Subgroups — #16952941

Order 4352.

342 conjugacy classes of subgroups.

The subgroup lattice has 196 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#14539298	C₂	1	1	yes	maximal, characteristic
2	#14528956	C₂	1	1	yes	maximal, characteristic
3	#14526880	C₂	1	4	yes	maximal
4	C₂×C₁₇×#346	C₂	1	1	yes	maximal, characteristic
5	#11763366	—	2	1	no	—
6	#11763214	C₄	1	2	yes	—
7	#11763211	—	2	1	no	—
8	#11763211	C₄	1	2	yes	—
9	#11763207	—	2	1	no	—
10	C₂×#10587490	—	2	1	no	—
11	C₂×#10587490	C₂²	1	2	yes	—
12	#11762847	—	2	1	no	—
13	#11762847	C₂²	1	2	yes	—
14	#11762833	—	2	1	no	—
15	C₂²×C₁₆×C₁₇	C₂²	1	2	yes	—
16	C₂×C₄×C₈×C₁₇	C₂²	1	1	yes	characteristic
17	C₁₇×#346	—	2	1	no	—
18	C₁₇×#346	—	2	1	no	—
19	C₂²×C₈×C₁₇	D₄	1	1	yes	characteristic
20	C₂²×C₈×C₁₇	C₂³	1	1	yes	characteristic
21	C₂×C₄²×C₁₇	D₄	1	1	yes	characteristic
22	#10587612	—	2	2	no	—
23	#10587610	—	4	2	no	—
24	C₂×#63223	—	2	2	no	—
25	#10587556	—	2	2	no	—
26	#10587555	—	2	2	no	—
27	C₄×Dic₃₄	—	4	2	no	—
28	C₂×C₁₆×C₁₇	—	2	2	no	—
29	C₂×C₁₆×C₁₇	—	2	2	no	—
30	C₂×C₁₆×C₁₇	Q₈	1	4	yes	complemented
31	C₄×C₈×C₁₇	C₂×C₄	1	1	yes	characteristic
32	C₄×C₈×C₁₇	C₂×C₄	1	1	yes	characteristic
33	C₄×C₈×C₁₇	D₄	1	1	yes	characteristic
34	C₄×C₈×C₁₇	D₄	1	1	yes	characteristic
35	#10587490	—	2	4	no	—
36	#10587490	—	4	2	no	—
37	#10587489	—	2	2	no	—
38	#10587489	C₂×C₄	1	4	yes	—
39	#10587488	—	2	2	no	—
40	#10587488	—	2	2	no	—
41	C₄⋊#3622	—	4	2	no	—
42	C₂²×C₄×C₁₇	SD₁₆	1	2	yes	—
43	C₂²×C₄×C₁₇	C₂×D₄	1	1	yes	characteristic
44	C₂×Dic₃₄	—	4	2	no	—
45	C₂×C₈×C₁₇	C₂²⋊C₄	1	1	yes	characteristic
46	C₂×C₈×C₁₇	C₂²⋊C₄	1	1	yes	characteristic
47	C₂×C₈×C₁₇	SD₁₆	1	2	yes	—
48	C₂×C₈×C₁₇	Q₁₆	1	2	yes	—
49	C₂×C₈×C₁₇	C₂²×C₄	1	1	yes	characteristic
50	C₂×C₈×C₁₇	C₂×D₄	1	1	yes	characteristic
51	C₂×C₈×C₁₇	C₂×Q₈	1	2	yes	—
52	C₂×C₈×C₁₇	#41	1	2	yes	—
53	C₄²×C₁₇	C₂²⋊C₄	1	1	yes	characteristic
54	C₄²×C₁₇	C₂²⋊C₄	1	1	yes	characteristic
55	C₄²×C₁₇	D₈	1	2	yes	—
56	C₂²⋊Dic₁₇	—	4	2	no	—
57	C₄⋊Dic₁₇	—	2	4	no	—
58	C₄⋊Dic₁₇	—	4	2	no	—
59	#63225	—	4	4	no	—
60	C₄×Dic₁₇	—	4	2	no	—
61	#63223	—	4	2	no	—
62	#63223	—	4	2	no	—
63	C₂×#3622	—	4	2	no	—
64	C₁₆×C₁₇	—	2	4	no	—
65	#13286	—	17	1	no	maximal, Sylow
66	C₂³×C₁₇	C₂×SD₁₆	1	1	yes	characteristic
67	C₂×C₄×C₁₇	—	4	2	no	—
68	C₂×C₄×C₁₇	#102	1	2	yes	—
69	C₂×C₄×C₁₇	#102	1	2	yes	—
70	C₂×C₄×C₁₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
71	C₂×C₄×C₁₇	#124	1	1	yes	characteristic
72	C₂×C₄×C₁₇	C₄⋊Q₈	1	1	yes	characteristic
73	C₂×C₄×C₁₇	C₂×D₈	1	1	yes	characteristic
74	C₂×C₄×C₁₇	C₂×SD₁₆	1	1	yes	characteristic
75	C₂×C₄×C₁₇	C₂×Q₁₆	1	1	yes	characteristic
76	C₂×Dic₁₇	—	4	4	no	—
77	Dic₃₄	—	8	4	no	—
78	C₈×C₁₇	#103	1	2	yes	—
79	C₈×C₁₇	#103	1	2	yes	—
80	C₈×C₁₇	C₄×Q₈	1	2	yes	—
81	C₈×C₁₇	C₂²⋊Q₈	1	2	yes	—
82	#3622	—	8	2	no	—
83	#3160	—	17	1	no	—
84	C₂×#346	D₁₇	1	1	yes	characteristic
85	#1848	—	17	1	no	—
86	#1382	—	17	4	no	—
87	C₂²×C₁₇	—	2	2	no	—
88	C₂²×C₁₇	C₂×#102	1	1	yes	characteristic
89	C₂²×C₁₇	#487	1	1	yes	characteristic
90	C₂²×C₁₇	#498	1	1	yes	characteristic
91	C₄×C₁₇	—	2	2	no	—
92	C₄×C₁₇	—	2	2	no	—
93	C₄×C₁₇	#391	1	1	yes	characteristic
94	C₄×C₁₇	C₂×#103	1	1	yes	characteristic
95	C₄×C₁₇	#486	1	1	yes	characteristic
96	C₄×C₁₇	#500	1	1	yes	characteristic
97	Dic₁₇	—	8	4	no	—
98	#533	—	34	1	no	—
99	C₂²×C₁₆	C₂×D₁₇	1	2	yes	—
100	#500	—	17	2	no	—
101	#499	—	17	2	no	—
102	#499	—	34	1	no	—
103	#498	—	34	1	no	—
104	C₂×#108	—	17	2	no	—
105	C₂×#108	—	34	1	no	—
106	#427	—	17	2	no	—
107	#427	—	34	1	no	—
108	#423	—	34	1	no	—
109	C₂×C₄×C₈	C₂×D₁₇	1	1	yes	characteristic
110	#346	—	2	1	no	—
111	#346	—	2	1	no	—
112	C₂×C₁₇	—	2	1	no	—
113	C₂×C₁₇	—	2	1	no	—
114	C₂×C₁₇	#1990	1	1	yes	characteristic
115	C₂×C₁₇	#2143	1	2	yes	—
116	C₂×M₁₆	—	34	2	no	—
117	C₂²×C₈	D₆₈	1	1	yes	characteristic
118	C₂²×C₈	C₂²×D₁₇	1	1	yes	characteristic
119	C₄⋊Q₈	—	34	2	no	—
120	#125	—	34	2	no	—
121	C₂²⋊Q₈	—	68	2	no	—
122	C₄×Q₈	—	68	2	no	—
123	#117	—	34	2	no	—
124	C₂×C₄²	D₆₈	1	1	yes	characteristic
125	C₂×C₁₆	—	2	2	no	—
126	C₂×C₁₆	—	2	2	no	—
127	C₂×C₁₆	Dic₃₄	1	4	yes	complemented
128	#108	—	34	4	no	—
129	#108	—	68	2	no	—
130	#107	—	17	4	no	—
131	#107	—	34	2	no	—
132	#106	—	34	2	no	—
133	#106	—	34	2	no	—
134	C₄⋊C₈	—	68	2	no	—
135	C₄×C₈	C₄×D₁₇	1	1	yes	characteristic
136	C₄×C₈	C₄×D₁₇	1	1	yes	characteristic
137	C₄×C₈	C₂²⋊D₁₇	1	1	yes	characteristic
138	C₄×C₈	C₂²⋊D₁₇	1	1	yes	characteristic
139	C₁₇	#13286	1	1	yes	characteristic, complemented, Sylow
140	C₂×Q₈	—	68	2	no	—
141	C₂²×C₄	#63219	1	2	yes	—
142	C₂²×C₄	C₂×D₆₈	1	1	yes	characteristic
143	M₁₆	—	68	2	no	—
144	M₁₆	—	68	2	no	—
145	C₂×C₈	—	68	2	no	—
146	C₂×C₈	#63219	1	2	yes	—
147	C₂×C₈	Dic₆₈	1	2	yes	—
148	C₂×C₈	#63227	1	1	yes	characteristic
149	C₂×C₈	#63227	1	1	yes	characteristic
150	C₂×C₈	C₂×Dic₃₄	1	2	yes	—
151	C₂×C₈	C₂×C₄×D₁₇	1	1	yes	characteristic
152	C₂×C₈	#63252	1	2	yes	—
153	C₂×C₈	C₂×C₂²⋊D₁₇	1	1	yes	characteristic
154	C₄⋊C₄	—	34	4	no	—
155	C₄⋊C₄	—	68	4	no	—
156	C₄⋊C₄	—	68	2	no	—
157	C₂²⋊C₄	—	68	2	no	—
158	C₄²	—	68	2	no	—
159	C₄²	D₁₃₆	1	2	yes	—
160	C₄²	#63227	1	1	yes	characteristic
161	C₄²	#63227	1	1	yes	characteristic
162	C₁₆	—	2	4	no	—
163	C₂³	C₂×#63219	1	1	yes	characteristic
164	Q₈	—	136	4	no	—
165	C₂×C₄	—	4	2	no	—
166	C₂×C₄	—	68	4	no	—
167	C₂×C₄	#10587492	1	2	yes	—
168	C₂×C₄	#10587492	1	2	yes	—
169	C₂×C₄	#10587555	1	1	yes	characteristic
170	C₂×C₄	#10587561	1	1	yes	characteristic
171	C₂×C₄	C₂×#63219	1	1	yes	characteristic
172	C₂×C₄	C₂×D₁₃₆	1	1	yes	characteristic
173	C₂×C₄	C₂×Dic₆₈	1	1	yes	characteristic
174	C₂×C₄	C₂×#63227	1	1	yes	characteristic
175	C₈	—	136	2	no	—
176	C₈	#10587487	1	2	yes	—
177	C₈	#10587487	1	2	yes	—
178	C₈	C₄×Dic₃₄	1	2	yes	—
179	C₈	#10587610	1	2	yes	—
180	C₂²	—	2	2	no	—
181	C₂²	C₂×#10587492	1	1	yes	characteristic
182	C₂²	#11763160	1	1	yes	characteristic
183	C₂²	#11763207	1	1	yes	characteristic
184	C₄	—	2	2	no	—
185	C₄	—	2	2	no	—
186	C₄	—	136	4	no	—
187	C₄	#11762715	1	1	yes	characteristic
188	C₄	C₂×#10587487	1	1	yes	characteristic
189	C₄	#11763156	1	1	yes	characteristic
190	C₄	#11763214	1	1	yes	characteristic
191	C₂	—	2	1	no	—
192	C₂	—	2	1	no	—
193	C₂	#14529745	1	1	yes	characteristic
194	C₂	#14530513	1	2	yes	—

	subgroup	G/H
center	C₂×C₄	C₂×D₁₃₆
commutator	C₂×C₈×C₁₇	C₂²×C₄
Frattini	C₂²×C₈	C₂²×D₁₇
Fitting	C₂×C₁₇×#346	C₂
socle	C₂²×C₁₇	C₂×#102
prime radical	C₂³×C₁₇	C₂×SD₁₆
cosocle	C₂²×C₈×C₁₇	C₂³
hypercenter	C₂×#346	D₁₇
hypocenter	C₁₇	#13286
LCS	G ▷ C₂×C₈×C₁₇ ▷ C₄×C₁₇ ▷ C₂×C₁₇ ▷ C₁₇
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ C₂×#346

Complements

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

N	H ≅ G/N	type
30. C₂×C₁₆×C₁₇	Q₈	semidirect
127. C₂×C₁₆	Dic₃₄	semidirect
139. C₁₇	#13286	semidirect

Maximal subgroups

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

subgroup	index
#14539298	2
#14528956	2
#14526880	2
C₂×C₁₇×#346	2
#13286	17