Subgroups — #17186896

Order 4352.

830 conjugacy classes of subgroups.

The subgroup lattice has 170 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#11763211	C₂	1	4	yes	maximal
2	#14537147	C₂	1	2	yes	maximal
3	#14528948	C₂	1	4	yes	maximal
4	#14528944	C₂	1	2	yes	maximal
5	#14528941	C₂	1	2	yes	maximal
6	C₄²×C₈×C₁₇	C₂	1	1	yes	maximal, characteristic
7	C₂×#10587556	C₂²	1	4	yes	—
8	#11763211	—	2	16	no	—
9	C₂×#10587489	C₂²	1	8	yes	—
10	C₂×#10587488	C₂²	1	8	yes	—
11	#11762707	—	2	8	no	—
12	#11762679	C₂²	1	4	yes	—
13	#11762675	C₂²	1	4	yes	—
14	C₂×C₄×C₈×C₁₇	C₂²	1	2	yes	—
15	C₂×C₄×C₈×C₁₇	C₂²	1	1	yes	characteristic
16	C₂×C₄×C₈×C₁₇	C₂²	1	1	yes	characteristic
17	C₂×C₄×C₈×C₁₇	C₂²	1	2	yes	—
18	C₄³×C₁₇	C₂²	1	1	yes	characteristic
19	C₂²×C₈×C₁₇	C₂³	1	2	yes	—
20	C₂²×C₈×C₁₇	C₂³	1	2	yes	—
21	C₂×C₄²×C₁₇	D₄	1	4	yes	—
22	C₂×C₄²×C₁₇	C₂³	1	1	yes	characteristic
23	C₂×C₄²×C₁₇	C₂³	1	1	yes	characteristic
24	C₂×C₄²×C₁₇	C₂³	1	1	yes	characteristic
25	C₂×C₄⋊Dic₁₇	—	2	8	no	—
26	C₂×C₄⋊Dic₁₇	C₂³	1	8	yes	—
27	C₂×#63225	—	2	8	no	—
28	#10587556	—	2	16	no	—
29	C₄×C₈×C₁₇	D₄	1	8	yes	—
30	C₄×C₈×C₁₇	Q₈	1	4	yes	—
31	C₄×C₈×C₁₇	Q₈	1	4	yes	—
32	C₄×C₈×C₁₇	Q₈	1	8	yes	—
33	#10587502	—	2	8	no	—
34	#10587489	—	2	16	no	—
35	#10587488	—	2	16	no	—
36	C₂²×C₄×C₁₇	C₂×D₄	1	2	yes	—
37	C₂²×C₄×C₁₇	C₂×D₄	1	2	yes	—
38	C₂²×C₄×C₁₇	C₂×D₄	1	2	yes	—
39	C₂²×C₄×C₁₇	C₂⁴	1	1	yes	characteristic
40	C₂²×Dic₁₇	—	2	8	no	—
41	C₂×C₈×C₁₇	C₂×D₄	1	4	yes	—
42	C₂×C₈×C₁₇	C₂×D₄	1	4	yes	—
43	C₂×C₈×C₁₇	C₂×Q₈	1	4	yes	—
44	C₂×C₈×C₁₇	C₂×Q₈	1	4	yes	—
45	C₂×C₈×C₁₇	C₂×Q₈	1	4	yes	—
46	C₂×C₈×C₁₇	C₂×Q₈	1	4	yes	—
47	C₄²×C₁₇	—	2	8	no	—
48	C₄²×C₁₇	C₂×D₄	1	4	yes	—
49	C₄²×C₁₇	C₂×Q₈	1	4	yes	—
50	C₄²×C₁₇	C₂×Q₈	1	2	yes	—
51	C₄²×C₁₇	C₂×Q₈	1	2	yes	—
52	C₄⋊Dic₁₇	—	2	16	no	—
53	C₄⋊Dic₁₇	—	4	16	no	—
54	#63225	—	4	16	no	—
55	#33311	—	17	1	no	maximal, Sylow
56	C₂³×C₁₇	C₂²×D₄	1	1	yes	characteristic
57	C₂×C₄×C₁₇	—	2	4	no	—
58	C₂×C₄×C₁₇	—	2	4	no	—
59	C₂×C₄×C₁₇	#127	1	4	yes	—
60	C₂×C₄×C₁₇	C₄⋊Q₈	1	4	yes	—
61	C₂×C₄×C₁₇	C₄⋊Q₈	1	4	yes	—
62	C₂×C₄×C₁₇	#135	1	8	yes	—
63	C₂×C₄×C₁₇	C₂²×D₄	1	2	yes	—
64	C₂×C₄×C₁₇	C₂²×Q₈	1	1	yes	characteristic
65	C₂×C₄×C₁₇	C₂²×Q₈	1	1	yes	characteristic
66	C₂×C₄×C₁₇	C₂²×Q₈	1	2	yes	—
67	C₂×Dic₁₇	—	4	8	no	—
68	C₂×Dic₁₇	—	4	16	no	—
69	C₈×C₁₇	C₄⋊Q₈	1	8	yes	—
70	C₈×C₁₇	C₄⋊Q₈	1	8	yes	—
71	C₂×#499	—	17	4	no	—
72	#2859	—	17	2	no	—
73	#1846	—	17	4	no	—
74	#1845	—	17	2	no	—
75	#1844	—	17	2	no	—
76	C₄²×C₈	D₁₇	1	1	yes	characteristic
77	C₂²×C₁₇	C₂×#127	1	1	yes	characteristic
78	C₂²×C₁₇	C₂×C₄⋊Q₈	1	1	yes	characteristic
79	C₂²×C₁₇	C₂×C₄⋊Q₈	1	1	yes	characteristic
80	C₂²×C₁₇	C₂×#135	1	4	yes	—
81	C₄×C₁₇	—	2	4	no	—
82	C₄×C₁₇	—	2	4	no	—
83	C₄×C₁₇	#499	1	8	yes	complemented
84	C₄×C₁₇	C₂×C₄⋊Q₈	1	2	yes	—
85	C₄×C₁₇	C₂×C₄⋊Q₈	1	2	yes	—
86	Dic₁₇	—	8	8	no	—
87	C₂×#125	—	17	4	no	—
88	#499	—	34	16	no	—
89	C₂×#107	—	17	8	no	—
90	C₂×#106	—	17	8	no	—
91	C₂×C₄×C₈	C₂×D₁₇	1	1	yes	characteristic
92	C₂×C₄×C₈	C₂×D₁₇	1	1	yes	characteristic
93	C₂×C₄×C₈	C₂×D₁₇	1	2	yes	—
94	C₂×C₄×C₈	C₂×D₁₇	1	2	yes	—
95	#389	—	34	8	no	—
96	#384	—	17	4	no	—
97	#382	—	17	4	no	—
98	C₄³	C₂×D₁₇	1	1	yes	characteristic
99	C₂×C₁₇	#2868	1	1	yes	characteristic
100	C₂×C₁₇	#3147	1	2	yes	—
101	C₂×C₁₇	C₂×#499	1	2	yes	—
102	C₂×C₁₇	#3160	1	2	yes	—
103	C₂²×C₈	C₂²×D₁₇	1	2	yes	—
104	C₂²×C₈	C₂²×D₁₇	1	2	yes	—
105	#125	—	34	16	no	—
106	C₂×C₄⋊C₄	—	17	8	no	—
107	C₂×C₄⋊C₄	—	34	8	no	—
108	C₂×C₄⋊C₄	—	34	8	no	—
109	C₂×C₄²	D₆₈	1	4	yes	—
110	C₂×C₄²	C₂²×D₁₇	1	1	yes	characteristic
111	C₂×C₄²	C₂²×D₁₇	1	1	yes	characteristic
112	C₂×C₄²	C₂²×D₁₇	1	1	yes	characteristic
113	#107	—	34	16	no	—
114	#106	—	34	16	no	—
115	C₄×C₈	Dic₃₄	1	4	yes	—
116	C₄×C₈	Dic₃₄	1	4	yes	—
117	C₄×C₈	Dic₃₄	1	8	yes	—
118	C₄×C₈	D₆₈	1	8	yes	—
119	#95	—	34	8	no	—
120	C₁₇	#33311	1	1	yes	characteristic, complemented, Sylow
121	C₂²×C₄	—	34	8	no	—
122	C₂²×C₄	C₂×D₆₈	1	2	yes	—
123	C₂²×C₄	C₂×D₆₈	1	2	yes	—
124	C₂²×C₄	C₂×D₆₈	1	2	yes	—
125	C₂²×C₄	C₂³×D₁₇	1	1	yes	characteristic
126	C₂×C₈	C₂×Dic₃₄	1	4	yes	—
127	C₂×C₈	C₂×Dic₃₄	1	4	yes	—
128	C₂×C₈	C₂×Dic₃₄	1	4	yes	—
129	C₂×C₈	C₂×Dic₃₄	1	4	yes	—
130	C₂×C₈	C₂×D₆₈	1	4	yes	—
131	C₂×C₈	C₂×D₆₈	1	4	yes	—
132	C₄⋊C₄	—	34	16	no	—
133	C₄⋊C₄	—	68	16	no	—
134	C₄⋊C₄	—	68	16	no	—
135	C₄²	—	2	8	no	—
136	C₄²	C₂×Dic₃₄	1	4	yes	—
137	C₄²	C₂×Dic₃₄	1	2	yes	—
138	C₄²	C₂×Dic₃₄	1	2	yes	—
139	C₄²	C₂×D₆₈	1	4	yes	—
140	C₂³	C₂²×D₆₈	1	1	yes	characteristic
141	C₂×C₄	—	2	4	no	—
142	C₂×C₄	—	2	4	no	—
143	C₂×C₄	—	68	16	no	—
144	C₂×C₄	—	68	8	no	—
145	C₂×C₄	#10587555	1	4	yes	—
146	C₂×C₄	#10587555	1	4	yes	—
147	C₂×C₄	#10587560	1	4	yes	—
148	C₂×C₄	#10587590	1	8	yes	—
149	C₂×C₄	C₂²×Dic₃₄	1	1	yes	characteristic
150	C₂×C₄	C₂²×Dic₃₄	1	1	yes	characteristic
151	C₂×C₄	C₂²×Dic₃₄	1	2	yes	—
152	C₂×C₄	C₂²×D₆₈	1	2	yes	—
153	C₈	#10587555	1	8	yes	—
154	C₈	#10587555	1	8	yes	—
155	C₂²	C₂×#10587560	1	1	yes	characteristic
156	C₂²	C₂×#10587555	1	1	yes	characteristic
157	C₂²	C₂×#10587555	1	1	yes	characteristic
158	C₂²	C₂×#10587590	1	4	yes	—
159	C₄	—	2	4	no	—
160	C₄	—	2	4	no	—
161	C₄	—	136	8	no	—
162	C₄	#11763211	1	8	yes	complemented
163	C₄	C₂×#10587555	1	2	yes	—
164	C₄	C₂×#10587555	1	2	yes	—
165	C₂	#14537182	1	1	yes	characteristic
166	C₂	#14539212	1	2	yes	—
167	C₂	C₂×#11763211	1	2	yes	—
168	C₂	#14539298	1	2	yes	—

	subgroup	G/H
center	C₂³	C₂²×D₆₈
commutator, cosocle	C₂²×C₄×C₁₇	C₂⁴
Frattini	C₂²×C₄	C₂³×D₁₇
Fitting	C₄²×C₈×C₁₇	C₂
socle, prime radical	C₂³×C₁₇	C₂²×D₄
hypercenter	C₄²×C₈	D₁₇
hypocenter	C₁₇	#33311
LCS	G ▷ C₂²×C₄×C₁₇ ▷ C₂×C₁₇ ▷ C₁₇
UCS	C₁ ◁ C₂³ ◁ C₄³ ◁ C₄²×C₈

Complements

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

N	H ≅ G/N	type
83. C₄×C₁₇	#499	semidirect
120. C₁₇	#33311	semidirect
162. C₄	#11763211	semidirect

Maximal subgroups

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

subgroup	index
C₂×#11763211	2
#14537147	2
#14528948	2
#14528944	2
#14528941	2
C₄²×C₈×C₁₇	2
#33311	17