Subgroups — #16933682

Order 4352.

394 conjugacy classes of subgroups.

The subgroup lattice has 272 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#14539298	C₂	1	1	yes	maximal, characteristic
2	#14527532	C₂	1	1	yes	maximal, characteristic
3	#14527282	C₂	1	2	yes	maximal
4	#14527186	C₂	1	2	yes	maximal
5	C₁₇×#1770	C₂	1	1	yes	maximal, characteristic
6	#11763366	C₂²	1	2	yes	—
7	#11763214	C₄	1	1	yes	characteristic, complemented
8	#11763214	C₄	1	1	yes	characteristic, complemented
9	#11763211	—	2	1	no	—
10	#11763211	—	2	1	no	—
11	#11763207	C₄	1	2	yes	complemented
12	#11762862	C₂²	1	2	yes	—
13	#11762847	—	2	1	no	—
14	#11762847	—	2	1	no	—
15	#11762562	—	2	1	no	—
16	#11762560	—	2	1	no	—
17	#11762545	—	2	4	no	—
18	#11762539	—	2	4	no	—
19	C₂×C₁₇×C₄⋊C₈	C₂²	1	1	yes	characteristic
20	C₂×C₁₇×#97	—	2	1	no	—
21	C₂×C₄×C₈×C₁₇	C₂²	1	1	yes	characteristic
22	C₄×C₄⋊C₄×C₁₇	C₂²	1	1	yes	characteristic
23	C₁₇×#336	—	2	1	no	—
24	C₂²×C₈×C₁₇	—	2	1	no	—
25	C₂²×C₈×C₁₇	D₄	1	1	yes	characteristic
26	C₂²×C₈×C₁₇	D₄	1	1	yes	characteristic
27	C₂×C₄⋊C₄×C₁₇	—	2	1	no	—
28	C₂×C₄²×C₁₇	—	2	1	no	—
29	C₂×C₄²×C₁₇	C₂³	1	1	yes	characteristic
30	#10587612	—	2	2	no	—
31	#10587610	—	2	2	no	—
32	#10587610	—	2	2	no	—
33	#10587556	—	2	2	no	—
34	#10587555	C₂×C₄	1	2	yes	—
35	#10587555	C₂×C₄	1	2	yes	—
36	C₄×Dic₃₄	—	2	2	no	—
37	C₄×Dic₃₄	—	2	2	no	—
38	C₁₇×C₄⋊C₈	—	2	1	no	—
39	C₁₇×C₄⋊C₈	D₄	1	2	yes	—
40	C₁₇×#97	—	4	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	C₁₇×#95	D₄	1	2	yes	—
46	C₂²⋊#3622	—	2	2	no	—
47	#10587489	—	2	1	no	—
48	#10587489	—	2	1	no	—
49	#10587489	—	2	1	no	—
50	#10587489	—	2	1	no	—
51	#10587488	—	2	2	no	—
52	#10587488	—	2	2	no	—
53	C₄⋊#3622	—	2	2	no	—
54	C₄×#3622	—	2	2	no	—
55	C₂²×C₄×C₁₇	—	2	2	no	—
56	C₂²×C₄×C₁₇	C₂×D₄	1	1	yes	characteristic
57	C₂²×C₄×C₁₇	C₂×D₄	1	1	yes	characteristic
58	C₂²×C₄×C₁₇	#41	1	1	yes	characteristic
59	C₂×Dic₃₄	—	2	2	no	—
60	C₂×Dic₃₄	—	2	2	no	—
61	C₂×C₈×C₁₇	—	2	1	no	—
62	C₂×C₈×C₁₇	—	2	1	no	—
63	C₂×C₈×C₁₇	—	2	2	no	—
64	C₂×C₈×C₁₇	—	4	1	no	—
65	C₂×C₈×C₁₇	—	4	1	no	—
66	C₂×C₈×C₁₇	C₂²⋊C₄	1	1	yes	characteristic
67	C₂×C₈×C₁₇	C₂²⋊C₄	1	1	yes	characteristic
68	C₂×C₈×C₁₇	C₂²⋊C₄	1	1	yes	characteristic
69	C₂×C₈×C₁₇	C₂²⋊C₄	1	1	yes	characteristic
70	C₂×C₈×C₁₇	SD₁₆	1	2	yes	—
71	C₂×C₈×C₁₇	Q₁₆	1	2	yes	—
72	C₄⋊C₄×C₁₇	—	4	2	no	—
73	C₄²×C₁₇	—	4	1	no	—
74	C₄²×C₁₇	—	4	1	no	—
75	C₄²×C₁₇	C₂²×C₄	1	1	yes	characteristic
76	C₄²×C₁₇	C₂×D₄	1	1	yes	characteristic
77	C₄²×C₁₇	C₂×D₄	1	1	yes	characteristic
78	C₄²×C₁₇	#41	1	1	yes	characteristic
79	C₂²⋊Dic₁₇	—	4	2	no	—
80	C₄⋊Dic₁₇	—	2	2	no	—
81	C₄⋊Dic₁₇	—	2	2	no	—
82	C₄⋊Dic₁₇	—	2	2	no	—
83	C₄⋊Dic₁₇	—	2	2	no	—
84	#63225	—	4	2	no	—
85	#63225	—	4	2	no	—
86	C₄×Dic₁₇	—	4	2	no	—
87	C₂×#3622	—	4	2	no	—
88	#9752	—	17	1	no	maximal, Sylow
89	C₂³×C₁₇	#121	1	1	yes	characteristic
90	C₂×C₄×C₁₇	—	2	2	no	—
91	C₂×C₄×C₁₇	—	4	1	no	—
92	C₂×C₄×C₁₇	—	4	2	no	—
93	C₂×C₄×C₁₇	—	4	2	no	—
94	C₂×C₄×C₁₇	—	4	2	no	—
95	C₂×C₄×C₁₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
96	C₂×C₄×C₁₇	C₄×D₄	1	1	yes	characteristic
97	C₂×C₄×C₁₇	C₄×D₄	1	1	yes	characteristic
98	C₂×C₄×C₁₇	#121	1	1	yes	characteristic
99	C₂×C₄×C₁₇	#124	1	1	yes	characteristic
100	C₂×C₄×C₁₇	#127	1	1	yes	characteristic
101	C₂×C₄×C₁₇	C₂×SD₁₆	1	1	yes	characteristic
102	C₂×C₄×C₁₇	C₂×Q₁₆	1	1	yes	characteristic
103	C₂×C₄×C₁₇	#137	1	1	yes	characteristic
104	C₂×C₄×C₁₇	#137	1	1	yes	characteristic
105	C₂×Dic₁₇	—	4	2	no	—
106	C₂×Dic₁₇	—	4	2	no	—
107	Dic₃₄	—	4	4	no	—
108	Dic₃₄	—	8	2	no	—
109	C₈×C₁₇	—	2	1	no	—
110	C₈×C₁₇	—	2	1	no	—
111	C₈×C₁₇	—	4	2	no	—
112	C₈×C₁₇	#103	1	2	yes	—
113	C₈×C₁₇	#103	1	2	yes	—
114	#3622	—	8	2	no	—
115	#3160	—	17	1	no	—
116	#1770	D₁₇	1	1	yes	characteristic
117	#1564	—	17	1	no	—
118	#1503	—	17	2	no	—
119	#1481	—	17	2	no	—
120	C₂²×C₁₇	—	2	1	no	—
121	C₂²×C₁₇	—	2	1	no	—
122	C₂²×C₁₇	#390	1	1	yes	characteristic
123	C₂²×C₁₇	#461	1	1	yes	characteristic
124	C₂²×C₁₇	#462	1	1	yes	characteristic
125	C₄×C₁₇	—	2	1	no	—
126	C₄×C₁₇	—	2	1	no	—
127	C₄×C₁₇	—	2	1	no	—
128	C₄×C₁₇	—	8	2	no	—
129	C₄×C₁₇	C₂×#103	1	1	yes	characteristic
130	C₄×C₁₇	#419	1	1	yes	characteristic
131	C₄×C₁₇	#444	1	2	yes	—
132	C₄×C₁₇	#461	1	1	yes	characteristic
133	C₄×C₁₇	#462	1	1	yes	characteristic
134	Dic₁₇	—	8	2	no	—
135	Dic₁₇	—	8	2	no	—
136	#533	—	17	2	no	—
137	#500	—	17	1	no	—
138	#500	—	17	1	no	—
139	#499	—	34	1	no	—
140	#499	—	34	1	no	—
141	#498	—	17	2	no	—
142	#431	—	17	2	no	—
143	#427	—	34	1	no	—
144	#427	—	34	1	no	—
145	C₂×C₄⋊C₈	C₂×D₁₇	1	1	yes	characteristic
146	C₂×#97	—	2	1	no	—
147	C₂×C₄×C₈	C₂×D₁₇	1	1	yes	characteristic
148	C₄×C₄⋊C₄	C₂×D₁₇	1	1	yes	characteristic
149	#336	—	2	1	no	—
150	#335	—	34	1	no	—
151	#334	—	34	1	no	—
152	#328	—	34	4	no	—
153	Q₈⋊C₈	—	34	4	no	—
154	C₂×C₁₇	—	2	1	no	—
155	C₂×C₁₇	—	2	1	no	—
156	C₂×C₁₇	#1975	1	1	yes	characteristic
157	C₂×C₁₇	#1976	1	1	yes	characteristic
158	C₂×C₁₇	#1981	1	1	yes	characteristic
159	C₂²×C₈	—	2	1	no	—
160	C₂²×C₈	C₂²⋊D₁₇	1	1	yes	characteristic
161	C₂²×C₈	C₂²⋊D₁₇	1	1	yes	characteristic
162	C₄⋊Q₈	—	17	2	no	—
163	C₄⋊Q₈	—	17	2	no	—
164	#125	—	34	2	no	—
165	C₂²⋊Q₈	—	34	2	no	—
166	C₂²⋊Q₈	—	34	2	no	—
167	C₄×Q₈	—	34	2	no	—
168	C₄×Q₈	—	34	2	no	—
169	#117	—	34	2	no	—
170	C₂×C₄⋊C₄	—	2	1	no	—
171	C₂×C₄²	—	2	1	no	—
172	C₂×C₄²	C₂²×D₁₇	1	1	yes	characteristic
173	#107	—	34	1	no	—
174	#107	—	34	1	no	—
175	#107	—	34	1	no	—
176	#107	—	34	1	no	—
177	#106	—	34	2	no	—
178	#106	—	34	2	no	—
179	C₄⋊C₈	—	2	1	no	—
180	C₄⋊C₈	—	34	2	no	—
181	C₄⋊C₈	D₆₈	1	2	yes	—
182	C₂²⋊C₈	—	34	2	no	—
183	#97	—	4	2	no	—
184	C₄×C₈	—	34	2	no	—
185	C₄×C₈	C₄×D₁₇	1	1	yes	characteristic
186	C₄×C₈	C₄×D₁₇	1	1	yes	characteristic
187	C₄×C₈	D₆₈	1	1	yes	characteristic
188	C₄×C₈	D₆₈	1	1	yes	characteristic
189	#95	D₆₈	1	2	yes	—
190	C₁₇	#9752	1	1	yes	characteristic, complemented, Sylow
191	C₂×Q₈	—	34	2	no	—
192	C₂×Q₈	—	34	2	no	—
193	C₂²×C₄	—	2	2	no	—
194	C₂²×C₄	C₂×D₆₈	1	1	yes	characteristic
195	C₂²×C₄	#63252	1	1	yes	characteristic
196	C₂²×C₄	C₂×C₂²⋊D₁₇	1	1	yes	characteristic
197	C₂×C₈	—	2	1	no	—
198	C₂×C₈	—	2	1	no	—
199	C₂×C₈	—	2	2	no	—
200	C₂×C₈	—	4	1	no	—
201	C₂×C₈	—	4	1	no	—
202	C₂×C₈	—	68	2	no	—
203	C₂×C₈	#63227	1	1	yes	characteristic
204	C₂×C₈	#63227	1	1	yes	characteristic
205	C₂×C₈	#63227	1	1	yes	characteristic
206	C₂×C₈	#63227	1	1	yes	characteristic
207	C₂×C₈	#63229	1	2	yes	—
208	C₂×C₈	#63231	1	2	yes	—
209	C₄⋊C₄	—	4	2	no	—
210	C₄⋊C₄	—	34	2	no	—
211	C₄⋊C₄	—	34	2	no	—
212	C₄⋊C₄	—	34	2	no	—
213	C₄⋊C₄	—	34	2	no	—
214	C₄⋊C₄	—	68	2	no	—
215	C₄⋊C₄	—	68	2	no	—
216	C₂²⋊C₄	—	68	2	no	—
217	C₄²	—	4	1	no	—
218	C₄²	—	4	1	no	—
219	C₄²	—	68	2	no	—
220	C₄²	C₂×C₄×D₁₇	1	1	yes	characteristic
221	C₄²	C₂×D₆₈	1	1	yes	characteristic
222	C₄²	C₂×D₆₈	1	1	yes	characteristic
223	C₄²	#63252	1	1	yes	characteristic
224	C₂³	#10587616	1	1	yes	characteristic
225	Q₈	—	68	4	no	—
226	Q₈	—	136	2	no	—
227	C₂×C₄	—	2	2	no	—
228	C₂×C₄	—	4	1	no	—
229	C₂×C₄	—	4	2	no	—
230	C₂×C₄	—	4	2	no	—
231	C₂×C₄	—	4	2	no	—
232	C₂×C₄	—	68	2	no	—
233	C₂×C₄	—	68	2	no	—
234	C₂×C₄	C₄×D₆₈	1	1	yes	characteristic
235	C₂×C₄	C₄×D₆₈	1	1	yes	characteristic
236	C₂×C₄	#10587560	1	1	yes	characteristic
237	C₂×C₄	#10587561	1	1	yes	characteristic
238	C₂×C₄	C₂×#63227	1	1	yes	characteristic
239	C₂×C₄	#10587616	1	1	yes	characteristic
240	C₂×C₄	C₂×#63229	1	1	yes	characteristic
241	C₂×C₄	C₂×#63231	1	1	yes	characteristic
242	C₂×C₄	#10587637	1	1	yes	characteristic
243	C₂×C₄	#10587637	1	1	yes	characteristic
244	C₈	—	2	1	no	—
245	C₈	—	2	1	no	—
246	C₈	—	4	2	no	—
247	C₈	—	136	2	no	—
248	C₈	#10587481	1	2	yes	—
249	C₈	#10587481	1	2	yes	—
250	C₂²	—	2	1	no	—
251	C₂²	—	2	1	no	—
252	C₂²	#11762711	1	1	yes	characteristic
253	C₂²	#11763014	1	1	yes	characteristic
254	C₂²	#11763019	1	1	yes	characteristic
255	C₄	—	2	1	no	—
256	C₄	—	2	1	no	—
257	C₄	—	2	1	no	—
258	C₄	—	8	2	no	—
259	C₄	—	136	2	no	—
260	C₄	—	136	2	no	—
261	C₄	C₂×#10587481	1	1	yes	characteristic
262	C₄	#11762817	1	1	yes	characteristic
263	C₄	#11762932	1	2	yes	—
264	C₄	#11763014	1	1	yes	characteristic
265	C₄	#11763019	1	1	yes	characteristic
266	C₂	—	2	1	no	—
267	C₂	—	2	1	no	—
268	C₂	#14529677	1	1	yes	characteristic
269	C₂	#14529682	1	1	yes	characteristic
270	C₂	#14529707	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	#10587616
commutator	C₄²×C₁₇	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₁₇
Fitting	C₁₇×#1770	C₂
socle	C₂²×C₁₇	#390
prime radical	C₂³×C₁₇	#121
cosocle	C₂×C₄²×C₁₇	C₂³
hypercenter	#1770	D₁₇
hypocenter	C₁₇	#9752
LCS	G ▷ C₄²×C₁₇ ▷ C₂²×C₁₇ ▷ C₁₇
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #1770

Complements

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

N	H ≅ G/N	type
7. #11763214	C₄	semidirect
8. #11763214	C₄	semidirect
11. #11763207	C₄	semidirect
190. C₁₇	#9752	semidirect

Maximal subgroups

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

subgroup	index
#14539298	2
#14527532	2
#14527282	2
#14527186	2
C₁₇×#1770	2
#9752	17