Subgroups — #16934458

Order 4352.

450 conjugacy classes of subgroups.

The subgroup lattice has 294 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#14537147	C₂	1	1	yes	maximal, characteristic
2	#14528936	C₂	1	1	yes	maximal, characteristic
3	#14526668	C₂	1	2	yes	maximal
4	#14526668	C₂	1	2	yes	maximal
5	C₄×C₁₇×C₄⋊C₈	C₂	1	1	yes	maximal, characteristic
6	C₂×#10587556	—	2	1	no	—
7	C₂×C₄⋊#3622	—	2	1	no	—
8	C₂×C₄⋊#3622	—	2	1	no	—
9	C₂×C₄⋊#3622	C₂²	1	2	yes	—
10	#11762707	—	2	1	no	—
11	#11762707	—	2	1	no	—
12	#11762707	C₄	1	2	yes	—
13	#11762707	C₄	1	2	yes	—
14	#11762679	C₂²	1	1	yes	characteristic
15	#11762679	C₂²	1	1	yes	characteristic
16	#11762675	—	2	1	no	—
17	C₂×C₁₇×C₄⋊C₈	C₂²	1	2	yes	—
18	C₂×C₄×C₈×C₁₇	—	2	1	no	—
19	C₄³×C₁₇	C₂²	1	1	yes	characteristic
20	C₁₇×#336	—	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	—	2	1	no	—
33	C₂×C₄⋊Dic₁₇	C₂×C₄	1	2	yes	—
34	C₂×C₄⋊Dic₁₇	C₂×C₄	1	2	yes	—
35	C₂×#63225	—	2	2	no	—
36	C₂×#63225	—	2	2	no	—
37	C₂²×#3622	—	2	2	no	—
38	#10587556	—	4	2	no	—
39	#10587556	—	4	2	no	—
40	C₁₇×C₄⋊C₈	—	2	2	no	—
41	C₁₇×C₄⋊C₈	Q₈	1	4	yes	—
42	C₄×C₈×C₁₇	—	4	2	no	—
43	#10587502	—	2	2	no	—
44	#10587502	—	2	2	no	—
45	C₄⋊#3622	—	2	4	no	—
46	C₄⋊#3622	—	4	2	no	—
47	C₄⋊#3622	—	4	2	no	—
48	C₂²×C₄×C₁₇	C₂²⋊C₄	1	1	yes	characteristic
49	C₂²×C₄×C₁₇	C₂²⋊C₄	1	1	yes	characteristic
50	C₂²×C₄×C₁₇	C₂²⋊C₄	1	1	yes	characteristic
51	C₂²×C₄×C₁₇	C₂²⋊C₄	1	1	yes	characteristic
52	C₂²×C₄×C₁₇	C₂²×C₄	1	1	yes	characteristic
53	C₂²×C₄×C₁₇	C₂×D₄	1	1	yes	characteristic
54	C₂²×C₄×C₁₇	C₂×D₄	1	1	yes	characteristic
55	C₂²×Dic₁₇	—	2	2	no	—
56	C₂²×Dic₁₇	—	2	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	1	no	—
62	C₄²×C₁₇	—	2	1	no	—
63	C₄²×C₁₇	—	4	1	no	—
64	C₄²×C₁₇	—	4	1	no	—
65	C₄²×C₁₇	D₈	1	2	yes	—
66	C₄²×C₁₇	SD₁₆	1	2	yes	—
67	C₄²×C₁₇	SD₁₆	1	2	yes	—
68	C₄²×C₁₇	Q₁₆	1	2	yes	—
69	C₄²×C₁₇	C₂×Q₈	1	2	yes	—
70	C₄²×C₁₇	#41	1	2	yes	—
71	C₄⋊Dic₁₇	—	4	1	no	—
72	C₄⋊Dic₁₇	—	4	1	no	—
73	C₄⋊Dic₁₇	—	4	1	no	—
74	C₄⋊Dic₁₇	—	4	1	no	—
75	C₄⋊Dic₁₇	—	4	2	no	—
76	C₄⋊Dic₁₇	—	4	1	no	—
77	C₄⋊Dic₁₇	—	4	1	no	—
78	C₄⋊Dic₁₇	—	4	2	no	—
79	C₄⋊Dic₁₇	—	4	2	no	—
80	#63225	—	4	4	no	—
81	#63225	—	4	4	no	—
82	C₂×#3622	—	4	2	no	—
83	C₂×#3622	—	4	2	no	—
84	C₂×#3622	—	4	2	no	—
85	#9900	—	17	1	no	maximal, Sylow
86	C₂³×C₁₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
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	—	2	1	no	—
95	C₂×C₄×C₁₇	—	2	1	no	—
96	C₂×C₄×C₁₇	—	2	1	no	—
97	C₂×C₄×C₁₇	#102	1	2	yes	—
98	C₂×C₄×C₁₇	#102	1	2	yes	—
99	C₂×C₄×C₁₇	#103	1	2	yes	—
100	C₂×C₄×C₁₇	#103	1	2	yes	—
101	C₂×C₄×C₁₇	C₄×Q₈	1	2	yes	—
102	C₂×C₄×C₁₇	C₂²⋊Q₈	1	2	yes	—
103	C₂×C₄×C₁₇	#124	1	1	yes	characteristic
104	C₂×C₄×C₁₇	C₄⋊Q₈	1	1	yes	characteristic
105	C₂×C₄×C₁₇	C₂×D₈	1	1	yes	characteristic
106	C₂×C₄×C₁₇	C₂×SD₁₆	1	1	yes	characteristic
107	C₂×C₄×C₁₇	C₂×SD₁₆	1	1	yes	characteristic
108	C₂×C₄×C₁₇	C₂×Q₁₆	1	1	yes	characteristic
109	C₂×C₄×C₁₇	#135	1	2	yes	—
110	C₂×C₄×C₁₇	AGL(1,ℤ/8)	1	1	yes	characteristic
111	C₂×C₄×C₁₇	#137	1	1	yes	characteristic
112	C₂×Dic₁₇	—	4	2	no	—
113	C₂×Dic₁₇	—	4	2	no	—
114	C₂×Dic₁₇	—	4	2	no	—
115	C₂×Dic₁₇	—	4	2	no	—
116	C₂×Dic₁₇	—	4	2	no	—
117	C₂×Dic₁₇	—	4	2	no	—
118	C₈×C₁₇	—	4	4	no	—
119	#3622	—	8	2	no	—
120	#2859	—	17	1	no	—
121	#1842	—	17	1	no	—
122	C₄×C₄⋊C₈	D₁₇	1	1	yes	characteristic
123	#1296	—	17	2	no	—
124	#1296	—	17	2	no	—
125	C₂²×C₁₇	#391	1	1	yes	characteristic
126	C₂²×C₁₇	#413	1	1	yes	characteristic
127	C₂²×C₁₇	#413	1	1	yes	characteristic
128	C₂²×C₁₇	C₂×#102	1	1	yes	characteristic
129	C₂²×C₁₇	C₂×#103	1	1	yes	characteristic
130	C₂²×C₁₇	#416	1	1	yes	characteristic
131	C₂²×C₁₇	#417	1	1	yes	characteristic
132	C₄×C₁₇	—	2	2	no	—
133	C₄×C₁₇	—	2	2	no	—
134	C₄×C₁₇	—	2	2	no	—
135	C₄×C₁₇	—	2	2	no	—
136	C₄×C₁₇	—	2	2	no	—
137	C₄×C₁₇	#478	1	2	yes	—
138	C₄×C₁₇	#479	1	2	yes	—
139	C₄×C₁₇	#486	1	1	yes	characteristic
140	C₄×C₁₇	#487	1	1	yes	characteristic
141	C₄×C₁₇	#498	1	1	yes	characteristic
142	C₄×C₁₇	#500	1	1	yes	characteristic
143	Dic₁₇	—	8	2	no	—
144	Dic₁₇	—	8	2	no	—
145	C₂×#125	—	34	1	no	—
146	C₂×C₄⋊C₈	—	17	2	no	—
147	C₂×C₄⋊C₈	—	34	1	no	—
148	C₂×C₄⋊C₈	—	34	1	no	—
149	C₂×C₄⋊C₈	C₂×D₁₇	1	2	yes	—
150	C₂×C₄×C₈	—	2	1	no	—
151	#389	—	17	2	no	—
152	#389	—	17	2	no	—
153	#389	—	34	1	no	—
154	#389	—	34	1	no	—
155	#384	—	17	1	no	—
156	#384	—	17	1	no	—
157	#382	—	34	1	no	—
158	C₄³	C₂×D₁₇	1	1	yes	characteristic
159	#336	—	2	1	no	—
160	C₂×C₁₇	#1540	1	1	yes	characteristic
161	C₂×C₁₇	#1542	1	1	yes	characteristic
162	C₂×C₁₇	#1547	1	1	yes	characteristic
163	C₂×C₁₇	#1550	1	1	yes	characteristic
164	C₂×C₁₇	#1989	1	1	yes	characteristic
165	C₂×C₁₇	#1990	1	1	yes	characteristic
166	C₂×C₁₇	#1998	1	1	yes	characteristic
167	C₂²×C₈	—	2	2	no	—
168	C₂²×C₈	—	34	2	no	—
169	#125	—	68	2	no	—
170	#125	—	68	2	no	—
171	C₂×C₄⋊C₄	—	17	2	no	—
172	C₂×C₄⋊C₄	—	17	2	no	—
173	C₂×C₄⋊C₄	—	34	1	no	—
174	C₂×C₄⋊C₄	—	34	1	no	—
175	C₂×C₄⋊C₄	—	34	1	no	—
176	C₂×C₄⋊C₄	—	34	1	no	—
177	C₂×C₄⋊C₄	—	34	2	no	—
178	C₂×C₄⋊C₄	—	34	2	no	—
179	C₂×C₄²	C₄×D₁₇	1	1	yes	characteristic
180	C₂×C₄²	C₄×D₁₇	1	1	yes	characteristic
181	C₂×C₄²	D₆₈	1	1	yes	characteristic
182	C₂×C₄²	D₆₈	1	1	yes	characteristic
183	C₂×C₄²	C₂²⋊D₁₇	1	1	yes	characteristic
184	C₂×C₄²	C₂²⋊D₁₇	1	1	yes	characteristic
185	C₂×C₄²	C₂²×D₁₇	1	1	yes	characteristic
186	C₄⋊C₈	—	2	2	no	—
187	C₄⋊C₈	—	34	4	no	—
188	C₄⋊C₈	—	68	2	no	—
189	C₄⋊C₈	—	68	2	no	—
190	C₄⋊C₈	Dic₃₄	1	4	yes	—
191	C₄×C₈	—	4	2	no	—
192	#95	—	34	2	no	—
193	#95	—	34	2	no	—
194	C₁₇	#9900	1	1	yes	characteristic, complemented, Sylow
195	C₂²×C₄	—	34	2	no	—
196	C₂²×C₄	—	34	2	no	—
197	C₂²×C₄	#63227	1	1	yes	characteristic
198	C₂²×C₄	#63227	1	1	yes	characteristic
199	C₂²×C₄	#63227	1	1	yes	characteristic
200	C₂²×C₄	#63227	1	1	yes	characteristic
201	C₂²×C₄	C₂×C₄×D₁₇	1	1	yes	characteristic
202	C₂²×C₄	C₂×D₆₈	1	1	yes	characteristic
203	C₂²×C₄	C₂×C₂²⋊D₁₇	1	1	yes	characteristic
204	C₂×C₈	—	2	4	no	—
205	C₂×C₈	—	4	2	no	—
206	C₂×C₈	—	4	2	no	—
207	C₂×C₈	—	68	2	no	—
208	C₂×C₈	—	68	2	no	—
209	C₂×C₈	—	68	2	no	—
210	C₄⋊C₄	—	68	2	no	—
211	C₄⋊C₄	—	68	2	no	—
212	C₄⋊C₄	—	68	4	no	—
213	C₄⋊C₄	—	68	4	no	—
214	C₄⋊C₄	—	68	1	no	—
215	C₄⋊C₄	—	68	1	no	—
216	C₄⋊C₄	—	68	1	no	—
217	C₄⋊C₄	—	68	1	no	—
218	C₄⋊C₄	—	68	1	no	—
219	C₄⋊C₄	—	68	1	no	—
220	C₄⋊C₄	—	68	2	no	—
221	C₄²	—	2	1	no	—
222	C₄²	—	2	1	no	—
223	C₄²	—	2	2	no	—
224	C₄²	—	4	1	no	—
225	C₄²	—	4	1	no	—
226	C₄²	#63219	1	2	yes	—
227	C₄²	#63219	1	2	yes	—
228	C₄²	D₁₃₆	1	2	yes	—
229	C₄²	Dic₆₈	1	2	yes	—
230	C₄²	C₂×Dic₃₄	1	2	yes	—
231	C₄²	#63252	1	2	yes	—
232	C₂³	C₂×#63227	1	1	yes	characteristic
233	C₂×C₄	—	2	1	no	—
234	C₂×C₄	—	2	1	no	—
235	C₂×C₄	—	2	1	no	—
236	C₂×C₄	—	2	1	no	—
237	C₂×C₄	—	2	1	no	—
238	C₂×C₄	—	2	1	no	—
239	C₂×C₄	—	2	1	no	—
240	C₂×C₄	—	2	1	no	—
241	C₂×C₄	—	2	1	no	—
242	C₂×C₄	—	2	1	no	—
243	C₂×C₄	—	68	2	no	—
244	C₂×C₄	—	68	2	no	—
245	C₂×C₄	—	68	2	no	—
246	C₂×C₄	—	68	2	no	—
247	C₂×C₄	—	68	2	no	—
248	C₂×C₄	—	68	2	no	—
249	C₂×C₄	#10587487	1	2	yes	—
250	C₂×C₄	#10587487	1	2	yes	—
251	C₂×C₄	#10587492	1	2	yes	—
252	C₂×C₄	#10587492	1	2	yes	—
253	C₂×C₄	C₄×Dic₃₄	1	2	yes	—
254	C₂×C₄	#10587555	1	1	yes	characteristic
255	C₂×C₄	#10587561	1	1	yes	characteristic
256	C₂×C₄	C₂×#63219	1	1	yes	characteristic
257	C₂×C₄	C₂×#63219	1	1	yes	characteristic
258	C₂×C₄	C₂×D₁₃₆	1	1	yes	characteristic
259	C₂×C₄	C₂×Dic₆₈	1	1	yes	characteristic
260	C₂×C₄	#10587610	1	2	yes	—
261	C₂×C₄	#10587618	1	1	yes	characteristic
262	C₂×C₄	#10587628	1	1	yes	characteristic
263	C₂×C₄	#10587636	1	2	yes	—
264	C₈	—	4	4	no	—
265	C₈	—	136	2	no	—
266	C₂²	#11762715	1	1	yes	characteristic
267	C₂²	#11762791	1	1	yes	characteristic
268	C₂²	#11762791	1	1	yes	characteristic
269	C₂²	C₂×#10587492	1	1	yes	characteristic
270	C₂²	C₂×#10587487	1	1	yes	characteristic
271	C₂²	#11762803	1	1	yes	characteristic
272	C₂²	#11762808	1	1	yes	characteristic
273	C₄	—	2	2	no	—
274	C₄	—	2	2	no	—
275	C₄	—	2	2	no	—
276	C₄	—	2	2	no	—
277	C₄	—	2	2	no	—
278	C₄	—	136	2	no	—
279	C₄	—	136	2	no	—
280	C₄	#11763109	1	2	yes	—
281	C₄	#11763116	1	2	yes	—
282	C₄	#11763156	1	1	yes	characteristic
283	C₄	#11763160	1	1	yes	characteristic
284	C₄	#11763207	1	1	yes	characteristic
285	C₄	#11763214	1	1	yes	characteristic
286	C₂	#14527431	1	1	yes	characteristic
287	C₂	#14527439	1	1	yes	characteristic
288	C₂	#14527459	1	1	yes	characteristic
289	C₂	#14527472	1	1	yes	characteristic
290	C₂	#14529741	1	1	yes	characteristic
291	C₂	#14529745	1	1	yes	characteristic
292	C₂	#14529777	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	C₂×#63227
commutator	C₂²×C₄×C₁₇	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₁₇
Fitting	C₄×C₁₇×C₄⋊C₈	C₂
socle, prime radical	C₂³×C₁₇	C₂×C₂²⋊C₄
cosocle	C₂×C₄²×C₁₇	C₂³
hypercenter	C₄×C₄⋊C₈	D₁₇
hypocenter	C₁₇	#9900
LCS	G ▷ C₂²×C₄×C₁₇ ▷ C₂²×C₁₇ ▷ C₁₇
UCS	C₁ ◁ C₂³ ◁ C₄³ ◁ C₄×C₄⋊C₈

Complements

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

N	H ≅ G/N	type
194. C₁₇	#9900	semidirect

Maximal subgroups

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

subgroup	index
#14537147	2
#14528936	2
#14526668	2
C₄×C₁₇×C₄⋊C₈	2
#9900	17