Subgroups — C₂×#14543852 #17960530

Order 4352.

586 conjugacy classes of subgroups.

The subgroup lattice has 284 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#11763819	C₂	1	2	yes	maximal
2	#14543852	C₂	1	4	yes	maximal, direct product factor
3	C₂²×#10587469	C₂	1	1	yes	maximal, characteristic
4	C₂×#10587677	—	2	2	no	—
5	#11763819	—	2	2	no	—
6	#11763819	—	2	2	no	—
7	#11763819	C₄	1	4	yes	complemented
8	#11763819	C₂²	1	4	yes	—
9	C₂×#10587534	—	2	2	no	—
10	C₂×#10587533	C₄	1	4	yes	complemented
11	C₂²×C₈×D₁₇	C₂²	1	1	yes	characteristic
12	C₂×#10587469	—	2	2	no	—
13	C₂×#10587469	—	2	2	no	—
14	C₂×#10587469	C₄	1	2	yes	complemented
15	C₂×#10587469	C₂²	1	2	yes	—
16	C₂²×#63214	C₄	1	1	yes	characteristic, complemented
17	C₂²×C₁₆×C₁₇	C₄	1	1	yes	characteristic, complemented
18	C₂²×C₄×D₁₇	D₄	1	1	yes	characteristic
19	C₂×C₂²⋊#595	—	4	2	no	—
20	#10587677	—	2	4	no	—
21	#10587677	—	2	4	no	—
22	#10587677	—	4	2	no	—
23	#10587677	—	4	2	no	—
24	C₂×C₄⋊#595	—	2	4	no	—
25	C₂×C₄×#595	—	4	2	no	—
26	C₂²×C₈×C₁₇	C₂×C₄	1	1	yes	characteristic
27	C₂²×#3622	Q₈	1	1	yes	characteristic
28	C₂×C₈×D₁₇	C₂×C₄	1	2	yes	—
29	C₂×C₈×D₁₇	C₂×C₄	1	2	yes	—
30	C₂×C₈×D₁₇	C₂×C₄	1	1	yes	characteristic
31	C₂×C₈×D₁₇	D₄	1	2	yes	—
32	C₂×C₈×D₁₇	D₄	1	2	yes	—
33	C₂×C₈×D₁₇	D₄	1	1	yes	characteristic
34	C₂×C₈×D₁₇	Q₈	1	1	yes	characteristic
35	C₂×C₈×D₁₇	C₂³	1	1	yes	characteristic
36	#10587534	—	2	4	no	—
37	#10587534	—	2	4	no	—
38	#10587533	—	2	4	no	—
39	#10587533	C₂×C₄	1	8	yes	complemented
40	C₂×C₁₆×C₁₇	—	2	1	no	—
41	C₂×C₁₆×C₁₇	—	2	1	no	—
42	C₂×C₁₆×C₁₇	C₂×C₄	1	2	yes	complemented
43	C₂×#63214	—	2	1	no	—
44	C₂×#63214	—	2	1	no	—
45	C₂×#63214	C₂×C₄	1	2	yes	complemented
46	#10587469	—	2	4	no	—
47	#10587469	—	2	4	no	—
48	C₂³×D₁₇	D₈	1	1	yes	characteristic
49	C₂²×#595	—	4	4	no	—
50	C₂²×C₄×C₁₇	C₄⋊C₄	1	1	yes	characteristic
51	C₂²×Dic₁₇	Q₁₆	1	1	yes	characteristic
52	C₂×C₄×D₁₇	C₂²⋊C₄	1	2	yes	—
53	C₂×C₄×D₁₇	C₂²⋊C₄	1	2	yes	—
54	C₂×C₄×D₁₇	C₄⋊C₄	1	1	yes	characteristic
55	C₂×C₄×D₁₇	D₈	1	1	yes	characteristic
56	C₂×C₄×D₁₇	SD₁₆	1	2	yes	—
57	C₂×C₄×D₁₇	SD₁₆	1	2	yes	—
58	C₂×C₄×D₁₇	Q₁₆	1	1	yes	characteristic
59	C₂×C₄×D₁₇	C₂×D₄	1	1	yes	characteristic
60	C₂²⋊#595	—	4	4	no	—
61	C₂²⋊#595	—	4	4	no	—
62	C₄⋊#595	—	2	8	no	—
63	C₄⋊#595	—	4	4	no	—
64	C₄×#595	—	4	4	no	—
65	C₄×#595	—	4	4	no	—
66	C₂×C₈×C₁₇	C₄²	1	1	yes	characteristic
67	C₂×C₈×C₁₇	C₄²	1	1	yes	characteristic
68	C₂×C₈×C₁₇	C₂²⋊C₄	1	1	yes	characteristic
69	C₂×C₈×C₁₇	C₂²⋊C₄	1	1	yes	characteristic
70	C₂×C₈×C₁₇	C₄⋊C₄	1	1	yes	characteristic
71	C₂×C₈×C₁₇	C₂²×C₄	1	1	yes	characteristic
72	C₂×#3622	C₄⋊C₄	1	1	yes	characteristic
73	C₂×#3622	C₄⋊C₄	1	2	yes	—
74	C₂×#3622	C₄⋊C₄	1	2	yes	—
75	C₂×#3622	C₂×Q₈	1	1	yes	characteristic
76	C₈×D₁₇	C₄²	1	2	yes	—
77	C₈×D₁₇	C₂²⋊C₄	1	2	yes	—
78	C₈×D₁₇	C₂²⋊C₄	1	2	yes	—
79	C₈×D₁₇	C₂²⋊C₄	1	2	yes	—
80	C₈×D₁₇	C₄⋊C₄	1	2	yes	—
81	C₈×D₁₇	C₄⋊C₄	1	2	yes	—
82	C₈×D₁₇	C₂²×C₄	1	2	yes	—
83	C₈×D₁₇	C₂×D₄	1	2	yes	—
84	C₁₆×C₁₇	—	2	2	no	—
85	#63214	—	2	2	no	—
86	C₂×#1384	—	17	1	no	maximal, Sylow
87	C₂³×C₁₇	#107	1	1	yes	characteristic
88	C₂²×D₁₇	—	4	2	no	—
89	C₂²×D₁₇	#102	1	2	yes	—
90	C₂²×D₁₇	#102	1	2	yes	—
91	C₂²×D₁₇	#107	1	1	yes	characteristic
92	C₂²×D₁₇	C₂×D₈	1	1	yes	characteristic
93	C₂×#595	—	4	8	no	—
94	C₂×#595	—	8	8	no	—
95	C₂×C₄×C₁₇	#95	1	1	yes	characteristic
96	C₂×C₄×C₁₇	#95	1	1	yes	characteristic
97	C₂×C₄×C₁₇	#106	1	1	yes	characteristic
98	C₂×C₄×C₁₇	#106	1	1	yes	characteristic
99	C₂×C₄×C₁₇	#107	1	1	yes	characteristic
100	C₂×C₄×C₁₇	C₂×C₄⋊C₄	1	1	yes	characteristic
101	C₂×Dic₁₇	#103	1	2	yes	—
102	C₂×Dic₁₇	#103	1	2	yes	—
103	C₂×Dic₁₇	#107	1	1	yes	characteristic
104	C₂×Dic₁₇	C₂×Q₁₆	1	1	yes	characteristic
105	C₄×D₁₇	#95	1	2	yes	—
106	C₄×D₁₇	#102	1	2	yes	—
107	C₄×D₁₇	#102	1	2	yes	—
108	C₄×D₁₇	#103	1	2	yes	—
109	C₄×D₁₇	#103	1	2	yes	—
110	C₄×D₁₇	#106	1	2	yes	—
111	C₄×D₁₇	C₂×C₂²⋊C₄	1	2	yes	—
112	C₄×D₁₇	C₂×SD₁₆	1	2	yes	—
113	C₈×C₁₇	#95	1	1	yes	characteristic
114	C₈×C₁₇	#95	1	1	yes	characteristic
115	C₈×C₁₇	C₂×C₄²	1	1	yes	characteristic
116	C₈×C₁₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
117	#3622	#95	1	2	yes	—
118	#3622	C₂×C₄⋊C₄	1	2	yes	—
119	C₂²×M₃₂	—	17	1	no	—
120	C₂×#427	—	17	2	no	—
121	#1384	—	17	4	no	—
122	C₂²×C₁₇	—	2	2	no	—
123	C₂²×C₁₇	#340	1	1	yes	characteristic
124	C₂²×C₁₇	#340	1	1	yes	characteristic
125	C₂²×C₁₇	C₂×#107	1	1	yes	characteristic
126	C₂×D₁₇	—	2	4	no	—
127	C₂×D₁₇	—	4	2	no	—
128	C₂×D₁₇	—	4	1	no	—
129	C₂×D₁₇	—	4	1	no	—
130	C₂×D₁₇	#340	1	2	yes	—
131	C₂×D₁₇	C₂×#102	1	2	yes	—
132	#595	—	8	8	no	—
133	C₄×C₁₇	#340	1	1	yes	characteristic
134	C₄×C₁₇	#340	1	1	yes	characteristic
135	C₄×C₁₇	C₂×#95	1	1	yes	characteristic
136	C₄×C₁₇	C₂×#106	1	1	yes	characteristic
137	Dic₁₇	#340	1	2	yes	—
138	Dic₁₇	C₂×#103	1	2	yes	—
139	C₂³×C₈	—	17	1	no	—
140	C₂×#117	—	34	2	no	—
141	C₂×M₃₂	—	17	2	no	—
142	C₂×M₃₂	—	17	2	no	—
143	C₂×M₃₂	—	34	2	no	—
144	C₂×M₃₂	—	34	2	no	—
145	C₂²×C₁₆	—	17	1	no	—
146	C₂²×C₁₆	#595	1	1	yes	characteristic, complemented
147	#427	—	17	4	no	—
148	#427	—	17	4	no	—
149	#427	—	34	2	no	—
150	#427	—	34	2	no	—
151	C₂×#107	—	34	2	no	—
152	C₂×#106	—	17	4	no	—
153	C₂×C₁₇	—	2	1	no	—
154	C₂×C₁₇	—	2	1	no	—
155	C₂×C₁₇	#1384	1	2	yes	complemented
156	C₂×C₁₇	C₂×#340	1	1	yes	characteristic
157	D₁₇	—	2	2	no	—
158	D₁₇	—	2	2	no	—
159	C₂³×C₄	—	17	1	no	—
160	C₂²×C₈	—	17	1	no	—
161	C₂²×C₈	—	17	2	no	—
162	C₂²×C₈	—	17	2	no	—
163	C₂²×C₈	—	17	1	no	—
164	C₂²×C₈	—	17	1	no	—
165	C₂²×C₈	—	17	2	no	—
166	C₂²×C₈	—	17	2	no	—
167	C₂²×C₈	—	17	1	no	—
168	C₂²×C₈	—	17	1	no	—
169	C₂²×C₈	C₂×#595	1	1	yes	characteristic
170	#117	—	34	4	no	—
171	#117	—	34	4	no	—
172	#117	—	68	2	no	—
173	#117	—	68	2	no	—
174	C₂×C₄⋊C₄	—	34	4	no	—
175	C₂×C₂²⋊C₄	—	68	2	no	—
176	C₂×C₄²	—	68	2	no	—
177	M₃₂	—	34	4	no	—
178	M₃₂	—	34	4	no	—
179	C₂×C₁₆	—	2	1	no	—
180	C₂×C₁₆	—	2	1	no	—
181	C₂×C₁₆	—	17	2	no	—
182	C₂×C₁₆	—	34	1	no	—
183	C₂×C₁₆	—	34	1	no	—
184	C₂×C₁₆	C₂×#595	1	2	yes	complemented
185	#107	—	34	4	no	—
186	#107	—	34	4	no	—
187	#106	—	17	8	no	—
188	#106	—	34	4	no	—
189	C₁₇	C₂×#1384	1	1	yes	characteristic, complemented, Sylow
190	C₂⁴	—	17	1	no	—
191	C₂²×C₄	—	17	1	no	—
192	C₂²×C₄	—	17	1	no	—
193	C₂²×C₄	—	17	1	no	—
194	C₂²×C₄	—	17	2	no	—
195	C₂²×C₄	—	17	2	no	—
196	C₂²×C₄	—	17	1	no	—
197	C₂²×C₄	—	17	1	no	—
198	C₂²×C₄	—	17	2	no	—
199	C₂²×C₄	—	17	2	no	—
200	C₂²×C₄	—	68	4	no	—
201	C₂²×C₄	C₄⋊#595	1	1	yes	characteristic
202	C₂×C₈	—	17	1	no	—
203	C₂×C₈	—	17	1	no	—
204	C₂×C₈	—	17	2	no	—
205	C₂×C₈	—	17	2	no	—
206	C₂×C₈	—	17	2	no	—
207	C₂×C₈	—	17	2	no	—
208	C₂×C₈	—	17	2	no	—
209	C₂×C₈	—	17	2	no	—
210	C₂×C₈	—	17	2	no	—
211	C₂×C₈	—	17	2	no	—
212	C₂×C₈	—	17	2	no	—
213	C₂×C₈	—	17	2	no	—
214	C₂×C₈	C₄×#595	1	1	yes	characteristic
215	C₂×C₈	C₄×#595	1	1	yes	characteristic
216	C₂×C₈	C₄⋊#595	1	1	yes	characteristic
217	C₂×C₈	C₂²⋊#595	1	1	yes	characteristic
218	C₂×C₈	C₂²⋊#595	1	1	yes	characteristic
219	C₂×C₈	C₂²×#595	1	1	yes	characteristic
220	C₄⋊C₄	—	34	8	no	—
221	C₄⋊C₄	—	68	4	no	—
222	C₂²⋊C₄	—	68	4	no	—
223	C₂²⋊C₄	—	68	4	no	—
224	C₄²	—	68	4	no	—
225	C₄²	—	68	4	no	—
226	C₁₆	—	2	2	no	—
227	C₁₆	—	34	2	no	—
228	C₂³	—	17	2	no	—
229	C₂³	—	17	1	no	—
230	C₂³	—	17	1	no	—
231	C₂³	—	17	2	no	—
232	C₂³	—	68	2	no	—
233	C₂³	#10587534	1	1	yes	characteristic
234	C₂×C₄	—	17	1	no	—
235	C₂×C₄	—	17	1	no	—
236	C₂×C₄	—	17	2	no	—
237	C₂×C₄	—	17	2	no	—
238	C₂×C₄	—	17	2	no	—
239	C₂×C₄	—	17	2	no	—
240	C₂×C₄	—	17	2	no	—
241	C₂×C₄	—	17	2	no	—
242	C₂×C₄	—	17	2	no	—
243	C₂×C₄	—	17	2	no	—
244	C₂×C₄	—	17	2	no	—
245	C₂×C₄	—	17	2	no	—
246	C₂×C₄	—	68	8	no	—
247	C₂×C₄	—	136	8	no	—
248	C₂×C₄	#10587533	1	1	yes	characteristic
249	C₂×C₄	#10587533	1	1	yes	characteristic
250	C₂×C₄	#10587534	1	1	yes	characteristic
251	C₂×C₄	(C₂×#595)⋊C₄	1	1	yes	characteristic
252	C₂×C₄	(C₂×#595)⋊C₄	1	1	yes	characteristic
253	C₂×C₄	C₂×C₄⋊#595	1	1	yes	characteristic
254	C₈	—	17	2	no	—
255	C₈	—	17	2	no	—
256	C₈	(C₂×#595)⋊C₄	1	1	yes	characteristic
257	C₈	(C₂×#595)⋊C₄	1	1	yes	characteristic
258	C₈	C₂×C₄×#595	1	1	yes	characteristic
259	C₈	C₂×C₂²⋊#595	1	1	yes	characteristic
260	C₂²	—	2	2	no	—
261	C₂²	—	17	2	no	—
262	C₂²	—	17	2	no	—
263	C₂²	—	34	4	no	—
264	C₂²	—	68	1	no	—
265	C₂²	—	68	1	no	—
266	C₂²	—	68	2	no	—
267	C₂²	#11763683	1	1	yes	characteristic
268	C₂²	#11763683	1	1	yes	characteristic
269	C₂²	C₂×#10587534	1	1	yes	characteristic
270	C₄	—	17	2	no	—
271	C₄	—	17	2	no	—
272	C₄	—	136	8	no	—
273	C₄	#11763683	1	1	yes	characteristic
274	C₄	#11763683	1	1	yes	characteristic
275	C₄	C₂×(C₂×#595)⋊C₄	1	1	yes	characteristic
276	C₄	C₂×#10587533	1	1	yes	characteristic
277	C₂	—	2	1	no	—
278	C₂	—	2	1	no	—
279	C₂	—	34	2	no	—
280	C₂	—	34	2	no	—
281	C₂	#14543852	1	2	yes	direct product factor
282	C₂	C₂×#11763683	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	#10587533
commutator	C₈×C₁₇	C₂×C₄²
Frattini	C₂×C₈	C₂²×#595
Fitting	C₂²×C₁₆×C₁₇	C₄
socle	C₂²×C₁₇	#340
prime radical	C₂³×D₁₇	D₈
cosocle	C₂×C₈×D₁₇	C₂³
hypercenter	C₂²×C₁₆	#595
hypocenter	C₁₇	C₂×#1384
LCS	G ▷ C₈×C₁₇ ▷ C₄×C₁₇ ▷ C₂×C₁₇ ▷ C₁₇
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄ ◁ C₂²×C₈ ◁ C₂²×C₁₆

Complements

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

N	H ≅ G/N	type
2. #14543852	C₂	direct (N × H)
7. #11763819	C₄	semidirect
10. C₂×#10587533	C₄	semidirect
14. C₂×#10587469	C₄	semidirect
16. C₂²×#63214	C₄	semidirect
17. C₂²×C₁₆×C₁₇	C₄	semidirect
39. #10587533	C₂×C₄	semidirect
42. C₂×C₁₆×C₁₇	C₂×C₄	semidirect
45. C₂×#63214	C₂×C₄	semidirect
146. C₂²×C₁₆	#595	semidirect
155. C₂×C₁₇	#1384	semidirect
184. C₂×C₁₆	C₂×#595	semidirect
189. C₁₇	C₂×#1384	semidirect
281. C₂	#14543852	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×#11763819	2
#14543852	2
C₂²×#10587469	2
C₂×#1384	17

Subgroups — C2×#14543852 #17960530

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#14543852 #17960530