Subgroups — #17189647

Order 4352.

590 conjugacy classes of subgroups.

The subgroup lattice has 268 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#14539298	C₂	1	1	yes	maximal, characteristic, complemented
2	#14531029	C₂	1	2	yes	maximal, complemented
3	#14531025	C₂	1	2	yes	maximal, complemented
4	#14531020	C₂	1	4	yes	maximal, complemented
5	#14530491	C₂	1	4	yes	maximal, complemented
6	C₂×#11762585	C₂	1	1	yes	maximal, characteristic, complemented
7	C₁₇×#3147	C₂	1	1	yes	maximal, characteristic
8	#11763366	—	2	1	no	—
9	C₂²×#63214	C₂²	1	2	yes	—
10	#11763214	C₂²	1	2	yes	—
11	#11763211	—	2	1	no	—
12	#11763211	C₂²	1	2	yes	—
13	#11763207	—	2	1	no	—
14	#11762847	—	2	1	no	—
15	#11762847	C₂²	1	2	yes	—
16	#11762613	C₂²	1	8	yes	—
17	#11762610	C₂²	1	8	yes	—
18	#11762585	C₂²	1	2	yes	—
19	#11762585	C₂²	1	2	yes	—
20	C₂×C₁₇×#135	—	2	1	no	—
21	C₂×C₁₇×#135	C₂²	1	2	yes	—
22	C₁₇×#532	—	2	1	no	—
23	C₁₇×#495	—	2	1	no	—
24	C₁₇×#495	C₂²	1	2	yes	—
25	C₁₇×#494	C₂²	1	1	yes	characteristic
26	C₁₇×#493	C₂²	1	1	yes	characteristic
27	C₁₇×#492	—	2	1	no	—
28	C₂×C₄×C₈×C₁₇	C₂²	1	1	yes	characteristic
29	C₂×#41×C₁₇	—	2	2	no	—
30	C₁₇×#135	—	2	8	no	—
31	C₁₇×#135	—	4	4	no	—
32	C₂×Q₁₆×C₁₇	—	2	1	no	—
33	C₂×Q₁₆×C₁₇	C₂³	1	2	yes	—
34	C₂×SD₁₆×C₁₇	—	2	2	no	—
35	C₂×SD₁₆×C₁₇	—	2	2	no	—
36	C₂×D₈×C₁₇	—	2	1	no	—
37	C₂×D₈×C₁₇	C₂³	1	2	yes	—
38	C₂²×C₈×C₁₇	D₄	1	1	yes	characteristic
39	C₂²×C₈×C₁₇	C₂³	1	1	yes	characteristic
40	C₁₇×C₄⋊Q₈	—	2	1	no	—
41	C₁₇×#127	—	2	1	no	—
42	C₁₇×#124	—	2	2	no	—
43	C₁₇×#121	—	4	2	no	—
44	C₄×D₄×C₁₇	—	4	2	no	—
45	C₂×C₄²×C₁₇	D₄	1	1	yes	characteristic
46	#10587612	—	2	2	no	—
47	#10587610	—	4	2	no	—
48	#10587556	—	2	2	no	—
49	#10587555	—	2	2	no	—
50	C₄×Dic₃₄	—	4	2	no	—
51	C₄×C₈×C₁₇	D₄	1	1	yes	characteristic
52	C₄×C₈×C₁₇	D₄	1	1	yes	characteristic
53	C₄×C₈×C₁₇	C₂³	1	1	yes	characteristic
54	C₄×C₈×C₁₇	C₂³	1	1	yes	characteristic
55	#10587489	—	2	2	no	—
56	#10587489	C₂³	1	4	yes	—
57	#10587488	—	2	2	no	—
58	#10587488	—	2	2	no	—
59	C₂×#63214	—	2	4	no	—
60	C₂×#63214	C₂³	1	4	yes	—
61	#41×C₁₇	—	4	4	no	—
62	#41×C₁₇	—	4	4	no	—
63	C₂×Q₈×C₁₇	—	2	2	no	—
64	C₂×D₄×C₁₇	—	2	2	no	—
65	C₂×D₄×C₁₇	—	4	2	no	—
66	C₂×D₄×C₁₇	—	4	2	no	—
67	C₂²×C₄×C₁₇	—	4	2	no	—
68	C₂²×C₄×C₁₇	D₈	1	2	yes	—
69	C₂²×C₄×C₁₇	C₂×D₄	1	1	yes	characteristic
70	C₂×Dic₃₄	—	4	2	no	—
71	Q₁₆×C₁₇	—	2	4	no	—
72	Q₁₆×C₁₇	—	4	2	no	—
73	SD₁₆×C₁₇	—	4	4	no	—
74	SD₁₆×C₁₇	—	4	4	no	—
75	D₈×C₁₇	—	2	4	no	—
76	D₈×C₁₇	—	4	2	no	—
77	C₂×C₈×C₁₇	SD₁₆	1	4	yes	—
78	C₂×C₈×C₁₇	C₂×D₄	1	1	yes	characteristic
79	C₂×C₈×C₁₇	C₂×D₄	1	1	yes	characteristic
80	C₂×C₈×C₁₇	C₂×D₄	1	1	yes	characteristic
81	C₂×C₈×C₁₇	#41	1	4	yes	—
82	C₂×C₈×C₁₇	C₂⁴	1	1	yes	characteristic
83	C₄⋊C₄×C₁₇	—	4	2	no	—
84	C₂²⋊C₄×C₁₇	—	4	4	no	—
85	C₄²×C₁₇	D₈	1	2	yes	—
86	C₄²×C₁₇	C₂×D₄	1	1	yes	characteristic
87	C₄²×C₁₇	C₂×D₄	1	1	yes	characteristic
88	C₂²⋊Dic₁₇	—	4	2	no	—
89	C₄⋊Dic₁₇	—	2	4	no	—
90	C₄⋊Dic₁₇	—	4	2	no	—
91	#63225	—	4	4	no	—
92	C₄×Dic₁₇	—	4	2	no	—
93	#63214	—	2	4	no	—
94	#33701	—	17	1	no	maximal, Sylow
95	C₂³×C₁₇	—	4	2	no	—
96	C₂³×C₁₇	C₂×D₈	1	1	yes	characteristic
97	Q₈×C₁₇	—	4	2	no	—
98	Q₈×C₁₇	—	4	2	no	—
99	D₄×C₁₇	—	4	2	no	—
100	D₄×C₁₇	—	4	2	no	—
101	D₄×C₁₇	—	8	4	no	—
102	D₄×C₁₇	—	8	4	no	—
103	C₂×C₄×C₁₇	—	4	2	no	—
104	C₂×C₄×C₁₇	—	4	2	no	—
105	C₂×C₄×C₁₇	—	8	4	no	—
106	C₂×C₄×C₁₇	#124	1	2	yes	—
107	C₂×C₄×C₁₇	C₂×D₈	1	2	yes	—
108	C₂×C₄×C₁₇	C₂×D₈	1	2	yes	—
109	C₂×C₄×C₁₇	C₂×D₈	1	1	yes	characteristic
110	C₂×C₄×C₁₇	C₂×SD₁₆	1	2	yes	—
111	C₂×C₄×C₁₇	C₂²×D₄	1	1	yes	characteristic
112	C₂×Dic₁₇	—	4	4	no	—
113	Dic₃₄	—	8	4	no	—
114	C₈×C₁₇	#124	1	2	yes	—
115	C₈×C₁₇	C₂×SD₁₆	1	2	yes	—
116	C₈×C₁₇	C₂×SD₁₆	1	2	yes	—
117	C₈×C₁₇	C₂×#41	1	2	yes	—
118	#3160	—	17	1	no	—
119	#3147	D₁₇	1	1	yes	characteristic
120	#2245	—	17	2	no	—
121	#2244	—	17	2	no	—
122	#2243	—	17	4	no	—
123	#2138	—	17	4	no	—
124	C₂×#346	—	17	1	no	—
125	C₂²×C₁₇	—	2	2	no	—
126	C₂²×C₁₇	—	8	2	no	—
127	C₂²×C₁₇	—	8	4	no	—
128	C₂²×C₁₇	#486	1	2	yes	—
129	C₂²×C₁₇	C₂²×D₈	1	1	yes	characteristic
130	C₄×C₁₇	—	2	2	no	—
131	C₄×C₁₇	—	2	2	no	—
132	C₄×C₁₇	—	8	2	no	—
133	C₄×C₁₇	#486	1	2	yes	—
134	C₄×C₁₇	C₂×#124	1	1	yes	characteristic
135	C₄×C₁₇	C₂²×SD₁₆	1	1	yes	characteristic
136	Dic₁₇	—	8	4	no	—
137	C₂×#135	—	2	1	no	—
138	C₂×#135	C₂×D₁₇	1	2	yes	—
139	#533	—	34	1	no	—
140	#532	—	2	1	no	—
141	C₂²×C₁₆	—	17	2	no	—
142	#500	—	17	2	no	—
143	#499	—	17	2	no	—
144	#499	—	34	1	no	—
145	#498	—	34	1	no	—
146	#495	—	2	1	no	—
147	#495	C₂×D₁₇	1	2	yes	—
148	#494	C₂×D₁₇	1	1	yes	characteristic
149	#493	C₂×D₁₇	1	1	yes	characteristic
150	#492	—	2	1	no	—
151	#427	—	17	2	no	—
152	#427	—	34	1	no	—
153	C₂×C₄×C₈	C₂×D₁₇	1	1	yes	characteristic
154	#358	—	17	8	no	—
155	#357	—	17	8	no	—
156	#346	—	17	2	no	—
157	#346	—	17	2	no	—
158	C₂×C₁₇	—	2	2	no	—
159	C₂×C₁₇	—	16	2	no	—
160	C₂×C₁₇	C₂×#486	1	1	yes	characteristic
161	C₂×C₁₇	#3414	1	2	yes	—
162	C₂×#41	—	2	2	no	—
163	#135	—	2	8	no	—
164	#135	—	4	4	no	—
165	C₂×Q₁₆	—	2	1	no	—
166	C₂×Q₁₆	C₂²×D₁₇	1	2	yes	—
167	C₂×SD₁₆	—	2	2	no	—
168	C₂×SD₁₆	—	2	2	no	—
169	C₂×D₈	—	2	1	no	—
170	C₂×D₈	C₂²×D₁₇	1	2	yes	—
171	C₂²×C₈	C₂²⋊D₁₇	1	1	yes	characteristic
172	C₂²×C₈	C₂²×D₁₇	1	1	yes	characteristic
173	C₄⋊Q₈	—	2	1	no	—
174	C₄⋊Q₈	—	34	2	no	—
175	#127	—	2	1	no	—
176	#125	—	34	2	no	—
177	#124	—	2	2	no	—
178	C₂²⋊Q₈	—	68	2	no	—
179	#121	—	4	2	no	—
180	C₄×Q₈	—	68	2	no	—
181	C₄×D₄	—	4	2	no	—
182	#117	—	34	2	no	—
183	C₂×C₄²	C₂²⋊D₁₇	1	1	yes	characteristic
184	C₂×C₁₆	—	17	4	no	—
185	C₂×C₁₆	—	34	4	no	—
186	#107	—	17	4	no	—
187	#107	—	34	2	no	—
188	#106	—	34	2	no	—
189	#106	—	34	2	no	—
190	C₄×C₈	C₂²⋊D₁₇	1	1	yes	characteristic
191	C₄×C₈	C₂²⋊D₁₇	1	1	yes	characteristic
192	C₄×C₈	C₂²×D₁₇	1	1	yes	characteristic
193	C₄×C₈	C₂²×D₁₇	1	1	yes	characteristic
194	C₁₇	#33701	1	1	yes	characteristic, complemented, Sylow
195	#41	—	4	4	no	—
196	#41	—	4	4	no	—
197	C₂×Q₈	—	2	2	no	—
198	C₂×Q₈	—	68	2	no	—
199	C₂×D₄	—	2	2	no	—
200	C₂×D₄	—	4	2	no	—
201	C₂×D₄	—	4	2	no	—
202	C₂²×C₄	—	4	2	no	—
203	C₂²×C₄	#63228	1	2	yes	—
204	C₂²×C₄	C₂×C₂²⋊D₁₇	1	1	yes	characteristic
205	Q₁₆	—	2	4	no	—
206	Q₁₆	—	4	2	no	—
207	SD₁₆	—	4	4	no	—
208	SD₁₆	—	4	4	no	—
209	D₈	—	2	4	no	—
210	D₈	—	4	2	no	—
211	C₂×C₈	#63229	1	4	yes	—
212	C₂×C₈	#63254	1	4	yes	—
213	C₂×C₈	C₂×C₂²⋊D₁₇	1	1	yes	characteristic
214	C₂×C₈	C₂×C₂²⋊D₁₇	1	1	yes	characteristic
215	C₂×C₈	C₂×C₂²⋊D₁₇	1	1	yes	characteristic
216	C₂×C₈	C₂³×D₁₇	1	1	yes	characteristic
217	C₄⋊C₄	—	4	2	no	—
218	C₄⋊C₄	—	34	4	no	—
219	C₄⋊C₄	—	68	4	no	—
220	C₄⋊C₄	—	68	2	no	—
221	C₂²⋊C₄	—	4	4	no	—
222	C₂²⋊C₄	—	68	2	no	—
223	C₄²	—	68	2	no	—
224	C₄²	#63228	1	2	yes	—
225	C₄²	C₂×C₂²⋊D₁₇	1	1	yes	characteristic
226	C₄²	C₂×C₂²⋊D₁₇	1	1	yes	characteristic
227	C₁₆	—	34	4	no	—
228	C₂³	—	4	2	no	—
229	C₂³	C₂×#63228	1	1	yes	characteristic
230	Q₈	—	4	2	no	—
231	Q₈	—	4	2	no	—
232	Q₈	—	136	4	no	—
233	D₄	—	4	2	no	—
234	D₄	—	4	2	no	—
235	D₄	—	8	4	no	—
236	D₄	—	8	4	no	—
237	C₂×C₄	—	4	2	no	—
238	C₂×C₄	—	4	2	no	—
239	C₂×C₄	—	8	4	no	—
240	C₂×C₄	—	68	4	no	—
241	C₂×C₄	C₂×#63228	1	2	yes	—
242	C₂×C₄	C₂×#63228	1	2	yes	—
243	C₂×C₄	C₂×#63228	1	1	yes	characteristic
244	C₂×C₄	C₂×#63229	1	2	yes	—
245	C₂×C₄	#10587622	1	2	yes	—
246	C₂×C₄	C₂²×C₂²⋊D₁₇	1	1	yes	characteristic
247	C₈	C₂×#63229	1	2	yes	—
248	C₈	C₂×#63229	1	2	yes	—
249	C₈	#10587622	1	2	yes	—
250	C₈	C₂×#63254	1	2	yes	—
251	C₂²	—	2	2	no	—
252	C₂²	—	8	2	no	—
253	C₂²	—	8	4	no	—
254	C₂²	#11763155	1	2	yes	—
255	C₂²	C₂²×#63228	1	1	yes	characteristic
256	C₄	—	2	2	no	—
257	C₄	—	2	2	no	—
258	C₄	—	8	2	no	—
259	C₄	—	136	4	no	—
260	C₄	#11763155	1	2	yes	—
261	C₄	C₂×#10587622	1	1	yes	characteristic
262	C₄	C₂²×#63229	1	1	yes	characteristic
263	C₂	—	2	2	no	—
264	C₂	—	16	2	no	—
265	C₂	C₂×#11763155	1	1	yes	characteristic
266	C₂	#14541918	1	2	yes	—

	subgroup	G/H
center	C₂×C₄	C₂×#63228
commutator, cosocle	C₂×C₈×C₁₇	C₂⁴
Frattini	C₂×C₈	C₂³×D₁₇
Fitting, prime radical	C₁₇×#3147	C₂
socle	C₂²×C₁₇	C₂²×D₈
hypercenter	#3147	D₁₇
hypocenter	C₁₇	#33701
LCS	G ▷ C₂×C₈×C₁₇ ▷ C₄×C₁₇ ▷ C₂×C₁₇ ▷ C₁₇
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ #3147

Complements

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

N	H ≅ G/N	type
1. #14539298	C₂	semidirect
2. #14531029	C₂	semidirect
3. #14531025	C₂	semidirect
4. #14531020	C₂	semidirect
5. #14530491	C₂	semidirect
6. C₂×#11762585	C₂	semidirect
194. C₁₇	#33701	semidirect

Maximal subgroups

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

subgroup	index
#14539298	2
#14531029	2
#14531025	2
#14531020	2
#14530491	2
C₂×#11762585	2
C₁₇×#3147	2
#33701	17