Subgroups — #17960429

Order 4352.

418 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	C₂×#11763700	C₂	1	2	yes	maximal
2	C₂×#11763699	C₂	1	2	yes	maximal
3	#14544991	C₂	1	1	yes	maximal, characteristic
4	C₄×#10587532	C₂	1	1	yes	maximal, characteristic
5	#14530377	C₂	1	1	yes	maximal, characteristic
6	C₂×#10587530	C₂²	1	2	yes	—
7	C₂×#10587532	C₄	1	1	yes	characteristic
8	C₂×#10587532	C₄	1	1	yes	characteristic
9	C₂×#10587532	C₂²	1	1	yes	characteristic
10	C₂×#10587532	C₂²	1	1	yes	characteristic
11	C₄×#63242	C₄	1	1	yes	characteristic
12	C₄²×#595	C₄	1	1	yes	characteristic
13	#11763700	C₄	1	8	yes	complemented
14	#11763699	C₄	1	8	yes	complemented
15	#11763697	C₄	1	2	yes	complemented
16	#11763695	C₄	1	2	yes	complemented
17	C₂×#10587469	C₂²	1	2	yes	—
18	C₄×C₈×D₁₇	C₂²	1	1	yes	characteristic
19	#11762585	C₄	1	1	yes	characteristic, complemented
20	C₁₆×Dic₁₇	C₄	1	2	yes	complemented
21	C₁₇×#346	C₄	1	1	yes	characteristic, complemented
22	C₂×C₄×#595	—	2	2	no	—
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	—	2	2	no	—
31	C₂×#63216	C₂×C₄	1	4	yes	—
32	#10587532	—	2	2	no	—
33	#10587532	—	2	2	no	—
34	#10587532	C₂×C₄	1	4	yes	—
35	#10587532	C₂×C₄	1	4	yes	—
36	#10587530	C₂×C₄	1	4	yes	—
37	#10587530	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	#10587469	C₂×C₄	1	4	yes	—
45	#10587469	C₂×C₄	1	4	yes	—
46	C₂²×#595	—	2	1	no	—
47	C₂²×#595	—	2	1	no	—
48	C₂×C₄×D₁₇	M₁₆	1	2	yes	—
49	C₂×C₄×D₁₇	C₂²×C₄	1	1	yes	characteristic
50	C₂×#3624	—	2	1	no	—
51	C₂×#3624	—	2	1	no	—
52	C₄×#595	—	2	1	no	—
53	C₄×#595	—	2	1	no	—
54	C₄×#595	—	4	4	no	—
55	C₄×#595	C₄²	1	2	yes	—
56	C₄×#595	C₄²	1	2	yes	—
57	#63242	—	2	1	no	—
58	#63242	—	2	1	no	—
59	#63242	C₄²	1	2	yes	—
60	#63242	C₄²	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	M₁₆	1	2	yes	—
66	C₂×#3622	C₄²	1	1	yes	characteristic
67	C₂×#3622	C₂²×C₄	1	1	yes	characteristic
68	C₈×D₁₇	M₁₆	1	2	yes	—
69	C₈×D₁₇	M₁₆	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×#347	—	17	1	no	maximal, Sylow
76	C₂²×D₁₇	C₂×M₁₆	1	1	yes	characteristic
77	C₂×#595	—	2	2	no	—
78	C₂×#595	—	2	2	no	—
79	C₂×#595	—	4	2	no	—
80	C₂×#595	—	4	2	no	—
81	C₂×C₄×C₁₇	#97	1	2	yes	—
82	C₂×C₄×C₁₇	C₂×C₄²	1	1	yes	characteristic
83	C₂×Dic₁₇	#97	1	2	yes	—
84	C₂×Dic₁₇	C₂×M₁₆	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	C₂×M₁₆	1	1	yes	characteristic
89	C₄×D₁₇	C₂×M₁₆	1	1	yes	characteristic
90	#3624	—	2	2	no	—
91	#3624	—	2	2	no	—
92	C₈×C₁₇	#97	1	2	yes	—
93	C₈×C₁₇	C₂×C₄²	1	2	yes	—
94	#3622	#97	1	2	yes	—
95	#3622	C₂×C₄²	1	2	yes	—
96	C₂×#347	—	17	2	no	—
97	C₂×#347	—	17	2	no	—
98	C₄×M₃₂	—	17	1	no	—
99	C₄×M₃₂	—	17	1	no	—
100	C₄×#97	—	17	1	no	—
101	C₂²×C₁₇	C₂×#97	1	1	yes	characteristic
102	C₂×D₁₇	—	2	2	no	—
103	C₂×D₁₇	C₄×M₁₆	1	1	yes	characteristic
104	C₂×D₁₇	C₄×M₁₆	1	1	yes	characteristic
105	#595	—	4	2	no	—
106	#595	—	4	2	no	—
107	C₄×C₁₇	—	2	2	no	—
108	C₄×C₁₇	C₄³	1	1	yes	characteristic
109	C₄×C₁₇	C₂×#97	1	1	yes	characteristic
110	Dic₁₇	#347	1	4	yes	complemented
111	Dic₁₇	C₄×M₁₆	1	1	yes	characteristic
112	Dic₁₇	C₄×M₁₆	1	1	yes	characteristic
113	C₂×M₃₂	—	17	2	no	—
114	C₂×M₃₂	—	17	2	no	—
115	C₂×#97	—	17	1	no	—
116	C₂×#97	—	17	1	no	—
117	C₂×#97	—	17	1	no	—
118	C₂×#97	—	17	1	no	—
119	C₂×C₄×C₈	—	17	1	no	—
120	C₂×C₄×C₈	—	17	1	no	—
121	C₄³	—	17	1	no	—
122	#347	—	17	8	no	—
123	#347	—	17	8	no	—
124	#346	—	17	2	no	—
125	#346	—	17	1	no	—
126	#346	#595	1	1	yes	characteristic, complemented
127	C₄×C₁₆	—	17	2	no	—
128	C₄×C₁₆	—	17	2	no	—
129	C₂×C₁₇	C₄×#97	1	1	yes	characteristic
130	C₂×C₁₇	C₂×#347	1	2	yes	—
131	D₁₇	—	2	1	no	—
132	D₁₇	—	2	1	no	—
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	—	34	2	no	—
141	M₃₂	—	17	4	no	—
142	M₃₂	—	17	4	no	—
143	M₃₂	—	17	4	no	—
144	M₃₂	—	17	4	no	—
145	C₂×C₁₆	—	17	4	no	—
146	C₂×C₁₆	—	17	2	no	—
147	C₂×C₁₆	C₂×#595	1	2	yes	—
148	#97	—	17	4	no	—
149	#97	—	17	4	no	—
150	#97	—	34	2	no	—
151	#97	—	34	2	no	—
152	C₄×C₈	—	17	1	no	—
153	C₄×C₈	—	17	1	no	—
154	C₄×C₈	—	17	1	no	—
155	C₄×C₈	—	34	2	no	—
156	C₄×C₈	C₂×#595	1	1	yes	characteristic
157	C₁₇	C₄×#347	1	1	yes	characteristic, complemented, Sylow
158	C₂²×C₄	—	17	1	no	—
159	C₂²×C₄	—	17	2	no	—
160	C₂²×C₄	—	34	1	no	—
161	C₂²×C₄	—	34	1	no	—
162	C₂×C₈	—	17	1	no	—
163	C₂×C₈	—	17	1	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	2	no	—
169	C₂×C₈	—	17	2	no	—
170	C₂×C₈	—	34	1	no	—
171	C₂×C₈	—	34	1	no	—
172	C₂×C₈	—	34	1	no	—
173	C₂×C₈	—	34	1	no	—
174	C₂×C₈	C₄×#595	1	1	yes	characteristic
175	C₂×C₈	C₂²×#595	1	1	yes	characteristic
176	C₄²	—	17	1	no	—
177	C₄²	—	17	2	no	—
178	C₄²	—	17	2	no	—
179	C₄²	—	17	2	no	—
180	C₄²	—	34	1	no	—
181	C₄²	—	34	1	no	—
182	C₄²	—	68	4	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	1	no	—
189	C₂×C₄	—	17	2	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	—	34	4	no	—
195	C₂×C₄	—	34	2	no	—
196	C₂×C₄	—	34	2	no	—
197	C₂×C₄	—	68	2	no	—
198	C₂×C₄	—	68	2	no	—
199	C₂×C₄	#10587532	1	2	yes	—
200	C₂×C₄	C₂×C₄×#595	1	1	yes	characteristic
201	C₈	—	17	2	no	—
202	C₈	—	17	2	no	—
203	C₈	—	34	2	no	—
204	C₈	—	34	2	no	—
205	C₈	#10587532	1	2	yes	—
206	C₈	C₂×C₄×#595	1	2	yes	—
207	C₂²	—	17	1	no	—
208	C₂²	—	17	1	no	—
209	C₂²	—	34	2	no	—
210	C₂²	C₂×#10587532	1	1	yes	characteristic
211	C₄	—	2	2	no	—
212	C₄	—	17	1	no	—
213	C₄	—	17	1	no	—
214	C₄	—	17	4	no	—
215	C₄	—	68	2	no	—
216	C₄	—	68	2	no	—
217	C₄	C₄²×#595	1	1	yes	characteristic
218	C₄	C₂×#10587532	1	1	yes	characteristic
219	C₂	—	34	1	no	—
220	C₂	—	34	1	no	—
221	C₂	C₄×#10587532	1	1	yes	characteristic
222	C₂	#14544999	1	2	yes	—

	subgroup	G/H
center	C₂×C₄	C₂×C₄×#595
commutator	C₄×C₁₇	C₄³
Frattini	C₂×C₈	C₂²×#595
Fitting	C₁₇×#346	C₄
socle	C₂²×C₁₇	C₂×#97
prime radical	C₂²×D₁₇	C₂×M₁₆
cosocle	C₂×C₈×D₁₇	C₂³
hypercenter	#346	#595
hypocenter	C₁₇	C₄×#347
LCS	G ▷ C₄×C₁₇ ▷ C₁₇
UCS	C₁ ◁ C₂×C₄ ◁ #346

Complements

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

N	H ≅ G/N	type
13. #11763700	C₄	semidirect
14. #11763699	C₄	semidirect
15. #11763697	C₄	semidirect
16. #11763695	C₄	semidirect
19. #11762585	C₄	semidirect
20. C₁₆×Dic₁₇	C₄	semidirect
21. C₁₇×#346	C₄	semidirect
72. #63216	C₄²	semidirect
73. C₁₆×C₁₇	C₄²	semidirect
74. #63214	C₄²	semidirect
110. Dic₁₇	#347	semidirect
126. #346	#595	semidirect
157. C₁₇	C₄×#347	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
C₂×#11763700	2
C₂×#11763699	2
#14544991	2
C₄×#10587532	2
#14530377	2
C₄×#347	17