Subgroups — C₂×#22562378 #31010559

Order 17152.

558 conjugacy classes of subgroups.

The subgroup lattice has 324 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#16863793	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×#16863513	C₂	1	1	yes	maximal, characteristic
3	#22562378	C₂	1	8	yes	maximal, direct product factor
4	C₂×#16863219	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂×#16863218	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×#16863216	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂×#16863206	C₂	1	1	yes	maximal, characteristic
8	C₂×#360×C₆₇	C₂	1	1	yes	maximal, characteristic
9	C₂×#14524050	—	2	1	no	—
10	C₂×#14524042	—	2	1	no	—
11	C₂×#14523983	C₂²	1	1	yes	characteristic
12	#16863793	—	2	2	no	—
13	#16863793	C₂²	1	4	yes	complemented
14	#16863513	—	2	2	no	—
15	#16863513	C₂²	1	4	yes	—
16	C₂×#14523979	C₂²	1	1	yes	characteristic
17	C₂×#14523975	C₂²	1	1	yes	characteristic
18	C₂×#14523974	—	2	1	no	—
19	C₂×#14523970	—	2	1	no	—
20	C₂×C₈×Dic₆₇	C₂²	1	1	yes	characteristic
21	#16863219	C₂²	1	4	yes	complemented
22	#16863218	C₂²	1	4	yes	complemented
23	#16863216	C₂²	1	4	yes	complemented
24	#16863206	C₂²	1	4	yes	—
25	C₂×#135×C₆₇	C₂²	1	1	yes	characteristic
26	C₂×M₃₂×C₆₇	C₂²	1	1	yes	characteristic
27	C₂×#106×C₆₇	C₂²	1	1	yes	characteristic
28	#360×C₆₇	C₂²	1	4	yes	—
29	C₂²×Dic₁₃₄	—	4	1	no	—
30	#14524050	—	2	4	no	—
31	#14524050	—	4	2	no	—
32	C₂×#11761612	—	2	1	no	—
33	C₂×#11761612	—	4	1	no	—
34	C₂²×C₆₇⋊C₈	—	4	1	no	—
35	C₂²×C₆₇⋊C₈	D₄	1	1	yes	characteristic
36	#14524042	—	2	4	no	—
37	#14524042	—	4	2	no	—
38	C₂×C₄⋊Dic₆₇	—	2	1	no	—
39	C₂×Dic₆₇⋊C₄	—	4	1	no	—
40	C₂×C₄×Dic₆₇	D₄	1	1	yes	characteristic
41	#14523983	—	2	1	no	—
42	#14523983	C₂³	1	2	yes	—
43	C₂×C₆₇⋊C₁₆	—	2	1	no	—
44	#14523979	—	2	1	no	—
45	#14523979	C₂³	1	2	yes	—
46	#14523975	—	2	1	no	—
47	#14523975	C₂³	1	2	yes	—
48	#14523974	—	2	4	no	—
49	#14523970	—	2	4	no	—
50	C₈×Dic₆₇	D₄	1	2	yes	—
51	C₈×Dic₆₇	C₂³	1	2	yes	—
52	C₂×#41×C₆₇	—	2	1	no	—
53	#135×C₆₇	—	2	1	no	—
54	#135×C₆₇	—	2	1	no	—
55	#135×C₆₇	Q₈	1	2	yes	complemented
56	#135×C₆₇	C₂³	1	2	yes	—
57	C₂×Q₁₆×C₆₇	Q₈	1	1	yes	characteristic, complemented
58	C₂×SD₁₆×C₆₇	—	2	1	no	—
59	C₂×D₈×C₆₇	Q₈	1	1	yes	characteristic, complemented
60	C₂²×C₈×C₆₇	C₂³	1	1	yes	characteristic
61	C₂×C₄⋊C₄×C₆₇	—	2	1	no	—
62	M₃₂×C₆₇	—	2	1	no	—
63	M₃₂×C₆₇	C₂³	1	2	yes	—
64	C₂×C₁₆×C₆₇	—	2	1	no	—
65	#106×C₆₇	—	2	1	no	—
66	#106×C₆₇	C₂³	1	2	yes	—
67	C₂×Dic₁₃₄	—	4	4	no	—
68	C₂×Dic₁₃₄	—	8	4	no	—
69	C₂²×Dic₆₇	—	2	1	no	—
70	C₂²×Dic₆₇	—	4	1	no	—
71	#11761612	—	2	2	no	—
72	#11761612	—	4	1	no	—
73	#11761612	—	4	4	no	—
74	C₂×C₆₇⋊C₈	—	4	1	no	—
75	C₂×C₆₇⋊C₈	—	4	4	no	—
76	C₂×C₆₇⋊C₈	SD₁₆	1	1	yes	characteristic
77	C₂×C₆₇⋊C₈	SD₁₆	1	1	yes	characteristic
78	C₂×C₆₇⋊C₈	SD₁₆	1	2	yes	—
79	C₂×C₆₇⋊C₈	C₂×D₄	1	2	yes	—
80	C₄⋊Dic₆₇	—	2	2	no	—
81	C₄⋊Dic₆₇	—	4	1	no	—
82	Dic₆₇⋊C₄	—	4	4	no	—
83	C₄×Dic₆₇	—	2	1	no	—
84	C₄×Dic₆₇	C₂×D₄	1	2	yes	—
85	C₆₇⋊C₁₆	—	2	2	no	—
86	#41×C₆₇	—	2	2	no	—
87	#41×C₆₇	—	2	2	no	—
88	#41×C₆₇	—	4	1	no	—
89	#41×C₆₇	—	4	1	no	—
90	C₂×Q₈×C₆₇	—	2	1	no	—
91	C₂×D₄×C₆₇	—	2	1	no	—
92	C₂×D₄×C₆₇	—	4	1	no	—
93	C₂²×C₄×C₆₇	—	4	1	no	—
94	C₂²×C₄×C₆₇	—	4	1	no	—
95	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
96	Q₁₆×C₆₇	—	2	1	no	—
97	Q₁₆×C₆₇	C₂×Q₈	1	2	yes	complemented
98	SD₁₆×C₆₇	—	2	2	no	—
99	SD₁₆×C₆₇	—	2	2	no	—
100	D₈×C₆₇	—	2	1	no	—
101	D₈×C₆₇	C₂×Q₈	1	2	yes	complemented
102	C₂×C₈×C₆₇	C₂×D₄	1	1	yes	characteristic
103	C₂×C₈×C₆₇	C₂×Q₈	1	1	yes	characteristic
104	C₂×C₈×C₆₇	C₂×Q₈	1	1	yes	characteristic
105	C₂×C₈×C₆₇	#41	1	1	yes	characteristic
106	C₂×C₈×C₆₇	#41	1	1	yes	characteristic
107	C₂×C₈×C₆₇	C₂⁴	1	1	yes	characteristic
108	C₄⋊C₄×C₆₇	—	2	2	no	—
109	C₄⋊C₄×C₆₇	—	4	1	no	—
110	C₁₆×C₆₇	—	2	2	no	—
111	C₂³×C₆₇	—	4	1	no	—
112	C₂³×C₆₇	AGL(1,ℤ/8)	1	1	yes	characteristic
113	Q₈×C₆₇	—	2	2	no	—
114	Q₈×C₆₇	—	4	1	no	—
115	D₄×C₆₇	—	2	2	no	—
116	D₄×C₆₇	—	4	2	no	—
117	D₄×C₆₇	—	4	2	no	—
118	D₄×C₆₇	—	4	1	no	—
119	C₂×C₄×C₆₇	—	4	1	no	—
120	C₂×C₄×C₆₇	—	4	2	no	—
121	C₂×C₄×C₆₇	—	4	2	no	—
122	C₂×C₄×C₆₇	—	4	2	no	—
123	C₂×C₄×C₆₇	—	8	2	no	—
124	C₂×C₄×C₆₇	C₂²⋊Q₈	1	1	yes	characteristic
125	C₂×C₄×C₆₇	C₂²⋊Q₈	1	1	yes	characteristic
126	C₂×C₄×C₆₇	C₂×SD₁₆	1	1	yes	characteristic
127	C₂×C₄×C₆₇	C₂×SD₁₆	1	1	yes	characteristic
128	C₂×C₄×C₆₇	AGL(1,ℤ/8)	1	1	yes	characteristic
129	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
130	C₂×Dic₆₇	—	2	4	no	—
131	C₂×Dic₆₇	—	2	2	no	—
132	C₂×Dic₆₇	—	4	2	no	—
133	C₂×Dic₆₇	—	8	2	no	—
134	Dic₁₃₄	—	8	8	no	—
135	C₈×C₆₇	C₂²⋊Q₈	1	1	yes	characteristic
136	C₈×C₆₇	C₂²⋊Q₈	1	1	yes	characteristic
137	C₈×C₆₇	C₂²×Q₈	1	1	yes	characteristic
138	C₈×C₆₇	C₂×#41	1	1	yes	characteristic
139	C₆₇⋊C₈	—	4	2	no	—
140	C₆₇⋊C₈	C₂×SD₁₆	1	2	yes	—
141	C₆₇⋊C₈	C₂×SD₁₆	1	2	yes	—
142	C₂²×C₆₇	—	2	2	no	—
143	C₂²×C₆₇	—	4	2	no	—
144	C₂²×C₆₇	—	8	2	no	—
145	C₂²×C₆₇	#476	1	1	yes	characteristic
146	C₂²×C₆₇	#476	1	1	yes	characteristic
147	C₂²×C₆₇	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
148	C₄×C₆₇	—	4	2	no	—
149	C₄×C₆₇	—	8	2	no	—
150	C₄×C₆₇	#476	1	1	yes	characteristic
151	C₄×C₆₇	#476	1	1	yes	characteristic
152	C₄×C₆₇	C₂×C₂²⋊Q₈	1	1	yes	characteristic
153	C₄×C₆₇	C₂²×SD₁₆	1	1	yes	characteristic
154	Dic₆₇	—	2	4	no	—
155	Dic₆₇	—	8	2	no	—
156	C₂×#2230	—	67	1	no	maximal, Sylow
157	C₂×C₆₇	—	2	1	no	—
158	C₂×C₆₇	—	2	1	no	—
159	C₂×C₆₇	—	8	2	no	—
160	C₂×C₆₇	#2230	1	2	yes	complemented
161	C₂×C₆₇	C₂×#476	1	1	yes	characteristic
162	C₂×#498	—	67	1	no	—
163	C₂×#443	—	67	1	no	—
164	#2230	—	67	8	no	—
165	C₂×#365	—	67	1	no	—
166	C₂×#365	—	67	1	no	—
167	C₂×#364	—	67	1	no	—
168	C₂×#360	—	67	1	no	—
169	C₂×#360	D₆₇	1	1	yes	characteristic
170	C₆₇	C₂×#2230	1	1	yes	characteristic, complemented, Sylow
171	C₂×#135	C₂×D₆₇	1	1	yes	characteristic
172	C₂×#131	—	134	1	no	—
173	C₂×C₄⋊Q₈	—	134	1	no	—
174	C₂×M₃₂	—	67	1	no	—
175	C₂×M₃₂	C₂×D₆₇	1	1	yes	characteristic
176	#498	—	67	4	no	—
177	#498	—	134	2	no	—
178	#443	—	67	4	no	—
179	#443	—	134	2	no	—
180	C₂×#108	—	67	1	no	—
181	C₂×#106	—	67	1	no	—
182	C₂×#106	—	134	1	no	—
183	C₂×#106	C₂×D₆₇	1	1	yes	characteristic
184	C₂×#104	—	134	1	no	—
185	C₂×C₄×C₈	—	67	1	no	—
186	#365	—	67	4	no	—
187	#365	—	67	4	no	—
188	#364	—	67	4	no	—
189	#360	—	67	4	no	—
190	#360	C₂×D₆₇	1	4	yes	—
191	C₂×#41	—	2	1	no	—
192	C₂²×Q₈	—	268	1	no	—
193	#135	—	2	1	no	—
194	#135	—	2	1	no	—
195	#135	Dic₁₃₄	1	2	yes	complemented
196	#135	C₂²×D₆₇	1	2	yes	—
197	C₂×Q₁₆	Dic₁₃₄	1	1	yes	characteristic, complemented
198	C₂×SD₁₆	—	2	1	no	—
199	C₂×D₈	Dic₁₃₄	1	1	yes	characteristic, complemented
200	#131	—	268	2	no	—
201	#131	—	134	4	no	—
202	C₂×M₁₆	—	268	1	no	—
203	C₂×M₁₆	—	134	1	no	—
204	C₂²×C₈	—	268	1	no	—
205	C₂²×C₈	—	67	1	no	—
206	C₂²×C₈	C₂²×D₆₇	1	1	yes	characteristic
207	C₄⋊Q₈	—	268	2	no	—
208	C₄⋊Q₈	—	134	4	no	—
209	C₂×C₄⋊C₄	—	2	1	no	—
210	C₂×C₄⋊C₄	—	268	1	no	—
211	C₂×C₄⋊C₄	—	134	1	no	—
212	C₂×C₄²	—	67	1	no	—
213	M₃₂	—	2	1	no	—
214	M₃₂	—	67	2	no	—
215	M₃₂	—	134	1	no	—
216	M₃₂	C₂²×D₆₇	1	2	yes	—
217	C₂×C₁₆	—	2	1	no	—
218	C₂×C₁₆	—	134	1	no	—
219	#108	—	67	2	no	—
220	#108	—	134	1	no	—
221	#106	—	2	1	no	—
222	#106	—	67	2	no	—
223	#106	—	134	4	no	—
224	#106	—	134	1	no	—
225	#106	C₂²×D₆₇	1	2	yes	—
226	#104	—	134	4	no	—
227	C₄×C₈	—	67	2	no	—
228	C₄×C₈	—	67	2	no	—
229	#41	—	2	2	no	—
230	#41	—	2	2	no	—
231	#41	—	4	1	no	—
232	#41	—	4	1	no	—
233	C₂×Q₈	—	2	1	no	—
234	C₂×Q₈	—	268	4	no	—
235	C₂×Q₈	—	536	4	no	—
236	C₂×D₄	—	2	1	no	—
237	C₂×D₄	—	4	1	no	—
238	C₂²×C₄	—	4	1	no	—
239	C₂²×C₄	—	4	1	no	—
240	C₂²×C₄	—	268	1	no	—
241	C₂²×C₄	—	134	1	no	—
242	C₂²×C₄	D₄×D₆₇	1	1	yes	characteristic
243	Q₁₆	—	2	1	no	—
244	Q₁₆	C₂×Dic₁₃₄	1	2	yes	complemented
245	SD₁₆	—	2	2	no	—
246	SD₁₆	—	2	2	no	—
247	D₈	—	2	1	no	—
248	D₈	C₂×Dic₁₃₄	1	2	yes	complemented
249	M₁₆	—	268	1	no	—
250	M₁₆	—	268	4	no	—
251	M₁₆	—	134	2	no	—
252	C₂×C₈	—	268	1	no	—
253	C₂×C₈	—	268	4	no	—
254	C₂×C₈	—	67	1	no	—
255	C₂×C₈	—	67	1	no	—
256	C₂×C₈	—	67	2	no	—
257	C₂×C₈	—	67	2	no	—
258	C₂×C₈	D₄×D₆₇	1	1	yes	characteristic
259	C₂×C₈	C₂×Dic₁₃₄	1	1	yes	characteristic
260	C₂×C₈	C₂×Dic₁₃₄	1	1	yes	characteristic
261	C₂×C₈	#11761627	1	1	yes	characteristic
262	C₂×C₈	#11761627	1	1	yes	characteristic
263	C₂×C₈	C₂³×D₆₇	1	1	yes	characteristic
264	C₄⋊C₄	—	2	2	no	—
265	C₄⋊C₄	—	4	1	no	—
266	C₄⋊C₄	—	268	4	no	—
267	C₄⋊C₄	—	268	1	no	—
268	C₄⋊C₄	—	134	2	no	—
269	C₄²	—	67	2	no	—
270	C₄²	—	134	1	no	—
271	C₁₆	—	2	2	no	—
272	C₁₆	—	134	2	no	—
273	C₂³	—	4	1	no	—
274	C₂³	#14524067	1	1	yes	characteristic
275	Q₈	—	2	2	no	—
276	Q₈	—	4	1	no	—
277	Q₈	—	536	8	no	—
278	D₄	—	2	2	no	—
279	D₄	—	4	2	no	—
280	D₄	—	4	2	no	—
281	D₄	—	4	1	no	—
282	C₂×C₄	—	4	1	no	—
283	C₂×C₄	—	4	2	no	—
284	C₂×C₄	—	4	2	no	—
285	C₂×C₄	—	4	2	no	—
286	C₂×C₄	—	8	2	no	—
287	C₂×C₄	—	268	2	no	—
288	C₂×C₄	—	536	2	no	—
289	C₂×C₄	—	134	4	no	—
290	C₂×C₄	—	134	2	no	—
291	C₂×C₄	#14524020	1	1	yes	characteristic
292	C₂×C₄	#14524020	1	1	yes	characteristic
293	C₂×C₄	SD₁₆×D₆₇	1	1	yes	characteristic
294	C₂×C₄	SD₁₆×D₆₇	1	1	yes	characteristic
295	C₂×C₄	#14524067	1	1	yes	characteristic
296	C₂×C₄	C₂×D₄×D₆₇	1	1	yes	characteristic
297	C₈	—	268	2	no	—
298	C₈	—	67	2	no	—
299	C₈	—	67	2	no	—
300	C₈	#14524020	1	1	yes	characteristic
301	C₈	#14524020	1	1	yes	characteristic
302	C₈	C₂²×Dic₁₃₄	1	1	yes	characteristic
303	C₈	C₂×#11761627	1	1	yes	characteristic
304	C₂²	—	2	2	no	—
305	C₂²	—	4	2	no	—
306	C₂²	—	8	2	no	—
307	C₂²	#16863684	1	1	yes	characteristic
308	C₂²	#16863684	1	1	yes	characteristic
309	C₂²	C₂×#14524067	1	1	yes	characteristic
310	C₄	—	4	2	no	—
311	C₄	—	8	2	no	—
312	C₄	—	536	2	no	—
313	C₄	—	134	4	no	—
314	C₄	#16863684	1	1	yes	characteristic
315	C₄	#16863684	1	1	yes	characteristic
316	C₄	C₂×#14524020	1	1	yes	characteristic
317	C₄	C₂×SD₁₆×D₆₇	1	1	yes	characteristic
318	C₂	—	2	1	no	—
319	C₂	—	2	1	no	—
320	C₂	—	8	2	no	—
321	C₂	#22562378	1	2	yes	direct product factor
322	C₂	C₂×#16863684	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	#16863684
commutator, cosocle	C₂×C₈×C₆₇	C₂⁴
Frattini	C₂×C₈	C₂³×D₆₇
Fitting	C₂×#360×C₆₇	C₂
socle	C₂²×C₆₇	#476
prime radical	C₂×#135×C₆₇	C₂²
hypercenter	C₂×#360	D₆₇
hypocenter	C₆₇	C₂×#2230
LCS	G ▷ C₂×C₈×C₆₇ ▷ C₄×C₆₇ ▷ C₂×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂²×C₈ ◁ C₂×#360

Complements

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

N	H ≅ G/N	type
1. C₂×#16863793	C₂	semidirect
3. #22562378	C₂	direct (N × H)
4. C₂×#16863219	C₂	semidirect
5. C₂×#16863218	C₂	semidirect
6. C₂×#16863216	C₂	semidirect
13. #16863793	C₂²	semidirect
21. #16863219	C₂²	semidirect
22. #16863218	C₂²	semidirect
23. #16863216	C₂²	semidirect
55. #135×C₆₇	Q₈	semidirect
57. C₂×Q₁₆×C₆₇	Q₈	semidirect
59. C₂×D₈×C₆₇	Q₈	semidirect
97. Q₁₆×C₆₇	C₂×Q₈	semidirect
101. D₈×C₆₇	C₂×Q₈	semidirect
160. C₂×C₆₇	#2230	semidirect
170. C₆₇	C₂×#2230	semidirect
195. #135	Dic₁₃₄	semidirect
197. C₂×Q₁₆	Dic₁₃₄	semidirect
199. C₂×D₈	Dic₁₃₄	semidirect
244. Q₁₆	C₂×Dic₁₃₄	semidirect
248. D₈	C₂×Dic₁₃₄	semidirect
321. C₂	#22562378	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×#16863793	2
C₂×#16863513	2
#22562378	2
C₂×#16863219	2
C₂×#16863218	2
C₂×#16863216	2
C₂×#16863206	2
C₂×#360×C₆₇	2
C₂×#2230	67

Subgroups — C2×#22562378 #31010559

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#22562378 #31010559