Subgroups — C₂×D₄⋊#6090 #15772618

Order 3328.

470 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	C₂×C₄⋊#6090	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×C₄×#6090	C₂	1	1	yes	maximal, characteristic, complemented
3	D₄⋊#6090	C₂	1	4	yes	maximal, direct product factor
4	C₂×C₈×D₄×C₁₃	C₂	1	1	yes	maximal, characteristic
5	C₂³×C₈×C₁₃	—	2	1	no	—
6	C₂×C₄×D₄×C₁₃	C₄	1	1	yes	characteristic
7	C₈×D₄×C₁₃	—	2	1	no	—
8	C₈×D₄×C₁₃	—	2	1	no	—
9	C₈×D₄×C₁₃	C₄	1	2	yes	—
10	C₈×D₄×C₁₃	C₂²	1	2	yes	—
11	C₂×C₁₃×C₄⋊C₈	C₄	1	1	yes	characteristic
12	C₂×C₁₃×C₂²⋊C₈	—	2	1	no	—
13	C₂×C₄×C₈×C₁₃	C₂²	1	1	yes	characteristic
14	C₂²×#6090	—	2	1	no	—
15	C₂²×#6090	—	2	1	no	—
16	C₄⋊#6090	—	2	1	no	—
17	C₄⋊#6090	C₂²	1	2	yes	complemented
18	C₄×#6090	—	2	1	no	—
19	C₄×#6090	C₂²	1	2	yes	complemented
20	C₂²×D₄×C₁₃	C₈	1	1	yes	characteristic
21	C₂³×C₄×C₁₃	—	2	1	no	—
22	C₂²×C₈×C₁₃	—	2	1	no	—
23	C₂²×C₈×C₁₃	—	2	2	no	—
24	C₂²×C₈×C₁₃	—	2	2	no	—
25	C₂²×C₈×C₁₃	—	4	2	no	—
26	C₂²×C₈×C₁₃	—	4	2	no	—
27	C₂²×C₈×C₁₃	D₄	1	1	yes	characteristic
28	C₂²×C₈×C₁₃	D₄	1	1	yes	characteristic
29	C₄×D₄×C₁₃	—	2	1	no	—
30	C₄×D₄×C₁₃	—	2	1	no	—
31	C₄×D₄×C₁₃	C₈	1	2	yes	—
32	C₄×D₄×C₁₃	C₂×C₄	1	2	yes	—
33	C₂×C₁₃×C₄⋊C₄	C₈	1	1	yes	characteristic
34	C₂×C₁₃×C₂²⋊C₄	—	2	1	no	—
35	C₂×C₄²×C₁₃	C₂×C₄	1	1	yes	characteristic
36	C₁₃×C₄⋊C₈	—	2	1	no	—
37	C₁₃×C₄⋊C₈	C₂×C₄	1	2	yes	—
38	C₁₃×C₂²⋊C₈	—	2	2	no	—
39	C₁₃×C₂²⋊C₈	—	2	2	no	—
40	C₄×C₈×C₁₃	C₂×C₄	1	1	yes	characteristic
41	C₄×C₈×C₁₃	D₄	1	1	yes	characteristic
42	C₄×C₈×C₁₃	D₄	1	1	yes	characteristic
43	C₄×C₈×C₁₃	C₂³	1	1	yes	characteristic
44	C₂×#6090	—	2	4	no	—
45	C₂×#6090	—	2	2	no	—
46	C₂×#6090	—	2	2	no	—
47	C₂×#6090	—	4	2	no	—
48	C₂×D₄⋊C₁₆	—	13	1	no	maximal, Sylow
49	C₂⁴×C₁₃	—	2	1	no	—
50	C₂×D₄×C₁₃	—	2	1	no	—
51	C₂×D₄×C₁₃	—	2	2	no	—
52	C₂×D₄×C₁₃	C₁₆	1	4	yes	complemented
53	C₂×D₄×C₁₃	C₂×C₈	1	2	yes	—
54	C₂²×C₄×C₁₃	—	2	1	no	—
55	C₂²×C₄×C₁₃	—	2	2	no	—
56	C₂²×C₄×C₁₃	—	2	2	no	—
57	C₂²×C₄×C₁₃	—	4	2	no	—
58	C₂²×C₄×C₁₃	—	4	2	no	—
59	C₂²×C₄×C₁₃	C₂²⋊C₄	1	1	yes	characteristic
60	C₂²×C₄×C₁₃	C₂×C₈	1	1	yes	characteristic
61	C₂²×C₄×C₁₃	M₁₆	1	1	yes	characteristic
62	C₂×C₈×C₁₃	—	2	2	no	—
63	C₂×C₈×C₁₃	—	2	2	no	—
64	C₂×C₈×C₁₃	—	4	2	no	—
65	C₂×C₈×C₁₃	—	4	4	no	—
66	C₂×C₈×C₁₃	—	4	2	no	—
67	C₂×C₈×C₁₃	—	4	2	no	—
68	C₂×C₈×C₁₃	C₂²⋊C₄	1	1	yes	characteristic
69	C₂×C₈×C₁₃	C₂²⋊C₄	1	1	yes	characteristic
70	C₂×C₈×C₁₃	D₈	1	2	yes	—
71	C₂×C₈×C₁₃	SD₁₆	1	2	yes	—
72	C₂×C₈×C₁₃	C₂×D₄	1	1	yes	characteristic
73	C₂×C₈×C₁₃	C₂×D₄	1	1	yes	characteristic
74	C₁₃×C₄⋊C₄	—	2	1	no	—
75	C₁₃×C₄⋊C₄	C₂×C₈	1	2	yes	—
76	C₁₃×C₂²⋊C₄	—	2	2	no	—
77	C₁₃×C₂²⋊C₄	—	2	2	no	—
78	C₄²×C₁₃	C₂²⋊C₄	1	1	yes	characteristic
79	C₄²×C₁₃	C₂×C₈	1	1	yes	characteristic
80	C₄²×C₁₃	M₁₆	1	1	yes	characteristic
81	C₄²×C₁₃	C₂²×C₄	1	1	yes	characteristic
82	#6090	—	2	4	no	—
83	#6090	—	4	2	no	—
84	C₂×C₈×D₄	D₁₃	1	1	yes	characteristic
85	C₂×C₄⋊C₁₆	—	13	1	no	—
86	C₂×C₄×C₁₆	—	13	1	no	—
87	D₄⋊C₁₆	—	13	4	no	—
88	C₂³×C₁₃	—	2	4	no	—
89	C₂³×C₁₃	—	2	2	no	—
90	C₂³×C₁₃	—	4	2	no	—
91	C₂³×C₁₃	—	4	2	no	—
92	C₂³×C₁₃	C₂²⋊C₈	1	1	yes	characteristic
93	D₄×C₁₃	—	2	4	no	—
94	D₄×C₁₃	—	2	2	no	—
95	D₄×C₁₃	C₂×C₁₆	1	4	yes	complemented
96	C₂×C₄×C₁₃	—	2	2	no	—
97	C₂×C₄×C₁₃	—	2	2	no	—
98	C₂×C₄×C₁₃	—	4	2	no	—
99	C₂×C₄×C₁₃	—	4	4	no	—
100	C₂×C₄×C₁₃	—	4	2	no	—
101	C₂×C₄×C₁₃	—	4	2	no	—
102	C₂×C₄×C₁₃	C₂²⋊C₈	1	1	yes	characteristic
103	C₂×C₄×C₁₃	C₂²⋊C₈	1	1	yes	characteristic
104	C₂×C₄×C₁₃	C₂²⋊C₈	1	1	yes	characteristic
105	C₂×C₄×C₁₃	#102	1	2	yes	—
106	C₂×C₄×C₁₃	#104	1	2	yes	—
107	C₂×C₄×C₁₃	C₂×C₁₆	1	2	yes	—
108	C₂×C₄×C₁₃	M₃₂	1	2	yes	—
109	C₂×C₄×C₁₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
110	C₂×C₄×C₁₃	C₂²×C₈	1	1	yes	characteristic
111	C₂×C₄×C₁₃	C₂×M₁₆	1	1	yes	characteristic
112	C₈×C₁₃	—	2	1	no	—
113	C₈×C₁₃	—	2	1	no	—
114	C₈×C₁₃	—	4	2	no	—
115	C₈×C₁₃	#102	1	2	yes	—
116	C₈×C₁₃	C₂×D₈	1	1	yes	characteristic
117	C₈×C₁₃	C₂×SD₁₆	1	1	yes	characteristic
118	C₂³×C₈	—	2	1	no	—
119	C₂×C₄×D₄	Dic₁₃	1	1	yes	characteristic
120	C₂²×C₁₆	—	26	1	no	—
121	C₂²×C₁₆	—	26	1	no	—
122	C₈×D₄	—	2	1	no	—
123	C₈×D₄	—	2	1	no	—
124	C₈×D₄	Dic₁₃	1	2	yes	—
125	C₈×D₄	C₂×D₁₃	1	2	yes	—
126	C₂×C₄⋊C₈	Dic₁₃	1	1	yes	characteristic
127	C₂×C₂²⋊C₈	—	2	1	no	—
128	C₂×C₄×C₈	C₂×D₁₃	1	1	yes	characteristic
129	C₄⋊C₁₆	—	13	2	no	—
130	C₄⋊C₁₆	—	26	1	no	—
131	C₄×C₁₆	—	13	2	no	—
132	C₄×C₁₆	—	26	1	no	—
133	C₂²×C₁₃	—	2	4	no	—
134	C₂²×C₁₃	—	4	8	no	—
135	C₂²×C₁₃	—	4	2	no	—
136	C₂²×C₁₃	—	4	2	no	—
137	C₂²×C₁₃	D₄⋊C₈	1	2	yes	—
138	C₂²×C₁₃	C₂²⋊C₁₆	1	2	yes	—
139	C₂²×C₁₃	#350	1	2	yes	—
140	C₂²×C₁₃	C₂×C₂²⋊C₈	1	1	yes	characteristic
141	C₄×C₁₃	—	2	1	no	—
142	C₄×C₁₃	—	2	1	no	—
143	C₄×C₁₃	—	4	2	no	—
144	C₄×C₁₃	D₄⋊C₈	1	2	yes	—
145	C₄×C₁₃	C₂²⋊C₁₆	1	2	yes	—
146	C₄×C₁₃	C₂×#102	1	1	yes	characteristic
147	C₄×C₁₃	C₂×#104	1	1	yes	characteristic
148	C₄×C₁₃	C₂²×C₁₆	1	1	yes	characteristic
149	C₄×C₁₃	C₂×M₃₂	1	1	yes	characteristic
150	C₂²×D₄	#1055	1	1	yes	characteristic
151	C₂³×C₄	—	2	1	no	—
152	C₂²×C₈	—	2	1	no	—
153	C₂²×C₈	—	2	2	no	—
154	C₂²×C₈	—	2	2	no	—
155	C₂²×C₈	—	4	2	no	—
156	C₂²×C₈	—	4	2	no	—
157	C₂²×C₈	C₂²⋊D₁₃	1	1	yes	characteristic
158	C₂²×C₈	C₂²⋊D₁₃	1	1	yes	characteristic
159	C₄×D₄	—	2	1	no	—
160	C₄×D₄	—	2	1	no	—
161	C₄×D₄	#1055	1	2	yes	—
162	C₄×D₄	C₂×Dic₁₃	1	2	yes	—
163	C₂×C₄⋊C₄	#1055	1	1	yes	characteristic
164	C₂×C₂²⋊C₄	—	2	1	no	—
165	C₂×C₄²	C₂×Dic₁₃	1	1	yes	characteristic
166	C₂×C₁₆	—	26	4	no	—
167	C₂×C₁₆	—	26	2	no	—
168	C₂×C₁₆	—	26	2	no	—
169	C₂×C₁₆	—	52	2	no	—
170	C₄⋊C₈	—	2	1	no	—
171	C₄⋊C₈	C₂×Dic₁₃	1	2	yes	—
172	C₂²⋊C₈	—	2	2	no	—
173	C₂²⋊C₈	—	2	2	no	—
174	C₄×C₈	C₂×Dic₁₃	1	1	yes	characteristic
175	C₄×C₈	C₂²⋊D₁₃	1	1	yes	characteristic
176	C₄×C₈	C₂²⋊D₁₃	1	1	yes	characteristic
177	C₄×C₈	C₂²×D₁₃	1	1	yes	characteristic
178	C₂×C₁₃	—	4	4	no	—
179	C₂×C₁₃	D₄⋊C₁₆	1	4	yes	complemented
180	C₂×C₁₃	C₂×D₄⋊C₈	1	1	yes	characteristic
181	C₂×C₁₃	C₂×C₂²⋊C₁₆	1	1	yes	characteristic
182	C₂×C₁₃	C₂×#350	1	1	yes	characteristic
183	C₂⁴	—	2	1	no	—
184	C₂×D₄	—	2	1	no	—
185	C₂×D₄	—	2	2	no	—
186	C₂×D₄	#6090	1	4	yes	complemented
187	C₂×D₄	C₂×#1055	1	2	yes	—
188	C₂²×C₄	—	2	1	no	—
189	C₂²×C₄	—	2	2	no	—
190	C₂²×C₄	—	2	2	no	—
191	C₂²×C₄	—	4	2	no	—
192	C₂²×C₄	—	4	2	no	—
193	C₂²×C₄	C₂×#1055	1	1	yes	characteristic
194	C₂²×C₄	#6099	1	1	yes	characteristic
195	C₂²×C₄	C₂²⋊Dic₁₃	1	1	yes	characteristic
196	C₂×C₈	—	2	2	no	—
197	C₂×C₈	—	2	2	no	—
198	C₂×C₈	—	4	2	no	—
199	C₂×C₈	—	4	4	no	—
200	C₂×C₈	—	4	2	no	—
201	C₂×C₈	—	4	2	no	—
202	C₂×C₈	#6104	1	2	yes	—
203	C₂×C₈	#6105	1	2	yes	—
204	C₂×C₈	C₂²⋊Dic₁₃	1	1	yes	characteristic
205	C₂×C₈	C₂²⋊Dic₁₃	1	1	yes	characteristic
206	C₂×C₈	C₂×C₂²⋊D₁₃	1	1	yes	characteristic
207	C₂×C₈	C₂×C₂²⋊D₁₃	1	1	yes	characteristic
208	C₄⋊C₄	—	2	1	no	—
209	C₄⋊C₄	C₂×#1055	1	2	yes	—
210	C₂²⋊C₄	—	2	2	no	—
211	C₂²⋊C₄	—	2	2	no	—
212	C₄²	C₂×#1055	1	1	yes	characteristic
213	C₄²	#6099	1	1	yes	characteristic
214	C₄²	C₂²⋊Dic₁₃	1	1	yes	characteristic
215	C₄²	C₂²×Dic₁₃	1	1	yes	characteristic
216	C₁₆	—	26	4	no	—
217	C₁₆	—	52	2	no	—
218	C₁₃	C₂×D₄⋊C₁₆	1	1	yes	characteristic, complemented, Sylow
219	C₂³	—	2	2	no	—
220	C₂³	—	2	4	no	—
221	C₂³	—	4	2	no	—
222	C₂³	—	4	2	no	—
223	C₂³	C₂²⋊#1055	1	1	yes	characteristic
224	D₄	—	2	2	no	—
225	D₄	—	2	4	no	—
226	D₄	C₂×#6090	1	4	yes	complemented
227	C₂×C₄	—	2	2	no	—
228	C₂×C₄	—	2	2	no	—
229	C₂×C₄	—	4	2	no	—
230	C₂×C₄	—	4	2	no	—
231	C₂×C₄	—	4	4	no	—
232	C₂×C₄	—	4	2	no	—
233	C₂×C₄	C₂×#6090	1	2	yes	—
234	C₂×C₄	#88052	1	2	yes	—
235	C₂×C₄	C₂²⋊#1055	1	1	yes	characteristic
236	C₂×C₄	C₂²⋊#1055	1	1	yes	characteristic
237	C₂×C₄	C₂²⋊#1055	1	1	yes	characteristic
238	C₂×C₄	#88072	1	2	yes	—
239	C₂×C₄	#88077	1	2	yes	—
240	C₂×C₄	C₂²×#1055	1	1	yes	characteristic
241	C₂×C₄	C₂×#6099	1	1	yes	characteristic
242	C₂×C₄	C₂×C₂²⋊Dic₁₃	1	1	yes	characteristic
243	C₈	—	2	1	no	—
244	C₈	—	2	1	no	—
245	C₈	—	4	2	no	—
246	C₈	#88072	1	2	yes	—
247	C₈	C₂×#6104	1	1	yes	characteristic
248	C₈	C₂×#6105	1	1	yes	characteristic
249	C₂²	—	2	4	no	—
250	C₂²	—	4	2	no	—
251	C₂²	—	4	2	no	—
252	C₂²	—	4	8	no	—
253	C₂²	D₄⋊#1055	1	2	yes	—
254	C₂²	C₂²⋊#6090	1	2	yes	—
255	C₂²	#11718153	1	2	yes	—
256	C₂²	C₂×C₂²⋊#1055	1	1	yes	characteristic
257	C₄	—	2	1	no	—
258	C₄	—	2	1	no	—
259	C₄	—	4	2	no	—
260	C₄	D₄⋊#1055	1	2	yes	—
261	C₄	C₂²⋊#6090	1	2	yes	—
262	C₄	C₂²×#6090	1	1	yes	characteristic
263	C₄	C₂×#88052	1	1	yes	characteristic
264	C₄	C₂×#88072	1	1	yes	characteristic
265	C₄	C₂×#88077	1	1	yes	characteristic
266	C₂	—	4	4	no	—
267	C₂	D₄⋊#6090	1	4	yes	direct product factor
268	C₂	C₂×D₄⋊#1055	1	1	yes	characteristic
269	C₂	C₂×C₂²⋊#6090	1	1	yes	characteristic
270	C₂	C₂×#11718153	1	1	yes	characteristic

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

Complements

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

N	H ≅ G/N	type
1. C₂×C₄⋊#6090	C₂	semidirect
2. C₂×C₄×#6090	C₂	semidirect
3. D₄⋊#6090	C₂	direct (N × H)
17. C₄⋊#6090	C₂²	semidirect
19. C₄×#6090	C₂²	semidirect
52. C₂×D₄×C₁₃	C₁₆	semidirect
95. D₄×C₁₃	C₂×C₁₆	semidirect
179. C₂×C₁₃	D₄⋊C₁₆	semidirect
186. C₂×D₄	#6090	semidirect
218. C₁₃	C₂×D₄⋊C₁₆	semidirect
226. D₄	C₂×#6090	semidirect
267. C₂	D₄⋊#6090	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂×C₄⋊#6090	2
C₂×C₄×#6090	2
D₄⋊#6090	2
C₂×C₈×D₄×C₁₃	2
C₂×D₄⋊C₁₆	13

Subgroups — C2×D4⋊#6090 #15772618

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×D₄⋊#6090 #15772618