Subgroups — C₂×C₄×D₂₇₂ #17187586

Order 4352.

806 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	C₂²×D₂₇₂	C₂	1	1	yes	maximal, characteristic
2	C₂×C₄×D₁₃₆	C₂	1	2	yes	maximal, complemented
3	C₄×D₂₇₂	C₂	1	8	yes	maximal, direct product factor
4	C₂×#11762635	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂×#11762611	C₂	1	2	yes	maximal, complemented
6	C₂×C₄×C₁₆×C₁₇	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂²×D₁₃₆	C₂²	1	2	yes	—
8	C₂×C₄×D₆₈	—	2	2	no	—
9	C₂×D₂₇₂	C₄	1	8	yes	direct product factor
10	C₂×D₂₇₂	C₂²	1	4	yes	—
11	C₄×D₁₃₆	—	2	4	no	—
12	C₄×D₁₃₆	C₂²	1	8	yes	complemented
13	C₂×#10587489	C₂²	1	2	yes	—
14	C₂×#10587492	—	2	2	no	—
15	#11762635	C₂²	1	4	yes	complemented
16	#11762611	C₂²	1	8	yes	complemented
17	C₂²×C₁₆×C₁₇	C₂²	1	1	yes	characteristic
18	C₂²×C₁₆×C₁₇	C₂²	1	1	yes	characteristic
19	C₂×C₄×C₈×C₁₇	C₂²	1	1	yes	characteristic
20	C₄×C₁₆×C₁₇	C₂²	1	4	yes	complemented
21	C₂²×D₆₈	—	2	2	no	—
22	C₂²×C₄×D₁₇	—	4	2	no	—
23	C₂²×C₈×C₁₇	D₄	1	1	yes	characteristic
24	C₂²×C₈×C₁₇	C₂³	1	1	yes	characteristic
25	C₂×C₄²×C₁₇	D₄	1	1	yes	characteristic
26	C₂×#63227	—	4	2	no	—
27	C₂×C₄⋊Dic₁₇	—	2	2	no	—
28	C₂×D₁₃₆	—	2	2	no	—
29	C₂×D₁₃₆	—	2	4	no	—
30	C₂×D₁₃₆	C₂×C₄	1	8	yes	complemented
31	C₂×D₁₃₆	C₂³	1	4	yes	—
32	C₄×D₆₈	—	2	8	no	—
33	C₄×D₆₈	—	4	4	no	—
34	C₂×C₁₆×C₁₇	—	2	2	no	—
35	C₂×C₁₆×C₁₇	C₂×C₄	1	4	yes	complemented
36	C₂×C₁₆×C₁₇	C₂³	1	2	yes	—
37	C₂×C₁₆×C₁₇	C₂³	1	2	yes	—
38	C₄×C₈×C₁₇	D₄	1	2	yes	—
39	C₄×C₈×C₁₇	C₂³	1	2	yes	—
40	#10587492	—	2	8	no	—
41	#10587489	—	2	2	no	—
42	#10587489	C₂³	1	4	yes	—
43	D₂₇₂	C₂×C₄	1	16	yes	direct product factor
44	C₂³×D₁₇	—	4	2	no	—
45	C₂²×C₄×C₁₇	D₈	1	1	yes	characteristic
46	C₂²×C₄×C₁₇	D₈	1	1	yes	characteristic
47	C₂²×C₄×C₁₇	C₂×D₄	1	1	yes	characteristic
48	C₂²×Dic₁₇	—	4	2	no	—
49	C₂×D₆₈	—	2	8	no	—
50	C₂×D₆₈	—	2	4	no	—
51	C₂×D₆₈	—	4	2	no	—
52	C₂×D₆₈	—	4	4	no	—
53	C₂×C₄×D₁₇	—	4	8	no	—
54	C₂×C₄×D₁₇	—	8	8	no	—
55	C₂×C₈×C₁₇	—	2	2	no	—
56	C₂×C₈×C₁₇	C₂²×C₄	1	2	yes	—
57	C₂×C₈×C₁₇	C₂×D₄	1	2	yes	—
58	C₂×C₈×C₁₇	C₂×D₄	1	1	yes	characteristic
59	C₂×C₈×C₁₇	#41	1	2	yes	—
60	C₂×C₈×C₁₇	C₂⁴	1	1	yes	characteristic
61	C₄²×C₁₇	D₈	1	2	yes	—
62	C₄²×C₁₇	C₂×D₄	1	2	yes	—
63	#63227	—	4	8	no	—
64	C₄⋊Dic₁₇	—	2	4	no	—
65	C₄⋊Dic₁₇	—	4	2	no	—
66	D₁₃₆	—	2	8	no	—
67	D₁₃₆	—	2	4	no	—
68	D₁₃₆	C₂²×C₄	1	8	yes	complemented
69	C₁₆×C₁₇	—	2	2	no	—
70	C₁₆×C₁₇	C₂²×C₄	1	4	yes	complemented
71	C₂×C₄×D₁₆	—	17	1	no	maximal, Sylow
72	C₂³×C₁₇	C₂×D₈	1	1	yes	characteristic
73	C₂²×D₁₇	—	4	4	no	—
74	C₂²×D₁₇	—	4	8	no	—
75	C₂²×D₁₇	—	8	4	no	—
76	C₂²×D₁₇	—	8	4	no	—
77	C₂×C₄×C₁₇	—	2	2	no	—
78	C₂×C₄×C₁₇	D₁₆	1	4	yes	complemented
79	C₂×C₄×C₁₇	C₄×D₄	1	2	yes	—
80	C₂×C₄×C₁₇	C₂×D₈	1	2	yes	—
81	C₂×C₄×C₁₇	C₂×D₈	1	1	yes	characteristic
82	C₂×C₄×C₁₇	C₂×D₈	1	2	yes	—
83	C₂×C₄×C₁₇	#135	1	2	yes	—
84	C₂×C₄×C₁₇	C₂²×D₄	1	1	yes	characteristic
85	C₂×Dic₁₇	—	4	4	no	—
86	C₂×Dic₁₇	—	8	4	no	—
87	D₆₈	—	2	8	no	—
88	D₆₈	—	4	4	no	—
89	D₆₈	—	4	8	no	—
90	C₄×D₁₇	—	8	16	no	—
91	C₈×C₁₇	—	2	2	no	—
92	C₈×C₁₇	C₄×D₄	1	2	yes	—
93	C₈×C₁₇	C₂³×C₄	1	1	yes	characteristic
94	C₈×C₁₇	C₂×#41	1	1	yes	characteristic
95	C₂²×D₁₆	—	17	1	no	—
96	C₂×C₄×D₈	—	17	2	no	—
97	C₄×D₁₆	—	17	8	no	—
98	C₂×#366	—	17	1	no	—
99	C₂×#357	—	17	2	no	—
100	C₂×C₄×C₁₆	D₁₇	1	1	yes	characteristic, complemented
101	C₂²×C₁₇	C₄×D₈	1	2	yes	—
102	C₂²×C₁₇	C₂×D₁₆	1	2	yes	—
103	C₂²×C₁₇	#508	1	2	yes	—
104	C₂²×C₁₇	C₂²×D₈	1	1	yes	characteristic
105	C₂×D₁₇	—	4	8	no	—
106	C₂×D₁₇	—	8	4	no	—
107	C₂×D₁₇	—	8	4	no	—
108	C₂×D₁₇	—	8	16	no	—
109	C₄×C₁₇	—	2	2	no	—
110	C₄×C₁₇	C₄×D₈	1	2	yes	—
111	C₄×C₁₇	C₂×D₁₆	1	4	yes	complemented
112	C₄×C₁₇	C₂×C₄×D₄	1	1	yes	characteristic
113	C₄×C₁₇	C₂×#135	1	1	yes	characteristic
114	Dic₁₇	—	8	4	no	—
115	C₂²×D₈	—	17	2	no	—
116	C₂×C₄×D₄	—	34	2	no	—
117	C₂×D₁₆	—	17	8	no	—
118	C₂×D₁₆	—	17	4	no	—
119	C₂²×C₁₆	C₂×D₁₇	1	1	yes	characteristic
120	C₂²×C₁₆	C₂×D₁₇	1	1	yes	characteristic
121	C₄×D₈	—	17	8	no	—
122	C₄×D₈	—	34	4	no	—
123	C₂×#107	—	17	2	no	—
124	C₂×#102	—	34	2	no	—
125	C₂×C₄×C₈	C₂×D₁₇	1	1	yes	characteristic
126	#366	—	17	4	no	—
127	#357	—	17	8	no	—
128	C₄×C₁₆	C₂×D₁₇	1	4	yes	complemented
129	C₂×C₁₇	C₄×D₁₆	1	4	yes	complemented
130	C₂×C₁₇	C₂×C₄×D₈	1	1	yes	characteristic
131	C₂×C₁₇	C₂²×D₁₆	1	1	yes	characteristic
132	C₂×C₁₇	C₂×#508	1	1	yes	characteristic
133	D₁₇	—	8	8	no	—
134	C₂²×D₄	—	34	2	no	—
135	C₂³×C₄	—	68	2	no	—
136	C₂×D₈	—	17	8	no	—
137	C₂×D₈	—	17	4	no	—
138	C₂×D₈	—	34	2	no	—
139	C₂×D₈	—	34	4	no	—
140	C₂²×C₈	D₆₈	1	1	yes	characteristic
141	C₂²×C₈	C₂²×D₁₇	1	1	yes	characteristic
142	C₄×D₄	—	34	8	no	—
143	C₄×D₄	—	68	4	no	—
144	C₂×C₄⋊C₄	—	34	2	no	—
145	C₂×C₂²⋊C₄	—	68	2	no	—
146	C₂×C₄²	D₆₈	1	1	yes	characteristic
147	D₁₆	—	17	16	no	—
148	C₂×C₁₆	—	2	2	no	—
149	C₂×C₁₆	C₄×D₁₇	1	4	yes	complemented
150	C₂×C₁₆	C₂²×D₁₇	1	2	yes	—
151	C₂×C₁₆	C₂²×D₁₇	1	2	yes	—
152	#107	—	17	4	no	—
153	#107	—	34	2	no	—
154	#102	—	34	8	no	—
155	C₄×C₈	D₆₈	1	2	yes	—
156	C₄×C₈	C₂²×D₁₇	1	2	yes	—
157	C₁₇	C₂×C₄×D₁₆	1	1	yes	characteristic, complemented, Sylow
158	C₂⁴	—	68	2	no	—
159	C₂×D₄	—	34	4	no	—
160	C₂×D₄	—	34	8	no	—
161	C₂×D₄	—	68	2	no	—
162	C₂×D₄	—	68	4	no	—
163	C₂²×C₄	—	68	2	no	—
164	C₂²×C₄	—	68	8	no	—
165	C₂²×C₄	—	136	8	no	—
166	C₂²×C₄	D₁₃₆	1	1	yes	characteristic
167	C₂²×C₄	D₁₃₆	1	1	yes	characteristic
168	C₂²×C₄	C₂×D₆₈	1	1	yes	characteristic
169	D₈	—	17	8	no	—
170	D₈	—	34	4	no	—
171	D₈	—	34	8	no	—
172	C₂×C₈	—	2	2	no	—
173	C₂×C₈	C₂×C₄×D₁₇	1	2	yes	—
174	C₂×C₈	C₂×D₆₈	1	2	yes	—
175	C₂×C₈	C₂×D₆₈	1	1	yes	characteristic
176	C₂×C₈	#63252	1	2	yes	—
177	C₂×C₈	C₂³×D₁₇	1	1	yes	characteristic
178	C₄⋊C₄	—	34	4	no	—
179	C₄⋊C₄	—	68	2	no	—
180	C₂²⋊C₄	—	68	8	no	—
181	C₄²	D₁₃₆	1	2	yes	—
182	C₄²	C₂×D₆₈	1	2	yes	—
183	C₁₆	—	2	2	no	—
184	C₁₆	C₂×C₄×D₁₇	1	4	yes	complemented
185	C₂³	—	68	4	no	—
186	C₂³	—	68	8	no	—
187	C₂³	—	136	4	no	—
188	C₂³	—	136	4	no	—
189	C₂³	C₂×D₁₃₆	1	1	yes	characteristic
190	D₄	—	34	8	no	—
191	D₄	—	68	4	no	—
192	D₄	—	68	8	no	—
193	C₂×C₄	—	2	2	no	—
194	C₂×C₄	—	68	4	no	—
195	C₂×C₄	—	136	4	no	—
196	C₂×C₄	—	136	16	no	—
197	C₂×C₄	D₂₇₂	1	4	yes	direct product factor
198	C₂×C₄	C₄×D₆₈	1	2	yes	—
199	C₂×C₄	C₂×D₁₃₆	1	2	yes	—
200	C₂×C₄	C₂×D₁₃₆	1	2	yes	—
201	C₂×C₄	C₂×D₁₃₆	1	1	yes	characteristic
202	C₂×C₄	#10587590	1	2	yes	—
203	C₂×C₄	C₂²×D₆₈	1	1	yes	characteristic
204	C₈	—	2	2	no	—
205	C₈	C₄×D₆₈	1	2	yes	—
206	C₈	C₂²×C₄×D₁₇	1	1	yes	characteristic
207	C₈	C₂×#63252	1	1	yes	characteristic
208	C₂²	—	68	8	no	—
209	C₂²	—	136	4	no	—
210	C₂²	—	136	16	no	—
211	C₂²	—	136	4	no	—
212	C₂²	C₄×D₁₃₆	1	2	yes	—
213	C₂²	C₂×D₂₇₂	1	2	yes	—
214	C₂²	#11763241	1	2	yes	—
215	C₂²	C₂²×D₁₃₆	1	1	yes	characteristic
216	C₄	—	2	2	no	—
217	C₄	—	136	4	no	—
218	C₄	C₄×D₁₃₆	1	2	yes	—
219	C₄	C₂×D₂₇₂	1	4	yes	direct product factor
220	C₄	C₂×C₄×D₆₈	1	1	yes	characteristic
221	C₄	C₂×#10587590	1	1	yes	characteristic
222	C₂	—	136	8	no	—
223	C₂	C₄×D₂₇₂	1	4	yes	direct product factor
224	C₂	C₂×C₄×D₁₃₆	1	1	yes	characteristic
225	C₂	C₂²×D₂₇₂	1	1	yes	characteristic
226	C₂	C₂×#11763241	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄	D₁₃₆
commutator	C₈×C₁₇	C₂³×C₄
Frattini	C₂×C₈	C₂³×D₁₇
Fitting	C₂×C₄×C₁₆×C₁₇	C₂
socle	C₂³×C₁₇	C₂×D₈
prime radical	C₂²×D₂₇₂	C₂
cosocle	C₂×C₈×C₁₇	C₂⁴
hypercenter	C₂×C₄×C₁₆	D₁₇
hypocenter	C₁₇	C₂×C₄×D₁₆
LCS	G ▷ C₈×C₁₇ ▷ C₄×C₁₇ ▷ C₂×C₁₇ ▷ C₁₇
UCS	C₁ ◁ C₂²×C₄ ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ C₂×C₄×C₁₆

Complements

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

N	H ≅ G/N	type
2. C₂×C₄×D₁₃₆	C₂	semidirect
3. C₄×D₂₇₂	C₂	direct (N × H)
4. C₂×#11762635	C₂	semidirect
5. C₂×#11762611	C₂	semidirect
6. C₂×C₄×C₁₆×C₁₇	C₂	semidirect
9. C₂×D₂₇₂	C₄	direct (N × H)
12. C₄×D₁₃₆	C₂²	semidirect
15. #11762635	C₂²	semidirect
16. #11762611	C₂²	semidirect
20. C₄×C₁₆×C₁₇	C₂²	semidirect
30. C₂×D₁₃₆	C₂×C₄	semidirect
35. C₂×C₁₆×C₁₇	C₂×C₄	semidirect
43. D₂₇₂	C₂×C₄	direct (N × H)
68. D₁₃₆	C₂²×C₄	semidirect
70. C₁₆×C₁₇	C₂²×C₄	semidirect
78. C₂×C₄×C₁₇	D₁₆	semidirect
100. C₂×C₄×C₁₆	D₁₇	semidirect
111. C₄×C₁₇	C₂×D₁₆	semidirect
128. C₄×C₁₆	C₂×D₁₇	semidirect
129. C₂×C₁₇	C₄×D₁₆	semidirect
149. C₂×C₁₆	C₄×D₁₇	semidirect
157. C₁₇	C₂×C₄×D₁₆	semidirect
184. C₁₆	C₂×C₄×D₁₇	semidirect
197. C₂×C₄	D₂₇₂	direct (N × H)
219. C₄	C₂×D₂₇₂	direct (N × H)
223. C₂	C₄×D₂₇₂	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×D₂₇₂	2
C₂×C₄×D₁₃₆	2
C₄×D₂₇₂	2
C₂×#11762635	2
C₂×#11762611	2
C₂×C₄×C₁₆×C₁₇	2
C₂×C₄×D₁₆	17

Subgroups — C2×C4×D272 #17187586

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₄×D₂₇₂ #17187586