Subgroups — #16934343

Order 4352.

490 conjugacy classes of subgroups.

The subgroup lattice has 278 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄×#10587555	C₂	1	1	yes	maximal, characteristic
2	#14528901	C₂	1	1	yes	maximal, characteristic
3	#14526624	C₂	1	4	yes	maximal
4	C₁₇×#1744	C₂	1	1	yes	maximal, characteristic
5	C₂×#10587555	C₄	1	1	yes	characteristic
6	C₂×C₄×Dic₃₄	—	2	1	no	—
7	C₂×C₄⋊#3622	C₂²	1	2	yes	—
8	#11762715	—	2	1	no	—
9	#11762707	—	2	1	no	—
10	#11762707	C₄	1	2	yes	—
11	#11762679	C₄	1	1	yes	characteristic
12	C₄×C₄⋊Dic₁₇	—	2	1	no	—
13	C₄×C₄⋊Dic₁₇	C₂²	1	2	yes	—
14	#11762564	—	2	1	no	—
15	#11762564	—	2	1	no	—
16	C₂×C₄×C₈×C₁₇	C₂²	1	2	yes	—
17	C₄³×C₁₇	C₂²	1	1	yes	characteristic
18	C₁₇×#336	—	2	1	no	—
19	C₁₇×#336	—	2	1	no	—
20	C₂²×Dic₃₄	—	2	1	no	—
21	C₂²×C₈×C₁₇	—	2	2	no	—
22	C₂×C₄²×C₁₇	C₂×C₄	1	1	yes	characteristic
23	C₂×C₄²×C₁₇	C₂×C₄	1	1	yes	characteristic
24	C₂×C₄²×C₁₇	D₄	1	1	yes	characteristic
25	C₂×C₄²×C₁₇	D₄	1	1	yes	characteristic
26	C₂×C₄²×C₁₇	D₄	1	1	yes	characteristic
27	C₂×C₄²×C₁₇	D₄	1	1	yes	characteristic
28	C₂×C₄²×C₁₇	C₂³	1	1	yes	characteristic
29	C₂×C₄⋊Dic₁₇	—	2	1	no	—
30	C₂×C₄⋊Dic₁₇	—	2	1	no	—
31	C₂×C₄⋊Dic₁₇	—	2	1	no	—
32	C₂×C₄⋊Dic₁₇	C₂×C₄	1	2	yes	—
33	C₂×C₄⋊Dic₁₇	C₂×C₄	1	2	yes	—
34	C₂×#63225	—	2	2	no	—
35	C₂×C₄×Dic₁₇	—	2	2	no	—
36	C₂²×#3622	—	2	2	no	—
37	#10587555	—	2	2	no	—
38	#10587555	C₈	1	4	yes	complemented
39	C₄×Dic₃₄	—	4	4	no	—
40	C₄×C₈×C₁₇	—	2	2	no	—
41	C₄×C₈×C₁₇	Q₈	1	4	yes	complemented
42	#10587502	—	2	2	no	—
43	#10587502	—	2	2	no	—
44	C₄⋊#3622	—	2	4	no	—
45	C₂²×C₄×C₁₇	C₂²⋊C₄	1	1	yes	characteristic
46	C₂²×C₄×C₁₇	C₂²⋊C₄	1	1	yes	characteristic
47	C₂²×C₄×C₁₇	C₂²⋊C₄	1	1	yes	characteristic
48	C₂²×C₄×C₁₇	C₂²⋊C₄	1	1	yes	characteristic
49	C₂²×C₄×C₁₇	C₂²×C₄	1	1	yes	characteristic
50	C₂²×C₄×C₁₇	C₂×D₄	1	1	yes	characteristic
51	C₂²×C₄×C₁₇	C₂×D₄	1	1	yes	characteristic
52	C₂²×Dic₁₇	—	2	2	no	—
53	C₂²×Dic₁₇	—	2	2	no	—
54	C₂×Dic₃₄	—	2	4	no	—
55	C₂×Dic₃₄	—	4	2	no	—
56	C₂×Dic₃₄	—	4	2	no	—
57	C₂×C₈×C₁₇	—	2	4	no	—
58	C₂×C₈×C₁₇	—	4	2	no	—
59	C₂×C₈×C₁₇	—	4	2	no	—
60	C₄²×C₁₇	—	2	2	no	—
61	C₄²×C₁₇	—	2	2	no	—
62	C₄²×C₁₇	—	2	2	no	—
63	C₄²×C₁₇	—	4	1	no	—
64	C₄²×C₁₇	—	4	1	no	—
65	C₄²×C₁₇	C₂×C₈	1	2	yes	—
66	C₄²×C₁₇	M₁₆	1	2	yes	—
67	C₄²×C₁₇	C₂×Q₈	1	2	yes	—
68	C₄²×C₁₇	#41	1	2	yes	—
69	C₄⋊Dic₁₇	—	2	4	no	—
70	C₄⋊Dic₁₇	—	2	2	no	—
71	C₄⋊Dic₁₇	—	2	4	no	—
72	C₄⋊Dic₁₇	—	4	2	no	—
73	C₄⋊Dic₁₇	—	4	2	no	—
74	C₄⋊Dic₁₇	C₂×C₈	1	4	yes	—
75	#63225	—	4	4	no	—
76	C₄×Dic₁₇	—	4	4	no	—
77	C₂×#3622	—	4	2	no	—
78	C₂×#3622	—	4	2	no	—
79	C₂×#3622	—	4	2	no	—
80	#9873	—	17	1	no	maximal, Sylow
81	C₂³×C₁₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
82	C₂×C₄×C₁₇	—	2	1	no	—
83	C₂×C₄×C₁₇	—	2	1	no	—
84	C₂×C₄×C₁₇	—	2	1	no	—
85	C₂×C₄×C₁₇	—	2	1	no	—
86	C₂×C₄×C₁₇	—	2	1	no	—
87	C₂×C₄×C₁₇	—	2	1	no	—
88	C₂×C₄×C₁₇	—	2	1	no	—
89	C₂×C₄×C₁₇	—	2	1	no	—
90	C₂×C₄×C₁₇	—	2	1	no	—
91	C₂×C₄×C₁₇	—	2	1	no	—
92	C₂×C₄×C₁₇	—	2	1	no	—
93	C₂×C₄×C₁₇	—	2	1	no	—
94	C₂×C₄×C₁₇	C₂²⋊C₈	1	2	yes	—
95	C₂×C₄×C₁₇	C₂²⋊C₈	1	2	yes	—
96	C₂×C₄×C₁₇	C₄×Q₈	1	2	yes	—
97	C₂×C₄×C₁₇	C₂²⋊Q₈	1	2	yes	—
98	C₂×C₄×C₁₇	#124	1	1	yes	characteristic
99	C₂×C₄×C₁₇	C₄⋊Q₈	1	1	yes	characteristic
100	C₂×C₄×C₁₇	C₂²×C₈	1	1	yes	characteristic
101	C₂×C₄×C₁₇	C₂×M₁₆	1	1	yes	characteristic
102	C₂×C₄×C₁₇	#131	1	2	yes	—
103	C₂×C₄×C₁₇	AGL(1,ℤ/8)	1	2	yes	—
104	C₂×C₄×C₁₇	#137	1	2	yes	—
105	C₂×Dic₁₇	—	2	4	no	—
106	C₂×Dic₁₇	—	4	2	no	—
107	C₂×Dic₁₇	—	4	2	no	—
108	C₂×Dic₁₇	—	4	2	no	—
109	C₂×Dic₁₇	—	4	2	no	—
110	C₂×Dic₁₇	—	4	2	no	—
111	Dic₃₄	—	4	8	no	—
112	C₈×C₁₇	—	4	4	no	—
113	#3622	—	8	2	no	—
114	C₄×C₄⋊Q₈	—	17	1	no	—
115	#1831	—	17	1	no	—
116	#1744	D₁₇	1	1	yes	characteristic
117	#1279	—	17	4	no	—
118	C₂²×C₁₇	#391	1	1	yes	characteristic
119	C₂²×C₁₇	C₂×C₂²⋊C₈	1	1	yes	characteristic
120	C₂²×C₁₇	#408	1	1	yes	characteristic
121	C₂²×C₁₇	#418	1	1	yes	characteristic
122	C₂²×C₁₇	#419	1	1	yes	characteristic
123	C₂²×C₁₇	#421	1	1	yes	characteristic
124	C₂²×C₁₇	#421	1	1	yes	characteristic
125	C₄×C₁₇	—	2	2	no	—
126	C₄×C₁₇	—	2	2	no	—
127	C₄×C₁₇	—	2	2	no	—
128	C₄×C₁₇	—	4	1	no	—
129	C₄×C₁₇	—	4	1	no	—
130	C₄×C₁₇	C₈×Q₈	1	2	yes	—
131	C₄×C₁₇	#446	1	2	yes	—
132	C₄×C₁₇	#489	1	2	yes	—
133	C₄×C₁₇	#501	1	2	yes	—
134	Dic₁₇	—	4	4	no	—
135	Dic₁₇	—	8	2	no	—
136	C₂×C₄⋊Q₈	—	17	1	no	—
137	C₂×C₄×Q₈	—	34	1	no	—
138	C₂×C₄⋊C₈	—	17	2	no	—
139	C₂×C₄×C₈	C₂×D₁₇	1	2	yes	—
140	#391	—	34	1	no	—
141	#389	—	17	2	no	—
142	#389	—	34	1	no	—
143	#384	—	17	1	no	—
144	C₄×C₄⋊C₄	—	17	2	no	—
145	C₄×C₄⋊C₄	—	34	1	no	—
146	C₄³	C₂×D₁₇	1	1	yes	characteristic
147	#336	—	2	1	no	—
148	#336	—	2	1	no	—
149	#336	—	34	1	no	—
150	#336	—	34	1	no	—
151	C₂×C₁₇	#1478	1	1	yes	characteristic
152	C₂×C₁₇	#1479	1	1	yes	characteristic
153	C₂×C₁₇	#1496	1	1	yes	characteristic
154	C₂×C₁₇	#1497	1	1	yes	characteristic
155	C₂×C₁₇	#1984	1	1	yes	characteristic
156	C₂×C₁₇	#1993	1	1	yes	characteristic
157	C₂×C₁₇	#1995	1	1	yes	characteristic
158	C₂²×Q₈	—	34	1	no	—
159	C₂²×C₈	—	2	2	no	—
160	C₂²×C₈	—	34	2	no	—
161	C₄⋊Q₈	—	17	4	no	—
162	C₄⋊Q₈	—	34	2	no	—
163	C₄×Q₈	—	68	4	no	—
164	C₂×C₄⋊C₄	—	17	2	no	—
165	C₂×C₄⋊C₄	—	17	2	no	—
166	C₂×C₄⋊C₄	—	34	1	no	—
167	C₂×C₄⋊C₄	—	34	1	no	—
168	C₂×C₄⋊C₄	—	34	2	no	—
169	C₂×C₄⋊C₄	—	34	1	no	—
170	C₂×C₄²	—	34	2	no	—
171	C₂×C₄²	C₄×D₁₇	1	1	yes	characteristic
172	C₂×C₄²	C₄×D₁₇	1	1	yes	characteristic
173	C₂×C₄²	D₆₈	1	1	yes	characteristic
174	C₂×C₄²	D₆₈	1	1	yes	characteristic
175	C₂×C₄²	C₂²⋊D₁₇	1	1	yes	characteristic
176	C₂×C₄²	C₂²⋊D₁₇	1	1	yes	characteristic
177	C₂×C₄²	C₂²×D₁₇	1	1	yes	characteristic
178	C₄⋊C₈	—	34	4	no	—
179	C₄×C₈	—	2	2	no	—
180	C₄×C₈	Dic₃₄	1	4	yes	complemented
181	#95	—	34	2	no	—
182	#95	—	34	2	no	—
183	C₁₇	#9873	1	1	yes	characteristic, complemented, Sylow
184	C₂×Q₈	—	34	4	no	—
185	C₂×Q₈	—	68	2	no	—
186	C₂×Q₈	—	68	2	no	—
187	C₂²×C₄	—	34	2	no	—
188	C₂²×C₄	—	34	2	no	—
189	C₂²×C₄	#63227	1	1	yes	characteristic
190	C₂²×C₄	#63227	1	1	yes	characteristic
191	C₂²×C₄	#63227	1	1	yes	characteristic
192	C₂²×C₄	#63227	1	1	yes	characteristic
193	C₂²×C₄	C₂×C₄×D₁₇	1	1	yes	characteristic
194	C₂²×C₄	C₂×D₆₈	1	1	yes	characteristic
195	C₂²×C₄	C₂×C₂²⋊D₁₇	1	1	yes	characteristic
196	C₂×C₈	—	2	4	no	—
197	C₂×C₈	—	4	2	no	—
198	C₂×C₈	—	4	2	no	—
199	C₂×C₈	—	68	2	no	—
200	C₂×C₈	—	68	2	no	—
201	C₂×C₈	—	68	2	no	—
202	C₄⋊C₄	—	17	4	no	—
203	C₄⋊C₄	—	34	4	no	—
204	C₄⋊C₄	—	34	2	no	—
205	C₄⋊C₄	—	34	4	no	—
206	C₄⋊C₄	—	68	4	no	—
207	C₄⋊C₄	—	68	2	no	—
208	C₄⋊C₄	—	68	2	no	—
209	C₄²	—	2	2	no	—
210	C₄²	—	2	2	no	—
211	C₄²	—	2	2	no	—
212	C₄²	—	4	1	no	—
213	C₄²	—	4	1	no	—
214	C₄²	—	68	4	no	—
215	C₄²	C₈×D₁₇	1	2	yes	—
216	C₄²	#63218	1	2	yes	—
217	C₄²	C₂×Dic₃₄	1	2	yes	—
218	C₄²	#63252	1	2	yes	—
219	C₂³	C₂×#63227	1	1	yes	characteristic
220	Q₈	—	68	8	no	—
221	C₂×C₄	—	2	1	no	—
222	C₂×C₄	—	2	1	no	—
223	C₂×C₄	—	2	1	no	—
224	C₂×C₄	—	2	1	no	—
225	C₂×C₄	—	2	1	no	—
226	C₂×C₄	—	2	1	no	—
227	C₂×C₄	—	2	1	no	—
228	C₂×C₄	—	2	1	no	—
229	C₂×C₄	—	2	1	no	—
230	C₂×C₄	—	2	1	no	—
231	C₂×C₄	—	2	1	no	—
232	C₂×C₄	—	2	1	no	—
233	C₂×C₄	—	34	4	no	—
234	C₂×C₄	—	68	2	no	—
235	C₂×C₄	—	68	2	no	—
236	C₂×C₄	—	68	2	no	—
237	C₂×C₄	—	68	2	no	—
238	C₂×C₄	—	68	2	no	—
239	C₂×C₄	#10587491	1	2	yes	—
240	C₂×C₄	#10587491	1	2	yes	—
241	C₂×C₄	C₄×Dic₃₄	1	2	yes	—
242	C₂×C₄	#10587555	1	1	yes	characteristic
243	C₂×C₄	#10587561	1	1	yes	characteristic
244	C₂×C₄	C₂×C₈×D₁₇	1	1	yes	characteristic
245	C₂×C₄	C₂×#63218	1	1	yes	characteristic
246	C₂×C₄	#10587587	1	2	yes	—
247	C₂×C₄	#10587594	1	2	yes	—
248	C₂×C₄	#10587595	1	2	yes	—
249	C₂×C₄	#10587610	1	2	yes	—
250	C₈	—	4	4	no	—
251	C₈	—	136	2	no	—
252	C₂²	#11762715	1	1	yes	characteristic
253	C₂²	C₂×#10587491	1	1	yes	characteristic
254	C₂²	#11762775	1	1	yes	characteristic
255	C₂²	#11762815	1	1	yes	characteristic
256	C₂²	#11762819	1	1	yes	characteristic
257	C₂²	#11762827	1	1	yes	characteristic
258	C₂²	#11762827	1	1	yes	characteristic
259	C₄	—	2	2	no	—
260	C₄	—	2	2	no	—
261	C₄	—	2	2	no	—
262	C₄	—	4	1	no	—
263	C₄	—	4	1	no	—
264	C₄	—	68	4	no	—
265	C₄	—	136	2	no	—
266	C₄	C₈×Dic₃₄	1	2	yes	—
267	C₄	#11762939	1	2	yes	—
268	C₄	#11763169	1	2	yes	—
269	C₄	#11763218	1	2	yes	—
270	C₂	#14527176	1	1	yes	characteristic
271	C₂	#14527180	1	1	yes	characteristic
272	C₂	#14527254	1	1	yes	characteristic
273	C₂	#14527258	1	1	yes	characteristic
274	C₂	#14529720	1	1	yes	characteristic
275	C₂	#14529757	1	1	yes	characteristic
276	C₂	#14529765	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄	C₂×D₆₈
commutator	C₂×C₄×C₁₇	C₂²×C₈
Frattini	C₂×C₄²	C₂²×D₁₇
Fitting	C₁₇×#1744	C₂
socle, prime radical	C₂³×C₁₇	C₂×C₂²⋊C₄
cosocle	C₂×C₄²×C₁₇	C₂³
hypercenter	#1744	D₁₇
hypocenter	C₁₇	#9873
LCS	G ▷ C₂×C₄×C₁₇ ▷ C₂×C₁₇ ▷ C₁₇
UCS	C₁ ◁ C₂²×C₄ ◁ C₄³ ◁ #1744

Complements

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

N	H ≅ G/N	type
38. #10587555	C₈	semidirect
41. C₄×C₈×C₁₇	Q₈	semidirect
180. C₄×C₈	Dic₃₄	semidirect
183. C₁₇	#9873	semidirect

Maximal subgroups

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

subgroup	index
C₄×#10587555	2
#14528901	2
#14526624	2
C₁₇×#1744	2
#9873	17