Subgroups — #346×#595 #17960417

Order 4352.

510 conjugacy classes of subgroups.

The subgroup lattice has 224 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#14544985	C₂	1	1	yes	maximal, characteristic
2	C₂×#11763696	C₂	1	2	yes	maximal
3	C₂×C₁₆×#595	C₂	1	2	yes	maximal
4	C₄×C₈×#595	C₂	1	1	yes	maximal, characteristic
5	D₁₇×#346	C₂	1	1	yes	maximal, characteristic
6	C₂×#10587529	C₂²	1	2	yes	—
7	C₄×#63242	C₄	1	1	yes	characteristic
8	C₂×C₈×#595	C₄	1	1	yes	characteristic
9	C₂×C₈×#595	C₄	1	1	yes	characteristic
10	C₂×C₈×#595	C₂²	1	1	yes	characteristic
11	C₂×C₈×#595	C₂²	1	1	yes	characteristic
12	C₄²×#595	C₄	1	1	yes	characteristic
13	#11763698	C₄	1	2	yes	complemented
14	#11763697	C₄	1	2	yes	complemented
15	#11763696	C₄	1	8	yes	complemented
16	C₁₆×#595	C₄	1	8	yes	complemented
17	C₂×C₁₆×D₁₇	C₂²	1	2	yes	—
18	C₄×C₈×D₁₇	C₂²	1	1	yes	characteristic
19	#11762585	C₄	1	1	yes	characteristic, complemented
20	#11762584	C₄	1	2	yes	complemented
21	C₁₇×#346	C₄	1	1	yes	characteristic, complemented
22	C₂×C₄×#595	C₈	1	4	yes	—
23	C₂×C₄×#595	C₂×C₄	1	1	yes	characteristic
24	C₂×C₄×#595	C₂×C₄	1	1	yes	characteristic
25	C₂×#63242	C₂×C₄	1	1	yes	characteristic
26	C₂×#63242	C₂×C₄	1	1	yes	characteristic
27	C₂×C₈×D₁₇	C₂×C₄	1	1	yes	characteristic
28	C₂×C₈×D₁₇	C₂³	1	1	yes	characteristic
29	C₄²×D₁₇	C₂×C₄	1	1	yes	characteristic
30	C₄×#3624	C₈	1	4	yes	—
31	C₂×#63216	C₂×C₄	1	4	yes	—
32	C₈×#595	C₈	1	4	yes	—
33	C₈×#595	C₈	1	4	yes	—
34	C₈×#595	C₂×C₄	1	4	yes	—
35	C₈×#595	C₂×C₄	1	4	yes	—
36	#10587529	C₂×C₄	1	4	yes	—
37	#10587529	C₂×C₄	1	4	yes	—
38	C₂×C₁₆×C₁₇	C₂×C₄	1	2	yes	—
39	C₄×C₈×C₁₇	C₂×C₄	1	1	yes	characteristic
40	C₈×Dic₁₇	C₂×C₄	1	1	yes	characteristic
41	C₈×Dic₁₇	C₂×C₄	1	1	yes	characteristic
42	C₂×#63214	C₂×C₄	1	2	yes	—
43	C₄×#3622	C₂×C₄	1	1	yes	characteristic
44	C₁₆×D₁₇	C₂×C₄	1	4	yes	—
45	C₁₆×D₁₇	C₂×C₄	1	4	yes	—
46	C₂²×#595	C₂×C₈	1	2	yes	—
47	C₂²×#595	C₂×C₈	1	2	yes	—
48	C₂×C₄×D₁₇	C₂×C₈	1	2	yes	—
49	C₂×C₄×D₁₇	C₂²×C₄	1	1	yes	characteristic
50	C₂×#3624	C₂×C₈	1	2	yes	—
51	C₂×#3624	C₂×C₈	1	2	yes	—
52	C₄×#595	—	2	8	no	—
53	C₄×#595	C₄²	1	2	yes	—
54	C₄×#595	C₄²	1	2	yes	—
55	C₄×#595	C₂×C₈	1	2	yes	—
56	C₄×#595	C₂×C₈	1	2	yes	—
57	#63242	C₄²	1	2	yes	—
58	#63242	C₄²	1	2	yes	—
59	#63242	C₂×C₈	1	2	yes	—
60	#63242	C₂×C₈	1	2	yes	—
61	C₂×C₈×C₁₇	C₄²	1	1	yes	characteristic
62	C₂×C₈×C₁₇	C₂²×C₄	1	1	yes	characteristic
63	C₄²×C₁₇	C₄²	1	1	yes	characteristic
64	C₄×Dic₁₇	C₄²	1	1	yes	characteristic
65	C₄×Dic₁₇	C₂×C₈	1	2	yes	—
66	C₂×#3622	C₄²	1	1	yes	characteristic
67	C₂×#3622	C₂²×C₄	1	1	yes	characteristic
68	C₈×D₁₇	C₂×C₈	1	2	yes	—
69	C₈×D₁₇	C₂×C₈	1	2	yes	—
70	C₈×D₁₇	C₂²×C₄	1	2	yes	—
71	C₈×D₁₇	C₂²×C₄	1	2	yes	—
72	#63216	C₄²	1	8	yes	complemented
73	C₁₆×C₁₇	C₄²	1	4	yes	complemented
74	#63214	C₄²	1	4	yes	complemented
75	C₄×#346	—	17	1	no	maximal, Sylow
76	C₂²×D₁₇	C₂²×C₈	1	1	yes	characteristic
77	C₂×#595	—	2	4	no	—
78	C₂×#595	C₄×C₈	1	4	yes	—
79	C₂×#595	C₄×C₈	1	4	yes	—
80	C₂×#595	M₃₂	1	8	yes	—
81	C₂×C₄×C₁₇	C₄×C₈	1	2	yes	—
82	C₂×C₄×C₁₇	C₂×C₄²	1	1	yes	characteristic
83	C₂×Dic₁₇	C₄×C₈	1	2	yes	—
84	C₂×Dic₁₇	C₂²×C₈	1	1	yes	characteristic
85	C₄×D₁₇	—	2	4	no	—
86	C₄×D₁₇	C₂×C₄²	1	1	yes	characteristic
87	C₄×D₁₇	C₂×C₄²	1	1	yes	characteristic
88	C₄×D₁₇	C₂²×C₈	1	1	yes	characteristic
89	C₄×D₁₇	C₂²×C₈	1	1	yes	characteristic
90	#3624	C₄×C₈	1	4	yes	—
91	#3624	C₄×C₈	1	4	yes	—
92	C₈×C₁₇	C₄×C₈	1	2	yes	—
93	C₈×C₁₇	C₂×C₄²	1	2	yes	—
94	#3622	C₄×C₈	1	2	yes	—
95	#3622	C₂×C₄²	1	2	yes	—
96	C₂×#346	—	17	2	no	—
97	C₂×#346	—	17	1	no	—
98	C₂×#346	—	17	1	no	—
99	C₂×C₄×C₁₆	—	17	2	no	—
100	C₄²×C₈	—	17	1	no	—
101	C₂²×C₁₇	C₂×C₄×C₈	1	1	yes	characteristic
102	C₂×D₁₇	C₂×C₄×C₈	1	1	yes	characteristic
103	C₂×D₁₇	C₂×C₄×C₈	1	1	yes	characteristic
104	C₂×D₁₇	C₂×M₃₂	1	4	yes	—
105	#595	—	2	4	no	—
106	#595	#346	1	8	yes	direct product factor
107	C₄×C₁₇	—	2	2	no	—
108	C₄×C₁₇	C₄³	1	1	yes	characteristic
109	C₄×C₁₇	C₂×C₄×C₈	1	1	yes	characteristic
110	Dic₁₇	—	2	2	no	—
111	Dic₁₇	C₂×C₄×C₈	1	1	yes	characteristic
112	Dic₁₇	C₂×C₄×C₈	1	1	yes	characteristic
113	C₂²×C₁₆	—	17	2	no	—
114	C₂²×C₁₆	—	17	2	no	—
115	C₂×C₄×C₈	—	17	1	no	—
116	C₂×C₄×C₈	—	17	1	no	—
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	C₄³	—	17	1	no	—
122	#346	—	17	8	no	—
123	#346	—	17	2	no	—
124	#346	—	17	2	no	—
125	#346	—	17	2	no	—
126	#346	—	17	1	no	—
127	#346	#595	1	1	yes	characteristic, direct product factor
128	C₄×C₁₆	—	17	8	no	—
129	C₂×C₁₇	C₄²×C₈	1	1	yes	characteristic
130	C₂×C₁₇	C₄×M₃₂	1	2	yes	—
131	D₁₇	C₂×#346	1	2	yes	—
132	D₁₇	C₄×M₃₂	1	2	yes	—
133	C₂²×C₈	—	17	1	no	—
134	C₂²×C₈	—	17	1	no	—
135	C₂²×C₈	—	17	1	no	—
136	C₂²×C₈	—	17	1	no	—
137	C₂×C₄²	—	17	1	no	—
138	C₂×C₄²	—	17	1	no	—
139	C₂×C₄²	—	17	1	no	—
140	C₂×C₄²	—	17	4	no	—
141	C₂×C₁₆	—	17	4	no	—
142	C₂×C₁₆	—	17	4	no	—
143	C₂×C₁₆	—	17	4	no	—
144	C₂×C₁₆	—	17	2	no	—
145	C₂×C₁₆	—	17	4	no	—
146	C₂×C₁₆	—	17	4	no	—
147	C₂×C₁₆	C₂×#595	1	2	yes	—
148	C₄×C₈	—	17	4	no	—
149	C₄×C₈	—	17	4	no	—
150	C₄×C₈	—	17	4	no	—
151	C₄×C₈	—	17	4	no	—
152	C₄×C₈	—	17	4	no	—
153	C₄×C₈	—	17	1	no	—
154	C₄×C₈	—	17	1	no	—
155	C₄×C₈	—	17	1	no	—
156	C₄×C₈	C₂×#595	1	1	yes	characteristic
157	C₁₇	C₄×#346	1	1	yes	characteristic, complemented, Sylow
158	C₂²×C₄	—	17	2	no	—
159	C₂²×C₄	—	17	1	no	—
160	C₂²×C₄	—	17	2	no	—
161	C₂²×C₄	—	17	2	no	—
162	C₂×C₈	—	17	2	no	—
163	C₂×C₈	—	17	2	no	—
164	C₂×C₈	—	17	2	no	—
165	C₂×C₈	—	17	2	no	—
166	C₂×C₈	—	17	2	no	—
167	C₂×C₈	—	17	2	no	—
168	C₂×C₈	—	17	1	no	—
169	C₂×C₈	—	17	1	no	—
170	C₂×C₈	—	17	2	no	—
171	C₂×C₈	—	17	2	no	—
172	C₂×C₈	—	17	2	no	—
173	C₂×C₈	—	17	2	no	—
174	C₂×C₈	C₄×#595	1	1	yes	characteristic
175	C₂×C₈	C₂²×#595	1	1	yes	characteristic
176	C₄²	—	17	2	no	—
177	C₄²	—	17	2	no	—
178	C₄²	—	17	2	no	—
179	C₄²	—	17	2	no	—
180	C₄²	—	17	1	no	—
181	C₄²	—	17	2	no	—
182	C₄²	—	34	8	no	—
183	C₄²	C₄×#595	1	1	yes	characteristic
184	C₁₆	—	17	8	no	—
185	C₁₆	—	17	4	no	—
186	C₁₆	C₄×#595	1	4	yes	complemented
187	C₂³	—	17	1	no	—
188	C₂×C₄	—	17	4	no	—
189	C₂×C₄	—	17	8	no	—
190	C₂×C₄	—	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	1	no	—
195	C₂×C₄	—	17	4	no	—
196	C₂×C₄	—	17	2	no	—
197	C₂×C₄	—	34	4	no	—
198	C₂×C₄	—	34	4	no	—
199	C₂×C₄	C₈×#595	1	2	yes	—
200	C₂×C₄	C₂×C₄×#595	1	1	yes	characteristic
201	C₈	—	17	4	no	—
202	C₈	—	17	4	no	—
203	C₈	—	17	2	no	—
204	C₈	—	17	2	no	—
205	C₈	C₈×#595	1	2	yes	—
206	C₈	C₂×C₄×#595	1	2	yes	—
207	C₂²	—	17	1	no	—
208	C₂²	—	17	1	no	—
209	C₂²	—	17	4	no	—
210	C₂²	C₂×C₈×#595	1	1	yes	characteristic
211	C₄	—	2	2	no	—
212	C₄	—	17	8	no	—
213	C₄	—	17	1	no	—
214	C₄	—	17	1	no	—
215	C₄	—	34	4	no	—
216	C₄	—	34	2	no	—
217	C₄	C₄²×#595	1	1	yes	characteristic
218	C₄	C₂×C₈×#595	1	1	yes	characteristic
219	C₂	—	17	2	no	—
220	C₂	—	17	2	no	—
221	C₂	C₄×C₈×#595	1	1	yes	characteristic
222	C₂	M₃₂×#595	1	2	yes	—

	subgroup	G/H
center, Frattini	C₂×C₈	C₂²×#595
commutator	C₂×C₁₇	C₄²×C₈
Fitting	C₁₇×#346	C₄
socle	C₂²×C₁₇	C₂×C₄×C₈
prime radical	C₂²×D₁₇	C₂²×C₈
cosocle	C₂×C₈×D₁₇	C₂³
hypercenter	#346	#595
hypocenter	C₁₇	C₄×#346
LCS	G ▷ C₂×C₁₇ ▷ C₁₇
UCS	C₁ ◁ C₂×C₈ ◁ #346

Complements

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

N	H ≅ G/N	type
13. #11763698	C₄	semidirect
14. #11763697	C₄	semidirect
15. #11763696	C₄	semidirect
16. C₁₆×#595	C₄	semidirect
19. #11762585	C₄	semidirect
20. #11762584	C₄	semidirect
21. C₁₇×#346	C₄	semidirect
72. #63216	C₄²	semidirect
73. C₁₆×C₁₇	C₄²	semidirect
74. #63214	C₄²	semidirect
106. #595	#346	direct (N × H)
127. #346	#595	direct (N × H)
157. C₁₇	C₄×#346	semidirect
186. C₁₆	C₄×#595	semidirect

Maximal subgroups

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

subgroup	index
#14544985	2
C₂×#11763696	2
C₂×C₁₆×#595	2
C₄×C₈×#595	2
D₁₇×#346	2
C₄×#346	17