Subgroups — #16940245

Order 4352.

714 conjugacy classes of subgroups.

The subgroup lattice has 268 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×#10587490	C₂	1	2	yes	maximal
2	#14528523	C₂	1	2	yes	maximal
3	C₂×#11762567	C₂	1	2	yes	maximal
4	C₂×C₁₇×#405	C₂	1	1	yes	maximal, characteristic
5	C₂²×#63223	C₂²	1	4	yes	—
6	C₂×#10587490	—	2	4	no	—
7	C₂×#10587490	—	2	4	no	—
8	C₂×#10587490	C₄	1	8	yes	complemented
9	#11762578	—	2	4	no	—
10	#11762567	—	2	4	no	—
11	C₂²×M₁₆×C₁₇	C₂²	1	1	yes	characteristic
12	C₂³×C₈×C₁₇	C₂²	1	1	yes	characteristic
13	C₂×C₁₇×#117	C₂²	1	1	yes	characteristic
14	C₁₇×#405	—	2	4	no	—
15	C₂×C₁₇×#97	C₄	1	2	yes	—
16	C₂×C₄×C₈×C₁₇	C₄	1	2	yes	—
17	C₂³×C₄×C₁₇	C₂³	1	1	yes	characteristic
18	C₂×M₁₆×C₁₇	—	2	2	no	—
19	C₂×M₁₆×C₁₇	—	2	2	no	—
20	C₂×M₁₆×C₁₇	D₄	1	2	yes	—
21	C₂×M₁₆×C₁₇	Q₈	1	2	yes	—
22	C₂²×C₈×C₁₇	C₂×C₄	1	2	yes	—
23	C₂²×C₈×C₁₇	C₂×C₄	1	4	yes	—
24	C₂²×C₈×C₁₇	C₂×C₄	1	1	yes	characteristic
25	C₂²×C₈×C₁₇	C₂×C₄	1	1	yes	characteristic
26	C₂²×C₈×C₁₇	D₄	1	2	yes	—
27	C₂²×C₈×C₁₇	D₄	1	4	yes	—
28	C₂²×C₈×C₁₇	Q₈	1	2	yes	—
29	C₁₇×#117	—	2	2	no	—
30	C₁₇×#117	—	2	2	no	—
31	C₂×C₄⋊C₄×C₁₇	D₄	1	2	yes	—
32	C₂×C₂²⋊C₄×C₁₇	Q₈	1	2	yes	—
33	C₂×C₄²×C₁₇	C₂×C₄	1	2	yes	—
34	C₂×#63223	—	2	4	no	—
35	C₂×#63223	—	2	4	no	—
36	C₂×#63223	—	2	4	no	—
37	C₂×#63223	—	2	4	no	—
38	C₂×#63223	—	2	4	no	—
39	C₂×#63223	C₂×C₄	1	8	yes	—
40	C₂²×#3622	—	2	4	no	—
41	C₁₇×#97	—	2	4	no	—
42	C₄×C₈×C₁₇	—	2	4	no	—
43	#10587490	—	2	16	no	—
44	C₂⁴×C₁₇	C₂×Q₈	1	1	yes	characteristic
45	C₂²×C₄×C₁₇	C₄⋊C₄	1	4	yes	—
46	C₂²×C₄×C₁₇	C₂²×C₄	1	2	yes	—
47	C₂²×C₄×C₁₇	C₂²×C₄	1	1	yes	characteristic
48	C₂²×C₄×C₁₇	C₂×D₄	1	1	yes	characteristic
49	C₂²×C₄×C₁₇	C₂×D₄	1	1	yes	characteristic
50	C₂²×C₄×C₁₇	C₂×D₄	1	1	yes	characteristic
51	C₂²×C₄×C₁₇	C₂×D₄	1	2	yes	—
52	C₂²×C₄×C₁₇	C₂×Q₈	1	1	yes	characteristic
53	C₂²×C₄×C₁₇	C₂×Q₈	1	1	yes	characteristic
54	C₂²×C₄×C₁₇	#41	1	2	yes	—
55	C₂²×C₄×C₁₇	#41	1	2	yes	—
56	M₁₆×C₁₇	—	2	4	no	—
57	M₁₆×C₁₇	—	2	4	no	—
58	C₂×C₈×C₁₇	—	2	2	no	—
59	C₂×C₈×C₁₇	—	2	2	no	—
60	C₂×C₈×C₁₇	C₄²	1	2	yes	—
61	C₂×C₈×C₁₇	C₄²	1	2	yes	—
62	C₂×C₈×C₁₇	C₂²⋊C₄	1	2	yes	—
63	C₂×C₈×C₁₇	C₂²⋊C₄	1	2	yes	—
64	C₂×C₈×C₁₇	C₂²⋊C₄	1	8	yes	—
65	C₂×C₈×C₁₇	C₄⋊C₄	1	8	yes	—
66	C₂×C₈×C₁₇	C₄⋊C₄	1	4	yes	—
67	C₂×C₈×C₁₇	C₄⋊C₄	1	2	yes	—
68	C₂×C₈×C₁₇	C₄⋊C₄	1	2	yes	—
69	C₄⋊C₄×C₁₇	—	2	4	no	—
70	C₂²⋊C₄×C₁₇	—	2	4	no	—
71	C₄²×C₁₇	—	2	2	no	—
72	C₄²×C₁₇	—	2	2	no	—
73	#63223	—	2	8	no	—
74	#63223	—	2	8	no	—
75	#63223	—	4	4	no	—
76	#63223	—	4	4	no	—
77	C₂×#3622	—	2	8	no	—
78	C₂×#3622	—	4	4	no	—
79	C₂×#3622	—	4	4	no	—
80	#10867	—	17	1	no	maximal, Sylow
81	C₂³×C₁₇	—	4	2	no	—
82	C₂³×C₁₇	C₂×C₄⋊C₄	1	1	yes	characteristic
83	C₂³×C₁₇	C₄×Q₈	1	2	yes	—
84	C₂³×C₁₇	C₂²⋊Q₈	1	2	yes	—
85	C₂³×C₁₇	C₄⋊Q₈	1	1	yes	characteristic
86	C₂³×C₁₇	C₄⋊Q₈	1	1	yes	characteristic
87	C₂×C₄×C₁₇	—	2	4	no	—
88	C₂×C₄×C₁₇	—	4	4	no	—
89	C₂×C₄×C₁₇	C₂×C₄²	1	1	yes	characteristic
90	C₂×C₄×C₁₇	C₂×C₂²⋊C₄	1	2	yes	—
91	C₂×C₄×C₁₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
92	C₂×C₄×C₁₇	C₂×C₄⋊C₄	1	2	yes	—
93	C₂×C₄×C₁₇	C₂×C₄⋊C₄	1	1	yes	characteristic
94	C₂×C₄×C₁₇	C₂×C₄⋊C₄	1	1	yes	characteristic
95	C₂×C₄×C₁₇	#117	1	1	yes	characteristic
96	C₂×C₄×C₁₇	#117	1	1	yes	characteristic
97	C₂×C₄×C₁₇	C₄×D₄	1	2	yes	—
98	C₂×C₄×C₁₇	C₄×D₄	1	2	yes	—
99	C₂×C₄×C₁₇	C₄×Q₈	1	2	yes	—
100	C₂×C₄×C₁₇	#121	1	2	yes	—
101	C₂×C₄×C₁₇	#121	1	2	yes	—
102	C₂×C₄×C₁₇	C₂²⋊Q₈	1	2	yes	—
103	C₂×C₄×C₁₇	#124	1	2	yes	—
104	C₂×C₄×C₁₇	#125	1	2	yes	—
105	C₂×C₄×C₁₇	#127	1	1	yes	characteristic
106	C₂×C₄×C₁₇	C₄⋊Q₈	1	1	yes	characteristic
107	C₈×C₁₇	—	2	4	no	—
108	C₈×C₁₇	#95	1	4	yes	—
109	C₈×C₁₇	#95	1	4	yes	—
110	#3622	—	4	8	no	—
111	C₂²×#108	—	17	2	no	—
112	C₂×#405	D₁₇	1	1	yes	characteristic
113	#1742	—	17	2	no	—
114	C₂×#338	—	17	2	no	—
115	C₂²×C₁₇	—	2	2	no	—
116	C₂²×C₁₇	—	2	4	no	—
117	C₂²×C₁₇	—	4	2	no	—
118	C₂²×C₁₇	—	4	1	no	—
119	C₂²×C₁₇	—	4	1	no	—
120	C₂²×C₁₇	C₄×C₄⋊C₄	1	1	yes	characteristic
121	C₂²×C₁₇	#380	1	1	yes	characteristic
122	C₂²×C₁₇	#384	1	1	yes	characteristic
123	C₂²×C₁₇	#389	1	2	yes	—
124	C₂²×C₁₇	#391	1	2	yes	—
125	C₄×C₁₇	—	4	4	no	—
126	C₄×C₁₇	C₂×#95	1	1	yes	characteristic
127	C₄×C₁₇	C₄×C₄⋊C₄	1	1	yes	characteristic
128	C₄×C₁₇	#380	1	1	yes	characteristic
129	C₄×C₁₇	#382	1	1	yes	characteristic
130	C₄×C₁₇	#389	1	2	yes	—
131	C₄×C₁₇	#390	1	2	yes	—
132	C₂²×M₁₆	—	17	4	no	—
133	C₂²×M₁₆	C₂×D₁₇	1	1	yes	characteristic
134	C₂³×C₈	C₂×D₁₇	1	1	yes	characteristic
135	C₂×#117	C₂×D₁₇	1	1	yes	characteristic
136	C₂×#108	—	17	8	no	—
137	C₂×#108	—	34	4	no	—
138	C₂×#108	—	34	4	no	—
139	#405	—	2	4	no	—
140	C₂×#97	Dic₁₇	1	2	yes	—
141	C₂×C₄×C₈	Dic₁₇	1	2	yes	—
142	#343	—	34	4	no	—
143	#338	—	34	4	no	—
144	C₂×C₁₇	—	2	1	no	—
145	C₂×C₁₇	—	2	1	no	—
146	C₂×C₁₇	—	2	2	no	—
147	C₂×C₁₇	—	2	2	no	—
148	C₂×C₁₇	C₄⋊#95	1	1	yes	characteristic
149	C₂×C₁₇	#1813	1	2	yes	—
150	C₂³×C₄	C₂²×D₁₇	1	1	yes	characteristic
151	C₂×M₁₆	—	2	2	no	—
152	C₂×M₁₆	—	2	2	no	—
153	C₂×M₁₆	—	17	8	no	—
154	C₂×M₁₆	—	34	4	no	—
155	C₂×M₁₆	—	34	4	no	—
156	C₂×M₁₆	—	34	4	no	—
157	C₂×M₁₆	—	34	4	no	—
158	C₂×M₁₆	—	34	4	no	—
159	C₂×M₁₆	Dic₃₄	1	2	yes	—
160	C₂×M₁₆	D₆₈	1	2	yes	—
161	C₂²×C₈	—	34	4	no	—
162	C₂²×C₈	Dic₃₄	1	2	yes	—
163	C₂²×C₈	C₄×D₁₇	1	4	yes	—
164	C₂²×C₈	D₆₈	1	2	yes	—
165	C₂²×C₈	C₂×Dic₁₇	1	2	yes	—
166	C₂²×C₈	C₂×Dic₁₇	1	1	yes	characteristic
167	C₂²×C₈	C₂×Dic₁₇	1	1	yes	characteristic
168	C₂²×C₈	C₂²⋊D₁₇	1	4	yes	—
169	#117	—	2	2	no	—
170	#117	—	2	2	no	—
171	C₂×C₄⋊C₄	D₆₈	1	2	yes	—
172	C₂×C₂²⋊C₄	Dic₃₄	1	2	yes	—
173	C₂×C₄²	C₂×Dic₁₇	1	2	yes	—
174	#108	—	34	16	no	—
175	#97	—	2	4	no	—
176	C₄×C₈	—	2	4	no	—
177	C₁₇	#10867	1	1	yes	characteristic, complemented, Sylow
178	C₂⁴	C₂×Dic₃₄	1	1	yes	characteristic
179	C₂²×C₄	C₄⋊Dic₁₇	1	4	yes	—
180	C₂²×C₄	C₂×Dic₃₄	1	1	yes	characteristic
181	C₂²×C₄	C₂×Dic₃₄	1	1	yes	characteristic
182	C₂²×C₄	C₂×C₄×D₁₇	1	2	yes	—
183	C₂²×C₄	C₂×D₆₈	1	1	yes	characteristic
184	C₂²×C₄	C₂×D₆₈	1	1	yes	characteristic
185	C₂²×C₄	C₂×D₆₈	1	1	yes	characteristic
186	C₂²×C₄	#63252	1	2	yes	—
187	C₂²×C₄	#63252	1	2	yes	—
188	C₂²×C₄	C₂²×Dic₁₇	1	1	yes	characteristic
189	C₂²×C₄	C₂×C₂²⋊D₁₇	1	2	yes	—
190	M₁₆	—	2	4	no	—
191	M₁₆	—	2	4	no	—
192	M₁₆	—	34	8	no	—
193	M₁₆	—	34	8	no	—
194	M₁₆	—	68	4	no	—
195	M₁₆	—	68	4	no	—
196	C₂×C₈	—	2	2	no	—
197	C₂×C₈	—	2	2	no	—
198	C₂×C₈	—	34	8	no	—
199	C₂×C₈	—	68	4	no	—
200	C₂×C₈	—	68	4	no	—
201	C₂×C₈	C₄×Dic₁₇	1	2	yes	—
202	C₂×C₈	C₄×Dic₁₇	1	2	yes	—
203	C₂×C₈	#63225	1	8	yes	—
204	C₂×C₈	C₄⋊Dic₁₇	1	4	yes	—
205	C₂×C₈	C₄⋊Dic₁₇	1	2	yes	—
206	C₂×C₈	C₄⋊Dic₁₇	1	2	yes	—
207	C₂×C₈	#63227	1	8	yes	—
208	C₂×C₈	C₂²⋊Dic₁₇	1	2	yes	—
209	C₂×C₈	C₂²⋊Dic₁₇	1	2	yes	—
210	C₄⋊C₄	—	2	4	no	—
211	C₂²⋊C₄	—	2	4	no	—
212	C₄²	—	2	2	no	—
213	C₄²	—	2	2	no	—
214	C₂³	—	4	2	no	—
215	C₂³	C₄×Dic₃₄	1	2	yes	—
216	C₂³	#10587555	1	1	yes	characteristic
217	C₂³	#10587555	1	1	yes	characteristic
218	C₂³	#10587610	1	2	yes	—
219	C₂³	C₂×C₄⋊Dic₁₇	1	1	yes	characteristic
220	C₂×C₄	—	2	4	no	—
221	C₂×C₄	—	4	4	no	—
222	C₂×C₄	C₄×Dic₃₄	1	2	yes	—
223	C₂×C₄	#10587555	1	1	yes	characteristic
224	C₂×C₄	#10587556	1	2	yes	—
225	C₂×C₄	C₄×D₆₈	1	2	yes	—
226	C₂×C₄	C₄×D₆₈	1	2	yes	—
227	C₂×C₄	#10587560	1	1	yes	characteristic
228	C₂×C₄	#10587561	1	2	yes	—
229	C₂×C₄	C₂×C₄×Dic₁₇	1	1	yes	characteristic
230	C₂×C₄	C₂×#63225	1	2	yes	—
231	C₂×C₄	#10587610	1	2	yes	—
232	C₂×C₄	C₂×C₄⋊Dic₁₇	1	1	yes	characteristic
233	C₂×C₄	C₂×C₄⋊Dic₁₇	1	1	yes	characteristic
234	C₂×C₄	#10587612	1	1	yes	characteristic
235	C₂×C₄	#10587612	1	1	yes	characteristic
236	C₂×C₄	C₂×#63227	1	2	yes	—
237	C₂×C₄	#10587616	1	2	yes	—
238	C₂×C₄	#10587616	1	2	yes	—
239	C₂×C₄	C₂×C₂²⋊Dic₁₇	1	1	yes	characteristic
240	C₈	—	2	4	no	—
241	C₈	—	68	8	no	—
242	C₈	#10587502	1	4	yes	—
243	C₈	#10587502	1	4	yes	—
244	C₂²	—	2	2	no	—
245	C₂²	—	2	4	no	—
246	C₂²	—	4	2	no	—
247	C₂²	—	4	1	no	—
248	C₂²	—	4	1	no	—
249	C₂²	C₄×C₄⋊Dic₁₇	1	1	yes	characteristic
250	C₂²	#11762669	1	1	yes	characteristic
251	C₂²	#11762679	1	1	yes	characteristic
252	C₂²	#11762707	1	2	yes	—
253	C₂²	#11762715	1	2	yes	—
254	C₄	—	4	4	no	—
255	C₄	C₂×#10587502	1	1	yes	characteristic
256	C₄	C₄×C₄⋊Dic₁₇	1	1	yes	characteristic
257	C₄	#11762669	1	1	yes	characteristic
258	C₄	#11762675	1	1	yes	characteristic
259	C₄	#11762707	1	2	yes	—
260	C₄	#11762711	1	2	yes	—
261	C₂	—	2	2	no	—
262	C₂	—	2	2	no	—
263	C₂	—	2	1	no	—
264	C₂	—	2	1	no	—
265	C₂	#14527014	1	1	yes	characteristic
266	C₂	#14528834	1	2	yes	—

	subgroup	G/H
center	C₂×C₄	#10587612
commutator	C₂×C₄×C₁₇	C₂×C₄²
Frattini	C₂³×C₄	C₂²×D₁₇
Fitting	C₂×C₁₇×#405	C₂
socle	C₂²×C₁₇	#380
prime radical	C₂⁴×C₁₇	C₂×Q₈
cosocle	C₂³×C₄×C₁₇	C₂³
hypercenter	C₂×#405	D₁₇
hypocenter	C₁₇	#10867
LCS	G ▷ C₂×C₄×C₁₇ ▷ C₂×C₁₇ ▷ C₁₇
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×M₁₆ ◁ C₂×#405

Complements

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

N	H ≅ G/N	type
8. C₂×#10587490	C₄	semidirect
177. C₁₇	#10867	semidirect

Maximal subgroups

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

subgroup	index
C₂²×#10587490	2
#14528523	2
C₂×#11762567	2
C₂×C₁₇×#405	2
#10867	17