Subgroups — #16952364

Order 4352.

318 conjugacy classes of subgroups.

The subgroup lattice has 198 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#11762637	C₂	1	1	yes	maximal, characteristic
2	C₂×#11762635	C₂	1	1	yes	maximal, characteristic
3	#14529517	C₂	1	2	yes	maximal
4	#14526876	C₂	1	2	yes	maximal
5	C₂×C₁₇×#346	C₂	1	1	yes	maximal, characteristic
6	C₂×#10587489	C₂²	1	2	yes	—
7	C₂×#10587489	C₂²	1	2	yes	—
8	#11762707	—	2	2	no	—
9	#11762637	C₄	1	4	yes	complemented
10	#11762635	—	2	2	no	—
11	#11762572	—	2	2	no	—
12	C₂²×C₁₆×C₁₇	C₂²	1	1	yes	characteristic
13	C₂²×C₁₆×C₁₇	C₂²	1	1	yes	characteristic
14	C₂×C₄×C₈×C₁₇	C₂²	1	1	yes	characteristic
15	C₁₇×#346	—	2	2	no	—
16	C₂²×C₈×C₁₇	D₄	1	1	yes	characteristic
17	C₂²×C₈×C₁₇	C₂³	1	1	yes	characteristic
18	C₂×C₄²×C₁₇	D₄	1	1	yes	characteristic
19	C₂×C₄⋊Dic₁₇	—	2	2	no	—
20	C₂×C₄⋊Dic₁₇	—	2	2	no	—
21	C₂×#63225	—	4	2	no	—
22	C₂×C₁₆×C₁₇	—	2	1	no	—
23	C₂×C₁₆×C₁₇	—	2	1	no	—
24	C₂×C₁₆×C₁₇	C₂×C₄	1	2	yes	—
25	C₂×C₁₆×C₁₇	D₄	1	2	yes	—
26	C₂×C₁₆×C₁₇	Q₈	1	2	yes	—
27	C₂×C₁₆×C₁₇	Q₈	1	2	yes	—
28	C₄×C₈×C₁₇	—	2	1	no	—
29	C₄×C₈×C₁₇	—	2	1	no	—
30	#10587502	—	4	2	no	—
31	#10587489	—	2	2	no	—
32	#10587489	—	2	2	no	—
33	#10587489	—	2	2	no	—
34	#10587489	C₂×C₄	1	4	yes	—
35	C₂²×C₄×C₁₇	D₈	1	1	yes	characteristic
36	C₂²×C₄×C₁₇	D₈	1	1	yes	characteristic
37	C₂²×C₄×C₁₇	C₂×D₄	1	1	yes	characteristic
38	C₂²×Dic₁₇	—	4	2	no	—
39	C₂²×Dic₁₇	—	4	2	no	—
40	C₂×C₈×C₁₇	—	2	1	no	—
41	C₂×C₈×C₁₇	—	2	1	no	—
42	C₂×C₈×C₁₇	Q₁₆	1	2	yes	—
43	C₂×C₈×C₁₇	C₂²×C₄	1	1	yes	characteristic
44	C₂×C₈×C₁₇	C₂×D₄	1	1	yes	characteristic
45	C₂×C₈×C₁₇	C₂×Q₈	1	1	yes	characteristic
46	C₂×C₈×C₁₇	C₂×Q₈	1	1	yes	characteristic
47	C₂×C₈×C₁₇	#41	1	1	yes	characteristic
48	C₂×C₈×C₁₇	#41	1	1	yes	characteristic
49	C₄²×C₁₇	—	2	1	no	—
50	C₄²×C₁₇	—	2	1	no	—
51	C₄⋊Dic₁₇	—	2	4	no	—
52	C₄⋊Dic₁₇	—	4	2	no	—
53	C₄⋊Dic₁₇	—	4	2	no	—
54	C₄⋊Dic₁₇	—	4	2	no	—
55	#63225	—	8	4	no	—
56	C₁₆×C₁₇	—	2	2	no	—
57	C₁₆×C₁₇	C₄⋊C₄	1	4	yes	complemented
58	#13167	—	17	1	no	maximal, Sylow
59	C₂³×C₁₇	C₂×D₈	1	1	yes	characteristic
60	C₂×C₄×C₁₇	—	2	1	no	—
61	C₂×C₄×C₁₇	—	2	1	no	—
62	C₂×C₄×C₁₇	—	2	2	no	—
63	C₂×C₄×C₁₇	—	2	2	no	—
64	C₂×C₄×C₁₇	C₄×D₄	1	1	yes	characteristic
65	C₂×C₄×C₁₇	#121	1	1	yes	characteristic
66	C₂×C₄×C₁₇	C₄⋊Q₈	1	1	yes	characteristic
67	C₂×C₄×C₁₇	C₂×Q₁₆	1	1	yes	characteristic
68	C₂×C₄×C₁₇	#135	1	1	yes	characteristic
69	C₂×C₄×C₁₇	#135	1	1	yes	characteristic
70	C₂×Dic₁₇	—	4	4	no	—
71	C₂×Dic₁₇	—	8	2	no	—
72	C₂×Dic₁₇	—	8	2	no	—
73	C₂×Dic₁₇	—	8	2	no	—
74	C₂×Dic₁₇	—	8	2	no	—
75	C₂×Dic₁₇	—	8	2	no	—
76	C₈×C₁₇	—	2	2	no	—
77	C₈×C₁₇	C₂×C₄⋊C₄	1	1	yes	characteristic
78	C₈×C₁₇	C₄×Q₈	1	1	yes	characteristic
79	C₈×C₁₇	C₂²⋊Q₈	1	1	yes	characteristic
80	C₈×C₁₇	#125	1	1	yes	characteristic
81	C₂×#367	—	17	1	no	—
82	C₂×#366	—	17	1	no	—
83	C₂×#346	D₁₇	1	1	yes	characteristic
84	#1942	—	17	2	no	—
85	#1380	—	17	2	no	—
86	C₂²×C₁₇	C₄×D₈	1	1	yes	characteristic
87	C₂²×C₁₇	#466	1	1	yes	characteristic
88	C₂²×C₁₇	#500	1	1	yes	characteristic
89	C₂²×C₁₇	#509	1	2	yes	—
90	C₂²×C₁₇	#510	1	2	yes	—
91	C₄×C₁₇	—	4	1	no	—
92	C₄×C₁₇	—	4	1	no	—
93	C₄×C₁₇	#389	1	1	yes	characteristic
94	C₄×C₁₇	C₄×Q₁₆	1	1	yes	characteristic
95	C₄×C₁₇	#467	1	1	yes	characteristic
96	C₄×C₁₇	#499	1	1	yes	characteristic
97	Dic₁₇	—	8	4	no	—
98	Dic₁₇	—	16	2	no	—
99	C₂²×C₁₆	C₂×D₁₇	1	1	yes	characteristic
100	C₂²×C₁₆	C₂×D₁₇	1	1	yes	characteristic
101	C₂×#107	—	17	2	no	—
102	C₂×#107	—	17	2	no	—
103	C₂×C₄×C₈	C₂×D₁₇	1	1	yes	characteristic
104	#389	—	34	2	no	—
105	#367	—	17	4	no	—
106	#366	—	34	2	no	—
107	#346	—	2	2	no	—
108	#340	—	34	2	no	—
109	C₂×C₁₇	#1942	1	1	yes	characteristic
110	C₂×C₁₇	#2175	1	2	yes	—
111	C₂×C₁₇	#2218	1	1	yes	characteristic
112	C₂×C₁₇	#2219	1	1	yes	characteristic
113	C₂×C₁₇	#2255	1	2	yes	—
114	C₂²×C₈	D₆₈	1	1	yes	characteristic
115	C₂²×C₈	C₂²×D₁₇	1	1	yes	characteristic
116	C₂×C₄⋊C₄	—	34	2	no	—
117	C₂×C₄⋊C₄	—	34	2	no	—
118	C₂×C₄⋊C₄	—	68	2	no	—
119	C₂×C₄²	D₆₈	1	1	yes	characteristic
120	C₂×C₁₆	—	2	1	no	—
121	C₂×C₁₆	—	2	1	no	—
122	C₂×C₁₆	Dic₃₄	1	2	yes	—
123	C₂×C₁₆	Dic₃₄	1	2	yes	—
124	C₂×C₁₆	C₄×D₁₇	1	2	yes	—
125	C₂×C₁₆	C₂²⋊D₁₇	1	2	yes	—
126	#107	—	17	4	no	—
127	#107	—	34	2	no	—
128	#107	—	34	2	no	—
129	#107	—	34	2	no	—
130	C₄×C₈	—	2	1	no	—
131	C₄×C₈	—	2	1	no	—
132	#95	—	68	2	no	—
133	C₁₇	#13167	1	1	yes	characteristic, complemented, Sylow
134	C₂²×C₄	—	68	2	no	—
135	C₂²×C₄	—	68	2	no	—
136	C₂²×C₄	D₁₃₆	1	1	yes	characteristic
137	C₂²×C₄	D₁₃₆	1	1	yes	characteristic
138	C₂²×C₄	C₂×D₆₈	1	1	yes	characteristic
139	C₂×C₈	—	2	1	no	—
140	C₂×C₈	—	2	1	no	—
141	C₂×C₈	Dic₆₈	1	2	yes	—
142	C₂×C₈	C₂×Dic₃₄	1	1	yes	characteristic
143	C₂×C₈	C₂×Dic₃₄	1	1	yes	characteristic
144	C₂×C₈	C₂×C₄×D₁₇	1	1	yes	characteristic
145	C₂×C₈	#63252	1	1	yes	characteristic
146	C₂×C₈	#63252	1	1	yes	characteristic
147	C₂×C₈	C₂×C₂²⋊D₁₇	1	1	yes	characteristic
148	C₄⋊C₄	—	34	4	no	—
149	C₄⋊C₄	—	68	2	no	—
150	C₄⋊C₄	—	68	2	no	—
151	C₄⋊C₄	—	68	2	no	—
152	C₄⋊C₄	—	136	4	no	—
153	C₄²	—	2	1	no	—
154	C₄²	—	2	1	no	—
155	C₁₆	—	2	2	no	—
156	C₁₆	#63225	1	4	yes	complemented
157	C₂³	C₂×D₁₃₆	1	1	yes	characteristic
158	C₂×C₄	—	2	2	no	—
159	C₂×C₄	—	2	2	no	—
160	C₂×C₄	—	2	1	no	—
161	C₂×C₄	—	2	1	no	—
162	C₂×C₄	—	68	4	no	—
163	C₂×C₄	—	136	2	no	—
164	C₂×C₄	—	136	2	no	—
165	C₂×C₄	—	136	2	no	—
166	C₂×C₄	—	136	2	no	—
167	C₂×C₄	—	136	2	no	—
168	C₂×C₄	#10587555	1	1	yes	characteristic
169	C₂×C₄	C₄×D₆₈	1	1	yes	characteristic
170	C₂×C₄	#10587590	1	1	yes	characteristic
171	C₂×C₄	#10587590	1	1	yes	characteristic
172	C₂×C₄	C₂×Dic₆₈	1	1	yes	characteristic
173	C₂×C₄	#10587616	1	1	yes	characteristic
174	C₈	—	2	2	no	—
175	C₈	C₄×Dic₃₄	1	1	yes	characteristic
176	C₈	#10587556	1	1	yes	characteristic
177	C₈	C₂×#63225	1	1	yes	characteristic
178	C₈	#10587610	1	1	yes	characteristic
179	C₂²	C₄×D₁₃₆	1	1	yes	characteristic
180	C₂²	#11763045	1	1	yes	characteristic
181	C₂²	#11763214	1	1	yes	characteristic
182	C₂²	#11763247	1	2	yes	—
183	C₂²	#11763252	1	2	yes	—
184	C₄	—	4	1	no	—
185	C₄	—	4	1	no	—
186	C₄	—	272	2	no	—
187	C₄	—	136	4	no	—
188	C₄	#11762707	1	1	yes	characteristic
189	C₄	C₄×Dic₆₈	1	1	yes	characteristic
190	C₄	#11763050	1	1	yes	characteristic
191	C₄	#11763211	1	1	yes	characteristic
192	C₂	#14529517	1	1	yes	characteristic
193	C₂	#14530650	1	2	yes	—
194	C₂	#14530887	1	1	yes	characteristic
195	C₂	#14530892	1	1	yes	characteristic
196	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	#13167
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
9. #11762637	C₄	semidirect
57. C₁₆×C₁₇	C₄⋊C₄	semidirect
133. C₁₇	#13167	semidirect
156. C₁₆	#63225	semidirect

Maximal subgroups

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

subgroup	index
C₂×#11762637	2
C₂×#11762635	2
#14529517	2
#14526876	2
C₂×C₁₇×#346	2
#13167	17