Subgroups — #17952349

Order 4352.

446 conjugacy classes of subgroups.

The subgroup lattice has 240 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄×#10587534	C₂	1	1	yes	maximal, characteristic
2	#14544577	C₂	1	2	yes	maximal
3	C₈×#63242	C₂	1	1	yes	maximal, characteristic
4	#14543618	C₂	1	2	yes	maximal
5	D₁₇×#335	C₂	1	1	yes	maximal, characteristic
6	C₂×#10587534	C₄	1	2	yes	—
7	C₄×#63242	C₂²	1	1	yes	characteristic
8	C₄×#63242	C₂²	1	1	yes	characteristic
9	C₂×C₈×#595	—	2	2	no	—
10	C₄×C₄⋊#595	C₂²	1	2	yes	—
11	#11763683	C₄	1	2	yes	—
12	#11763674	—	2	2	no	—
13	C₈×#3624	C₄	1	2	yes	complemented
14	C₈×#3624	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	4	yes	—
21	C₂×C₄×#595	—	2	2	no	—
22	C₂×#63242	—	2	1	no	—
23	C₂×#63242	D₄	1	1	yes	characteristic
24	C₂×#63242	D₄	1	1	yes	characteristic
25	C₂×C₈×D₁₇	—	2	2	no	—
26	C₂×C₈×D₁₇	C₂×C₄	1	1	yes	characteristic
27	C₂×C₈×D₁₇	C₂×C₄	1	1	yes	characteristic
28	C₄²×D₁₇	C₂³	1	1	yes	characteristic
29	(C₂×#595)⋊C₄	—	2	2	no	—
30	C₄×#3624	—	2	1	no	—
31	C₄×#3624	C₂×C₄	1	2	yes	—
32	C₄×#3624	C₂×C₄	1	2	yes	—
33	C₄×#3624	Q₈	1	2	yes	—
34	#10587534	C₈	1	8	yes	complemented
35	C₈×#595	—	4	4	no	—
36	C₁₇×C₄⋊C₈	C₂×C₄	1	2	yes	—
37	C₄×C₈×C₁₇	C₂×C₄	1	1	yes	characteristic
38	#10587485	—	2	2	no	—
39	C₈×Dic₁₇	C₂×C₄	1	1	yes	characteristic
40	C₈×Dic₁₇	C₂×C₄	1	1	yes	characteristic
41	C₄⋊#3622	C₂×C₄	1	2	yes	—
42	C₄×#3622	C₂×C₄	1	1	yes	characteristic
43	C₂²×#595	—	2	4	no	—
44	C₂×C₄×D₁₇	C₂²×C₄	1	1	yes	characteristic
45	C₂×C₄×D₁₇	C₂×D₄	1	1	yes	characteristic
46	C₂×C₄×D₁₇	#41	1	1	yes	characteristic
47	C₂×#3624	—	2	2	no	—
48	C₂×#3624	C₄⋊C₄	1	2	yes	—
49	C₂×#3624	C₄⋊C₄	1	2	yes	—
50	C₄⋊#595	—	2	4	no	—
51	C₄⋊#595	C₂×C₈	1	8	yes	—
52	C₄×#595	—	4	2	no	—
53	C₄×#595	—	4	2	no	—
54	#63242	—	2	4	no	—
55	#63242	—	2	1	no	—
56	#63242	—	2	1	no	—
57	#63242	D₈	1	2	yes	—
58	#63242	Q₁₆	1	2	yes	—
59	C₂×C₈×C₁₇	—	2	2	no	—
60	C₂×C₈×C₁₇	C₄²	1	1	yes	characteristic
61	C₂×C₈×C₁₇	C₄²	1	1	yes	characteristic
62	C₄²×C₁₇	C₂²×C₄	1	1	yes	characteristic
63	C₄×Dic₁₇	C₂²×C₄	1	1	yes	characteristic
64	C₄×Dic₁₇	C₂×Q₈	1	1	yes	characteristic
65	C₄×Dic₁₇	#41	1	1	yes	characteristic
66	C₂×#3622	—	2	2	no	—
67	C₂×#3622	C₄²	1	1	yes	characteristic
68	C₂×#3622	C₄²	1	1	yes	characteristic
69	C₈×D₁₇	—	2	2	no	—
70	C₈×D₁₇	—	4	2	no	—
71	C₈×D₁₇	—	4	2	no	—
72	C₈×D₁₇	C₂×C₈	1	4	yes	—
73	C₈×#107	—	17	1	no	maximal, Sylow
74	C₂²×D₁₇	C₄×D₄	1	1	yes	characteristic
75	C₂×#595	—	2	8	no	—
76	C₂×#595	—	4	4	no	—
77	C₂×#595	—	4	4	no	—
78	C₂×C₄×C₁₇	C₂×C₄²	1	1	yes	characteristic
79	C₂×C₄×C₁₇	C₄×D₄	1	1	yes	characteristic
80	C₂×C₄×C₁₇	C₄×Q₈	1	1	yes	characteristic
81	C₂×Dic₁₇	C₂×C₄⋊C₄	1	1	yes	characteristic
82	C₂×Dic₁₇	#117	1	1	yes	characteristic
83	C₂×Dic₁₇	C₄×Q₈	1	1	yes	characteristic
84	C₄×D₁₇	—	2	2	no	—
85	C₄×D₁₇	C₂²×C₈	1	2	yes	—
86	C₄×D₁₇	#131	1	2	yes	—
87	C₄×D₁₇	C₂×D₈	1	1	yes	characteristic
88	C₄×D₁₇	C₂×Q₁₆	1	1	yes	characteristic
89	C₄×D₁₇	#135	1	1	yes	characteristic
90	C₄×D₁₇	#135	1	1	yes	characteristic
91	#3624	—	2	4	no	—
92	#3624	—	2	2	no	—
93	#3624	#107	1	4	yes	complemented
94	C₈×C₁₇	—	2	1	no	—
95	C₈×C₁₇	—	4	2	no	—
96	C₈×C₁₇	C₄×C₈	1	2	yes	complemented
97	#3622	—	2	1	no	—
98	#3622	—	4	2	no	—
99	#3622	C₄×C₈	1	2	yes	complemented
100	C₄×#107	—	17	1	no	—
101	C₈×C₄⋊C₄	—	17	2	no	—
102	C₂×#335	—	17	1	no	—
103	C₂×C₈²	—	17	1	no	—
104	#1279	—	17	2	no	—
105	C₂²×C₁₇	C₄×C₄⋊C₄	1	1	yes	characteristic
106	C₂×D₁₇	C₈×D₄	1	2	yes	—
107	C₂×D₁₇	C₄×D₈	1	1	yes	characteristic
108	C₂×D₁₇	C₄×Q₁₆	1	1	yes	characteristic
109	C₂×D₁₇	#443	1	2	yes	—
110	#595	—	4	8	no	—
111	C₄×C₁₇	—	2	1	no	—
112	C₄×C₁₇	C₂×C₄×C₈	1	1	yes	characteristic
113	C₄×C₁₇	#405	1	1	yes	characteristic
114	C₄×C₁₇	C₄×D₈	1	1	yes	characteristic
115	C₄×C₁₇	C₄×Q₁₆	1	1	yes	characteristic
116	Dic₁₇	—	2	1	no	—
117	Dic₁₇	C₂×#107	1	1	yes	characteristic
118	Dic₁₇	#427	1	1	yes	characteristic
119	Dic₁₇	C₈×Q₈	1	2	yes	—
120	C₂×#107	—	17	2	no	—
121	C₂×C₄⋊C₈	—	17	2	no	—
122	C₂×C₄×C₈	—	17	1	no	—
123	C₂×C₄×C₈	—	17	1	no	—
124	C₂×C₄×C₈	—	17	1	no	—
125	C₂×C₄×C₈	—	34	2	no	—
126	C₄×C₄⋊C₄	—	17	2	no	—
127	#340	—	17	2	no	—
128	#336	—	34	2	no	—
129	#335	—	17	2	no	—
130	#335	—	17	1	no	—
131	#335	#595	1	1	yes	characteristic, complemented
132	C₈²	—	17	2	no	—
133	C₈²	—	17	2	no	—
134	C₂×C₁₇	C₈×C₄⋊C₄	1	1	yes	characteristic
135	C₂×C₁₇	#1772	1	1	yes	characteristic
136	C₂×C₁₇	C₄×#107	1	1	yes	characteristic
137	D₁₇	C₈×D₈	1	2	yes	—
138	D₁₇	C₈×Q₁₆	1	2	yes	—
139	C₂²×C₈	—	17	1	no	—
140	C₂²×C₈	—	17	1	no	—
141	C₂²×C₈	—	17	1	no	—
142	C₂²×C₈	—	17	1	no	—
143	C₂²×C₈	—	34	2	no	—
144	C₂²×C₈	—	34	1	no	—
145	C₂×C₄⋊C₄	—	17	4	no	—
146	C₂×C₄²	—	17	1	no	—
147	C₂×C₄²	—	34	2	no	—
148	#107	—	17	8	no	—
149	C₄⋊C₈	—	17	2	no	—
150	C₄⋊C₈	—	34	2	no	—
151	C₄⋊C₈	C₂×#595	1	2	yes	—
152	C₄×C₈	—	17	2	no	—
153	C₄×C₈	—	17	2	no	—
154	C₄×C₈	—	17	1	no	—
155	C₄×C₈	—	17	1	no	—
156	C₄×C₈	—	17	2	no	—
157	C₄×C₈	—	17	1	no	—
158	C₄×C₈	—	34	1	no	—
159	C₄×C₈	—	68	4	no	—
160	C₄×C₈	C₂×#595	1	1	yes	characteristic
161	#95	—	34	2	no	—
162	C₁₇	C₈×#107	1	1	yes	characteristic, complemented, Sylow
163	C₂²×C₄	—	17	1	no	—
164	C₂²×C₄	—	17	1	no	—
165	C₂²×C₄	—	17	1	no	—
166	C₂²×C₄	—	34	4	no	—
167	C₂×C₈	—	2	2	no	—
168	C₂×C₈	—	17	2	no	—
169	C₂×C₈	—	17	2	no	—
170	C₂×C₈	—	17	2	no	—
171	C₂×C₈	—	17	1	no	—
172	C₂×C₈	—	17	4	no	—
173	C₂×C₈	—	17	2	no	—
174	C₂×C₈	—	17	1	no	—
175	C₂×C₈	—	34	2	no	—
176	C₂×C₈	—	34	1	no	—
177	C₂×C₈	—	34	1	no	—
178	C₂×C₈	—	34	2	no	—
179	C₂×C₈	—	34	4	no	—
180	C₂×C₈	—	34	2	no	—
181	C₂×C₈	—	68	2	no	—
182	C₂×C₈	—	68	2	no	—
183	C₂×C₈	C₄×#595	1	1	yes	characteristic
184	C₂×C₈	C₄×#595	1	1	yes	characteristic
185	C₄⋊C₄	—	17	8	no	—
186	C₄⋊C₄	—	34	4	no	—
187	C₄²	—	17	1	no	—
188	C₄²	—	17	1	no	—
189	C₄²	—	17	1	no	—
190	C₄²	—	68	2	no	—
191	C₄²	—	68	2	no	—
192	C₄²	C₂²×#595	1	1	yes	characteristic
193	C₂³	—	17	1	no	—
194	C₂×C₄	—	17	1	no	—
195	C₂×C₄	—	17	1	no	—
196	C₂×C₄	—	17	2	no	—
197	C₂×C₄	—	17	2	no	—
198	C₂×C₄	—	17	1	no	—
199	C₂×C₄	—	17	1	no	—
200	C₂×C₄	—	17	1	no	—
201	C₂×C₄	—	17	1	no	—
202	C₂×C₄	—	17	1	no	—
203	C₂×C₄	—	34	2	no	—
204	C₂×C₄	—	34	8	no	—
205	C₂×C₄	—	68	4	no	—
206	C₂×C₄	—	68	4	no	—
207	C₂×C₄	C₂×C₄×#595	1	1	yes	characteristic
208	C₂×C₄	D₄×#595	1	1	yes	characteristic
209	C₂×C₄	Q₈×#595	1	1	yes	characteristic
210	C₈	—	2	1	no	—
211	C₈	—	4	2	no	—
212	C₈	—	17	4	no	—
213	C₈	—	17	2	no	—
214	C₈	—	34	2	no	—
215	C₈	—	34	4	no	—
216	C₈	—	34	1	no	—
217	C₈	—	68	2	no	—
218	C₈	C₈×#595	1	2	yes	complemented
219	C₂²	—	17	2	no	—
220	C₂²	—	17	2	no	—
221	C₂²	—	17	1	no	—
222	C₂²	—	17	1	no	—
223	C₂²	C₄⋊C₄×#595	1	1	yes	characteristic
224	C₄	—	2	1	no	—
225	C₄	—	17	2	no	—
226	C₄	—	17	1	no	—
227	C₄	—	17	1	no	—
228	C₄	—	34	1	no	—
229	C₄	—	68	8	no	—
230	C₄	C₂×C₈×#595	1	1	yes	characteristic
231	C₄	#11763775	1	1	yes	characteristic
232	C₄	D₈×#595	1	1	yes	characteristic
233	C₄	Q₁₆×#595	1	1	yes	characteristic
234	C₂	—	17	2	no	—
235	C₂	—	17	2	no	—
236	C₂	#14544577	1	1	yes	characteristic
237	C₂	#14544589	1	1	yes	characteristic
238	C₂	#107×#595	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	D₄×#595
commutator	C₄×C₁₇	C₂×C₄×C₈
Frattini	C₄²	C₂²×#595
Fitting	C₁₇×#335	C₄
socle	C₂²×C₁₇	C₄×C₄⋊C₄
prime radical	C₂²×D₁₇	C₄×D₄
cosocle	C₄²×D₁₇	C₂³
hypercenter	#335	#595
hypocenter	C₁₇	C₈×#107
LCS	G ▷ 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. C₈×#3624	C₄	semidirect
14. C₈×#3624	C₄	semidirect
17. #11762562	C₄	semidirect
18. #11762561	C₄	semidirect
19. C₁₇×#335	C₄	semidirect
34. #10587534	C₈	semidirect
93. #3624	#107	semidirect
96. C₈×C₁₇	C₄×C₈	semidirect
99. #3622	C₄×C₈	semidirect
131. #335	#595	semidirect
162. C₁₇	C₈×#107	semidirect
218. C₈	C₈×#595	semidirect

Maximal subgroups

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

subgroup	index
C₄×#10587534	2
#14544577	2
C₈×#63242	2
#14543618	2
D₁₇×#335	2
C₈×#107	17