Subgroups — #16950750

Order 4352.

334 conjugacy classes of subgroups.

The subgroup lattice has 176 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#11762632	C₂	1	4	yes	maximal
2	#14528956	C₂	1	1	yes	maximal, characteristic
3	#14528941	C₂	1	1	yes	maximal, characteristic
4	C₄×C₁₇×M₃₂	C₂	1	1	yes	maximal, characteristic
5	C₂×#10587490	—	2	1	no	—
6	C₂×#10587490	C₂²	1	2	yes	—
7	C₂×#10587488	—	2	1	no	—
8	C₂×#10587488	C₂²	1	2	yes	—
9	#11762833	—	2	1	no	—
10	#11762679	—	2	1	no	—
11	#11762632	—	2	8	no	—
12	C₂×C₁₇×M₃₂	C₂²	1	2	yes	—
13	C₂×C₄×C₈×C₁₇	C₂²	1	1	yes	characteristic
14	C₁₇×#346	C₄	1	2	yes	complemented
15	C₄×C₁₆×C₁₇	C₄	1	2	yes	complemented
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	—	4	2	no	—
21	C₂×#63223	—	2	2	no	—
22	C₁₇×M₃₂	D₄	1	4	yes	—
23	C₁₇×M₃₂	Q₈	1	4	yes	—
24	C₂×C₁₆×C₁₇	C₂×C₄	1	4	yes	—
25	C₄×C₈×C₁₇	C₂×C₄	1	1	yes	characteristic
26	C₄×C₈×C₁₇	C₂×C₄	1	1	yes	characteristic
27	C₄×C₈×C₁₇	Q₈	1	1	yes	characteristic
28	C₄×C₈×C₁₇	Q₈	1	1	yes	characteristic
29	#10587490	—	2	4	no	—
30	#10587490	—	4	2	no	—
31	#10587488	—	2	4	no	—
32	#10587488	—	4	2	no	—
33	C₄⋊#3622	—	4	2	no	—
34	C₂²×C₄×C₁₇	SD₁₆	1	2	yes	—
35	C₂²×C₄×C₁₇	C₂×D₄	1	1	yes	characteristic
36	C₂²×Dic₁₇	—	4	2	no	—
37	C₂×C₈×C₁₇	C₄⋊C₄	1	1	yes	characteristic
38	C₂×C₈×C₁₇	C₄⋊C₄	1	1	yes	characteristic
39	C₂×C₈×C₁₇	SD₁₆	1	2	yes	—
40	C₂×C₈×C₁₇	SD₁₆	1	2	yes	—
41	C₂×C₈×C₁₇	C₂²×C₄	1	1	yes	characteristic
42	C₂×C₈×C₁₇	C₂×D₄	1	2	yes	—
43	C₂×C₈×C₁₇	C₂×Q₈	1	2	yes	—
44	C₂×C₈×C₁₇	C₂×Q₈	1	1	yes	characteristic
45	C₄²×C₁₇	C₄⋊C₄	1	1	yes	characteristic
46	C₄²×C₁₇	C₄⋊C₄	1	1	yes	characteristic
47	C₄²×C₁₇	SD₁₆	1	2	yes	—
48	C₄⋊Dic₁₇	—	4	2	no	—
49	C₄⋊Dic₁₇	—	4	2	no	—
50	C₄⋊Dic₁₇	—	8	4	no	—
51	#63223	—	4	2	no	—
52	#63223	—	4	2	no	—
53	C₂×#3622	—	4	2	no	—
54	C₁₆×C₁₇	C₄⋊C₄	1	8	yes	complemented
55	#12903	—	17	1	no	maximal, Sylow
56	C₂³×C₁₇	C₂×SD₁₆	1	1	yes	characteristic
57	C₂×C₄×C₁₇	—	2	4	no	—
58	C₂×C₄×C₁₇	#106	1	2	yes	—
59	C₂×C₄×C₁₇	#106	1	2	yes	—
60	C₂×C₄×C₁₇	C₂×C₄⋊C₄	1	1	yes	characteristic
61	C₂×C₄×C₁₇	#127	1	1	yes	characteristic
62	C₂×C₄×C₁₇	C₄⋊Q₈	1	1	yes	characteristic
63	C₂×C₄×C₁₇	C₂×SD₁₆	1	1	yes	characteristic
64	C₂×C₄×C₁₇	C₂×SD₁₆	1	1	yes	characteristic
65	C₂×C₄×C₁₇	C₂×SD₁₆	1	1	yes	characteristic
66	C₂×Dic₁₇	—	8	4	no	—
67	C₂×Dic₁₇	—	8	2	no	—
68	C₈×C₁₇	#106	1	2	yes	—
69	C₈×C₁₇	#106	1	2	yes	—
70	C₈×C₁₇	C₂×C₄⋊C₄	1	2	yes	—
71	C₈×C₁₇	C₄⋊Q₈	1	2	yes	—
72	#3622	—	8	2	no	—
73	C₂×#365	—	17	4	no	—
74	C₄×M₃₂	D₁₇	1	1	yes	characteristic
75	#1848	—	17	1	no	—
76	#1844	—	17	1	no	—
77	C₂²×C₁₇	—	2	2	no	—
78	C₂²×C₁₇	C₂×#106	1	1	yes	characteristic
79	C₂²×C₁₇	#492	1	1	yes	characteristic
80	C₂²×C₁₇	#498	1	1	yes	characteristic
81	C₄×C₁₇	—	2	2	no	—
82	C₄×C₁₇	#365	1	4	yes	complemented
83	C₄×C₁₇	#384	1	1	yes	characteristic
84	C₄×C₁₇	C₂×#106	1	1	yes	characteristic
85	C₄×C₁₇	#492	1	1	yes	characteristic
86	C₄×C₁₇	#498	1	1	yes	characteristic
87	Dic₁₇	—	16	2	no	—
88	C₂×M₃₂	C₂×D₁₇	1	2	yes	—
89	C₂×#108	—	17	2	no	—
90	C₂×#108	—	34	1	no	—
91	C₂×#106	—	17	2	no	—
92	C₂×#106	—	34	1	no	—
93	#423	—	34	1	no	—
94	C₂×C₄×C₈	C₂×D₁₇	1	1	yes	characteristic
95	#384	—	34	1	no	—
96	#365	—	34	8	no	—
97	#346	Dic₁₇	1	2	yes	complemented
98	C₄×C₁₆	Dic₁₇	1	2	yes	complemented
99	C₂×C₁₇	—	2	1	no	—
100	C₂×C₁₇	—	2	1	no	—
101	C₂×C₁₇	#1844	1	1	yes	characteristic
102	C₂×C₁₇	C₂×#365	1	2	yes	—
103	C₂×M₁₆	—	34	2	no	—
104	C₂²×C₈	D₆₈	1	1	yes	characteristic
105	C₂²×C₈	C₂²×D₁₇	1	1	yes	characteristic
106	C₂×C₄⋊C₄	—	34	2	no	—
107	C₂×C₄⋊C₄	—	68	2	no	—
108	C₂×C₄²	D₆₈	1	1	yes	characteristic
109	M₃₂	Dic₃₄	1	4	yes	—
110	M₃₂	D₆₈	1	4	yes	—
111	C₂×C₁₆	C₂×Dic₁₇	1	4	yes	—
112	#108	—	34	4	no	—
113	#108	—	68	2	no	—
114	#106	—	34	4	no	—
115	#106	—	68	2	no	—
116	C₄⋊C₈	—	68	2	no	—
117	C₄×C₈	Dic₃₄	1	1	yes	characteristic
118	C₄×C₈	Dic₃₄	1	1	yes	characteristic
119	C₄×C₈	C₂×Dic₁₇	1	1	yes	characteristic
120	C₄×C₈	C₂×Dic₁₇	1	1	yes	characteristic
121	C₁₇	#12903	1	1	yes	characteristic, complemented, Sylow
122	C₂²×C₄	—	68	2	no	—
123	C₂²×C₄	#63219	1	2	yes	—
124	C₂²×C₄	C₂×D₆₈	1	1	yes	characteristic
125	M₁₆	—	68	2	no	—
126	M₁₆	—	68	2	no	—
127	C₂×C₈	—	68	2	no	—
128	C₂×C₈	#63219	1	2	yes	—
129	C₂×C₈	#63219	1	2	yes	—
130	C₂×C₈	C₄⋊Dic₁₇	1	1	yes	characteristic
131	C₂×C₈	C₄⋊Dic₁₇	1	1	yes	characteristic
132	C₂×C₈	C₂×Dic₃₄	1	1	yes	characteristic
133	C₂×C₈	C₂×Dic₃₄	1	2	yes	—
134	C₂×C₈	C₂×D₆₈	1	2	yes	—
135	C₂×C₈	C₂²×Dic₁₇	1	1	yes	characteristic
136	C₄⋊C₄	—	68	2	no	—
137	C₄⋊C₄	—	68	2	no	—
138	C₄⋊C₄	—	136	4	no	—
139	C₄²	#63219	1	2	yes	—
140	C₄²	C₄⋊Dic₁₇	1	1	yes	characteristic
141	C₄²	C₄⋊Dic₁₇	1	1	yes	characteristic
142	C₁₆	C₄⋊Dic₁₇	1	8	yes	complemented
143	C₂³	C₂×#63219	1	1	yes	characteristic
144	C₂×C₄	—	2	4	no	—
145	C₂×C₄	—	136	4	no	—
146	C₂×C₄	—	136	2	no	—
147	C₂×C₄	#10587488	1	2	yes	—
148	C₂×C₄	#10587488	1	2	yes	—
149	C₂×C₄	#10587555	1	1	yes	characteristic
150	C₂×C₄	#10587560	1	1	yes	characteristic
151	C₂×C₄	C₂×#63219	1	1	yes	characteristic
152	C₂×C₄	C₂×#63219	1	1	yes	characteristic
153	C₂×C₄	C₂×#63219	1	1	yes	characteristic
154	C₂×C₄	C₂×C₄⋊Dic₁₇	1	1	yes	characteristic
155	C₈	—	136	2	no	—
156	C₈	#10587488	1	2	yes	—
157	C₈	#10587488	1	2	yes	—
158	C₈	#10587555	1	2	yes	—
159	C₈	C₂×C₄⋊Dic₁₇	1	2	yes	—
160	C₂²	—	2	2	no	—
161	C₂²	C₂×#10587488	1	1	yes	characteristic
162	C₂²	#11763182	1	1	yes	characteristic
163	C₂²	#11763207	1	1	yes	characteristic
164	C₄	—	2	2	no	—
165	C₄	—	272	2	no	—
166	C₄	#11762632	1	4	yes	complemented
167	C₄	#11762679	1	1	yes	characteristic
168	C₄	C₂×#10587488	1	1	yes	characteristic
169	C₄	#11763182	1	1	yes	characteristic
170	C₄	#11763207	1	1	yes	characteristic
171	C₂	—	2	1	no	—
172	C₂	—	2	1	no	—
173	C₂	#14528941	1	1	yes	characteristic
174	C₂	C₂×#11762632	1	2	yes	—

	subgroup	G/H
center	C₂²	C₂×#10587488
commutator	C₂×C₈×C₁₇	C₂²×C₄
Frattini	C₂²×C₈	C₂²×D₁₇
Fitting	C₄×C₁₇×M₃₂	C₂
socle	C₂²×C₁₇	C₂×#106
prime radical	C₂³×C₁₇	C₂×SD₁₆
cosocle	C₂²×C₈×C₁₇	C₂³
hypercenter	C₄×M₃₂	D₁₇
hypocenter	C₁₇	#12903
LCS	G ▷ C₂×C₈×C₁₇ ▷ C₄×C₁₇ ▷ C₂×C₁₇ ▷ C₁₇
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ C₄×M₃₂

Complements

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

N	H ≅ G/N	type
14. C₁₇×#346	C₄	semidirect
15. C₄×C₁₆×C₁₇	C₄	semidirect
54. C₁₆×C₁₇	C₄⋊C₄	semidirect
82. C₄×C₁₇	#365	semidirect
97. #346	Dic₁₇	semidirect
98. C₄×C₁₆	Dic₁₇	semidirect
121. C₁₇	#12903	semidirect
142. C₁₆	C₄⋊Dic₁₇	semidirect
166. C₄	#11762632	semidirect

Maximal subgroups

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

subgroup	index
C₂×#11762632	2
#14528956	2
#14528941	2
C₄×C₁₇×M₃₂	2
#12903	17