Subgroups — #17189232

Order 4352.

534 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	#14538435	C₂	1	1	yes	maximal, characteristic
2	#14537392	C₂	1	1	yes	maximal, characteristic
3	#14531002	C₂	1	8	yes	maximal, complemented
4	#14530558	C₂	1	1	yes	maximal, characteristic, complemented
5	#14530557	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×#11762625	C₂	1	1	yes	maximal, characteristic, complemented
7	#14530404	C₂	1	1	yes	maximal, characteristic
8	C₂×C₁₇×#362	C₂	1	1	yes	maximal, characteristic, complemented
9	#11763575	—	2	1	no	—
10	C₂×#10587587	C₂²	1	1	yes	characteristic
11	#11763383	—	2	1	no	—
12	C₂×#10587483	C₂²	1	1	yes	characteristic
13	#11763064	C₂²	1	4	yes	—
14	#11763047	—	2	2	no	—
15	#11762857	—	2	1	no	—
16	#11762856	C₂²	1	4	yes	—
17	C₂×#10587490	C₂²	1	1	yes	characteristic
18	#108×D₁₇	—	2	2	no	—
19	#11762847	C₂²	1	1	yes	characteristic
20	#11762818	—	2	1	no	—
21	#11762633	C₂²	1	4	yes	—
22	#11762632	C₂²	1	4	yes	—
23	#11762625	C₂²	1	4	yes	—
24	#11762590	C₂²	1	4	yes	—
25	C₂²×Q₁₆×C₁₇	C₂²	1	1	yes	characteristic
26	C₂×C₁₇×M₃₂	C₂²	1	1	yes	characteristic
27	C₂×C₁₇×#108	C₂²	1	1	yes	characteristic
28	C₁₇×#362	C₂²	1	4	yes	—
29	C₂²×Q₈×C₁₇	—	2	1	no	—
30	C₂×#63252	D₄	1	1	yes	characteristic
31	C₂×Q₁₆×C₁₇	—	2	1	no	—
32	C₂×Q₁₆×C₁₇	—	2	1	no	—
33	C₂×Q₁₆×C₁₇	—	2	1	no	—
34	C₂×Q₁₆×C₁₇	—	2	1	no	—
35	C₂×Q₁₆×C₁₇	—	2	1	no	—
36	C₂×Q₁₆×C₁₇	C₂³	1	2	yes	—
37	C₂×M₁₆×C₁₇	—	2	1	no	—
38	C₂²×C₈×C₁₇	C₂³	1	1	yes	characteristic
39	#10587633	—	4	1	no	—
40	#10587632	—	2	4	no	—
41	#10587630	—	4	1	no	—
42	#10587615	—	4	2	no	—
43	#10587612	—	2	1	no	—
44	C₂×#63223	—	2	1	no	—
45	C₂×#63223	D₄	1	1	yes	characteristic
46	#10587593	—	4	2	no	—
47	M₁₆×D₁₇	—	2	4	no	—
48	#10587587	—	2	1	no	—
49	#10587587	—	2	1	no	—
50	#10587587	—	2	1	no	—
51	#10587587	—	2	1	no	—
52	#10587587	—	4	2	no	—
53	C₂×#63218	C₂³	1	2	yes	—
54	C₂×C₈×D₁₇	D₄	1	2	yes	—
55	C₁₇×M₃₂	—	2	1	no	—
56	C₁₇×M₃₂	C₂³	1	2	yes	—
57	C₂×C₁₆×C₁₇	—	2	1	no	—
58	C₁₇×#108	—	2	1	no	—
59	C₁₇×#108	C₂³	1	2	yes	—
60	#10587506	—	2	4	no	—
61	#10587493	—	2	4	no	—
62	#10587490	—	2	1	no	—
63	#10587490	C₂³	1	2	yes	—
64	#10587489	—	2	1	no	—
65	#10587488	C₂³	1	2	yes	—
66	#10587483	—	2	1	no	—
67	#10587483	C₂³	1	2	yes	—
68	C₂×#63214	—	2	1	no	—
69	C₂×Q₈×C₁₇	—	2	2	no	—
70	C₂×Q₈×C₁₇	—	4	1	no	—
71	C₂×Q₈×C₁₇	—	4	1	no	—
72	C₂×Q₈×C₁₇	—	4	1	no	—
73	C₂×Q₈×C₁₇	—	4	1	no	—
74	C₂×Q₈×C₁₇	—	4	1	no	—
75	C₂²×C₄×C₁₇	—	4	1	no	—
76	C₂²×C₄×C₁₇	C₂×D₄	1	1	yes	characteristic
77	C₂×C₂²⋊D₁₇	SD₁₆	1	2	yes	—
78	#63252	—	2	1	no	—
79	#63252	—	2	1	no	—
80	#63252	—	2	1	no	—
81	#63252	—	2	1	no	—
82	C₂×D₆₈	SD₁₆	1	1	yes	characteristic
83	C₂×C₄×D₁₇	C₂×D₄	1	2	yes	—
84	C₂×Dic₃₄	SD₁₆	1	1	yes	characteristic
85	Q₁₆×C₁₇	—	2	2	no	—
86	Q₁₆×C₁₇	—	2	2	no	—
87	Q₁₆×C₁₇	—	4	1	no	—
88	Q₁₆×C₁₇	—	4	1	no	—
89	M₁₆×C₁₇	—	2	2	no	—
90	M₁₆×C₁₇	—	4	1	no	—
91	C₂×C₈×C₁₇	—	4	1	no	—
92	C₂×C₈×C₁₇	C₂×D₄	1	1	yes	characteristic
93	C₂×C₈×C₁₇	#41	1	1	yes	characteristic
94	C₂×C₈×C₁₇	#41	1	1	yes	characteristic
95	C₂×C₈×C₁₇	#41	1	1	yes	characteristic
96	C₂×C₈×C₁₇	#41	1	1	yes	characteristic
97	C₂×C₈×C₁₇	C₂⁴	1	1	yes	characteristic
98	C₂²⋊Dic₁₇	—	4	1	no	—
99	#63227	—	4	4	no	—
100	C₄⋊Dic₁₇	—	2	2	no	—
101	#63225	—	4	4	no	—
102	C₄×Dic₁₇	—	4	1	no	—
103	#63223	—	2	1	no	—
104	#63223	—	2	1	no	—
105	#63223	—	2	2	no	—
106	#63223	—	4	1	no	—
107	C₂×#3622	—	4	1	no	—
108	C₂×#3622	C₂×D₄	1	2	yes	—
109	#63218	—	2	2	no	—
110	#63218	—	2	2	no	—
111	#63218	—	4	4	no	—
112	C₈×D₁₇	—	2	2	no	—
113	C₈×D₁₇	—	2	2	no	—
114	C₈×D₁₇	—	4	4	no	—
115	C₁₆×C₁₇	—	2	2	no	—
116	#63214	—	2	2	no	—
117	#33658	—	17	1	no	maximal, Sylow
118	C₂³×C₁₇	C₂×SD₁₆	1	1	yes	characteristic
119	C₂²×D₁₇	C₂×SD₁₆	1	2	yes	—
120	Q₈×C₁₇	—	4	2	no	—
121	Q₈×C₁₇	—	4	2	no	—
122	Q₈×C₁₇	—	8	1	no	—
123	Q₈×C₁₇	—	8	1	no	—
124	C₂×C₄×C₁₇	—	4	2	no	—
125	C₂×C₄×C₁₇	—	8	1	no	—
126	C₂×C₄×C₁₇	—	8	1	no	—
127	C₂×C₄×C₁₇	#123	1	1	yes	characteristic
128	C₂×C₄×C₁₇	#123	1	1	yes	characteristic
129	C₂×C₄×C₁₇	C₂×SD₁₆	1	1	yes	characteristic
130	C₂×C₄×C₁₇	AGL(1,ℤ/8)	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	C₂²⋊D₁₇	—	2	2	no	—
134	C₂²⋊D₁₇	—	2	2	no	—
135	C₂×Dic₁₇	—	4	2	no	—
136	C₂×Dic₁₇	C₂×SD₁₆	1	2	yes	—
137	D₆₈	—	2	1	no	—
138	D₆₈	—	2	1	no	—
139	C₄×D₁₇	—	2	2	no	—
140	C₄×D₁₇	—	2	2	no	—
141	Dic₃₄	—	2	1	no	—
142	Dic₃₄	—	2	1	no	—
143	C₈×C₁₇	—	4	2	no	—
144	C₈×C₁₇	#123	1	1	yes	characteristic
145	C₈×C₁₇	#123	1	1	yes	characteristic
146	C₈×C₁₇	C₂×#41	1	1	yes	characteristic
147	C₈×C₁₇	C₂×#41	1	1	yes	characteristic
148	#3622	—	2	2	no	—
149	#3622	—	4	2	no	—
150	#3059	—	17	1	no	—
151	#2916	—	17	1	no	—
152	#2239	—	17	8	no	—
153	#2155	—	17	1	no	—
154	#2155	—	17	1	no	—
155	C₂×#362	—	17	1	no	—
156	C₂×#362	D₁₇	1	1	yes	characteristic, complemented
157	#2115	—	17	1	no	—
158	C₂²×C₁₇	—	4	1	no	—
159	C₂²×C₁₇	#481	1	1	yes	characteristic
160	C₂²×C₁₇	#481	1	1	yes	characteristic
161	C₂²×C₁₇	C₂²×SD₁₆	1	1	yes	characteristic
162	C₂×D₁₇	—	2	2	no	—
163	C₂×D₁₇	—	4	2	no	—
164	C₄×C₁₇	—	8	2	no	—
165	C₄×C₁₇	#481	1	1	yes	characteristic
166	C₄×C₁₇	#481	1	1	yes	characteristic
167	C₄×C₁₇	C₂×#123	1	1	yes	characteristic
168	C₄×C₁₇	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
169	Dic₁₇	—	2	2	no	—
170	Dic₁₇	—	8	2	no	—
171	C₂²×Q₁₆	C₂×D₁₇	1	1	yes	characteristic
172	#568	—	34	1	no	—
173	C₂×#131	—	17	1	no	—
174	#536	—	34	1	no	—
175	C₂×M₃₂	—	17	1	no	—
176	C₂×M₃₂	C₂×D₁₇	1	1	yes	characteristic
177	#470	—	17	4	no	—
178	#467	—	34	2	no	—
179	#430	—	17	4	no	—
180	#430	—	34	1	no	—
181	C₂×#108	—	17	1	no	—
182	C₂×#108	—	34	2	no	—
183	C₂×#108	C₂×D₁₇	1	1	yes	characteristic
184	#427	—	17	1	no	—
185	#419	—	34	1	no	—
186	#365	—	17	4	no	—
187	#365	—	17	4	no	—
188	#362	—	17	4	no	—
189	#362	C₂×D₁₇	1	4	yes	—
190	#349	—	17	4	no	—
191	C₂×C₁₇	—	2	1	no	—
192	C₂×C₁₇	—	2	1	no	—
193	C₂×C₁₇	—	2	1	no	—
194	C₂×C₁₇	C₂×#481	1	1	yes	characteristic
195	D₁₇	—	4	2	no	—
196	C₂×#41	—	17	1	no	—
197	C₂²×Q₈	—	2	1	no	—
198	C₂×Q₁₆	—	2	1	no	—
199	C₂×Q₁₆	—	2	1	no	—
200	C₂×Q₁₆	—	2	1	no	—
201	C₂×Q₁₆	—	2	1	no	—
202	C₂×Q₁₆	—	2	1	no	—
203	C₂×Q₁₆	C₂²×D₁₇	1	2	yes	—
204	#131	—	34	1	no	—
205	#131	—	34	1	no	—
206	#131	—	34	1	no	—
207	#131	—	34	1	no	—
208	#131	—	68	2	no	—
209	#131	—	68	2	no	—
210	C₂×M₁₆	—	2	1	no	—
211	C₂×M₁₆	—	17	1	no	—
212	C₂×M₁₆	—	17	2	no	—
213	C₂×M₁₆	—	34	4	no	—
214	C₂×M₁₆	—	34	1	no	—
215	C₂²×C₈	—	17	2	no	—
216	C₂²×C₈	C₂²×D₁₇	1	1	yes	characteristic
217	C₄⋊Q₈	—	68	1	no	—
218	#124	—	68	1	no	—
219	#123	—	68	2	no	—
220	C₂²⋊Q₈	—	34	4	no	—
221	#117	—	34	1	no	—
222	M₃₂	—	2	1	no	—
223	M₃₂	—	17	2	no	—
224	M₃₂	—	34	1	no	—
225	M₃₂	C₂²×D₁₇	1	2	yes	—
226	C₂×C₁₆	—	2	1	no	—
227	C₂×C₁₆	—	34	1	no	—
228	#108	—	2	1	no	—
229	#108	—	17	2	no	—
230	#108	—	34	4	no	—
231	#108	—	34	1	no	—
232	#108	C₂²×D₁₇	1	2	yes	—
233	#107	—	34	1	no	—
234	#106	—	17	2	no	—
235	#103	—	34	4	no	—
236	C₁₇	#33658	1	1	yes	characteristic, complemented, Sylow
237	#41	—	34	1	no	—
238	#41	—	34	1	no	—
239	#41	—	34	1	no	—
240	#41	—	34	1	no	—
241	C₂×Q₈	—	2	2	no	—
242	C₂×Q₈	—	4	1	no	—
243	C₂×Q₈	—	4	1	no	—
244	C₂×Q₈	—	4	1	no	—
245	C₂×Q₈	—	4	1	no	—
246	C₂×Q₈	—	4	1	no	—
247	C₂×Q₈	—	17	1	no	—
248	C₂×D₄	—	17	2	no	—
249	C₂×D₄	—	17	1	no	—
250	C₂²×C₄	—	4	1	no	—
251	C₂²×C₄	—	17	2	no	—
252	C₂²×C₄	D₄×D₁₇	1	1	yes	characteristic
253	Q₁₆	—	2	2	no	—
254	Q₁₆	—	2	2	no	—
255	Q₁₆	—	4	1	no	—
256	Q₁₆	—	4	1	no	—
257	M₁₆	—	2	2	no	—
258	M₁₆	—	4	1	no	—
259	M₁₆	—	34	1	no	—
260	M₁₆	—	34	1	no	—
261	M₁₆	—	34	2	no	—
262	M₁₆	—	34	2	no	—
263	M₁₆	—	34	2	no	—
264	M₁₆	—	68	4	no	—
265	M₁₆	—	68	1	no	—
266	C₂×C₈	—	4	1	no	—
267	C₂×C₈	—	17	2	no	—
268	C₂×C₈	—	34	2	no	—
269	C₂×C₈	—	34	2	no	—
270	C₂×C₈	—	68	4	no	—
271	C₂×C₈	—	68	1	no	—
272	C₂×C₈	#63252	1	1	yes	characteristic
273	C₂×C₈	#63252	1	1	yes	characteristic
274	C₂×C₈	D₄×D₁₇	1	1	yes	characteristic
275	C₂×C₈	#63254	1	1	yes	characteristic
276	C₂×C₈	#63254	1	1	yes	characteristic
277	C₂×C₈	C₂³×D₁₇	1	1	yes	characteristic
278	C₄⋊C₄	—	34	2	no	—
279	C₄⋊C₄	—	68	4	no	—
280	C₂²⋊C₄	—	68	4	no	—
281	C₂²⋊C₄	—	68	1	no	—
282	C₄²	—	68	1	no	—
283	C₁₆	—	2	2	no	—
284	C₁₆	—	34	2	no	—
285	C₂³	—	17	2	no	—
286	C₂³	SD₁₆×D₁₇	1	1	yes	characteristic
287	Q₈	—	4	2	no	—
288	Q₈	—	4	2	no	—
289	Q₈	—	8	1	no	—
290	Q₈	—	8	1	no	—
291	Q₈	—	34	1	no	—
292	Q₈	—	34	1	no	—
293	D₄	—	34	2	no	—
294	D₄	—	34	2	no	—
295	D₄	—	34	1	no	—
296	D₄	—	34	1	no	—
297	C₂×C₄	—	4	2	no	—
298	C₂×C₄	—	8	1	no	—
299	C₂×C₄	—	8	1	no	—
300	C₂×C₄	—	17	2	no	—
301	C₂×C₄	—	34	2	no	—
302	C₂×C₄	—	34	2	no	—
303	C₂×C₄	—	68	2	no	—
304	C₂×C₄	#10587569	1	1	yes	characteristic
305	C₂×C₄	#10587569	1	1	yes	characteristic
306	C₂×C₄	#10587597	1	1	yes	characteristic
307	C₂×C₄	#10587597	1	1	yes	characteristic
308	C₂×C₄	SD₁₆×D₁₇	1	1	yes	characteristic
309	C₂×C₄	C₂×D₄×D₁₇	1	1	yes	characteristic
310	C₈	—	4	2	no	—
311	C₈	—	34	2	no	—
312	C₈	—	68	2	no	—
313	C₈	#10587569	1	1	yes	characteristic
314	C₈	#10587569	1	1	yes	characteristic
315	C₈	C₂×#63252	1	1	yes	characteristic
316	C₈	C₂×#63254	1	1	yes	characteristic
317	C₂²	—	4	1	no	—
318	C₂²	—	34	2	no	—
319	C₂²	—	68	2	no	—
320	C₂²	#11763126	1	1	yes	characteristic
321	C₂²	#11763126	1	1	yes	characteristic
322	C₂²	C₂×SD₁₆×D₁₇	1	1	yes	characteristic
323	C₄	—	8	2	no	—
324	C₄	—	34	2	no	—
325	C₄	—	136	2	no	—
326	C₄	#11763126	1	1	yes	characteristic
327	C₄	#11763126	1	1	yes	characteristic
328	C₄	C₂×#10587569	1	1	yes	characteristic
329	C₄	C₂×#10587597	1	1	yes	characteristic
330	C₂	—	2	1	no	—
331	C₂	—	2	1	no	—
332	C₂	—	2	1	no	—
333	C₂	—	68	2	no	—
334	C₂	C₂×#11763126	1	1	yes	characteristic

	subgroup	G/H
center	C₂	C₂×#11763126
commutator, cosocle	C₂×C₈×C₁₇	C₂⁴
Frattini	C₂×C₈	C₂³×D₁₇
Fitting	C₂×C₁₇×#362	C₂
socle	C₂×C₁₇	C₂×#481
prime radical	C₂×#63252	D₄
hypercenter	C₂×#362	D₁₇
hypocenter	C₁₇	#33658
LCS	G ▷ C₂×C₈×C₁₇ ▷ C₄×C₁₇ ▷ C₂×C₁₇ ▷ C₁₇
UCS	C₁ ◁ C₂ ◁ C₂²×C₄ ◁ C₂²×C₈ ◁ C₂×#362

Complements

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

N	H ≅ G/N	type
3. #14531002	C₂	semidirect
4. #14530558	C₂	semidirect
5. #14530557	C₂	semidirect
6. C₂×#11762625	C₂	semidirect
8. C₂×C₁₇×#362	C₂	semidirect
156. C₂×#362	D₁₇	semidirect
236. C₁₇	#33658	semidirect

Maximal subgroups

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

subgroup	index
#14538435	2
#14537392	2
#14531002	2
#14530558	2
#14530557	2
C₂×#11762625	2
#14530404	2
C₂×C₁₇×#362	2
#33658	17