Subgroups — #17053276

Order 4352.

822 conjugacy classes of subgroups.

The subgroup lattice has 188 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	Q₈×C₄⋊Dic₁₇	C₂	1	2	yes	maximal
2	#14529543	C₂	1	6	yes	maximal
3	#14528912	C₂	1	3	yes	maximal
4	C₄×#10587489	C₂	1	3	yes	maximal
5	C₂×C₈×Q₈×C₁₇	C₂	1	1	yes	maximal, characteristic
6	C₂×Q₈×Dic₁₇	—	2	2	no	—
7	C₂×#10587489	C₂²	1	6	yes	—
8	#11762708	—	2	6	no	—
9	#11762679	C₂²	1	6	yes	—
10	C₄×C₄⋊Dic₁₇	C₂²	1	6	yes	—
11	#11762572	C₂²	1	6	yes	—
12	#11762572	C₂²	1	3	yes	—
13	#11762572	C₂²	1	1	yes	characteristic
14	C₂×C₄×Q₈×C₁₇	C₂²	1	1	yes	characteristic
15	C₈×Q₈×C₁₇	C₄	1	8	yes	complemented
16	C₂×C₁₇×C₄⋊C₈	C₂²	1	3	yes	—
17	C₂×C₄×C₈×C₁₇	C₂²	1	3	yes	—
18	C₂²×Q₈×C₁₇	D₄	1	1	yes	characteristic
19	C₂²×C₈×C₁₇	C₂³	1	1	yes	characteristic
20	C₂²×C₈×C₁₇	C₂³	1	3	yes	—
21	C₄×Q₈×C₁₇	C₂×C₄	1	4	yes	—
22	C₄×Q₈×C₁₇	Q₈	1	4	yes	—
23	C₂×C₄⋊C₄×C₁₇	D₄	1	3	yes	—
24	C₂×C₄²×C₁₇	C₂³	1	3	yes	—
25	Q₈×Dic₁₇	—	4	8	no	—
26	C₂×C₄⋊Dic₁₇	—	2	6	no	—
27	C₂×C₄⋊Dic₁₇	—	2	6	no	—
28	C₂×C₄⋊Dic₁₇	C₂³	1	6	yes	—
29	C₂×C₄⋊Dic₁₇	C₂³	1	2	yes	—
30	C₂×C₄×Dic₁₇	—	2	6	no	—
31	C₁₇×C₄⋊C₈	C₂×C₄	1	12	yes	—
32	C₄×C₈×C₁₇	C₂×C₄	1	12	yes	—
33	#10587502	—	2	6	no	—
34	#10587489	—	2	12	no	—
35	C₂×Q₈×C₁₇	C₄⋊C₄	1	4	yes	—
36	C₂×Q₈×C₁₇	D₈	1	4	yes	—
37	C₂×Q₈×C₁₇	Q₁₆	1	4	yes	—
38	C₂²×C₄×C₁₇	C₂×D₄	1	3	yes	—
39	C₂²×C₄×C₁₇	C₂×D₄	1	3	yes	—
40	C₂²×C₄×C₁₇	C₂⁴	1	1	yes	characteristic
41	C₂²×Dic₁₇	—	2	2	no	—
42	C₂²×Dic₁₇	—	2	6	no	—
43	C₂×C₈×C₁₇	—	2	6	no	—
44	C₂×C₈×C₁₇	—	2	2	no	—
45	C₂×C₈×C₁₇	C₂²×C₄	1	2	yes	—
46	C₂×C₈×C₁₇	C₂²×C₄	1	6	yes	—
47	C₄⋊C₄×C₁₇	C₄⋊C₄	1	12	yes	—
48	C₄²×C₁₇	C₂²×C₄	1	6	yes	—
49	C₄²×C₁₇	C₂×Q₈	1	6	yes	—
50	C₄⋊Dic₁₇	—	2	2	no	—
51	C₄⋊Dic₁₇	—	2	6	no	—
52	C₄⋊Dic₁₇	—	2	6	no	—
53	C₄⋊Dic₁₇	—	4	12	no	—
54	C₄⋊Dic₁₇	—	8	6	no	—
55	C₄⋊Dic₁₇	#41	1	4	yes	—
56	C₄×Dic₁₇	—	4	12	no	—
57	#22740	—	17	1	no	maximal, Sylow
58	C₂³×C₁₇	C₂²×D₄	1	1	yes	characteristic
59	Q₈×C₁₇	#107	1	16	yes	complemented
60	C₂×C₄×C₁₇	—	2	6	no	—
61	C₂×C₄×C₁₇	C₂×C₄⋊C₄	1	6	yes	—
62	C₂×C₄×C₁₇	C₂×C₄⋊C₄	1	6	yes	—
63	C₂×C₄×C₁₇	C₂×D₈	1	6	yes	—
64	C₂×C₄×C₁₇	C₂×Q₁₆	1	6	yes	—
65	C₂×C₄×C₁₇	C₂³×C₄	1	1	yes	characteristic
66	C₂×C₄×C₁₇	C₂²×Q₈	1	1	yes	characteristic
67	C₂×C₄×C₁₇	C₂×#41	1	2	yes	—
68	C₂×C₄×C₁₇	2₋¹⁺⁴	1	2	yes	—
69	C₂×Dic₁₇	—	2	4	no	—
70	C₂×Dic₁₇	—	4	2	no	—
71	C₂×Dic₁₇	—	4	2	no	—
72	C₂×Dic₁₇	—	4	6	no	—
73	C₂×Dic₁₇	—	4	6	no	—
74	C₂×Dic₁₇	—	4	6	no	—
75	C₈×C₁₇	—	2	6	no	—
76	C₈×C₁₇	—	2	2	no	—
77	C₂×C₈×Q₈	D₁₇	1	1	yes	characteristic
78	Q₈×C₄⋊C₄	—	17	2	no	—
79	#1947	—	17	6	no	—
80	#1834	—	17	3	no	—
81	C₄×#107	—	17	3	no	—
82	C₂²×C₁₇	C₂²×C₄⋊C₄	1	1	yes	characteristic
83	C₂²×C₁₇	#548	1	2	yes	—
84	C₂²×C₁₇	C₂²×D₈	1	1	yes	characteristic
85	C₂²×C₁₇	C₂²×Q₁₆	1	1	yes	characteristic
86	C₂²×C₁₇	#577	1	2	yes	—
87	C₄×C₁₇	—	2	6	no	—
88	C₄×C₁₇	C₂×#107	1	12	yes	—
89	C₄×C₁₇	C₄×#41	1	1	yes	characteristic
90	C₄×C₁₇	#519	1	1	yes	characteristic
91	C₄×C₁₇	#557	1	2	yes	—
92	Dic₁₇	—	4	4	no	—
93	Dic₁₇	—	8	6	no	—
94	C₂×C₄×Q₈	—	34	2	no	—
95	C₂×C₄×Q₈	C₂×D₁₇	1	1	yes	characteristic
96	C₈×Q₈	Dic₁₇	1	8	yes	complemented
97	C₂×#107	—	17	6	no	—
98	C₂×C₄⋊C₈	C₂×D₁₇	1	3	yes	—
99	C₂×C₄×C₈	C₂×D₁₇	1	3	yes	—
100	#389	—	34	6	no	—
101	#384	—	17	6	no	—
102	C₄×C₄⋊C₄	—	17	6	no	—
103	#340	—	17	3	no	—
104	#340	—	17	6	no	—
105	#340	—	17	1	no	—
106	C₂×C₁₇	#2351	1	1	yes	characteristic
107	C₂×C₁₇	C₂²×#107	1	1	yes	characteristic
108	C₂×C₁₇	#2912	1	1	yes	characteristic
109	C₂×C₁₇	#3358	1	2	yes	—
110	C₂×C₁₇	#3363	1	2	yes	—
111	C₂²×Q₈	D₆₈	1	1	yes	characteristic
112	C₂²×C₈	C₂²×D₁₇	1	1	yes	characteristic
113	C₂²×C₈	C₂²×D₁₇	1	3	yes	—
114	C₄×Q₈	—	68	8	no	—
115	C₄×Q₈	Dic₃₄	1	4	yes	—
116	C₄×Q₈	C₂×Dic₁₇	1	4	yes	—
117	C₂×C₄⋊C₄	—	17	2	no	—
118	C₂×C₄⋊C₄	—	17	6	no	—
119	C₂×C₄⋊C₄	—	34	6	no	—
120	C₂×C₄⋊C₄	—	34	6	no	—
121	C₂×C₄⋊C₄	D₆₈	1	3	yes	—
122	C₂×C₄²	—	34	6	no	—
123	C₂×C₄²	C₂²×D₁₇	1	3	yes	—
124	#107	—	34	12	no	—
125	C₄⋊C₈	C₂×Dic₁₇	1	12	yes	—
126	C₄×C₈	C₂×Dic₁₇	1	12	yes	—
127	#95	—	34	6	no	—
128	C₁₇	#22740	1	1	yes	characteristic, complemented, Sylow
129	C₂×Q₈	D₁₃₆	1	4	yes	—
130	C₂×Q₈	Dic₆₈	1	4	yes	—
131	C₂×Q₈	C₄⋊Dic₁₇	1	4	yes	—
132	C₂²×C₄	—	34	2	no	—
133	C₂²×C₄	—	34	6	no	—
134	C₂²×C₄	C₂×D₆₈	1	3	yes	—
135	C₂²×C₄	C₂×D₆₈	1	3	yes	—
136	C₂²×C₄	C₂³×D₁₇	1	1	yes	characteristic
137	C₂×C₈	—	2	2	no	—
138	C₂×C₈	—	2	6	no	—
139	C₂×C₈	C₂²×Dic₁₇	1	2	yes	—
140	C₂×C₈	C₂²×Dic₁₇	1	6	yes	—
141	C₄⋊C₄	—	17	4	no	—
142	C₄⋊C₄	—	34	2	no	—
143	C₄⋊C₄	—	34	6	no	—
144	C₄⋊C₄	—	34	6	no	—
145	C₄⋊C₄	—	68	12	no	—
146	C₄⋊C₄	—	136	6	no	—
147	C₄⋊C₄	C₄⋊Dic₁₇	1	12	yes	—
148	C₄²	—	68	12	no	—
149	C₄²	C₂×Dic₃₄	1	6	yes	—
150	C₄²	C₂²×Dic₁₇	1	6	yes	—
151	C₂³	C₂²×D₆₈	1	1	yes	characteristic
152	Q₈	#10587489	1	16	yes	complemented
153	C₂×C₄	—	2	6	no	—
154	C₂×C₄	—	34	4	no	—
155	C₂×C₄	—	68	2	no	—
156	C₂×C₄	—	68	2	no	—
157	C₂×C₄	—	68	6	no	—
158	C₂×C₄	—	68	6	no	—
159	C₂×C₄	—	68	6	no	—
160	C₂×C₄	C₂×D₁₃₆	1	6	yes	—
161	C₂×C₄	C₂×Dic₆₈	1	6	yes	—
162	C₂×C₄	C₂×C₄⋊Dic₁₇	1	6	yes	—
163	C₂×C₄	C₂×C₄⋊Dic₁₇	1	6	yes	—
164	C₂×C₄	C₂²×Dic₃₄	1	1	yes	characteristic
165	C₂×C₄	#10587694	1	2	yes	—
166	C₂×C₄	#41×D₁₇	1	2	yes	—
167	C₂×C₄	C₂³×Dic₁₇	1	1	yes	characteristic
168	C₈	—	2	6	no	—
169	C₈	—	2	2	no	—
170	C₂²	C₂²×C₄⋊Dic₁₇	1	1	yes	characteristic
171	C₂²	#11763471	1	2	yes	—
172	C₂²	C₂²×D₁₃₆	1	1	yes	characteristic
173	C₂²	C₂²×Dic₆₈	1	1	yes	characteristic
174	C₂²	#11763621	1	2	yes	—
175	C₄	—	2	6	no	—
176	C₄	—	68	4	no	—
177	C₄	—	136	6	no	—
178	C₄	C₂×#10587489	1	12	yes	—
179	C₄	#41×Dic₁₇	1	1	yes	characteristic
180	C₄	#11763287	1	1	yes	characteristic
181	C₄	#11763535	1	2	yes	—
182	C₂	#14531705	1	1	yes	characteristic
183	C₂	C₂²×#10587489	1	1	yes	characteristic
184	C₂	#14537379	1	1	yes	characteristic
185	C₂	#14541420	1	2	yes	—
186	C₂	#14541463	1	2	yes	—

	subgroup	G/H
center	C₂³	C₂²×D₆₈
commutator	C₂×C₄×C₁₇	C₂³×C₄
Frattini	C₂²×C₄	C₂³×D₁₇
Fitting	C₂×C₈×Q₈×C₁₇	C₂
socle, prime radical	C₂³×C₁₇	C₂²×D₄
cosocle	C₂²×C₄×C₁₇	C₂⁴
hypercenter	C₂×C₈×Q₈	D₁₇
hypocenter	C₁₇	#22740
LCS	G ▷ C₂×C₄×C₁₇ ▷ C₂×C₁₇ ▷ C₁₇
UCS	C₁ ◁ C₂³ ◁ C₂×C₄×Q₈ ◁ C₂×C₈×Q₈

Complements

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

N	H ≅ G/N	type
15. C₈×Q₈×C₁₇	C₄	semidirect
59. Q₈×C₁₇	#107	semidirect
96. C₈×Q₈	Dic₁₇	semidirect
128. C₁₇	#22740	semidirect
152. Q₈	#10587489	semidirect

Maximal subgroups

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

subgroup	index
Q₈×C₄⋊Dic₁₇	2
#14529543	2
#14528912	2
C₄×#10587489	2
C₂×C₈×Q₈×C₁₇	2
#22740	17