Subgroups — C₂×#22562384 #31010566

Order 17152.

526 conjugacy classes of subgroups.

The subgroup lattice has 320 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#16863796	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×#16863512	C₂	1	1	yes	maximal, characteristic
3	#22562384	C₂	1	8	yes	maximal, direct product factor
4	C₂×#16863220	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂×#16863216	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×#16863207	C₂	1	1	yes	maximal, characteristic
7	C₂×#16863205	C₂	1	1	yes	maximal, characteristic
8	C₂×#365×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂×#14524065	C₂²	1	1	yes	characteristic
10	C₂×#14524051	—	2	1	no	—
11	C₂×#14524034	—	2	1	no	—
12	C₂×#14523983	C₂²	1	1	yes	characteristic
13	#16863796	—	2	2	no	—
14	#16863796	C₂²	1	4	yes	complemented
15	#16863512	—	2	2	no	—
16	#16863512	C₂²	1	4	yes	—
17	C₂×#14523976	—	2	1	no	—
18	C₂×#14523975	C₂²	1	1	yes	characteristic
19	C₂×#14523969	—	2	1	no	—
20	C₂×C₈×Dic₆₇	C₂²	1	1	yes	characteristic
21	#16863220	C₂²	1	4	yes	complemented
22	#16863216	C₂²	1	4	yes	complemented
23	#16863207	C₂²	1	4	yes	—
24	#16863205	C₂²	1	4	yes	—
25	C₂×M₃₂×C₆₇	C₂²	1	1	yes	characteristic
26	C₂×#108×C₆₇	C₂²	1	1	yes	characteristic
27	C₂×#106×C₆₇	C₂²	1	1	yes	characteristic
28	#365×C₆₇	C₂²	1	4	yes	complemented
29	C₂×#11761628	—	2	1	no	—
30	#14524065	—	2	1	no	—
31	#14524065	—	2	1	no	—
32	#14524065	Q₈	1	2	yes	—
33	#14524065	C₂³	1	2	yes	—
34	C₂×Dic₂₆₈	Q₈	1	1	yes	characteristic
35	C₂×#11761617	—	2	1	no	—
36	C₂×D₅₃₆	Q₈	1	1	yes	characteristic
37	#14524051	—	2	4	no	—
38	#14524051	—	4	2	no	—
39	C₂×#11761611	—	4	1	no	—
40	C₂²×C₆₇⋊C₈	D₄	1	1	yes	characteristic
41	C₂×C₈×D₆₇	—	4	1	no	—
42	#14524034	—	2	4	no	—
43	#14524034	—	4	2	no	—
44	C₂×C₄⋊Dic₆₇	—	2	1	no	—
45	C₂×C₄⋊Dic₆₇	—	4	1	no	—
46	C₂×Dic₆₇⋊C₄	—	4	1	no	—
47	C₂×C₄×Dic₆₇	D₄	1	1	yes	characteristic
48	#14523983	—	2	1	no	—
49	#14523983	C₂³	1	2	yes	—
50	C₂×C₆₇⋊C₁₆	—	2	1	no	—
51	#14523976	—	2	4	no	—
52	#14523975	—	2	1	no	—
53	#14523975	C₂³	1	2	yes	—
54	#14523969	—	2	4	no	—
55	C₈×Dic₆₇	D₄	1	2	yes	—
56	C₈×Dic₆₇	C₂³	1	2	yes	—
57	C₂×M₁₆×C₆₇	—	2	1	no	—
58	C₂²×C₈×C₆₇	C₂³	1	1	yes	characteristic
59	C₂×C₄⋊C₄×C₆₇	—	2	1	no	—
60	M₃₂×C₆₇	—	2	1	no	—
61	M₃₂×C₆₇	C₂³	1	2	yes	—
62	C₂×C₁₆×C₆₇	—	2	1	no	—
63	#108×C₆₇	—	2	1	no	—
64	#108×C₆₇	C₂³	1	2	yes	—
65	#106×C₆₇	—	2	1	no	—
66	#106×C₆₇	C₂³	1	2	yes	—
67	#11761628	—	2	2	no	—
68	#11761628	—	2	2	no	—
69	#11761628	—	4	1	no	—
70	#11761628	—	4	1	no	—
71	C₂×Dic₁₃₄	—	2	1	no	—
72	C₂×D₂₆₈	—	2	1	no	—
73	C₂×C₂²⋊D₆₇	—	4	1	no	—
74	C₂²×Dic₆₇	—	2	1	no	—
75	C₂²×Dic₆₇	—	4	1	no	—
76	C₂×C₄×D₆₇	—	4	1	no	—
77	Dic₂₆₈	—	2	1	no	—
78	Dic₂₆₈	C₂×Q₈	1	2	yes	—
79	#11761617	—	2	2	no	—
80	#11761617	—	2	2	no	—
81	D₅₃₆	—	2	1	no	—
82	D₅₃₆	C₂×Q₈	1	2	yes	—
83	#11761611	—	4	4	no	—
84	C₂×C₆₇⋊C₈	SD₁₆	1	1	yes	characteristic
85	C₂×C₆₇⋊C₈	SD₁₆	1	1	yes	characteristic
86	C₂×C₆₇⋊C₈	SD₁₆	1	2	yes	—
87	C₂×C₆₇⋊C₈	C₂×D₄	1	2	yes	—
88	C₈×D₆₇	—	4	4	no	—
89	C₄⋊Dic₆₇	—	2	2	no	—
90	C₄⋊Dic₆₇	—	4	4	no	—
91	C₄⋊Dic₆₇	—	4	1	no	—
92	Dic₆₇⋊C₄	—	4	4	no	—
93	C₄×Dic₆₇	—	2	1	no	—
94	C₄×Dic₆₇	C₂×D₄	1	2	yes	—
95	C₆₇⋊C₁₆	—	2	2	no	—
96	C₂²×C₄×C₆₇	—	4	1	no	—
97	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
98	M₁₆×C₆₇	—	2	2	no	—
99	M₁₆×C₆₇	—	4	1	no	—
100	C₂×C₈×C₆₇	—	4	1	no	—
101	C₂×C₈×C₆₇	C₂×D₄	1	1	yes	characteristic
102	C₂×C₈×C₆₇	C₂×Q₈	1	1	yes	characteristic
103	C₂×C₈×C₆₇	C₂×Q₈	1	1	yes	characteristic
104	C₂×C₈×C₆₇	#41	1	1	yes	characteristic
105	C₂×C₈×C₆₇	#41	1	1	yes	characteristic
106	C₂×C₈×C₆₇	C₂⁴	1	1	yes	characteristic
107	C₄⋊C₄×C₆₇	—	2	2	no	—
108	C₄⋊C₄×C₆₇	—	4	1	no	—
109	C₁₆×C₆₇	—	2	2	no	—
110	C₂³×C₆₇	AGL(1,ℤ/8)	1	1	yes	characteristic
111	C₂²×D₆₇	—	4	1	no	—
112	C₂×C₄×C₆₇	—	4	2	no	—
113	C₂×C₄×C₆₇	—	8	2	no	—
114	C₂×C₄×C₆₇	C₂²⋊Q₈	1	1	yes	characteristic
115	C₂×C₄×C₆₇	C₂²⋊Q₈	1	1	yes	characteristic
116	C₂×C₄×C₆₇	C₂×SD₁₆	1	1	yes	characteristic
117	C₂×C₄×C₆₇	C₂×SD₁₆	1	1	yes	characteristic
118	C₂×C₄×C₆₇	AGL(1,ℤ/8)	1	1	yes	characteristic
119	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
120	C₂²⋊D₆₇	—	4	2	no	—
121	C₂²⋊D₆₇	—	4	2	no	—
122	C₂×Dic₆₇	—	2	2	no	—
123	C₂×Dic₆₇	—	4	2	no	—
124	C₂×Dic₆₇	—	4	1	no	—
125	C₂×Dic₆₇	—	4	2	no	—
126	C₂×Dic₆₇	—	8	2	no	—
127	D₂₆₈	—	2	2	no	—
128	D₂₆₈	—	4	1	no	—
129	C₄×D₆₇	—	4	2	no	—
130	C₄×D₆₇	—	4	2	no	—
131	Dic₁₃₄	—	2	2	no	—
132	Dic₁₃₄	—	4	1	no	—
133	C₈×C₆₇	—	4	2	no	—
134	C₈×C₆₇	C₂²⋊Q₈	1	1	yes	characteristic
135	C₈×C₆₇	C₂²⋊Q₈	1	1	yes	characteristic
136	C₈×C₆₇	C₂²×Q₈	1	1	yes	characteristic
137	C₈×C₆₇	C₂×#41	1	1	yes	characteristic
138	C₆₇⋊C₈	C₂×SD₁₆	1	2	yes	—
139	C₆₇⋊C₈	C₂×SD₁₆	1	2	yes	—
140	C₂²×C₆₇	—	2	2	no	—
141	C₂²×C₆₇	#476	1	1	yes	characteristic
142	C₂²×C₆₇	#476	1	1	yes	characteristic
143	C₂²×C₆₇	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
144	C₂×D₆₇	—	4	2	no	—
145	C₂×D₆₇	—	8	2	no	—
146	C₄×C₆₇	—	8	2	no	—
147	C₄×C₆₇	#476	1	1	yes	characteristic
148	C₄×C₆₇	#476	1	1	yes	characteristic
149	C₄×C₆₇	C₂×C₂²⋊Q₈	1	1	yes	characteristic
150	C₄×C₆₇	C₂²×SD₁₆	1	1	yes	characteristic
151	Dic₆₇	—	4	2	no	—
152	Dic₆₇	—	4	2	no	—
153	Dic₆₇	—	8	2	no	—
154	C₂×#2231	—	67	1	no	maximal, Sylow
155	C₂×C₆₇	—	2	1	no	—
156	C₂×C₆₇	—	2	1	no	—
157	C₂×C₆₇	#2231	1	2	yes	complemented
158	C₂×C₆₇	C₂×#476	1	1	yes	characteristic
159	D₆₇	—	8	2	no	—
160	C₂×#499	—	67	1	no	—
161	C₂×#443	—	67	1	no	—
162	#2231	—	67	8	no	—
163	C₂×#365	—	67	1	no	—
164	C₂×#365	D₆₇	1	1	yes	characteristic, complemented
165	C₂×#364	—	67	1	no	—
166	C₂×#360	—	67	1	no	—
167	C₂×#360	—	67	1	no	—
168	C₆₇	C₂×#2231	1	1	yes	characteristic, complemented, Sylow
169	C₂×#135	—	67	1	no	—
170	C₂×#131	—	134	1	no	—
171	C₂×#125	—	134	1	no	—
172	C₂×M₃₂	—	67	1	no	—
173	C₂×M₃₂	C₂×D₆₇	1	1	yes	characteristic
174	#499	—	67	4	no	—
175	#499	—	134	2	no	—
176	#443	—	67	4	no	—
177	#443	—	134	2	no	—
178	C₂×#108	C₂×D₆₇	1	1	yes	characteristic
179	C₂×#107	—	134	1	no	—
180	C₂×#106	—	67	1	no	—
181	C₂×#106	C₂×D₆₇	1	1	yes	characteristic
182	C₂×#104	—	134	1	no	—
183	C₂×C₄×C₈	—	67	1	no	—
184	#365	—	67	4	no	—
185	#365	C₂×D₆₇	1	4	yes	complemented
186	#364	—	67	4	no	—
187	#360	—	67	4	no	—
188	#360	—	67	4	no	—
189	C₂×#41	—	134	1	no	—
190	#135	—	67	2	no	—
191	#135	—	67	2	no	—
192	#135	—	134	1	no	—
193	#135	—	134	1	no	—
194	C₂×Q₁₆	—	67	1	no	—
195	C₂×SD₁₆	—	134	1	no	—
196	C₂×D₈	—	67	1	no	—
197	#131	—	268	2	no	—
198	#131	—	134	4	no	—
199	C₂×M₁₆	—	2	1	no	—
200	C₂×M₁₆	—	268	1	no	—
201	C₂²×C₈	—	268	1	no	—
202	C₂²×C₈	—	67	1	no	—
203	C₂²×C₈	C₂²×D₆₇	1	1	yes	characteristic
204	#125	—	268	2	no	—
205	#125	—	134	4	no	—
206	C₂×C₄⋊C₄	—	2	1	no	—
207	C₂×C₄⋊C₄	—	268	1	no	—
208	C₂×C₄⋊C₄	—	268	1	no	—
209	C₂×C₄⋊C₄	—	134	1	no	—
210	C₂×C₄²	—	67	1	no	—
211	M₃₂	—	2	1	no	—
212	M₃₂	—	67	2	no	—
213	M₃₂	—	134	1	no	—
214	M₃₂	C₂²×D₆₇	1	2	yes	—
215	C₂×C₁₆	—	2	1	no	—
216	C₂×C₁₆	—	134	1	no	—
217	#108	—	2	1	no	—
218	#108	C₂²×D₆₇	1	2	yes	—
219	#107	—	134	4	no	—
220	#106	—	2	1	no	—
221	#106	—	67	2	no	—
222	#106	—	134	1	no	—
223	#106	C₂²×D₆₇	1	2	yes	—
224	#104	—	134	4	no	—
225	C₄×C₈	—	67	2	no	—
226	C₄×C₈	—	67	2	no	—
227	#41	—	268	1	no	—
228	#41	—	268	1	no	—
229	#41	—	134	2	no	—
230	#41	—	134	2	no	—
231	C₂×Q₈	—	134	1	no	—
232	C₂×D₄	—	268	1	no	—
233	C₂×D₄	—	134	1	no	—
234	C₂²×C₄	—	4	1	no	—
235	C₂²×C₄	—	268	1	no	—
236	C₂²×C₄	—	268	1	no	—
237	C₂²×C₄	—	134	1	no	—
238	C₂²×C₄	D₄×D₆₇	1	1	yes	characteristic
239	Q₁₆	—	67	2	no	—
240	Q₁₆	—	134	1	no	—
241	SD₁₆	—	134	2	no	—
242	SD₁₆	—	134	2	no	—
243	D₈	—	67	2	no	—
244	D₈	—	134	1	no	—
245	M₁₆	—	2	2	no	—
246	M₁₆	—	4	1	no	—
247	M₁₆	—	268	4	no	—
248	C₂×C₈	—	4	1	no	—
249	C₂×C₈	—	268	4	no	—
250	C₂×C₈	—	67	1	no	—
251	C₂×C₈	—	67	1	no	—
252	C₂×C₈	—	67	2	no	—
253	C₂×C₈	—	67	2	no	—
254	C₂×C₈	D₄×D₆₇	1	1	yes	characteristic
255	C₂×C₈	Q₈×D₆₇	1	1	yes	characteristic
256	C₂×C₈	Q₈×D₆₇	1	1	yes	characteristic
257	C₂×C₈	#11761628	1	1	yes	characteristic
258	C₂×C₈	#11761628	1	1	yes	characteristic
259	C₂×C₈	C₂³×D₆₇	1	1	yes	characteristic
260	C₄⋊C₄	—	2	2	no	—
261	C₄⋊C₄	—	4	1	no	—
262	C₄⋊C₄	—	268	4	no	—
263	C₄⋊C₄	—	268	4	no	—
264	C₄⋊C₄	—	268	1	no	—
265	C₄⋊C₄	—	134	2	no	—
266	C₄²	—	67	2	no	—
267	C₄²	—	134	1	no	—
268	C₁₆	—	2	2	no	—
269	C₁₆	—	134	2	no	—
270	C₂³	—	268	1	no	—
271	C₂³	#14524070	1	1	yes	characteristic
272	Q₈	—	268	1	no	—
273	Q₈	—	134	2	no	—
274	D₄	—	268	2	no	—
275	D₄	—	268	2	no	—
276	D₄	—	268	1	no	—
277	D₄	—	134	2	no	—
278	C₂×C₄	—	4	2	no	—
279	C₂×C₄	—	8	2	no	—
280	C₂×C₄	—	268	2	no	—
281	C₂×C₄	—	268	1	no	—
282	C₂×C₄	—	268	2	no	—
283	C₂×C₄	—	268	2	no	—
284	C₂×C₄	—	268	2	no	—
285	C₂×C₄	—	536	2	no	—
286	C₂×C₄	—	134	2	no	—
287	C₂×C₄	#14524021	1	1	yes	characteristic
288	C₂×C₄	#14524021	1	1	yes	characteristic
289	C₂×C₄	SD₁₆×D₆₇	1	1	yes	characteristic
290	C₂×C₄	SD₁₆×D₆₇	1	1	yes	characteristic
291	C₂×C₄	#14524070	1	1	yes	characteristic
292	C₂×C₄	C₂×D₄×D₆₇	1	1	yes	characteristic
293	C₈	—	4	2	no	—
294	C₈	—	67	2	no	—
295	C₈	—	67	2	no	—
296	C₈	#14524021	1	1	yes	characteristic
297	C₈	#14524021	1	1	yes	characteristic
298	C₈	C₂×Q₈×D₆₇	1	1	yes	characteristic
299	C₈	C₂×#11761628	1	1	yes	characteristic
300	C₂²	—	2	2	no	—
301	C₂²	—	268	2	no	—
302	C₂²	—	536	2	no	—
303	C₂²	#16863685	1	1	yes	characteristic
304	C₂²	#16863685	1	1	yes	characteristic
305	C₂²	C₂×#14524070	1	1	yes	characteristic
306	C₄	—	8	2	no	—
307	C₄	—	268	2	no	—
308	C₄	—	268	2	no	—
309	C₄	—	536	2	no	—
310	C₄	#16863685	1	1	yes	characteristic
311	C₄	#16863685	1	1	yes	characteristic
312	C₄	C₂×#14524021	1	1	yes	characteristic
313	C₄	C₂×SD₁₆×D₆₇	1	1	yes	characteristic
314	C₂	—	2	1	no	—
315	C₂	—	2	1	no	—
316	C₂	—	536	2	no	—
317	C₂	#22562384	1	2	yes	direct product factor
318	C₂	C₂×#16863685	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	#16863685
commutator, cosocle	C₂×C₈×C₆₇	C₂⁴
Frattini	C₂×C₈	C₂³×D₆₇
Fitting	C₂×#365×C₆₇	C₂
socle	C₂²×C₆₇	#476
prime radical	C₂×#14524065	C₂²
hypercenter	C₂×#365	D₆₇
hypocenter	C₆₇	C₂×#2231
LCS	G ▷ C₂×C₈×C₆₇ ▷ C₄×C₆₇ ▷ C₂×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂²×C₈ ◁ C₂×#365

Complements

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

N	H ≅ G/N	type
1. C₂×#16863796	C₂	semidirect
3. #22562384	C₂	direct (N × H)
4. C₂×#16863220	C₂	semidirect
5. C₂×#16863216	C₂	semidirect
8. C₂×#365×C₆₇	C₂	semidirect
14. #16863796	C₂²	semidirect
21. #16863220	C₂²	semidirect
22. #16863216	C₂²	semidirect
28. #365×C₆₇	C₂²	semidirect
157. C₂×C₆₇	#2231	semidirect
164. C₂×#365	D₆₇	semidirect
168. C₆₇	C₂×#2231	semidirect
185. #365	C₂×D₆₇	semidirect
317. C₂	#22562384	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×#16863796	2
C₂×#16863512	2
#22562384	2
C₂×#16863220	2
C₂×#16863216	2
C₂×#16863207	2
C₂×#16863205	2
C₂×#365×C₆₇	2
C₂×#2231	67

Subgroups — C2×#22562384 #31010566

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#22562384 #31010566