Subgroups — #16944265

Order 4352.

342 conjugacy classes of subgroups.

The subgroup lattice has 300 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₈×Q₈×D₁₇	C₂	1	1	yes	maximal, characteristic
2	#14530582	C₂	1	1	yes	maximal, characteristic
3	#14530581	C₂	1	1	yes	maximal, characteristic
4	#14526764	C₂	1	1	yes	maximal, characteristic, complemented
5	Q₈⋊#63214	C₂	1	1	yes	maximal, characteristic, complemented
6	#14526762	C₂	1	1	yes	maximal, characteristic, complemented
7	C₁₇×Q₈⋊C₁₆	C₂	1	1	yes	maximal, characteristic, complemented
8	C₄×Q₈×D₁₇	C₄	1	1	yes	characteristic
9	Q₈×#3622	C₄	1	1	yes	characteristic
10	C₈×Dic₃₄	—	2	1	no	—
11	C₈×Dic₃₄	C₂²	1	1	yes	characteristic
12	#11762933	—	2	1	no	—
13	#11762933	C₄	1	1	yes	characteristic
14	C₄⋊C₈×D₁₇	—	2	1	no	—
15	C₄⋊C₈×D₁₇	C₄	1	1	yes	characteristic
16	C₄×C₈×D₁₇	—	2	1	no	—
17	C₄×C₈×D₁₇	C₂²	1	1	yes	characteristic
18	C₄⋊#63214	C₂²	1	1	yes	characteristic
19	#11762627	—	2	1	no	—
20	#11762588	—	2	1	no	—
21	#11762588	—	2	1	no	—
22	C₄×#63214	C₂²	1	1	yes	characteristic
23	C₁₆×Dic₁₇	—	2	1	no	—
24	C₈×Q₈×C₁₇	C₂²	1	1	yes	characteristic
25	C₁₇×C₄⋊C₁₆	C₂²	1	1	yes	characteristic
26	C₄×C₁₆×C₁₇	C₂²	1	1	yes	characteristic
27	C₂×Q₈×D₁₇	C₈	1	1	yes	characteristic
28	C₄×Q₈×C₁₇	C₂×C₄	1	1	yes	characteristic
29	Q₈×Dic₁₇	C₈	1	1	yes	characteristic
30	C₂×C₈×D₁₇	—	2	1	no	—
31	C₂×C₈×D₁₇	D₄	1	1	yes	characteristic
32	C₂×C₈×D₁₇	D₄	1	1	yes	characteristic
33	C₄⋊C₄×D₁₇	—	2	1	no	—
34	C₄⋊C₄×D₁₇	C₈	1	1	yes	characteristic
35	#10587573	—	2	1	no	—
36	#10587573	C₈	1	1	yes	characteristic
37	C₄²×D₁₇	—	2	1	no	—
38	C₄²×D₁₇	C₂×C₄	1	1	yes	characteristic
39	C₄×Dic₃₄	—	2	1	no	—
40	C₄×Dic₃₄	C₂×C₄	1	1	yes	characteristic
41	C₂×C₁₆×C₁₇	—	2	1	no	—
42	C₂×C₁₆×C₁₇	—	2	1	no	—
43	C₁₇×C₄⋊C₈	—	2	1	no	—
44	C₁₇×C₄⋊C₈	C₂×C₄	1	1	yes	characteristic
45	C₄×C₈×C₁₇	—	2	1	no	—
46	C₄×C₈×C₁₇	C₂³	1	1	yes	characteristic
47	#10587485	—	2	1	no	—
48	#10587485	—	2	1	no	—
49	#10587485	—	2	1	no	—
50	C₈×Dic₁₇	—	2	1	no	—
51	C₈×Dic₁₇	—	2	1	no	—
52	C₈×Dic₁₇	D₄	1	1	yes	characteristic
53	C₈×Dic₁₇	D₄	1	1	yes	characteristic
54	C₂×#63214	—	2	1	no	—
55	C₂×#63214	—	2	1	no	—
56	C₄⋊#3622	—	2	1	no	—
57	C₄⋊#3622	C₂×C₄	1	1	yes	characteristic
58	C₄×#3622	—	2	1	no	—
59	C₄×#3622	C₂×C₄	1	1	yes	characteristic
60	C₂×Q₈×C₁₇	C₂×C₈	1	1	yes	characteristic
61	Q₈×D₁₇	—	2	2	no	—
62	Q₈×D₁₇	C₁₆	1	4	yes	complemented
63	C₂×C₄×D₁₇	—	2	1	no	—
64	C₂×C₄×D₁₇	—	2	1	no	—
65	C₂×C₄×D₁₇	C₂²⋊C₄	1	1	yes	characteristic
66	C₂×C₄×D₁₇	C₂×C₈	1	1	yes	characteristic
67	C₂×C₄×D₁₇	M₁₆	1	1	yes	characteristic
68	C₂×Dic₃₄	—	2	1	no	—
69	C₂×Dic₃₄	C₂×C₈	1	1	yes	characteristic
70	C₂×C₈×C₁₇	—	2	1	no	—
71	C₂×C₈×C₁₇	C₂×D₄	1	1	yes	characteristic
72	C₂×C₈×C₁₇	C₂×D₄	1	1	yes	characteristic
73	C₄⋊C₄×C₁₇	—	2	1	no	—
74	C₄⋊C₄×C₁₇	C₂×C₈	1	1	yes	characteristic
75	C₄²×C₁₇	—	2	1	no	—
76	C₄²×C₁₇	C₂²×C₄	1	1	yes	characteristic
77	C₄⋊Dic₁₇	—	2	1	no	—
78	C₄⋊Dic₁₇	C₂×C₈	1	1	yes	characteristic
79	#63225	—	2	1	no	—
80	#63225	—	2	1	no	—
81	#63225	—	2	1	no	—
82	C₄×Dic₁₇	—	2	1	no	—
83	C₄×Dic₁₇	—	2	1	no	—
84	C₄×Dic₁₇	—	2	1	no	—
85	C₄×Dic₁₇	C₂²⋊C₄	1	1	yes	characteristic
86	C₄×Dic₁₇	C₂×C₈	1	1	yes	characteristic
87	C₄×Dic₁₇	M₁₆	1	1	yes	characteristic
88	C₂×#3622	—	2	1	no	—
89	C₂×#3622	C₂²⋊C₄	1	1	yes	characteristic
90	C₂×#3622	C₂²⋊C₄	1	1	yes	characteristic
91	C₈×D₁₇	—	2	1	no	—
92	C₈×D₁₇	—	2	1	no	—
93	C₈×D₁₇	—	4	1	no	—
94	C₈×D₁₇	—	4	2	no	—
95	C₁₆×C₁₇	—	4	1	no	—
96	C₁₆×C₁₇	—	4	1	no	—
97	#63214	—	4	1	no	—
98	#63214	—	4	1	no	—
99	#11637	—	17	1	no	maximal, Sylow
100	C₂²×D₁₇	C₂²⋊C₈	1	1	yes	characteristic
101	Q₈×C₁₇	—	2	1	no	—
102	Q₈×C₁₇	C₂×C₁₆	1	2	yes	—
103	C₂×C₄×C₁₇	—	2	1	no	—
104	C₂×C₄×C₁₇	—	2	1	no	—
105	C₂×C₄×C₁₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
106	C₂×C₄×C₁₇	C₂²×C₈	1	1	yes	characteristic
107	C₂×C₄×C₁₇	C₂×M₁₆	1	1	yes	characteristic
108	C₂×Dic₁₇	—	2	1	no	—
109	C₂×Dic₁₇	—	2	1	no	—
110	C₂×Dic₁₇	C₂²⋊C₈	1	1	yes	characteristic
111	C₂×Dic₁₇	C₂²⋊C₈	1	1	yes	characteristic
112	C₂×Dic₁₇	C₂²⋊C₈	1	1	yes	characteristic
113	C₄×D₁₇	—	2	4	no	—
114	C₄×D₁₇	—	2	1	no	—
115	C₄×D₁₇	—	2	1	no	—
116	C₄×D₁₇	—	4	1	no	—
117	C₄×D₁₇	—	4	2	no	—
118	C₄×D₁₇	C₂×C₁₆	1	2	yes	—
119	C₄×D₁₇	M₃₂	1	2	yes	—
120	Dic₃₄	—	2	1	no	—
121	Dic₃₄	—	2	4	no	—
122	Dic₃₄	C₂×C₁₆	1	2	yes	—
123	C₈×C₁₇	—	2	1	no	—
124	C₈×C₁₇	—	4	1	no	—
125	C₈×C₁₇	#137	1	1	yes	characteristic
126	C₈×C₁₇	#137	1	1	yes	characteristic
127	#3622	—	2	1	no	—
128	#3622	—	2	1	no	—
129	#3622	—	4	1	no	—
130	C₂×C₈×Q₈	—	17	1	no	—
131	#2162	—	17	1	no	—
132	#2162	—	17	1	no	—
133	Q₈⋊C₁₆	—	17	1	no	—
134	Q₈⋊C₁₆	—	17	1	no	—
135	Q₈⋊C₁₆	—	17	1	no	—
136	Q₈⋊C₁₆	D₁₇	1	1	yes	characteristic, complemented
137	C₂²×C₁₇	C₂×C₂²⋊C₈	1	1	yes	characteristic
138	C₂×D₁₇	—	2	1	no	—
139	C₂×D₁₇	—	2	1	no	—
140	C₂×D₁₇	C₂²⋊C₁₆	1	2	yes	—
141	C₄×C₁₇	—	2	2	no	—
142	C₄×C₁₇	—	2	1	no	—
143	C₄×C₁₇	—	4	1	no	—
144	C₄×C₁₇	#419	1	1	yes	characteristic
145	C₄×C₁₇	#421	1	1	yes	characteristic
146	C₄×C₁₇	C₂²×C₁₆	1	1	yes	characteristic
147	C₄×C₁₇	C₂×M₃₂	1	1	yes	characteristic
148	Dic₁₇	—	2	2	no	—
149	Dic₁₇	—	2	1	no	—
150	Dic₁₇	—	2	1	no	—
151	Dic₁₇	—	4	1	no	—
152	Dic₁₇	C₂²⋊C₁₆	1	2	yes	—
153	C₂×C₄×Q₈	—	17	1	no	—
154	C₈×Q₈	—	17	1	no	—
155	C₈×Q₈	—	17	1	no	—
156	C₈×Q₈	—	17	1	no	—
157	C₈×Q₈	—	34	1	no	—
158	C₈×Q₈	—	34	1	no	—
159	C₈×Q₈	C₂×D₁₇	1	1	yes	characteristic
160	C₂×C₄⋊C₈	—	17	1	no	—
161	C₂×C₄⋊C₈	—	34	1	no	—
162	C₂×C₄×C₈	—	17	1	no	—
163	C₂×C₄×C₈	—	34	1	no	—
164	C₄⋊C₁₆	—	17	1	no	—
165	C₄⋊C₁₆	—	34	1	no	—
166	C₄⋊C₁₆	C₂×D₁₇	1	1	yes	characteristic
167	C₂²⋊C₁₆	—	34	1	no	—
168	C₂²⋊C₁₆	—	34	1	no	—
169	C₄×C₁₆	—	17	1	no	—
170	C₄×C₁₆	—	34	1	no	—
171	C₄×C₁₆	C₂×D₁₇	1	1	yes	characteristic
172	C₂×C₁₇	#1479	1	1	yes	characteristic
173	C₂×C₁₇	C₂×C₂²⋊C₁₆	1	1	yes	characteristic
174	C₂×C₁₇	#2117	1	1	yes	characteristic
175	D₁₇	—	2	2	no	—
176	C₂²×Q₈	—	17	1	no	—
177	C₂²×C₈	—	17	1	no	—
178	C₂²×C₈	—	17	1	no	—
179	C₂²×C₈	—	34	1	no	—
180	C₄×Q₈	—	17	1	no	—
181	C₄×Q₈	—	17	1	no	—
182	C₄×Q₈	—	17	1	no	—
183	C₄×Q₈	—	34	1	no	—
184	C₄×Q₈	—	34	1	no	—
185	C₄×Q₈	C₄×D₁₇	1	1	yes	characteristic
186	C₂×C₄⋊C₄	—	17	1	no	—
187	C₂×C₄⋊C₄	—	34	1	no	—
188	C₂×C₄²	—	17	1	no	—
189	C₂×C₄²	—	34	1	no	—
190	C₂×C₁₆	—	2	1	no	—
191	C₂×C₁₆	—	2	1	no	—
192	C₂×C₁₆	—	34	1	no	—
193	C₂×C₁₆	—	34	1	no	—
194	C₄⋊C₈	—	2	1	no	—
195	C₄⋊C₈	—	17	1	no	—
196	C₄⋊C₈	—	34	1	no	—
197	C₄⋊C₈	—	34	1	no	—
198	C₄⋊C₈	—	34	1	no	—
199	C₄⋊C₈	—	34	1	no	—
200	C₄⋊C₈	C₄×D₁₇	1	1	yes	characteristic
201	C₄×C₈	—	2	1	no	—
202	C₄×C₈	—	17	1	no	—
203	C₄×C₈	—	17	1	no	—
204	C₄×C₈	—	17	1	no	—
205	C₄×C₈	—	34	1	no	—
206	C₄×C₈	—	34	1	no	—
207	C₄×C₈	—	34	1	no	—
208	C₄×C₈	C₂²×D₁₇	1	1	yes	characteristic
209	C₁₇	#11637	1	1	yes	characteristic, complemented, Sylow
210	C₂×Q₈	—	17	4	no	—
211	C₂×Q₈	—	17	1	no	—
212	C₂×Q₈	—	34	1	no	—
213	C₂×Q₈	—	34	2	no	—
214	C₂×Q₈	C₈×D₁₇	1	1	yes	characteristic
215	C₂²×C₄	—	17	1	no	—
216	C₂²×C₄	—	17	1	no	—
217	C₂²×C₄	—	17	1	no	—
218	C₂²×C₄	—	34	1	no	—
219	C₂²×C₄	—	34	1	no	—
220	C₂×C₈	—	2	1	no	—
221	C₂×C₈	—	17	1	no	—
222	C₂×C₈	—	17	1	no	—
223	C₂×C₈	—	34	1	no	—
224	C₂×C₈	—	34	1	no	—
225	C₂×C₈	—	34	1	no	—
226	C₂×C₈	—	68	1	no	—
227	C₂×C₈	—	68	2	no	—
228	C₂×C₈	D₄×D₁₇	1	1	yes	characteristic
229	C₂×C₈	D₄×D₁₇	1	1	yes	characteristic
230	C₄⋊C₄	—	2	1	no	—
231	C₄⋊C₄	—	17	1	no	—
232	C₄⋊C₄	—	34	1	no	—
233	C₄⋊C₄	—	34	1	no	—
234	C₄⋊C₄	—	34	1	no	—
235	C₄⋊C₄	—	34	1	no	—
236	C₄⋊C₄	C₈×D₁₇	1	1	yes	characteristic
237	C₄²	—	2	1	no	—
238	C₄²	—	17	1	no	—
239	C₄²	—	17	1	no	—
240	C₄²	—	17	1	no	—
241	C₄²	—	34	1	no	—
242	C₄²	—	34	1	no	—
243	C₄²	—	34	1	no	—
244	C₄²	C₂×C₄×D₁₇	1	1	yes	characteristic
245	C₁₆	—	4	1	no	—
246	C₁₆	—	4	1	no	—
247	C₁₆	—	68	1	no	—
248	C₁₆	—	68	1	no	—
249	C₂³	—	17	1	no	—
250	Q₈	—	2	1	no	—
251	Q₈	—	17	2	no	—
252	Q₈	—	34	4	no	—
253	Q₈	—	34	1	no	—
254	Q₈	C₁₆×D₁₇	1	2	yes	—
255	C₂×C₄	—	2	1	no	—
256	C₂×C₄	—	2	1	no	—
257	C₂×C₄	—	17	1	no	—
258	C₂×C₄	—	17	2	no	—
259	C₂×C₄	—	17	2	no	—
260	C₂×C₄	—	17	1	no	—
261	C₂×C₄	—	17	1	no	—
262	C₂×C₄	—	34	4	no	—
263	C₂×C₄	—	34	1	no	—
264	C₂×C₄	—	34	1	no	—
265	C₂×C₄	—	34	1	no	—
266	C₂×C₄	—	34	1	no	—
267	C₂×C₄	—	68	1	no	—
268	C₂×C₄	—	68	2	no	—
269	C₂×C₄	C₂²⋊C₄×D₁₇	1	1	yes	characteristic
270	C₂×C₄	C₂×C₈×D₁₇	1	1	yes	characteristic
271	C₂×C₄	M₁₆×D₁₇	1	1	yes	characteristic
272	C₈	—	2	1	no	—
273	C₈	—	4	1	no	—
274	C₈	—	34	1	no	—
275	C₈	—	34	1	no	—
276	C₈	—	68	1	no	—
277	C₈	#10587601	1	1	yes	characteristic
278	C₈	#10587604	1	1	yes	characteristic
279	C₂²	—	17	2	no	—
280	C₂²	—	34	1	no	—
281	C₂²	—	34	1	no	—
282	C₂²	C₂²⋊C₈×D₁₇	1	1	yes	characteristic
283	C₄	—	2	2	no	—
284	C₄	—	2	1	no	—
285	C₄	—	4	1	no	—
286	C₄	—	17	2	no	—
287	C₄	—	34	2	no	—
288	C₄	—	34	1	no	—
289	C₄	—	34	1	no	—
290	C₄	—	68	1	no	—
291	C₄	#11762816	1	1	yes	characteristic
292	C₄	#11762824	1	1	yes	characteristic
293	C₄	C₂×C₁₆×D₁₇	1	1	yes	characteristic
294	C₄	M₃₂×D₁₇	1	1	yes	characteristic
295	C₂	—	34	2	no	—
296	C₂	#14527177	1	1	yes	characteristic
297	C₂	D₁₇×C₂²⋊C₁₆	1	1	yes	characteristic
298	C₂	#14530410	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₈	D₄×D₁₇
commutator	C₄×C₁₇	C₂²×C₁₆
Frattini	C₄×C₈	C₂²×D₁₇
Fitting	C₁₇×Q₈⋊C₁₆	C₂
socle	C₂²×C₁₇	C₂×C₂²⋊C₈
prime radical	C₂²×D₁₇	C₂²⋊C₈
cosocle	C₄×C₈×C₁₇	C₂³
hypercenter	Q₈⋊C₁₆	D₁₇
hypocenter	C₁₇	#11637
LCS	G ▷ C₄×C₁₇ ▷ C₂×C₁₇ ▷ C₁₇
UCS	C₁ ◁ C₂×C₈ ◁ C₄×C₈ ◁ Q₈⋊C₁₆

Complements

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

N	H ≅ G/N	type
4. #14526764	C₂	semidirect
5. Q₈⋊#63214	C₂	semidirect
6. #14526762	C₂	semidirect
7. C₁₇×Q₈⋊C₁₆	C₂	semidirect
62. Q₈×D₁₇	C₁₆	semidirect
136. Q₈⋊C₁₆	D₁₇	semidirect
209. C₁₇	#11637	semidirect

Maximal subgroups

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

subgroup	index
C₈×Q₈×D₁₇	2
#14530582	2
#14530581	2
#14526764	2
Q₈⋊#63214	2
#14526762	2
C₁₇×Q₈⋊C₁₆	2
#11637	17