Subgroups — #17961019

Order 4352.

374 conjugacy classes of subgroups.

The subgroup lattice has 288 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#14544985	C₂	1	1	yes	maximal, characteristic
2	C₂×#11763696	C₂	1	1	yes	maximal, characteristic
3	C₂×C₁₆×#595	C₂	1	1	yes	maximal, characteristic
4	C₈×C₄⋊#595	C₂	1	1	yes	maximal, characteristic
5	#14543821	C₂	1	1	yes	maximal, characteristic
6	#14543821	C₂	1	1	yes	maximal, characteristic
7	D₁₇×C₄⋊C₁₆	C₂	1	1	yes	maximal, characteristic
8	C₂×#10587529	C₂²	1	1	yes	characteristic
9	C₂×#10587529	C₂²	1	1	yes	characteristic
10	#11763810	C₄	1	1	yes	characteristic
11	C₂×C₈×#595	C₂²	1	1	yes	characteristic
12	C₂×C₈×#595	C₂²	1	1	yes	characteristic
13	C₄×C₄⋊#595	C₄	1	1	yes	characteristic
14	#11763698	C₄	1	2	yes	complemented
15	#11763697	C₄	1	2	yes	complemented
16	#11763696	—	2	2	no	—
17	C₁₆×#595	—	2	2	no	—
18	#11763674	C₄	1	1	yes	characteristic
19	#11763674	C₄	1	1	yes	characteristic
20	C₂×C₁₆×D₁₇	C₂²	1	1	yes	characteristic
21	C₂×C₁₆×D₁₇	C₂²	1	1	yes	characteristic
22	C₄×C₈×D₁₇	C₂²	1	1	yes	characteristic
23	C₄⋊#63214	C₄	1	1	yes	characteristic, complemented
24	#11762627	C₄	1	1	yes	characteristic, complemented
25	#11762627	C₄	1	1	yes	characteristic, complemented
26	C₁₇×C₄⋊C₁₆	C₄	1	1	yes	characteristic, complemented
27	C₂×C₄⋊#595	C₈	1	2	yes	—
28	C₂×C₄×#595	C₂×C₄	1	1	yes	characteristic
29	C₂×C₄×#595	C₂×C₄	1	1	yes	characteristic
30	C₂×#63242	C₂×C₄	1	1	yes	characteristic
31	C₂×#63242	C₂×C₄	1	1	yes	characteristic
32	C₂×C₈×D₁₇	C₂×C₄	1	1	yes	characteristic
33	C₂×C₈×D₁₇	C₂³	1	1	yes	characteristic
34	C₄²×D₁₇	C₂×C₄	1	1	yes	characteristic
35	(C₂×#595)⋊C₄	C₈	1	2	yes	—
36	#10587544	C₈	1	2	yes	—
37	C₄⋊#3624	C₈	1	2	yes	—
38	C₂×#63216	C₂×C₄	1	2	yes	—
39	C₂×#63216	C₂×C₄	1	2	yes	—
40	C₈×#595	—	2	1	no	—
41	C₈×#595	—	2	1	no	—
42	C₈×#595	—	2	1	no	—
43	C₈×#595	—	2	1	no	—
44	#10587529	—	2	1	no	—
45	#10587529	—	2	1	no	—
46	#10587529	D₄	1	2	yes	—
47	#10587529	Q₈	1	2	yes	—
48	C₂×C₁₆×C₁₇	C₂×C₄	1	1	yes	characteristic
49	C₂×C₁₆×C₁₇	C₂×C₄	1	1	yes	characteristic
50	C₄×C₈×C₁₇	C₂×C₄	1	1	yes	characteristic
51	C₈×Dic₁₇	C₂×C₄	1	1	yes	characteristic
52	C₈×Dic₁₇	C₂×C₄	1	1	yes	characteristic
53	C₂×#63214	C₂×C₄	1	1	yes	characteristic
54	C₂×#63214	C₂×C₄	1	1	yes	characteristic
55	C₄×#3622	C₂×C₄	1	1	yes	characteristic
56	C₁₆×D₁₇	—	2	1	no	—
57	C₁₆×D₁₇	—	2	1	no	—
58	C₁₆×D₁₇	—	2	1	no	—
59	C₁₆×D₁₇	—	2	1	no	—
60	C₂²×#595	C₂×C₈	1	2	yes	—
61	C₂²×#595	C₂×C₈	1	2	yes	—
62	C₂×C₄×D₁₇	C₂×C₈	1	1	yes	characteristic
63	C₂×C₄×D₁₇	C₂×C₈	1	1	yes	characteristic
64	C₂×C₄×D₁₇	C₂²×C₄	1	1	yes	characteristic
65	C₂×#3624	C₂×C₈	1	2	yes	—
66	C₂×#3624	C₂×C₈	1	2	yes	—
67	C₄⋊#595	—	2	4	no	—
68	C₄×#595	—	2	1	no	—
69	C₄×#595	—	2	1	no	—
70	C₄×#595	—	2	1	no	—
71	C₄×#595	—	2	1	no	—
72	#63242	—	2	1	no	—
73	#63242	—	2	1	no	—
74	#63242	—	2	1	no	—
75	#63242	—	2	1	no	—
76	C₂×C₈×C₁₇	C₄²	1	1	yes	characteristic
77	C₂×C₈×C₁₇	C₂²×C₄	1	1	yes	characteristic
78	C₄²×C₁₇	C₄²	1	1	yes	characteristic
79	C₄×Dic₁₇	C₄²	1	1	yes	characteristic
80	C₄×Dic₁₇	C₂×C₈	1	1	yes	characteristic
81	C₄×Dic₁₇	C₂×C₈	1	1	yes	characteristic
82	C₂×#3622	C₄²	1	1	yes	characteristic
83	C₂×#3622	C₂²×C₄	1	1	yes	characteristic
84	C₈×D₁₇	—	2	1	no	—
85	C₈×D₁₇	—	2	1	no	—
86	C₈×D₁₇	C₂×D₄	1	1	yes	characteristic
87	C₈×D₁₇	C₂×Q₈	1	1	yes	characteristic
88	C₈×D₁₇	#41	1	1	yes	characteristic
89	C₈×D₁₇	#41	1	1	yes	characteristic
90	#63216	—	2	2	no	—
91	#63216	C₄⋊C₄	1	4	yes	complemented
92	C₁₆×C₁₇	—	2	1	no	—
93	C₁₆×C₁₇	—	2	1	no	—
94	#63214	—	2	1	no	—
95	#63214	—	2	1	no	—
96	#12663	—	17	1	no	maximal, Sylow
97	C₂²×D₁₇	C₂²×C₈	1	1	yes	characteristic
98	C₂×#595	—	2	2	no	—
99	C₂×#595	—	2	2	no	—
100	C₂×#595	—	2	2	no	—
101	C₂×#595	—	2	2	no	—
102	C₂×#595	—	2	2	no	—
103	C₂×#595	M₃₂	1	4	yes	—
104	C₂×C₄×C₁₇	C₄×C₈	1	1	yes	characteristic
105	C₂×C₄×C₁₇	C₄×C₈	1	1	yes	characteristic
106	C₂×C₄×C₁₇	C₂×C₄²	1	1	yes	characteristic
107	C₂×Dic₁₇	C₄×C₈	1	1	yes	characteristic
108	C₂×Dic₁₇	C₄×C₈	1	1	yes	characteristic
109	C₂×Dic₁₇	C₂²×C₈	1	1	yes	characteristic
110	C₄×D₁₇	—	2	2	no	—
111	C₄×D₁₇	—	2	2	no	—
112	C₄×D₁₇	C₄×D₄	1	1	yes	characteristic
113	C₄×D₁₇	C₄×Q₈	1	1	yes	characteristic
114	C₄×D₁₇	#131	1	1	yes	characteristic
115	C₄×D₁₇	#131	1	1	yes	characteristic
116	#3624	—	2	2	no	—
117	#3624	—	2	2	no	—
118	C₈×C₁₇	—	2	1	no	—
119	C₈×C₁₇	C₄×D₄	1	1	yes	characteristic
120	C₈×C₁₇	C₄×Q₈	1	1	yes	characteristic
121	#3622	—	2	1	no	—
122	#3622	C₂×C₄⋊C₄	1	1	yes	characteristic
123	#3622	#117	1	1	yes	characteristic
124	C₂×C₄⋊C₁₆	—	17	1	no	—
125	C₂×#346	—	17	1	no	—
126	C₂×#346	—	17	1	no	—
127	C₂×C₄×C₁₆	—	17	1	no	—
128	C₈×C₄⋊C₄	—	17	1	no	—
129	#1374	—	17	1	no	—
130	#1374	—	17	1	no	—
131	C₂²×C₁₇	C₂×C₄×C₈	1	1	yes	characteristic
132	C₂×D₁₇	C₈×D₄	1	1	yes	characteristic
133	C₂×D₁₇	C₈×Q₈	1	1	yes	characteristic
134	C₂×D₁₇	C₂×M₃₂	1	2	yes	—
135	C₂×D₁₇	#504	1	2	yes	—
136	#595	—	2	4	no	—
137	#595	—	4	2	no	—
138	C₄×C₁₇	—	2	1	no	—
139	C₄×C₁₇	—	2	1	no	—
140	C₄×C₁₇	C₄×C₄⋊C₄	1	1	yes	characteristic
141	C₄×C₁₇	#405	1	1	yes	characteristic
142	Dic₁₇	—	2	1	no	—
143	Dic₁₇	—	2	1	no	—
144	Dic₁₇	C₈×D₄	1	1	yes	characteristic
145	Dic₁₇	C₈×Q₈	1	1	yes	characteristic
146	C₂²×C₁₆	—	17	1	no	—
147	C₂²×C₁₆	—	17	1	no	—
148	C₂²×C₁₆	—	17	1	no	—
149	C₂²×C₁₆	—	17	1	no	—
150	C₂×C₄⋊C₈	—	17	1	no	—
151	C₂×C₄×C₈	—	17	1	no	—
152	C₂×C₄×C₈	—	17	1	no	—
153	C₂×C₄×C₈	—	17	1	no	—
154	C₄×C₄⋊C₄	—	17	1	no	—
155	C₄⋊C₁₆	—	17	1	no	—
156	C₄⋊C₁₆	—	17	1	no	—
157	C₄⋊C₁₆	—	17	1	no	—
158	C₄⋊C₁₆	#595	1	1	yes	characteristic, complemented
159	#346	—	17	2	no	—
160	#346	—	17	2	no	—
161	#346	—	34	2	no	—
162	C₄×C₁₆	—	34	2	no	—
163	#336	—	17	1	no	—
164	#336	—	17	1	no	—
165	C₂×C₁₇	C₈×C₄⋊C₄	1	1	yes	characteristic
166	C₂×C₁₇	C₄×M₃₂	1	1	yes	characteristic
167	C₂×C₁₇	#2108	1	1	yes	characteristic
168	D₁₇	#2168	1	2	yes	—
169	D₁₇	#2183	1	2	yes	—
170	C₂²×C₈	—	17	1	no	—
171	C₂²×C₈	—	17	1	no	—
172	C₂²×C₈	—	17	1	no	—
173	C₂²×C₈	—	17	1	no	—
174	C₂×C₄⋊C₄	—	17	2	no	—
175	C₂×C₄²	—	17	1	no	—
176	C₂×C₄²	—	17	1	no	—
177	C₂×C₄²	—	17	1	no	—
178	C₂×C₁₆	—	17	2	no	—
179	C₂×C₁₆	—	17	2	no	—
180	C₂×C₁₆	—	17	2	no	—
181	C₂×C₁₆	—	17	2	no	—
182	C₂×C₁₆	—	17	1	no	—
183	C₂×C₁₆	—	17	1	no	—
184	C₂×C₁₆	—	34	1	no	—
185	C₂×C₁₆	—	34	1	no	—
186	C₂×C₁₆	—	34	1	no	—
187	C₂×C₁₆	—	34	1	no	—
188	C₂×C₁₆	—	34	1	no	—
189	C₂×C₁₆	—	34	1	no	—
190	C₂×C₁₆	C₂×#595	1	1	yes	characteristic
191	C₂×C₁₆	C₂×#595	1	1	yes	characteristic
192	C₄⋊C₈	—	17	2	no	—
193	C₄⋊C₈	—	17	2	no	—
194	C₄×C₈	—	17	1	no	—
195	C₄×C₈	—	17	1	no	—
196	C₄×C₈	—	17	1	no	—
197	C₄×C₈	—	34	1	no	—
198	C₄×C₈	—	34	1	no	—
199	C₄×C₈	—	34	1	no	—
200	C₄×C₈	—	34	1	no	—
201	C₄×C₈	C₂×#595	1	1	yes	characteristic
202	#95	—	17	2	no	—
203	C₁₇	#12663	1	1	yes	characteristic, complemented, Sylow
204	C₂²×C₄	—	17	1	no	—
205	C₂²×C₄	—	17	2	no	—
206	C₂²×C₄	—	17	1	no	—
207	C₂²×C₄	—	17	1	no	—
208	C₂²×C₄	—	17	2	no	—
209	C₂×C₈	—	17	1	no	—
210	C₂×C₈	—	17	1	no	—
211	C₂×C₈	—	17	1	no	—
212	C₂×C₈	—	17	1	no	—
213	C₂×C₈	—	17	1	no	—
214	C₂×C₈	—	17	2	no	—
215	C₂×C₈	—	17	2	no	—
216	C₂×C₈	—	17	1	no	—
217	C₂×C₈	—	34	1	no	—
218	C₂×C₈	—	34	1	no	—
219	C₂×C₈	—	34	1	no	—
220	C₂×C₈	—	34	1	no	—
221	C₂×C₈	—	34	1	no	—
222	C₂×C₈	—	34	1	no	—
223	C₂×C₈	C₄×#595	1	1	yes	characteristic
224	C₂×C₈	C₂²×#595	1	1	yes	characteristic
225	C₄⋊C₄	—	34	4	no	—
226	C₄²	—	17	1	no	—
227	C₄²	—	17	1	no	—
228	C₄²	—	17	1	no	—
229	C₄²	—	34	1	no	—
230	C₄²	—	34	1	no	—
231	C₄²	—	34	1	no	—
232	C₄²	—	34	1	no	—
233	C₄²	C₄×#595	1	1	yes	characteristic
234	C₁₆	—	2	1	no	—
235	C₁₆	—	2	1	no	—
236	C₁₆	—	17	4	no	—
237	C₁₆	—	34	2	no	—
238	C₁₆	—	34	1	no	—
239	C₁₆	—	34	1	no	—
240	C₂³	—	17	1	no	—
241	C₂×C₄	—	17	1	no	—
242	C₂×C₄	—	17	1	no	—
243	C₂×C₄	—	17	1	no	—
244	C₂×C₄	—	17	1	no	—
245	C₂×C₄	—	17	1	no	—
246	C₂×C₄	—	17	4	no	—
247	C₂×C₄	—	17	1	no	—
248	C₂×C₄	—	17	1	no	—
249	C₂×C₄	—	34	2	no	—
250	C₂×C₄	—	34	2	no	—
251	C₂×C₄	—	34	2	no	—
252	C₂×C₄	—	34	2	no	—
253	C₂×C₄	—	34	2	no	—
254	C₂×C₄	—	34	2	no	—
255	C₂×C₄	—	34	2	no	—
256	C₂×C₄	C₈×#595	1	1	yes	characteristic
257	C₂×C₄	C₈×#595	1	1	yes	characteristic
258	C₂×C₄	C₂×C₄×#595	1	1	yes	characteristic
259	C₈	—	2	1	no	—
260	C₈	—	17	1	no	—
261	C₈	—	17	1	no	—
262	C₈	—	34	2	no	—
263	C₈	—	34	2	no	—
264	C₈	—	34	1	no	—
265	C₈	D₄×#595	1	1	yes	characteristic
266	C₈	Q₈×#595	1	1	yes	characteristic
267	C₂²	—	17	2	no	—
268	C₂²	—	17	2	no	—
269	C₂²	—	17	1	no	—
270	C₂²	—	17	1	no	—
271	C₂²	C₂×C₈×#595	1	1	yes	characteristic
272	C₄	—	2	1	no	—
273	C₄	—	2	1	no	—
274	C₄	—	17	1	no	—
275	C₄	—	17	1	no	—
276	C₄	—	34	4	no	—
277	C₄	—	34	1	no	—
278	C₄	—	34	1	no	—
279	C₄	—	68	2	no	—
280	C₄	C₄⋊C₄×#595	1	1	yes	characteristic
281	C₄	#11763775	1	1	yes	characteristic
282	C₂	—	17	2	no	—
283	C₂	—	17	2	no	—
284	C₂	#14544577	1	1	yes	characteristic
285	C₂	M₃₂×#595	1	1	yes	characteristic
286	C₂	#14544992	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂×C₈	C₂²×#595
commutator, socle	C₂²×C₁₇	C₂×C₄×C₈
Fitting	C₁₇×C₄⋊C₁₆	C₄
prime radical	C₂²×D₁₇	C₂²×C₈
cosocle	C₂×C₈×D₁₇	C₂³
hypercenter	C₄⋊C₁₆	#595
hypocenter	C₁₇	#12663
LCS	G ▷ C₂²×C₁₇ ▷ C₁₇
UCS	C₁ ◁ C₂×C₈ ◁ C₄⋊C₁₆

Complements

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

N	H ≅ G/N	type
14. #11763698	C₄	semidirect
15. #11763697	C₄	semidirect
23. C₄⋊#63214	C₄	semidirect
24. #11762627	C₄	semidirect
25. #11762627	C₄	semidirect
26. C₁₇×C₄⋊C₁₆	C₄	semidirect
91. #63216	C₄⋊C₄	semidirect
158. C₄⋊C₁₆	#595	semidirect
203. C₁₇	#12663	semidirect

Maximal subgroups

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

subgroup	index
#14544985	2
C₂×#11763696	2
C₂×C₁₆×#595	2
C₈×C₄⋊#595	2
#14543821	2
D₁₇×C₄⋊C₁₆	2
#12663	17