Subgroups — #17189830

Order 4352.

474 conjugacy classes of subgroups.

The subgroup lattice has 220 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#14539320	C₂	1	2	yes	maximal
2	#14531068	C₂	1	4	yes	maximal, complemented
3	#14531061	C₂	1	2	yes	maximal, complemented
4	#14531058	C₂	1	2	yes	maximal, complemented
5	#14530570	C₂	1	2	yes	maximal
6	#14530568	C₂	1	1	yes	maximal, characteristic
7	#14530380	C₂	1	1	yes	maximal, characteristic
8	C₂×C₁₇×#366	C₂	1	1	yes	maximal, characteristic
9	#11763440	—	2	2	no	—
10	C₂×#10587483	C₂²	1	1	yes	characteristic
11	#11763217	—	2	4	no	—
12	#11763213	C₂²	1	4	yes	—
13	#11763210	C₂²	1	4	yes	—
14	#11762849	—	2	2	no	—
15	#11762847	C₂²	1	2	yes	—
16	#11762764	C₂²	1	1	yes	characteristic
17	#11762637	C₂²	1	2	yes	—
18	#11762636	C₂²	1	4	yes	—
19	#11762635	C₂²	1	1	yes	characteristic
20	#11762635	C₂²	1	1	yes	characteristic
21	#11762634	C₂²	1	4	yes	—
22	#11762584	C₂²	1	2	yes	—
23	C₁₆×Dic₁₇	C₂²	1	2	yes	—
24	C₂²×C₁₆×C₁₇	C₂²	1	1	yes	characteristic
25	C₂×C₁₇×#107	C₂²	1	2	yes	—
26	C₁₇×#366	C₂²	1	2	yes	—
27	C₁₇×#366	C₂²	1	2	yes	—
28	C₂²×C₈×C₁₇	C₂³	1	1	yes	characteristic
29	C₂×C₄⋊C₄×C₁₇	—	2	2	no	—
30	#10587612	—	2	2	no	—
31	#10587612	D₄	1	1	yes	characteristic
32	#10587610	—	4	4	no	—
33	C₂×#63223	D₄	1	1	yes	characteristic
34	#10587576	—	2	4	no	—
35	#10587576	—	4	2	no	—
36	#10587574	—	2	4	no	—
37	#10587574	—	4	2	no	—
38	C₂×C₁₆×C₁₇	Q₈	1	4	yes	complemented
39	C₂×C₁₆×C₁₇	C₂³	1	1	yes	characteristic
40	C₂×C₁₆×C₁₇	C₂³	1	1	yes	characteristic
41	C₁₇×#107	—	2	2	no	—
42	C₁₇×#107	C₂³	1	4	yes	—
43	#10587489	C₂³	1	2	yes	—
44	#10587489	C₂³	1	2	yes	—
45	#10587488	—	2	2	no	—
46	#10587486	D₄	1	2	yes	—
47	C₈×Dic₁₇	C₂³	1	2	yes	—
48	#10587483	Q₈	1	4	yes	complemented
49	C₂×#63214	C₂³	1	2	yes	—
50	#10587479	—	2	4	no	—
51	#10587478	—	2	4	no	—
52	C₂²×C₄×C₁₇	—	4	2	no	—
53	C₂²×C₄×C₁₇	C₂×D₄	1	1	yes	characteristic
54	C₂×Dic₃₄	—	4	4	no	—
55	C₂×C₈×C₁₇	C₂×D₄	1	1	yes	characteristic
56	C₂×C₈×C₁₇	C₂×Q₈	1	2	yes	—
57	C₂×C₈×C₁₇	C₂×Q₈	1	2	yes	—
58	C₂×C₈×C₁₇	C₂⁴	1	1	yes	characteristic
59	C₄⋊C₄×C₁₇	—	2	4	no	—
60	C₄⋊C₄×C₁₇	—	4	2	no	—
61	C₂²⋊Dic₁₇	—	4	2	no	—
62	C₂²⋊Dic₁₇	D₈	1	2	yes	—
63	C₄⋊Dic₁₇	—	2	2	no	—
64	C₄⋊Dic₁₇	—	2	2	no	—
65	C₄⋊Dic₁₇	—	4	4	no	—
66	C₄⋊Dic₁₇	D₈	1	1	yes	characteristic
67	C₄⋊Dic₁₇	D₈	1	1	yes	characteristic
68	#63225	—	4	4	no	—
69	#63225	—	4	4	no	—
70	C₄×Dic₁₇	—	4	2	no	—
71	C₄×Dic₁₇	C₂×D₄	1	2	yes	—
72	#63223	Q₁₆	1	4	yes	—
73	C₂×#3622	C₂×D₄	1	2	yes	—
74	C₁₆×C₁₇	C₂×Q₈	1	2	yes	—
75	C₁₆×C₁₇	C₂×Q₈	1	2	yes	—
76	#63214	C₂×Q₈	1	4	yes	—
77	#33729	—	17	1	no	maximal, Sylow
78	C₂³×C₁₇	C₂×D₈	1	1	yes	characteristic
79	C₂×C₄×C₁₇	—	4	4	no	—
80	C₂×C₄×C₁₇	—	16	2	no	—
81	C₂×C₄×C₁₇	C₄⋊Q₈	1	2	yes	—
82	C₂×C₄×C₁₇	C₂×D₈	1	1	yes	characteristic
83	C₂×C₄×C₁₇	C₂×Q₁₆	1	2	yes	—
84	C₂×C₄×C₁₇	C₂²×D₄	1	1	yes	characteristic
85	C₂×Dic₁₇	—	4	2	no	—
86	C₂×Dic₁₇	—	4	2	no	—
87	C₂×Dic₁₇	C₂×D₈	1	2	yes	—
88	C₂×Dic₁₇	C₂×D₈	1	2	yes	—
89	Dic₃₄	—	16	4	no	—
90	C₈×C₁₇	C₄⋊Q₈	1	2	yes	—
91	C₈×C₁₇	C₂²×Q₈	1	1	yes	characteristic
92	C₈×C₁₇	C₂²×Q₈	1	1	yes	characteristic
93	#3622	C₂×Q₁₆	1	4	yes	—
94	#3164	—	17	2	no	—
95	#2255	—	17	4	no	—
96	#2253	—	17	2	no	—
97	#2252	—	17	2	no	—
98	#2159	—	17	2	no	—
99	#2158	—	17	1	no	—
100	C₂×#366	D₁₇	1	1	yes	characteristic
101	#2108	—	17	1	no	—
102	C₂²×C₁₇	—	2	1	no	—
103	C₂²×C₁₇	—	2	1	no	—
104	C₂²×C₁₇	#500	1	2	yes	—
105	C₂²×C₁₇	C₂²×D₈	1	1	yes	characteristic
106	C₄×C₁₇	—	8	4	no	—
107	C₄×C₁₇	#500	1	2	yes	—
108	C₄×C₁₇	C₂×C₄⋊Q₈	1	1	yes	characteristic
109	C₄×C₁₇	C₂²×Q₁₆	1	1	yes	characteristic
110	Dic₁₇	—	2	2	no	—
111	Dic₁₇	—	2	2	no	—
112	Dic₁₇	—	8	2	no	—
113	Dic₁₇	—	8	2	no	—
114	#544	—	34	2	no	—
115	C₂×M₃₂	—	17	1	no	—
116	C₂²×C₁₆	C₂×D₁₇	1	1	yes	characteristic
117	#501	—	34	4	no	—
118	#500	—	17	4	no	—
119	#499	—	17	4	no	—
120	#428	—	34	2	no	—
121	#427	—	17	2	no	—
122	C₂×#107	C₂×D₁₇	1	2	yes	—
123	#405	—	17	1	no	—
124	#367	—	17	2	no	—
125	#367	—	17	4	no	—
126	#366	—	17	1	no	—
127	#366	—	17	1	no	—
128	#366	—	17	4	no	—
129	#366	C₂×D₁₇	1	2	yes	—
130	#366	C₂×D₁₇	1	2	yes	—
131	#346	—	17	2	no	—
132	C₄×C₁₆	—	17	2	no	—
133	C₂×C₁₇	—	2	2	no	—
134	C₂×C₁₇	C₂×#500	1	1	yes	characteristic
135	C₂×C₁₇	#3415	1	1	yes	characteristic
136	C₂×C₁₇	#3417	1	1	yes	characteristic
137	C₂×M₁₆	—	17	1	no	—
138	C₂²×C₈	C₂²×D₁₇	1	1	yes	characteristic
139	C₄⋊Q₈	—	34	4	no	—
140	C₄⋊Q₈	—	68	2	no	—
141	#125	—	34	4	no	—
142	#125	—	68	2	no	—
143	C₂²⋊Q₈	—	68	4	no	—
144	#117	—	17	1	no	—
145	#117	—	34	2	no	—
146	C₂×C₄⋊C₄	—	2	2	no	—
147	M₃₂	—	17	4	no	—
148	C₂×C₁₆	—	17	2	no	—
149	C₂×C₁₆	Dic₃₄	1	4	yes	complemented
150	C₂×C₁₆	C₂²×D₁₇	1	1	yes	characteristic
151	C₂×C₁₆	C₂²×D₁₇	1	1	yes	characteristic
152	#107	—	2	2	no	—
153	#107	—	17	2	no	—
154	#107	—	17	2	no	—
155	#107	—	34	4	no	—
156	#107	C₂²×D₁₇	1	4	yes	—
157	#106	—	34	4	no	—
158	#106	—	34	2	no	—
159	#97	—	17	2	no	—
160	C₄×C₈	—	17	2	no	—
161	C₁₇	#33729	1	1	yes	characteristic, complemented, Sylow
162	C₂×Q₈	—	68	4	no	—
163	C₂²×C₄	—	4	2	no	—
164	C₂²×C₄	D₄×D₁₇	1	1	yes	characteristic
165	M₁₆	—	17	4	no	—
166	C₂×C₈	—	17	2	no	—
167	C₂×C₈	C₂×Dic₃₄	1	2	yes	—
168	C₂×C₈	D₄×D₁₇	1	1	yes	characteristic
169	C₂×C₈	Q₈×D₁₇	1	2	yes	—
170	C₂×C₈	C₂³×D₁₇	1	1	yes	characteristic
171	C₄⋊C₄	—	2	4	no	—
172	C₄⋊C₄	—	4	2	no	—
173	C₄⋊C₄	—	17	1	no	—
174	C₄⋊C₄	—	17	1	no	—
175	C₄⋊C₄	—	34	2	no	—
176	C₄⋊C₄	—	34	2	no	—
177	C₄⋊C₄	—	68	4	no	—
178	C₄⋊C₄	—	68	4	no	—
179	C₄⋊C₄	—	68	4	no	—
180	C₂²⋊C₄	—	17	2	no	—
181	C₂²⋊C₄	—	68	2	no	—
182	C₄²	—	17	2	no	—
183	C₄²	—	68	2	no	—
184	C₁₆	—	17	4	no	—
185	C₁₆	C₂×Dic₃₄	1	2	yes	—
186	C₁₆	C₂×Dic₃₄	1	2	yes	—
187	C₂³	D₈×D₁₇	1	1	yes	characteristic
188	Q₈	—	272	4	no	—
189	C₂×C₄	—	4	4	no	—
190	C₂×C₄	—	16	2	no	—
191	C₂×C₄	—	17	2	no	—
192	C₂×C₄	—	17	2	no	—
193	C₂×C₄	—	68	2	no	—
194	C₂×C₄	—	68	2	no	—
195	C₂×C₄	#10587574	1	2	yes	—
196	C₂×C₄	D₈×D₁₇	1	1	yes	characteristic
197	C₂×C₄	Q₁₆×D₁₇	1	2	yes	—
198	C₂×C₄	C₂×D₄×D₁₇	1	1	yes	characteristic
199	C₈	—	17	4	no	—
200	C₈	#10587574	1	2	yes	—
201	C₈	C₂²×Dic₃₄	1	1	yes	characteristic
202	C₈	C₂×Q₈×D₁₇	1	1	yes	characteristic
203	C₂²	—	2	1	no	—
204	C₂²	—	2	1	no	—
205	C₂²	#11763213	1	2	yes	—
206	C₂²	C₂×D₈×D₁₇	1	1	yes	characteristic
207	C₄	—	8	4	no	—
208	C₄	—	34	2	no	—
209	C₄	—	34	2	no	—
210	C₄	—	136	2	no	—
211	C₄	—	136	2	no	—
212	C₄	#11763213	1	2	yes	—
213	C₄	C₂×#10587574	1	1	yes	characteristic
214	C₄	C₂×Q₁₆×D₁₇	1	1	yes	characteristic
215	C₂	—	2	2	no	—
216	C₂	C₂×#11763213	1	1	yes	characteristic
217	C₂	#14541921	1	1	yes	characteristic
218	C₂	#14541933	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×D₈×D₁₇
commutator, cosocle	C₂×C₈×C₁₇	C₂⁴
Frattini	C₂×C₈	C₂³×D₁₇
Fitting	C₂×C₁₇×#366	C₂
socle	C₂²×C₁₇	C₂²×D₈
prime radical	C₂³×C₁₇	C₂×D₈
hypercenter	C₂×#366	D₁₇
hypocenter	C₁₇	#33729
LCS	G ▷ C₂×C₈×C₁₇ ▷ C₄×C₁₇ ▷ C₂×C₁₇ ▷ C₁₇
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂²×C₈ ◁ C₂×#366

Complements

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

N	H ≅ G/N	type
2. #14531068	C₂	semidirect
3. #14531061	C₂	semidirect
4. #14531058	C₂	semidirect
38. C₂×C₁₆×C₁₇	Q₈	semidirect
48. #10587483	Q₈	semidirect
149. C₂×C₁₆	Dic₃₄	semidirect
161. C₁₇	#33729	semidirect

Maximal subgroups

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

subgroup	index
#14539320	2
#14531068	2
#14531061	2
#14531058	2
#14530570	2
#14530568	2
#14530380	2
C₂×C₁₇×#366	2
#33729	17