Subgroups — C₂×#14526890 #16949239

Order 4352.

630 conjugacy classes of subgroups.

The subgroup lattice has 322 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#11762847	C₂	1	1	yes	maximal, characteristic
2	C₂²×#10587488	C₂	1	1	yes	maximal, characteristic
3	#14526890	C₂	1	4	yes	maximal, direct product factor
4	C₂²×C₁₇×M₃₂	C₂	1	1	yes	maximal, characteristic
5	C₂×#10587612	—	2	1	no	—
6	C₂²×C₄⋊Dic₁₇	—	2	1	no	—
7	#11762847	—	2	1	no	—
8	#11762847	—	2	1	no	—
9	#11762847	C₄	1	2	yes	complemented
10	#11762847	C₂²	1	2	yes	—
11	C₂×#10587489	C₄	1	2	yes	complemented
12	C₂×#10587488	—	2	1	no	—
13	C₂×#10587488	—	2	2	no	—
14	C₂×#10587488	—	2	1	no	—
15	C₂×#10587488	C₄	1	4	yes	complemented
16	C₂×#10587488	C₂²	1	2	yes	—
17	C₂³×C₈×C₁₇	C₂²	1	1	yes	characteristic
18	C₂×C₁₇×M₃₂	—	2	2	no	—
19	C₂×C₁₇×M₃₂	—	2	1	no	—
20	C₂×C₁₇×M₃₂	C₄	1	4	yes	complemented
21	C₂×C₁₇×M₃₂	C₂²	1	2	yes	—
22	C₂²×C₁₆×C₁₇	—	2	1	no	—
23	C₂³×C₄×C₁₇	D₄	1	1	yes	characteristic
24	C₂³×Dic₁₇	—	4	1	no	—
25	C₂²×C₈×C₁₇	C₂×C₄	1	2	yes	—
26	C₂²×C₈×C₁₇	C₂×C₄	1	2	yes	—
27	C₂²×C₈×C₁₇	C₂×C₄	1	1	yes	characteristic
28	C₂²×C₈×C₁₇	C₂×C₄	1	1	yes	characteristic
29	C₂²×C₈×C₁₇	D₄	1	1	yes	characteristic
30	C₂²×C₈×C₁₇	D₄	1	1	yes	characteristic
31	C₂²×C₈×C₁₇	D₄	1	2	yes	—
32	C₂²×C₈×C₁₇	D₄	1	1	yes	characteristic
33	C₂²×C₈×C₁₇	Q₈	1	2	yes	—
34	C₂²×C₈×C₁₇	C₂³	1	1	yes	characteristic
35	C₂×C₂²⋊Dic₁₇	—	4	1	no	—
36	#10587612	—	2	2	no	—
37	#10587612	—	2	2	no	—
38	#10587612	—	4	1	no	—
39	#10587612	—	4	1	no	—
40	C₂×C₄⋊Dic₁₇	—	2	4	no	—
41	C₂×C₄⋊Dic₁₇	—	2	2	no	—
42	C₂×C₄⋊Dic₁₇	—	2	2	no	—
43	C₂×C₄⋊Dic₁₇	—	4	1	no	—
44	C₂×C₄⋊Dic₁₇	—	4	2	no	—
45	C₂×C₄×Dic₁₇	—	4	1	no	—
46	C₁₇×M₃₂	—	2	4	no	—
47	C₁₇×M₃₂	—	2	2	no	—
48	C₁₇×M₃₂	C₂×C₄	1	4	yes	complemented
49	C₂×C₁₆×C₁₇	—	2	4	no	—
50	C₂×C₁₆×C₁₇	—	2	2	no	—
51	#10587489	—	2	2	no	—
52	#10587489	C₂×C₄	1	4	yes	complemented
53	#10587488	—	2	4	no	—
54	#10587488	—	2	2	no	—
55	#10587488	—	2	2	no	—
56	#10587488	—	2	2	no	—
57	#10587488	C₂×C₄	1	4	yes	complemented
58	C₂⁴×C₁₇	D₈	1	1	yes	characteristic
59	C₂²×C₄×C₁₇	C₂²⋊C₄	1	2	yes	—
60	C₂²×C₄×C₁₇	C₂²⋊C₄	1	1	yes	characteristic
61	C₂²×C₄×C₁₇	C₂²⋊C₄	1	1	yes	characteristic
62	C₂²×C₄×C₁₇	C₄⋊C₄	1	2	yes	—
63	C₂²×C₄×C₁₇	D₈	1	1	yes	characteristic
64	C₂²×C₄×C₁₇	SD₁₆	1	1	yes	characteristic
65	C₂²×C₄×C₁₇	SD₁₆	1	2	yes	—
66	C₂²×C₄×C₁₇	SD₁₆	1	1	yes	characteristic
67	C₂²×C₄×C₁₇	Q₁₆	1	2	yes	—
68	C₂²×C₄×C₁₇	C₂×D₄	1	1	yes	characteristic
69	C₂²×Dic₁₇	—	4	4	no	—
70	C₂²×Dic₁₇	—	4	2	no	—
71	C₂²×Dic₁₇	—	4	2	no	—
72	C₂²×Dic₁₇	—	8	4	no	—
73	C₂×C₈×C₁₇	C₄²	1	2	yes	—
74	C₂×C₈×C₁₇	C₄²	1	2	yes	—
75	C₂×C₈×C₁₇	C₂²⋊C₄	1	2	yes	—
76	C₂×C₈×C₁₇	C₂²⋊C₄	1	2	yes	—
77	C₂×C₈×C₁₇	C₂²⋊C₄	1	2	yes	—
78	C₂×C₈×C₁₇	C₂²⋊C₄	1	1	yes	characteristic
79	C₂×C₈×C₁₇	C₂²⋊C₄	1	1	yes	characteristic
80	C₂×C₈×C₁₇	C₄⋊C₄	1	2	yes	—
81	C₂×C₈×C₁₇	C₄⋊C₄	1	2	yes	—
82	C₂×C₈×C₁₇	C₄⋊C₄	1	2	yes	—
83	C₂×C₈×C₁₇	C₄⋊C₄	1	2	yes	—
84	C₂×C₈×C₁₇	C₄⋊C₄	1	2	yes	—
85	C₂×C₈×C₁₇	C₂²×C₄	1	1	yes	characteristic
86	C₂×C₈×C₁₇	C₂²×C₄	1	1	yes	characteristic
87	C₂×C₈×C₁₇	C₂²×C₄	1	1	yes	characteristic
88	C₂×C₈×C₁₇	C₂×D₄	1	1	yes	characteristic
89	C₂×C₈×C₁₇	C₂×D₄	1	1	yes	characteristic
90	C₂×C₈×C₁₇	C₂×Q₈	1	1	yes	characteristic
91	C₂²⋊Dic₁₇	—	4	2	no	—
92	C₂²⋊Dic₁₇	—	4	2	no	—
93	C₄⋊Dic₁₇	—	2	4	no	—
94	C₄⋊Dic₁₇	—	2	4	no	—
95	C₄⋊Dic₁₇	—	4	4	no	—
96	C₄⋊Dic₁₇	—	4	2	no	—
97	C₄⋊Dic₁₇	—	4	2	no	—
98	C₄×Dic₁₇	—	4	2	no	—
99	C₄×Dic₁₇	—	4	2	no	—
100	C₁₆×C₁₇	—	2	4	no	—
101	C₂×#1386	—	17	1	no	maximal, Sylow
102	C₂³×C₁₇	—	2	4	no	—
103	C₂³×C₁₇	#102	1	1	yes	characteristic
104	C₂³×C₁₇	#102	1	2	yes	—
105	C₂³×C₁₇	#102	1	1	yes	characteristic
106	C₂³×C₁₇	#107	1	2	yes	—
107	C₂³×C₁₇	C₂×D₈	1	1	yes	characteristic
108	C₂×C₄×C₁₇	#95	1	2	yes	—
109	C₂×C₄×C₁₇	#95	1	2	yes	—
110	C₂×C₄×C₁₇	#102	1	1	yes	characteristic
111	C₂×C₄×C₁₇	#102	1	2	yes	—
112	C₂×C₄×C₁₇	#102	1	1	yes	characteristic
113	C₂×C₄×C₁₇	#103	1	2	yes	—
114	C₂×C₄×C₁₇	#103	1	2	yes	—
115	C₂×C₄×C₁₇	#103	1	2	yes	—
116	C₂×C₄×C₁₇	#103	1	2	yes	—
117	C₂×C₄×C₁₇	#106	1	2	yes	—
118	C₂×C₄×C₁₇	#106	1	2	yes	—
119	C₂×C₄×C₁₇	#107	1	2	yes	—
120	C₂×C₄×C₁₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
121	C₂×C₄×C₁₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
122	C₂×C₄×C₁₇	C₂×C₄⋊C₄	1	1	yes	characteristic
123	C₂×C₄×C₁₇	C₂×SD₁₆	1	1	yes	characteristic
124	C₂×C₄×C₁₇	C₂×SD₁₆	1	1	yes	characteristic
125	C₂×C₄×C₁₇	C₂×Q₁₆	1	1	yes	characteristic
126	C₂×Dic₁₇	—	4	4	no	—
127	C₂×Dic₁₇	—	4	4	no	—
128	C₂×Dic₁₇	—	8	4	no	—
129	C₂×Dic₁₇	—	8	8	no	—
130	C₂×Dic₁₇	—	8	4	no	—
131	C₈×C₁₇	#95	1	2	yes	—
132	C₈×C₁₇	#95	1	2	yes	—
133	C₈×C₁₇	C₂×C₄²	1	1	yes	characteristic
134	C₈×C₁₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
135	C₈×C₁₇	C₂×C₄⋊C₄	1	1	yes	characteristic
136	C₈×C₁₇	C₂×C₄⋊C₄	1	1	yes	characteristic
137	C₂²×M₃₂	D₁₇	1	1	yes	characteristic
138	C₂×#427	—	17	1	no	—
139	C₂²×#106	—	17	1	no	—
140	#1386	—	17	4	no	—
141	C₂²×C₁₇	—	2	2	no	—
142	C₂²×C₁₇	—	2	2	no	—
143	C₂²×C₁₇	—	2	4	no	—
144	C₂²×C₁₇	—	2	4	no	—
145	C₂²×C₁₇	#340	1	2	yes	—
146	C₂²×C₁₇	#340	1	2	yes	—
147	C₂²×C₁₇	#360	1	4	yes	—
148	C₂²×C₁₇	C₂×#102	1	1	yes	characteristic
149	C₂²×C₁₇	C₂×#102	1	1	yes	characteristic
150	C₂²×C₁₇	C₂×#107	1	1	yes	characteristic
151	C₄×C₁₇	#340	1	2	yes	—
152	C₄×C₁₇	#340	1	2	yes	—
153	C₄×C₁₇	C₂×#95	1	1	yes	characteristic
154	C₄×C₁₇	C₂×#103	1	1	yes	characteristic
155	C₄×C₁₇	C₂×#103	1	1	yes	characteristic
156	C₄×C₁₇	C₂×#106	1	1	yes	characteristic
157	Dic₁₇	—	8	4	no	—
158	Dic₁₇	—	8	4	no	—
159	C₂³×C₈	C₂×D₁₇	1	1	yes	characteristic
160	C₂×#117	—	34	1	no	—
161	C₂²×C₄⋊C₄	—	34	1	no	—
162	C₂×M₃₂	—	2	2	no	—
163	C₂×M₃₂	—	2	1	no	—
164	C₂×M₃₂	Dic₁₇	1	4	yes	complemented
165	C₂×M₃₂	C₂×D₁₇	1	2	yes	—
166	C₂²×C₁₆	—	2	1	no	—
167	#427	—	17	2	no	—
168	#427	—	17	2	no	—
169	#427	—	34	1	no	—
170	#427	—	34	1	no	—
171	C₂×#107	—	17	2	no	—
172	C₂×#106	—	17	4	no	—
173	C₂×#106	—	17	2	no	—
174	C₂×#106	—	34	1	no	—
175	C₂×#106	—	34	2	no	—
176	C₂×#106	—	34	1	no	—
177	C₂×C₁₇	—	2	1	no	—
178	C₂×C₁₇	—	2	2	no	—
179	C₂×C₁₇	—	2	1	no	—
180	C₂×C₁₇	#1386	1	4	yes	complemented
181	C₂×C₁₇	C₂×#340	1	1	yes	characteristic
182	C₂×C₁₇	C₂×#360	1	2	yes	—
183	C₂³×C₄	—	68	1	no	—
184	C₂³×C₄	D₆₈	1	1	yes	characteristic
185	C₂²×C₈	Dic₃₄	1	2	yes	—
186	C₂²×C₈	C₄×D₁₇	1	2	yes	—
187	C₂²×C₈	C₄×D₁₇	1	1	yes	characteristic
188	C₂²×C₈	C₄×D₁₇	1	1	yes	characteristic
189	C₂²×C₈	D₆₈	1	1	yes	characteristic
190	C₂²×C₈	C₂×Dic₁₇	1	2	yes	—
191	C₂²×C₈	C₂²⋊D₁₇	1	1	yes	characteristic
192	C₂²×C₈	C₂²⋊D₁₇	1	2	yes	—
193	C₂²×C₈	C₂²⋊D₁₇	1	1	yes	characteristic
194	C₂²×C₈	C₂²×D₁₇	1	1	yes	characteristic
195	#117	—	34	2	no	—
196	#117	—	34	2	no	—
197	#117	—	68	1	no	—
198	#117	—	68	1	no	—
199	C₂×C₄⋊C₄	—	34	2	no	—
200	C₂×C₄⋊C₄	—	34	4	no	—
201	C₂×C₄⋊C₄	—	34	2	no	—
202	C₂×C₄⋊C₄	—	68	2	no	—
203	C₂×C₄⋊C₄	—	68	1	no	—
204	C₂×C₂²⋊C₄	—	68	1	no	—
205	C₂×C₄²	—	68	1	no	—
206	M₃₂	—	2	4	no	—
207	M₃₂	—	2	2	no	—
208	M₃₂	C₂×Dic₁₇	1	4	yes	complemented
209	C₂×C₁₆	—	2	4	no	—
210	C₂×C₁₆	—	2	2	no	—
211	#107	—	17	4	no	—
212	#107	—	34	2	no	—
213	#106	—	17	4	no	—
214	#106	—	34	2	no	—
215	#106	—	34	2	no	—
216	#106	—	34	4	no	—
217	#106	—	34	2	no	—
218	C₁₇	C₂×#1386	1	1	yes	characteristic, complemented, Sylow
219	C₂⁴	D₁₃₆	1	1	yes	characteristic
220	C₂²×C₄	—	68	4	no	—
221	C₂²×C₄	—	68	2	no	—
222	C₂²×C₄	—	68	2	no	—
223	C₂²×C₄	—	136	4	no	—
224	C₂²×C₄	#63219	1	2	yes	—
225	C₂²×C₄	#63219	1	1	yes	characteristic
226	C₂²×C₄	#63219	1	1	yes	characteristic
227	C₂²×C₄	D₁₃₆	1	1	yes	characteristic
228	C₂²×C₄	Dic₆₈	1	2	yes	—
229	C₂²×C₄	C₄⋊Dic₁₇	1	2	yes	—
230	C₂²×C₄	#63227	1	2	yes	—
231	C₂²×C₄	#63227	1	1	yes	characteristic
232	C₂²×C₄	#63227	1	1	yes	characteristic
233	C₂²×C₄	C₂×D₆₈	1	1	yes	characteristic
234	C₂×C₈	C₄×Dic₁₇	1	2	yes	—
235	C₂×C₈	C₄×Dic₁₇	1	2	yes	—
236	C₂×C₈	#63225	1	2	yes	—
237	C₂×C₈	#63225	1	2	yes	—
238	C₂×C₈	#63225	1	2	yes	—
239	C₂×C₈	#63225	1	2	yes	—
240	C₂×C₈	C₄⋊Dic₁₇	1	2	yes	—
241	C₂×C₈	#63227	1	2	yes	—
242	C₂×C₈	#63227	1	1	yes	characteristic
243	C₂×C₈	#63227	1	1	yes	characteristic
244	C₂×C₈	C₂²⋊Dic₁₇	1	2	yes	—
245	C₂×C₈	C₂²⋊Dic₁₇	1	2	yes	—
246	C₂×C₈	C₂×Dic₃₄	1	1	yes	characteristic
247	C₂×C₈	C₂×C₄×D₁₇	1	1	yes	characteristic
248	C₂×C₈	C₂×C₄×D₁₇	1	1	yes	characteristic
249	C₂×C₈	C₂²×Dic₁₇	1	1	yes	characteristic
250	C₂×C₈	C₂×C₂²⋊D₁₇	1	1	yes	characteristic
251	C₂×C₈	C₂×C₂²⋊D₁₇	1	1	yes	characteristic
252	C₄⋊C₄	—	34	4	no	—
253	C₄⋊C₄	—	34	4	no	—
254	C₄⋊C₄	—	68	4	no	—
255	C₄⋊C₄	—	68	2	no	—
256	C₄⋊C₄	—	68	2	no	—
257	C₂²⋊C₄	—	68	2	no	—
258	C₂²⋊C₄	—	68	2	no	—
259	C₄²	—	68	2	no	—
260	C₄²	—	68	2	no	—
261	C₁₆	—	2	4	no	—
262	C₂³	—	2	4	no	—
263	C₂³	#10587489	1	2	yes	—
264	C₂³	#10587492	1	1	yes	characteristic
265	C₂³	#10587492	1	2	yes	—
266	C₂³	#10587492	1	1	yes	characteristic
267	C₂³	C₂×D₁₃₆	1	1	yes	characteristic
268	C₂×C₄	—	68	4	no	—
269	C₂×C₄	—	68	4	no	—
270	C₂×C₄	—	136	8	no	—
271	C₂×C₄	—	136	4	no	—
272	C₂×C₄	—	136	4	no	—
273	C₂×C₄	#10587487	1	2	yes	—
274	C₂×C₄	#10587487	1	2	yes	—
275	C₂×C₄	#10587487	1	2	yes	—
276	C₂×C₄	#10587487	1	2	yes	—
277	C₂×C₄	#10587488	1	2	yes	—
278	C₂×C₄	#10587488	1	2	yes	—
279	C₂×C₄	#10587489	1	2	yes	—
280	C₂×C₄	#10587492	1	1	yes	characteristic
281	C₂×C₄	#10587492	1	2	yes	—
282	C₂×C₄	#10587492	1	1	yes	characteristic
283	C₂×C₄	#10587502	1	2	yes	—
284	C₂×C₄	#10587502	1	2	yes	—
285	C₂×C₄	C₂×#63219	1	1	yes	characteristic
286	C₂×C₄	C₂×#63219	1	1	yes	characteristic
287	C₂×C₄	C₂×Dic₆₈	1	1	yes	characteristic
288	C₂×C₄	C₂×C₄⋊Dic₁₇	1	1	yes	characteristic
289	C₂×C₄	C₂×#63227	1	1	yes	characteristic
290	C₂×C₄	C₂×#63227	1	1	yes	characteristic
291	C₈	#10587502	1	2	yes	—
292	C₈	#10587502	1	2	yes	—
293	C₈	C₂×C₄×Dic₁₇	1	1	yes	characteristic
294	C₈	C₂×#63225	1	1	yes	characteristic
295	C₈	C₂×#63225	1	1	yes	characteristic
296	C₈	C₂×C₂²⋊Dic₁₇	1	1	yes	characteristic
297	C₂²	—	2	4	no	—
298	C₂²	—	2	4	no	—
299	C₂²	—	2	2	no	—
300	C₂²	—	2	2	no	—
301	C₂²	#11762572	1	2	yes	—
302	C₂²	#11762572	1	2	yes	—
303	C₂²	#11762620	1	4	yes	—
304	C₂²	C₂×#10587492	1	1	yes	characteristic
305	C₂²	C₂×#10587492	1	1	yes	characteristic
306	C₂²	C₂×#10587489	1	1	yes	characteristic
307	C₄	—	136	4	no	—
308	C₄	—	136	4	no	—
309	C₄	#11762572	1	2	yes	—
310	C₄	#11762572	1	2	yes	—
311	C₄	C₂×#10587502	1	1	yes	characteristic
312	C₄	C₂×#10587487	1	1	yes	characteristic
313	C₄	C₂×#10587487	1	1	yes	characteristic
314	C₄	C₂×#10587488	1	1	yes	characteristic
315	C₂	—	2	1	no	—
316	C₂	—	2	2	no	—
317	C₂	—	2	1	no	—
318	C₂	#14526890	1	4	yes	direct product factor
319	C₂	C₂×#11762572	1	1	yes	characteristic
320	C₂	C₂×#11762620	1	2	yes	—

	subgroup	G/H
center	C₂³	#10587492
commutator	C₈×C₁₇	C₂×C₄²
Frattini	C₂²×C₈	C₂²×D₁₇
Fitting	C₂²×C₁₇×M₃₂	C₂
socle	C₂³×C₁₇	#102
prime radical	C₂⁴×C₁₇	D₈
cosocle	C₂²×C₈×C₁₇	C₂³
hypercenter	C₂²×M₃₂	D₁₇
hypocenter	C₁₇	C₂×#1386
LCS	G ▷ 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
3. #14526890	C₂	direct (N × H)
9. #11762847	C₄	semidirect
11. C₂×#10587489	C₄	semidirect
15. C₂×#10587488	C₄	semidirect
20. C₂×C₁₇×M₃₂	C₄	semidirect
48. C₁₇×M₃₂	C₂×C₄	semidirect
52. #10587489	C₂×C₄	semidirect
57. #10587488	C₂×C₄	semidirect
164. C₂×M₃₂	Dic₁₇	semidirect
180. C₂×C₁₇	#1386	semidirect
208. M₃₂	C₂×Dic₁₇	semidirect
218. C₁₇	C₂×#1386	semidirect
318. C₂	#14526890	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂×#11762847	2
C₂²×#10587488	2
#14526890	2
C₂²×C₁₇×M₃₂	2
C₂×#1386	17

Subgroups — C2×#14526890 #16949239

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#14526890 #16949239