Subgroups — #17960503

Order 4352.

606 conjugacy classes of subgroups.

The subgroup lattice has 224 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#11763819	C₂	1	2	yes	maximal
2	#14543842	C₂	1	4	yes	maximal, complemented
3	C₂²×C₁₆×D₁₇	C₂	1	1	yes	maximal, characteristic
4	C₂×#10587677	—	2	2	no	—
5	#11763819	—	2	4	no	—
6	#11763819	C₄	1	8	yes	complemented
7	C₂×#10587534	C₂²	1	4	yes	—
8	C₂×#10587533	—	2	2	no	—
9	C₂²×C₈×D₁₇	C₂²	1	1	yes	characteristic
10	C₂²×#63214	C₄	1	1	yes	characteristic, complemented
11	C₂×C₁₆×D₁₇	—	2	4	no	—
12	C₂×C₁₆×D₁₇	C₄	1	2	yes	complemented
13	C₂×C₁₆×D₁₇	C₂²	1	2	yes	—
14	C₂²×C₁₆×C₁₇	C₄	1	1	yes	characteristic, complemented
15	C₂²×C₄×D₁₇	D₄	1	1	yes	characteristic
16	C₂×C₂²⋊#595	—	4	2	no	—
17	#10587677	—	2	8	no	—
18	#10587677	—	4	4	no	—
19	C₂×C₄⋊#595	—	2	4	no	—
20	C₂×C₄×#595	—	4	2	no	—
21	C₂²×C₈×C₁₇	C₂×C₄	1	1	yes	characteristic
22	C₂²×#3622	Q₈	1	1	yes	characteristic
23	C₂×C₈×D₁₇	C₂×C₄	1	4	yes	—
24	C₂×C₈×D₁₇	C₂×C₄	1	1	yes	characteristic
25	C₂×C₈×D₁₇	D₄	1	4	yes	—
26	C₂×C₈×D₁₇	D₄	1	1	yes	characteristic
27	C₂×C₈×D₁₇	Q₈	1	1	yes	characteristic
28	C₂×C₈×D₁₇	C₂³	1	1	yes	characteristic
29	#10587534	—	2	4	no	—
30	#10587534	C₂×C₄	1	8	yes	—
31	#10587533	—	2	8	no	—
32	C₂×C₁₆×C₁₇	—	2	2	no	—
33	C₂×C₁₆×C₁₇	C₂×C₄	1	2	yes	—
34	C₂×#63214	—	2	2	no	—
35	C₂×#63214	C₂×C₄	1	2	yes	—
36	C₁₆×D₁₇	—	2	4	no	—
37	C₁₆×D₁₇	—	2	4	no	—
38	C₂³×D₁₇	D₈	1	1	yes	characteristic
39	C₂²×#595	—	4	4	no	—
40	C₂²×C₄×C₁₇	C₄⋊C₄	1	1	yes	characteristic
41	C₂²×Dic₁₇	Q₁₆	1	1	yes	characteristic
42	C₂×C₄×D₁₇	C₂²⋊C₄	1	4	yes	—
43	C₂×C₄×D₁₇	C₄⋊C₄	1	1	yes	characteristic
44	C₂×C₄×D₁₇	D₈	1	1	yes	characteristic
45	C₂×C₄×D₁₇	SD₁₆	1	4	yes	—
46	C₂×C₄×D₁₇	Q₁₆	1	1	yes	characteristic
47	C₂×C₄×D₁₇	C₂×D₄	1	1	yes	characteristic
48	C₂²⋊#595	—	4	8	no	—
49	C₄⋊#595	—	2	8	no	—
50	C₄⋊#595	—	4	4	no	—
51	C₄×#595	—	4	8	no	—
52	C₂×C₈×C₁₇	C₄²	1	2	yes	—
53	C₂×C₈×C₁₇	C₂²⋊C₄	1	2	yes	—
54	C₂×C₈×C₁₇	C₄⋊C₄	1	1	yes	characteristic
55	C₂×C₈×C₁₇	C₂²×C₄	1	1	yes	characteristic
56	C₂×#3622	C₄⋊C₄	1	1	yes	characteristic
57	C₂×#3622	C₄⋊C₄	1	4	yes	—
58	C₂×#3622	C₂×Q₈	1	1	yes	characteristic
59	C₈×D₁₇	C₄²	1	2	yes	—
60	C₈×D₁₇	C₂²⋊C₄	1	4	yes	—
61	C₈×D₁₇	C₂²⋊C₄	1	2	yes	—
62	C₈×D₁₇	C₄⋊C₄	1	4	yes	—
63	C₈×D₁₇	C₂²×C₄	1	2	yes	—
64	C₈×D₁₇	C₂×D₄	1	2	yes	—
65	C₁₆×C₁₇	—	2	2	no	—
66	#63214	—	2	2	no	—
67	#12556	—	17	1	no	maximal, Sylow
68	C₂³×C₁₇	#107	1	1	yes	characteristic
69	C₂²×D₁₇	—	2	4	no	—
70	C₂²×D₁₇	#102	1	4	yes	—
71	C₂²×D₁₇	#107	1	1	yes	characteristic
72	C₂²×D₁₇	C₂×D₈	1	1	yes	characteristic
73	C₂×#595	—	4	8	no	—
74	C₂×#595	—	8	8	no	—
75	C₂×C₄×C₁₇	#95	1	2	yes	—
76	C₂×C₄×C₁₇	#106	1	2	yes	—
77	C₂×C₄×C₁₇	#107	1	1	yes	characteristic
78	C₂×C₄×C₁₇	C₂×C₄⋊C₄	1	1	yes	characteristic
79	C₂×Dic₁₇	#103	1	4	yes	—
80	C₂×Dic₁₇	#107	1	1	yes	characteristic
81	C₂×Dic₁₇	C₂×Q₁₆	1	1	yes	characteristic
82	C₄×D₁₇	#95	1	2	yes	—
83	C₄×D₁₇	#102	1	4	yes	—
84	C₄×D₁₇	#103	1	4	yes	—
85	C₄×D₁₇	#106	1	2	yes	—
86	C₄×D₁₇	C₂×C₂²⋊C₄	1	2	yes	—
87	C₄×D₁₇	C₂×SD₁₆	1	2	yes	—
88	C₈×C₁₇	#95	1	2	yes	—
89	C₈×C₁₇	C₂×C₄²	1	1	yes	characteristic
90	C₈×C₁₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
91	#3622	#95	1	2	yes	—
92	#3622	C₂×C₄⋊C₄	1	2	yes	—
93	C₂³×C₁₆	—	17	1	no	—
94	C₂×#427	—	17	2	no	—
95	#1380	—	17	4	no	—
96	C₂²×C₁₇	—	2	2	no	—
97	C₂²×C₁₇	#340	1	2	yes	—
98	C₂²×C₁₇	C₂×#107	1	1	yes	characteristic
99	C₂×D₁₇	—	2	2	no	—
100	C₂×D₁₇	—	2	8	no	—
101	C₂×D₁₇	#340	1	2	yes	—
102	C₂×D₁₇	C₂×#102	1	2	yes	—
103	C₂×D₁₇	#508	1	4	yes	—
104	#595	—	8	8	no	—
105	C₄×C₁₇	#340	1	2	yes	—
106	C₄×C₁₇	C₂×#95	1	1	yes	characteristic
107	C₄×C₁₇	C₂×#106	1	1	yes	characteristic
108	Dic₁₇	#340	1	2	yes	—
109	Dic₁₇	C₂×#103	1	2	yes	—
110	C₂³×C₈	—	17	1	no	—
111	C₂×#117	—	34	2	no	—
112	C₂²×C₁₆	—	17	1	no	—
113	C₂²×C₁₆	—	17	2	no	—
114	C₂²×C₁₆	—	17	2	no	—
115	C₂²×C₁₆	—	34	4	no	—
116	C₂²×C₁₆	#595	1	1	yes	characteristic, complemented
117	#427	—	17	8	no	—
118	#427	—	34	4	no	—
119	C₂×#107	—	17	4	no	—
120	C₂×#106	—	34	2	no	—
121	C₂×C₁₇	—	2	2	no	—
122	C₂×C₁₇	C₂×#340	1	1	yes	characteristic
123	C₂×C₁₇	#2158	1	2	yes	—
124	D₁₇	—	2	2	no	—
125	D₁₇	#2138	1	4	yes	—
126	C₂³×C₄	—	17	1	no	—
127	C₂²×C₈	—	17	1	no	—
128	C₂²×C₈	—	17	4	no	—
129	C₂²×C₈	—	17	1	no	—
130	C₂²×C₈	—	17	1	no	—
131	C₂²×C₈	—	17	4	no	—
132	C₂²×C₈	—	17	1	no	—
133	C₂²×C₈	—	17	1	no	—
134	C₂²×C₈	C₂×#595	1	1	yes	characteristic
135	#117	—	34	8	no	—
136	#117	—	68	4	no	—
137	C₂×C₄⋊C₄	—	34	4	no	—
138	C₂×C₂²⋊C₄	—	68	2	no	—
139	C₂×C₄²	—	68	2	no	—
140	C₂×C₁₆	—	2	2	no	—
141	C₂×C₁₆	—	17	2	no	—
142	C₂×C₁₆	—	34	4	no	—
143	C₂×C₁₆	—	34	4	no	—
144	C₂×C₁₆	—	34	2	no	—
145	C₂×C₁₆	C₂×#595	1	2	yes	—
146	#107	—	17	8	no	—
147	#107	—	34	4	no	—
148	#106	—	34	8	no	—
149	C₁₇	#12556	1	1	yes	characteristic, complemented, Sylow
150	C₂⁴	—	17	1	no	—
151	C₂²×C₄	—	17	1	no	—
152	C₂²×C₄	—	17	1	no	—
153	C₂²×C₄	—	17	1	no	—
154	C₂²×C₄	—	17	4	no	—
155	C₂²×C₄	—	17	1	no	—
156	C₂²×C₄	—	17	1	no	—
157	C₂²×C₄	—	17	4	no	—
158	C₂²×C₄	—	68	4	no	—
159	C₂²×C₄	C₄⋊#595	1	1	yes	characteristic
160	C₂×C₈	—	17	1	no	—
161	C₂×C₈	—	17	1	no	—
162	C₂×C₈	—	17	4	no	—
163	C₂×C₈	—	17	2	no	—
164	C₂×C₈	—	17	2	no	—
165	C₂×C₈	—	17	2	no	—
166	C₂×C₈	—	17	2	no	—
167	C₂×C₈	—	17	4	no	—
168	C₂×C₈	—	17	4	no	—
169	C₂×C₈	C₄×#595	1	2	yes	—
170	C₂×C₈	C₄⋊#595	1	1	yes	characteristic
171	C₂×C₈	C₂²⋊#595	1	2	yes	—
172	C₂×C₈	C₂²×#595	1	1	yes	characteristic
173	C₄⋊C₄	—	34	8	no	—
174	C₄⋊C₄	—	68	4	no	—
175	C₂²⋊C₄	—	68	8	no	—
176	C₄²	—	68	8	no	—
177	C₁₆	—	2	2	no	—
178	C₁₆	—	34	2	no	—
179	C₂³	—	17	1	no	—
180	C₂³	—	17	1	no	—
181	C₂³	—	17	4	no	—
182	C₂³	—	34	4	no	—
183	C₂³	#10587534	1	1	yes	characteristic
184	C₂×C₄	—	17	1	no	—
185	C₂×C₄	—	17	1	no	—
186	C₂×C₄	—	17	4	no	—
187	C₂×C₄	—	17	2	no	—
188	C₂×C₄	—	17	2	no	—
189	C₂×C₄	—	17	2	no	—
190	C₂×C₄	—	17	2	no	—
191	C₂×C₄	—	17	4	no	—
192	C₂×C₄	—	17	4	no	—
193	C₂×C₄	—	68	8	no	—
194	C₂×C₄	—	136	8	no	—
195	C₂×C₄	#10587533	1	2	yes	—
196	C₂×C₄	#10587534	1	1	yes	characteristic
197	C₂×C₄	(C₂×#595)⋊C₄	1	2	yes	—
198	C₂×C₄	C₂×C₄⋊#595	1	1	yes	characteristic
199	C₈	—	17	2	no	—
200	C₈	—	17	2	no	—
201	C₈	(C₂×#595)⋊C₄	1	2	yes	—
202	C₈	C₂×C₄×#595	1	1	yes	characteristic
203	C₈	C₂×C₂²⋊#595	1	1	yes	characteristic
204	C₂²	—	2	2	no	—
205	C₂²	—	17	2	no	—
206	C₂²	—	17	4	no	—
207	C₂²	—	17	2	no	—
208	C₂²	—	34	8	no	—
209	C₂²	—	34	2	no	—
210	C₂²	#11763683	1	2	yes	—
211	C₂²	C₂×#10587534	1	1	yes	characteristic
212	C₄	—	17	2	no	—
213	C₄	—	17	2	no	—
214	C₄	—	136	8	no	—
215	C₄	#11763683	1	2	yes	—
216	C₄	C₂×(C₂×#595)⋊C₄	1	1	yes	characteristic
217	C₄	C₂×#10587533	1	1	yes	characteristic
218	C₂	—	2	2	no	—
219	C₂	—	17	4	no	—
220	C₂	—	34	2	no	—
221	C₂	C₂×#11763683	1	1	yes	characteristic
222	C₂	#14545135	1	2	yes	—

	subgroup	G/H
center	C₂×C₄	#10587534
commutator	C₈×C₁₇	C₂×C₄²
Frattini	C₂×C₈	C₂²×#595
Fitting	C₂²×C₁₆×C₁₇	C₄
socle	C₂²×C₁₇	C₂×#107
prime radical	C₂³×D₁₇	D₈
cosocle	C₂×C₈×D₁₇	C₂³
hypercenter	C₂²×C₁₆	#595
hypocenter	C₁₇	#12556
LCS	G ▷ C₈×C₁₇ ▷ C₄×C₁₇ ▷ C₂×C₁₇ ▷ C₁₇
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄ ◁ C₂²×C₈ ◁ C₂²×C₁₆

Complements

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

N	H ≅ G/N	type
2. #14543842	C₂	semidirect
6. #11763819	C₄	semidirect
10. C₂²×#63214	C₄	semidirect
12. C₂×C₁₆×D₁₇	C₄	semidirect
14. C₂²×C₁₆×C₁₇	C₄	semidirect
116. C₂²×C₁₆	#595	semidirect
149. C₁₇	#12556	semidirect

Maximal subgroups

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

subgroup	index
C₂×#11763819	2
#14543842	2
C₂²×C₁₆×D₁₇	2
#12556	17