Subgroups — #16938587

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	C₂×#11763214	C₂	1	1	yes	maximal, characteristic
2	#14529707	C₂	1	2	yes	maximal
3	C₂×#11762554	C₂	1	2	yes	maximal
4	#14526619	C₂	1	1	yes	maximal, characteristic
5	C₁₇×#1939	C₂	1	1	yes	maximal, characteristic
6	C₂×#10587555	C₂²	1	2	yes	—
7	#11763214	—	2	4	no	—
8	C₂×#10587489	—	2	1	no	—
9	C₂×#10587489	—	2	1	no	—
10	C₂×C₄⋊#3622	C₂²	1	2	yes	—
11	C₂×#10587481	—	2	2	no	—
12	C₂×#10587481	—	2	2	no	—
13	#11762554	—	2	4	no	—
14	C₂×C₁₇×C₄⋊C₈	—	2	1	no	—
15	C₂×C₁₇×C₄⋊C₈	C₂²	1	1	yes	characteristic
16	C₂×C₄×C₈×C₁₇	C₂²	1	1	yes	characteristic
17	C₄×C₄⋊C₄×C₁₇	C₂²	1	1	yes	characteristic
18	C₁₇×#336	—	2	1	no	—
19	C₂²×Dic₃₄	—	2	2	no	—
20	C₂²×C₈×C₁₇	—	2	1	no	—
21	C₂²×C₈×C₁₇	D₄	1	1	yes	characteristic
22	C₂²×C₈×C₁₇	D₄	1	1	yes	characteristic
23	C₂×C₄⋊C₄×C₁₇	D₄	1	1	yes	characteristic
24	C₂×C₄⋊C₄×C₁₇	D₄	1	1	yes	characteristic
25	C₂×C₄²×C₁₇	—	2	1	no	—
26	C₂×C₄²×C₁₇	C₂³	1	1	yes	characteristic
27	C₂×C₄⋊Dic₁₇	—	2	2	no	—
28	C₂×C₄⋊Dic₁₇	—	2	2	no	—
29	C₂²×#3622	—	2	2	no	—
30	#10587555	—	2	2	no	—
31	#10587555	—	2	2	no	—
32	#10587555	—	2	2	no	—
33	#10587555	D₄	1	4	yes	—
34	C₁₇×C₄⋊C₈	—	2	2	no	—
35	C₁₇×C₄⋊C₈	—	4	2	no	—
36	C₄×C₈×C₁₇	—	2	2	no	—
37	C₁₇×#95	—	2	1	no	—
38	#10587489	—	2	4	no	—
39	#10587489	—	2	4	no	—
40	#10587481	—	4	4	no	—
41	#10587481	—	4	4	no	—
42	C₄⋊#3622	—	2	2	no	—
43	C₄⋊#3622	—	2	2	no	—
44	C₂²×C₄×C₁₇	—	2	1	no	—
45	C₂²×C₄×C₁₇	—	2	1	no	—
46	C₂²×C₄×C₁₇	C₂×D₄	1	1	yes	characteristic
47	C₂²×C₄×C₁₇	C₂×D₄	1	1	yes	characteristic
48	C₂²×C₄×C₁₇	#41	1	1	yes	characteristic
49	C₂²×Dic₁₇	—	4	2	no	—
50	C₂×Dic₃₄	—	2	4	no	—
51	C₂×Dic₃₄	—	4	2	no	—
52	C₂×Dic₃₄	—	4	2	no	—
53	C₂×Dic₃₄	—	4	2	no	—
54	C₂×Dic₃₄	—	4	2	no	—
55	C₂×Dic₃₄	—	4	2	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	—	4	1	no	—
61	C₂×C₈×C₁₇	—	4	1	no	—
62	C₂×C₈×C₁₇	—	4	1	no	—
63	C₂×C₈×C₁₇	SD₁₆	1	2	yes	—
64	C₂×C₈×C₁₇	Q₁₆	1	2	yes	—
65	C₄⋊C₄×C₁₇	—	2	1	no	—
66	C₄⋊C₄×C₁₇	—	2	2	no	—
67	C₄⋊C₄×C₁₇	D₈	1	2	yes	—
68	C₄²×C₁₇	—	4	1	no	—
69	C₄²×C₁₇	—	4	1	no	—
70	C₄²×C₁₇	C₂×D₄	1	2	yes	—
71	C₄²×C₁₇	#41	1	2	yes	—
72	C₄⋊Dic₁₇	—	2	4	no	—
73	C₄⋊Dic₁₇	—	2	4	no	—
74	C₄⋊Dic₁₇	—	4	2	no	—
75	C₄⋊Dic₁₇	—	4	2	no	—
76	C₂×#3622	—	4	2	no	—
77	C₂×#3622	—	4	2	no	—
78	C₂×#3622	—	4	2	no	—
79	#10545	—	17	1	no	maximal, Sylow
80	C₂³×C₁₇	#121	1	1	yes	characteristic
81	C₂×C₄×C₁₇	—	2	1	no	—
82	C₂×C₄×C₁₇	—	2	1	no	—
83	C₂×C₄×C₁₇	—	2	2	no	—
84	C₂×C₄×C₁₇	—	4	1	no	—
85	C₂×C₄×C₁₇	—	4	1	no	—
86	C₂×C₄×C₁₇	—	4	1	no	—
87	C₂×C₄×C₁₇	—	4	1	no	—
88	C₂×C₄×C₁₇	—	4	1	no	—
89	C₂×C₄×C₁₇	#120	1	1	yes	characteristic
90	C₂×C₄×C₁₇	#121	1	2	yes	—
91	C₂×C₄×C₁₇	#123	1	2	yes	—
92	C₂×C₄×C₁₇	#124	1	1	yes	characteristic
93	C₂×C₄×C₁₇	C₂×D₈	1	1	yes	characteristic
94	C₂×C₄×C₁₇	C₂×SD₁₆	1	1	yes	characteristic
95	C₂×C₄×C₁₇	C₂×Q₁₆	1	1	yes	characteristic
96	C₂×C₄×C₁₇	#135	1	1	yes	characteristic
97	C₂×C₄×C₁₇	AGL(1,ℤ/8)	1	2	yes	—
98	C₂×C₄×C₁₇	#137	1	1	yes	characteristic
99	C₂×C₄×C₁₇	#137	1	1	yes	characteristic
100	C₂×Dic₁₇	—	4	4	no	—
101	C₂×Dic₁₇	—	8	2	no	—
102	C₂×Dic₁₇	—	8	2	no	—
103	Dic₃₄	—	4	4	no	—
104	Dic₃₄	—	4	4	no	—
105	Dic₃₄	—	8	2	no	—
106	Dic₃₄	—	8	2	no	—
107	C₈×C₁₇	—	2	2	no	—
108	C₈×C₁₇	—	2	2	no	—
109	C₈×C₁₇	—	8	1	no	—
110	#3622	—	8	2	no	—
111	C₂×#500	—	17	1	no	—
112	#1981	—	17	2	no	—
113	#1939	D₁₇	1	1	yes	characteristic
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	—	4	2	no	—
125	C₄×C₁₇	—	8	1	no	—
126	C₄×C₁₇	#448	1	1	yes	characteristic
127	C₄×C₁₇	#451	1	1	yes	characteristic
128	C₄×C₁₇	#459	1	2	yes	—
129	C₄×C₁₇	#463	1	2	yes	—
130	C₄×C₁₇	#487	1	1	yes	characteristic
131	C₄×C₁₇	#491	1	1	yes	characteristic
132	Dic₁₇	—	8	4	no	—
133	C₂×C₄⋊Q₈	—	17	2	no	—
134	#500	—	34	4	no	—
135	C₂×#107	—	34	1	no	—
136	C₂×#107	—	34	1	no	—
137	C₂×C₄⋊C₈	—	2	1	no	—
138	C₂×C₄⋊C₈	—	17	2	no	—
139	C₂×C₄⋊C₈	C₂×D₁₇	1	1	yes	characteristic
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	C₂×D₁₇	1	1	yes	characteristic
144	#336	—	2	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	—	34	2	no	—
154	C₂²×C₈	—	2	1	no	—
155	C₂²×C₈	—	34	2	no	—
156	C₂²×C₈	C₂²⋊D₁₇	1	1	yes	characteristic
157	C₂²×C₈	C₂²⋊D₁₇	1	1	yes	characteristic
158	C₄⋊Q₈	—	17	4	no	—
159	C₄⋊Q₈	—	34	2	no	—
160	C₄⋊Q₈	—	34	2	no	—
161	C₄⋊Q₈	—	34	2	no	—
162	C₂×C₄⋊C₄	—	34	2	no	—
163	C₂×C₄⋊C₄	—	34	2	no	—
164	C₂×C₄⋊C₄	D₆₈	1	1	yes	characteristic
165	C₂×C₄⋊C₄	D₆₈	1	1	yes	characteristic
166	C₂×C₄²	—	2	1	no	—
167	C₂×C₄²	C₂²×D₁₇	1	1	yes	characteristic
168	#107	—	34	4	no	—
169	#107	—	34	4	no	—
170	C₄⋊C₈	—	2	2	no	—
171	C₄⋊C₈	—	4	2	no	—
172	C₄⋊C₈	—	34	2	no	—
173	C₄⋊C₈	—	34	2	no	—
174	#103	—	68	4	no	—
175	#103	—	68	4	no	—
176	C₄×C₈	—	2	2	no	—
177	#95	—	2	1	no	—
178	C₁₇	#10545	1	1	yes	characteristic, complemented, Sylow
179	C₂×Q₈	—	34	4	no	—
180	C₂×Q₈	—	68	2	no	—
181	C₂×Q₈	—	68	2	no	—
182	C₂×Q₈	—	68	2	no	—
183	C₂×Q₈	—	68	2	no	—
184	C₂×Q₈	—	68	2	no	—
185	C₂²×C₄	—	2	1	no	—
186	C₂²×C₄	—	2	1	no	—
187	C₂²×C₄	—	68	2	no	—
188	C₂²×C₄	C₂×D₆₈	1	1	yes	characteristic
189	C₂²×C₄	#63252	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	—	4	1	no	—
196	C₂×C₈	—	4	1	no	—
197	C₂×C₈	—	4	1	no	—
198	C₂×C₈	—	68	2	no	—
199	C₂×C₈	—	68	2	no	—
200	C₂×C₈	—	68	2	no	—
201	C₂×C₈	#63229	1	2	yes	—
202	C₂×C₈	#63231	1	2	yes	—
203	C₄⋊C₄	—	2	1	no	—
204	C₄⋊C₄	—	2	2	no	—
205	C₄⋊C₄	—	34	4	no	—
206	C₄⋊C₄	—	34	4	no	—
207	C₄⋊C₄	—	68	2	no	—
208	C₄⋊C₄	—	68	2	no	—
209	C₄⋊C₄	D₁₃₆	1	2	yes	—
210	C₄²	—	4	1	no	—
211	C₄²	—	4	1	no	—
212	C₄²	D₄×D₁₇	1	2	yes	—
213	C₄²	#63256	1	2	yes	—
214	C₂³	#10587616	1	1	yes	characteristic
215	Q₈	—	68	4	no	—
216	Q₈	—	68	4	no	—
217	Q₈	—	136	2	no	—
218	Q₈	—	136	2	no	—
219	C₂×C₄	—	2	1	no	—
220	C₂×C₄	—	2	1	no	—
221	C₂×C₄	—	2	2	no	—
222	C₂×C₄	—	4	1	no	—
223	C₂×C₄	—	4	1	no	—
224	C₂×C₄	—	4	1	no	—
225	C₂×C₄	—	4	1	no	—
226	C₂×C₄	—	4	1	no	—
227	C₂×C₄	—	68	4	no	—
228	C₂×C₄	—	136	2	no	—
229	C₂×C₄	—	136	2	no	—
230	C₂×C₄	#10587580	1	2	yes	—
231	C₂×C₄	#10587581	1	2	yes	—
232	C₂×C₄	C₂×D₁₃₆	1	1	yes	characteristic
233	C₂×C₄	#10587590	1	1	yes	characteristic
234	C₂×C₄	#10587594	1	2	yes	—
235	C₂×C₄	C₂×#63229	1	1	yes	characteristic
236	C₂×C₄	#10587623	1	1	yes	characteristic
237	C₂×C₄	C₂×#63231	1	1	yes	characteristic
238	C₂×C₄	#10587633	1	1	yes	characteristic
239	C₂×C₄	#10587637	1	1	yes	characteristic
240	C₂×C₄	#10587637	1	1	yes	characteristic
241	C₈	—	2	2	no	—
242	C₈	—	2	2	no	—
243	C₈	—	8	1	no	—
244	C₈	—	136	2	no	—
245	C₂²	#11762726	1	1	yes	characteristic
246	C₂²	#11763014	1	1	yes	characteristic
247	C₂²	#11763019	1	1	yes	characteristic
248	C₂²	#11763045	1	1	yes	characteristic
249	C₂²	#11763062	1	1	yes	characteristic
250	C₂²	#11763084	1	1	yes	characteristic
251	C₂²	#11763084	1	1	yes	characteristic
252	C₄	—	2	2	no	—
253	C₄	—	4	2	no	—
254	C₄	—	8	1	no	—
255	C₄	—	136	4	no	—
256	C₄	#11762947	1	1	yes	characteristic
257	C₄	#11762964	1	1	yes	characteristic
258	C₄	#11763005	1	2	yes	—
259	C₄	#11763025	1	2	yes	—
260	C₄	#11763161	1	1	yes	characteristic
261	C₄	#11763179	1	1	yes	characteristic
262	C₂	#14528173	1	1	yes	characteristic
263	C₂	#14528178	1	1	yes	characteristic
264	C₂	#14528297	1	1	yes	characteristic
265	C₂	#14528302	1	1	yes	characteristic
266	C₂	#14530030	1	1	yes	characteristic
267	C₂	#14530052	1	1	yes	characteristic
268	C₂	#14530095	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	#10587616
commutator, cosocle	C₂×C₄²×C₁₇	C₂³
Frattini	C₂×C₄²	C₂²×D₁₇
Fitting	C₁₇×#1939	C₂
socle, prime radical	C₂³×C₁₇	#121
hypercenter	#1939	D₁₇
hypocenter	C₁₇	#10545
LCS	G ▷ C₂×C₄²×C₁₇ ▷ C₂²×C₁₇ ▷ C₁₇
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ #1939

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
C₂×#11763214	2
#14529707	2
C₂×#11762554	2
#14526619	2
C₁₇×#1939	2
#10545	17