Subgroups — #16952929

Order 4352.

390 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	C₂×#11763214	C₂	1	1	yes	maximal, characteristic
2	#14528948	C₂	1	1	yes	maximal, characteristic
3	#14526876	C₂	1	4	yes	maximal
4	C₂×C₁₇×#346	C₂	1	1	yes	maximal, characteristic
5	C₂×#10587555	—	2	1	no	—
6	#11763214	—	2	2	no	—
7	#11763214	C₄	1	4	yes	complemented
8	C₂×#10587489	—	2	1	no	—
9	C₂×#10587489	C₂²	1	2	yes	—
10	C₂×#10587489	C₂²	1	2	yes	—
11	C₂×#10587488	—	2	1	no	—
12	#11762675	—	2	1	no	—
13	C₂²×C₁₆×C₁₇	C₂²	1	2	yes	—
14	C₂×C₄×C₈×C₁₇	C₂²	1	1	yes	characteristic
15	C₁₇×#346	—	2	2	no	—
16	C₂²×Dic₃₄	—	4	1	no	—
17	C₂²×C₈×C₁₇	D₄	1	1	yes	characteristic
18	C₂²×C₈×C₁₇	C₂³	1	1	yes	characteristic
19	C₂×C₄²×C₁₇	D₄	1	1	yes	characteristic
20	C₂×C₄⋊Dic₁₇	—	2	2	no	—
21	C₂×C₄⋊Dic₁₇	—	2	2	no	—
22	C₂×C₄⋊Dic₁₇	—	4	1	no	—
23	#10587555	—	2	4	no	—
24	#10587555	—	4	2	no	—
25	C₂×C₁₆×C₁₇	—	2	2	no	—
26	C₂×C₁₆×C₁₇	—	2	2	no	—
27	C₂×C₁₆×C₁₇	Q₈	1	4	yes	complemented
28	C₄×C₈×C₁₇	C₂×C₄	1	2	yes	—
29	C₄×C₈×C₁₇	D₄	1	2	yes	—
30	#10587502	—	4	1	no	—
31	#10587502	—	4	1	no	—
32	#10587489	—	2	4	no	—
33	#10587489	—	2	2	no	—
34	#10587489	—	2	4	no	—
35	#10587489	C₂×C₄	1	4	yes	—
36	#10587488	—	4	2	no	—
37	C₂²×C₄×C₁₇	D₈	1	1	yes	characteristic
38	C₂²×C₄×C₁₇	D₈	1	1	yes	characteristic
39	C₂²×C₄×C₁₇	C₂×D₄	1	1	yes	characteristic
40	C₂²×Dic₁₇	—	4	2	no	—
41	C₂²×Dic₁₇	—	4	2	no	—
42	C₂×Dic₃₄	—	4	4	no	—
43	C₂×Dic₃₄	—	8	4	no	—
44	C₂×C₈×C₁₇	C₂²⋊C₄	1	2	yes	—
45	C₂×C₈×C₁₇	SD₁₆	1	2	yes	—
46	C₂×C₈×C₁₇	Q₁₆	1	2	yes	—
47	C₂×C₈×C₁₇	C₂²×C₄	1	1	yes	characteristic
48	C₂×C₈×C₁₇	C₂×D₄	1	1	yes	characteristic
49	C₂×C₈×C₁₇	C₂×Q₈	1	2	yes	—
50	C₂×C₈×C₁₇	#41	1	2	yes	—
51	C₄²×C₁₇	C₂²⋊C₄	1	2	yes	—
52	C₄²×C₁₇	SD₁₆	1	2	yes	—
53	C₄⋊Dic₁₇	—	2	4	no	—
54	C₄⋊Dic₁₇	—	4	4	no	—
55	C₄⋊Dic₁₇	—	4	2	no	—
56	C₄⋊Dic₁₇	—	4	2	no	—
57	C₄⋊Dic₁₇	—	4	2	no	—
58	C₁₆×C₁₇	—	2	4	no	—
59	#13283	—	17	1	no	maximal, Sylow
60	C₂³×C₁₇	C₂×D₈	1	1	yes	characteristic
61	C₂×C₄×C₁₇	—	2	2	no	—
62	C₂×C₄×C₁₇	—	2	2	no	—
63	C₂×C₄×C₁₇	#102	1	2	yes	—
64	C₂×C₄×C₁₇	#102	1	2	yes	—
65	C₂×C₄×C₁₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
66	C₂×C₄×C₁₇	#124	1	1	yes	characteristic
67	C₂×C₄×C₁₇	C₄⋊Q₈	1	1	yes	characteristic
68	C₂×C₄×C₁₇	C₂×SD₁₆	1	1	yes	characteristic
69	C₂×C₄×C₁₇	C₂×SD₁₆	1	1	yes	characteristic
70	C₂×C₄×C₁₇	C₂×Q₁₆	1	1	yes	characteristic
71	C₂×Dic₁₇	—	4	4	no	—
72	C₂×Dic₁₇	—	8	4	no	—
73	C₂×Dic₁₇	—	8	2	no	—
74	C₂×Dic₁₇	—	8	2	no	—
75	C₂×Dic₁₇	—	8	2	no	—
76	Dic₃₄	—	8	8	no	—
77	C₈×C₁₇	#103	1	4	yes	—
78	C₈×C₁₇	C₄×Q₈	1	2	yes	—
79	C₈×C₁₇	C₂²⋊Q₈	1	2	yes	—
80	C₂×#500	—	17	1	no	—
81	C₂×#346	D₁₇	1	1	yes	characteristic
82	#1846	—	17	1	no	—
83	#1380	—	17	4	no	—
84	C₂²×C₁₇	C₂×#102	1	1	yes	characteristic
85	C₂²×C₁₇	#486	1	1	yes	characteristic
86	C₂²×C₁₇	#500	1	1	yes	characteristic
87	C₂²×C₁₇	#509	1	2	yes	—
88	C₂²×C₁₇	#510	1	2	yes	—
89	C₄×C₁₇	—	2	2	no	—
90	C₄×C₁₇	—	2	2	no	—
91	C₄×C₁₇	#391	1	1	yes	characteristic
92	C₄×C₁₇	C₂×#103	1	1	yes	characteristic
93	C₄×C₁₇	#487	1	1	yes	characteristic
94	C₄×C₁₇	#498	1	1	yes	characteristic
95	Dic₁₇	—	8	4	no	—
96	Dic₁₇	—	16	2	no	—
97	C₂×C₄⋊Q₈	—	34	1	no	—
98	C₂²×C₁₆	C₂×D₁₇	1	2	yes	—
99	#500	—	17	4	no	—
100	#500	—	34	2	no	—
101	C₂×#107	—	17	2	no	—
102	C₂×#107	—	17	2	no	—
103	C₂×#107	—	34	1	no	—
104	C₂×#106	—	34	1	no	—
105	C₂×C₄×C₈	C₂×D₁₇	1	1	yes	characteristic
106	#382	—	34	1	no	—
107	#346	—	2	2	no	—
108	C₂×C₁₇	#1990	1	1	yes	characteristic
109	C₂×C₁₇	#2140	1	1	yes	characteristic
110	C₂×C₁₇	#2141	1	1	yes	characteristic
111	C₂×C₁₇	#2243	1	2	yes	—
112	C₂×C₁₇	#2255	1	2	yes	—
113	C₂²×Q₈	—	68	1	no	—
114	C₂²×C₈	D₆₈	1	1	yes	characteristic
115	C₂²×C₈	C₂²×D₁₇	1	1	yes	characteristic
116	C₄⋊Q₈	—	34	4	no	—
117	C₄⋊Q₈	—	68	2	no	—
118	C₂×C₄⋊C₄	—	34	2	no	—
119	C₂×C₄⋊C₄	—	34	2	no	—
120	C₂×C₄⋊C₄	—	68	1	no	—
121	C₂×C₄²	D₆₈	1	1	yes	characteristic
122	C₂×C₁₆	—	2	2	no	—
123	C₂×C₁₆	—	2	2	no	—
124	C₂×C₁₆	Dic₃₄	1	4	yes	complemented
125	#107	—	17	4	no	—
126	#107	—	34	2	no	—
127	#107	—	34	4	no	—
128	#107	—	34	4	no	—
129	#106	—	68	2	no	—
130	C₄×C₈	C₄×D₁₇	1	2	yes	—
131	C₄×C₈	C₂²⋊D₁₇	1	2	yes	—
132	#95	—	68	1	no	—
133	#95	—	68	1	no	—
134	C₁₇	#13283	1	1	yes	characteristic, complemented, Sylow
135	C₂×Q₈	—	68	4	no	—
136	C₂×Q₈	—	136	4	no	—
137	C₂²×C₄	—	68	2	no	—
138	C₂²×C₄	—	68	2	no	—
139	C₂²×C₄	D₁₃₆	1	1	yes	characteristic
140	C₂²×C₄	D₁₃₆	1	1	yes	characteristic
141	C₂²×C₄	C₂×D₆₈	1	1	yes	characteristic
142	C₂×C₈	#63219	1	2	yes	—
143	C₂×C₈	Dic₆₈	1	2	yes	—
144	C₂×C₈	#63227	1	2	yes	—
145	C₂×C₈	C₂×Dic₃₄	1	2	yes	—
146	C₂×C₈	C₂×C₄×D₁₇	1	1	yes	characteristic
147	C₂×C₈	#63252	1	2	yes	—
148	C₂×C₈	C₂×C₂²⋊D₁₇	1	1	yes	characteristic
149	C₄⋊C₄	—	34	4	no	—
150	C₄⋊C₄	—	68	4	no	—
151	C₄⋊C₄	—	68	2	no	—
152	C₄⋊C₄	—	68	2	no	—
153	C₄⋊C₄	—	68	2	no	—
154	C₄²	#63219	1	2	yes	—
155	C₄²	#63227	1	2	yes	—
156	C₁₆	—	2	4	no	—
157	C₂³	C₂×D₁₃₆	1	1	yes	characteristic
158	Q₈	—	136	8	no	—
159	C₂×C₄	—	2	2	no	—
160	C₂×C₄	—	2	2	no	—
161	C₂×C₄	—	68	4	no	—
162	C₂×C₄	—	136	2	no	—
163	C₂×C₄	—	136	2	no	—
164	C₂×C₄	—	136	2	no	—
165	C₂×C₄	—	136	4	no	—
166	C₂×C₄	#10587492	1	2	yes	—
167	C₂×C₄	#10587492	1	2	yes	—
168	C₂×C₄	#10587555	1	1	yes	characteristic
169	C₂×C₄	#10587561	1	1	yes	characteristic
170	C₂×C₄	C₂×#63219	1	1	yes	characteristic
171	C₂×C₄	C₂×#63219	1	1	yes	characteristic
172	C₂×C₄	C₂×Dic₆₈	1	1	yes	characteristic
173	C₂×C₄	C₂×#63227	1	1	yes	characteristic
174	C₈	#10587487	1	4	yes	—
175	C₈	C₄×Dic₃₄	1	2	yes	—
176	C₈	#10587610	1	2	yes	—
177	C₂²	C₂×#10587492	1	1	yes	characteristic
178	C₂²	#11763156	1	1	yes	characteristic
179	C₂²	#11763214	1	1	yes	characteristic
180	C₂²	#11763247	1	2	yes	—
181	C₂²	#11763252	1	2	yes	—
182	C₄	—	2	2	no	—
183	C₄	—	2	2	no	—
184	C₄	—	272	2	no	—
185	C₄	—	136	4	no	—
186	C₄	#11762715	1	1	yes	characteristic
187	C₄	C₂×#10587487	1	1	yes	characteristic
188	C₄	#11763160	1	1	yes	characteristic
189	C₄	#11763207	1	1	yes	characteristic
190	C₂	#14529745	1	1	yes	characteristic
191	C₂	#14530501	1	1	yes	characteristic
192	C₂	#14530505	1	1	yes	characteristic
193	C₂	#14531021	1	2	yes	—
194	C₂	#14531070	1	2	yes	—

	subgroup	G/H
center	C₂³	C₂×D₁₃₆
commutator	C₂×C₈×C₁₇	C₂²×C₄
Frattini	C₂²×C₈	C₂²×D₁₇
Fitting	C₂×C₁₇×#346	C₂
socle, prime radical	C₂³×C₁₇	C₂×D₈
cosocle	C₂²×C₈×C₁₇	C₂³
hypercenter	C₂×#346	D₁₇
hypocenter	C₁₇	#13283
LCS	G ▷ C₂×C₈×C₁₇ ▷ C₄×C₁₇ ▷ C₂×C₁₇ ▷ C₁₇
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ C₂×#346

Complements

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

N	H ≅ G/N	type
7. #11763214	C₄	semidirect
27. C₂×C₁₆×C₁₇	Q₈	semidirect
124. C₂×C₁₆	Dic₃₄	semidirect
134. C₁₇	#13283	semidirect

Maximal subgroups

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

subgroup	index
C₂×#11763214	2
#14528948	2
#14526876	2
C₂×C₁₇×#346	2
#13283	17