Subgroups — #16952949

Order 4352.

354 conjugacy classes of subgroups.

The subgroup lattice has 200 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#14539298	C₂	1	1	yes	maximal, characteristic
2	#14528948	C₂	1	1	yes	maximal, characteristic
3	#14526890	C₂	1	4	yes	maximal
4	C₄×C₁₇×M₃₂	C₂	1	1	yes	maximal, characteristic
5	#11763366	—	2	1	no	—
6	#11763214	C₄	1	2	yes	complemented
7	#11763211	—	2	1	no	—
8	#11763211	C₄	1	2	yes	complemented
9	#11763207	—	2	1	no	—
10	#11762847	—	2	1	no	—
11	#11762847	C₂²	1	2	yes	—
12	C₂×#10587489	—	2	1	no	—
13	C₂×#10587488	C₂²	1	2	yes	—
14	#11762675	—	2	1	no	—
15	C₂×C₁₇×M₃₂	C₂²	1	2	yes	—
16	C₂×C₄×C₈×C₁₇	C₂²	1	1	yes	characteristic
17	C₁₇×#346	—	2	1	no	—
18	C₄×C₁₆×C₁₇	—	2	1	no	—
19	C₂²×C₈×C₁₇	D₄	1	1	yes	characteristic
20	C₂²×C₈×C₁₇	C₂³	1	1	yes	characteristic
21	C₂×C₄²×C₁₇	D₄	1	1	yes	characteristic
22	#10587612	—	2	2	no	—
23	C₂×C₄⋊Dic₁₇	—	2	2	no	—
24	#10587610	—	4	2	no	—
25	#10587556	—	2	2	no	—
26	#10587555	—	2	2	no	—
27	C₄×Dic₃₄	—	4	2	no	—
28	C₁₇×M₃₂	—	2	2	no	—
29	C₁₇×M₃₂	Q₈	1	4	yes	complemented
30	C₂×C₁₆×C₁₇	—	2	2	no	—
31	C₄×C₈×C₁₇	C₂×C₄	1	1	yes	characteristic
32	C₄×C₈×C₁₇	C₂×C₄	1	1	yes	characteristic
33	C₄×C₈×C₁₇	D₄	1	1	yes	characteristic
34	C₄×C₈×C₁₇	D₄	1	1	yes	characteristic
35	#10587502	—	4	2	no	—
36	#10587489	—	2	2	no	—
37	#10587489	—	4	2	no	—
38	#10587489	C₂×C₄	1	4	yes	—
39	#10587488	—	2	2	no	—
40	#10587488	—	2	2	no	—
41	#10587488	—	2	4	no	—
42	C₂²×C₄×C₁₇	D₈	1	2	yes	—
43	C₂²×C₄×C₁₇	C₂×D₄	1	1	yes	characteristic
44	C₂²×Dic₁₇	—	4	2	no	—
45	C₂×Dic₃₄	—	4	2	no	—
46	C₂×C₈×C₁₇	C₂²⋊C₄	1	1	yes	characteristic
47	C₂×C₈×C₁₇	C₂²⋊C₄	1	1	yes	characteristic
48	C₂×C₈×C₁₇	SD₁₆	1	2	yes	—
49	C₂×C₈×C₁₇	Q₁₆	1	2	yes	—
50	C₂×C₈×C₁₇	C₂²×C₄	1	1	yes	characteristic
51	C₂×C₈×C₁₇	C₂×D₄	1	1	yes	characteristic
52	C₂×C₈×C₁₇	C₂×Q₈	1	2	yes	—
53	C₂×C₈×C₁₇	#41	1	2	yes	—
54	C₄²×C₁₇	C₂²⋊C₄	1	1	yes	characteristic
55	C₄²×C₁₇	C₂²⋊C₄	1	1	yes	characteristic
56	C₄²×C₁₇	SD₁₆	1	2	yes	—
57	C₂²⋊Dic₁₇	—	4	2	no	—
58	C₄⋊Dic₁₇	—	2	4	no	—
59	C₄⋊Dic₁₇	—	4	2	no	—
60	C₄⋊Dic₁₇	—	4	2	no	—
61	C₄⋊Dic₁₇	—	4	2	no	—
62	#63225	—	4	4	no	—
63	C₄×Dic₁₇	—	4	2	no	—
64	C₁₆×C₁₇	—	2	4	no	—
65	#13288	—	17	1	no	maximal, Sylow
66	C₂³×C₁₇	C₂×D₈	1	1	yes	characteristic
67	C₂×C₄×C₁₇	—	4	2	no	—
68	C₂×C₄×C₁₇	#102	1	2	yes	—
69	C₂×C₄×C₁₇	#102	1	2	yes	—
70	C₂×C₄×C₁₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
71	C₂×C₄×C₁₇	#124	1	1	yes	characteristic
72	C₂×C₄×C₁₇	C₄⋊Q₈	1	1	yes	characteristic
73	C₂×C₄×C₁₇	C₂×SD₁₆	1	1	yes	characteristic
74	C₂×C₄×C₁₇	C₂×SD₁₆	1	1	yes	characteristic
75	C₂×C₄×C₁₇	C₂×Q₁₆	1	1	yes	characteristic
76	C₂×Dic₁₇	—	4	4	no	—
77	C₂×Dic₁₇	—	8	4	no	—
78	C₂×Dic₁₇	—	8	2	no	—
79	Dic₃₄	—	8	4	no	—
80	C₈×C₁₇	#103	1	2	yes	—
81	C₈×C₁₇	#103	1	2	yes	—
82	C₈×C₁₇	C₄×Q₈	1	2	yes	—
83	C₈×C₁₇	C₂²⋊Q₈	1	2	yes	—
84	#3160	—	17	1	no	—
85	C₄×M₃₂	D₁₇	1	1	yes	characteristic
86	#1846	—	17	1	no	—
87	#1386	—	17	4	no	—
88	C₂²×C₁₇	—	2	2	no	—
89	C₂²×C₁₇	C₂×#102	1	1	yes	characteristic
90	C₂²×C₁₇	#486	1	1	yes	characteristic
91	C₂²×C₁₇	#500	1	1	yes	characteristic
92	C₄×C₁₇	—	2	2	no	—
93	C₄×C₁₇	—	2	2	no	—
94	C₄×C₁₇	#391	1	1	yes	characteristic
95	C₄×C₁₇	C₂×#103	1	1	yes	characteristic
96	C₄×C₁₇	#487	1	1	yes	characteristic
97	C₄×C₁₇	#498	1	1	yes	characteristic
98	Dic₁₇	—	8	4	no	—
99	Dic₁₇	—	16	2	no	—
100	#533	—	34	1	no	—
101	C₂×M₃₂	C₂×D₁₇	1	2	yes	—
102	#500	—	17	2	no	—
103	#499	—	17	2	no	—
104	#499	—	34	1	no	—
105	#498	—	34	1	no	—
106	#427	—	17	2	no	—
107	#427	—	34	1	no	—
108	C₂×#107	—	34	1	no	—
109	C₂×#106	—	17	2	no	—
110	C₂×C₄×C₈	C₂×D₁₇	1	1	yes	characteristic
111	#382	—	34	1	no	—
112	#346	—	2	1	no	—
113	C₄×C₁₆	—	2	1	no	—
114	C₂×C₁₇	—	2	1	no	—
115	C₂×C₁₇	—	2	1	no	—
116	C₂×C₁₇	#1990	1	1	yes	characteristic
117	C₂×C₁₇	#2145	1	2	yes	—
118	C₂²×C₈	D₆₈	1	1	yes	characteristic
119	C₂²×C₈	C₂²×D₁₇	1	1	yes	characteristic
120	C₄⋊Q₈	—	34	2	no	—
121	#125	—	34	2	no	—
122	C₂²⋊Q₈	—	68	2	no	—
123	C₄×Q₈	—	68	2	no	—
124	#117	—	34	2	no	—
125	C₂×C₄⋊C₄	—	34	2	no	—
126	C₂×C₄²	D₆₈	1	1	yes	characteristic
127	M₃₂	—	2	2	no	—
128	M₃₂	Dic₃₄	1	4	yes	complemented
129	C₂×C₁₆	—	2	2	no	—
130	#107	—	17	4	no	—
131	#107	—	34	2	no	—
132	#107	—	68	2	no	—
133	#106	—	34	2	no	—
134	#106	—	34	2	no	—
135	#106	—	34	4	no	—
136	C₄×C₈	C₄×D₁₇	1	1	yes	characteristic
137	C₄×C₈	C₄×D₁₇	1	1	yes	characteristic
138	C₄×C₈	C₂²⋊D₁₇	1	1	yes	characteristic
139	C₄×C₈	C₂²⋊D₁₇	1	1	yes	characteristic
140	#95	—	68	2	no	—
141	C₁₇	#13288	1	1	yes	characteristic, complemented, Sylow
142	C₂×Q₈	—	68	2	no	—
143	C₂²×C₄	—	68	2	no	—
144	C₂²×C₄	D₁₃₆	1	2	yes	—
145	C₂²×C₄	C₂×D₆₈	1	1	yes	characteristic
146	C₂×C₈	#63219	1	2	yes	—
147	C₂×C₈	Dic₆₈	1	2	yes	—
148	C₂×C₈	#63227	1	1	yes	characteristic
149	C₂×C₈	#63227	1	1	yes	characteristic
150	C₂×C₈	C₂×Dic₃₄	1	2	yes	—
151	C₂×C₈	C₂×C₄×D₁₇	1	1	yes	characteristic
152	C₂×C₈	#63252	1	2	yes	—
153	C₂×C₈	C₂×C₂²⋊D₁₇	1	1	yes	characteristic
154	C₄⋊C₄	—	34	4	no	—
155	C₄⋊C₄	—	68	4	no	—
156	C₄⋊C₄	—	68	2	no	—
157	C₄⋊C₄	—	68	2	no	—
158	C₄⋊C₄	—	68	2	no	—
159	C₂²⋊C₄	—	68	2	no	—
160	C₄²	—	68	2	no	—
161	C₄²	#63219	1	2	yes	—
162	C₄²	#63227	1	1	yes	characteristic
163	C₄²	#63227	1	1	yes	characteristic
164	C₁₆	—	2	4	no	—
165	C₂³	C₂×D₁₃₆	1	1	yes	characteristic
166	Q₈	—	136	4	no	—
167	C₂×C₄	—	4	2	no	—
168	C₂×C₄	—	68	4	no	—
169	C₂×C₄	—	136	4	no	—
170	C₂×C₄	—	136	2	no	—
171	C₂×C₄	#10587492	1	2	yes	—
172	C₂×C₄	#10587492	1	2	yes	—
173	C₂×C₄	#10587555	1	1	yes	characteristic
174	C₂×C₄	#10587561	1	1	yes	characteristic
175	C₂×C₄	C₂×#63219	1	1	yes	characteristic
176	C₂×C₄	C₂×#63219	1	1	yes	characteristic
177	C₂×C₄	C₂×Dic₆₈	1	1	yes	characteristic
178	C₂×C₄	C₂×#63227	1	1	yes	characteristic
179	C₈	#10587487	1	2	yes	—
180	C₈	#10587487	1	2	yes	—
181	C₈	C₄×Dic₃₄	1	2	yes	—
182	C₈	#10587610	1	2	yes	—
183	C₂²	—	2	2	no	—
184	C₂²	C₂×#10587492	1	1	yes	characteristic
185	C₂²	#11763156	1	1	yes	characteristic
186	C₂²	#11763214	1	1	yes	characteristic
187	C₄	—	2	2	no	—
188	C₄	—	2	2	no	—
189	C₄	—	272	2	no	—
190	C₄	—	136	4	no	—
191	C₄	#11762715	1	1	yes	characteristic
192	C₄	C₂×#10587487	1	1	yes	characteristic
193	C₄	#11763160	1	1	yes	characteristic
194	C₄	#11763207	1	1	yes	characteristic
195	C₂	—	2	1	no	—
196	C₂	—	2	1	no	—
197	C₂	#14529745	1	1	yes	characteristic
198	C₂	#14530521	1	2	yes	—

	subgroup	G/H
center	C₂²	C₂×#10587492
commutator	C₂×C₈×C₁₇	C₂²×C₄
Frattini	C₂²×C₈	C₂²×D₁₇
Fitting	C₄×C₁₇×M₃₂	C₂
socle	C₂²×C₁₇	C₂×#102
prime radical	C₂³×C₁₇	C₂×D₈
cosocle	C₂²×C₈×C₁₇	C₂³
hypercenter	C₄×M₃₂	D₁₇
hypocenter	C₁₇	#13288
LCS	G ▷ C₂×C₈×C₁₇ ▷ C₄×C₁₇ ▷ C₂×C₁₇ ▷ C₁₇
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ C₄×M₃₂

Complements

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

N	H ≅ G/N	type
6. #11763214	C₄	semidirect
8. #11763211	C₄	semidirect
29. C₁₇×M₃₂	Q₈	semidirect
128. M₃₂	Dic₃₄	semidirect
141. C₁₇	#13288	semidirect

Maximal subgroups

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

subgroup	index
#14539298	2
#14528948	2
#14526890	2
C₄×C₁₇×M₃₂	2
#13288	17