Subgroups — #17950285

Order 4352.

518 conjugacy classes of subgroups.

The subgroup lattice has 210 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#14544499	C₂	1	2	yes	maximal
2	C₂×#11763651	C₂	1	2	yes	maximal
3	C₂×#11763650	C₂	1	1	yes	maximal, characteristic
4	C₂×C₈×#3624	C₂	1	1	yes	maximal, characteristic
5	#97×Dic₁₇	C₂	1	1	yes	maximal, characteristic
6	C₂×C₄×#3624	C₂²	1	2	yes	—
7	C₂×C₄×#3624	C₂²	1	2	yes	—
8	#11763675	C₄	1	8	yes	—
9	#11763651	C₄	1	8	yes	complemented
10	#11763650	C₄	1	4	yes	complemented
11	C₈×#3624	C₄	1	4	yes	complemented
12	C₂×#10587474	C₄	1	1	yes	characteristic
13	C₂×#10587486	C₄	1	1	yes	characteristic
14	C₂×#10587486	C₄	1	1	yes	characteristic
15	C₂×C₈×Dic₁₇	C₂²	1	1	yes	characteristic
16	C₂×C₈×Dic₁₇	C₂²	1	1	yes	characteristic
17	C₄²×Dic₁₇	C₂²	1	1	yes	characteristic
18	C₂×C₁₇×#97	C₄	1	1	yes	characteristic
19	C₂²×#3624	C₂×C₄	1	4	yes	—
20	C₂²×#3624	C₂×C₄	1	4	yes	—
21	C₂²×C₈×C₁₇	C₂×C₄	1	1	yes	characteristic
22	C₂²×C₈×C₁₇	C₂×C₄	1	1	yes	characteristic
23	C₂×C₄²×C₁₇	C₂×C₄	1	1	yes	characteristic
24	C₂×C₄×Dic₁₇	C₂×C₄	1	1	yes	characteristic
25	C₂×C₄×Dic₁₇	C₂×C₄	1	4	yes	—
26	C₂×C₄×Dic₁₇	C₂³	1	1	yes	characteristic
27	C₂²×#3622	C₂×C₄	1	1	yes	characteristic
28	C₂²×#3622	C₂×C₄	1	1	yes	characteristic
29	C₄×#3624	C₂×C₄	1	4	yes	—
30	C₄×#3624	C₂×C₄	1	4	yes	—
31	C₄×#3624	C₂×C₄	1	4	yes	—
32	C₄×#3624	C₂×C₄	1	4	yes	—
33	C₁₇×#97	—	2	2	no	—
34	#10587486	—	2	2	no	—
35	#10587486	—	2	2	no	—
36	C₈×Dic₁₇	C₂×C₄	1	2	yes	—
37	C₈×Dic₁₇	C₂×C₄	1	2	yes	—
38	C₈×Dic₁₇	C₂×C₄	1	4	yes	—
39	#10587474	—	2	2	no	—
40	C₂²×C₄×C₁₇	C₄²	1	2	yes	—
41	C₂²×C₄×C₁₇	C₂²×C₄	1	1	yes	characteristic
42	C₂²×Dic₁₇	C₄²	1	2	yes	—
43	C₂²×Dic₁₇	C₂²×C₄	1	2	yes	—
44	C₂×#3624	—	2	8	no	—
45	C₂×#3624	—	2	8	no	—
46	C₂×#3624	C₄²	1	8	yes	—
47	C₂×#3624	C₄²	1	8	yes	—
48	C₂×C₈×C₁₇	—	2	1	no	—
49	C₂×C₈×C₁₇	—	2	1	no	—
50	C₂×C₈×C₁₇	C₄²	1	2	yes	—
51	C₂×C₈×C₁₇	C₄²	1	2	yes	—
52	C₂×C₈×C₁₇	M₁₆	1	2	yes	—
53	C₂×C₈×C₁₇	M₁₆	1	2	yes	—
54	C₄²×C₁₇	—	2	1	no	—
55	C₄²×C₁₇	—	2	1	no	—
56	C₄×Dic₁₇	—	2	1	no	—
57	C₄×Dic₁₇	—	2	1	no	—
58	C₄×Dic₁₇	—	2	8	no	—
59	C₄×Dic₁₇	C₂²×C₄	1	1	yes	characteristic
60	C₄×Dic₁₇	C₂²×C₄	1	1	yes	characteristic
61	C₄×Dic₁₇	C₂²×C₄	1	2	yes	—
62	C₂×#3622	—	2	1	no	—
63	C₂×#3622	—	2	1	no	—
64	C₂×#3622	C₄²	1	2	yes	—
65	C₂×#3622	C₄²	1	2	yes	—
66	C₂×#3622	M₁₆	1	2	yes	—
67	C₂×#3622	M₁₆	1	2	yes	—
68	#7844	—	17	1	no	maximal, Sylow
69	C₂³×C₁₇	C₂×C₄²	1	1	yes	characteristic
70	C₂×C₄×C₁₇	—	2	2	no	—
71	C₂×C₄×C₁₇	—	2	4	no	—
72	C₂×C₄×C₁₇	C₂×C₄²	1	1	yes	characteristic
73	C₂×C₄×C₁₇	C₂×C₄²	1	1	yes	characteristic
74	C₂×C₄×C₁₇	C₂×M₁₆	1	1	yes	characteristic
75	C₂×C₄×C₁₇	C₂×M₁₆	1	1	yes	characteristic
76	C₂×C₄×C₁₇	C₂×M₁₆	1	1	yes	characteristic
77	C₂×C₄×C₁₇	C₂×M₁₆	1	1	yes	characteristic
78	C₂×Dic₁₇	—	2	2	no	—
79	C₂×Dic₁₇	—	2	4	no	—
80	C₂×Dic₁₇	C₂×C₄²	1	2	yes	—
81	C₂×Dic₁₇	C₂×C₄²	1	2	yes	—
82	C₂×Dic₁₇	#131	1	8	yes	—
83	#3624	—	2	8	no	—
84	#3624	—	2	8	no	—
85	C₈×C₁₇	—	2	2	no	—
86	C₈×C₁₇	#97	1	4	yes	—
87	#3622	—	2	2	no	—
88	#3622	#97	1	4	yes	—
89	#1744	—	17	2	no	—
90	C₄×#97	—	17	1	no	—
91	C₂×#322	—	17	1	no	—
92	C₂×#322	—	17	2	no	—
93	C₂×C₈²	—	17	1	no	—
94	C₂²×C₁₇	C₄³	1	1	yes	characteristic
95	C₂²×C₁₇	C₄×M₁₆	1	1	yes	characteristic
96	C₂²×C₁₇	C₄×M₁₆	1	1	yes	characteristic
97	C₂²×C₁₇	#405	1	4	yes	—
98	C₄×C₁₇	—	4	2	no	—
99	C₄×C₁₇	C₂×#97	1	1	yes	characteristic
100	C₄×C₁₇	C₂×#97	1	1	yes	characteristic
101	C₄×C₁₇	C₄×M₁₆	1	1	yes	characteristic
102	C₄×C₁₇	C₄×M₁₆	1	1	yes	characteristic
103	Dic₁₇	—	4	2	no	—
104	Dic₁₇	#405	1	4	yes	—
105	Dic₁₇	#405	1	4	yes	—
106	C₂×#97	—	17	1	no	—
107	C₂×#97	—	17	1	no	—
108	C₂×#97	—	17	1	no	—
109	C₂×#97	#595	1	1	yes	characteristic
110	C₂×C₄×C₈	—	17	2	no	—
111	C₂×C₄×C₈	—	17	2	no	—
112	C₂×C₄×C₈	—	17	1	no	—
113	C₂×C₄×C₈	—	17	1	no	—
114	C₄³	—	17	1	no	—
115	#336	—	17	8	no	—
116	#322	—	17	4	no	—
117	#322	—	17	8	no	—
118	C₈²	—	17	4	no	—
119	C₂×C₁₇	#1453	1	2	yes	—
120	C₂×C₁₇	#1453	1	2	yes	—
121	C₂×C₁₇	C₄×#97	1	1	yes	characteristic
122	C₂×C₁₇	#1726	1	2	yes	—
123	C₂²×C₈	—	17	1	no	—
124	C₂²×C₈	—	17	4	no	—
125	C₂²×C₈	—	17	4	no	—
126	C₂²×C₈	—	17	1	no	—
127	C₂²×C₈	C₂×#595	1	1	yes	characteristic
128	C₂²×C₈	C₂×#595	1	1	yes	characteristic
129	C₂×C₄²	—	17	1	no	—
130	C₂×C₄²	—	17	4	no	—
131	C₂×C₄²	—	17	1	no	—
132	C₂×C₄²	C₂×#595	1	1	yes	characteristic
133	#97	—	2	2	no	—
134	#97	—	34	2	no	—
135	#97	—	34	2	no	—
136	#97	—	34	2	no	—
137	C₄×C₈	—	17	4	no	—
138	C₄×C₈	—	17	4	no	—
139	C₄×C₈	—	17	4	no	—
140	C₄×C₈	—	17	4	no	—
141	C₄×C₈	—	17	2	no	—
142	C₄×C₈	—	17	2	no	—
143	C₄×C₈	—	17	4	no	—
144	C₁₇	#7844	1	1	yes	characteristic, complemented, Sylow
145	C₂²×C₄	—	17	2	no	—
146	C₂²×C₄	—	17	2	no	—
147	C₂²×C₄	C₄×#595	1	2	yes	—
148	C₂²×C₄	C₂²×#595	1	1	yes	characteristic
149	C₂×C₈	—	2	1	no	—
150	C₂×C₈	—	2	1	no	—
151	C₂×C₈	—	17	2	no	—
152	C₂×C₈	—	17	2	no	—
153	C₂×C₈	—	17	2	no	—
154	C₂×C₈	—	17	8	no	—
155	C₂×C₈	—	17	8	no	—
156	C₂×C₈	—	17	2	no	—
157	C₂×C₈	—	34	8	no	—
158	C₂×C₈	—	34	8	no	—
159	C₂×C₈	—	34	1	no	—
160	C₂×C₈	—	34	1	no	—
161	C₂×C₈	#63243	1	2	yes	—
162	C₂×C₈	C₄×#595	1	2	yes	—
163	C₂×C₈	C₄×#595	1	2	yes	—
164	C₂×C₈	#63247	1	2	yes	—
165	C₄²	—	2	1	no	—
166	C₄²	—	2	1	no	—
167	C₄²	—	17	1	no	—
168	C₄²	—	17	1	no	—
169	C₄²	—	17	2	no	—
170	C₄²	—	34	8	no	—
171	C₄²	—	34	1	no	—
172	C₄²	—	34	1	no	—
173	C₂³	C₂×C₄×#595	1	1	yes	characteristic
174	C₂×C₄	—	2	2	no	—
175	C₂×C₄	—	2	4	no	—
176	C₂×C₄	—	17	2	no	—
177	C₂×C₄	—	17	2	no	—
178	C₂×C₄	—	17	8	no	—
179	C₂×C₄	—	34	2	no	—
180	C₂×C₄	—	34	4	no	—
181	C₂×C₄	C₂×#63243	1	1	yes	characteristic
182	C₂×C₄	D₁₇⋊M₁₆	1	1	yes	characteristic
183	C₂×C₄	D₁₇⋊M₁₆	1	1	yes	characteristic
184	C₂×C₄	C₂×C₄×#595	1	1	yes	characteristic
185	C₂×C₄	C₂×C₄×#595	1	1	yes	characteristic
186	C₂×C₄	C₂×#63247	1	1	yes	characteristic
187	C₈	—	2	2	no	—
188	C₈	—	17	4	no	—
189	C₈	—	34	8	no	—
190	C₈	—	34	8	no	—
191	C₈	—	34	2	no	—
192	C₈	#10587542	1	4	yes	—
193	C₂²	C₄²×#595	1	1	yes	characteristic
194	C₂²	C₄×#63247	1	1	yes	characteristic
195	C₂²	C₄×#63243	1	1	yes	characteristic
196	C₂²	#11763775	1	4	yes	—
197	C₄	—	4	2	no	—
198	C₄	—	17	4	no	—
199	C₄	—	17	4	no	—
200	C₄	—	68	2	no	—
201	C₄	C₂×#10587542	1	1	yes	characteristic
202	C₄	#11763768	1	1	yes	characteristic
203	C₄	C₄×#63247	1	1	yes	characteristic
204	C₄	C₄×#63243	1	1	yes	characteristic
205	C₂	#14544003	1	2	yes	—
206	C₂	#14544004	1	2	yes	—
207	C₂	C₄×#10587542	1	1	yes	characteristic
208	C₂	#14544442	1	2	yes	—

	subgroup	G/H
center, Frattini	C₂²×C₄	C₂²×#595
commutator	C₂²×C₁₇	C₄³
Fitting	C₂×C₁₇×#97	C₄
socle, prime radical	C₂³×C₁₇	C₂×C₄²
cosocle	C₂×C₄×Dic₁₇	C₂³
hypercenter	C₂×#97	#595
hypocenter	C₁₇	#7844
LCS	G ▷ C₂²×C₁₇ ▷ C₁₇
UCS	C₁ ◁ C₂²×C₄ ◁ C₂×#97

Complements

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

N	H ≅ G/N	type
9. #11763651	C₄	semidirect
10. #11763650	C₄	semidirect
11. C₈×#3624	C₄	semidirect
144. C₁₇	#7844	semidirect

Maximal subgroups

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

subgroup	index
#14544499	2
C₂×#11763651	2
C₂×#11763650	2
C₂×C₈×#3624	2
#97×Dic₁₇	2
#7844	17