Subgroups — C₂×#22562330 #31010389

Order 17152.

678 conjugacy classes of subgroups.

The subgroup lattice has 310 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×D₁₀₇₂	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×#16863812	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×#16863667	C₂	1	2	yes	maximal, complemented
4	#22562330	C₂	1	8	yes	maximal, direct product factor
5	C₂×#16863208	C₂	1	2	yes	maximal, complemented
6	C₂×#368×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂×#14524071	—	2	2	no	—
8	C₂²×D₅₃₆	C₂²	1	2	yes	—
9	C₂×#14524049	C₂²	1	1	yes	characteristic
10	C₂×D₁₀₇₂	—	2	1	no	—
11	C₂×D₁₀₇₂	—	2	1	no	—
12	C₂×D₁₀₇₂	—	2	2	no	—
13	C₂×D₁₀₇₂	C₂²	1	4	yes	complemented
14	#16863812	—	2	2	no	—
15	#16863812	C₂²	1	4	yes	complemented
16	C₂×#14523983	C₂²	1	1	yes	characteristic
17	C₂×#14523982	—	2	1	no	—
18	C₂×C₁₆×D₆₇	—	2	1	no	—
19	#16863667	—	2	4	no	—
20	#16863667	C₂²	1	8	yes	complemented
21	C₂×#14523959	—	2	2	no	—
22	#16863208	C₂²	1	8	yes	complemented
23	C₂²×C₁₆×C₆₇	C₂²	1	1	yes	characteristic
24	C₂×#108×C₆₇	C₂²	1	2	yes	—
25	#368×C₆₇	C₂²	1	4	yes	complemented
26	C₂×#11761628	D₄	1	1	yes	characteristic
27	C₂²×D₂₆₈	—	2	2	no	—
28	#14524071	—	2	8	no	—
29	#14524071	—	4	4	no	—
30	C₂×#11761617	—	4	2	no	—
31	C₂×D₅₃₆	—	2	2	no	—
32	C₂×D₅₃₆	—	2	2	no	—
33	C₂×D₅₃₆	—	2	2	no	—
34	C₂×D₅₃₆	—	2	2	no	—
35	C₂×D₅₃₆	—	2	2	no	—
36	C₂×D₅₃₆	—	4	2	no	—
37	C₂×D₅₃₆	C₂³	1	4	yes	—
38	#14524049	—	2	1	no	—
39	#14524049	—	2	1	no	—
40	#14524049	D₄	1	2	yes	—
41	#14524049	C₂³	1	2	yes	—
42	C₂×#11761612	D₄	1	1	yes	characteristic
43	C₂×#11761611	—	2	1	no	—
44	C₂×C₈×D₆₇	—	2	1	no	—
45	D₁₀₇₂	—	2	4	no	—
46	D₁₀₇₂	—	2	4	no	—
47	#14523983	—	2	1	no	—
48	#14523983	C₂³	1	2	yes	—
49	#14523982	—	2	4	no	—
50	C₂×C₆₇⋊C₁₆	—	2	1	no	—
51	C₁₆×D₆₇	—	2	4	no	—
52	#14523959	—	2	8	no	—
53	C₂×M₁₆×C₆₇	—	2	2	no	—
54	C₂²×C₈×C₆₇	C₂³	1	1	yes	characteristic
55	C₂×C₁₆×C₆₇	D₄	1	1	yes	characteristic
56	C₂×C₁₆×C₆₇	D₄	1	1	yes	characteristic
57	C₂×C₁₆×C₆₇	D₄	1	2	yes	—
58	C₂×C₁₆×C₆₇	C₂³	1	2	yes	—
59	#108×C₆₇	—	2	2	no	—
60	#108×C₆₇	C₂³	1	4	yes	—
61	C₂³×D₆₇	—	4	2	no	—
62	#11761628	—	2	1	no	—
63	#11761628	—	2	1	no	—
64	#11761628	D₈	1	2	yes	—
65	#11761628	C₂×D₄	1	2	yes	—
66	C₂×Dic₁₃₄	D₈	1	1	yes	characteristic
67	C₂×D₂₆₈	—	2	4	no	—
68	C₂×D₂₆₈	—	4	2	no	—
69	C₂×D₂₆₈	—	4	2	no	—
70	C₂×D₂₆₈	—	4	2	no	—
71	C₂×D₂₆₈	—	4	2	no	—
72	C₂×D₂₆₈	—	4	2	no	—
73	C₂×D₂₆₈	D₈	1	1	yes	characteristic
74	C₂×C₂²⋊D₆₇	—	2	1	no	—
75	C₂×C₄×D₆₇	—	2	1	no	—
76	#11761617	—	4	8	no	—
77	D₅₃₆	—	2	4	no	—
78	D₅₃₆	—	2	4	no	—
79	D₅₃₆	—	4	8	no	—
80	D₅₃₆	—	4	2	no	—
81	D₅₃₆	—	4	2	no	—
82	#11761612	—	2	1	no	—
83	#11761612	C₂×D₄	1	2	yes	—
84	#11761611	—	2	2	no	—
85	#11761611	—	2	2	no	—
86	C₂×C₆₇⋊C₈	—	2	1	no	—
87	C₈×D₆₇	—	2	2	no	—
88	C₈×D₆₇	—	2	2	no	—
89	C₆₇⋊C₁₆	—	2	2	no	—
90	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
91	M₁₆×C₆₇	—	2	4	no	—
92	M₁₆×C₆₇	—	4	2	no	—
93	C₂×C₈×C₆₇	—	4	2	no	—
94	C₂×C₈×C₆₇	C₂×D₄	1	1	yes	characteristic
95	C₂×C₈×C₆₇	C₂×D₄	1	1	yes	characteristic
96	C₂×C₈×C₆₇	C₂×D₄	1	1	yes	characteristic
97	C₂×C₈×C₆₇	#41	1	1	yes	characteristic
98	C₂×C₈×C₆₇	#41	1	1	yes	characteristic
99	C₂×C₈×C₆₇	C₂⁴	1	1	yes	characteristic
100	C₁₆×C₆₇	C₂×D₄	1	2	yes	—
101	C₁₆×C₆₇	C₂×D₄	1	2	yes	—
102	C₂³×C₆₇	#135	1	1	yes	characteristic
103	C₂²×D₆₇	—	2	1	no	—
104	C₂²×D₆₇	—	4	4	no	—
105	C₂²×D₆₇	—	8	2	no	—
106	C₂²×D₆₇	—	8	2	no	—
107	C₂²×D₆₇	—	16	4	no	—
108	C₂×C₄×C₆₇	#121	1	1	yes	characteristic
109	C₂×C₄×C₆₇	#121	1	1	yes	characteristic
110	C₂×C₄×C₆₇	C₂×D₈	1	1	yes	characteristic
111	C₂×C₄×C₆₇	C₂×D₈	1	1	yes	characteristic
112	C₂×C₄×C₆₇	#135	1	1	yes	characteristic
113	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
114	C₂²⋊D₆₇	—	2	2	no	—
115	C₂²⋊D₆₇	—	2	2	no	—
116	C₂×Dic₆₇	—	2	1	no	—
117	D₂₆₈	—	2	1	no	—
118	D₂₆₈	—	4	4	no	—
119	D₂₆₈	—	4	4	no	—
120	D₂₆₈	—	8	2	no	—
121	D₂₆₈	—	8	2	no	—
122	D₂₆₈	C₂×D₈	1	2	yes	—
123	C₄×D₆₇	—	2	2	no	—
124	C₄×D₆₇	—	2	2	no	—
125	Dic₁₃₄	—	2	1	no	—
126	Dic₁₃₄	C₂×D₈	1	2	yes	—
127	C₈×C₆₇	—	4	4	no	—
128	C₈×C₆₇	#121	1	1	yes	characteristic
129	C₈×C₆₇	#121	1	1	yes	characteristic
130	C₈×C₆₇	C₂²×D₄	1	1	yes	characteristic
131	C₈×C₆₇	C₂×#41	1	1	yes	characteristic
132	C₆₇⋊C₈	—	2	2	no	—
133	C₂²×C₆₇	—	2	2	no	—
134	C₂²×C₆₇	#466	1	1	yes	characteristic
135	C₂²×C₆₇	#466	1	1	yes	characteristic
136	C₂²×C₆₇	C₂×#135	1	1	yes	characteristic
137	C₂×D₆₇	—	2	2	no	—
138	C₂×D₆₇	—	4	2	no	—
139	C₂×D₆₇	—	8	4	no	—
140	C₂×D₆₇	—	16	4	no	—
141	C₂×D₆₇	—	16	4	no	—
142	C₂×D₆₇	—	16	4	no	—
143	C₄×C₆₇	#466	1	1	yes	characteristic
144	C₄×C₆₇	#466	1	1	yes	characteristic
145	C₄×C₆₇	C₂×#121	1	1	yes	characteristic
146	C₄×C₆₇	C₂²×D₈	1	1	yes	characteristic
147	Dic₆₇	—	2	2	no	—
148	C₂×#2221	—	67	1	no	maximal, Sylow
149	C₂×C₆₇	—	2	1	no	—
150	C₂×C₆₇	—	2	1	no	—
151	C₂×C₆₇	#2221	1	2	yes	complemented
152	C₂×C₆₇	C₂×#466	1	1	yes	characteristic
153	D₆₇	—	4	2	no	—
154	D₆₇	—	16	4	no	—
155	C₂²×D₁₆	—	67	1	no	—
156	C₂×#504	—	67	1	no	—
157	C₂×#472	—	67	2	no	—
158	#2221	—	67	8	no	—
159	C₂×#368	D₆₇	1	1	yes	characteristic, complemented
160	C₂×#361	—	67	2	no	—
161	C₆₇	C₂×#2221	1	1	yes	characteristic, complemented, Sylow
162	C₂×AGL(1,ℤ/8)	—	134	2	no	—
163	C₂²×D₈	—	67	2	no	—
164	C₂×#131	—	67	1	no	—
165	C₂×D₁₆	—	67	4	no	—
166	C₂×D₁₆	—	134	1	no	—
167	C₂×D₁₆	—	134	1	no	—
168	C₂×D₁₆	—	134	2	no	—
169	#504	—	67	4	no	—
170	#504	—	134	2	no	—
171	C₂×M₃₂	—	67	1	no	—
172	C₂×M₃₂	—	134	1	no	—
173	C₂²×C₁₆	—	134	1	no	—
174	C₂²×C₁₆	C₂×D₆₇	1	1	yes	characteristic
175	#472	—	67	8	no	—
176	#472	—	134	4	no	—
177	C₂×#108	C₂×D₆₇	1	2	yes	—
178	C₂×#100	—	134	2	no	—
179	#368	C₂×D₆₇	1	4	yes	complemented
180	#361	—	67	8	no	—
181	C₂×#41	—	67	1	no	—
182	C₂²×D₄	—	134	2	no	—
183	AGL(1,ℤ/8)	—	268	4	no	—
184	AGL(1,ℤ/8)	—	134	8	no	—
185	C₂×SD₁₆	—	268	2	no	—
186	C₂×D₈	—	268	2	no	—
187	C₂×D₈	—	67	4	no	—
188	C₂×D₈	—	134	2	no	—
189	C₂×D₈	—	134	2	no	—
190	C₂×D₈	—	134	2	no	—
191	C₂×D₈	—	134	2	no	—
192	C₂×D₈	—	134	2	no	—
193	#131	—	67	2	no	—
194	#131	—	67	2	no	—
195	#131	—	134	1	no	—
196	#131	—	134	1	no	—
197	C₂×M₁₆	—	2	2	no	—
198	C₂×M₁₆	—	67	1	no	—
199	C₂×M₁₆	—	134	1	no	—
200	C₂²×C₈	—	134	1	no	—
201	C₂²×C₈	C₂²×D₆₇	1	1	yes	characteristic
202	D₁₆	—	134	4	no	—
203	D₁₆	—	134	4	no	—
204	M₃₂	—	67	2	no	—
205	M₃₂	—	134	1	no	—
206	M₃₂	—	134	4	no	—
207	C₂×C₁₆	—	134	1	no	—
208	C₂×C₁₆	—	134	4	no	—
209	C₂×C₁₆	D₂₆₈	1	1	yes	characteristic
210	C₂×C₁₆	D₂₆₈	1	1	yes	characteristic
211	C₂×C₁₆	D₂₆₈	1	2	yes	—
212	C₂×C₁₆	C₂²×D₆₇	1	2	yes	—
213	#108	—	2	2	no	—
214	#108	C₂²×D₆₇	1	4	yes	—
215	#100	—	134	8	no	—
216	C₂⁴	—	268	2	no	—
217	#41	—	67	2	no	—
218	#41	—	67	2	no	—
219	#41	—	134	1	no	—
220	#41	—	134	1	no	—
221	C₂×Q₈	—	67	1	no	—
222	C₂×D₄	—	268	2	no	—
223	C₂×D₄	—	268	2	no	—
224	C₂×D₄	—	268	2	no	—
225	C₂×D₄	—	268	2	no	—
226	C₂×D₄	—	268	2	no	—
227	C₂×D₄	—	67	1	no	—
228	C₂×D₄	—	134	1	no	—
229	C₂×D₄	—	134	4	no	—
230	C₂²×C₄	—	134	1	no	—
231	C₂²×C₄	D₄×D₆₇	1	1	yes	characteristic
232	SD₁₆	—	268	8	no	—
233	D₈	—	268	8	no	—
234	D₈	—	268	2	no	—
235	D₈	—	268	2	no	—
236	D₈	—	134	4	no	—
237	D₈	—	134	4	no	—
238	M₁₆	—	2	4	no	—
239	M₁₆	—	4	2	no	—
240	M₁₆	—	67	2	no	—
241	M₁₆	—	134	1	no	—
242	M₁₆	—	134	2	no	—
243	M₁₆	—	134	2	no	—
244	C₂×C₈	—	4	2	no	—
245	C₂×C₈	—	134	1	no	—
246	C₂×C₈	—	134	2	no	—
247	C₂×C₈	—	134	2	no	—
248	C₂×C₈	D₄×D₆₇	1	1	yes	characteristic
249	C₂×C₈	C₂×D₂₆₈	1	1	yes	characteristic
250	C₂×C₈	C₂×D₂₆₈	1	1	yes	characteristic
251	C₂×C₈	#11761629	1	1	yes	characteristic
252	C₂×C₈	#11761629	1	1	yes	characteristic
253	C₂×C₈	C₂³×D₆₇	1	1	yes	characteristic
254	C₁₆	—	134	2	no	—
255	C₁₆	C₂×D₂₆₈	1	2	yes	—
256	C₁₆	C₂×D₂₆₈	1	2	yes	—
257	C₂³	—	268	4	no	—
258	C₂³	—	536	2	no	—
259	C₂³	—	536	2	no	—
260	C₂³	—	1072	4	no	—
261	C₂³	—	134	1	no	—
262	C₂³	#14524066	1	1	yes	characteristic
263	Q₈	—	67	2	no	—
264	Q₈	—	134	1	no	—
265	D₄	—	268	4	no	—
266	D₄	—	268	4	no	—
267	D₄	—	536	2	no	—
268	D₄	—	536	2	no	—
269	D₄	—	67	2	no	—
270	D₄	—	134	2	no	—
271	D₄	—	134	2	no	—
272	D₄	—	134	1	no	—
273	C₂×C₄	—	134	1	no	—
274	C₂×C₄	—	134	2	no	—
275	C₂×C₄	—	134	2	no	—
276	C₂×C₄	#14524018	1	1	yes	characteristic
277	C₂×C₄	#14524018	1	1	yes	characteristic
278	C₂×C₄	D₈×D₆₇	1	1	yes	characteristic
279	C₂×C₄	D₈×D₆₇	1	1	yes	characteristic
280	C₂×C₄	#14524066	1	1	yes	characteristic
281	C₂×C₄	C₂×D₄×D₆₇	1	1	yes	characteristic
282	C₈	—	4	4	no	—
283	C₈	—	134	2	no	—
284	C₈	#14524018	1	1	yes	characteristic
285	C₈	#14524018	1	1	yes	characteristic
286	C₈	C₂²×D₂₆₈	1	1	yes	characteristic
287	C₈	C₂×#11761629	1	1	yes	characteristic
288	C₂²	—	2	2	no	—
289	C₂²	—	268	2	no	—
290	C₂²	—	536	4	no	—
291	C₂²	—	1072	4	no	—
292	C₂²	—	1072	4	no	—
293	C₂²	—	1072	4	no	—
294	C₂²	—	134	2	no	—
295	C₂²	#16863633	1	1	yes	characteristic
296	C₂²	#16863633	1	1	yes	characteristic
297	C₂²	C₂×#14524066	1	1	yes	characteristic
298	C₄	—	134	2	no	—
299	C₄	#16863633	1	1	yes	characteristic
300	C₄	#16863633	1	1	yes	characteristic
301	C₄	C₂×#14524018	1	1	yes	characteristic
302	C₄	C₂×D₈×D₆₇	1	1	yes	characteristic
303	C₂	—	2	1	no	—
304	C₂	—	2	1	no	—
305	C₂	—	268	2	no	—
306	C₂	—	1072	4	no	—
307	C₂	#22562330	1	2	yes	direct product factor
308	C₂	C₂×#16863633	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	#16863633
commutator, cosocle	C₂×C₈×C₆₇	C₂⁴
Frattini	C₂×C₈	C₂³×D₆₇
Fitting	C₂×#368×C₆₇	C₂
socle	C₂²×C₆₇	#466
hypercenter	C₂×#368	D₆₇
hypocenter	C₆₇	C₂×#2221
LCS	G ▷ C₂×C₈×C₆₇ ▷ C₄×C₆₇ ▷ C₂×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂²×C₈ ◁ C₂×#368

Complements

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

N	H ≅ G/N	type
1. C₂²×D₁₀₇₂	C₂	semidirect
2. C₂×#16863812	C₂	semidirect
3. C₂×#16863667	C₂	semidirect
4. #22562330	C₂	direct (N × H)
5. C₂×#16863208	C₂	semidirect
6. C₂×#368×C₆₇	C₂	semidirect
13. C₂×D₁₀₇₂	C₂²	semidirect
15. #16863812	C₂²	semidirect
20. #16863667	C₂²	semidirect
22. #16863208	C₂²	semidirect
25. #368×C₆₇	C₂²	semidirect
151. C₂×C₆₇	#2221	semidirect
159. C₂×#368	D₆₇	semidirect
161. C₆₇	C₂×#2221	semidirect
179. #368	C₂×D₆₇	semidirect
307. C₂	#22562330	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×D₁₀₇₂	2
C₂×#16863812	2
C₂×#16863667	2
#22562330	2
C₂×#16863208	2
C₂×#368×C₆₇	2
C₂×#2221	67

Subgroups — C2×#22562330 #31010389

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#22562330 #31010389