Subgroups — #17187579

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	#14541887	C₂	1	2	yes	maximal, complemented
2	#14530626	C₂	1	4	yes	maximal, complemented
3	#14530621	C₂	1	4	yes	maximal, complemented
4	C₂×#11762630	C₂	1	1	yes	maximal, characteristic, complemented
5	#14530412	C₂	1	2	yes	maximal, complemented
6	C₄×#10587483	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂×C₁₇×#443	C₂	1	1	yes	maximal, characteristic
8	#11763226	—	2	4	no	—
9	#11763226	C₂²	1	8	yes	complemented
10	C₂×#10587483	—	2	4	no	—
11	C₂×#10587483	C₂²	1	2	yes	—
12	C₂×#10587483	C₂²	1	1	yes	characteristic
13	C₂×#10587483	C₂²	1	1	yes	characteristic
14	#11762630	C₂²	1	2	yes	complemented
15	#11762630	C₂²	1	2	yes	complemented
16	#11762593	C₂²	1	8	yes	complemented
17	#11762585	C₂²	1	2	yes	complemented
18	C₄×#63214	C₂²	1	2	yes	complemented
19	C₂×C₁₇×#135	C₄	1	1	yes	characteristic
20	C₂×C₁₇×#131	C₂²	1	2	yes	—
21	C₁₇×#443	C₄	1	2	yes	—
22	C₁₇×#443	C₄	1	2	yes	—
23	C₁₇×#443	C₂²	1	2	yes	—
24	C₁₇×#443	C₂²	1	2	yes	—
25	C₂×C₁₇×#108	C₄	1	1	yes	characteristic
26	C₂×C₁₇×#104	C₄	1	2	yes	—
27	C₂×C₄×C₈×C₁₇	C₂²	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	C₁₇×#131	—	2	2	no	—
37	C₁₇×#131	—	2	2	no	—
38	C₁₇×#131	C₂×C₄	1	4	yes	—
39	C₁₇×#131	C₂³	1	4	yes	—
40	C₂×M₁₆×C₁₇	—	2	2	no	—
41	C₂×M₁₆×C₁₇	C₂×C₄	1	2	yes	—
42	C₂²×C₈×C₁₇	—	2	2	no	—
43	C₂²×C₈×C₁₇	C₂×C₄	1	1	yes	characteristic
44	C₂²×C₈×C₁₇	C₂³	1	1	yes	characteristic
45	C₂×C₄²×C₁₇	C₂×C₄	1	1	yes	characteristic
46	C₁₇×#108	C₂×C₄	1	2	yes	—
47	C₁₇×#108	C₂×C₄	1	2	yes	—
48	C₁₇×#104	C₂×C₄	1	4	yes	—
49	C₁₇×#104	C₂×C₄	1	4	yes	—
50	C₄×C₈×C₁₇	C₂×C₄	1	1	yes	characteristic
51	C₄×C₈×C₁₇	C₂×C₄	1	1	yes	characteristic
52	C₄×C₈×C₁₇	C₂³	1	1	yes	characteristic
53	C₄×C₈×C₁₇	C₂³	1	1	yes	characteristic
54	#10587483	—	2	8	no	—
55	#10587483	—	2	2	no	—
56	#10587483	—	2	1	no	—
57	#10587483	—	2	1	no	—
58	#10587483	D₄	1	2	yes	—
59	#10587483	D₄	1	2	yes	—
60	#10587483	C₂³	1	4	yes	—
61	C₂×#63214	—	2	8	no	—
62	C₂×#63214	—	2	2	no	—
63	C₂×#63214	C₂³	1	2	yes	—
64	C₂×#63214	C₂³	1	2	yes	—
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	—	2	1	no	—
74	C₂²×C₄×C₁₇	C₂²×C₄	1	1	yes	characteristic
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	M₁₆×C₁₇	—	2	2	no	—
82	M₁₆×C₁₇	—	2	4	no	—
83	M₁₆×C₁₇	—	2	4	no	—
84	M₁₆×C₁₇	C₂²×C₄	1	4	yes	—
85	C₂×C₈×C₁₇	—	2	2	no	—
86	C₂×C₈×C₁₇	—	2	4	no	—
87	C₂×C₈×C₁₇	—	2	4	no	—
88	C₂×C₈×C₁₇	C₂×C₈	1	1	yes	characteristic
89	C₂×C₈×C₁₇	C₂×C₈	1	1	yes	characteristic
90	C₂×C₈×C₁₇	C₂×C₈	1	1	yes	characteristic
91	C₂×C₈×C₁₇	C₂×C₈	1	1	yes	characteristic
92	C₂×C₈×C₁₇	C₂²×C₄	1	1	yes	characteristic
93	C₂×C₈×C₁₇	C₂²×C₄	1	1	yes	characteristic
94	C₂×C₈×C₁₇	C₂²×C₄	1	1	yes	characteristic
95	C₂×C₈×C₁₇	C₂×D₄	1	1	yes	characteristic
96	C₂×C₈×C₁₇	C₂×D₄	1	1	yes	characteristic
97	C₂×C₈×C₁₇	#41	1	1	yes	characteristic
98	C₂×C₈×C₁₇	#41	1	1	yes	characteristic
99	C₂×C₈×C₁₇	C₂⁴	1	1	yes	characteristic
100	C₄²×C₁₇	—	2	1	no	—
101	C₄²×C₁₇	C₂²×C₄	1	1	yes	characteristic
102	C₄²×C₁₇	C₂²×C₄	1	1	yes	characteristic
103	#63214	—	2	4	no	—
104	#63214	—	2	2	no	—
105	#63214	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	1	no	—
116	C₂×C₄×C₁₇	—	2	2	no	—
117	C₂×C₄×C₁₇	—	2	4	no	—
118	C₂×C₄×C₁₇	—	2	4	no	—
119	C₂×C₄×C₁₇	—	2	1	no	—
120	C₂×C₄×C₁₇	—	4	2	no	—
121	C₂×C₄×C₁₇	C₄×D₄	1	1	yes	characteristic
122	C₂×C₄×C₁₇	C₄×D₄	1	1	yes	characteristic
123	C₂×C₄×C₁₇	C₂²×C₈	1	1	yes	characteristic
124	C₂×C₄×C₁₇	C₂²×C₈	1	1	yes	characteristic
125	C₂×C₄×C₁₇	#131	1	1	yes	characteristic
126	C₂×C₄×C₁₇	C₂³×C₄	1	1	yes	characteristic
127	C₈×C₁₇	—	2	4	no	—
128	C₈×C₁₇	C₄×D₄	1	1	yes	characteristic
129	C₈×C₁₇	C₄×D₄	1	1	yes	characteristic
130	C₈×C₁₇	C₂²×C₈	1	1	yes	characteristic
131	C₈×C₁₇	C₂²×C₈	1	1	yes	characteristic
132	C₈×C₁₇	C₂²×C₈	1	1	yes	characteristic
133	C₈×C₁₇	C₂²×C₈	1	1	yes	characteristic
134	C₈×C₁₇	C₂²×D₄	1	1	yes	characteristic
135	C₈×C₁₇	C₂×#41	1	1	yes	characteristic
136	#3407	—	17	2	no	—
137	C₂×#443	D₁₇	1	1	yes	characteristic
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	2	no	—
150	C₄×C₁₇	—	2	4	no	—
151	C₄×C₁₇	—	4	1	no	—
152	C₄×C₁₇	C₈×D₄	1	1	yes	characteristic
153	C₄×C₁₇	C₈×D₄	1	1	yes	characteristic
154	C₄×C₁₇	C₂×C₄×D₄	1	1	yes	characteristic
155	C₄×C₁₇	C₂³×C₈	1	1	yes	characteristic
156	C₂×#135	Dic₁₇	1	1	yes	characteristic
157	C₂×#131	C₂×D₁₇	1	2	yes	—
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	Dic₁₇	1	2	yes	—
165	#443	Dic₁₇	1	2	yes	—
166	#443	C₂×D₁₇	1	2	yes	—
167	#443	C₂×D₁₇	1	2	yes	—
168	C₂×#108	Dic₁₇	1	1	yes	characteristic
169	C₂×#104	Dic₁₇	1	2	yes	—
170	C₂×C₄×C₈	C₂×D₁₇	1	1	yes	characteristic
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×Dic₁₇	1	2	yes	—
182	#135	#3622	1	2	yes	—
183	#135	#3622	1	2	yes	—
184	#135	C₂×Dic₁₇	1	2	yes	—
185	#135	C₂×Dic₁₇	1	2	yes	—
186	C₂×Q₁₆	#3622	1	1	yes	characteristic
187	C₂×SD₁₆	#3622	1	2	yes	—
188	C₂×D₈	#3622	1	1	yes	characteristic
189	#131	—	2	2	no	—
190	#131	—	2	2	no	—
191	#131	C₂×Dic₁₇	1	4	yes	—
192	#131	C₂²×D₁₇	1	4	yes	—
193	C₂×M₁₆	—	2	2	no	—
194	C₂×M₁₆	C₂×Dic₁₇	1	2	yes	—
195	C₂²×C₈	—	2	2	no	—
196	C₂²×C₈	C₂×Dic₁₇	1	1	yes	characteristic
197	C₂²×C₈	C₂²×D₁₇	1	1	yes	characteristic
198	C₂×C₄²	C₂×Dic₁₇	1	1	yes	characteristic
199	M₃₂	—	17	2	no	—
200	M₃₂	—	17	2	no	—
201	M₃₂	—	17	4	no	—
202	M₃₂	—	34	8	no	—
203	M₃₂	—	34	2	no	—
204	M₃₂	—	34	1	no	—
205	M₃₂	—	34	1	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	C₂×Dic₁₇	1	2	yes	—
211	#108	C₂×Dic₁₇	1	2	yes	—
212	#104	C₂×Dic₁₇	1	4	yes	—
213	#104	C₂×Dic₁₇	1	4	yes	—
214	C₄×C₈	C₂×Dic₁₇	1	1	yes	characteristic
215	C₄×C₈	C₂×Dic₁₇	1	1	yes	characteristic
216	C₄×C₈	C₂²×D₁₇	1	1	yes	characteristic
217	C₄×C₈	C₂²×D₁₇	1	1	yes	characteristic
218	C₁₇	#33443	1	1	yes	characteristic, complemented, Sylow
219	#41	—	2	2	no	—
220	#41	—	2	2	no	—
221	#41	C₂×#3622	1	4	yes	—
222	#41	C₂²×Dic₁₇	1	4	yes	—
223	C₂×Q₈	C₂×#3622	1	2	yes	—
224	C₂×D₄	—	2	2	no	—
225	C₂×D₄	C₂×#3622	1	2	yes	—
226	C₂²×C₄	—	2	2	no	—
227	C₂²×C₄	—	2	1	no	—
228	C₂²×C₄	C₂²×Dic₁₇	1	1	yes	characteristic
229	Q₁₆	C₂×#3622	1	2	yes	—
230	Q₁₆	C₂×#3622	1	2	yes	—
231	SD₁₆	C₂×#3622	1	4	yes	—
232	SD₁₆	C₂×#3622	1	4	yes	—
233	D₈	C₂×#3622	1	2	yes	—
234	D₈	C₂×#3622	1	2	yes	—
235	M₁₆	—	2	2	no	—
236	M₁₆	—	2	4	no	—
237	M₁₆	—	2	4	no	—
238	M₁₆	C₂²×Dic₁₇	1	4	yes	—
239	C₂×C₈	—	2	2	no	—
240	C₂×C₈	—	2	4	no	—
241	C₂×C₈	—	2	4	no	—
242	C₂×C₈	C₂×#3622	1	1	yes	characteristic
243	C₂×C₈	C₂×#3622	1	1	yes	characteristic
244	C₂×C₈	C₂×#3622	1	1	yes	characteristic
245	C₂×C₈	C₂×#3622	1	1	yes	characteristic
246	C₂×C₈	D₄×D₁₇	1	1	yes	characteristic
247	C₂×C₈	D₄×D₁₇	1	1	yes	characteristic
248	C₂×C₈	#63254	1	1	yes	characteristic
249	C₂×C₈	#63254	1	1	yes	characteristic
250	C₂×C₈	C₂²×Dic₁₇	1	1	yes	characteristic
251	C₂×C₈	C₂²×Dic₁₇	1	1	yes	characteristic
252	C₂×C₈	C₂²×Dic₁₇	1	1	yes	characteristic
253	C₂×C₈	C₂³×D₁₇	1	1	yes	characteristic
254	C₄²	—	2	1	no	—
255	C₄²	C₂²×Dic₁₇	1	1	yes	characteristic
256	C₄²	C₂²×Dic₁₇	1	1	yes	characteristic
257	C₁₆	—	17	4	no	—
258	C₁₆	—	34	4	no	—
259	C₁₆	—	34	2	no	—
260	C₂³	—	2	2	no	—
261	C₂³	#10587634	1	1	yes	characteristic
262	Q₈	—	2	2	no	—
263	Q₈	C₂²×#3622	1	4	yes	—
264	D₄	—	2	4	no	—
265	D₄	—	2	4	no	—
266	D₄	—	2	2	no	—
267	D₄	C₂²×#3622	1	4	yes	—
268	C₂×C₄	—	2	1	no	—
269	C₂×C₄	—	2	2	no	—
270	C₂×C₄	—	2	4	no	—
271	C₂×C₄	—	2	4	no	—
272	C₂×C₄	—	2	1	no	—
273	C₂×C₄	—	4	2	no	—
274	C₂×C₄	C₂²×#3622	1	1	yes	characteristic
275	C₂×C₄	C₂²×#3622	1	1	yes	characteristic
276	C₂×C₄	D₄×Dic₁₇	1	1	yes	characteristic
277	C₂×C₄	D₄×Dic₁₇	1	1	yes	characteristic
278	C₂×C₄	#10587634	1	1	yes	characteristic
279	C₂×C₄	C₂³×Dic₁₇	1	1	yes	characteristic
280	C₈	—	2	4	no	—
281	C₈	C₂²×#3622	1	1	yes	characteristic
282	C₈	C₂²×#3622	1	1	yes	characteristic
283	C₈	C₂²×#3622	1	1	yes	characteristic
284	C₈	C₂²×#3622	1	1	yes	characteristic
285	C₈	D₄×Dic₁₇	1	1	yes	characteristic
286	C₈	D₄×Dic₁₇	1	1	yes	characteristic
287	C₈	C₂×D₄×D₁₇	1	1	yes	characteristic
288	C₈	C₂×#63254	1	1	yes	characteristic
289	C₂²	—	2	4	no	—
290	C₂²	—	4	1	no	—
291	C₂²	—	8	2	no	—
292	C₂²	D₄×#3622	1	1	yes	characteristic
293	C₂²	D₄×#3622	1	1	yes	characteristic
294	C₂²	C₂×#10587634	1	1	yes	characteristic
295	C₄	—	2	2	no	—
296	C₄	—	2	4	no	—
297	C₄	—	4	1	no	—
298	C₄	D₄×#3622	1	1	yes	characteristic
299	C₄	D₄×#3622	1	1	yes	characteristic
300	C₄	C₂×D₄×Dic₁₇	1	1	yes	characteristic
301	C₄	C₂³×#3622	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×#3622	1	1	yes	characteristic

	subgroup	G/H
center	C₈	C₂×D₄×D₁₇
commutator	C₄×C₁₇	C₂³×C₈
Frattini	C₂×C₈	C₂³×D₁₇
Fitting	C₂×C₁₇×#443	C₂
socle	C₂×C₁₇	C₂×C₈×D₄
prime radical	C₂×C₁₇×#135	C₄
cosocle	C₂×C₈×C₁₇	C₂⁴
hypercenter	C₂×#443	D₁₇
hypocenter	C₁₇	#33443
LCS	G ▷ C₄×C₁₇ ▷ C₂×C₁₇ ▷ C₁₇
UCS	C₁ ◁ C₈ ◁ C₂²×C₈ ◁ C₂×#443

Complements

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

N	H ≅ G/N	type
1. #14541887	C₂	semidirect
2. #14530626	C₂	semidirect
3. #14530621	C₂	semidirect
4. C₂×#11762630	C₂	semidirect
5. #14530412	C₂	semidirect
6. C₄×#10587483	C₂	semidirect
9. #11763226	C₂²	semidirect
14. #11762630	C₂²	semidirect
15. #11762630	C₂²	semidirect
16. #11762593	C₂²	semidirect
17. #11762585	C₂²	semidirect
18. C₄×#63214	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
#14541887	2
#14530626	2
#14530621	2
C₂×#11762630	2
#14530412	2
C₄×#10587483	2
C₂×C₁₇×#443	2
#33443	17