Subgroups — #16933012

Order 4352.

370 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	#14529548	C₂	1	2	yes	maximal
2	#14528944	C₂	1	1	yes	maximal, characteristic
3	#14528651	C₂	1	1	yes	maximal, characteristic
4	#14526668	C₂	1	2	yes	maximal
5	C₂×C₁₇×#334	C₂	1	1	yes	maximal, characteristic
6	C₂×#10587489	—	2	1	no	—
7	C₂×#10587489	—	2	1	no	—
8	C₂×#10587488	—	2	2	no	—
9	C₂×C₄⋊#3622	C₂²	1	1	yes	characteristic
10	C₂×#10587486	—	2	1	no	—
11	#11762679	C₂²	1	2	yes	—
12	C₄×C₄⋊Dic₁₇	C₂²	1	1	yes	characteristic
13	#11762572	—	2	2	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	C₁₇×#334	C₄	1	4	yes	complemented
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×C₄⋊Dic₁₇	D₄	1	1	yes	characteristic
27	C₂×C₄×Dic₁₇	—	2	1	no	—
28	C₂²×#3622	—	2	1	no	—
29	C₁₇×C₄⋊C₈	—	2	2	no	—
30	C₁₇×C₄⋊C₈	C₂×C₄	1	4	yes	—
31	C₄×C₈×C₁₇	C₂×C₄	1	2	yes	—
32	C₄×C₈×C₁₇	Q₈	1	2	yes	—
33	#10587502	—	2	1	no	—
34	#10587489	—	4	2	no	—
35	#10587489	—	4	2	no	—
36	#10587488	—	4	4	no	—
37	#10587486	—	4	2	no	—
38	C₄⋊#3622	—	2	1	no	—
39	C₄⋊#3622	Q₈	1	2	yes	—
40	C₂²×C₄×C₁₇	C₂×D₄	1	1	yes	characteristic
41	C₂²×C₄×C₁₇	C₂×D₄	1	1	yes	characteristic
42	C₂²×C₄×C₁₇	#41	1	1	yes	characteristic
43	C₂²×Dic₁₇	—	2	1	no	—
44	C₂²×Dic₁₇	—	2	1	no	—
45	C₂²×Dic₁₇	—	4	2	no	—
46	C₂×C₈×C₁₇	—	2	1	no	—
47	C₂×C₈×C₁₇	—	2	1	no	—
48	C₂×C₈×C₁₇	—	2	4	no	—
49	C₂×C₈×C₁₇	—	4	2	no	—
50	C₂×C₈×C₁₇	—	4	2	no	—
51	C₂×C₈×C₁₇	C₄⋊C₄	1	2	yes	—
52	C₂×C₈×C₁₇	C₄⋊C₄	1	2	yes	—
53	C₂×C₈×C₁₇	SD₁₆	1	2	yes	—
54	C₂×C₈×C₁₇	SD₁₆	1	2	yes	—
55	C₄²×C₁₇	C₂²×C₄	1	1	yes	characteristic
56	C₄²×C₁₇	C₂×Q₈	1	1	yes	characteristic
57	C₄²×C₁₇	C₂×Q₈	1	1	yes	characteristic
58	C₄²×C₁₇	#41	1	1	yes	characteristic
59	C₄⋊Dic₁₇	—	2	2	no	—
60	C₄⋊Dic₁₇	—	2	2	no	—
61	C₄⋊Dic₁₇	—	4	2	no	—
62	C₄⋊Dic₁₇	—	4	2	no	—
63	C₄⋊Dic₁₇	—	4	2	no	—
64	C₄⋊Dic₁₇	—	4	2	no	—
65	C₄⋊Dic₁₇	—	8	2	no	—
66	C₄×Dic₁₇	—	4	1	no	—
67	C₄×Dic₁₇	—	4	1	no	—
68	C₂×#3622	—	2	2	no	—
69	C₂×#3622	—	4	1	no	—
70	C₂×#3622	—	4	1	no	—
71	#9621	—	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	C₂×C₄⋊C₄	1	1	yes	characteristic
76	C₂×C₄×C₁₇	C₄×D₄	1	1	yes	characteristic
77	C₂×C₄×C₁₇	C₄×Q₈	1	1	yes	characteristic
78	C₂×C₄×C₁₇	C₂²⋊Q₈	1	1	yes	characteristic
79	C₂×C₄×C₁₇	#125	1	1	yes	characteristic
80	C₂×C₄×C₁₇	C₄⋊Q₈	1	1	yes	characteristic
81	C₂×C₄×C₁₇	C₂×SD₁₆	1	1	yes	characteristic
82	C₂×C₄×C₁₇	C₂×SD₁₆	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×C₄×C₁₇	#137	1	2	yes	—
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	—	4	1	no	—
93	C₂×Dic₁₇	—	8	2	no	—
94	C₂×Dic₁₇	—	8	2	no	—
95	C₂×Dic₁₇	—	8	2	no	—
96	C₈×C₁₇	—	2	2	no	—
97	C₈×C₁₇	—	4	4	no	—
98	C₈×C₁₇	#106	1	4	yes	complemented
99	#3622	—	4	2	no	—
100	#1948	—	17	2	no	—
101	#1845	—	17	1	no	—
102	#1770	—	17	1	no	—
103	C₂×#334	D₁₇	1	1	yes	characteristic
104	#1296	—	17	2	no	—
105	C₂²×C₁₇	#389	1	1	yes	characteristic
106	C₂²×C₁₇	#460	1	1	yes	characteristic
107	C₂²×C₁₇	#461	1	1	yes	characteristic
108	C₂²×C₁₇	#469	1	1	yes	characteristic
109	C₂²×C₁₇	#470	1	1	yes	characteristic
110	C₂²×C₁₇	#472	1	1	yes	characteristic
111	C₂²×C₁₇	#473	1	1	yes	characteristic
112	C₄×C₁₇	—	2	1	no	—
113	C₄×C₁₇	—	2	1	no	—
114	C₄×C₁₇	C₂×#106	1	1	yes	characteristic
115	C₄×C₁₇	#428	1	1	yes	characteristic
116	C₄×C₁₇	#440	1	2	yes	—
117	C₄×C₁₇	#475	1	1	yes	characteristic
118	C₄×C₁₇	#476	1	1	yes	characteristic
119	C₄×C₁₇	#501	1	2	yes	—
120	Dic₁₇	—	8	1	no	—
121	Dic₁₇	—	8	1	no	—
122	Dic₁₇	—	16	2	no	—
123	C₂×#107	—	34	1	no	—
124	C₂×#107	—	34	1	no	—
125	C₂×#106	—	34	2	no	—
126	C₂×C₄⋊C₈	—	17	1	no	—
127	C₂×C₄⋊C₈	C₂×D₁₇	1	2	yes	—
128	C₂×#97	—	34	1	no	—
129	C₂×C₄×C₈	C₂×D₁₇	1	1	yes	characteristic
130	#384	—	17	2	no	—
131	C₄×C₄⋊C₄	—	17	1	no	—
132	#340	—	34	2	no	—
133	#336	—	34	1	no	—
134	#334	Dic₁₇	1	4	yes	complemented
135	C₂×C₁₇	#1688	1	1	yes	characteristic
136	C₂×C₁₇	#1690	1	1	yes	characteristic
137	C₂×C₁₇	#1693	1	1	yes	characteristic
138	C₂×C₁₇	#1695	1	1	yes	characteristic
139	C₂×C₁₇	#1944	1	1	yes	characteristic
140	C₂×C₁₇	#1948	1	1	yes	characteristic
141	C₂×C₁₇	#1952	1	1	yes	characteristic
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	C₂×C₄⋊C₄	—	17	1	no	—
147	C₂×C₄⋊C₄	—	17	1	no	—
148	C₂×C₄⋊C₄	—	34	2	no	—
149	C₂×C₄⋊C₄	—	34	2	no	—
150	C₂×C₄⋊C₄	—	34	2	no	—
151	C₂×C₄²	—	34	1	no	—
152	C₂×C₄²	C₂²×D₁₇	1	1	yes	characteristic
153	#107	—	68	2	no	—
154	#107	—	68	2	no	—
155	#106	—	68	4	no	—
156	C₄⋊C₈	—	2	2	no	—
157	C₄⋊C₈	—	17	2	no	—
158	C₄⋊C₈	—	34	1	no	—
159	C₄⋊C₈	C₂×Dic₁₇	1	4	yes	—
160	#97	—	68	2	no	—
161	C₄×C₈	Dic₃₄	1	2	yes	—
162	C₄×C₈	C₂×Dic₁₇	1	2	yes	—
163	#95	—	34	1	no	—
164	C₁₇	#9621	1	1	yes	characteristic, complemented, Sylow
165	C₂²×C₄	—	34	1	no	—
166	C₂²×C₄	—	34	1	no	—
167	C₂²×C₄	—	68	2	no	—
168	C₂²×C₄	C₂×D₆₈	1	1	yes	characteristic
169	C₂²×C₄	D₄×D₁₇	1	1	yes	characteristic
170	C₂²×C₄	#63256	1	1	yes	characteristic
171	C₂×C₈	—	2	1	no	—
172	C₂×C₈	—	2	1	no	—
173	C₂×C₈	—	2	4	no	—
174	C₂×C₈	—	4	2	no	—
175	C₂×C₈	—	4	2	no	—
176	C₂×C₈	—	34	2	no	—
177	C₂×C₈	—	68	1	no	—
178	C₂×C₈	—	68	1	no	—
179	C₂×C₈	#63219	1	2	yes	—
180	C₂×C₈	#63219	1	2	yes	—
181	C₂×C₈	C₄⋊Dic₁₇	1	2	yes	—
182	C₂×C₈	C₄⋊Dic₁₇	1	2	yes	—
183	C₄⋊C₄	—	34	2	no	—
184	C₄⋊C₄	—	34	2	no	—
185	C₄⋊C₄	—	68	2	no	—
186	C₄⋊C₄	—	68	2	no	—
187	C₄⋊C₄	—	68	2	no	—
188	C₄⋊C₄	—	68	2	no	—
189	C₄⋊C₄	—	136	2	no	—
190	C₄²	—	68	1	no	—
191	C₄²	—	68	1	no	—
192	C₄²	C₂×Dic₃₄	1	1	yes	characteristic
193	C₄²	#63254	1	1	yes	characteristic
194	C₄²	Q₈×D₁₇	1	1	yes	characteristic
195	C₄²	C₂²×Dic₁₇	1	1	yes	characteristic
196	C₂³	#10587581	1	1	yes	characteristic
197	C₂×C₄	—	2	1	no	—
198	C₂×C₄	—	2	1	no	—
199	C₂×C₄	—	68	1	no	—
200	C₂×C₄	—	68	1	no	—
201	C₂×C₄	—	68	1	no	—
202	C₂×C₄	—	68	1	no	—
203	C₂×C₄	—	68	1	no	—
204	C₂×C₄	—	68	1	no	—
205	C₂×C₄	—	136	2	no	—
206	C₂×C₄	—	136	2	no	—
207	C₂×C₄	—	136	2	no	—
208	C₂×C₄	#10587574	1	1	yes	characteristic
209	C₂×C₄	#10587576	1	1	yes	characteristic
210	C₂×C₄	#10587583	1	1	yes	characteristic
211	C₂×C₄	C₂×#63219	1	1	yes	characteristic
212	C₂×C₄	C₂×#63219	1	1	yes	characteristic
213	C₂×C₄	#10587594	1	1	yes	characteristic
214	C₂×C₄	#10587595	1	1	yes	characteristic
215	C₂×C₄	#10587600	1	2	yes	—
216	C₂×C₄	#10587601	1	2	yes	—
217	C₂×C₄	C₂×C₄⋊Dic₁₇	1	1	yes	characteristic
218	C₂×C₄	D₄×Dic₁₇	1	1	yes	characteristic
219	C₂×C₄	Q₈×Dic₁₇	1	1	yes	characteristic
220	C₈	—	2	2	no	—
221	C₈	—	4	4	no	—
222	C₈	—	68	2	no	—
223	C₈	#10587488	1	4	yes	complemented
224	C₂²	#11762708	1	1	yes	characteristic
225	C₂²	#11763010	1	1	yes	characteristic
226	C₂²	#11763015	1	1	yes	characteristic
227	C₂²	#11763063	1	1	yes	characteristic
228	C₂²	#11763068	1	1	yes	characteristic
229	C₂²	#11763080	1	1	yes	characteristic
230	C₂²	#11763085	1	1	yes	characteristic
231	C₄	—	2	1	no	—
232	C₄	—	2	1	no	—
233	C₄	—	272	2	no	—
234	C₄	—	136	1	no	—
235	C₄	—	136	1	no	—
236	C₄	C₂×#10587488	1	1	yes	characteristic
237	C₄	#11762851	1	1	yes	characteristic
238	C₄	#11762910	1	2	yes	—
239	C₄	#11763095	1	1	yes	characteristic
240	C₄	#11763100	1	1	yes	characteristic
241	C₄	#11763216	1	2	yes	—
242	C₂	#14528269	1	1	yes	characteristic
243	C₂	#14528279	1	1	yes	characteristic
244	C₂	#14528298	1	1	yes	characteristic
245	C₂	#14528308	1	1	yes	characteristic
246	C₂	#14529528	1	1	yes	characteristic
247	C₂	#14529548	1	1	yes	characteristic
248	C₂	#14529570	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	#10587581
commutator	C₄²×C₁₇	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₁₇
Fitting	C₂×C₁₇×#334	C₂
socle, prime radical	C₂³×C₁₇	#121
cosocle	C₂×C₄²×C₁₇	C₂³
hypercenter	C₂×#334	D₁₇
hypocenter	C₁₇	#9621
LCS	G ▷ C₄²×C₁₇ ▷ C₂²×C₁₇ ▷ C₁₇
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×#334

Complements

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

N	H ≅ G/N	type
17. C₁₇×#334	C₄	semidirect
98. C₈×C₁₇	#106	semidirect
134. #334	Dic₁₇	semidirect
164. C₁₇	#9621	semidirect
223. C₈	#10587488	semidirect

Maximal subgroups

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

subgroup	index
#14529548	2
#14528944	2
#14528651	2
#14526668	2
C₂×C₁₇×#334	2
#9621	17