Subgroups — #347×#595 #17960426

Order 4352.

418 conjugacy classes of subgroups.

The subgroup lattice has 242 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#14545001	C₂	1	1	yes	maximal, characteristic
2	#14544999	C₂	1	2	yes	maximal
3	M₃₂×#595	C₂	1	2	yes	maximal
4	#97×#595	C₂	1	1	yes	maximal, characteristic
5	D₁₇×#347	C₂	1	1	yes	maximal, characteristic
6	#11763865	C₂²	1	2	yes	—
7	#11763768	C₄	1	1	yes	characteristic
8	C₂×#10587532	C₄	1	1	yes	characteristic
9	C₂×#10587532	C₂²	1	1	yes	characteristic
10	C₂×C₈×#595	C₄	1	1	yes	characteristic
11	C₂×C₈×#595	C₂²	1	1	yes	characteristic
12	C₄²×#595	C₄	1	1	yes	characteristic
13	#11763702	C₄	1	2	yes	complemented
14	#11763701	C₄	1	2	yes	complemented
15	#11763700	C₄	1	4	yes	complemented
16	#11763699	C₄	1	4	yes	complemented
17	#11763696	C₄	1	4	yes	complemented
18	C₁₆×#595	C₄	1	4	yes	complemented
19	M₃₂×D₁₇	C₂²	1	2	yes	—
20	#97×D₁₇	C₂²	1	1	yes	characteristic
21	#11762587	C₄	1	1	yes	characteristic, complemented
22	#11762586	C₄	1	2	yes	complemented
23	C₁₇×#347	C₄	1	1	yes	characteristic, complemented
24	C₂×C₄×#595	—	2	2	no	—
25	C₂×C₄×#595	C₂×C₄	1	1	yes	characteristic
26	C₂×C₄×#595	C₂×C₄	1	1	yes	characteristic
27	C₂×#63242	C₂×C₄	1	1	yes	characteristic
28	C₂×#63242	C₂×C₄	1	1	yes	characteristic
29	C₂×C₈×D₁₇	C₂×C₄	1	1	yes	characteristic
30	C₂×C₈×D₁₇	C₂³	1	1	yes	characteristic
31	C₄²×D₁₇	C₂×C₄	1	1	yes	characteristic
32	#10587542	—	2	2	no	—
33	#10587538	C₂×C₄	1	4	yes	—
34	#10587532	—	2	2	no	—
35	#10587532	C₂×C₄	1	2	yes	—
36	#10587532	C₂×C₄	1	2	yes	—
37	C₈×#595	—	2	2	no	—
38	C₈×#595	C₂×C₄	1	2	yes	—
39	C₈×#595	C₂×C₄	1	2	yes	—
40	#10587530	C₂×C₄	1	4	yes	—
41	#10587529	C₂×C₄	1	4	yes	—
42	C₁₇×M₃₂	C₂×C₄	1	2	yes	—
43	C₁₇×#97	C₂×C₄	1	1	yes	characteristic
44	#10587486	C₂×C₄	1	1	yes	characteristic
45	#10587486	C₂×C₄	1	1	yes	characteristic
46	#10587483	C₂×C₄	1	2	yes	—
47	#10587474	C₂×C₄	1	1	yes	characteristic
48	#10587469	C₂×C₄	1	4	yes	—
49	C₁₆×D₁₇	C₂×C₄	1	4	yes	—
50	C₂²×#595	—	2	1	no	—
51	C₂²×#595	M₁₆	1	2	yes	—
52	C₂×C₄×D₁₇	—	2	1	no	—
53	C₂×C₄×D₁₇	C₂²×C₄	1	1	yes	characteristic
54	C₂×#3624	—	2	1	no	—
55	C₂×#3624	M₁₆	1	2	yes	—
56	C₄×#595	—	2	1	no	—
57	C₄×#595	—	4	4	no	—
58	C₄×#595	C₄²	1	2	yes	—
59	C₄×#595	C₄²	1	1	yes	characteristic
60	C₄×#595	C₄²	1	1	yes	characteristic
61	C₄×#595	M₁₆	1	2	yes	—
62	#63242	—	2	1	no	—
63	#63242	C₄²	1	1	yes	characteristic
64	#63242	C₄²	1	1	yes	characteristic
65	#63242	C₄²	1	2	yes	—
66	#63242	M₁₆	1	2	yes	—
67	C₂×C₈×C₁₇	C₄²	1	1	yes	characteristic
68	C₂×C₈×C₁₇	C₂²×C₄	1	1	yes	characteristic
69	C₄²×C₁₇	C₄²	1	1	yes	characteristic
70	C₄×Dic₁₇	—	2	1	no	—
71	C₄×Dic₁₇	C₄²	1	1	yes	characteristic
72	C₂×#3622	C₄²	1	1	yes	characteristic
73	C₂×#3622	C₂²×C₄	1	1	yes	characteristic
74	C₈×D₁₇	—	2	1	no	—
75	C₈×D₁₇	—	2	1	no	—
76	C₈×D₁₇	C₂²×C₄	1	2	yes	—
77	C₈×D₁₇	C₂²×C₄	1	2	yes	—
78	#63216	C₄²	1	8	yes	complemented
79	C₁₆×C₁₇	C₄²	1	4	yes	complemented
80	#63214	C₄²	1	4	yes	complemented
81	C₄×#347	—	17	1	no	maximal, Sylow
82	C₂²×D₁₇	C₂×M₁₆	1	1	yes	characteristic
83	C₂×#595	—	2	4	no	—
84	C₂×#595	—	2	2	no	—
85	C₂×#595	—	4	2	no	—
86	C₂×#595	#97	1	2	yes	—
87	C₂×#595	#97	1	2	yes	—
88	C₂×C₄×C₁₇	—	2	1	no	—
89	C₂×C₄×C₁₇	C₂×C₄²	1	1	yes	characteristic
90	C₂×Dic₁₇	—	2	1	no	—
91	C₂×Dic₁₇	C₂×M₁₆	1	1	yes	characteristic
92	C₄×D₁₇	—	4	2	no	—
93	C₄×D₁₇	C₂×C₄²	1	1	yes	characteristic
94	C₄×D₁₇	C₂×C₄²	1	1	yes	characteristic
95	C₄×D₁₇	C₂×M₁₆	1	1	yes	characteristic
96	C₄×D₁₇	C₂×M₁₆	1	1	yes	characteristic
97	#3624	—	2	1	no	—
98	#3624	—	2	1	no	—
99	#3624	#97	1	4	yes	—
100	C₈×C₁₇	—	2	1	no	—
101	C₈×C₁₇	C₂×C₄²	1	2	yes	—
102	#3622	—	2	1	no	—
103	#3622	C₂×C₄²	1	2	yes	—
104	C₂×#347	—	17	2	no	—
105	C₂×#347	—	17	1	no	—
106	C₂×#347	—	17	1	no	—
107	C₄×M₃₂	—	17	2	no	—
108	C₄×#97	—	17	1	no	—
109	C₂²×C₁₇	C₄×M₁₆	1	1	yes	characteristic
110	C₂×D₁₇	—	2	2	no	—
111	C₂×D₁₇	C₂×#97	1	1	yes	characteristic
112	C₂×D₁₇	C₄×M₁₆	1	1	yes	characteristic
113	#595	—	2	2	no	—
114	#595	—	4	2	no	—
115	#595	#347	1	4	yes	direct product factor
116	C₄×C₁₇	—	4	1	no	—
117	C₄×C₁₇	C₄³	1	1	yes	characteristic
118	C₄×C₁₇	C₄×M₁₆	1	1	yes	characteristic
119	Dic₁₇	—	4	1	no	—
120	Dic₁₇	C₂×#97	1	1	yes	characteristic
121	Dic₁₇	C₄×M₁₆	1	1	yes	characteristic
122	C₂×M₃₂	—	17	2	no	—
123	C₂×M₃₂	—	17	2	no	—
124	C₂×#97	—	17	1	no	—
125	C₂×#97	—	17	1	no	—
126	C₂×#97	—	17	1	no	—
127	C₂×#97	—	17	1	no	—
128	C₂×C₄×C₈	—	17	1	no	—
129	C₂×C₄×C₈	—	17	1	no	—
130	C₄³	—	17	1	no	—
131	#347	—	17	4	no	—
132	#347	—	17	4	no	—
133	#347	—	17	2	no	—
134	#347	—	17	2	no	—
135	#347	—	17	2	no	—
136	#347	—	17	1	no	—
137	#347	#595	1	1	yes	characteristic, direct product factor
138	#346	—	17	4	no	—
139	C₄×C₁₆	—	17	4	no	—
140	C₂×C₁₇	—	2	1	no	—
141	C₂×C₁₇	C₄×#97	1	1	yes	characteristic
142	D₁₇	—	2	1	no	—
143	D₁₇	C₂×#347	1	2	yes	—
144	C₂²×C₈	—	17	1	no	—
145	C₂²×C₈	—	17	1	no	—
146	C₂²×C₈	—	17	1	no	—
147	C₂²×C₈	—	17	1	no	—
148	C₂×C₄²	—	17	1	no	—
149	C₂×C₄²	—	17	1	no	—
150	C₂×C₄²	—	17	1	no	—
151	C₂×C₄²	—	34	2	no	—
152	M₃₂	—	17	4	no	—
153	M₃₂	—	17	4	no	—
154	M₃₂	—	17	2	no	—
155	M₃₂	—	17	4	no	—
156	M₃₂	C₂×#595	1	2	yes	—
157	C₂×C₁₆	—	17	4	no	—
158	C₂×C₁₆	—	17	4	no	—
159	#97	—	17	2	no	—
160	#97	—	17	2	no	—
161	#97	—	17	1	no	—
162	#97	—	17	1	no	—
163	#97	—	17	1	no	—
164	#97	—	34	2	no	—
165	#97	—	34	2	no	—
166	#97	C₂×#595	1	1	yes	characteristic
167	C₄×C₈	—	17	2	no	—
168	C₄×C₈	—	17	2	no	—
169	C₄×C₈	—	34	2	no	—
170	C₁₇	C₄×#347	1	1	yes	characteristic, complemented, Sylow
171	C₂²×C₄	—	17	1	no	—
172	C₂²×C₄	—	17	2	no	—
173	C₂²×C₄	—	34	1	no	—
174	C₂²×C₄	—	34	1	no	—
175	C₂×C₈	—	17	2	no	—
176	C₂×C₈	—	17	2	no	—
177	C₂×C₈	—	17	2	no	—
178	C₂×C₈	—	17	1	no	—
179	C₂×C₈	—	17	2	no	—
180	C₂×C₈	—	17	2	no	—
181	C₂×C₈	—	17	1	no	—
182	C₂×C₈	—	17	1	no	—
183	C₂×C₈	—	17	1	no	—
184	C₂×C₈	—	34	1	no	—
185	C₂×C₈	—	34	1	no	—
186	C₂×C₈	—	34	1	no	—
187	C₂×C₈	—	34	1	no	—
188	C₂×C₈	C₄×#595	1	1	yes	characteristic
189	C₂×C₈	C₂²×#595	1	1	yes	characteristic
190	C₄²	—	17	1	no	—
191	C₄²	—	17	1	no	—
192	C₄²	—	17	2	no	—
193	C₄²	—	17	2	no	—
194	C₄²	—	17	1	no	—
195	C₄²	—	34	1	no	—
196	C₄²	—	34	1	no	—
197	C₄²	—	68	4	no	—
198	C₄²	C₄×#595	1	1	yes	characteristic
199	C₁₆	—	17	8	no	—
200	C₁₆	—	17	4	no	—
201	C₁₆	C₄×#595	1	4	yes	complemented
202	C₂³	—	17	1	no	—
203	C₂×C₄	—	2	1	no	—
204	C₂×C₄	—	17	2	no	—
205	C₂×C₄	—	17	1	no	—
206	C₂×C₄	—	17	1	no	—
207	C₂×C₄	—	17	1	no	—
208	C₂×C₄	—	17	1	no	—
209	C₂×C₄	—	17	1	no	—
210	C₂×C₄	—	17	2	no	—
211	C₂×C₄	—	34	4	no	—
212	C₂×C₄	—	34	2	no	—
213	C₂×C₄	—	34	1	no	—
214	C₂×C₄	—	68	2	no	—
215	C₂×C₄	—	68	2	no	—
216	C₂×C₄	C₂×C₄×#595	1	1	yes	characteristic
217	C₈	—	2	1	no	—
218	C₈	—	17	4	no	—
219	C₈	—	17	2	no	—
220	C₈	—	34	1	no	—
221	C₈	—	34	1	no	—
222	C₈	—	34	1	no	—
223	C₈	C₂×C₄×#595	1	2	yes	—
224	C₂²	—	17	1	no	—
225	C₂²	—	17	1	no	—
226	C₂²	—	34	2	no	—
227	C₂²	M₁₆×#595	1	1	yes	characteristic
228	C₄	—	4	1	no	—
229	C₄	—	17	4	no	—
230	C₄	—	17	1	no	—
231	C₄	—	17	1	no	—
232	C₄	—	34	2	no	—
233	C₄	—	68	1	no	—
234	C₄	—	68	2	no	—
235	C₄	C₄²×#595	1	1	yes	characteristic
236	C₄	M₁₆×#595	1	1	yes	characteristic
237	C₂	—	2	1	no	—
238	C₂	—	17	2	no	—
239	C₂	—	34	1	no	—
240	C₂	#97×#595	1	1	yes	characteristic

	subgroup	G/H
center	C₄	C₄²×#595
commutator	C₄×C₁₇	C₄³
Frattini	C₂×C₈	C₂²×#595
Fitting	C₁₇×#347	C₄
socle	C₂×C₁₇	C₄×#97
prime radical	C₂²×D₁₇	C₂×M₁₆
cosocle	C₂×C₈×D₁₇	C₂³
hypercenter	#347	#595
hypocenter	C₁₇	C₄×#347
LCS	G ▷ C₄×C₁₇ ▷ C₁₇
UCS	C₁ ◁ C₄ ◁ #347

Complements

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

N	H ≅ G/N	type
13. #11763702	C₄	semidirect
14. #11763701	C₄	semidirect
15. #11763700	C₄	semidirect
16. #11763699	C₄	semidirect
17. #11763696	C₄	semidirect
18. C₁₆×#595	C₄	semidirect
21. #11762587	C₄	semidirect
22. #11762586	C₄	semidirect
23. C₁₇×#347	C₄	semidirect
78. #63216	C₄²	semidirect
79. C₁₆×C₁₇	C₄²	semidirect
80. #63214	C₄²	semidirect
115. #595	#347	direct (N × H)
137. #347	#595	direct (N × H)
170. C₁₇	C₄×#347	semidirect
201. 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
#14545001	2
#14544999	2
M₃₂×#595	2
#97×#595	2
D₁₇×#347	2
C₄×#347	17