Subgroups — C₂×#14543842 #17960501

Order 4352.

694 conjugacy classes of subgroups.

The subgroup lattice has 250 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×#10587534	C₂	1	2	yes	maximal
2	#14543842	C₂	1	4	yes	maximal, direct product factor
3	C₂²×C₁₆×D₁₇	C₂	1	1	yes	maximal, characteristic
4	C₂²×C₄⋊#595	—	2	2	no	—
5	C₂×#10587534	—	2	4	no	—
6	C₂×#10587534	—	2	2	no	—
7	C₂×#10587534	C₄	1	8	yes	complemented
8	C₂×#10587534	C₂²	1	4	yes	—
9	C₂²×C₈×D₁₇	C₂²	1	1	yes	characteristic
10	C₂²×#63214	C₄	1	1	yes	characteristic, complemented
11	C₂×C₁₆×D₁₇	—	2	4	no	—
12	C₂×C₁₆×D₁₇	C₄	1	2	yes	complemented
13	C₂×C₁₆×D₁₇	C₂²	1	2	yes	—
14	C₂²×C₁₆×C₁₇	C₄	1	1	yes	characteristic, complemented
15	C₂³×#595	—	4	2	no	—
16	C₂²×C₄×D₁₇	D₄	1	1	yes	characteristic
17	C₂×C₄⋊#595	—	2	8	no	—
18	C₂×C₄⋊#595	—	2	4	no	—
19	C₂×C₄⋊#595	—	4	2	no	—
20	C₂×C₄⋊#595	—	4	4	no	—
21	C₂²×C₈×C₁₇	C₂×C₄	1	1	yes	characteristic
22	C₂²×#3622	Q₈	1	1	yes	characteristic
23	C₂×C₈×D₁₇	C₂×C₄	1	4	yes	—
24	C₂×C₈×D₁₇	C₂×C₄	1	1	yes	characteristic
25	C₂×C₈×D₁₇	D₄	1	4	yes	—
26	C₂×C₈×D₁₇	D₄	1	1	yes	characteristic
27	C₂×C₈×D₁₇	Q₈	1	1	yes	characteristic
28	C₂×C₈×D₁₇	C₂³	1	1	yes	characteristic
29	#10587534	—	2	8	no	—
30	#10587534	—	2	4	no	—
31	#10587534	C₂×C₄	1	8	yes	complemented
32	C₂×C₁₆×C₁₇	—	2	2	no	—
33	C₂×C₁₆×C₁₇	C₂×C₄	1	2	yes	complemented
34	C₂×#63214	—	2	2	no	—
35	C₂×#63214	C₂×C₄	1	2	yes	complemented
36	C₁₆×D₁₇	—	2	4	no	—
37	C₁₆×D₁₇	—	2	4	no	—
38	C₂³×D₁₇	D₈	1	1	yes	characteristic
39	C₂²×#595	—	4	8	no	—
40	C₂²×#595	—	4	4	no	—
41	C₂²×#595	—	8	4	no	—
42	C₂²×#595	—	8	4	no	—
43	C₂²×C₄×C₁₇	C₄⋊C₄	1	1	yes	characteristic
44	C₂²×Dic₁₇	Q₁₆	1	1	yes	characteristic
45	C₂×C₄×D₁₇	C₂²⋊C₄	1	4	yes	—
46	C₂×C₄×D₁₇	C₄⋊C₄	1	1	yes	characteristic
47	C₂×C₄×D₁₇	D₈	1	1	yes	characteristic
48	C₂×C₄×D₁₇	SD₁₆	1	4	yes	—
49	C₂×C₄×D₁₇	Q₁₆	1	1	yes	characteristic
50	C₂×C₄×D₁₇	C₂×D₄	1	1	yes	characteristic
51	C₄⋊#595	—	2	8	no	—
52	C₄⋊#595	—	4	8	no	—
53	C₄⋊#595	—	4	4	no	—
54	C₂×C₈×C₁₇	C₄²	1	2	yes	—
55	C₂×C₈×C₁₇	C₂²⋊C₄	1	2	yes	—
56	C₂×C₈×C₁₇	C₄⋊C₄	1	1	yes	characteristic
57	C₂×C₈×C₁₇	C₂²×C₄	1	1	yes	characteristic
58	C₂×#3622	C₄⋊C₄	1	1	yes	characteristic
59	C₂×#3622	C₄⋊C₄	1	4	yes	—
60	C₂×#3622	C₂×Q₈	1	1	yes	characteristic
61	C₈×D₁₇	C₄²	1	2	yes	—
62	C₈×D₁₇	C₂²⋊C₄	1	4	yes	—
63	C₈×D₁₇	C₂²⋊C₄	1	2	yes	—
64	C₈×D₁₇	C₄⋊C₄	1	4	yes	—
65	C₈×D₁₇	C₂²×C₄	1	2	yes	—
66	C₈×D₁₇	C₂×D₄	1	2	yes	—
67	C₁₆×C₁₇	—	2	2	no	—
68	#63214	—	2	2	no	—
69	C₂×#1380	—	17	1	no	maximal, Sylow
70	C₂³×C₁₇	#107	1	1	yes	characteristic
71	C₂²×D₁₇	#102	1	4	yes	—
72	C₂²×D₁₇	#107	1	1	yes	characteristic
73	C₂²×D₁₇	D₁₆	1	2	yes	—
74	C₂²×D₁₇	SD₃₂	1	4	yes	—
75	C₂²×D₁₇	Q₃₂	1	2	yes	—
76	C₂²×D₁₇	C₂×D₈	1	1	yes	characteristic
77	C₂×#595	—	4	8	no	—
78	C₂×#595	—	8	16	no	—
79	C₂×#595	—	8	4	no	—
80	C₂×#595	—	8	4	no	—
81	C₂×C₄×C₁₇	#95	1	2	yes	—
82	C₂×C₄×C₁₇	#106	1	2	yes	—
83	C₂×C₄×C₁₇	#107	1	1	yes	characteristic
84	C₂×C₄×C₁₇	C₂×C₄⋊C₄	1	1	yes	characteristic
85	C₂×Dic₁₇	#103	1	4	yes	—
86	C₂×Dic₁₇	#107	1	1	yes	characteristic
87	C₂×Dic₁₇	C₂×Q₁₆	1	1	yes	characteristic
88	C₄×D₁₇	#95	1	2	yes	—
89	C₄×D₁₇	#102	1	4	yes	—
90	C₄×D₁₇	#103	1	4	yes	—
91	C₄×D₁₇	#106	1	2	yes	—
92	C₄×D₁₇	C₂×C₂²⋊C₄	1	2	yes	—
93	C₄×D₁₇	C₂×SD₁₆	1	2	yes	—
94	C₈×C₁₇	#95	1	2	yes	—
95	C₈×C₁₇	C₂×C₄²	1	1	yes	characteristic
96	C₈×C₁₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
97	#3622	#95	1	2	yes	—
98	#3622	C₂×C₄⋊C₄	1	2	yes	—
99	C₂³×C₁₆	—	17	1	no	—
100	C₂²×#107	—	17	2	no	—
101	#1380	—	17	4	no	—
102	C₂²×C₁₇	#340	1	2	yes	—
103	C₂²×C₁₇	#366	1	2	yes	—
104	C₂²×C₁₇	#367	1	2	yes	—
105	C₂²×C₁₇	C₂×#107	1	1	yes	characteristic
106	C₂×D₁₇	#340	1	2	yes	—
107	C₂×D₁₇	#357	1	8	yes	—
108	C₂×D₁₇	#358	1	8	yes	—
109	C₂×D₁₇	#366	1	2	yes	—
110	C₂×D₁₇	#367	1	2	yes	—
111	C₂×D₁₇	C₂×#102	1	2	yes	—
112	C₂×D₁₇	C₂×D₁₆	1	1	yes	characteristic
113	C₂×D₁₇	C₂×SD₃₂	1	2	yes	—
114	C₂×D₁₇	C₂×Q₃₂	1	1	yes	characteristic
115	#595	—	8	8	no	—
116	C₄×C₁₇	#340	1	2	yes	—
117	C₄×C₁₇	C₂×#95	1	1	yes	characteristic
118	C₄×C₁₇	C₂×#106	1	1	yes	characteristic
119	Dic₁₇	#340	1	2	yes	—
120	Dic₁₇	C₂×#103	1	2	yes	—
121	C₂³×C₈	—	17	1	no	—
122	C₂²×C₄⋊C₄	—	34	2	no	—
123	C₂²×C₁₆	—	17	1	no	—
124	C₂²×C₁₆	—	17	2	no	—
125	C₂²×C₁₆	—	17	2	no	—
126	C₂²×C₁₆	—	34	4	no	—
127	C₂²×C₁₆	#595	1	1	yes	characteristic, complemented
128	C₂×#107	—	17	4	no	—
129	C₂×#107	—	17	8	no	—
130	C₂×#107	—	34	2	no	—
131	C₂×#107	—	34	4	no	—
132	C₂×C₁₇	#1380	1	4	yes	complemented
133	C₂×C₁₇	C₂×#340	1	1	yes	characteristic
134	C₂×C₁₇	C₂×#366	1	1	yes	characteristic
135	C₂×C₁₇	C₂×#367	1	1	yes	characteristic
136	D₁₇	#1380	1	4	yes	complemented
137	D₁₇	C₂×#357	1	2	yes	—
138	D₁₇	C₂×#358	1	2	yes	—
139	C₂³×C₄	—	17	1	no	—
140	C₂³×C₄	—	68	2	no	—
141	C₂²×C₈	—	17	1	no	—
142	C₂²×C₈	—	17	4	no	—
143	C₂²×C₈	—	17	1	no	—
144	C₂²×C₈	—	17	1	no	—
145	C₂²×C₈	—	17	4	no	—
146	C₂²×C₈	—	17	1	no	—
147	C₂²×C₈	—	17	1	no	—
148	C₂²×C₈	C₂×#595	1	1	yes	characteristic
149	C₂×C₄⋊C₄	—	34	4	no	—
150	C₂×C₄⋊C₄	—	34	8	no	—
151	C₂×C₄⋊C₄	—	68	4	no	—
152	C₂×C₄⋊C₄	—	68	2	no	—
153	C₂×C₁₆	—	2	2	no	—
154	C₂×C₁₆	—	17	2	no	—
155	C₂×C₁₆	—	34	4	no	—
156	C₂×C₁₆	—	34	4	no	—
157	C₂×C₁₆	—	34	2	no	—
158	C₂×C₁₆	C₂×#595	1	2	yes	complemented
159	#107	—	17	8	no	—
160	#107	—	34	4	no	—
161	#107	—	34	8	no	—
162	C₁₇	C₂×#1380	1	1	yes	characteristic, complemented, Sylow
163	C₂⁴	—	17	1	no	—
164	C₂²×C₄	—	17	1	no	—
165	C₂²×C₄	—	17	1	no	—
166	C₂²×C₄	—	17	1	no	—
167	C₂²×C₄	—	17	4	no	—
168	C₂²×C₄	—	17	1	no	—
169	C₂²×C₄	—	17	1	no	—
170	C₂²×C₄	—	17	4	no	—
171	C₂²×C₄	—	68	4	no	—
172	C₂²×C₄	—	68	8	no	—
173	C₂²×C₄	—	136	4	no	—
174	C₂²×C₄	—	136	4	no	—
175	C₂²×C₄	C₄⋊#595	1	1	yes	characteristic
176	C₂×C₈	—	17	1	no	—
177	C₂×C₈	—	17	1	no	—
178	C₂×C₈	—	17	4	no	—
179	C₂×C₈	—	17	2	no	—
180	C₂×C₈	—	17	2	no	—
181	C₂×C₈	—	17	2	no	—
182	C₂×C₈	—	17	2	no	—
183	C₂×C₈	—	17	4	no	—
184	C₂×C₈	—	17	4	no	—
185	C₂×C₈	C₄×#595	1	2	yes	—
186	C₂×C₈	C₄⋊#595	1	1	yes	characteristic
187	C₂×C₈	C₂²⋊#595	1	2	yes	—
188	C₂×C₈	C₂²×#595	1	1	yes	characteristic
189	C₄⋊C₄	—	34	8	no	—
190	C₄⋊C₄	—	68	4	no	—
191	C₄⋊C₄	—	68	8	no	—
192	C₁₆	—	2	2	no	—
193	C₁₆	—	34	2	no	—
194	C₂³	—	17	2	no	—
195	C₂³	—	17	2	no	—
196	C₂³	—	17	4	no	—
197	C₂³	—	17	1	no	—
198	C₂³	—	17	1	no	—
199	C₂³	—	17	4	no	—
200	C₂³	#10587534	1	1	yes	characteristic
201	C₂×C₄	—	17	1	no	—
202	C₂×C₄	—	17	1	no	—
203	C₂×C₄	—	17	4	no	—
204	C₂×C₄	—	17	2	no	—
205	C₂×C₄	—	17	2	no	—
206	C₂×C₄	—	17	2	no	—
207	C₂×C₄	—	17	2	no	—
208	C₂×C₄	—	17	4	no	—
209	C₂×C₄	—	17	4	no	—
210	C₂×C₄	—	68	8	no	—
211	C₂×C₄	—	136	16	no	—
212	C₂×C₄	—	136	4	no	—
213	C₂×C₄	—	136	4	no	—
214	C₂×C₄	#10587533	1	2	yes	—
215	C₂×C₄	#10587534	1	1	yes	characteristic
216	C₂×C₄	(C₂×#595)⋊C₄	1	2	yes	—
217	C₂×C₄	C₂×C₄⋊#595	1	1	yes	characteristic
218	C₈	—	17	2	no	—
219	C₈	—	17	2	no	—
220	C₈	(C₂×#595)⋊C₄	1	2	yes	—
221	C₈	C₂×C₄×#595	1	1	yes	characteristic
222	C₈	C₂×C₂²⋊#595	1	1	yes	characteristic
223	C₂²	—	17	2	no	—
224	C₂²	—	17	2	no	—
225	C₂²	—	17	8	no	—
226	C₂²	—	17	8	no	—
227	C₂²	—	17	1	no	—
228	C₂²	—	17	2	no	—
229	C₂²	—	17	1	no	—
230	C₂²	—	17	2	no	—
231	C₂²	—	17	2	no	—
232	C₂²	#11763683	1	2	yes	—
233	C₂²	#11763735	1	2	yes	—
234	C₂²	#11763736	1	2	yes	—
235	C₂²	C₂×#10587534	1	1	yes	characteristic
236	C₄	—	17	2	no	—
237	C₄	—	17	2	no	—
238	C₄	—	136	8	no	—
239	C₄	#11763683	1	2	yes	—
240	C₄	C₂×(C₂×#595)⋊C₄	1	1	yes	characteristic
241	C₄	C₂×#10587533	1	1	yes	characteristic
242	C₂	—	17	2	no	—
243	C₂	—	17	2	no	—
244	C₂	—	17	4	no	—
245	C₂	#14543842	1	4	yes	direct product factor
246	C₂	C₂×#11763683	1	1	yes	characteristic
247	C₂	C₂×#11763735	1	1	yes	characteristic
248	C₂	C₂×#11763736	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	#10587534
commutator	C₈×C₁₇	C₂×C₄²
Frattini	C₂×C₈	C₂²×#595
Fitting	C₂²×C₁₆×C₁₇	C₄
socle	C₂³×C₁₇	#107
prime radical	C₂³×D₁₇	D₈
cosocle	C₂×C₈×D₁₇	C₂³
hypercenter	C₂²×C₁₆	#595
hypocenter	C₁₇	C₂×#1380
LCS	G ▷ C₈×C₁₇ ▷ C₄×C₁₇ ▷ C₂×C₁₇ ▷ C₁₇
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄ ◁ C₂²×C₈ ◁ C₂²×C₁₆

Complements

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

N	H ≅ G/N	type
2. #14543842	C₂	direct (N × H)
7. C₂×#10587534	C₄	semidirect
10. C₂²×#63214	C₄	semidirect
12. C₂×C₁₆×D₁₇	C₄	semidirect
14. C₂²×C₁₆×C₁₇	C₄	semidirect
31. #10587534	C₂×C₄	semidirect
33. C₂×C₁₆×C₁₇	C₂×C₄	semidirect
35. C₂×#63214	C₂×C₄	semidirect
127. C₂²×C₁₆	#595	semidirect
132. C₂×C₁₇	#1380	semidirect
136. D₁₇	#1380	semidirect
158. C₂×C₁₆	C₂×#595	semidirect
162. C₁₇	C₂×#1380	semidirect
245. C₂	#14543842	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×#10587534	2
#14543842	2
C₂²×C₁₆×D₁₇	2
C₂×#1380	17

Subgroups — C2×#14543842 #17960501

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#14543842 #17960501