Subgroups — #17978655

Order 4352.

542 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	#14545201	C₂	1	4	yes	maximal, complemented
2	#14545200	C₂	1	2	yes	maximal, complemented
3	#14545199	C₂	1	2	yes	maximal, complemented
4	#14545114	C₂	1	1	yes	maximal, characteristic
5	#14545114	C₂	1	2	yes	maximal
6	#14544985	C₂	1	2	yes	maximal
7	C₄×#10587529	C₂	1	1	yes	maximal, characteristic
8	C₈×Q₈×D₁₇	C₂	1	1	yes	maximal, characteristic
9	C₂×#10587529	C₂²	1	1	yes	characteristic
10	C₂×#10587529	C₂²	1	2	yes	—
11	C₂×#10587529	C₂²	1	1	yes	characteristic
12	#11763730	C₂²	1	4	yes	—
13	#11763730	C₂²	1	2	yes	—
14	C₄⋊#63216	C₂²	1	2	yes	—
15	C₄⋊#63216	C₂²	1	4	yes	—
16	#11763698	C₂²	1	4	yes	—
17	#11763697	C₂²	1	4	yes	—
18	#11763695	C₂²	1	2	yes	—
19	C₄×#63216	C₂²	1	2	yes	—
20	C₄×Q₈×D₁₇	C₄	1	1	yes	characteristic
21	Q₈×#3622	C₂²	1	1	yes	characteristic
22	C₈×Dic₃₄	C₄	1	2	yes	—
23	C₈×Dic₃₄	C₄	1	1	yes	characteristic
24	#11762933	C₂²	1	1	yes	characteristic
25	#11762933	C₂²	1	2	yes	—
26	C₄⋊C₈×D₁₇	C₄	1	2	yes	—
27	C₄⋊C₈×D₁₇	C₄	1	1	yes	characteristic
28	C₄×C₈×D₁₇	C₂²	1	1	yes	characteristic
29	C₄×C₈×D₁₇	C₂²	1	2	yes	—
30	C₈×Q₈×C₁₇	C₄	1	1	yes	characteristic
31	C₂×Q₈×D₁₇	C₈	1	1	yes	characteristic
32	C₄×Q₈×C₁₇	C₂×C₄	1	1	yes	characteristic
33	Q₈×Dic₁₇	C₈	1	1	yes	characteristic
34	C₂×C₈×D₁₇	C₂×C₄	1	1	yes	characteristic
35	C₂×C₈×D₁₇	C₂×C₄	1	2	yes	—
36	C₂×C₈×D₁₇	C₂³	1	1	yes	characteristic
37	C₄⋊C₄×D₁₇	C₈	1	2	yes	—
38	C₄⋊C₄×D₁₇	C₈	1	1	yes	characteristic
39	#10587573	C₈	1	2	yes	—
40	#10587573	C₈	1	1	yes	characteristic
41	C₄²×D₁₇	C₂×C₄	1	2	yes	—
42	C₄²×D₁₇	C₂×C₄	1	1	yes	characteristic
43	C₄×Dic₃₄	C₂×C₄	1	1	yes	characteristic
44	C₄×Dic₃₄	C₂×C₄	1	2	yes	—
45	C₂×#63216	C₂³	1	2	yes	—
46	C₂×#63216	C₂³	1	4	yes	—
47	C₂×#63216	C₂³	1	2	yes	—
48	#10587529	—	2	2	no	—
49	#10587529	—	2	4	no	—
50	#10587529	Q₈	1	4	yes	complemented
51	C₁₇×C₄⋊C₈	C₂×C₄	1	1	yes	characteristic
52	C₁₇×C₄⋊C₈	C₂×C₄	1	2	yes	—
53	C₄×C₈×C₁₇	C₂×C₄	1	1	yes	characteristic
54	C₄×C₈×C₁₇	C₂×C₄	1	2	yes	—
55	#10587485	C₂×C₄	1	2	yes	—
56	#10587485	C₂×C₄	1	2	yes	—
57	#10587485	C₂×C₄	1	2	yes	—
58	C₈×Dic₁₇	C₂×C₄	1	1	yes	characteristic
59	C₈×Dic₁₇	C₂×C₄	1	2	yes	—
60	C₈×Dic₁₇	C₂³	1	1	yes	characteristic
61	C₈×Dic₁₇	C₂³	1	2	yes	—
62	C₄⋊#3622	C₂×C₄	1	2	yes	—
63	C₄⋊#3622	C₂×C₄	1	1	yes	characteristic
64	C₄×#3622	C₂³	1	1	yes	characteristic
65	C₄×#3622	C₂³	1	2	yes	—
66	C₂×Q₈×C₁₇	C₂×C₈	1	1	yes	characteristic
67	Q₈×D₁₇	—	2	4	no	—
68	C₂×C₄×D₁₇	C₂×C₈	1	2	yes	—
69	C₂×C₄×D₁₇	C₂×C₈	1	2	yes	—
70	C₂×C₄×D₁₇	C₂×C₈	1	1	yes	characteristic
71	C₂×C₄×D₁₇	C₂×C₈	1	1	yes	characteristic
72	C₂×C₄×D₁₇	C₂²×C₄	1	1	yes	characteristic
73	C₂×Dic₃₄	C₂×C₈	1	1	yes	characteristic
74	C₂×Dic₃₄	C₂×C₈	1	2	yes	—
75	C₂×C₈×C₁₇	C₂²×C₄	1	1	yes	characteristic
76	C₂×C₈×C₁₇	C₂²×C₄	1	2	yes	—
77	C₂×C₈×C₁₇	C₂²×C₄	1	1	yes	characteristic
78	C₄⋊C₄×C₁₇	C₂×C₈	1	2	yes	—
79	C₄⋊C₄×C₁₇	C₂×C₈	1	1	yes	characteristic
80	C₄²×C₁₇	C₂²×C₄	1	2	yes	—
81	C₄²×C₁₇	C₂²×C₄	1	1	yes	characteristic
82	C₄⋊Dic₁₇	C₂×C₈	1	2	yes	—
83	C₄⋊Dic₁₇	C₂×C₈	1	1	yes	characteristic
84	#63225	C₂×C₈	1	2	yes	—
85	#63225	C₂×C₈	1	2	yes	—
86	#63225	C₂×C₈	1	2	yes	—
87	C₄×Dic₁₇	C₂×C₈	1	2	yes	—
88	C₄×Dic₁₇	C₂×C₈	1	1	yes	characteristic
89	C₄×Dic₁₇	C₂×C₈	1	2	yes	—
90	C₄×Dic₁₇	C₂×C₈	1	1	yes	characteristic
91	C₄×Dic₁₇	C₂²×C₄	1	2	yes	—
92	C₄×Dic₁₇	C₂²×C₄	1	1	yes	characteristic
93	C₂×#3622	C₂²×C₄	1	1	yes	characteristic
94	C₂×#3622	C₂²×C₄	1	2	yes	—
95	C₂×#3622	C₂⁴	1	1	yes	characteristic
96	C₈×D₁₇	—	2	2	no	—
97	C₈×D₁₇	—	2	4	no	—
98	C₈×D₁₇	C₂×Q₈	1	2	yes	—
99	C₈×D₁₇	#41	1	2	yes	—
100	#63216	—	2	2	no	—
101	#63216	—	2	4	no	—
102	#63216	C₂×Q₈	1	4	yes	complemented
103	C₂×#2183	—	17	1	no	maximal, Sylow
104	C₂²×D₁₇	C₂²×C₈	1	1	yes	characteristic
105	Q₈×C₁₇	—	2	2	no	—
106	C₂×C₄×C₁₇	C₂²×C₈	1	2	yes	—
107	C₂×C₄×C₁₇	C₂²×C₈	1	1	yes	characteristic
108	C₂×C₄×C₁₇	C₂²×C₈	1	2	yes	—
109	C₂×C₄×C₁₇	C₂²×C₈	1	1	yes	characteristic
110	C₂×C₄×C₁₇	C₂³×C₄	1	1	yes	characteristic
111	C₂×Dic₁₇	C₂²×C₈	1	2	yes	—
112	C₂×Dic₁₇	C₂²×C₈	1	2	yes	—
113	C₂×Dic₁₇	C₂²×C₈	1	1	yes	characteristic
114	C₂×Dic₁₇	C₂²×C₈	1	1	yes	characteristic
115	C₂×Dic₁₇	C₂²×C₈	1	1	yes	characteristic
116	C₄×D₁₇	—	2	4	no	—
117	C₄×D₁₇	—	2	4	no	—
118	C₄×D₁₇	—	2	2	no	—
119	C₄×D₁₇	M₃₂	1	4	yes	—
120	C₄×D₁₇	C₄×Q₈	1	2	yes	—
121	C₄×D₁₇	#131	1	2	yes	—
122	Dic₃₄	—	2	2	no	—
123	Dic₃₄	—	2	4	no	—
124	C₈×C₁₇	—	2	1	no	—
125	C₈×C₁₇	—	2	2	no	—
126	C₈×C₁₇	C₄×Q₈	1	2	yes	—
127	#3622	—	2	1	no	—
128	#3622	—	2	2	no	—
129	#3622	C₂²×Q₈	1	1	yes	characteristic
130	#3622	C₂×#41	1	1	yes	characteristic
131	C₂×C₈×Q₈	—	17	1	no	—
132	#2183	—	17	2	no	—
133	#2183	—	17	2	no	—
134	#2183	—	17	4	no	—
135	C₂×C₄⋊C₁₆	—	17	1	no	—
136	C₂×C₄⋊C₁₆	—	17	2	no	—
137	C₂×#346	—	17	2	no	—
138	C₂×C₄×C₁₆	—	17	1	no	—
139	C₂²×C₁₇	C₂³×C₈	1	1	yes	characteristic
140	C₂×D₁₇	C₈×Q₈	1	2	yes	—
141	C₂×D₁₇	C₂×M₃₂	1	2	yes	—
142	C₂×D₁₇	#504	1	2	yes	—
143	C₄×C₁₇	—	2	2	no	—
144	C₄×C₁₇	—	2	2	no	—
145	C₄×C₁₇	—	2	1	no	—
146	C₄×C₁₇	C₂×M₃₂	1	2	yes	—
147	C₄×C₁₇	C₂×C₄×Q₈	1	1	yes	characteristic
148	C₄×C₁₇	C₂×#131	1	1	yes	characteristic
149	Dic₁₇	—	2	2	no	—
150	Dic₁₇	—	2	2	no	—
151	Dic₁₇	—	2	1	no	—
152	Dic₁₇	C₈×Q₈	1	2	yes	—
153	Dic₁₇	C₂×M₃₂	1	2	yes	—
154	C₂×C₄×Q₈	—	17	1	no	—
155	C₂²×C₁₆	—	17	1	no	—
156	C₂²×C₁₆	—	17	1	no	—
157	C₂²×C₁₆	—	17	2	no	—
158	C₈×Q₈	—	17	1	no	—
159	C₈×Q₈	—	17	2	no	—
160	C₈×Q₈	—	17	1	no	—
161	C₈×Q₈	—	17	2	no	—
162	C₈×Q₈	—	17	1	no	—
163	C₈×Q₈	#595	1	1	yes	characteristic
164	C₂×C₄⋊C₈	—	17	1	no	—
165	C₂×C₄⋊C₈	—	17	2	no	—
166	C₂×C₄×C₈	—	17	1	no	—
167	C₂×C₄×C₈	—	17	2	no	—
168	C₄⋊C₁₆	—	17	2	no	—
169	C₄⋊C₁₆	—	17	4	no	—
170	C₄⋊C₁₆	—	17	2	no	—
171	C₄⋊C₁₆	—	17	4	no	—
172	#346	—	17	4	no	—
173	#346	—	17	4	no	—
174	C₄×C₁₆	—	17	2	no	—
175	C₄×C₁₆	—	17	2	no	—
176	C₂×C₁₇	C₂×C₈×Q₈	1	1	yes	characteristic
177	C₂×C₁₇	C₂²×M₃₂	1	1	yes	characteristic
178	C₂×C₁₇	C₂×#504	1	1	yes	characteristic
179	D₁₇	#2183	1	4	yes	complemented
180	C₂²×Q₈	—	17	1	no	—
181	C₂²×C₈	—	17	1	no	—
182	C₂²×C₈	—	17	1	no	—
183	C₂²×C₈	—	17	2	no	—
184	C₄×Q₈	—	17	1	no	—
185	C₄×Q₈	—	17	2	no	—
186	C₄×Q₈	—	17	1	no	—
187	C₄×Q₈	—	17	2	no	—
188	C₄×Q₈	—	17	1	no	—
189	C₄×Q₈	C₂×#595	1	1	yes	characteristic
190	C₂×C₄⋊C₄	—	17	1	no	—
191	C₂×C₄⋊C₄	—	17	2	no	—
192	C₂×C₄²	—	17	1	no	—
193	C₂×C₄²	—	17	2	no	—
194	C₂×C₁₆	—	17	2	no	—
195	C₂×C₁₆	—	17	4	no	—
196	C₂×C₁₆	—	17	2	no	—
197	C₂×C₁₆	—	17	4	no	—
198	C₂×C₁₆	—	34	2	no	—
199	C₂×C₁₆	—	34	4	no	—
200	C₄⋊C₈	—	17	2	no	—
201	C₄⋊C₈	—	17	2	no	—
202	C₄⋊C₈	—	17	2	no	—
203	C₄⋊C₈	—	17	2	no	—
204	C₄⋊C₈	—	17	1	no	—
205	C₄⋊C₈	C₂×#595	1	2	yes	—
206	C₄⋊C₈	C₂×#595	1	1	yes	characteristic
207	C₄×C₈	—	17	1	no	—
208	C₄×C₈	—	17	2	no	—
209	C₄×C₈	—	17	1	no	—
210	C₄×C₈	—	17	2	no	—
211	C₄×C₈	—	17	1	no	—
212	C₄×C₈	—	17	2	no	—
213	C₄×C₈	C₂×#595	1	1	yes	characteristic
214	C₄×C₈	C₂×#595	1	2	yes	—
215	C₁₇	C₂×#2183	1	1	yes	characteristic, complemented, Sylow
216	C₂×Q₈	—	17	2	no	—
217	C₂×Q₈	—	17	1	no	—
218	C₂×Q₈	—	34	4	no	—
219	C₂×Q₈	#63242	1	1	yes	characteristic
220	C₂²×C₄	—	17	1	no	—
221	C₂²×C₄	—	17	1	no	—
222	C₂²×C₄	—	17	1	no	—
223	C₂²×C₄	—	17	2	no	—
224	C₂²×C₄	—	17	2	no	—
225	C₂×C₈	—	17	1	no	—
226	C₂×C₈	—	17	2	no	—
227	C₂×C₈	—	17	2	no	—
228	C₂×C₈	—	17	1	no	—
229	C₂×C₈	—	17	2	no	—
230	C₂×C₈	—	34	2	no	—
231	C₂×C₈	—	34	4	no	—
232	C₂×C₈	C₂²×#595	1	1	yes	characteristic
233	C₂×C₈	C₂²×#595	1	2	yes	—
234	C₂×C₈	C₂²×#595	1	1	yes	characteristic
235	C₄⋊C₄	—	17	2	no	—
236	C₄⋊C₄	—	17	2	no	—
237	C₄⋊C₄	—	17	2	no	—
238	C₄⋊C₄	—	17	2	no	—
239	C₄⋊C₄	—	17	1	no	—
240	C₄⋊C₄	#63242	1	2	yes	—
241	C₄⋊C₄	#63242	1	1	yes	characteristic
242	C₄²	—	17	1	no	—
243	C₄²	—	17	2	no	—
244	C₄²	—	17	1	no	—
245	C₄²	—	17	2	no	—
246	C₄²	—	17	1	no	—
247	C₄²	—	17	2	no	—
248	C₄²	C₂²×#595	1	2	yes	—
249	C₄²	C₂²×#595	1	1	yes	characteristic
250	C₁₆	—	17	4	no	—
251	C₁₆	—	34	2	no	—
252	C₁₆	—	34	4	no	—
253	C₂³	—	17	1	no	—
254	Q₈	—	2	2	no	—
255	Q₈	—	34	4	no	—
256	Q₈	—	34	2	no	—
257	C₂×C₄	—	17	1	no	—
258	C₂×C₄	—	17	1	no	—
259	C₂×C₄	—	17	4	no	—
260	C₂×C₄	—	17	2	no	—
261	C₂×C₄	—	17	2	no	—
262	C₂×C₄	—	17	2	no	—
263	C₂×C₄	—	17	1	no	—
264	C₂×C₄	—	17	2	no	—
265	C₂×C₄	—	34	4	no	—
266	C₂×C₄	—	34	2	no	—
267	C₂×C₄	—	34	4	no	—
268	C₂×C₄	C₂×#63242	1	1	yes	characteristic
269	C₂×C₄	C₂×#63242	1	2	yes	—
270	C₂×C₄	C₂×#63242	1	2	yes	—
271	C₂×C₄	C₂×#63242	1	1	yes	characteristic
272	C₂×C₄	C₂³×#595	1	1	yes	characteristic
273	C₈	—	2	1	no	—
274	C₈	—	2	2	no	—
275	C₈	—	17	1	no	—
276	C₈	—	17	1	no	—
277	C₈	—	34	1	no	—
278	C₈	—	34	2	no	—
279	C₈	Q₈×#595	1	2	yes	—
280	C₂²	—	17	2	no	—
281	C₂²	—	17	2	no	—
282	C₂²	—	17	2	no	—
283	C₂²	C₂²×#63242	1	1	yes	characteristic
284	C₄	—	2	2	no	—
285	C₄	—	2	1	no	—
286	C₄	—	2	2	no	—
287	C₄	—	17	2	no	—
288	C₄	—	17	2	no	—
289	C₄	—	34	2	no	—
290	C₄	—	34	2	no	—
291	C₄	—	34	1	no	—
292	C₄	#11763865	1	2	yes	—
293	C₄	C₂×Q₈×#595	1	1	yes	characteristic
294	C₄	D₁₇⋊#131	1	1	yes	characteristic
295	C₂	—	17	4	no	—
296	C₂	Q₈×#63242	1	1	yes	characteristic
297	C₂	C₂×#11763865	1	1	yes	characteristic
298	C₂	#14545749	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₈	C₂²×#595
commutator, socle	C₂²×C₁₇	C₂³×C₈
Frattini	C₂×C₄	C₂³×#595
Fitting	C₈×Q₈×C₁₇	C₄
prime radical	C₂²×D₁₇	C₂²×C₈
cosocle	C₂×#3622	C₂⁴
hypercenter	C₈×Q₈	#595
hypocenter	C₁₇	C₂×#2183
LCS	G ▷ C₂²×C₁₇ ▷ C₁₇
UCS	C₁ ◁ C₂×C₈ ◁ C₈×Q₈

Complements

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

N	H ≅ G/N	type
1. #14545201	C₂	semidirect
2. #14545200	C₂	semidirect
3. #14545199	C₂	semidirect
50. #10587529	Q₈	semidirect
102. #63216	C₂×Q₈	semidirect
179. D₁₇	#2183	semidirect
215. C₁₇	C₂×#2183	semidirect

Maximal subgroups

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

subgroup	index
#14545201	2
#14545200	2
#14545199	2
#14545114	2
#14544985	2
C₄×#10587529	2
C₈×Q₈×D₁₇	2
C₂×#2183	17