Subgroups — #17950289

Order 4352.

486 conjugacy classes of subgroups.

The subgroup lattice has 228 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#14544497	C₂	1	1	yes	maximal, characteristic
2	#14544497	C₂	1	1	yes	maximal, characteristic
3	#14544497	C₂	1	1	yes	maximal, characteristic
4	#14543980	C₂	1	1	yes	maximal, characteristic
5	#14543978	C₂	1	1	yes	maximal, characteristic
6	C₈×#63242	C₂	1	1	yes	maximal, characteristic
7	D₁₇×#322	C₂	1	1	yes	maximal, characteristic
8	C₄×#63242	C₂²	1	1	yes	characteristic
9	C₄×#63242	C₂²	1	1	yes	characteristic
10	C₄×#63242	C₂²	1	1	yes	characteristic
11	C₄²×#595	C₂²	1	1	yes	characteristic
12	#11763674	C₄	1	4	yes	—
13	#11763674	C₄	1	4	yes	—
14	#11763674	C₄	1	4	yes	—
15	#11763651	C₄	1	4	yes	complemented
16	#11763650	C₄	1	4	yes	complemented
17	C₈×#3624	C₄	1	4	yes	complemented
18	C₄×C₈×D₁₇	C₂²	1	1	yes	characteristic
19	C₄×C₈×D₁₇	C₂²	1	1	yes	characteristic
20	C₄×C₈×D₁₇	C₂²	1	1	yes	characteristic
21	#11762531	C₄	1	1	yes	characteristic, complemented
22	#11762531	C₄	1	1	yes	characteristic, complemented
23	#11762530	C₄	1	1	yes	characteristic, complemented
24	C₁₇×#322	C₄	1	1	yes	characteristic, complemented
25	C₂×C₄×#595	C₂×C₄	1	2	yes	—
26	C₂×C₄×#595	C₂×C₄	1	2	yes	—
27	C₂×C₄×#595	C₂×C₄	1	2	yes	—
28	C₂×#63242	C₂×C₄	1	2	yes	—
29	C₂×#63242	C₂×C₄	1	2	yes	—
30	C₂×#63242	C₂×C₄	1	2	yes	—
31	C₂×C₈×D₁₇	C₂×C₄	1	2	yes	—
32	C₂×C₈×D₁₇	C₂×C₄	1	2	yes	—
33	C₂×C₈×D₁₇	C₂×C₄	1	2	yes	—
34	C₄²×D₁₇	C₂³	1	1	yes	characteristic
35	C₄×#3624	C₂×C₄	1	2	yes	—
36	C₄×#3624	C₂×C₄	1	2	yes	—
37	C₄×#3624	C₂×C₄	1	2	yes	—
38	C₄×#3624	C₂×C₄	1	2	yes	—
39	C₄×#3624	C₂×C₄	1	2	yes	—
40	C₄×#3624	C₂×C₄	1	2	yes	—
41	C₄×C₈×C₁₇	C₂×C₄	1	1	yes	characteristic
42	C₄×C₈×C₁₇	C₂×C₄	1	1	yes	characteristic
43	C₄×C₈×C₁₇	C₂×C₄	1	1	yes	characteristic
44	C₈×Dic₁₇	C₂×C₄	1	2	yes	—
45	C₈×Dic₁₇	C₂×C₄	1	2	yes	—
46	C₈×Dic₁₇	C₂×C₄	1	2	yes	—
47	C₄×#3622	C₂×C₄	1	1	yes	characteristic
48	C₄×#3622	C₂×C₄	1	1	yes	characteristic
49	C₄×#3622	C₂×C₄	1	1	yes	characteristic
50	C₂²×#595	C₄²	1	4	yes	—
51	C₂×C₄×D₁₇	C₂²×C₄	1	1	yes	characteristic
52	C₂×C₄×D₁₇	C₂²×C₄	1	1	yes	characteristic
53	C₂×C₄×D₁₇	C₂²×C₄	1	1	yes	characteristic
54	C₂×#3624	C₄²	1	4	yes	—
55	C₂×#3624	C₄²	1	4	yes	—
56	C₂×#3624	C₄²	1	4	yes	—
57	C₄×#595	—	2	4	no	—
58	C₄×#595	—	2	4	no	—
59	C₄×#595	—	2	4	no	—
60	#63242	—	2	4	no	—
61	#63242	—	2	4	no	—
62	#63242	—	2	4	no	—
63	C₂×C₈×C₁₇	C₄²	1	2	yes	—
64	C₂×C₈×C₁₇	C₄²	1	2	yes	—
65	C₂×C₈×C₁₇	C₄²	1	2	yes	—
66	C₄²×C₁₇	C₂²×C₄	1	1	yes	characteristic
67	C₄×Dic₁₇	C₂²×C₄	1	1	yes	characteristic
68	C₄×Dic₁₇	C₂²×C₄	1	1	yes	characteristic
69	C₄×Dic₁₇	C₂²×C₄	1	1	yes	characteristic
70	C₂×#3622	C₄²	1	2	yes	—
71	C₂×#3622	C₄²	1	2	yes	—
72	C₂×#3622	C₄²	1	2	yes	—
73	C₈×D₁₇	—	2	4	no	—
74	C₈×D₁₇	—	2	4	no	—
75	C₈×D₁₇	—	2	4	no	—
76	#7845	—	17	1	no	maximal, Sylow
77	C₂²×D₁₇	C₂×C₄²	1	1	yes	characteristic
78	C₂×#595	—	2	4	no	—
79	C₂×#595	—	2	4	no	—
80	C₂×#595	—	2	4	no	—
81	C₂×C₄×C₁₇	C₂×C₄²	1	1	yes	characteristic
82	C₂×C₄×C₁₇	C₂×C₄²	1	1	yes	characteristic
83	C₂×C₄×C₁₇	C₂×C₄²	1	1	yes	characteristic
84	C₂×Dic₁₇	C₂×C₄²	1	1	yes	characteristic
85	C₂×Dic₁₇	C₂×C₄²	1	1	yes	characteristic
86	C₂×Dic₁₇	C₂×C₄²	1	1	yes	characteristic
87	C₄×D₁₇	#131	1	4	yes	—
88	C₄×D₁₇	#131	1	4	yes	—
89	C₄×D₁₇	#131	1	4	yes	—
90	#3624	—	2	4	no	—
91	#3624	—	2	4	no	—
92	#3624	—	2	4	no	—
93	C₈×C₁₇	—	2	2	no	—
94	C₈×C₁₇	—	2	2	no	—
95	C₈×C₁₇	—	2	2	no	—
96	#3622	—	2	2	no	—
97	#3622	—	2	2	no	—
98	#3622	—	2	2	no	—
99	#1744	—	17	1	no	—
100	#1744	—	17	1	no	—
101	#1744	—	17	1	no	—
102	C₂×#322	—	17	1	no	—
103	C₂×#322	—	17	1	no	—
104	C₂×#322	—	17	1	no	—
105	C₂×C₈²	—	17	1	no	—
106	C₂²×C₁₇	C₄³	1	1	yes	characteristic
107	C₂×D₁₇	#405	1	2	yes	—
108	C₂×D₁₇	#405	1	2	yes	—
109	C₂×D₁₇	#405	1	2	yes	—
110	#595	—	4	4	no	—
111	C₄×C₁₇	#405	1	2	yes	—
112	C₄×C₁₇	#405	1	2	yes	—
113	C₄×C₁₇	#405	1	2	yes	—
114	Dic₁₇	#405	1	2	yes	—
115	Dic₁₇	#405	1	2	yes	—
116	Dic₁₇	#405	1	2	yes	—
117	C₂×C₄×C₈	—	17	1	no	—
118	C₂×C₄×C₈	—	17	1	no	—
119	C₂×C₄×C₈	—	17	1	no	—
120	C₂×C₄×C₈	—	17	1	no	—
121	C₂×C₄×C₈	—	17	1	no	—
122	C₂×C₄×C₈	—	17	1	no	—
123	C₄³	—	17	1	no	—
124	#336	—	17	4	no	—
125	#336	—	17	4	no	—
126	#336	—	17	4	no	—
127	#322	—	17	4	no	—
128	#322	—	17	4	no	—
129	#322	—	17	1	no	—
130	#322	—	17	1	no	—
131	#322	—	17	1	no	—
132	#322	#595	1	1	yes	characteristic, complemented
133	C₈²	—	17	4	no	—
134	C₂×C₁₇	#1726	1	1	yes	characteristic
135	C₂×C₁₇	#1726	1	1	yes	characteristic
136	C₂×C₁₇	#1726	1	1	yes	characteristic
137	D₁₇	#1454	1	4	yes	—
138	C₂²×C₈	—	17	2	no	—
139	C₂²×C₈	—	17	2	no	—
140	C₂²×C₈	—	17	2	no	—
141	C₂²×C₈	—	17	2	no	—
142	C₂²×C₈	—	17	2	no	—
143	C₂²×C₈	—	17	2	no	—
144	C₂×C₄²	—	17	1	no	—
145	C₂×C₄²	—	17	2	no	—
146	C₂×C₄²	—	17	2	no	—
147	C₂×C₄²	—	17	2	no	—
148	C₄×C₈	—	17	2	no	—
149	C₄×C₈	—	17	2	no	—
150	C₄×C₈	—	17	2	no	—
151	C₄×C₈	—	17	2	no	—
152	C₄×C₈	—	17	2	no	—
153	C₄×C₈	—	17	2	no	—
154	C₄×C₈	—	17	2	no	—
155	C₄×C₈	—	17	2	no	—
156	C₄×C₈	—	17	2	no	—
157	C₄×C₈	—	17	1	no	—
158	C₄×C₈	—	17	1	no	—
159	C₄×C₈	—	17	1	no	—
160	C₄×C₈	C₂×#595	1	1	yes	characteristic
161	C₄×C₈	C₂×#595	1	1	yes	characteristic
162	C₄×C₈	C₂×#595	1	1	yes	characteristic
163	C₁₇	#7845	1	1	yes	characteristic, complemented, Sylow
164	C₂²×C₄	—	17	1	no	—
165	C₂²×C₄	—	17	1	no	—
166	C₂²×C₄	—	17	1	no	—
167	C₂²×C₄	—	17	4	no	—
168	C₂×C₈	—	17	4	no	—
169	C₂×C₈	—	17	4	no	—
170	C₂×C₈	—	17	4	no	—
171	C₂×C₈	—	17	2	no	—
172	C₂×C₈	—	17	2	no	—
173	C₂×C₈	—	17	2	no	—
174	C₂×C₈	—	34	4	no	—
175	C₂×C₈	—	34	4	no	—
176	C₂×C₈	—	34	4	no	—
177	C₂×C₈	—	34	4	no	—
178	C₂×C₈	—	34	4	no	—
179	C₂×C₈	—	34	4	no	—
180	C₂×C₈	C₄×#595	1	2	yes	—
181	C₂×C₈	C₄×#595	1	2	yes	—
182	C₂×C₈	C₄×#595	1	2	yes	—
183	C₄²	—	17	1	no	—
184	C₄²	—	17	1	no	—
185	C₄²	—	17	1	no	—
186	C₄²	—	34	4	no	—
187	C₄²	—	34	4	no	—
188	C₄²	—	34	4	no	—
189	C₄²	C₂²×#595	1	1	yes	characteristic
190	C₂³	—	17	1	no	—
191	C₂×C₄	—	17	1	no	—
192	C₂×C₄	—	17	1	no	—
193	C₂×C₄	—	17	1	no	—
194	C₂×C₄	—	17	4	no	—
195	C₂×C₄	—	17	4	no	—
196	C₂×C₄	—	17	4	no	—
197	C₂×C₄	—	34	4	no	—
198	C₂×C₄	—	34	4	no	—
199	C₂×C₄	—	34	4	no	—
200	C₂×C₄	C₂×C₄×#595	1	1	yes	characteristic
201	C₂×C₄	C₂×C₄×#595	1	1	yes	characteristic
202	C₂×C₄	C₂×C₄×#595	1	1	yes	characteristic
203	C₈	—	2	2	no	—
204	C₈	—	2	2	no	—
205	C₈	—	2	2	no	—
206	C₈	—	34	4	no	—
207	C₈	—	34	4	no	—
208	C₈	—	34	4	no	—
209	C₈	—	34	2	no	—
210	C₈	—	34	2	no	—
211	C₈	—	34	2	no	—
212	C₂²	—	17	2	no	—
213	C₂²	—	17	2	no	—
214	C₂²	—	17	2	no	—
215	C₂²	C₄²×#595	1	1	yes	characteristic
216	C₄	—	17	2	no	—
217	C₄	—	17	2	no	—
218	C₄	—	17	2	no	—
219	C₄	—	68	4	no	—
220	C₄	#11763772	1	2	yes	—
221	C₄	#11763775	1	2	yes	—
222	C₄	#11763775	1	2	yes	—
223	C₂	—	17	4	no	—
224	C₂	#14544440	1	1	yes	characteristic
225	C₂	#14544442	1	1	yes	characteristic
226	C₂	#14544442	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₄²	C₂²×#595
commutator, socle	C₂²×C₁₇	C₄³
Fitting	C₁₇×#322	C₄
prime radical	C₂²×D₁₇	C₂×C₄²
cosocle	C₄²×D₁₇	C₂³
hypercenter	#322	#595
hypocenter	C₁₇	#7845
LCS	G ▷ C₂²×C₁₇ ▷ C₁₇
UCS	C₁ ◁ C₄² ◁ #322

Complements

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

N	H ≅ G/N	type
15. #11763651	C₄	semidirect
16. #11763650	C₄	semidirect
17. C₈×#3624	C₄	semidirect
21. #11762531	C₄	semidirect
22. #11762531	C₄	semidirect
23. #11762530	C₄	semidirect
24. C₁₇×#322	C₄	semidirect
132. #322	#595	semidirect
163. C₁₇	#7845	semidirect

Maximal subgroups

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

subgroup	index
#14544497	2
#14543980	2
#14543978	2
C₈×#63242	2
D₁₇×#322	2
#7845	17