Subgroups — #16950769

Order 4352.

350 conjugacy classes of subgroups.

The subgroup lattice has 194 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	2	yes	maximal
2	C₂×#11762635	C₂	1	2	yes	maximal
3	#14528948	C₂	1	1	yes	maximal, characteristic
4	#14528944	C₂	1	1	yes	maximal, characteristic
5	C₂×C₁₇×#346	C₂	1	1	yes	maximal, characteristic
6	C₂×#10587489	—	2	1	no	—
7	C₂×#10587489	C₂²	1	1	yes	characteristic
8	C₂×#10587489	C₂²	1	2	yes	—
9	C₂×#10587489	C₂²	1	1	yes	characteristic
10	C₂×#10587488	—	2	1	no	—
11	#11762679	—	2	1	no	—
12	#11762675	—	2	1	no	—
13	#11762637	—	2	4	no	—
14	#11762635	—	2	4	no	—
15	C₂²×C₁₆×C₁₇	C₂²	1	2	yes	—
16	C₂×C₄×C₈×C₁₇	C₂²	1	1	yes	characteristic
17	C₁₇×#346	C₄	1	4	yes	complemented
18	C₂²×C₈×C₁₇	D₄	1	1	yes	characteristic
19	C₂²×C₈×C₁₇	C₂³	1	1	yes	characteristic
20	C₂×C₄²×C₁₇	D₄	1	1	yes	characteristic
21	C₂×C₄⋊Dic₁₇	—	2	1	no	—
22	C₂×C₄⋊Dic₁₇	—	2	1	no	—
23	C₂×C₄⋊Dic₁₇	—	2	2	no	—
24	C₂×C₄⋊Dic₁₇	—	4	2	no	—
25	C₂×C₁₆×C₁₇	C₂×C₄	1	4	yes	—
26	C₂×C₁₆×C₁₇	D₄	1	4	yes	—
27	C₂×C₁₆×C₁₇	Q₈	1	4	yes	—
28	C₄×C₈×C₁₇	C₂×C₄	1	2	yes	—
29	C₄×C₈×C₁₇	Q₈	1	2	yes	—
30	#10587502	—	4	2	no	—
31	#10587489	—	2	2	no	—
32	#10587489	—	2	2	no	—
33	#10587489	—	2	2	no	—
34	#10587489	—	2	2	no	—
35	#10587489	—	4	2	no	—
36	#10587488	—	4	2	no	—
37	C₂²×C₄×C₁₇	D₈	1	2	yes	—
38	C₂²×C₄×C₁₇	C₂×D₄	1	1	yes	characteristic
39	C₂²×Dic₁₇	—	4	1	no	—
40	C₂²×Dic₁₇	—	4	1	no	—
41	C₂²×Dic₁₇	—	4	2	no	—
42	C₂×C₈×C₁₇	C₄⋊C₄	1	2	yes	—
43	C₂×C₈×C₁₇	D₈	1	2	yes	—
44	C₂×C₈×C₁₇	Q₁₆	1	2	yes	—
45	C₂×C₈×C₁₇	C₂²×C₄	1	1	yes	characteristic
46	C₂×C₈×C₁₇	C₂×D₄	1	2	yes	—
47	C₂×C₈×C₁₇	C₂×Q₈	1	2	yes	—
48	C₂×C₈×C₁₇	C₂×Q₈	1	1	yes	characteristic
49	C₄²×C₁₇	C₄⋊C₄	1	2	yes	—
50	C₄²×C₁₇	Q₁₆	1	2	yes	—
51	C₄⋊Dic₁₇	—	4	1	no	—
52	C₄⋊Dic₁₇	—	4	2	no	—
53	C₄⋊Dic₁₇	—	4	1	no	—
54	C₄⋊Dic₁₇	—	4	1	no	—
55	C₄⋊Dic₁₇	—	4	1	no	—
56	C₄⋊Dic₁₇	—	4	2	no	—
57	C₄⋊Dic₁₇	—	8	4	no	—
58	C₁₆×C₁₇	C₄⋊C₄	1	8	yes	complemented
59	#12908	—	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	#107	1	4	yes	—
64	C₂×C₄×C₁₇	C₂×C₄⋊C₄	1	1	yes	characteristic
65	C₂×C₄×C₁₇	#127	1	1	yes	characteristic
66	C₂×C₄×C₁₇	C₄⋊Q₈	1	1	yes	characteristic
67	C₂×C₄×C₁₇	C₂×D₈	1	1	yes	characteristic
68	C₂×C₄×C₁₇	C₂×Q₁₆	1	1	yes	characteristic
69	C₂×C₄×C₁₇	C₂×Q₁₆	1	1	yes	characteristic
70	C₂×Dic₁₇	—	8	1	no	—
71	C₂×Dic₁₇	—	8	2	no	—
72	C₂×Dic₁₇	—	8	1	no	—
73	C₂×Dic₁₇	—	8	2	no	—
74	C₂×Dic₁₇	—	8	4	no	—
75	C₂×Dic₁₇	—	8	2	no	—
76	C₈×C₁₇	#107	1	4	yes	—
77	C₈×C₁₇	C₂×C₄⋊C₄	1	2	yes	—
78	C₈×C₁₇	C₄⋊Q₈	1	2	yes	—
79	C₂×#367	—	17	2	no	—
80	C₂×#366	—	17	2	no	—
81	C₂×#346	D₁₇	1	1	yes	characteristic
82	#1846	—	17	1	no	—
83	#1845	—	17	1	no	—
84	C₂²×C₁₇	C₂×#107	1	1	yes	characteristic
85	C₂²×C₁₇	#493	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	4	no	—
90	C₄×C₁₇	#384	1	1	yes	characteristic
91	C₄×C₁₇	C₂×#107	1	1	yes	characteristic
92	C₄×C₁₇	#494	1	1	yes	characteristic
93	C₄×C₁₇	#500	1	1	yes	characteristic
94	Dic₁₇	—	16	1	no	—
95	Dic₁₇	—	16	2	no	—
96	Dic₁₇	—	16	1	no	—
97	C₂²×C₁₆	C₂×D₁₇	1	2	yes	—
98	C₂×#107	—	17	2	no	—
99	C₂×#107	—	17	1	no	—
100	C₂×#107	—	17	1	no	—
101	C₂×#107	—	34	1	no	—
102	C₂×#106	—	34	1	no	—
103	C₂×C₄×C₈	C₂×D₁₇	1	1	yes	characteristic
104	#384	—	34	1	no	—
105	#382	—	34	1	no	—
106	#367	—	34	4	no	—
107	#366	—	34	4	no	—
108	#346	Dic₁₇	1	4	yes	complemented
109	C₂×C₁₇	#1845	1	1	yes	characteristic
110	C₂×C₁₇	#2159	1	2	yes	—
111	C₂×C₁₇	#2250	1	1	yes	characteristic
112	C₂×C₁₇	#2251	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	1	no	—
117	C₂×C₄⋊C₄	—	34	1	no	—
118	C₂×C₄⋊C₄	—	34	2	no	—
119	C₂×C₄⋊C₄	—	68	2	no	—
120	C₂×C₄²	D₆₈	1	1	yes	characteristic
121	C₂×C₁₆	Dic₃₄	1	4	yes	—
122	C₂×C₁₆	D₆₈	1	4	yes	—
123	C₂×C₁₆	C₂×Dic₁₇	1	4	yes	—
124	#107	—	34	2	no	—
125	#107	—	34	2	no	—
126	#107	—	34	2	no	—
127	#107	—	34	2	no	—
128	#107	—	68	2	no	—
129	#106	—	68	2	no	—
130	C₄×C₈	Dic₃₄	1	2	yes	—
131	C₄×C₈	C₂×Dic₁₇	1	2	yes	—
132	#95	—	68	2	no	—
133	C₁₇	#12908	1	1	yes	characteristic, complemented, Sylow
134	C₂²×C₄	—	68	1	no	—
135	C₂²×C₄	—	68	1	no	—
136	C₂²×C₄	—	68	2	no	—
137	C₂²×C₄	D₁₃₆	1	2	yes	—
138	C₂²×C₄	C₂×D₆₈	1	1	yes	characteristic
139	C₂×C₈	D₁₃₆	1	2	yes	—
140	C₂×C₈	Dic₆₈	1	2	yes	—
141	C₂×C₈	C₄⋊Dic₁₇	1	2	yes	—
142	C₂×C₈	C₂×Dic₃₄	1	1	yes	characteristic
143	C₂×C₈	C₂×Dic₃₄	1	2	yes	—
144	C₂×C₈	C₂×D₆₈	1	2	yes	—
145	C₂×C₈	C₂²×Dic₁₇	1	1	yes	characteristic
146	C₄⋊C₄	—	68	1	no	—
147	C₄⋊C₄	—	68	2	no	—
148	C₄⋊C₄	—	68	1	no	—
149	C₄⋊C₄	—	68	1	no	—
150	C₄⋊C₄	—	68	1	no	—
151	C₄⋊C₄	—	68	2	no	—
152	C₄⋊C₄	—	136	4	no	—
153	C₄²	Dic₆₈	1	2	yes	—
154	C₄²	C₄⋊Dic₁₇	1	2	yes	—
155	C₁₆	C₄⋊Dic₁₇	1	8	yes	complemented
156	C₂³	C₂×D₁₃₆	1	1	yes	characteristic
157	C₂×C₄	—	2	2	no	—
158	C₂×C₄	—	2	2	no	—
159	C₂×C₄	—	136	4	no	—
160	C₂×C₄	—	136	2	no	—
161	C₂×C₄	—	136	2	no	—
162	C₂×C₄	—	136	1	no	—
163	C₂×C₄	—	136	2	no	—
164	C₂×C₄	—	136	1	no	—
165	C₂×C₄	#10587489	1	4	yes	—
166	C₂×C₄	#10587555	1	1	yes	characteristic
167	C₂×C₄	#10587560	1	1	yes	characteristic
168	C₂×C₄	C₂×D₁₃₆	1	1	yes	characteristic
169	C₂×C₄	C₂×Dic₆₈	1	1	yes	characteristic
170	C₂×C₄	C₂×Dic₆₈	1	1	yes	characteristic
171	C₂×C₄	C₂×C₄⋊Dic₁₇	1	1	yes	characteristic
172	C₈	#10587489	1	4	yes	—
173	C₈	#10587555	1	2	yes	—
174	C₈	C₂×C₄⋊Dic₁₇	1	2	yes	—
175	C₂²	C₂×#10587489	1	1	yes	characteristic
176	C₂²	#11763186	1	1	yes	characteristic
177	C₂²	#11763214	1	1	yes	characteristic
178	C₂²	#11763247	1	2	yes	—
179	C₂²	#11763252	1	2	yes	—
180	C₄	—	2	4	no	—
181	C₄	—	272	2	no	—
182	C₄	—	272	1	no	—
183	C₄	—	272	1	no	—
184	C₄	#11762679	1	1	yes	characteristic
185	C₄	C₂×#10587489	1	1	yes	characteristic
186	C₄	#11763189	1	1	yes	characteristic
187	C₄	#11763214	1	1	yes	characteristic
188	C₂	#14528944	1	1	yes	characteristic
189	C₂	#14530572	1	2	yes	—
190	C₂	#14531050	1	1	yes	characteristic
191	C₂	#14531055	1	1	yes	characteristic
192	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	#12908
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
17. C₁₇×#346	C₄	semidirect
58. C₁₆×C₁₇	C₄⋊C₄	semidirect
108. #346	Dic₁₇	semidirect
133. C₁₇	#12908	semidirect
155. C₁₆	C₄⋊Dic₁₇	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
#14528948	2
#14528944	2
C₂×C₁₇×#346	2
#12908	17