Subgroups — #13136263

Order 1728.

620 conjugacy classes of subgroups.

The subgroup lattice has 228 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	#11720142	C₂	1	2	yes	maximal, complemented
5	#11720136	C₂	1	2	yes	maximal, complemented
6	#6171⋊C₄	C₂	1	4	yes	maximal, complemented
7	#11720121	C₂	1	2	yes	maximal, complemented
8	C₄²×D₂₇	C₂	1	2	yes	maximal, complemented
9	#10589624	—	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	2	yes	—
19	C₂²⋊C₄×C₂₇	C₂²	1	2	yes	—
20	C₄²×C₂₇	C₂²	1	2	yes	—
21	(C₂×D₂₇)⋊C₄	C₂²	1	4	yes	—
22	Dic₂₇⋊C₄	C₂²	1	4	yes	—
23	C₄×Dic₂₇	C₂²	1	1	yes	characteristic
24	C₄×Dic₂₇	C₂²	1	4	yes	—
25	C₄×Dic₂₇	C₂²	1	1	yes	characteristic
26	C₄×Dic₂₇	C₂²	1	2	yes	—
27	C₂×#3697	—	3	1	no	—
28	C₉×#117	D₃	1	1	yes	characteristic
29	C₂×C₄×Dic₉	—	3	1	no	—
30	#63455	—	3	2	no	—
31	#63449	—	3	2	no	—
32	#611⋊C₄	—	3	4	no	—
33	#63434	—	3	2	no	—
34	C₄²×D₉	—	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	4	yes	—
39	C₂×C₄×C₂₇	C₂³	1	1	yes	characteristic
40	C₂×C₄×C₂₇	C₂³	1	1	yes	characteristic
41	#6171	C₂×C₄	1	8	yes	complemented
42	C₂×Dic₂₇	—	2	4	no	—
43	C₂×Dic₂₇	C₂³	1	2	yes	—
44	C₂×Dic₂₇	C₂³	1	2	yes	—
45	C₂×Dic₂₇	C₂³	1	2	yes	—
46	D₁₀₈	C₂×C₄	1	4	yes	complemented
47	C₄×D₂₇	—	2	8	no	—
48	C₄×D₂₇	C₂×C₄	1	8	yes	complemented
49	Dic₅₄	C₂×C₄	1	4	yes	complemented
50	#5522	—	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	2	yes	—
60	C₉×C₂²⋊C₄	C₂×D₃	1	2	yes	—
61	C₄²×C₉	C₂×D₃	1	2	yes	—
62	(C₂×D₉)⋊C₄	—	3	4	no	—
63	Dic₉⋊C₄	—	3	4	no	—
64	C₄×Dic₉	—	3	1	no	—
65	C₄×Dic₉	—	3	4	no	—
66	C₄×Dic₉	—	3	1	no	—
67	C₄×Dic₉	—	3	2	no	—
68	C₂²×C₂₇	—	2	2	no	—
69	C₂²×C₂₇	C₂²×C₄	1	2	yes	—
70	C₂²×C₂₇	C₂⁴	1	1	yes	characteristic
71	C₂×D₂₇	—	2	4	no	—
72	C₂×D₂₇	C₂²×C₄	1	4	yes	—
73	C₄×C₂₇	—	2	4	no	—
74	C₄×C₂₇	C₂²×C₄	1	2	yes	—
75	C₄×C₂₇	C₂²×C₄	1	2	yes	—
76	Dic₂₇	—	2	2	no	—
77	Dic₂₇	C₂²×C₄	1	4	yes	—
78	Dic₂₇	#41	1	4	yes	complemented
79	C₂×#235	—	9	1	no	—
80	C₃×#117	D₉	1	1	yes	characteristic
81	C₂×C₄×Dic₃	—	9	1	no	—
82	#893	—	9	2	no	—
83	#887	—	9	2	no	—
84	#67⋊C₄	—	9	4	no	—
85	#872	—	9	2	no	—
86	C₄²×D₃	—	9	2	no	—
87	C₂³×C₉	C₂²×D₃	1	1	yes	characteristic
88	C₂²×D₉	—	3	2	no	—
89	C₂×C₄×C₉	C₄×D₃	1	4	yes	—
90	C₂×C₄×C₉	C₂²×D₃	1	4	yes	—
91	C₂×C₄×C₉	C₂²×D₃	1	1	yes	characteristic
92	C₂×C₄×C₉	C₂²×D₃	1	1	yes	characteristic
93	#611	—	3	8	no	—
94	C₂×Dic₉	—	3	2	no	—
95	C₂×Dic₉	—	3	2	no	—
96	C₂×Dic₉	—	3	2	no	—
97	C₂×Dic₉	—	6	4	no	—
98	D₃₆	—	3	4	no	—
99	C₄×D₉	—	3	8	no	—
100	C₄×D₉	—	6	8	no	—
101	Dic₁₈	—	3	4	no	—
102	C₄×#41	—	27	1	no	Sylow
103	C₂×C₂₇	—	2	2	no	—
104	C₂×C₂₇	C₂³×C₄	1	1	yes	characteristic
105	C₂×C₂₇	C₂×#41	1	2	yes	—
106	D₂₇	—	2	4	no	—
107	C₂²×C₃×C₄	C₂×D₉	1	1	yes	characteristic
108	C₂×#67	—	9	2	no	—
109	C₂²×Dic₃	—	9	2	no	—
110	#235	—	9	8	no	—
111	C₂×D₁₂	—	9	1	no	—
112	C₂×C₄×D₃	—	9	2	no	—
113	C₂×C₄×D₃	—	9	4	no	—
114	C₂×Dic₆	—	9	1	no	—
115	C₃×C₄⋊C₄	C₂×D₉	1	2	yes	—
116	C₃×C₂²⋊C₄	C₂×D₉	1	2	yes	—
117	C₃×C₄²	C₂×D₉	1	2	yes	—
118	(C₂×D₃)⋊C₄	—	9	4	no	—
119	Dic₃⋊C₄	—	9	4	no	—
120	C₄×Dic₃	—	9	1	no	—
121	C₄×Dic₃	—	9	4	no	—
122	C₄×Dic₃	—	9	1	no	—
123	C₄×Dic₃	—	9	2	no	—
124	C₂²×C₉	—	2	2	no	—
125	C₂²×C₉	C₂×C₄×D₃	1	2	yes	—
126	C₂²×C₉	C₂³×D₃	1	1	yes	characteristic
127	C₂×D₉	—	3	4	no	—
128	C₂×D₉	—	6	4	no	—
129	C₄×C₉	—	2	4	no	—
130	C₄×C₉	C₂×C₄×D₃	1	2	yes	—
131	C₄×C₉	C₂×C₄×D₃	1	2	yes	—
132	Dic₉	—	3	4	no	—
133	Dic₉	—	3	4	no	—
134	Dic₉	—	6	2	no	—
135	C₂×#41	—	27	1	no	—
136	C₄×Q₈	—	27	2	no	—
137	C₄×D₄	—	27	2	no	—
138	C₄×D₄	—	27	4	no	—
139	#117	—	27	2	no	—
140	#117	D₂₇	1	1	yes	characteristic, complemented
141	C₂×C₄²	—	27	1	no	—
142	C₂×C₄²	—	27	2	no	—
143	C₂₇	C₄×#41	1	1	yes	characteristic, complemented, Sylow
144	C₂³×C₃	C₂²×D₉	1	1	yes	characteristic
145	C₂²×D₃	—	9	2	no	—
146	C₂×C₃×C₄	C₄×D₉	1	4	yes	—
147	C₂×C₃×C₄	C₂²×D₉	1	4	yes	—
148	C₂×C₃×C₄	C₂²×D₉	1	1	yes	characteristic
149	C₂×C₃×C₄	C₂²×D₉	1	1	yes	characteristic
150	#67	—	9	8	no	—
151	C₂×Dic₃	—	9	2	no	—
152	C₂×Dic₃	—	9	2	no	—
153	C₂×Dic₃	—	9	2	no	—
154	C₂×Dic₃	—	18	4	no	—
155	D₁₂	—	9	4	no	—
156	C₄×D₃	—	9	8	no	—
157	C₄×D₃	—	18	8	no	—
158	Dic₆	—	9	4	no	—
159	C₂×C₉	—	2	2	no	—
160	C₂×C₉	C₂²×C₄×D₃	1	1	yes	characteristic
161	C₂×C₉	D₃×#41	1	2	yes	—
162	D₉	—	6	4	no	—
163	#41	—	27	8	no	—
164	C₂×Q₈	—	27	1	no	—
165	C₂×D₄	—	27	2	no	—
166	C₂×D₄	—	27	1	no	—
167	C₂²×C₄	—	27	2	no	—
168	C₂²×C₄	—	27	2	no	—
169	C₂²×C₄	—	27	4	no	—
170	C₂²×C₄	C₂×D₂₇	1	1	yes	characteristic
171	C₄⋊C₄	—	27	4	no	—
172	C₄⋊C₄	C₂×D₂₇	1	2	yes	—
173	C₂²⋊C₄	—	27	4	no	—
174	C₂²⋊C₄	C₂×D₂₇	1	2	yes	—
175	C₄²	—	27	4	no	—
176	C₄²	—	27	2	no	—
177	C₄²	—	27	1	no	—
178	C₄²	—	27	1	no	—
179	C₄²	C₂×D₂₇	1	2	yes	—
180	C₂²×C₃	—	2	2	no	—
181	C₂²×C₃	C₂×C₄×D₉	1	2	yes	—
182	C₂²×C₃	C₂³×D₉	1	1	yes	characteristic
183	C₂×D₃	—	9	4	no	—
184	C₂×D₃	—	18	4	no	—
185	C₃×C₄	—	2	4	no	—
186	C₃×C₄	C₂×C₄×D₉	1	2	yes	—
187	C₃×C₄	C₂×C₄×D₉	1	2	yes	—
188	Dic₃	—	9	4	no	—
189	Dic₃	—	9	4	no	—
190	Dic₃	—	18	2	no	—
191	C₉	#5522	1	1	yes	characteristic
192	C₂³	—	27	2	no	—
193	C₂³	C₂²×D₂₇	1	1	yes	characteristic
194	Q₈	—	27	4	no	—
195	D₄	—	27	8	no	—
196	D₄	—	27	4	no	—
197	C₂×C₄	—	27	2	no	—
198	C₂×C₄	—	27	2	no	—
199	C₂×C₄	—	27	8	no	—
200	C₂×C₄	—	27	2	no	—
201	C₂×C₄	—	54	8	no	—
202	C₂×C₄	—	54	4	no	—
203	C₂×C₄	C₄×D₂₇	1	4	yes	complemented
204	C₂×C₄	C₂²×D₂₇	1	4	yes	—
205	C₂×C₄	C₂²×D₂₇	1	1	yes	characteristic
206	C₂×C₄	C₂²×D₂₇	1	1	yes	characteristic
207	C₂×C₃	—	2	2	no	—
208	C₂×C₃	C₂²×C₄×D₉	1	1	yes	characteristic
209	C₂×C₃	#41×D₉	1	2	yes	—
210	D₃	—	18	4	no	—
211	C₂²	—	2	2	no	—
212	C₂²	—	27	4	no	—
213	C₂²	—	54	4	no	—
214	C₂²	C₂×C₄×D₂₇	1	2	yes	—
215	C₂²	C₂³×D₂₇	1	1	yes	characteristic
216	C₄	—	2	4	no	—
217	C₄	—	27	4	no	—
218	C₄	—	27	4	no	—
219	C₄	—	54	2	no	—
220	C₄	C₂×C₄×D₂₇	1	2	yes	—
221	C₄	C₂×C₄×D₂₇	1	2	yes	—
222	C₃	#10589624	1	1	yes	characteristic
223	C₂	—	2	2	no	—
224	C₂	—	54	4	no	—
225	C₂	C₂²×C₄×D₂₇	1	1	yes	characteristic
226	C₂	#41×D₂₇	1	2	yes	—

	subgroup	G/H
center	C₂×C₄	C₂²×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	C₄×#41
LCS	G ▷ C₂×C₂₇ ▷ C₂₇
UCS	C₁ ◁ C₂×C₄ ◁ #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. #11720142	C₂	semidirect
5. #11720136	C₂	semidirect
6. #6171⋊C₄	C₂	semidirect
7. #11720121	C₂	semidirect
8. C₄²×D₂₇	C₂	semidirect
13. #88399	C₄	semidirect
37. C₂×C₄×C₂₇	C₂×C₄	semidirect
41. #6171	C₂×C₄	semidirect
46. D₁₀₈	C₂×C₄	semidirect
48. C₄×D₂₇	C₂×C₄	semidirect
49. Dic₅₄	C₂×C₄	semidirect
78. Dic₂₇	#41	semidirect
140. #117	D₂₇	semidirect
143. C₂₇	C₄×#41	semidirect
203. 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
#11720142	2
#11720136	2
#6171⋊C₄	2
#11720121	2
C₄²×D₂₇	2
#10589624	3