Subgroups — #16952925

Order 4352.

414 conjugacy classes of subgroups.

The subgroup lattice has 216 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#11763214	C₂	1	1	yes	maximal, characteristic
2	#14528944	C₂	1	1	yes	maximal, characteristic
3	#14526876	C₂	1	4	yes	maximal
4	C₂×C₄×C₁₆×C₁₇	C₂	1	1	yes	maximal, characteristic
5	C₂×#10587555	—	2	1	no	—
6	#11763214	—	2	2	no	—
7	#11763214	C₄	1	4	yes	complemented
8	C₂×#10587489	—	2	1	no	—
9	C₂×#10587489	—	2	1	no	—
10	C₂×#10587489	C₂²	1	2	yes	—
11	C₂×#10587489	C₂²	1	2	yes	—
12	#11762679	—	2	1	no	—
13	C₂²×C₁₆×C₁₇	C₂²	1	2	yes	—
14	C₂×C₄×C₈×C₁₇	C₂²	1	1	yes	characteristic
15	C₄×C₁₆×C₁₇	—	2	2	no	—
16	C₂²×Dic₃₄	—	4	1	no	—
17	C₂²×C₈×C₁₇	D₄	1	1	yes	characteristic
18	C₂²×C₈×C₁₇	C₂³	1	1	yes	characteristic
19	C₂×C₄²×C₁₇	D₄	1	1	yes	characteristic
20	C₂×C₄⋊Dic₁₇	—	2	2	no	—
21	C₂×C₄⋊Dic₁₇	—	2	2	no	—
22	C₂×C₄⋊Dic₁₇	—	4	1	no	—
23	C₂×C₄⋊Dic₁₇	—	4	1	no	—
24	C₂×C₄⋊Dic₁₇	—	4	1	no	—
25	#10587555	—	2	4	no	—
26	#10587555	—	4	2	no	—
27	C₂×C₁₆×C₁₇	—	2	2	no	—
28	C₂×C₁₆×C₁₇	—	2	2	no	—
29	C₂×C₁₆×C₁₇	Q₈	1	4	yes	complemented
30	C₄×C₈×C₁₇	C₂×C₄	1	2	yes	—
31	C₄×C₈×C₁₇	D₄	1	2	yes	—
32	#10587489	—	2	4	no	—
33	#10587489	—	2	2	no	—
34	#10587489	—	2	4	no	—
35	#10587489	—	4	2	no	—
36	#10587489	C₂×C₄	1	4	yes	—
37	C₂²×C₄×C₁₇	D₈	1	1	yes	characteristic
38	C₂²×C₄×C₁₇	D₈	1	1	yes	characteristic
39	C₂²×C₄×C₁₇	C₂×D₄	1	1	yes	characteristic
40	C₂²×Dic₁₇	—	4	2	no	—
41	C₂²×Dic₁₇	—	4	2	no	—
42	C₂×Dic₃₄	—	4	4	no	—
43	C₂×Dic₃₄	—	8	2	no	—
44	C₂×Dic₃₄	—	8	2	no	—
45	C₂×C₈×C₁₇	C₂²⋊C₄	1	2	yes	—
46	C₂×C₈×C₁₇	SD₁₆	1	2	yes	—
47	C₂×C₈×C₁₇	Q₁₆	1	2	yes	—
48	C₂×C₈×C₁₇	C₂²×C₄	1	1	yes	characteristic
49	C₂×C₈×C₁₇	C₂×D₄	1	1	yes	characteristic
50	C₂×C₈×C₁₇	C₂×Q₈	1	2	yes	—
51	C₂×C₈×C₁₇	#41	1	2	yes	—
52	C₄²×C₁₇	C₂²⋊C₄	1	2	yes	—
53	C₄²×C₁₇	SD₁₆	1	2	yes	—
54	C₄⋊Dic₁₇	—	2	4	no	—
55	C₄⋊Dic₁₇	—	4	4	no	—
56	C₄⋊Dic₁₇	—	4	2	no	—
57	C₄⋊Dic₁₇	—	4	2	no	—
58	C₄⋊Dic₁₇	—	4	2	no	—
59	C₄⋊Dic₁₇	—	8	2	no	—
60	C₄⋊Dic₁₇	—	8	2	no	—
61	C₁₆×C₁₇	—	2	4	no	—
62	#13282	—	17	1	no	maximal, Sylow
63	C₂³×C₁₇	C₂×D₈	1	1	yes	characteristic
64	C₂×C₄×C₁₇	#102	1	2	yes	—
65	C₂×C₄×C₁₇	#102	1	2	yes	—
66	C₂×C₄×C₁₇	D₁₆	1	2	yes	—
67	C₂×C₄×C₁₇	SD₃₂	1	2	yes	—
68	C₂×C₄×C₁₇	SD₃₂	1	2	yes	—
69	C₂×C₄×C₁₇	Q₃₂	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	2	no	—
78	C₂×Dic₁₇	—	8	2	no	—
79	C₂×Dic₁₇	—	8	2	no	—
80	C₂×Dic₁₇	—	8	2	no	—
81	C₂×Dic₁₇	—	8	2	no	—
82	Dic₃₄	—	8	8	no	—
83	C₈×C₁₇	#103	1	4	yes	—
84	C₈×C₁₇	C₄×Q₈	1	2	yes	—
85	C₈×C₁₇	C₂²⋊Q₈	1	2	yes	—
86	C₂×#500	—	17	1	no	—
87	C₂×C₄×C₁₆	D₁₇	1	1	yes	characteristic
88	#1845	—	17	1	no	—
89	#1380	—	17	4	no	—
90	C₂²×C₁₇	C₂×#102	1	1	yes	characteristic
91	C₂²×C₁₇	#486	1	1	yes	characteristic
92	C₂²×C₁₇	#500	1	1	yes	characteristic
93	C₂²×C₁₇	C₂×D₁₆	1	1	yes	characteristic
94	C₂²×C₁₇	C₂×SD₃₂	1	1	yes	characteristic
95	C₂²×C₁₇	C₂×SD₃₂	1	1	yes	characteristic
96	C₂²×C₁₇	C₂×Q₃₂	1	1	yes	characteristic
97	C₄×C₁₇	#357	1	4	yes	—
98	C₄×C₁₇	#358	1	4	yes	—
99	C₄×C₁₇	#391	1	1	yes	characteristic
100	C₄×C₁₇	C₂×#103	1	1	yes	characteristic
101	C₄×C₁₇	#487	1	1	yes	characteristic
102	C₄×C₁₇	#498	1	1	yes	characteristic
103	Dic₁₇	—	8	4	no	—
104	Dic₁₇	—	16	2	no	—
105	C₂×C₄⋊Q₈	—	34	1	no	—
106	C₂²×C₁₆	C₂×D₁₇	1	2	yes	—
107	#500	—	17	4	no	—
108	#500	—	34	2	no	—
109	C₂×#107	—	17	2	no	—
110	C₂×#107	—	17	2	no	—
111	C₂×#107	—	34	1	no	—
112	C₂×#107	—	34	1	no	—
113	C₂×C₄×C₈	C₂×D₁₇	1	1	yes	characteristic
114	#384	—	34	1	no	—
115	C₄×C₁₆	—	2	2	no	—
116	C₂×C₁₇	#1990	1	1	yes	characteristic
117	C₂×C₁₇	C₂×#357	1	1	yes	characteristic
118	C₂×C₁₇	C₂×#358	1	1	yes	characteristic
119	C₂×C₁₇	#2240	1	1	yes	characteristic
120	C₂×C₁₇	#2241	1	1	yes	characteristic
121	C₂×C₁₇	#2252	1	1	yes	characteristic
122	C₂×C₁₇	#2254	1	1	yes	characteristic
123	C₂²×Q₈	—	68	1	no	—
124	C₂²×C₈	D₆₈	1	1	yes	characteristic
125	C₂²×C₈	C₂²×D₁₇	1	1	yes	characteristic
126	C₄⋊Q₈	—	34	4	no	—
127	C₄⋊Q₈	—	68	2	no	—
128	C₂×C₄⋊C₄	—	34	2	no	—
129	C₂×C₄⋊C₄	—	34	2	no	—
130	C₂×C₄⋊C₄	—	68	1	no	—
131	C₂×C₄⋊C₄	—	68	1	no	—
132	C₂×C₄⋊C₄	—	68	1	no	—
133	C₂×C₄²	D₆₈	1	1	yes	characteristic
134	C₂×C₁₆	—	2	2	no	—
135	C₂×C₁₆	—	2	2	no	—
136	C₂×C₁₆	Dic₃₄	1	4	yes	complemented
137	#107	—	17	4	no	—
138	#107	—	34	2	no	—
139	#107	—	34	4	no	—
140	#107	—	34	4	no	—
141	#107	—	68	2	no	—
142	C₄×C₈	C₄×D₁₇	1	2	yes	—
143	C₄×C₈	C₂²⋊D₁₇	1	2	yes	—
144	C₁₇	#13282	1	1	yes	characteristic, complemented, Sylow
145	C₂×Q₈	—	68	4	no	—
146	C₂×Q₈	—	136	2	no	—
147	C₂×Q₈	—	136	2	no	—
148	C₂²×C₄	—	68	2	no	—
149	C₂²×C₄	—	68	2	no	—
150	C₂²×C₄	D₁₃₆	1	1	yes	characteristic
151	C₂²×C₄	D₁₃₆	1	1	yes	characteristic
152	C₂²×C₄	C₂×D₆₈	1	1	yes	characteristic
153	C₂×C₈	#63219	1	2	yes	—
154	C₂×C₈	Dic₆₈	1	2	yes	—
155	C₂×C₈	#63227	1	2	yes	—
156	C₂×C₈	C₂×Dic₃₄	1	2	yes	—
157	C₂×C₈	C₂×C₄×D₁₇	1	1	yes	characteristic
158	C₂×C₈	#63252	1	2	yes	—
159	C₂×C₈	C₂×C₂²⋊D₁₇	1	1	yes	characteristic
160	C₄⋊C₄	—	34	4	no	—
161	C₄⋊C₄	—	68	4	no	—
162	C₄⋊C₄	—	68	2	no	—
163	C₄⋊C₄	—	68	2	no	—
164	C₄⋊C₄	—	68	2	no	—
165	C₄⋊C₄	—	136	2	no	—
166	C₄⋊C₄	—	136	2	no	—
167	C₄²	#63219	1	2	yes	—
168	C₄²	#63227	1	2	yes	—
169	C₁₆	—	2	4	no	—
170	C₂³	C₂×D₁₃₆	1	1	yes	characteristic
171	Q₈	—	136	8	no	—
172	C₂×C₄	—	68	4	no	—
173	C₂×C₄	—	136	2	no	—
174	C₂×C₄	—	136	2	no	—
175	C₂×C₄	—	136	2	no	—
176	C₂×C₄	—	136	2	no	—
177	C₂×C₄	—	136	2	no	—
178	C₂×C₄	D₂₇₂	1	2	yes	—
179	C₂×C₄	#10587471	1	2	yes	—
180	C₂×C₄	#10587471	1	2	yes	—
181	C₂×C₄	Dic₁₃₆	1	2	yes	—
182	C₂×C₄	#10587492	1	2	yes	—
183	C₂×C₄	#10587492	1	2	yes	—
184	C₂×C₄	#10587555	1	1	yes	characteristic
185	C₂×C₄	#10587561	1	1	yes	characteristic
186	C₂×C₄	C₂×#63219	1	1	yes	characteristic
187	C₂×C₄	C₂×#63219	1	1	yes	characteristic
188	C₂×C₄	C₂×Dic₆₈	1	1	yes	characteristic
189	C₂×C₄	C₂×#63227	1	1	yes	characteristic
190	C₈	#10587487	1	4	yes	—
191	C₈	C₄×Dic₃₄	1	2	yes	—
192	C₈	#10587610	1	2	yes	—
193	C₂²	C₂×#10587492	1	1	yes	characteristic
194	C₂²	#11763156	1	1	yes	characteristic
195	C₂²	#11763214	1	1	yes	characteristic
196	C₂²	C₂×D₂₇₂	1	1	yes	characteristic
197	C₂²	C₂×#10587471	1	1	yes	characteristic
198	C₂²	C₂×#10587471	1	1	yes	characteristic
199	C₂²	C₂×Dic₁₃₆	1	1	yes	characteristic
200	C₄	—	272	2	no	—
201	C₄	—	136	4	no	—
202	C₄	#11762611	1	4	yes	—
203	C₄	#11762614	1	4	yes	—
204	C₄	#11762715	1	1	yes	characteristic
205	C₄	C₂×#10587487	1	1	yes	characteristic
206	C₄	#11763160	1	1	yes	characteristic
207	C₄	#11763207	1	1	yes	characteristic
208	C₂	#14529745	1	1	yes	characteristic
209	C₂	C₂×#11762611	1	1	yes	characteristic
210	C₂	C₂×#11762614	1	1	yes	characteristic
211	C₂	#14531008	1	1	yes	characteristic
212	C₂	#14531012	1	1	yes	characteristic
213	C₂	#14531059	1	1	yes	characteristic
214	C₂	#14531066	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	C₂×D₁₃₆
commutator	C₂×C₈×C₁₇	C₂²×C₄
Frattini	C₂²×C₈	C₂²×D₁₇
Fitting	C₂×C₄×C₁₆×C₁₇	C₂
socle, prime radical	C₂³×C₁₇	C₂×D₈
cosocle	C₂²×C₈×C₁₇	C₂³
hypercenter	C₂×C₄×C₁₆	D₁₇
hypocenter	C₁₇	#13282
LCS	G ▷ C₂×C₈×C₁₇ ▷ C₄×C₁₇ ▷ C₂×C₁₇ ▷ C₁₇
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ C₂×C₄×C₁₆

Complements

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

N	H ≅ G/N	type
7. #11763214	C₄	semidirect
29. C₂×C₁₆×C₁₇	Q₈	semidirect
136. C₂×C₁₆	Dic₃₄	semidirect
144. C₁₇	#13282	semidirect

Maximal subgroups

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

subgroup	index
C₂×#11763214	2
#14528944	2
#14526876	2
C₂×C₄×C₁₆×C₁₇	2
#13282	17