Subgroups — C₂×#22562091 #31009232

Order 17152.

662 conjugacy classes of subgroups.

The subgroup lattice has 270 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#16863828	C₂	1	1	yes	maximal, characteristic
2	C₂×#16863514	C₂	1	2	yes	maximal, complemented
3	#22562091	C₂	1	8	yes	maximal, direct product factor
4	C₂×#16863226	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂×#16863204	C₂	1	2	yes	maximal, complemented
6	C₂×C₄×C₁₆×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂×#14524065	C₂²	1	2	yes	—
8	C₂×#14524049	—	2	2	no	—
9	#16863828	C₄	1	4	yes	complemented
10	#16863828	C₂²	1	4	yes	—
11	C₂×Dic₅₃₆	C₄	1	1	yes	characteristic, complemented
12	C₂×#14523988	C₄	1	2	yes	complemented
13	C₂×D₁₀₇₂	C₄	1	1	yes	characteristic, complemented
14	#16863514	—	2	4	no	—
15	#16863514	C₂²	1	8	yes	complemented
16	C₂×#14523979	C₂²	1	2	yes	—
17	C₂×#14523971	—	2	2	no	—
18	#16863226	C₂²	1	4	yes	complemented
19	#16863204	C₂²	1	8	yes	complemented
20	C₂²×C₁₆×C₆₇	C₂²	1	1	yes	characteristic
21	C₂²×C₁₆×C₆₇	C₂²	1	1	yes	characteristic
22	C₂×C₄×C₈×C₆₇	C₂²	1	1	yes	characteristic
23	C₄×C₁₆×C₆₇	C₂²	1	4	yes	complemented
24	C₂×#11761628	—	2	2	no	—
25	#14524065	—	2	2	no	—
26	#14524065	—	2	2	no	—
27	#14524065	C₂×C₄	1	4	yes	—
28	#14524065	C₂³	1	4	yes	—
29	C₂×Dic₂₆₈	C₂×C₄	1	2	yes	—
30	C₂×#11761617	—	2	2	no	—
31	C₂×D₅₃₆	C₂×C₄	1	2	yes	—
32	#14524049	—	2	8	no	—
33	#14524049	—	4	4	no	—
34	C₂×#11761612	—	2	2	no	—
35	C₂×#11761611	—	4	2	no	—
36	C₂×C₈×D₆₇	—	4	2	no	—
37	Dic₅₃₆	C₂×C₄	1	4	yes	complemented
38	#14523988	C₂×C₄	1	8	yes	complemented
39	D₁₀₇₂	C₂×C₄	1	4	yes	complemented
40	#14523979	—	2	2	no	—
41	#14523979	C₂³	1	4	yes	—
42	#14523971	—	2	8	no	—
43	C₂²×C₈×C₆₇	D₄	1	1	yes	characteristic
44	C₂²×C₈×C₆₇	C₂³	1	1	yes	characteristic
45	C₂×C₄²×C₆₇	D₄	1	1	yes	characteristic
46	C₂×C₁₆×C₆₇	—	2	1	no	—
47	C₂×C₁₆×C₆₇	—	2	1	no	—
48	C₂×C₁₆×C₆₇	C₂×C₄	1	1	yes	characteristic
49	C₂×C₁₆×C₆₇	C₂×C₄	1	1	yes	characteristic
50	C₂×C₁₆×C₆₇	C₂×C₄	1	2	yes	—
51	C₂×C₁₆×C₆₇	C₂³	1	2	yes	—
52	C₂×C₁₆×C₆₇	C₂³	1	2	yes	—
53	C₄×C₈×C₆₇	D₄	1	2	yes	—
54	C₄×C₈×C₆₇	C₂³	1	2	yes	—
55	#11761628	—	2	4	no	—
56	#11761628	—	2	4	no	—
57	#11761628	—	4	2	no	—
58	#11761628	—	4	2	no	—
59	C₂×Dic₁₃₄	—	2	2	no	—
60	C₂×D₂₆₈	—	2	2	no	—
61	C₂×C₂²⋊D₆₇	—	4	2	no	—
62	C₂×C₄×D₆₇	—	4	2	no	—
63	Dic₂₆₈	—	2	2	no	—
64	Dic₂₆₈	C₂²×C₄	1	4	yes	—
65	#11761617	—	2	4	no	—
66	#11761617	—	2	4	no	—
67	D₅₃₆	—	2	2	no	—
68	D₅₃₆	C₂²×C₄	1	4	yes	—
69	#11761612	—	2	4	no	—
70	#11761612	—	4	2	no	—
71	#11761611	—	4	8	no	—
72	C₂×C₆₇⋊C₈	—	4	2	no	—
73	C₈×D₆₇	—	4	8	no	—
74	C₂²×C₄×C₆₇	—	2	1	no	—
75	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
76	C₂×C₈×C₆₇	D₈	1	1	yes	characteristic
77	C₂×C₈×C₆₇	D₈	1	2	yes	—
78	C₂×C₈×C₆₇	D₈	1	1	yes	characteristic
79	C₂×C₈×C₆₇	C₂²×C₄	1	1	yes	characteristic
80	C₂×C₈×C₆₇	C₂²×C₄	1	1	yes	characteristic
81	C₂×C₈×C₆₇	C₂×D₄	1	1	yes	characteristic
82	C₂×C₈×C₆₇	C₂×D₄	1	2	yes	—
83	C₂×C₈×C₆₇	#41	1	1	yes	characteristic
84	C₂×C₈×C₆₇	#41	1	1	yes	characteristic
85	C₂×C₈×C₆₇	C₂⁴	1	1	yes	characteristic
86	C₄²×C₆₇	—	2	1	no	—
87	C₄²×C₆₇	C₂×D₄	1	2	yes	—
88	C₁₆×C₆₇	—	2	2	no	—
89	C₁₆×C₆₇	C₂²×C₄	1	2	yes	—
90	C₁₆×C₆₇	C₂²×C₄	1	2	yes	—
91	C₂³×C₆₇	#135	1	1	yes	characteristic
92	C₂²×D₆₇	—	4	2	no	—
93	C₂×C₄×C₆₇	—	2	4	no	—
94	C₂×C₄×C₆₇	—	2	2	no	—
95	C₂×C₄×C₆₇	C₄×D₄	1	1	yes	characteristic
96	C₂×C₄×C₆₇	C₄×D₄	1	1	yes	characteristic
97	C₂×C₄×C₆₇	C₂×D₈	1	1	yes	characteristic
98	C₂×C₄×C₆₇	C₂×D₈	1	1	yes	characteristic
99	C₂×C₄×C₆₇	#135	1	1	yes	characteristic
100	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
101	C₂²⋊D₆₇	—	4	4	no	—
102	C₂²⋊D₆₇	—	4	4	no	—
103	C₂×Dic₆₇	—	4	2	no	—
104	D₂₆₈	—	2	4	no	—
105	D₂₆₈	—	4	2	no	—
106	C₄×D₆₇	—	4	4	no	—
107	C₄×D₆₇	—	4	4	no	—
108	Dic₁₃₄	—	2	4	no	—
109	Dic₁₃₄	—	4	2	no	—
110	C₈×C₆₇	C₄×D₄	1	1	yes	characteristic
111	C₈×C₆₇	C₄×D₄	1	1	yes	characteristic
112	C₈×C₆₇	C₂×D₈	1	2	yes	—
113	C₈×C₆₇	C₂×D₈	1	2	yes	—
114	C₈×C₆₇	C₂³×C₄	1	1	yes	characteristic
115	C₈×C₆₇	C₂×#41	1	1	yes	characteristic
116	C₆₇⋊C₈	—	4	4	no	—
117	C₂²×C₆₇	—	2	2	no	—
118	C₂²×C₆₇	C₄×D₈	1	1	yes	characteristic
119	C₂²×C₆₇	C₄×D₈	1	1	yes	characteristic
120	C₂²×C₆₇	C₂×#135	1	1	yes	characteristic
121	C₂×D₆₇	—	4	4	no	—
122	C₂×D₆₇	—	8	4	no	—
123	C₄×C₆₇	—	2	4	no	—
124	C₄×C₆₇	C₄×D₈	1	1	yes	characteristic
125	C₄×C₆₇	C₄×D₈	1	1	yes	characteristic
126	C₄×C₆₇	C₂×C₄×D₄	1	1	yes	characteristic
127	C₄×C₆₇	C₂²×D₈	1	1	yes	characteristic
128	Dic₆₇	—	4	4	no	—
129	C₂×#2178	—	67	1	no	maximal, Sylow
130	C₂×C₆₇	—	2	1	no	—
131	C₂×C₆₇	—	2	1	no	—
132	C₂×C₆₇	#2178	1	2	yes	complemented
133	C₂×C₆₇	C₂×C₄×D₈	1	1	yes	characteristic
134	D₆₇	—	8	4	no	—
135	C₂×#508	—	67	1	no	—
136	C₂×#443	—	67	2	no	—
137	#2178	—	67	8	no	—
138	C₂×#368	—	67	1	no	—
139	C₂×#359	—	67	2	no	—
140	C₂×C₄×C₁₆	D₆₇	1	1	yes	characteristic, complemented
141	C₆₇	C₂×#2178	1	1	yes	characteristic, complemented, Sylow
142	C₂×#135	—	67	2	no	—
143	C₂×#131	—	134	2	no	—
144	#508	—	67	4	no	—
145	#508	—	67	4	no	—
146	C₂×Q₃₂	—	67	1	no	—
147	C₂×SD₃₂	—	67	2	no	—
148	C₂×D₁₆	—	67	1	no	—
149	C₂²×C₁₆	C₂×D₆₇	1	1	yes	characteristic
150	C₂²×C₁₆	C₂×D₆₇	1	1	yes	characteristic
151	#443	—	67	8	no	—
152	#443	—	134	4	no	—
153	C₂×#108	—	67	2	no	—
154	C₂×#104	—	134	2	no	—
155	C₂×C₄×C₈	C₂×D₆₇	1	1	yes	characteristic
156	#368	—	67	4	no	—
157	#359	—	67	8	no	—
158	C₄×C₁₆	C₂×D₆₇	1	4	yes	complemented
159	C₂×#41	—	134	2	no	—
160	#135	—	67	4	no	—
161	#135	—	67	4	no	—
162	#135	—	134	2	no	—
163	#135	—	134	2	no	—
164	C₂×Q₁₆	—	67	2	no	—
165	C₂×SD₁₆	—	134	2	no	—
166	C₂×D₈	—	67	2	no	—
167	#131	—	268	4	no	—
168	#131	—	134	8	no	—
169	C₂×M₁₆	—	268	2	no	—
170	C₂×M₁₆	—	134	2	no	—
171	C₂²×C₈	—	268	2	no	—
172	C₂²×C₈	D₂₆₈	1	1	yes	characteristic
173	C₂²×C₈	C₂²×D₆₇	1	1	yes	characteristic
174	C₂×C₄²	D₂₆₈	1	1	yes	characteristic
175	Q₃₂	—	67	4	no	—
176	SD₃₂	—	67	8	no	—
177	D₁₆	—	67	4	no	—
178	C₂×C₁₆	—	2	1	no	—
179	C₂×C₁₆	—	2	1	no	—
180	C₂×C₁₆	C₄×D₆₇	1	1	yes	characteristic
181	C₂×C₁₆	C₄×D₆₇	1	1	yes	characteristic
182	C₂×C₁₆	C₄×D₆₇	1	2	yes	—
183	C₂×C₁₆	C₂²×D₆₇	1	2	yes	—
184	C₂×C₁₆	C₂²×D₆₇	1	2	yes	—
185	#108	—	67	4	no	—
186	#108	—	134	2	no	—
187	#104	—	134	8	no	—
188	C₄×C₈	D₂₆₈	1	2	yes	—
189	C₄×C₈	C₂²×D₆₇	1	2	yes	—
190	#41	—	268	2	no	—
191	#41	—	268	2	no	—
192	#41	—	134	4	no	—
193	#41	—	134	4	no	—
194	C₂×Q₈	—	134	2	no	—
195	C₂×D₄	—	268	2	no	—
196	C₂×D₄	—	134	2	no	—
197	C₂²×C₄	—	2	1	no	—
198	C₂²×C₄	—	268	2	no	—
199	C₂²×C₄	C₂×D₂₆₈	1	1	yes	characteristic
200	Q₁₆	—	67	4	no	—
201	Q₁₆	—	134	2	no	—
202	SD₁₆	—	134	4	no	—
203	SD₁₆	—	134	4	no	—
204	D₈	—	67	4	no	—
205	D₈	—	134	2	no	—
206	M₁₆	—	268	2	no	—
207	M₁₆	—	268	8	no	—
208	M₁₆	—	134	4	no	—
209	C₂×C₈	—	268	2	no	—
210	C₂×C₈	—	268	8	no	—
211	C₂×C₈	D₅₃₆	1	1	yes	characteristic
212	C₂×C₈	D₅₃₆	1	2	yes	—
213	C₂×C₈	D₅₃₆	1	1	yes	characteristic
214	C₂×C₈	C₂×C₄×D₆₇	1	1	yes	characteristic
215	C₂×C₈	C₂×C₄×D₆₇	1	1	yes	characteristic
216	C₂×C₈	C₂×D₂₆₈	1	1	yes	characteristic
217	C₂×C₈	C₂×D₂₆₈	1	2	yes	—
218	C₂×C₈	#11761628	1	1	yes	characteristic
219	C₂×C₈	#11761628	1	1	yes	characteristic
220	C₂×C₈	C₂³×D₆₇	1	1	yes	characteristic
221	C₄²	—	2	1	no	—
222	C₄²	C₂×D₂₆₈	1	2	yes	—
223	C₁₆	—	2	2	no	—
224	C₁₆	C₂×C₄×D₆₇	1	2	yes	—
225	C₁₆	C₂×C₄×D₆₇	1	2	yes	—
226	C₂³	—	268	2	no	—
227	C₂³	#14524065	1	1	yes	characteristic
228	Q₈	—	268	2	no	—
229	Q₈	—	134	4	no	—
230	D₄	—	268	4	no	—
231	D₄	—	268	4	no	—
232	D₄	—	268	2	no	—
233	D₄	—	134	4	no	—
234	C₂×C₄	—	2	4	no	—
235	C₂×C₄	—	2	2	no	—
236	C₂×C₄	—	268	2	no	—
237	C₂×C₄	—	268	4	no	—
238	C₂×C₄	—	268	4	no	—
239	C₂×C₄	C₄×D₂₆₈	1	1	yes	characteristic
240	C₂×C₄	C₄×D₂₆₈	1	1	yes	characteristic
241	C₂×C₄	C₂×D₅₃₆	1	1	yes	characteristic
242	C₂×C₄	C₂×D₅₃₆	1	1	yes	characteristic
243	C₂×C₄	#14524065	1	1	yes	characteristic
244	C₂×C₄	C₂²×D₂₆₈	1	1	yes	characteristic
245	C₈	—	268	4	no	—
246	C₈	C₄×D₂₆₈	1	1	yes	characteristic
247	C₈	C₄×D₂₆₈	1	1	yes	characteristic
248	C₈	C₂×D₅₃₆	1	2	yes	—
249	C₈	C₂×D₅₃₆	1	2	yes	—
250	C₈	C₂²×C₄×D₆₇	1	1	yes	characteristic
251	C₈	C₂×#11761628	1	1	yes	characteristic
252	C₂²	—	2	2	no	—
253	C₂²	—	268	4	no	—
254	C₂²	—	536	4	no	—
255	C₂²	C₄×D₅₃₆	1	1	yes	characteristic
256	C₂²	C₄×D₅₃₆	1	1	yes	characteristic
257	C₂²	C₂×#14524065	1	1	yes	characteristic
258	C₄	—	2	4	no	—
259	C₄	—	268	4	no	—
260	C₄	C₄×D₅₃₆	1	1	yes	characteristic
261	C₄	C₄×D₅₃₆	1	1	yes	characteristic
262	C₄	C₂×C₄×D₂₆₈	1	1	yes	characteristic
263	C₄	C₂²×D₅₃₆	1	1	yes	characteristic
264	C₂	—	2	1	no	—
265	C₂	—	2	1	no	—
266	C₂	—	536	4	no	—
267	C₂	#22562091	1	2	yes	direct product factor
268	C₂	C₂×C₄×D₅₃₆	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₈	D₅₃₆
commutator	C₈×C₆₇	C₂³×C₄
Frattini	C₂×C₈	C₂³×D₆₇
Fitting	C₂×C₄×C₁₆×C₆₇	C₂
socle	C₂²×C₆₇	C₄×D₈
prime radical	C₂×#16863828	C₂
cosocle	C₂×C₈×C₆₇	C₂⁴
hypercenter	C₂×C₄×C₁₆	D₆₇
hypocenter	C₆₇	C₂×#2178
LCS	G ▷ C₈×C₆₇ ▷ C₄×C₆₇ ▷ C₂×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂×C₈ ◁ C₂²×C₈ ◁ C₂×C₄×C₈ ◁ C₂×C₄×C₁₆

Complements

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

N	H ≅ G/N	type
2. C₂×#16863514	C₂	semidirect
3. #22562091	C₂	direct (N × H)
4. C₂×#16863226	C₂	semidirect
5. C₂×#16863204	C₂	semidirect
6. C₂×C₄×C₁₆×C₆₇	C₂	semidirect
9. #16863828	C₄	semidirect
11. C₂×Dic₅₃₆	C₄	semidirect
12. C₂×#14523988	C₄	semidirect
13. C₂×D₁₀₇₂	C₄	semidirect
15. #16863514	C₂²	semidirect
18. #16863226	C₂²	semidirect
19. #16863204	C₂²	semidirect
23. C₄×C₁₆×C₆₇	C₂²	semidirect
37. Dic₅₃₆	C₂×C₄	semidirect
38. #14523988	C₂×C₄	semidirect
39. D₁₀₇₂	C₂×C₄	semidirect
132. C₂×C₆₇	#2178	semidirect
140. C₂×C₄×C₁₆	D₆₇	semidirect
141. C₆₇	C₂×#2178	semidirect
158. C₄×C₁₆	C₂×D₆₇	semidirect
267. C₂	#22562091	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×#16863828	2
C₂×#16863514	2
#22562091	2
C₂×#16863226	2
C₂×#16863204	2
C₂×C₄×C₁₆×C₆₇	2
C₂×#2178	67

Subgroups — C2×#22562091 #31009232

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#22562091 #31009232