Subgroups — C₂×#21447988 #29834741

Order 15616.

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×#16840159	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×#16839879	C₂	1	1	yes	maximal, characteristic
3	#21447988	C₂	1	8	yes	maximal, direct product factor
4	C₂×#16839585	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂×#16839584	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×#16839582	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂×#16839572	C₂	1	1	yes	maximal, characteristic
8	C₂×C₆₁×#360	C₂	1	1	yes	maximal, characteristic
9	C₂×#14519784	—	2	1	no	—
10	C₂×#14519776	—	2	1	no	—
11	C₂×#14519717	C₂²	1	1	yes	characteristic
12	#16840159	—	2	2	no	—
13	#16840159	C₂²	1	4	yes	complemented
14	#16839879	—	2	2	no	—
15	#16839879	C₂²	1	4	yes	—
16	C₂×#14519713	C₂²	1	1	yes	characteristic
17	C₂×#14519709	C₂²	1	1	yes	characteristic
18	C₂×#14519708	—	2	1	no	—
19	C₂×#14519704	—	2	1	no	—
20	C₂×C₈×Dic₆₁	C₂²	1	1	yes	characteristic
21	#16839585	C₂²	1	4	yes	complemented
22	#16839584	C₂²	1	4	yes	complemented
23	#16839582	C₂²	1	4	yes	complemented
24	#16839572	C₂²	1	4	yes	—
25	C₂×#135×C₆₁	C₂²	1	1	yes	characteristic
26	C₂×M₃₂×C₆₁	C₂²	1	1	yes	characteristic
27	C₂×#106×C₆₁	C₂²	1	1	yes	characteristic
28	C₆₁×#360	C₂²	1	4	yes	—
29	C₂²×Dic₁₂₂	—	4	1	no	—
30	#14519784	—	2	4	no	—
31	#14519784	—	4	2	no	—
32	C₂×#11758194	—	2	1	no	—
33	C₂×#11758194	—	4	1	no	—
34	C₂²×#92418	—	4	1	no	—
35	C₂²×#92418	D₄	1	1	yes	characteristic
36	#14519776	—	2	4	no	—
37	#14519776	—	4	2	no	—
38	C₂×C₄⋊Dic₆₁	—	2	1	no	—
39	C₂×#11758196	—	4	1	no	—
40	C₂×C₄×Dic₆₁	D₄	1	1	yes	characteristic
41	#14519717	—	2	1	no	—
42	#14519717	C₂³	1	2	yes	—
43	C₂×#11758185	—	2	1	no	—
44	#14519713	—	2	1	no	—
45	#14519713	C₂³	1	2	yes	—
46	#14519709	—	2	1	no	—
47	#14519709	C₂³	1	2	yes	—
48	#14519708	—	2	4	no	—
49	#14519704	—	2	4	no	—
50	C₈×Dic₆₁	D₄	1	2	yes	—
51	C₈×Dic₆₁	C₂³	1	2	yes	—
52	C₂×#41×C₆₁	—	2	1	no	—
53	#135×C₆₁	—	2	1	no	—
54	#135×C₆₁	—	2	1	no	—
55	#135×C₆₁	Q₈	1	2	yes	complemented
56	#135×C₆₁	C₂³	1	2	yes	—
57	C₂×Q₁₆×C₆₁	Q₈	1	1	yes	characteristic, complemented
58	C₂×SD₁₆×C₆₁	—	2	1	no	—
59	C₂×D₈×C₆₁	Q₈	1	1	yes	characteristic, complemented
60	C₂²×C₈×C₆₁	C₂³	1	1	yes	characteristic
61	C₂×C₄⋊C₄×C₆₁	—	2	1	no	—
62	M₃₂×C₆₁	—	2	1	no	—
63	M₃₂×C₆₁	C₂³	1	2	yes	—
64	C₂×C₁₆×C₆₁	—	2	1	no	—
65	#106×C₆₁	—	2	1	no	—
66	#106×C₆₁	C₂³	1	2	yes	—
67	#41×C₆₁	—	2	2	no	—
68	#41×C₆₁	—	2	2	no	—
69	#41×C₆₁	—	4	1	no	—
70	#41×C₆₁	—	4	1	no	—
71	C₂×Q₈×C₆₁	—	2	1	no	—
72	C₂×D₄×C₆₁	—	2	1	no	—
73	C₂×D₄×C₆₁	—	4	1	no	—
74	C₂²×C₄×C₆₁	—	4	1	no	—
75	C₂²×C₄×C₆₁	—	4	1	no	—
76	C₂²×C₄×C₆₁	C₂×D₄	1	1	yes	characteristic
77	C₂²×Dic₆₁	—	2	1	no	—
78	C₂²×Dic₆₁	—	4	1	no	—
79	C₂×Dic₁₂₂	—	4	4	no	—
80	C₂×Dic₁₂₂	—	8	4	no	—
81	Q₁₆×C₆₁	—	2	1	no	—
82	Q₁₆×C₆₁	C₂×Q₈	1	2	yes	complemented
83	SD₁₆×C₆₁	—	2	2	no	—
84	SD₁₆×C₆₁	—	2	2	no	—
85	D₈×C₆₁	—	2	1	no	—
86	D₈×C₆₁	C₂×Q₈	1	2	yes	complemented
87	C₂×C₈×C₆₁	C₂×D₄	1	1	yes	characteristic
88	C₂×C₈×C₆₁	C₂×Q₈	1	1	yes	characteristic
89	C₂×C₈×C₆₁	C₂×Q₈	1	1	yes	characteristic
90	C₂×C₈×C₆₁	#41	1	1	yes	characteristic
91	C₂×C₈×C₆₁	#41	1	1	yes	characteristic
92	C₂×C₈×C₆₁	C₂⁴	1	1	yes	characteristic
93	C₄⋊C₄×C₆₁	—	2	2	no	—
94	C₄⋊C₄×C₆₁	—	4	1	no	—
95	C₄⋊Dic₆₁	—	2	2	no	—
96	C₄⋊Dic₆₁	—	4	1	no	—
97	#11758196	—	4	4	no	—
98	C₄×Dic₆₁	—	2	1	no	—
99	C₄×Dic₆₁	C₂×D₄	1	2	yes	—
100	#11758194	—	2	2	no	—
101	#11758194	—	4	1	no	—
102	#11758194	—	4	4	no	—
103	C₂×#92418	—	4	1	no	—
104	C₂×#92418	—	4	4	no	—
105	C₂×#92418	SD₁₆	1	1	yes	characteristic
106	C₂×#92418	SD₁₆	1	1	yes	characteristic
107	C₂×#92418	SD₁₆	1	2	yes	—
108	C₂×#92418	C₂×D₄	1	2	yes	—
109	C₁₆×C₆₁	—	2	2	no	—
110	#11758185	—	2	2	no	—
111	C₂³×C₆₁	—	4	1	no	—
112	C₂³×C₆₁	AGL(1,ℤ/8)	1	1	yes	characteristic
113	Q₈×C₆₁	—	2	2	no	—
114	Q₈×C₆₁	—	4	1	no	—
115	D₄×C₆₁	—	2	2	no	—
116	D₄×C₆₁	—	4	2	no	—
117	D₄×C₆₁	—	4	2	no	—
118	D₄×C₆₁	—	4	1	no	—
119	C₂×C₄×C₆₁	—	4	1	no	—
120	C₂×C₄×C₆₁	—	4	2	no	—
121	C₂×C₄×C₆₁	—	4	2	no	—
122	C₂×C₄×C₆₁	—	4	2	no	—
123	C₂×C₄×C₆₁	—	8	2	no	—
124	C₂×C₄×C₆₁	C₂²⋊Q₈	1	1	yes	characteristic
125	C₂×C₄×C₆₁	C₂²⋊Q₈	1	1	yes	characteristic
126	C₂×C₄×C₆₁	C₂×SD₁₆	1	1	yes	characteristic
127	C₂×C₄×C₆₁	C₂×SD₁₆	1	1	yes	characteristic
128	C₂×C₄×C₆₁	AGL(1,ℤ/8)	1	1	yes	characteristic
129	C₂×C₄×C₆₁	C₂²×D₄	1	1	yes	characteristic
130	C₂×Dic₆₁	—	2	4	no	—
131	C₂×Dic₆₁	—	2	2	no	—
132	C₂×Dic₆₁	—	4	2	no	—
133	C₂×Dic₆₁	—	8	2	no	—
134	Dic₁₂₂	—	8	8	no	—
135	C₈×C₆₁	C₂²⋊Q₈	1	1	yes	characteristic
136	C₈×C₆₁	C₂²⋊Q₈	1	1	yes	characteristic
137	C₈×C₆₁	C₂²×Q₈	1	1	yes	characteristic
138	C₈×C₆₁	C₂×#41	1	1	yes	characteristic
139	#92418	—	4	2	no	—
140	#92418	C₂×SD₁₆	1	2	yes	—
141	#92418	C₂×SD₁₆	1	2	yes	—
142	C₂×#2230	—	61	1	no	maximal, Sylow
143	C₂²×C₆₁	—	2	2	no	—
144	C₂²×C₆₁	—	4	2	no	—
145	C₂²×C₆₁	—	8	2	no	—
146	C₂²×C₆₁	#476	1	1	yes	characteristic
147	C₂²×C₆₁	#476	1	1	yes	characteristic
148	C₂²×C₆₁	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
149	C₄×C₆₁	—	4	2	no	—
150	C₄×C₆₁	—	8	2	no	—
151	C₄×C₆₁	#476	1	1	yes	characteristic
152	C₄×C₆₁	#476	1	1	yes	characteristic
153	C₄×C₆₁	C₂×C₂²⋊Q₈	1	1	yes	characteristic
154	C₄×C₆₁	C₂²×SD₁₆	1	1	yes	characteristic
155	Dic₆₁	—	2	4	no	—
156	Dic₆₁	—	8	2	no	—
157	C₂×#498	—	61	1	no	—
158	C₂×#443	—	61	1	no	—
159	#2230	—	61	8	no	—
160	C₂×#365	—	61	1	no	—
161	C₂×#365	—	61	1	no	—
162	C₂×#364	—	61	1	no	—
163	C₂×#360	—	61	1	no	—
164	C₂×#360	D₆₁	1	1	yes	characteristic
165	C₂×C₆₁	—	2	1	no	—
166	C₂×C₆₁	—	2	1	no	—
167	C₂×C₆₁	—	8	2	no	—
168	C₂×C₆₁	#2230	1	2	yes	complemented
169	C₂×C₆₁	C₂×#476	1	1	yes	characteristic
170	C₂×#135	C₂×D₆₁	1	1	yes	characteristic
171	C₂×#131	—	122	1	no	—
172	C₂×C₄⋊Q₈	—	122	1	no	—
173	C₂×M₃₂	—	61	1	no	—
174	C₂×M₃₂	C₂×D₆₁	1	1	yes	characteristic
175	#498	—	61	4	no	—
176	#498	—	122	2	no	—
177	#443	—	61	4	no	—
178	#443	—	122	2	no	—
179	C₂×#108	—	61	1	no	—
180	C₂×#106	—	61	1	no	—
181	C₂×#106	—	122	1	no	—
182	C₂×#106	C₂×D₆₁	1	1	yes	characteristic
183	C₂×#104	—	122	1	no	—
184	C₂×C₄×C₈	—	61	1	no	—
185	#365	—	61	4	no	—
186	#365	—	61	4	no	—
187	#364	—	61	4	no	—
188	#360	—	61	4	no	—
189	#360	C₂×D₆₁	1	4	yes	—
190	C₆₁	C₂×#2230	1	1	yes	characteristic, complemented, Sylow
191	C₂×#41	—	2	1	no	—
192	C₂²×Q₈	—	244	1	no	—
193	#135	—	2	1	no	—
194	#135	—	2	1	no	—
195	#135	Dic₁₂₂	1	2	yes	complemented
196	#135	C₂²×D₆₁	1	2	yes	—
197	C₂×Q₁₆	Dic₁₂₂	1	1	yes	characteristic, complemented
198	C₂×SD₁₆	—	2	1	no	—
199	C₂×D₈	Dic₁₂₂	1	1	yes	characteristic, complemented
200	#131	—	122	4	no	—
201	#131	—	244	2	no	—
202	C₂×M₁₆	—	122	1	no	—
203	C₂×M₁₆	—	244	1	no	—
204	C₂²×C₈	—	61	1	no	—
205	C₂²×C₈	—	244	1	no	—
206	C₂²×C₈	C₂²×D₆₁	1	1	yes	characteristic
207	C₄⋊Q₈	—	122	4	no	—
208	C₄⋊Q₈	—	244	2	no	—
209	C₂×C₄⋊C₄	—	2	1	no	—
210	C₂×C₄⋊C₄	—	122	1	no	—
211	C₂×C₄⋊C₄	—	244	1	no	—
212	C₂×C₄²	—	61	1	no	—
213	M₃₂	—	2	1	no	—
214	M₃₂	—	61	2	no	—
215	M₃₂	—	122	1	no	—
216	M₃₂	C₂²×D₆₁	1	2	yes	—
217	C₂×C₁₆	—	2	1	no	—
218	C₂×C₁₆	—	122	1	no	—
219	#108	—	61	2	no	—
220	#108	—	122	1	no	—
221	#106	—	2	1	no	—
222	#106	—	61	2	no	—
223	#106	—	122	4	no	—
224	#106	—	122	1	no	—
225	#106	C₂²×D₆₁	1	2	yes	—
226	#104	—	122	4	no	—
227	C₄×C₈	—	61	2	no	—
228	C₄×C₈	—	61	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	—	488	4	no	—
235	C₂×Q₈	—	244	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	—	122	1	no	—
241	C₂²×C₄	—	244	1	no	—
242	C₂²×C₄	D₄×D₆₁	1	1	yes	characteristic
243	Q₁₆	—	2	1	no	—
244	Q₁₆	C₂×Dic₁₂₂	1	2	yes	complemented
245	SD₁₆	—	2	2	no	—
246	SD₁₆	—	2	2	no	—
247	D₈	—	2	1	no	—
248	D₈	C₂×Dic₁₂₂	1	2	yes	complemented
249	M₁₆	—	122	2	no	—
250	M₁₆	—	244	4	no	—
251	M₁₆	—	244	1	no	—
252	C₂×C₈	—	61	2	no	—
253	C₂×C₈	—	61	1	no	—
254	C₂×C₈	—	61	1	no	—
255	C₂×C₈	—	61	2	no	—
256	C₂×C₈	—	244	4	no	—
257	C₂×C₈	—	244	1	no	—
258	C₂×C₈	C₂×Dic₁₂₂	1	1	yes	characteristic
259	C₂×C₈	C₂×Dic₁₂₂	1	1	yes	characteristic
260	C₂×C₈	D₄×D₆₁	1	1	yes	characteristic
261	C₂×C₈	#11758224	1	1	yes	characteristic
262	C₂×C₈	#11758224	1	1	yes	characteristic
263	C₂×C₈	C₂³×D₆₁	1	1	yes	characteristic
264	C₄⋊C₄	—	2	2	no	—
265	C₄⋊C₄	—	4	1	no	—
266	C₄⋊C₄	—	122	2	no	—
267	C₄⋊C₄	—	244	4	no	—
268	C₄⋊C₄	—	244	1	no	—
269	C₄²	—	61	2	no	—
270	C₄²	—	122	1	no	—
271	C₁₆	—	2	2	no	—
272	C₁₆	—	122	2	no	—
273	C₂³	—	4	1	no	—
274	C₂³	#14519801	1	1	yes	characteristic
275	Q₈	—	2	2	no	—
276	Q₈	—	4	1	no	—
277	Q₈	—	488	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	—	122	4	no	—
288	C₂×C₄	—	122	2	no	—
289	C₂×C₄	—	488	2	no	—
290	C₂×C₄	—	244	2	no	—
291	C₂×C₄	#14519754	1	1	yes	characteristic
292	C₂×C₄	#14519754	1	1	yes	characteristic
293	C₂×C₄	SD₁₆×D₆₁	1	1	yes	characteristic
294	C₂×C₄	SD₁₆×D₆₁	1	1	yes	characteristic
295	C₂×C₄	#14519801	1	1	yes	characteristic
296	C₂×C₄	C₂×D₄×D₆₁	1	1	yes	characteristic
297	C₈	—	61	2	no	—
298	C₈	—	61	2	no	—
299	C₈	—	244	2	no	—
300	C₈	#14519754	1	1	yes	characteristic
301	C₈	#14519754	1	1	yes	characteristic
302	C₈	C₂²×Dic₁₂₂	1	1	yes	characteristic
303	C₈	C₂×#11758224	1	1	yes	characteristic
304	C₂²	—	2	2	no	—
305	C₂²	—	4	2	no	—
306	C₂²	—	8	2	no	—
307	C₂²	#16840050	1	1	yes	characteristic
308	C₂²	#16840050	1	1	yes	characteristic
309	C₂²	C₂×#14519801	1	1	yes	characteristic
310	C₄	—	4	2	no	—
311	C₄	—	8	2	no	—
312	C₄	—	122	4	no	—
313	C₄	—	488	2	no	—
314	C₄	#16840050	1	1	yes	characteristic
315	C₄	#16840050	1	1	yes	characteristic
316	C₄	C₂×#14519754	1	1	yes	characteristic
317	C₄	C₂×SD₁₆×D₆₁	1	1	yes	characteristic
318	C₂	—	2	1	no	—
319	C₂	—	2	1	no	—
320	C₂	—	8	2	no	—
321	C₂	#21447988	1	2	yes	direct product factor
322	C₂	C₂×#16840050	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	#16840050
commutator, cosocle	C₂×C₈×C₆₁	C₂⁴
Frattini	C₂×C₈	C₂³×D₆₁
Fitting	C₂×C₆₁×#360	C₂
socle	C₂²×C₆₁	#476
prime radical	C₂×#135×C₆₁	C₂²
hypercenter	C₂×#360	D₆₁
hypocenter	C₆₁	C₂×#2230
LCS	G ▷ C₂×C₈×C₆₁ ▷ C₄×C₆₁ ▷ C₂×C₆₁ ▷ C₆₁
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×#16840159	C₂	semidirect
3. #21447988	C₂	direct (N × H)
4. C₂×#16839585	C₂	semidirect
5. C₂×#16839584	C₂	semidirect
6. C₂×#16839582	C₂	semidirect
13. #16840159	C₂²	semidirect
21. #16839585	C₂²	semidirect
22. #16839584	C₂²	semidirect
23. #16839582	C₂²	semidirect
55. #135×C₆₁	Q₈	semidirect
57. C₂×Q₁₆×C₆₁	Q₈	semidirect
59. C₂×D₈×C₆₁	Q₈	semidirect
82. Q₁₆×C₆₁	C₂×Q₈	semidirect
86. D₈×C₆₁	C₂×Q₈	semidirect
168. C₂×C₆₁	#2230	semidirect
190. C₆₁	C₂×#2230	semidirect
195. #135	Dic₁₂₂	semidirect
197. C₂×Q₁₆	Dic₁₂₂	semidirect
199. C₂×D₈	Dic₁₂₂	semidirect
244. Q₁₆	C₂×Dic₁₂₂	semidirect
248. D₈	C₂×Dic₁₂₂	semidirect
321. C₂	#21447988	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×#16840159	2
C₂×#16839879	2
#21447988	2
C₂×#16839585	2
C₂×#16839584	2
C₂×#16839582	2
C₂×#16839572	2
C₂×C₆₁×#360	2
C₂×#2230	61

Subgroups — C2×#21447988 #29834741

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#21447988 #29834741