Subgroups — D₄×Dic₅₄ #13136274

Order 1728.

560 conjugacy classes of subgroups.

The subgroup lattice has 224 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×Dic₅₄	C₂	1	2	yes	maximal, complemented
2	C₄×D₄×C₂₇	C₂	1	1	yes	maximal, characteristic
3	D₄×Dic₂₇	C₂	1	2	yes	maximal
4	#11720173	C₂	1	2	yes	maximal, complemented
5	#11720137	C₂	1	2	yes	maximal, complemented
6	#11720127	C₂	1	4	yes	maximal, complemented
7	#11720118	C₂	1	1	yes	maximal, characteristic, complemented
8	C₄×Dic₅₄	C₂	1	1	yes	maximal, characteristic, complemented
9	D₄×Dic₁₈	—	3	1	no	maximal
10	C₂×D₄×C₂₇	C₂²	1	1	yes	characteristic
11	C₂²×C₄×C₂₇	C₂²	1	2	yes	—
12	C₂²×Dic₂₇	C₂²	1	4	yes	—
13	C₂×Dic₅₄	—	2	8	no	—
14	C₂×Dic₅₄	C₂²	1	2	yes	—
15	C₂×Dic₅₄	C₂²	1	4	yes	—
16	C₂×Dic₅₄	C₂²	1	1	yes	characteristic
17	C₄⋊C₄×C₂₇	C₂²	1	1	yes	characteristic
18	C₂²⋊C₄×C₂₇	C₂²	1	2	yes	—
19	C₄²×C₂₇	C₂²	1	1	yes	characteristic
20	#88372	C₂²	1	4	yes	—
21	C₄⋊Dic₂₇	C₂²	1	1	yes	characteristic
22	C₄⋊Dic₂₇	C₂²	1	2	yes	—
23	C₄⋊Dic₂₇	C₂²	1	2	yes	—
24	Dic₂₇⋊C₄	C₂²	1	2	yes	—
25	Dic₂₇⋊C₄	C₂²	1	4	yes	—
26	C₄×Dic₂₇	C₂²	1	2	yes	—
27	C₂²×Dic₁₈	—	3	2	no	—
28	C₄×D₄×C₉	D₃	1	1	yes	characteristic
29	D₄×Dic₉	—	3	2	no	—
30	#63486	—	3	2	no	—
31	#63450	—	3	2	no	—
32	#63440	—	3	4	no	—
33	#63431	—	3	1	no	—
34	C₄×Dic₁₈	—	3	1	no	—
35	C₂³×C₂₇	C₂³	1	2	yes	—
36	D₄×C₂₇	Q₈	1	4	yes	complemented
37	C₂×C₄×C₂₇	—	2	4	no	—
38	C₂×C₄×C₂₇	C₂³	1	1	yes	characteristic
39	C₂×C₄×C₂₇	C₂³	1	2	yes	—
40	C₂×C₄×C₂₇	C₂³	1	1	yes	characteristic
41	C₂×C₄×C₂₇	C₂³	1	1	yes	characteristic
42	C₂×Dic₂₇	—	2	8	no	—
43	C₂×Dic₂₇	C₂³	1	2	yes	—
44	C₂×Dic₂₇	C₂³	1	4	yes	—
45	C₂×Dic₂₇	C₂³	1	2	yes	—
46	Dic₅₄	—	2	4	no	—
47	Dic₅₄	—	2	8	no	—
48	Dic₅₄	D₄	1	4	yes	direct product factor
49	D₄×Dic₆	—	9	1	no	—
50	C₂×D₄×C₉	C₂×D₃	1	1	yes	characteristic
51	C₂²×C₄×C₉	C₂×D₃	1	2	yes	—
52	C₂²×Dic₉	—	3	4	no	—
53	C₂×Dic₁₈	—	3	1	no	—
54	C₂×Dic₁₈	—	3	2	no	—
55	C₂×Dic₁₈	—	3	4	no	—
56	C₂×Dic₁₈	—	6	8	no	—
57	C₉×C₄⋊C₄	C₂×D₃	1	1	yes	characteristic
58	C₉×C₂²⋊C₄	C₂×D₃	1	2	yes	—
59	C₄²×C₉	C₂×D₃	1	1	yes	characteristic
60	#3676	—	3	4	no	—
61	C₄⋊Dic₉	—	3	2	no	—
62	C₄⋊Dic₉	—	3	1	no	—
63	C₄⋊Dic₉	—	3	2	no	—
64	Dic₉⋊C₄	—	3	2	no	—
65	Dic₉⋊C₄	—	3	4	no	—
66	C₄×Dic₉	—	3	2	no	—
67	C₂²×C₂₇	—	2	2	no	—
68	C₂²×C₂₇	—	2	2	no	—
69	C₂²×C₂₇	C₂×Q₈	1	4	yes	complemented
70	C₂²×C₂₇	C₂⁴	1	1	yes	characteristic
71	C₄×C₂₇	—	2	2	no	—
72	C₄×C₂₇	—	2	1	no	—
73	C₄×C₂₇	C₂×D₄	1	2	yes	—
74	C₄×C₂₇	C₂×Q₈	1	2	yes	complemented
75	Dic₂₇	—	2	4	no	—
76	Dic₂₇	—	2	2	no	—
77	Dic₂₇	C₂×D₄	1	4	yes	—
78	C₂²×Dic₆	—	9	2	no	—
79	C₃×C₄×D₄	D₉	1	1	yes	characteristic
80	D₄×Dic₃	—	9	2	no	—
81	#924	—	9	2	no	—
82	#888	—	9	2	no	—
83	#878	—	9	4	no	—
84	#869	—	9	1	no	—
85	C₄×Dic₆	—	9	1	no	—
86	C₂³×C₉	C₂²×D₃	1	2	yes	—
87	D₄×C₉	Dic₆	1	4	yes	—
88	C₂×C₄×C₉	—	2	4	no	—
89	C₂×C₄×C₉	C₂²×D₃	1	1	yes	characteristic
90	C₂×C₄×C₉	C₂²×D₃	1	2	yes	—
91	C₂×C₄×C₉	C₂²×D₃	1	1	yes	characteristic
92	C₂×C₄×C₉	C₂²×D₃	1	1	yes	characteristic
93	C₂×Dic₉	—	3	2	no	—
94	C₂×Dic₉	—	3	2	no	—
95	C₂×Dic₉	—	3	4	no	—
96	C₂×Dic₉	—	6	8	no	—
97	Dic₁₈	—	3	4	no	—
98	Dic₁₈	—	6	8	no	—
99	Dic₁₈	—	6	4	no	—
100	D₄×Q₈	—	27	1	no	Sylow
101	C₂×C₂₇	—	2	4	no	—
102	C₂×C₂₇	C₂²×D₄	1	1	yes	characteristic
103	C₂×C₂₇	C₂²×Q₈	1	1	yes	characteristic
104	C₂×C₂₇	2₋¹⁺⁴	1	1	yes	characteristic
105	C₂×C₃×D₄	C₂×D₉	1	1	yes	characteristic
106	C₂²×C₃×C₄	C₂×D₉	1	2	yes	—
107	C₂²×Dic₃	—	9	4	no	—
108	C₂×Dic₆	—	9	1	no	—
109	C₂×Dic₆	—	9	2	no	—
110	C₂×Dic₆	—	9	4	no	—
111	C₂×Dic₆	—	18	8	no	—
112	C₃×C₄⋊C₄	C₂×D₉	1	1	yes	characteristic
113	C₃×C₂²⋊C₄	C₂×D₉	1	2	yes	—
114	C₃×C₄²	C₂×D₉	1	1	yes	characteristic
115	#217	—	9	4	no	—
116	C₄⋊Dic₃	—	9	1	no	—
117	C₄⋊Dic₃	—	9	2	no	—
118	C₄⋊Dic₃	—	9	2	no	—
119	Dic₃⋊C₄	—	9	2	no	—
120	Dic₃⋊C₄	—	9	4	no	—
121	C₄×Dic₃	—	9	2	no	—
122	C₂²×C₉	—	2	2	no	—
123	C₂²×C₉	—	2	2	no	—
124	C₂²×C₉	C₂×Dic₆	1	4	yes	—
125	C₂²×C₉	C₂³×D₃	1	1	yes	characteristic
126	C₄×C₉	—	2	2	no	—
127	C₄×C₉	—	2	1	no	—
128	C₄×C₉	C₂×Dic₆	1	2	yes	—
129	C₄×C₉	D₃×D₄	1	2	yes	—
130	Dic₉	—	3	4	no	—
131	Dic₉	—	6	4	no	—
132	Dic₉	—	6	2	no	—
133	C₂²×Q₈	—	27	2	no	—
134	C₄⋊Q₈	—	27	2	no	—
135	C₄⋊Q₈	—	27	1	no	—
136	C₂²⋊Q₈	—	27	2	no	—
137	C₂²⋊Q₈	—	27	4	no	—
138	C₄×Q₈	—	27	1	no	—
139	C₄×D₄	—	27	2	no	—
140	C₄×D₄	D₂₇	1	1	yes	characteristic
141	C₂₇	D₄×Q₈	1	1	yes	characteristic, complemented, Sylow
142	C₂³×C₃	C₂²×D₉	1	2	yes	—
143	C₃×D₄	Dic₁₈	1	4	yes	—
144	C₂×C₃×C₄	—	2	4	no	—
145	C₂×C₃×C₄	C₂²×D₉	1	1	yes	characteristic
146	C₂×C₃×C₄	C₂²×D₉	1	2	yes	—
147	C₂×C₃×C₄	C₂²×D₉	1	1	yes	characteristic
148	C₂×C₃×C₄	C₂²×D₉	1	1	yes	characteristic
149	C₂×Dic₃	—	9	2	no	—
150	C₂×Dic₃	—	9	2	no	—
151	C₂×Dic₃	—	9	4	no	—
152	C₂×Dic₃	—	18	8	no	—
153	Dic₆	—	9	4	no	—
154	Dic₆	—	18	8	no	—
155	Dic₆	—	18	4	no	—
156	C₂×C₉	—	2	4	no	—
157	C₂×C₉	C₂²×Dic₆	1	1	yes	characteristic
158	C₂×C₉	C₂×D₃×D₄	1	1	yes	characteristic
159	C₂×C₉	#1010	1	1	yes	characteristic
160	C₂×Q₈	—	27	1	no	—
161	C₂×Q₈	—	27	4	no	—
162	C₂×Q₈	—	27	2	no	—
163	C₂×Q₈	—	54	8	no	—
164	C₂×D₄	C₂×D₂₇	1	1	yes	characteristic
165	C₂²×C₄	—	27	4	no	—
166	C₂²×C₄	C₂×D₂₇	1	2	yes	—
167	C₄⋊C₄	—	27	2	no	—
168	C₄⋊C₄	—	27	2	no	—
169	C₄⋊C₄	—	27	1	no	—
170	C₄⋊C₄	—	27	4	no	—
171	C₄⋊C₄	—	27	2	no	—
172	C₄⋊C₄	C₂×D₂₇	1	1	yes	characteristic
173	C₂²⋊C₄	—	27	4	no	—
174	C₂²⋊C₄	C₂×D₂₇	1	2	yes	—
175	C₄²	—	27	2	no	—
176	C₄²	C₂×D₂₇	1	1	yes	characteristic
177	C₂²×C₃	—	2	2	no	—
178	C₂²×C₃	—	2	2	no	—
179	C₂²×C₃	C₂×Dic₁₈	1	4	yes	—
180	C₂²×C₃	C₂³×D₉	1	1	yes	characteristic
181	C₃×C₄	—	2	2	no	—
182	C₃×C₄	—	2	1	no	—
183	C₃×C₄	C₂×Dic₁₈	1	2	yes	—
184	C₃×C₄	D₄×D₉	1	2	yes	—
185	Dic₃	—	9	4	no	—
186	Dic₃	—	18	4	no	—
187	Dic₃	—	18	2	no	—
188	C₉	D₄×Dic₆	1	1	yes	characteristic
189	C₂³	C₂²×D₂₇	1	2	yes	—
190	Q₈	—	27	4	no	—
191	Q₈	—	54	8	no	—
192	Q₈	—	54	4	no	—
193	D₄	Dic₅₄	1	4	yes	direct product factor
194	C₂×C₄	—	2	4	no	—
195	C₂×C₄	—	27	2	no	—
196	C₂×C₄	—	27	2	no	—
197	C₂×C₄	—	27	4	no	—
198	C₂×C₄	—	54	8	no	—
199	C₂×C₄	C₂²×D₂₇	1	1	yes	characteristic
200	C₂×C₄	C₂²×D₂₇	1	2	yes	—
201	C₂×C₄	C₂²×D₂₇	1	1	yes	characteristic
202	C₂×C₄	C₂²×D₂₇	1	1	yes	characteristic
203	C₂×C₃	—	2	4	no	—
204	C₂×C₃	C₂²×Dic₁₈	1	1	yes	characteristic
205	C₂×C₃	C₂×D₄×D₉	1	1	yes	characteristic
206	C₂×C₃	#63716	1	1	yes	characteristic
207	C₂²	—	2	2	no	—
208	C₂²	—	2	2	no	—
209	C₂²	C₂×Dic₅₄	1	4	yes	complemented
210	C₂²	C₂³×D₂₇	1	1	yes	characteristic
211	C₄	—	2	2	no	—
212	C₄	—	2	1	no	—
213	C₄	—	27	4	no	—
214	C₄	—	54	4	no	—
215	C₄	—	54	2	no	—
216	C₄	C₂×Dic₅₄	1	2	yes	complemented
217	C₄	D₄×D₂₇	1	2	yes	—
218	C₃	D₄×Dic₁₈	1	1	yes	characteristic
219	C₂	—	2	4	no	—
220	C₂	C₂²×Dic₅₄	1	1	yes	characteristic
221	C₂	C₂×D₄×D₂₇	1	1	yes	characteristic
222	C₂	#11720676	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂³×D₂₇
commutator, cosocle	C₂²×C₂₇	C₂⁴
Frattini	C₂²×C₉	C₂³×D₃
Fitting	C₄×D₄×C₂₇	C₂
socle	C₂²×C₃	C₂³×D₉
prime radical	C₂×C₃×D₄	C₂×D₉
hypercenter	C₄×D₄	D₂₇
hypocenter	C₂₇	D₄×Q₈
LCS	G ▷ C₂²×C₂₇ ▷ C₂₇
UCS	C₁ ◁ C₂² ◁ C₄×D₄

Complements

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

N	H ≅ G/N	type
1. C₂²×Dic₅₄	C₂	semidirect
4. #11720173	C₂	semidirect
5. #11720137	C₂	semidirect
6. #11720127	C₂	semidirect
7. #11720118	C₂	semidirect
8. C₄×Dic₅₄	C₂	semidirect
36. D₄×C₂₇	Q₈	semidirect
48. Dic₅₄	D₄	direct (N × H)
69. C₂²×C₂₇	C₂×Q₈	semidirect
74. C₄×C₂₇	C₂×Q₈	semidirect
141. C₂₇	D₄×Q₈	semidirect
193. D₄	Dic₅₄	direct (N × H)
209. C₂²	C₂×Dic₅₄	semidirect
216. C₄	C₂×Dic₅₄	semidirect

Maximal subgroups

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

subgroup	index
C₂²×Dic₅₄	2
C₄×D₄×C₂₇	2
D₄×Dic₂₇	2
#11720173	2
#11720137	2
#11720127	2
#11720118	2
C₄×Dic₅₄	2
D₄×Dic₁₈	3

Subgroups — D4×Dic54 #13136274

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — D₄×Dic₅₄ #13136274