Subgroups — C₂×D₄×D₄₀₄ #27184572

Order 12928.

2536 conjugacy classes of subgroups.

The subgroup lattice has 280 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	2	yes	maximal, complemented
2	C₂²×D₄×D₁₀₁	C₂	1	2	yes	maximal, complemented
3	D₄×D₄₀₄	C₂	1	16	yes	maximal, direct product factor
4	C₂×#15692252	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂×#15692230	C₂	1	2	yes	maximal, complemented
6	C₂×#15692229	C₂	1	2	yes	maximal, complemented
7	C₂×#15692225	C₂	1	4	yes	maximal, complemented
8	C₂×C₄×D₄₀₄	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂×C₄×D₄×C₁₀₁	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂⁴×D₁₀₁	C₂²	1	4	yes	complemented
11	C₂²×D₄₀₄	—	2	16	no	—
12	C₂²×D₄₀₄	C₂²	1	1	yes	characteristic
13	C₂²×D₄₀₄	C₂²	1	2	yes	complemented
14	C₂²×D₄₀₄	C₂²	1	4	yes	complemented
15	C₂²×D₄₀₄	C₂²	1	2	yes	complemented
16	C₂²×D₄₀₄	C₂²	1	2	yes	complemented
17	C₂²×D₄₀₄	C₂²	1	16	yes	complemented
18	C₂²×C₂²⋊D₁₀₁	C₂²	1	4	yes	complemented
19	C₂×D₄×D₁₀₁	—	2	16	no	—
20	C₂×D₄×D₁₀₁	C₂²	1	16	yes	complemented
21	C₂²×C₄×D₁₀₁	C₂²	1	2	yes	complemented
22	#15692252	C₂²	1	8	yes	complemented
23	#15692230	C₂²	1	16	yes	complemented
24	#15692229	C₂²	1	16	yes	complemented
25	#15692225	C₂²	1	32	yes	complemented
26	C₄×D₄₀₄	C₂²	1	8	yes	complemented
27	C₂×C₄⋊Dic₁₀₁	C₂²	1	1	yes	characteristic, complemented
28	C₂×#13108051	C₂²	1	2	yes	complemented
29	C₂×#13108051	C₂²	1	4	yes	complemented
30	C₂²×D₄×C₁₀₁	C₂²	1	1	yes	characteristic
31	C₂³×C₄×C₁₀₁	C₂²	1	2	yes	complemented
32	C₄×D₄×C₁₀₁	C₂²	1	8	yes	complemented
33	C₂×C₄⋊C₄×C₁₀₁	C₂²	1	1	yes	characteristic, complemented
34	C₂×C₂²⋊C₄×C₁₀₁	C₂²	1	2	yes	complemented
35	C₂×C₄²×C₁₀₁	C₂²	1	1	yes	characteristic, complemented
36	C₂³×D₁₀₁	—	2	16	no	—
37	C₂³×D₁₀₁	—	2	16	no	—
38	C₂³×D₁₀₁	—	4	8	no	—
39	C₂³×D₁₀₁	C₂³	1	4	yes	—
40	C₂³×D₁₀₁	C₂³	1	2	yes	—
41	C₂³×D₁₀₁	C₂³	1	16	yes	complemented
42	C₂×D₄₀₄	—	2	16	no	—
43	C₂×D₄₀₄	—	2	8	no	—
44	C₂×D₄₀₄	—	2	8	no	—
45	C₂×D₄₀₄	—	2	64	no	—
46	C₂×D₄₀₄	—	4	4	no	—
47	C₂×D₄₀₄	D₄	1	8	yes	direct product factor
48	C₂×D₄₀₄	C₂³	1	4	yes	—
49	C₂×D₄₀₄	C₂³	1	8	yes	complemented
50	C₂×D₄₀₄	C₂³	1	16	yes	complemented
51	C₂×D₄₀₄	C₂³	1	8	yes	complemented
52	C₂×D₄₀₄	C₂³	1	8	yes	complemented
53	C₂×C₂²⋊D₁₀₁	—	2	16	no	—
54	C₂×C₂²⋊D₁₀₁	C₂³	1	16	yes	complemented
55	D₄×D₁₀₁	—	2	64	no	—
56	C₂²×Dic₁₀₁	C₂³	1	2	yes	—
57	C₂×C₄×D₁₀₁	—	2	8	no	—
58	C₂×C₄×D₁₀₁	C₂³	1	8	yes	complemented
59	C₄⋊Dic₁₀₁	C₂³	1	4	yes	complemented
60	#13108051	C₂³	1	8	yes	complemented
61	#13108051	C₂³	1	16	yes	complemented
62	C₂⁴×C₁₀₁	C₂³	1	2	yes	—
63	C₂×D₄×C₁₀₁	D₄	1	8	yes	complemented
64	C₂×D₄×C₁₀₁	C₂³	1	4	yes	—
65	C₂²×C₄×C₁₀₁	—	2	8	no	—
66	C₂²×C₄×C₁₀₁	C₂³	1	1	yes	characteristic
67	C₂²×C₄×C₁₀₁	C₂³	1	2	yes	—
68	C₂²×C₄×C₁₀₁	C₂³	1	1	yes	characteristic
69	C₂²×C₄×C₁₀₁	C₂³	1	1	yes	characteristic
70	C₂²×C₄×C₁₀₁	C₂³	1	8	yes	complemented
71	C₄⋊C₄×C₁₀₁	C₂³	1	4	yes	complemented
72	C₂²⋊C₄×C₁₀₁	C₂³	1	8	yes	complemented
73	C₄²×C₁₀₁	C₂³	1	4	yes	complemented
74	C₂³×C₁₀₁	—	2	4	no	—
75	C₂³×C₁₀₁	—	2	4	no	—
76	C₂³×C₁₀₁	C₂×D₄	1	8	yes	complemented
77	C₂³×C₁₀₁	C₂⁴	1	4	yes	—
78	C₂³×C₁₀₁	C₂⁴	1	1	yes	characteristic
79	C₂²×D₁₀₁	—	2	4	no	—
80	C₂²×D₁₀₁	—	2	4	no	—
81	C₂²×D₁₀₁	—	2	8	no	—
82	C₂²×D₁₀₁	—	2	8	no	—
83	C₂²×D₁₀₁	—	2	32	no	—
84	C₂²×D₁₀₁	—	2	8	no	—
85	C₂²×D₁₀₁	—	2	32	no	—
86	C₂²×D₁₀₁	—	4	16	no	—
87	C₂²×D₁₀₁	—	4	64	no	—
88	C₂²×D₁₀₁	C₂×D₄	1	8	yes	complemented
89	C₂²×D₁₀₁	C₂⁴	1	8	yes	—
90	C₂²×D₁₀₁	C₂⁴	1	4	yes	—
91	D₄×C₁₀₁	C₂×D₄	1	16	yes	complemented
92	C₂×C₄×C₁₀₁	—	2	4	no	—
93	C₂×C₄×C₁₀₁	—	2	2	no	—
94	C₂×C₄×C₁₀₁	—	2	16	no	—
95	C₂×C₄×C₁₀₁	C₂×D₄	1	4	yes	complemented
96	C₂×C₄×C₁₀₁	C₂×D₄	1	4	yes	complemented
97	C₂×C₄×C₁₀₁	C₂⁴	1	2	yes	—
98	C₂×C₄×C₁₀₁	C₂⁴	1	4	yes	—
99	C₂×C₄×C₁₀₁	C₂⁴	1	2	yes	—
100	C₂×C₄×C₁₀₁	C₂⁴	1	2	yes	—
101	C₂²⋊D₁₀₁	—	2	32	no	—
102	C₂×Dic₁₀₁	—	2	4	no	—
103	C₂×Dic₁₀₁	C₂⁴	1	4	yes	—
104	D₄₀₄	—	2	32	no	—
105	D₄₀₄	—	2	16	no	—
106	D₄₀₄	—	2	16	no	—
107	D₄₀₄	—	4	8	no	—
108	D₄₀₄	C₂×D₄	1	16	yes	direct product factor
109	C₄×D₁₀₁	—	2	16	no	—
110	C₂²×C₁₀₁	—	2	16	no	—
111	C₂²×C₁₀₁	—	2	4	no	—
112	C₂²×C₁₀₁	—	2	4	no	—
113	C₂²×C₁₀₁	C₂²×D₄	1	2	yes	—
114	C₂²×C₁₀₁	C₂²×D₄	1	2	yes	—
115	C₂²×C₁₀₁	C₂²×D₄	1	8	yes	complemented
116	C₂²×C₁₀₁	H(5,𝔽₂)	1	2	yes	—
117	C₂²×C₁₀₁	C₂⁵	1	1	yes	characteristic
118	C₂×D₁₀₁	—	2	16	no	—
119	C₂×D₁₀₁	—	2	4	no	—
120	C₂×D₁₀₁	—	2	4	no	—
121	C₂×D₁₀₁	—	2	8	no	—
122	C₂×D₁₀₁	—	2	8	no	—
123	C₂×D₁₀₁	—	2	8	no	—
124	C₂×D₁₀₁	—	4	64	no	—
125	C₂×D₁₀₁	—	4	16	no	—
126	C₂×D₁₀₁	C₂²×D₄	1	8	yes	complemented
127	C₄×C₁₀₁	—	2	4	no	—
128	C₄×C₁₀₁	—	2	2	no	—
129	C₄×C₁₀₁	C₂²×D₄	1	4	yes	complemented
130	C₄×C₁₀₁	C₂²×D₄	1	4	yes	complemented
131	Dic₁₀₁	—	2	4	no	—
132	C₂×C₁₀₁	—	2	8	no	—
133	C₂×C₁₀₁	D₄²	1	4	yes	complemented
134	C₂×C₁₀₁	C₂³×D₄	1	1	yes	characteristic
135	C₂×C₁₀₁	C₂³×D₄	1	1	yes	characteristic
136	C₂×C₁₀₁	C₂×H(5,𝔽₂)	1	1	yes	characteristic
137	D₁₀₁	—	2	8	no	—
138	D₁₀₁	—	4	8	no	—
139	C₂×D₄²	—	101	1	no	maximal, Sylow
140	C₁₀₁	C₂×D₄²	1	1	yes	characteristic, complemented, Sylow
141	C₂³×D₄	—	101	2	no	—
142	C₂³×D₄	—	101	2	no	—
143	D₄²	—	101	16	no	—
144	C₂×#127	—	101	1	no	—
145	C₂×#121	—	101	2	no	—
146	C₂×#121	—	101	2	no	—
147	C₂×#120	—	101	4	no	—
148	C₂×C₄×D₄	—	101	1	no	—
149	C₂×C₄×D₄	D₁₀₁	1	1	yes	characteristic, complemented
150	C₂⁵	—	101	4	no	—
151	C₂²×D₄	—	101	4	no	—
152	C₂²×D₄	—	101	16	no	—
153	C₂²×D₄	—	101	1	no	—
154	C₂²×D₄	—	101	2	no	—
155	C₂²×D₄	—	101	4	no	—
156	C₂²×D₄	—	101	2	no	—
157	C₂²×D₄	—	101	2	no	—
158	C₂²×D₄	—	101	16	no	—
159	C₂²×D₄	—	202	16	no	—
160	C₂²×D₄	—	202	16	no	—
161	C₂²×D₄	C₂×D₁₀₁	1	1	yes	characteristic
162	C₂³×C₄	—	101	2	no	—
163	C₂³×C₄	C₂×D₁₀₁	1	2	yes	complemented
164	#127	—	101	8	no	—
165	#121	—	101	16	no	—
166	#121	—	101	16	no	—
167	#120	—	101	32	no	—
168	C₄×D₄	—	101	8	no	—
169	C₄×D₄	C₂×D₁₀₁	1	8	yes	complemented
170	C₂×C₄⋊C₄	—	101	1	no	—
171	C₂×C₄⋊C₄	C₂×D₁₀₁	1	1	yes	characteristic, complemented
172	C₂×C₂²⋊C₄	—	101	2	no	—
173	C₂×C₂²⋊C₄	—	101	4	no	—
174	C₂×C₂²⋊C₄	C₂×D₁₀₁	1	2	yes	complemented
175	C₂×C₄²	C₂×D₁₀₁	1	1	yes	characteristic, complemented
176	C₂⁴	—	101	2	no	—
177	C₂⁴	—	101	16	no	—
178	C₂⁴	—	101	4	no	—
179	C₂⁴	—	404	8	no	—
180	C₂⁴	—	202	16	no	—
181	C₂⁴	—	202	16	no	—
182	C₂⁴	C₂²×D₁₀₁	1	2	yes	—
183	C₂×D₄	—	101	16	no	—
184	C₂×D₄	—	101	8	no	—
185	C₂×D₄	—	101	4	no	—
186	C₂×D₄	—	101	8	no	—
187	C₂×D₄	—	101	16	no	—
188	C₂×D₄	—	101	8	no	—
189	C₂×D₄	—	101	8	no	—
190	C₂×D₄	—	404	4	no	—
191	C₂×D₄	—	202	64	no	—
192	C₂×D₄	—	202	16	no	—
193	C₂×D₄	—	202	16	no	—
194	C₂×D₄	—	202	64	no	—
195	C₂×D₄	—	202	8	no	—
196	C₂×D₄	—	202	8	no	—
197	C₂×D₄	D₄₀₄	1	8	yes	direct product factor
198	C₂×D₄	C₂²×D₁₀₁	1	4	yes	—
199	C₂²×C₄	—	2	8	no	—
200	C₂²×C₄	—	101	2	no	—
201	C₂²×C₄	—	101	8	no	—
202	C₂²×C₄	—	202	8	no	—
203	C₂²×C₄	C₂²×D₁₀₁	1	1	yes	characteristic
204	C₂²×C₄	C₂²×D₁₀₁	1	2	yes	—
205	C₂²×C₄	C₂²×D₁₀₁	1	1	yes	characteristic
206	C₂²×C₄	C₂²×D₁₀₁	1	1	yes	characteristic
207	C₂²×C₄	C₂²×D₁₀₁	1	8	yes	complemented
208	C₄⋊C₄	—	101	4	no	—
209	C₄⋊C₄	C₂²×D₁₀₁	1	4	yes	complemented
210	C₂²⋊C₄	—	101	8	no	—
211	C₂²⋊C₄	—	101	16	no	—
212	C₂²⋊C₄	C₂²×D₁₀₁	1	8	yes	complemented
213	C₄²	C₂²×D₁₀₁	1	4	yes	complemented
214	C₂³	—	2	4	no	—
215	C₂³	—	2	4	no	—
216	C₂³	—	101	8	no	—
217	C₂³	—	101	4	no	—
218	C₂³	—	101	8	no	—
219	C₂³	—	404	64	no	—
220	C₂³	—	404	16	no	—
221	C₂³	—	202	32	no	—
222	C₂³	—	202	8	no	—
223	C₂³	—	202	8	no	—
224	C₂³	—	202	32	no	—
225	C₂³	—	202	8	no	—
226	C₂³	—	202	4	no	—
227	C₂³	—	202	4	no	—
228	C₂³	C₂×D₄₀₄	1	8	yes	complemented
229	C₂³	C₂³×D₁₀₁	1	4	yes	—
230	C₂³	C₂³×D₁₀₁	1	1	yes	characteristic
231	D₄	—	101	16	no	—
232	D₄	—	404	8	no	—
233	D₄	—	202	32	no	—
234	D₄	—	202	32	no	—
235	D₄	—	202	16	no	—
236	D₄	—	202	16	no	—
237	D₄	C₂×D₄₀₄	1	16	yes	direct product factor
238	C₂×C₄	—	2	4	no	—
239	C₂×C₄	—	2	2	no	—
240	C₂×C₄	—	2	16	no	—
241	C₂×C₄	—	101	4	no	—
242	C₂×C₄	—	202	4	no	—
243	C₂×C₄	—	202	16	no	—
244	C₂×C₄	D₄×D₁₀₁	1	4	yes	complemented
245	C₂×C₄	C₂×D₄₀₄	1	4	yes	complemented
246	C₂×C₄	C₂³×D₁₀₁	1	2	yes	—
247	C₂×C₄	C₂³×D₁₀₁	1	4	yes	—
248	C₂×C₄	C₂³×D₁₀₁	1	2	yes	—
249	C₂×C₄	C₂³×D₁₀₁	1	2	yes	—
250	C₂²	—	2	4	no	—
251	C₂²	—	2	4	no	—
252	C₂²	—	2	16	no	—
253	C₂²	—	101	8	no	—
254	C₂²	—	404	64	no	—
255	C₂²	—	404	16	no	—
256	C₂²	—	202	8	no	—
257	C₂²	—	202	8	no	—
258	C₂²	—	202	8	no	—
259	C₂²	—	202	4	no	—
260	C₂²	—	202	4	no	—
261	C₂²	—	202	16	no	—
262	C₂²	C₂×D₄×D₁₀₁	1	2	yes	—
263	C₂²	C₂²×D₄₀₄	1	2	yes	—
264	C₂²	C₂²×D₄₀₄	1	8	yes	complemented
265	C₂²	#15692301	1	2	yes	—
266	C₂²	C₂⁴×D₁₀₁	1	1	yes	characteristic
267	C₄	—	2	4	no	—
268	C₄	—	2	2	no	—
269	C₄	—	202	4	no	—
270	C₄	C₂×D₄×D₁₀₁	1	4	yes	complemented
271	C₄	C₂²×D₄₀₄	1	4	yes	complemented
272	C₂	—	2	8	no	—
273	C₂	—	404	8	no	—
274	C₂	—	202	8	no	—
275	C₂	D₄×D₄₀₄	1	4	yes	direct product factor
276	C₂	C₂²×D₄×D₁₀₁	1	1	yes	characteristic
277	C₂	C₂³×D₄₀₄	1	1	yes	characteristic
278	C₂	C₂×#15692301	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	C₂³×D₁₀₁
commutator, cosocle	C₂²×C₁₀₁	C₂⁵
Frattini	C₂²	C₂⁴×D₁₀₁
Fitting	C₂×C₄×D₄×C₁₀₁	C₂
socle	C₂³×C₁₀₁	C₂⁴
hypercenter	C₂×C₄×D₄	D₁₀₁
hypocenter	C₁₀₁	C₂×D₄²
LCS	G ▷ C₂²×C₁₀₁ ▷ C₁₀₁
UCS	C₁ ◁ C₂³ ◁ C₂×C₄×D₄

Complements

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

N	H ≅ G/N	type
1. C₂³×D₄₀₄	C₂	semidirect
2. C₂²×D₄×D₁₀₁	C₂	semidirect
3. D₄×D₄₀₄	C₂	direct (N × H)
4. C₂×#15692252	C₂	semidirect
5. C₂×#15692230	C₂	semidirect
6. C₂×#15692229	C₂	semidirect
7. C₂×#15692225	C₂	semidirect
8. C₂×C₄×D₄₀₄	C₂	semidirect
9. C₂×C₄×D₄×C₁₀₁	C₂	semidirect
10. C₂⁴×D₁₀₁	C₂²	semidirect
13. C₂²×D₄₀₄	C₂²	semidirect
14. C₂²×D₄₀₄	C₂²	semidirect
15. C₂²×D₄₀₄	C₂²	semidirect
16. C₂²×D₄₀₄	C₂²	semidirect
17. C₂²×D₄₀₄	C₂²	semidirect
18. C₂²×C₂²⋊D₁₀₁	C₂²	semidirect
20. C₂×D₄×D₁₀₁	C₂²	semidirect
21. C₂²×C₄×D₁₀₁	C₂²	semidirect
22. #15692252	C₂²	semidirect
23. #15692230	C₂²	semidirect
24. #15692229	C₂²	semidirect
25. #15692225	C₂²	semidirect
26. C₄×D₄₀₄	C₂²	semidirect
27. C₂×C₄⋊Dic₁₀₁	C₂²	semidirect
28. C₂×#13108051	C₂²	semidirect
29. C₂×#13108051	C₂²	semidirect
31. C₂³×C₄×C₁₀₁	C₂²	semidirect
32. C₄×D₄×C₁₀₁	C₂²	semidirect
33. C₂×C₄⋊C₄×C₁₀₁	C₂²	semidirect
34. C₂×C₂²⋊C₄×C₁₀₁	C₂²	semidirect
35. C₂×C₄²×C₁₀₁	C₂²	semidirect
41. C₂³×D₁₀₁	C₂³	semidirect
47. C₂×D₄₀₄	D₄	direct (N × H)
49. C₂×D₄₀₄	C₂³	semidirect
50. C₂×D₄₀₄	C₂³	semidirect
51. C₂×D₄₀₄	C₂³	semidirect
52. C₂×D₄₀₄	C₂³	semidirect
54. C₂×C₂²⋊D₁₀₁	C₂³	semidirect
58. C₂×C₄×D₁₀₁	C₂³	semidirect
59. C₄⋊Dic₁₀₁	C₂³	semidirect
60. #13108051	C₂³	semidirect
61. #13108051	C₂³	semidirect
63. C₂×D₄×C₁₀₁	D₄	semidirect
70. C₂²×C₄×C₁₀₁	C₂³	semidirect
71. C₄⋊C₄×C₁₀₁	C₂³	semidirect
72. C₂²⋊C₄×C₁₀₁	C₂³	semidirect
73. C₄²×C₁₀₁	C₂³	semidirect
76. C₂³×C₁₀₁	C₂×D₄	semidirect
88. C₂²×D₁₀₁	C₂×D₄	semidirect
91. D₄×C₁₀₁	C₂×D₄	semidirect
95. C₂×C₄×C₁₀₁	C₂×D₄	semidirect
96. C₂×C₄×C₁₀₁	C₂×D₄	semidirect
108. D₄₀₄	C₂×D₄	direct (N × H)
115. C₂²×C₁₀₁	C₂²×D₄	semidirect
126. C₂×D₁₀₁	C₂²×D₄	semidirect
129. C₄×C₁₀₁	C₂²×D₄	semidirect
130. C₄×C₁₀₁	C₂²×D₄	semidirect
133. C₂×C₁₀₁	D₄²	semidirect
140. C₁₀₁	C₂×D₄²	semidirect
149. C₂×C₄×D₄	D₁₀₁	semidirect
163. C₂³×C₄	C₂×D₁₀₁	semidirect
169. C₄×D₄	C₂×D₁₀₁	semidirect
171. C₂×C₄⋊C₄	C₂×D₁₀₁	semidirect
174. C₂×C₂²⋊C₄	C₂×D₁₀₁	semidirect
175. C₂×C₄²	C₂×D₁₀₁	semidirect
197. C₂×D₄	D₄₀₄	direct (N × H)
207. C₂²×C₄	C₂²×D₁₀₁	semidirect
209. C₄⋊C₄	C₂²×D₁₀₁	semidirect
212. C₂²⋊C₄	C₂²×D₁₀₁	semidirect
213. C₄²	C₂²×D₁₀₁	semidirect
228. C₂³	C₂×D₄₀₄	semidirect
237. D₄	C₂×D₄₀₄	direct (N × H)
244. C₂×C₄	D₄×D₁₀₁	semidirect
245. C₂×C₄	C₂×D₄₀₄	semidirect
264. C₂²	C₂²×D₄₀₄	semidirect
270. C₄	C₂×D₄×D₁₀₁	semidirect
271. C₄	C₂²×D₄₀₄	semidirect
275. C₂	D₄×D₄₀₄	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂³×D₄₀₄	2
C₂²×D₄×D₁₀₁	2
D₄×D₄₀₄	2
C₂×#15692252	2
C₂×#15692230	2
C₂×#15692229	2
C₂×#15692225	2
C₂×C₄×D₄₀₄	2
C₂×C₄×D₄×C₁₀₁	2
C₂×D₄²	101

Subgroups — C2×D4×D404 #27184572

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×D₄×D₄₀₄ #27184572