Finite Group Databasebeta

Subgroups — #17960440

Order 4352.

350 conjugacy classes of subgroups; only the 168 normal subgroups are shown.

normal subgroups only

This group has 350 conjugacy classes of subgroups — too many to draw. Only its 168 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#14545007	C₂	1	1	yes	maximal, characteristic
2	#14545005	C₂	1	2	yes	maximal
3	#14544995	C₂	1	1	yes	maximal, characteristic
4	#14544993	C₂	1	2	yes	maximal
5	#14528538	C₂	1	1	yes	maximal, characteristic
6	C₂×#10587538	C₂²	1	2	yes	—
7	C₂×#10587538	C₂²	1	1	yes	characteristic
8	C₂²×#63216	C₂²	1	1	yes	characteristic
9	#11763702	C₄	1	4	yes	complemented
10	#11763701	C₄	1	8	yes	complemented
11	#11763698	C₄	1	2	yes	complemented
12	#11763697	C₄	1	4	yes	complemented
13	#11763695	C₄	1	2	yes	complemented
14	C₄×#63216	C₄	1	4	yes	complemented
15	#11762767	C₂²	1	1	yes	characteristic
16	#11762765	C₂²	1	2	yes	—
17	#11762567	C₄	1	1	yes	characteristic
18	#11762565	C₄	1	2	yes	—
19	C₁₇×H(3,ℤ/4)	C₄	1	1	yes	characteristic
20	C₂×M₁₆×C₁₇	C₂×C₄	1	1	yes	characteristic
21	C₁₇×#117	C₂×C₄	1	2	yes	—
22	#10587612	C₂×C₄	1	1	yes	characteristic
23	C₂×#63223	C₂×C₄	1	2	yes	—
24	C₂²×#3622	C₂³	1	1	yes	characteristic
25	#10587538	C₂×C₄	1	4	yes	—
26	#10587538	C₂×C₄	1	4	yes	—
27	#10587538	C₂×C₄	1	4	yes	—
28	C₂×#63216	C₂×C₄	1	2	yes	—
29	C₂×#63216	C₂×C₄	1	1	yes	characteristic
30	C₂×#63216	C₂×C₄	1	2	yes	—
31	C₂×#63216	C₂×C₄	1	1	yes	characteristic
32	C₂×#63216	C₂×C₄	1	4	yes	—
33	C₂×#63216	C₂×C₄	1	2	yes	—
34	#10587486	C₂×C₄	1	2	yes	—
35	C₈×Dic₁₇	C₂×C₄	1	2	yes	—
36	#10587474	C₂×C₄	1	4	yes	—
37	C₄×#3622	C₂×C₄	1	4	yes	—
38	C₂²×C₄×C₁₇	C₂²×C₄	1	1	yes	characteristic
39	M₁₆×C₁₇	—	2	2	no	—
40	C₂×C₈×C₁₇	C₄²	1	2	yes	—
41	C₄⋊C₄×C₁₇	—	2	2	no	—
42	C₂²⋊C₄×C₁₇	—	2	2	no	—
43	C₄²×C₁₇	C₄²	1	4	yes	—
44	C₂²⋊Dic₁₇	—	2	1	no	—
45	C₄⋊Dic₁₇	—	2	1	no	—
46	C₄×Dic₁₇	C₄²	1	2	yes	—
47	#63223	—	2	2	no	—
48	#63223	—	2	2	no	—
49	C₂×#3622	C₄²	1	4	yes	—
50	C₂×#3622	C₂²×C₄	1	2	yes	—
51	C₂×#3622	C₂²×C₄	1	2	yes	—
52	C₂×#3622	C₂²×C₄	1	1	yes	characteristic
53	C₂×#3622	C₂²×C₄	1	1	yes	characteristic
54	#63216	C₄²	1	1	yes	characteristic
55	#63216	C₄²	1	2	yes	—
56	#63216	C₄²	1	1	yes	characteristic
57	#63216	C₄²	1	8	yes	—
58	#63216	C₄²	1	4	yes	—
59	#12542	—	17	1	no	maximal, Sylow
60	C₂³×C₁₇	#131	1	1	yes	characteristic
61	C₂×C₄×C₁₇	—	2	4	no	—
62	C₂×C₄×C₁₇	C₂×C₄²	1	2	yes	—
63	C₂×C₄×C₁₇	C₂×C₄²	1	1	yes	characteristic
64	C₂×C₄×C₁₇	#131	1	2	yes	—
65	C₂×C₄×C₁₇	#131	1	1	yes	characteristic
66	C₂×Dic₁₇	—	2	2	no	—
67	C₈×C₁₇	—	2	2	no	—
68	#3622	—	2	4	no	—
69	#3622	C₂×C₄²	1	1	yes	characteristic
70	#3622	C₂×C₄²	1	2	yes	—
71	#3622	C₂×C₄²	1	1	yes	characteristic
72	#2110	—	17	2	no	—
73	#2110	—	17	1	no	—
74	#2108	—	17	2	no	—
75	#2108	—	17	1	no	—
76	#1747	—	17	1	no	—
77	C₂²×C₁₇	—	4	1	no	—
78	C₂²×C₁₇	#405	1	2	yes	—
79	C₂²×C₁₇	#405	1	1	yes	characteristic
80	C₄×C₁₇	—	4	4	no	—
81	C₄×C₁₇	C₄³	1	1	yes	characteristic
82	C₄×C₁₇	#405	1	1	yes	characteristic
83	C₄×C₁₇	#405	1	2	yes	—
84	Dic₁₇	—	4	2	no	—
85	C₂×M₃₂	—	17	2	no	—
86	C₂×M₃₂	—	17	1	no	—
87	C₂²×C₁₆	—	17	1	no	—
88	#405	—	17	1	no	—
89	#405	—	17	2	no	—
90	#347	—	17	8	no	—
91	#347	—	17	4	no	—
92	#346	—	17	4	no	—
93	#346	—	17	2	no	—
94	C₄×C₁₆	—	17	4	no	—
95	C₄×C₁₆	—	17	2	no	—
96	#338	—	17	1	no	—
97	H(3,ℤ/4)	—	17	2	no	—
98	H(3,ℤ/4)	#595	1	1	yes	characteristic
99	C₂×C₁₇	—	2	1	no	—
100	C₂×C₁₇	—	2	2	no	—
101	C₂×C₁₇	#1726	1	1	yes	characteristic
102	C₂×M₁₆	—	17	2	no	—
103	C₂×M₁₆	C₂×#595	1	1	yes	characteristic
104	C₂²×C₈	—	17	1	no	—
105	#117	—	17	1	no	—
106	#117	C₂×#595	1	2	yes	—
107	M₃₂	—	17	4	no	—
108	M₃₂	—	17	4	no	—
109	M₃₂	—	17	4	no	—
110	C₂×C₁₆	—	17	2	no	—
111	C₂×C₁₆	—	17	1	no	—
112	C₂×C₁₆	—	17	4	no	—
113	C₂×C₁₆	—	17	2	no	—
114	C₂×C₁₆	—	17	2	no	—
115	C₂×C₁₆	—	17	1	no	—
116	#97	—	17	2	no	—
117	#97	—	17	4	no	—
118	C₄×C₈	—	17	2	no	—
119	C₄×C₈	—	17	4	no	—
120	C₁₇	#12542	1	1	yes	characteristic, complemented, Sylow
121	C₂²×C₄	C₂²×#595	1	1	yes	characteristic
122	M₁₆	—	2	2	no	—
123	M₁₆	—	34	2	no	—
124	M₁₆	—	34	2	no	—
125	C₂×C₈	—	17	1	no	—
126	C₂×C₈	—	17	2	no	—
127	C₂×C₈	—	17	1	no	—
128	C₂×C₈	—	17	2	no	—
129	C₂×C₈	—	17	4	no	—
130	C₂×C₈	C₄×#595	1	2	yes	—
131	C₄⋊C₄	—	2	2	no	—
132	C₄⋊C₄	—	34	1	no	—
133	C₂²⋊C₄	—	2	2	no	—
134	C₂²⋊C₄	—	34	1	no	—
135	C₄²	—	17	2	no	—
136	C₄²	C₄×#595	1	4	yes	—
137	C₁₆	—	17	1	no	—
138	C₁₆	—	17	2	no	—
139	C₁₆	—	17	1	no	—
140	C₁₆	—	17	8	no	—
141	C₁₆	—	17	4	no	—
142	C₂³	#10587678	1	1	yes	characteristic
143	C₂×C₄	—	2	4	no	—
144	C₂×C₄	—	34	2	no	—
145	C₂×C₄	C₂×C₄×#595	1	1	yes	characteristic
146	C₂×C₄	C₂×C₄×#595	1	2	yes	—
147	C₂×C₄	#10587678	1	2	yes	—
148	C₂×C₄	#10587680	1	1	yes	characteristic
149	C₈	—	2	2	no	—
150	C₈	—	17	1	no	—
151	C₈	—	17	1	no	—
152	C₈	—	17	2	no	—
153	C₈	—	34	4	no	—
154	C₂²	—	4	1	no	—
155	C₂²	#11763773	1	2	yes	—
156	C₂²	#11763774	1	1	yes	characteristic
157	C₄	—	4	4	no	—
158	C₄	—	68	2	no	—
159	C₄	C₄²×#595	1	1	yes	characteristic
160	C₄	#11763773	1	1	yes	characteristic
161	C₄	#11763774	1	2	yes	—
162	C₂	—	2	2	no	—
163	C₂	—	2	1	no	—
164	C₂	#14544441	1	1	yes	characteristic

	subgroup	G/H
center	C₄	C₄²×#595
commutator	C₄×C₁₇	C₄³
Frattini	C₂²×C₄	C₂²×#595
Fitting	C₁₇×H(3,ℤ/4)	C₄
socle	C₂×C₁₇	#1726
prime radical	C₂³×C₁₇	#131
cosocle	C₂²×#3622	C₂³
hypercenter	H(3,ℤ/4)	#595
hypocenter	C₁₇	#12542
LCS	G ▷ C₄×C₁₇ ▷ C₁₇
UCS	C₁ ◁ C₄ ◁ H(3,ℤ/4)

Complements

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

N	H ≅ G/N	type
9. #11763702	C₄	semidirect
10. #11763701	C₄	semidirect
11. #11763698	C₄	semidirect
12. #11763697	C₄	semidirect
13. #11763695	C₄	semidirect
14. C₄×#63216	C₄	semidirect
120. C₁₇	#12542	semidirect

Maximal subgroups

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

subgroup	index
#14545007	2
#14545005	2
#14544995	2
#14544993	2
#14528538	2
#12542	17