Subgroups — #16938584

Order 4352.

454 conjugacy classes of subgroups.

The subgroup lattice has 270 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#14529708	C₂	1	2	yes	maximal
2	#14529502	C₂	1	1	yes	maximal, characteristic
3	C₂×#11762555	C₂	1	2	yes	maximal
4	#14526619	C₂	1	1	yes	maximal, characteristic
5	C₂×C₁₇×#500	C₂	1	1	yes	maximal, characteristic
6	C₂×C₄⋊#3622	C₂²	1	2	yes	—
7	C₂×C₄⋊#3622	C₂²	1	1	yes	characteristic
8	C₂×#10587485	—	2	1	no	—
9	C₂×#10587506	—	2	2	no	—
10	C₂×#10587506	—	2	2	no	—
11	C₄×C₄⋊Dic₁₇	C₂²	1	1	yes	characteristic
12	#11762563	—	2	1	no	—
13	#11762555	—	2	4	no	—
14	C₂×C₁₇×C₄⋊Q₈	C₂²	1	2	yes	—
15	C₁₇×#500	—	2	4	no	—
16	C₂×C₁₇×#107	—	2	1	no	—
17	C₂×C₁₇×#107	—	2	1	no	—
18	C₂×C₄×C₈×C₁₇	C₂²	1	1	yes	characteristic
19	C₂²×Q₈×C₁₇	—	2	2	no	—
20	C₂²×C₈×C₁₇	D₄	1	1	yes	characteristic
21	C₂²×C₈×C₁₇	D₄	1	1	yes	characteristic
22	C₁₇×C₄⋊Q₈	—	2	2	no	—
23	C₁₇×C₄⋊Q₈	—	2	2	no	—
24	C₁₇×C₄⋊Q₈	—	2	2	no	—
25	C₁₇×C₄⋊Q₈	D₄	1	4	yes	—
26	C₂×C₄⋊C₄×C₁₇	—	2	2	no	—
27	C₂×C₄⋊C₄×C₁₇	—	2	2	no	—
28	C₂×C₄²×C₁₇	C₂³	1	1	yes	characteristic
29	C₂×C₄⋊Dic₁₇	D₄	1	1	yes	characteristic
30	C₂×C₄⋊Dic₁₇	D₄	1	1	yes	characteristic
31	C₂×C₄×Dic₁₇	—	2	1	no	—
32	C₂²×#3622	—	2	2	no	—
33	C₂²×#3622	—	2	1	no	—
34	C₁₇×#107	—	2	4	no	—
35	C₁₇×#107	—	2	4	no	—
36	C₄×C₈×C₁₇	—	2	2	no	—
37	#10587506	—	4	4	no	—
38	#10587506	—	4	4	no	—
39	#10587502	—	2	1	no	—
40	#10587485	—	4	2	no	—
41	C₄⋊#3622	—	2	2	no	—
42	C₄⋊#3622	—	2	2	no	—
43	C₄⋊#3622	—	2	2	no	—
44	C₂×Q₈×C₁₇	—	2	4	no	—
45	C₂×Q₈×C₁₇	—	4	2	no	—
46	C₂×Q₈×C₁₇	—	4	2	no	—
47	C₂×Q₈×C₁₇	—	4	2	no	—
48	C₂×Q₈×C₁₇	—	4	2	no	—
49	C₂×Q₈×C₁₇	—	4	2	no	—
50	C₂²×C₄×C₁₇	—	4	2	no	—
51	C₂²×C₄×C₁₇	C₂×D₄	1	1	yes	characteristic
52	C₂²×C₄×C₁₇	C₂×D₄	1	1	yes	characteristic
53	C₂²×C₄×C₁₇	#41	1	1	yes	characteristic
54	C₂²×Dic₁₇	—	2	1	no	—
55	C₂²×Dic₁₇	—	2	1	no	—
56	C₂×C₈×C₁₇	—	2	1	no	—
57	C₂×C₈×C₁₇	—	2	1	no	—
58	C₂×C₈×C₁₇	—	2	1	no	—
59	C₂×C₈×C₁₇	—	2	1	no	—
60	C₂×C₈×C₁₇	SD₁₆	1	2	yes	—
61	C₂×C₈×C₁₇	Q₁₆	1	2	yes	—
62	C₄⋊C₄×C₁₇	—	2	4	no	—
63	C₄⋊C₄×C₁₇	—	2	4	no	—
64	C₄⋊C₄×C₁₇	—	4	2	no	—
65	C₄⋊C₄×C₁₇	—	4	2	no	—
66	C₄²×C₁₇	C₂×D₄	1	2	yes	—
67	C₄²×C₁₇	#41	1	2	yes	—
68	C₄⋊Dic₁₇	—	2	1	no	—
69	C₄⋊Dic₁₇	—	2	2	no	—
70	C₄⋊Dic₁₇	D₈	1	2	yes	—
71	C₄×Dic₁₇	—	4	1	no	—
72	C₄×Dic₁₇	—	4	1	no	—
73	C₂×#3622	—	4	1	no	—
74	C₂×#3622	—	4	2	no	—
75	C₂×#3622	—	4	2	no	—
76	C₂×#3622	—	4	1	no	—
77	C₂×#3622	—	4	2	no	—
78	C₂×#3622	—	4	1	no	—
79	#10545	—	17	1	no	maximal, Sylow
80	C₂³×C₁₇	#121	1	1	yes	characteristic
81	Q₈×C₁₇	—	4	4	no	—
82	Q₈×C₁₇	—	4	4	no	—
83	Q₈×C₁₇	—	8	2	no	—
84	Q₈×C₁₇	—	8	2	no	—
85	C₂×C₄×C₁₇	—	2	1	no	—
86	C₂×C₄×C₁₇	—	2	1	no	—
87	C₂×C₄×C₁₇	—	4	4	no	—
88	C₂×C₄×C₁₇	—	8	2	no	—
89	C₂×C₄×C₁₇	—	8	2	no	—
90	C₂×C₄×C₁₇	#120	1	1	yes	characteristic
91	C₂×C₄×C₁₇	#121	1	2	yes	—
92	C₂×C₄×C₁₇	#123	1	2	yes	—
93	C₂×C₄×C₁₇	#124	1	1	yes	characteristic
94	C₂×C₄×C₁₇	C₂×D₈	1	1	yes	characteristic
95	C₂×C₄×C₁₇	C₂×SD₁₆	1	1	yes	characteristic
96	C₂×C₄×C₁₇	C₂×Q₁₆	1	1	yes	characteristic
97	C₂×C₄×C₁₇	#135	1	1	yes	characteristic
98	C₂×C₄×C₁₇	AGL(1,ℤ/8)	1	2	yes	—
99	C₂×C₄×C₁₇	#137	1	1	yes	characteristic
100	C₂×C₄×C₁₇	#137	1	1	yes	characteristic
101	C₂×Dic₁₇	—	2	2	no	—
102	C₂×Dic₁₇	—	4	1	no	—
103	C₂×Dic₁₇	—	4	1	no	—
104	C₂×Dic₁₇	—	4	1	no	—
105	C₂×Dic₁₇	—	4	1	no	—
106	C₂×Dic₁₇	—	4	1	no	—
107	C₈×C₁₇	—	2	2	no	—
108	C₈×C₁₇	—	2	2	no	—
109	#3622	—	8	1	no	—
110	#3622	—	8	2	no	—
111	C₂×#500	D₁₇	1	1	yes	characteristic
112	#1981	—	17	2	no	—
113	#1939	—	17	1	no	—
114	C₂×#332	—	17	2	no	—
115	#1277	—	17	1	no	—
116	C₂²×C₁₇	#394	1	1	yes	characteristic
117	C₂²×C₁₇	#461	1	1	yes	characteristic
118	C₂²×C₁₇	#462	1	1	yes	characteristic
119	C₂²×C₁₇	#466	1	1	yes	characteristic
120	C₂²×C₁₇	#469	1	1	yes	characteristic
121	C₂²×C₁₇	#473	1	1	yes	characteristic
122	C₂²×C₁₇	#473	1	1	yes	characteristic
123	C₄×C₁₇	—	2	2	no	—
124	C₄×C₁₇	—	8	4	no	—
125	C₄×C₁₇	#448	1	1	yes	characteristic
126	C₄×C₁₇	#451	1	1	yes	characteristic
127	C₄×C₁₇	#459	1	2	yes	—
128	C₄×C₁₇	#463	1	2	yes	—
129	C₄×C₁₇	#487	1	1	yes	characteristic
130	C₄×C₁₇	#491	1	1	yes	characteristic
131	Dic₁₇	—	4	2	no	—
132	Dic₁₇	—	8	1	no	—
133	C₂×C₄⋊Q₈	C₂×D₁₇	1	2	yes	—
134	#500	—	2	4	no	—
135	C₂×#107	—	2	1	no	—
136	C₂×#107	—	2	1	no	—
137	C₂×C₄⋊C₈	—	17	1	no	—
138	C₂×C₄⋊C₈	—	17	2	no	—
139	C₂×C₄⋊C₈	—	34	1	no	—
140	C₂×#103	—	34	2	no	—
141	C₂×#103	—	34	2	no	—
142	C₂×C₄×C₈	C₂×D₁₇	1	1	yes	characteristic
143	C₄×C₄⋊C₄	—	17	1	no	—
144	#336	—	34	1	no	—
145	#332	—	34	4	no	—
146	C₂×C₁₇	#1673	1	1	yes	characteristic
147	C₂×C₁₇	#1674	1	1	yes	characteristic
148	C₂×C₁₇	#1693	1	1	yes	characteristic
149	C₂×C₁₇	#1694	1	1	yes	characteristic
150	C₂×C₁₇	#2043	1	1	yes	characteristic
151	C₂×C₁₇	#2047	1	1	yes	characteristic
152	C₂×C₁₇	#2054	1	1	yes	characteristic
153	C₂²×Q₈	—	2	2	no	—
154	C₂²×C₈	—	34	2	no	—
155	C₂²×C₈	—	34	1	no	—
156	C₂²×C₈	C₂²⋊D₁₇	1	1	yes	characteristic
157	C₂²×C₈	C₂²⋊D₁₇	1	1	yes	characteristic
158	C₄⋊Q₈	—	2	2	no	—
159	C₄⋊Q₈	—	2	2	no	—
160	C₄⋊Q₈	—	2	2	no	—
161	C₄⋊Q₈	C₂²⋊D₁₇	1	4	yes	—
162	C₂×C₄⋊C₄	—	2	2	no	—
163	C₂×C₄⋊C₄	—	2	2	no	—
164	C₂×C₄⋊C₄	—	17	1	no	—
165	C₂×C₄⋊C₄	—	17	1	no	—
166	C₂×C₄²	—	34	1	no	—
167	C₂×C₄²	C₂²×D₁₇	1	1	yes	characteristic
168	#107	—	2	4	no	—
169	#107	—	2	4	no	—
170	C₄⋊C₈	—	34	2	no	—
171	C₄⋊C₈	—	34	2	no	—
172	C₄⋊C₈	—	34	2	no	—
173	C₄⋊C₈	—	68	2	no	—
174	#103	—	68	4	no	—
175	#103	—	68	4	no	—
176	C₄×C₈	—	2	2	no	—
177	#95	—	34	1	no	—
178	C₁₇	#10545	1	1	yes	characteristic, complemented, Sylow
179	C₂×Q₈	—	2	4	no	—
180	C₂×Q₈	—	4	2	no	—
181	C₂×Q₈	—	4	2	no	—
182	C₂×Q₈	—	4	2	no	—
183	C₂×Q₈	—	4	2	no	—
184	C₂×Q₈	—	4	2	no	—
185	C₂²×C₄	—	4	2	no	—
186	C₂²×C₄	—	34	1	no	—
187	C₂²×C₄	—	34	1	no	—
188	C₂²×C₄	D₄×D₁₇	1	1	yes	characteristic
189	C₂²×C₄	#63254	1	1	yes	characteristic
190	C₂²×C₄	C₂×C₂²⋊D₁₇	1	1	yes	characteristic
191	C₂×C₈	—	2	1	no	—
192	C₂×C₈	—	2	1	no	—
193	C₂×C₈	—	2	1	no	—
194	C₂×C₈	—	2	1	no	—
195	C₂×C₈	—	68	2	no	—
196	C₂×C₈	—	68	2	no	—
197	C₂×C₈	—	68	2	no	—
198	C₂×C₈	—	68	1	no	—
199	C₂×C₈	—	68	1	no	—
200	C₂×C₈	—	68	1	no	—
201	C₂×C₈	#63230	1	2	yes	—
202	C₂×C₈	#63231	1	2	yes	—
203	C₄⋊C₄	—	2	4	no	—
204	C₄⋊C₄	—	2	4	no	—
205	C₄⋊C₄	—	4	2	no	—
206	C₄⋊C₄	—	4	2	no	—
207	C₄⋊C₄	—	17	2	no	—
208	C₄⋊C₄	—	34	1	no	—
209	C₄⋊C₄	—	34	2	no	—
210	C₄²	—	68	1	no	—
211	C₄²	—	68	1	no	—
212	C₄²	#63254	1	2	yes	—
213	C₄²	C₂×C₂²⋊D₁₇	1	2	yes	—
214	C₂³	#10587624	1	1	yes	characteristic
215	Q₈	—	4	4	no	—
216	Q₈	—	4	4	no	—
217	Q₈	—	8	2	no	—
218	Q₈	—	8	2	no	—
219	C₂×C₄	—	2	1	no	—
220	C₂×C₄	—	2	1	no	—
221	C₂×C₄	—	4	4	no	—
222	C₂×C₄	—	8	2	no	—
223	C₂×C₄	—	8	2	no	—
224	C₂×C₄	—	34	2	no	—
225	C₂×C₄	—	68	1	no	—
226	C₂×C₄	—	68	1	no	—
227	C₂×C₄	—	68	1	no	—
228	C₂×C₄	—	68	1	no	—
229	C₂×C₄	—	68	1	no	—
230	C₂×C₄	D₈×D₁₇	1	1	yes	characteristic
231	C₂×C₄	#10587597	1	2	yes	—
232	C₂×C₄	#10587598	1	1	yes	characteristic
233	C₂×C₄	#10587621	1	2	yes	—
234	C₂×C₄	#10587622	1	1	yes	characteristic
235	C₂×C₄	#10587625	1	2	yes	—
236	C₂×C₄	C₂×#63230	1	1	yes	characteristic
237	C₂×C₄	#10587628	1	1	yes	characteristic
238	C₂×C₄	#10587628	1	1	yes	characteristic
239	C₂×C₄	C₂×#63231	1	1	yes	characteristic
240	C₂×C₄	#10587639	1	1	yes	characteristic
241	C₈	—	2	2	no	—
242	C₈	—	2	2	no	—
243	C₈	—	136	1	no	—
244	C₈	—	136	2	no	—
245	C₂²	#11762723	1	1	yes	characteristic
246	C₂²	#11763011	1	1	yes	characteristic
247	C₂²	#11763016	1	1	yes	characteristic
248	C₂²	#11763042	1	1	yes	characteristic
249	C₂²	#11763059	1	1	yes	characteristic
250	C₂²	#11763081	1	1	yes	characteristic
251	C₂²	#11763081	1	1	yes	characteristic
252	C₄	—	2	2	no	—
253	C₄	—	8	4	no	—
254	C₄	—	68	2	no	—
255	C₄	—	136	1	no	—
256	C₄	#11762948	1	1	yes	characteristic
257	C₄	#11762965	1	1	yes	characteristic
258	C₄	#11763002	1	2	yes	—
259	C₄	#11763022	1	2	yes	—
260	C₄	#11763159	1	1	yes	characteristic
261	C₄	#11763177	1	1	yes	characteristic
262	C₂	#14528170	1	1	yes	characteristic
263	C₂	#14528175	1	1	yes	characteristic
264	C₂	#14528294	1	1	yes	characteristic
265	C₂	#14528299	1	1	yes	characteristic
266	C₂	#14530027	1	1	yes	characteristic
267	C₂	#14530049	1	1	yes	characteristic
268	C₂	#14530092	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	#10587624
commutator, cosocle	C₂×C₄²×C₁₇	C₂³
Frattini	C₂×C₄²	C₂²×D₁₇
Fitting	C₂×C₁₇×#500	C₂
socle, prime radical	C₂³×C₁₇	#121
hypercenter	C₂×#500	D₁₇
hypocenter	C₁₇	#10545
LCS	G ▷ C₂×C₄²×C₁₇ ▷ C₂²×C₁₇ ▷ C₁₇
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×#500

Complements

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

N	H ≅ G/N	type
178. C₁₇	#10545	semidirect

Maximal subgroups

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

subgroup	index
#14529708	2
#14529502	2
C₂×#11762555	2
#14526619	2
C₂×C₁₇×#500	2
#10545	17