Subgroups — C₂×#22562250 #31010076

Order 17152.

806 conjugacy classes of subgroups.

The subgroup lattice has 288 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×#14523988	C₂	1	2	yes	maximal, complemented
2	C₂×#16863773	C₂	1	1	yes	maximal, characteristic
3	C₂×C₄×Dic₂₆₈	C₂	1	1	yes	maximal, characteristic, complemented
4	#22562250	C₂	1	8	yes	maximal, direct product factor
5	C₂×#16863198	C₂	1	2	yes	maximal, complemented
6	C₂×C₄⋊C₁₆×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂²×Dic₂₆₈	C₂²	1	1	yes	characteristic
8	C₂²×D₅₃₆	—	2	1	no	—
9	C₂²×D₅₃₆	C₂²	1	2	yes	—
10	C₂×#14524040	—	2	1	no	—
11	C₂×C₄×Dic₁₃₄	—	2	1	no	—
12	C₂×#14523988	—	2	8	no	—
13	C₂×#14523988	C₂²	1	8	yes	complemented
14	#16863773	—	2	2	no	—
15	#16863773	C₂²	1	4	yes	—
16	C₄×Dic₂₆₈	—	2	2	no	—
17	C₄×Dic₂₆₈	C₂²	1	4	yes	complemented
18	C₂×#14523977	C₂²	1	1	yes	characteristic
19	C₂×#14523968	—	2	1	no	—
20	#16863198	C₂²	1	8	yes	complemented
21	C₂²×C₁₆×C₆₇	C₂²	1	2	yes	—
22	C₂×C₄×C₈×C₆₇	C₂²	1	1	yes	characteristic
23	C₄⋊C₁₆×C₆₇	C₂²	1	4	yes	complemented
24	C₂²×Dic₁₃₄	—	2	1	no	—
25	C₂²×D₂₆₈	—	2	2	no	—
26	C₂²×D₂₆₈	—	4	1	no	—
27	C₂²×D₂₆₈	—	4	1	no	—
28	C₂×Dic₂₆₈	—	2	1	no	—
29	C₂×Dic₂₆₈	—	2	2	no	—
30	C₂×Dic₂₆₈	D₄	1	4	yes	complemented
31	C₂×Dic₂₆₈	C₂³	1	2	yes	—
32	C₂×D₅₃₆	—	2	4	no	—
33	C₂×D₅₃₆	—	2	2	no	—
34	C₂×D₅₃₆	—	2	4	no	—
35	C₂×D₅₃₆	—	2	4	no	—
36	C₂×D₅₃₆	—	4	4	no	—
37	C₂×D₅₃₆	C₂³	1	4	yes	—
38	#14524040	—	2	4	no	—
39	#14524040	—	4	2	no	—
40	C₄×Dic₁₃₄	—	2	4	no	—
41	C₄×Dic₁₃₄	—	4	2	no	—
42	C₂×C₄⋊Dic₆₇	—	2	1	no	—
43	C₂×Dic₆₇⋊C₄	—	4	1	no	—
44	C₂×C₄×Dic₆₇	—	4	1	no	—
45	#14523988	—	2	16	no	—
46	#14523977	—	2	1	no	—
47	#14523977	C₂³	1	2	yes	—
48	#14523968	—	2	4	no	—
49	C₂²×C₈×C₆₇	D₄	1	1	yes	characteristic
50	C₂²×C₈×C₆₇	C₂³	1	1	yes	characteristic
51	C₂×C₄²×C₆₇	D₄	1	1	yes	characteristic
52	C₂×C₁₆×C₆₇	—	2	4	no	—
53	C₂×C₁₆×C₆₇	C₂³	1	4	yes	—
54	C₄×C₈×C₆₇	D₄	1	2	yes	—
55	C₄×C₈×C₆₇	C₂³	1	2	yes	—
56	C₂³×D₆₇	—	4	2	no	—
57	C₂×Dic₁₃₄	—	2	4	no	—
58	C₂×Dic₁₃₄	—	2	2	no	—
59	C₂×Dic₁₃₄	—	4	1	no	—
60	C₂×Dic₁₃₄	—	4	2	no	—
61	C₂×D₂₆₈	—	2	4	no	—
62	C₂×D₂₆₈	—	4	4	no	—
63	C₂×D₂₆₈	—	4	4	no	—
64	C₂×D₂₆₈	—	4	4	no	—
65	C₂×D₂₆₈	—	4	4	no	—
66	C₂×D₂₆₈	—	4	2	no	—
67	C₂×D₂₆₈	—	8	4	no	—
68	C₂×D₂₆₈	—	8	4	no	—
69	C₂²×Dic₆₇	—	4	1	no	—
70	C₂²×Dic₆₇	—	4	1	no	—
71	Dic₂₆₈	—	2	4	no	—
72	Dic₂₆₈	—	2	2	no	—
73	Dic₂₆₈	C₂×D₄	1	4	yes	complemented
74	D₅₃₆	—	2	4	no	—
75	D₅₃₆	—	2	4	no	—
76	D₅₃₆	—	4	8	no	—
77	D₅₃₆	—	4	4	no	—
78	C₄⋊Dic₆₇	—	2	2	no	—
79	C₄⋊Dic₆₇	—	4	1	no	—
80	Dic₆₇⋊C₄	—	4	4	no	—
81	C₄×Dic₆₇	—	4	4	no	—
82	C₂²×C₄×C₆₇	D₈	1	1	yes	characteristic
83	C₂²×C₄×C₆₇	D₈	1	1	yes	characteristic
84	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
85	C₂×C₈×C₆₇	—	2	2	no	—
86	C₂×C₈×C₆₇	C₂×D₄	1	2	yes	—
87	C₂×C₈×C₆₇	C₂×D₄	1	2	yes	—
88	C₂×C₈×C₆₇	C₂×D₄	1	1	yes	characteristic
89	C₂×C₈×C₆₇	#41	1	2	yes	—
90	C₂×C₈×C₆₇	C₂⁴	1	1	yes	characteristic
91	C₄²×C₆₇	D₈	1	2	yes	—
92	C₄²×C₆₇	C₂×D₄	1	2	yes	—
93	C₁₆×C₆₇	—	2	4	no	—
94	C₂³×C₆₇	C₂×D₈	1	1	yes	characteristic
95	C₂²×D₆₇	—	4	4	no	—
96	C₂²×D₆₇	—	8	4	no	—
97	C₂²×D₆₇	—	8	4	no	—
98	C₂²×D₆₇	—	8	4	no	—
99	C₂×C₄×C₆₇	—	2	2	no	—
100	C₂×C₄×C₆₇	SD₃₂	1	4	yes	complemented
101	C₂×C₄×C₆₇	#121	1	2	yes	—
102	C₂×C₄×C₆₇	C₂×D₈	1	2	yes	—
103	C₂×C₄×C₆₇	C₂×D₈	1	1	yes	characteristic
104	C₂×C₄×C₆₇	C₂×D₈	1	2	yes	—
105	C₂×C₄×C₆₇	AGL(1,ℤ/8)	1	2	yes	—
106	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
107	C₂×Dic₆₇	—	4	2	no	—
108	C₂×Dic₆₇	—	4	4	no	—
109	C₂×Dic₆₇	—	4	2	no	—
110	C₂×Dic₆₇	—	8	2	no	—
111	D₂₆₈	—	4	4	no	—
112	D₂₆₈	—	4	4	no	—
113	D₂₆₈	—	8	8	no	—
114	D₂₆₈	—	8	4	no	—
115	D₂₆₈	—	8	8	no	—
116	Dic₁₃₄	—	2	4	no	—
117	Dic₁₃₄	—	4	2	no	—
118	Dic₁₃₄	—	4	4	no	—
119	C₈×C₆₇	—	2	2	no	—
120	C₈×C₆₇	#121	1	2	yes	—
121	C₈×C₆₇	C₂²×D₄	1	1	yes	characteristic
122	C₈×C₆₇	C₂×#41	1	1	yes	characteristic
123	C₂²×C₆₇	#459	1	2	yes	—
124	C₂²×C₆₇	C₂×SD₃₂	1	2	yes	—
125	C₂²×C₆₇	#509	1	2	yes	—
126	C₂²×C₆₇	C₂²×D₈	1	1	yes	characteristic
127	C₂×D₆₇	—	8	4	no	—
128	C₂×D₆₇	—	8	4	no	—
129	C₂×D₆₇	—	8	4	no	—
130	C₂×D₆₇	—	16	8	no	—
131	C₄×C₆₇	—	2	2	no	—
132	C₄×C₆₇	#459	1	2	yes	—
133	C₄×C₆₇	C₂×SD₃₂	1	4	yes	complemented
134	C₄×C₆₇	C₂×#121	1	1	yes	characteristic
135	C₄×C₆₇	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
136	Dic₆₇	—	4	4	no	—
137	Dic₆₇	—	8	2	no	—
138	C₂×#2207	—	67	1	no	maximal, Sylow
139	C₂×C₆₇	#2207	1	4	yes	complemented
140	C₂×C₆₇	C₂×#459	1	1	yes	characteristic
141	C₂×C₆₇	C₂²×SD₃₂	1	1	yes	characteristic
142	C₂×C₆₇	C₂×#509	1	1	yes	characteristic
143	D₆₇	—	16	4	no	—
144	C₂²×SD₃₂	—	67	2	no	—
145	C₂×#493	—	67	1	no	—
146	C₂×C₄×Q₁₆	—	67	1	no	—
147	#2207	—	67	8	no	—
148	C₂×C₄⋊C₁₆	D₆₇	1	1	yes	characteristic, complemented
149	C₂×#357	—	67	2	no	—
150	C₆₇	C₂×#2207	1	1	yes	characteristic, complemented, Sylow
151	C₂²×Q₁₆	—	67	1	no	—
152	C₂²×D₈	—	67	2	no	—
153	C₂²×D₈	—	134	1	no	—
154	C₂×#127	—	134	1	no	—
155	C₂×C₄×Q₈	—	134	1	no	—
156	C₂×SD₃₂	—	67	8	no	—
157	C₂×SD₃₂	—	134	8	no	—
158	C₂²×C₁₆	C₂×D₆₇	1	2	yes	—
159	#493	—	67	4	no	—
160	#493	—	134	2	no	—
161	C₄×Q₁₆	—	67	4	no	—
162	C₄×Q₁₆	—	134	2	no	—
163	C₂×#107	—	67	1	no	—
164	C₂×#103	—	134	1	no	—
165	C₂×C₄×C₈	C₂×D₆₇	1	1	yes	characteristic
166	C₄⋊C₁₆	C₂×D₆₇	1	4	yes	complemented
167	#357	—	67	8	no	—
168	C₂²×Q₈	—	134	1	no	—
169	C₂²×D₄	—	268	1	no	—
170	C₂²×D₄	—	268	1	no	—
171	C₂²×D₄	—	134	2	no	—
172	C₂×Q₁₆	—	67	4	no	—
173	C₂×Q₁₆	—	67	2	no	—
174	C₂×Q₁₆	—	134	1	no	—
175	C₂×Q₁₆	—	134	2	no	—
176	C₂×D₈	—	268	4	no	—
177	C₂×D₈	—	67	4	no	—
178	C₂×D₈	—	134	4	no	—
179	C₂×D₈	—	134	2	no	—
180	C₂×D₈	—	134	4	no	—
181	C₂×D₈	—	134	4	no	—
182	C₂²×C₈	D₂₆₈	1	1	yes	characteristic
183	C₂²×C₈	C₂²×D₆₇	1	1	yes	characteristic
184	#127	—	268	2	no	—
185	#127	—	134	4	no	—
186	C₄×Q₈	—	268	2	no	—
187	C₄×Q₈	—	134	4	no	—
188	C₂×C₄⋊C₄	—	268	1	no	—
189	C₂×C₄⋊C₄	—	134	1	no	—
190	C₂×C₄²	—	268	1	no	—
191	C₂×C₄²	D₂₆₈	1	1	yes	characteristic
192	SD₃₂	—	134	16	no	—
193	C₂×C₁₆	—	2	4	no	—
194	C₂×C₁₆	C₂²×D₆₇	1	4	yes	—
195	#107	—	67	2	no	—
196	#107	—	134	1	no	—
197	#103	—	134	4	no	—
198	C₄×C₈	D₂₆₈	1	2	yes	—
199	C₄×C₈	C₂²×D₆₇	1	2	yes	—
200	C₂⁴	—	268	2	no	—
201	C₂×Q₈	—	268	1	no	—
202	C₂×Q₈	—	268	2	no	—
203	C₂×Q₈	—	134	4	no	—
204	C₂×Q₈	—	134	2	no	—
205	C₂×D₄	—	268	4	no	—
206	C₂×D₄	—	268	2	no	—
207	C₂×D₄	—	268	4	no	—
208	C₂×D₄	—	268	4	no	—
209	C₂×D₄	—	268	4	no	—
210	C₂×D₄	—	536	4	no	—
211	C₂×D₄	—	536	4	no	—
212	C₂×D₄	—	134	4	no	—
213	C₂²×C₄	—	268	1	no	—
214	C₂²×C₄	—	268	1	no	—
215	C₂²×C₄	D₅₃₆	1	1	yes	characteristic
216	C₂²×C₄	D₅₃₆	1	1	yes	characteristic
217	C₂²×C₄	C₂×D₂₆₈	1	1	yes	characteristic
218	Q₁₆	—	67	4	no	—
219	Q₁₆	—	134	4	no	—
220	Q₁₆	—	134	2	no	—
221	D₈	—	268	8	no	—
222	D₈	—	268	4	no	—
223	D₈	—	134	4	no	—
224	D₈	—	134	4	no	—
225	C₂×C₈	—	2	2	no	—
226	C₂×C₈	D₄×D₆₇	1	2	yes	—
227	C₂×C₈	C₂×D₂₆₈	1	2	yes	—
228	C₂×C₈	C₂×D₂₆₈	1	1	yes	characteristic
229	C₂×C₈	#11761629	1	2	yes	—
230	C₂×C₈	C₂³×D₆₇	1	1	yes	characteristic
231	C₄⋊C₄	—	268	4	no	—
232	C₄⋊C₄	—	268	1	no	—
233	C₄⋊C₄	—	134	2	no	—
234	C₄²	—	268	4	no	—
235	C₄²	D₅₃₆	1	2	yes	—
236	C₄²	C₂×D₂₆₈	1	2	yes	—
237	C₁₆	—	2	4	no	—
238	C₂³	—	268	4	no	—
239	C₂³	—	536	4	no	—
240	C₂³	—	536	4	no	—
241	C₂³	—	536	4	no	—
242	C₂³	C₂×D₅₃₆	1	1	yes	characteristic
243	Q₈	—	268	4	no	—
244	Q₈	—	268	2	no	—
245	Q₈	—	134	4	no	—
246	D₄	—	268	4	no	—
247	D₄	—	268	4	no	—
248	D₄	—	536	8	no	—
249	D₄	—	536	8	no	—
250	D₄	—	536	4	no	—
251	C₂×C₄	—	2	2	no	—
252	C₂×C₄	—	268	4	no	—
253	C₂×C₄	—	268	2	no	—
254	C₂×C₄	—	268	2	no	—
255	C₂×C₄	—	536	2	no	—
256	C₂×C₄	#14523988	1	4	yes	complemented
257	C₂×C₄	#14524018	1	2	yes	—
258	C₂×C₄	C₂×D₅₃₆	1	2	yes	—
259	C₂×C₄	C₂×D₅₃₆	1	2	yes	—
260	C₂×C₄	C₂×D₅₃₆	1	1	yes	characteristic
261	C₂×C₄	#14524071	1	2	yes	—
262	C₂×C₄	C₂²×D₂₆₈	1	1	yes	characteristic
263	C₈	—	2	2	no	—
264	C₈	#14524018	1	2	yes	—
265	C₈	C₂×D₄×D₆₇	1	1	yes	characteristic
266	C₈	C₂×#11761629	1	1	yes	characteristic
267	C₂²	—	536	4	no	—
268	C₂²	—	536	4	no	—
269	C₂²	—	536	4	no	—
270	C₂²	—	1072	8	no	—
271	C₂²	#16863592	1	2	yes	—
272	C₂²	C₂×#14523988	1	2	yes	—
273	C₂²	#16863834	1	2	yes	—
274	C₂²	C₂²×D₅₃₆	1	1	yes	characteristic
275	C₄	—	2	2	no	—
276	C₄	—	268	4	no	—
277	C₄	—	536	2	no	—
278	C₄	#16863592	1	2	yes	—
279	C₄	C₂×#14523988	1	4	yes	complemented
280	C₄	C₂×#14524018	1	1	yes	characteristic
281	C₄	C₂×#14524071	1	1	yes	characteristic
282	C₂	—	1072	4	no	—
283	C₂	#22562250	1	4	yes	direct product factor
284	C₂	C₂×#16863592	1	1	yes	characteristic
285	C₂	C₂²×#14523988	1	1	yes	characteristic
286	C₂	C₂×#16863834	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	C₂×D₅₃₆
commutator, cosocle	C₂×C₈×C₆₇	C₂⁴
Frattini	C₂×C₈	C₂³×D₆₇
Fitting	C₂×C₄⋊C₁₆×C₆₇	C₂
socle	C₂³×C₆₇	C₂×D₈
prime radical	C₂×#16863773	C₂
hypercenter	C₂×C₄⋊C₁₆	D₆₇
hypocenter	C₆₇	C₂×#2207
LCS	G ▷ C₂×C₈×C₆₇ ▷ C₄×C₆₇ ▷ C₂×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ C₂×C₄⋊C₁₆

Complements

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

N	H ≅ G/N	type
1. C₂²×#14523988	C₂	semidirect
3. C₂×C₄×Dic₂₆₈	C₂	semidirect
4. #22562250	C₂	direct (N × H)
5. C₂×#16863198	C₂	semidirect
6. C₂×C₄⋊C₁₆×C₆₇	C₂	semidirect
13. C₂×#14523988	C₂²	semidirect
17. C₄×Dic₂₆₈	C₂²	semidirect
20. #16863198	C₂²	semidirect
23. C₄⋊C₁₆×C₆₇	C₂²	semidirect
30. C₂×Dic₂₆₈	D₄	semidirect
73. Dic₂₆₈	C₂×D₄	semidirect
100. C₂×C₄×C₆₇	SD₃₂	semidirect
133. C₄×C₆₇	C₂×SD₃₂	semidirect
139. C₂×C₆₇	#2207	semidirect
148. C₂×C₄⋊C₁₆	D₆₇	semidirect
150. C₆₇	C₂×#2207	semidirect
166. C₄⋊C₁₆	C₂×D₆₇	semidirect
256. C₂×C₄	#14523988	semidirect
279. C₄	C₂×#14523988	semidirect
283. C₂	#22562250	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂²×#14523988	2
C₂×#16863773	2
C₂×C₄×Dic₂₆₈	2
#22562250	2
C₂×#16863198	2
C₂×C₄⋊C₁₆×C₆₇	2
C₂×#2207	67

Subgroups — C2×#22562250 #31010076

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#22562250 #31010076