Subgroups — #13136264

Order 1728.

588 conjugacy classes of subgroups.

The subgroup lattice has 240 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#88399	C₂	1	1	yes	maximal, characteristic
2	C₂₇×#117	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×C₄⋊Dic₂₇	C₂	1	1	yes	maximal, characteristic, complemented
4	#11720141	C₂	1	2	yes	maximal, complemented
5	C₄⋊C₄×D₂₇	C₂	1	2	yes	maximal, complemented
6	#6171⋊C₄	C₂	1	4	yes	maximal, complemented
7	C₄×D₁₀₈	C₂	1	2	yes	maximal, complemented
8	C₄×Dic₅₄	C₂	1	2	yes	maximal, complemented
9	#10589625	—	3	1	no	maximal
10	C₂²×C₄×C₂₇	C₂²	1	1	yes	characteristic
11	C₂×#6171	C₂²	1	2	yes	—
12	C₂²×Dic₂₇	C₂²	1	2	yes	—
13	#88399	C₄	1	8	yes	complemented
14	C₂×D₁₀₈	C₂²	1	1	yes	characteristic
15	C₂×C₄×D₂₇	C₂²	1	4	yes	—
16	C₂×C₄×D₂₇	C₂²	1	2	yes	—
17	C₂×Dic₅₄	C₂²	1	1	yes	characteristic
18	C₄⋊C₄×C₂₇	C₂²	1	1	yes	characteristic
19	C₄⋊C₄×C₂₇	C₂²	1	1	yes	characteristic
20	C₂²⋊C₄×C₂₇	C₂²	1	2	yes	—
21	C₄²×C₂₇	C₂²	1	2	yes	—
22	(C₂×D₂₇)⋊C₄	C₂²	1	4	yes	—
23	C₄⋊Dic₂₇	C₂²	1	2	yes	—
24	C₄⋊Dic₂₇	C₂²	1	2	yes	—
25	Dic₂₇⋊C₄	C₂²	1	4	yes	—
26	C₄×Dic₂₇	C₂²	1	4	yes	—
27	C₂×#3697	—	3	1	no	—
28	C₉×#117	D₃	1	1	yes	characteristic
29	C₂×C₄⋊Dic₉	—	3	1	no	—
30	#63454	—	3	2	no	—
31	C₄⋊C₄×D₉	—	3	2	no	—
32	#611⋊C₄	—	3	4	no	—
33	C₄×D₃₆	—	3	2	no	—
34	C₄×Dic₁₈	—	3	2	no	—
35	C₂³×C₂₇	C₂³	1	1	yes	characteristic
36	C₂²×D₂₇	C₂³	1	2	yes	—
37	C₂×C₄×C₂₇	C₂×C₄	1	4	yes	complemented
38	C₂×C₄×C₂₇	C₂³	1	2	yes	—
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	#6171	C₂×C₄	1	8	yes	complemented
43	C₂×Dic₂₇	—	4	2	no	—
44	C₂×Dic₂₇	C₂³	1	4	yes	—
45	C₂×Dic₂₇	C₂³	1	2	yes	—
46	D₁₀₈	C₂×C₄	1	4	yes	complemented
47	C₄×D₂₇	—	4	4	no	—
48	C₄×D₂₇	C₂×C₄	1	8	yes	complemented
49	Dic₅₄	C₂×C₄	1	4	yes	complemented
50	#5523	—	9	1	no	—
51	C₂²×C₄×C₉	C₂×D₃	1	1	yes	characteristic
52	C₂×#611	—	3	2	no	—
53	C₂²×Dic₉	—	3	2	no	—
54	#3697	—	3	8	no	—
55	C₂×D₃₆	—	3	1	no	—
56	C₂×C₄×D₉	—	3	2	no	—
57	C₂×C₄×D₉	—	3	4	no	—
58	C₂×Dic₁₈	—	3	1	no	—
59	C₉×C₄⋊C₄	C₂×D₃	1	1	yes	characteristic
60	C₉×C₄⋊C₄	C₂×D₃	1	1	yes	characteristic
61	C₉×C₂²⋊C₄	C₂×D₃	1	2	yes	—
62	C₄²×C₉	C₂×D₃	1	2	yes	—
63	(C₂×D₉)⋊C₄	—	3	4	no	—
64	C₄⋊Dic₉	—	3	2	no	—
65	C₄⋊Dic₉	—	3	2	no	—
66	Dic₉⋊C₄	—	3	4	no	—
67	C₄×Dic₉	—	3	4	no	—
68	C₂²×C₂₇	—	2	1	no	—
69	C₂²×C₂₇	—	2	1	no	—
70	C₂²×C₂₇	C₂²×C₄	1	2	yes	—
71	C₂²×C₂₇	C₂⁴	1	1	yes	characteristic
72	C₂×D₂₇	—	2	2	no	—
73	C₂×D₂₇	—	2	2	no	—
74	C₂×D₂₇	C₂²×C₄	1	4	yes	—
75	C₄×C₂₇	—	2	2	no	—
76	C₄×C₂₇	—	2	2	no	—
77	C₄×C₂₇	C₂²×C₄	1	2	yes	—
78	C₄×C₂₇	C₂²×C₄	1	2	yes	—
79	Dic₂₇	—	2	4	no	—
80	Dic₂₇	C₂²×C₄	1	4	yes	—
81	C₂×#235	—	9	1	no	—
82	C₃×#117	D₉	1	1	yes	characteristic
83	C₂×C₄⋊Dic₃	—	9	1	no	—
84	#892	—	9	2	no	—
85	D₃×C₄⋊C₄	—	9	2	no	—
86	#67⋊C₄	—	9	4	no	—
87	C₄×D₁₂	—	9	2	no	—
88	C₄×Dic₆	—	9	2	no	—
89	C₂³×C₉	C₂²×D₃	1	1	yes	characteristic
90	C₂²×D₉	—	3	2	no	—
91	C₂×C₄×C₉	C₄×D₃	1	4	yes	—
92	C₂×C₄×C₉	C₂²×D₃	1	2	yes	—
93	C₂×C₄×C₉	C₂²×D₃	1	2	yes	—
94	C₂×C₄×C₉	C₂²×D₃	1	1	yes	characteristic
95	C₂×C₄×C₉	C₂²×D₃	1	1	yes	characteristic
96	#611	—	3	8	no	—
97	C₂×Dic₉	—	3	2	no	—
98	C₂×Dic₉	—	3	4	no	—
99	C₂×Dic₉	—	12	2	no	—
100	D₃₆	—	3	4	no	—
101	C₄×D₉	—	3	8	no	—
102	C₄×D₉	—	12	4	no	—
103	Dic₁₈	—	3	4	no	—
104	#520	—	27	1	no	Sylow
105	C₂×C₂₇	—	2	2	no	—
106	C₂×C₂₇	C₂³×C₄	1	1	yes	characteristic
107	C₂×C₂₇	H(5,𝔽₂)	1	1	yes	characteristic
108	C₂×C₂₇	2₋¹⁺⁴	1	1	yes	characteristic
109	D₂₇	—	2	4	no	—
110	C₂²×C₃×C₄	C₂×D₉	1	1	yes	characteristic
111	C₂×#67	—	9	2	no	—
112	C₂²×Dic₃	—	9	2	no	—
113	#235	—	9	8	no	—
114	C₂×D₁₂	—	9	1	no	—
115	C₂×C₄×D₃	—	9	2	no	—
116	C₂×C₄×D₃	—	9	4	no	—
117	C₂×Dic₆	—	9	1	no	—
118	C₃×C₄⋊C₄	C₂×D₉	1	1	yes	characteristic
119	C₃×C₄⋊C₄	C₂×D₉	1	1	yes	characteristic
120	C₃×C₂²⋊C₄	C₂×D₉	1	2	yes	—
121	C₃×C₄²	C₂×D₉	1	2	yes	—
122	(C₂×D₃)⋊C₄	—	9	4	no	—
123	C₄⋊Dic₃	—	9	2	no	—
124	C₄⋊Dic₃	—	9	2	no	—
125	Dic₃⋊C₄	—	9	4	no	—
126	C₄×Dic₃	—	9	4	no	—
127	C₂²×C₉	—	2	1	no	—
128	C₂²×C₉	—	2	1	no	—
129	C₂²×C₉	C₂×C₄×D₃	1	2	yes	—
130	C₂²×C₉	C₂³×D₃	1	1	yes	characteristic
131	C₂×D₉	—	3	4	no	—
132	C₂×D₉	—	6	2	no	—
133	C₂×D₉	—	6	2	no	—
134	C₄×C₉	—	2	2	no	—
135	C₄×C₉	—	2	2	no	—
136	C₄×C₉	C₂×C₄×D₃	1	2	yes	—
137	C₄×C₉	C₂×C₄×D₃	1	2	yes	—
138	Dic₉	—	3	4	no	—
139	Dic₉	—	6	4	no	—
140	C₂×#41	—	27	1	no	—
141	C₄×Q₈	—	27	2	no	—
142	C₄×D₄	—	27	4	no	—
143	C₄×D₄	—	27	2	no	—
144	#117	—	27	2	no	—
145	#117	D₂₇	1	1	yes	characteristic, complemented
146	C₂×C₄⋊C₄	—	27	1	no	—
147	C₂×C₄⋊C₄	—	27	2	no	—
148	C₂₇	#520	1	1	yes	characteristic, complemented, Sylow
149	C₂³×C₃	C₂²×D₉	1	1	yes	characteristic
150	C₂²×D₃	—	9	2	no	—
151	C₂×C₃×C₄	C₄×D₉	1	4	yes	—
152	C₂×C₃×C₄	C₂²×D₉	1	2	yes	—
153	C₂×C₃×C₄	C₂²×D₉	1	2	yes	—
154	C₂×C₃×C₄	C₂²×D₉	1	1	yes	characteristic
155	C₂×C₃×C₄	C₂²×D₉	1	1	yes	characteristic
156	#67	—	9	8	no	—
157	C₂×Dic₃	—	9	2	no	—
158	C₂×Dic₃	—	9	4	no	—
159	C₂×Dic₃	—	36	2	no	—
160	D₁₂	—	9	4	no	—
161	C₄×D₃	—	9	8	no	—
162	C₄×D₃	—	36	4	no	—
163	Dic₆	—	9	4	no	—
164	C₂×C₉	—	2	2	no	—
165	C₂×C₉	C₂²×C₄×D₃	1	1	yes	characteristic
166	C₂×C₉	#1009	1	1	yes	characteristic
167	C₂×C₉	#1010	1	1	yes	characteristic
168	D₉	—	6	4	no	—
169	#41	—	27	8	no	—
170	C₂×Q₈	—	27	1	no	—
171	C₂×D₄	—	27	2	no	—
172	C₂×D₄	—	27	1	no	—
173	C₂²×C₄	—	27	2	no	—
174	C₂²×C₄	—	27	2	no	—
175	C₂²×C₄	—	27	4	no	—
176	C₂²×C₄	C₂×D₂₇	1	1	yes	characteristic
177	C₄⋊C₄	—	27	2	no	—
178	C₄⋊C₄	—	27	4	no	—
179	C₄⋊C₄	—	27	2	no	—
180	C₄⋊C₄	C₂×D₂₇	1	1	yes	characteristic
181	C₄⋊C₄	C₂×D₂₇	1	1	yes	characteristic
182	C₂²⋊C₄	—	27	4	no	—
183	C₂²⋊C₄	C₂×D₂₇	1	2	yes	—
184	C₄²	—	27	4	no	—
185	C₄²	C₂×D₂₇	1	2	yes	—
186	C₂²×C₃	—	2	1	no	—
187	C₂²×C₃	—	2	1	no	—
188	C₂²×C₃	C₂×C₄×D₉	1	2	yes	—
189	C₂²×C₃	C₂³×D₉	1	1	yes	characteristic
190	C₂×D₃	—	9	4	no	—
191	C₂×D₃	—	18	2	no	—
192	C₂×D₃	—	18	2	no	—
193	C₃×C₄	—	2	2	no	—
194	C₃×C₄	—	2	2	no	—
195	C₃×C₄	C₂×C₄×D₉	1	2	yes	—
196	C₃×C₄	C₂×C₄×D₉	1	2	yes	—
197	Dic₃	—	9	4	no	—
198	Dic₃	—	18	4	no	—
199	C₉	#5523	1	1	yes	characteristic
200	C₂³	—	27	2	no	—
201	C₂³	C₂²×D₂₇	1	1	yes	characteristic
202	Q₈	—	27	4	no	—
203	D₄	—	27	8	no	—
204	D₄	—	27	4	no	—
205	C₂×C₄	—	27	2	no	—
206	C₂×C₄	—	27	8	no	—
207	C₂×C₄	—	27	4	no	—
208	C₂×C₄	—	108	4	no	—
209	C₂×C₄	—	108	2	no	—
210	C₂×C₄	C₄×D₂₇	1	4	yes	complemented
211	C₂×C₄	C₂²×D₂₇	1	2	yes	—
212	C₂×C₄	C₂²×D₂₇	1	2	yes	—
213	C₂×C₄	C₂²×D₂₇	1	1	yes	characteristic
214	C₂×C₄	C₂²×D₂₇	1	1	yes	characteristic
215	C₂×C₃	—	2	2	no	—
216	C₂×C₃	C₂²×C₄×D₉	1	1	yes	characteristic
217	C₂×C₃	#63715	1	1	yes	characteristic
218	C₂×C₃	#63716	1	1	yes	characteristic
219	D₃	—	18	4	no	—
220	C₂²	—	2	1	no	—
221	C₂²	—	2	1	no	—
222	C₂²	—	27	4	no	—
223	C₂²	—	54	2	no	—
224	C₂²	—	54	2	no	—
225	C₂²	C₂×C₄×D₂₇	1	2	yes	—
226	C₂²	C₂³×D₂₇	1	1	yes	characteristic
227	C₄	—	2	2	no	—
228	C₄	—	2	2	no	—
229	C₄	—	27	4	no	—
230	C₄	—	54	4	no	—
231	C₄	C₂×C₄×D₂₇	1	2	yes	—
232	C₄	C₂×C₄×D₂₇	1	2	yes	—
233	C₃	#10589625	1	1	yes	characteristic
234	C₂	—	2	2	no	—
235	C₂	—	54	4	no	—
236	C₂	C₂²×C₄×D₂₇	1	1	yes	characteristic
237	C₂	#11720675	1	1	yes	characteristic
238	C₂	#11720676	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂³×D₂₇
commutator	C₂×C₂₇	C₂³×C₄
Frattini	C₂²×C₉	C₂³×D₃
Fitting	C₂₇×#117	C₂
socle	C₂²×C₃	C₂³×D₉
prime radical	C₂×#88399	C₂
cosocle	C₂²×C₂₇	C₂⁴
hypercenter	#117	D₂₇
hypocenter	C₂₇	#520
LCS	G ▷ C₂×C₂₇ ▷ C₂₇
UCS	C₁ ◁ C₂² ◁ #117

Complements

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

N	H ≅ G/N	type
2. C₂₇×#117	C₂	semidirect
3. C₂×C₄⋊Dic₂₇	C₂	semidirect
4. #11720141	C₂	semidirect
5. C₄⋊C₄×D₂₇	C₂	semidirect
6. #6171⋊C₄	C₂	semidirect
7. C₄×D₁₀₈	C₂	semidirect
8. C₄×Dic₅₄	C₂	semidirect
13. #88399	C₄	semidirect
37. C₂×C₄×C₂₇	C₂×C₄	semidirect
42. #6171	C₂×C₄	semidirect
46. D₁₀₈	C₂×C₄	semidirect
48. C₄×D₂₇	C₂×C₄	semidirect
49. Dic₅₄	C₂×C₄	semidirect
145. #117	D₂₇	semidirect
148. C₂₇	#520	semidirect
210. C₂×C₄	C₄×D₂₇	semidirect

Maximal subgroups

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

subgroup	index
C₂×#88399	2
C₂₇×#117	2
C₂×C₄⋊Dic₂₇	2
#11720141	2
C₄⋊C₄×D₂₇	2
#6171⋊C₄	2
C₄×D₁₀₈	2
C₄×Dic₅₄	2
#10589625	3