Subgroups — C₄×#11762632 #16949708

Order 4352.

330 conjugacy classes of subgroups.

The subgroup lattice has 198 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#11762632	C₂	1	2	yes	maximal
2	C₄×#10587490	C₂	1	1	yes	maximal, characteristic
3	C₄×#10587488	C₂	1	1	yes	maximal, characteristic
4	#14526883	C₂	1	2	yes	maximal
5	C₄×C₁₇×M₃₂	C₂	1	1	yes	maximal, characteristic
6	C₂×#10587490	C₂²	1	2	yes	—
7	C₂×#10587488	C₂²	1	2	yes	—
8	C₄×#63223	—	2	1	no	—
9	C₄×C₄⋊Dic₁₇	—	2	1	no	—
10	#11762632	C₄	1	8	yes	direct product factor
11	#11762574	—	2	1	no	—
12	#11762572	—	2	1	no	—
13	C₂×C₁₇×M₃₂	C₂²	1	1	yes	characteristic
14	C₂×C₁₇×M₃₂	C₂²	1	1	yes	characteristic
15	C₂×C₄×C₈×C₁₇	C₂²	1	1	yes	characteristic
16	C₁₇×#346	C₄	1	2	yes	complemented
17	C₄×C₁₆×C₁₇	C₄	1	2	yes	complemented
18	C₂²×C₈×C₁₇	D₄	1	1	yes	characteristic
19	C₂²×C₈×C₁₇	C₂³	1	1	yes	characteristic
20	C₂×C₄²×C₁₇	D₄	1	1	yes	characteristic
21	C₂×C₄⋊Dic₁₇	—	2	2	no	—
22	C₂×C₄×Dic₁₇	—	4	1	no	—
23	C₂×#63223	—	2	2	no	—
24	C₁₇×M₃₂	—	2	2	no	—
25	C₁₇×M₃₂	C₂×C₄	1	4	yes	—
26	C₂×C₁₆×C₁₇	C₂×C₄	1	2	yes	—
27	C₂×C₁₆×C₁₇	C₂×C₄	1	2	yes	—
28	C₄×C₈×C₁₇	C₂×C₄	1	1	yes	characteristic
29	C₄×C₈×C₁₇	C₂×C₄	1	1	yes	characteristic
30	C₄×C₈×C₁₇	Q₈	1	1	yes	characteristic
31	C₄×C₈×C₁₇	Q₈	1	1	yes	characteristic
32	#10587502	—	4	1	no	—
33	#10587490	—	2	2	no	—
34	#10587490	C₂×C₄	1	4	yes	—
35	#10587488	—	2	2	no	—
36	#10587488	C₂×C₄	1	4	yes	—
37	#10587474	—	4	1	no	—
38	C₄×#3622	—	4	1	no	—
39	C₂²×C₄×C₁₇	SD₁₆	1	1	yes	characteristic
40	C₂²×C₄×C₁₇	SD₁₆	1	1	yes	characteristic
41	C₂²×C₄×C₁₇	C₂×D₄	1	1	yes	characteristic
42	C₂²×Dic₁₇	—	4	2	no	—
43	C₂×C₈×C₁₇	—	2	2	no	—
44	C₂×C₈×C₁₇	C₄⋊C₄	1	1	yes	characteristic
45	C₂×C₈×C₁₇	C₄⋊C₄	1	1	yes	characteristic
46	C₂×C₈×C₁₇	C₂²×C₄	1	2	yes	—
47	C₂×C₈×C₁₇	C₂²×C₄	1	1	yes	characteristic
48	C₂×C₈×C₁₇	C₂×Q₈	1	1	yes	characteristic
49	C₂×C₈×C₁₇	#41	1	2	yes	—
50	C₄²×C₁₇	C₄⋊C₄	1	1	yes	characteristic
51	C₄²×C₁₇	C₄⋊C₄	1	1	yes	characteristic
52	C₄²×C₁₇	SD₁₆	1	1	yes	characteristic
53	C₄²×C₁₇	SD₁₆	1	1	yes	characteristic
54	C₄⋊Dic₁₇	—	2	4	no	—
55	C₄⋊Dic₁₇	—	4	2	no	—
56	C₄×Dic₁₇	—	8	2	no	—
57	#63223	—	2	4	no	—
58	#63223	—	4	2	no	—
59	C₂×#3622	—	4	2	no	—
60	C₁₆×C₁₇	—	2	2	no	—
61	C₁₆×C₁₇	C₄²	1	4	yes	complemented
62	C₄×#365	—	17	1	no	maximal, Sylow
63	C₂³×C₁₇	C₂×SD₁₆	1	1	yes	characteristic
64	C₂×C₄×C₁₇	—	2	2	no	—
65	C₂×C₄×C₁₇	—	4	1	no	—
66	C₂×C₄×C₁₇	#106	1	1	yes	characteristic
67	C₂×C₄×C₁₇	#106	1	1	yes	characteristic
68	C₂×C₄×C₁₇	#106	1	1	yes	characteristic
69	C₂×C₄×C₁₇	#106	1	1	yes	characteristic
70	C₂×C₄×C₁₇	C₂×C₄⋊C₄	1	1	yes	characteristic
71	C₂×C₄×C₁₇	C₄×D₄	1	2	yes	—
72	C₂×C₄×C₁₇	C₂×SD₁₆	1	1	yes	characteristic
73	C₂×C₄×C₁₇	#135	1	2	yes	—
74	C₂×Dic₁₇	—	4	4	no	—
75	C₂×Dic₁₇	—	8	2	no	—
76	C₂×Dic₁₇	—	8	2	no	—
77	C₈×C₁₇	—	2	1	no	—
78	C₈×C₁₇	—	2	1	no	—
79	C₈×C₁₇	C₂×C₄²	1	1	yes	characteristic
80	C₈×C₁₇	#117	1	1	yes	characteristic
81	C₈×C₁₇	C₄×Q₈	1	2	yes	—
82	#3622	—	4	4	no	—
83	C₂×#365	—	17	2	no	—
84	C₄×M₃₂	D₁₇	1	1	yes	characteristic
85	C₄×#108	—	17	1	no	—
86	C₄×#106	—	17	1	no	—
87	#1383	—	17	2	no	—
88	C₂²×C₁₇	—	2	1	no	—
89	C₂²×C₁₇	—	2	1	no	—
90	C₂²×C₁₇	C₂×#106	1	1	yes	characteristic
91	C₂²×C₁₇	C₄×SD₁₆	1	2	yes	—
92	C₄×C₁₇	—	2	1	no	—
93	C₄×C₁₇	—	2	1	no	—
94	C₄×C₁₇	—	2	1	no	—
95	C₄×C₁₇	#365	1	2	yes	complemented
96	C₄×C₁₇	C₄×C₄⋊C₄	1	1	yes	characteristic
97	C₄×C₁₇	#427	1	1	yes	characteristic
98	C₄×C₁₇	C₄×SD₁₆	1	2	yes	—
99	Dic₁₇	—	8	4	no	—
100	C₂×M₃₂	C₂×D₁₇	1	1	yes	characteristic
101	C₂×M₃₂	C₂×D₁₇	1	1	yes	characteristic
102	C₂×#108	—	17	2	no	—
103	C₂×#106	—	17	2	no	—
104	C₄×M₁₆	—	34	1	no	—
105	C₂×C₄×C₈	C₂×D₁₇	1	1	yes	characteristic
106	C₄×C₄⋊C₄	—	34	1	no	—
107	#365	—	17	8	no	—
108	#346	Dic₁₇	1	2	yes	complemented
109	C₄×C₁₆	Dic₁₇	1	2	yes	complemented
110	#341	—	34	1	no	—
111	#340	—	34	1	no	—
112	C₂×C₁₇	—	2	2	no	—
113	C₂×C₁₇	C₄×#106	1	1	yes	characteristic
114	C₂×C₁₇	C₂×#365	1	1	yes	characteristic
115	C₂×C₁₇	#2155	1	1	yes	characteristic
116	C₂×M₁₆	—	34	2	no	—
117	C₂²×C₈	D₆₈	1	1	yes	characteristic
118	C₂²×C₈	C₂²×D₁₇	1	1	yes	characteristic
119	C₂×C₄⋊C₄	—	34	2	no	—
120	C₂×C₄²	—	68	1	no	—
121	C₂×C₄²	D₆₈	1	1	yes	characteristic
122	M₃₂	—	2	2	no	—
123	M₃₂	C₄×D₁₇	1	4	yes	—
124	C₂×C₁₆	C₂×Dic₁₇	1	2	yes	—
125	C₂×C₁₆	C₂×Dic₁₇	1	2	yes	—
126	#108	—	17	4	no	—
127	#108	—	34	2	no	—
128	#106	—	17	4	no	—
129	#106	—	34	2	no	—
130	#97	—	68	1	no	—
131	C₄×C₈	—	68	1	no	—
132	C₄×C₈	Dic₃₄	1	1	yes	characteristic
133	C₄×C₈	Dic₃₄	1	1	yes	characteristic
134	C₄×C₈	C₂×Dic₁₇	1	1	yes	characteristic
135	C₄×C₈	C₂×Dic₁₇	1	1	yes	characteristic
136	#95	—	68	1	no	—
137	C₁₇	C₄×#365	1	1	yes	characteristic, complemented, Sylow
138	C₂²×C₄	—	68	2	no	—
139	C₂²×C₄	#63219	1	1	yes	characteristic
140	C₂²×C₄	#63219	1	1	yes	characteristic
141	C₂²×C₄	C₂×D₆₈	1	1	yes	characteristic
142	M₁₆	—	34	4	no	—
143	M₁₆	—	68	2	no	—
144	C₂×C₈	—	2	2	no	—
145	C₂×C₈	—	68	2	no	—
146	C₂×C₈	C₄⋊Dic₁₇	1	1	yes	characteristic
147	C₂×C₈	C₄⋊Dic₁₇	1	1	yes	characteristic
148	C₂×C₈	C₂×Dic₃₄	1	1	yes	characteristic
149	C₂×C₈	C₂×C₄×D₁₇	1	2	yes	—
150	C₂×C₈	#63252	1	2	yes	—
151	C₂×C₈	C₂²×Dic₁₇	1	1	yes	characteristic
152	C₄⋊C₄	—	34	4	no	—
153	C₄⋊C₄	—	68	2	no	—
154	C₄²	—	136	2	no	—
155	C₄²	#63219	1	1	yes	characteristic
156	C₄²	#63219	1	1	yes	characteristic
157	C₄²	C₄⋊Dic₁₇	1	1	yes	characteristic
158	C₄²	C₄⋊Dic₁₇	1	1	yes	characteristic
159	C₁₆	—	2	2	no	—
160	C₁₆	C₄×Dic₁₇	1	4	yes	complemented
161	C₂³	C₂×#63219	1	1	yes	characteristic
162	C₂×C₄	—	2	2	no	—
163	C₂×C₄	—	4	1	no	—
164	C₂×C₄	—	68	4	no	—
165	C₂×C₄	—	136	2	no	—
166	C₂×C₄	—	136	2	no	—
167	C₂×C₄	#10587488	1	1	yes	characteristic
168	C₂×C₄	#10587488	1	1	yes	characteristic
169	C₂×C₄	#10587488	1	1	yes	characteristic
170	C₂×C₄	#10587488	1	1	yes	characteristic
171	C₂×C₄	C₄×D₆₈	1	2	yes	—
172	C₂×C₄	C₂×#63219	1	1	yes	characteristic
173	C₂×C₄	#10587590	1	2	yes	—
174	C₂×C₄	C₂×C₄⋊Dic₁₇	1	1	yes	characteristic
175	C₈	—	2	1	no	—
176	C₈	—	2	1	no	—
177	C₈	—	68	4	no	—
178	C₈	C₄×Dic₃₄	1	2	yes	—
179	C₈	C₂×C₄×Dic₁₇	1	1	yes	characteristic
180	C₈	#10587612	1	1	yes	characteristic
181	C₂²	—	2	1	no	—
182	C₂²	—	2	1	no	—
183	C₂²	C₂×#10587488	1	1	yes	characteristic
184	C₂²	C₄×#63219	1	2	yes	—
185	C₄	—	2	1	no	—
186	C₄	—	2	1	no	—
187	C₄	—	2	1	no	—
188	C₄	—	136	4	no	—
189	C₄	#11762632	1	2	yes	direct product factor
190	C₄	C₄×C₄⋊Dic₁₇	1	1	yes	characteristic
191	C₄	#11762847	1	1	yes	characteristic
192	C₄	C₄×#63219	1	2	yes	—
193	C₂	—	2	2	no	—
194	C₂	C₄×#10587488	1	1	yes	characteristic
195	C₂	C₂×#11762632	1	1	yes	characteristic
196	C₂	#14530557	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	#10587488
commutator	C₈×C₁₇	C₂×C₄²
Frattini	C₂²×C₈	C₂²×D₁₇
Fitting	C₄×C₁₇×M₃₂	C₂
socle	C₂²×C₁₇	C₂×#106
prime radical	C₂³×C₁₇	C₂×SD₁₆
cosocle	C₂²×C₈×C₁₇	C₂³
hypercenter	C₄×M₃₂	D₁₇
hypocenter	C₁₇	C₄×#365
LCS	G ▷ C₈×C₁₇ ▷ C₄×C₁₇ ▷ C₂×C₁₇ ▷ C₁₇
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ C₄×M₃₂

Complements

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

N	H ≅ G/N	type
10. #11762632	C₄	direct (N × H)
16. C₁₇×#346	C₄	semidirect
17. C₄×C₁₆×C₁₇	C₄	semidirect
61. C₁₆×C₁₇	C₄²	semidirect
95. C₄×C₁₇	#365	semidirect
108. #346	Dic₁₇	semidirect
109. C₄×C₁₆	Dic₁₇	semidirect
137. C₁₇	C₄×#365	semidirect
160. C₁₆	C₄×Dic₁₇	semidirect
189. C₄	#11762632	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂×#11762632	2
C₄×#10587490	2
C₄×#10587488	2
#14526883	2
C₄×C₁₇×M₃₂	2
C₄×#365	17

Subgroups — C4×#11762632 #16949708

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₄×#11762632 #16949708