Subgroups — #17189007

Order 4352.

556 conjugacy classes of subgroups.

The subgroup lattice has 346 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#14537637	C₂	1	1	yes	maximal, characteristic
2	#14530938	C₂	1	2	yes	maximal, complemented
3	#14530938	C₂	1	2	yes	maximal, complemented
4	#14530928	C₂	1	2	yes	maximal, complemented
5	#14530918	C₂	1	2	yes	maximal, complemented
6	#14530570	C₂	1	1	yes	maximal, characteristic, complemented
7	#14530566	C₂	1	1	yes	maximal, characteristic, complemented
8	#14530543	C₂	1	1	yes	maximal, characteristic, complemented
9	#14530499	C₂	1	1	yes	maximal, characteristic
10	C₂×#11762609	C₂	1	1	yes	maximal, characteristic
11	C₁₇×#3164	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂²×D₁₃₆	C₂²	1	1	yes	characteristic
13	#11763283	—	2	1	no	—
14	C₂×#10587483	C₂²	1	1	yes	characteristic
15	C₂²×#63214	C₂²	1	1	yes	characteristic
16	#11762922	—	2	2	no	—
17	C₄×D₁₃₆	C₂²	1	2	yes	—
18	C₄×D₁₃₆	C₂²	1	2	yes	—
19	#11762847	C₂²	1	1	yes	characteristic
20	#11762815	—	2	1	no	—
21	#11762636	C₂²	1	1	yes	characteristic, complemented
22	#11762636	C₂²	1	1	yes	characteristic, complemented
23	#11762636	C₂²	1	2	yes	complemented
24	#11762634	C₂²	1	1	yes	characteristic, complemented
25	#11762634	C₂²	1	1	yes	characteristic, complemented
26	#11762634	C₂²	1	2	yes	complemented
27	C₄⋊#63214	C₂²	1	2	yes	complemented
28	C₄⋊#63214	C₂²	1	2	yes	complemented
29	#11762609	C₂²	1	2	yes	—
30	#11762609	C₂²	1	2	yes	—
31	#11762609	C₂²	1	2	yes	—
32	#11762609	C₂²	1	2	yes	—
33	C₁₇×#544	—	2	1	no	—
34	C₁₇×#501	—	2	2	no	—
35	C₁₇×#500	C₂²	1	2	yes	complemented
36	C₁₇×#499	C₂²	1	2	yes	complemented
37	C₁₇×#428	—	2	1	no	—
38	C₁₇×#427	C₂²	1	1	yes	characteristic
39	C₂×C₁₇×#107	C₂²	1	1	yes	characteristic
40	C₁₇×#405	C₂²	1	1	yes	characteristic
41	C₂²×D₆₈	—	2	1	no	—
42	C₂×M₁₆×C₁₇	D₄	1	1	yes	characteristic
43	C₂²×C₈×C₁₇	C₂³	1	1	yes	characteristic
44	C₁₇×C₄⋊Q₈	—	2	2	no	—
45	C₁₇×C₄⋊Q₈	—	4	1	no	—
46	C₁₇×#125	—	2	2	no	—
47	C₁₇×#125	—	4	1	no	—
48	C₁₇×C₂²⋊Q₈	—	4	2	no	—
49	C₁₇×#117	—	2	1	no	—
50	C₁₇×#117	D₄	1	1	yes	characteristic
51	C₂×C₄⋊C₄×C₁₇	—	2	1	no	—
52	#10587612	—	2	1	no	—
53	C₂×D₁₃₆	—	2	1	no	—
54	C₂×D₁₃₆	—	2	2	no	—
55	C₂×D₁₃₆	Q₈	1	4	yes	complemented
56	C₂×D₁₃₆	C₂³	1	2	yes	—
57	#10587579	—	4	1	no	—
58	#10587579	—	4	1	no	—
59	C₂²⋊C₄×D₁₇	—	4	2	no	—
60	C₄×D₆₈	—	2	2	no	—
61	C₄×D₆₈	—	2	2	no	—
62	C₁₇×#107	—	2	2	no	—
63	C₁₇×#107	—	2	1	no	—
64	C₁₇×#107	C₂³	1	1	yes	characteristic
65	C₁₇×#107	C₂³	1	2	yes	—
66	C₁₇×#107	C₂³	1	1	yes	characteristic
67	C₁₇×#106	—	2	2	no	—
68	C₁₇×#106	—	2	1	no	—
69	C₁₇×#97	D₄	1	2	yes	—
70	C₄×C₈×C₁₇	C₂³	1	2	yes	—
71	#10587492	—	2	2	no	—
72	#10587492	—	2	2	no	—
73	#10587489	C₂³	1	1	yes	characteristic
74	#10587489	C₂³	1	1	yes	characteristic
75	#10587488	—	2	1	no	—
76	#10587483	—	2	2	no	—
77	C₂×#63214	—	2	2	no	—
78	C₂×#63214	C₂³	1	1	yes	characteristic
79	C₂×#63214	C₂³	1	1	yes	characteristic
80	C₂×#63214	C₂³	1	2	yes	—
81	C₂³×D₁₇	—	4	1	no	—
82	C₂×Q₈×C₁₇	—	4	2	no	—
83	C₂²×C₄×C₁₇	—	4	1	no	—
84	C₂²×C₄×C₁₇	C₂×D₄	1	1	yes	characteristic
85	C₂×D₆₈	—	2	4	no	—
86	C₂×D₆₈	—	2	2	no	—
87	C₂×D₆₈	—	4	1	no	—
88	C₂×D₆₈	—	4	2	no	—
89	C₂×C₄×D₁₇	—	4	2	no	—
90	C₂×C₄×D₁₇	—	4	2	no	—
91	M₁₆×C₁₇	—	2	2	no	—
92	C₂×C₈×C₁₇	C₂×D₄	1	2	yes	—
93	C₂×C₈×C₁₇	C₂×D₄	1	1	yes	characteristic
94	C₂×C₈×C₁₇	C₂×Q₈	1	2	yes	—
95	C₂×C₈×C₁₇	#41	1	2	yes	—
96	C₂×C₈×C₁₇	C₂⁴	1	1	yes	characteristic
97	C₄⋊C₄×C₁₇	—	2	1	no	—
98	C₄⋊C₄×C₁₇	—	2	2	no	—
99	C₄⋊C₄×C₁₇	—	2	1	no	—
100	C₄⋊C₄×C₁₇	—	4	2	no	—
101	C₄⋊C₄×C₁₇	—	4	2	no	—
102	C₄⋊C₄×C₁₇	—	4	2	no	—
103	C₄⋊C₄×C₁₇	—	4	1	no	—
104	C₄⋊C₄×C₁₇	D₈	1	1	yes	characteristic
105	C₄⋊C₄×C₁₇	D₈	1	1	yes	characteristic
106	C₂²⋊C₄×C₁₇	—	4	1	no	—
107	C₂²⋊C₄×C₁₇	D₈	1	2	yes	—
108	C₄²×C₁₇	—	4	1	no	—
109	C₄²×C₁₇	C₂×D₄	1	2	yes	—
110	C₂²⋊Dic₁₇	—	4	1	no	—
111	#63227	—	4	2	no	—
112	#63227	—	4	2	no	—
113	C₄⋊Dic₁₇	—	2	1	no	—
114	C₄⋊Dic₁₇	—	2	1	no	—
115	C₄×Dic₁₇	—	4	1	no	—
116	D₁₃₆	—	2	4	no	—
117	D₁₃₆	—	2	2	no	—
118	D₁₃₆	C₂×Q₈	1	4	yes	—
119	#63214	—	2	1	no	—
120	#63214	—	2	1	no	—
121	#63214	—	2	2	no	—
122	#33631	—	17	1	no	maximal, Sylow
123	C₂³×C₁₇	C₂×D₈	1	1	yes	characteristic
124	C₂²×D₁₇	—	4	2	no	—
125	C₂²×D₁₇	—	4	4	no	—
126	C₂²×D₁₇	—	16	1	no	—
127	C₂²×D₁₇	—	16	1	no	—
128	Q₈×C₁₇	—	16	2	no	—
129	C₂×C₄×C₁₇	—	4	1	no	—
130	C₂×C₄×C₁₇	—	4	1	no	—
131	C₂×C₄×C₁₇	—	4	2	no	—
132	C₂×C₄×C₁₇	—	16	1	no	—
133	C₂×C₄×C₁₇	C₂²⋊Q₈	1	2	yes	—
134	C₂×C₄×C₁₇	C₂×D₈	1	2	yes	—
135	C₂×C₄×C₁₇	C₂×D₈	1	2	yes	—
136	C₂×C₄×C₁₇	C₂×D₈	1	1	yes	characteristic
137	C₂×C₄×C₁₇	#137	1	2	yes	—
138	C₂×C₄×C₁₇	C₂²×D₄	1	1	yes	characteristic
139	C₂×Dic₁₇	—	4	1	no	—
140	C₂×Dic₁₇	—	4	1	no	—
141	D₆₈	—	2	4	no	—
142	D₆₈	—	4	2	no	—
143	D₆₈	—	4	4	no	—
144	C₄×D₁₇	—	16	2	no	—
145	C₄×D₁₇	—	16	2	no	—
146	C₈×C₁₇	—	2	2	no	—
147	C₈×C₁₇	C₂²⋊Q₈	1	2	yes	—
148	C₈×C₁₇	C₂²×Q₈	1	1	yes	characteristic
149	C₈×C₁₇	C₂×#41	1	1	yes	characteristic
150	#3164	D₁₇	1	1	yes	characteristic, complemented
151	#2948	—	17	1	no	—
152	#2228	—	17	2	no	—
153	#2228	—	17	2	no	—
154	#2226	—	17	2	no	—
155	#2224	—	17	2	no	—
156	#2159	—	17	1	no	—
157	#2158	—	17	1	no	—
158	#2151	—	17	1	no	—
159	#2140	—	17	1	no	—
160	C₂×#357	—	17	1	no	—
161	C₂²×C₁₇	—	2	1	no	—
162	C₂²×C₁₇	—	2	1	no	—
163	C₂²×C₁₇	#474	1	2	yes	—
164	C₂²×C₁₇	C₂²×D₈	1	1	yes	characteristic
165	C₂×D₁₇	—	4	4	no	—
166	C₂×D₁₇	—	8	2	no	—
167	C₂×D₁₇	—	8	2	no	—
168	C₂×D₁₇	—	16	4	no	—
169	C₄×C₁₇	—	2	2	no	—
170	C₄×C₁₇	—	2	2	no	—
171	C₄×C₁₇	—	8	2	no	—
172	C₄×C₁₇	—	8	1	no	—
173	C₄×C₁₇	—	8	1	no	—
174	C₄×C₁₇	#474	1	2	yes	—
175	C₄×C₁₇	C₂×C₂²⋊Q₈	1	1	yes	characteristic
176	C₄×C₁₇	C₂×#137	1	1	yes	characteristic
177	Dic₁₇	—	8	1	no	—
178	Dic₁₇	—	8	1	no	—
179	C₂²×D₈	—	17	1	no	—
180	#544	—	2	1	no	—
181	#518	—	34	1	no	—
182	C₂×M₃₂	—	17	1	no	—
183	C₂²×C₁₆	—	17	1	no	—
184	#501	—	2	2	no	—
185	#500	C₂×D₁₇	1	2	yes	complemented
186	#499	C₂×D₁₇	1	2	yes	complemented
187	#442	—	34	2	no	—
188	C₄×D₈	—	17	2	no	—
189	C₄×D₈	—	17	2	no	—
190	#428	—	2	1	no	—
191	#427	—	17	1	no	—
192	#427	C₂×D₁₇	1	1	yes	characteristic
193	C₂×#107	C₂×D₁₇	1	1	yes	characteristic
194	#418	—	34	1	no	—
195	#405	C₂×D₁₇	1	1	yes	characteristic
196	#367	—	17	1	no	—
197	#367	—	17	1	no	—
198	#367	—	17	2	no	—
199	#366	—	17	1	no	—
200	#366	—	17	1	no	—
201	#366	—	17	2	no	—
202	C₄⋊C₁₆	—	17	2	no	—
203	C₄⋊C₁₆	—	17	2	no	—
204	#357	—	17	2	no	—
205	#357	—	17	2	no	—
206	#357	—	17	2	no	—
207	#357	—	17	2	no	—
208	C₂×C₁₇	—	2	2	no	—
209	C₂×C₁₇	C₂×#474	1	1	yes	characteristic
210	C₂×C₁₇	#3415	1	1	yes	characteristic
211	C₂×C₁₇	#3416	1	1	yes	characteristic
212	D₁₇	—	8	4	no	—
213	C₂²×D₄	—	34	1	no	—
214	C₂×D₈	—	17	4	no	—
215	C₂×D₈	—	17	2	no	—
216	C₂×D₈	—	34	2	no	—
217	C₂×D₈	—	34	1	no	—
218	C₂×M₁₆	C₂²⋊D₁₇	1	1	yes	characteristic
219	C₂²×C₈	C₂²×D₁₇	1	1	yes	characteristic
220	C₄⋊Q₈	—	2	2	no	—
221	C₄⋊Q₈	—	4	1	no	—
222	#125	—	2	2	no	—
223	#125	—	4	1	no	—
224	C₂²⋊Q₈	—	4	2	no	—
225	C₄×D₄	—	34	2	no	—
226	C₄×D₄	—	34	2	no	—
227	C₄×D₄	—	68	1	no	—
228	C₄×D₄	—	68	1	no	—
229	#117	—	2	1	no	—
230	#117	—	34	1	no	—
231	#117	C₂²⋊D₁₇	1	1	yes	characteristic
232	C₂×C₄⋊C₄	—	2	1	no	—
233	C₂×C₂²⋊C₄	—	68	2	no	—
234	M₃₂	—	34	2	no	—
235	C₂×C₁₆	—	17	1	no	—
236	C₂×C₁₆	—	17	2	no	—
237	C₂×C₁₆	—	17	1	no	—
238	C₂×C₁₆	—	34	2	no	—
239	#107	—	2	1	no	—
240	#107	—	2	2	no	—
241	#107	—	17	1	no	—
242	#107	—	17	1	no	—
243	#107	C₂²×D₁₇	1	2	yes	—
244	#107	C₂²×D₁₇	1	1	yes	characteristic
245	#107	C₂²×D₁₇	1	1	yes	characteristic
246	#106	—	2	1	no	—
247	#106	—	2	2	no	—
248	#106	—	34	1	no	—
249	#102	—	34	2	no	—
250	#102	—	34	2	no	—
251	#97	C₂²⋊D₁₇	1	2	yes	—
252	C₄×C₈	C₂²×D₁₇	1	2	yes	—
253	C₁₇	#33631	1	1	yes	characteristic, complemented, Sylow
254	C₂⁴	—	68	1	no	—
255	C₂×Q₈	—	4	2	no	—
256	C₂×D₄	—	34	2	no	—
257	C₂×D₄	—	34	4	no	—
258	C₂×D₄	—	68	1	no	—
259	C₂×D₄	—	68	2	no	—
260	C₂²×C₄	—	4	1	no	—
261	C₂²×C₄	—	68	2	no	—
262	C₂²×C₄	—	68	2	no	—
263	C₂²×C₄	C₂×C₂²⋊D₁₇	1	1	yes	characteristic
264	D₈	—	17	4	no	—
265	D₈	—	34	2	no	—
266	D₈	—	34	4	no	—
267	M₁₆	—	2	2	no	—
268	C₂×C₈	Q₈×D₁₇	1	2	yes	—
269	C₂×C₈	#63256	1	2	yes	—
270	C₂×C₈	C₂×C₂²⋊D₁₇	1	2	yes	—
271	C₂×C₈	C₂×C₂²⋊D₁₇	1	1	yes	characteristic
272	C₂×C₈	C₂³×D₁₇	1	1	yes	characteristic
273	C₄⋊C₄	—	2	1	no	—
274	C₄⋊C₄	—	2	2	no	—
275	C₄⋊C₄	—	2	1	no	—
276	C₄⋊C₄	—	4	1	no	—
277	C₄⋊C₄	—	4	2	no	—
278	C₄⋊C₄	—	4	2	no	—
279	C₄⋊C₄	—	4	2	no	—
280	C₄⋊C₄	—	34	1	no	—
281	C₄⋊C₄	—	34	1	no	—
282	C₄⋊C₄	#63228	1	1	yes	characteristic
283	C₄⋊C₄	#63228	1	1	yes	characteristic
284	C₂²⋊C₄	—	4	1	no	—
285	C₂²⋊C₄	—	68	1	no	—
286	C₂²⋊C₄	—	68	2	no	—
287	C₂²⋊C₄	—	68	2	no	—
288	C₂²⋊C₄	#63228	1	2	yes	—
289	C₄²	—	4	1	no	—
290	C₄²	—	68	1	no	—
291	C₄²	C₂×C₂²⋊D₁₇	1	2	yes	—
292	C₁₆	—	34	1	no	—
293	C₁₆	—	34	1	no	—
294	C₁₆	—	34	2	no	—
295	C₂³	—	272	1	no	—
296	C₂³	—	272	1	no	—
297	C₂³	—	68	4	no	—
298	C₂³	—	68	2	no	—
299	C₂³	C₂×#63228	1	1	yes	characteristic
300	Q₈	—	16	2	no	—
301	D₄	—	34	4	no	—
302	D₄	—	68	4	no	—
303	D₄	—	68	2	no	—
304	C₂×C₄	—	4	1	no	—
305	C₂×C₄	—	4	1	no	—
306	C₂×C₄	—	4	2	no	—
307	C₂×C₄	—	16	1	no	—
308	C₂×C₄	—	272	2	no	—
309	C₂×C₄	—	272	2	no	—
310	C₂×C₄	—	68	1	no	—
311	C₂×C₄	—	68	1	no	—
312	C₂×C₄	C₂×#63228	1	2	yes	—
313	C₂×C₄	C₂×#63228	1	2	yes	—
314	C₂×C₄	C₂×#63228	1	1	yes	characteristic
315	C₂×C₄	#10587628	1	2	yes	—
316	C₂×C₄	#10587632	1	2	yes	—
317	C₂×C₄	C₂²×C₂²⋊D₁₇	1	1	yes	characteristic
318	C₈	—	2	2	no	—
319	C₈	#10587632	1	2	yes	—
320	C₈	C₂×Q₈×D₁₇	1	1	yes	characteristic
321	C₈	C₂×#63256	1	1	yes	characteristic
322	C₂²	—	2	1	no	—
323	C₂²	—	2	1	no	—
324	C₂²	—	272	4	no	—
325	C₂²	—	68	4	no	—
326	C₂²	—	136	2	no	—
327	C₂²	—	136	2	no	—
328	C₂²	#11763086	1	2	yes	—
329	C₂²	C₂²×#63228	1	1	yes	characteristic
330	C₄	—	2	2	no	—
331	C₄	—	2	2	no	—
332	C₄	—	8	2	no	—
333	C₄	—	8	1	no	—
334	C₄	—	8	1	no	—
335	C₄	—	136	1	no	—
336	C₄	—	136	1	no	—
337	C₄	#11763086	1	2	yes	—
338	C₄	C₂×#10587632	1	1	yes	characteristic
339	C₄	C₂×#10587628	1	1	yes	characteristic
340	C₂	—	2	2	no	—
341	C₂	—	136	4	no	—
342	C₂	C₂×#11763086	1	1	yes	characteristic
343	C₂	#14541923	1	1	yes	characteristic
344	C₂	#14541928	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂²×#63228
commutator, cosocle	C₂×C₈×C₁₇	C₂⁴
Frattini	C₂×C₈	C₂³×D₁₇
Fitting	C₁₇×#3164	C₂
socle	C₂²×C₁₇	C₂²×D₈
prime radical	C₂²×D₁₃₆	C₂²
hypercenter	#3164	D₁₇
hypocenter	C₁₇	#33631
LCS	G ▷ C₂×C₈×C₁₇ ▷ C₄×C₁₇ ▷ C₂×C₁₇ ▷ C₁₇
UCS	C₁ ◁ C₂² ◁ #117 ◁ #405 ◁ #3164

Complements

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

N	H ≅ G/N	type
2. #14530938	C₂	semidirect
3. #14530938	C₂	semidirect
4. #14530928	C₂	semidirect
5. #14530918	C₂	semidirect
6. #14530570	C₂	semidirect
7. #14530566	C₂	semidirect
8. #14530543	C₂	semidirect
11. C₁₇×#3164	C₂	semidirect
21. #11762636	C₂²	semidirect
22. #11762636	C₂²	semidirect
23. #11762636	C₂²	semidirect
24. #11762634	C₂²	semidirect
25. #11762634	C₂²	semidirect
26. #11762634	C₂²	semidirect
27. C₄⋊#63214	C₂²	semidirect
28. C₄⋊#63214	C₂²	semidirect
35. C₁₇×#500	C₂²	semidirect
36. C₁₇×#499	C₂²	semidirect
55. C₂×D₁₃₆	Q₈	semidirect
150. #3164	D₁₇	semidirect
185. #500	C₂×D₁₇	semidirect
186. #499	C₂×D₁₇	semidirect
253. C₁₇	#33631	semidirect

Maximal subgroups

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

subgroup	index
#14537637	2
#14530938	2
#14530928	2
#14530918	2
#14530570	2
#14530566	2
#14530543	2
#14530499	2
C₂×#11762609	2
C₁₇×#3164	2
#33631	17