Subgroups — #17948542

Order 4352.

242 conjugacy classes of subgroups.

The subgroup lattice has 170 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#11763672	C₂	1	2	yes	maximal
2	C₂×C₁₆×Dic₁₇	C₂	1	1	yes	maximal, characteristic
3	C₂×C₄⋊#3624	—	2	2	no	—
4	#11763672	—	2	2	no	—
5	#11763672	C₄	1	4	yes	—
6	C₂²×#63214	C₄	1	1	yes	characteristic
7	C₂×C₈×Dic₁₇	C₂²	1	1	yes	characteristic
8	C₁₆×Dic₁₇	—	2	2	no	—
9	C₂²×C₁₆×C₁₇	C₄	1	1	yes	characteristic
10	C₂²×#3624	—	4	2	no	—
11	C₂²×C₈×C₁₇	C₂×C₄	1	1	yes	characteristic
12	C₂×C₄×Dic₁₇	D₄	1	1	yes	characteristic
13	C₂²×#3622	Q₈	1	1	yes	characteristic
14	C₄⋊#3624	—	2	4	no	—
15	C₄⋊#3624	—	4	2	no	—
16	C₂×C₁₆×C₁₇	—	2	1	no	—
17	C₂×C₁₆×C₁₇	—	2	1	no	—
18	C₂×C₁₆×C₁₇	C₈	1	2	yes	complemented
19	C₈×Dic₁₇	C₂×C₄	1	2	yes	—
20	C₈×Dic₁₇	D₄	1	2	yes	—
21	C₂×#63214	—	2	1	no	—
22	C₂×#63214	—	2	1	no	—
23	C₂×#63214	C₈	1	2	yes	complemented
24	C₂²×C₄×C₁₇	C₄⋊C₄	1	1	yes	characteristic
25	C₂²×Dic₁₇	D₈	1	1	yes	characteristic
26	C₂²×Dic₁₇	Q₁₆	1	1	yes	characteristic
27	C₂×#3624	—	4	4	no	—
28	C₂×#3624	—	8	2	no	—
29	C₂×#3624	—	8	2	no	—
30	C₂×C₈×C₁₇	C₄²	1	1	yes	characteristic
31	C₂×C₈×C₁₇	C₂²⋊C₄	1	1	yes	characteristic
32	C₂×C₈×C₁₇	C₂×C₈	1	1	yes	characteristic
33	C₂×C₈×C₁₇	C₂×C₈	1	1	yes	characteristic
34	C₂×C₈×C₁₇	M₁₆	1	1	yes	characteristic
35	C₂×C₈×C₁₇	M₁₆	1	1	yes	characteristic
36	C₄×Dic₁₇	C₂²⋊C₄	1	2	yes	—
37	C₄×Dic₁₇	SD₁₆	1	2	yes	—
38	C₂×#3622	—	2	1	no	—
39	C₂×#3622	—	2	1	no	—
40	C₂×#3622	C₄⋊C₄	1	2	yes	—
41	C₁₆×C₁₇	—	2	2	no	—
42	#63214	—	2	2	no	—
43	#7154	—	17	1	no	maximal, Sylow
44	C₂³×C₁₇	#107	1	1	yes	characteristic
45	C₂×C₄×C₁₇	#95	1	1	yes	characteristic
46	C₂×C₄×C₁₇	C₄⋊C₈	1	1	yes	characteristic
47	C₂×C₄×C₁₇	C₄⋊C₈	1	1	yes	characteristic
48	C₂×C₄×C₁₇	#106	1	1	yes	characteristic
49	C₂×C₄×C₁₇	#108	1	1	yes	characteristic
50	C₂×C₄×C₁₇	#108	1	1	yes	characteristic
51	C₂×Dic₁₇	—	2	1	no	—
52	C₂×Dic₁₇	—	2	1	no	—
53	C₂×Dic₁₇	#102	1	2	yes	—
54	C₂×Dic₁₇	#103	1	2	yes	—
55	C₂×Dic₁₇	D₁₆	1	1	yes	characteristic
56	C₂×Dic₁₇	SD₃₂	1	2	yes	—
57	C₂×Dic₁₇	Q₃₂	1	1	yes	characteristic
58	#3624	—	8	4	no	—
59	C₈×C₁₇	C₄×C₈	1	1	yes	characteristic
60	C₈×C₁₇	#97	1	1	yes	characteristic
61	C₈×C₁₇	C₂²⋊C₈	1	1	yes	characteristic
62	C₈×C₁₇	C₂²⋊C₈	1	1	yes	characteristic
63	#3622	—	2	2	no	—
64	C₂×C₄×C₁₆	—	17	1	no	—
65	C₂×#335	—	17	2	no	—
66	C₂²×C₁₇	#335	1	1	yes	characteristic
67	C₂²×C₁₇	#335	1	1	yes	characteristic
68	C₂²×C₁₇	#340	1	1	yes	characteristic
69	C₂²×C₁₇	#366	1	1	yes	characteristic
70	C₂²×C₁₇	#367	1	1	yes	characteristic
71	C₂²×C₁₇	#368	1	1	yes	characteristic
72	C₂²×C₁₇	#368	1	1	yes	characteristic
73	C₄×C₁₇	#334	1	1	yes	characteristic
74	C₄×C₁₇	#334	1	1	yes	characteristic
75	C₄×C₁₇	#336	1	1	yes	characteristic
76	C₄×C₁₇	#341	1	1	yes	characteristic
77	Dic₁₇	—	2	2	no	—
78	Dic₁₇	#357	1	2	yes	—
79	Dic₁₇	#358	1	2	yes	—
80	C₂²×C₁₆	—	17	1	no	—
81	C₂²×C₁₆	#595	1	1	yes	characteristic
82	C₂×C₄⋊C₈	—	34	2	no	—
83	C₂×C₄×C₈	—	17	1	no	—
84	C₄×C₁₆	—	34	2	no	—
85	#335	—	17	4	no	—
86	#335	—	34	2	no	—
87	C₂×C₁₇	#1277	1	1	yes	characteristic
88	C₂×C₁₇	#1371	1	1	yes	characteristic
89	C₂×C₁₇	#1371	1	1	yes	characteristic
90	C₂×C₁₇	#1372	1	1	yes	characteristic
91	C₂×C₁₇	#1372	1	1	yes	characteristic
92	C₂×C₁₇	#1380	1	1	yes	characteristic
93	C₂×C₁₇	#1381	1	1	yes	characteristic
94	C₂²×C₈	—	17	1	no	—
95	C₂²×C₈	—	68	2	no	—
96	C₂²×C₈	C₂×#595	1	1	yes	characteristic
97	C₂×C₄²	—	17	1	no	—
98	C₂×C₁₆	—	2	1	no	—
99	C₂×C₁₆	—	2	1	no	—
100	C₂×C₁₆	—	17	2	no	—
101	C₂×C₁₆	—	34	1	no	—
102	C₂×C₁₆	—	34	1	no	—
103	C₂×C₁₆	#3624	1	2	yes	complemented
104	C₄⋊C₈	—	34	4	no	—
105	C₄⋊C₈	—	68	2	no	—
106	C₄×C₈	—	17	2	no	—
107	C₄×C₈	—	17	2	no	—
108	C₁₇	#7154	1	1	yes	characteristic, complemented, Sylow
109	C₂²×C₄	—	17	1	no	—
110	C₂²×C₄	—	17	1	no	—
111	C₂²×C₄	C₄⋊#595	1	1	yes	characteristic
112	C₂×C₈	—	17	2	no	—
113	C₂×C₈	—	34	1	no	—
114	C₂×C₈	—	34	1	no	—
115	C₂×C₈	—	68	4	no	—
116	C₂×C₈	—	136	2	no	—
117	C₂×C₈	—	136	2	no	—
118	C₂×C₈	#63242	1	1	yes	characteristic
119	C₂×C₈	#63243	1	1	yes	characteristic
120	C₂×C₈	C₄×#595	1	1	yes	characteristic
121	C₂×C₈	C₂×#3624	1	1	yes	characteristic
122	C₂×C₈	#63247	1	1	yes	characteristic
123	C₂×C₈	C₂²⋊#595	1	1	yes	characteristic
124	C₄²	—	17	2	no	—
125	C₄²	—	17	2	no	—
126	C₁₆	—	2	2	no	—
127	C₁₆	—	34	2	no	—
128	C₂³	#10587534	1	1	yes	characteristic
129	C₂×C₄	—	17	1	no	—
130	C₂×C₄	—	17	1	no	—
131	C₂×C₄	—	17	2	no	—
132	C₂×C₄	—	17	2	no	—
133	C₂×C₄	—	17	2	no	—
134	C₂×C₄	—	34	1	no	—
135	C₂×C₄	—	34	1	no	—
136	C₂×C₄	#10587533	1	1	yes	characteristic
137	C₂×C₄	#10587535	1	1	yes	characteristic
138	C₂×C₄	#10587536	1	1	yes	characteristic
139	C₂×C₄	C₄⋊#3624	1	1	yes	characteristic
140	C₂×C₄	#10587544	1	1	yes	characteristic
141	C₂×C₄	(C₂×#595)⋊C₄	1	1	yes	characteristic
142	C₈	—	34	2	no	—
143	C₈	—	136	4	no	—
144	C₈	C₄×#3624	1	1	yes	characteristic
145	C₈	#10587542	1	1	yes	characteristic
146	C₈	#10587543	1	1	yes	characteristic
147	C₈	C₂²⋊#3624	1	1	yes	characteristic
148	C₂²	#11763672	1	1	yes	characteristic
149	C₂²	#11763673	1	1	yes	characteristic
150	C₂²	#11763683	1	1	yes	characteristic
151	C₂²	#11763735	1	1	yes	characteristic
152	C₂²	#11763736	1	1	yes	characteristic
153	C₂²	#11763737	1	1	yes	characteristic
154	C₂²	#11763738	1	1	yes	characteristic
155	C₄	—	17	2	no	—
156	C₄	—	17	2	no	—
157	C₄	—	34	2	no	—
158	C₄	#11763670	1	1	yes	characteristic
159	C₄	#11763671	1	1	yes	characteristic
160	C₄	#11763675	1	1	yes	characteristic
161	C₄	#11763685	1	1	yes	characteristic
162	C₂	#14543614	1	1	yes	characteristic
163	C₂	#14543815	1	1	yes	characteristic
164	C₂	#14543816	1	1	yes	characteristic
165	C₂	#14543817	1	1	yes	characteristic
166	C₂	#14543818	1	1	yes	characteristic
167	C₂	#14543842	1	1	yes	characteristic
168	C₂	#14543844	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	#10587534
commutator	C₈×C₁₇	C₄×C₈
Frattini	C₂²×C₈	C₂×#595
Fitting	C₂²×C₁₆×C₁₇	C₄
socle, prime radical	C₂³×C₁₇	#107
cosocle	C₂×C₈×Dic₁₇	C₂²
hypercenter	C₂²×C₁₆	#595
hypocenter	C₁₇	#7154
LCS	G ▷ C₈×C₁₇ ▷ C₄×C₁₇ ▷ C₂×C₁₇ ▷ C₁₇
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄ ◁ C₂²×C₈ ◁ C₂²×C₁₆

Complements

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

N	H ≅ G/N	type
18. C₂×C₁₆×C₁₇	C₈	semidirect
23. C₂×#63214	C₈	semidirect
103. C₂×C₁₆	#3624	semidirect
108. C₁₇	#7154	semidirect

Maximal subgroups

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

subgroup	index
C₂×#11763672	2
C₂×C₁₆×Dic₁₇	2
#7154	17