Subgroups — #17955836

Order 4352.

404 conjugacy classes of subgroups.

The subgroup lattice has 286 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#14544716	C₂	1	1	yes	maximal, characteristic
2	C₄×C₄⋊#3624	C₂	1	1	yes	maximal, characteristic
3	C₂×#11763672	C₂	1	2	yes	maximal
4	#14543614	C₂	1	2	yes	maximal
5	C₄⋊C₈×Dic₁₇	C₂	1	1	yes	maximal, characteristic
6	C₂×C₄⋊#3624	—	2	1	no	—
7	C₂×C₄⋊#3624	—	2	1	no	—
8	C₂×C₄⋊#3624	C₂²	1	2	yes	—
9	C₂×C₄⋊#3624	C₂²	1	1	yes	characteristic
10	C₂×C₄⋊#3624	C₂²	1	1	yes	characteristic
11	C₂×C₄×#3624	—	2	1	no	—
12	#11763675	—	2	1	no	—
13	#11763672	—	2	4	no	—
14	C₂×C₄⋊#3622	C₄	1	1	yes	characteristic
15	C₂×C₈×Dic₁₇	C₂²	1	2	yes	—
16	C₄²×Dic₁₇	C₂²	1	1	yes	characteristic
17	#11762563	C₄	1	2	yes	—
18	C₂×C₁₇×C₄⋊C₈	C₄	1	1	yes	characteristic
19	C₂²×#3624	—	2	1	no	—
20	C₂²×#3624	—	2	1	no	—
21	C₂²×#3624	—	2	2	no	—
22	C₂²×C₈×C₁₇	C₂×C₄	1	2	yes	—
23	C₂×C₄²×C₁₇	C₂×C₄	1	1	yes	characteristic
24	C₂×C₄×Dic₁₇	C₂×C₄	1	1	yes	characteristic
25	C₂×C₄×Dic₁₇	D₄	1	1	yes	characteristic
26	C₂×C₄×Dic₁₇	D₄	1	1	yes	characteristic
27	C₂×C₄×Dic₁₇	Q₈	1	1	yes	characteristic
28	C₂×C₄×Dic₁₇	Q₈	1	1	yes	characteristic
29	C₂×C₄×Dic₁₇	C₂³	1	1	yes	characteristic
30	C₂²×#3622	C₂×C₄	1	2	yes	—
31	C₄⋊#3624	—	2	1	no	—
32	C₄⋊#3624	—	2	1	no	—
33	C₄⋊#3624	—	2	2	no	—
34	C₄⋊#3624	—	2	2	no	—
35	C₄⋊#3624	—	4	2	no	—
36	C₄⋊#3624	—	4	2	no	—
37	C₄⋊#3624	D₄	1	2	yes	—
38	C₄⋊#3624	Q₈	1	2	yes	—
39	C₄×#3624	—	4	2	no	—
40	C₁₇×C₄⋊C₈	C₈	1	4	yes	complemented
41	C₈×Dic₁₇	—	2	2	no	—
42	C₈×Dic₁₇	—	2	2	no	—
43	C₄⋊#3622	—	2	2	no	—
44	C₂²×C₄×C₁₇	C₄⋊C₄	1	1	yes	characteristic
45	C₂²×C₄×C₁₇	C₄⋊C₄	1	1	yes	characteristic
46	C₂²×C₄×C₁₇	C₂²×C₄	1	1	yes	characteristic
47	C₂²×Dic₁₇	C₄⋊C₄	1	1	yes	characteristic
48	C₂²×Dic₁₇	C₄⋊C₄	1	1	yes	characteristic
49	C₂²×Dic₁₇	C₂×D₄	1	1	yes	characteristic
50	C₂²×Dic₁₇	C₂×Q₈	1	1	yes	characteristic
51	C₂×#3624	—	2	2	no	—
52	C₂×#3624	—	2	2	no	—
53	C₂×#3624	—	4	1	no	—
54	C₂×#3624	—	4	1	no	—
55	C₂×#3624	—	4	1	no	—
56	C₂×#3624	—	4	1	no	—
57	C₂×#3624	—	4	2	no	—
58	C₂×#3624	—	4	2	no	—
59	C₂×#3624	—	4	2	no	—
60	C₂×C₈×C₁₇	—	2	2	no	—
61	C₂×C₈×C₁₇	—	2	2	no	—
62	C₂×C₈×C₁₇	C₂×C₈	1	4	yes	—
63	C₄²×C₁₇	C₂×C₈	1	2	yes	—
64	C₄²×C₁₇	M₁₆	1	2	yes	—
65	C₄×Dic₁₇	—	2	2	no	—
66	C₄×Dic₁₇	—	2	1	no	—
67	C₄×Dic₁₇	—	2	1	no	—
68	C₄×Dic₁₇	—	2	1	no	—
69	C₄×Dic₁₇	—	2	1	no	—
70	C₄×Dic₁₇	—	4	1	no	—
71	C₄×Dic₁₇	D₈	1	2	yes	—
72	C₄×Dic₁₇	Q₁₆	1	2	yes	—
73	C₄×Dic₁₇	C₂×D₄	1	1	yes	characteristic
74	C₄×Dic₁₇	C₂×Q₈	1	1	yes	characteristic
75	C₄×Dic₁₇	#41	1	1	yes	characteristic
76	C₄×Dic₁₇	#41	1	1	yes	characteristic
77	C₂×#3622	—	2	2	no	—
78	C₂×#3622	—	2	2	no	—
79	C₂×#3622	—	2	2	no	—
80	#9601	—	17	1	no	maximal, Sylow
81	C₂³×C₁₇	C₂×C₄⋊C₄	1	1	yes	characteristic
82	C₂×C₄×C₁₇	—	2	1	no	—
83	C₂×C₄×C₁₇	—	2	1	no	—
84	C₂×C₄×C₁₇	C₄⋊C₈	1	2	yes	—
85	C₂×C₄×C₁₇	C₄⋊C₈	1	2	yes	—
86	C₂×C₄×C₁₇	#107	1	2	yes	—
87	C₂×C₄×C₁₇	#108	1	2	yes	—
88	C₂×C₄×C₁₇	C₄×D₄	1	1	yes	characteristic
89	C₂×C₄×C₁₇	C₄×Q₈	1	1	yes	characteristic
90	C₂×C₄×C₁₇	C₂²×C₈	1	1	yes	characteristic
91	C₂×C₄×C₁₇	C₂×M₁₆	1	1	yes	characteristic
92	C₂×C₄×C₁₇	#131	1	1	yes	characteristic
93	C₂×C₄×C₁₇	#131	1	1	yes	characteristic
94	C₂×Dic₁₇	—	2	1	no	—
95	C₂×Dic₁₇	—	2	1	no	—
96	C₂×Dic₁₇	—	2	1	no	—
97	C₂×Dic₁₇	—	2	1	no	—
98	C₂×Dic₁₇	—	2	1	no	—
99	C₂×Dic₁₇	—	2	1	no	—
100	C₂×Dic₁₇	—	2	1	no	—
101	C₂×Dic₁₇	#107	1	2	yes	—
102	C₂×Dic₁₇	#121	1	1	yes	characteristic
103	C₂×Dic₁₇	C₂²⋊Q₈	1	1	yes	characteristic
104	C₂×Dic₁₇	#125	1	1	yes	characteristic
105	C₂×Dic₁₇	C₄⋊Q₈	1	1	yes	characteristic
106	C₂×Dic₁₇	C₂×D₈	1	1	yes	characteristic
107	C₂×Dic₁₇	C₂×Q₁₆	1	1	yes	characteristic
108	C₂×Dic₁₇	AGL(1,ℤ/8)	1	1	yes	characteristic
109	C₂×Dic₁₇	#137	1	1	yes	characteristic
110	#3624	—	4	2	no	—
111	#3624	—	4	2	no	—
112	#3624	—	8	2	no	—
113	C₈×C₁₇	—	2	4	no	—
114	#3622	—	4	2	no	—
115	#1842	—	17	1	no	—
116	C₄×C₄⋊C₈	—	17	1	no	—
117	C₄×C₄⋊C₈	—	17	1	no	—
118	C₂×#335	—	17	2	no	—
119	#1277	—	17	2	no	—
120	C₂²×C₁₇	#389	1	1	yes	characteristic
121	C₂²×C₁₇	C₂×C₄⋊C₈	1	1	yes	characteristic
122	C₂²×C₁₇	#424	1	1	yes	characteristic
123	C₂²×C₁₇	C₂×#107	1	1	yes	characteristic
124	C₂²×C₁₇	#428	1	1	yes	characteristic
125	C₂²×C₁₇	C₂×#108	1	1	yes	characteristic
126	C₂²×C₁₇	#430	1	1	yes	characteristic
127	C₄×C₁₇	—	2	2	no	—
128	C₄×C₁₇	#335	1	4	yes	complemented
129	C₄×C₁₇	C₈×D₄	1	1	yes	characteristic
130	C₄×C₁₇	#436	1	1	yes	characteristic
131	C₄×C₁₇	C₈×Q₈	1	1	yes	characteristic
132	C₄×C₁₇	#446	1	1	yes	characteristic
133	Dic₁₇	—	2	2	no	—
134	Dic₁₇	—	2	1	no	—
135	Dic₁₇	—	2	1	no	—
136	Dic₁₇	—	4	1	no	—
137	Dic₁₇	#459	1	1	yes	characteristic
138	Dic₁₇	#462	1	1	yes	characteristic
139	Dic₁₇	#474	1	1	yes	characteristic
140	Dic₁₇	#477	1	1	yes	characteristic
141	C₂×C₄⋊C₈	—	17	1	no	—
142	C₂×C₄⋊C₈	—	17	1	no	—
143	C₂×C₄⋊C₈	—	17	2	no	—
144	C₂×C₄⋊C₈	—	17	1	no	—
145	C₂×C₄⋊C₈	—	34	1	no	—
146	C₂×C₄⋊C₈	—	34	1	no	—
147	C₂×C₄⋊C₈	#595	1	1	yes	characteristic
148	C₂×C₄×C₈	—	17	2	no	—
149	C₂×C₄×C₈	—	34	1	no	—
150	C₄³	—	17	1	no	—
151	#336	—	17	2	no	—
152	#336	—	34	1	no	—
153	#335	—	34	4	no	—
154	C₂×C₁₇	C₂×#335	1	1	yes	characteristic
155	C₂×C₁₇	#1565	1	1	yes	characteristic
156	C₂×C₁₇	#1571	1	1	yes	characteristic
157	C₂×C₁₇	#1574	1	1	yes	characteristic
158	C₂×C₁₇	#1939	1	1	yes	characteristic
159	C₂×C₁₇	#1947	1	1	yes	characteristic
160	C₂×C₁₇	#1950	1	1	yes	characteristic
161	C₂²×C₈	—	17	2	no	—
162	C₂²×C₈	—	34	1	no	—
163	C₂²×C₈	—	34	1	no	—
164	C₂²×C₈	—	34	2	no	—
165	C₂²×C₈	C₂×#595	1	2	yes	—
166	C₂×C₄²	—	17	1	no	—
167	C₂×C₄²	—	17	1	no	—
168	C₂×C₄²	—	17	1	no	—
169	C₂×C₄²	—	17	1	no	—
170	C₂×C₄²	—	17	1	no	—
171	C₂×C₄²	—	17	1	no	—
172	C₂×C₄²	C₂×#595	1	1	yes	characteristic
173	C₄⋊C₈	—	17	2	no	—
174	C₄⋊C₈	—	17	2	no	—
175	C₄⋊C₈	—	34	1	no	—
176	C₄⋊C₈	—	34	1	no	—
177	C₄⋊C₈	—	34	2	no	—
178	C₄⋊C₈	—	34	2	no	—
179	C₄⋊C₈	—	34	2	no	—
180	C₄⋊C₈	—	68	2	no	—
181	C₄⋊C₈	—	68	2	no	—
182	C₄⋊C₈	#3624	1	4	yes	complemented
183	C₄×C₈	—	34	2	no	—
184	C₄×C₈	—	34	2	no	—
185	C₄×C₈	—	68	2	no	—
186	C₁₇	#9601	1	1	yes	characteristic, complemented, Sylow
187	C₂²×C₄	—	17	1	no	—
188	C₂²×C₄	—	17	1	no	—
189	C₂²×C₄	—	17	1	no	—
190	C₂²×C₄	—	17	1	no	—
191	C₂²×C₄	C₄⋊#595	1	1	yes	characteristic
192	C₂²×C₄	C₄⋊#595	1	1	yes	characteristic
193	C₂²×C₄	C₂²×#595	1	1	yes	characteristic
194	C₂×C₈	—	2	2	no	—
195	C₂×C₈	—	2	2	no	—
196	C₂×C₈	—	34	2	no	—
197	C₂×C₈	—	34	2	no	—
198	C₂×C₈	—	34	2	no	—
199	C₂×C₈	—	34	2	no	—
200	C₂×C₈	—	34	2	no	—
201	C₂×C₈	—	68	1	no	—
202	C₂×C₈	—	68	1	no	—
203	C₂×C₈	—	68	1	no	—
204	C₂×C₈	—	68	1	no	—
205	C₂×C₈	—	68	2	no	—
206	C₂×C₈	—	68	2	no	—
207	C₂×C₈	—	68	2	no	—
208	C₂×C₈	C₂×#3624	1	4	yes	—
209	C₄²	—	17	1	no	—
210	C₄²	—	17	1	no	—
211	C₄²	—	17	1	no	—
212	C₄²	—	17	1	no	—
213	C₄²	—	17	2	no	—
214	C₄²	—	17	2	no	—
215	C₄²	—	34	1	no	—
216	C₄²	—	34	1	no	—
217	C₄²	—	34	1	no	—
218	C₄²	—	34	1	no	—
219	C₄²	—	34	2	no	—
220	C₄²	—	68	1	no	—
221	C₄²	#63243	1	2	yes	—
222	C₄²	C₂×#3624	1	2	yes	—
223	C₂³	C₂×C₄⋊#595	1	1	yes	characteristic
224	C₂×C₄	—	2	1	no	—
225	C₂×C₄	—	2	1	no	—
226	C₂×C₄	—	17	1	no	—
227	C₂×C₄	—	17	1	no	—
228	C₂×C₄	—	17	1	no	—
229	C₂×C₄	—	17	1	no	—
230	C₂×C₄	—	17	1	no	—
231	C₂×C₄	—	17	1	no	—
232	C₂×C₄	—	17	1	no	—
233	C₂×C₄	—	17	1	no	—
234	C₂×C₄	—	17	2	no	—
235	C₂×C₄	—	34	1	no	—
236	C₂×C₄	—	34	1	no	—
237	C₂×C₄	—	34	1	no	—
238	C₂×C₄	—	34	1	no	—
239	C₂×C₄	—	34	1	no	—
240	C₂×C₄	—	34	1	no	—
241	C₂×C₄	—	34	1	no	—
242	C₂×C₄	#10587534	1	2	yes	—
243	C₂×C₄	#10587536	1	2	yes	—
244	C₂×C₄	C₄⋊#3624	1	2	yes	—
245	C₂×C₄	C₄⋊#3624	1	2	yes	—
246	C₂×C₄	C₂×#63243	1	1	yes	characteristic
247	C₂×C₄	#10587678	1	1	yes	characteristic
248	C₂×C₄	D₄×#595	1	1	yes	characteristic
249	C₂×C₄	#10587680	1	1	yes	characteristic
250	C₂×C₄	Q₈×#595	1	1	yes	characteristic
251	C₂×C₄	C₂²×#3624	1	1	yes	characteristic
252	C₈	—	2	4	no	—
253	C₈	—	68	2	no	—
254	C₈	—	68	2	no	—
255	C₈	—	68	2	no	—
256	C₈	—	136	2	no	—
257	C₂²	#11763760	1	1	yes	characteristic
258	C₂²	C₂×C₄⋊#3624	1	1	yes	characteristic
259	C₂²	#11763816	1	1	yes	characteristic
260	C₂²	C₂×#10587534	1	1	yes	characteristic
261	C₂²	#11763820	1	1	yes	characteristic
262	C₂²	C₂×#10587536	1	1	yes	characteristic
263	C₂²	#11763824	1	1	yes	characteristic
264	C₄	—	2	2	no	—
265	C₄	—	17	1	no	—
266	C₄	—	17	1	no	—
267	C₄	—	17	1	no	—
268	C₄	—	17	1	no	—
269	C₄	—	34	1	no	—
270	C₄	—	34	1	no	—
271	C₄	—	34	2	no	—
272	C₄	—	68	1	no	—
273	C₄	#11763672	1	4	yes	complemented
274	C₄	D₄×#3624	1	1	yes	characteristic
275	C₄	#11763843	1	1	yes	characteristic
276	C₄	Q₈×#3624	1	1	yes	characteristic
277	C₄	#11763859	1	1	yes	characteristic
278	C₂	C₂×#11763672	1	1	yes	characteristic
279	C₂	#14544335	1	1	yes	characteristic
280	C₂	#14544354	1	1	yes	characteristic
281	C₂	#14544362	1	1	yes	characteristic
282	C₂	#14544881	1	1	yes	characteristic
283	C₂	#14544895	1	1	yes	characteristic
284	C₂	#14544898	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	C₂×C₄⋊#595
commutator	C₂×C₄×C₁₇	C₂²×C₈
Frattini	C₂²×C₄	C₂²×#595
Fitting	C₂×C₁₇×C₄⋊C₈	C₄
socle, prime radical	C₂³×C₁₇	C₂×C₄⋊C₄
cosocle	C₂×C₄×Dic₁₇	C₂³
hypercenter	C₂×C₄⋊C₈	#595
hypocenter	C₁₇	#9601
LCS	G ▷ C₂×C₄×C₁₇ ▷ C₂×C₁₇ ▷ C₁₇
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×C₄⋊C₈

Complements

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

N	H ≅ G/N	type
40. C₁₇×C₄⋊C₈	C₈	semidirect
128. C₄×C₁₇	#335	semidirect
182. C₄⋊C₈	#3624	semidirect
186. C₁₇	#9601	semidirect
273. C₄	#11763672	semidirect

Maximal subgroups

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

subgroup	index
#14544716	2
C₄×C₄⋊#3624	2
C₂×#11763672	2
#14543614	2
C₄⋊C₈×Dic₁₇	2
#9601	17