Subgroups — #16950755

Order 4352.

318 conjugacy classes of subgroups.

The subgroup lattice has 174 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#11762633	C₂	1	4	yes	maximal
2	C₄×#10587483	C₂	1	1	yes	maximal, characteristic
3	#14528948	C₂	1	1	yes	maximal, characteristic
4	C₁₇×#1848	C₂	1	1	yes	maximal, characteristic
5	C₂×#10587483	C₂²	1	2	yes	—
6	C₂×#10587489	—	2	1	no	—
7	C₂×#10587488	C₂²	1	2	yes	—
8	#11762675	—	2	1	no	—
9	#11762633	—	2	8	no	—
10	#11762585	C₄	1	2	yes	complemented
11	C₄×#63214	C₄	1	2	yes	complemented
12	C₂×C₁₇×#108	—	2	1	no	—
13	C₂×C₁₇×#108	C₂²	1	2	yes	—
14	C₁₇×#423	—	2	1	no	—
15	C₂×C₄×C₈×C₁₇	C₂²	1	1	yes	characteristic
16	C₂×M₁₆×C₁₇	—	2	2	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	C₁₇×#108	—	2	4	no	—
22	C₁₇×#108	—	4	2	no	—
23	C₁₇×C₄⋊C₈	—	4	2	no	—
24	C₄×C₈×C₁₇	C₂×C₄	1	1	yes	characteristic
25	C₄×C₈×C₁₇	C₂×C₄	1	1	yes	characteristic
26	C₄×C₈×C₁₇	Q₈	1	1	yes	characteristic
27	C₄×C₈×C₁₇	Q₈	1	1	yes	characteristic
28	#10587502	—	4	2	no	—
29	#10587489	—	4	2	no	—
30	#10587488	—	2	4	no	—
31	#10587483	D₄	1	4	yes	—
32	#10587483	Q₈	1	4	yes	—
33	C₂×#63214	C₂×C₄	1	4	yes	—
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	M₁₆×C₁₇	—	4	2	no	—
38	M₁₆×C₁₇	—	4	2	no	—
39	C₂×C₈×C₁₇	—	4	2	no	—
40	C₂×C₈×C₁₇	C₄⋊C₄	1	1	yes	characteristic
41	C₂×C₈×C₁₇	C₄⋊C₄	1	1	yes	characteristic
42	C₂×C₈×C₁₇	SD₁₆	1	2	yes	—
43	C₂×C₈×C₁₇	SD₁₆	1	2	yes	—
44	C₂×C₈×C₁₇	C₂²×C₄	1	1	yes	characteristic
45	C₂×C₈×C₁₇	C₂×D₄	1	2	yes	—
46	C₂×C₈×C₁₇	C₂×Q₈	1	1	yes	characteristic
47	C₂×C₈×C₁₇	C₂×Q₈	1	2	yes	—
48	C₄²×C₁₇	C₄⋊C₄	1	1	yes	characteristic
49	C₄²×C₁₇	C₄⋊C₄	1	1	yes	characteristic
50	C₄²×C₁₇	SD₁₆	1	2	yes	—
51	C₄⋊Dic₁₇	—	4	2	no	—
52	C₄⋊Dic₁₇	—	4	2	no	—
53	#63214	C₄⋊C₄	1	8	yes	—
54	#12904	—	17	1	no	maximal, Sylow
55	C₂³×C₁₇	C₂×SD₁₆	1	1	yes	characteristic
56	C₂×C₄×C₁₇	—	4	2	no	—
57	C₂×C₄×C₁₇	#106	1	2	yes	—
58	C₂×C₄×C₁₇	#106	1	2	yes	—
59	C₂×C₄×C₁₇	C₂×C₄⋊C₄	1	1	yes	characteristic
60	C₂×C₄×C₁₇	#127	1	1	yes	characteristic
61	C₂×C₄×C₁₇	C₄⋊Q₈	1	1	yes	characteristic
62	C₂×C₄×C₁₇	C₂×SD₁₆	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×Dic₁₇	—	8	4	no	—
66	C₂×Dic₁₇	—	8	2	no	—
67	C₈×C₁₇	—	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	C₂×#365	—	17	4	no	—
73	C₄×M₃₂	—	17	1	no	—
74	#1848	D₁₇	1	1	yes	characteristic
75	#1846	—	17	1	no	—
76	C₂²×C₁₇	—	2	2	no	—
77	C₂²×C₁₇	C₂×#106	1	1	yes	characteristic
78	C₂²×C₁₇	#492	1	1	yes	characteristic
79	C₂²×C₁₇	#498	1	1	yes	characteristic
80	C₄×C₁₇	—	2	2	no	—
81	C₄×C₁₇	—	2	2	no	—
82	C₄×C₁₇	#384	1	1	yes	characteristic
83	C₄×C₁₇	C₂×#106	1	1	yes	characteristic
84	C₄×C₁₇	#492	1	1	yes	characteristic
85	C₄×C₁₇	#498	1	1	yes	characteristic
86	Dic₁₇	—	16	2	no	—
87	C₂×M₃₂	—	17	2	no	—
88	C₂×#108	—	2	1	no	—
89	C₂×#108	C₂×D₁₇	1	2	yes	—
90	C₂×#107	—	34	1	no	—
91	C₂×#106	—	17	2	no	—
92	#423	—	2	1	no	—
93	C₂×C₄×C₈	C₂×D₁₇	1	1	yes	characteristic
94	#382	—	34	1	no	—
95	#365	—	34	8	no	—
96	#346	—	17	2	no	—
97	C₄×C₁₆	—	17	2	no	—
98	C₂×C₁₇	—	2	1	no	—
99	C₂×C₁₇	—	2	1	no	—
100	C₂×C₁₇	#1844	1	1	yes	characteristic
101	C₂×C₁₇	#2155	1	2	yes	—
102	C₂×M₁₆	—	2	2	no	—
103	C₂²×C₈	C₂²⋊D₁₇	1	1	yes	characteristic
104	C₂²×C₈	C₂²×D₁₇	1	1	yes	characteristic
105	C₂×C₄⋊C₄	—	34	2	no	—
106	C₂×C₄²	C₂²⋊D₁₇	1	1	yes	characteristic
107	M₃₂	—	17	4	no	—
108	M₃₂	—	17	4	no	—
109	C₂×C₁₆	—	17	4	no	—
110	#108	—	2	4	no	—
111	#108	—	4	2	no	—
112	#107	—	68	2	no	—
113	#106	—	34	4	no	—
114	C₄⋊C₈	—	4	2	no	—
115	C₄×C₈	Dic₃₄	1	1	yes	characteristic
116	C₄×C₈	Dic₃₄	1	1	yes	characteristic
117	C₄×C₈	C₄×D₁₇	1	1	yes	characteristic
118	C₄×C₈	C₄×D₁₇	1	1	yes	characteristic
119	#95	—	68	2	no	—
120	C₁₇	#12904	1	1	yes	characteristic, complemented, Sylow
121	C₂²×C₄	—	68	2	no	—
122	C₂²×C₄	#63230	1	2	yes	—
123	C₂²×C₄	C₂×C₂²⋊D₁₇	1	1	yes	characteristic
124	M₁₆	—	4	2	no	—
125	M₁₆	—	4	2	no	—
126	C₂×C₈	—	4	2	no	—
127	C₂×C₈	#63225	1	1	yes	characteristic
128	C₂×C₈	#63225	1	1	yes	characteristic
129	C₂×C₈	#63229	1	2	yes	—
130	C₂×C₈	#63230	1	2	yes	—
131	C₂×C₈	C₂×Dic₃₄	1	1	yes	characteristic
132	C₂×C₈	C₂×C₄×D₁₇	1	1	yes	characteristic
133	C₂×C₈	D₄×D₁₇	1	2	yes	—
134	C₂×C₈	Q₈×D₁₇	1	2	yes	—
135	C₄⋊C₄	—	68	2	no	—
136	C₄⋊C₄	—	68	2	no	—
137	C₄²	#63225	1	1	yes	characteristic
138	C₄²	#63225	1	1	yes	characteristic
139	C₄²	#63229	1	2	yes	—
140	C₁₆	—	17	8	no	—
141	C₂³	C₂×#63230	1	1	yes	characteristic
142	C₂×C₄	—	4	2	no	—
143	C₂×C₄	—	136	4	no	—
144	C₂×C₄	—	136	2	no	—
145	C₂×C₄	#10587479	1	2	yes	—
146	C₂×C₄	#10587479	1	2	yes	—
147	C₂×C₄	C₂×#63225	1	1	yes	characteristic
148	C₂×C₄	C₂×#63229	1	1	yes	characteristic
149	C₂×C₄	C₂×#63229	1	1	yes	characteristic
150	C₂×C₄	#10587626	1	1	yes	characteristic
151	C₂×C₄	C₂×#63230	1	1	yes	characteristic
152	C₂×C₄	#10587630	1	1	yes	characteristic
153	C₈	—	8	2	no	—
154	C₈	#10587479	1	2	yes	—
155	C₈	#10587479	1	2	yes	—
156	C₈	#10587574	1	2	yes	—
157	C₈	C₄⋊C₄×D₁₇	1	2	yes	—
158	C₂²	—	2	2	no	—
159	C₂²	C₂×#10587479	1	1	yes	characteristic
160	C₂²	#11763183	1	1	yes	characteristic
161	C₂²	#11763205	1	1	yes	characteristic
162	C₄	—	2	2	no	—
163	C₄	—	2	2	no	—
164	C₄	—	272	2	no	—
165	C₄	#11762678	1	1	yes	characteristic
166	C₄	C₂×#10587479	1	1	yes	characteristic
167	C₄	#11763181	1	1	yes	characteristic
168	C₄	#11763205	1	1	yes	characteristic
169	C₂	—	2	1	no	—
170	C₂	—	2	1	no	—
171	C₂	#14528940	1	1	yes	characteristic
172	C₂	#14530558	1	2	yes	—

	subgroup	G/H
center	C₂²	C₂×#10587479
commutator	C₂×C₈×C₁₇	C₂²×C₄
Frattini	C₂²×C₈	C₂²×D₁₇
Fitting	C₁₇×#1848	C₂
socle	C₂²×C₁₇	C₂×#106
prime radical	C₂³×C₁₇	C₂×SD₁₆
cosocle	C₂²×C₈×C₁₇	C₂³
hypercenter	#1848	D₁₇
hypocenter	C₁₇	#12904
LCS	G ▷ C₂×C₈×C₁₇ ▷ C₄×C₁₇ ▷ C₂×C₁₇ ▷ C₁₇
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ #1848

Complements

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

N	H ≅ G/N	type
10. #11762585	C₄	semidirect
11. C₄×#63214	C₄	semidirect
120. C₁₇	#12904	semidirect

Maximal subgroups

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

subgroup	index
C₂×#11762633	2
C₄×#10587483	2
#14528948	2
C₁₇×#1848	2
#12904	17