Subgroups — C₂×#14544719 #17969672

Order 4352.

1358 conjugacy classes of subgroups.

The subgroup lattice has 258 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×#10587533	C₂	1	4	yes	maximal
2	C₂×#11763755	C₂	1	2	yes	maximal
3	#14544719	C₂	1	8	yes	maximal, direct product factor
4	C₂×C₄×C₈×D₁₇	C₂	1	1	yes	maximal, characteristic
5	C₂²×C₄⋊#595	—	2	4	no	—
6	C₂²×C₄⋊#595	C₂²	1	4	yes	—
7	C₂×#10587533	—	2	16	no	—
8	C₂×#10587533	C₂²	1	16	yes	—
9	#11763755	—	2	4	no	—
10	#11763755	C₂²	1	8	yes	—
11	C₂²×C₈×D₁₇	C₂²	1	2	yes	—
12	C₂×C₄²×D₁₇	C₂²	1	1	yes	characteristic
13	C₂×C₄×#3622	C₄	1	1	yes	characteristic, complemented
14	C₂×C₈×Dic₁₇	C₄	1	2	yes	complemented
15	C₄×C₈×D₁₇	C₄	1	4	yes	complemented
16	C₄×C₈×D₁₇	C₂²	1	4	yes	—
17	C₂×C₄×C₈×C₁₇	C₄	1	1	yes	characteristic, complemented
18	C₂³×#595	—	2	4	no	—
19	C₂²×C₄×D₁₇	D₄	1	2	yes	—
20	C₂²×C₄×D₁₇	C₂³	1	1	yes	characteristic
21	C₂×C₄⋊#595	—	2	16	no	—
22	C₂×C₄⋊#595	—	2	4	no	—
23	C₂×C₄⋊#595	—	2	16	no	—
24	C₂×C₄⋊#595	—	4	16	no	—
25	C₂×C₄⋊#595	C₂³	1	8	yes	—
26	C₂²×C₈×C₁₇	C₂×C₄	1	2	yes	—
27	C₂×C₄²×C₁₇	C₂×C₄	1	1	yes	characteristic
28	C₂×C₄×Dic₁₇	C₂×C₄	1	1	yes	characteristic
29	C₂×C₄×Dic₁₇	Q₈	1	2	yes	—
30	C₂²×#3622	C₂×C₄	1	2	yes	—
31	C₂×C₈×D₁₇	C₂×C₄	1	4	yes	—
32	C₂×C₈×D₁₇	D₄	1	8	yes	—
33	C₂×C₈×D₁₇	Q₈	1	8	yes	—
34	C₂×C₈×D₁₇	C₂³	1	4	yes	—
35	C₄²×D₁₇	C₂×C₄	1	2	yes	—
36	C₄²×D₁₇	D₄	1	2	yes	—
37	C₄²×D₁₇	Q₈	1	2	yes	—
38	C₄²×D₁₇	C₂³	1	2	yes	—
39	#10587533	—	2	32	no	—
40	C₄×C₈×C₁₇	C₂×C₄	1	4	yes	complemented
41	C₈×Dic₁₇	C₂×C₄	1	8	yes	complemented
42	C₄×#3622	C₂×C₄	1	4	yes	complemented
43	C₂³×D₁₇	C₂×D₄	1	1	yes	characteristic
44	C₂²×#595	—	2	8	no	—
45	C₂²×#595	—	4	8	no	—
46	C₂²×#595	—	4	16	no	—
47	C₂²×C₄×C₁₇	C₄⋊C₄	1	2	yes	—
48	C₂²×C₄×C₁₇	C₂²×C₄	1	1	yes	characteristic
49	C₂²×Dic₁₇	C₄⋊C₄	1	2	yes	—
50	C₂²×Dic₁₇	C₂×Q₈	1	1	yes	characteristic
51	C₂×C₄×D₁₇	C₄⋊C₄	1	4	yes	—
52	C₂×C₄×D₁₇	SD₁₆	1	16	yes	—
53	C₂×C₄×D₁₇	C₂²×C₄	1	1	yes	characteristic
54	C₂×C₄×D₁₇	C₂×D₄	1	4	yes	—
55	C₂×C₄×D₁₇	C₂×D₄	1	1	yes	characteristic
56	C₂×C₄×D₁₇	C₂×D₄	1	4	yes	—
57	C₂×C₄×D₁₇	C₂×Q₈	1	4	yes	—
58	C₂×C₄×D₁₇	C₂×Q₈	1	1	yes	characteristic
59	C₂×C₄×D₁₇	C₂⁴	1	1	yes	characteristic
60	C₄⋊#595	—	2	16	no	—
61	C₄⋊#595	—	4	8	no	—
62	C₄⋊#595	—	4	32	no	—
63	C₂×C₈×C₁₇	C₄⋊C₄	1	8	yes	complemented
64	C₂×C₈×C₁₇	C₂²×C₄	1	4	yes	—
65	C₄²×C₁₇	C₄⋊C₄	1	2	yes	—
66	C₄²×C₁₇	C₂²×C₄	1	2	yes	—
67	C₄×Dic₁₇	C₄⋊C₄	1	4	yes	—
68	C₄×Dic₁₇	C₄⋊C₄	1	2	yes	—
69	C₄×Dic₁₇	C₂²×C₄	1	2	yes	—
70	C₄×Dic₁₇	C₂×Q₈	1	4	yes	—
71	C₂×#3622	C₄⋊C₄	1	8	yes	complemented
72	C₂×#3622	C₂²×C₄	1	4	yes	—
73	C₈×D₁₇	C₄⋊C₄	1	16	yes	complemented
74	C₈×D₁₇	C₂×D₄	1	8	yes	—
75	C₈×D₁₇	C₂×Q₈	1	8	yes	—
76	C₂×#1844	—	17	1	no	maximal, Sylow
77	C₂³×C₁₇	C₂×C₄⋊C₄	1	1	yes	characteristic
78	C₂²×D₁₇	C₂×C₄⋊C₄	1	1	yes	characteristic
79	C₂²×D₁₇	#127	1	2	yes	—
80	C₂²×D₁₇	C₄⋊Q₈	1	2	yes	—
81	C₂²×D₁₇	C₂×SD₁₆	1	8	yes	—
82	C₂²×D₁₇	C₂²×D₄	1	1	yes	characteristic
83	C₂×#595	—	4	8	no	—
84	C₂×#595	—	4	16	no	—
85	C₂×#595	—	8	16	no	—
86	C₂×C₄×C₁₇	#106	1	8	yes	complemented
87	C₂×C₄×C₁₇	C₂×C₄⋊C₄	1	4	yes	—
88	C₂×C₄×C₁₇	C₂×C₄⋊C₄	1	1	yes	characteristic
89	C₂×C₄×C₁₇	C₂×C₄⋊C₄	1	4	yes	—
90	C₂×C₄×C₁₇	C₂³×C₄	1	1	yes	characteristic
91	C₂×Dic₁₇	#106	1	8	yes	complemented
92	C₂×Dic₁₇	C₂×C₄⋊C₄	1	1	yes	characteristic
93	C₂×Dic₁₇	C₂×C₄⋊C₄	1	4	yes	—
94	C₂×Dic₁₇	C₄⋊Q₈	1	4	yes	—
95	C₂×Dic₁₇	C₂²×Q₈	1	1	yes	characteristic
96	C₄×D₁₇	#106	1	16	yes	complemented
97	C₄×D₁₇	C₂×C₄⋊C₄	1	4	yes	—
98	C₄×D₁₇	#127	1	2	yes	—
99	C₄×D₁₇	C₄⋊Q₈	1	4	yes	—
100	C₄×D₁₇	C₄⋊Q₈	1	2	yes	—
101	C₄×D₁₇	C₂×SD₁₆	1	16	yes	—
102	C₄×D₁₇	C₂²×D₄	1	2	yes	—
103	C₄×D₁₇	C₂²×Q₈	1	2	yes	—
104	C₈×C₁₇	C₂×C₄⋊C₄	1	8	yes	complemented
105	#3622	C₂×C₄⋊C₄	1	8	yes	complemented
106	C₂²×#106	—	17	4	no	—
107	C₂²×C₄×C₈	—	17	1	no	—
108	C₂×#384	—	17	2	no	—
109	#1844	—	17	8	no	—
110	C₂²×C₁₇	#384	1	2	yes	—
111	C₂²×C₁₇	C₂×#106	1	4	yes	—
112	C₂²×C₁₇	C₂²×C₄⋊C₄	1	1	yes	characteristic
113	C₂×D₁₇	#384	1	2	yes	—
114	C₂×D₁₇	C₂×#106	1	4	yes	—
115	C₂×D₁₇	#492	1	8	yes	—
116	C₂×D₁₇	#498	1	8	yes	—
117	C₂×D₁₇	C₂×#127	1	1	yes	characteristic
118	C₂×D₁₇	C₂×C₄⋊Q₈	1	1	yes	characteristic
119	C₂×D₁₇	C₂²×SD₁₆	1	4	yes	—
120	#595	—	8	8	no	—
121	C₄×C₁₇	#384	1	2	yes	—
122	C₄×C₁₇	C₂×#106	1	8	yes	complemented
123	C₄×C₁₇	C₂²×C₄⋊C₄	1	2	yes	—
124	Dic₁₇	#384	1	2	yes	—
125	Dic₁₇	C₂×#106	1	8	yes	complemented
126	Dic₁₇	C₂×C₄⋊Q₈	1	2	yes	—
127	C₂³×C₈	—	17	2	no	—
128	C₂²×C₄⋊C₄	—	17	4	no	—
129	C₂²×C₄⋊C₄	—	34	4	no	—
130	C₂²×C₄²	—	17	1	no	—
131	C₂×#106	—	17	16	no	—
132	C₂×#106	—	34	16	no	—
133	C₂×C₄×C₈	—	17	2	no	—
134	C₂×C₄×C₈	—	17	1	no	—
135	C₂×C₄×C₈	—	17	4	no	—
136	C₂×C₄×C₈	—	17	4	no	—
137	C₂×C₄×C₈	#595	1	1	yes	characteristic, complemented
138	#384	—	17	8	no	—
139	#384	—	34	4	no	—
140	C₂×C₁₇	#1844	1	4	yes	complemented
141	C₂×C₁₇	C₂×#384	1	1	yes	characteristic
142	C₂×C₁₇	C₂²×#106	1	2	yes	—
143	D₁₇	#1844	1	4	yes	complemented
144	D₁₇	C₂×#492	1	2	yes	—
145	D₁₇	C₂×#498	1	2	yes	—
146	C₂³×C₄	—	17	2	no	—
147	C₂³×C₄	—	17	1	no	—
148	C₂³×C₄	—	34	4	no	—
149	C₂²×C₈	—	17	2	no	—
150	C₂²×C₈	—	17	8	no	—
151	C₂²×C₈	—	17	8	no	—
152	C₂²×C₈	—	17	4	no	—
153	C₂²×C₈	—	17	4	no	—
154	C₂²×C₈	C₂×#595	1	2	yes	—
155	C₂×C₄⋊C₄	—	17	8	no	—
156	C₂×C₄⋊C₄	—	34	16	no	—
157	C₂×C₄⋊C₄	—	34	4	no	—
158	C₂×C₄⋊C₄	—	34	16	no	—
159	C₂×C₄⋊C₄	—	68	16	no	—
160	C₂×C₄²	—	17	2	no	—
161	C₂×C₄²	—	17	1	no	—
162	C₂×C₄²	—	17	2	no	—
163	C₂×C₄²	—	17	2	no	—
164	C₂×C₄²	—	17	2	no	—
165	C₂×C₄²	—	17	2	no	—
166	C₂×C₄²	C₂×#595	1	1	yes	characteristic
167	#106	—	34	32	no	—
168	C₄×C₈	—	17	8	no	—
169	C₄×C₈	—	17	4	no	—
170	C₄×C₈	C₂×#595	1	4	yes	complemented
171	C₁₇	C₂×#1844	1	1	yes	characteristic, complemented, Sylow
172	C₂⁴	—	17	1	no	—
173	C₂²×C₄	—	17	2	no	—
174	C₂²×C₄	—	17	1	no	—
175	C₂²×C₄	—	17	4	no	—
176	C₂²×C₄	—	17	1	no	—
177	C₂²×C₄	—	17	4	no	—
178	C₂²×C₄	—	17	4	no	—
179	C₂²×C₄	—	17	1	no	—
180	C₂²×C₄	—	17	4	no	—
181	C₂²×C₄	—	17	1	no	—
182	C₂²×C₄	—	17	1	no	—
183	C₂²×C₄	—	17	16	no	—
184	C₂²×C₄	—	34	8	no	—
185	C₂²×C₄	—	68	8	no	—
186	C₂²×C₄	—	68	16	no	—
187	C₂²×C₄	C₄⋊#595	1	2	yes	—
188	C₂²×C₄	C₂²×#595	1	1	yes	characteristic
189	C₂×C₈	—	17	8	no	—
190	C₂×C₈	—	17	4	no	—
191	C₂×C₈	—	17	8	no	—
192	C₂×C₈	—	17	16	no	—
193	C₂×C₈	—	17	8	no	—
194	C₂×C₈	C₄⋊#595	1	8	yes	complemented
195	C₂×C₈	C₂²×#595	1	4	yes	—
196	C₄⋊C₄	—	34	16	no	—
197	C₄⋊C₄	—	68	8	no	—
198	C₄⋊C₄	—	68	32	no	—
199	C₄²	—	17	2	no	—
200	C₄²	—	17	4	no	—
201	C₄²	—	17	4	no	—
202	C₄²	—	17	2	no	—
203	C₄²	C₄⋊#595	1	2	yes	—
204	C₄²	C₂²×#595	1	2	yes	—
205	C₂³	—	17	2	no	—
206	C₂³	—	17	2	no	—
207	C₂³	—	17	8	no	—
208	C₂³	—	17	1	no	—
209	C₂³	—	17	1	no	—
210	C₂³	C₂×C₄⋊#595	1	1	yes	characteristic
211	C₂×C₄	—	17	4	no	—
212	C₂×C₄	—	17	8	no	—
213	C₂×C₄	—	17	4	no	—
214	C₂×C₄	—	17	1	no	—
215	C₂×C₄	—	17	1	no	—
216	C₂×C₄	—	17	2	no	—
217	C₂×C₄	—	17	4	no	—
218	C₂×C₄	—	17	2	no	—
219	C₂×C₄	—	17	16	no	—
220	C₂×C₄	—	17	16	no	—
221	C₂×C₄	—	17	2	no	—
222	C₂×C₄	—	17	4	no	—
223	C₂×C₄	—	17	2	no	—
224	C₂×C₄	—	68	8	no	—
225	C₂×C₄	—	68	16	no	—
226	C₂×C₄	—	136	16	no	—
227	C₂×C₄	#10587533	1	8	yes	complemented
228	C₂×C₄	C₂×C₄⋊#595	1	1	yes	characteristic
229	C₂×C₄	C₂×C₄⋊#595	1	4	yes	—
230	C₂×C₄	C₂×C₄⋊#595	1	4	yes	—
231	C₂×C₄	C₂³×#595	1	1	yes	characteristic
232	C₈	—	17	8	no	—
233	C₈	C₂×C₄⋊#595	1	8	yes	complemented
234	C₂²	—	17	4	no	—
235	C₂²	—	17	4	no	—
236	C₂²	—	17	8	no	—
237	C₂²	—	17	8	no	—
238	C₂²	—	17	1	no	—
239	C₂²	—	17	1	no	—
240	C₂²	—	17	2	no	—
241	C₂²	#11763755	1	2	yes	—
242	C₂²	C₂×#10587533	1	4	yes	—
243	C₂²	C₂²×C₄⋊#595	1	1	yes	characteristic
244	C₄	—	17	8	no	—
245	C₄	—	17	2	no	—
246	C₄	—	17	2	no	—
247	C₄	—	136	8	no	—
248	C₄	#11763755	1	2	yes	—
249	C₄	C₂×#10587533	1	8	yes	complemented
250	C₄	C₂²×C₄⋊#595	1	2	yes	—
251	C₂	—	17	2	no	—
252	C₂	—	17	4	no	—
253	C₂	—	17	2	no	—
254	C₂	#14544719	1	4	yes	direct product factor
255	C₂	C₂×#11763755	1	1	yes	characteristic
256	C₂	C₂²×#10587533	1	2	yes	—

	subgroup	G/H
center	C₂³	C₂×C₄⋊#595
commutator	C₂×C₄×C₁₇	C₂³×C₄
Frattini	C₂×C₄	C₂³×#595
Fitting	C₂×C₄×C₈×C₁₇	C₄
socle	C₂³×C₁₇	C₂×C₄⋊C₄
prime radical	C₂³×D₁₇	C₂×D₄
cosocle	C₂×C₄×D₁₇	C₂⁴
hypercenter	C₂×C₄×C₈	#595
hypocenter	C₁₇	C₂×#1844
LCS	G ▷ C₂×C₄×C₁₇ ▷ C₂×C₁₇ ▷ C₁₇
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×C₄×C₈

Complements

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

N	H ≅ G/N	type
3. #14544719	C₂	direct (N × H)
13. C₂×C₄×#3622	C₄	semidirect
14. C₂×C₈×Dic₁₇	C₄	semidirect
15. C₄×C₈×D₁₇	C₄	semidirect
17. C₂×C₄×C₈×C₁₇	C₄	semidirect
40. C₄×C₈×C₁₇	C₂×C₄	semidirect
41. C₈×Dic₁₇	C₂×C₄	semidirect
42. C₄×#3622	C₂×C₄	semidirect
63. C₂×C₈×C₁₇	C₄⋊C₄	semidirect
71. C₂×#3622	C₄⋊C₄	semidirect
73. C₈×D₁₇	C₄⋊C₄	semidirect
86. C₂×C₄×C₁₇	#106	semidirect
91. C₂×Dic₁₇	#106	semidirect
96. C₄×D₁₇	#106	semidirect
104. C₈×C₁₇	C₂×C₄⋊C₄	semidirect
105. #3622	C₂×C₄⋊C₄	semidirect
122. C₄×C₁₇	C₂×#106	semidirect
125. Dic₁₇	C₂×#106	semidirect
137. C₂×C₄×C₈	#595	semidirect
140. C₂×C₁₇	#1844	semidirect
143. D₁₇	#1844	semidirect
170. C₄×C₈	C₂×#595	semidirect
171. C₁₇	C₂×#1844	semidirect
194. C₂×C₈	C₄⋊#595	semidirect
227. C₂×C₄	#10587533	semidirect
233. C₈	C₂×C₄⋊#595	semidirect
249. C₄	C₂×#10587533	semidirect
254. C₂	#14544719	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×#10587533	2
C₂×#11763755	2
#14544719	2
C₂×C₄×C₈×D₁₇	2
C₂×#1844	17

Subgroups — C2×#14544719 #17969672

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#14544719 #17969672