Subgroups — #16933688

Order 4352.

502 conjugacy classes of subgroups.

The subgroup lattice has 322 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#14528651	C₂	1	1	yes	maximal, characteristic, complemented
2	#14527532	C₂	1	1	yes	maximal, characteristic, complemented
3	#14527279	C₂	1	2	yes	maximal, complemented
4	#14527183	C₂	1	2	yes	maximal, complemented
5	C₁₇×#3147	C₂	1	1	yes	maximal, characteristic
6	#11762862	C₂²	1	2	yes	—
7	C₂×C₄⋊#3622	C₂²	1	1	yes	characteristic
8	C₂×#10587486	—	2	1	no	—
9	C₄×C₄⋊Dic₁₇	C₂²	1	1	yes	characteristic
10	#11762563	—	2	1	no	—
11	#11762562	—	2	1	no	—
12	#11762560	—	2	1	no	—
13	#11762543	—	2	4	no	—
14	D₄⋊#3622	—	2	4	no	—
15	C₂×C₁₇×#135	—	2	1	no	—
16	C₂×C₁₇×#135	—	2	1	no	—
17	C₁₇×#532	C₂²	1	2	yes	—
18	C₁₇×#495	—	2	1	no	—
19	C₁₇×#495	—	2	1	no	—
20	C₁₇×#494	C₄	1	1	yes	characteristic, complemented
21	C₁₇×#493	C₄	1	1	yes	characteristic, complemented
22	C₁₇×#492	C₄	1	2	yes	complemented
23	C₂×C₄×C₈×C₁₇	C₂²	1	1	yes	characteristic
24	C₂×#41×C₁₇	—	2	2	no	—
25	C₁₇×#135	—	4	2	no	—
26	C₁₇×#135	—	4	2	no	—
27	C₁₇×#135	—	4	2	no	—
28	C₁₇×#135	—	4	2	no	—
29	C₂×Q₁₆×C₁₇	—	2	1	no	—
30	C₂×Q₁₆×C₁₇	—	2	1	no	—
31	C₂×SD₁₆×C₁₇	—	2	2	no	—
32	C₂×SD₁₆×C₁₇	—	2	2	no	—
33	C₂×D₈×C₁₇	—	2	1	no	—
34	C₂×D₈×C₁₇	—	2	1	no	—
35	C₂²×C₈×C₁₇	D₄	1	1	yes	characteristic
36	C₂²×C₈×C₁₇	D₄	1	1	yes	characteristic
37	C₁₇×C₄⋊Q₈	C₂×C₄	1	2	yes	—
38	C₁₇×#127	C₂×C₄	1	2	yes	—
39	C₁₇×#124	—	2	2	no	—
40	C₁₇×#121	—	2	2	no	—
41	C₁₇×#121	—	2	2	no	—
42	C₄×D₄×C₁₇	—	2	2	no	—
43	C₄×D₄×C₁₇	—	2	2	no	—
44	C₂×C₄²×C₁₇	C₂³	1	1	yes	characteristic
45	C₂×C₄⋊Dic₁₇	—	2	1	no	—
46	C₂×C₄×Dic₁₇	—	2	1	no	—
47	C₂²×#3622	—	2	1	no	—
48	C₄×C₈×C₁₇	C₂×C₄	1	1	yes	characteristic
49	C₄×C₈×C₁₇	C₂×C₄	1	1	yes	characteristic
50	C₄×C₈×C₁₇	D₄	1	1	yes	characteristic
51	C₄×C₈×C₁₇	D₄	1	1	yes	characteristic
52	#10587502	D₄	1	2	yes	—
53	C₂²⋊#3622	—	2	2	no	—
54	#10587486	—	4	2	no	—
55	C₄⋊#3622	—	2	2	no	—
56	C₄⋊#3622	—	2	1	no	—
57	C₄⋊#3622	D₄	1	2	yes	—
58	C₄×#3622	—	2	2	no	—
59	#41×C₁₇	—	4	2	no	—
60	#41×C₁₇	—	4	2	no	—
61	#41×C₁₇	—	4	2	no	—
62	#41×C₁₇	—	4	2	no	—
63	C₂×Q₈×C₁₇	—	2	2	no	—
64	C₂×D₄×C₁₇	—	2	2	no	—
65	C₂×D₄×C₁₇	—	2	2	no	—
66	C₂×D₄×C₁₇	—	2	2	no	—
67	C₂×D₄×C₁₇	—	4	2	no	—
68	C₂²×C₄×C₁₇	—	4	2	no	—
69	C₂²×C₄×C₁₇	C₂×D₄	1	1	yes	characteristic
70	C₂²×C₄×C₁₇	C₂×D₄	1	1	yes	characteristic
71	C₂²×C₄×C₁₇	#41	1	1	yes	characteristic
72	C₂²×Dic₁₇	—	2	2	no	—
73	Q₁₆×C₁₇	—	4	2	no	—
74	Q₁₆×C₁₇	—	4	2	no	—
75	SD₁₆×C₁₇	—	4	4	no	—
76	SD₁₆×C₁₇	—	4	4	no	—
77	D₈×C₁₇	—	4	2	no	—
78	D₈×C₁₇	—	4	2	no	—
79	C₂×C₈×C₁₇	—	2	1	no	—
80	C₂×C₈×C₁₇	—	2	1	no	—
81	C₂×C₈×C₁₇	C₂²⋊C₄	1	1	yes	characteristic
82	C₂×C₈×C₁₇	C₂²⋊C₄	1	1	yes	characteristic
83	C₂×C₈×C₁₇	C₂²⋊C₄	1	1	yes	characteristic
84	C₂×C₈×C₁₇	C₂²⋊C₄	1	1	yes	characteristic
85	C₂×C₈×C₁₇	D₈	1	2	yes	—
86	C₂×C₈×C₁₇	SD₁₆	1	2	yes	—
87	C₄⋊C₄×C₁₇	—	2	2	no	—
88	C₄⋊C₄×C₁₇	—	2	2	no	—
89	C₂²⋊C₄×C₁₇	—	4	2	no	—
90	C₂²⋊C₄×C₁₇	—	4	2	no	—
91	C₄²×C₁₇	C₂²×C₄	1	1	yes	characteristic
92	C₄²×C₁₇	C₂×D₄	1	1	yes	characteristic
93	C₄²×C₁₇	C₂×D₄	1	1	yes	characteristic
94	C₄²×C₁₇	#41	1	1	yes	characteristic
95	C₄⋊Dic₁₇	—	4	2	no	—
96	C₄×Dic₁₇	—	4	1	no	—
97	C₄×Dic₁₇	—	4	1	no	—
98	C₂×#3622	—	2	2	no	—
99	C₂×#3622	—	4	1	no	—
100	C₂×#3622	—	4	1	no	—
101	C₂×#3622	—	4	2	no	—
102	#9753	—	17	1	no	maximal, Sylow
103	C₂³×C₁₇	—	4	2	no	—
104	C₂³×C₁₇	#121	1	1	yes	characteristic
105	Q₈×C₁₇	—	4	2	no	—
106	Q₈×C₁₇	—	4	2	no	—
107	D₄×C₁₇	—	4	2	no	—
108	D₄×C₁₇	—	4	2	no	—
109	D₄×C₁₇	—	4	4	no	—
110	D₄×C₁₇	—	8	2	no	—
111	D₄×C₁₇	—	8	2	no	—
112	D₄×C₁₇	—	8	2	no	—
113	C₂×C₄×C₁₇	—	2	2	no	—
114	C₂×C₄×C₁₇	—	4	1	no	—
115	C₂×C₄×C₁₇	—	4	2	no	—
116	C₂×C₄×C₁₇	—	8	2	no	—
117	C₂×C₄×C₁₇	—	8	2	no	—
118	C₂×C₄×C₁₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
119	C₂×C₄×C₁₇	C₄×D₄	1	1	yes	characteristic
120	C₂×C₄×C₁₇	C₄×D₄	1	1	yes	characteristic
121	C₂×C₄×C₁₇	#121	1	1	yes	characteristic
122	C₂×C₄×C₁₇	#124	1	1	yes	characteristic
123	C₂×C₄×C₁₇	#127	1	1	yes	characteristic
124	C₂×C₄×C₁₇	C₂×D₈	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	AGL(1,ℤ/8)	1	1	yes	characteristic
128	C₂×Dic₁₇	—	4	2	no	—
129	C₂×Dic₁₇	—	4	2	no	—
130	C₂×Dic₁₇	—	4	2	no	—
131	C₈×C₁₇	—	2	1	no	—
132	C₈×C₁₇	—	2	1	no	—
133	C₈×C₁₇	#102	1	2	yes	—
134	C₈×C₁₇	#102	1	2	yes	—
135	#3622	—	4	2	no	—
136	#3622	—	8	2	no	—
137	#3147	D₁₇	1	1	yes	characteristic
138	#1770	—	17	1	no	—
139	#1564	—	17	1	no	—
140	#1502	—	17	2	no	—
141	#1480	—	17	2	no	—
142	C₂²×C₁₇	—	2	1	no	—
143	C₂²×C₁₇	—	2	1	no	—
144	C₂²×C₁₇	—	8	2	no	—
145	C₂²×C₁₇	—	8	2	no	—
146	C₂²×C₁₇	—	8	2	no	—
147	C₂²×C₁₇	#390	1	1	yes	characteristic
148	C₂²×C₁₇	#459	1	1	yes	characteristic
149	C₂²×C₁₇	#460	1	1	yes	characteristic
150	C₄×C₁₇	—	2	1	no	—
151	C₄×C₁₇	—	2	1	no	—
152	C₄×C₁₇	—	2	1	no	—
153	C₄×C₁₇	—	8	2	no	—
154	C₄×C₁₇	C₂×#102	1	1	yes	characteristic
155	C₄×C₁₇	#418	1	1	yes	characteristic
156	C₄×C₁₇	#444	1	2	yes	—
157	C₄×C₁₇	#459	1	1	yes	characteristic
158	C₄×C₁₇	#460	1	1	yes	characteristic
159	Dic₁₇	—	8	2	no	—
160	C₂×#135	—	2	1	no	—
161	C₂×#135	—	2	1	no	—
162	#532	C₂×D₁₇	1	2	yes	—
163	#495	—	2	1	no	—
164	#495	—	2	1	no	—
165	#494	Dic₁₇	1	1	yes	characteristic, complemented
166	#493	Dic₁₇	1	1	yes	characteristic, complemented
167	#492	Dic₁₇	1	2	yes	complemented
168	#431	—	17	2	no	—
169	C₂×C₄⋊C₈	—	17	1	no	—
170	C₂×#97	—	34	1	no	—
171	C₂×C₄×C₈	C₂×D₁₇	1	1	yes	characteristic
172	C₄×C₄⋊C₄	—	17	1	no	—
173	#336	—	34	1	no	—
174	#335	—	34	1	no	—
175	#334	—	34	1	no	—
176	#327	—	34	4	no	—
177	D₄⋊C₈	—	34	4	no	—
178	C₂×C₁₇	—	2	1	no	—
179	C₂×C₁₇	—	2	1	no	—
180	C₂×C₁₇	—	16	2	no	—
181	C₂×C₁₇	#1975	1	1	yes	characteristic
182	C₂×C₁₇	#1976	1	1	yes	characteristic
183	C₂×C₁₇	#1982	1	1	yes	characteristic
184	C₂×#41	—	2	2	no	—
185	#135	—	4	2	no	—
186	#135	—	4	2	no	—
187	#135	—	4	2	no	—
188	#135	—	4	2	no	—
189	C₂×Q₁₆	—	2	1	no	—
190	C₂×Q₁₆	—	2	1	no	—
191	C₂×SD₁₆	—	2	2	no	—
192	C₂×SD₁₆	—	2	2	no	—
193	C₂×D₈	—	2	1	no	—
194	C₂×D₈	—	2	1	no	—
195	C₂²×C₈	—	34	1	no	—
196	C₂²×C₈	C₂²⋊D₁₇	1	1	yes	characteristic
197	C₂²×C₈	C₂²⋊D₁₇	1	1	yes	characteristic
198	C₄⋊Q₈	C₂×Dic₁₇	1	2	yes	—
199	#127	C₂×Dic₁₇	1	2	yes	—
200	#124	—	2	2	no	—
201	#121	—	2	2	no	—
202	#121	—	2	2	no	—
203	C₄×D₄	—	2	2	no	—
204	C₄×D₄	—	2	2	no	—
205	C₂×C₄⋊C₄	—	34	1	no	—
206	C₂×C₄²	—	34	1	no	—
207	C₂×C₄²	C₂²×D₁₇	1	1	yes	characteristic
208	C₄⋊C₈	—	17	2	no	—
209	C₄⋊C₈	—	34	1	no	—
210	C₄⋊C₈	—	34	2	no	—
211	C₂²⋊C₈	—	34	2	no	—
212	#97	—	68	2	no	—
213	C₄×C₈	—	34	2	no	—
214	C₄×C₈	C₂×Dic₁₇	1	1	yes	characteristic
215	C₄×C₈	C₂×Dic₁₇	1	1	yes	characteristic
216	C₄×C₈	C₂²⋊D₁₇	1	1	yes	characteristic
217	C₄×C₈	C₂²⋊D₁₇	1	1	yes	characteristic
218	#95	—	17	2	no	—
219	C₁₇	#9753	1	1	yes	characteristic, complemented, Sylow
220	#41	—	4	2	no	—
221	#41	—	4	2	no	—
222	#41	—	4	2	no	—
223	#41	—	4	2	no	—
224	C₂×Q₈	—	2	2	no	—
225	C₂×D₄	—	2	2	no	—
226	C₂×D₄	—	2	2	no	—
227	C₂×D₄	—	2	2	no	—
228	C₂×D₄	—	4	2	no	—
229	C₂²×C₄	—	4	2	no	—
230	C₂²×C₄	—	34	2	no	—
231	C₂²×C₄	D₄×D₁₇	1	1	yes	characteristic
232	C₂²×C₄	#63254	1	1	yes	characteristic
233	C₂²×C₄	C₂×C₂²⋊D₁₇	1	1	yes	characteristic
234	Q₁₆	—	4	2	no	—
235	Q₁₆	—	4	2	no	—
236	SD₁₆	—	4	4	no	—
237	SD₁₆	—	4	4	no	—
238	D₈	—	4	2	no	—
239	D₈	—	4	2	no	—
240	C₂×C₈	—	2	1	no	—
241	C₂×C₈	—	2	1	no	—
242	C₂×C₈	—	34	2	no	—
243	C₂×C₈	—	68	1	no	—
244	C₂×C₈	—	68	1	no	—
245	C₂×C₈	—	68	2	no	—
246	C₂×C₈	#63228	1	2	yes	—
247	C₂×C₈	#63229	1	2	yes	—
248	C₂×C₈	C₂²⋊Dic₁₇	1	1	yes	characteristic
249	C₂×C₈	C₂²⋊Dic₁₇	1	1	yes	characteristic
250	C₂×C₈	C₂²⋊Dic₁₇	1	1	yes	characteristic
251	C₂×C₈	C₂²⋊Dic₁₇	1	1	yes	characteristic
252	C₄⋊C₄	—	2	2	no	—
253	C₄⋊C₄	—	2	2	no	—
254	C₄⋊C₄	—	68	2	no	—
255	C₂²⋊C₄	—	4	2	no	—
256	C₂²⋊C₄	—	4	2	no	—
257	C₄²	—	68	1	no	—
258	C₄²	—	68	1	no	—
259	C₄²	D₄×D₁₇	1	1	yes	characteristic
260	C₄²	#63254	1	1	yes	characteristic
261	C₄²	C₂²×Dic₁₇	1	1	yes	characteristic
262	C₄²	C₂×C₂²⋊D₁₇	1	1	yes	characteristic
263	C₂³	—	4	2	no	—
264	C₂³	#10587624	1	1	yes	characteristic
265	Q₈	—	4	2	no	—
266	Q₈	—	4	2	no	—
267	D₄	—	4	2	no	—
268	D₄	—	4	2	no	—
269	D₄	—	4	4	no	—
270	D₄	—	8	2	no	—
271	D₄	—	8	2	no	—
272	D₄	—	8	2	no	—
273	C₂×C₄	—	2	2	no	—
274	C₂×C₄	—	4	1	no	—
275	C₂×C₄	—	4	2	no	—
276	C₂×C₄	—	8	2	no	—
277	C₂×C₄	—	8	2	no	—
278	C₂×C₄	—	68	2	no	—
279	C₂×C₄	—	68	2	no	—
280	C₂×C₄	—	68	2	no	—
281	C₂×C₄	C₂×#63228	1	1	yes	characteristic
282	C₂×C₄	#10587618	1	1	yes	characteristic
283	C₂×C₄	#10587618	1	1	yes	characteristic
284	C₂×C₄	C₂×#63229	1	1	yes	characteristic
285	C₂×C₄	D₄×Dic₁₇	1	1	yes	characteristic
286	C₂×C₄	D₄×Dic₁₇	1	1	yes	characteristic
287	C₂×C₄	#10587622	1	1	yes	characteristic
288	C₂×C₄	#10587624	1	1	yes	characteristic
289	C₂×C₄	#10587626	1	1	yes	characteristic
290	C₂×C₄	C₂×C₂²⋊Dic₁₇	1	1	yes	characteristic
291	C₈	—	2	1	no	—
292	C₈	—	2	1	no	—
293	C₈	—	68	2	no	—
294	C₈	—	136	2	no	—
295	C₈	#10587503	1	2	yes	—
296	C₈	#10587503	1	2	yes	—
297	C₂²	—	2	1	no	—
298	C₂²	—	2	1	no	—
299	C₂²	—	8	2	no	—
300	C₂²	—	8	2	no	—
301	C₂²	—	8	2	no	—
302	C₂²	#11762712	1	1	yes	characteristic
303	C₂²	#11763001	1	1	yes	characteristic
304	C₂²	#11763006	1	1	yes	characteristic
305	C₄	—	2	1	no	—
306	C₄	—	2	1	no	—
307	C₄	—	2	1	no	—
308	C₄	—	8	2	no	—
309	C₄	—	136	2	no	—
310	C₄	C₂×#10587503	1	1	yes	characteristic
311	C₄	#11762814	1	1	yes	characteristic
312	C₄	#11762931	1	2	yes	—
313	C₄	#11763001	1	1	yes	characteristic
314	C₄	#11763006	1	1	yes	characteristic
315	C₂	—	2	1	no	—
316	C₂	—	2	1	no	—
317	C₂	—	16	2	no	—
318	C₂	#14529678	1	1	yes	characteristic
319	C₂	#14529683	1	1	yes	characteristic
320	C₂	#14529712	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	#10587624
commutator	C₄²×C₁₇	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₁₇
Fitting, prime radical	C₁₇×#3147	C₂
socle	C₂²×C₁₇	#390
cosocle	C₂×C₄²×C₁₇	C₂³
hypercenter	#3147	D₁₇
hypocenter	C₁₇	#9753
LCS	G ▷ C₄²×C₁₇ ▷ C₂²×C₁₇ ▷ C₁₇
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #3147

Complements

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

N	H ≅ G/N	type
1. #14528651	C₂	semidirect
2. #14527532	C₂	semidirect
3. #14527279	C₂	semidirect
4. #14527183	C₂	semidirect
20. C₁₇×#494	C₄	semidirect
21. C₁₇×#493	C₄	semidirect
22. C₁₇×#492	C₄	semidirect
165. #494	Dic₁₇	semidirect
166. #493	Dic₁₇	semidirect
167. #492	Dic₁₇	semidirect
219. C₁₇	#9753	semidirect

Maximal subgroups

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

subgroup	index
#14528651	2
#14527532	2
#14527279	2
#14527183	2
C₁₇×#3147	2
#9753	17