Subgroups — #17189075

Order 4352.

514 conjugacy classes of subgroups.

The subgroup lattice has 336 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#14539315	C₂	1	1	yes	maximal, characteristic, complemented
2	#14537702	C₂	1	1	yes	maximal, characteristic
3	#14530961	C₂	1	4	yes	maximal, complemented
4	#14530956	C₂	1	4	yes	maximal, complemented
5	#14530557	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×#11762632	C₂	1	1	yes	maximal, characteristic, complemented
7	#14530521	C₂	1	1	yes	maximal, characteristic
8	C₂×#11762620	C₂	1	1	yes	maximal, characteristic
9	C₁₇×#2153	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂×#10587590	C₂²	1	1	yes	characteristic
11	#11763575	—	2	1	no	—
12	#11763437	—	2	1	no	—
13	#11763218	—	2	2	no	—
14	#11763211	C₂²	1	2	yes	complemented
15	#11763207	C₂²	1	2	yes	complemented
16	#11762932	—	2	2	no	—
17	#11762927	C₂²	1	4	yes	—
18	C₂×#10587490	C₂²	1	1	yes	characteristic
19	#11762851	—	2	1	no	—
20	#11762847	C₂²	1	1	yes	characteristic
21	C₂×#10587488	C₂²	1	1	yes	characteristic
22	#11762827	—	2	1	no	—
23	#11762632	C₂²	1	2	yes	complemented
24	#11762632	C₂²	1	4	yes	complemented
25	#11762632	C₂²	1	2	yes	complemented
26	#11762620	C₂²	1	2	yes	—
27	#11762620	C₂²	1	4	yes	—
28	#11762620	C₂²	1	2	yes	—
29	C₂×C₁₇×M₃₂	C₂²	1	1	yes	characteristic
30	C₂×C₁₇×M₃₂	C₂²	1	1	yes	characteristic
31	C₁₇×#405	C₂²	1	1	yes	characteristic
32	C₁₇×#364	C₂²	1	4	yes	complemented
33	C₂×#63252	—	2	1	no	—
34	C₂×M₁₆×C₁₇	D₄	1	1	yes	characteristic
35	C₂²×C₈×C₁₇	C₂³	1	1	yes	characteristic
36	C₁₇×#117	D₄	1	1	yes	characteristic
37	#10587612	—	2	1	no	—
38	C₂×C₄⋊Dic₁₇	—	2	1	no	—
39	C₂×#63223	—	2	1	no	—
40	#10587593	—	4	2	no	—
41	M₁₆×D₁₇	—	4	2	no	—
42	C₂×Dic₆₈	Q₈	1	1	yes	characteristic, complemented
43	#10587590	—	2	1	no	—
44	#10587590	—	2	1	no	—
45	#10587590	Q₈	1	2	yes	complemented
46	#10587590	C₂³	1	2	yes	—
47	C₂×D₁₃₆	Q₈	1	1	yes	characteristic, complemented
48	C₂×#63219	—	2	1	no	—
49	#10587587	—	2	4	no	—
50	#10587576	—	4	1	no	—
51	#10587574	—	4	1	no	—
52	#10587564	—	4	2	no	—
53	#10587556	—	2	2	no	—
54	#10587555	—	2	2	no	—
55	C₁₇×M₃₂	—	2	1	no	—
56	C₁₇×M₃₂	—	2	1	no	—
57	C₁₇×M₃₂	C₂³	1	2	yes	—
58	C₁₇×M₃₂	C₂³	1	2	yes	—
59	C₂×C₁₆×C₁₇	—	2	1	no	—
60	C₂×C₁₆×C₁₇	—	2	1	no	—
61	C₁₇×#97	D₄	1	2	yes	—
62	C₄×C₈×C₁₇	C₂³	1	2	yes	—
63	#10587490	—	2	1	no	—
64	#10587490	C₂³	1	2	yes	—
65	#10587489	—	2	2	no	—
66	#10587489	—	2	1	no	—
67	#10587488	—	2	2	no	—
68	#10587488	—	2	1	no	—
69	#10587488	C₂³	1	1	yes	characteristic
70	#10587488	C₂³	1	2	yes	—
71	#10587488	C₂³	1	1	yes	characteristic
72	#10587476	—	2	4	no	—
73	C₂²×C₄×C₁₇	C₂×D₄	1	1	yes	characteristic
74	C₂×C₂²⋊D₁₇	—	4	1	no	—
75	C₂²×Dic₁₇	—	4	1	no	—
76	#63252	—	2	2	no	—
77	#63252	—	2	2	no	—
78	#63252	—	4	1	no	—
79	#63252	—	4	1	no	—
80	C₂×D₆₈	—	2	1	no	—
81	C₂×C₄×D₁₇	—	4	1	no	—
82	C₂×Dic₃₄	—	2	1	no	—
83	C₂×Dic₃₄	—	4	2	no	—
84	M₁₆×C₁₇	SD₁₆	1	2	yes	—
85	M₁₆×C₁₇	SD₁₆	1	2	yes	—
86	C₂×C₈×C₁₇	C₂×D₄	1	1	yes	characteristic
87	C₂×C₈×C₁₇	C₂×D₄	1	2	yes	—
88	C₂×C₈×C₁₇	C₂×Q₈	1	1	yes	characteristic
89	C₂×C₈×C₁₇	C₂×Q₈	1	1	yes	characteristic
90	C₂×C₈×C₁₇	#41	1	1	yes	characteristic
91	C₂×C₈×C₁₇	#41	1	1	yes	characteristic
92	C₂×C₈×C₁₇	C₂⁴	1	1	yes	characteristic
93	C₄⋊C₄×C₁₇	—	2	1	no	—
94	C₂²⋊C₄×C₁₇	—	2	1	no	—
95	C₄²×C₁₇	C₂×D₄	1	2	yes	—
96	C₂²⋊Dic₁₇	—	4	1	no	—
97	C₄⋊Dic₁₇	—	2	1	no	—
98	C₄⋊Dic₁₇	—	2	2	no	—
99	C₄⋊Dic₁₇	—	2	1	no	—
100	C₄⋊Dic₁₇	—	4	2	no	—
101	C₄⋊Dic₁₇	—	4	1	no	—
102	#63225	—	4	2	no	—
103	#63225	—	4	2	no	—
104	C₄×Dic₁₇	—	4	1	no	—
105	#63223	—	2	2	no	—
106	#63223	—	4	1	no	—
107	C₂×#3622	—	4	1	no	—
108	Dic₆₈	—	2	1	no	—
109	Dic₆₈	C₂×Q₈	1	2	yes	—
110	D₁₃₆	—	2	1	no	—
111	D₁₃₆	C₂×Q₈	1	2	yes	—
112	#63219	—	2	2	no	—
113	#63219	—	2	2	no	—
114	#63218	—	4	4	no	—
115	C₈×D₁₇	—	4	4	no	—
116	C₁₆×C₁₇	—	2	2	no	—
117	C₁₆×C₁₇	—	2	2	no	—
118	#33638	—	17	1	no	maximal, Sylow
119	C₂³×C₁₇	AGL(1,ℤ/8)	1	1	yes	characteristic
120	C₂²×D₁₇	—	4	1	no	—
121	C₂×C₄×C₁₇	—	2	2	no	—
122	C₂×C₄×C₁₇	C₂²⋊Q₈	1	1	yes	characteristic
123	C₂×C₄×C₁₇	C₂²⋊Q₈	1	1	yes	characteristic
124	C₂×C₄×C₁₇	C₂×SD₁₆	1	1	yes	characteristic
125	C₂×C₄×C₁₇	C₂×SD₁₆	1	1	yes	characteristic
126	C₂×C₄×C₁₇	AGL(1,ℤ/8)	1	1	yes	characteristic
127	C₂×C₄×C₁₇	C₂²×D₄	1	1	yes	characteristic
128	C₂²⋊D₁₇	—	4	2	no	—
129	C₂²⋊D₁₇	—	4	2	no	—
130	C₂×Dic₁₇	—	4	1	no	—
131	C₂×Dic₁₇	—	4	1	no	—
132	C₂×Dic₁₇	—	4	1	no	—
133	C₂×Dic₁₇	—	4	2	no	—
134	C₂×Dic₁₇	—	16	1	no	—
135	D₆₈	—	2	2	no	—
136	D₆₈	—	4	1	no	—
137	C₄×D₁₇	—	4	2	no	—
138	C₄×D₁₇	—	4	2	no	—
139	Dic₃₄	—	2	2	no	—
140	Dic₃₄	—	4	1	no	—
141	Dic₃₄	—	16	2	no	—
142	C₈×C₁₇	C₂²⋊Q₈	1	1	yes	characteristic
143	C₈×C₁₇	C₂²⋊Q₈	1	1	yes	characteristic
144	C₈×C₁₇	C₂×SD₁₆	1	4	yes	—
145	C₈×C₁₇	C₂²×Q₈	1	1	yes	characteristic
146	C₈×C₁₇	C₂×#41	1	1	yes	characteristic
147	#3622	—	4	2	no	—
148	#3163	—	17	1	no	—
149	#2957	—	17	1	no	—
150	#2231	—	17	4	no	—
151	#2230	—	17	4	no	—
152	#2155	—	17	1	no	—
153	C₂×#365	—	17	1	no	—
154	#2153	D₁₇	1	1	yes	characteristic, complemented
155	#2145	—	17	1	no	—
156	C₂×#360	—	17	1	no	—
157	C₂²×C₁₇	—	4	1	no	—
158	C₂²×C₁₇	#476	1	1	yes	characteristic
159	C₂²×C₁₇	#476	1	1	yes	characteristic
160	C₂²×C₁₇	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
161	C₂×D₁₇	—	4	2	no	—
162	C₂×D₁₇	—	16	1	no	—
163	C₄×C₁₇	—	4	2	no	—
164	C₄×C₁₇	#476	1	1	yes	characteristic
165	C₄×C₁₇	#476	1	1	yes	characteristic
166	C₄×C₁₇	C₂×C₂²⋊Q₈	1	1	yes	characteristic
167	C₄×C₁₇	C₂²×SD₁₆	1	1	yes	characteristic
168	Dic₁₇	—	4	2	no	—
169	Dic₁₇	—	8	1	no	—
170	Dic₁₇	—	8	2	no	—
171	Dic₁₇	—	8	1	no	—
172	C₂×#135	—	17	1	no	—
173	#568	—	34	1	no	—
174	#544	—	34	1	no	—
175	C₂×M₃₂	C₂×D₁₇	1	1	yes	characteristic
176	C₂×M₃₂	C₂×D₁₇	1	1	yes	characteristic
177	#501	—	34	2	no	—
178	#499	—	17	2	no	—
179	#498	—	17	2	no	—
180	#444	—	34	2	no	—
181	#443	—	17	4	no	—
182	C₂×#108	—	17	1	no	—
183	#428	—	34	1	no	—
184	#427	—	17	1	no	—
185	C₂×#106	—	17	1	no	—
186	#421	—	34	1	no	—
187	#405	C₂×D₁₇	1	1	yes	characteristic
188	#365	—	17	4	no	—
189	#365	—	17	2	no	—
190	#365	—	17	2	no	—
191	#364	C₂×D₁₇	1	4	yes	complemented
192	#360	—	17	4	no	—
193	#360	—	17	2	no	—
194	#360	—	17	2	no	—
195	C₂×C₁₇	—	2	1	no	—
196	C₂×C₁₇	—	2	1	no	—
197	C₂×C₁₇	—	2	1	no	—
198	C₂×C₁₇	C₂×#476	1	1	yes	characteristic
199	D₁₇	—	8	2	no	—
200	C₂×#41	—	34	1	no	—
201	#135	—	17	2	no	—
202	#135	—	17	2	no	—
203	#135	—	34	1	no	—
204	#135	—	34	1	no	—
205	C₂×Q₁₆	—	17	1	no	—
206	C₂×SD₁₆	—	34	1	no	—
207	C₂×D₈	—	17	1	no	—
208	#131	—	34	4	no	—
209	#131	—	68	2	no	—
210	C₂×M₁₆	—	34	1	no	—
211	C₂×M₁₆	—	68	2	no	—
212	C₂×M₁₆	D₆₈	1	1	yes	characteristic
213	C₂²×C₈	C₂²×D₁₇	1	1	yes	characteristic
214	C₄⋊Q₈	—	34	2	no	—
215	C₄⋊Q₈	—	68	1	no	—
216	#125	—	34	2	no	—
217	#125	—	68	1	no	—
218	C₂²⋊Q₈	—	68	2	no	—
219	#117	—	34	1	no	—
220	#117	D₆₈	1	1	yes	characteristic
221	C₂×C₄⋊C₄	—	34	1	no	—
222	M₃₂	—	2	1	no	—
223	M₃₂	—	2	1	no	—
224	M₃₂	C₂²×D₁₇	1	2	yes	—
225	M₃₂	C₂²×D₁₇	1	2	yes	—
226	C₂×C₁₆	—	2	1	no	—
227	C₂×C₁₆	—	2	1	no	—
228	#108	—	17	2	no	—
229	#108	—	34	1	no	—
230	#107	—	34	1	no	—
231	#107	—	34	2	no	—
232	#106	—	17	2	no	—
233	#106	—	17	1	no	—
234	#106	—	17	1	no	—
235	#106	—	34	1	no	—
236	#106	—	34	2	no	—
237	#104	—	34	4	no	—
238	#97	D₆₈	1	2	yes	—
239	C₄×C₈	C₂²×D₁₇	1	2	yes	—
240	C₁₇	#33638	1	1	yes	characteristic, complemented, Sylow
241	#41	—	34	2	no	—
242	#41	—	34	2	no	—
243	#41	—	68	1	no	—
244	#41	—	68	1	no	—
245	C₂×Q₈	—	34	1	no	—
246	C₂×Q₈	—	68	2	no	—
247	C₂×D₄	—	34	1	no	—
248	C₂×D₄	—	68	1	no	—
249	C₂²×C₄	—	68	1	no	—
250	C₂²×C₄	—	68	1	no	—
251	C₂²×C₄	C₂×D₆₈	1	1	yes	characteristic
252	Q₁₆	—	17	2	no	—
253	Q₁₆	—	34	1	no	—
254	SD₁₆	—	34	2	no	—
255	SD₁₆	—	34	2	no	—
256	D₈	—	17	2	no	—
257	D₈	—	34	1	no	—
258	M₁₆	—	34	2	no	—
259	M₁₆	—	68	1	no	—
260	M₁₆	—	68	4	no	—
261	M₁₆	#63219	1	2	yes	—
262	M₁₆	#63219	1	2	yes	—
263	C₂×C₈	—	68	1	no	—
264	C₂×C₈	—	68	4	no	—
265	C₂×C₈	C₂×D₆₈	1	1	yes	characteristic
266	C₂×C₈	C₂×D₆₈	1	2	yes	—
267	C₂×C₈	#63254	1	1	yes	characteristic
268	C₂×C₈	#63254	1	1	yes	characteristic
269	C₂×C₈	Q₈×D₁₇	1	1	yes	characteristic
270	C₂×C₈	Q₈×D₁₇	1	1	yes	characteristic
271	C₂×C₈	C₂³×D₁₇	1	1	yes	characteristic
272	C₄⋊C₄	—	2	1	no	—
273	C₄⋊C₄	—	34	2	no	—
274	C₄⋊C₄	—	34	1	no	—
275	C₄⋊C₄	—	34	1	no	—
276	C₄⋊C₄	—	68	2	no	—
277	C₄⋊C₄	—	68	2	no	—
278	C₄⋊C₄	—	68	2	no	—
279	C₄⋊C₄	—	68	1	no	—
280	C₂²⋊C₄	—	2	1	no	—
281	C₂²⋊C₄	—	68	1	no	—
282	C₄²	—	68	1	no	—
283	C₄²	C₂×D₆₈	1	2	yes	—
284	C₁₆	—	2	2	no	—
285	C₁₆	—	2	2	no	—
286	C₂³	—	68	1	no	—
287	C₂³	#10587594	1	1	yes	characteristic
288	Q₈	—	272	2	no	—
289	Q₈	—	34	2	no	—
290	Q₈	—	68	1	no	—
291	D₄	—	34	2	no	—
292	D₄	—	68	2	no	—
293	D₄	—	68	2	no	—
294	D₄	—	68	1	no	—
295	C₂×C₄	—	2	2	no	—
296	C₂×C₄	—	272	1	no	—
297	C₂×C₄	—	68	1	no	—
298	C₂×C₄	—	68	2	no	—
299	C₂×C₄	—	68	2	no	—
300	C₂×C₄	—	68	1	no	—
301	C₂×C₄	—	68	1	no	—
302	C₂×C₄	—	68	2	no	—
303	C₂×C₄	#10587583	1	1	yes	characteristic
304	C₂×C₄	#10587583	1	1	yes	characteristic
305	C₂×C₄	C₂×#63219	1	1	yes	characteristic
306	C₂×C₄	C₂×#63219	1	1	yes	characteristic
307	C₂×C₄	#10587594	1	1	yes	characteristic
308	C₂×C₄	C₂²×D₆₈	1	1	yes	characteristic
309	C₈	—	68	2	no	—
310	C₈	#10587583	1	1	yes	characteristic
311	C₈	#10587583	1	1	yes	characteristic
312	C₈	C₂×#63219	1	4	yes	—
313	C₈	C₂×#63254	1	1	yes	characteristic
314	C₈	C₂×Q₈×D₁₇	1	1	yes	characteristic
315	C₂²	—	4	1	no	—
316	C₂²	—	272	1	no	—
317	C₂²	—	68	2	no	—
318	C₂²	#11763100	1	1	yes	characteristic
319	C₂²	#11763100	1	1	yes	characteristic
320	C₂²	C₂×#10587594	1	1	yes	characteristic
321	C₄	—	4	2	no	—
322	C₄	—	68	2	no	—
323	C₄	—	136	2	no	—
324	C₄	—	136	1	no	—
325	C₄	—	136	1	no	—
326	C₄	#11763100	1	1	yes	characteristic
327	C₄	#11763100	1	1	yes	characteristic
328	C₄	C₂×#10587583	1	1	yes	characteristic
329	C₄	C₂²×#63219	1	1	yes	characteristic
330	C₂	—	2	1	no	—
331	C₂	—	2	1	no	—
332	C₂	—	2	1	no	—
333	C₂	—	136	2	no	—
334	C₂	C₂×#11763100	1	1	yes	characteristic

	subgroup	G/H
center	C₂	C₂×#11763100
commutator, cosocle	C₂×C₈×C₁₇	C₂⁴
Frattini	C₂×C₈	C₂³×D₁₇
Fitting	C₁₇×#2153	C₂
socle	C₂×C₁₇	C₂×#476
prime radical	C₂×#10587590	C₂²
hypercenter	#2153	D₁₇
hypocenter	C₁₇	#33638
LCS	G ▷ C₂×C₈×C₁₇ ▷ C₄×C₁₇ ▷ C₂×C₁₇ ▷ C₁₇
UCS	C₁ ◁ C₂ ◁ C₂²×C₄ ◁ #405 ◁ #2153

Complements

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

N	H ≅ G/N	type
1. #14539315	C₂	semidirect
3. #14530961	C₂	semidirect
4. #14530956	C₂	semidirect
5. #14530557	C₂	semidirect
6. C₂×#11762632	C₂	semidirect
9. C₁₇×#2153	C₂	semidirect
14. #11763211	C₂²	semidirect
15. #11763207	C₂²	semidirect
23. #11762632	C₂²	semidirect
24. #11762632	C₂²	semidirect
25. #11762632	C₂²	semidirect
32. C₁₇×#364	C₂²	semidirect
42. C₂×Dic₆₈	Q₈	semidirect
45. #10587590	Q₈	semidirect
47. C₂×D₁₃₆	Q₈	semidirect
154. #2153	D₁₇	semidirect
191. #364	C₂×D₁₇	semidirect
240. C₁₇	#33638	semidirect

Maximal subgroups

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

subgroup	index
#14539315	2
#14537702	2
#14530961	2
#14530956	2
#14530557	2
C₂×#11762632	2
#14530521	2
C₂×#11762620	2
C₁₇×#2153	2
#33638	17