Subgroups — #16921230

Order 4352.

254 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	#14530540	C₂	1	1	yes	maximal, characteristic
2	#14528651	C₂	1	1	yes	maximal, characteristic
3	C₄×C₁₇×M₃₂	C₂	1	1	yes	maximal, characteristic
4	C₂×#10587483	—	2	1	no	—
5	C₂×C₄⋊#3622	C₄	1	1	yes	characteristic
6	C₂×#10587486	—	2	1	no	—
7	C₄×C₄⋊Dic₁₇	C₄	1	1	yes	characteristic
8	C₄⋊#63214	—	2	1	no	—
9	C₄⋊#63214	C₄	1	2	yes	complemented
10	#11762563	—	2	1	no	—
11	C₂×C₁₇×M₃₂	—	2	1	no	—
12	C₂×C₄×C₈×C₁₇	C₂²	1	1	yes	characteristic
13	C₁₇×#346	—	2	1	no	—
14	C₄×C₁₆×C₁₇	C₄	1	2	yes	complemented
15	C₂²×C₈×C₁₇	D₄	1	1	yes	characteristic
16	C₂²×C₈×C₁₇	D₄	1	1	yes	characteristic
17	C₂×C₄²×C₁₇	C₂×C₄	1	1	yes	characteristic
18	C₂×C₄⋊Dic₁₇	—	2	1	no	—
19	C₂×C₄×Dic₁₇	—	2	1	no	—
20	C₂²×#3622	—	2	1	no	—
21	C₁₇×M₃₂	—	2	2	no	—
22	C₁₇×M₃₂	—	2	2	no	—
23	C₂×C₁₆×C₁₇	—	2	2	no	—
24	C₄×C₈×C₁₇	C₂×C₄	1	1	yes	characteristic
25	C₄×C₈×C₁₇	C₂×C₄	1	1	yes	characteristic
26	C₄×C₈×C₁₇	D₄	1	1	yes	characteristic
27	C₄×C₈×C₁₇	Q₈	1	1	yes	characteristic
28	#10587502	C₈	1	2	yes	—
29	#10587486	—	4	1	no	—
30	#10587486	—	4	1	no	—
31	#10587483	—	4	1	no	—
32	#10587483	—	4	1	no	—
33	C₂×#63214	—	2	2	no	—
34	C₄⋊#3622	—	2	1	no	—
35	C₄⋊#3622	C₈	1	2	yes	—
36	C₂²×C₄×C₁₇	C₂²⋊C₄	1	1	yes	characteristic
37	C₂²×C₄×C₁₇	C₂×C₈	1	1	yes	characteristic
38	C₂²×C₄×C₁₇	M₁₆	1	1	yes	characteristic
39	C₂²×Dic₁₇	—	2	2	no	—
40	C₂×C₈×C₁₇	—	2	1	no	—
41	C₂×C₈×C₁₇	—	2	1	no	—
42	C₂×C₈×C₁₇	C₂²⋊C₄	1	1	yes	characteristic
43	C₂×C₈×C₁₇	C₂²⋊C₄	1	1	yes	characteristic
44	C₂×C₈×C₁₇	C₄⋊C₄	1	1	yes	characteristic
45	C₂×C₈×C₁₇	C₄⋊C₄	1	1	yes	characteristic
46	C₂×C₈×C₁₇	D₈	1	1	yes	characteristic
47	C₂×C₈×C₁₇	SD₁₆	1	1	yes	characteristic
48	C₂×C₈×C₁₇	SD₁₆	1	1	yes	characteristic
49	C₂×C₈×C₁₇	Q₁₆	1	1	yes	characteristic
50	C₄²×C₁₇	C₄²	1	1	yes	characteristic
51	C₄²×C₁₇	C₄⋊C₄	1	1	yes	characteristic
52	C₄²×C₁₇	C₂×C₈	1	1	yes	characteristic
53	C₄²×C₁₇	M₁₆	1	1	yes	characteristic
54	C₄⋊Dic₁₇	—	4	2	no	—
55	C₄×Dic₁₇	—	4	1	no	—
56	C₄×Dic₁₇	—	4	1	no	—
57	C₂×#3622	—	2	2	no	—
58	C₂×#3622	—	4	1	no	—
59	C₂×#3622	—	4	1	no	—
60	C₁₆×C₁₇	—	2	4	no	—
61	#63214	—	4	2	no	—
62	#7168	—	17	1	no	maximal, Sylow
63	C₂³×C₁₇	C₂²⋊C₈	1	1	yes	characteristic
64	C₂×C₄×C₁₇	—	2	2	no	—
65	C₂×C₄×C₁₇	—	4	1	no	—
66	C₂×C₄×C₁₇	#95	1	1	yes	characteristic
67	C₂×C₄×C₁₇	C₄×C₈	1	1	yes	characteristic
68	C₂×C₄×C₁₇	#97	1	1	yes	characteristic
69	C₂×C₄×C₁₇	C₂²⋊C₈	1	1	yes	characteristic
70	C₂×C₄×C₁₇	#102	1	1	yes	characteristic
71	C₂×C₄×C₁₇	#103	1	1	yes	characteristic
72	C₂×C₄×C₁₇	#104	1	1	yes	characteristic
73	C₂×C₄×C₁₇	#104	1	1	yes	characteristic
74	C₂×C₄×C₁₇	C₄⋊C₈	1	1	yes	characteristic
75	C₂×C₄×C₁₇	C₄⋊C₈	1	1	yes	characteristic
76	C₂×Dic₁₇	—	4	2	no	—
77	C₂×Dic₁₇	—	4	2	no	—
78	C₂×Dic₁₇	—	4	2	no	—
79	C₈×C₁₇	—	2	1	no	—
80	C₈×C₁₇	—	2	1	no	—
81	C₈×C₁₇	#102	1	1	yes	characteristic
82	C₈×C₁₇	#103	1	1	yes	characteristic
83	C₈×C₁₇	#106	1	1	yes	characteristic
84	C₈×C₁₇	#107	1	1	yes	characteristic
85	#3622	—	4	2	no	—
86	#2150	—	17	1	no	—
87	C₄×M₃₂	D₁₇	1	1	yes	characteristic
88	#1770	—	17	1	no	—
89	C₂²×C₁₇	—	2	1	no	—
90	C₂²×C₁₇	—	2	1	no	—
91	C₂²×C₁₇	D₄⋊C₈	1	1	yes	characteristic
92	C₂²×C₁₇	Q₈⋊C₈	1	1	yes	characteristic
93	C₂²×C₁₇	#336	1	1	yes	characteristic
94	C₄×C₁₇	—	2	1	no	—
95	C₄×C₁₇	—	2	1	no	—
96	C₄×C₁₇	—	2	1	no	—
97	C₄×C₁₇	D₄⋊C₈	1	1	yes	characteristic
98	C₄×C₁₇	Q₈⋊C₈	1	1	yes	characteristic
99	C₄×C₁₇	#339	1	1	yes	characteristic
100	C₄×C₁₇	#340	1	1	yes	characteristic
101	C₄×C₁₇	#347	1	2	yes	—
102	Dic₁₇	—	8	2	no	—
103	C₂×M₃₂	—	2	1	no	—
104	C₂×M₃₂	—	34	1	no	—
105	C₂×C₄⋊C₈	—	17	1	no	—
106	C₂×#97	—	34	1	no	—
107	C₂×C₄×C₈	C₂×D₁₇	1	1	yes	characteristic
108	C₄×C₄⋊C₄	—	17	1	no	—
109	C₄⋊C₁₆	—	17	2	no	—
110	C₄⋊C₁₆	—	34	1	no	—
111	#346	—	2	1	no	—
112	C₄×C₁₆	Dic₁₇	1	2	yes	complemented
113	#336	—	34	1	no	—
114	C₂×C₁₇	—	2	1	no	—
115	C₂×C₁₇	—	2	1	no	—
116	C₂×C₁₇	#1279	1	1	yes	characteristic
117	C₂×C₁₇	#1375	1	1	yes	characteristic
118	C₂×C₁₇	#1378	1	1	yes	characteristic
119	C₂²×C₈	—	34	1	no	—
120	C₂²×C₈	D₆₈	1	1	yes	characteristic
121	C₂²×C₈	C₂²⋊D₁₇	1	1	yes	characteristic
122	C₂×C₄⋊C₄	—	34	1	no	—
123	C₂×C₄²	—	34	1	no	—
124	C₂×C₄²	C₄×D₁₇	1	1	yes	characteristic
125	M₃₂	—	2	2	no	—
126	M₃₂	—	2	2	no	—
127	M₃₂	—	68	1	no	—
128	M₃₂	—	68	1	no	—
129	C₂×C₁₆	—	2	2	no	—
130	C₂×C₁₆	—	34	2	no	—
131	C₄⋊C₈	—	17	2	no	—
132	C₄⋊C₈	—	34	1	no	—
133	#97	—	68	1	no	—
134	#97	—	68	1	no	—
135	C₄×C₈	Dic₃₄	1	1	yes	characteristic
136	C₄×C₈	C₄×D₁₇	1	1	yes	characteristic
137	C₄×C₈	C₂×Dic₁₇	1	1	yes	characteristic
138	C₄×C₈	C₂²⋊D₁₇	1	1	yes	characteristic
139	#95	—	17	2	no	—
140	C₁₇	#7168	1	1	yes	characteristic, complemented, Sylow
141	C₂²×C₄	—	34	2	no	—
142	C₂²×C₄	C₈×D₁₇	1	1	yes	characteristic
143	C₂²×C₄	#63218	1	1	yes	characteristic
144	C₂²×C₄	#63227	1	1	yes	characteristic
145	C₂×C₈	—	2	1	no	—
146	C₂×C₈	—	2	1	no	—
147	C₂×C₈	—	34	2	no	—
148	C₂×C₈	—	68	1	no	—
149	C₂×C₈	—	68	1	no	—
150	C₂×C₈	#63219	1	1	yes	characteristic
151	C₂×C₈	#63219	1	1	yes	characteristic
152	C₂×C₈	D₁₃₆	1	1	yes	characteristic
153	C₂×C₈	Dic₆₈	1	1	yes	characteristic
154	C₂×C₈	#63225	1	1	yes	characteristic
155	C₂×C₈	C₄⋊Dic₁₇	1	1	yes	characteristic
156	C₂×C₈	#63227	1	1	yes	characteristic
157	C₂×C₈	C₂²⋊Dic₁₇	1	1	yes	characteristic
158	C₄⋊C₄	—	68	2	no	—
159	C₄²	—	68	1	no	—
160	C₄²	—	68	1	no	—
161	C₄²	C₈×D₁₇	1	1	yes	characteristic
162	C₄²	#63218	1	1	yes	characteristic
163	C₄²	C₄×Dic₁₇	1	1	yes	characteristic
164	C₄²	#63225	1	1	yes	characteristic
165	C₁₆	—	2	4	no	—
166	C₁₆	—	68	2	no	—
167	C₂³	#10587491	1	1	yes	characteristic
168	C₂×C₄	—	2	2	no	—
169	C₂×C₄	—	4	1	no	—
170	C₂×C₄	—	68	2	no	—
171	C₂×C₄	—	68	2	no	—
172	C₂×C₄	—	68	2	no	—
173	C₂×C₄	#10587476	1	1	yes	characteristic
174	C₂×C₄	#10587476	1	1	yes	characteristic
175	C₂×C₄	C₈×Dic₁₇	1	1	yes	characteristic
176	C₂×C₄	#10587485	1	1	yes	characteristic
177	C₂×C₄	#10587485	1	1	yes	characteristic
178	C₂×C₄	#10587486	1	1	yes	characteristic
179	C₂×C₄	#10587487	1	1	yes	characteristic
180	C₂×C₄	#10587491	1	1	yes	characteristic
181	C₂×C₄	#10587492	1	1	yes	characteristic
182	C₂×C₄	#10587502	1	1	yes	characteristic
183	C₈	—	2	1	no	—
184	C₈	—	2	1	no	—
185	C₈	—	68	2	no	—
186	C₈	#10587487	1	1	yes	characteristic
187	C₈	#10587488	1	1	yes	characteristic
188	C₈	#10587489	1	1	yes	characteristic
189	C₈	#10587492	1	1	yes	characteristic
190	C₂²	—	2	1	no	—
191	C₂²	—	2	1	no	—
192	C₂²	#11762538	1	1	yes	characteristic
193	C₂²	#11762541	1	1	yes	characteristic
194	C₂²	#11762563	1	1	yes	characteristic
195	C₄	—	2	1	no	—
196	C₄	—	2	1	no	—
197	C₄	—	2	1	no	—
198	C₄	—	136	2	no	—
199	C₄	#11762538	1	1	yes	characteristic
200	C₄	#11762541	1	1	yes	characteristic
201	C₄	#11762569	1	1	yes	characteristic
202	C₄	#11762572	1	1	yes	characteristic
203	C₄	#11762586	1	2	yes	—
204	C₂	—	2	1	no	—
205	C₂	—	2	1	no	—
206	C₂	#14526624	1	1	yes	characteristic
207	C₂	#14526863	1	1	yes	characteristic
208	C₂	#14526869	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	#10587502
commutator	C₂×C₄×C₁₇	C₄×C₈
Frattini	C₂×C₄×C₈	C₂×D₁₇
Fitting	C₄×C₁₇×M₃₂	C₂
socle	C₂²×C₁₇	#336
prime radical	C₂³×C₁₇	C₂²⋊C₈
cosocle	C₂×C₄×C₈×C₁₇	C₂²
hypercenter	C₄×M₃₂	D₁₇
hypocenter	C₁₇	#7168
LCS	G ▷ C₂×C₄×C₁₇ ▷ C₂×C₁₇ ▷ C₁₇
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄×C₈ ◁ C₄×M₃₂

Complements

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

N	H ≅ G/N	type
9. C₄⋊#63214	C₄	semidirect
14. C₄×C₁₆×C₁₇	C₄	semidirect
112. C₄×C₁₆	Dic₁₇	semidirect
140. C₁₇	#7168	semidirect

Maximal subgroups

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

subgroup	index
#14530540	2
#14528651	2
C₄×C₁₇×M₃₂	2
#7168	17