Subgroups — C₂×#22590181 #32150685

Order 18176.

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×#17986417	C₂	1	2	yes	maximal, complemented
2	C₂×#17986117	C₂	1	1	yes	maximal, characteristic
3	#22590181	C₂	1	8	yes	maximal, direct product factor
4	C₂×#17985784	C₂	1	2	yes	maximal, complemented
5	C₂×#17985775	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×#364×C₇₁	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂×#14556150	C₄	1	1	yes	characteristic
8	C₂×#14556134	C₂²	1	2	yes	—
9	#17986417	—	2	4	no	—
10	#17986417	C₂²	1	8	yes	complemented
11	C₂×#14556067	—	2	2	no	—
12	C₂²×C₇₁⋊C₁₆	C₂²	1	1	yes	characteristic
13	C₂²×C₇₁⋊C₁₆	C₂²	1	1	yes	characteristic
14	C₂×C₁₆×D₇₁	—	2	2	no	—
15	#17986117	C₄	1	4	yes	—
16	#17986117	C₂²	1	4	yes	—
17	C₂×#14556064	C₄	1	1	yes	characteristic
18	C₂×#14556056	C₄	1	2	yes	—
19	#17985784	C₂²	1	8	yes	complemented
20	#17985775	C₂²	1	4	yes	complemented
21	C₂×M₃₂×C₇₁	C₂²	1	2	yes	—
22	C₂×C₄×C₈×C₇₁	C₂²	1	1	yes	characteristic
23	#364×C₇₁	C₂²	1	4	yes	complemented
24	C₂×#11765865	C₂×C₄	1	2	yes	—
25	#14556150	C₈	1	4	yes	—
26	#14556150	C₂×C₄	1	4	yes	—
27	C₂×Dic₂₈₄	C₈	1	1	yes	characteristic
28	C₂×#11765854	C₈	1	2	yes	—
29	C₂×D₅₆₈	C₈	1	1	yes	characteristic
30	#14556134	—	2	2	no	—
31	#14556134	—	2	2	no	—
32	#14556134	C₂×C₄	1	4	yes	—
33	#14556134	C₂³	1	4	yes	—
34	C₂×#11765849	C₂×C₄	1	2	yes	—
35	C₂×#11765848	—	2	2	no	—
36	C₂×C₈×D₇₁	—	2	2	no	—
37	#14556067	—	2	8	no	—
38	C₂×C₇₁⋊C₁₆	—	2	1	no	—
39	C₂×C₇₁⋊C₁₆	—	2	1	no	—
40	C₂×C₇₁⋊C₁₆	D₄	1	2	yes	—
41	C₂×C₇₁⋊C₁₆	D₄	1	2	yes	—
42	C₂×C₇₁⋊C₁₆	C₂³	1	2	yes	—
43	C₂×C₇₁⋊C₁₆	C₂³	1	2	yes	—
44	C₁₆×D₇₁	—	2	8	no	—
45	#14556064	C₂×C₄	1	4	yes	—
46	#14556056	C₂×C₄	1	8	yes	—
47	C₂²×C₈×C₇₁	C₂×C₄	1	1	yes	characteristic
48	C₂²×C₈×C₇₁	C₂³	1	1	yes	characteristic
49	C₂×C₄²×C₇₁	C₂×C₄	1	1	yes	characteristic
50	M₃₂×C₇₁	—	2	2	no	—
51	M₃₂×C₇₁	C₂³	1	4	yes	—
52	C₂×C₁₆×C₇₁	—	2	2	no	—
53	C₄×C₈×C₇₁	C₂×C₄	1	2	yes	—
54	C₄×C₈×C₇₁	C₂³	1	2	yes	—
55	#11765865	—	2	2	no	—
56	#11765865	—	2	2	no	—
57	#11765865	C₂×C₈	1	4	yes	—
58	#11765865	C₂²×C₄	1	4	yes	—
59	C₂×Dic₁₄₂	C₂×C₈	1	2	yes	—
60	C₂×D₂₈₄	C₂×C₈	1	2	yes	—
61	C₂×C₂²⋊D₇₁	—	2	2	no	—
62	C₂×C₄×D₇₁	—	2	2	no	—
63	Dic₂₈₄	C₂×C₈	1	4	yes	—
64	#11765854	C₂×C₈	1	8	yes	—
65	D₅₆₈	C₂×C₈	1	4	yes	—
66	#11765849	—	2	2	no	—
67	#11765849	C₂²×C₄	1	4	yes	—
68	#11765848	—	2	4	no	—
69	#11765848	—	2	4	no	—
70	C₂×C₇₁⋊C₈	—	2	2	no	—
71	C₈×D₇₁	—	2	4	no	—
72	C₈×D₇₁	—	2	4	no	—
73	C₇₁⋊C₁₆	—	2	2	no	—
74	C₇₁⋊C₁₆	C₂×D₄	1	4	yes	—
75	C₂²×C₄×C₇₁	—	2	1	no	—
76	C₂²×C₄×C₇₁	C₂²×C₄	1	1	yes	characteristic
77	C₂×C₈×C₇₁	C₂×C₈	1	1	yes	characteristic
78	C₂×C₈×C₇₁	C₂×C₈	1	1	yes	characteristic
79	C₂×C₈×C₇₁	C₂×C₈	1	2	yes	—
80	C₂×C₈×C₇₁	C₂²×C₄	1	1	yes	characteristic
81	C₂×C₈×C₇₁	C₂²×C₄	1	2	yes	—
82	C₂×C₈×C₇₁	C₂×D₄	1	1	yes	characteristic
83	C₂×C₈×C₇₁	C₂×D₄	1	1	yes	characteristic
84	C₂×C₈×C₇₁	#41	1	1	yes	characteristic
85	C₂×C₈×C₇₁	#41	1	1	yes	characteristic
86	C₂×C₈×C₇₁	C₂⁴	1	1	yes	characteristic
87	C₄²×C₇₁	—	2	1	no	—
88	C₄²×C₇₁	C₂²×C₄	1	2	yes	—
89	C₁₆×C₇₁	—	2	4	no	—
90	C₂³×C₇₁	#131	1	1	yes	characteristic
91	C₂²×D₇₁	—	2	2	no	—
92	C₂×C₄×C₇₁	—	2	2	no	—
93	C₂×C₄×C₇₁	—	4	2	no	—
94	C₂×C₄×C₇₁	C₄×D₄	1	1	yes	characteristic
95	C₂×C₄×C₇₁	C₄×D₄	1	1	yes	characteristic
96	C₂×C₄×C₇₁	C₂²×C₈	1	1	yes	characteristic
97	C₂×C₄×C₇₁	C₂²×C₈	1	1	yes	characteristic
98	C₂×C₄×C₇₁	#131	1	1	yes	characteristic
99	C₂×C₄×C₇₁	C₂³×C₄	1	1	yes	characteristic
100	C₂²⋊D₇₁	—	2	4	no	—
101	C₂²⋊D₇₁	—	2	4	no	—
102	C₂×Dic₇₁	—	2	2	no	—
103	D₂₈₄	—	2	2	no	—
104	D₂₈₄	C₂²×C₈	1	4	yes	—
105	C₄×D₇₁	—	2	4	no	—
106	C₄×D₇₁	—	2	4	no	—
107	Dic₁₄₂	—	2	2	no	—
108	Dic₁₄₂	C₂²×C₈	1	4	yes	—
109	C₈×C₇₁	C₄×D₄	1	1	yes	characteristic
110	C₈×C₇₁	C₄×D₄	1	1	yes	characteristic
111	C₈×C₇₁	C₂²×C₈	1	2	yes	—
112	C₈×C₇₁	C₂²×C₈	1	2	yes	—
113	C₈×C₇₁	C₂²×D₄	1	1	yes	characteristic
114	C₈×C₇₁	C₂×#41	1	1	yes	characteristic
115	C₇₁⋊C₈	—	2	4	no	—
116	C₂²×C₇₁	—	2	2	no	—
117	C₂²×C₇₁	C₈×D₄	1	1	yes	characteristic
118	C₂²×C₇₁	C₈×D₄	1	1	yes	characteristic
119	C₂²×C₇₁	C₂×#131	1	1	yes	characteristic
120	C₂×D₇₁	—	2	4	no	—
121	C₂×D₇₁	—	4	4	no	—
122	C₄×C₇₁	—	4	2	no	—
123	C₄×C₇₁	C₈×D₄	1	1	yes	characteristic
124	C₄×C₇₁	C₈×D₄	1	1	yes	characteristic
125	C₄×C₇₁	C₂×C₄×D₄	1	1	yes	characteristic
126	C₄×C₇₁	C₂³×C₈	1	1	yes	characteristic
127	Dic₇₁	—	2	4	no	—
128	C₂×#2171	—	71	1	no	maximal, Sylow
129	C₂×C₇₁	—	2	1	no	—
130	C₂×C₇₁	—	2	1	no	—
131	C₂×C₇₁	#2171	1	2	yes	complemented
132	C₂×C₇₁	C₂×C₈×D₄	1	1	yes	characteristic
133	D₇₁	—	4	4	no	—
134	C₂×#504	—	71	2	no	—
135	C₂×#443	—	71	1	no	—
136	#2171	—	71	8	no	—
137	C₂×#364	D₇₁	1	1	yes	characteristic, complemented
138	C₂×#350	—	71	2	no	—
139	C₂×#346	—	71	1	no	—
140	C₇₁	C₂×#2171	1	1	yes	characteristic, complemented, Sylow
141	C₂×#135	—	71	1	no	—
142	C₂×#131	—	71	2	no	—
143	#504	—	71	8	no	—
144	#504	—	142	4	no	—
145	C₂×M₃₂	—	142	2	no	—
146	C₂×M₃₂	C₂×D₇₁	1	2	yes	—
147	C₂²×C₁₆	—	71	1	no	—
148	C₂²×C₁₆	—	71	1	no	—
149	C₂²×C₁₆	—	142	2	no	—
150	#443	—	71	4	no	—
151	#443	—	71	4	no	—
152	C₂×#108	—	71	1	no	—
153	C₂×#104	—	71	2	no	—
154	C₂×C₄×C₈	C₂×D₇₁	1	1	yes	characteristic
155	#364	C₂×D₇₁	1	4	yes	complemented
156	#350	—	71	8	no	—
157	#346	—	71	4	no	—
158	C₂×#41	—	71	2	no	—
159	#135	—	71	4	no	—
160	#135	—	71	4	no	—
161	C₂×Q₁₆	—	71	1	no	—
162	C₂×SD₁₆	—	71	2	no	—
163	C₂×D₈	—	71	1	no	—
164	#131	—	71	4	no	—
165	#131	—	71	4	no	—
166	#131	—	142	2	no	—
167	#131	—	142	2	no	—
168	C₂×M₁₆	—	71	2	no	—
169	C₂×M₁₆	—	142	2	no	—
170	C₂²×C₈	—	142	2	no	—
171	C₂²×C₈	C₄×D₇₁	1	1	yes	characteristic
172	C₂²×C₈	C₂²×D₇₁	1	1	yes	characteristic
173	C₂×C₄²	C₄×D₇₁	1	1	yes	characteristic
174	M₃₂	—	2	2	no	—
175	M₃₂	—	142	8	no	—
176	M₃₂	C₂²×D₇₁	1	4	yes	—
177	C₂×C₁₆	—	2	2	no	—
178	C₂×C₁₆	—	71	2	no	—
179	C₂×C₁₆	—	71	2	no	—
180	C₂×C₁₆	—	71	2	no	—
181	C₂×C₁₆	—	71	2	no	—
182	C₂×C₁₆	—	142	8	no	—
183	C₂×C₁₆	—	142	1	no	—
184	C₂×C₁₆	—	142	1	no	—
185	#108	—	71	4	no	—
186	#104	—	71	8	no	—
187	C₄×C₈	C₄×D₇₁	1	2	yes	—
188	C₄×C₈	C₂²×D₇₁	1	2	yes	—
189	#41	—	71	4	no	—
190	#41	—	71	4	no	—
191	#41	—	142	2	no	—
192	#41	—	142	2	no	—
193	C₂×Q₈	—	71	2	no	—
194	C₂×D₄	—	71	2	no	—
195	C₂×D₄	—	142	2	no	—
196	C₂²×C₄	—	2	1	no	—
197	C₂²×C₄	—	142	2	no	—
198	C₂²×C₄	C₂×C₄×D₇₁	1	1	yes	characteristic
199	Q₁₆	—	71	4	no	—
200	SD₁₆	—	71	8	no	—
201	D₈	—	71	4	no	—
202	M₁₆	—	71	4	no	—
203	M₁₆	—	142	2	no	—
204	M₁₆	—	142	4	no	—
205	M₁₆	—	142	4	no	—
206	C₂×C₈	—	142	2	no	—
207	C₂×C₈	—	142	4	no	—
208	C₂×C₈	—	142	4	no	—
209	C₂×C₈	C₈×D₇₁	1	1	yes	characteristic
210	C₂×C₈	C₈×D₇₁	1	1	yes	characteristic
211	C₂×C₈	C₈×D₇₁	1	2	yes	—
212	C₂×C₈	C₂×C₄×D₇₁	1	1	yes	characteristic
213	C₂×C₈	C₂×C₄×D₇₁	1	2	yes	—
214	C₂×C₈	D₄×D₇₁	1	1	yes	characteristic
215	C₂×C₈	D₄×D₇₁	1	1	yes	characteristic
216	C₂×C₈	#11765866	1	1	yes	characteristic
217	C₂×C₈	#11765866	1	1	yes	characteristic
218	C₂×C₈	C₂³×D₇₁	1	1	yes	characteristic
219	C₄²	—	2	1	no	—
220	C₄²	C₂×C₄×D₇₁	1	2	yes	—
221	C₁₆	—	2	4	no	—
222	C₁₆	—	71	4	no	—
223	C₁₆	—	142	2	no	—
224	C₂³	—	142	2	no	—
225	C₂³	#14556136	1	1	yes	characteristic
226	Q₈	—	71	4	no	—
227	Q₈	—	142	2	no	—
228	D₄	—	71	4	no	—
229	D₄	—	142	4	no	—
230	D₄	—	142	4	no	—
231	D₄	—	142	2	no	—
232	C₂×C₄	—	2	2	no	—
233	C₂×C₄	—	4	2	no	—
234	C₂×C₄	—	142	2	no	—
235	C₂×C₄	—	142	4	no	—
236	C₂×C₄	—	142	4	no	—
237	C₂×C₄	#14556090	1	1	yes	characteristic
238	C₂×C₄	#14556090	1	1	yes	characteristic
239	C₂×C₄	C₂×C₈×D₇₁	1	1	yes	characteristic
240	C₂×C₄	C₂×C₈×D₇₁	1	1	yes	characteristic
241	C₂×C₄	#14556136	1	1	yes	characteristic
242	C₂×C₄	C₂²×C₄×D₇₁	1	1	yes	characteristic
243	C₈	—	142	4	no	—
244	C₈	#14556090	1	1	yes	characteristic
245	C₈	#14556090	1	1	yes	characteristic
246	C₈	C₂×C₈×D₇₁	1	2	yes	—
247	C₈	C₂×C₈×D₇₁	1	2	yes	—
248	C₈	C₂×D₄×D₇₁	1	1	yes	characteristic
249	C₈	C₂×#11765866	1	1	yes	characteristic
250	C₂²	—	2	2	no	—
251	C₂²	—	284	4	no	—
252	C₂²	—	142	4	no	—
253	C₂²	#17986064	1	1	yes	characteristic
254	C₂²	#17986064	1	1	yes	characteristic
255	C₂²	C₂×#14556136	1	1	yes	characteristic
256	C₄	—	4	2	no	—
257	C₄	—	142	4	no	—
258	C₄	#17986064	1	1	yes	characteristic
259	C₄	#17986064	1	1	yes	characteristic
260	C₄	C₂×#14556090	1	1	yes	characteristic
261	C₄	C₂²×C₈×D₇₁	1	1	yes	characteristic
262	C₂	—	2	1	no	—
263	C₂	—	2	1	no	—
264	C₂	—	284	4	no	—
265	C₂	#22590181	1	2	yes	direct product factor
266	C₂	C₂×#17986064	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₈	D₄×D₇₁
commutator	C₄×C₇₁	C₂³×C₈
Frattini	C₂×C₈	C₂³×D₇₁
Fitting	C₂×#364×C₇₁	C₂
socle	C₂²×C₇₁	C₈×D₄
prime radical	C₂×#14556150	C₄
cosocle	C₂×C₈×C₇₁	C₂⁴
hypercenter	C₂×#364	D₇₁
hypocenter	C₇₁	C₂×#2171
LCS	G ▷ C₄×C₇₁ ▷ C₂×C₇₁ ▷ C₇₁
UCS	C₁ ◁ C₂×C₈ ◁ C₂²×C₈ ◁ C₂×#364

Complements

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

N	H ≅ G/N	type
1. C₂×#17986417	C₂	semidirect
3. #22590181	C₂	direct (N × H)
4. C₂×#17985784	C₂	semidirect
5. C₂×#17985775	C₂	semidirect
6. C₂×#364×C₇₁	C₂	semidirect
10. #17986417	C₂²	semidirect
19. #17985784	C₂²	semidirect
20. #17985775	C₂²	semidirect
23. #364×C₇₁	C₂²	semidirect
131. C₂×C₇₁	#2171	semidirect
137. C₂×#364	D₇₁	semidirect
140. C₇₁	C₂×#2171	semidirect
155. #364	C₂×D₇₁	semidirect
265. C₂	#22590181	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×#17986417	2
C₂×#17986117	2
#22590181	2
C₂×#17985784	2
C₂×#17985775	2
C₂×#364×C₇₁	2
C₂×#2171	71

Subgroups — C2×#22590181 #32150685

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#22590181 #32150685