Subgroups — #334×#595 #17952316

Order 4352.

470 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⋊C₈×#595	C₂	1	2	yes	maximal
2	C₄×C₈×#595	C₂	1	1	yes	maximal, characteristic
3	#14544326	C₂	1	1	yes	maximal, characteristic
4	#14543613	C₂	1	2	yes	maximal
5	D₁₇×#334	C₂	1	1	yes	maximal, characteristic
6	#11763810	C₂²	1	2	yes	—
7	C₄×#63242	C₂²	1	1	yes	characteristic
8	C₂×C₈×#595	—	2	2	no	—
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	#11763671	C₄	1	2	yes	complemented
14	#11763670	C₄	1	2	yes	complemented
15	C₄⋊C₈×D₁₇	C₂²	1	2	yes	—
16	C₄×C₈×D₁₇	C₂²	1	1	yes	characteristic
17	#11762560	C₄	1	1	yes	characteristic, complemented
18	#11762559	C₄	1	2	yes	complemented
19	C₁₇×#334	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	C₈×#595	—	2	4	no	—
35	C₈×#595	—	4	4	no	—
36	C₈×#595	C₈	1	8	yes	complemented
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	C₂×C₈	1	4	yes	—
54	C₄×#595	M₁₆	1	4	yes	—
55	C₄×#595	SD₁₆	1	4	yes	—
56	#63242	—	2	2	no	—
57	#63242	—	4	2	no	—
58	#63242	—	4	2	no	—
59	#63242	C₂×C₈	1	4	yes	—
60	C₂×C₈×C₁₇	—	2	2	no	—
61	C₂×C₈×C₁₇	C₄²	1	1	yes	characteristic
62	C₂×C₈×C₁₇	C₄²	1	1	yes	characteristic
63	C₄²×C₁₇	C₂²×C₄	1	1	yes	characteristic
64	C₄×Dic₁₇	C₂²×C₄	1	1	yes	characteristic
65	C₄×Dic₁₇	#41	1	1	yes	characteristic
66	C₄×Dic₁₇	#41	1	1	yes	characteristic
67	C₂×#3622	—	2	2	no	—
68	C₂×#3622	C₄²	1	1	yes	characteristic
69	C₂×#3622	C₄²	1	1	yes	characteristic
70	C₈×D₁₇	—	2	2	no	—
71	C₈×D₁₇	—	4	2	no	—
72	C₈×D₁₇	—	4	2	no	—
73	C₈×D₁₇	C₂×C₈	1	4	yes	—
74	C₄×#334	—	17	1	no	maximal, Sylow
75	C₂²×D₁₇	C₂×C₄⋊C₄	1	1	yes	characteristic
76	C₂×#595	—	2	2	no	—
77	C₂×#595	—	2	2	no	—
78	C₂×#595	C₄⋊C₈	1	4	yes	—
79	C₂×#595	C₄⋊C₈	1	4	yes	—
80	C₂×#595	#106	1	4	yes	—
81	C₂×#595	#108	1	4	yes	—
82	C₂×C₄×C₁₇	C₂×C₄²	1	1	yes	characteristic
83	C₂×C₄×C₁₇	C₄×D₄	1	1	yes	characteristic
84	C₂×C₄×C₁₇	C₄×Q₈	1	1	yes	characteristic
85	C₂×Dic₁₇	#117	1	1	yes	characteristic
86	C₂×Dic₁₇	C₄×D₄	1	1	yes	characteristic
87	C₂×Dic₁₇	C₄×Q₈	1	1	yes	characteristic
88	C₄×D₁₇	—	2	2	no	—
89	C₄×D₁₇	C₂²×C₈	1	2	yes	—
90	C₄×D₁₇	C₂×M₁₆	1	2	yes	—
91	C₄×D₁₇	C₂×SD₁₆	1	2	yes	—
92	C₄×D₁₇	#135	1	2	yes	—
93	#3624	—	2	4	no	—
94	#3624	—	4	4	no	—
95	C₈×C₁₇	—	2	1	no	—
96	C₈×C₁₇	—	4	2	no	—
97	C₈×C₁₇	C₄×C₈	1	2	yes	complemented
98	#3622	—	2	1	no	—
99	#3622	—	4	2	no	—
100	#3622	C₄×C₈	1	2	yes	complemented
101	C₄×C₄⋊C₈	—	17	2	no	—
102	C₄²×C₈	—	17	1	no	—
103	C₂×#334	—	17	1	no	—
104	C₂×#334	—	17	1	no	—
105	#1277	—	17	2	no	—
106	C₂²×C₁₇	C₄×C₄⋊C₄	1	1	yes	characteristic
107	C₂×D₁₇	C₂×C₄⋊C₈	1	2	yes	—
108	C₂×D₁₇	C₂×#106	1	1	yes	characteristic
109	C₂×D₁₇	#427	1	1	yes	characteristic
110	C₂×D₁₇	C₂×#108	1	2	yes	—
111	#595	—	2	4	no	—
112	#595	#334	1	8	yes	direct product factor
113	C₄×C₁₇	—	2	1	no	—
114	C₄×C₁₇	C₂×C₄×C₈	1	1	yes	characteristic
115	C₄×C₁₇	C₄×M₁₆	1	1	yes	characteristic
116	C₄×C₁₇	C₄×SD₁₆	1	2	yes	—
117	Dic₁₇	—	2	1	no	—
118	Dic₁₇	#431	1	2	yes	—
119	Dic₁₇	C₄×SD₁₆	1	2	yes	—
120	C₂×C₄⋊C₈	—	17	2	no	—
121	C₂×C₄⋊C₈	—	17	2	no	—
122	C₂×C₄×C₈	—	17	2	no	—
123	C₂×C₄×C₈	—	17	1	no	—
124	C₂×C₄×C₈	—	17	2	no	—
125	C₂×C₄×C₈	—	17	1	no	—
126	C₂×C₄×C₈	—	34	2	no	—
127	C₄³	—	17	1	no	—
128	#336	—	34	2	no	—
129	#334	—	17	2	no	—
130	#334	—	17	2	no	—
131	#334	—	17	2	no	—
132	#334	—	17	1	no	—
133	#334	#595	1	1	yes	characteristic, direct product factor
134	C₂×C₁₇	C₄×C₄⋊C₈	1	1	yes	characteristic
135	C₂×C₁₇	C₄×#106	1	1	yes	characteristic
136	C₂×C₁₇	C₄×#108	1	1	yes	characteristic
137	D₁₇	C₂×#334	1	2	yes	—
138	D₁₇	#1564	1	2	yes	—
139	C₂²×C₈	—	17	1	no	—
140	C₂²×C₈	—	17	2	no	—
141	C₂²×C₈	—	17	1	no	—
142	C₂²×C₈	—	34	2	no	—
143	C₂²×C₈	—	34	2	no	—
144	C₂×C₄²	—	17	2	no	—
145	C₂×C₄²	—	17	1	no	—
146	C₂×C₄²	—	17	1	no	—
147	C₂×C₄²	—	17	1	no	—
148	C₂×C₄²	—	17	2	no	—
149	C₄⋊C₈	—	17	4	no	—
150	C₄⋊C₈	—	17	2	no	—
151	C₄⋊C₈	—	34	2	no	—
152	C₄⋊C₈	—	34	2	no	—
153	C₄⋊C₈	C₂×#595	1	2	yes	—
154	C₄×C₈	—	17	8	no	—
155	C₄×C₈	—	17	1	no	—
156	C₄×C₈	—	17	1	no	—
157	C₄×C₈	—	17	1	no	—
158	C₄×C₈	—	34	4	no	—
159	C₄×C₈	—	34	1	no	—
160	C₄×C₈	—	34	1	no	—
161	C₄×C₈	—	68	4	no	—
162	C₄×C₈	C₂×#595	1	1	yes	characteristic
163	C₁₇	C₄×#334	1	1	yes	characteristic, complemented, Sylow
164	C₂²×C₄	—	17	2	no	—
165	C₂²×C₄	—	17	1	no	—
166	C₂²×C₄	—	17	1	no	—
167	C₂²×C₄	—	17	2	no	—
168	C₂²×C₄	—	17	1	no	—
169	C₂×C₈	—	2	2	no	—
170	C₂×C₈	—	17	1	no	—
171	C₂×C₈	—	17	4	no	—
172	C₂×C₈	—	17	4	no	—
173	C₂×C₈	—	17	1	no	—
174	C₂×C₈	—	34	2	no	—
175	C₂×C₈	—	34	2	no	—
176	C₂×C₈	—	34	2	no	—
177	C₂×C₈	—	34	4	no	—
178	C₂×C₈	—	34	2	no	—
179	C₂×C₈	—	68	2	no	—
180	C₂×C₈	—	68	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	C₄²	—	17	4	no	—
186	C₄²	—	17	4	no	—
187	C₄²	—	17	4	no	—
188	C₄²	—	17	1	no	—
189	C₄²	—	17	1	no	—
190	C₄²	—	17	1	no	—
191	C₄²	—	34	4	no	—
192	C₄²	—	34	2	no	—
193	C₄²	C₂²×#595	1	1	yes	characteristic
194	C₂³	—	17	1	no	—
195	C₂×C₄	—	17	4	no	—
196	C₂×C₄	—	17	4	no	—
197	C₂×C₄	—	17	4	no	—
198	C₂×C₄	—	17	1	no	—
199	C₂×C₄	—	17	1	no	—
200	C₂×C₄	—	17	2	no	—
201	C₂×C₄	—	17	2	no	—
202	C₂×C₄	—	17	2	no	—
203	C₂×C₄	—	17	2	no	—
204	C₂×C₄	—	17	4	no	—
205	C₂×C₄	—	17	1	no	—
206	C₂×C₄	—	34	2	no	—
207	C₂×C₄	—	34	2	no	—
208	C₂×C₄	—	34	2	no	—
209	C₂×C₄	C₂×C₄×#595	1	1	yes	characteristic
210	C₂×C₄	D₄×#595	1	1	yes	characteristic
211	C₂×C₄	Q₈×#595	1	1	yes	characteristic
212	C₈	—	2	1	no	—
213	C₈	—	4	2	no	—
214	C₈	—	17	2	no	—
215	C₈	—	34	4	no	—
216	C₈	—	34	1	no	—
217	C₈	—	68	4	no	—
218	C₈	—	68	2	no	—
219	C₈	C₈×#595	1	2	yes	complemented
220	C₂²	—	17	2	no	—
221	C₂²	—	17	1	no	—
222	C₂²	—	17	1	no	—
223	C₂²	—	17	2	no	—
224	C₂²	C₄⋊C₄×#595	1	1	yes	characteristic
225	C₄	—	2	1	no	—
226	C₄	—	17	8	no	—
227	C₄	—	17	2	no	—
228	C₄	—	17	2	no	—
229	C₄	—	34	4	no	—
230	C₄	—	34	1	no	—
231	C₄	C₂×C₈×#595	1	1	yes	characteristic
232	C₄	M₁₆×#595	1	1	yes	characteristic
233	C₄	SD₁₆×#595	1	2	yes	—
234	C₂	—	17	2	no	—
235	C₂	—	17	2	no	—
236	C₂	C₄⋊C₈×#595	1	1	yes	characteristic
237	C₂	#106×#595	1	1	yes	characteristic
238	C₂	#108×#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×#334	C₄
socle	C₂²×C₁₇	C₄×C₄⋊C₄
prime radical	C₂²×D₁₇	C₂×C₄⋊C₄
cosocle	C₄²×D₁₇	C₂³
hypercenter	#334	#595
hypocenter	C₁₇	C₄×#334
LCS	G ▷ C₄×C₁₇ ▷ C₂×C₁₇ ▷ C₁₇
UCS	C₁ ◁ C₂×C₄ ◁ C₄² ◁ #334

Complements

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

N	H ≅ G/N	type
13. #11763671	C₄	semidirect
14. #11763670	C₄	semidirect
17. #11762560	C₄	semidirect
18. #11762559	C₄	semidirect
19. C₁₇×#334	C₄	semidirect
36. C₈×#595	C₈	semidirect
97. C₈×C₁₇	C₄×C₈	semidirect
100. #3622	C₄×C₈	semidirect
112. #595	#334	direct (N × H)
133. #334	#595	direct (N × H)
163. C₁₇	C₄×#334	semidirect
219. 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⋊C₈×#595	2
C₄×C₈×#595	2
#14544326	2
#14543613	2
D₁₇×#334	2
C₄×#334	17