Subgroups — #17189539

Order 4352.

544 conjugacy classes of subgroups.

The subgroup lattice has 314 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#14539434	C₂	1	1	yes	maximal, characteristic
2	#14530975	C₂	1	2	yes	maximal, complemented
3	#14530948	C₂	1	2	yes	maximal, complemented
4	#14530824	C₂	1	2	yes	maximal, complemented
5	#14530701	C₂	1	2	yes	maximal, complemented
6	#14530646	C₂	1	2	yes	maximal, complemented
7	C₄×#10587499	C₂	1	2	yes	maximal, complemented
8	#14530591	C₂	1	1	yes	maximal, characteristic, complemented
9	C₁₇×#2950	C₂	1	1	yes	maximal, characteristic, complemented
10	#11763414	—	2	1	no	—
11	C₂×#10587499	C₂²	1	2	yes	—
12	C₂×#10587499	C₂²	1	2	yes	—
13	#11763211	C₂²	1	1	yes	characteristic
14	#11763186	C₂²	1	1	yes	characteristic
15	#11763174	—	2	1	no	—
16	#11763062	—	2	1	no	—
17	#11763045	C₂²	1	2	yes	—
18	#11762922	—	2	1	no	—
19	C₄×D₁₃₆	C₂²	1	2	yes	—
20	#11762636	C₂²	1	2	yes	—
21	#11762634	C₂²	1	2	yes	—
22	C₄⋊#63214	C₂²	1	2	yes	—
23	#11762613	C₂²	1	2	yes	—
24	#11762613	C₂²	1	2	yes	—
25	#11762609	C₂²	1	2	yes	—
26	#11762609	C₂²	1	2	yes	—
27	C₂²⋊#63214	C₂²	1	2	yes	—
28	#11762585	C₂²	1	1	yes	characteristic
29	C₄×#63214	C₂²	1	1	yes	characteristic
30	C₂²×Q₁₆×C₁₇	C₂²	1	1	yes	characteristic
31	C₁₇×#519	—	2	1	no	—
32	C₁₇×#441	—	2	1	no	—
33	C₁₇×#441	—	2	1	no	—
34	C₄×Q₁₆×C₁₇	C₂²	1	2	yes	—
35	C₄×Q₁₆×C₁₇	C₂²	1	2	yes	—
36	C₁₇×#427	C₂²	1	1	yes	characteristic
37	C₁₇×#419	—	2	1	no	—
38	C₁₇×#405	C₂²	1	1	yes	characteristic
39	C₂²×Q₈×C₁₇	—	2	1	no	—
40	C₂×Q₁₆×C₁₇	—	2	4	no	—
41	C₂×Q₁₆×C₁₇	—	2	1	no	—
42	C₂×Q₁₆×C₁₇	C₂³	1	2	yes	—
43	C₂×M₁₆×C₁₇	D₄	1	1	yes	characteristic
44	C₂²×C₈×C₁₇	C₂³	1	1	yes	characteristic
45	C₄×Q₈×C₁₇	—	2	2	no	—
46	C₄×Q₈×C₁₇	—	2	2	no	—
47	C₄×Q₈×C₁₇	—	4	2	no	—
48	C₁₇×#117	—	2	1	no	—
49	C₁₇×#117	—	4	2	no	—
50	C₁₇×#117	D₄	1	1	yes	characteristic
51	#10587616	—	2	2	no	—
52	C₂×D₁₃₆	—	2	1	no	—
53	C₂×D₁₃₆	C₂³	1	2	yes	—
54	#10587580	—	4	2	no	—
55	#10587570	—	4	2	no	—
56	#10587560	—	2	1	no	—
57	C₄×D₆₈	—	2	2	no	—
58	#10587556	—	2	1	no	—
59	C₁₇×#107	C₂³	1	2	yes	—
60	C₁₇×#106	—	2	1	no	—
61	C₁₇×#103	—	2	2	no	—
62	C₁₇×#103	—	2	2	no	—
63	C₁₇×#97	D₄	1	1	yes	characteristic
64	C₁₇×#97	D₄	1	1	yes	characteristic
65	C₄×C₈×C₁₇	C₂³	1	1	yes	characteristic
66	C₄×C₈×C₁₇	C₂³	1	1	yes	characteristic
67	#10587499	—	2	4	no	—
68	#10587499	—	2	4	no	—
69	#10587492	—	2	2	no	—
70	#10587492	—	2	2	no	—
71	#10587489	C₂³	1	2	yes	—
72	#10587488	—	2	1	no	—
73	C₂×#63214	C₂³	1	2	yes	—
74	C₂×#63214	C₂³	1	2	yes	—
75	C₂×Q₈×C₁₇	—	2	4	no	—
76	C₂×Q₈×C₁₇	—	2	2	no	—
77	C₂×Q₈×C₁₇	—	4	2	no	—
78	C₂×Q₈×C₁₇	—	4	1	no	—
79	C₂²×C₄×C₁₇	—	4	1	no	—
80	C₂²×C₄×C₁₇	C₂×D₄	1	1	yes	characteristic
81	C₂×C₂²⋊D₁₇	—	4	2	no	—
82	C₂×D₆₈	—	2	2	no	—
83	C₂×D₆₈	—	4	2	no	—
84	C₂×C₄×D₁₇	—	4	2	no	—
85	Q₁₆×C₁₇	—	2	4	no	—
86	Q₁₆×C₁₇	—	2	2	no	—
87	Q₁₆×C₁₇	#41	1	4	yes	—
88	M₁₆×C₁₇	—	4	1	no	—
89	C₂×C₈×C₁₇	—	2	2	no	—
90	C₂×C₈×C₁₇	C₂×D₄	1	1	yes	characteristic
91	C₂×C₈×C₁₇	C₂×D₄	1	1	yes	characteristic
92	C₂×C₈×C₁₇	C₂×D₄	1	1	yes	characteristic
93	C₂×C₈×C₁₇	C₂⁴	1	1	yes	characteristic
94	C₄⋊C₄×C₁₇	—	2	2	no	—
95	C₄⋊C₄×C₁₇	—	4	2	no	—
96	C₄⋊C₄×C₁₇	—	4	2	no	—
97	C₄⋊C₄×C₁₇	D₈	1	2	yes	—
98	C₂²⋊C₄×C₁₇	—	4	1	no	—
99	C₂²⋊C₄×C₁₇	D₈	1	2	yes	—
100	C₄²×C₁₇	—	4	1	no	—
101	C₄²×C₁₇	—	4	2	no	—
102	C₄²×C₁₇	—	4	2	no	—
103	C₄²×C₁₇	C₂×D₄	1	1	yes	characteristic
104	C₄²×C₁₇	C₂×D₄	1	1	yes	characteristic
105	#63227	—	4	2	no	—
106	#63227	—	4	2	no	—
107	C₄⋊Dic₁₇	—	2	2	no	—
108	#63225	—	4	2	no	—
109	D₁₃₆	—	2	2	no	—
110	D₁₃₆	—	2	2	no	—
111	D₁₃₆	—	4	2	no	—
112	#63214	—	2	2	no	—
113	#63214	—	2	2	no	—
114	#33686	—	17	1	no	maximal, Sylow
115	C₂³×C₁₇	C₂×D₈	1	1	yes	characteristic
116	C₂²×D₁₇	—	4	2	no	—
117	Q₈×C₁₇	—	2	4	no	—
118	Q₈×C₁₇	—	4	4	no	—
119	Q₈×C₁₇	—	4	2	no	—
120	C₂×C₄×C₁₇	—	2	1	no	—
121	C₂×C₄×C₁₇	—	2	1	no	—
122	C₂×C₄×C₁₇	—	4	4	no	—
123	C₂×C₄×C₁₇	—	4	2	no	—
124	C₂×C₄×C₁₇	—	4	2	no	—
125	C₂×C₄×C₁₇	C₂×D₈	1	2	yes	—
126	C₂×C₄×C₁₇	C₂×D₈	1	2	yes	—
127	C₂×C₄×C₁₇	C₂×D₈	1	1	yes	characteristic
128	C₂×C₄×C₁₇	C₂²×D₄	1	1	yes	characteristic
129	C₂²⋊D₁₇	—	16	2	no	—
130	C₂×Dic₁₇	—	4	2	no	—
131	D₆₈	—	4	2	no	—
132	D₆₈	—	4	2	no	—
133	D₆₈	—	16	2	no	—
134	C₄×D₁₇	—	16	2	no	—
135	C₈×C₁₇	—	2	1	no	—
136	C₈×C₁₇	—	2	1	no	—
137	C₈×C₁₇	—	2	1	no	—
138	C₈×C₁₇	C₂×#41	1	2	yes	—
139	#3176	—	17	1	no	—
140	#2950	D₁₇	1	1	yes	characteristic, complemented
141	#2234	—	17	2	no	—
142	#2229	—	17	2	no	—
143	#2207	—	17	2	no	—
144	#2185	—	17	2	no	—
145	#2175	—	17	2	no	—
146	C₄×SD₃₂	—	17	2	no	—
147	#2164	—	17	1	no	—
148	C₂²×C₁₇	—	2	1	no	—
149	C₂²×C₁₇	—	2	2	no	—
150	C₂²×C₁₇	C₂²×D₈	1	1	yes	characteristic
151	C₂×D₁₇	—	8	2	no	—
152	C₂×D₁₇	—	8	2	no	—
153	C₂×D₁₇	—	8	2	no	—
154	C₄×C₁₇	—	2	1	no	—
155	C₄×C₁₇	—	2	2	no	—
156	C₄×C₁₇	—	2	2	no	—
157	C₄×C₁₇	—	4	4	no	—
158	C₄×C₁₇	—	8	2	no	—
159	C₄×C₁₇	#525	1	1	yes	characteristic
160	C₄×C₁₇	#575	1	1	yes	characteristic
161	Dic₁₇	—	8	2	no	—
162	C₂²×Q₁₆	C₂×D₁₇	1	1	yes	characteristic
163	#540	—	34	1	no	—
164	#519	—	2	1	no	—
165	C₂×SD₃₂	—	17	2	no	—
166	C₂×SD₃₂	—	17	2	no	—
167	#499	—	17	1	no	—
168	#493	—	17	1	no	—
169	#490	—	34	1	no	—
170	#469	—	34	1	no	—
171	#466	—	17	2	no	—
172	#442	—	34	1	no	—
173	#441	—	2	1	no	—
174	#441	—	2	1	no	—
175	C₄×Q₁₆	C₂×D₁₇	1	2	yes	—
176	C₄×Q₁₆	C₂×D₁₇	1	2	yes	—
177	C₄×D₈	—	17	2	no	—
178	#427	C₂×D₁₇	1	1	yes	characteristic
179	#419	—	2	1	no	—
180	#405	C₂×D₁₇	1	1	yes	characteristic
181	#367	—	17	2	no	—
182	#366	—	17	2	no	—
183	C₄⋊C₁₆	—	17	2	no	—
184	#358	—	17	2	no	—
185	#358	—	17	2	no	—
186	#357	—	17	2	no	—
187	#357	—	17	2	no	—
188	C₂²⋊C₁₆	—	17	2	no	—
189	#346	—	17	1	no	—
190	C₄×C₁₆	—	17	1	no	—
191	C₂×C₁₇	—	4	1	no	—
192	C₂×C₁₇	#3100	1	1	yes	characteristic
193	C₂×C₁₇	#3416	1	2	yes	—
194	D₁₇	—	16	2	no	—
195	C₂²×Q₈	—	2	1	no	—
196	C₂×Q₁₆	—	2	4	no	—
197	C₂×Q₁₆	—	2	1	no	—
198	C₂×Q₁₆	C₂²×D₁₇	1	2	yes	—
199	C₂×D₈	—	17	2	no	—
200	C₂×D₈	—	34	1	no	—
201	C₂×M₁₆	C₂²⋊D₁₇	1	1	yes	characteristic
202	C₂²×C₈	C₂²×D₁₇	1	1	yes	characteristic
203	#127	—	34	1	no	—
204	#125	—	34	1	no	—
205	#123	—	68	2	no	—
206	#121	—	34	2	no	—
207	#121	—	68	2	no	—
208	C₄×Q₈	—	2	2	no	—
209	C₄×Q₈	—	2	2	no	—
210	C₄×Q₈	—	4	2	no	—
211	C₄×D₄	—	34	2	no	—
212	#117	—	2	1	no	—
213	#117	—	4	2	no	—
214	#117	C₂²⋊D₁₇	1	1	yes	characteristic
215	SD₃₂	—	34	4	no	—
216	SD₃₂	—	34	4	no	—
217	C₂×C₁₆	—	17	2	no	—
218	C₂×C₁₆	—	17	2	no	—
219	#107	—	17	2	no	—
220	#107	C₂²×D₁₇	1	2	yes	—
221	#106	—	2	1	no	—
222	#106	—	34	1	no	—
223	#103	—	2	2	no	—
224	#103	—	2	2	no	—
225	#102	—	34	2	no	—
226	#102	—	34	2	no	—
227	#97	C₂²⋊D₁₇	1	1	yes	characteristic
228	#97	C₂²⋊D₁₇	1	1	yes	characteristic
229	C₄×C₈	C₂²×D₁₇	1	1	yes	characteristic
230	C₄×C₈	C₂²×D₁₇	1	1	yes	characteristic
231	C₁₇	#33686	1	1	yes	characteristic, complemented, Sylow
232	C₂×Q₈	—	2	4	no	—
233	C₂×Q₈	—	2	2	no	—
234	C₂×Q₈	—	4	2	no	—
235	C₂×Q₈	—	4	1	no	—
236	C₂×D₄	—	34	2	no	—
237	C₂×D₄	—	68	2	no	—
238	C₂×D₄	—	68	2	no	—
239	C₂²×C₄	—	4	1	no	—
240	C₂²×C₄	—	68	2	no	—
241	C₂²×C₄	C₂×C₂²⋊D₁₇	1	1	yes	characteristic
242	Q₁₆	—	2	4	no	—
243	Q₁₆	—	2	2	no	—
244	Q₁₆	#63252	1	4	yes	—
245	D₈	—	34	2	no	—
246	D₈	—	34	2	no	—
247	D₈	—	68	2	no	—
248	M₁₆	—	4	1	no	—
249	C₂×C₈	—	2	2	no	—
250	C₂×C₈	C₂×C₂²⋊D₁₇	1	1	yes	characteristic
251	C₂×C₈	C₂×C₂²⋊D₁₇	1	1	yes	characteristic
252	C₂×C₈	C₂×C₂²⋊D₁₇	1	1	yes	characteristic
253	C₂×C₈	C₂³×D₁₇	1	1	yes	characteristic
254	C₄⋊C₄	—	2	2	no	—
255	C₄⋊C₄	—	4	2	no	—
256	C₄⋊C₄	—	4	2	no	—
257	C₄⋊C₄	—	34	2	no	—
258	C₄⋊C₄	—	68	2	no	—
259	C₄⋊C₄	#63228	1	2	yes	—
260	C₂²⋊C₄	—	4	1	no	—
261	C₂²⋊C₄	—	68	2	no	—
262	C₂²⋊C₄	—	68	2	no	—
263	C₂²⋊C₄	#63228	1	2	yes	—
264	C₄²	—	4	1	no	—
265	C₄²	—	4	2	no	—
266	C₄²	—	4	2	no	—
267	C₄²	C₂×C₂²⋊D₁₇	1	1	yes	characteristic
268	C₄²	C₂×C₂²⋊D₁₇	1	1	yes	characteristic
269	C₁₆	—	34	2	no	—
270	C₁₆	—	34	2	no	—
271	C₂³	—	68	2	no	—
272	C₂³	C₂×#63228	1	1	yes	characteristic
273	Q₈	—	2	4	no	—
274	Q₈	—	4	4	no	—
275	Q₈	—	4	2	no	—
276	D₄	—	272	2	no	—
277	D₄	—	272	2	no	—
278	D₄	—	68	2	no	—
279	D₄	—	68	2	no	—
280	C₂×C₄	—	2	1	no	—
281	C₂×C₄	—	2	1	no	—
282	C₂×C₄	—	4	4	no	—
283	C₂×C₄	—	4	2	no	—
284	C₂×C₄	—	4	2	no	—
285	C₂×C₄	—	272	2	no	—
286	C₂×C₄	—	68	2	no	—
287	C₂×C₄	C₂×#63228	1	2	yes	—
288	C₂×C₄	C₂×#63228	1	2	yes	—
289	C₂×C₄	C₂×#63228	1	1	yes	characteristic
290	C₂×C₄	C₂²×C₂²⋊D₁₇	1	1	yes	characteristic
291	C₈	—	2	1	no	—
292	C₈	—	2	1	no	—
293	C₈	—	2	1	no	—
294	C₈	C₂×#63252	1	2	yes	—
295	C₂²	—	2	1	no	—
296	C₂²	—	2	2	no	—
297	C₂²	—	136	2	no	—
298	C₂²	—	136	2	no	—
299	C₂²	—	136	2	no	—
300	C₂²	C₂²×#63228	1	1	yes	characteristic
301	C₄	—	2	1	no	—
302	C₄	—	2	2	no	—
303	C₄	—	2	2	no	—
304	C₄	—	4	4	no	—
305	C₄	—	8	2	no	—
306	C₄	—	136	2	no	—
307	C₄	#11763320	1	1	yes	characteristic
308	C₄	#11763607	1	1	yes	characteristic
309	C₂	—	4	1	no	—
310	C₂	—	272	2	no	—
311	C₂	#14538784	1	1	yes	characteristic
312	C₂	#14541928	1	2	yes	—

	subgroup	G/H
center	C₂²	C₂²×#63228
commutator, cosocle	C₂×C₈×C₁₇	C₂⁴
Frattini	C₂×C₈	C₂³×D₁₇
Fitting	C₁₇×#2950	C₂
socle	C₂²×C₁₇	C₂²×D₈
prime radical	#14539434	C₂
hypercenter	#2950	D₁₇
hypocenter	C₁₇	#33686
LCS	G ▷ C₂×C₈×C₁₇ ▷ C₄×C₁₇ ▷ C₂×C₁₇ ▷ C₁₇
UCS	C₁ ◁ C₂² ◁ #117 ◁ #405 ◁ #2950

Complements

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

N	H ≅ G/N	type
2. #14530975	C₂	semidirect
3. #14530948	C₂	semidirect
4. #14530824	C₂	semidirect
5. #14530701	C₂	semidirect
6. #14530646	C₂	semidirect
7. C₄×#10587499	C₂	semidirect
8. #14530591	C₂	semidirect
9. C₁₇×#2950	C₂	semidirect
140. #2950	D₁₇	semidirect
231. C₁₇	#33686	semidirect

Maximal subgroups

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

subgroup	index
#14539434	2
#14530975	2
#14530948	2
#14530824	2
#14530701	2
#14530646	2
C₄×#10587499	2
#14530591	2
C₁₇×#2950	2
#33686	17