Subgroups — C₂²⋊C₄×D₄₀₄ #27173955

Order 12928.

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×D₄₀₄	C₂	1	1	yes	maximal, characteristic
2	C₂×C₄×D₄₀₄	C₂	1	2	yes	maximal, complemented
3	C₂×C₂²⋊C₄×D₁₀₁	C₂	1	2	yes	maximal, complemented
4	#20265079	C₂	1	2	yes	maximal, complemented
5	#20265055	C₂	1	4	yes	maximal, complemented
6	#20265037	C₂	1	1	yes	maximal, characteristic, complemented
7	#20265032	C₂	1	2	yes	maximal, complemented
8	C₄×C₂²⋊C₄×C₁₀₁	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂⁴×D₁₀₁	C₂²	1	2	yes	—
10	C₂²×D₄₀₄	—	2	8	no	—
11	C₂²×D₄₀₄	C₄	1	8	yes	complemented
12	C₂²×D₄₀₄	C₂²	1	2	yes	—
13	C₂²×D₄₀₄	C₂²	1	1	yes	characteristic
14	C₂²×D₄₀₄	C₂²	1	1	yes	characteristic
15	C₂²×C₄×D₁₀₁	C₂²	1	4	yes	complemented
16	C₄×D₄₀₄	—	2	8	no	—
17	C₂×C₄⋊Dic₁₀₁	C₂²	1	2	yes	complemented
18	C₂×C₂²⋊Dic₁₀₁	C₂²	1	2	yes	—
19	C₂×#13108051	C₂²	1	4	yes	complemented
20	C₂×#13108051	C₂²	1	4	yes	complemented
21	C₂×#13108051	C₂²	1	4	yes	complemented
22	C₂²⋊C₄×D₁₀₁	—	2	8	no	—
23	#15692162	C₂²	1	2	yes	complemented
24	C₂³×C₄×C₁₀₁	C₂²	1	1	yes	characteristic
25	C₂×C₂²⋊C₄×C₁₀₁	C₂²	1	1	yes	characteristic
26	C₂×C₂²⋊C₄×C₁₀₁	C₂²	1	1	yes	characteristic
27	C₂×C₄²×C₁₀₁	C₂²	1	2	yes	complemented
28	#95×C₁₀₁	C₂²	1	2	yes	complemented
29	C₂³×D₁₀₁	—	2	4	no	—
30	C₂³×D₁₀₁	—	2	4	no	—
31	C₂³×D₁₀₁	—	2	8	no	—
32	C₂³×D₁₀₁	—	4	4	no	—
33	C₂³×D₁₀₁	C₂×C₄	1	8	yes	complemented
34	C₂³×D₁₀₁	C₂³	1	2	yes	—
35	C₂³×D₁₀₁	C₂³	1	2	yes	—
36	C₂×D₄₀₄	—	2	8	no	—
37	C₂×D₄₀₄	—	2	4	no	—
38	C₂×D₄₀₄	—	2	32	no	—
39	C₂×D₄₀₄	C₂×C₄	1	4	yes	—
40	C₂×D₄₀₄	C₂×C₄	1	8	yes	complemented
41	C₂×D₄₀₄	C₂×C₄	1	4	yes	complemented
42	C₂×D₄₀₄	D₄	1	8	yes	—
43	C₂²×Dic₁₀₁	C₂³	1	4	yes	—
44	C₂×C₄×D₁₀₁	—	2	8	no	—
45	C₂×C₄×D₁₀₁	—	2	8	no	—
46	C₂×C₄×D₁₀₁	—	4	4	no	—
47	C₄⋊Dic₁₀₁	—	2	4	no	—
48	C₂²⋊Dic₁₀₁	—	2	4	no	—
49	#13108051	—	2	8	no	—
50	#13108051	—	2	8	no	—
51	#13108051	—	4	4	no	—
52	C₂⁴×C₁₀₁	C₂³	1	1	yes	characteristic
53	C₂²×C₄×C₁₀₁	—	2	4	no	—
54	C₂²×C₄×C₁₀₁	C₂×C₄	1	4	yes	complemented
55	C₂²×C₄×C₁₀₁	C₂³	1	2	yes	—
56	C₂²×C₄×C₁₀₁	C₂³	1	1	yes	characteristic
57	C₂²×C₄×C₁₀₁	C₂³	1	2	yes	—
58	C₂²×C₄×C₁₀₁	C₂³	1	1	yes	characteristic
59	C₂²⋊C₄×C₁₀₁	—	2	2	no	—
60	C₂²⋊C₄×C₁₀₁	D₄	1	4	yes	complemented
61	C₄²×C₁₀₁	—	2	4	no	—
62	C₂³×C₁₀₁	—	2	2	no	—
63	C₂³×C₁₀₁	—	2	2	no	—
64	C₂³×C₁₀₁	C₂²×C₄	1	2	yes	—
65	C₂³×C₁₀₁	C₂×D₄	1	2	yes	—
66	C₂³×C₁₀₁	#41	1	2	yes	—
67	C₂³×C₁₀₁	C₂⁴	1	1	yes	characteristic
68	C₂²×D₁₀₁	—	2	2	no	—
69	C₂²×D₁₀₁	—	2	2	no	—
70	C₂²×D₁₀₁	—	2	4	no	—
71	C₂²×D₁₀₁	—	2	16	no	—
72	C₂²×D₁₀₁	—	2	2	no	—
73	C₂²×D₁₀₁	—	2	4	no	—
74	C₂²×D₁₀₁	—	2	2	no	—
75	C₂²×D₁₀₁	—	2	4	no	—
76	C₂²×D₁₀₁	—	2	16	no	—
77	C₂²×D₁₀₁	—	4	8	no	—
78	C₂²×D₁₀₁	—	4	16	no	—
79	C₂²×D₁₀₁	—	4	16	no	—
80	C₂²×D₁₀₁	C₂²×C₄	1	4	yes	—
81	C₂²×D₁₀₁	C₂²×C₄	1	4	yes	—
82	C₂²×D₁₀₁	C₂×D₄	1	8	yes	—
83	C₂×C₄×C₁₀₁	—	2	2	no	—
84	C₂×C₄×C₁₀₁	—	2	2	no	—
85	C₂×C₄×C₁₀₁	—	2	2	no	—
86	C₂×C₄×C₁₀₁	—	2	8	no	—
87	C₂×C₄×C₁₀₁	—	2	2	no	—
88	C₂×C₄×C₁₀₁	—	2	2	no	—
89	C₂×C₄×C₁₀₁	—	2	2	no	—
90	C₂×C₄×C₁₀₁	C₂²×C₄	1	2	yes	—
91	C₂×C₄×C₁₀₁	C₂²×C₄	1	2	yes	—
92	C₂×C₄×C₁₀₁	C₂×D₄	1	4	yes	—
93	C₂×C₄×C₁₀₁	C₂×D₄	1	4	yes	complemented
94	C₂×Dic₁₀₁	—	2	4	no	—
95	C₂×Dic₁₀₁	—	2	4	no	—
96	C₂×Dic₁₀₁	—	2	4	no	—
97	D₄₀₄	—	2	16	no	—
98	D₄₀₄	—	2	8	no	—
99	D₄₀₄	C₂²⋊C₄	1	16	yes	direct product factor
100	C₄×D₁₀₁	—	4	16	no	—
101	C₂²×C₁₀₁	—	2	2	no	—
102	C₂²×C₁₀₁	—	2	2	no	—
103	C₂²×C₁₀₁	—	2	4	no	—
104	C₂²×C₁₀₁	—	2	4	no	—
105	C₂²×C₁₀₁	C₄×D₄	1	4	yes	complemented
106	C₂²×C₁₀₁	C₂³×C₄	1	1	yes	characteristic
107	C₂²×C₁₀₁	C₂²×D₄	1	2	yes	—
108	C₂²×C₁₀₁	C₂²×D₄	1	1	yes	characteristic
109	C₂²×C₁₀₁	C₂×#41	1	1	yes	characteristic
110	C₂²×C₁₀₁	H(5,𝔽₂)	1	2	yes	—
111	C₂×D₁₀₁	—	2	4	no	—
112	C₂×D₁₀₁	—	2	4	no	—
113	C₂×D₁₀₁	—	2	8	no	—
114	C₂×D₁₀₁	—	2	8	no	—
115	C₂×D₁₀₁	—	2	4	no	—
116	C₂×D₁₀₁	—	2	4	no	—
117	C₂×D₁₀₁	—	2	4	no	—
118	C₂×D₁₀₁	—	4	32	no	—
119	C₂×D₁₀₁	—	4	4	no	—
120	C₂×D₁₀₁	—	4	4	no	—
121	C₂×D₁₀₁	C₂×C₂²⋊C₄	1	8	yes	complemented
122	C₄×C₁₀₁	—	2	2	no	—
123	C₄×C₁₀₁	—	2	4	no	—
124	C₄×C₁₀₁	—	4	2	no	—
125	C₄×C₁₀₁	C₂×C₂²⋊C₄	1	4	yes	complemented
126	Dic₁₀₁	—	4	4	no	—
127	C₂×C₁₀₁	—	2	4	no	—
128	C₂×C₁₀₁	C₂²×C₂²⋊C₄	1	1	yes	characteristic
129	C₂×C₁₀₁	C₂×C₄×D₄	1	1	yes	characteristic
130	C₂×C₁₀₁	#518	1	1	yes	characteristic
131	C₂×C₁₀₁	D₄²	1	2	yes	—
132	C₂×C₁₀₁	#546	1	2	yes	—
133	D₁₀₁	—	2	8	no	—
134	D₁₀₁	—	4	4	no	—
135	D₄×C₂²⋊C₄	—	101	1	no	maximal, Sylow
136	C₁₀₁	D₄×C₂²⋊C₄	1	1	yes	characteristic, complemented, Sylow
137	C₂³×D₄	—	101	1	no	—
138	C₂×C₄×D₄	—	101	2	no	—
139	C₂²×C₂²⋊C₄	—	101	2	no	—
140	#390	—	101	2	no	—
141	#386	—	101	4	no	—
142	#380	—	101	1	no	—
143	#379	—	101	2	no	—
144	C₄×C₂²⋊C₄	D₁₀₁	1	1	yes	characteristic, complemented
145	C₂⁵	—	101	2	no	—
146	C₂²×D₄	—	101	8	no	—
147	C₂²×D₄	—	101	1	no	—
148	C₂²×D₄	—	101	2	no	—
149	C₂²×D₄	—	101	1	no	—
150	C₂²×D₄	—	202	8	no	—
151	C₂³×C₄	—	101	4	no	—
152	C₂³×C₄	C₂×D₁₀₁	1	1	yes	characteristic
153	C₄×D₄	—	202	8	no	—
154	C₂×C₄⋊C₄	—	101	2	no	—
155	C₂×C₂²⋊C₄	—	101	4	no	—
156	C₂×C₂²⋊C₄	—	101	2	no	—
157	C₂×C₂²⋊C₄	—	101	4	no	—
158	C₂×C₂²⋊C₄	—	101	4	no	—
159	C₂×C₂²⋊C₄	—	202	8	no	—
160	C₂×C₂²⋊C₄	C₂×D₁₀₁	1	1	yes	characteristic
161	C₂×C₂²⋊C₄	C₂×D₁₀₁	1	1	yes	characteristic
162	C₂×C₄²	C₂×D₁₀₁	1	2	yes	complemented
163	#95	—	101	2	no	—
164	#95	C₂×D₁₀₁	1	2	yes	complemented
165	C₂⁴	—	101	2	no	—
166	C₂⁴	—	101	8	no	—
167	C₂⁴	—	101	2	no	—
168	C₂⁴	—	404	4	no	—
169	C₂⁴	—	202	8	no	—
170	C₂⁴	—	202	4	no	—
171	C₂⁴	—	202	4	no	—
172	C₂⁴	C₂²×D₁₀₁	1	1	yes	characteristic
173	C₂×D₄	—	101	4	no	—
174	C₂×D₄	—	101	8	no	—
175	C₂×D₄	—	101	4	no	—
176	C₂×D₄	—	101	8	no	—
177	C₂×D₄	—	202	32	no	—
178	C₂×D₄	—	202	8	no	—
179	C₂×D₄	—	202	4	no	—
180	C₂²×C₄	—	2	4	no	—
181	C₂²×C₄	—	101	4	no	—
182	C₂²×C₄	—	404	4	no	—
183	C₂²×C₄	—	202	8	no	—
184	C₂²×C₄	—	202	8	no	—
185	C₂²×C₄	C₄×D₁₀₁	1	4	yes	complemented
186	C₂²×C₄	C₂²×D₁₀₁	1	2	yes	—
187	C₂²×C₄	C₂²×D₁₀₁	1	1	yes	characteristic
188	C₂²×C₄	C₂²×D₁₀₁	1	2	yes	—
189	C₂²×C₄	C₂²×D₁₀₁	1	1	yes	characteristic
190	C₄⋊C₄	—	202	4	no	—
191	C₂²⋊C₄	—	2	2	no	—
192	C₂²⋊C₄	—	404	4	no	—
193	C₂²⋊C₄	—	202	8	no	—
194	C₂²⋊C₄	—	202	4	no	—
195	C₂²⋊C₄	—	202	8	no	—
196	C₂²⋊C₄	D₄₀₄	1	4	yes	direct product factor
197	C₄²	—	2	4	no	—
198	C₂³	—	2	2	no	—
199	C₂³	—	2	2	no	—
200	C₂³	—	101	4	no	—
201	C₂³	—	101	8	no	—
202	C₂³	—	101	4	no	—
203	C₂³	—	404	16	no	—
204	C₂³	—	404	16	no	—
205	C₂³	—	404	8	no	—
206	C₂³	—	202	16	no	—
207	C₂³	—	202	4	no	—
208	C₂³	—	202	2	no	—
209	C₂³	—	202	2	no	—
210	C₂³	—	202	4	no	—
211	C₂³	—	202	16	no	—
212	C₂³	—	202	2	no	—
213	C₂³	—	202	2	no	—
214	C₂³	—	202	4	no	—
215	C₂³	C₂×C₄×D₁₀₁	1	2	yes	—
216	C₂³	C₂×D₄₀₄	1	2	yes	—
217	C₂³	#13108074	1	2	yes	—
218	C₂³	C₂³×D₁₀₁	1	1	yes	characteristic
219	D₄	—	101	16	no	—
220	D₄	—	202	16	no	—
221	D₄	—	202	8	no	—
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	2	no	—
228	C₂×C₄	—	2	2	no	—
229	C₂×C₄	—	404	16	no	—
230	C₂×C₄	—	202	4	no	—
231	C₂×C₄	—	202	4	no	—
232	C₂×C₄	—	202	4	no	—
233	C₂×C₄	C₂×C₄×D₁₀₁	1	2	yes	—
234	C₂×C₄	C₂×C₄×D₁₀₁	1	2	yes	—
235	C₂×C₄	D₄×D₁₀₁	1	4	yes	—
236	C₂×C₄	C₂×D₄₀₄	1	4	yes	complemented
237	C₂²	—	2	4	no	—
238	C₂²	—	2	4	no	—
239	C₂²	—	2	2	no	—
240	C₂²	—	2	2	no	—
241	C₂²	—	101	8	no	—
242	C₂²	—	404	32	no	—
243	C₂²	—	404	4	no	—
244	C₂²	—	404	4	no	—
245	C₂²	—	202	4	no	—
246	C₂²	—	202	4	no	—
247	C₂²	—	202	4	no	—
248	C₂²	—	202	8	no	—
249	C₂²	—	202	8	no	—
250	C₂²	—	202	4	no	—
251	C₂²	—	202	4	no	—
252	C₂²	C₄×D₄₀₄	1	4	yes	complemented
253	C₂²	C₂²×C₄×D₁₀₁	1	1	yes	characteristic
254	C₂²	C₂×D₄×D₁₀₁	1	2	yes	—
255	C₂²	C₂²×D₄₀₄	1	1	yes	characteristic
256	C₂²	C₂×#13108074	1	1	yes	characteristic
257	C₂²	#15692301	1	2	yes	—
258	C₄	—	2	4	no	—
259	C₄	—	2	2	no	—
260	C₄	—	4	2	no	—
261	C₄	—	404	4	no	—
262	C₄	C₂²⋊C₄×D₁₀₁	1	4	yes	complemented
263	C₂	—	2	4	no	—
264	C₂	—	404	4	no	—
265	C₂	—	202	8	no	—
266	C₂	C₂×C₂²⋊C₄×D₁₀₁	1	1	yes	characteristic
267	C₂	C₂×C₄×D₄₀₄	1	1	yes	characteristic
268	C₂	#20265651	1	1	yes	characteristic
269	C₂	D₄×D₄₀₄	1	2	yes	—
270	C₂	#20265823	1	2	yes	—

	subgroup	G/H
center, Frattini	C₂³	C₂³×D₁₀₁
commutator	C₂²×C₁₀₁	C₂³×C₄
Fitting	C₄×C₂²⋊C₄×C₁₀₁	C₂
socle, cosocle	C₂³×C₁₀₁	C₂⁴
prime radical	C₂³×D₄₀₄	C₂
hypercenter	C₄×C₂²⋊C₄	D₁₀₁
hypocenter	C₁₀₁	D₄×C₂²⋊C₄
LCS	G ▷ C₂²×C₁₀₁ ▷ C₁₀₁
UCS	C₁ ◁ C₂³ ◁ C₄×C₂²⋊C₄

Complements

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

N	H ≅ G/N	type
2. C₂×C₄×D₄₀₄	C₂	semidirect
3. C₂×C₂²⋊C₄×D₁₀₁	C₂	semidirect
4. #20265079	C₂	semidirect
5. #20265055	C₂	semidirect
6. #20265037	C₂	semidirect
7. #20265032	C₂	semidirect
8. C₄×C₂²⋊C₄×C₁₀₁	C₂	semidirect
11. C₂²×D₄₀₄	C₄	semidirect
15. C₂²×C₄×D₁₀₁	C₂²	semidirect
17. C₂×C₄⋊Dic₁₀₁	C₂²	semidirect
19. C₂×#13108051	C₂²	semidirect
20. C₂×#13108051	C₂²	semidirect
21. C₂×#13108051	C₂²	semidirect
23. #15692162	C₂²	semidirect
27. C₂×C₄²×C₁₀₁	C₂²	semidirect
28. #95×C₁₀₁	C₂²	semidirect
33. C₂³×D₁₀₁	C₂×C₄	semidirect
40. C₂×D₄₀₄	C₂×C₄	semidirect
41. C₂×D₄₀₄	C₂×C₄	semidirect
54. C₂²×C₄×C₁₀₁	C₂×C₄	semidirect
60. C₂²⋊C₄×C₁₀₁	D₄	semidirect
93. C₂×C₄×C₁₀₁	C₂×D₄	semidirect
99. D₄₀₄	C₂²⋊C₄	direct (N × H)
105. C₂²×C₁₀₁	C₄×D₄	semidirect
121. C₂×D₁₀₁	C₂×C₂²⋊C₄	semidirect
125. C₄×C₁₀₁	C₂×C₂²⋊C₄	semidirect
136. C₁₀₁	D₄×C₂²⋊C₄	semidirect
144. C₄×C₂²⋊C₄	D₁₀₁	semidirect
162. C₂×C₄²	C₂×D₁₀₁	semidirect
164. #95	C₂×D₁₀₁	semidirect
185. C₂²×C₄	C₄×D₁₀₁	semidirect
196. C₂²⋊C₄	D₄₀₄	direct (N × H)
236. C₂×C₄	C₂×D₄₀₄	semidirect
252. C₂²	C₄×D₄₀₄	semidirect
262. C₄	C₂²⋊C₄×D₁₀₁	semidirect

Maximal subgroups

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

subgroup	index
C₂³×D₄₀₄	2
C₂×C₄×D₄₀₄	2
C₂×C₂²⋊C₄×D₁₀₁	2
#20265079	2
#20265055	2
#20265037	2
#20265032	2
C₄×C₂²⋊C₄×C₁₀₁	2
D₄×C₂²⋊C₄	101

Subgroups — C22⋊C4×D404 #27173955

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂²⋊C₄×D₄₀₄ #27173955