Subgroups — #16950780

Order 4352.

266 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	C₂×#11762639	C₂	1	1	yes	maximal, characteristic
2	#14530568	C₂	1	1	yes	maximal, characteristic
3	#14530557	C₂	1	2	yes	maximal
4	#14528908	C₂	1	2	yes	maximal
5	C₁₇×#2108	C₂	1	1	yes	maximal, characteristic
6	C₂×#10587490	C₂²	1	2	yes	—
7	#11762847	C₂²	1	2	yes	—
8	#11762639	—	2	1	no	—
9	#11762639	—	2	1	no	—
10	#11762637	—	2	1	no	—
11	#11762635	—	2	1	no	—
12	#11762632	—	2	4	no	—
13	#11762580	—	2	2	no	—
14	#11762565	—	2	2	no	—
15	C₂×C₁₇×M₃₂	C₂²	1	1	yes	characteristic
16	C₂²×C₁₆×C₁₇	C₂²	1	1	yes	characteristic
17	C₁₇×#405	C₂²	1	1	yes	characteristic
18	C₁₇×#346	C₄	1	2	yes	complemented
19	C₄×C₁₆×C₁₇	C₄	1	2	yes	complemented
20	C₂×M₁₆×C₁₇	D₄	1	1	yes	characteristic
21	C₂²×C₈×C₁₇	C₂³	1	1	yes	characteristic
22	C₁₇×#117	D₄	1	1	yes	characteristic
23	#10587612	—	2	2	no	—
24	C₂×#63223	—	2	2	no	—
25	C₁₇×M₃₂	D₄	1	2	yes	—
26	C₁₇×M₃₂	Q₈	1	2	yes	—
27	C₂×C₁₆×C₁₇	C₂×C₄	1	1	yes	characteristic
28	C₂×C₁₆×C₁₇	C₂×C₄	1	1	yes	characteristic
29	C₂×C₁₆×C₁₇	C₂×C₄	1	2	yes	—
30	C₂×C₁₆×C₁₇	D₄	1	1	yes	characteristic
31	C₂×C₁₆×C₁₇	D₄	1	1	yes	characteristic
32	C₂×C₁₆×C₁₇	Q₈	1	1	yes	characteristic
33	C₂×C₁₆×C₁₇	Q₈	1	1	yes	characteristic
34	C₁₇×#97	Q₈	1	2	yes	—
35	C₄×C₈×C₁₇	C₂×C₄	1	2	yes	—
36	#10587490	—	2	2	no	—
37	#10587490	—	2	2	no	—
38	#10587489	—	2	2	no	—
39	#10587488	—	2	2	no	—
40	C₂²×C₄×C₁₇	C₂×D₄	1	1	yes	characteristic
41	M₁₆×C₁₇	—	2	1	no	—
42	M₁₆×C₁₇	—	2	1	no	—
43	C₂×C₈×C₁₇	C₄⋊C₄	1	2	yes	—
44	C₂×C₈×C₁₇	C₂²×C₄	1	1	yes	characteristic
45	C₂×C₈×C₁₇	C₂×D₄	1	1	yes	characteristic
46	C₂×C₈×C₁₇	C₂×D₄	1	1	yes	characteristic
47	C₂×C₈×C₁₇	C₂×Q₈	1	1	yes	characteristic
48	C₂×C₈×C₁₇	C₂×Q₈	1	1	yes	characteristic
49	C₂×C₈×C₁₇	C₂×Q₈	1	1	yes	characteristic
50	C₄⋊C₄×C₁₇	—	2	1	no	—
51	C₂²⋊C₄×C₁₇	—	2	1	no	—
52	C₄²×C₁₇	C₄⋊C₄	1	2	yes	—
53	C₂²⋊Dic₁₇	—	4	2	no	—
54	C₄⋊Dic₁₇	—	4	2	no	—
55	C₄×Dic₁₇	—	4	2	no	—
56	#63223	—	4	2	no	—
57	#63223	—	4	2	no	—
58	C₂×#3622	—	4	2	no	—
59	C₁₆×C₁₇	C₄⋊C₄	1	1	yes	characteristic
60	C₁₆×C₁₇	C₄⋊C₄	1	1	yes	characteristic
61	C₁₆×C₁₇	C₄⋊C₄	1	1	yes	characteristic
62	C₁₆×C₁₇	C₄⋊C₄	1	1	yes	characteristic
63	C₁₆×C₁₇	C₄⋊C₄	1	4	yes	—
64	#12911	—	17	1	no	maximal, Sylow
65	C₂³×C₁₇	#135	1	1	yes	characteristic
66	C₂×C₄×C₁₇	—	2	2	no	—
67	C₂×C₄×C₁₇	C₂×C₄⋊C₄	1	1	yes	characteristic
68	C₂×C₄×C₁₇	#127	1	1	yes	characteristic
69	C₂×C₄×C₁₇	C₄⋊Q₈	1	1	yes	characteristic
70	C₂×C₄×C₁₇	#135	1	1	yes	characteristic
71	C₂×C₄×C₁₇	#135	1	1	yes	characteristic
72	C₂×C₄×C₁₇	#135	1	1	yes	characteristic
73	C₂×Dic₁₇	—	8	2	no	—
74	C₈×C₁₇	—	2	2	no	—
75	C₈×C₁₇	C₂×C₄⋊C₄	1	1	yes	characteristic
76	C₈×C₁₇	C₂×C₄⋊C₄	1	1	yes	characteristic
77	C₈×C₁₇	C₄⋊Q₈	1	1	yes	characteristic
78	C₈×C₁₇	C₄⋊Q₈	1	1	yes	characteristic
79	#3622	—	8	2	no	—
80	C₂×#368	—	17	1	no	—
81	#2158	—	17	1	no	—
82	#2155	—	17	2	no	—
83	#2108	D₁₇	1	1	yes	characteristic
84	#1833	—	17	2	no	—
85	C₂²×C₁₇	—	4	1	no	—
86	C₂²×C₁₇	#427	1	1	yes	characteristic
87	C₂²×C₁₇	#495	1	1	yes	characteristic
88	C₂²×C₁₇	#499	1	1	yes	characteristic
89	C₄×C₁₇	—	4	2	no	—
90	C₄×C₁₇	#384	1	1	yes	characteristic
91	C₄×C₁₇	#427	1	1	yes	characteristic
92	C₄×C₁₇	#495	1	1	yes	characteristic
93	C₄×C₁₇	#499	1	1	yes	characteristic
94	Dic₁₇	—	16	2	no	—
95	C₂×M₃₂	C₂×D₁₇	1	1	yes	characteristic
96	C₂²×C₁₆	C₂×D₁₇	1	1	yes	characteristic
97	C₂×#108	—	17	2	no	—
98	#427	—	17	2	no	—
99	#405	C₂×D₁₇	1	1	yes	characteristic
100	#368	—	34	1	no	—
101	#368	—	34	1	no	—
102	#367	—	34	1	no	—
103	#366	—	34	1	no	—
104	#365	—	34	4	no	—
105	#346	Dic₁₇	1	2	yes	complemented
106	C₄×C₁₆	Dic₁₇	1	2	yes	complemented
107	#344	—	34	2	no	—
108	H(3,ℤ/4)	—	34	2	no	—
109	C₂×C₁₇	—	2	1	no	—
110	C₂×C₁₇	—	2	1	no	—
111	C₂×C₁₇	—	2	1	no	—
112	C₂×C₁₇	#1846	1	1	yes	characteristic
113	C₂×M₁₆	—	34	2	no	—
114	C₂×M₁₆	D₆₈	1	1	yes	characteristic
115	C₂²×C₈	C₂²×D₁₇	1	1	yes	characteristic
116	#117	—	34	2	no	—
117	#117	D₆₈	1	1	yes	characteristic
118	M₃₂	Dic₃₄	1	2	yes	—
119	M₃₂	D₆₈	1	2	yes	—
120	C₂×C₁₆	Dic₃₄	1	1	yes	characteristic
121	C₂×C₁₆	Dic₃₄	1	1	yes	characteristic
122	C₂×C₁₆	D₆₈	1	1	yes	characteristic
123	C₂×C₁₆	D₆₈	1	1	yes	characteristic
124	C₂×C₁₆	C₂×Dic₁₇	1	1	yes	characteristic
125	C₂×C₁₆	C₂×Dic₁₇	1	2	yes	—
126	C₂×C₁₆	C₂×Dic₁₇	1	1	yes	characteristic
127	#108	—	34	2	no	—
128	#108	—	34	2	no	—
129	#107	—	34	2	no	—
130	#106	—	34	2	no	—
131	#97	Dic₃₄	1	2	yes	—
132	C₄×C₈	C₂×Dic₁₇	1	2	yes	—
133	C₁₇	#12911	1	1	yes	characteristic, complemented, Sylow
134	C₂²×C₄	C₂×D₆₈	1	1	yes	characteristic
135	M₁₆	—	2	1	no	—
136	M₁₆	—	2	1	no	—
137	M₁₆	—	68	2	no	—
138	M₁₆	—	68	2	no	—
139	C₂×C₈	—	68	2	no	—
140	C₂×C₈	C₄⋊Dic₁₇	1	2	yes	—
141	C₂×C₈	C₂×Dic₃₄	1	1	yes	characteristic
142	C₂×C₈	C₂×Dic₃₄	1	1	yes	characteristic
143	C₂×C₈	C₂×Dic₃₄	1	1	yes	characteristic
144	C₂×C₈	C₂×D₆₈	1	1	yes	characteristic
145	C₂×C₈	C₂×D₆₈	1	1	yes	characteristic
146	C₂×C₈	C₂²×Dic₁₇	1	1	yes	characteristic
147	C₄⋊C₄	—	2	1	no	—
148	C₄⋊C₄	—	68	2	no	—
149	C₂²⋊C₄	—	2	1	no	—
150	C₂²⋊C₄	—	68	2	no	—
151	C₄²	—	68	2	no	—
152	C₄²	C₄⋊Dic₁₇	1	2	yes	—
153	C₁₆	C₄⋊Dic₁₇	1	1	yes	characteristic
154	C₁₆	C₄⋊Dic₁₇	1	1	yes	characteristic
155	C₁₆	C₄⋊Dic₁₇	1	1	yes	characteristic
156	C₁₆	C₄⋊Dic₁₇	1	1	yes	characteristic
157	C₁₆	C₄⋊Dic₁₇	1	4	yes	—
158	C₂³	#10587590	1	1	yes	characteristic
159	C₂×C₄	—	2	2	no	—
160	C₂×C₄	—	136	2	no	—
161	C₂×C₄	#10587555	1	1	yes	characteristic
162	C₂×C₄	#10587560	1	1	yes	characteristic
163	C₂×C₄	#10587590	1	1	yes	characteristic
164	C₂×C₄	#10587590	1	1	yes	characteristic
165	C₂×C₄	#10587590	1	1	yes	characteristic
166	C₂×C₄	C₂×C₄⋊Dic₁₇	1	1	yes	characteristic
167	C₈	—	2	2	no	—
168	C₈	—	136	2	no	—
169	C₈	#10587555	1	1	yes	characteristic
170	C₈	#10587555	1	1	yes	characteristic
171	C₈	C₂×C₄⋊Dic₁₇	1	1	yes	characteristic
172	C₈	C₂×C₄⋊Dic₁₇	1	1	yes	characteristic
173	C₂²	—	4	1	no	—
174	C₂²	#11762847	1	1	yes	characteristic
175	C₂²	#11763193	1	1	yes	characteristic
176	C₂²	#11763211	1	1	yes	characteristic
177	C₄	—	4	2	no	—
178	C₄	—	272	2	no	—
179	C₄	#11762679	1	1	yes	characteristic
180	C₄	#11762847	1	1	yes	characteristic
181	C₄	#11763193	1	1	yes	characteristic
182	C₄	#11763211	1	1	yes	characteristic
183	C₂	—	2	1	no	—
184	C₂	—	2	1	no	—
185	C₂	—	2	1	no	—
186	C₂	#14528948	1	1	yes	characteristic

	subgroup	G/H
center	C₄	#11762847
commutator	C₂×C₈×C₁₇	C₂²×C₄
Frattini	C₂²×C₈	C₂²×D₁₇
Fitting	C₁₇×#2108	C₂
socle	C₂×C₁₇	#1846
prime radical	C₂³×C₁₇	#135
cosocle	C₂²×C₈×C₁₇	C₂³
hypercenter	#2108	D₁₇
hypocenter	C₁₇	#12911
LCS	G ▷ C₂×C₈×C₁₇ ▷ C₄×C₁₇ ▷ C₂×C₁₇ ▷ C₁₇
UCS	C₁ ◁ C₄ ◁ #117 ◁ #405 ◁ #2108

Complements

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

N	H ≅ G/N	type
18. C₁₇×#346	C₄	semidirect
19. C₄×C₁₆×C₁₇	C₄	semidirect
105. #346	Dic₁₇	semidirect
106. C₄×C₁₆	Dic₁₇	semidirect
133. C₁₇	#12911	semidirect

Maximal subgroups

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

subgroup	index
C₂×#11762639	2
#14530568	2
#14530557	2
#14528908	2
C₁₇×#2108	2
#12911	17