Subgroups — C₂×#14528941 #17034722

Order 4352.

1358 conjugacy classes of subgroups.

The subgroup lattice has 192 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×#10587488	C₂	1	4	yes	maximal
2	C₂×#11762679	C₂	1	2	yes	maximal
3	#14528941	C₂	1	8	yes	maximal, direct product factor
4	C₂²×C₄×C₈×C₁₇	C₂	1	1	yes	maximal, characteristic
5	C₂²×C₄⋊Dic₁₇	—	2	4	no	—
6	C₂²×C₄⋊Dic₁₇	C₂²	1	4	yes	—
7	C₂×#10587488	—	2	16	no	—
8	C₂×#10587488	C₂²	1	16	yes	—
9	#11762679	—	2	4	no	—
10	#11762679	C₂²	1	8	yes	—
11	C₂³×C₈×C₁₇	C₂²	1	2	yes	—
12	C₂²×C₄²×C₁₇	C₂²	1	1	yes	characteristic
13	C₂×C₄×C₈×C₁₇	C₄	1	8	yes	complemented
14	C₂×C₄×C₈×C₁₇	C₂²	1	4	yes	—
15	C₂³×C₄×C₁₇	D₄	1	2	yes	—
16	C₂³×C₄×C₁₇	C₂³	1	1	yes	characteristic
17	C₂³×Dic₁₇	—	2	4	no	—
18	C₂²×C₈×C₁₇	C₂×C₄	1	8	yes	—
19	C₂²×C₈×C₁₇	D₄	1	8	yes	—
20	C₂²×C₈×C₁₇	Q₈	1	8	yes	—
21	C₂²×C₈×C₁₇	C₂³	1	4	yes	—
22	C₂×C₄²×C₁₇	C₂×C₄	1	4	yes	—
23	C₂×C₄²×C₁₇	D₄	1	2	yes	—
24	C₂×C₄²×C₁₇	Q₈	1	4	yes	—
25	C₂×C₄²×C₁₇	C₂³	1	2	yes	—
26	C₂×C₄⋊Dic₁₇	—	2	16	no	—
27	C₂×C₄⋊Dic₁₇	—	2	4	no	—
28	C₂×C₄⋊Dic₁₇	—	2	16	no	—
29	C₂×C₄⋊Dic₁₇	—	4	16	no	—
30	C₂×C₄⋊Dic₁₇	C₂³	1	8	yes	—
31	C₄×C₈×C₁₇	C₂×C₄	1	16	yes	complemented
32	#10587488	—	2	32	no	—
33	C₂⁴×C₁₇	C₂×D₄	1	1	yes	characteristic
34	C₂²×C₄×C₁₇	C₄⋊C₄	1	8	yes	—
35	C₂²×C₄×C₁₇	SD₁₆	1	16	yes	—
36	C₂²×C₄×C₁₇	C₂²×C₄	1	2	yes	—
37	C₂²×C₄×C₁₇	C₂×D₄	1	1	yes	characteristic
38	C₂²×C₄×C₁₇	C₂×D₄	1	4	yes	—
39	C₂²×C₄×C₁₇	C₂×D₄	1	4	yes	—
40	C₂²×C₄×C₁₇	C₂×Q₈	1	4	yes	—
41	C₂²×C₄×C₁₇	C₂×Q₈	1	2	yes	—
42	C₂²×C₄×C₁₇	C₂⁴	1	1	yes	characteristic
43	C₂²×Dic₁₇	—	2	8	no	—
44	C₂²×Dic₁₇	—	4	8	no	—
45	C₂²×Dic₁₇	—	4	16	no	—
46	C₂×C₈×C₁₇	C₄⋊C₄	1	32	yes	complemented
47	C₂×C₈×C₁₇	C₂²×C₄	1	8	yes	—
48	C₂×C₈×C₁₇	C₂×D₄	1	8	yes	—
49	C₂×C₈×C₁₇	C₂×Q₈	1	8	yes	—
50	C₄²×C₁₇	C₄⋊C₄	1	8	yes	—
51	C₄²×C₁₇	C₂²×C₄	1	4	yes	—
52	C₄²×C₁₇	C₂×Q₈	1	4	yes	—
53	C₄⋊Dic₁₇	—	2	16	no	—
54	C₄⋊Dic₁₇	—	4	8	no	—
55	C₄⋊Dic₁₇	—	4	32	no	—
56	C₂×#1844	—	17	1	no	maximal, Sylow
57	C₂³×C₁₇	C₂×C₄⋊C₄	1	2	yes	—
58	C₂³×C₁₇	#127	1	2	yes	—
59	C₂³×C₁₇	C₄⋊Q₈	1	2	yes	—
60	C₂³×C₁₇	C₂×SD₁₆	1	8	yes	—
61	C₂³×C₁₇	C₂²×D₄	1	1	yes	characteristic
62	C₂×C₄×C₁₇	#106	1	32	yes	complemented
63	C₂×C₄×C₁₇	C₂×C₄⋊C₄	1	8	yes	—
64	C₂×C₄×C₁₇	C₂×C₄⋊C₄	1	2	yes	—
65	C₂×C₄×C₁₇	C₂×C₄⋊C₄	1	8	yes	—
66	C₂×C₄×C₁₇	#127	1	2	yes	—
67	C₂×C₄×C₁₇	C₄⋊Q₈	1	8	yes	—
68	C₂×C₄×C₁₇	C₄⋊Q₈	1	2	yes	—
69	C₂×C₄×C₁₇	C₂×SD₁₆	1	16	yes	—
70	C₂×C₄×C₁₇	C₂³×C₄	1	1	yes	characteristic
71	C₂×C₄×C₁₇	C₂²×D₄	1	2	yes	—
72	C₂×C₄×C₁₇	C₂²×Q₈	1	2	yes	—
73	C₂×C₄×C₁₇	C₂²×Q₈	1	1	yes	characteristic
74	C₂×Dic₁₇	—	4	8	no	—
75	C₂×Dic₁₇	—	4	16	no	—
76	C₂×Dic₁₇	—	8	16	no	—
77	C₈×C₁₇	C₂×C₄⋊C₄	1	16	yes	complemented
78	C₂²×#106	—	17	4	no	—
79	C₂²×C₄×C₈	D₁₇	1	1	yes	characteristic
80	C₂×#384	—	17	2	no	—
81	#1844	—	17	8	no	—
82	C₂²×C₁₇	#384	1	4	yes	—
83	C₂²×C₁₇	C₂×#106	1	8	yes	—
84	C₂²×C₁₇	#492	1	8	yes	—
85	C₂²×C₁₇	#498	1	8	yes	—
86	C₂²×C₁₇	C₂²×C₄⋊C₄	1	1	yes	characteristic
87	C₂²×C₁₇	C₂×#127	1	1	yes	characteristic
88	C₂²×C₁₇	C₂×C₄⋊Q₈	1	1	yes	characteristic
89	C₂²×C₁₇	C₂²×SD₁₆	1	4	yes	—
90	C₄×C₁₇	#384	1	4	yes	—
91	C₄×C₁₇	C₂×#106	1	16	yes	complemented
92	C₄×C₁₇	C₂²×C₄⋊C₄	1	2	yes	—
93	C₄×C₁₇	C₂×C₄⋊Q₈	1	2	yes	—
94	Dic₁₇	—	8	8	no	—
95	C₂³×C₈	C₂×D₁₇	1	2	yes	—
96	C₂²×C₄⋊C₄	—	17	4	no	—
97	C₂²×C₄⋊C₄	—	34	4	no	—
98	C₂²×C₄²	C₂×D₁₇	1	1	yes	characteristic
99	C₂×#106	—	17	16	no	—
100	C₂×#106	—	34	16	no	—
101	C₂×C₄×C₈	Dic₁₇	1	8	yes	complemented
102	C₂×C₄×C₈	C₂×D₁₇	1	4	yes	—
103	#384	—	17	8	no	—
104	#384	—	34	4	no	—
105	C₂×C₁₇	#1844	1	8	yes	complemented
106	C₂×C₁₇	C₂×#384	1	1	yes	characteristic
107	C₂×C₁₇	C₂²×#106	1	2	yes	—
108	C₂×C₁₇	C₂×#492	1	2	yes	—
109	C₂×C₁₇	C₂×#498	1	2	yes	—
110	C₂³×C₄	—	34	4	no	—
111	C₂³×C₄	D₆₈	1	2	yes	—
112	C₂³×C₄	C₂²×D₁₇	1	1	yes	characteristic
113	C₂²×C₈	Dic₃₄	1	8	yes	—
114	C₂²×C₈	D₆₈	1	8	yes	—
115	C₂²×C₈	C₂×Dic₁₇	1	8	yes	—
116	C₂²×C₈	C₂²×D₁₇	1	4	yes	—
117	C₂×C₄⋊C₄	—	17	8	no	—
118	C₂×C₄⋊C₄	—	34	16	no	—
119	C₂×C₄⋊C₄	—	34	4	no	—
120	C₂×C₄⋊C₄	—	34	16	no	—
121	C₂×C₄⋊C₄	—	68	16	no	—
122	C₂×C₄²	Dic₃₄	1	4	yes	—
123	C₂×C₄²	D₆₈	1	2	yes	—
124	C₂×C₄²	C₂×Dic₁₇	1	4	yes	—
125	C₂×C₄²	C₂²×D₁₇	1	2	yes	—
126	#106	—	34	32	no	—
127	C₄×C₈	C₂×Dic₁₇	1	16	yes	complemented
128	C₁₇	C₂×#1844	1	1	yes	characteristic, complemented, Sylow
129	C₂⁴	C₂×D₆₈	1	1	yes	characteristic
130	C₂²×C₄	—	34	8	no	—
131	C₂²×C₄	—	68	8	no	—
132	C₂²×C₄	—	68	16	no	—
133	C₂²×C₄	#63219	1	16	yes	—
134	C₂²×C₄	C₄⋊Dic₁₇	1	8	yes	—
135	C₂²×C₄	C₂×Dic₃₄	1	2	yes	—
136	C₂²×C₄	C₂×Dic₃₄	1	4	yes	—
137	C₂²×C₄	C₂×D₆₈	1	4	yes	—
138	C₂²×C₄	C₂×D₆₈	1	1	yes	characteristic
139	C₂²×C₄	C₂×D₆₈	1	4	yes	—
140	C₂²×C₄	C₂²×Dic₁₇	1	2	yes	—
141	C₂²×C₄	C₂³×D₁₇	1	1	yes	characteristic
142	C₂×C₈	C₄⋊Dic₁₇	1	32	yes	complemented
143	C₂×C₈	C₂×Dic₃₄	1	8	yes	—
144	C₂×C₈	C₂×D₆₈	1	8	yes	—
145	C₂×C₈	C₂²×Dic₁₇	1	8	yes	—
146	C₄⋊C₄	—	34	16	no	—
147	C₄⋊C₄	—	68	8	no	—
148	C₄⋊C₄	—	68	32	no	—
149	C₄²	C₄⋊Dic₁₇	1	8	yes	—
150	C₄²	C₂×Dic₃₄	1	4	yes	—
151	C₄²	C₂²×Dic₁₇	1	4	yes	—
152	C₂³	#10587555	1	2	yes	—
153	C₂³	#10587560	1	2	yes	—
154	C₂³	C₂×#63219	1	8	yes	—
155	C₂³	C₂×C₄⋊Dic₁₇	1	2	yes	—
156	C₂³	C₂²×D₆₈	1	1	yes	characteristic
157	C₂×C₄	—	68	16	no	—
158	C₂×C₄	—	68	8	no	—
159	C₂×C₄	—	136	16	no	—
160	C₂×C₄	#10587488	1	32	yes	complemented
161	C₂×C₄	#10587555	1	8	yes	—
162	C₂×C₄	#10587555	1	2	yes	—
163	C₂×C₄	#10587560	1	2	yes	—
164	C₂×C₄	C₂×#63219	1	16	yes	—
165	C₂×C₄	C₂×C₄⋊Dic₁₇	1	8	yes	—
166	C₂×C₄	C₂×C₄⋊Dic₁₇	1	8	yes	—
167	C₂×C₄	C₂×C₄⋊Dic₁₇	1	2	yes	—
168	C₂×C₄	C₂²×Dic₃₄	1	2	yes	—
169	C₂×C₄	C₂²×Dic₃₄	1	1	yes	characteristic
170	C₂×C₄	C₂²×D₆₈	1	2	yes	—
171	C₂×C₄	C₂³×Dic₁₇	1	1	yes	characteristic
172	C₈	C₂×C₄⋊Dic₁₇	1	16	yes	complemented
173	C₂²	#11762679	1	4	yes	—
174	C₂²	C₂×#10587488	1	8	yes	—
175	C₂²	#11763182	1	8	yes	—
176	C₂²	#11763207	1	8	yes	—
177	C₂²	C₂²×C₄⋊Dic₁₇	1	1	yes	characteristic
178	C₂²	C₂×#10587560	1	1	yes	characteristic
179	C₂²	C₂×#10587555	1	1	yes	characteristic
180	C₂²	C₂²×#63219	1	4	yes	—
181	C₄	—	136	8	no	—
182	C₄	#11762679	1	4	yes	—
183	C₄	C₂×#10587488	1	16	yes	complemented
184	C₄	C₂²×C₄⋊Dic₁₇	1	2	yes	—
185	C₄	C₂×#10587555	1	2	yes	—
186	C₂	#14528941	1	8	yes	direct product factor
187	C₂	C₂×#11762679	1	1	yes	characteristic
188	C₂	C₂²×#10587488	1	2	yes	—
189	C₂	C₂×#11763182	1	2	yes	—
190	C₂	C₂×#11763207	1	2	yes	—

	subgroup	G/H
center	C₂⁴	C₂×D₆₈
commutator	C₂×C₄×C₁₇	C₂³×C₄
Frattini	C₂²×C₄	C₂³×D₁₇
Fitting	C₂²×C₄×C₈×C₁₇	C₂
socle, prime radical	C₂⁴×C₁₇	C₂×D₄
cosocle	C₂²×C₄×C₁₇	C₂⁴
hypercenter	C₂²×C₄×C₈	D₁₇
hypocenter	C₁₇	C₂×#1844
LCS	G ▷ C₂×C₄×C₁₇ ▷ C₂×C₁₇ ▷ C₁₇
UCS	C₁ ◁ C₂⁴ ◁ C₂²×C₄² ◁ C₂²×C₄×C₈

Complements

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

N	H ≅ G/N	type
3. #14528941	C₂	direct (N × H)
13. C₂×C₄×C₈×C₁₇	C₄	semidirect
31. C₄×C₈×C₁₇	C₂×C₄	semidirect
46. C₂×C₈×C₁₇	C₄⋊C₄	semidirect
62. C₂×C₄×C₁₇	#106	semidirect
77. C₈×C₁₇	C₂×C₄⋊C₄	semidirect
91. C₄×C₁₇	C₂×#106	semidirect
101. C₂×C₄×C₈	Dic₁₇	semidirect
105. C₂×C₁₇	#1844	semidirect
127. C₄×C₈	C₂×Dic₁₇	semidirect
128. C₁₇	C₂×#1844	semidirect
142. C₂×C₈	C₄⋊Dic₁₇	semidirect
160. C₂×C₄	#10587488	semidirect
172. C₈	C₂×C₄⋊Dic₁₇	semidirect
183. C₄	C₂×#10587488	semidirect
186. C₂	#14528941	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×#10587488	2
C₂×#11762679	2
#14528941	2
C₂²×C₄×C₈×C₁₇	2
C₂×#1844	17

Subgroups — C2×#14528941 #17034722

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#14528941 #17034722