Subgroups — #17981757

Order 4352.

404 conjugacy classes of subgroups.

The subgroup lattice has 308 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#14545928	C₂	1	1	yes	maximal, characteristic
2	C₄×C₄⋊#3633	C₂	1	1	yes	maximal, characteristic
3	C₂×#11763898	C₂	1	2	yes	maximal
4	#14545790	C₂	1	2	yes	maximal
5	C₄⋊C₈×#595	C₂	1	1	yes	maximal, characteristic
6	C₂×C₄⋊#3633	—	2	1	no	—
7	C₂×C₄⋊#3633	—	2	1	no	—
8	C₂×C₄⋊#3633	C₂²	1	2	yes	—
9	C₂×C₄⋊#3633	C₂²	1	1	yes	characteristic
10	C₂×C₄⋊#3633	C₂²	1	1	yes	characteristic
11	C₂×C₄×#3633	—	2	1	no	—
12	(C₂×#3633)⋊C₄	—	2	1	no	—
13	#11763898	—	2	4	no	—
14	#11763810	C₄	1	1	yes	characteristic
15	C₂×C₈×#595	C₂²	1	2	yes	—
16	C₄²×#595	C₂²	1	1	yes	characteristic
17	#11763674	C₄	1	2	yes	—
18	C₄⋊C₈×D₁₇	C₄	1	1	yes	characteristic
19	C₂²×#3633	—	2	1	no	—
20	C₂²×#3633	—	2	1	no	—
21	C₂²×#3633	—	2	2	no	—
22	C₂×C₄×#595	C₂×C₄	1	1	yes	characteristic
23	C₂×C₄×#595	D₄	1	1	yes	characteristic
24	C₂×C₄×#595	D₄	1	1	yes	characteristic
25	C₂×C₄×#595	Q₈	1	1	yes	characteristic
26	C₂×C₄×#595	Q₈	1	1	yes	characteristic
27	C₂×C₄×#595	C₂³	1	1	yes	characteristic
28	C₂×#63242	C₂×C₄	1	2	yes	—
29	C₄⋊#3633	—	2	2	no	—
30	C₄⋊#3633	—	2	1	no	—
31	C₄⋊#3633	—	2	1	no	—
32	C₄⋊#3633	—	2	2	no	—
33	C₄⋊#3633	—	4	2	no	—
34	C₄⋊#3633	—	4	2	no	—
35	C₄⋊#3633	D₄	1	2	yes	—
36	C₄⋊#3633	Q₈	1	2	yes	—
37	C₄×#3633	—	4	2	no	—
38	C₂×C₈×D₁₇	C₂×C₄	1	2	yes	—
39	C₄²×D₁₇	C₂×C₄	1	1	yes	characteristic
40	#10587544	—	2	1	no	—
41	C₄⋊#3624	—	2	1	no	—
42	C₈×#595	—	2	2	no	—
43	C₈×#595	—	2	2	no	—
44	C₁₇×C₄⋊C₈	C₈	1	1	yes	characteristic, complemented
45	#10587485	C₈	1	2	yes	complemented
46	C₄⋊#3622	C₈	1	1	yes	characteristic, complemented
47	C₂²×#595	C₄⋊C₄	1	1	yes	characteristic
48	C₂²×#595	C₄⋊C₄	1	1	yes	characteristic
49	C₂²×#595	C₂×D₄	1	1	yes	characteristic
50	C₂²×#595	C₂×Q₈	1	1	yes	characteristic
51	C₂×#3633	—	2	2	no	—
52	C₂×#3633	—	2	2	no	—
53	C₂×#3633	—	4	1	no	—
54	C₂×#3633	—	4	1	no	—
55	C₂×#3633	—	4	1	no	—
56	C₂×#3633	—	4	1	no	—
57	C₂×#3633	—	4	2	no	—
58	C₂×#3633	—	4	2	no	—
59	C₂×#3633	—	4	2	no	—
60	C₂×C₄×D₁₇	C₄⋊C₄	1	1	yes	characteristic
61	C₂×C₄×D₁₇	C₄⋊C₄	1	1	yes	characteristic
62	C₂×C₄×D₁₇	C₂²×C₄	1	1	yes	characteristic
63	C₂×#3624	—	2	2	no	—
64	C₄×#595	—	2	1	no	—
65	C₄×#595	—	2	1	no	—
66	C₄×#595	—	2	2	no	—
67	C₄×#595	—	2	1	no	—
68	C₄×#595	—	2	1	no	—
69	C₄×#595	—	4	1	no	—
70	C₄×#595	SD₁₆	1	2	yes	—
71	C₄×#595	SD₁₆	1	2	yes	—
72	C₄×#595	C₂×D₄	1	1	yes	characteristic
73	C₄×#595	C₂×Q₈	1	1	yes	characteristic
74	C₄×#595	#41	1	1	yes	characteristic
75	C₄×#595	#41	1	1	yes	characteristic
76	#63242	—	2	2	no	—
77	#63242	—	2	2	no	—
78	C₂×C₈×C₁₇	C₂×C₈	1	2	yes	—
79	C₄²×C₁₇	C₂×C₈	1	1	yes	characteristic
80	C₄×Dic₁₇	C₂×C₈	1	1	yes	characteristic
81	C₄×Dic₁₇	M₁₆	1	1	yes	characteristic
82	C₄×Dic₁₇	M₁₆	1	1	yes	characteristic
83	C₂×#3622	C₂×C₈	1	2	yes	—
84	C₈×D₁₇	—	2	2	no	—
85	C₈×D₁₇	—	2	2	no	—
86	#9602	—	17	1	no	maximal, Sylow
87	C₂²×D₁₇	C₂×C₄⋊C₄	1	1	yes	characteristic
88	C₂×#595	—	2	1	no	—
89	C₂×#595	—	2	1	no	—
90	C₂×#595	—	2	1	no	—
91	C₂×#595	—	2	1	no	—
92	C₂×#595	—	2	1	no	—
93	C₂×#595	—	2	1	no	—
94	C₂×#595	—	2	1	no	—
95	C₂×#595	#106	1	2	yes	—
96	C₂×#595	#121	1	1	yes	characteristic
97	C₂×#595	C₂²⋊Q₈	1	1	yes	characteristic
98	C₂×#595	#125	1	1	yes	characteristic
99	C₂×#595	C₄⋊Q₈	1	1	yes	characteristic
100	C₂×#595	C₂×SD₁₆	1	1	yes	characteristic
101	C₂×#595	C₂×SD₁₆	1	1	yes	characteristic
102	C₂×#595	AGL(1,ℤ/8)	1	1	yes	characteristic
103	C₂×#595	#137	1	1	yes	characteristic
104	#3633	—	4	2	no	—
105	#3633	—	4	2	no	—
106	#3633	—	8	2	no	—
107	C₂×C₄×C₁₇	C₄⋊C₈	1	1	yes	characteristic
108	C₂×C₄×C₁₇	C₄⋊C₈	1	1	yes	characteristic
109	C₂×C₄×C₁₇	C₂²×C₈	1	1	yes	characteristic
110	C₂×Dic₁₇	C₄⋊C₈	1	1	yes	characteristic
111	C₂×Dic₁₇	C₄⋊C₈	1	1	yes	characteristic
112	C₂×Dic₁₇	C₂×M₁₆	1	1	yes	characteristic
113	C₄×D₁₇	—	2	1	no	—
114	C₄×D₁₇	—	2	1	no	—
115	C₄×D₁₇	#106	1	2	yes	—
116	C₄×D₁₇	#108	1	2	yes	—
117	C₄×D₁₇	C₄×D₄	1	1	yes	characteristic
118	C₄×D₁₇	C₄×Q₈	1	1	yes	characteristic
119	C₄×D₁₇	#131	1	1	yes	characteristic
120	C₄×D₁₇	#131	1	1	yes	characteristic
121	#3624	—	4	2	no	—
122	C₈×C₁₇	—	2	2	no	—
123	#3622	—	2	2	no	—
124	#1842	—	17	1	no	—
125	C₄×C₄⋊C₈	—	17	1	no	—
126	C₄×C₄⋊C₈	—	17	1	no	—
127	C₂×#334	—	17	2	no	—
128	#1277	—	17	2	no	—
129	C₂²×C₁₇	C₂×C₄⋊C₈	1	1	yes	characteristic
130	C₂×D₁₇	#389	1	1	yes	characteristic
131	C₂×D₁₇	#424	1	1	yes	characteristic
132	C₂×D₁₇	C₂×#106	1	1	yes	characteristic
133	C₂×D₁₇	#428	1	1	yes	characteristic
134	C₂×D₁₇	C₂×#108	1	1	yes	characteristic
135	C₂×D₁₇	#430	1	1	yes	characteristic
136	#595	—	2	2	no	—
137	#595	—	2	1	no	—
138	#595	—	2	1	no	—
139	#595	—	4	1	no	—
140	#595	#460	1	1	yes	characteristic
141	#595	#461	1	1	yes	characteristic
142	#595	#475	1	1	yes	characteristic
143	#595	#476	1	1	yes	characteristic
144	C₄×C₁₇	—	2	1	no	—
145	C₄×C₁₇	#334	1	2	yes	complemented
146	C₄×C₁₇	C₈×D₄	1	1	yes	characteristic
147	C₄×C₁₇	C₈×Q₈	1	1	yes	characteristic
148	Dic₁₇	—	2	1	no	—
149	Dic₁₇	#334	1	2	yes	complemented
150	Dic₁₇	#436	1	1	yes	characteristic
151	Dic₁₇	#446	1	1	yes	characteristic
152	C₂×C₄⋊C₈	—	17	1	no	—
153	C₂×C₄⋊C₈	—	17	1	no	—
154	C₂×C₄⋊C₈	—	17	1	no	—
155	C₂×C₄⋊C₈	—	17	1	no	—
156	C₂×C₄⋊C₈	—	17	2	no	—
157	C₂×C₄⋊C₈	—	34	1	no	—
158	C₂×C₄⋊C₈	—	34	1	no	—
159	C₂×C₄×C₈	—	17	2	no	—
160	C₂×C₄×C₈	—	34	1	no	—
161	C₄³	—	17	1	no	—
162	#336	—	17	2	no	—
163	#336	—	34	1	no	—
164	#334	—	34	4	no	—
165	C₂×C₁₇	C₂×#334	1	1	yes	characteristic
166	C₂×C₁₇	#1565	1	1	yes	characteristic
167	C₂×C₁₇	#1939	1	1	yes	characteristic
168	D₁₇	#1570	1	1	yes	characteristic
169	D₁₇	#1574	1	1	yes	characteristic
170	D₁₇	#1948	1	1	yes	characteristic
171	D₁₇	#1950	1	1	yes	characteristic
172	C₂²×C₈	—	17	2	no	—
173	C₂²×C₈	—	17	2	no	—
174	C₂²×C₈	—	34	1	no	—
175	C₂²×C₈	—	34	1	no	—
176	C₂²×C₈	—	34	2	no	—
177	C₂×C₄²	—	17	1	no	—
178	C₂×C₄²	—	17	1	no	—
179	C₂×C₄²	—	17	1	no	—
180	C₂×C₄²	—	17	1	no	—
181	C₂×C₄²	—	17	1	no	—
182	C₂×C₄²	—	17	1	no	—
183	C₂×C₄²	—	17	1	no	—
184	C₄⋊C₈	—	17	2	no	—
185	C₄⋊C₈	—	17	2	no	—
186	C₄⋊C₈	—	17	2	no	—
187	C₄⋊C₈	—	17	1	no	—
188	C₄⋊C₈	—	34	1	no	—
189	C₄⋊C₈	—	34	1	no	—
190	C₄⋊C₈	—	34	1	no	—
191	C₄⋊C₈	—	34	1	no	—
192	C₄⋊C₈	—	34	2	no	—
193	C₄⋊C₈	—	34	2	no	—
194	C₄⋊C₈	—	68	2	no	—
195	C₄⋊C₈	—	68	2	no	—
196	C₄⋊C₈	#3633	1	1	yes	characteristic, complemented
197	C₄×C₈	—	34	2	no	—
198	C₄×C₈	—	34	2	no	—
199	C₄×C₈	—	68	2	no	—
200	C₁₇	#9602	1	1	yes	characteristic, complemented, Sylow
201	C₂²×C₄	—	17	1	no	—
202	C₂²×C₄	—	17	1	no	—
203	C₂²×C₄	—	17	1	no	—
204	C₂²×C₄	—	17	1	no	—
205	C₂²×C₄	—	17	1	no	—
206	C₂²×C₄	—	17	1	no	—
207	C₂²×C₄	—	17	1	no	—
208	C₂×C₈	—	17	2	no	—
209	C₂×C₈	—	34	2	no	—
210	C₂×C₈	—	34	2	no	—
211	C₂×C₈	—	34	2	no	—
212	C₂×C₈	—	34	2	no	—
213	C₂×C₈	—	34	2	no	—
214	C₂×C₈	—	34	2	no	—
215	C₂×C₈	—	34	2	no	—
216	C₂×C₈	—	68	1	no	—
217	C₂×C₈	—	68	1	no	—
218	C₂×C₈	—	68	1	no	—
219	C₂×C₈	—	68	1	no	—
220	C₂×C₈	—	68	2	no	—
221	C₂×C₈	—	68	2	no	—
222	C₂×C₈	—	68	2	no	—
223	C₂×C₈	C₂×#3633	1	2	yes	—
224	C₄²	—	17	2	no	—
225	C₄²	—	17	2	no	—
226	C₄²	—	17	1	no	—
227	C₄²	—	17	1	no	—
228	C₄²	—	17	1	no	—
229	C₄²	—	17	1	no	—
230	C₄²	—	17	1	no	—
231	C₄²	—	17	1	no	—
232	C₄²	—	17	1	no	—
233	C₄²	—	34	1	no	—
234	C₄²	—	34	1	no	—
235	C₄²	—	34	2	no	—
236	C₄²	—	34	1	no	—
237	C₄²	—	34	1	no	—
238	C₄²	—	68	1	no	—
239	C₄²	C₂×#3633	1	1	yes	characteristic
240	C₂³	—	17	1	no	—
241	C₂×C₄	—	17	1	no	—
242	C₂×C₄	—	17	1	no	—
243	C₂×C₄	—	17	1	no	—
244	C₂×C₄	—	17	1	no	—
245	C₂×C₄	—	17	1	no	—
246	C₂×C₄	—	17	1	no	—
247	C₂×C₄	—	17	1	no	—
248	C₂×C₄	—	17	1	no	—
249	C₂×C₄	—	17	2	no	—
250	C₂×C₄	—	17	2	no	—
251	C₂×C₄	—	17	1	no	—
252	C₂×C₄	—	17	1	no	—
253	C₂×C₄	—	17	1	no	—
254	C₂×C₄	—	17	1	no	—
255	C₂×C₄	—	17	2	no	—
256	C₂×C₄	—	17	1	no	—
257	C₂×C₄	—	17	1	no	—
258	C₂×C₄	—	17	1	no	—
259	C₂×C₄	—	34	1	no	—
260	C₂×C₄	—	34	1	no	—
261	C₂×C₄	—	34	1	no	—
262	C₂×C₄	—	34	1	no	—
263	C₂×C₄	—	34	1	no	—
264	C₂×C₄	—	34	1	no	—
265	C₂×C₄	—	34	1	no	—
266	C₂×C₄	—	34	1	no	—
267	C₂×C₄	—	34	1	no	—
268	C₂×C₄	C₄⋊#3633	1	1	yes	characteristic
269	C₂×C₄	C₄⋊#3633	1	1	yes	characteristic
270	C₂×C₄	C₂²×#3633	1	1	yes	characteristic
271	C₈	—	2	2	no	—
272	C₈	—	34	2	no	—
273	C₈	—	68	2	no	—
274	C₈	—	68	2	no	—
275	C₈	—	68	2	no	—
276	C₈	—	136	2	no	—
277	C₂²	—	17	1	no	—
278	C₂²	—	17	1	no	—
279	C₂²	—	17	1	no	—
280	C₂²	—	17	1	no	—
281	C₂²	—	17	1	no	—
282	C₂²	—	17	1	no	—
283	C₂²	C₂×C₄⋊#3633	1	1	yes	characteristic
284	C₄	—	2	1	no	—
285	C₄	—	17	1	no	—
286	C₄	—	17	1	no	—
287	C₄	—	17	1	no	—
288	C₄	—	17	1	no	—
289	C₄	—	17	1	no	—
290	C₄	—	17	1	no	—
291	C₄	—	17	2	no	—
292	C₄	—	34	2	no	—
293	C₄	—	34	1	no	—
294	C₄	—	34	1	no	—
295	C₄	—	34	1	no	—
296	C₄	—	68	1	no	—
297	C₄	#11763898	1	2	yes	complemented
298	C₄	D₄×#3633	1	1	yes	characteristic
299	C₄	Q₈×#3633	1	1	yes	characteristic
300	C₂	—	17	1	no	—
301	C₂	—	17	1	no	—
302	C₂	—	17	1	no	—
303	C₂	—	17	1	no	—
304	C₂	C₂×#11763898	1	1	yes	characteristic
305	C₂	#14545909	1	1	yes	characteristic
306	C₂	#14545933	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×C₄⋊#3633
commutator	C₂×C₄×C₁₇	C₂²×C₈
Frattini	C₂×C₄	C₂²×#3633
Fitting	C₁₇×C₄⋊C₈	C₈
socle	C₂²×C₁₇	C₂×C₄⋊C₈
prime radical	C₂²×D₁₇	C₂×C₄⋊C₄
cosocle	C₂×C₄×#595	C₂³
hypercenter	C₄⋊C₈	#3633
hypocenter	C₁₇	#9602
LCS	G ▷ C₂×C₄×C₁₇ ▷ C₂×C₁₇ ▷ C₁₇
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₄⋊C₈

Complements

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

N	H ≅ G/N	type
44. C₁₇×C₄⋊C₈	C₈	semidirect
45. #10587485	C₈	semidirect
46. C₄⋊#3622	C₈	semidirect
145. C₄×C₁₇	#334	semidirect
149. Dic₁₇	#334	semidirect
196. C₄⋊C₈	#3633	semidirect
200. C₁₇	#9602	semidirect
297. C₄	#11763898	semidirect

Maximal subgroups

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

subgroup	index
#14545928	2
C₄×C₄⋊#3633	2
C₂×#11763898	2
#14545790	2
C₄⋊C₈×#595	2
#9602	17