Subgroups — #16933703

Order 4352.

502 conjugacy classes of subgroups.

The subgroup lattice has 266 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#14528948	C₂	1	1	yes	maximal, characteristic, complemented
2	#14527344	C₂	1	4	yes	maximal, complemented
3	#14527067	C₂	1	1	yes	maximal, characteristic, complemented
4	C₁₇×#3147	C₂	1	1	yes	maximal, characteristic
5	#11762862	C₂²	1	2	yes	—
6	C₂×#10587489	—	2	1	no	—
7	C₂×#10587488	—	2	1	no	—
8	#11762675	C₂²	1	2	yes	—
9	#11762543	—	2	8	no	—
10	#11762530	—	2	1	no	—
11	C₈×#3622	—	2	1	no	—
12	C₂×C₁₇×#135	—	2	1	no	—
13	C₂×C₁₇×#135	—	2	1	no	—
14	C₁₇×#532	C₂²	1	2	yes	—
15	C₁₇×#495	—	2	1	no	—
16	C₁₇×#495	—	2	1	no	—
17	C₁₇×#494	C₄	1	1	yes	characteristic, complemented
18	C₁₇×#493	C₄	1	1	yes	characteristic, complemented
19	C₁₇×#492	C₄	1	2	yes	complemented
20	C₂×C₄×C₈×C₁₇	C₂²	1	1	yes	characteristic
21	C₂×#41×C₁₇	—	2	2	no	—
22	C₁₇×#135	—	4	2	no	—
23	C₁₇×#135	—	4	2	no	—
24	C₁₇×#135	—	4	2	no	—
25	C₁₇×#135	—	4	2	no	—
26	C₂×Q₁₆×C₁₇	—	2	1	no	—
27	C₂×Q₁₆×C₁₇	—	2	1	no	—
28	C₂×SD₁₆×C₁₇	—	2	2	no	—
29	C₂×SD₁₆×C₁₇	—	2	2	no	—
30	C₂×D₈×C₁₇	—	2	1	no	—
31	C₂×D₈×C₁₇	—	2	1	no	—
32	C₂²×C₈×C₁₇	D₄	1	1	yes	characteristic
33	C₂²×C₈×C₁₇	D₄	1	1	yes	characteristic
34	C₁₇×C₄⋊Q₈	C₂×C₄	1	2	yes	—
35	C₁₇×#127	C₂×C₄	1	2	yes	—
36	C₁₇×#124	—	2	2	no	—
37	C₁₇×#121	—	2	4	no	—
38	C₄×D₄×C₁₇	—	2	4	no	—
39	C₂×C₄²×C₁₇	C₂³	1	1	yes	characteristic
40	C₂×C₄⋊Dic₁₇	—	2	2	no	—
41	C₄×C₈×C₁₇	C₂×C₄	1	1	yes	characteristic
42	C₄×C₈×C₁₇	C₂×C₄	1	1	yes	characteristic
43	C₄×C₈×C₁₇	D₄	1	1	yes	characteristic
44	C₄×C₈×C₁₇	D₄	1	1	yes	characteristic
45	#10587502	—	2	4	no	—
46	C₂²⋊#3622	D₄	1	4	yes	—
47	#10587489	—	4	2	no	—
48	#10587488	—	4	2	no	—
49	C₄⋊#3622	—	2	2	no	—
50	C₄×#3622	—	2	2	no	—
51	#41×C₁₇	—	4	2	no	—
52	#41×C₁₇	—	4	2	no	—
53	#41×C₁₇	—	4	2	no	—
54	#41×C₁₇	—	4	2	no	—
55	C₂×Q₈×C₁₇	—	2	2	no	—
56	C₂×D₄×C₁₇	—	2	2	no	—
57	C₂×D₄×C₁₇	—	2	4	no	—
58	C₂×D₄×C₁₇	—	4	2	no	—
59	C₂²×C₄×C₁₇	—	4	2	no	—
60	C₂²×C₄×C₁₇	C₂×D₄	1	2	yes	—
61	C₂²×C₄×C₁₇	C₂×D₄	1	1	yes	characteristic
62	C₂²×Dic₁₇	—	4	2	no	—
63	Q₁₆×C₁₇	—	4	2	no	—
64	Q₁₆×C₁₇	—	4	2	no	—
65	SD₁₆×C₁₇	—	4	4	no	—
66	SD₁₆×C₁₇	—	4	4	no	—
67	D₈×C₁₇	—	4	2	no	—
68	D₈×C₁₇	—	4	2	no	—
69	C₂×C₈×C₁₇	C₂²⋊C₄	1	1	yes	characteristic
70	C₂×C₈×C₁₇	C₂²⋊C₄	1	1	yes	characteristic
71	C₂×C₈×C₁₇	C₂²⋊C₄	1	1	yes	characteristic
72	C₂×C₈×C₁₇	C₂²⋊C₄	1	1	yes	characteristic
73	C₂×C₈×C₁₇	D₈	1	2	yes	—
74	C₂×C₈×C₁₇	D₈	1	2	yes	—
75	C₂×C₈×C₁₇	SD₁₆	1	2	yes	—
76	C₂×C₈×C₁₇	SD₁₆	1	2	yes	—
77	C₄⋊C₄×C₁₇	—	2	4	no	—
78	C₂²⋊C₄×C₁₇	—	4	4	no	—
79	C₄²×C₁₇	C₂²×C₄	1	1	yes	characteristic
80	C₄²×C₁₇	C₂×D₄	1	1	yes	characteristic
81	C₄²×C₁₇	#41	1	2	yes	—
82	C₄⋊Dic₁₇	—	4	2	no	—
83	C₄⋊Dic₁₇	—	4	2	no	—
84	C₂×#3622	—	2	4	no	—
85	#9756	—	17	1	no	maximal, Sylow
86	C₂³×C₁₇	—	4	2	no	—
87	C₂³×C₁₇	#127	1	1	yes	characteristic
88	Q₈×C₁₇	—	4	2	no	—
89	Q₈×C₁₇	—	4	2	no	—
90	D₄×C₁₇	—	4	2	no	—
91	D₄×C₁₇	—	4	2	no	—
92	D₄×C₁₇	—	8	4	no	—
93	D₄×C₁₇	—	8	2	no	—
94	D₄×C₁₇	—	8	2	no	—
95	C₂×C₄×C₁₇	—	4	2	no	—
96	C₂×C₄×C₁₇	—	4	2	no	—
97	C₂×C₄×C₁₇	—	8	2	no	—
98	C₂×C₄×C₁₇	—	8	2	no	—
99	C₂×C₄×C₁₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
100	C₂×C₄×C₁₇	C₄×D₄	1	2	yes	—
101	C₂×C₄×C₁₇	#121	1	2	yes	—
102	C₂×C₄×C₁₇	#124	1	1	yes	characteristic
103	C₂×C₄×C₁₇	C₂×D₈	1	1	yes	characteristic
104	C₂×C₄×C₁₇	C₂×D₈	1	1	yes	characteristic
105	C₂×C₄×C₁₇	C₂×SD₁₆	1	1	yes	characteristic
106	C₂×C₄×C₁₇	C₂×SD₁₆	1	1	yes	characteristic
107	C₂×Dic₁₇	—	8	4	no	—
108	C₂×Dic₁₇	—	8	2	no	—
109	C₈×C₁₇	#102	1	2	yes	—
110	C₈×C₁₇	#102	1	2	yes	—
111	C₈×C₁₇	#102	1	2	yes	—
112	C₈×C₁₇	#102	1	2	yes	—
113	#3622	—	4	4	no	—
114	#3147	D₁₇	1	1	yes	characteristic
115	#1846	—	17	1	no	—
116	#1517	—	17	4	no	—
117	#1450	—	17	1	no	—
118	C₂²×C₁₇	—	2	2	no	—
119	C₂²×C₁₇	—	8	2	no	—
120	C₂²×C₁₇	—	8	4	no	—
121	C₂²×C₁₇	#390	1	1	yes	characteristic
122	C₂²×C₁₇	#492	1	1	yes	characteristic
123	C₂²×C₁₇	#493	1	1	yes	characteristic
124	C₄×C₁₇	—	2	2	no	—
125	C₄×C₁₇	—	2	2	no	—
126	C₄×C₁₇	—	8	2	no	—
127	C₄×C₁₇	C₂×#102	1	1	yes	characteristic
128	C₄×C₁₇	C₂×#102	1	1	yes	characteristic
129	C₄×C₁₇	#486	1	1	yes	characteristic
130	C₄×C₁₇	#486	1	1	yes	characteristic
131	Dic₁₇	—	16	2	no	—
132	C₂×#135	—	2	1	no	—
133	C₂×#135	—	2	1	no	—
134	#532	C₂×D₁₇	1	2	yes	—
135	#495	—	2	1	no	—
136	#495	—	2	1	no	—
137	#494	Dic₁₇	1	1	yes	characteristic, complemented
138	#493	Dic₁₇	1	1	yes	characteristic, complemented
139	#492	Dic₁₇	1	2	yes	complemented
140	#431	—	17	2	no	—
141	C₂×#107	—	34	1	no	—
142	C₂×#106	—	34	1	no	—
143	C₂×C₄×C₈	C₂×D₁₇	1	1	yes	characteristic
144	#382	—	17	2	no	—
145	#327	—	34	8	no	—
146	#322	—	34	1	no	—
147	C₈²	—	34	1	no	—
148	C₂×C₁₇	—	2	1	no	—
149	C₂×C₁₇	—	2	1	no	—
150	C₂×C₁₇	—	16	2	no	—
151	C₂×C₁₇	#1957	1	1	yes	characteristic
152	C₂×C₁₇	#1976	1	2	yes	—
153	C₂×#41	—	2	2	no	—
154	#135	—	4	2	no	—
155	#135	—	4	2	no	—
156	#135	—	4	2	no	—
157	#135	—	4	2	no	—
158	C₂×Q₁₆	—	2	1	no	—
159	C₂×Q₁₆	—	2	1	no	—
160	C₂×SD₁₆	—	2	2	no	—
161	C₂×SD₁₆	—	2	2	no	—
162	C₂×D₈	—	2	1	no	—
163	C₂×D₈	—	2	1	no	—
164	C₂²×C₈	C₂²⋊D₁₇	1	1	yes	characteristic
165	C₂²×C₈	C₂²⋊D₁₇	1	1	yes	characteristic
166	C₄⋊Q₈	C₂×Dic₁₇	1	2	yes	—
167	#127	C₂×Dic₁₇	1	2	yes	—
168	#124	—	2	2	no	—
169	#121	—	2	4	no	—
170	C₄×D₄	—	2	4	no	—
171	C₂×C₄⋊C₄	—	34	2	no	—
172	C₂×C₄²	C₂²×D₁₇	1	1	yes	characteristic
173	#107	—	68	2	no	—
174	#106	—	68	2	no	—
175	C₄⋊C₈	—	34	2	no	—
176	C₂²⋊C₈	—	17	4	no	—
177	C₄×C₈	—	34	2	no	—
178	C₄×C₈	C₂×Dic₁₇	1	1	yes	characteristic
179	C₄×C₈	C₂×Dic₁₇	1	1	yes	characteristic
180	C₄×C₈	C₂²⋊D₁₇	1	1	yes	characteristic
181	C₄×C₈	C₂²⋊D₁₇	1	1	yes	characteristic
182	#95	—	34	4	no	—
183	C₁₇	#9756	1	1	yes	characteristic, complemented, Sylow
184	#41	—	4	2	no	—
185	#41	—	4	2	no	—
186	#41	—	4	2	no	—
187	#41	—	4	2	no	—
188	C₂×Q₈	—	2	2	no	—
189	C₂×D₄	—	2	4	no	—
190	C₂×D₄	—	2	2	no	—
191	C₂×D₄	—	4	2	no	—
192	C₂²×C₄	—	4	2	no	—
193	C₂²×C₄	—	68	2	no	—
194	C₂²×C₄	D₄×D₁₇	1	2	yes	—
195	C₂²×C₄	C₂×C₂²⋊D₁₇	1	1	yes	characteristic
196	Q₁₆	—	4	2	no	—
197	Q₁₆	—	4	2	no	—
198	SD₁₆	—	4	4	no	—
199	SD₁₆	—	4	4	no	—
200	D₈	—	4	2	no	—
201	D₈	—	4	2	no	—
202	C₂×C₈	—	34	4	no	—
203	C₂×C₈	#63228	1	2	yes	—
204	C₂×C₈	#63228	1	2	yes	—
205	C₂×C₈	#63229	1	2	yes	—
206	C₂×C₈	#63229	1	2	yes	—
207	C₂×C₈	C₂²⋊Dic₁₇	1	1	yes	characteristic
208	C₂×C₈	C₂²⋊Dic₁₇	1	1	yes	characteristic
209	C₂×C₈	C₂²⋊Dic₁₇	1	1	yes	characteristic
210	C₂×C₈	C₂²⋊Dic₁₇	1	1	yes	characteristic
211	C₄⋊C₄	—	2	4	no	—
212	C₄⋊C₄	—	68	2	no	—
213	C₄⋊C₄	—	68	2	no	—
214	C₂²⋊C₄	—	4	4	no	—
215	C₄²	#63254	1	2	yes	—
216	C₄²	C₂²×Dic₁₇	1	1	yes	characteristic
217	C₄²	C₂×C₂²⋊D₁₇	1	1	yes	characteristic
218	C₂³	—	4	2	no	—
219	C₂³	#10587626	1	1	yes	characteristic
220	Q₈	—	4	2	no	—
221	Q₈	—	4	2	no	—
222	D₄	—	4	2	no	—
223	D₄	—	4	2	no	—
224	D₄	—	8	4	no	—
225	D₄	—	8	2	no	—
226	D₄	—	8	2	no	—
227	C₂×C₄	—	4	2	no	—
228	C₂×C₄	—	4	2	no	—
229	C₂×C₄	—	8	2	no	—
230	C₂×C₄	—	8	2	no	—
231	C₂×C₄	—	136	4	no	—
232	C₂×C₄	—	136	2	no	—
233	C₂×C₄	C₂×#63228	1	1	yes	characteristic
234	C₂×C₄	C₂×#63228	1	1	yes	characteristic
235	C₂×C₄	C₂×#63229	1	1	yes	characteristic
236	C₂×C₄	C₂×#63229	1	1	yes	characteristic
237	C₂×C₄	D₄×Dic₁₇	1	2	yes	—
238	C₂×C₄	#10587622	1	1	yes	characteristic
239	C₂×C₄	#10587624	1	2	yes	—
240	C₂×C₄	C₂×C₂²⋊Dic₁₇	1	1	yes	characteristic
241	C₈	—	68	4	no	—
242	C₈	#10587503	1	2	yes	—
243	C₈	#10587503	1	2	yes	—
244	C₈	#10587503	1	2	yes	—
245	C₈	#10587503	1	2	yes	—
246	C₂²	—	2	2	no	—
247	C₂²	—	8	2	no	—
248	C₂²	—	8	4	no	—
249	C₂²	#11762712	1	1	yes	characteristic
250	C₂²	#11763181	1	1	yes	characteristic
251	C₂²	#11763185	1	1	yes	characteristic
252	C₄	—	2	2	no	—
253	C₄	—	2	2	no	—
254	C₄	—	8	2	no	—
255	C₄	—	272	2	no	—
256	C₄	C₂×#10587503	1	1	yes	characteristic
257	C₄	C₂×#10587503	1	1	yes	characteristic
258	C₄	#11763155	1	1	yes	characteristic
259	C₄	#11763155	1	1	yes	characteristic
260	C₂	—	2	1	no	—
261	C₂	—	2	1	no	—
262	C₂	—	16	2	no	—
263	C₂	#14529591	1	1	yes	characteristic
264	C₂	#14529683	1	2	yes	—

	subgroup	G/H
center	C₂²	#11762712
commutator	C₄²×C₁₇	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₁₇
Fitting, prime radical	C₁₇×#3147	C₂
socle	C₂²×C₁₇	#390
cosocle	C₂×C₄²×C₁₇	C₂³
hypercenter	#3147	D₁₇
hypocenter	C₁₇	#9756
LCS	G ▷ C₄²×C₁₇ ▷ C₂²×C₁₇ ▷ C₁₇
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #3147

Complements

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

N	H ≅ G/N	type
1. #14528948	C₂	semidirect
2. #14527344	C₂	semidirect
3. #14527067	C₂	semidirect
17. C₁₇×#494	C₄	semidirect
18. C₁₇×#493	C₄	semidirect
19. C₁₇×#492	C₄	semidirect
137. #494	Dic₁₇	semidirect
138. #493	Dic₁₇	semidirect
139. #492	Dic₁₇	semidirect
183. C₁₇	#9756	semidirect

Maximal subgroups

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

subgroup	index
#14528948	2
#14527344	2
#14527067	2
C₁₇×#3147	2
#9756	17