Subgroups — #17978435

Order 4352.

674 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	C₂×#11763819	C₂	1	2	yes	maximal
2	#14545136	C₂	1	8	yes	maximal, complemented
3	#14545135	C₂	1	4	yes	maximal, complemented
4	C₂×M₃₂×D₁₇	C₂	1	1	yes	maximal, characteristic
5	C₂×#10587677	—	2	2	no	—
6	#11763819	—	2	2	no	—
7	#11763819	—	2	2	no	—
8	#11763819	C₂²	1	4	yes	—
9	#11763819	C₂²	1	4	yes	—
10	C₂×#10587534	C₂²	1	4	yes	—
11	C₂×#10587533	—	2	2	no	—
12	#11763736	C₂²	1	8	yes	complemented
13	#11763735	C₂²	1	8	yes	complemented
14	C₂²×C₈×D₁₇	C₂²	1	1	yes	characteristic
15	C₂×#10587483	C₄	1	1	yes	characteristic, complemented
16	C₂×#10587469	C₄	1	2	yes	complemented
17	M₃₂×D₁₇	C₄	1	4	yes	complemented
18	M₃₂×D₁₇	C₂²	1	4	yes	—
19	C₂×C₁₆×D₁₇	C₂²	1	2	yes	—
20	C₂×C₁₇×M₃₂	C₄	1	1	yes	characteristic, complemented
21	C₂²×C₄×D₁₇	D₄	1	1	yes	characteristic
22	C₂×C₂²⋊#595	—	4	2	no	—
23	#10587677	—	2	4	no	—
24	#10587677	—	2	4	no	—
25	#10587677	—	4	2	no	—
26	#10587677	—	4	2	no	—
27	C₂×C₄⋊#595	—	2	4	no	—
28	C₂×C₄×#595	—	4	2	no	—
29	C₂²×C₈×C₁₇	C₂×C₄	1	1	yes	characteristic
30	C₂²×#3622	Q₈	1	1	yes	characteristic
31	C₂×C₈×D₁₇	C₂×C₄	1	2	yes	—
32	C₂×C₈×D₁₇	C₂×C₄	1	1	yes	characteristic
33	C₂×C₈×D₁₇	D₄	1	2	yes	—
34	C₂×C₈×D₁₇	D₄	1	1	yes	characteristic
35	C₂×C₈×D₁₇	Q₈	1	2	yes	—
36	C₂×C₈×D₁₇	Q₈	1	1	yes	characteristic
37	C₂×C₈×D₁₇	C₂³	1	1	yes	characteristic
38	C₂×C₈×D₁₇	C₂³	1	2	yes	—
39	#10587534	—	2	4	no	—
40	#10587534	C₂³	1	8	yes	—
41	#10587533	—	2	4	no	—
42	#10587533	—	2	4	no	—
43	C₁₇×M₃₂	C₂×C₄	1	4	yes	complemented
44	C₂×C₁₆×C₁₇	C₂×C₄	1	2	yes	complemented
45	#10587483	C₂×C₄	1	4	yes	complemented
46	C₂×#63214	C₂×C₄	1	2	yes	complemented
47	#10587469	C₂×C₄	1	8	yes	complemented
48	C₁₆×D₁₇	C₂×C₄	1	4	yes	complemented
49	C₁₆×D₁₇	C₂³	1	4	yes	—
50	C₂³×D₁₇	D₈	1	1	yes	characteristic
51	C₂²×#595	—	4	4	no	—
52	C₂²×C₄×C₁₇	C₄⋊C₄	1	1	yes	characteristic
53	C₂²×Dic₁₇	Q₁₆	1	1	yes	characteristic
54	C₂×C₄×D₁₇	C₄⋊C₄	1	2	yes	—
55	C₂×C₄×D₁₇	C₄⋊C₄	1	1	yes	characteristic
56	C₂×C₄×D₁₇	D₈	1	2	yes	—
57	C₂×C₄×D₁₇	D₈	1	1	yes	characteristic
58	C₂×C₄×D₁₇	Q₁₆	1	1	yes	characteristic
59	C₂×C₄×D₁₇	Q₁₆	1	2	yes	—
60	C₂×C₄×D₁₇	C₂×D₄	1	2	yes	—
61	C₂×C₄×D₁₇	C₂×D₄	1	1	yes	characteristic
62	C₂²⋊#595	—	4	4	no	—
63	C₂²⋊#595	—	4	4	no	—
64	C₄⋊#595	—	2	8	no	—
65	C₄⋊#595	—	4	4	no	—
66	C₄×#595	—	4	4	no	—
67	C₄×#595	—	4	4	no	—
68	C₂×C₈×C₁₇	C₄⋊C₄	1	2	yes	—
69	C₂×C₈×C₁₇	C₄⋊C₄	1	1	yes	characteristic
70	C₂×C₈×C₁₇	C₂²×C₄	1	1	yes	characteristic
71	C₂×C₈×C₁₇	C₂²×C₄	1	2	yes	—
72	C₂×#3622	C₄⋊C₄	1	2	yes	—
73	C₂×#3622	C₄⋊C₄	1	1	yes	characteristic
74	C₂×#3622	C₂×Q₈	1	1	yes	characteristic
75	C₂×#3622	C₂×Q₈	1	2	yes	—
76	C₈×D₁₇	C₄⋊C₄	1	2	yes	—
77	C₈×D₁₇	C₄⋊C₄	1	2	yes	—
78	C₈×D₁₇	C₄⋊C₄	1	2	yes	—
79	C₈×D₁₇	C₂²×C₄	1	1	yes	characteristic
80	C₈×D₁₇	C₂²×C₄	1	2	yes	—
81	C₈×D₁₇	C₂×D₄	1	1	yes	characteristic
82	C₈×D₁₇	C₂×D₄	1	2	yes	—
83	C₈×D₁₇	C₂×Q₈	1	1	yes	characteristic
84	C₈×D₁₇	C₂×Q₈	1	2	yes	—
85	C₈×D₁₇	C₂⁴	1	1	yes	characteristic
86	C₁₆×C₁₇	C₂²×C₄	1	4	yes	—
87	#63214	C₂²×C₄	1	4	yes	—
88	#33409	—	17	1	no	maximal, Sylow
89	C₂³×C₁₇	#107	1	1	yes	characteristic
90	C₂²×D₁₇	—	4	2	no	—
91	C₂²×D₁₇	#107	1	2	yes	—
92	C₂²×D₁₇	#107	1	1	yes	characteristic
93	C₂²×D₁₇	C₂×D₈	1	1	yes	characteristic
94	C₂²×D₁₇	C₂×D₈	1	2	yes	—
95	C₂×#595	—	4	8	no	—
96	C₂×#595	—	8	8	no	—
97	C₂×C₄×C₁₇	#107	1	1	yes	characteristic
98	C₂×C₄×C₁₇	#107	1	2	yes	—
99	C₂×C₄×C₁₇	C₂×C₄⋊C₄	1	2	yes	—
100	C₂×C₄×C₁₇	C₂×C₄⋊C₄	1	1	yes	characteristic
101	C₂×Dic₁₇	#107	1	2	yes	—
102	C₂×Dic₁₇	#107	1	1	yes	characteristic
103	C₂×Dic₁₇	C₂×Q₁₆	1	2	yes	—
104	C₂×Dic₁₇	C₂×Q₁₆	1	1	yes	characteristic
105	C₄×D₁₇	#107	1	2	yes	—
106	C₄×D₁₇	#107	1	2	yes	—
107	C₄×D₁₇	#107	1	2	yes	—
108	C₄×D₁₇	C₂×C₄⋊C₄	1	2	yes	—
109	C₄×D₁₇	C₂×C₄⋊C₄	1	1	yes	characteristic
110	C₄×D₁₇	C₂×D₈	1	1	yes	characteristic
111	C₄×D₁₇	C₂×D₈	1	2	yes	—
112	C₄×D₁₇	C₂×Q₁₆	1	2	yes	—
113	C₄×D₁₇	C₂×Q₁₆	1	1	yes	characteristic
114	C₄×D₁₇	C₂²×D₄	1	1	yes	characteristic
115	C₈×C₁₇	C₂×C₄⋊C₄	1	1	yes	characteristic
116	C₈×C₁₇	C₂×C₄⋊C₄	1	2	yes	—
117	C₈×C₁₇	C₂³×C₄	1	1	yes	characteristic
118	#3622	C₂×C₄⋊C₄	1	1	yes	characteristic
119	#3622	C₂×C₄⋊C₄	1	2	yes	—
120	#3622	C₂²×Q₈	1	1	yes	characteristic
121	C₂²×M₃₂	—	17	1	no	—
122	C₂×#427	—	17	2	no	—
123	#2159	—	17	8	no	—
124	#2158	—	17	4	no	—
125	C₂²×C₁₇	—	4	1	no	—
126	C₂²×C₁₇	C₂×#107	1	1	yes	characteristic
127	C₂²×C₁₇	C₂×#107	1	2	yes	—
128	C₂×D₁₇	—	2	4	no	—
129	C₂×D₁₇	—	2	2	no	—
130	C₂×D₁₇	—	4	1	no	—
131	C₂×D₁₇	—	4	2	no	—
132	C₂×D₁₇	C₂×#107	1	1	yes	characteristic
133	C₂×D₁₇	C₂×#107	1	2	yes	—
134	C₂×D₁₇	C₂²×D₈	1	1	yes	characteristic
135	#595	—	8	8	no	—
136	C₄×C₁₇	C₂×#107	1	2	yes	—
137	C₄×C₁₇	C₂×#107	1	1	yes	characteristic
138	C₄×C₁₇	C₂²×C₄⋊C₄	1	1	yes	characteristic
139	Dic₁₇	C₂×#107	1	1	yes	characteristic
140	Dic₁₇	C₂×#107	1	2	yes	—
141	Dic₁₇	C₂²×Q₁₆	1	1	yes	characteristic
142	C₂³×C₈	—	17	1	no	—
143	C₂×#117	—	34	2	no	—
144	C₂×M₃₂	—	17	1	no	—
145	C₂×M₃₂	—	17	4	no	—
146	C₂×M₃₂	—	17	4	no	—
147	C₂×M₃₂	—	17	2	no	—
148	C₂×M₃₂	#595	1	1	yes	characteristic, complemented
149	C₂²×C₁₆	—	17	2	no	—
150	#427	—	17	4	no	—
151	#427	—	17	4	no	—
152	#427	—	34	2	no	—
153	#427	—	34	2	no	—
154	C₂×#107	—	17	4	no	—
155	C₂×#106	—	34	2	no	—
156	#367	—	17	8	no	—
157	#366	—	17	8	no	—
158	C₂×C₁₇	—	2	2	no	—
159	C₂×C₁₇	—	2	1	no	—
160	C₂×C₁₇	C₂²×#107	1	1	yes	characteristic
161	D₁₇	—	2	2	no	—
162	D₁₇	—	2	1	no	—
163	D₁₇	#3414	1	2	yes	—
164	C₂³×C₄	—	17	1	no	—
165	C₂²×C₈	—	17	1	no	—
166	C₂²×C₈	—	17	2	no	—
167	C₂²×C₈	—	17	1	no	—
168	C₂²×C₈	—	17	2	no	—
169	C₂²×C₈	—	17	1	no	—
170	C₂²×C₈	—	17	2	no	—
171	C₂²×C₈	—	17	1	no	—
172	C₂²×C₈	—	17	2	no	—
173	C₂²×C₈	—	17	1	no	—
174	C₂²×C₈	C₂×#595	1	1	yes	characteristic
175	#117	—	34	4	no	—
176	#117	—	34	4	no	—
177	#117	—	68	2	no	—
178	#117	—	68	2	no	—
179	C₂×C₄⋊C₄	—	34	4	no	—
180	C₂×C₂²⋊C₄	—	68	2	no	—
181	C₂×C₄²	—	68	2	no	—
182	M₃₂	—	17	4	no	—
183	M₃₂	—	17	8	no	—
184	M₃₂	C₂×#595	1	4	yes	complemented
185	C₂×C₁₆	—	17	2	no	—
186	C₂×C₁₆	—	17	4	no	—
187	C₂×C₁₆	—	17	4	no	—
188	C₂×C₁₆	C₂×#595	1	2	yes	complemented
189	#107	—	17	8	no	—
190	#107	—	34	4	no	—
191	#106	—	34	4	no	—
192	#106	—	34	4	no	—
193	C₁₇	#33409	1	1	yes	characteristic, complemented, Sylow
194	C₂⁴	—	17	1	no	—
195	C₂²×C₄	—	17	1	no	—
196	C₂²×C₄	—	17	2	no	—
197	C₂²×C₄	—	17	1	no	—
198	C₂²×C₄	—	17	2	no	—
199	C₂²×C₄	—	17	1	no	—
200	C₂²×C₄	—	17	2	no	—
201	C₂²×C₄	—	17	1	no	—
202	C₂²×C₄	—	17	1	no	—
203	C₂²×C₄	—	17	2	no	—
204	C₂²×C₄	—	68	4	no	—
205	C₂²×C₄	C₄⋊#595	1	1	yes	characteristic
206	C₂×C₈	—	17	2	no	—
207	C₂×C₈	—	17	1	no	—
208	C₂×C₈	—	17	2	no	—
209	C₂×C₈	—	17	1	no	—
210	C₂×C₈	—	17	1	no	—
211	C₂×C₈	—	17	2	no	—
212	C₂×C₈	—	17	2	no	—
213	C₂×C₈	—	17	2	no	—
214	C₂×C₈	—	17	2	no	—
215	C₂×C₈	—	17	2	no	—
216	C₂×C₈	—	17	1	no	—
217	C₂×C₈	—	17	1	no	—
218	C₂×C₈	—	17	1	no	—
219	C₂×C₈	—	17	2	no	—
220	C₂×C₈	C₄⋊#595	1	2	yes	—
221	C₂×C₈	C₄⋊#595	1	1	yes	characteristic
222	C₂×C₈	C₂²×#595	1	1	yes	characteristic
223	C₂×C₈	C₂²×#595	1	2	yes	—
224	C₄⋊C₄	—	34	8	no	—
225	C₄⋊C₄	—	68	4	no	—
226	C₂²⋊C₄	—	68	4	no	—
227	C₂²⋊C₄	—	68	4	no	—
228	C₄²	—	68	4	no	—
229	C₄²	—	68	4	no	—
230	C₁₆	—	17	4	no	—
231	C₁₆	C₂²×#595	1	4	yes	—
232	C₂³	—	17	1	no	—
233	C₂³	—	17	2	no	—
234	C₂³	—	17	2	no	—
235	C₂³	—	17	1	no	—
236	C₂³	—	68	2	no	—
237	C₂³	#10587534	1	1	yes	characteristic
238	C₂×C₄	—	17	2	no	—
239	C₂×C₄	—	17	1	no	—
240	C₂×C₄	—	17	2	no	—
241	C₂×C₄	—	17	1	no	—
242	C₂×C₄	—	17	2	no	—
243	C₂×C₄	—	17	2	no	—
244	C₂×C₄	—	17	2	no	—
245	C₂×C₄	—	17	2	no	—
246	C₂×C₄	—	17	1	no	—
247	C₂×C₄	—	17	2	no	—
248	C₂×C₄	—	17	1	no	—
249	C₂×C₄	—	17	1	no	—
250	C₂×C₄	—	17	1	no	—
251	C₂×C₄	—	17	2	no	—
252	C₂×C₄	—	68	8	no	—
253	C₂×C₄	—	136	8	no	—
254	C₂×C₄	#10587534	1	1	yes	characteristic
255	C₂×C₄	#10587534	1	2	yes	—
256	C₂×C₄	C₂×C₄⋊#595	1	2	yes	—
257	C₂×C₄	C₂×C₄⋊#595	1	1	yes	characteristic
258	C₈	—	17	1	no	—
259	C₈	—	17	2	no	—
260	C₈	—	17	1	no	—
261	C₈	C₂×C₄⋊#595	1	1	yes	characteristic
262	C₈	C₂×C₄⋊#595	1	2	yes	—
263	C₈	C₂³×#595	1	1	yes	characteristic
264	C₂²	—	4	1	no	—
265	C₂²	—	17	1	no	—
266	C₂²	—	17	1	no	—
267	C₂²	—	17	2	no	—
268	C₂²	—	34	4	no	—
269	C₂²	—	34	2	no	—
270	C₂²	—	68	2	no	—
271	C₂²	—	68	1	no	—
272	C₂²	C₂×#10587534	1	2	yes	—
273	C₂²	C₂×#10587534	1	1	yes	characteristic
274	C₄	—	17	1	no	—
275	C₄	—	17	2	no	—
276	C₄	—	17	1	no	—
277	C₄	—	136	8	no	—
278	C₄	C₂×#10587534	1	2	yes	—
279	C₄	C₂×#10587534	1	1	yes	characteristic
280	C₄	C₂²×C₄⋊#595	1	1	yes	characteristic
281	C₂	—	2	2	no	—
282	C₂	—	2	1	no	—
283	C₂	—	17	2	no	—
284	C₂	—	34	2	no	—
285	C₂	—	34	1	no	—
286	C₂	C₂²×#10587534	1	1	yes	characteristic

	subgroup	G/H
center	C₄	C₂×#10587534
commutator	C₈×C₁₇	C₂³×C₄
Frattini	C₈	C₂³×#595
Fitting	C₂×C₁₇×M₃₂	C₄
socle	C₂×C₁₇	C₂²×#107
prime radical	C₂³×D₁₇	D₈
cosocle	C₈×D₁₇	C₂⁴
hypercenter	C₂×M₃₂	#595
hypocenter	C₁₇	#33409
LCS	G ▷ C₈×C₁₇ ▷ C₄×C₁₇ ▷ C₂×C₁₇ ▷ C₁₇
UCS	C₁ ◁ C₄ ◁ C₂²×C₄ ◁ C₂²×C₈ ◁ C₂×M₃₂

Complements

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

N	H ≅ G/N	type
2. #14545136	C₂	semidirect
3. #14545135	C₂	semidirect
12. #11763736	C₂²	semidirect
13. #11763735	C₂²	semidirect
15. C₂×#10587483	C₄	semidirect
16. C₂×#10587469	C₄	semidirect
17. M₃₂×D₁₇	C₄	semidirect
20. C₂×C₁₇×M₃₂	C₄	semidirect
43. C₁₇×M₃₂	C₂×C₄	semidirect
44. C₂×C₁₆×C₁₇	C₂×C₄	semidirect
45. #10587483	C₂×C₄	semidirect
46. C₂×#63214	C₂×C₄	semidirect
47. #10587469	C₂×C₄	semidirect
48. C₁₆×D₁₇	C₂×C₄	semidirect
148. C₂×M₃₂	#595	semidirect
184. M₃₂	C₂×#595	semidirect
188. C₂×C₁₆	C₂×#595	semidirect
193. C₁₇	#33409	semidirect

Maximal subgroups

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

subgroup	index
C₂×#11763819	2
#14545136	2
#14545135	2
C₂×M₃₂×D₁₇	2
#33409	17