Subgroups — #16934154

Order 4352.

526 conjugacy classes of subgroups.

The subgroup lattice has 214 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#14529746	C₂	1	2	yes	maximal
2	#14529746	C₂	1	2	yes	maximal
3	#14528912	C₂	1	1	yes	maximal, characteristic
4	C₂×#11762531	C₂	1	1	yes	maximal, characteristic
5	C₂×C₁₇×#494	C₂	1	1	yes	maximal, characteristic
6	C₂×C₄×#3622	C₂²	1	1	yes	characteristic
7	C₂×C₄×#3622	C₂²	1	1	yes	characteristic
8	#11762679	C₂²	1	2	yes	—
9	#11762572	—	2	2	no	—
10	#11762571	—	2	4	no	—
11	#11762571	—	2	4	no	—
12	#11762531	—	2	2	no	—
13	C₂²×Q₁₆×C₁₇	—	2	2	no	—
14	C₂×C₁₇×C₄⋊Q₈	C₂²	1	2	yes	—
15	C₁₇×#494	—	2	2	no	—
16	C₁₇×#494	C₄	1	4	yes	complemented
17	C₂×C₄×C₈×C₁₇	C₂²	1	1	yes	characteristic
18	C₂²×Q₈×C₁₇	—	2	2	no	—
19	C₂×Q₁₆×C₁₇	—	2	8	no	—
20	C₂×Q₁₆×C₁₇	—	4	8	no	—
21	C₂²×C₈×C₁₇	D₄	1	2	yes	—
22	C₁₇×C₄⋊Q₈	—	2	2	no	—
23	C₁₇×C₄⋊Q₈	—	2	2	no	—
24	C₁₇×C₄⋊Q₈	—	2	2	no	—
25	C₁₇×C₄⋊Q₈	C₂×C₄	1	4	yes	—
26	C₂×C₄⋊C₄×C₁₇	—	2	2	no	—
27	C₂×C₄⋊C₄×C₁₇	—	2	2	no	—
28	C₂×C₄²×C₁₇	C₂³	1	1	yes	characteristic
29	C₂×C₄⋊Dic₁₇	—	2	2	no	—
30	C₂×C₄⋊Dic₁₇	—	2	2	no	—
31	C₂×C₄⋊Dic₁₇	—	2	2	no	—
32	C₂²×#3622	D₄	1	2	yes	—
33	C₂²×#3622	D₄	1	2	yes	—
34	C₄×C₈×C₁₇	C₂×C₄	1	2	yes	—
35	C₄×C₈×C₁₇	D₄	1	2	yes	—
36	C₄×#3622	—	2	1	no	—
37	C₄×#3622	—	2	1	no	—
38	C₄×#3622	—	2	1	no	—
39	C₄×#3622	—	2	1	no	—
40	C₂×Q₈×C₁₇	—	2	4	no	—
41	C₂×Q₈×C₁₇	—	4	2	no	—
42	C₂×Q₈×C₁₇	—	4	8	no	—
43	C₂²×C₄×C₁₇	—	4	2	no	—
44	C₂²×C₄×C₁₇	C₂×D₄	1	1	yes	characteristic
45	C₂²×C₄×C₁₇	C₂×D₄	1	1	yes	characteristic
46	C₂²×C₄×C₁₇	C₂×D₄	1	1	yes	characteristic
47	C₂²×Dic₁₇	—	4	2	no	—
48	Q₁₆×C₁₇	—	4	16	no	—
49	C₂×C₈×C₁₇	—	2	4	no	—
50	C₂×C₈×C₁₇	C₂²⋊C₄	1	4	yes	—
51	C₄⋊C₄×C₁₇	—	2	4	no	—
52	C₄⋊C₄×C₁₇	—	2	4	no	—
53	C₄⋊C₄×C₁₇	—	4	2	no	—
54	C₄⋊C₄×C₁₇	—	4	2	no	—
55	C₄²×C₁₇	C₂²×C₄	1	1	yes	characteristic
56	C₄²×C₁₇	C₂×D₄	1	1	yes	characteristic
57	C₄²×C₁₇	#41	1	1	yes	characteristic
58	C₄²×C₁₇	#41	1	1	yes	characteristic
59	C₄⋊Dic₁₇	—	4	4	no	—
60	C₄⋊Dic₁₇	—	4	4	no	—
61	C₄⋊Dic₁₇	—	4	4	no	—
62	C₂×#3622	—	2	2	no	—
63	C₂×#3622	—	2	2	no	—
64	C₂×#3622	—	2	2	no	—
65	C₂×#3622	—	2	4	no	—
66	C₂×#3622	Q₁₆	1	4	yes	complemented
67	#9835	—	17	1	no	maximal, Sylow
68	C₂³×C₁₇	#127	1	1	yes	characteristic
69	Q₈×C₁₇	—	4	8	no	—
70	Q₈×C₁₇	—	8	4	no	—
71	C₂×C₄×C₁₇	—	4	4	no	—
72	C₂×C₄×C₁₇	—	8	4	no	—
73	C₂×C₄×C₁₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
74	C₂×C₄×C₁₇	C₄×D₄	1	1	yes	characteristic
75	C₂×C₄×C₁₇	C₄×D₄	1	1	yes	characteristic
76	C₂×C₄×C₁₇	#121	1	1	yes	characteristic
77	C₂×C₄×C₁₇	#121	1	1	yes	characteristic
78	C₂×C₄×C₁₇	#124	1	1	yes	characteristic
79	C₂×C₄×C₁₇	C₂×Q₁₆	1	2	yes	—
80	C₂×C₄×C₁₇	#135	1	2	yes	—
81	C₂×C₄×C₁₇	AGL(1,ℤ/8)	1	4	yes	—
82	C₂×C₄×C₁₇	#137	1	4	yes	—
83	C₂×Dic₁₇	—	8	2	no	—
84	C₂×Dic₁₇	—	8	2	no	—
85	C₂×Dic₁₇	—	8	2	no	—
86	C₈×C₁₇	—	2	4	no	—
87	#3622	—	2	4	no	—
88	#3622	—	4	2	no	—
89	C₂×#494	D₁₇	1	1	yes	characteristic
90	#1990	—	17	2	no	—
91	#1990	—	17	2	no	—
92	#1834	—	17	1	no	—
93	C₂×#322	—	17	1	no	—
94	C₂²×C₁₇	#390	1	1	yes	characteristic
95	C₂²×C₁₇	#494	1	1	yes	characteristic
96	C₂²×C₁₇	#495	1	1	yes	characteristic
97	C₂²×C₁₇	#496	1	2	yes	—
98	C₂²×C₁₇	#497	1	2	yes	—
99	C₄×C₁₇	—	8	4	no	—
100	C₄×C₁₇	#418	1	2	yes	—
101	C₄×C₁₇	C₄×Q₁₆	1	2	yes	—
102	C₄×C₁₇	#441	1	2	yes	—
103	C₄×C₁₇	#467	1	2	yes	—
104	C₄×C₁₇	#470	1	2	yes	—
105	C₄×C₁₇	#490	1	2	yes	—
106	Dic₁₇	—	16	2	no	—
107	C₂²×Q₁₆	—	2	2	no	—
108	C₂×C₄⋊Q₈	C₂×D₁₇	1	2	yes	—
109	#494	—	2	2	no	—
110	#494	Dic₁₇	1	4	yes	complemented
111	C₂×C₄×C₈	—	17	1	no	—
112	C₂×C₄×C₈	—	17	1	no	—
113	C₂×C₄×C₈	C₂×D₁₇	1	1	yes	characteristic
114	#384	—	17	2	no	—
115	#340	—	34	4	no	—
116	#340	—	34	4	no	—
117	#340	—	34	2	no	—
118	#322	—	34	2	no	—
119	C₂×C₁₇	#1720	1	2	yes	—
120	C₂×C₁₇	#1723	1	2	yes	—
121	C₂×C₁₇	#1961	1	1	yes	characteristic
122	C₂×C₁₇	#1969	1	1	yes	characteristic
123	C₂×C₁₇	#1971	1	1	yes	characteristic
124	C₂²×Q₈	—	2	2	no	—
125	C₂×Q₁₆	—	2	8	no	—
126	C₂×Q₁₆	—	4	8	no	—
127	C₂²×C₈	—	17	2	no	—
128	C₂²×C₈	—	17	2	no	—
129	C₂²×C₈	C₂²⋊D₁₇	1	2	yes	—
130	C₄⋊Q₈	—	2	2	no	—
131	C₄⋊Q₈	—	2	2	no	—
132	C₄⋊Q₈	—	2	2	no	—
133	C₄⋊Q₈	C₂×Dic₁₇	1	4	yes	—
134	C₂×C₄⋊C₄	—	2	2	no	—
135	C₂×C₄⋊C₄	—	2	2	no	—
136	C₂×C₄⋊C₄	—	34	2	no	—
137	C₂×C₄⋊C₄	—	34	2	no	—
138	C₂×C₄⋊C₄	—	34	2	no	—
139	C₂×C₄²	C₂²×D₁₇	1	1	yes	characteristic
140	C₄×C₈	—	34	1	no	—
141	C₄×C₈	—	34	1	no	—
142	C₄×C₈	—	34	1	no	—
143	C₄×C₈	—	34	1	no	—
144	C₄×C₈	C₂×Dic₁₇	1	2	yes	—
145	C₄×C₈	C₂²⋊D₁₇	1	2	yes	—
146	C₁₇	#9835	1	1	yes	characteristic, complemented, Sylow
147	C₂×Q₈	—	2	4	no	—
148	C₂×Q₈	—	4	2	no	—
149	C₂×Q₈	—	4	8	no	—
150	C₂²×C₄	—	4	2	no	—
151	C₂²×C₄	—	68	2	no	—
152	C₂²×C₄	D₄×D₁₇	1	1	yes	characteristic
153	C₂²×C₄	D₄×D₁₇	1	1	yes	characteristic
154	C₂²×C₄	C₂×C₂²⋊D₁₇	1	1	yes	characteristic
155	Q₁₆	—	4	16	no	—
156	C₂×C₈	—	2	4	no	—
157	C₂×C₈	—	17	4	no	—
158	C₂×C₈	—	34	2	no	—
159	C₂×C₈	—	34	2	no	—
160	C₂×C₈	—	34	4	no	—
161	C₂×C₈	—	34	2	no	—
162	C₂×C₈	C₂²⋊Dic₁₇	1	4	yes	—
163	C₄⋊C₄	—	2	4	no	—
164	C₄⋊C₄	—	2	4	no	—
165	C₄⋊C₄	—	4	2	no	—
166	C₄⋊C₄	—	4	2	no	—
167	C₄⋊C₄	—	68	4	no	—
168	C₄⋊C₄	—	68	4	no	—
169	C₄⋊C₄	—	68	4	no	—
170	C₄²	#63254	1	1	yes	characteristic
171	C₄²	#63254	1	1	yes	characteristic
172	C₄²	C₂²×Dic₁₇	1	1	yes	characteristic
173	C₄²	C₂×C₂²⋊D₁₇	1	1	yes	characteristic
174	C₂³	#10587626	1	1	yes	characteristic
175	Q₈	—	4	8	no	—
176	Q₈	—	8	4	no	—
177	C₂×C₄	—	4	4	no	—
178	C₂×C₄	—	8	4	no	—
179	C₂×C₄	—	136	2	no	—
180	C₂×C₄	—	136	2	no	—
181	C₂×C₄	—	136	2	no	—
182	C₂×C₄	Q₁₆×D₁₇	1	2	yes	—
183	C₂×C₄	#10587604	1	4	yes	—
184	C₂×C₄	#10587605	1	2	yes	—
185	C₂×C₄	#10587618	1	4	yes	—
186	C₂×C₄	D₄×Dic₁₇	1	1	yes	characteristic
187	C₂×C₄	D₄×Dic₁₇	1	1	yes	characteristic
188	C₂×C₄	#10587622	1	1	yes	characteristic
189	C₂×C₄	#10587624	1	1	yes	characteristic
190	C₂×C₄	#10587624	1	1	yes	characteristic
191	C₂×C₄	C₂×C₂²⋊Dic₁₇	1	1	yes	characteristic
192	C₈	—	2	4	no	—
193	C₈	—	34	4	no	—
194	C₈	—	68	2	no	—
195	C₂²	#11762712	1	1	yes	characteristic
196	C₂²	#11763187	1	1	yes	characteristic
197	C₂²	#11763194	1	1	yes	characteristic
198	C₂²	#11763196	1	2	yes	—
199	C₂²	#11763202	1	2	yes	—
200	C₄	—	8	4	no	—
201	C₄	—	272	2	no	—
202	C₄	#11762814	1	2	yes	—
203	C₄	Q₁₆×Dic₁₇	1	2	yes	—
204	C₄	#11762916	1	2	yes	—
205	C₄	#11763047	1	2	yes	—
206	C₄	#11763064	1	2	yes	—
207	C₄	#11763173	1	2	yes	—
208	C₂	#14528447	1	2	yes	—
209	C₂	#14528459	1	2	yes	—
210	C₂	#14529609	1	1	yes	characteristic
211	C₂	#14529646	1	1	yes	characteristic
212	C₂	#14529656	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	#10587626
commutator	C₄²×C₁₇	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₁₇
Fitting	C₂×C₁₇×#494	C₂
socle, prime radical	C₂³×C₁₇	#127
cosocle	C₂×C₄²×C₁₇	C₂³
hypercenter	C₂×#494	D₁₇
hypocenter	C₁₇	#9835
LCS	G ▷ C₄²×C₁₇ ▷ C₂²×C₁₇ ▷ C₁₇
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×#494

Complements

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

N	H ≅ G/N	type
16. C₁₇×#494	C₄	semidirect
66. C₂×#3622	Q₁₆	semidirect
110. #494	Dic₁₇	semidirect
146. C₁₇	#9835	semidirect

Maximal subgroups

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

subgroup	index
#14529746	2
#14528912	2
C₂×#11762531	2
C₂×C₁₇×#494	2
#9835	17