Subgroups — #16928107

Order 4352.

438 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	#14528943	C₂	1	1	yes	maximal, characteristic
2	#14528940	C₂	1	1	yes	maximal, characteristic
3	#14528940	C₂	1	1	yes	maximal, characteristic
4	#14528912	C₂	1	1	yes	maximal, characteristic
5	#14528911	C₂	1	1	yes	maximal, characteristic
6	C₂×#11762531	C₂	1	1	yes	maximal, characteristic
7	C₁₇×#1844	C₂	1	1	yes	maximal, characteristic
8	C₂×#10587478	—	2	2	no	—
9	C₂×#10587479	—	2	2	no	—
10	C₂×#10587479	—	2	2	no	—
11	C₂×C₄×#3622	C₂²	1	1	yes	characteristic
12	C₂×C₄×#3622	C₂²	1	1	yes	characteristic
13	#11762679	C₂²	1	1	yes	characteristic
14	#11762679	C₂²	1	1	yes	characteristic
15	#11762572	—	2	2	no	—
16	#11762571	—	2	2	no	—
17	#11762531	C₄	1	4	yes	complemented
18	C₂×C₁₇×#106	—	2	2	no	—
19	C₂×C₄×C₈×C₁₇	C₂²	1	1	yes	characteristic
20	C₁₇×#384	C₂²	1	1	yes	characteristic
21	C₁₇×#384	C₂²	1	1	yes	characteristic
22	C₂²×C₈×C₁₇	D₄	1	2	yes	—
23	C₂×C₄⋊C₄×C₁₇	—	2	1	no	—
24	C₂×C₄⋊C₄×C₁₇	—	2	1	no	—
25	C₂×C₄⋊C₄×C₁₇	—	2	1	no	—
26	C₂×C₄⋊C₄×C₁₇	—	2	1	no	—
27	C₂×C₄⋊C₄×C₁₇	—	2	1	no	—
28	C₂×C₄⋊C₄×C₁₇	—	2	1	no	—
29	C₂×C₄²×C₁₇	C₂³	1	1	yes	characteristic
30	C₂×C₄⋊Dic₁₇	—	2	1	no	—
31	C₂×C₄⋊Dic₁₇	—	2	1	no	—
32	C₂×C₄⋊Dic₁₇	—	2	1	no	—
33	C₂×C₄⋊Dic₁₇	—	2	1	no	—
34	C₂×C₄⋊Dic₁₇	—	2	1	no	—
35	C₂×C₄⋊Dic₁₇	—	2	1	no	—
36	C₂²×#3622	D₄	1	2	yes	—
37	C₂²×#3622	D₄	1	2	yes	—
38	C₁₇×#106	—	4	4	no	—
39	C₄×C₈×C₁₇	C₂×C₄	1	2	yes	—
40	C₄×C₈×C₁₇	Q₈	1	2	yes	—
41	#10587479	—	4	4	no	—
42	#10587479	—	4	4	no	—
43	#10587478	—	4	4	no	—
44	C₄×#3622	C₂×C₄	1	2	yes	—
45	C₄×#3622	C₂×C₄	1	2	yes	—
46	C₄×#3622	Q₈	1	2	yes	—
47	C₄×#3622	Q₈	1	2	yes	—
48	C₂²×C₄×C₁₇	—	4	1	no	—
49	C₂²×C₄×C₁₇	—	4	1	no	—
50	C₂²×C₄×C₁₇	C₂×D₄	1	1	yes	characteristic
51	C₂²×C₄×C₁₇	C₂×D₄	1	1	yes	characteristic
52	C₂²×C₄×C₁₇	C₂×D₄	1	1	yes	characteristic
53	C₂²×Dic₁₇	—	4	1	no	—
54	C₂²×Dic₁₇	—	4	1	no	—
55	C₂×C₈×C₁₇	—	2	2	no	—
56	C₂×C₈×C₁₇	—	2	2	no	—
57	C₂×C₈×C₁₇	C₄⋊C₄	1	4	yes	—
58	C₄⋊C₄×C₁₇	—	4	2	no	—
59	C₄⋊C₄×C₁₇	—	4	2	no	—
60	C₄⋊C₄×C₁₇	—	4	2	no	—
61	C₄⋊C₄×C₁₇	—	4	2	no	—
62	C₄⋊C₄×C₁₇	—	4	2	no	—
63	C₄⋊C₄×C₁₇	—	4	2	no	—
64	C₄²×C₁₇	C₂²×C₄	1	1	yes	characteristic
65	C₄²×C₁₇	C₂×Q₈	1	1	yes	characteristic
66	C₄²×C₁₇	C₂×Q₈	1	1	yes	characteristic
67	C₄²×C₁₇	C₂×Q₈	1	1	yes	characteristic
68	C₄⋊Dic₁₇	—	4	2	no	—
69	C₄⋊Dic₁₇	—	4	2	no	—
70	C₄⋊Dic₁₇	—	4	2	no	—
71	C₄⋊Dic₁₇	—	4	2	no	—
72	C₄⋊Dic₁₇	—	4	2	no	—
73	C₄⋊Dic₁₇	—	4	2	no	—
74	C₂×#3622	—	2	2	no	—
75	C₂×#3622	—	2	2	no	—
76	C₂×#3622	C₄⋊C₄	1	4	yes	—
77	C₂×#3622	C₄⋊C₄	1	4	yes	—
78	C₂×#3622	SD₁₆	1	4	yes	—
79	C₂×#3622	SD₁₆	1	4	yes	—
80	#8819	—	17	1	no	maximal, Sylow
81	C₂³×C₁₇	#127	1	1	yes	characteristic
82	C₂×C₄×C₁₇	—	8	1	no	—
83	C₂×C₄×C₁₇	—	8	1	no	—
84	C₂×C₄×C₁₇	—	8	1	no	—
85	C₂×C₄×C₁₇	—	8	1	no	—
86	C₂×C₄×C₁₇	—	8	1	no	—
87	C₂×C₄×C₁₇	—	8	1	no	—
88	C₂×C₄×C₁₇	C₂×C₄⋊C₄	1	1	yes	characteristic
89	C₂×C₄×C₁₇	C₂×C₄⋊C₄	1	1	yes	characteristic
90	C₂×C₄×C₁₇	C₂×C₄⋊C₄	1	1	yes	characteristic
91	C₂×C₄×C₁₇	C₄⋊Q₈	1	1	yes	characteristic
92	C₂×C₄×C₁₇	C₄⋊Q₈	1	1	yes	characteristic
93	C₂×C₄×C₁₇	C₄⋊Q₈	1	1	yes	characteristic
94	C₂×C₄×C₁₇	C₂×SD₁₆	1	2	yes	—
95	C₂×C₄×C₁₇	C₂×SD₁₆	1	2	yes	—
96	C₂×C₄×C₁₇	AGL(1,ℤ/8)	1	2	yes	—
97	C₂×C₄×C₁₇	AGL(1,ℤ/8)	1	2	yes	—
98	C₂×C₄×C₁₇	#137	1	2	yes	—
99	C₂×C₄×C₁₇	#137	1	2	yes	—
100	C₂×Dic₁₇	—	8	1	no	—
101	C₂×Dic₁₇	—	8	1	no	—
102	C₂×Dic₁₇	—	8	1	no	—
103	C₂×Dic₁₇	—	8	1	no	—
104	C₂×Dic₁₇	—	8	1	no	—
105	C₂×Dic₁₇	—	8	1	no	—
106	C₈×C₁₇	—	2	4	no	—
107	#3622	—	2	4	no	—
108	#3622	#106	1	8	yes	complemented
109	#1845	—	17	1	no	—
110	#1844	—	17	1	no	—
111	#1844	—	17	1	no	—
112	#1844	D₁₇	1	1	yes	characteristic
113	#1834	—	17	1	no	—
114	#1834	—	17	1	no	—
115	C₂×#322	—	17	1	no	—
116	C₂²×C₁₇	#384	1	1	yes	characteristic
117	C₂²×C₁₇	#492	1	1	yes	characteristic
118	C₂²×C₁₇	#492	1	1	yes	characteristic
119	C₂²×C₁₇	#496	1	1	yes	characteristic
120	C₂²×C₁₇	#496	1	1	yes	characteristic
121	C₂²×C₁₇	#497	1	1	yes	characteristic
122	C₂²×C₁₇	#497	1	1	yes	characteristic
123	C₄×C₁₇	—	16	1	no	—
124	C₄×C₁₇	—	16	1	no	—
125	C₄×C₁₇	C₂×#106	1	2	yes	—
126	C₄×C₁₇	#428	1	2	yes	—
127	C₄×C₁₇	#428	1	2	yes	—
128	C₄×C₁₇	#498	1	2	yes	—
129	C₄×C₁₇	#501	1	2	yes	—
130	C₄×C₁₇	#501	1	2	yes	—
131	Dic₁₇	—	16	1	no	—
132	Dic₁₇	—	16	1	no	—
133	C₂×#107	—	34	2	no	—
134	C₂×#106	—	2	2	no	—
135	C₂×#106	—	34	2	no	—
136	C₂×#106	—	34	2	no	—
137	C₂×C₄×C₈	—	17	1	no	—
138	C₂×C₄×C₈	—	17	1	no	—
139	C₂×C₄×C₈	C₂×D₁₇	1	1	yes	characteristic
140	#384	—	17	1	no	—
141	#384	—	17	1	no	—
142	#384	C₂×D₁₇	1	1	yes	characteristic
143	#384	C₂×D₁₇	1	1	yes	characteristic
144	#340	—	34	2	no	—
145	#340	—	34	2	no	—
146	#322	—	17	4	no	—
147	C₂×C₁₇	#1714	1	1	yes	characteristic
148	C₂×C₁₇	#1714	1	1	yes	characteristic
149	C₂×C₁₇	#1715	1	1	yes	characteristic
150	C₂×C₁₇	#1716	1	1	yes	characteristic
151	C₂×C₁₇	#1844	1	1	yes	characteristic
152	C₂×C₁₇	#1847	1	1	yes	characteristic
153	C₂×C₁₇	#1847	1	1	yes	characteristic
154	C₂²×C₈	—	17	2	no	—
155	C₂²×C₈	—	17	2	no	—
156	C₂²×C₈	C₂²⋊D₁₇	1	2	yes	—
157	C₂×C₄⋊C₄	—	2	1	no	—
158	C₂×C₄⋊C₄	—	2	1	no	—
159	C₂×C₄⋊C₄	—	2	1	no	—
160	C₂×C₄⋊C₄	—	2	1	no	—
161	C₂×C₄⋊C₄	—	2	1	no	—
162	C₂×C₄⋊C₄	—	2	1	no	—
163	C₂×C₄⋊C₄	—	34	1	no	—
164	C₂×C₄⋊C₄	—	34	1	no	—
165	C₂×C₄⋊C₄	—	34	1	no	—
166	C₂×C₄⋊C₄	—	34	1	no	—
167	C₂×C₄⋊C₄	—	34	1	no	—
168	C₂×C₄⋊C₄	—	34	1	no	—
169	C₂×C₄²	C₂²×D₁₇	1	1	yes	characteristic
170	#107	—	68	4	no	—
171	#106	—	4	4	no	—
172	#106	—	68	4	no	—
173	#106	—	68	4	no	—
174	C₄×C₈	—	17	2	no	—
175	C₄×C₈	—	17	2	no	—
176	C₄×C₈	—	17	2	no	—
177	C₄×C₈	—	17	2	no	—
178	C₄×C₈	Dic₃₄	1	2	yes	—
179	C₄×C₈	C₄×D₁₇	1	2	yes	—
180	C₁₇	#8819	1	1	yes	characteristic, complemented, Sylow
181	C₂²×C₄	—	4	1	no	—
182	C₂²×C₄	—	4	1	no	—
183	C₂²×C₄	—	68	1	no	—
184	C₂²×C₄	—	68	1	no	—
185	C₂²×C₄	D₄×D₁₇	1	1	yes	characteristic
186	C₂²×C₄	D₄×D₁₇	1	1	yes	characteristic
187	C₂²×C₄	C₂×C₂²⋊D₁₇	1	1	yes	characteristic
188	C₂×C₈	—	2	2	no	—
189	C₂×C₈	—	2	2	no	—
190	C₂×C₈	—	17	4	no	—
191	C₂×C₈	—	17	4	no	—
192	C₂×C₈	—	17	4	no	—
193	C₂×C₈	—	17	4	no	—
194	C₂×C₈	—	34	2	no	—
195	C₂×C₈	—	34	2	no	—
196	C₂×C₈	#63225	1	4	yes	—
197	C₄⋊C₄	—	4	2	no	—
198	C₄⋊C₄	—	4	2	no	—
199	C₄⋊C₄	—	4	2	no	—
200	C₄⋊C₄	—	4	2	no	—
201	C₄⋊C₄	—	4	2	no	—
202	C₄⋊C₄	—	4	2	no	—
203	C₄⋊C₄	—	68	2	no	—
204	C₄⋊C₄	—	68	2	no	—
205	C₄⋊C₄	—	68	2	no	—
206	C₄⋊C₄	—	68	2	no	—
207	C₄⋊C₄	—	68	2	no	—
208	C₄⋊C₄	—	68	2	no	—
209	C₄²	C₂×Dic₃₄	1	1	yes	characteristic
210	C₄²	C₂×C₄×D₁₇	1	1	yes	characteristic
211	C₄²	Q₈×D₁₇	1	1	yes	characteristic
212	C₄²	Q₈×D₁₇	1	1	yes	characteristic
213	C₂³	#10587626	1	1	yes	characteristic
214	C₂×C₄	—	8	1	no	—
215	C₂×C₄	—	8	1	no	—
216	C₂×C₄	—	8	1	no	—
217	C₂×C₄	—	8	1	no	—
218	C₂×C₄	—	8	1	no	—
219	C₂×C₄	—	8	1	no	—
220	C₂×C₄	—	136	1	no	—
221	C₂×C₄	—	136	1	no	—
222	C₂×C₄	—	136	1	no	—
223	C₂×C₄	—	136	1	no	—
224	C₂×C₄	—	136	1	no	—
225	C₂×C₄	—	136	1	no	—
226	C₂×C₄	#10587574	1	1	yes	characteristic
227	C₂×C₄	#10587574	1	1	yes	characteristic
228	C₂×C₄	C₄⋊C₄×D₁₇	1	1	yes	characteristic
229	C₂×C₄	C₄⋊C₄×D₁₇	1	1	yes	characteristic
230	C₂×C₄	SD₁₆×D₁₇	1	2	yes	—
231	C₂×C₄	SD₁₆×D₁₇	1	2	yes	—
232	C₂×C₄	#10587600	1	2	yes	—
233	C₂×C₄	#10587601	1	2	yes	—
234	C₂×C₄	C₂×#63225	1	1	yes	characteristic
235	C₂×C₄	#10587618	1	2	yes	—
236	C₂×C₄	#10587628	1	2	yes	—
237	C₂×C₄	#10587630	1	1	yes	characteristic
238	C₈	—	2	4	no	—
239	C₈	—	17	8	no	—
240	C₈	—	34	4	no	—
241	C₂²	#11762678	1	1	yes	characteristic
242	C₂²	#11763180	1	1	yes	characteristic
243	C₂²	#11763180	1	1	yes	characteristic
244	C₂²	#11763195	1	1	yes	characteristic
245	C₂²	#11763196	1	1	yes	characteristic
246	C₂²	#11763200	1	1	yes	characteristic
247	C₂²	#11763201	1	1	yes	characteristic
248	C₄	—	16	1	no	—
249	C₄	—	16	1	no	—
250	C₄	—	272	1	no	—
251	C₄	—	272	1	no	—
252	C₄	#106×D₁₇	1	2	yes	—
253	C₄	#11762848	1	2	yes	—
254	C₄	#11762849	1	2	yes	—
255	C₄	#11763206	1	2	yes	—
256	C₄	#11763215	1	2	yes	—
257	C₄	#11763216	1	2	yes	—
258	C₂	#14528404	1	1	yes	characteristic
259	C₂	#14528409	1	1	yes	characteristic
260	C₂	#14528411	1	1	yes	characteristic
261	C₂	#14528416	1	1	yes	characteristic
262	C₂	#14528939	1	1	yes	characteristic
263	C₂	#14528949	1	1	yes	characteristic
264	C₂	#14528950	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	#10587626
commutator	C₄²×C₁₇	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₁₇
Fitting	C₁₇×#1844	C₂
socle, prime radical	C₂³×C₁₇	#127
cosocle	C₂×C₄²×C₁₇	C₂³
hypercenter	#1844	D₁₇
hypocenter	C₁₇	#8819
LCS	G ▷ C₄²×C₁₇ ▷ C₂²×C₁₇ ▷ C₁₇
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ #1844

Complements

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

N	H ≅ G/N	type
17. #11762531	C₄	semidirect
108. #3622	#106	semidirect
180. C₁₇	#8819	semidirect

Maximal subgroups

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

subgroup	index
#14528943	2
#14528940	2
#14528912	2
#14528911	2
C₂×#11762531	2
C₁₇×#1844	2
#8819	17