Subgroups — #16927623

Order 4352.

358 conjugacy classes of subgroups.

The subgroup lattice has 298 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄×#10587490	C₂	1	1	yes	maximal, characteristic, complemented
2	#14527547	C₂	1	1	yes	maximal, characteristic, complemented
3	#14527391	C₂	1	2	yes	maximal
4	#14526655	C₂	1	2	yes	maximal, complemented
5	C₁₇×#2966	C₂	1	1	yes	maximal, characteristic
6	#11762866	C₂²	1	2	yes	—
7	C₂×#10587490	—	2	1	no	—
8	C₄×#63223	C₂²	1	2	yes	—
9	#11762574	—	2	1	no	—
10	#11762562	—	2	1	no	—
11	#11762560	—	2	1	no	—
12	#11762551	—	2	2	no	—
13	#11762549	—	2	2	no	—
14	C₁₇×#529	C₂²	1	1	yes	characteristic
15	C₁₇×#446	C₄	1	1	yes	characteristic
16	C₈×Q₈×C₁₇	C₄	1	1	yes	characteristic
17	C₁₇×#436	C₄	1	1	yes	characteristic
18	C₁₇×#435	—	2	1	no	—
19	C₈×D₄×C₁₇	—	2	1	no	—
20	C₈×D₄×C₁₇	C₄	1	1	yes	characteristic
21	C₁₇×#433	—	2	1	no	—
22	C₁₇×#432	C₂²	1	1	yes	characteristic
23	C₁₇×#405	—	2	1	no	—
24	C₂×C₄×C₈×C₁₇	C₂²	1	1	yes	characteristic
25	C₂×M₁₆×C₁₇	—	2	1	no	—
26	C₂²×C₈×C₁₇	—	2	1	no	—
27	C₂²×C₈×C₁₇	D₄	1	1	yes	characteristic
28	C₂²×C₈×C₁₇	D₄	1	1	yes	characteristic
29	C₁₇×#126	—	2	1	no	—
30	C₁₇×#125	D₄	1	1	yes	characteristic
31	C₁₇×#124	D₄	1	1	yes	characteristic
32	C₁₇×#123	—	2	1	no	—
33	C₁₇×C₂²⋊Q₈	Q₈	1	1	yes	characteristic
34	C₁₇×#121	Q₈	1	1	yes	characteristic
35	C₄×Q₈×C₁₇	C₂×C₄	1	1	yes	characteristic
36	C₄×D₄×C₁₇	—	2	1	no	—
37	C₄×D₄×C₁₇	C₂×C₄	1	1	yes	characteristic
38	C₁₇×#117	—	2	1	no	—
39	C₂×C₄²×C₁₇	C₂³	1	1	yes	characteristic
40	C₂×#63223	—	2	2	no	—
41	C₁₇×C₄⋊C₈	—	2	1	no	—
42	C₁₇×C₄⋊C₈	—	2	1	no	—
43	C₁₇×C₄⋊C₈	C₂×C₄	1	1	yes	characteristic
44	C₁₇×C₄⋊C₈	C₂×C₄	1	1	yes	characteristic
45	C₁₇×C₂²⋊C₈	—	2	1	no	—
46	C₁₇×C₂²⋊C₈	—	2	1	no	—
47	C₁₇×C₂²⋊C₈	—	2	1	no	—
48	C₁₇×#97	—	2	1	no	—
49	C₁₇×#97	—	2	1	no	—
50	C₄×C₈×C₁₇	—	2	1	no	—
51	C₄×C₈×C₁₇	C₂×C₄	1	1	yes	characteristic
52	C₄×C₈×C₁₇	C₂×C₄	1	1	yes	characteristic
53	C₄×C₈×C₁₇	D₄	1	1	yes	characteristic
54	C₄×C₈×C₁₇	D₄	1	1	yes	characteristic
55	C₂²⋊#3622	—	2	2	no	—
56	#10587490	—	2	4	no	—
57	C₄⋊#3622	—	2	2	no	—
58	#10587474	—	2	2	no	—
59	#10587474	—	2	2	no	—
60	C₄×#3622	—	2	2	no	—
61	C₂×Q₈×C₁₇	C₄⋊C₄	1	1	yes	characteristic
62	C₂×D₄×C₁₇	—	2	1	no	—
63	C₂×D₄×C₁₇	C₄⋊C₄	1	1	yes	characteristic
64	C₂²×C₄×C₁₇	—	2	1	no	—
65	C₂²×C₄×C₁₇	C₂×D₄	1	1	yes	characteristic
66	C₂²×C₄×C₁₇	C₂×Q₈	1	1	yes	characteristic
67	C₂²×C₄×C₁₇	#41	1	1	yes	characteristic
68	M₁₆×C₁₇	—	4	2	no	—
69	C₂×C₈×C₁₇	—	2	2	no	—
70	C₂×C₈×C₁₇	—	2	1	no	—
71	C₂×C₈×C₁₇	—	2	1	no	—
72	C₂×C₈×C₁₇	—	4	1	no	—
73	C₂×C₈×C₁₇	—	4	2	no	—
74	C₂×C₈×C₁₇	C₂²⋊C₄	1	1	yes	characteristic
75	C₂×C₈×C₁₇	C₂²⋊C₄	1	1	yes	characteristic
76	C₂×C₈×C₁₇	C₂²⋊C₄	1	1	yes	characteristic
77	C₂×C₈×C₁₇	C₂²⋊C₄	1	1	yes	characteristic
78	C₄⋊C₄×C₁₇	—	2	1	no	—
79	C₄⋊C₄×C₁₇	—	2	1	no	—
80	C₄⋊C₄×C₁₇	—	2	1	no	—
81	C₄⋊C₄×C₁₇	—	2	1	no	—
82	C₄⋊C₄×C₁₇	C₄⋊C₄	1	1	yes	characteristic
83	C₄⋊C₄×C₁₇	C₄⋊C₄	1	1	yes	characteristic
84	C₂²⋊C₄×C₁₇	—	2	1	no	—
85	C₂²⋊C₄×C₁₇	—	2	1	no	—
86	C₂²⋊C₄×C₁₇	—	2	1	no	—
87	C₂²⋊C₄×C₁₇	—	2	1	no	—
88	C₂²⋊C₄×C₁₇	—	2	1	no	—
89	C₄²×C₁₇	—	2	1	no	—
90	C₄²×C₁₇	C₂²×C₄	1	1	yes	characteristic
91	C₄²×C₁₇	C₂×D₄	1	1	yes	characteristic
92	C₄²×C₁₇	C₂×D₄	1	1	yes	characteristic
93	C₄²×C₁₇	#41	1	1	yes	characteristic
94	#63223	—	2	4	no	—
95	#63223	—	4	2	no	—
96	C₂×#3622	—	4	2	no	—
97	C₂×#3622	—	4	2	no	—
98	#8693	—	17	1	no	maximal, Sylow
99	C₂³×C₁₇	—	2	1	no	—
100	C₂³×C₁₇	C₂²⋊Q₈	1	1	yes	characteristic
101	Q₈×C₁₇	—	4	1	no	—
102	D₄×C₁₇	—	4	1	no	—
103	D₄×C₁₇	—	4	2	no	—
104	C₂×C₄×C₁₇	—	2	1	no	—
105	C₂×C₄×C₁₇	—	2	1	no	—
106	C₂×C₄×C₁₇	—	2	1	no	—
107	C₂×C₄×C₁₇	—	2	1	no	—
108	C₂×C₄×C₁₇	—	2	1	no	—
109	C₂×C₄×C₁₇	—	4	1	no	—
110	C₂×C₄×C₁₇	—	4	1	no	—
111	C₂×C₄×C₁₇	—	4	1	no	—
112	C₂×C₄×C₁₇	—	4	1	no	—
113	C₂×C₄×C₁₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
114	C₂×C₄×C₁₇	C₂×C₄⋊C₄	1	1	yes	characteristic
115	C₂×C₄×C₁₇	#117	1	1	yes	characteristic
116	C₂×C₄×C₁₇	#121	1	1	yes	characteristic
117	C₂×C₄×C₁₇	#121	1	1	yes	characteristic
118	C₂×C₄×C₁₇	C₂²⋊Q₈	1	1	yes	characteristic
119	C₈×C₁₇	—	2	1	no	—
120	C₈×C₁₇	—	2	1	no	—
121	C₈×C₁₇	—	2	1	no	—
122	C₈×C₁₇	—	4	1	no	—
123	C₈×C₁₇	—	4	1	no	—
124	C₈×C₁₇	#104	1	2	yes	—
125	#3622	—	4	4	no	—
126	#3622	—	8	2	no	—
127	#2966	D₁₇	1	1	yes	characteristic
128	C₄×#108	—	17	1	no	—
129	#1568	—	17	1	no	—
130	#1529	—	17	2	no	—
131	#1291	—	17	2	no	—
132	C₂²×C₁₇	—	2	1	no	—
133	C₂²×C₁₇	—	2	1	no	—
134	C₂²×C₁₇	—	2	1	no	—
135	C₂²×C₁₇	—	4	1	no	—
136	C₂²×C₁₇	—	4	1	no	—
137	C₂²×C₁₇	—	4	1	no	—
138	C₂²×C₁₇	#380	1	1	yes	characteristic
139	C₄×C₁₇	—	2	1	no	—
140	C₄×C₁₇	—	2	1	no	—
141	C₄×C₁₇	—	2	1	no	—
142	C₄×C₁₇	—	2	1	no	—
143	C₄×C₁₇	—	2	1	no	—
144	C₄×C₁₇	—	4	1	no	—
145	C₄×C₁₇	—	4	1	no	—
146	C₄×C₁₇	—	4	1	no	—
147	C₄×C₁₇	C₂×#104	1	1	yes	characteristic
148	C₄×C₁₇	#421	1	1	yes	characteristic
149	#529	C₂×D₁₇	1	1	yes	characteristic
150	#446	Dic₁₇	1	1	yes	characteristic
151	C₈×Q₈	Dic₁₇	1	1	yes	characteristic
152	#436	Dic₁₇	1	1	yes	characteristic
153	#435	—	2	1	no	—
154	C₈×D₄	—	2	1	no	—
155	C₈×D₄	Dic₁₇	1	1	yes	characteristic
156	#433	—	2	1	no	—
157	#432	—	17	2	no	—
158	#432	C₂×D₁₇	1	1	yes	characteristic
159	C₂×#108	—	34	1	no	—
160	#405	—	2	1	no	—
161	C₄×M₁₆	—	17	2	no	—
162	C₂×C₄×C₈	C₂×D₁₇	1	1	yes	characteristic
163	#341	—	34	1	no	—
164	#335	—	34	1	no	—
165	#334	—	34	1	no	—
166	#330	—	34	2	no	—
167	#329	—	34	2	no	—
168	C₂×C₁₇	—	4	1	no	—
169	C₂×C₁₇	—	8	1	no	—
170	C₂×C₁₇	#1806	1	1	yes	characteristic
171	C₂×C₁₇	#1813	1	1	yes	characteristic
172	C₂×C₁₇	#1814	1	1	yes	characteristic
173	C₂×M₁₆	—	2	1	no	—
174	C₂×M₁₆	—	34	2	no	—
175	C₂²×C₈	—	2	1	no	—
176	C₂²×C₈	C₂²⋊D₁₇	1	1	yes	characteristic
177	C₂²×C₈	C₂²⋊D₁₇	1	1	yes	characteristic
178	#126	—	2	1	no	—
179	#125	D₆₈	1	1	yes	characteristic
180	#124	D₆₈	1	1	yes	characteristic
181	#123	—	2	1	no	—
182	C₂²⋊Q₈	Dic₃₄	1	1	yes	characteristic
183	#121	Dic₃₄	1	1	yes	characteristic
184	C₄×Q₈	C₂×Dic₁₇	1	1	yes	characteristic
185	C₄×D₄	—	2	1	no	—
186	C₄×D₄	C₂×Dic₁₇	1	1	yes	characteristic
187	#117	—	2	1	no	—
188	C₂×C₄²	C₂²×D₁₇	1	1	yes	characteristic
189	#108	—	34	4	no	—
190	C₄⋊C₈	—	2	1	no	—
191	C₄⋊C₈	—	2	1	no	—
192	C₄⋊C₈	—	34	2	no	—
193	C₄⋊C₈	C₂×Dic₁₇	1	1	yes	characteristic
194	C₄⋊C₈	C₂×Dic₁₇	1	1	yes	characteristic
195	C₂²⋊C₈	—	2	1	no	—
196	C₂²⋊C₈	—	2	1	no	—
197	C₂²⋊C₈	—	2	1	no	—
198	C₂²⋊C₈	—	34	2	no	—
199	#97	—	2	1	no	—
200	#97	—	2	1	no	—
201	#97	—	34	2	no	—
202	#97	—	34	2	no	—
203	C₄×C₈	—	2	1	no	—
204	C₄×C₈	—	34	2	no	—
205	C₄×C₈	C₂×Dic₁₇	1	1	yes	characteristic
206	C₄×C₈	C₂×Dic₁₇	1	1	yes	characteristic
207	C₄×C₈	C₂²⋊D₁₇	1	1	yes	characteristic
208	C₄×C₈	C₂²⋊D₁₇	1	1	yes	characteristic
209	C₁₇	#8693	1	1	yes	characteristic, complemented, Sylow
210	C₂×Q₈	C₄⋊Dic₁₇	1	1	yes	characteristic
211	C₂×D₄	—	2	1	no	—
212	C₂×D₄	C₄⋊Dic₁₇	1	1	yes	characteristic
213	C₂²×C₄	—	2	1	no	—
214	C₂²×C₄	C₂×Dic₃₄	1	1	yes	characteristic
215	C₂²×C₄	#63252	1	1	yes	characteristic
216	C₂²×C₄	C₂×C₂²⋊D₁₇	1	1	yes	characteristic
217	M₁₆	—	4	2	no	—
218	M₁₆	—	34	4	no	—
219	M₁₆	—	68	2	no	—
220	C₂×C₈	—	2	2	no	—
221	C₂×C₈	—	2	1	no	—
222	C₂×C₈	—	2	1	no	—
223	C₂×C₈	—	4	1	no	—
224	C₂×C₈	—	4	2	no	—
225	C₂×C₈	—	68	2	no	—
226	C₂×C₈	—	68	2	no	—
227	C₂×C₈	C₂²⋊Dic₁₇	1	1	yes	characteristic
228	C₂×C₈	C₂²⋊Dic₁₇	1	1	yes	characteristic
229	C₂×C₈	C₂²⋊Dic₁₇	1	1	yes	characteristic
230	C₂×C₈	C₂²⋊Dic₁₇	1	1	yes	characteristic
231	C₄⋊C₄	—	2	1	no	—
232	C₄⋊C₄	—	2	1	no	—
233	C₄⋊C₄	—	2	1	no	—
234	C₄⋊C₄	—	2	1	no	—
235	C₄⋊C₄	C₄⋊Dic₁₇	1	1	yes	characteristic
236	C₄⋊C₄	C₄⋊Dic₁₇	1	1	yes	characteristic
237	C₂²⋊C₄	—	2	1	no	—
238	C₂²⋊C₄	—	2	1	no	—
239	C₂²⋊C₄	—	2	1	no	—
240	C₂²⋊C₄	—	2	1	no	—
241	C₂²⋊C₄	—	2	1	no	—
242	C₄²	—	2	1	no	—
243	C₄²	C₂×D₆₈	1	1	yes	characteristic
244	C₄²	#63252	1	1	yes	characteristic
245	C₄²	C₂²×Dic₁₇	1	1	yes	characteristic
246	C₄²	C₂×C₂²⋊D₁₇	1	1	yes	characteristic
247	C₂³	—	2	1	no	—
248	C₂³	#10587610	1	1	yes	characteristic
249	Q₈	—	4	1	no	—
250	D₄	—	4	1	no	—
251	D₄	—	4	2	no	—
252	C₂×C₄	—	2	1	no	—
253	C₂×C₄	—	2	1	no	—
254	C₂×C₄	—	2	1	no	—
255	C₂×C₄	—	2	1	no	—
256	C₂×C₄	—	2	1	no	—
257	C₂×C₄	—	4	1	no	—
258	C₂×C₄	—	4	1	no	—
259	C₂×C₄	—	4	1	no	—
260	C₂×C₄	—	4	1	no	—
261	C₂×C₄	#10587610	1	1	yes	characteristic
262	C₂×C₄	C₂×C₄⋊Dic₁₇	1	1	yes	characteristic
263	C₂×C₄	#10587612	1	1	yes	characteristic
264	C₂×C₄	#10587616	1	1	yes	characteristic
265	C₂×C₄	#10587616	1	1	yes	characteristic
266	C₂×C₄	C₂×C₂²⋊Dic₁₇	1	1	yes	characteristic
267	C₈	—	2	1	no	—
268	C₈	—	2	1	no	—
269	C₈	—	2	1	no	—
270	C₈	—	4	1	no	—
271	C₈	—	4	1	no	—
272	C₈	—	68	4	no	—
273	C₈	—	136	2	no	—
274	C₈	#10587508	1	2	yes	—
275	C₂²	—	2	1	no	—
276	C₂²	—	2	1	no	—
277	C₂²	—	2	1	no	—
278	C₂²	—	4	1	no	—
279	C₂²	—	4	1	no	—
280	C₂²	—	4	1	no	—
281	C₂²	#11762669	1	1	yes	characteristic
282	C₄	—	2	1	no	—
283	C₄	—	2	1	no	—
284	C₄	—	2	1	no	—
285	C₄	—	2	1	no	—
286	C₄	—	2	1	no	—
287	C₄	—	4	1	no	—
288	C₄	—	4	1	no	—
289	C₄	—	4	1	no	—
290	C₄	C₂×#10587508	1	1	yes	characteristic
291	C₄	#11762826	1	1	yes	characteristic
292	C₂	—	4	1	no	—
293	C₂	—	8	1	no	—
294	C₂	#14528804	1	1	yes	characteristic
295	C₂	#14528834	1	1	yes	characteristic
296	C₂	#14528838	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	#11762669
commutator	C₄²×C₁₇	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₁₇
Fitting	C₁₇×#2966	C₂
socle	C₂²×C₁₇	#380
prime radical	C₁₇×#121	Q₈
cosocle	C₂×C₄²×C₁₇	C₂³
hypercenter	#2966	D₁₇
hypocenter	C₁₇	#8693
LCS	G ▷ C₄²×C₁₇ ▷ C₂²×C₁₇ ▷ C₁₇
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #2966

Complements

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

N	H ≅ G/N	type
1. C₄×#10587490	C₂	semidirect
2. #14527547	C₂	semidirect
4. #14526655	C₂	semidirect
209. C₁₇	#8693	semidirect

Maximal subgroups

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

subgroup	index
C₄×#10587490	2
#14527547	2
#14527391	2
#14526655	2
C₁₇×#2966	2
#8693	17