Subgroups — #17955847

Order 4352.

388 conjugacy classes of subgroups.

The subgroup lattice has 272 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	#14544570	C₂	1	1	yes	maximal, characteristic
3	C₂×#11763673	C₂	1	2	yes	maximal
4	#14543614	C₂	1	2	yes	maximal
5	#14528632	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×#10587542	—	2	1	no	—
12	#11763675	—	2	1	no	—
13	#11763673	—	2	4	no	—
14	C₂×#10587485	C₄	1	2	yes	—
15	C₂×C₈×Dic₁₇	C₂²	1	2	yes	—
16	C₄²×Dic₁₇	C₂²	1	1	yes	characteristic
17	#11762564	C₄	1	1	yes	characteristic
18	C₁₇×#336	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	#10587542	—	4	2	no	—
32	C₄⋊#3624	—	2	1	no	—
33	C₄⋊#3624	—	2	1	no	—
34	C₄⋊#3624	—	2	2	no	—
35	C₄⋊#3624	—	2	2	no	—
36	C₄⋊#3624	—	4	2	no	—
37	C₄⋊#3624	—	4	2	no	—
38	C₄⋊#3624	D₄	1	2	yes	—
39	C₄⋊#3624	Q₈	1	2	yes	—
40	#10587485	—	2	4	no	—
41	C₈×Dic₁₇	—	2	2	no	—
42	C₈×Dic₁₇	—	2	2	no	—
43	C₂²×C₄×C₁₇	C₄⋊C₄	1	1	yes	characteristic
44	C₂²×C₄×C₁₇	C₄⋊C₄	1	1	yes	characteristic
45	C₂²×C₄×C₁₇	C₂²×C₄	1	1	yes	characteristic
46	C₂²×Dic₁₇	C₄⋊C₄	1	1	yes	characteristic
47	C₂²×Dic₁₇	C₄⋊C₄	1	1	yes	characteristic
48	C₂²×Dic₁₇	C₂×D₄	1	1	yes	characteristic
49	C₂²×Dic₁₇	C₂×Q₈	1	1	yes	characteristic
50	C₂×#3624	—	2	2	no	—
51	C₂×#3624	—	2	2	no	—
52	C₂×#3624	—	4	1	no	—
53	C₂×#3624	—	4	1	no	—
54	C₂×#3624	—	4	1	no	—
55	C₂×#3624	—	4	1	no	—
56	C₂×#3624	—	4	2	no	—
57	C₂×#3624	—	4	2	no	—
58	C₂×#3624	—	4	2	no	—
59	C₂×C₈×C₁₇	—	2	2	no	—
60	C₂×C₈×C₁₇	—	2	2	no	—
61	C₂×C₈×C₁₇	—	2	2	no	—
62	C₄²×C₁₇	—	2	2	no	—
63	C₄×Dic₁₇	—	2	2	no	—
64	C₄×Dic₁₇	—	2	2	no	—
65	C₄×Dic₁₇	—	2	1	no	—
66	C₄×Dic₁₇	—	2	1	no	—
67	C₄×Dic₁₇	—	4	1	no	—
68	C₄×Dic₁₇	D₈	1	2	yes	—
69	C₄×Dic₁₇	Q₁₆	1	2	yes	—
70	C₄×Dic₁₇	C₂×D₄	1	1	yes	characteristic
71	C₄×Dic₁₇	C₂×Q₈	1	1	yes	characteristic
72	C₄×Dic₁₇	#41	1	1	yes	characteristic
73	C₄×Dic₁₇	#41	1	1	yes	characteristic
74	C₂×#3622	—	2	2	no	—
75	C₂×#3622	—	2	2	no	—
76	C₂×#3622	M₁₆	1	4	yes	—
77	#9604	—	17	1	no	maximal, Sylow
78	C₂³×C₁₇	C₂×C₄⋊C₄	1	1	yes	characteristic
79	C₂×C₄×C₁₇	—	2	1	no	—
80	C₂×C₄×C₁₇	—	2	1	no	—
81	C₂×C₄×C₁₇	—	2	1	no	—
82	C₂×C₄×C₁₇	—	2	1	no	—
83	C₂×C₄×C₁₇	#107	1	2	yes	—
84	C₂×C₄×C₁₇	#108	1	2	yes	—
85	C₂×C₄×C₁₇	C₄×D₄	1	1	yes	characteristic
86	C₂×C₄×C₁₇	C₄×Q₈	1	1	yes	characteristic
87	C₂×C₄×C₁₇	C₂×M₁₆	1	2	yes	—
88	C₂×C₄×C₁₇	#131	1	2	yes	—
89	C₂×Dic₁₇	—	2	1	no	—
90	C₂×Dic₁₇	—	2	1	no	—
91	C₂×Dic₁₇	—	2	1	no	—
92	C₂×Dic₁₇	—	2	1	no	—
93	C₂×Dic₁₇	—	2	1	no	—
94	C₂×Dic₁₇	—	2	1	no	—
95	C₂×Dic₁₇	—	2	1	no	—
96	C₂×Dic₁₇	#107	1	2	yes	—
97	C₂×Dic₁₇	#121	1	1	yes	characteristic
98	C₂×Dic₁₇	C₂²⋊Q₈	1	1	yes	characteristic
99	C₂×Dic₁₇	#125	1	1	yes	characteristic
100	C₂×Dic₁₇	C₄⋊Q₈	1	1	yes	characteristic
101	C₂×Dic₁₇	C₂×D₈	1	1	yes	characteristic
102	C₂×Dic₁₇	C₂×Q₁₆	1	1	yes	characteristic
103	C₂×Dic₁₇	AGL(1,ℤ/8)	1	1	yes	characteristic
104	C₂×Dic₁₇	#137	1	1	yes	characteristic
105	#3624	—	4	2	no	—
106	#3624	—	4	2	no	—
107	#3624	—	8	2	no	—
108	C₈×C₁₇	—	4	2	no	—
109	#3622	—	2	4	no	—
110	#1842	—	17	1	no	—
111	#1767	—	17	1	no	—
112	C₄×C₄⋊C₈	—	17	1	no	—
113	C₂×#335	—	17	2	no	—
114	#1277	—	17	2	no	—
115	C₂²×C₁₇	#389	1	1	yes	characteristic
116	C₂²×C₁₇	#423	1	1	yes	characteristic
117	C₂²×C₁₇	#424	1	1	yes	characteristic
118	C₂²×C₁₇	C₂×#107	1	1	yes	characteristic
119	C₂²×C₁₇	#428	1	1	yes	characteristic
120	C₂²×C₁₇	C₂×#108	1	1	yes	characteristic
121	C₂²×C₁₇	#430	1	1	yes	characteristic
122	C₄×C₁₇	—	2	2	no	—
123	C₄×C₁₇	—	2	2	no	—
124	C₄×C₁₇	#435	1	2	yes	—
125	C₄×C₁₇	#446	1	2	yes	—
126	Dic₁₇	—	2	2	no	—
127	Dic₁₇	—	2	1	no	—
128	Dic₁₇	—	2	1	no	—
129	Dic₁₇	—	4	1	no	—
130	Dic₁₇	#459	1	1	yes	characteristic
131	Dic₁₇	#462	1	1	yes	characteristic
132	Dic₁₇	#474	1	1	yes	characteristic
133	Dic₁₇	#477	1	1	yes	characteristic
134	C₂×C₄⋊C₈	—	17	1	no	—
135	C₂×C₄⋊C₈	—	17	1	no	—
136	C₂×C₄⋊C₈	—	17	2	no	—
137	C₂×C₄⋊C₈	—	17	2	no	—
138	C₂×C₄⋊C₈	—	34	1	no	—
139	C₂×C₄⋊C₈	—	34	1	no	—
140	C₂×#97	—	34	1	no	—
141	C₂×C₄×C₈	—	17	2	no	—
142	C₄³	—	17	1	no	—
143	#336	—	17	1	no	—
144	#336	—	34	1	no	—
145	#336	#595	1	1	yes	characteristic
146	#335	—	34	4	no	—
147	C₂×C₁₇	#1567	1	1	yes	characteristic
148	C₂×C₁₇	#1569	1	1	yes	characteristic
149	C₂×C₁₇	#1571	1	1	yes	characteristic
150	C₂×C₁₇	#1573	1	1	yes	characteristic
151	C₂×C₁₇	#1940	1	1	yes	characteristic
152	C₂×C₁₇	#1947	1	1	yes	characteristic
153	C₂×C₁₇	#1950	1	1	yes	characteristic
154	C₂²×C₈	—	17	2	no	—
155	C₂²×C₈	—	34	1	no	—
156	C₂²×C₈	—	34	1	no	—
157	C₂²×C₈	—	34	2	no	—
158	C₂²×C₈	C₂×#595	1	2	yes	—
159	C₂×C₄²	—	17	1	no	—
160	C₂×C₄²	—	17	1	no	—
161	C₂×C₄²	—	17	1	no	—
162	C₂×C₄²	—	17	1	no	—
163	C₂×C₄²	—	17	1	no	—
164	C₂×C₄²	—	17	1	no	—
165	C₂×C₄²	C₂×#595	1	1	yes	characteristic
166	C₄⋊C₈	—	17	2	no	—
167	C₄⋊C₈	—	17	2	no	—
168	C₄⋊C₈	—	34	1	no	—
169	C₄⋊C₈	—	34	1	no	—
170	C₄⋊C₈	—	34	2	no	—
171	C₄⋊C₈	—	34	2	no	—
172	C₄⋊C₈	—	34	4	no	—
173	C₄⋊C₈	—	68	2	no	—
174	C₄⋊C₈	—	68	2	no	—
175	#97	—	68	2	no	—
176	C₄×C₈	—	34	2	no	—
177	C₄×C₈	—	34	2	no	—
178	C₁₇	#9604	1	1	yes	characteristic, complemented, Sylow
179	C₂²×C₄	—	17	1	no	—
180	C₂²×C₄	—	17	1	no	—
181	C₂²×C₄	—	17	1	no	—
182	C₂²×C₄	—	17	1	no	—
183	C₂²×C₄	C₄⋊#595	1	1	yes	characteristic
184	C₂²×C₄	C₄⋊#595	1	1	yes	characteristic
185	C₂²×C₄	C₂²×#595	1	1	yes	characteristic
186	C₂×C₈	—	2	2	no	—
187	C₂×C₈	—	2	2	no	—
188	C₂×C₈	—	2	2	no	—
189	C₂×C₈	—	17	4	no	—
190	C₂×C₈	—	34	2	no	—
191	C₂×C₈	—	34	2	no	—
192	C₂×C₈	—	34	2	no	—
193	C₂×C₈	—	34	2	no	—
194	C₂×C₈	—	68	1	no	—
195	C₂×C₈	—	68	1	no	—
196	C₂×C₈	—	68	1	no	—
197	C₂×C₈	—	68	1	no	—
198	C₂×C₈	—	68	2	no	—
199	C₂×C₈	—	68	2	no	—
200	C₂×C₈	—	68	2	no	—
201	C₄²	—	2	2	no	—
202	C₄²	—	17	1	no	—
203	C₄²	—	17	1	no	—
204	C₄²	—	17	1	no	—
205	C₄²	—	17	1	no	—
206	C₄²	—	17	2	no	—
207	C₄²	—	17	2	no	—
208	C₄²	—	34	1	no	—
209	C₄²	—	34	1	no	—
210	C₄²	—	34	2	no	—
211	C₄²	—	34	2	no	—
212	C₄²	—	68	1	no	—
213	C₂³	C₂×C₄⋊#595	1	1	yes	characteristic
214	C₂×C₄	—	2	1	no	—
215	C₂×C₄	—	2	1	no	—
216	C₂×C₄	—	2	1	no	—
217	C₂×C₄	—	2	1	no	—
218	C₂×C₄	—	17	1	no	—
219	C₂×C₄	—	17	1	no	—
220	C₂×C₄	—	17	1	no	—
221	C₂×C₄	—	17	1	no	—
222	C₂×C₄	—	17	1	no	—
223	C₂×C₄	—	17	1	no	—
224	C₂×C₄	—	17	1	no	—
225	C₂×C₄	—	17	1	no	—
226	C₂×C₄	—	17	2	no	—
227	C₂×C₄	—	34	1	no	—
228	C₂×C₄	—	34	1	no	—
229	C₂×C₄	—	34	1	no	—
230	C₂×C₄	—	34	1	no	—
231	C₂×C₄	—	34	1	no	—
232	C₂×C₄	—	34	1	no	—
233	C₂×C₄	—	34	1	no	—
234	C₂×C₄	#10587534	1	2	yes	—
235	C₂×C₄	#10587535	1	2	yes	—
236	C₂×C₄	D₁₇⋊M₁₆	1	2	yes	—
237	C₂×C₄	#10587678	1	2	yes	—
238	C₂×C₄	D₄×#595	1	1	yes	characteristic
239	C₂×C₄	Q₈×#595	1	1	yes	characteristic
240	C₈	—	4	2	no	—
241	C₈	—	34	4	no	—
242	C₈	—	68	2	no	—
243	C₈	—	68	2	no	—
244	C₈	—	136	2	no	—
245	C₂²	#11763760	1	1	yes	characteristic
246	C₂²	#11763814	1	1	yes	characteristic
247	C₂²	#11763815	1	1	yes	characteristic
248	C₂²	C₂×#10587534	1	1	yes	characteristic
249	C₂²	#11763820	1	1	yes	characteristic
250	C₂²	C₂×#10587535	1	1	yes	characteristic
251	C₂²	#11763824	1	1	yes	characteristic
252	C₄	—	2	2	no	—
253	C₄	—	2	2	no	—
254	C₄	—	17	1	no	—
255	C₄	—	17	1	no	—
256	C₄	—	17	1	no	—
257	C₄	—	17	1	no	—
258	C₄	—	34	1	no	—
259	C₄	—	34	1	no	—
260	C₄	—	34	2	no	—
261	C₄	—	68	1	no	—
262	C₄	#11763841	1	2	yes	—
263	C₄	#11763861	1	2	yes	—
264	C₂	#14544343	1	1	yes	characteristic
265	C₂	#14544350	1	1	yes	characteristic
266	C₂	#14544353	1	1	yes	characteristic
267	C₂	#14544357	1	1	yes	characteristic
268	C₂	#14544885	1	1	yes	characteristic
269	C₂	#14544895	1	1	yes	characteristic
270	C₂	#14544899	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	C₂×C₄⋊#595
commutator	C₂²×C₄×C₁₇	C₂²×C₄
Frattini	C₂²×C₄	C₂²×#595
Fitting	C₁₇×#336	C₄
socle, prime radical	C₂³×C₁₇	C₂×C₄⋊C₄
cosocle	C₂×C₄×Dic₁₇	C₂³
hypercenter	#336	#595
hypocenter	C₁₇	#9604
LCS	G ▷ C₂²×C₄×C₁₇ ▷ C₂×C₁₇ ▷ C₁₇
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ #336

Complements

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

N	H ≅ G/N	type
178. C₁₇	#9604	semidirect

Maximal subgroups

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

subgroup	index
#14544716	2
#14544570	2
C₂×#11763673	2
#14543614	2
#14528632	2
#9604	17