Finite Group Databasebeta

Subgroups — #17961641

Order 4352.

234 conjugacy classes of subgroups; only the 76 normal subgroups are shown.

normal subgroups only

This group has 234 conjugacy classes of subgroups — too many to draw. Only its 76 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	#14544703	C₂	1	2	yes	maximal
2	#14543854	C₂	1	1	yes	maximal, characteristic
3	#14543852	C₂	1	1	yes	maximal, characteristic
4	#14543846	C₂	1	2	yes	maximal
5	#14530382	C₂	1	1	yes	maximal, characteristic
6	D₁₇⋊#108	C₂²	1	2	yes	—
7	#11763819	C₂²	1	2	yes	—
8	#11763691	—	2	2	no	—
9	#11763676	—	2	2	no	—
10	C₂×#10587469	C₂²	1	1	yes	characteristic
11	C₂×C₁₆×D₁₇	C₂²	1	1	yes	characteristic
12	#11762763	C₂²	1	1	yes	characteristic
13	#11762584	C₄	1	1	yes	characteristic, complemented
14	C₄×#63214	C₄	1	1	yes	characteristic, complemented
15	C₁₆×Dic₁₇	C₄	1	1	yes	characteristic, complemented
16	C₁₇×#346	C₄	1	1	yes	characteristic, complemented
17	#10587677	—	2	2	no	—
18	D₁₇⋊M₁₆	—	2	2	no	—
19	C₂×#63218	D₄	1	1	yes	characteristic
20	C₂×C₈×D₁₇	C₂³	1	1	yes	characteristic
21	#10587558	D₄	1	1	yes	characteristic
22	#10587536	—	2	2	no	—
23	#10587535	—	2	2	no	—
24	#10587534	—	2	2	no	—
25	#10587533	—	2	2	no	—
26	C₂×C₁₆×C₁₇	C₂×C₄	1	1	yes	characteristic
27	C₂×C₁₆×C₁₇	C₂×C₄	1	1	yes	characteristic
28	C₄×C₈×C₁₇	C₂×C₄	1	1	yes	characteristic
29	#10587486	Q₈	1	1	yes	characteristic
30	C₈×Dic₁₇	C₂×C₄	1	1	yes	characteristic
31	C₂×#63214	C₂×C₄	1	1	yes	characteristic
32	C₂×#63214	C₂×C₄	1	1	yes	characteristic
33	#10587474	Q₈	1	1	yes	characteristic
34	#10587469	—	2	2	no	—
35	C₁₆×D₁₇	—	2	2	no	—
36	C₂×C₄×D₁₇	C₂×D₄	1	1	yes	characteristic
37	C₂²⋊#595	—	4	2	no	—
38	#63247	—	4	2	no	—
39	C₄⋊#595	—	4	2	no	—
40	C₄×#595	—	4	2	no	—
41	#63243	—	4	2	no	—
42	#63242	—	4	2	no	—
43	C₂×C₈×C₁₇	C₄⋊C₄	1	1	yes	characteristic
44	C₂×C₈×C₁₇	C₂²×C₄	1	1	yes	characteristic
45	C₄²×C₁₇	C₄⋊C₄	1	1	yes	characteristic
46	#63227	—	2	1	no	—
47	#63225	—	2	1	no	—
48	C₄×Dic₁₇	C₄⋊C₄	1	1	yes	characteristic
49	C₂×#3622	C₄⋊C₄	1	1	yes	characteristic
50	C₂×#3622	C₂×Q₈	1	1	yes	characteristic
51	#63218	—	2	2	no	—
52	C₈×D₁₇	#41	1	2	yes	—
53	C₈×D₁₇	#41	1	2	yes	—
54	C₁₆×C₁₇	—	2	1	no	—
55	C₁₆×C₁₇	—	2	1	no	—
56	#63214	—	2	1	no	—
57	#63214	—	2	1	no	—
58	#12893	—	17	1	no	maximal, Sylow
59	C₂²×D₁₇	#135	1	1	yes	characteristic
60	C₂×#595	—	8	2	no	—
61	C₂×C₄×C₁₇	—	2	1	no	—
62	C₂×C₄×C₁₇	C₂×C₄⋊C₄	1	1	yes	characteristic
63	C₂×Dic₁₇	—	2	1	no	—
64	C₂×Dic₁₇	#135	1	1	yes	characteristic
65	C₄×D₁₇	#124	1	2	yes	—
66	C₄×D₁₇	#135	1	2	yes	—
67	#3624	—	8	2	no	—
68	C₈×C₁₇	—	2	1	no	—
69	C₈×C₁₇	#117	1	1	yes	characteristic
70	C₈×C₁₇	#117	1	1	yes	characteristic
71	#3622	—	2	1	no	—
72	#3622	#125	1	2	yes	—
73	#2108	—	17	1	no	—
74	#1833	—	17	2	no	—
75	#1385	—	17	1	no	—
76	#1384	—	17	1	no	—
77	#1382	—	17	2	no	—
78	C₂²×C₁₇	#427	1	1	yes	characteristic
79	C₂×D₁₇	—	4	1	no	—
80	C₂×D₁₇	#488	1	2	yes	—
81	#595	—	16	2	no	—
82	C₄×C₁₇	—	4	1	no	—
83	C₄×C₁₇	#382	1	1	yes	characteristic
84	C₄×C₁₇	#427	1	1	yes	characteristic
85	Dic₁₇	—	4	1	no	—
86	Dic₁₇	#488	1	2	yes	—
87	C₂×M₃₂	—	17	1	no	—
88	C₂²×C₁₆	—	17	1	no	—
89	C₂×#108	—	17	2	no	—
90	#427	—	17	2	no	—
91	#405	—	17	1	no	—
92	#346	—	17	1	no	—
93	#346	#595	1	1	yes	characteristic, complemented
94	C₄×C₁₆	—	17	1	no	—
95	C₄×C₁₆	—	17	1	no	—
96	#344	—	34	2	no	—
97	H(3,ℤ/4)	—	34	2	no	—
98	C₂×C₁₇	—	2	1	no	—
99	C₂×C₁₇	#1835	1	1	yes	characteristic
100	D₁₇	—	2	2	no	—
101	C₂×M₁₆	—	17	1	no	—
102	C₂×M₁₆	—	34	2	no	—
103	C₂²×C₈	—	17	1	no	—
104	#117	—	17	1	no	—
105	#117	—	34	2	no	—
106	M₃₂	—	34	2	no	—
107	C₂×C₁₆	—	17	1	no	—
108	C₂×C₁₆	—	17	1	no	—
109	C₂×C₁₆	—	34	2	no	—
110	C₂×C₁₆	C₂×#595	1	1	yes	characteristic
111	C₂×C₁₆	C₂×#595	1	1	yes	characteristic
112	#108	—	34	2	no	—
113	#108	—	34	2	no	—
114	#107	—	34	2	no	—
115	#106	—	34	2	no	—
116	#97	—	17	1	no	—
117	#97	—	17	1	no	—
118	C₄×C₈	—	17	1	no	—
119	C₄×C₈	C₂×#595	1	1	yes	characteristic
120	C₁₇	#12893	1	1	yes	characteristic, complemented, Sylow
121	C₂²×C₄	—	17	1	no	—
122	M₁₆	—	34	2	no	—
123	M₁₆	—	68	2	no	—
124	M₁₆	—	68	2	no	—
125	C₂×C₈	—	17	1	no	—
126	C₂×C₈	—	17	2	no	—
127	C₂×C₈	—	17	2	no	—
128	C₂×C₈	—	17	1	no	—
129	C₂×C₈	—	68	2	no	—
130	C₂×C₈	C₄⋊#595	1	1	yes	characteristic
131	C₂×C₈	C₂²×#595	1	1	yes	characteristic
132	C₄⋊C₄	—	34	1	no	—
133	C₄⋊C₄	—	68	2	no	—
134	C₂²⋊C₄	—	34	1	no	—
135	C₂²⋊C₄	—	68	2	no	—
136	C₄²	—	17	1	no	—
137	C₄²	—	68	2	no	—
138	C₄²	C₄⋊#595	1	1	yes	characteristic
139	C₁₆	—	2	1	no	—
140	C₁₆	—	2	1	no	—
141	C₁₆	—	34	1	no	—
142	C₁₆	—	34	1	no	—
143	C₂³	—	17	1	no	—
144	C₂×C₄	—	2	1	no	—
145	C₂×C₄	—	17	1	no	—
146	C₂×C₄	—	17	2	no	—
147	C₂×C₄	—	17	2	no	—
148	C₂×C₄	—	34	1	no	—
149	C₂×C₄	—	136	2	no	—
150	C₂×C₄	C₂×C₄⋊#595	1	1	yes	characteristic
151	C₈	—	2	1	no	—
152	C₈	—	17	2	no	—
153	C₈	—	34	1	no	—
154	C₈	—	136	2	no	—
155	C₈	#10587677	1	1	yes	characteristic
156	C₈	#10587677	1	1	yes	characteristic
157	C₂²	—	17	2	no	—
158	C₂²	—	68	1	no	—
159	C₂²	#11763819	1	1	yes	characteristic
160	C₄	—	4	1	no	—
161	C₄	—	272	2	no	—
162	C₄	—	17	2	no	—
163	C₄	—	68	1	no	—
164	C₄	#11763753	1	1	yes	characteristic
165	C₄	#11763819	1	1	yes	characteristic
166	C₂	—	2	1	no	—
167	C₂	—	34	2	no	—
168	C₂	#14544706	1	1	yes	characteristic

	subgroup	G/H
center	C₄	#11763819
commutator	C₂×C₈×C₁₇	C₂²×C₄
Frattini	C₂×C₈	C₂²×#595
Fitting	C₁₇×#346	C₄
socle	C₂×C₁₇	#1835
prime radical	C₂²×D₁₇	#135
cosocle	C₂×C₈×D₁₇	C₂³
hypercenter	#346	#595
hypocenter	C₁₇	#12893
LCS	G ▷ C₂×C₈×C₁₇ ▷ C₄×C₁₇ ▷ C₂×C₁₇ ▷ C₁₇
UCS	C₁ ◁ C₄ ◁ C₄² ◁ C₄×C₈ ◁ #346

Complements

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

N	H ≅ G/N	type
13. #11762584	C₄	semidirect
14. C₄×#63214	C₄	semidirect
15. C₁₆×Dic₁₇	C₄	semidirect
16. C₁₇×#346	C₄	semidirect
93. #346	#595	semidirect
120. C₁₇	#12893	semidirect

Maximal subgroups

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

subgroup	index
#14544703	2
#14543854	2
#14543852	2
#14543846	2
#14530382	2
#12893	17