Subgroups — #16950531

Order 4352.

366 conjugacy classes of subgroups.

The subgroup lattice has 240 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#11762639	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×#11762637	C₂	1	1	yes	maximal, characteristic, complemented
3	#14528848	C₂	1	2	yes	maximal
4	#14526893	C₂	1	2	yes	maximal, complemented
5	C₂×C₁₇×#504	C₂	1	1	yes	maximal, characteristic
6	C₂×#10587490	C₂²	1	2	yes	—
7	C₂×#10587489	C₂²	1	2	yes	—
8	#11762810	—	2	2	no	—
9	#11762639	—	2	2	no	—
10	#11762637	—	2	1	no	—
11	#11762637	—	2	1	no	—
12	#11762577	—	2	2	no	—
13	C₂×C₁₇×#131	C₂²	1	1	yes	characteristic
14	C₁₇×#504	—	2	2	no	—
15	C₁₇×#504	C₄	1	4	yes	complemented
16	C₂×C₁₇×M₃₂	—	2	1	no	—
17	C₂×C₁₇×M₃₂	C₂²	1	1	yes	characteristic
18	C₂²×C₁₆×C₁₇	—	2	1	no	—
19	C₂²×C₁₆×C₁₇	C₂²	1	1	yes	characteristic
20	C₂×#41×C₁₇	D₄	1	1	yes	characteristic
21	C₁₇×#131	—	2	1	no	—
22	C₁₇×#131	—	2	1	no	—
23	C₁₇×#131	C₂×C₄	1	2	yes	—
24	C₁₇×#131	Q₈	1	2	yes	—
25	C₂×M₁₆×C₁₇	—	2	1	no	—
26	C₂×M₁₆×C₁₇	D₄	1	1	yes	characteristic
27	C₂²×C₈×C₁₇	—	2	1	no	—
28	C₂²×C₈×C₁₇	C₂³	1	1	yes	characteristic
29	C₂×C₄⋊Dic₁₇	—	2	2	no	—
30	C₂×#63223	—	2	2	no	—
31	C₁₇×M₃₂	—	2	4	no	—
32	C₁₇×M₃₂	—	2	1	no	—
33	C₁₇×M₃₂	C₂×C₄	1	2	yes	—
34	C₂×C₁₆×C₁₇	—	2	4	no	—
35	C₂×C₁₆×C₁₇	—	2	1	no	—
36	C₂×C₁₆×C₁₇	C₂×C₄	1	2	yes	—
37	C₂×C₁₆×C₁₇	D₄	1	1	yes	characteristic
38	C₂×C₁₆×C₁₇	D₄	1	1	yes	characteristic
39	C₂×C₁₆×C₁₇	D₄	1	2	yes	—
40	#10587506	—	4	2	no	—
41	#10587503	—	4	2	no	—
42	#10587490	—	2	2	no	—
43	#10587490	—	2	2	no	—
44	#10587489	—	2	2	no	—
45	#10587489	—	2	2	no	—
46	#41×C₁₇	—	2	1	no	—
47	#41×C₁₇	—	2	1	no	—
48	#41×C₁₇	C₄⋊C₄	1	2	yes	—
49	#41×C₁₇	Q₁₆	1	2	yes	—
50	C₂×Q₈×C₁₇	D₈	1	1	yes	characteristic
51	C₂×D₄×C₁₇	—	2	1	no	—
52	C₂×D₄×C₁₇	D₈	1	1	yes	characteristic
53	C₂²×C₄×C₁₇	—	2	1	no	—
54	C₂²×C₄×C₁₇	C₂×D₄	1	1	yes	characteristic
55	C₂²×Dic₁₇	—	4	2	no	—
56	M₁₆×C₁₇	—	2	1	no	—
57	M₁₆×C₁₇	—	2	2	no	—
58	M₁₆×C₁₇	—	2	2	no	—
59	M₁₆×C₁₇	C₄⋊C₄	1	2	yes	—
60	C₂×C₈×C₁₇	—	2	1	no	—
61	C₂×C₈×C₁₇	—	2	2	no	—
62	C₂×C₈×C₁₇	—	2	2	no	—
63	C₂×C₈×C₁₇	C₂²×C₄	1	1	yes	characteristic
64	C₂×C₈×C₁₇	C₂×D₄	1	1	yes	characteristic
65	C₂×C₈×C₁₇	C₂×D₄	1	1	yes	characteristic
66	C₂×C₈×C₁₇	C₂×Q₈	1	1	yes	characteristic
67	C₂×C₈×C₁₇	#41	1	1	yes	characteristic
68	C₂×C₈×C₁₇	#41	1	1	yes	characteristic
69	C₄⋊Dic₁₇	—	4	2	no	—
70	C₄⋊Dic₁₇	—	4	2	no	—
71	#63223	—	4	2	no	—
72	#63223	—	4	2	no	—
73	C₂×#3622	—	4	2	no	—
74	C₁₆×C₁₇	—	2	2	no	—
75	C₁₆×C₁₇	C₂²⋊C₄	1	2	yes	—
76	C₁₆×C₁₇	C₂²⋊C₄	1	2	yes	—
77	#12843	—	17	1	no	maximal, Sylow
78	C₂³×C₁₇	—	2	1	no	—
79	C₂³×C₁₇	#135	1	1	yes	characteristic
80	Q₈×C₁₇	—	2	1	no	—
81	Q₈×C₁₇	#107	1	2	yes	—
82	D₄×C₁₇	—	2	2	no	—
83	D₄×C₁₇	—	2	2	no	—
84	D₄×C₁₇	—	2	1	no	—
85	D₄×C₁₇	#107	1	2	yes	—
86	C₂×C₄×C₁₇	—	2	1	no	—
87	C₂×C₄×C₁₇	—	2	2	no	—
88	C₂×C₄×C₁₇	—	2	2	no	—
89	C₂×C₄×C₁₇	C₂×C₄⋊C₄	1	1	yes	characteristic
90	C₂×C₄×C₁₇	#121	1	1	yes	characteristic
91	C₂×C₄×C₁₇	#121	1	1	yes	characteristic
92	C₂×C₄×C₁₇	C₂×D₈	1	1	yes	characteristic
93	C₂×C₄×C₁₇	C₂×Q₁₆	1	1	yes	characteristic
94	C₂×C₄×C₁₇	#135	1	1	yes	characteristic
95	C₂×Dic₁₇	—	8	2	no	—
96	C₂×Dic₁₇	—	16	2	no	—
97	C₈×C₁₇	—	2	2	no	—
98	C₈×C₁₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
99	C₈×C₁₇	#117	1	1	yes	characteristic
100	C₈×C₁₇	C₂²⋊Q₈	1	1	yes	characteristic
101	C₈×C₁₇	C₂²⋊Q₈	1	1	yes	characteristic
102	#3622	—	8	2	no	—
103	C₂×#504	D₁₇	1	1	yes	characteristic
104	C₂×#368	—	17	1	no	—
105	C₂×#367	—	17	1	no	—
106	#1816	—	17	2	no	—
107	#1387	—	17	2	no	—
108	C₂²×C₁₇	—	2	2	no	—
109	C₂²×C₁₇	—	2	2	no	—
110	C₂²×C₁₇	—	4	2	no	—
111	C₂²×C₁₇	#427	1	1	yes	characteristic
112	C₂²×C₁₇	#466	1	1	yes	characteristic
113	C₂²×C₁₇	#467	1	1	yes	characteristic
114	C₄×C₁₇	—	2	2	no	—
115	C₄×C₁₇	#380	1	1	yes	characteristic
116	C₄×C₁₇	C₂×#107	1	1	yes	characteristic
117	C₄×C₁₇	#466	1	1	yes	characteristic
118	C₄×C₁₇	#467	1	1	yes	characteristic
119	Dic₁₇	—	16	2	no	—
120	C₂×#131	C₂×D₁₇	1	1	yes	characteristic
121	#504	—	2	2	no	—
122	#504	Dic₁₇	1	4	yes	complemented
123	C₂×M₃₂	—	2	1	no	—
124	C₂×M₃₂	C₂×D₁₇	1	1	yes	characteristic
125	C₂²×C₁₆	—	2	1	no	—
126	C₂²×C₁₆	C₂×D₁₇	1	1	yes	characteristic
127	C₂×#108	—	17	2	no	—
128	C₂×#107	—	17	2	no	—
129	#417	—	34	2	no	—
130	#368	—	34	2	no	—
131	#367	—	34	1	no	—
132	#367	—	34	1	no	—
133	#343	—	34	2	no	—
134	C₂×C₁₇	—	2	1	no	—
135	C₂×C₁₇	—	2	1	no	—
136	C₂×C₁₇	—	4	2	no	—
137	C₂×C₁₇	#1808	1	1	yes	characteristic
138	C₂×C₁₇	#2223	1	2	yes	—
139	C₂×#41	D₆₈	1	1	yes	characteristic
140	#131	—	2	1	no	—
141	#131	—	2	1	no	—
142	#131	Dic₃₄	1	2	yes	—
143	#131	C₂×Dic₁₇	1	2	yes	—
144	C₂×M₁₆	—	2	1	no	—
145	C₂×M₁₆	—	34	2	no	—
146	C₂×M₁₆	D₆₈	1	1	yes	characteristic
147	C₂²×C₈	—	2	1	no	—
148	C₂²×C₈	C₂²×D₁₇	1	1	yes	characteristic
149	C₂×C₄⋊C₄	—	34	2	no	—
150	M₃₂	—	2	1	no	—
151	M₃₂	—	2	4	no	—
152	M₃₂	C₂×Dic₁₇	1	2	yes	—
153	C₂×C₁₆	—	2	1	no	—
154	C₂×C₁₆	—	2	4	no	—
155	C₂×C₁₆	C₂×Dic₁₇	1	2	yes	—
156	C₂×C₁₆	C₂²⋊D₁₇	1	1	yes	characteristic
157	C₂×C₁₆	C₂²⋊D₁₇	1	1	yes	characteristic
158	C₂×C₁₆	C₂²⋊D₁₇	1	2	yes	—
159	#108	—	34	2	no	—
160	#108	—	34	2	no	—
161	#107	—	34	2	no	—
162	#107	—	34	2	no	—
163	#103	—	68	2	no	—
164	#102	—	68	2	no	—
165	C₁₇	#12843	1	1	yes	characteristic, complemented, Sylow
166	#41	—	2	1	no	—
167	#41	—	2	1	no	—
168	#41	Dic₆₈	1	2	yes	—
169	#41	C₄⋊Dic₁₇	1	2	yes	—
170	C₂×Q₈	D₁₃₆	1	1	yes	characteristic
171	C₂×D₄	—	2	1	no	—
172	C₂×D₄	D₁₃₆	1	1	yes	characteristic
173	C₂²×C₄	—	2	1	no	—
174	C₂²×C₄	—	68	2	no	—
175	C₂²×C₄	C₂×D₆₈	1	1	yes	characteristic
176	M₁₆	—	2	1	no	—
177	M₁₆	—	2	2	no	—
178	M₁₆	—	2	2	no	—
179	M₁₆	—	68	2	no	—
180	M₁₆	—	68	2	no	—
181	M₁₆	C₄⋊Dic₁₇	1	2	yes	—
182	C₂×C₈	—	2	1	no	—
183	C₂×C₈	—	2	2	no	—
184	C₂×C₈	—	2	2	no	—
185	C₂×C₈	—	68	2	no	—
186	C₂×C₈	C₂×Dic₃₄	1	1	yes	characteristic
187	C₂×C₈	#63252	1	1	yes	characteristic
188	C₂×C₈	#63252	1	1	yes	characteristic
189	C₂×C₈	C₂²×Dic₁₇	1	1	yes	characteristic
190	C₂×C₈	C₂×C₂²⋊D₁₇	1	1	yes	characteristic
191	C₂×C₈	C₂×C₂²⋊D₁₇	1	1	yes	characteristic
192	C₄⋊C₄	—	68	2	no	—
193	C₄⋊C₄	—	68	2	no	—
194	C₁₆	—	2	2	no	—
195	C₁₆	C₂²⋊Dic₁₇	1	2	yes	—
196	C₁₆	C₂²⋊Dic₁₇	1	2	yes	—
197	C₂³	—	2	1	no	—
198	C₂³	#10587590	1	1	yes	characteristic
199	Q₈	—	2	1	no	—
200	Q₈	#10587489	1	2	yes	—
201	D₄	—	2	2	no	—
202	D₄	—	2	2	no	—
203	D₄	—	2	1	no	—
204	D₄	#10587489	1	2	yes	—
205	C₂×C₄	—	2	1	no	—
206	C₂×C₄	—	2	2	no	—
207	C₂×C₄	—	2	2	no	—
208	C₂×C₄	—	272	2	no	—
209	C₂×C₄	—	136	2	no	—
210	C₂×C₄	C₂×D₁₃₆	1	1	yes	characteristic
211	C₂×C₄	#10587590	1	1	yes	characteristic
212	C₂×C₄	C₂×Dic₆₈	1	1	yes	characteristic
213	C₂×C₄	C₂×C₄⋊Dic₁₇	1	1	yes	characteristic
214	C₂×C₄	#10587616	1	1	yes	characteristic
215	C₂×C₄	#10587616	1	1	yes	characteristic
216	C₈	—	2	2	no	—
217	C₈	—	136	2	no	—
218	C₈	#10587610	1	1	yes	characteristic
219	C₈	#10587610	1	1	yes	characteristic
220	C₈	#10587612	1	1	yes	characteristic
221	C₈	C₂×C₂²⋊Dic₁₇	1	1	yes	characteristic
222	C₂²	—	2	2	no	—
223	C₂²	—	2	2	no	—
224	C₂²	—	4	2	no	—
225	C₂²	#11762847	1	1	yes	characteristic
226	C₂²	#11763045	1	1	yes	characteristic
227	C₂²	#11763050	1	1	yes	characteristic
228	C₄	—	2	2	no	—
229	C₄	—	272	2	no	—
230	C₄	#11762669	1	1	yes	characteristic
231	C₄	C₂×#10587489	1	1	yes	characteristic
232	C₄	#11763045	1	1	yes	characteristic
233	C₄	#11763050	1	1	yes	characteristic
234	C₂	—	2	1	no	—
235	C₂	—	2	1	no	—
236	C₂	—	4	2	no	—
237	C₂	#14528813	1	1	yes	characteristic
238	C₂	#14530914	1	2	yes	—

	subgroup	G/H
center	C₂²	#11763045
commutator	C₂×C₈×C₁₇	C₂²×C₄
Frattini	C₂²×C₈	C₂²×D₁₇
Fitting	C₂×C₁₇×#504	C₂
socle	C₂²×C₁₇	#466
prime radical	C₂×#41×C₁₇	D₄
cosocle	C₂²×C₈×C₁₇	C₂³
hypercenter	C₂×#504	D₁₇
hypocenter	C₁₇	#12843
LCS	G ▷ C₂×C₈×C₁₇ ▷ C₄×C₁₇ ▷ C₂×C₁₇ ▷ C₁₇
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×#131 ◁ C₂×#504

Complements

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

N	H ≅ G/N	type
1. C₂×#11762639	C₂	semidirect
2. C₂×#11762637	C₂	semidirect
4. #14526893	C₂	semidirect
15. C₁₇×#504	C₄	semidirect
122. #504	Dic₁₇	semidirect
165. C₁₇	#12843	semidirect

Maximal subgroups

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

subgroup	index
C₂×#11762639	2
C₂×#11762637	2
#14528848	2
#14526893	2
C₂×C₁₇×#504	2
#12843	17