	2	yes	—
234	C₂×C₄	C₂²×C₄×C₂₂₃	1	2	yes	—
235	C₂×C₄	C₂×D₄×C₂₂₃	1	4	yes	—
236	C₂×C₄	C₂×D₄×C₂₂₃	1	4	yes	complemented
237	C₂²	—	2	8	no	—
238	C₂²	—	2	4	no	—
239	C₂²	—	2	4	no	—
240	C₂²	—	2	8	no	—
241	C₂²	—	2	2	no	—
242	C₂²	—	2	4	no	—
243	C₂²	—	2	2	no	—
244	C₂²	—	2	4	no	—
245	C₂²	—	2	4	no	—
246	C₂²	—	2	4	no	—
247	C₂²	—	2	4	no	—
248	C₂²	—	4	32	no	—
249	C₂²	—	4	4	no	—
250	C₂²	—	4	4	no	—
251	C₂²	C₂×C₂²⋊C₄×C₂₂₃	1	8	yes	complemented
252	C₂²	C₄×D₄×C₂₂₃	1	4	yes	complemented
253	C₂²	C₂³×C₄×C₂₂₃	1	1	yes	characteristic
254	C₂²	C₂²×D₄×C₂₂₃	1	1	yes	characteristic
255	C₂²	C₂²×D₄×C₂₂₃	1	2	yes	—
256	C₂²	C₂×#41×C₂₂₃	1	1	yes	characteristic
257	C₂²	H(5,𝔽₂)×C₂₂₃	1	2	yes	—
258	C₄	—	2	4	no	—
259	C₄	—	2	2	no	—
260	C₄	—	4	4	no	—
261	C₄	—	4	2	no	—
262	C₄	C₂×C₂²⋊C₄×C₂₂₃	1	4	yes	complemented
263	C₂	—	2	4	no	—
264	C₂	—	2	8	no	—
265	C₂	—	4	4	no	—
266	C₂	C₂²×C₂²⋊C₄×C₂₂₃	1	1	yes	characteristic
267	C₂	C₂×C₄×D₄×C₂₂₃	1	1	yes	characteristic
268	C₂	#518×C₂₂₃	1	1	yes	characteristic
269	C₂	D₄²×C₂₂₃	1	2	yes	—
270	C₂	#546×C₂₂₃	1	2	yes	—

	subgroup	G/H
center, socle	C₂³×C₂₂₃	C₂⁴
commutator	C₂²	C₂³×C₄×C₂₂₃
Frattini, cosocle	C₂³	C₂⁴×C₂₂₃
prime radical	C₂³×D₄×C₂₂₃	C₂
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³×C₂₂₃ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₂×C₄×D₄×C₂₂₃	C₂	semidirect
3. C₂²×C₂²⋊C₄×C₂₂₃	C₂	semidirect
4. #390×C₂₂₃	C₂	semidirect
5. #386×C₂₂₃	C₂	semidirect
6. #380×C₂₂₃	C₂	semidirect
7. #379×C₂₂₃	C₂	semidirect
8. C₄×C₂²⋊C₄×C₂₂₃	C₂	semidirect
11. C₂²×D₄×C₂₂₃	C₄	semidirect
16. C₂³×C₄×C₂₂₃	C₂²	semidirect
18. C₂×C₄⋊C₄×C₂₂₃	C₂²	semidirect
23. C₂×C₂²⋊C₄×C₂₂₃	C₂²	semidirect
24. C₂×C₂²⋊C₄×C₂₂₃	C₂²	semidirect
25. C₂×C₂²⋊C₄×C₂₂₃	C₂²	semidirect
26. C₂×C₄²×C₂₂₃	C₂²	semidirect
27. #95×C₂₂₃	C₂²	semidirect
28. #95×C₂₂₃	C₂²	semidirect
33. C₂⁴×C₂₂₃	C₂×C₄	semidirect
41. C₂×D₄×C₂₂₃	C₂×C₄	semidirect
42. C₂×D₄×C₂₂₃	C₂×C₄	semidirect
48. C₂²×C₄×C₂₂₃	C₂×C₄	semidirect
60. C₂²⋊C₄×C₂₂₃	D₄	direct (N × H)
85. D₄×C₂₂₃	C₂²⋊C₄	direct (N × H)
100. C₂×C₄×C₂₂₃	C₂×D₄	semidirect
115. C₂²×C₂₂₃	C₂×C₂²⋊C₄	semidirect
116. C₂²×C₂₂₃	C₄×D₄	semidirect
126. C₄×C₂₂₃	C₂×C₂²⋊C₄	semidirect
135. C₂₂₃	D₄×C₂²⋊C₄	direct (N × H)
136. D₄×C₂²⋊C₄	C₂₂₃	direct (N × H)
138. C₂×C₄×D₄	C₂×C₂₂₃	semidirect
139. C₂²×C₂²⋊C₄	C₂×C₂₂₃	semidirect
140. #390	C₂×C₂₂₃	semidirect
141. #386	C₂×C₂₂₃	semidirect
142. #380	C₂×C₂₂₃	semidirect
143. #379	C₂×C₂₂₃	semidirect
144. C₄×C₂²⋊C₄	C₂×C₂₂₃	semidirect
147. C₂²×D₄	C₄×C₂₂₃	semidirect
152. C₂³×C₄	C₂²×C₂₂₃	semidirect
154. C₂×C₄⋊C₄	C₂²×C₂₂₃	semidirect
159. C₂×C₂²⋊C₄	C₂²×C₂₂₃	semidirect
160. C₂×C₂²⋊C₄	C₂²×C₂₂₃	semidirect
161. C₂×C₂²⋊C₄	C₂²×C₂₂₃	semidirect
162. C₂×C₄²	C₂²×C₂₂₃	semidirect
163. #95	C₂²×C₂₂₃	semidirect
164. #95	C₂²×C₂₂₃	semidirect
169. C₂⁴	C₂×C₄×C₂₂₃	semidirect
177. C₂×D₄	C₂×C₄×C₂₂₃	semidirect
178. C₂×D₄	C₂×C₄×C₂₂₃	semidirect
184. C₂²×C₄	C₂×C₄×C₂₂₃	semidirect
196. C₂²⋊C₄	D₄×C₂₂₃	direct (N × H)
221. D₄	C₂²⋊C₄×C₂₂₃	direct (N × H)
236. C₂×C₄	C₂×D₄×C₂₂₃	semidirect
251. C₂²	C₂×C₂²⋊C₄×C₂₂₃	semidirect
252. C₂²	C₄×D₄×C₂₂₃	semidirect
262. C₄	C₂×C₂²⋊C₄×C₂₂₃	semidirect

Maximal subgroups

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

subgroup	index
C₂³×D₄×C₂₂₃	2
C₂×C₄×D₄×C₂₂₃	2
C₂²×C₂²⋊C₄×C₂₂₃	2
#390×C₂₂₃	2
#386×C₂₂₃	2
#380×C₂₂₃	2
#379×C₂₂₃	2
C₄×C₂²⋊C₄×C₂₂₃	2
D₄×C₂²⋊C₄	223

Subgroups — D4×C22⋊C4×C223 #43251353

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — D₄×C₂²⋊C₄×C₂₂₃ #43251353