Subgroups — #16939041

Order 4352.

366 conjugacy classes of subgroups.

The subgroup lattice has 250 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#14529720	C₂	1	1	yes	maximal, characteristic
2	#14528944	C₂	1	1	yes	maximal, characteristic
3	#14528790	C₂	1	2	yes	maximal
4	#14526668	C₂	1	2	yes	maximal
5	C₁₇×#1277	C₂	1	1	yes	maximal, characteristic
6	C₂×C₄×Dic₃₄	C₂²	1	1	yes	characteristic
7	C₂×#10587489	—	2	1	no	—
8	C₂×#10587489	—	2	1	no	—
9	C₂×C₄⋊#3622	C₂²	1	1	yes	characteristic
10	C₂×#10587487	—	2	2	no	—
11	#11762679	C₂²	1	2	yes	—
12	#11762572	—	2	2	no	—
13	#11762563	—	2	1	no	—
14	#11762563	—	2	1	no	—
15	C₂×C₁₇×C₄⋊C₈	C₂²	1	2	yes	—
16	C₂×C₄×C₈×C₁₇	C₂²	1	1	yes	characteristic
17	C₂²×Dic₃₄	D₄	1	1	yes	characteristic
18	C₂²×C₈×C₁₇	—	2	2	no	—
19	C₂²×C₈×C₁₇	D₄	1	1	yes	characteristic
20	C₂²×C₈×C₁₇	D₄	1	1	yes	characteristic
21	C₂×C₄²×C₁₇	C₂³	1	1	yes	characteristic
22	C₂×C₄⋊Dic₁₇	—	2	2	no	—
23	C₂×C₄⋊Dic₁₇	—	2	2	no	—
24	C₂×C₄⋊Dic₁₇	—	2	2	no	—
25	C₂×C₄⋊Dic₁₇	D₄	1	1	yes	characteristic
26	C₂×#63225	—	2	1	no	—
27	C₂×C₄×Dic₁₇	—	2	1	no	—
28	C₂²×#3622	—	2	1	no	—
29	C₄×Dic₃₄	—	2	2	no	—
30	C₄×Dic₃₄	—	2	2	no	—
31	C₁₇×C₄⋊C₈	—	2	2	no	—
32	C₁₇×C₄⋊C₈	—	2	2	no	—
33	C₄×C₈×C₁₇	—	2	2	no	—
34	#10587489	—	4	2	no	—
35	#10587489	—	4	2	no	—
36	#10587487	—	4	4	no	—
37	C₄⋊#3622	—	2	2	no	—
38	C₂²×C₄×C₁₇	C₂×D₄	1	1	yes	characteristic
39	C₂²×C₄×C₁₇	C₂×D₄	1	1	yes	characteristic
40	C₂²×C₄×C₁₇	#41	1	1	yes	characteristic
41	C₂²×Dic₁₇	—	2	1	no	—
42	C₂²×Dic₁₇	—	2	1	no	—
43	C₂²×Dic₁₇	—	4	2	no	—
44	C₂×Dic₃₄	—	2	2	no	—
45	C₂×Dic₃₄	—	2	2	no	—
46	C₂×Dic₃₄	—	2	1	no	—
47	C₂×Dic₃₄	Q₁₆	1	2	yes	—
48	C₂×C₈×C₁₇	—	2	1	no	—
49	C₂×C₈×C₁₇	—	2	1	no	—
50	C₂×C₈×C₁₇	—	2	1	no	—
51	C₂×C₈×C₁₇	—	2	1	no	—
52	C₂×C₈×C₁₇	—	4	2	no	—
53	C₂×C₈×C₁₇	—	4	2	no	—
54	C₂×C₈×C₁₇	—	4	2	no	—
55	C₂×C₈×C₁₇	SD₁₆	1	2	yes	—
56	C₂×C₈×C₁₇	Q₁₆	1	2	yes	—
57	C₄²×C₁₇	#41	1	2	yes	—
58	C₄²×C₁₇	#41	1	2	yes	—
59	C₄⋊Dic₁₇	—	2	2	no	—
60	C₄⋊Dic₁₇	—	4	2	no	—
61	C₄⋊Dic₁₇	—	4	2	no	—
62	C₄⋊Dic₁₇	—	4	2	no	—
63	C₄⋊Dic₁₇	—	4	2	no	—
64	C₄⋊Dic₁₇	—	8	2	no	—
65	#63225	—	4	2	no	—
66	C₄×Dic₁₇	—	4	1	no	—
67	C₄×Dic₁₇	—	4	1	no	—
68	C₂×#3622	—	4	1	no	—
69	C₂×#3622	—	4	1	no	—
70	C₂×#3622	—	4	1	no	—
71	#10618	—	17	1	no	maximal, Sylow
72	C₂³×C₁₇	#121	1	1	yes	characteristic
73	C₂×C₄×C₁₇	—	2	1	no	—
74	C₂×C₄×C₁₇	—	2	1	no	—
75	C₂×C₄×C₁₇	#123	1	2	yes	—
76	C₂×C₄×C₁₇	#123	1	1	yes	characteristic
77	C₂×C₄×C₁₇	#124	1	1	yes	characteristic
78	C₂×C₄×C₁₇	#126	1	2	yes	—
79	C₂×C₄×C₁₇	C₂×SD₁₆	1	1	yes	characteristic
80	C₂×C₄×C₁₇	C₂×Q₁₆	1	1	yes	characteristic
81	C₂×C₄×C₁₇	C₂×Q₁₆	1	1	yes	characteristic
82	C₂×C₄×C₁₇	#135	1	1	yes	characteristic
83	C₂×C₄×C₁₇	AGL(1,ℤ/8)	1	1	yes	characteristic
84	C₂×C₄×C₁₇	AGL(1,ℤ/8)	1	2	yes	—
85	C₂×C₄×C₁₇	#137	1	1	yes	characteristic
86	C₂×Dic₁₇	—	2	2	no	—
87	C₂×Dic₁₇	—	4	1	no	—
88	C₂×Dic₁₇	—	4	1	no	—
89	C₂×Dic₁₇	—	4	1	no	—
90	C₂×Dic₁₇	—	4	1	no	—
91	C₂×Dic₁₇	—	4	1	no	—
92	C₂×Dic₁₇	—	8	2	no	—
93	C₂×Dic₁₇	—	8	2	no	—
94	C₂×Dic₁₇	—	8	2	no	—
95	Dic₃₄	—	2	4	no	—
96	Dic₃₄	—	4	2	no	—
97	C₈×C₁₇	—	2	2	no	—
98	C₈×C₁₇	—	2	2	no	—
99	C₈×C₁₇	—	8	2	no	—
100	#3622	—	8	1	no	—
101	#1984	—	17	1	no	—
102	#1845	—	17	1	no	—
103	#1803	—	17	2	no	—
104	#1296	—	17	2	no	—
105	#1277	D₁₇	1	1	yes	characteristic
106	C₂²×C₁₇	#397	1	1	yes	characteristic
107	C₂²×C₁₇	#460	1	1	yes	characteristic
108	C₂²×C₁₇	#462	1	1	yes	characteristic
109	C₂²×C₁₇	#467	1	1	yes	characteristic
110	C₂²×C₁₇	#469	1	1	yes	characteristic
111	C₂²×C₁₇	#472	1	1	yes	characteristic
112	C₂²×C₁₇	#473	1	1	yes	characteristic
113	C₄×C₁₇	—	2	2	no	—
114	C₄×C₁₇	#482	1	2	yes	—
115	C₄×C₁₇	#483	1	1	yes	characteristic
116	C₄×C₁₇	#484	1	2	yes	—
117	C₄×C₁₇	#484	1	1	yes	characteristic
118	C₄×C₁₇	#487	1	1	yes	characteristic
119	C₄×C₁₇	#489	1	1	yes	characteristic
120	Dic₁₇	—	4	2	no	—
121	Dic₁₇	—	8	1	no	—
122	Dic₁₇	—	16	2	no	—
123	C₂×C₄×Q₈	—	17	1	no	—
124	C₂×#107	—	34	1	no	—
125	C₂×#107	—	34	1	no	—
126	C₂×C₄⋊C₈	—	17	1	no	—
127	C₂×C₄⋊C₈	C₂×D₁₇	1	2	yes	—
128	C₂×#103	—	34	2	no	—
129	C₂×C₄×C₈	C₂×D₁₇	1	1	yes	characteristic
130	#384	—	17	2	no	—
131	#340	—	34	2	no	—
132	#336	—	34	1	no	—
133	#336	—	34	1	no	—
134	C₂×C₁₇	#1670	1	1	yes	characteristic
135	C₂×C₁₇	#1676	1	1	yes	characteristic
136	C₂×C₁₇	#1688	1	1	yes	characteristic
137	C₂×C₁₇	#1694	1	1	yes	characteristic
138	C₂×C₁₇	#2072	1	1	yes	characteristic
139	C₂×C₁₇	#2076	1	1	yes	characteristic
140	C₂×C₁₇	#2078	1	1	yes	characteristic
141	C₂²×Q₈	—	17	1	no	—
142	C₂²×C₈	—	2	2	no	—
143	C₂²×C₈	—	34	1	no	—
144	C₂²×C₈	D₆₈	1	1	yes	characteristic
145	C₂²×C₈	D₆₈	1	1	yes	characteristic
146	C₄×Q₈	—	34	2	no	—
147	C₄×Q₈	—	34	2	no	—
148	C₂×C₄⋊C₄	—	17	1	no	—
149	C₂×C₄⋊C₄	—	34	1	no	—
150	C₂×C₄⋊C₄	—	34	2	no	—
151	C₂×C₄⋊C₄	—	34	2	no	—
152	C₂×C₄⋊C₄	—	34	2	no	—
153	C₂×C₄²	—	34	1	no	—
154	C₂×C₄²	C₂²×D₁₇	1	1	yes	characteristic
155	#107	—	68	2	no	—
156	#107	—	68	2	no	—
157	C₄⋊C₈	—	2	2	no	—
158	C₄⋊C₈	—	2	2	no	—
159	C₄⋊C₈	—	34	2	no	—
160	#103	—	68	4	no	—
161	C₄×C₈	—	2	2	no	—
162	C₁₇	#10618	1	1	yes	characteristic, complemented, Sylow
163	C₂×Q₈	—	17	2	no	—
164	C₂×Q₈	—	34	1	no	—
165	C₂×Q₈	—	34	2	no	—
166	C₂×Q₈	—	34	2	no	—
167	C₂²×C₄	—	34	1	no	—
168	C₂²×C₄	—	34	1	no	—
169	C₂²×C₄	—	68	2	no	—
170	C₂²×C₄	C₂×D₆₈	1	1	yes	characteristic
171	C₂²×C₄	D₄×D₁₇	1	1	yes	characteristic
172	C₂²×C₄	#63256	1	1	yes	characteristic
173	C₂×C₈	—	2	1	no	—
174	C₂×C₈	—	2	1	no	—
175	C₂×C₈	—	2	1	no	—
176	C₂×C₈	—	2	1	no	—
177	C₂×C₈	—	4	2	no	—
178	C₂×C₈	—	4	2	no	—
179	C₂×C₈	—	4	2	no	—
180	C₂×C₈	—	68	1	no	—
181	C₂×C₈	—	68	1	no	—
182	C₂×C₈	—	68	1	no	—
183	C₂×C₈	#63219	1	2	yes	—
184	C₂×C₈	Dic₆₈	1	2	yes	—
185	C₄⋊C₄	—	34	2	no	—
186	C₄⋊C₄	—	68	2	no	—
187	C₄⋊C₄	—	68	2	no	—
188	C₄⋊C₄	—	68	2	no	—
189	C₄⋊C₄	—	68	2	no	—
190	C₄⋊C₄	—	68	2	no	—
191	C₄⋊C₄	—	136	2	no	—
192	C₄²	—	68	1	no	—
193	C₄²	—	68	1	no	—
194	C₄²	#63252	1	2	yes	—
195	C₄²	#63254	1	2	yes	—
196	C₂³	#10587581	1	1	yes	characteristic
197	Q₈	—	34	4	no	—
198	Q₈	—	68	2	no	—
199	C₂×C₄	—	2	1	no	—
200	C₂×C₄	—	2	1	no	—
201	C₂×C₄	—	34	2	no	—
202	C₂×C₄	—	68	1	no	—
203	C₂×C₄	—	68	1	no	—
204	C₂×C₄	—	68	1	no	—
205	C₂×C₄	—	68	1	no	—
206	C₂×C₄	—	68	1	no	—
207	C₂×C₄	—	136	2	no	—
208	C₂×C₄	—	136	2	no	—
209	C₂×C₄	—	136	2	no	—
210	C₂×C₄	#10587561	1	1	yes	characteristic
211	C₂×C₄	#10587569	1	2	yes	—
212	C₂×C₄	#10587572	1	1	yes	characteristic
213	C₂×C₄	#10587584	1	2	yes	—
214	C₂×C₄	C₂×#63219	1	1	yes	characteristic
215	C₂×C₄	C₂×Dic₆₈	1	1	yes	characteristic
216	C₂×C₄	#10587594	1	1	yes	characteristic
217	C₂×C₄	#10587595	1	1	yes	characteristic
218	C₂×C₄	#10587598	1	1	yes	characteristic
219	C₂×C₄	#10587600	1	2	yes	—
220	C₂×C₄	Q₁₆×D₁₇	1	1	yes	characteristic
221	C₈	—	2	2	no	—
222	C₈	—	2	2	no	—
223	C₈	—	8	2	no	—
224	C₈	—	136	1	no	—
225	C₂²	#11762736	1	1	yes	characteristic
226	C₂²	#11763010	1	1	yes	characteristic
227	C₂²	#11763020	1	1	yes	characteristic
228	C₂²	#11763051	1	1	yes	characteristic
229	C₂²	#11763063	1	1	yes	characteristic
230	C₂²	#11763080	1	1	yes	characteristic
231	C₂²	#11763085	1	1	yes	characteristic
232	C₄	—	2	2	no	—
233	C₄	—	272	2	no	—
234	C₄	—	68	2	no	—
235	C₄	—	136	1	no	—
236	C₄	#11763131	1	2	yes	—
237	C₄	#11763140	1	1	yes	characteristic
238	C₄	#11763143	1	2	yes	—
239	C₄	#11763145	1	1	yes	characteristic
240	C₄	#11763160	1	1	yes	characteristic
241	C₄	#11763169	1	1	yes	characteristic
242	C₂	#14528155	1	1	yes	characteristic
243	C₂	#14528189	1	1	yes	characteristic
244	C₂	#14528269	1	1	yes	characteristic
245	C₂	#14528303	1	1	yes	characteristic
246	C₂	#14530194	1	1	yes	characteristic
247	C₂	#14530214	1	1	yes	characteristic
248	C₂	#14530226	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	#10587581
commutator, cosocle	C₂×C₄²×C₁₇	C₂³
Frattini	C₂×C₄²	C₂²×D₁₇
Fitting	C₁₇×#1277	C₂
socle, prime radical	C₂³×C₁₇	#121
hypercenter	#1277	D₁₇
hypocenter	C₁₇	#10618
LCS	G ▷ C₂×C₄²×C₁₇ ▷ C₂²×C₁₇ ▷ C₁₇
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ #1277

Complements

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

N	H ≅ G/N	type
162. C₁₇	#10618	semidirect

Maximal subgroups

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

subgroup	index
#14529720	2
#14528944	2
#14528790	2
#14526668	2
C₁₇×#1277	2
#10618	17