Subgroups — C₂×#14530626 #17187558

Order 4352.

654 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	C₂×#11763226	C₂	1	2	yes	maximal, complemented
2	#14530626	C₂	1	8	yes	maximal, direct product factor
3	C₂×#11762630	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂×#11762593	C₂	1	2	yes	maximal, complemented
5	C₂×#11762585	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×C₁₇×#443	C₂	1	1	yes	maximal, characteristic
7	#11763226	—	2	4	no	—
8	#11763226	C₂²	1	8	yes	complemented
9	C₂×#10587483	—	2	2	no	—
10	C₂×#10587483	C₂²	1	2	yes	—
11	C₂²×#63214	—	2	2	no	—
12	C₂²×#63214	C₂²	1	1	yes	characteristic
13	C₂²×#63214	C₂²	1	1	yes	characteristic
14	#11762630	C₂²	1	4	yes	complemented
15	#11762593	C₂²	1	8	yes	complemented
16	#11762585	C₂²	1	4	yes	complemented
17	C₂×C₁₇×#135	C₄	1	1	yes	characteristic
18	C₂×C₁₇×#131	C₂²	1	2	yes	—
19	C₁₇×#443	C₄	1	4	yes	—
20	C₁₇×#443	C₂²	1	4	yes	—
21	C₂×C₁₇×#108	C₄	1	1	yes	characteristic
22	C₂×C₁₇×#104	C₄	1	2	yes	—
23	C₂×C₄×C₈×C₁₇	C₂²	1	1	yes	characteristic
24	C₂×#41×C₁₇	C₂×C₄	1	2	yes	—
25	C₁₇×#135	C₈	1	4	yes	—
26	C₁₇×#135	C₂×C₄	1	4	yes	—
27	C₂×Q₁₆×C₁₇	C₈	1	1	yes	characteristic
28	C₂×SD₁₆×C₁₇	C₈	1	2	yes	—
29	C₂×D₈×C₁₇	C₈	1	1	yes	characteristic
30	C₁₇×#131	—	2	2	no	—
31	C₁₇×#131	—	2	2	no	—
32	C₁₇×#131	C₂×C₄	1	4	yes	—
33	C₁₇×#131	C₂³	1	4	yes	—
34	C₂×M₁₆×C₁₇	—	2	2	no	—
35	C₂×M₁₆×C₁₇	C₂×C₄	1	2	yes	—
36	C₂²×C₈×C₁₇	—	2	2	no	—
37	C₂²×C₈×C₁₇	C₂×C₄	1	1	yes	characteristic
38	C₂²×C₈×C₁₇	C₂³	1	1	yes	characteristic
39	C₂×C₄²×C₁₇	C₂×C₄	1	1	yes	characteristic
40	C₁₇×#108	C₂×C₄	1	4	yes	—
41	C₁₇×#104	C₂×C₄	1	8	yes	—
42	C₄×C₈×C₁₇	C₂×C₄	1	2	yes	—
43	C₄×C₈×C₁₇	C₂³	1	2	yes	—
44	#10587483	—	2	8	no	—
45	#10587483	—	2	2	no	—
46	#10587483	C₂³	1	4	yes	—
47	C₂×#63214	—	2	8	no	—
48	C₂×#63214	—	2	2	no	—
49	C₂×#63214	—	2	1	no	—
50	C₂×#63214	—	2	1	no	—
51	C₂×#63214	D₄	1	2	yes	—
52	C₂×#63214	D₄	1	2	yes	—
53	C₂×#63214	C₂³	1	2	yes	—
54	C₂×#63214	C₂³	1	2	yes	—
55	#41×C₁₇	—	2	2	no	—
56	#41×C₁₇	—	2	2	no	—
57	#41×C₁₇	C₂×C₈	1	4	yes	—
58	#41×C₁₇	C₂²×C₄	1	4	yes	—
59	C₂×Q₈×C₁₇	C₂×C₈	1	2	yes	—
60	C₂×D₄×C₁₇	—	2	2	no	—
61	C₂×D₄×C₁₇	C₂×C₈	1	2	yes	—
62	C₂²×C₄×C₁₇	—	2	2	no	—
63	C₂²×C₄×C₁₇	—	2	1	no	—
64	C₂²×C₄×C₁₇	C₂²×C₄	1	1	yes	characteristic
65	Q₁₆×C₁₇	C₂×C₈	1	4	yes	—
66	SD₁₆×C₁₇	C₂×C₈	1	8	yes	—
67	D₈×C₁₇	C₂×C₈	1	4	yes	—
68	M₁₆×C₁₇	—	2	2	no	—
69	M₁₆×C₁₇	—	2	4	no	—
70	M₁₆×C₁₇	—	2	4	no	—
71	M₁₆×C₁₇	C₂²×C₄	1	4	yes	—
72	C₂×C₈×C₁₇	—	2	2	no	—
73	C₂×C₈×C₁₇	—	2	4	no	—
74	C₂×C₈×C₁₇	—	2	4	no	—
75	C₂×C₈×C₁₇	C₂×C₈	1	1	yes	characteristic
76	C₂×C₈×C₁₇	C₂×C₈	1	1	yes	characteristic
77	C₂×C₈×C₁₇	C₂×C₈	1	2	yes	—
78	C₂×C₈×C₁₇	C₂²×C₄	1	1	yes	characteristic
79	C₂×C₈×C₁₇	C₂²×C₄	1	2	yes	—
80	C₂×C₈×C₁₇	C₂×D₄	1	1	yes	characteristic
81	C₂×C₈×C₁₇	C₂×D₄	1	1	yes	characteristic
82	C₂×C₈×C₁₇	#41	1	1	yes	characteristic
83	C₂×C₈×C₁₇	#41	1	1	yes	characteristic
84	C₂×C₈×C₁₇	C₂⁴	1	1	yes	characteristic
85	C₄²×C₁₇	—	2	1	no	—
86	C₄²×C₁₇	C₂²×C₄	1	2	yes	—
87	#63214	—	2	4	no	—
88	#63214	—	2	2	no	—
89	#63214	C₂×D₄	1	4	yes	—
90	C₂×#2171	—	17	1	no	maximal, Sylow
91	C₂³×C₁₇	—	2	2	no	—
92	C₂³×C₁₇	#131	1	1	yes	characteristic
93	Q₈×C₁₇	—	2	2	no	—
94	Q₈×C₁₇	C₂²×C₈	1	4	yes	—
95	D₄×C₁₇	—	2	4	no	—
96	D₄×C₁₇	—	2	4	no	—
97	D₄×C₁₇	—	2	2	no	—
98	D₄×C₁₇	C₂²×C₈	1	4	yes	—
99	C₂×C₄×C₁₇	—	2	2	no	—
100	C₂×C₄×C₁₇	—	2	4	no	—
101	C₂×C₄×C₁₇	—	2	4	no	—
102	C₂×C₄×C₁₇	—	2	2	no	—
103	C₂×C₄×C₁₇	—	4	2	no	—
104	C₂×C₄×C₁₇	C₄×D₄	1	1	yes	characteristic
105	C₂×C₄×C₁₇	C₄×D₄	1	1	yes	characteristic
106	C₂×C₄×C₁₇	C₂²×C₈	1	1	yes	characteristic
107	C₂×C₄×C₁₇	C₂²×C₈	1	1	yes	characteristic
108	C₂×C₄×C₁₇	#131	1	1	yes	characteristic
109	C₂×C₄×C₁₇	C₂³×C₄	1	1	yes	characteristic
110	C₈×C₁₇	—	2	4	no	—
111	C₈×C₁₇	C₄×D₄	1	1	yes	characteristic
112	C₈×C₁₇	C₄×D₄	1	1	yes	characteristic
113	C₈×C₁₇	C₂²×C₈	1	2	yes	—
114	C₈×C₁₇	C₂²×C₈	1	2	yes	—
115	C₈×C₁₇	C₂²×D₄	1	1	yes	characteristic
116	C₈×C₁₇	C₂×#41	1	1	yes	characteristic
117	C₂×#504	—	17	2	no	—
118	C₂×#443	D₁₇	1	1	yes	characteristic
119	#2171	—	17	8	no	—
120	C₂×#364	—	17	1	no	—
121	C₂×#350	—	17	2	no	—
122	C₂×#346	—	17	1	no	—
123	C₂²×C₁₇	—	2	2	no	—
124	C₂²×C₁₇	—	2	4	no	—
125	C₂²×C₁₇	—	4	4	no	—
126	C₂²×C₁₇	C₈×D₄	1	1	yes	characteristic
127	C₂²×C₁₇	C₈×D₄	1	1	yes	characteristic
128	C₂²×C₁₇	C₂×#131	1	1	yes	characteristic
129	C₄×C₁₇	—	2	4	no	—
130	C₄×C₁₇	—	4	2	no	—
131	C₄×C₁₇	C₈×D₄	1	1	yes	characteristic
132	C₄×C₁₇	C₈×D₄	1	1	yes	characteristic
133	C₄×C₁₇	C₂×C₄×D₄	1	1	yes	characteristic
134	C₄×C₁₇	C₂³×C₈	1	1	yes	characteristic
135	C₂×#135	Dic₁₇	1	1	yes	characteristic
136	C₂×#131	C₂×D₁₇	1	2	yes	—
137	#504	—	17	8	no	—
138	#504	—	34	4	no	—
139	C₂×M₃₂	—	17	2	no	—
140	C₂×M₃₂	—	34	2	no	—
141	C₂²×C₁₆	—	17	1	no	—
142	C₂²×C₁₆	—	17	1	no	—
143	C₂²×C₁₆	—	34	2	no	—
144	#443	Dic₁₇	1	4	yes	—
145	#443	C₂×D₁₇	1	4	yes	—
146	C₂×#108	Dic₁₇	1	1	yes	characteristic
147	C₂×#104	Dic₁₇	1	2	yes	—
148	C₂×C₄×C₈	C₂×D₁₇	1	1	yes	characteristic
149	#364	—	17	4	no	—
150	#350	—	17	8	no	—
151	#346	—	17	4	no	—
152	C₂×C₁₇	—	2	1	no	—
153	C₂×C₁₇	—	2	1	no	—
154	C₂×C₁₇	—	4	4	no	—
155	C₂×C₁₇	#2171	1	2	yes	complemented
156	C₂×C₁₇	C₂×C₈×D₄	1	1	yes	characteristic
157	C₂×#41	C₂×Dic₁₇	1	2	yes	—
158	#135	#3622	1	4	yes	—
159	#135	C₂×Dic₁₇	1	4	yes	—
160	C₂×Q₁₆	#3622	1	1	yes	characteristic
161	C₂×SD₁₆	#3622	1	2	yes	—
162	C₂×D₈	#3622	1	1	yes	characteristic
163	#131	—	2	2	no	—
164	#131	—	2	2	no	—
165	#131	C₂×Dic₁₇	1	4	yes	—
166	#131	C₂²×D₁₇	1	4	yes	—
167	C₂×M₁₆	—	2	2	no	—
168	C₂×M₁₆	C₂×Dic₁₇	1	2	yes	—
169	C₂²×C₈	—	2	2	no	—
170	C₂²×C₈	C₂×Dic₁₇	1	1	yes	characteristic
171	C₂²×C₈	C₂²×D₁₇	1	1	yes	characteristic
172	C₂×C₄²	C₂×Dic₁₇	1	1	yes	characteristic
173	M₃₂	—	17	4	no	—
174	M₃₂	—	34	8	no	—
175	M₃₂	—	34	2	no	—
176	C₂×C₁₆	—	17	2	no	—
177	C₂×C₁₆	—	17	2	no	—
178	C₂×C₁₆	—	17	2	no	—
179	C₂×C₁₆	—	17	2	no	—
180	C₂×C₁₆	—	34	8	no	—
181	C₂×C₁₆	—	34	2	no	—
182	C₂×C₁₆	—	34	1	no	—
183	C₂×C₁₆	—	34	1	no	—
184	#108	C₂×Dic₁₇	1	4	yes	—
185	#104	C₂×Dic₁₇	1	8	yes	—
186	C₄×C₈	C₂×Dic₁₇	1	2	yes	—
187	C₄×C₈	C₂²×D₁₇	1	2	yes	—
188	C₁₇	C₂×#2171	1	1	yes	characteristic, complemented, Sylow
189	#41	—	2	2	no	—
190	#41	—	2	2	no	—
191	#41	C₂×#3622	1	4	yes	—
192	#41	C₂²×Dic₁₇	1	4	yes	—
193	C₂×Q₈	C₂×#3622	1	2	yes	—
194	C₂×D₄	—	2	2	no	—
195	C₂×D₄	C₂×#3622	1	2	yes	—
196	C₂²×C₄	—	2	2	no	—
197	C₂²×C₄	—	2	1	no	—
198	C₂²×C₄	C₂²×Dic₁₇	1	1	yes	characteristic
199	Q₁₆	C₂×#3622	1	4	yes	—
200	SD₁₆	C₂×#3622	1	8	yes	—
201	D₈	C₂×#3622	1	4	yes	—
202	M₁₆	—	2	2	no	—
203	M₁₆	—	2	4	no	—
204	M₁₆	—	2	4	no	—
205	M₁₆	C₂²×Dic₁₇	1	4	yes	—
206	C₂×C₈	—	2	2	no	—
207	C₂×C₈	—	2	4	no	—
208	C₂×C₈	—	2	4	no	—
209	C₂×C₈	C₂×#3622	1	1	yes	characteristic
210	C₂×C₈	C₂×#3622	1	1	yes	characteristic
211	C₂×C₈	C₂×#3622	1	2	yes	—
212	C₂×C₈	D₄×D₁₇	1	1	yes	characteristic
213	C₂×C₈	D₄×D₁₇	1	1	yes	characteristic
214	C₂×C₈	#63254	1	1	yes	characteristic
215	C₂×C₈	#63254	1	1	yes	characteristic
216	C₂×C₈	C₂²×Dic₁₇	1	2	yes	—
217	C₂×C₈	C₂²×Dic₁₇	1	1	yes	characteristic
218	C₂×C₈	C₂³×D₁₇	1	1	yes	characteristic
219	C₄²	—	2	1	no	—
220	C₄²	C₂²×Dic₁₇	1	2	yes	—
221	C₁₆	—	17	4	no	—
222	C₁₆	—	34	4	no	—
223	C₁₆	—	34	2	no	—
224	C₂³	—	2	2	no	—
225	C₂³	#10587634	1	1	yes	characteristic
226	Q₈	—	2	2	no	—
227	Q₈	C₂²×#3622	1	4	yes	—
228	D₄	—	2	4	no	—
229	D₄	—	2	4	no	—
230	D₄	—	2	2	no	—
231	D₄	C₂²×#3622	1	4	yes	—
232	C₂×C₄	—	2	2	no	—
233	C₂×C₄	—	2	4	no	—
234	C₂×C₄	—	2	4	no	—
235	C₂×C₄	—	2	2	no	—
236	C₂×C₄	—	4	2	no	—
237	C₂×C₄	C₂²×#3622	1	1	yes	characteristic
238	C₂×C₄	C₂²×#3622	1	1	yes	characteristic
239	C₂×C₄	D₄×Dic₁₇	1	1	yes	characteristic
240	C₂×C₄	D₄×Dic₁₇	1	1	yes	characteristic
241	C₂×C₄	#10587634	1	1	yes	characteristic
242	C₂×C₄	C₂³×Dic₁₇	1	1	yes	characteristic
243	C₈	—	2	4	no	—
244	C₈	C₂²×#3622	1	2	yes	—
245	C₈	C₂²×#3622	1	2	yes	—
246	C₈	D₄×Dic₁₇	1	1	yes	characteristic
247	C₈	D₄×Dic₁₇	1	1	yes	characteristic
248	C₈	C₂×D₄×D₁₇	1	1	yes	characteristic
249	C₈	C₂×#63254	1	1	yes	characteristic
250	C₂²	—	2	2	no	—
251	C₂²	—	2	4	no	—
252	C₂²	—	4	4	no	—
253	C₂²	D₄×#3622	1	1	yes	characteristic
254	C₂²	D₄×#3622	1	1	yes	characteristic
255	C₂²	C₂×#10587634	1	1	yes	characteristic
256	C₄	—	2	4	no	—
257	C₄	—	4	2	no	—
258	C₄	D₄×#3622	1	1	yes	characteristic
259	C₄	D₄×#3622	1	1	yes	characteristic
260	C₄	C₂×D₄×Dic₁₇	1	1	yes	characteristic
261	C₄	C₂³×#3622	1	1	yes	characteristic
262	C₂	—	2	1	no	—
263	C₂	—	2	1	no	—
264	C₂	—	4	4	no	—
265	C₂	#14530626	1	2	yes	direct product factor
266	C₂	C₂×D₄×#3622	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₈	D₄×D₁₇
commutator	C₄×C₁₇	C₂³×C₈
Frattini	C₂×C₈	C₂³×D₁₇
Fitting	C₂×C₁₇×#443	C₂
socle	C₂²×C₁₇	C₈×D₄
prime radical	C₂×C₁₇×#135	C₄
cosocle	C₂×C₈×C₁₇	C₂⁴
hypercenter	C₂×#443	D₁₇
hypocenter	C₁₇	C₂×#2171
LCS	G ▷ C₄×C₁₇ ▷ C₂×C₁₇ ▷ C₁₇
UCS	C₁ ◁ C₂×C₈ ◁ C₂²×C₈ ◁ C₂×#443

Complements

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

N	H ≅ G/N	type
1. C₂×#11763226	C₂	semidirect
2. #14530626	C₂	direct (N × H)
3. C₂×#11762630	C₂	semidirect
4. C₂×#11762593	C₂	semidirect
5. C₂×#11762585	C₂	semidirect
8. #11763226	C₂²	semidirect
14. #11762630	C₂²	semidirect
15. #11762593	C₂²	semidirect
16. #11762585	C₂²	semidirect
155. C₂×C₁₇	#2171	semidirect
188. C₁₇	C₂×#2171	semidirect
265. C₂	#14530626	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×#11763226	2
#14530626	2
C₂×#11762630	2
C₂×#11762593	2
C₂×#11762585	2
C₂×C₁₇×#443	2
C₂×#2171	17

Subgroups — C2×#14530626 #17187558

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#14530626 #17187558