Subgroups — C₂×M₁₆×D₂₇ #13136481

Order 1728.

596 conjugacy classes of subgroups.

The subgroup lattice has 244 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₄×D₂₇	C₂	1	1	yes	maximal, characteristic
2	C₂×M₁₆×C₂₇	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×#88363	C₂	1	1	yes	maximal, characteristic, complemented
4	M₁₆×D₂₇	C₂	1	8	yes	maximal, direct product factor
5	C₂×#88359	C₂	1	2	yes	maximal, complemented
6	C₂×C₈×D₂₇	C₂	1	2	yes	maximal, complemented
7	C₂×M₁₆×D₉	—	3	1	no	maximal
8	C₂³×D₂₇	C₄	1	1	yes	characteristic
9	C₂²×C₄×C₂₇	C₂²	1	1	yes	characteristic
10	C₂²×Dic₂₇	C₄	1	1	yes	characteristic
11	C₂×C₄×D₂₇	C₄	1	4	yes	—
12	C₂×C₄×D₂₇	C₄	1	1	yes	characteristic
13	C₂×C₄×D₂₇	C₄	1	1	yes	characteristic
14	C₂×C₄×D₂₇	C₂²	1	1	yes	characteristic
15	C₂×C₄×D₂₇	C₂²	1	1	yes	characteristic
16	C₂×C₄×D₂₇	C₂²	1	4	yes	—
17	M₁₆×C₂₇	C₂²	1	4	yes	complemented
18	C₂×C₈×C₂₇	C₂²	1	2	yes	complemented
19	#88363	C₂²	1	4	yes	complemented
20	C₂×C₂₇⋊C₈	C₂²	1	2	yes	complemented
21	#88359	C₂²	1	8	yes	complemented
22	C₈×D₂₇	C₂²	1	8	yes	complemented
23	C₂²×C₄×D₉	—	3	1	no	—
24	C₂×C₉×M₁₆	D₃	1	1	yes	characteristic
25	C₂×#3667	—	3	1	no	—
26	M₁₆×D₉	—	3	8	no	—
27	C₂×#3663	—	3	2	no	—
28	C₂×C₈×D₉	—	3	2	no	—
29	C₂³×C₂₇	C₂×C₄	1	1	yes	characteristic
30	C₂²×D₂₇	—	2	4	no	—
31	C₂²×D₂₇	C₂×C₄	1	1	yes	characteristic
32	C₂²×D₂₇	C₂×C₄	1	4	yes	—
33	C₂²×D₂₇	C₂×C₄	1	1	yes	characteristic
34	C₂×C₄×C₂₇	C₂×C₄	1	2	yes	—
35	C₂×C₄×C₂₇	C₂×C₄	1	1	yes	characteristic
36	C₂×C₄×C₂₇	C₂³	1	1	yes	characteristic
37	C₂×C₄×C₂₇	C₂³	1	2	yes	—
38	C₂×Dic₂₇	C₂×C₄	1	4	yes	—
39	C₂×Dic₂₇	C₂×C₄	1	1	yes	characteristic
40	C₂×Dic₂₇	C₂×C₄	1	1	yes	characteristic
41	C₄×D₂₇	C₂×C₄	1	2	yes	—
42	C₄×D₂₇	C₂×C₄	1	4	yes	—
43	C₄×D₂₇	C₂×C₄	1	2	yes	—
44	C₄×D₂₇	C₂×C₄	1	4	yes	—
45	C₄×D₂₇	C₂³	1	2	yes	—
46	C₄×D₂₇	C₂³	1	2	yes	—
47	C₈×C₂₇	C₂³	1	4	yes	complemented
48	C₂₇⋊C₈	C₂³	1	4	yes	complemented
49	C₂×D₃×M₁₆	—	9	1	no	—
50	C₂³×D₉	—	3	1	no	—
51	C₂²×C₄×C₉	C₂×D₃	1	1	yes	characteristic
52	C₂²×Dic₉	—	3	1	no	—
53	C₂×C₄×D₉	—	3	1	no	—
54	C₂×C₄×D₉	—	3	4	no	—
55	C₂×C₄×D₉	—	3	1	no	—
56	C₂×C₄×D₉	—	3	1	no	—
57	C₂×C₄×D₉	—	3	1	no	—
58	C₂×C₄×D₉	—	3	4	no	—
59	C₉×M₁₆	C₂×D₃	1	4	yes	—
60	C₂×C₈×C₉	C₂×D₃	1	2	yes	—
61	#3667	—	3	4	no	—
62	C₂×C₉⋊C₈	—	3	2	no	—
63	#3663	—	3	8	no	—
64	C₈×D₉	—	3	8	no	—
65	C₂²×C₂₇	—	2	2	no	—
66	C₂²×C₂₇	C₂²×C₄	1	2	yes	—
67	C₂²×C₂₇	C₂²×C₄	1	1	yes	characteristic
68	C₂×D₂₇	—	2	2	no	—
69	C₂×D₂₇	—	2	8	no	—
70	C₂×D₂₇	M₁₆	1	4	yes	direct product factor
71	C₂×D₂₇	C₂²×C₄	1	2	yes	—
72	C₂×D₂₇	C₂²×C₄	1	2	yes	—
73	C₄×C₂₇	C₂²×C₄	1	1	yes	characteristic
74	C₄×C₂₇	C₂²×C₄	1	2	yes	—
75	C₄×C₂₇	C₂⁴	1	1	yes	characteristic
76	Dic₂₇	C₂²×C₄	1	2	yes	—
77	Dic₂₇	C₂²×C₄	1	2	yes	—
78	C₂²×C₄×D₃	—	9	1	no	—
79	C₂×C₃×M₁₆	D₉	1	1	yes	characteristic
80	C₂×#208	—	9	1	no	—
81	D₃×M₁₆	—	9	8	no	—
82	C₂×ΓL(1,𝔽₂₅)	—	9	2	no	—
83	C₂×D₃×C₈	—	9	2	no	—
84	C₂³×C₉	C₄×D₃	1	1	yes	characteristic
85	C₂²×D₉	—	3	1	no	—
86	C₂²×D₉	—	3	4	no	—
87	C₂²×D₉	—	3	1	no	—
88	C₂²×D₉	—	6	4	no	—
89	C₂×C₄×C₉	C₄×D₃	1	2	yes	—
90	C₂×C₄×C₉	C₄×D₃	1	1	yes	characteristic
91	C₂×C₄×C₉	C₂²×D₃	1	1	yes	characteristic
92	C₂×C₄×C₉	C₂²×D₃	1	2	yes	—
93	C₂×Dic₉	—	3	4	no	—
94	C₂×Dic₉	—	3	1	no	—
95	C₂×Dic₉	—	3	1	no	—
96	C₄×D₉	—	3	2	no	—
97	C₄×D₉	—	3	4	no	—
98	C₄×D₉	—	3	2	no	—
99	C₄×D₉	—	3	4	no	—
100	C₄×D₉	—	3	2	no	—
101	C₄×D₉	—	3	2	no	—
102	C₈×C₉	C₂²×D₃	1	4	yes	—
103	C₉⋊C₈	—	3	4	no	—
104	C₂²×M₁₆	—	27	1	no	Sylow
105	C₂×C₂₇	—	2	2	no	—
106	C₂×C₂₇	C₂×M₁₆	1	2	yes	complemented
107	C₂×C₂₇	C₂³×C₄	1	1	yes	characteristic
108	D₂₇	—	2	2	no	—
109	D₂₇	C₂×M₁₆	1	4	yes	direct product factor
110	C₂³×D₃	—	9	1	no	—
111	C₂²×C₃×C₄	C₂×D₉	1	1	yes	characteristic
112	C₂²×Dic₃	—	9	1	no	—
113	C₂×C₄×D₃	—	9	1	no	—
114	C₂×C₄×D₃	—	9	4	no	—
115	C₂×C₄×D₃	—	9	1	no	—
116	C₂×C₄×D₃	—	9	1	no	—
117	C₂×C₄×D₃	—	9	1	no	—
118	C₂×C₄×D₃	—	9	4	no	—
119	C₃×M₁₆	C₂×D₉	1	4	yes	—
120	C₂×C₃×C₈	C₂×D₉	1	2	yes	—
121	#208	—	9	4	no	—
122	C₂×C₃⋊C₈	—	9	2	no	—
123	ΓL(1,𝔽₂₅)	—	9	8	no	—
124	D₃×C₈	—	9	8	no	—
125	C₂²×C₉	—	2	2	no	—
126	C₂²×C₉	C₂×C₄×D₃	1	2	yes	—
127	C₂²×C₉	C₂×C₄×D₃	1	1	yes	characteristic
128	C₂×D₉	—	3	4	no	—
129	C₂×D₉	—	3	2	no	—
130	C₂×D₉	—	3	2	no	—
131	C₂×D₉	—	6	8	no	—
132	C₂×D₉	—	6	2	no	—
133	C₄×C₉	C₂×C₄×D₃	1	1	yes	characteristic
134	C₄×C₉	C₂×C₄×D₃	1	2	yes	—
135	C₄×C₉	C₂³×D₃	1	1	yes	characteristic
136	Dic₉	—	3	2	no	—
137	Dic₉	—	3	2	no	—
138	C₂³×C₄	—	27	1	no	—
139	C₂×M₁₆	—	27	1	no	—
140	C₂×M₁₆	—	27	8	no	—
141	C₂×M₁₆	—	27	2	no	—
142	C₂×M₁₆	D₂₇	1	1	yes	characteristic, direct product factor
143	C₂²×C₈	—	27	2	no	—
144	C₂₇	C₂²×M₁₆	1	1	yes	characteristic, complemented, Sylow
145	C₂³×C₃	C₄×D₉	1	1	yes	characteristic
146	C₂²×D₃	—	9	1	no	—
147	C₂²×D₃	—	9	4	no	—
148	C₂²×D₃	—	9	1	no	—
149	C₂²×D₃	—	18	4	no	—
150	C₂×C₃×C₄	C₄×D₉	1	2	yes	—
151	C₂×C₃×C₄	C₄×D₉	1	1	yes	characteristic
152	C₂×C₃×C₄	C₂²×D₉	1	2	yes	—
153	C₂×C₃×C₄	C₂²×D₉	1	1	yes	characteristic
154	C₂×Dic₃	—	9	4	no	—
155	C₂×Dic₃	—	9	1	no	—
156	C₂×Dic₃	—	9	1	no	—
157	C₄×D₃	—	9	2	no	—
158	C₄×D₃	—	9	4	no	—
159	C₄×D₃	—	9	2	no	—
160	C₄×D₃	—	9	4	no	—
161	C₄×D₃	—	9	2	no	—
162	C₄×D₃	—	9	2	no	—
163	C₃×C₈	C₂²×D₉	1	4	yes	—
164	C₃⋊C₈	—	9	4	no	—
165	C₂×C₉	—	2	2	no	—
166	C₂×C₉	D₃×M₁₆	1	2	yes	—
167	C₂×C₉	C₂²×C₄×D₃	1	1	yes	characteristic
168	D₉	—	3	4	no	—
169	D₉	—	6	2	no	—
170	C₂⁴	—	27	1	no	—
171	C₂²×C₄	—	27	1	no	—
172	C₂²×C₄	—	27	1	no	—
173	C₂²×C₄	—	27	4	no	—
174	C₂²×C₄	—	27	1	no	—
175	C₂²×C₄	—	27	4	no	—
176	C₂²×C₄	—	27	1	no	—
177	C₂²×C₄	—	27	1	no	—
178	C₂²×C₄	C₂×D₂₇	1	1	yes	characteristic
179	M₁₆	—	27	4	no	—
180	M₁₆	—	27	8	no	—
181	M₁₆	C₂×D₂₇	1	4	yes	direct product factor
182	C₂×C₈	—	27	2	no	—
183	C₂×C₈	—	27	8	no	—
184	C₂×C₈	C₂×D₂₇	1	2	yes	complemented
185	C₂²×C₃	—	2	2	no	—
186	C₂²×C₃	C₂×C₄×D₉	1	2	yes	—
187	C₂²×C₃	C₂×C₄×D₉	1	1	yes	characteristic
188	C₂×D₃	—	9	2	no	—
189	C₂×D₃	—	9	4	no	—
190	C₂×D₃	—	9	2	no	—
191	C₂×D₃	—	18	8	no	—
192	C₂×D₃	—	18	2	no	—
193	C₃×C₄	C₂×C₄×D₉	1	2	yes	—
194	C₃×C₄	C₂×C₄×D₉	1	1	yes	characteristic
195	C₃×C₄	C₂³×D₉	1	1	yes	characteristic
196	Dic₃	—	9	2	no	—
197	Dic₃	—	9	2	no	—
198	C₉	C₂×D₃×M₁₆	1	1	yes	characteristic
199	C₂³	—	27	1	no	—
200	C₂³	—	27	4	no	—
201	C₂³	—	27	1	no	—
202	C₂³	—	54	4	no	—
203	C₂³	C₄×D₂₇	1	1	yes	characteristic
204	C₂×C₄	—	27	4	no	—
205	C₂×C₄	—	27	1	no	—
206	C₂×C₄	—	27	1	no	—
207	C₂×C₄	—	27	2	no	—
208	C₂×C₄	—	27	2	no	—
209	C₂×C₄	—	27	4	no	—
210	C₂×C₄	—	27	4	no	—
211	C₂×C₄	—	27	2	no	—
212	C₂×C₄	—	27	2	no	—
213	C₂×C₄	C₄×D₂₇	1	1	yes	characteristic
214	C₂×C₄	C₄×D₂₇	1	2	yes	—
215	C₂×C₄	C₂²×D₂₇	1	2	yes	—
216	C₂×C₄	C₂²×D₂₇	1	1	yes	characteristic
217	C₈	—	27	4	no	—
218	C₈	C₂²×D₂₇	1	4	yes	complemented
219	C₂×C₃	—	2	2	no	—
220	C₂×C₃	M₁₆×D₉	1	2	yes	—
221	C₂×C₃	C₂²×C₄×D₉	1	1	yes	characteristic
222	D₃	—	9	4	no	—
223	D₃	—	18	2	no	—
224	C₂²	—	2	2	no	—
225	C₂²	—	27	2	no	—
226	C₂²	—	27	4	no	—
227	C₂²	—	27	2	no	—
228	C₂²	—	54	8	no	—
229	C₂²	—	54	2	no	—
230	C₂²	C₂×C₄×D₂₇	1	2	yes	—
231	C₂²	C₂×C₄×D₂₇	1	1	yes	characteristic
232	C₄	—	27	2	no	—
233	C₄	—	27	2	no	—
234	C₄	C₂×C₄×D₂₇	1	2	yes	—
235	C₄	C₂×C₄×D₂₇	1	1	yes	characteristic
236	C₄	C₂³×D₂₇	1	1	yes	characteristic
237	C₃	C₂×M₁₆×D₉	1	1	yes	characteristic
238	C₂	—	2	2	no	—
239	C₂	—	27	4	no	—
240	C₂	—	54	2	no	—
241	C₂	M₁₆×D₂₇	1	2	yes	direct product factor
242	C₂	C₂²×C₄×D₂₇	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₂²×D₂₇
commutator	C₂×C₂₇	C₂³×C₄
Frattini	C₄×C₉	C₂³×D₃
Fitting	C₂×M₁₆×C₂₇	C₂
socle	C₂²×C₃	C₂×C₄×D₉
prime radical	C₂³×D₂₇	C₄
cosocle	C₄×C₂₇	C₂⁴
hypercenter	C₂×M₁₆	D₂₇
hypocenter	C₂₇	C₂²×M₁₆
LCS	G ▷ C₂×C₂₇ ▷ C₂₇
UCS	C₁ ◁ C₂×C₄ ◁ C₂×M₁₆

Complements

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

N	H ≅ G/N	type
2. C₂×M₁₆×C₂₇	C₂	semidirect
3. C₂×#88363	C₂	semidirect
4. M₁₆×D₂₇	C₂	direct (N × H)
5. C₂×#88359	C₂	semidirect
6. C₂×C₈×D₂₇	C₂	semidirect
17. M₁₆×C₂₇	C₂²	semidirect
18. C₂×C₈×C₂₇	C₂²	semidirect
19. #88363	C₂²	semidirect
20. C₂×C₂₇⋊C₈	C₂²	semidirect
21. #88359	C₂²	semidirect
22. C₈×D₂₇	C₂²	semidirect
47. C₈×C₂₇	C₂³	semidirect
48. C₂₇⋊C₈	C₂³	semidirect
70. C₂×D₂₇	M₁₆	direct (N × H)
106. C₂×C₂₇	C₂×M₁₆	semidirect
109. D₂₇	C₂×M₁₆	direct (N × H)
142. C₂×M₁₆	D₂₇	direct (N × H)
144. C₂₇	C₂²×M₁₆	semidirect
181. M₁₆	C₂×D₂₇	direct (N × H)
184. C₂×C₈	C₂×D₂₇	semidirect
218. C₈	C₂²×D₂₇	semidirect
241. C₂	M₁₆×D₂₇	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂²×C₄×D₂₇	2
C₂×M₁₆×C₂₇	2
C₂×#88363	2
M₁₆×D₂₇	2
C₂×#88359	2
C₂×C₈×D₂₇	2
C₂×M₁₆×D₉	3

Subgroups — C2×M16×D27 #13136481

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×M₁₆×D₂₇ #13136481