Subgroups — #17189067

Order 4352.

524 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	#14539298	C₂	1	1	yes	maximal, characteristic, complemented
2	#14530960	C₂	1	4	yes	maximal, complemented
3	#14530955	C₂	1	4	yes	maximal, complemented
4	#14530558	C₂	1	2	yes	maximal, complemented
5	C₂×#11762630	C₂	1	1	yes	maximal, characteristic, complemented
6	#14530520	C₂	1	2	yes	maximal
7	C₂×C₁₇×#443	C₂	1	1	yes	maximal, characteristic
8	#11763366	—	2	1	no	—
9	C₂×#10587483	C₂²	1	2	yes	—
10	#11763214	—	2	1	no	—
11	#11763211	—	2	1	no	—
12	#11763211	C₂²	1	2	yes	complemented
13	#11763207	C₂²	1	2	yes	complemented
14	#11762847	—	2	1	no	—
15	#11762847	C₂²	1	2	yes	—
16	#11762633	C₂²	1	8	yes	complemented
17	#11762630	C₂²	1	2	yes	complemented
18	#11762630	C₂²	1	2	yes	complemented
19	#11762619	C₂²	1	8	yes	—
20	C₂×C₁₇×#135	C₂²	1	1	yes	characteristic
21	C₂×C₁₇×#131	—	2	1	no	—
22	C₁₇×#443	—	2	1	no	—
23	C₁₇×#443	—	2	1	no	—
24	C₁₇×#443	C₂²	1	2	yes	—
25	C₁₇×#443	C₂²	1	2	yes	—
26	C₂×C₁₇×#108	C₂²	1	1	yes	characteristic
27	C₂×C₁₇×#104	—	2	1	no	—
28	C₂×C₄×C₈×C₁₇	C₂²	1	1	yes	characteristic
29	C₂×#41×C₁₇	—	2	1	no	—
30	C₁₇×#135	—	2	1	no	—
31	C₁₇×#135	—	2	1	no	—
32	C₁₇×#135	Q₈	1	2	yes	complemented
33	C₁₇×#135	C₂³	1	2	yes	—
34	C₂×Q₁₆×C₁₇	Q₈	1	1	yes	characteristic, complemented
35	C₂×SD₁₆×C₁₇	—	2	1	no	—
36	C₂×D₈×C₁₇	Q₈	1	1	yes	characteristic, complemented
37	C₁₇×#131	—	2	2	no	—
38	C₁₇×#131	—	2	2	no	—
39	C₁₇×#131	—	4	1	no	—
40	C₁₇×#131	—	4	1	no	—
41	C₂×M₁₆×C₁₇	—	2	1	no	—
42	C₂×M₁₆×C₁₇	—	4	1	no	—
43	C₂²×C₈×C₁₇	—	4	1	no	—
44	C₂²×C₈×C₁₇	D₄	1	1	yes	characteristic
45	C₂²×C₈×C₁₇	C₂³	1	1	yes	characteristic
46	C₂×C₄²×C₁₇	D₄	1	1	yes	characteristic
47	#10587612	—	2	2	no	—
48	#10587610	—	4	2	no	—
49	#10587556	—	2	2	no	—
50	#10587555	—	2	2	no	—
51	C₄×Dic₃₄	—	4	2	no	—
52	C₁₇×#108	—	2	1	no	—
53	C₁₇×#108	C₂³	1	2	yes	—
54	C₁₇×#104	—	2	2	no	—
55	C₁₇×#104	—	2	2	no	—
56	C₄×C₈×C₁₇	D₄	1	1	yes	characteristic
57	C₄×C₈×C₁₇	D₄	1	1	yes	characteristic
58	C₄×C₈×C₁₇	C₂³	1	1	yes	characteristic
59	C₄×C₈×C₁₇	C₂³	1	1	yes	characteristic
60	#10587489	—	2	2	no	—
61	#10587489	—	2	2	no	—
62	#10587488	—	2	2	no	—
63	#10587488	C₂³	1	4	yes	—
64	#10587483	—	2	2	no	—
65	#10587483	C₂³	1	4	yes	—
66	C₂×#63214	—	2	2	no	—
67	#41×C₁₇	—	2	2	no	—
68	#41×C₁₇	—	2	2	no	—
69	#41×C₁₇	—	4	1	no	—
70	#41×C₁₇	—	4	1	no	—
71	C₂×Q₈×C₁₇	—	2	1	no	—
72	C₂×D₄×C₁₇	—	2	1	no	—
73	C₂×D₄×C₁₇	—	4	1	no	—
74	C₂²×C₄×C₁₇	—	2	1	no	—
75	C₂²×C₄×C₁₇	—	4	1	no	—
76	C₂²×C₄×C₁₇	C₂×D₄	1	1	yes	characteristic
77	C₂×Dic₃₄	—	4	2	no	—
78	Q₁₆×C₁₇	—	2	1	no	—
79	Q₁₆×C₁₇	C₂×Q₈	1	2	yes	—
80	SD₁₆×C₁₇	—	2	2	no	—
81	SD₁₆×C₁₇	—	2	2	no	—
82	D₈×C₁₇	—	2	1	no	—
83	D₈×C₁₇	C₂×Q₈	1	2	yes	—
84	M₁₆×C₁₇	—	2	2	no	—
85	M₁₆×C₁₇	—	4	1	no	—
86	M₁₆×C₁₇	—	4	2	no	—
87	M₁₆×C₁₇	—	4	2	no	—
88	C₂×C₈×C₁₇	—	4	1	no	—
89	C₂×C₈×C₁₇	—	4	2	no	—
90	C₂×C₈×C₁₇	—	4	2	no	—
91	C₂×C₈×C₁₇	SD₁₆	1	1	yes	characteristic
92	C₂×C₈×C₁₇	SD₁₆	1	1	yes	characteristic
93	C₂×C₈×C₁₇	SD₁₆	1	1	yes	characteristic
94	C₂×C₈×C₁₇	SD₁₆	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	C₂×D₄	1	1	yes	characteristic
98	C₂×C₈×C₁₇	C₂×Q₈	1	1	yes	characteristic
99	C₂×C₈×C₁₇	C₂×Q₈	1	1	yes	characteristic
100	C₂×C₈×C₁₇	#41	1	1	yes	characteristic
101	C₂×C₈×C₁₇	#41	1	1	yes	characteristic
102	C₂×C₈×C₁₇	C₂⁴	1	1	yes	characteristic
103	C₄²×C₁₇	—	2	1	no	—
104	C₄²×C₁₇	C₂×D₄	1	1	yes	characteristic
105	C₄²×C₁₇	C₂×D₄	1	1	yes	characteristic
106	C₂²⋊Dic₁₇	—	4	2	no	—
107	C₄⋊Dic₁₇	—	2	4	no	—
108	C₄⋊Dic₁₇	—	4	2	no	—
109	#63225	—	4	4	no	—
110	C₄×Dic₁₇	—	4	2	no	—
111	#63214	—	2	4	no	—
112	#33637	—	17	1	no	maximal, Sylow
113	C₂³×C₁₇	—	4	1	no	—
114	C₂³×C₁₇	AGL(1,ℤ/8)	1	1	yes	characteristic
115	Q₈×C₁₇	—	2	2	no	—
116	Q₈×C₁₇	—	4	1	no	—
117	D₄×C₁₇	—	2	2	no	—
118	D₄×C₁₇	—	4	2	no	—
119	D₄×C₁₇	—	4	2	no	—
120	D₄×C₁₇	—	4	1	no	—
121	C₂×C₄×C₁₇	—	2	1	no	—
122	C₂×C₄×C₁₇	—	2	1	no	—
123	C₂×C₄×C₁₇	—	4	2	no	—
124	C₂×C₄×C₁₇	—	4	1	no	—
125	C₂×C₄×C₁₇	—	4	2	no	—
126	C₂×C₄×C₁₇	—	4	2	no	—
127	C₂×C₄×C₁₇	C₂²⋊Q₈	1	1	yes	characteristic
128	C₂×C₄×C₁₇	C₂²⋊Q₈	1	1	yes	characteristic
129	C₂×C₄×C₁₇	C₂×SD₁₆	1	1	yes	characteristic
130	C₂×C₄×C₁₇	C₂×SD₁₆	1	1	yes	characteristic
131	C₂×C₄×C₁₇	AGL(1,ℤ/8)	1	1	yes	characteristic
132	C₂×C₄×C₁₇	C₂²×D₄	1	1	yes	characteristic
133	C₂×Dic₁₇	—	4	4	no	—
134	Dic₃₄	—	8	4	no	—
135	C₈×C₁₇	—	4	2	no	—
136	C₈×C₁₇	C₂²⋊Q₈	1	1	yes	characteristic
137	C₈×C₁₇	C₂²⋊Q₈	1	1	yes	characteristic
138	C₈×C₁₇	C₂×SD₁₆	1	1	yes	characteristic
139	C₈×C₁₇	C₂×SD₁₆	1	1	yes	characteristic
140	C₈×C₁₇	C₂×SD₁₆	1	1	yes	characteristic
141	C₈×C₁₇	C₂×SD₁₆	1	1	yes	characteristic
142	C₈×C₁₇	C₂²×Q₈	1	1	yes	characteristic
143	C₈×C₁₇	C₂×#41	1	1	yes	characteristic
144	#3160	—	17	1	no	—
145	C₂×#443	D₁₇	1	1	yes	characteristic
146	#2231	—	17	4	no	—
147	#2230	—	17	4	no	—
148	#2155	—	17	2	no	—
149	C₂×#364	—	17	1	no	—
150	#2145	—	17	2	no	—
151	C₂²×C₁₇	—	4	1	no	—
152	C₂²×C₁₇	—	4	2	no	—
153	C₂²×C₁₇	—	8	2	no	—
154	C₂²×C₁₇	#476	1	1	yes	characteristic
155	C₂²×C₁₇	#476	1	1	yes	characteristic
156	C₂²×C₁₇	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
157	C₄×C₁₇	—	2	2	no	—
158	C₄×C₁₇	—	4	1	no	—
159	C₄×C₁₇	—	4	2	no	—
160	C₄×C₁₇	#476	1	1	yes	characteristic
161	C₄×C₁₇	#476	1	1	yes	characteristic
162	C₄×C₁₇	C₂×C₂²⋊Q₈	1	1	yes	characteristic
163	C₄×C₁₇	C₂²×SD₁₆	1	1	yes	characteristic
164	Dic₁₇	—	8	4	no	—
165	C₂×#135	C₂×D₁₇	1	1	yes	characteristic
166	C₂×#131	—	2	1	no	—
167	#533	—	34	1	no	—
168	C₂×M₃₂	—	17	2	no	—
169	#500	—	34	1	no	—
170	#499	—	17	2	no	—
171	#499	—	34	1	no	—
172	#498	—	17	2	no	—
173	#443	—	2	1	no	—
174	#443	—	2	1	no	—
175	#443	C₂×D₁₇	1	2	yes	—
176	#443	C₂×D₁₇	1	2	yes	—
177	C₂×#108	C₂×D₁₇	1	1	yes	characteristic
178	#427	—	17	2	no	—
179	#427	—	34	1	no	—
180	C₂×#104	—	2	1	no	—
181	C₂×C₄×C₈	C₂×D₁₇	1	1	yes	characteristic
182	#365	—	17	8	no	—
183	#364	—	17	2	no	—
184	#364	—	17	2	no	—
185	#360	—	17	8	no	—
186	C₂×C₁₇	—	2	1	no	—
187	C₂×C₁₇	—	2	1	no	—
188	C₂×C₁₇	—	2	1	no	—
189	C₂×C₁₇	—	8	2	no	—
190	C₂×C₁₇	C₂×#476	1	1	yes	characteristic
191	C₂×#41	—	2	1	no	—
192	#135	—	2	1	no	—
193	#135	—	2	1	no	—
194	#135	Dic₃₄	1	2	yes	complemented
195	#135	C₂²×D₁₇	1	2	yes	—
196	C₂×Q₁₆	Dic₃₄	1	1	yes	characteristic, complemented
197	C₂×SD₁₆	—	2	1	no	—
198	C₂×D₈	Dic₃₄	1	1	yes	characteristic, complemented
199	#131	—	2	2	no	—
200	#131	—	2	2	no	—
201	#131	—	4	1	no	—
202	#131	—	4	1	no	—
203	C₂×M₁₆	—	2	1	no	—
204	C₂×M₁₆	—	4	1	no	—
205	C₂²×C₈	—	4	1	no	—
206	C₂²×C₈	C₂²⋊D₁₇	1	1	yes	characteristic
207	C₂²×C₈	C₂²×D₁₇	1	1	yes	characteristic
208	C₄⋊Q₈	—	34	2	no	—
209	#125	—	34	2	no	—
210	C₂²⋊Q₈	—	68	2	no	—
211	C₄×Q₈	—	68	2	no	—
212	#117	—	34	2	no	—
213	C₂×C₄²	C₂²⋊D₁₇	1	1	yes	characteristic
214	M₃₂	—	17	4	no	—
215	M₃₂	—	34	2	no	—
216	C₂×C₁₆	—	34	2	no	—
217	#108	—	2	1	no	—
218	#108	C₂²×D₁₇	1	2	yes	—
219	#107	—	34	2	no	—
220	#107	—	34	2	no	—
221	#106	—	17	4	no	—
222	#106	—	34	2	no	—
223	#104	—	2	2	no	—
224	#104	—	2	2	no	—
225	C₄×C₈	C₂²⋊D₁₇	1	1	yes	characteristic
226	C₄×C₈	C₂²⋊D₁₇	1	1	yes	characteristic
227	C₄×C₈	C₂²×D₁₇	1	1	yes	characteristic
228	C₄×C₈	C₂²×D₁₇	1	1	yes	characteristic
229	C₁₇	#33637	1	1	yes	characteristic, complemented, Sylow
230	#41	—	2	2	no	—
231	#41	—	2	2	no	—
232	#41	—	4	1	no	—
233	#41	—	4	1	no	—
234	C₂×Q₈	—	2	1	no	—
235	C₂×Q₈	—	68	2	no	—
236	C₂×D₄	—	2	1	no	—
237	C₂×D₄	—	4	1	no	—
238	C₂²×C₄	—	2	1	no	—
239	C₂²×C₄	—	4	1	no	—
240	C₂²×C₄	C₂×C₂²⋊D₁₇	1	1	yes	characteristic
241	Q₁₆	—	2	1	no	—
242	Q₁₆	C₂×Dic₃₄	1	2	yes	—
243	SD₁₆	—	2	2	no	—
244	SD₁₆	—	2	2	no	—
245	D₈	—	2	1	no	—
246	D₈	C₂×Dic₃₄	1	2	yes	—
247	M₁₆	—	2	2	no	—
248	M₁₆	—	4	1	no	—
249	M₁₆	—	4	2	no	—
250	M₁₆	—	4	2	no	—
251	C₂×C₈	—	4	1	no	—
252	C₂×C₈	—	4	2	no	—
253	C₂×C₈	—	4	2	no	—
254	C₂×C₈	#63229	1	1	yes	characteristic
255	C₂×C₈	#63229	1	1	yes	characteristic
256	C₂×C₈	#63229	1	1	yes	characteristic
257	C₂×C₈	#63229	1	1	yes	characteristic
258	C₂×C₈	C₂×Dic₃₄	1	1	yes	characteristic
259	C₂×C₈	C₂×Dic₃₄	1	1	yes	characteristic
260	C₂×C₈	#63252	1	1	yes	characteristic
261	C₂×C₈	#63252	1	1	yes	characteristic
262	C₂×C₈	C₂×C₂²⋊D₁₇	1	1	yes	characteristic
263	C₂×C₈	C₂×C₂²⋊D₁₇	1	1	yes	characteristic
264	C₂×C₈	C₂×C₂²⋊D₁₇	1	1	yes	characteristic
265	C₂×C₈	C₂³×D₁₇	1	1	yes	characteristic
266	C₄⋊C₄	—	34	4	no	—
267	C₄⋊C₄	—	68	4	no	—
268	C₄⋊C₄	—	68	2	no	—
269	C₂²⋊C₄	—	68	2	no	—
270	C₄²	—	2	1	no	—
271	C₄²	—	68	2	no	—
272	C₄²	C₂×C₂²⋊D₁₇	1	1	yes	characteristic
273	C₄²	C₂×C₂²⋊D₁₇	1	1	yes	characteristic
274	C₁₆	—	34	4	no	—
275	C₂³	—	4	1	no	—
276	C₂³	#10587635	1	1	yes	characteristic
277	Q₈	—	2	2	no	—
278	Q₈	—	4	1	no	—
279	Q₈	—	136	4	no	—
280	D₄	—	2	2	no	—
281	D₄	—	4	2	no	—
282	D₄	—	4	2	no	—
283	D₄	—	4	1	no	—
284	C₂×C₄	—	2	1	no	—
285	C₂×C₄	—	2	1	no	—
286	C₂×C₄	—	4	2	no	—
287	C₂×C₄	—	4	1	no	—
288	C₂×C₄	—	4	2	no	—
289	C₂×C₄	—	4	2	no	—
290	C₂×C₄	—	68	4	no	—
291	C₂×C₄	#10587610	1	1	yes	characteristic
292	C₂×C₄	#10587610	1	1	yes	characteristic
293	C₂×C₄	C₂×#63229	1	1	yes	characteristic
294	C₂×C₄	C₂×#63229	1	1	yes	characteristic
295	C₂×C₄	#10587635	1	1	yes	characteristic
296	C₂×C₄	C₂²×C₂²⋊D₁₇	1	1	yes	characteristic
297	C₈	—	4	2	no	—
298	C₈	#10587610	1	1	yes	characteristic
299	C₈	#10587610	1	1	yes	characteristic
300	C₈	C₂×#63229	1	1	yes	characteristic
301	C₈	C₂×#63229	1	1	yes	characteristic
302	C₈	C₂×#63229	1	1	yes	characteristic
303	C₈	C₂×#63229	1	1	yes	characteristic
304	C₈	C₂²×Dic₃₄	1	1	yes	characteristic
305	C₈	C₂×#63252	1	1	yes	characteristic
306	C₂²	—	4	1	no	—
307	C₂²	—	4	2	no	—
308	C₂²	—	8	2	no	—
309	C₂²	#11763099	1	1	yes	characteristic
310	C₂²	#11763099	1	1	yes	characteristic
311	C₂²	C₂×#10587635	1	1	yes	characteristic
312	C₄	—	2	2	no	—
313	C₄	—	4	1	no	—
314	C₄	—	4	2	no	—
315	C₄	—	136	4	no	—
316	C₄	#11763099	1	1	yes	characteristic
317	C₄	#11763099	1	1	yes	characteristic
318	C₄	C₂×#10587610	1	1	yes	characteristic
319	C₄	C₂²×#63229	1	1	yes	characteristic
320	C₂	—	2	1	no	—
321	C₂	—	2	1	no	—
322	C₂	—	2	1	no	—
323	C₂	—	8	2	no	—
324	C₂	C₂×#11763099	1	1	yes	characteristic

	subgroup	G/H
center	C₄	#11763099
commutator, cosocle	C₂×C₈×C₁₇	C₂⁴
Frattini	C₂×C₈	C₂³×D₁₇
Fitting	C₂×C₁₇×#443	C₂
socle	C₂×C₁₇	C₂×#476
prime radical	C₂×C₁₇×#135	C₂²
hypercenter	C₂×#443	D₁₇
hypocenter	C₁₇	#33637
LCS	G ▷ C₂×C₈×C₁₇ ▷ C₄×C₁₇ ▷ C₂×C₁₇ ▷ C₁₇
UCS	C₁ ◁ C₄ ◁ C₂²×C₄ ◁ C₂×C₄×C₈ ◁ C₂×#443

Complements

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

N	H ≅ G/N	type
1. #14539298	C₂	semidirect
2. #14530960	C₂	semidirect
3. #14530955	C₂	semidirect
4. #14530558	C₂	semidirect
5. C₂×#11762630	C₂	semidirect
12. #11763211	C₂²	semidirect
13. #11763207	C₂²	semidirect
16. #11762633	C₂²	semidirect
17. #11762630	C₂²	semidirect
18. #11762630	C₂²	semidirect
32. C₁₇×#135	Q₈	semidirect
34. C₂×Q₁₆×C₁₇	Q₈	semidirect
36. C₂×D₈×C₁₇	Q₈	semidirect
194. #135	Dic₃₄	semidirect
196. C₂×Q₁₆	Dic₃₄	semidirect
198. C₂×D₈	Dic₃₄	semidirect
229. C₁₇	#33637	semidirect

Maximal subgroups

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

subgroup	index
#14539298	2
#14530960	2
#14530955	2
#14530558	2
C₂×#11762630	2
#14530520	2
C₂×C₁₇×#443	2
#33637	17