Subgroups — #17978587

Order 4352.

648 conjugacy classes of subgroups.

The subgroup lattice has 308 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#14545751	C₂	1	2	yes	maximal, complemented
2	#14545174	C₂	1	4	yes	maximal, complemented
3	#14545171	C₂	1	4	yes	maximal, complemented
4	C₂×#11763731	C₂	1	1	yes	maximal, characteristic, complemented
5	#14545026	C₂	1	2	yes	maximal, complemented
6	#14544990	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂×#11762926	C₂	1	1	yes	maximal, characteristic
8	#11763867	—	2	4	no	—
9	#11763867	C₂²	1	8	yes	complemented
10	C₂×#10587538	—	2	4	no	—
11	C₂×#10587538	C₂²	1	2	yes	—
12	C₂×#10587538	C₂²	1	1	yes	characteristic
13	C₂×#10587538	C₂²	1	1	yes	characteristic
14	#11763731	C₂²	1	2	yes	complemented
15	#11763731	C₂²	1	2	yes	complemented
16	#11763707	C₂²	1	8	yes	complemented
17	#11763698	C₂²	1	2	yes	complemented
18	#11763695	C₂²	1	2	yes	complemented
19	C₂×#10587634	C₂²	1	2	yes	—
20	#11762926	C₄	1	2	yes	—
21	#11762926	C₄	1	2	yes	—
22	#11762926	C₂²	1	2	yes	—
23	#11762926	C₂²	1	2	yes	—
24	C₂×#10587490	C₄	1	1	yes	characteristic
25	C₂×#10587508	C₄	1	2	yes	—
26	C₂×C₈×Dic₁₇	C₂²	1	1	yes	characteristic
27	C₂×C₁₇×#135	C₄	1	1	yes	characteristic
28	C₂×#41×C₁₇	C₂×C₄	1	2	yes	—
29	C₁₇×#135	C₈	1	2	yes	—
30	C₁₇×#135	C₈	1	2	yes	—
31	C₁₇×#135	C₂×C₄	1	2	yes	—
32	C₁₇×#135	C₂×C₄	1	2	yes	—
33	C₂×Q₁₆×C₁₇	C₈	1	1	yes	characteristic
34	C₂×SD₁₆×C₁₇	C₈	1	2	yes	—
35	C₂×D₈×C₁₇	C₈	1	1	yes	characteristic
36	C₂²×C₈×C₁₇	C₂×C₄	1	1	yes	characteristic
37	#10587634	—	2	2	no	—
38	#10587634	—	2	2	no	—
39	#10587634	C₂×C₄	1	4	yes	—
40	#10587634	C₂³	1	4	yes	—
41	C₂×C₄×Dic₁₇	C₂×C₄	1	1	yes	characteristic
42	C₂×#63223	—	2	2	no	—
43	C₂×#63223	C₂×C₄	1	2	yes	—
44	C₂²×#3622	—	2	2	no	—
45	C₂²×#3622	C₂³	1	1	yes	characteristic
46	#10587538	—	2	2	no	—
47	#10587538	—	2	1	no	—
48	#10587538	—	2	1	no	—
49	#10587538	—	2	8	no	—
50	#10587538	D₄	1	2	yes	—
51	#10587538	D₄	1	2	yes	—
52	#10587538	C₂³	1	4	yes	—
53	C₂×#63216	—	2	2	no	—
54	C₂×#63216	—	2	8	no	—
55	C₂×#63216	C₂³	1	2	yes	—
56	C₂×#63216	C₂³	1	2	yes	—
57	#10587508	C₂×C₄	1	4	yes	—
58	#10587508	C₂×C₄	1	4	yes	—
59	#10587490	C₂×C₄	1	2	yes	—
60	#10587490	C₂×C₄	1	2	yes	—
61	C₈×Dic₁₇	C₂×C₄	1	1	yes	characteristic
62	C₈×Dic₁₇	C₂×C₄	1	1	yes	characteristic
63	C₈×Dic₁₇	C₂³	1	1	yes	characteristic
64	C₈×Dic₁₇	C₂³	1	1	yes	characteristic
65	#41×C₁₇	—	2	2	no	—
66	#41×C₁₇	—	2	2	no	—
67	#41×C₁₇	C₂×C₈	1	4	yes	—
68	#41×C₁₇	C₂²×C₄	1	4	yes	—
69	C₂×Q₈×C₁₇	C₂×C₈	1	2	yes	—
70	C₂×D₄×C₁₇	—	2	2	no	—
71	C₂×D₄×C₁₇	C₂×C₈	1	2	yes	—
72	C₂²×C₄×C₁₇	—	2	2	no	—
73	C₂²×C₄×C₁₇	C₂²×C₄	1	1	yes	characteristic
74	C₂²×Dic₁₇	—	2	1	no	—
75	Q₁₆×C₁₇	C₂×C₈	1	2	yes	—
76	Q₁₆×C₁₇	C₂×C₈	1	2	yes	—
77	SD₁₆×C₁₇	C₂×C₈	1	4	yes	—
78	SD₁₆×C₁₇	C₂×C₈	1	4	yes	—
79	D₈×C₁₇	C₂×C₈	1	2	yes	—
80	D₈×C₁₇	C₂×C₈	1	2	yes	—
81	C₂×C₈×C₁₇	C₂×C₈	1	1	yes	characteristic
82	C₂×C₈×C₁₇	C₂×C₈	1	1	yes	characteristic
83	C₂×C₈×C₁₇	C₂×C₈	1	1	yes	characteristic
84	C₂×C₈×C₁₇	C₂×C₈	1	1	yes	characteristic
85	C₂×C₈×C₁₇	C₂²×C₄	1	1	yes	characteristic
86	C₂×C₈×C₁₇	C₂²×C₄	1	1	yes	characteristic
87	C₄×Dic₁₇	—	2	1	no	—
88	C₄×Dic₁₇	C₂²×C₄	1	1	yes	characteristic
89	C₄×Dic₁₇	C₂²×C₄	1	1	yes	characteristic
90	#63223	—	2	2	no	—
91	#63223	—	2	4	no	—
92	#63223	—	2	4	no	—
93	#63223	C₂²×C₄	1	4	yes	—
94	C₂×#3622	—	2	2	no	—
95	C₂×#3622	—	2	4	no	—
96	C₂×#3622	—	2	4	no	—
97	C₂×#3622	C₂²×C₄	1	1	yes	characteristic
98	C₂×#3622	C₂×D₄	1	1	yes	characteristic
99	C₂×#3622	C₂×D₄	1	1	yes	characteristic
100	C₂×#3622	#41	1	1	yes	characteristic
101	C₂×#3622	#41	1	1	yes	characteristic
102	C₂×#3622	C₂⁴	1	1	yes	characteristic
103	#63216	—	2	4	no	—
104	#63216	—	2	2	no	—
105	#63216	C₂×D₄	1	4	yes	—
106	#33443	—	17	1	no	maximal, Sylow
107	C₂³×C₁₇	—	2	2	no	—
108	C₂³×C₁₇	#131	1	1	yes	characteristic
109	Q₈×C₁₇	—	2	2	no	—
110	Q₈×C₁₇	C₂²×C₈	1	4	yes	—
111	D₄×C₁₇	—	2	4	no	—
112	D₄×C₁₇	—	2	4	no	—
113	D₄×C₁₇	—	2	2	no	—
114	D₄×C₁₇	C₂²×C₈	1	4	yes	—
115	C₂×C₄×C₁₇	—	2	2	no	—
116	C₂×C₄×C₁₇	—	2	4	no	—
117	C₂×C₄×C₁₇	—	2	4	no	—
118	C₂×C₄×C₁₇	C₄×D₄	1	1	yes	characteristic
119	C₂×C₄×C₁₇	C₄×D₄	1	1	yes	characteristic
120	C₂×C₄×C₁₇	C₂²×C₈	1	1	yes	characteristic
121	C₂×C₄×C₁₇	C₂²×C₈	1	1	yes	characteristic
122	C₂×C₄×C₁₇	#131	1	1	yes	characteristic
123	C₂×C₄×C₁₇	C₂³×C₄	1	1	yes	characteristic
124	C₂×Dic₁₇	—	2	1	no	—
125	C₂×Dic₁₇	—	2	1	no	—
126	C₂×Dic₁₇	—	4	2	no	—
127	C₈×C₁₇	C₂²×C₈	1	1	yes	characteristic
128	C₈×C₁₇	C₂²×C₈	1	1	yes	characteristic
129	C₈×C₁₇	C₂²×C₈	1	1	yes	characteristic
130	C₈×C₁₇	C₂²×C₈	1	1	yes	characteristic
131	#3622	—	2	4	no	—
132	#3622	C₄×D₄	1	1	yes	characteristic
133	#3622	C₄×D₄	1	1	yes	characteristic
134	#3622	C₂²×D₄	1	1	yes	characteristic
135	#3622	C₂×#41	1	1	yes	characteristic
136	#3407	—	17	2	no	—
137	C₂×#443	—	17	1	no	—
138	#2171	—	17	4	no	—
139	#2170	—	17	4	no	—
140	C₂×#364	—	17	1	no	—
141	#2117	—	17	2	no	—
142	C₄×M₃₂	—	17	1	no	—
143	C₂²×C₁₇	—	2	4	no	—
144	C₂²×C₁₇	—	4	1	no	—
145	C₂²×C₁₇	—	8	2	no	—
146	C₂²×C₁₇	C₈×D₄	1	1	yes	characteristic
147	C₂²×C₁₇	C₈×D₄	1	1	yes	characteristic
148	C₂²×C₁₇	C₂×#131	1	1	yes	characteristic
149	C₄×C₁₇	—	2	4	no	—
150	C₄×C₁₇	C₈×D₄	1	1	yes	characteristic
151	C₄×C₁₇	C₈×D₄	1	1	yes	characteristic
152	C₄×C₁₇	C₂×C₄×D₄	1	1	yes	characteristic
153	C₄×C₁₇	C₂³×C₈	1	1	yes	characteristic
154	Dic₁₇	—	2	2	no	—
155	Dic₁₇	—	4	1	no	—
156	C₂×#135	#595	1	1	yes	characteristic
157	C₂×#131	—	17	2	no	—
158	#504	—	17	8	no	—
159	#504	—	34	4	no	—
160	C₂×M₃₂	—	17	1	no	—
161	C₂×M₃₂	—	17	1	no	—
162	C₂×M₃₂	—	17	2	no	—
163	C₂×M₃₂	—	34	4	no	—
164	#443	—	17	2	no	—
165	#443	—	17	2	no	—
166	#443	—	17	2	no	—
167	#443	—	17	2	no	—
168	C₂×#108	—	17	1	no	—
169	C₂×#104	—	17	2	no	—
170	C₂×C₄×C₈	—	17	1	no	—
171	#364	—	17	2	no	—
172	#364	—	17	2	no	—
173	#350	—	17	8	no	—
174	#346	—	17	2	no	—
175	C₄×C₁₆	—	17	2	no	—
176	C₂×C₁₇	—	2	1	no	—
177	C₂×C₁₇	—	2	1	no	—
178	C₂×C₁₇	—	2	1	no	—
179	C₂×C₁₇	—	4	4	no	—
180	C₂×C₁₇	C₂×C₈×D₄	1	1	yes	characteristic
181	C₂×#41	C₂×#595	1	2	yes	—
182	#135	#3624	1	2	yes	—
183	#135	#3624	1	2	yes	—
184	#135	C₂×#595	1	2	yes	—
185	#135	C₂×#595	1	2	yes	—
186	C₂×Q₁₆	#3624	1	1	yes	characteristic
187	C₂×SD₁₆	#3624	1	2	yes	—
188	C₂×D₈	#3624	1	1	yes	characteristic
189	#131	—	17	4	no	—
190	#131	—	17	4	no	—
191	#131	—	34	2	no	—
192	#131	—	34	2	no	—
193	C₂×M₁₆	—	17	2	no	—
194	C₂×M₁₆	—	34	2	no	—
195	C₂²×C₈	—	17	1	no	—
196	C₂²×C₈	—	34	2	no	—
197	C₂²×C₈	C₂×#595	1	1	yes	characteristic
198	C₂×C₄²	—	17	1	no	—
199	M₃₂	—	17	2	no	—
200	M₃₂	—	17	2	no	—
201	M₃₂	—	17	4	no	—
202	M₃₂	—	34	8	no	—
203	M₃₂	—	34	1	no	—
204	M₃₂	—	34	1	no	—
205	M₃₂	—	34	2	no	—
206	C₂×C₁₆	—	17	2	no	—
207	C₂×C₁₆	—	17	2	no	—
208	C₂×C₁₆	—	34	8	no	—
209	C₂×C₁₆	—	34	2	no	—
210	#108	—	17	2	no	—
211	#108	—	17	2	no	—
212	#104	—	17	4	no	—
213	#104	—	17	4	no	—
214	C₄×C₈	—	17	1	no	—
215	C₄×C₈	—	17	1	no	—
216	C₄×C₈	—	17	1	no	—
217	C₄×C₈	—	17	1	no	—
218	C₁₇	#33443	1	1	yes	characteristic, complemented, Sylow
219	#41	—	2	2	no	—
220	#41	—	2	2	no	—
221	#41	C₂×#3624	1	4	yes	—
222	#41	C₂²×#595	1	4	yes	—
223	C₂×Q₈	C₂×#3624	1	2	yes	—
224	C₂×D₄	—	2	2	no	—
225	C₂×D₄	C₂×#3624	1	2	yes	—
226	C₂²×C₄	—	2	2	no	—
227	C₂²×C₄	—	34	1	no	—
228	C₂²×C₄	C₂²×#595	1	1	yes	characteristic
229	Q₁₆	C₂×#3624	1	2	yes	—
230	Q₁₆	C₂×#3624	1	2	yes	—
231	SD₁₆	C₂×#3624	1	4	yes	—
232	SD₁₆	C₂×#3624	1	4	yes	—
233	D₈	C₂×#3624	1	2	yes	—
234	D₈	C₂×#3624	1	2	yes	—
235	M₁₆	—	17	4	no	—
236	M₁₆	—	34	4	no	—
237	M₁₆	—	34	4	no	—
238	M₁₆	—	34	2	no	—
239	C₂×C₈	—	17	1	no	—
240	C₂×C₈	—	17	1	no	—
241	C₂×C₈	—	17	1	no	—
242	C₂×C₈	—	17	1	no	—
243	C₂×C₈	—	17	1	no	—
244	C₂×C₈	—	17	1	no	—
245	C₂×C₈	—	34	4	no	—
246	C₂×C₈	—	34	4	no	—
247	C₂×C₈	—	34	2	no	—
248	C₂×C₈	C₂×#3624	1	1	yes	characteristic
249	C₂×C₈	C₂×#3624	1	1	yes	characteristic
250	C₂×C₈	C₂×#3624	1	1	yes	characteristic
251	C₂×C₈	C₂×#3624	1	1	yes	characteristic
252	C₂×C₈	C₂²×#595	1	1	yes	characteristic
253	C₂×C₈	C₂²×#595	1	1	yes	characteristic
254	C₄²	—	17	1	no	—
255	C₄²	—	17	1	no	—
256	C₄²	—	34	1	no	—
257	C₁₆	—	17	4	no	—
258	C₁₆	—	34	2	no	—
259	C₁₆	—	34	4	no	—
260	C₂³	—	2	2	no	—
261	C₂³	#10587678	1	1	yes	characteristic
262	Q₈	—	2	2	no	—
263	Q₈	C₂²×#3624	1	4	yes	—
264	D₄	—	2	4	no	—
265	D₄	—	2	4	no	—
266	D₄	—	2	2	no	—
267	D₄	C₂²×#3624	1	4	yes	—
268	C₂×C₄	—	2	2	no	—
269	C₂×C₄	—	2	4	no	—
270	C₂×C₄	—	2	4	no	—
271	C₂×C₄	—	34	1	no	—
272	C₂×C₄	—	34	1	no	—
273	C₂×C₄	—	68	2	no	—
274	C₂×C₄	#10587678	1	1	yes	characteristic
275	C₂×C₄	D₄×#595	1	1	yes	characteristic
276	C₂×C₄	D₄×#595	1	1	yes	characteristic
277	C₂×C₄	C₂²×#3624	1	1	yes	characteristic
278	C₂×C₄	C₂²×#3624	1	1	yes	characteristic
279	C₂×C₄	C₂³×#595	1	1	yes	characteristic
280	C₈	—	17	1	no	—
281	C₈	—	17	1	no	—
282	C₈	—	17	1	no	—
283	C₈	—	17	1	no	—
284	C₈	—	34	4	no	—
285	C₈	C₂²×#3624	1	1	yes	characteristic
286	C₈	C₂²×#3624	1	1	yes	characteristic
287	C₈	C₂²×#3624	1	1	yes	characteristic
288	C₈	C₂²×#3624	1	1	yes	characteristic
289	C₂²	—	2	4	no	—
290	C₂²	—	4	1	no	—
291	C₂²	—	8	2	no	—
292	C₂²	D₄×#3624	1	1	yes	characteristic
293	C₂²	D₄×#3624	1	1	yes	characteristic
294	C₂²	C₂×#10587678	1	1	yes	characteristic
295	C₄	—	2	4	no	—
296	C₄	—	34	2	no	—
297	C₄	—	68	1	no	—
298	C₄	D₄×#3624	1	1	yes	characteristic
299	C₄	D₄×#3624	1	1	yes	characteristic
300	C₄	C₂×D₄×#595	1	1	yes	characteristic
301	C₄	C₂³×#3624	1	1	yes	characteristic
302	C₂	—	2	1	no	—
303	C₂	—	2	1	no	—
304	C₂	—	2	1	no	—
305	C₂	—	4	4	no	—
306	C₂	C₂×D₄×#3624	1	1	yes	characteristic

	subgroup	G/H
center	C₄	C₂×D₄×#595
commutator	C₄×C₁₇	C₂³×C₈
Frattini	C₂×C₄	C₂³×#595
Fitting, prime radical	C₂×C₁₇×#135	C₄
socle	C₂×C₁₇	C₂×C₈×D₄
cosocle	C₂×#3622	C₂⁴
hypercenter	C₂×#135	#595
hypocenter	C₁₇	#33443
LCS	G ▷ C₄×C₁₇ ▷ C₂×C₁₇ ▷ C₁₇
UCS	C₁ ◁ C₄ ◁ C₂²×C₄ ◁ C₂×#135

Complements

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

N	H ≅ G/N	type
1. #14545751	C₂	semidirect
2. #14545174	C₂	semidirect
3. #14545171	C₂	semidirect
4. C₂×#11763731	C₂	semidirect
5. #14545026	C₂	semidirect
6. #14544990	C₂	semidirect
9. #11763867	C₂²	semidirect
14. #11763731	C₂²	semidirect
15. #11763731	C₂²	semidirect
16. #11763707	C₂²	semidirect
17. #11763698	C₂²	semidirect
18. #11763695	C₂²	semidirect
218. C₁₇	#33443	semidirect

Maximal subgroups

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

subgroup	index
#14545751	2
#14545174	2
#14545171	2
C₂×#11763731	2
#14545026	2
#14544990	2
C₂×#11762926	2
#33443	17