Subgroups — #16933554

Order 4352.

486 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	#14529708	C₂	1	2	yes	maximal
2	C₈×C₄⋊Dic₁₇	C₂	1	1	yes	maximal, characteristic
3	C₂×#11762562	C₂	1	1	yes	maximal, characteristic
4	C₂×#11762555	C₂	1	2	yes	maximal
5	C₂×C₁₇×#500	C₂	1	1	yes	maximal, characteristic
6	C₂×C₄⋊#3622	C₂²	1	2	yes	—
7	C₂×C₄⋊#3622	C₂²	1	1	yes	characteristic
8	C₂×#10587506	—	2	4	no	—
9	C₂×C₈×Dic₁₇	—	2	1	no	—
10	C₄×C₄⋊Dic₁₇	C₂²	1	1	yes	characteristic
11	#11762563	—	2	1	no	—
12	#11762562	—	2	2	no	—
13	#11762555	—	2	4	no	—
14	C₂×C₁₇×C₄⋊Q₈	C₂²	1	2	yes	—
15	C₁₇×#500	—	2	2	no	—
16	C₁₇×#500	C₄	1	4	yes	complemented
17	C₂×C₁₇×#107	—	2	1	no	—
18	C₂×C₁₇×#107	—	2	1	no	—
19	C₂×C₄×C₈×C₁₇	C₂²	1	1	yes	characteristic
20	C₂²×Q₈×C₁₇	—	2	2	no	—
21	C₂²×C₈×C₁₇	D₄	1	1	yes	characteristic
22	C₂²×C₈×C₁₇	D₄	1	1	yes	characteristic
23	C₁₇×C₄⋊Q₈	—	2	2	no	—
24	C₁₇×C₄⋊Q₈	—	2	2	no	—
25	C₁₇×C₄⋊Q₈	—	2	2	no	—
26	C₁₇×C₄⋊Q₈	C₂×C₄	1	4	yes	—
27	C₂×C₄⋊C₄×C₁₇	—	2	2	no	—
28	C₂×C₄⋊C₄×C₁₇	—	2	2	no	—
29	C₂×C₄²×C₁₇	C₂³	1	1	yes	characteristic
30	C₂×C₄⋊Dic₁₇	D₄	1	2	yes	—
31	C₂×C₄×Dic₁₇	—	2	1	no	—
32	C₂²×#3622	—	2	1	no	—
33	C₂²×#3622	—	2	2	no	—
34	C₁₇×#107	—	2	4	no	—
35	C₁₇×#107	—	2	4	no	—
36	C₄×C₈×C₁₇	C₂×C₄	1	2	yes	—
37	C₄×C₈×C₁₇	D₄	1	2	yes	—
38	#10587506	—	4	8	no	—
39	#10587502	—	2	1	no	—
40	C₈×Dic₁₇	—	4	2	no	—
41	C₄⋊#3622	—	2	2	no	—
42	C₄⋊#3622	—	2	2	no	—
43	C₄⋊#3622	—	2	1	no	—
44	C₄⋊#3622	D₄	1	2	yes	—
45	C₂×Q₈×C₁₇	—	2	4	no	—
46	C₂×Q₈×C₁₇	—	4	2	no	—
47	C₂×Q₈×C₁₇	—	4	4	no	—
48	C₂×Q₈×C₁₇	—	4	4	no	—
49	C₂²×C₄×C₁₇	—	4	2	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	#41	1	1	yes	characteristic
53	C₂²×Dic₁₇	—	2	2	no	—
54	C₂×C₈×C₁₇	—	2	1	no	—
55	C₂×C₈×C₁₇	—	2	1	no	—
56	C₂×C₈×C₁₇	C₂²⋊C₄	1	2	yes	—
57	C₂×C₈×C₁₇	C₂²⋊C₄	1	2	yes	—
58	C₂×C₈×C₁₇	SD₁₆	1	2	yes	—
59	C₂×C₈×C₁₇	Q₁₆	1	2	yes	—
60	C₄⋊C₄×C₁₇	—	2	4	no	—
61	C₄⋊C₄×C₁₇	—	2	4	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×D₄	1	1	yes	characteristic
66	C₄²×C₁₇	C₂×D₄	1	1	yes	characteristic
67	C₄²×C₁₇	#41	1	1	yes	characteristic
68	C₄⋊Dic₁₇	—	2	2	no	—
69	C₄⋊Dic₁₇	D₈	1	4	yes	—
70	C₄×Dic₁₇	—	4	1	no	—
71	C₄×Dic₁₇	—	4	1	no	—
72	C₂×#3622	—	2	2	no	—
73	C₂×#3622	—	4	1	no	—
74	C₂×#3622	—	4	1	no	—
75	C₂×#3622	—	4	2	no	—
76	C₂×#3622	—	4	4	no	—
77	#9731	—	17	1	no	maximal, Sylow
78	C₂³×C₁₇	#121	1	1	yes	characteristic
79	Q₈×C₁₇	—	4	8	no	—
80	Q₈×C₁₇	—	8	2	no	—
81	Q₈×C₁₇	—	8	2	no	—
82	C₂×C₄×C₁₇	—	2	2	no	—
83	C₂×C₄×C₁₇	—	4	4	no	—
84	C₂×C₄×C₁₇	—	8	4	no	—
85	C₂×C₄×C₁₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
86	C₂×C₄×C₁₇	C₄×D₄	1	1	yes	characteristic
87	C₂×C₄×C₁₇	C₄×D₄	1	1	yes	characteristic
88	C₂×C₄×C₁₇	#121	1	1	yes	characteristic
89	C₂×C₄×C₁₇	#124	1	1	yes	characteristic
90	C₂×C₄×C₁₇	#127	1	1	yes	characteristic
91	C₂×C₄×C₁₇	C₂×D₈	1	2	yes	—
92	C₂×C₄×C₁₇	C₂×SD₁₆	1	1	yes	characteristic
93	C₂×C₄×C₁₇	C₂×Q₁₆	1	1	yes	characteristic
94	C₂×C₄×C₁₇	#135	1	2	yes	—
95	C₂×C₄×C₁₇	#137	1	1	yes	characteristic
96	C₂×C₄×C₁₇	#137	1	1	yes	characteristic
97	C₂×Dic₁₇	—	2	4	no	—
98	C₂×Dic₁₇	—	4	2	no	—
99	C₂×Dic₁₇	—	4	2	no	—
100	C₈×C₁₇	—	2	2	no	—
101	C₈×C₁₇	#103	1	4	yes	complemented
102	#3622	—	4	2	no	—
103	#3622	—	8	2	no	—
104	C₂×#500	D₁₇	1	1	yes	characteristic
105	#1981	—	17	2	no	—
106	C₈×C₄⋊C₄	—	17	1	no	—
107	C₂×#335	—	17	1	no	—
108	C₂×#332	—	17	2	no	—
109	C₂²×C₁₇	#390	1	1	yes	characteristic
110	C₂²×C₁₇	#461	1	1	yes	characteristic
111	C₂²×C₁₇	#462	1	1	yes	characteristic
112	C₂²×C₁₇	#466	1	2	yes	—
113	C₂²×C₁₇	#473	1	2	yes	—
114	C₄×C₁₇	—	2	1	no	—
115	C₄×C₁₇	—	2	1	no	—
116	C₄×C₁₇	—	8	4	no	—
117	C₄×C₁₇	C₂×#103	1	1	yes	characteristic
118	C₄×C₁₇	#419	1	1	yes	characteristic
119	C₄×C₁₇	C₄×D₈	1	2	yes	—
120	C₄×C₁₇	#461	1	1	yes	characteristic
121	C₄×C₁₇	#462	1	1	yes	characteristic
122	C₄×C₁₇	#493	1	1	yes	characteristic
123	C₄×C₁₇	#495	1	1	yes	characteristic
124	Dic₁₇	—	4	4	no	—
125	C₂×C₄⋊Q₈	C₂×D₁₇	1	2	yes	—
126	#500	—	2	2	no	—
127	#500	Dic₁₇	1	4	yes	complemented
128	C₂×#107	—	2	1	no	—
129	C₂×#107	—	2	1	no	—
130	C₂×C₄⋊C₈	—	17	1	no	—
131	C₂×C₄⋊C₈	—	17	2	no	—
132	C₂×#103	—	34	4	no	—
133	C₂×C₄×C₈	—	34	1	no	—
134	C₂×C₄×C₈	C₂×D₁₇	1	1	yes	characteristic
135	C₄×C₄⋊C₄	—	17	1	no	—
136	#336	—	34	1	no	—
137	#335	—	34	2	no	—
138	#332	—	34	4	no	—
139	C₂×C₁₇	#1673	1	2	yes	—
140	C₂×C₁₇	#1674	1	2	yes	—
141	C₂×C₁₇	#1970	1	1	yes	characteristic
142	C₂×C₁₇	#1979	1	1	yes	characteristic
143	C₂×C₁₇	#1981	1	1	yes	characteristic
144	C₂²×Q₈	—	2	2	no	—
145	C₂²×C₈	—	34	1	no	—
146	C₂²×C₈	—	34	2	no	—
147	C₂²×C₈	C₂²⋊D₁₇	1	1	yes	characteristic
148	C₂²×C₈	C₂²⋊D₁₇	1	1	yes	characteristic
149	C₄⋊Q₈	—	2	2	no	—
150	C₄⋊Q₈	—	2	2	no	—
151	C₄⋊Q₈	—	2	2	no	—
152	C₄⋊Q₈	C₂×Dic₁₇	1	4	yes	—
153	C₂×C₄⋊C₄	—	2	2	no	—
154	C₂×C₄⋊C₄	—	2	2	no	—
155	C₂×C₄⋊C₄	—	17	2	no	—
156	C₂×C₄²	—	34	1	no	—
157	C₂×C₄²	C₂²×D₁₇	1	1	yes	characteristic
158	#107	—	2	4	no	—
159	#107	—	2	4	no	—
160	C₄⋊C₈	—	17	2	no	—
161	C₄⋊C₈	—	34	2	no	—
162	C₄⋊C₈	—	34	1	no	—
163	C₄⋊C₈	—	34	2	no	—
164	#103	—	68	8	no	—
165	C₄×C₈	—	68	2	no	—
166	C₄×C₈	C₂×Dic₁₇	1	2	yes	—
167	C₄×C₈	C₂²⋊D₁₇	1	2	yes	—
168	#95	—	34	1	no	—
169	C₁₇	#9731	1	1	yes	characteristic, complemented, Sylow
170	C₂×Q₈	—	2	4	no	—
171	C₂×Q₈	—	4	2	no	—
172	C₂×Q₈	—	4	4	no	—
173	C₂×Q₈	—	4	4	no	—
174	C₂²×C₄	—	4	2	no	—
175	C₂²×C₄	—	34	2	no	—
176	C₂²×C₄	D₄×D₁₇	1	1	yes	characteristic
177	C₂²×C₄	#63254	1	1	yes	characteristic
178	C₂²×C₄	C₂×C₂²⋊D₁₇	1	1	yes	characteristic
179	C₂×C₈	—	2	1	no	—
180	C₂×C₈	—	2	1	no	—
181	C₂×C₈	—	34	2	no	—
182	C₂×C₈	—	68	1	no	—
183	C₂×C₈	—	68	1	no	—
184	C₂×C₈	—	68	4	no	—
185	C₂×C₈	—	68	2	no	—
186	C₂×C₈	#63230	1	2	yes	—
187	C₂×C₈	#63231	1	2	yes	—
188	C₂×C₈	C₂²⋊Dic₁₇	1	2	yes	—
189	C₂×C₈	C₂²⋊Dic₁₇	1	2	yes	—
190	C₄⋊C₄	—	2	4	no	—
191	C₄⋊C₄	—	2	4	no	—
192	C₄⋊C₄	—	4	2	no	—
193	C₄⋊C₄	—	4	2	no	—
194	C₄⋊C₄	—	17	4	no	—
195	C₄⋊C₄	—	34	2	no	—
196	C₄²	—	68	1	no	—
197	C₄²	—	68	1	no	—
198	C₄²	D₄×D₁₇	1	1	yes	characteristic
199	C₄²	#63254	1	1	yes	characteristic
200	C₄²	C₂²×Dic₁₇	1	1	yes	characteristic
201	C₄²	C₂×C₂²⋊D₁₇	1	1	yes	characteristic
202	C₂³	#10587624	1	1	yes	characteristic
203	Q₈	—	4	8	no	—
204	Q₈	—	8	2	no	—
205	Q₈	—	8	2	no	—
206	C₂×C₄	—	2	2	no	—
207	C₂×C₄	—	4	4	no	—
208	C₂×C₄	—	8	4	no	—
209	C₂×C₄	—	34	4	no	—
210	C₂×C₄	—	68	2	no	—
211	C₂×C₄	—	68	2	no	—
212	C₂×C₄	D₈×D₁₇	1	2	yes	—
213	C₂×C₄	#10587598	1	2	yes	—
214	C₂×C₄	D₄×Dic₁₇	1	1	yes	characteristic
215	C₂×C₄	D₄×Dic₁₇	1	1	yes	characteristic
216	C₂×C₄	#10587622	1	1	yes	characteristic
217	C₂×C₄	#10587624	1	1	yes	characteristic
218	C₂×C₄	#10587626	1	1	yes	characteristic
219	C₂×C₄	C₂×#63230	1	1	yes	characteristic
220	C₂×C₄	#10587628	1	1	yes	characteristic
221	C₂×C₄	#10587628	1	1	yes	characteristic
222	C₂×C₄	C₂×#63231	1	1	yes	characteristic
223	C₂×C₄	C₂×C₂²⋊Dic₁₇	1	1	yes	characteristic
224	C₈	—	2	2	no	—
225	C₈	—	68	2	no	—
226	C₈	—	136	2	no	—
227	C₈	#10587506	1	4	yes	complemented
228	C₂²	#11762712	1	1	yes	characteristic
229	C₂²	#11763011	1	1	yes	characteristic
230	C₂²	#11763016	1	1	yes	characteristic
231	C₂²	#11763042	1	2	yes	—
232	C₂²	#11763081	1	2	yes	—
233	C₄	—	2	1	no	—
234	C₄	—	2	1	no	—
235	C₄	—	8	4	no	—
236	C₄	—	68	4	no	—
237	C₄	C₂×#10587506	1	1	yes	characteristic
238	C₄	#11762818	1	1	yes	characteristic
239	C₄	D₈×Dic₁₇	1	2	yes	—
240	C₄	#11763011	1	1	yes	characteristic
241	C₄	#11763016	1	1	yes	characteristic
242	C₄	#11763184	1	1	yes	characteristic
243	C₄	#11763192	1	1	yes	characteristic
244	C₂	#14528170	1	2	yes	—
245	C₂	#14528175	1	2	yes	—
246	C₂	#14529651	1	1	yes	characteristic
247	C₂	#14529700	1	1	yes	characteristic
248	C₂	#14529708	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	#10587624
commutator	C₄²×C₁₇	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₁₇
Fitting	C₂×C₁₇×#500	C₂
socle, prime radical	C₂³×C₁₇	#121
cosocle	C₂×C₄²×C₁₇	C₂³
hypercenter	C₂×#500	D₁₇
hypocenter	C₁₇	#9731
LCS	G ▷ C₄²×C₁₇ ▷ C₂²×C₁₇ ▷ C₁₇
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×#500

Complements

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

N	H ≅ G/N	type
16. C₁₇×#500	C₄	semidirect
101. C₈×C₁₇	#103	semidirect
127. #500	Dic₁₇	semidirect
169. C₁₇	#9731	semidirect
227. C₈	#10587506	semidirect

Maximal subgroups

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

subgroup	index
#14529708	2
C₈×C₄⋊Dic₁₇	2
C₂×#11762562	2
C₂×#11762555	2
C₂×C₁₇×#500	2
#9731	17