Subgroups — #17952325

Order 4352.

438 conjugacy classes of subgroups.

The subgroup lattice has 252 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#14544571	C₂	1	2	yes	maximal
2	C₄×C₈×#595	C₂	1	1	yes	maximal, characteristic
3	#14544329	C₂	1	1	yes	maximal, characteristic
4	#14543613	C₂	1	2	yes	maximal
5	D₁₇×#335	C₂	1	1	yes	maximal, characteristic
6	#11763810	C₂²	1	2	yes	—
7	C₂×#10587532	—	2	2	no	—
8	C₄×#63242	C₂²	1	1	yes	characteristic
9	C₂×C₈×#595	C₄	1	2	yes	—
10	C₂×C₈×#595	C₄	1	2	yes	—
11	C₄²×#595	C₂²	1	1	yes	characteristic
12	#11763674	—	2	2	no	—
13	#11763673	C₄	1	2	yes	complemented
14	#11763672	C₄	1	2	yes	complemented
15	C₄⋊C₈×D₁₇	C₂²	1	2	yes	—
16	C₄×C₈×D₁₇	C₂²	1	1	yes	characteristic
17	#11762562	C₄	1	1	yes	characteristic, complemented
18	#11762561	C₄	1	2	yes	complemented
19	C₁₇×#335	C₄	1	1	yes	characteristic, complemented
20	C₂×C₄×#595	C₂×C₄	1	2	yes	—
21	C₂×C₄×#595	D₄	1	1	yes	characteristic
22	C₂×C₄×#595	D₄	1	1	yes	characteristic
23	C₂×C₄×#595	Q₈	1	2	yes	—
24	C₂×#63242	—	2	2	no	—
25	C₂×#63242	C₂×C₄	1	2	yes	—
26	C₂×C₈×D₁₇	—	2	2	no	—
27	C₂×C₈×D₁₇	C₂×C₄	1	1	yes	characteristic
28	C₂×C₈×D₁₇	C₂×C₄	1	1	yes	characteristic
29	C₄²×D₁₇	C₂³	1	1	yes	characteristic
30	#10587544	—	2	2	no	—
31	C₄⋊#3624	C₂×C₄	1	4	yes	—
32	C₄×#3624	—	2	1	no	—
33	C₄×#3624	—	2	1	no	—
34	#10587532	—	4	4	no	—
35	C₈×#595	—	2	4	no	—
36	C₈×#595	—	2	4	no	—
37	C₁₇×C₄⋊C₈	C₂×C₄	1	2	yes	—
38	C₄×C₈×C₁₇	C₂×C₄	1	1	yes	characteristic
39	#10587485	—	2	2	no	—
40	C₈×Dic₁₇	C₂×C₄	1	1	yes	characteristic
41	C₈×Dic₁₇	C₂×C₄	1	1	yes	characteristic
42	C₄⋊#3622	C₂×C₄	1	2	yes	—
43	C₄×#3622	C₂×C₄	1	1	yes	characteristic
44	C₂²×#595	C₄⋊C₄	1	2	yes	—
45	C₂²×#595	C₄⋊C₄	1	2	yes	—
46	C₂×C₄×D₁₇	C₂²×C₄	1	1	yes	characteristic
47	C₂×C₄×D₁₇	C₂×D₄	1	1	yes	characteristic
48	C₂×C₄×D₁₇	C₂×Q₈	1	1	yes	characteristic
49	C₂×#3624	—	2	2	no	—
50	C₂×#3624	—	2	4	no	—
51	C₄×#595	—	2	4	no	—
52	C₄×#595	—	2	2	no	—
53	C₄×#595	—	2	2	no	—
54	C₄×#595	—	2	1	no	—
55	C₄×#595	—	2	1	no	—
56	C₄×#595	D₈	1	2	yes	—
57	C₄×#595	Q₁₆	1	2	yes	—
58	#63242	—	2	2	no	—
59	#63242	—	4	2	no	—
60	#63242	—	4	2	no	—
61	#63242	M₁₆	1	4	yes	—
62	C₂×C₈×C₁₇	—	2	2	no	—
63	C₂×C₈×C₁₇	C₄²	1	1	yes	characteristic
64	C₂×C₈×C₁₇	C₄²	1	1	yes	characteristic
65	C₄²×C₁₇	C₂²×C₄	1	1	yes	characteristic
66	C₄×Dic₁₇	C₂²×C₄	1	1	yes	characteristic
67	C₄×Dic₁₇	#41	1	1	yes	characteristic
68	C₄×Dic₁₇	#41	1	1	yes	characteristic
69	C₂×#3622	—	2	2	no	—
70	C₂×#3622	C₄²	1	1	yes	characteristic
71	C₂×#3622	C₄²	1	1	yes	characteristic
72	C₈×D₁₇	—	2	2	no	—
73	C₈×D₁₇	—	4	2	no	—
74	C₈×D₁₇	—	4	2	no	—
75	C₈×D₁₇	M₁₆	1	4	yes	—
76	#8274	—	17	1	no	maximal, Sylow
77	C₂²×D₁₇	C₂×C₄⋊C₄	1	1	yes	characteristic
78	C₂×#595	—	2	2	no	—
79	C₂×#595	—	2	2	no	—
80	C₂×#595	—	2	2	no	—
81	C₂×#595	—	2	2	no	—
82	C₂×#595	#107	1	4	yes	—
83	C₂×#595	#108	1	4	yes	—
84	C₂×C₄×C₁₇	C₂×C₄²	1	1	yes	characteristic
85	C₂×C₄×C₁₇	C₄×D₄	1	1	yes	characteristic
86	C₂×C₄×C₁₇	C₄×Q₈	1	1	yes	characteristic
87	C₂×Dic₁₇	#117	1	1	yes	characteristic
88	C₂×Dic₁₇	C₄×D₄	1	1	yes	characteristic
89	C₂×Dic₁₇	C₄×Q₈	1	1	yes	characteristic
90	C₄×D₁₇	—	2	2	no	—
91	C₄×D₁₇	C₂×M₁₆	1	2	yes	—
92	C₄×D₁₇	C₂×M₁₆	1	2	yes	—
93	C₄×D₁₇	C₂×D₈	1	1	yes	characteristic
94	C₄×D₁₇	C₂×Q₁₆	1	1	yes	characteristic
95	C₄×D₁₇	#135	1	1	yes	characteristic
96	C₄×D₁₇	#135	1	1	yes	characteristic
97	#3624	—	2	4	no	—
98	#3624	—	4	4	no	—
99	C₈×C₁₇	—	2	1	no	—
100	C₈×C₁₇	—	4	2	no	—
101	C₈×C₁₇	#97	1	2	yes	complemented
102	#3622	—	2	1	no	—
103	#3622	—	4	2	no	—
104	#3622	#97	1	2	yes	complemented
105	#1767	—	17	2	no	—
106	C₄²×C₈	—	17	1	no	—
107	C₂×#335	—	17	1	no	—
108	C₂×#335	—	17	1	no	—
109	#1277	—	17	2	no	—
110	C₂²×C₁₇	C₄×C₄⋊C₄	1	1	yes	characteristic
111	C₂×D₁₇	#423	1	2	yes	—
112	C₂×D₁₇	C₂×#107	1	1	yes	characteristic
113	C₂×D₁₇	#427	1	1	yes	characteristic
114	C₂×D₁₇	C₂×#108	1	2	yes	—
115	#595	—	2	4	no	—
116	#595	—	2	4	no	—
117	C₄×C₁₇	—	2	1	no	—
118	C₄×C₁₇	C₂×#97	1	1	yes	characteristic
119	C₄×C₁₇	C₄×M₁₆	1	1	yes	characteristic
120	C₄×C₁₇	C₄×D₈	1	1	yes	characteristic
121	C₄×C₁₇	C₄×Q₁₆	1	1	yes	characteristic
122	Dic₁₇	—	2	1	no	—
123	Dic₁₇	#432	1	2	yes	—
124	Dic₁₇	C₄×D₈	1	1	yes	characteristic
125	Dic₁₇	C₄×Q₁₆	1	1	yes	characteristic
126	C₂×C₄⋊C₈	—	17	2	no	—
127	C₂×C₄⋊C₈	—	17	2	no	—
128	C₂×#97	—	34	2	no	—
129	C₂×C₄×C₈	—	17	1	no	—
130	C₂×C₄×C₈	—	17	2	no	—
131	C₂×C₄×C₈	—	17	2	no	—
132	C₂×C₄×C₈	—	17	1	no	—
133	C₄³	—	17	1	no	—
134	#336	—	34	2	no	—
135	#335	—	17	2	no	—
136	#335	—	17	2	no	—
137	#335	—	17	2	no	—
138	#335	—	17	1	no	—
139	#335	#595	1	1	yes	characteristic, complemented
140	C₂×C₁₇	#1767	1	1	yes	characteristic
141	C₂×C₁₇	C₄×#107	1	1	yes	characteristic
142	C₂×C₁₇	C₄×#108	1	1	yes	characteristic
143	D₁₇	#1567	1	2	yes	—
144	D₁₇	#1568	1	2	yes	—
145	C₂²×C₈	—	17	1	no	—
146	C₂²×C₈	—	17	2	no	—
147	C₂²×C₈	—	17	1	no	—
148	C₂²×C₈	—	34	2	no	—
149	C₂²×C₈	—	34	2	no	—
150	C₂×C₄²	—	17	1	no	—
151	C₂×C₄²	—	17	1	no	—
152	C₂×C₄²	—	17	2	no	—
153	C₂×C₄²	—	17	1	no	—
154	C₂×C₄²	—	17	2	no	—
155	C₄⋊C₈	—	17	4	no	—
156	C₄⋊C₈	—	17	2	no	—
157	C₄⋊C₈	—	34	2	no	—
158	C₄⋊C₈	—	34	2	no	—
159	C₄⋊C₈	C₂×#595	1	2	yes	—
160	#97	—	68	4	no	—
161	C₄×C₈	—	17	1	no	—
162	C₄×C₈	—	17	1	no	—
163	C₄×C₈	—	17	1	no	—
164	C₄×C₈	—	34	1	no	—
165	C₄×C₈	—	34	1	no	—
166	C₄×C₈	—	34	4	no	—
167	C₄×C₈	—	34	4	no	—
168	C₄×C₈	C₂×#595	1	1	yes	characteristic
169	C₁₇	#8274	1	1	yes	characteristic, complemented, Sylow
170	C₂²×C₄	—	17	1	no	—
171	C₂²×C₄	—	17	1	no	—
172	C₂²×C₄	—	17	2	no	—
173	C₂²×C₄	—	17	2	no	—
174	C₂²×C₄	—	17	1	no	—
175	C₂×C₈	—	2	2	no	—
176	C₂×C₈	—	17	1	no	—
177	C₂×C₈	—	17	4	no	—
178	C₂×C₈	—	17	4	no	—
179	C₂×C₈	—	17	1	no	—
180	C₂×C₈	—	34	2	no	—
181	C₂×C₈	—	34	2	no	—
182	C₂×C₈	—	34	2	no	—
183	C₂×C₈	—	34	4	no	—
184	C₂×C₈	—	34	2	no	—
185	C₂×C₈	—	68	2	no	—
186	C₂×C₈	—	68	2	no	—
187	C₂×C₈	—	68	2	no	—
188	C₂×C₈	—	68	2	no	—
189	C₂×C₈	C₄×#595	1	1	yes	characteristic
190	C₂×C₈	C₄×#595	1	1	yes	characteristic
191	C₄²	—	17	2	no	—
192	C₄²	—	17	2	no	—
193	C₄²	—	17	1	no	—
194	C₄²	—	17	1	no	—
195	C₄²	—	17	1	no	—
196	C₄²	—	34	2	no	—
197	C₄²	—	34	2	no	—
198	C₄²	—	34	1	no	—
199	C₄²	—	34	1	no	—
200	C₄²	—	34	4	no	—
201	C₄²	C₂²×#595	1	1	yes	characteristic
202	C₂³	—	17	1	no	—
203	C₂×C₄	—	17	1	no	—
204	C₂×C₄	—	17	1	no	—
205	C₂×C₄	—	17	2	no	—
206	C₂×C₄	—	17	2	no	—
207	C₂×C₄	—	17	1	no	—
208	C₂×C₄	—	17	1	no	—
209	C₂×C₄	—	17	1	no	—
210	C₂×C₄	—	17	1	no	—
211	C₂×C₄	—	17	4	no	—
212	C₂×C₄	—	17	4	no	—
213	C₂×C₄	—	17	1	no	—
214	C₂×C₄	—	34	2	no	—
215	C₂×C₄	—	34	2	no	—
216	C₂×C₄	—	34	2	no	—
217	C₂×C₄	—	34	2	no	—
218	C₂×C₄	—	34	2	no	—
219	C₂×C₄	C₂×C₄×#595	1	1	yes	characteristic
220	C₂×C₄	D₄×#595	1	1	yes	characteristic
221	C₂×C₄	Q₈×#595	1	1	yes	characteristic
222	C₈	—	2	1	no	—
223	C₈	—	4	2	no	—
224	C₈	—	17	2	no	—
225	C₈	—	34	4	no	—
226	C₈	—	34	1	no	—
227	C₈	—	68	4	no	—
228	C₈	—	68	2	no	—
229	C₈	#10587532	1	2	yes	complemented
230	C₂²	—	17	2	no	—
231	C₂²	—	17	1	no	—
232	C₂²	—	17	1	no	—
233	C₂²	—	17	2	no	—
234	C₂²	C₄⋊C₄×#595	1	1	yes	characteristic
235	C₄	—	2	1	no	—
236	C₄	—	17	2	no	—
237	C₄	—	17	1	no	—
238	C₄	—	17	1	no	—
239	C₄	—	34	4	no	—
240	C₄	—	34	4	no	—
241	C₄	—	34	1	no	—
242	C₄	C₂×#10587532	1	1	yes	characteristic
243	C₄	M₁₆×#595	1	1	yes	characteristic
244	C₄	D₈×#595	1	1	yes	characteristic
245	C₄	Q₁₆×#595	1	1	yes	characteristic
246	C₂	—	17	2	no	—
247	C₂	—	17	2	no	—
248	C₂	#14544571	1	1	yes	characteristic
249	C₂	#107×#595	1	1	yes	characteristic
250	C₂	#108×#595	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	D₄×#595
commutator	C₂×C₄×C₁₇	C₂×C₄²
Frattini	C₄²	C₂²×#595
Fitting	C₁₇×#335	C₄
socle	C₂²×C₁₇	C₄×C₄⋊C₄
prime radical	C₂²×D₁₇	C₂×C₄⋊C₄
cosocle	C₄²×D₁₇	C₂³
hypercenter	#335	#595
hypocenter	C₁₇	#8274
LCS	G ▷ C₂×C₄×C₁₇ ▷ C₂×C₁₇ ▷ C₁₇
UCS	C₁ ◁ C₂×C₄ ◁ C₄² ◁ #335

Complements

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

N	H ≅ G/N	type
13. #11763673	C₄	semidirect
14. #11763672	C₄	semidirect
17. #11762562	C₄	semidirect
18. #11762561	C₄	semidirect
19. C₁₇×#335	C₄	semidirect
101. C₈×C₁₇	#97	semidirect
104. #3622	#97	semidirect
139. #335	#595	semidirect
169. C₁₇	#8274	semidirect
229. C₈	#10587532	semidirect

Maximal subgroups

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

subgroup	index
#14544571	2
C₄×C₈×#595	2
#14544329	2
#14543613	2
D₁₇×#335	2
#8274	17