Subgroups — C₂×#22562244 #31010055

Order 17152.

870 conjugacy classes of subgroups.

The subgroup lattice has 304 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×#14523984	C₂	1	2	yes	maximal, complemented
2	C₂×#16863773	C₂	1	1	yes	maximal, characteristic, complemented
3	#22562244	C₂	1	8	yes	maximal, direct product factor
4	C₂×C₄⋊(C₆₇⋊C₁₆)	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂×#16863196	C₂	1	2	yes	maximal, complemented
6	C₂×C₄×D₈×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂²×D₅₃₆	—	2	1	no	—
8	C₂²×D₅₃₆	C₂²	1	2	yes	—
9	C₂×#14524040	—	2	1	no	—
10	C₂×#14523984	—	2	8	no	—
11	C₂×#14523984	C₂²	1	8	yes	complemented
12	C₂²×C₆₇⋊C₁₆	C₂²	1	2	yes	—
13	#16863773	—	2	2	no	—
14	#16863773	C₂²	1	4	yes	complemented
15	C₄⋊(C₆₇⋊C₁₆)	C₂²	1	4	yes	complemented
16	#16863196	C₂²	1	8	yes	complemented
17	C₂²×D₈×C₆₇	C₂²	1	1	yes	characteristic
18	C₂×C₄×D₄×C₆₇	—	2	1	no	—
19	C₄×D₈×C₆₇	—	2	2	no	—
20	C₄×D₈×C₆₇	C₂²	1	4	yes	complemented
21	C₂×#107×C₆₇	C₂²	1	1	yes	characteristic
22	C₂×#102×C₆₇	—	2	1	no	—
23	C₂×C₄×C₈×C₆₇	C₂²	1	1	yes	characteristic
24	C₂²×D₂₆₈	—	2	2	no	—
25	C₂²×D₂₆₈	—	4	1	no	—
26	C₂²×D₂₆₈	—	4	1	no	—
27	C₂×D₅₃₆	—	2	4	no	—
28	C₂×D₅₃₆	—	2	2	no	—
29	C₂×D₅₃₆	—	2	4	no	—
30	C₂×D₅₃₆	—	2	4	no	—
31	C₂×D₅₃₆	—	4	4	no	—
32	C₂×D₅₃₆	C₂³	1	4	yes	—
33	#14524040	—	2	4	no	—
34	#14524040	—	4	2	no	—
35	#14523984	—	2	16	no	—
36	C₂×C₆₇⋊C₁₆	—	2	4	no	—
37	C₂×C₆₇⋊C₁₆	C₂³	1	4	yes	—
38	C₂²×D₄×C₆₇	—	2	1	no	—
39	C₂³×C₄×C₆₇	—	4	1	no	—
40	C₂×D₈×C₆₇	—	2	1	no	—
41	C₂×D₈×C₆₇	—	2	2	no	—
42	C₂×D₈×C₆₇	D₄	1	4	yes	complemented
43	C₂×D₈×C₆₇	C₂³	1	2	yes	—
44	C₂²×C₈×C₆₇	D₄	1	1	yes	characteristic
45	C₂²×C₈×C₆₇	C₂³	1	1	yes	characteristic
46	C₄×D₄×C₆₇	—	2	4	no	—
47	C₄×D₄×C₆₇	—	4	2	no	—
48	C₂×C₄⋊C₄×C₆₇	—	2	1	no	—
49	C₂×C₂²⋊C₄×C₆₇	—	4	1	no	—
50	C₂×C₄²×C₆₇	D₄	1	1	yes	characteristic
51	#107×C₆₇	—	2	1	no	—
52	#107×C₆₇	C₂³	1	2	yes	—
53	#102×C₆₇	—	2	4	no	—
54	C₄×C₈×C₆₇	D₄	1	2	yes	—
55	C₄×C₈×C₆₇	C₂³	1	2	yes	—
56	C₂³×D₆₇	—	4	2	no	—
57	C₂×D₂₆₈	—	2	4	no	—
58	C₂×D₂₆₈	—	4	4	no	—
59	C₂×D₂₆₈	—	4	4	no	—
60	C₂×D₂₆₈	—	4	4	no	—
61	C₂×D₂₆₈	—	4	4	no	—
62	C₂×D₂₆₈	—	4	2	no	—
63	C₂×D₂₆₈	—	8	4	no	—
64	C₂×D₂₆₈	—	8	4	no	—
65	D₅₃₆	—	2	4	no	—
66	D₅₃₆	—	2	4	no	—
67	D₅₃₆	—	4	8	no	—
68	D₅₃₆	—	4	4	no	—
69	C₆₇⋊C₁₆	—	2	4	no	—
70	C₂⁴×C₆₇	—	4	1	no	—
71	C₂×D₄×C₆₇	—	2	4	no	—
72	C₂×D₄×C₆₇	—	2	2	no	—
73	C₂×D₄×C₆₇	—	4	1	no	—
74	C₂×D₄×C₆₇	—	4	2	no	—
75	C₂²×C₄×C₆₇	—	4	1	no	—
76	C₂²×C₄×C₆₇	—	4	4	no	—
77	C₂²×C₄×C₆₇	—	8	4	no	—
78	C₂²×C₄×C₆₇	D₈	1	1	yes	characteristic
79	C₂²×C₄×C₆₇	D₈	1	1	yes	characteristic
80	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
81	D₈×C₆₇	—	2	4	no	—
82	D₈×C₆₇	—	2	2	no	—
83	D₈×C₆₇	C₂×D₄	1	4	yes	complemented
84	C₂×C₈×C₆₇	—	2	2	no	—
85	C₂×C₈×C₆₇	C₂×D₄	1	2	yes	—
86	C₂×C₈×C₆₇	C₂×D₄	1	1	yes	characteristic
87	C₂×C₈×C₆₇	C₂×D₄	1	2	yes	—
88	C₂×C₈×C₆₇	#41	1	2	yes	—
89	C₂×C₈×C₆₇	C₂⁴	1	1	yes	characteristic
90	C₄⋊C₄×C₆₇	—	2	2	no	—
91	C₄⋊C₄×C₆₇	—	4	1	no	—
92	C₂²⋊C₄×C₆₇	—	4	4	no	—
93	C₄²×C₆₇	D₈	1	2	yes	—
94	C₄²×C₆₇	C₂×D₄	1	2	yes	—
95	C₂³×C₆₇	—	4	2	no	—
96	C₂³×C₆₇	—	4	4	no	—
97	C₂³×C₆₇	—	8	2	no	—
98	C₂³×C₆₇	—	8	2	no	—
99	C₂³×C₆₇	C₂×D₈	1	1	yes	characteristic
100	C₂²×D₆₇	—	4	4	no	—
101	C₂²×D₆₇	—	8	4	no	—
102	C₂²×D₆₇	—	8	4	no	—
103	C₂²×D₆₇	—	8	4	no	—
104	D₄×C₆₇	—	2	4	no	—
105	D₄×C₆₇	—	4	2	no	—
106	D₄×C₆₇	—	4	4	no	—
107	C₂×C₄×C₆₇	—	2	2	no	—
108	C₂×C₄×C₆₇	—	4	2	no	—
109	C₂×C₄×C₆₇	—	8	2	no	—
110	C₂×C₄×C₆₇	—	8	8	no	—
111	C₂×C₄×C₆₇	D₁₆	1	4	yes	complemented
112	C₂×C₄×C₆₇	#121	1	2	yes	—
113	C₂×C₄×C₆₇	C₂×D₈	1	2	yes	—
114	C₂×C₄×C₆₇	C₂×D₈	1	1	yes	characteristic
115	C₂×C₄×C₆₇	C₂×D₈	1	2	yes	—
116	C₂×C₄×C₆₇	AGL(1,ℤ/8)	1	2	yes	—
117	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
118	D₂₆₈	—	4	4	no	—
119	D₂₆₈	—	4	4	no	—
120	D₂₆₈	—	8	8	no	—
121	D₂₆₈	—	8	4	no	—
122	D₂₆₈	—	8	8	no	—
123	C₈×C₆₇	—	2	2	no	—
124	C₈×C₆₇	#121	1	2	yes	—
125	C₈×C₆₇	C₂²×D₄	1	1	yes	characteristic
126	C₈×C₆₇	C₂×#41	1	1	yes	characteristic
127	C₂²×C₆₇	—	4	4	no	—
128	C₂²×C₆₇	—	8	2	no	—
129	C₂²×C₆₇	—	8	2	no	—
130	C₂²×C₆₇	—	8	8	no	—
131	C₂²×C₆₇	#459	1	2	yes	—
132	C₂²×C₆₇	C₂×D₁₆	1	2	yes	—
133	C₂²×C₆₇	#509	1	2	yes	—
134	C₂²×C₆₇	C₂²×D₈	1	1	yes	characteristic
135	C₂×D₆₇	—	8	4	no	—
136	C₂×D₆₇	—	8	4	no	—
137	C₂×D₆₇	—	8	4	no	—
138	C₂×D₆₇	—	16	8	no	—
139	C₄×C₆₇	—	2	2	no	—
140	C₄×C₆₇	—	8	2	no	—
141	C₄×C₆₇	#459	1	2	yes	—
142	C₄×C₆₇	C₂×D₁₆	1	4	yes	complemented
143	C₄×C₆₇	C₂×#121	1	1	yes	characteristic
144	C₄×C₆₇	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
145	C₂×#2206	—	67	1	no	maximal, Sylow
146	C₂×C₆₇	—	8	4	no	—
147	C₂×C₆₇	#2206	1	4	yes	complemented
148	C₂×C₆₇	C₂×#459	1	1	yes	characteristic
149	C₂×C₆₇	C₂²×D₁₆	1	1	yes	characteristic
150	C₂×C₆₇	C₂×#509	1	1	yes	characteristic
151	D₆₇	—	16	4	no	—
152	C₂²×D₁₆	—	67	2	no	—
153	C₂×#493	—	67	1	no	—
154	C₂×C₄×D₈	D₆₇	1	1	yes	characteristic, complemented
155	#2206	—	67	8	no	—
156	C₂×C₄⋊C₁₆	—	67	1	no	—
157	C₂×#357	—	67	2	no	—
158	C₆₇	C₂×#2206	1	1	yes	characteristic, complemented, Sylow
159	C₂²×D₈	—	67	2	no	—
160	C₂²×D₈	—	134	1	no	—
161	C₂²×D₈	C₂×D₆₇	1	1	yes	characteristic
162	C₂×#127	—	134	1	no	—
163	C₂×C₄×D₄	—	2	1	no	—
164	C₂×D₁₆	—	67	8	no	—
165	C₂×D₁₆	—	134	8	no	—
166	C₂²×C₁₆	—	67	2	no	—
167	#493	—	67	4	no	—
168	#493	—	134	2	no	—
169	C₄×D₈	—	2	2	no	—
170	C₄×D₈	C₂×D₆₇	1	4	yes	complemented
171	C₂×#107	C₂×D₆₇	1	1	yes	characteristic
172	C₂×#102	—	2	1	no	—
173	C₂×C₄×C₈	C₂×D₆₇	1	1	yes	characteristic
174	C₄⋊C₁₆	—	67	4	no	—
175	#357	—	67	8	no	—
176	C₂²×D₄	—	2	1	no	—
177	C₂²×D₄	—	268	1	no	—
178	C₂²×D₄	—	268	1	no	—
179	C₂²×D₄	—	134	2	no	—
180	C₂³×C₄	—	4	1	no	—
181	C₂×D₈	—	2	1	no	—
182	C₂×D₈	—	2	2	no	—
183	C₂×D₈	—	268	4	no	—
184	C₂×D₈	—	67	4	no	—
185	C₂×D₈	—	134	2	no	—
186	C₂×D₈	—	134	4	no	—
187	C₂×D₈	—	134	4	no	—
188	C₂×D₈	—	134	4	no	—
189	C₂×D₈	D₂₆₈	1	4	yes	complemented
190	C₂×D₈	C₂²×D₆₇	1	2	yes	—
191	C₂²×C₈	C₂²⋊D₆₇	1	1	yes	characteristic
192	C₂²×C₈	C₂²×D₆₇	1	1	yes	characteristic
193	#127	—	268	2	no	—
194	#127	—	134	4	no	—
195	C₄×D₄	—	2	4	no	—
196	C₄×D₄	—	4	2	no	—
197	C₂×C₄⋊C₄	—	2	1	no	—
198	C₂×C₂²⋊C₄	—	4	1	no	—
199	C₂×C₄²	C₂²⋊D₆₇	1	1	yes	characteristic
200	D₁₆	—	134	16	no	—
201	C₂×C₁₆	—	67	4	no	—
202	C₂×C₁₆	—	134	4	no	—
203	#107	—	2	1	no	—
204	#107	C₂²×D₆₇	1	2	yes	—
205	#102	—	2	4	no	—
206	C₄×C₈	C₂²⋊D₆₇	1	2	yes	—
207	C₄×C₈	C₂²×D₆₇	1	2	yes	—
208	C₂⁴	—	4	1	no	—
209	C₂⁴	—	268	2	no	—
210	C₂×D₄	—	2	4	no	—
211	C₂×D₄	—	2	2	no	—
212	C₂×D₄	—	4	1	no	—
213	C₂×D₄	—	4	2	no	—
214	C₂×D₄	—	268	4	no	—
215	C₂×D₄	—	268	2	no	—
216	C₂×D₄	—	268	4	no	—
217	C₂×D₄	—	268	4	no	—
218	C₂×D₄	—	268	4	no	—
219	C₂×D₄	—	536	4	no	—
220	C₂×D₄	—	536	4	no	—
221	C₂×D₄	—	134	4	no	—
222	C₂²×C₄	—	4	1	no	—
223	C₂²×C₄	—	4	4	no	—
224	C₂²×C₄	—	8	4	no	—
225	C₂²×C₄	#11761613	1	1	yes	characteristic
226	C₂²×C₄	#11761613	1	1	yes	characteristic
227	C₂²×C₄	C₂×C₂²⋊D₆₇	1	1	yes	characteristic
228	D₈	—	2	4	no	—
229	D₈	—	2	2	no	—
230	D₈	—	268	8	no	—
231	D₈	—	268	4	no	—
232	D₈	—	134	4	no	—
233	D₈	—	134	4	no	—
234	D₈	C₂×D₂₆₈	1	4	yes	complemented
235	C₂×C₈	—	2	2	no	—
236	C₂×C₈	C₂×C₂²⋊D₆₇	1	2	yes	—
237	C₂×C₈	C₂×C₂²⋊D₆₇	1	1	yes	characteristic
238	C₂×C₈	C₂×D₂₆₈	1	2	yes	—
239	C₂×C₈	#11761628	1	2	yes	—
240	C₂×C₈	C₂³×D₆₇	1	1	yes	characteristic
241	C₄⋊C₄	—	2	2	no	—
242	C₄⋊C₄	—	4	1	no	—
243	C₂²⋊C₄	—	4	4	no	—
244	C₄²	#11761613	1	2	yes	—
245	C₄²	C₂×C₂²⋊D₆₇	1	2	yes	—
246	C₁₆	—	134	4	no	—
247	C₂³	—	4	4	no	—
248	C₂³	—	4	2	no	—
249	C₂³	—	8	2	no	—
250	C₂³	—	8	2	no	—
251	C₂³	—	268	4	no	—
252	C₂³	—	536	4	no	—
253	C₂³	—	536	4	no	—
254	C₂³	—	536	4	no	—
255	C₂³	C₂×#11761613	1	1	yes	characteristic
256	D₄	—	2	4	no	—
257	D₄	—	4	4	no	—
258	D₄	—	4	2	no	—
259	D₄	—	268	4	no	—
260	D₄	—	268	4	no	—
261	D₄	—	536	8	no	—
262	D₄	—	536	8	no	—
263	D₄	—	536	4	no	—
264	C₂×C₄	—	2	2	no	—
265	C₂×C₄	—	4	2	no	—
266	C₂×C₄	—	8	2	no	—
267	C₂×C₄	—	8	8	no	—
268	C₂×C₄	#14523984	1	4	yes	complemented
269	C₂×C₄	#14524017	1	2	yes	—
270	C₂×C₄	C₂×#11761613	1	2	yes	—
271	C₂×C₄	C₂×#11761613	1	2	yes	—
272	C₂×C₄	C₂×#11761613	1	1	yes	characteristic
273	C₂×C₄	#14524068	1	2	yes	—
274	C₂×C₄	C₂²×C₂²⋊D₆₇	1	1	yes	characteristic
275	C₈	—	2	2	no	—
276	C₈	#14524017	1	2	yes	—
277	C₈	C₂²×D₂₆₈	1	1	yes	characteristic
278	C₈	C₂×#11761628	1	1	yes	characteristic
279	C₂²	—	4	4	no	—
280	C₂²	—	8	2	no	—
281	C₂²	—	8	2	no	—
282	C₂²	—	8	8	no	—
283	C₂²	—	536	4	no	—
284	C₂²	—	536	4	no	—
285	C₂²	—	536	4	no	—
286	C₂²	—	1072	8	no	—
287	C₂²	#16863591	1	2	yes	—
288	C₂²	C₂×#14523984	1	2	yes	—
289	C₂²	#16863831	1	2	yes	—
290	C₂²	C₂²×#11761613	1	1	yes	characteristic
291	C₄	—	2	2	no	—
292	C₄	—	8	2	no	—
293	C₄	#16863591	1	2	yes	—
294	C₄	C₂×#14523984	1	4	yes	complemented
295	C₄	C₂×#14524017	1	1	yes	characteristic
296	C₄	C₂×#14524068	1	1	yes	characteristic
297	C₂	—	8	4	no	—
298	C₂	—	1072	4	no	—
299	C₂	#22562244	1	4	yes	direct product factor
300	C₂	C₂×#16863591	1	1	yes	characteristic
301	C₂	C₂²×#14523984	1	1	yes	characteristic
302	C₂	C₂×#16863831	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	C₂×#11761613
commutator, cosocle	C₂×C₈×C₆₇	C₂⁴
Frattini	C₂×C₈	C₂³×D₆₇
Fitting	C₂×C₄×D₈×C₆₇	C₂
socle	C₂³×C₆₇	C₂×D₈
hypercenter	C₂×C₄×D₈	D₆₇
hypocenter	C₆₇	C₂×#2206
LCS	G ▷ C₂×C₈×C₆₇ ▷ C₄×C₆₇ ▷ C₂×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ C₂×C₄×D₈

Complements

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

N	H ≅ G/N	type
1. C₂²×#14523984	C₂	semidirect
2. C₂×#16863773	C₂	semidirect
3. #22562244	C₂	direct (N × H)
4. C₂×C₄⋊(C₆₇⋊C₁₆)	C₂	semidirect
5. C₂×#16863196	C₂	semidirect
6. C₂×C₄×D₈×C₆₇	C₂	semidirect
11. C₂×#14523984	C₂²	semidirect
14. #16863773	C₂²	semidirect
15. C₄⋊(C₆₇⋊C₁₆)	C₂²	semidirect
16. #16863196	C₂²	semidirect
20. C₄×D₈×C₆₇	C₂²	semidirect
42. C₂×D₈×C₆₇	D₄	semidirect
83. D₈×C₆₇	C₂×D₄	semidirect
111. C₂×C₄×C₆₇	D₁₆	semidirect
142. C₄×C₆₇	C₂×D₁₆	semidirect
147. C₂×C₆₇	#2206	semidirect
154. C₂×C₄×D₈	D₆₇	semidirect
158. C₆₇	C₂×#2206	semidirect
170. C₄×D₈	C₂×D₆₇	semidirect
189. C₂×D₈	D₂₆₈	semidirect
234. D₈	C₂×D₂₆₈	semidirect
268. C₂×C₄	#14523984	semidirect
294. C₄	C₂×#14523984	semidirect
299. C₂	#22562244	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×#14523984	2
C₂×#16863773	2
#22562244	2
C₂×C₄⋊(C₆₇⋊C₁₆)	2
C₂×#16863196	2
C₂×C₄×D₈×C₆₇	2
C₂×#2206	67

Subgroups — C2×#22562244 #31010055

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#22562244 #31010055