Subgroups — #17187373

Order 4352.

658 conjugacy classes of subgroups.

The subgroup lattice has 156 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#14537384	C₂	1	2	yes	maximal
2	#14530572	C₂	1	6	yes	maximal, complemented
3	#14530568	C₂	1	3	yes	maximal, complemented
4	C₂×#11762637	C₂	1	3	yes	maximal, complemented
5	C₂×C₁₇×#504	C₂	1	1	yes	maximal, characteristic
6	#11763292	—	2	2	no	—
7	#11762851	—	2	6	no	—
8	#11762847	C₂²	1	6	yes	—
9	C₂×#10587489	C₂²	1	6	yes	—
10	#11762637	C₂²	1	3	yes	—
11	#11762637	C₂²	1	6	yes	—
12	#11762637	C₂²	1	1	yes	characteristic
13	#11762635	C₂²	1	6	yes	—
14	C₂×C₁₇×#131	C₂²	1	1	yes	characteristic
15	C₁₇×#504	C₄	1	8	yes	complemented
16	C₂×C₁₇×M₃₂	C₂²	1	3	yes	—
17	C₂²×C₁₆×C₁₇	C₂²	1	3	yes	—
18	C₂×#41×C₁₇	D₄	1	1	yes	characteristic
19	C₁₇×#131	C₂×C₄	1	4	yes	—
20	C₁₇×#131	Q₈	1	4	yes	—
21	C₂×M₁₆×C₁₇	D₄	1	3	yes	—
22	C₂²×C₈×C₁₇	C₂³	1	3	yes	—
23	Q₈×Dic₁₇	—	4	2	no	—
24	D₄×Dic₁₇	—	4	6	no	—
25	#10587612	—	2	6	no	—
26	C₂×C₄⋊Dic₁₇	—	2	6	no	—
27	C₁₇×M₃₂	C₂×C₄	1	12	yes	complemented
28	C₂×C₁₆×C₁₇	C₂×C₄	1	12	yes	complemented
29	C₂×C₁₆×C₁₇	C₂³	1	1	yes	characteristic
30	C₂×C₁₆×C₁₇	C₂³	1	3	yes	—
31	#10587489	—	2	6	no	—
32	#10587489	C₂³	1	6	yes	—
33	#10587489	C₂³	1	2	yes	—
34	#10587488	—	2	6	no	—
35	#41×C₁₇	C₄⋊C₄	1	4	yes	—
36	#41×C₁₇	Q₁₆	1	4	yes	—
37	C₂×Q₈×C₁₇	D₈	1	1	yes	characteristic
38	C₂×D₄×C₁₇	D₈	1	3	yes	—
39	C₂²×C₄×C₁₇	C₂×D₄	1	3	yes	—
40	C₂²×Dic₁₇	—	4	6	no	—
41	M₁₆×C₁₇	C₄⋊C₄	1	12	yes	—
42	C₂×C₈×C₁₇	C₂²×C₄	1	6	yes	—
43	C₂×C₈×C₁₇	C₂×D₄	1	3	yes	—
44	C₂×C₈×C₁₇	C₂×Q₈	1	6	yes	—
45	C₂×C₈×C₁₇	C₂⁴	1	1	yes	characteristic
46	C₂²⋊Dic₁₇	—	4	6	no	—
47	C₄⋊Dic₁₇	—	2	2	no	—
48	C₄⋊Dic₁₇	—	2	6	no	—
49	C₄⋊Dic₁₇	—	4	6	no	—
50	C₄×Dic₁₇	—	4	6	no	—
51	C₁₆×C₁₇	C₂²×C₄	1	2	yes	—
52	C₁₆×C₁₇	C₂²×C₄	1	6	yes	—
53	#33411	—	17	1	no	maximal, Sylow
54	C₂³×C₁₇	C₂×D₈	1	3	yes	—
55	Q₈×C₁₇	#107	1	4	yes	—
56	D₄×C₁₇	#107	1	12	yes	—
57	C₂×C₄×C₁₇	C₂×C₄⋊C₄	1	6	yes	—
58	C₂×C₄×C₁₇	C₂×D₈	1	3	yes	—
59	C₂×C₄×C₁₇	C₂×Q₁₆	1	6	yes	—
60	C₂×C₄×C₁₇	C₂²×D₄	1	1	yes	characteristic
61	C₂×Dic₁₇	—	4	2	no	—
62	C₂×Dic₁₇	—	4	6	no	—
63	C₂×Dic₁₇	—	16	6	no	—
64	C₈×C₁₇	C₂×C₄⋊C₄	1	6	yes	—
65	C₈×C₁₇	C₂³×C₄	1	1	yes	characteristic
66	C₈×C₁₇	C₂²×Q₈	1	1	yes	characteristic
67	C₂×#504	D₁₇	1	1	yes	characteristic
68	#2913	—	17	2	no	—
69	#2159	—	17	6	no	—
70	#2158	—	17	3	no	—
71	C₂×#367	—	17	3	no	—
72	C₂²×C₁₇	—	2	6	no	—
73	C₂²×C₁₇	C₂×#107	1	6	yes	—
74	C₂²×C₁₇	C₂²×D₈	1	1	yes	characteristic
75	C₄×C₁₇	C₂×#107	1	6	yes	—
76	C₄×C₁₇	C₂²×C₄⋊C₄	1	1	yes	characteristic
77	C₄×C₁₇	C₂²×Q₁₆	1	1	yes	characteristic
78	Dic₁₇	—	8	2	no	—
79	Dic₁₇	—	8	6	no	—
80	C₂×#131	C₂×D₁₇	1	1	yes	characteristic
81	#520	—	34	2	no	—
82	#504	Dic₁₇	1	8	yes	complemented
83	C₂×M₃₂	C₂×D₁₇	1	3	yes	—
84	C₂²×C₁₆	C₂×D₁₇	1	3	yes	—
85	#428	—	34	6	no	—
86	#427	—	17	6	no	—
87	C₂×#107	—	17	6	no	—
88	#367	—	17	3	no	—
89	#367	—	17	6	no	—
90	#367	—	17	1	no	—
91	#366	—	17	6	no	—
92	C₂×C₁₇	—	2	6	no	—
93	C₂×C₁₇	C₂²×#107	1	1	yes	characteristic
94	C₂×C₁₇	#3416	1	2	yes	—
95	C₂×#41	D₆₈	1	1	yes	characteristic
96	#131	Dic₃₄	1	4	yes	—
97	#131	C₂×Dic₁₇	1	4	yes	—
98	C₂×M₁₆	D₆₈	1	3	yes	—
99	C₂²×C₈	C₂²×D₁₇	1	3	yes	—
100	C₄×Q₈	—	68	2	no	—
101	C₄×D₄	—	68	6	no	—
102	#117	—	34	6	no	—
103	C₂×C₄⋊C₄	—	34	6	no	—
104	M₃₂	C₂×Dic₁₇	1	12	yes	complemented
105	C₂×C₁₆	C₂×Dic₁₇	1	12	yes	complemented
106	C₂×C₁₆	C₂²×D₁₇	1	3	yes	—
107	C₂×C₁₆	C₂²×D₁₇	1	1	yes	characteristic
108	#107	—	17	2	no	—
109	#107	—	17	6	no	—
110	#107	—	34	6	no	—
111	#106	—	34	6	no	—
112	C₁₇	#33411	1	1	yes	characteristic, complemented, Sylow
113	#41	Dic₆₈	1	4	yes	—
114	#41	C₄⋊Dic₁₇	1	4	yes	—
115	C₂×Q₈	D₁₃₆	1	1	yes	characteristic
116	C₂×D₄	D₁₃₆	1	3	yes	—
117	C₂²×C₄	—	68	6	no	—
118	C₂²×C₄	C₂×D₆₈	1	3	yes	—
119	M₁₆	C₄⋊Dic₁₇	1	12	yes	—
120	C₂×C₈	C₂×Dic₃₄	1	6	yes	—
121	C₂×C₈	C₂×D₆₈	1	3	yes	—
122	C₂×C₈	C₂²×Dic₁₇	1	6	yes	—
123	C₂×C₈	C₂³×D₁₇	1	1	yes	characteristic
124	C₄⋊C₄	—	34	2	no	—
125	C₄⋊C₄	—	34	6	no	—
126	C₄⋊C₄	—	68	6	no	—
127	C₂²⋊C₄	—	68	6	no	—
128	C₄²	—	68	6	no	—
129	C₁₆	C₂²×Dic₁₇	1	2	yes	—
130	C₁₆	C₂²×Dic₁₇	1	6	yes	—
131	C₂³	C₂×D₁₃₆	1	3	yes	—
132	Q₈	#10587489	1	4	yes	—
133	D₄	#10587489	1	12	yes	—
134	C₂×C₄	—	272	6	no	—
135	C₂×C₄	—	68	2	no	—
136	C₂×C₄	—	68	6	no	—
137	C₂×C₄	C₂×D₁₃₆	1	3	yes	—
138	C₂×C₄	C₂×Dic₆₈	1	6	yes	—
139	C₂×C₄	C₂×C₄⋊Dic₁₇	1	6	yes	—
140	C₂×C₄	C₂²×D₆₈	1	1	yes	characteristic
141	C₈	C₂×C₄⋊Dic₁₇	1	6	yes	—
142	C₈	C₂²×Dic₃₄	1	1	yes	characteristic
143	C₈	C₂³×Dic₁₇	1	1	yes	characteristic
144	C₂²	—	2	6	no	—
145	C₂²	C₂×#10587489	1	6	yes	—
146	C₂²	C₂²×D₁₃₆	1	1	yes	characteristic
147	C₄	—	136	6	no	—
148	C₄	—	136	2	no	—
149	C₄	C₂×#10587489	1	6	yes	—
150	C₄	C₂²×C₄⋊Dic₁₇	1	1	yes	characteristic
151	C₄	C₂²×Dic₆₈	1	1	yes	characteristic
152	C₂	—	2	6	no	—
153	C₂	C₂²×#10587489	1	1	yes	characteristic
154	C₂	#14541932	1	2	yes	—

	subgroup	G/H
center	C₂²	C₂²×D₁₃₆
commutator	C₈×C₁₇	C₂³×C₄
Frattini	C₂×C₈	C₂³×D₁₇
Fitting	C₂×C₁₇×#504	C₂
socle	C₂²×C₁₇	C₂²×D₈
prime radical	C₂×#41×C₁₇	D₄
cosocle	C₂×C₈×C₁₇	C₂⁴
hypercenter	C₂×#504	D₁₇
hypocenter	C₁₇	#33411
LCS	G ▷ C₈×C₁₇ ▷ C₄×C₁₇ ▷ C₂×C₁₇ ▷ C₁₇
UCS	C₁ ◁ C₂² ◁ C₂×#41 ◁ C₂×#131 ◁ C₂×#504

Complements

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

N	H ≅ G/N	type
2. #14530572	C₂	semidirect
3. #14530568	C₂	semidirect
4. C₂×#11762637	C₂	semidirect
15. C₁₇×#504	C₄	semidirect
27. C₁₇×M₃₂	C₂×C₄	semidirect
28. C₂×C₁₆×C₁₇	C₂×C₄	semidirect
82. #504	Dic₁₇	semidirect
104. M₃₂	C₂×Dic₁₇	semidirect
105. C₂×C₁₆	C₂×Dic₁₇	semidirect
112. C₁₇	#33411	semidirect

Maximal subgroups

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

subgroup	index
#14537384	2
#14530572	2
#14530568	2
C₂×#11762637	2
C₂×C₁₇×#504	2
#33411	17