Subgroups — #17189824

Order 4352.

482 conjugacy classes of subgroups.

The subgroup lattice has 216 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#14539298	C₂	1	1	yes	maximal, characteristic
2	#14531071	C₂	1	4	yes	maximal, complemented
3	#14531067	C₂	1	4	yes	maximal, complemented
4	#14530566	C₂	1	4	yes	maximal
5	C₂×#11762585	C₂	1	1	yes	maximal, characteristic
6	C₁₇×#3160	C₂	1	1	yes	maximal, characteristic
7	#11763366	—	2	1	no	—
8	C₂²×#63214	C₂²	1	2	yes	—
9	#11763214	C₂²	1	2	yes	—
10	#11763211	—	2	1	no	—
11	#11763211	C₂²	1	2	yes	—
12	#11763207	—	2	1	no	—
13	#11762847	—	2	1	no	—
14	#11762847	C₂²	1	2	yes	—
15	#11762636	C₂²	1	8	yes	—
16	#11762634	C₂²	1	8	yes	—
17	#11762585	C₂²	1	2	yes	—
18	#11762585	C₂²	1	2	yes	—
19	C₁₇×#533	—	2	1	no	—
20	C₁₇×#500	C₂²	1	2	yes	—
21	C₁₇×#499	—	2	1	no	—
22	C₁₇×#499	C₂²	1	2	yes	—
23	C₁₇×#498	—	2	1	no	—
24	C₁₇×#427	—	2	1	no	—
25	C₁₇×#427	C₂²	1	2	yes	—
26	C₂×C₄×C₈×C₁₇	C₂²	1	1	yes	characteristic
27	C₂²×C₈×C₁₇	D₄	1	1	yes	characteristic
28	C₂²×C₈×C₁₇	C₂³	1	1	yes	characteristic
29	C₁₇×C₄⋊Q₈	—	2	2	no	—
30	C₁₇×#125	—	2	2	no	—
31	C₁₇×C₂²⋊Q₈	—	4	2	no	—
32	C₄×Q₈×C₁₇	—	4	2	no	—
33	C₁₇×#117	—	2	2	no	—
34	C₂×C₄²×C₁₇	D₄	1	1	yes	characteristic
35	#10587612	—	2	2	no	—
36	#10587610	—	4	2	no	—
37	#10587556	—	2	2	no	—
38	#10587555	—	2	2	no	—
39	C₄×Dic₃₄	—	4	2	no	—
40	C₁₇×#107	—	2	2	no	—
41	C₁₇×#107	C₂³	1	4	yes	—
42	C₁₇×#106	—	2	2	no	—
43	C₁₇×#106	—	2	2	no	—
44	C₄×C₈×C₁₇	D₄	1	1	yes	characteristic
45	C₄×C₈×C₁₇	D₄	1	1	yes	characteristic
46	C₄×C₈×C₁₇	C₂³	1	1	yes	characteristic
47	C₄×C₈×C₁₇	C₂³	1	1	yes	characteristic
48	#10587489	—	2	2	no	—
49	#10587489	C₂³	1	4	yes	—
50	#10587488	—	2	2	no	—
51	#10587488	—	2	2	no	—
52	C₂×#63214	Q₈	1	8	yes	complemented
53	C₂×#63214	C₂³	1	4	yes	—
54	C₂×Q₈×C₁₇	—	4	2	no	—
55	C₂²×C₄×C₁₇	D₈	1	2	yes	—
56	C₂²×C₄×C₁₇	C₂×D₄	1	1	yes	characteristic
57	C₂×Dic₃₄	—	4	2	no	—
58	C₂×C₈×C₁₇	Q₁₆	1	4	yes	—
59	C₂×C₈×C₁₇	C₂×D₄	1	1	yes	characteristic
60	C₂×C₈×C₁₇	C₂×D₄	1	1	yes	characteristic
61	C₂×C₈×C₁₇	C₂×D₄	1	1	yes	characteristic
62	C₂×C₈×C₁₇	C₂×Q₈	1	4	yes	—
63	C₂×C₈×C₁₇	C₂⁴	1	1	yes	characteristic
64	C₄⋊C₄×C₁₇	—	2	4	no	—
65	C₄⋊C₄×C₁₇	—	4	4	no	—
66	C₄⋊C₄×C₁₇	—	4	2	no	—
67	C₂²⋊C₄×C₁₇	—	4	2	no	—
68	C₄²×C₁₇	—	4	2	no	—
69	C₄²×C₁₇	D₈	1	2	yes	—
70	C₄²×C₁₇	C₂×D₄	1	1	yes	characteristic
71	C₄²×C₁₇	C₂×D₄	1	1	yes	characteristic
72	C₂²⋊Dic₁₇	—	4	2	no	—
73	C₄⋊Dic₁₇	—	2	4	no	—
74	C₄⋊Dic₁₇	—	4	2	no	—
75	#63225	—	4	4	no	—
76	C₄×Dic₁₇	—	4	2	no	—
77	#63214	C₂×Q₈	1	8	yes	—
78	#33728	—	17	1	no	maximal, Sylow
79	C₂³×C₁₇	C₂×D₈	1	1	yes	characteristic
80	Q₈×C₁₇	—	8	4	no	—
81	C₂×C₄×C₁₇	—	4	2	no	—
82	C₂×C₄×C₁₇	—	4	4	no	—
83	C₂×C₄×C₁₇	C₄⋊Q₈	1	2	yes	—
84	C₂×C₄×C₁₇	C₂×D₈	1	2	yes	—
85	C₂×C₄×C₁₇	C₂×D₈	1	2	yes	—
86	C₂×C₄×C₁₇	C₂×D₈	1	1	yes	characteristic
87	C₂×C₄×C₁₇	C₂×Q₁₆	1	2	yes	—
88	C₂×C₄×C₁₇	C₂²×D₄	1	1	yes	characteristic
89	C₂×Dic₁₇	—	4	4	no	—
90	Dic₃₄	—	8	4	no	—
91	C₈×C₁₇	C₄⋊Q₈	1	2	yes	—
92	C₈×C₁₇	C₂×Q₁₆	1	2	yes	—
93	C₈×C₁₇	C₂×Q₁₆	1	2	yes	—
94	C₈×C₁₇	C₂²×Q₈	1	2	yes	—
95	#3160	—	17	1	no	—
96	#3160	D₁₇	1	1	yes	characteristic
97	#2255	—	17	4	no	—
98	#2255	—	17	4	no	—
99	#2158	—	17	4	no	—
100	C₂×#346	—	17	1	no	—
101	C₂²×C₁₇	—	2	2	no	—
102	C₂²×C₁₇	#500	1	2	yes	—
103	C₂²×C₁₇	C₂²×D₈	1	1	yes	characteristic
104	C₄×C₁₇	—	2	2	no	—
105	C₄×C₁₇	—	2	2	no	—
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	—	8	4	no	—
111	#533	—	2	1	no	—
112	#533	—	34	1	no	—
113	C₂²×C₁₆	—	17	2	no	—
114	#500	—	17	2	no	—
115	#500	C₂×D₁₇	1	2	yes	—
116	#499	—	2	1	no	—
117	#499	—	17	2	no	—
118	#499	—	34	1	no	—
119	#499	C₂×D₁₇	1	2	yes	—
120	#498	—	2	1	no	—
121	#498	—	34	1	no	—
122	#427	—	2	1	no	—
123	#427	—	17	2	no	—
124	#427	—	34	1	no	—
125	#427	C₂×D₁₇	1	2	yes	—
126	C₂×C₄×C₈	C₂×D₁₇	1	1	yes	characteristic
127	#367	—	17	8	no	—
128	#366	—	17	8	no	—
129	#346	—	17	2	no	—
130	#346	—	17	2	no	—
131	C₂×C₁₇	—	2	2	no	—
132	C₂×C₁₇	C₂×#500	1	1	yes	characteristic
133	C₂×C₁₇	#3414	1	2	yes	—
134	C₂²×C₈	C₂²⋊D₁₇	1	1	yes	characteristic
135	C₂²×C₈	C₂²×D₁₇	1	1	yes	characteristic
136	C₄⋊Q₈	—	2	2	no	—
137	C₄⋊Q₈	—	34	2	no	—
138	#125	—	2	2	no	—
139	#125	—	34	2	no	—
140	C₂²⋊Q₈	—	4	2	no	—
141	C₂²⋊Q₈	—	68	2	no	—
142	C₄×Q₈	—	4	2	no	—
143	C₄×Q₈	—	68	2	no	—
144	#117	—	2	2	no	—
145	#117	—	34	2	no	—
146	C₂×C₄²	C₂²⋊D₁₇	1	1	yes	characteristic
147	C₂×C₁₆	—	17	8	no	—
148	C₂×C₁₆	—	17	4	no	—
149	#107	—	2	2	no	—
150	#107	—	17	4	no	—
151	#107	—	34	2	no	—
152	#107	C₂²×D₁₇	1	4	yes	—
153	#106	—	2	2	no	—
154	#106	—	2	2	no	—
155	#106	—	34	2	no	—
156	#106	—	34	2	no	—
157	C₄×C₈	C₂²⋊D₁₇	1	1	yes	characteristic
158	C₄×C₈	C₂²⋊D₁₇	1	1	yes	characteristic
159	C₄×C₈	C₂²×D₁₇	1	1	yes	characteristic
160	C₄×C₈	C₂²×D₁₇	1	1	yes	characteristic
161	C₁₇	#33728	1	1	yes	characteristic, complemented, Sylow
162	C₂×Q₈	—	4	2	no	—
163	C₂×Q₈	—	68	2	no	—
164	C₂²×C₄	#63228	1	2	yes	—
165	C₂²×C₄	C₂×C₂²⋊D₁₇	1	1	yes	characteristic
166	C₂×C₈	#63231	1	4	yes	—
167	C₂×C₈	Q₈×D₁₇	1	4	yes	—
168	C₂×C₈	C₂×C₂²⋊D₁₇	1	1	yes	characteristic
169	C₂×C₈	C₂×C₂²⋊D₁₇	1	1	yes	characteristic
170	C₂×C₈	C₂×C₂²⋊D₁₇	1	1	yes	characteristic
171	C₂×C₈	C₂³×D₁₇	1	1	yes	characteristic
172	C₄⋊C₄	—	2	4	no	—
173	C₄⋊C₄	—	4	4	no	—
174	C₄⋊C₄	—	4	2	no	—
175	C₄⋊C₄	—	34	4	no	—
176	C₄⋊C₄	—	68	4	no	—
177	C₄⋊C₄	—	68	2	no	—
178	C₂²⋊C₄	—	4	2	no	—
179	C₂²⋊C₄	—	68	2	no	—
180	C₄²	—	4	2	no	—
181	C₄²	—	68	2	no	—
182	C₄²	#63228	1	2	yes	—
183	C₄²	C₂×C₂²⋊D₁₇	1	1	yes	characteristic
184	C₄²	C₂×C₂²⋊D₁₇	1	1	yes	characteristic
185	C₁₆	—	17	8	no	—
186	C₂³	C₂×#63228	1	1	yes	characteristic
187	Q₈	—	8	4	no	—
188	Q₈	—	136	4	no	—
189	C₂×C₄	—	4	2	no	—
190	C₂×C₄	—	4	4	no	—
191	C₂×C₄	—	68	4	no	—
192	C₂×C₄	C₂×#63228	1	2	yes	—
193	C₂×C₄	C₂×#63228	1	2	yes	—
194	C₂×C₄	C₂×#63228	1	1	yes	characteristic
195	C₂×C₄	C₂×#63231	1	2	yes	—
196	C₂×C₄	#10587630	1	2	yes	—
197	C₂×C₄	C₂²×C₂²⋊D₁₇	1	1	yes	characteristic
198	C₈	C₂×#63231	1	2	yes	—
199	C₈	C₂×#63231	1	2	yes	—
200	C₈	#10587630	1	2	yes	—
201	C₈	C₂×Q₈×D₁₇	1	2	yes	—
202	C₂²	—	2	2	no	—
203	C₂²	#11763212	1	2	yes	—
204	C₂²	C₂²×#63228	1	1	yes	characteristic
205	C₄	—	2	2	no	—
206	C₄	—	2	2	no	—
207	C₄	—	8	4	no	—
208	C₄	—	136	4	no	—
209	C₄	#11763212	1	2	yes	—
210	C₄	C₂×#10587630	1	1	yes	characteristic
211	C₄	C₂²×#63231	1	1	yes	characteristic
212	C₂	—	2	2	no	—
213	C₂	C₂×#11763212	1	1	yes	characteristic
214	C₂	#14541918	1	2	yes	—

	subgroup	G/H
center	C₂×C₄	C₂×#63228
commutator, cosocle	C₂×C₈×C₁₇	C₂⁴
Frattini	C₂×C₈	C₂³×D₁₇
Fitting	C₁₇×#3160	C₂
socle	C₂²×C₁₇	C₂²×D₈
prime radical	C₂³×C₁₇	C₂×D₈
hypercenter	#3160	D₁₇
hypocenter	C₁₇	#33728
LCS	G ▷ C₂×C₈×C₁₇ ▷ C₄×C₁₇ ▷ C₂×C₁₇ ▷ C₁₇
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ #3160

Complements

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

N	H ≅ G/N	type
2. #14531071	C₂	semidirect
3. #14531067	C₂	semidirect
52. C₂×#63214	Q₈	semidirect
161. C₁₇	#33728	semidirect

Maximal subgroups

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

subgroup	index
#14539298	2
#14531071	2
#14531067	2
#14530566	2
C₂×#11762585	2
C₁₇×#3160	2
#33728	17