Subgroups — #16950500

Order 4352.

418 conjugacy classes of subgroups.

The subgroup lattice has 254 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#11762637	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×#11762635	C₂	1	1	yes	maximal, characteristic, complemented
3	#14528813	C₂	1	2	yes	maximal
4	#14526876	C₂	1	2	yes	maximal, complemented
5	C₂²×C₁₇×M₃₂	C₂	1	1	yes	maximal, characteristic
6	C₂×#10587489	C₂²	1	2	yes	—
7	C₂×#10587489	C₂²	1	2	yes	—
8	#11762669	—	2	2	no	—
9	#11762637	—	2	2	no	—
10	#11762635	—	2	2	no	—
11	#11762572	—	2	2	no	—
12	C₂³×C₈×C₁₇	C₂²	1	1	yes	characteristic
13	C₂×C₁₇×M₃₂	—	2	2	no	—
14	C₂×C₁₇×M₃₂	—	2	2	no	—
15	C₂×C₁₇×M₃₂	C₄	1	4	yes	complemented
16	C₂²×C₁₆×C₁₇	C₂²	1	1	yes	characteristic
17	C₂²×C₁₆×C₁₇	C₂²	1	1	yes	characteristic
18	C₂³×C₄×C₁₇	D₄	1	1	yes	characteristic
19	C₂²×C₈×C₁₇	—	2	1	no	—
20	C₂²×C₈×C₁₇	—	2	1	no	—
21	C₂²×C₈×C₁₇	—	2	1	no	—
22	C₂²×C₈×C₁₇	—	2	1	no	—
23	C₂²×C₈×C₁₇	C₂×C₄	1	2	yes	—
24	C₂²×C₈×C₁₇	D₄	1	1	yes	characteristic
25	C₂²×C₈×C₁₇	Q₈	1	2	yes	—
26	C₂²×C₈×C₁₇	C₂³	1	1	yes	characteristic
27	C₂×C₂²⋊Dic₁₇	—	4	2	no	—
28	C₂×C₄⋊Dic₁₇	—	2	2	no	—
29	C₂×C₄⋊Dic₁₇	—	2	2	no	—
30	C₁₇×M₃₂	—	2	8	no	—
31	C₂×C₁₆×C₁₇	—	2	1	no	—
32	C₂×C₁₆×C₁₇	—	2	1	no	—
33	C₂×C₁₆×C₁₇	C₂×C₄	1	2	yes	—
34	C₂×C₁₆×C₁₇	C₂×C₄	1	2	yes	—
35	C₂×C₁₆×C₁₇	D₄	1	2	yes	—
36	C₂×C₁₆×C₁₇	D₄	1	2	yes	—
37	#10587502	—	4	2	no	—
38	#10587489	—	2	2	no	—
39	#10587489	—	2	2	no	—
40	#10587489	—	2	2	no	—
41	#10587489	—	2	2	no	—
42	C₂⁴×C₁₇	D₈	1	1	yes	characteristic
43	C₂²×C₄×C₁₇	—	2	1	no	—
44	C₂²×C₄×C₁₇	—	2	1	no	—
45	C₂²×C₄×C₁₇	—	2	1	no	—
46	C₂²×C₄×C₁₇	—	2	1	no	—
47	C₂²×C₄×C₁₇	C₄⋊C₄	1	2	yes	—
48	C₂²×C₄×C₁₇	D₈	1	1	yes	characteristic
49	C₂²×C₄×C₁₇	Q₁₆	1	2	yes	—
50	C₂²×C₄×C₁₇	C₂×D₄	1	1	yes	characteristic
51	C₂²×Dic₁₇	—	4	2	no	—
52	C₂²×Dic₁₇	—	4	2	no	—
53	C₂×C₈×C₁₇	—	2	1	no	—
54	C₂×C₈×C₁₇	—	2	1	no	—
55	C₂×C₈×C₁₇	—	2	2	no	—
56	C₂×C₈×C₁₇	—	2	2	no	—
57	C₂×C₈×C₁₇	—	2	2	no	—
58	C₂×C₈×C₁₇	—	2	2	no	—
59	C₂×C₈×C₁₇	C₄⋊C₄	1	2	yes	—
60	C₂×C₈×C₁₇	C₂²×C₄	1	1	yes	characteristic
61	C₂×C₈×C₁₇	C₂×D₄	1	1	yes	characteristic
62	C₂×C₈×C₁₇	C₂×D₄	1	1	yes	characteristic
63	C₂×C₈×C₁₇	C₂×Q₈	1	1	yes	characteristic
64	C₂×C₈×C₁₇	#41	1	1	yes	characteristic
65	C₂×C₈×C₁₇	#41	1	1	yes	characteristic
66	C₂²⋊Dic₁₇	—	8	4	no	—
67	C₄⋊Dic₁₇	—	4	2	no	—
68	C₄⋊Dic₁₇	—	4	2	no	—
69	C₄⋊Dic₁₇	—	4	2	no	—
70	C₄⋊Dic₁₇	—	4	2	no	—
71	C₁₆×C₁₇	—	2	2	no	—
72	C₁₆×C₁₇	C₂²⋊C₄	1	4	yes	complemented
73	#12836	—	17	1	no	maximal, Sylow
74	C₂³×C₁₇	—	2	1	no	—
75	C₂³×C₁₇	—	2	1	no	—
76	C₂³×C₁₇	—	2	2	no	—
77	C₂³×C₁₇	—	2	2	no	—
78	C₂³×C₁₇	#107	1	2	yes	—
79	C₂³×C₁₇	C₂×D₈	1	1	yes	characteristic
80	C₂×C₄×C₁₇	—	2	1	no	—
81	C₂×C₄×C₁₇	—	2	1	no	—
82	C₂×C₄×C₁₇	—	2	2	no	—
83	C₂×C₄×C₁₇	—	2	2	no	—
84	C₂×C₄×C₁₇	—	2	2	no	—
85	C₂×C₄×C₁₇	—	2	2	no	—
86	C₂×C₄×C₁₇	#107	1	2	yes	—
87	C₂×C₄×C₁₇	C₂×C₄⋊C₄	1	1	yes	characteristic
88	C₂×C₄×C₁₇	#121	1	1	yes	characteristic
89	C₂×C₄×C₁₇	#121	1	1	yes	characteristic
90	C₂×C₄×C₁₇	C₂×Q₁₆	1	1	yes	characteristic
91	C₂×C₄×C₁₇	#135	1	1	yes	characteristic
92	C₂×C₄×C₁₇	#135	1	1	yes	characteristic
93	C₂×Dic₁₇	—	8	2	no	—
94	C₂×Dic₁₇	—	8	2	no	—
95	C₂×Dic₁₇	—	8	2	no	—
96	C₂×Dic₁₇	—	8	2	no	—
97	C₂×Dic₁₇	—	8	2	no	—
98	C₂×Dic₁₇	—	8	2	no	—
99	C₈×C₁₇	—	2	2	no	—
100	C₈×C₁₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
101	C₈×C₁₇	#117	1	1	yes	characteristic
102	C₈×C₁₇	C₂²⋊Q₈	1	1	yes	characteristic
103	C₈×C₁₇	C₂²⋊Q₈	1	1	yes	characteristic
104	C₂²×M₃₂	D₁₇	1	1	yes	characteristic
105	C₂×#367	—	17	1	no	—
106	C₂×#366	—	17	1	no	—
107	#1808	—	17	2	no	—
108	#1380	—	17	2	no	—
109	C₂²×C₁₇	—	2	2	no	—
110	C₂²×C₁₇	—	2	4	no	—
111	C₂²×C₁₇	—	4	1	no	—
112	C₂²×C₁₇	—	4	1	no	—
113	C₂²×C₁₇	—	4	2	no	—
114	C₂²×C₁₇	C₂×#107	1	1	yes	characteristic
115	C₂²×C₁₇	#466	1	1	yes	characteristic
116	C₂²×C₁₇	#466	1	1	yes	characteristic
117	C₂²×C₁₇	#509	1	2	yes	—
118	C₂²×C₁₇	#510	1	2	yes	—
119	C₄×C₁₇	—	2	2	no	—
120	C₄×C₁₇	#380	1	1	yes	characteristic
121	C₄×C₁₇	#427	1	1	yes	characteristic
122	C₄×C₁₇	#467	1	1	yes	characteristic
123	C₄×C₁₇	#467	1	1	yes	characteristic
124	Dic₁₇	—	16	2	no	—
125	Dic₁₇	—	16	2	no	—
126	C₂³×C₈	C₂×D₁₇	1	1	yes	characteristic
127	C₂×M₃₂	—	2	2	no	—
128	C₂×M₃₂	—	2	2	no	—
129	C₂×M₃₂	Dic₁₇	1	4	yes	complemented
130	C₂²×C₁₆	C₂×D₁₇	1	1	yes	characteristic
131	C₂²×C₁₆	C₂×D₁₇	1	1	yes	characteristic
132	C₂×#107	—	17	2	no	—
133	C₂×#107	—	17	2	no	—
134	#380	—	34	2	no	—
135	#367	—	34	2	no	—
136	#366	—	34	2	no	—
137	#340	—	34	2	no	—
138	C₂×C₁₇	—	4	2	no	—
139	C₂×C₁₇	#1808	1	1	yes	characteristic
140	C₂×C₁₇	#2159	1	2	yes	—
141	C₂×C₁₇	#2218	1	1	yes	characteristic
142	C₂×C₁₇	#2219	1	1	yes	characteristic
143	C₂×C₁₇	#2220	1	2	yes	—
144	C₂³×C₄	D₆₈	1	1	yes	characteristic
145	C₂²×C₈	—	2	1	no	—
146	C₂²×C₈	—	2	1	no	—
147	C₂²×C₈	—	2	1	no	—
148	C₂²×C₈	—	2	1	no	—
149	C₂²×C₈	Dic₃₄	1	2	yes	—
150	C₂²×C₈	D₆₈	1	1	yes	characteristic
151	C₂²×C₈	C₂×Dic₁₇	1	2	yes	—
152	C₂²×C₈	C₂²×D₁₇	1	1	yes	characteristic
153	C₂×C₄⋊C₄	—	34	2	no	—
154	C₂×C₄⋊C₄	—	34	2	no	—
155	C₂×C₂²⋊C₄	—	68	2	no	—
156	M₃₂	—	2	8	no	—
157	C₂×C₁₆	—	2	1	no	—
158	C₂×C₁₆	—	2	1	no	—
159	C₂×C₁₆	C₂×Dic₁₇	1	2	yes	—
160	C₂×C₁₆	C₂×Dic₁₇	1	2	yes	—
161	C₂×C₁₆	C₂²⋊D₁₇	1	2	yes	—
162	C₂×C₁₆	C₂²⋊D₁₇	1	2	yes	—
163	#107	—	34	2	no	—
164	#107	—	34	2	no	—
165	#107	—	34	2	no	—
166	#107	—	34	2	no	—
167	#95	—	68	2	no	—
168	C₁₇	#12836	1	1	yes	characteristic, complemented, Sylow
169	C₂⁴	D₁₃₆	1	1	yes	characteristic
170	C₂²×C₄	—	2	1	no	—
171	C₂²×C₄	—	2	1	no	—
172	C₂²×C₄	—	2	1	no	—
173	C₂²×C₄	—	2	1	no	—
174	C₂²×C₄	—	68	2	no	—
175	C₂²×C₄	—	68	2	no	—
176	C₂²×C₄	D₁₃₆	1	1	yes	characteristic
177	C₂²×C₄	Dic₆₈	1	2	yes	—
178	C₂²×C₄	C₄⋊Dic₁₇	1	2	yes	—
179	C₂²×C₄	C₂×D₆₈	1	1	yes	characteristic
180	C₂×C₈	—	2	1	no	—
181	C₂×C₈	—	2	1	no	—
182	C₂×C₈	—	2	2	no	—
183	C₂×C₈	—	2	2	no	—
184	C₂×C₈	—	2	2	no	—
185	C₂×C₈	—	2	2	no	—
186	C₂×C₈	C₄⋊Dic₁₇	1	2	yes	—
187	C₂×C₈	C₂×Dic₃₄	1	1	yes	characteristic
188	C₂×C₈	#63252	1	1	yes	characteristic
189	C₂×C₈	#63252	1	1	yes	characteristic
190	C₂×C₈	C₂²×Dic₁₇	1	1	yes	characteristic
191	C₂×C₈	C₂×C₂²⋊D₁₇	1	1	yes	characteristic
192	C₂×C₈	C₂×C₂²⋊D₁₇	1	1	yes	characteristic
193	C₄⋊C₄	—	68	2	no	—
194	C₄⋊C₄	—	68	2	no	—
195	C₄⋊C₄	—	68	2	no	—
196	C₄⋊C₄	—	68	2	no	—
197	C₂²⋊C₄	—	136	4	no	—
198	C₁₆	—	2	2	no	—
199	C₁₆	C₂²⋊Dic₁₇	1	4	yes	complemented
200	C₂³	—	2	2	no	—
201	C₂³	—	2	2	no	—
202	C₂³	—	2	1	no	—
203	C₂³	—	2	1	no	—
204	C₂³	#10587489	1	2	yes	—
205	C₂³	C₂×D₁₃₆	1	1	yes	characteristic
206	C₂×C₄	—	2	1	no	—
207	C₂×C₄	—	2	1	no	—
208	C₂×C₄	—	2	2	no	—
209	C₂×C₄	—	2	2	no	—
210	C₂×C₄	—	2	2	no	—
211	C₂×C₄	—	2	2	no	—
212	C₂×C₄	—	136	2	no	—
213	C₂×C₄	—	136	2	no	—
214	C₂×C₄	—	136	2	no	—
215	C₂×C₄	—	136	2	no	—
216	C₂×C₄	—	136	2	no	—
217	C₂×C₄	—	136	2	no	—
218	C₂×C₄	#10587489	1	2	yes	—
219	C₂×C₄	#10587590	1	1	yes	characteristic
220	C₂×C₄	#10587590	1	1	yes	characteristic
221	C₂×C₄	C₂×Dic₆₈	1	1	yes	characteristic
222	C₂×C₄	C₂×C₄⋊Dic₁₇	1	1	yes	characteristic
223	C₂×C₄	#10587616	1	1	yes	characteristic
224	C₂×C₄	#10587616	1	1	yes	characteristic
225	C₈	—	2	2	no	—
226	C₈	#10587610	1	1	yes	characteristic
227	C₈	#10587610	1	1	yes	characteristic
228	C₈	#10587612	1	1	yes	characteristic
229	C₈	C₂×C₂²⋊Dic₁₇	1	1	yes	characteristic
230	C₂²	—	2	4	no	—
231	C₂²	—	2	2	no	—
232	C₂²	—	4	1	no	—
233	C₂²	—	4	1	no	—
234	C₂²	—	4	2	no	—
235	C₂²	C₂×#10587489	1	1	yes	characteristic
236	C₂²	#11763045	1	1	yes	characteristic
237	C₂²	#11763045	1	1	yes	characteristic
238	C₂²	#11763247	1	2	yes	—
239	C₂²	#11763252	1	2	yes	—
240	C₄	—	2	2	no	—
241	C₄	—	272	2	no	—
242	C₄	—	272	2	no	—
243	C₄	#11762669	1	1	yes	characteristic
244	C₄	#11762847	1	1	yes	characteristic
245	C₄	#11763050	1	1	yes	characteristic
246	C₄	#11763050	1	1	yes	characteristic
247	C₂	—	4	2	no	—
248	C₂	#14528813	1	1	yes	characteristic
249	C₂	#14530572	1	2	yes	—
250	C₂	#14530887	1	1	yes	characteristic
251	C₂	#14530892	1	1	yes	characteristic
252	C₂	#14530897	1	2	yes	—

	subgroup	G/H
center	C₂³	C₂×D₁₃₆
commutator	C₂×C₈×C₁₇	C₂²×C₄
Frattini	C₂²×C₈	C₂²×D₁₇
Fitting	C₂²×C₁₇×M₃₂	C₂
socle	C₂³×C₁₇	C₂×D₈
prime radical	C₂⁴×C₁₇	D₈
cosocle	C₂²×C₈×C₁₇	C₂³
hypercenter	C₂²×M₃₂	D₁₇
hypocenter	C₁₇	#12836
LCS	G ▷ C₂×C₈×C₁₇ ▷ C₄×C₁₇ ▷ C₂×C₁₇ ▷ C₁₇
UCS	C₁ ◁ C₂³ ◁ C₂³×C₄ ◁ C₂³×C₈ ◁ C₂²×M₃₂

Complements

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

N	H ≅ G/N	type
1. C₂×#11762637	C₂	semidirect
2. C₂×#11762635	C₂	semidirect
4. #14526876	C₂	semidirect
15. C₂×C₁₇×M₃₂	C₄	semidirect
72. C₁₆×C₁₇	C₂²⋊C₄	semidirect
129. C₂×M₃₂	Dic₁₇	semidirect
168. C₁₇	#12836	semidirect
199. C₁₆	C₂²⋊Dic₁₇	semidirect

Maximal subgroups

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

subgroup	index
C₂×#11762637	2
C₂×#11762635	2
#14528813	2
#14526876	2
C₂²×C₁₇×M₃₂	2
#12836	17