Subgroups — C₂×#14544720 #17969673

Order 4352.

1358 conjugacy classes of subgroups.

The subgroup lattice has 270 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	4	yes	maximal
2	C₂×#11763755	C₂	1	2	yes	maximal
3	#14544720	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×#10587534	—	2	16	no	—
8	C₂×#10587534	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	#10587534	—	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	D₈	1	8	yes	—
53	C₂×C₄×D₁₇	Q₁₆	1	8	yes	—
54	C₂×C₄×D₁₇	C₂²×C₄	1	1	yes	characteristic
55	C₂×C₄×D₁₇	C₂×D₄	1	4	yes	—
56	C₂×C₄×D₁₇	C₂×D₄	1	1	yes	characteristic
57	C₂×C₄×D₁₇	C₂×D₄	1	4	yes	—
58	C₂×C₄×D₁₇	C₂×Q₈	1	4	yes	—
59	C₂×C₄×D₁₇	C₂×Q₈	1	1	yes	characteristic
60	C₂×C₄×D₁₇	C₂⁴	1	1	yes	characteristic
61	C₄⋊#595	—	2	16	no	—
62	C₄⋊#595	—	4	8	no	—
63	C₄⋊#595	—	4	32	no	—
64	C₂×C₈×C₁₇	C₄⋊C₄	1	8	yes	complemented
65	C₂×C₈×C₁₇	C₂²×C₄	1	4	yes	—
66	C₄²×C₁₇	C₄⋊C₄	1	2	yes	—
67	C₄²×C₁₇	C₂²×C₄	1	2	yes	—
68	C₄×Dic₁₇	C₄⋊C₄	1	4	yes	—
69	C₄×Dic₁₇	C₄⋊C₄	1	2	yes	—
70	C₄×Dic₁₇	C₂²×C₄	1	2	yes	—
71	C₄×Dic₁₇	C₂×Q₈	1	4	yes	—
72	C₂×#3622	C₄⋊C₄	1	8	yes	complemented
73	C₂×#3622	C₂²×C₄	1	4	yes	—
74	C₈×D₁₇	C₄⋊C₄	1	16	yes	complemented
75	C₈×D₁₇	C₂×D₄	1	8	yes	—
76	C₈×D₁₇	C₂×Q₈	1	8	yes	—
77	C₂×#1845	—	17	1	no	maximal, Sylow
78	C₂³×C₁₇	C₂×C₄⋊C₄	1	1	yes	characteristic
79	C₂²×D₁₇	C₂×C₄⋊C₄	1	1	yes	characteristic
80	C₂²×D₁₇	#127	1	2	yes	—
81	C₂²×D₁₇	C₄⋊Q₈	1	2	yes	—
82	C₂²×D₁₇	C₂×D₈	1	4	yes	—
83	C₂²×D₁₇	C₂×Q₁₆	1	4	yes	—
84	C₂²×D₁₇	C₂²×D₄	1	1	yes	characteristic
85	C₂×#595	—	4	8	no	—
86	C₂×#595	—	4	16	no	—
87	C₂×#595	—	8	16	no	—
88	C₂×C₄×C₁₇	#107	1	8	yes	complemented
89	C₂×C₄×C₁₇	C₂×C₄⋊C₄	1	4	yes	—
90	C₂×C₄×C₁₇	C₂×C₄⋊C₄	1	1	yes	characteristic
91	C₂×C₄×C₁₇	C₂×C₄⋊C₄	1	4	yes	—
92	C₂×C₄×C₁₇	C₂³×C₄	1	1	yes	characteristic
93	C₂×Dic₁₇	#107	1	8	yes	complemented
94	C₂×Dic₁₇	C₂×C₄⋊C₄	1	1	yes	characteristic
95	C₂×Dic₁₇	C₂×C₄⋊C₄	1	4	yes	—
96	C₂×Dic₁₇	C₄⋊Q₈	1	4	yes	—
97	C₂×Dic₁₇	C₂²×Q₈	1	1	yes	characteristic
98	C₄×D₁₇	#107	1	16	yes	complemented
99	C₄×D₁₇	C₂×C₄⋊C₄	1	4	yes	—
100	C₄×D₁₇	#127	1	2	yes	—
101	C₄×D₁₇	C₄⋊Q₈	1	4	yes	—
102	C₄×D₁₇	C₄⋊Q₈	1	2	yes	—
103	C₄×D₁₇	C₂×D₈	1	8	yes	—
104	C₄×D₁₇	C₂×Q₁₆	1	8	yes	—
105	C₄×D₁₇	C₂²×D₄	1	2	yes	—
106	C₄×D₁₇	C₂²×Q₈	1	2	yes	—
107	C₈×C₁₇	C₂×C₄⋊C₄	1	8	yes	complemented
108	#3622	C₂×C₄⋊C₄	1	8	yes	complemented
109	C₂²×#107	—	17	4	no	—
110	C₂²×C₄×C₈	—	17	1	no	—
111	C₂×#384	—	17	2	no	—
112	#1845	—	17	8	no	—
113	C₂²×C₁₇	#384	1	2	yes	—
114	C₂²×C₁₇	C₂×#107	1	4	yes	—
115	C₂²×C₁₇	C₂²×C₄⋊C₄	1	1	yes	characteristic
116	C₂×D₁₇	#384	1	2	yes	—
117	C₂×D₁₇	C₂×#107	1	4	yes	—
118	C₂×D₁₇	#493	1	4	yes	—
119	C₂×D₁₇	#494	1	4	yes	—
120	C₂×D₁₇	#500	1	8	yes	—
121	C₂×D₁₇	C₂×#127	1	1	yes	characteristic
122	C₂×D₁₇	C₂×C₄⋊Q₈	1	1	yes	characteristic
123	C₂×D₁₇	C₂²×D₈	1	2	yes	—
124	C₂×D₁₇	C₂²×Q₁₆	1	2	yes	—
125	#595	—	8	8	no	—
126	C₄×C₁₇	#384	1	2	yes	—
127	C₄×C₁₇	C₂×#107	1	8	yes	complemented
128	C₄×C₁₇	C₂²×C₄⋊C₄	1	2	yes	—
129	Dic₁₇	#384	1	2	yes	—
130	Dic₁₇	C₂×#107	1	8	yes	complemented
131	Dic₁₇	C₂×C₄⋊Q₈	1	2	yes	—
132	C₂³×C₈	—	17	2	no	—
133	C₂²×C₄⋊C₄	—	17	4	no	—
134	C₂²×C₄⋊C₄	—	34	4	no	—
135	C₂²×C₄²	—	17	1	no	—
136	C₂×#107	—	17	16	no	—
137	C₂×#107	—	34	16	no	—
138	C₂×C₄×C₈	—	17	2	no	—
139	C₂×C₄×C₈	—	17	1	no	—
140	C₂×C₄×C₈	—	17	4	no	—
141	C₂×C₄×C₈	—	17	4	no	—
142	C₂×C₄×C₈	#595	1	1	yes	characteristic, complemented
143	#384	—	17	8	no	—
144	#384	—	34	4	no	—
145	C₂×C₁₇	#1845	1	4	yes	complemented
146	C₂×C₁₇	C₂×#384	1	1	yes	characteristic
147	C₂×C₁₇	C₂²×#107	1	2	yes	—
148	D₁₇	#1845	1	4	yes	complemented
149	D₁₇	C₂×#493	1	1	yes	characteristic
150	D₁₇	C₂×#494	1	1	yes	characteristic
151	D₁₇	C₂×#500	1	2	yes	—
152	C₂³×C₄	—	17	2	no	—
153	C₂³×C₄	—	17	1	no	—
154	C₂³×C₄	—	34	4	no	—
155	C₂²×C₈	—	17	2	no	—
156	C₂²×C₈	—	17	8	no	—
157	C₂²×C₈	—	17	8	no	—
158	C₂²×C₈	—	17	4	no	—
159	C₂²×C₈	—	17	4	no	—
160	C₂²×C₈	C₂×#595	1	2	yes	—
161	C₂×C₄⋊C₄	—	17	8	no	—
162	C₂×C₄⋊C₄	—	34	16	no	—
163	C₂×C₄⋊C₄	—	34	4	no	—
164	C₂×C₄⋊C₄	—	34	16	no	—
165	C₂×C₄⋊C₄	—	68	16	no	—
166	C₂×C₄²	—	17	2	no	—
167	C₂×C₄²	—	17	1	no	—
168	C₂×C₄²	—	17	2	no	—
169	C₂×C₄²	—	17	2	no	—
170	C₂×C₄²	—	17	2	no	—
171	C₂×C₄²	—	17	2	no	—
172	C₂×C₄²	C₂×#595	1	1	yes	characteristic
173	#107	—	34	32	no	—
174	C₄×C₈	—	17	8	no	—
175	C₄×C₈	—	17	4	no	—
176	C₄×C₈	C₂×#595	1	4	yes	complemented
177	C₁₇	C₂×#1845	1	1	yes	characteristic, complemented, Sylow
178	C₂⁴	—	17	1	no	—
179	C₂²×C₄	—	17	2	no	—
180	C₂²×C₄	—	17	1	no	—
181	C₂²×C₄	—	17	4	no	—
182	C₂²×C₄	—	17	1	no	—
183	C₂²×C₄	—	17	4	no	—
184	C₂²×C₄	—	17	4	no	—
185	C₂²×C₄	—	17	1	no	—
186	C₂²×C₄	—	17	4	no	—
187	C₂²×C₄	—	17	1	no	—
188	C₂²×C₄	—	17	1	no	—
189	C₂²×C₄	—	17	8	no	—
190	C₂²×C₄	—	17	8	no	—
191	C₂²×C₄	—	34	8	no	—
192	C₂²×C₄	—	68	8	no	—
193	C₂²×C₄	—	68	16	no	—
194	C₂²×C₄	C₄⋊#595	1	2	yes	—
195	C₂²×C₄	C₂²×#595	1	1	yes	characteristic
196	C₂×C₈	—	17	8	no	—
197	C₂×C₈	—	17	4	no	—
198	C₂×C₈	—	17	8	no	—
199	C₂×C₈	—	17	16	no	—
200	C₂×C₈	—	17	8	no	—
201	C₂×C₈	C₄⋊#595	1	8	yes	complemented
202	C₂×C₈	C₂²×#595	1	4	yes	—
203	C₄⋊C₄	—	34	16	no	—
204	C₄⋊C₄	—	68	8	no	—
205	C₄⋊C₄	—	68	32	no	—
206	C₄²	—	17	2	no	—
207	C₄²	—	17	4	no	—
208	C₄²	—	17	4	no	—
209	C₄²	—	17	2	no	—
210	C₄²	C₄⋊#595	1	2	yes	—
211	C₄²	C₂²×#595	1	2	yes	—
212	C₂³	—	17	2	no	—
213	C₂³	—	17	2	no	—
214	C₂³	—	17	4	no	—
215	C₂³	—	17	4	no	—
216	C₂³	—	17	1	no	—
217	C₂³	—	17	1	no	—
218	C₂³	C₂×C₄⋊#595	1	1	yes	characteristic
219	C₂×C₄	—	17	4	no	—
220	C₂×C₄	—	17	8	no	—
221	C₂×C₄	—	17	4	no	—
222	C₂×C₄	—	17	1	no	—
223	C₂×C₄	—	17	1	no	—
224	C₂×C₄	—	17	2	no	—
225	C₂×C₄	—	17	4	no	—
226	C₂×C₄	—	17	2	no	—
227	C₂×C₄	—	17	8	no	—
228	C₂×C₄	—	17	16	no	—
229	C₂×C₄	—	17	8	no	—
230	C₂×C₄	—	17	2	no	—
231	C₂×C₄	—	17	4	no	—
232	C₂×C₄	—	17	2	no	—
233	C₂×C₄	—	68	8	no	—
234	C₂×C₄	—	68	16	no	—
235	C₂×C₄	—	136	16	no	—
236	C₂×C₄	#10587534	1	8	yes	complemented
237	C₂×C₄	C₂×C₄⋊#595	1	1	yes	characteristic
238	C₂×C₄	C₂×C₄⋊#595	1	4	yes	—
239	C₂×C₄	C₂×C₄⋊#595	1	4	yes	—
240	C₂×C₄	C₂³×#595	1	1	yes	characteristic
241	C₈	—	17	8	no	—
242	C₈	C₂×C₄⋊#595	1	8	yes	complemented
243	C₂²	—	17	2	no	—
244	C₂²	—	17	2	no	—
245	C₂²	—	17	4	no	—
246	C₂²	—	17	4	no	—
247	C₂²	—	17	4	no	—
248	C₂²	—	17	8	no	—
249	C₂²	—	17	1	no	—
250	C₂²	—	17	1	no	—
251	C₂²	—	17	2	no	—
252	C₂²	#11763755	1	2	yes	—
253	C₂²	C₂×#10587534	1	4	yes	—
254	C₂²	C₂²×C₄⋊#595	1	1	yes	characteristic
255	C₄	—	17	8	no	—
256	C₄	—	17	2	no	—
257	C₄	—	17	2	no	—
258	C₄	—	136	8	no	—
259	C₄	#11763755	1	2	yes	—
260	C₄	C₂×#10587534	1	8	yes	complemented
261	C₄	C₂²×C₄⋊#595	1	2	yes	—
262	C₂	—	17	1	no	—
263	C₂	—	17	4	no	—
264	C₂	—	17	1	no	—
265	C₂	—	17	2	no	—
266	C₂	#14544720	1	4	yes	direct product factor
267	C₂	C₂×#11763755	1	1	yes	characteristic
268	C₂	C₂²×#10587534	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×#1845
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. #14544720	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
64. C₂×C₈×C₁₇	C₄⋊C₄	semidirect
72. C₂×#3622	C₄⋊C₄	semidirect
74. C₈×D₁₇	C₄⋊C₄	semidirect
88. C₂×C₄×C₁₇	#107	semidirect
93. C₂×Dic₁₇	#107	semidirect
98. C₄×D₁₇	#107	semidirect
107. C₈×C₁₇	C₂×C₄⋊C₄	semidirect
108. #3622	C₂×C₄⋊C₄	semidirect
127. C₄×C₁₇	C₂×#107	semidirect
130. Dic₁₇	C₂×#107	semidirect
142. C₂×C₄×C₈	#595	semidirect
145. C₂×C₁₇	#1845	semidirect
148. D₁₇	#1845	semidirect
176. C₄×C₈	C₂×#595	semidirect
177. C₁₇	C₂×#1845	semidirect
201. C₂×C₈	C₄⋊#595	semidirect
236. C₂×C₄	#10587534	semidirect
242. C₈	C₂×C₄⋊#595	semidirect
260. C₄	C₂×#10587534	semidirect
266. C₂	#14544720	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
C₂×#11763755	2
#14544720	2
C₂×C₄×C₈×D₁₇	2
C₂×#1845	17

Subgroups — C2×#14544720 #17969673

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#14544720 #17969673