Subgroups — D₄×Dic₁₀₄ #16019649

Order 3328.

664 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	C₂²×Dic₁₀₄	C₂	1	2	yes	maximal, complemented
2	D₄×Dic₅₂	C₂	1	2	yes	maximal
3	#13117089	C₂	1	1	yes	maximal, characteristic, complemented
4	#13116927	C₂	1	2	yes	maximal, complemented
5	#13116887	C₂	1	2	yes	maximal, complemented
6	#13116772	C₂	1	4	yes	maximal, complemented
7	C₄×Dic₁₀₄	C₂	1	1	yes	maximal, characteristic, complemented
8	D₄×C₁₃×C₁₆	C₂	1	1	yes	maximal, characteristic
9	C₂²×Dic₅₂	C₂²	1	4	yes	—
10	D₄×Dic₂₆	—	2	2	no	—
11	C₂²×C₁₃×C₁₆	C₂²	1	2	yes	—
12	C₈×D₄×C₁₃	C₂²	1	1	yes	characteristic
13	#11718777	C₂²	1	4	yes	—
14	C₂×Dic₁₀₄	—	2	8	no	—
15	C₂×Dic₁₀₄	C₂²	1	4	yes	—
16	C₂×Dic₁₀₄	C₂²	1	2	yes	—
17	C₂×Dic₁₀₄	C₂²	1	2	yes	—
18	C₂×Dic₁₀₄	C₂²	1	1	yes	characteristic
19	#11718510	—	2	2	no	—
20	#11718400	—	2	4	no	—
21	#11718360	C₂²	1	2	yes	—
22	C₄×Dic₅₂	C₂²	1	2	yes	—
23	C₁₃×C₄⋊C₁₆	C₂²	1	1	yes	characteristic
24	C₁₃×C₂²⋊C₁₆	C₂²	1	2	yes	—
25	C₄×C₁₃×C₁₆	C₂²	1	1	yes	characteristic
26	#11718096	C₂²	1	1	yes	characteristic
27	#11718095	C₂²	1	2	yes	—
28	#11718095	C₂²	1	4	yes	—
29	C₂²×Dic₂₆	—	2	4	no	—
30	C₂²×C₈×C₁₃	C₂³	1	2	yes	—
31	C₄×D₄×C₁₃	D₄	1	1	yes	characteristic
32	D₄×Dic₁₃	—	4	2	no	—
33	#88178	—	2	4	no	—
34	C₂×Dic₅₂	—	2	8	no	—
35	C₂×Dic₅₂	—	2	4	no	—
36	C₂×Dic₅₂	—	2	2	no	—
37	C₂×Dic₅₂	—	4	4	no	—
38	C₂×Dic₅₂	C₂³	1	4	yes	—
39	C₂×Dic₅₂	C₂³	1	2	yes	—
40	#88142	—	4	2	no	—
41	#88132	—	4	4	no	—
42	#88123	—	2	2	no	—
43	C₄×Dic₂₆	—	2	2	no	—
44	C₂×C₁₃×C₁₆	—	2	4	no	—
45	C₂×C₁₃×C₁₆	C₂³	1	1	yes	characteristic
46	C₂×C₁₃×C₁₆	C₂³	1	2	yes	—
47	C₂×C₁₃×C₁₆	C₂³	1	1	yes	characteristic
48	C₁₃×C₄⋊C₈	D₄	1	1	yes	characteristic
49	C₁₃×C₂²⋊C₈	D₄	1	2	yes	—
50	C₄×C₈×C₁₃	C₂³	1	1	yes	characteristic
51	#88058	C₂³	1	2	yes	—
52	#88056	—	2	2	no	—
53	#88056	—	2	4	no	—
54	Dic₁₀₄	—	2	8	no	—
55	Dic₁₀₄	—	2	4	no	—
56	Dic₁₀₄	—	4	2	no	—
57	Dic₁₀₄	D₄	1	4	yes	direct product factor
58	D₄×Q₃₂	—	13	1	no	maximal, Sylow
59	C₂×D₄×C₁₃	D₈	1	1	yes	characteristic
60	C₂²×C₄×C₁₃	C₂×D₄	1	2	yes	—
61	C₂²×Dic₁₃	—	4	4	no	—
62	C₂×Dic₂₆	—	2	2	no	—
63	C₂×Dic₂₆	—	2	4	no	—
64	C₂×Dic₂₆	—	4	8	no	—
65	C₂×Dic₂₆	—	4	4	no	—
66	C₂×Dic₂₆	—	8	4	no	—
67	C₂×C₈×C₁₃	—	2	2	no	—
68	C₂×C₈×C₁₃	—	2	2	no	—
69	C₂×C₈×C₁₃	C₂×D₄	1	1	yes	characteristic
70	C₂×C₈×C₁₃	C₂×D₄	1	2	yes	—
71	C₂×C₈×C₁₃	C₂⁴	1	1	yes	characteristic
72	C₁₃×C₄⋊C₄	D₈	1	1	yes	characteristic
73	C₁₃×C₂²⋊C₄	D₈	1	2	yes	—
74	C₄²×C₁₃	C₂×D₄	1	1	yes	characteristic
75	C₂²⋊Dic₁₃	—	4	4	no	—
76	C₄⋊Dic₁₃	—	2	2	no	—
77	C₄⋊Dic₁₃	—	4	2	no	—
78	C₄⋊Dic₁₃	—	4	2	no	—
79	#6101	—	4	2	no	—
80	#6101	—	4	4	no	—
81	C₄×Dic₁₃	—	4	2	no	—
82	Dic₅₂	—	2	4	no	—
83	Dic₅₂	—	2	2	no	—
84	Dic₅₂	—	2	4	no	—
85	Dic₅₂	—	4	8	no	—
86	Dic₅₂	—	8	2	no	—
87	Dic₅₂	C₂×D₄	1	4	yes	—
88	C₁₃×C₁₆	—	2	2	no	—
89	C₁₃×C₁₆	—	2	1	no	—
90	C₁₃×C₁₆	C₂×D₄	1	2	yes	—
91	C₂²×Q₃₂	—	13	2	no	—
92	D₄×Q₁₆	—	13	2	no	—
93	#2247	—	13	1	no	—
94	#2216	—	13	2	no	—
95	#2209	—	13	2	no	—
96	#2188	—	13	4	no	—
97	C₄×Q₃₂	—	13	1	no	—
98	D₄×C₁₆	D₁₃	1	1	yes	characteristic
99	C₂³×C₁₃	C₂×D₈	1	2	yes	—
100	D₄×C₁₃	Q₃₂	1	4	yes	complemented
101	C₂×C₄×C₁₃	—	2	2	no	—
102	C₂×C₄×C₁₃	—	2	2	no	—
103	C₂×C₄×C₁₃	C₂×D₈	1	1	yes	characteristic
104	C₂×C₄×C₁₃	C₂×D₈	1	2	yes	—
105	C₂×C₄×C₁₃	C₂×D₈	1	1	yes	characteristic
106	C₂×C₄×C₁₃	C₂²×D₄	1	1	yes	characteristic
107	C₂×Dic₁₃	—	4	2	no	—
108	C₂×Dic₁₃	—	4	2	no	—
109	C₂×Dic₁₃	—	4	4	no	—
110	C₂×Dic₁₃	—	8	8	no	—
111	Dic₂₆	—	2	4	no	—
112	Dic₂₆	—	4	2	no	—
113	Dic₂₆	—	4	4	no	—
114	Dic₂₆	—	4	4	no	—
115	Dic₂₆	—	8	8	no	—
116	C₈×C₁₃	—	2	2	no	—
117	C₈×C₁₃	—	2	1	no	—
118	C₈×C₁₃	C₂²×D₄	1	1	yes	characteristic
119	C₈×C₁₃	H(5,𝔽₂)	1	1	yes	characteristic
120	C₂²×Q₁₆	—	13	4	no	—
121	D₄×Q₈	—	26	2	no	—
122	C₂×Q₃₂	—	13	1	no	—
123	C₂×Q₃₂	—	13	4	no	—
124	C₂×Q₃₂	—	13	2	no	—
125	C₂×Q₃₂	—	13	2	no	—
126	C₂×Q₃₂	—	26	8	no	—
127	C₂²×C₁₆	C₂×D₁₃	1	2	yes	—
128	#494	—	13	2	no	—
129	#467	—	13	4	no	—
130	#462	—	26	2	no	—
131	#451	—	26	4	no	—
132	C₄×Q₁₆	—	13	2	no	—
133	C₈×D₄	C₂×D₁₃	1	1	yes	characteristic
134	#366	—	13	1	no	—
135	C₄⋊C₁₆	C₂×D₁₃	1	1	yes	characteristic
136	#358	—	13	2	no	—
137	#358	—	13	4	no	—
138	C₂²⋊C₁₆	C₂×D₁₃	1	2	yes	—
139	C₄×C₁₆	C₂×D₁₃	1	1	yes	characteristic
140	C₂²×C₁₃	—	2	2	no	—
141	C₂²×C₁₃	—	2	2	no	—
142	C₂²×C₁₃	C₂×Q₃₂	1	4	yes	complemented
143	C₂²×C₁₃	C₂²×D₈	1	1	yes	characteristic
144	C₄×C₁₃	—	2	1	no	—
145	C₄×C₁₃	—	2	2	no	—
146	C₄×C₁₃	C₂×Q₃₂	1	2	yes	complemented
147	C₄×C₁₃	D₄²	1	1	yes	characteristic
148	C₄×C₁₃	#576	1	1	yes	characteristic
149	Dic₁₃	—	4	4	no	—
150	Dic₁₃	—	8	2	no	—
151	Dic₁₃	—	8	4	no	—
152	C₂²×Q₈	—	26	4	no	—
153	C₂×Q₁₆	—	13	2	no	—
154	C₂×Q₁₆	—	13	4	no	—
155	C₂×Q₁₆	—	26	8	no	—
156	C₂×Q₁₆	—	26	4	no	—
157	C₂×Q₁₆	—	26	2	no	—
158	C₂×Q₁₆	—	52	4	no	—
159	C₂²×C₈	C₂²×D₁₃	1	2	yes	—
160	C₄⋊Q₈	—	26	2	no	—
161	C₄⋊Q₈	—	52	2	no	—
162	C₂²⋊Q₈	—	26	4	no	—
163	C₂²⋊Q₈	—	52	4	no	—
164	C₄×Q₈	—	26	2	no	—
165	C₄×D₄	—	52	2	no	—
166	C₄×D₄	D₅₂	1	1	yes	characteristic
167	Q₃₂	—	13	4	no	—
168	Q₃₂	—	26	8	no	—
169	Q₃₂	—	26	4	no	—
170	Q₃₂	—	52	2	no	—
171	C₂×C₁₆	—	2	4	no	—
172	C₂×C₁₆	C₂²×D₁₃	1	1	yes	characteristic
173	C₂×C₁₆	C₂²×D₁₃	1	2	yes	—
174	C₂×C₁₆	C₂²×D₁₃	1	1	yes	characteristic
175	#107	—	13	2	no	—
176	C₄⋊C₈	D₅₂	1	1	yes	characteristic
177	#103	—	26	2	no	—
178	#103	—	26	4	no	—
179	C₂²⋊C₈	D₅₂	1	2	yes	—
180	C₄×C₈	C₂²×D₁₃	1	1	yes	characteristic
181	C₂×C₁₃	—	2	4	no	—
182	C₂×C₁₃	D₄×D₈	1	1	yes	characteristic
183	C₂×C₁₃	C₂²×Q₃₂	1	1	yes	characteristic
184	C₂×C₁₃	#3417	1	1	yes	characteristic
185	C₂×Q₈	—	26	2	no	—
186	C₂×Q₈	—	26	4	no	—
187	C₂×Q₈	—	52	8	no	—
188	C₂×Q₈	—	52	4	no	—
189	C₂×Q₈	—	104	4	no	—
190	C₂×D₄	D₁₀₄	1	1	yes	characteristic
191	C₂²×C₄	—	52	4	no	—
192	C₂²×C₄	C₂×D₅₂	1	2	yes	—
193	Q₁₆	—	13	4	no	—
194	Q₁₆	—	26	4	no	—
195	Q₁₆	—	26	2	no	—
196	Q₁₆	—	26	4	no	—
197	Q₁₆	—	52	8	no	—
198	Q₁₆	—	104	2	no	—
199	C₂×C₈	—	2	2	no	—
200	C₂×C₈	—	2	2	no	—
201	C₂×C₈	C₂×D₅₂	1	1	yes	characteristic
202	C₂×C₈	C₂×D₅₂	1	2	yes	—
203	C₂×C₈	C₂³×D₁₃	1	1	yes	characteristic
204	C₄⋊C₄	—	26	2	no	—
205	C₄⋊C₄	—	52	2	no	—
206	C₄⋊C₄	—	52	2	no	—
207	C₄⋊C₄	—	52	4	no	—
208	C₄⋊C₄	—	52	2	no	—
209	C₄⋊C₄	D₁₀₄	1	1	yes	characteristic
210	C₂²⋊C₄	—	52	4	no	—
211	C₂²⋊C₄	D₁₀₄	1	2	yes	—
212	C₄²	—	52	2	no	—
213	C₄²	C₂×D₅₂	1	1	yes	characteristic
214	C₁₆	—	2	2	no	—
215	C₁₆	—	2	1	no	—
216	C₁₆	D₄×D₁₃	1	2	yes	—
217	C₁₃	D₄×Q₃₂	1	1	yes	characteristic, complemented, Sylow
218	C₂³	C₂×D₁₀₄	1	2	yes	—
219	Q₈	—	26	4	no	—
220	Q₈	—	52	2	no	—
221	Q₈	—	52	4	no	—
222	Q₈	—	52	4	no	—
223	Q₈	—	104	8	no	—
224	D₄	Dic₁₀₄	1	4	yes	direct product factor
225	C₂×C₄	—	2	2	no	—
226	C₂×C₄	—	2	2	no	—
227	C₂×C₄	—	52	2	no	—
228	C₂×C₄	—	52	2	no	—
229	C₂×C₄	—	52	4	no	—
230	C₂×C₄	—	104	8	no	—
231	C₂×C₄	C₂×D₁₀₄	1	1	yes	characteristic
232	C₂×C₄	C₂×D₁₀₄	1	2	yes	—
233	C₂×C₄	C₂×D₁₀₄	1	1	yes	characteristic
234	C₂×C₄	C₂²×D₅₂	1	1	yes	characteristic
235	C₈	—	2	1	no	—
236	C₈	—	2	2	no	—
237	C₈	C₂×D₄×D₁₃	1	1	yes	characteristic
238	C₈	#88256	1	1	yes	characteristic
239	C₂²	—	2	2	no	—
240	C₂²	—	2	2	no	—
241	C₂²	C₂×Dic₁₀₄	1	4	yes	complemented
242	C₂²	C₂²×D₁₀₄	1	1	yes	characteristic
243	C₄	—	2	1	no	—
244	C₄	—	2	2	no	—
245	C₄	—	52	4	no	—
246	C₄	—	104	2	no	—
247	C₄	—	104	4	no	—
248	C₄	C₂×Dic₁₀₄	1	2	yes	complemented
249	C₄	D₄×D₅₂	1	1	yes	characteristic
250	C₄	#11719457	1	1	yes	characteristic
251	C₂	—	2	4	no	—
252	C₂	D₄×D₁₀₄	1	1	yes	characteristic
253	C₂	C₂²×Dic₁₀₄	1	1	yes	characteristic
254	C₂	#13127989	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂²×D₁₀₄
commutator, cosocle	C₂×C₈×C₁₃	C₂⁴
Frattini	C₂×C₈	C₂³×D₁₃
Fitting	D₄×C₁₃×C₁₆	C₂
socle	C₂²×C₁₃	C₂²×D₈
prime radical	C₂×D₄×C₁₃	D₈
hypercenter	D₄×C₁₆	D₁₃
hypocenter	C₁₃	D₄×Q₃₂
LCS	G ▷ C₂×C₈×C₁₃ ▷ C₄×C₁₃ ▷ C₂×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂² ◁ C₄×D₄ ◁ C₈×D₄ ◁ D₄×C₁₆

Complements

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

N	H ≅ G/N	type
1. C₂²×Dic₁₀₄	C₂	semidirect
3. #13117089	C₂	semidirect
4. #13116927	C₂	semidirect
5. #13116887	C₂	semidirect
6. #13116772	C₂	semidirect
7. C₄×Dic₁₀₄	C₂	semidirect
57. Dic₁₀₄	D₄	direct (N × H)
100. D₄×C₁₃	Q₃₂	semidirect
142. C₂²×C₁₃	C₂×Q₃₂	semidirect
146. C₄×C₁₃	C₂×Q₃₂	semidirect
217. C₁₃	D₄×Q₃₂	semidirect
224. D₄	Dic₁₀₄	direct (N × H)
241. C₂²	C₂×Dic₁₀₄	semidirect
248. C₄	C₂×Dic₁₀₄	semidirect

Maximal subgroups

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

subgroup	index
C₂²×Dic₁₀₄	2
D₄×Dic₅₂	2
#13117089	2
#13116927	2
#13116887	2
#13116772	2
C₄×Dic₁₀₄	2
D₄×C₁₃×C₁₆	2
D₄×Q₃₂	13

Subgroups — D4×Dic104 #16019649

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — D₄×Dic₁₀₄ #16019649