Subgroups — #17191951

Order 4352.

470 conjugacy classes of subgroups.

The subgroup lattice has 326 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#14541216	C₂	1	1	yes	maximal, characteristic
2	#14541178	C₂	1	1	yes	maximal, characteristic
3	#14531066	C₂	1	1	yes	maximal, characteristic, complemented
4	#14530990	C₂	1	2	yes	maximal, complemented
5	#14530980	C₂	1	2	yes	maximal, complemented
6	#14530936	C₂	1	1	yes	maximal, characteristic, complemented
7	#14530926	C₂	1	1	yes	maximal, characteristic, complemented
8	#14530889	C₂	1	2	yes	maximal, complemented
9	#14530617	C₂	1	1	yes	maximal, characteristic
10	#14530569	C₂	1	2	yes	maximal, complemented
11	C₄×C₁₇×Q₃₂	C₂	1	1	yes	maximal, characteristic, complemented
12	#11763532	—	2	1	no	—
13	#11763519	—	2	1	no	—
14	C₂×#10587469	C₂²	1	2	yes	—
15	#11763214	C₂²	1	1	yes	characteristic
16	#11763214	C₂²	1	1	yes	characteristic
17	#11763149	—	2	2	no	—
18	#11763143	—	2	2	no	—
19	#11763104	—	2	1	no	—
20	#11763094	—	2	1	no	—
21	#11763047	C₂²	1	2	yes	—
22	#11763047	C₂²	1	2	yes	—
23	C₈×D₆₈	C₂²	1	1	yes	characteristic
24	#11762846	C₂²	1	2	yes	—
25	#107×D₁₇	C₂²	1	2	yes	—
26	#11762637	C₂²	1	2	yes	—
27	#11762637	C₂²	1	2	yes	—
28	#11762636	C₂²	1	2	yes	—
29	#11762634	C₂²	1	2	yes	—
30	C₄⋊#63214	C₂²	1	1	yes	characteristic
31	#11762613	C₂²	1	2	yes	—
32	#11762613	C₂²	1	2	yes	—
33	#11762588	C₂²	1	2	yes	—
34	C₂×C₁₇×Q₃₂	C₂²	1	1	yes	characteristic
35	C₄×Q₁₆×C₁₇	C₂²	1	1	yes	characteristic
36	C₄×Q₁₆×C₁₇	C₂²	1	1	yes	characteristic
37	C₁₇×#366	C₂²	1	1	yes	characteristic
38	C₁₇×#358	C₂²	1	1	yes	characteristic
39	C₁₇×#358	C₂²	1	1	yes	characteristic
40	C₄×C₁₆×C₁₇	C₂²	1	1	yes	characteristic
41	C₂×Q₁₆×C₁₇	C₂³	1	1	yes	characteristic
42	C₂×Q₁₆×C₁₇	C₂³	1	1	yes	characteristic
43	C₄×Q₈×C₁₇	—	2	1	no	—
44	C₄×Q₈×C₁₇	—	2	1	no	—
45	#10587632	—	2	2	no	—
46	#10587632	—	2	2	no	—
47	C₂×C₈×D₁₇	C₂³	1	2	yes	—
48	#10587584	—	4	2	no	—
49	#10587582	—	4	2	no	—
50	#10587578	—	2	2	no	—
51	C₄⋊C₄×D₁₇	—	2	2	no	—
52	#10587561	—	4	1	no	—
53	C₄×D₆₈	—	4	1	no	—
54	C₄×D₆₈	D₄	1	1	yes	characteristic
55	#10587556	—	4	1	no	—
56	#10587555	—	2	1	no	—
57	#10587555	—	2	1	no	—
58	C₄×Dic₃₄	—	4	1	no	—
59	C₁₇×Q₃₂	—	2	2	no	—
60	C₂×C₁₆×C₁₇	C₂³	1	1	yes	characteristic
61	C₂×C₁₆×C₁₇	C₂³	1	1	yes	characteristic
62	C₁₇×#107	C₂³	1	1	yes	characteristic
63	C₁₇×#107	C₂³	1	1	yes	characteristic
64	C₁₇×#103	—	2	1	no	—
65	C₁₇×#103	—	2	1	no	—
66	C₄×C₈×C₁₇	C₂³	1	1	yes	characteristic
67	#10587506	—	2	2	no	—
68	#10587506	—	2	2	no	—
69	#10587491	D₄	1	2	yes	—
70	#10587489	—	2	1	no	—
71	#10587489	—	2	1	no	—
72	#10587489	C₂³	1	2	yes	—
73	#10587489	C₂³	1	2	yes	—
74	C₂×#63214	C₂³	1	2	yes	—
75	#10587479	—	2	2	no	—
76	#10587478	—	2	2	no	—
77	C₄⋊#3622	D₄	1	1	yes	characteristic
78	#10587469	—	2	4	no	—
79	C₂×Q₈×C₁₇	—	2	1	no	—
80	C₂×Q₈×C₁₇	—	2	1	no	—
81	C₂×D₆₈	D₈	1	1	yes	characteristic
82	C₂×C₄×D₁₇	—	4	2	no	—
83	C₂×C₄×D₁₇	C₂×D₄	1	2	yes	—
84	C₂×Dic₃₄	—	4	1	no	—
85	C₂×Dic₃₄	—	4	1	no	—
86	Q₁₆×C₁₇	—	2	1	no	—
87	Q₁₆×C₁₇	—	2	1	no	—
88	Q₁₆×C₁₇	—	2	1	no	—
89	Q₁₆×C₁₇	—	2	1	no	—
90	C₂×C₈×C₁₇	C₂×D₄	1	1	yes	characteristic
91	C₂×C₈×C₁₇	C₂⁴	1	1	yes	characteristic
92	C₄⋊C₄×C₁₇	—	2	1	no	—
93	C₄⋊C₄×C₁₇	—	2	1	no	—
94	C₄⋊C₄×C₁₇	—	4	1	no	—
95	C₄⋊C₄×C₁₇	—	4	1	no	—
96	C₄²×C₁₇	—	4	1	no	—
97	C₄²×C₁₇	—	4	1	no	—
98	C₄²×C₁₇	C₂×D₄	1	1	yes	characteristic
99	#63227	—	4	2	no	—
100	#63227	—	4	2	no	—
101	#63227	—	4	2	no	—
102	#63227	D₈	1	2	yes	—
103	C₄⋊Dic₁₇	—	2	2	no	—
104	C₄⋊Dic₁₇	—	2	2	no	—
105	C₄⋊Dic₁₇	—	4	1	no	—
106	C₄⋊Dic₁₇	—	4	1	no	—
107	C₄⋊Dic₁₇	D₈	1	1	yes	characteristic
108	#63225	—	4	2	no	—
109	#63225	—	4	2	no	—
110	#63225	—	4	2	no	—
111	C₄×Dic₁₇	—	4	2	no	—
112	C₂×#3622	C₂×D₄	1	2	yes	—
113	C₈×D₁₇	—	2	2	no	—
114	C₈×D₁₇	—	2	2	no	—
115	C₁₆×C₁₇	—	2	1	no	—
116	C₁₆×C₁₇	#41	1	2	yes	—
117	#63214	—	2	2	no	—
118	#33923	—	17	1	no	maximal, Sylow
119	C₂²×D₁₇	C₂×D₈	1	2	yes	—
120	Q₈×C₁₇	—	4	1	no	—
121	Q₈×C₁₇	—	4	1	no	—
122	Q₈×C₁₇	—	4	1	no	—
123	Q₈×C₁₇	—	4	1	no	—
124	C₂×C₄×C₁₇	—	4	1	no	—
125	C₂×C₄×C₁₇	—	4	1	no	—
126	C₂×C₄×C₁₇	—	4	1	no	—
127	C₂×C₄×C₁₇	—	4	1	no	—
128	C₂×C₄×C₁₇	C₂×D₈	1	1	yes	characteristic
129	C₂×C₄×C₁₇	C₂×D₈	1	1	yes	characteristic
130	C₂×C₄×C₁₇	C₂²×D₄	1	1	yes	characteristic
131	C₂×Dic₁₇	—	4	2	no	—
132	C₂×Dic₁₇	—	4	2	no	—
133	C₂×Dic₁₇	C₂×D₈	1	2	yes	—
134	D₆₈	—	2	2	no	—
135	C₄×D₁₇	—	2	2	no	—
136	C₄×D₁₇	—	2	2	no	—
137	C₄×D₁₇	—	16	2	no	—
138	Dic₃₄	—	8	2	no	—
139	Dic₃₄	—	16	1	no	—
140	C₈×C₁₇	—	2	1	no	—
141	C₈×C₁₇	C₂×#41	1	1	yes	characteristic
142	C₈×C₁₇	2₋¹⁺⁴	1	1	yes	characteristic
143	#3622	—	2	2	no	—
144	#3342	—	17	1	no	—
145	#3338	—	17	1	no	—
146	#2254	—	17	1	no	—
147	#2237	—	17	2	no	—
148	#2235	—	17	2	no	—
149	#2227	—	17	1	no	—
150	#2225	—	17	1	no	—
151	#2219	—	17	2	no	—
152	C₄×Q₃₂	D₁₇	1	1	yes	characteristic, complemented
153	#2169	—	17	1	no	—
154	#2159	—	17	2	no	—
155	C₂²×C₁₇	C₂²×D₈	1	1	yes	characteristic
156	C₂×D₁₇	—	2	2	no	—
157	C₂×D₁₇	—	2	2	no	—
158	C₂×D₁₇	—	2	2	no	—
159	C₄×C₁₇	—	2	1	no	—
160	C₄×C₁₇	—	8	1	no	—
161	C₄×C₁₇	—	8	1	no	—
162	C₄×C₁₇	—	8	1	no	—
163	C₄×C₁₇	—	8	1	no	—
164	C₄×C₁₇	#510	1	2	yes	—
165	C₄×C₁₇	#547	1	1	yes	characteristic
166	C₄×C₁₇	#578	1	1	yes	characteristic
167	Dic₁₇	—	2	2	no	—
168	Dic₁₇	—	8	2	no	—
169	Dic₁₇	—	8	2	no	—
170	#556	—	34	1	no	—
171	#554	—	34	1	no	—
172	C₂×Q₃₂	C₂×D₁₇	1	1	yes	characteristic
173	C₂×M₃₂	—	17	2	no	—
174	#500	—	17	1	no	—
175	#500	—	17	1	no	—
176	#485	—	34	2	no	—
177	#484	—	34	2	no	—
178	#477	—	34	1	no	—
179	#475	—	34	1	no	—
180	#467	—	17	2	no	—
181	#467	—	17	2	no	—
182	C₄×Q₁₆	C₂×D₁₇	1	1	yes	characteristic
183	C₄×Q₁₆	C₂×D₁₇	1	1	yes	characteristic
184	C₈×D₄	—	17	1	no	—
185	#427	—	17	2	no	—
186	C₂×#107	—	17	2	no	—
187	#367	—	17	2	no	—
188	#367	—	17	2	no	—
189	#367	—	17	2	no	—
190	#366	—	17	2	no	—
191	#366	C₂×D₁₇	1	1	yes	characteristic
192	C₄⋊C₁₆	—	17	1	no	—
193	#358	—	17	2	no	—
194	#358	—	17	2	no	—
195	#358	C₂×D₁₇	1	1	yes	characteristic
196	#358	C₂×D₁₇	1	1	yes	characteristic
197	C₂²⋊C₁₆	—	17	2	no	—
198	C₄×C₁₆	C₂×D₁₇	1	1	yes	characteristic
199	C₂×C₁₇	#3334	1	1	yes	characteristic
200	C₂×C₁₇	C₂×#510	1	1	yes	characteristic
201	C₂×C₁₇	#3416	1	1	yes	characteristic
202	D₁₇	—	4	2	no	—
203	C₂×Q₁₆	C₂²×D₁₇	1	1	yes	characteristic
204	C₂×Q₁₆	C₂²×D₁₇	1	1	yes	characteristic
205	C₂²×C₈	—	17	2	no	—
206	C₄⋊Q₈	—	34	1	no	—
207	C₄⋊Q₈	—	34	1	no	—
208	#126	—	68	2	no	—
209	#125	—	68	1	no	—
210	#124	—	68	1	no	—
211	C₂²⋊Q₈	—	34	2	no	—
212	C₂²⋊Q₈	—	34	2	no	—
213	C₂²⋊Q₈	—	68	2	no	—
214	C₄×Q₈	—	2	1	no	—
215	C₄×Q₈	—	2	1	no	—
216	C₄×Q₈	—	68	1	no	—
217	C₄×D₄	—	17	1	no	—
218	C₄×D₄	—	68	1	no	—
219	#117	—	34	2	no	—
220	C₂×C₄⋊C₄	—	34	2	no	—
221	Q₃₂	—	2	2	no	—
222	M₃₂	—	34	4	no	—
223	C₂×C₁₆	—	17	2	no	—
224	C₂×C₁₆	C₂²×D₁₇	1	1	yes	characteristic
225	C₂×C₁₆	C₂²×D₁₇	1	1	yes	characteristic
226	#107	—	17	2	no	—
227	#107	—	17	2	no	—
228	#107	—	34	2	no	—
229	#107	—	34	1	no	—
230	#107	—	34	1	no	—
231	#107	C₂²×D₁₇	1	1	yes	characteristic
232	#107	C₂²×D₁₇	1	1	yes	characteristic
233	#106	—	34	2	no	—
234	C₄⋊C₈	—	17	1	no	—
235	#103	—	2	1	no	—
236	#103	—	2	1	no	—
237	#103	—	34	2	no	—
238	#103	—	34	2	no	—
239	C₂²⋊C₈	—	17	2	no	—
240	C₄×C₈	C₂²×D₁₇	1	1	yes	characteristic
241	C₁₇	#33923	1	1	yes	characteristic, complemented, Sylow
242	C₂×Q₈	—	2	1	no	—
243	C₂×Q₈	—	2	1	no	—
244	C₂×Q₈	—	68	1	no	—
245	C₂×Q₈	—	68	1	no	—
246	C₂×D₄	—	17	1	no	—
247	C₂²×C₄	—	17	2	no	—
248	C₂²×C₄	—	68	2	no	—
249	Q₁₆	—	2	1	no	—
250	Q₁₆	—	2	1	no	—
251	Q₁₆	—	2	1	no	—
252	Q₁₆	—	2	1	no	—
253	C₂×C₈	—	17	2	no	—
254	C₂×C₈	—	34	2	no	—
255	C₂×C₈	—	34	2	no	—
256	C₂×C₈	D₄×D₁₇	1	1	yes	characteristic
257	C₂×C₈	C₂³×D₁₇	1	1	yes	characteristic
258	C₄⋊C₄	—	2	1	no	—
259	C₄⋊C₄	—	2	1	no	—
260	C₄⋊C₄	—	4	1	no	—
261	C₄⋊C₄	—	4	1	no	—
262	C₄⋊C₄	—	17	1	no	—
263	C₄⋊C₄	—	34	2	no	—
264	C₄⋊C₄	—	34	2	no	—
265	C₄⋊C₄	—	68	2	no	—
266	C₄⋊C₄	—	68	2	no	—
267	C₄⋊C₄	—	68	2	no	—
268	C₄⋊C₄	—	68	1	no	—
269	C₄⋊C₄	—	68	1	no	—
270	C₂²⋊C₄	—	17	2	no	—
271	C₂²⋊C₄	—	68	2	no	—
272	C₂²⋊C₄	—	68	2	no	—
273	C₂²⋊C₄	—	68	2	no	—
274	C₄²	—	4	1	no	—
275	C₄²	—	4	1	no	—
276	C₄²	—	68	2	no	—
277	C₄²	D₄×D₁₇	1	1	yes	characteristic
278	C₁₆	—	2	1	no	—
279	C₁₆	—	34	2	no	—
280	C₁₆	#63252	1	2	yes	—
281	C₂³	—	17	2	no	—
282	Q₈	—	4	1	no	—
283	Q₈	—	4	1	no	—
284	Q₈	—	4	1	no	—
285	Q₈	—	4	1	no	—
286	Q₈	—	272	1	no	—
287	Q₈	—	136	2	no	—
288	D₄	—	34	2	no	—
289	C₂×C₄	—	4	1	no	—
290	C₂×C₄	—	4	1	no	—
291	C₂×C₄	—	4	1	no	—
292	C₂×C₄	—	4	1	no	—
293	C₂×C₄	—	272	2	no	—
294	C₂×C₄	—	17	2	no	—
295	C₂×C₄	—	34	2	no	—
296	C₂×C₄	—	34	2	no	—
297	C₂×C₄	—	68	2	no	—
298	C₂×C₄	—	68	2	no	—
299	C₂×C₄	D₈×D₁₇	1	1	yes	characteristic
300	C₂×C₄	D₈×D₁₇	1	1	yes	characteristic
301	C₂×C₄	C₂×D₄×D₁₇	1	1	yes	characteristic
302	C₈	—	2	1	no	—
303	C₈	—	34	2	no	—
304	C₈	C₂×#63252	1	1	yes	characteristic
305	C₈	#10587697	1	1	yes	characteristic
306	C₂²	—	34	2	no	—
307	C₂²	—	34	2	no	—
308	C₂²	—	34	2	no	—
309	C₂²	C₂×D₈×D₁₇	1	1	yes	characteristic
310	C₄	—	2	1	no	—
311	C₄	—	8	1	no	—
312	C₄	—	8	1	no	—
313	C₄	—	8	1	no	—
314	C₄	—	8	1	no	—
315	C₄	—	34	2	no	—
316	C₄	—	136	2	no	—
317	C₄	—	136	2	no	—
318	C₄	#11763251	1	2	yes	—
319	C₄	#11763464	1	1	yes	characteristic
320	C₄	#11763623	1	1	yes	characteristic
321	C₂	—	68	2	no	—
322	C₂	#14541133	1	1	yes	characteristic
323	C₂	C₂×#11763251	1	1	yes	characteristic
324	C₂	#14541927	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×D₈×D₁₇
commutator, cosocle	C₂×C₈×C₁₇	C₂⁴
Frattini	C₂×C₈	C₂³×D₁₇
Fitting	C₄×C₁₇×Q₃₂	C₂
socle	C₂²×C₁₇	C₂²×D₈
prime radical	C₂×D₆₈	D₈
hypercenter	C₄×Q₃₂	D₁₇
hypocenter	C₁₇	#33923
LCS	G ▷ C₂×C₈×C₁₇ ▷ C₄×C₁₇ ▷ C₂×C₁₇ ▷ C₁₇
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₄×C₈ ◁ C₄×Q₃₂

Complements

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

N	H ≅ G/N	type
3. #14531066	C₂	semidirect
4. #14530990	C₂	semidirect
5. #14530980	C₂	semidirect
6. #14530936	C₂	semidirect
7. #14530926	C₂	semidirect
8. #14530889	C₂	semidirect
10. #14530569	C₂	semidirect
11. C₄×C₁₇×Q₃₂	C₂	semidirect
152. C₄×Q₃₂	D₁₇	semidirect
241. C₁₇	#33923	semidirect

Maximal subgroups

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

subgroup	index
#14541216	2
#14541178	2
#14531066	2
#14530990	2
#14530980	2
#14530936	2
#14530926	2
#14530889	2
#14530617	2
#14530569	2
C₄×C₁₇×Q₃₂	2
#33923	17