Subgroups — C₂×#21447895 #29834393

Order 15616.

790 conjugacy classes of subgroups.

The subgroup lattice has 346 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#16840197	C₂	1	2	yes	maximal, complemented
2	C₂×#16840139	C₂	1	1	yes	maximal, characteristic, complemented
3	#21447895	C₂	1	8	yes	maximal, direct product factor
4	C₂×#16839583	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂×#16839574	C₂	1	2	yes	maximal, complemented
6	C₂×C₆₁×#443	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂²×D₄₈₈	—	2	1	no	—
8	C₂²×D₄₈₈	C₂²	1	2	yes	—
9	C₂×#14519774	—	2	1	no	—
10	#16840197	—	2	4	no	—
11	#16840197	C₂²	1	8	yes	complemented
12	C₂×#14519720	—	2	2	no	—
13	C₂×#14519718	—	2	2	no	—
14	C₂×#14519717	C₂²	1	2	yes	—
15	#16840139	—	2	2	no	—
16	#16840139	C₂²	1	4	yes	complemented
17	#16839583	C₂²	1	4	yes	complemented
18	#16839574	C₂²	1	8	yes	complemented
19	C₂×#135×C₆₁	C₂²	1	1	yes	characteristic
20	C₂×#131×C₆₁	—	2	1	no	—
21	C₆₁×#443	—	2	2	no	—
22	C₆₁×#443	C₂²	1	4	yes	complemented
23	C₂×#108×C₆₁	C₂²	1	1	yes	characteristic
24	C₂×#104×C₆₁	—	2	1	no	—
25	C₂×C₄×C₈×C₆₁	C₂²	1	1	yes	characteristic
26	C₂²×D₂₄₄	—	2	2	no	—
27	C₂²×D₂₄₄	—	4	2	no	—
28	C₂×D₄₈₈	—	2	2	no	—
29	C₂×D₄₈₈	—	2	2	no	—
30	C₂×D₄₈₈	—	2	2	no	—
31	C₂×D₄₈₈	—	2	2	no	—
32	C₂×D₄₈₈	—	2	4	no	—
33	C₂×D₄₈₈	—	2	2	no	—
34	C₂×D₄₈₈	—	4	1	no	—
35	C₂×D₄₈₈	—	4	1	no	—
36	C₂×D₄₈₈	—	4	2	no	—
37	C₂×D₄₈₈	C₂³	1	4	yes	—
38	#14519774	—	2	4	no	—
39	#14519774	—	4	2	no	—
40	#14519720	—	2	8	no	—
41	#14519718	—	2	8	no	—
42	#14519717	—	2	2	no	—
43	#14519717	C₂³	1	4	yes	—
44	C₂×#11758185	—	2	2	no	—
45	C₂×#41×C₆₁	—	2	1	no	—
46	#135×C₆₁	—	2	1	no	—
47	#135×C₆₁	—	2	1	no	—
48	#135×C₆₁	D₄	1	2	yes	complemented
49	#135×C₆₁	C₂³	1	2	yes	—
50	C₂×Q₁₆×C₆₁	D₄	1	1	yes	characteristic, complemented
51	C₂×SD₁₆×C₆₁	—	2	1	no	—
52	C₂×D₈×C₆₁	D₄	1	1	yes	characteristic, complemented
53	#131×C₆₁	—	2	4	no	—
54	#131×C₆₁	—	4	2	no	—
55	C₂×M₁₆×C₆₁	—	2	1	no	—
56	C₂×M₁₆×C₆₁	—	4	1	no	—
57	C₂²×C₈×C₆₁	—	4	1	no	—
58	C₂²×C₈×C₆₁	D₄	1	1	yes	characteristic
59	C₂²×C₈×C₆₁	C₂³	1	1	yes	characteristic
60	C₂×C₄²×C₆₁	D₄	1	1	yes	characteristic
61	#108×C₆₁	—	2	1	no	—
62	#108×C₆₁	C₂³	1	2	yes	—
63	#104×C₆₁	—	2	4	no	—
64	C₄×C₈×C₆₁	D₄	1	2	yes	—
65	C₄×C₈×C₆₁	C₂³	1	2	yes	—
66	C₂³×D₆₁	—	4	2	no	—
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	—	2	1	no	—
75	C₂²×C₄×C₆₁	—	4	1	no	—
76	C₂²×C₄×C₆₁	C₂×D₄	1	1	yes	characteristic
77	C₂×D₂₄₄	—	2	4	no	—
78	C₂×D₂₄₄	—	4	2	no	—
79	C₂×D₂₄₄	—	4	2	no	—
80	C₂×D₂₄₄	—	4	2	no	—
81	C₂×D₂₄₄	—	4	2	no	—
82	C₂×D₂₄₄	—	4	8	no	—
83	C₂×D₂₄₄	—	4	2	no	—
84	C₂×D₂₄₄	—	8	8	no	—
85	Q₁₆×C₆₁	—	2	1	no	—
86	Q₁₆×C₆₁	C₂×D₄	1	2	yes	complemented
87	SD₁₆×C₆₁	—	2	2	no	—
88	SD₁₆×C₆₁	—	2	2	no	—
89	D₈×C₆₁	—	2	1	no	—
90	D₈×C₆₁	C₂×D₄	1	2	yes	complemented
91	M₁₆×C₆₁	—	2	2	no	—
92	M₁₆×C₆₁	—	4	1	no	—
93	M₁₆×C₆₁	—	4	4	no	—
94	C₂×C₈×C₆₁	—	4	1	no	—
95	C₂×C₈×C₆₁	—	4	4	no	—
96	C₂×C₈×C₆₁	D₈	1	1	yes	characteristic
97	C₂×C₈×C₆₁	D₈	1	1	yes	characteristic
98	C₂×C₈×C₆₁	D₈	1	2	yes	—
99	C₂×C₈×C₆₁	C₂×D₄	1	1	yes	characteristic
100	C₂×C₈×C₆₁	C₂×D₄	1	1	yes	characteristic
101	C₂×C₈×C₆₁	C₂×D₄	1	1	yes	characteristic
102	C₂×C₈×C₆₁	C₂×D₄	1	2	yes	—
103	C₂×C₈×C₆₁	#41	1	1	yes	characteristic
104	C₂×C₈×C₆₁	#41	1	1	yes	characteristic
105	C₂×C₈×C₆₁	C₂⁴	1	1	yes	characteristic
106	C₄²×C₆₁	—	2	1	no	—
107	C₄²×C₆₁	C₂×D₄	1	2	yes	—
108	D₄₈₈	—	2	4	no	—
109	D₄₈₈	—	2	4	no	—
110	D₄₈₈	—	4	2	no	—
111	D₄₈₈	—	4	2	no	—
112	D₄₈₈	—	4	4	no	—
113	D₄₈₈	—	4	4	no	—
114	#11758185	—	2	4	no	—
115	C₂³×C₆₁	—	4	1	no	—
116	C₂³×C₆₁	AGL(1,ℤ/8)	1	1	yes	characteristic
117	C₂²×D₆₁	—	4	4	no	—
118	C₂²×D₆₁	—	8	8	no	—
119	C₂²×D₆₁	—	8	2	no	—
120	C₂²×D₆₁	—	8	2	no	—
121	Q₈×C₆₁	—	2	2	no	—
122	Q₈×C₆₁	—	4	1	no	—
123	D₄×C₆₁	—	2	2	no	—
124	D₄×C₆₁	—	4	2	no	—
125	D₄×C₆₁	—	4	2	no	—
126	D₄×C₆₁	—	4	1	no	—
127	C₂×C₄×C₆₁	—	2	4	no	—
128	C₂×C₄×C₆₁	—	2	2	no	—
129	C₂×C₄×C₆₁	—	4	1	no	—
130	C₂×C₄×C₆₁	—	4	2	no	—
131	C₂×C₄×C₆₁	—	4	2	no	—
132	C₂×C₄×C₆₁	#121	1	1	yes	characteristic
133	C₂×C₄×C₆₁	#121	1	1	yes	characteristic
134	C₂×C₄×C₆₁	C₂×D₈	1	1	yes	characteristic
135	C₂×C₄×C₆₁	C₂×D₈	1	1	yes	characteristic
136	C₂×C₄×C₆₁	AGL(1,ℤ/8)	1	1	yes	characteristic
137	C₂×C₄×C₆₁	C₂²×D₄	1	1	yes	characteristic
138	D₂₄₄	—	4	4	no	—
139	D₂₄₄	—	4	4	no	—
140	D₂₄₄	—	8	16	no	—
141	D₂₄₄	—	8	2	no	—
142	D₂₄₄	—	8	2	no	—
143	C₈×C₆₁	—	4	2	no	—
144	C₈×C₆₁	#121	1	1	yes	characteristic
145	C₈×C₆₁	#121	1	1	yes	characteristic
146	C₈×C₆₁	C₂×D₈	1	2	yes	—
147	C₈×C₆₁	C₂×D₈	1	2	yes	—
148	C₈×C₆₁	C₂²×D₄	1	1	yes	characteristic
149	C₈×C₆₁	C₂×#41	1	1	yes	characteristic
150	C₂×#2213	—	61	1	no	maximal, Sylow
151	C₂²×C₆₁	—	2	2	no	—
152	C₂²×C₆₁	—	4	2	no	—
153	C₂²×C₆₁	—	8	2	no	—
154	C₂²×C₆₁	#459	1	1	yes	characteristic
155	C₂²×C₆₁	#459	1	1	yes	characteristic
156	C₂²×C₆₁	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
157	C₂×D₆₁	—	8	8	no	—
158	C₂×D₆₁	—	8	4	no	—
159	C₂×D₆₁	—	16	4	no	—
160	C₂×D₆₁	—	16	4	no	—
161	C₄×C₆₁	—	2	4	no	—
162	C₄×C₆₁	—	4	2	no	—
163	C₄×C₆₁	#459	1	1	yes	characteristic
164	C₄×C₆₁	#459	1	1	yes	characteristic
165	C₄×C₆₁	C₂×#121	1	1	yes	characteristic
166	C₄×C₆₁	C₂²×D₈	1	1	yes	characteristic
167	C₂×#509	—	61	2	no	—
168	C₂×#493	—	61	1	no	—
169	C₂×#443	D₆₁	1	1	yes	characteristic, complemented
170	#2213	—	61	8	no	—
171	C₂×#364	—	61	1	no	—
172	C₂×#361	—	61	2	no	—
173	C₂×C₆₁	—	2	1	no	—
174	C₂×C₆₁	—	2	1	no	—
175	C₂×C₆₁	—	8	2	no	—
176	C₂×C₆₁	#2213	1	2	yes	complemented
177	C₂×C₆₁	C₂×#459	1	1	yes	characteristic
178	D₆₁	—	16	4	no	—
179	C₂×#135	C₂×D₆₁	1	1	yes	characteristic
180	C₂²×D₈	—	61	2	no	—
181	C₂²×D₈	—	122	1	no	—
182	C₂×#131	—	2	1	no	—
183	C₂×#127	—	122	1	no	—
184	#509	—	61	8	no	—
185	#509	—	122	4	no	—
186	C₂×SD₃₂	—	122	2	no	—
187	C₂×D₁₆	—	122	2	no	—
188	C₂×M₃₂	—	61	2	no	—
189	#493	—	61	4	no	—
190	#493	—	122	2	no	—
191	#443	—	2	2	no	—
192	#443	C₂×D₆₁	1	4	yes	complemented
193	C₂×#108	C₂×D₆₁	1	1	yes	characteristic
194	C₂×#104	—	2	1	no	—
195	C₂×C₄×C₈	C₂×D₆₁	1	1	yes	characteristic
196	#364	—	61	4	no	—
197	#361	—	61	8	no	—
198	C₆₁	C₂×#2213	1	1	yes	characteristic, complemented, Sylow
199	C₂×#41	—	2	1	no	—
200	C₂²×D₄	—	122	2	no	—
201	C₂²×D₄	—	244	2	no	—
202	#135	—	2	1	no	—
203	#135	—	2	1	no	—
204	#135	D₂₄₄	1	2	yes	complemented
205	#135	C₂²×D₆₁	1	2	yes	—
206	C₂×Q₁₆	D₂₄₄	1	1	yes	characteristic, complemented
207	C₂×SD₁₆	—	2	1	no	—
208	C₂×D₈	—	61	4	no	—
209	C₂×D₈	—	122	2	no	—
210	C₂×D₈	—	122	2	no	—
211	C₂×D₈	—	122	2	no	—
212	C₂×D₈	—	122	2	no	—
213	C₂×D₈	—	122	2	no	—
214	C₂×D₈	—	122	4	no	—
215	C₂×D₈	—	244	1	no	—
216	C₂×D₈	—	244	1	no	—
217	C₂×D₈	—	244	2	no	—
218	C₂×D₈	D₂₄₄	1	1	yes	characteristic, complemented
219	#131	—	2	4	no	—
220	#131	—	4	2	no	—
221	C₂×M₁₆	—	2	1	no	—
222	C₂×M₁₆	—	4	1	no	—
223	C₂²×C₈	—	4	1	no	—
224	C₂²×C₈	C₂²⋊D₆₁	1	1	yes	characteristic
225	C₂²×C₈	C₂²×D₆₁	1	1	yes	characteristic
226	#127	—	122	4	no	—
227	#127	—	244	2	no	—
228	C₂×C₄²	C₂²⋊D₆₁	1	1	yes	characteristic
229	SD₃₂	—	122	8	no	—
230	D₁₆	—	122	8	no	—
231	M₃₂	—	61	4	no	—
232	M₃₂	—	122	2	no	—
233	C₂×C₁₆	—	122	2	no	—
234	#108	—	2	1	no	—
235	#108	C₂²×D₆₁	1	2	yes	—
236	#104	—	2	4	no	—
237	C₄×C₈	C₂²⋊D₆₁	1	2	yes	—
238	C₄×C₈	C₂²×D₆₁	1	2	yes	—
239	C₂⁴	—	244	2	no	—
240	#41	—	2	2	no	—
241	#41	—	2	2	no	—
242	#41	—	4	1	no	—
243	#41	—	4	1	no	—
244	C₂×Q₈	—	2	1	no	—
245	C₂×D₄	—	2	1	no	—
246	C₂×D₄	—	4	1	no	—
247	C₂×D₄	—	122	4	no	—
248	C₂×D₄	—	488	8	no	—
249	C₂×D₄	—	244	8	no	—
250	C₂×D₄	—	244	2	no	—
251	C₂×D₄	—	244	2	no	—
252	C₂×D₄	—	244	2	no	—
253	C₂×D₄	—	244	2	no	—
254	C₂×D₄	—	244	2	no	—
255	C₂²×C₄	—	2	1	no	—
256	C₂²×C₄	—	4	1	no	—
257	C₂²×C₄	C₂×C₂²⋊D₆₁	1	1	yes	characteristic
258	Q₁₆	—	2	1	no	—
259	Q₁₆	C₂×D₂₄₄	1	2	yes	complemented
260	SD₁₆	—	2	2	no	—
261	SD₁₆	—	2	2	no	—
262	D₈	—	2	1	no	—
263	D₈	—	122	4	no	—
264	D₈	—	122	4	no	—
265	D₈	—	244	2	no	—
266	D₈	—	244	2	no	—
267	D₈	—	244	4	no	—
268	D₈	—	244	4	no	—
269	D₈	C₂×D₂₄₄	1	2	yes	complemented
270	M₁₆	—	2	2	no	—
271	M₁₆	—	4	1	no	—
272	M₁₆	—	4	4	no	—
273	C₂×C₈	—	4	1	no	—
274	C₂×C₈	—	4	4	no	—
275	C₂×C₈	#11758199	1	1	yes	characteristic
276	C₂×C₈	#11758199	1	1	yes	characteristic
277	C₂×C₈	#11758199	1	2	yes	—
278	C₂×C₈	C₂×D₂₄₄	1	1	yes	characteristic
279	C₂×C₈	C₂×D₂₄₄	1	1	yes	characteristic
280	C₂×C₈	#11758222	1	1	yes	characteristic
281	C₂×C₈	#11758222	1	1	yes	characteristic
282	C₂×C₈	C₂×C₂²⋊D₆₁	1	1	yes	characteristic
283	C₂×C₈	C₂×C₂²⋊D₆₁	1	2	yes	—
284	C₂×C₈	C₂³×D₆₁	1	1	yes	characteristic
285	C₄²	—	2	1	no	—
286	C₄²	C₂×C₂²⋊D₆₁	1	2	yes	—
287	C₁₆	—	122	4	no	—
288	C₂³	—	4	1	no	—
289	C₂³	—	488	8	no	—
290	C₂³	—	488	2	no	—
291	C₂³	—	488	2	no	—
292	C₂³	—	244	4	no	—
293	C₂³	#14519802	1	1	yes	characteristic
294	Q₈	—	2	2	no	—
295	Q₈	—	4	1	no	—
296	D₄	—	2	2	no	—
297	D₄	—	4	2	no	—
298	D₄	—	4	2	no	—
299	D₄	—	4	1	no	—
300	D₄	—	488	16	no	—
301	D₄	—	488	2	no	—
302	D₄	—	488	2	no	—
303	D₄	—	244	4	no	—
304	D₄	—	244	4	no	—
305	C₂×C₄	—	2	4	no	—
306	C₂×C₄	—	2	2	no	—
307	C₂×C₄	—	4	1	no	—
308	C₂×C₄	—	4	2	no	—
309	C₂×C₄	—	4	2	no	—
310	C₂×C₄	#14519751	1	1	yes	characteristic
311	C₂×C₄	#14519751	1	1	yes	characteristic
312	C₂×C₄	C₂×#11758199	1	1	yes	characteristic
313	C₂×C₄	C₂×#11758199	1	1	yes	characteristic
314	C₂×C₄	#14519802	1	1	yes	characteristic
315	C₂×C₄	C₂²×C₂²⋊D₆₁	1	1	yes	characteristic
316	C₈	—	4	2	no	—
317	C₈	#14519751	1	1	yes	characteristic
318	C₈	#14519751	1	1	yes	characteristic
319	C₈	C₂×#11758199	1	2	yes	—
320	C₈	C₂×#11758199	1	2	yes	—
321	C₈	C₂²×D₂₄₄	1	1	yes	characteristic
322	C₈	C₂×#11758222	1	1	yes	characteristic
323	C₂²	—	2	2	no	—
324	C₂²	—	4	2	no	—
325	C₂²	—	8	2	no	—
326	C₂²	—	976	4	no	—
327	C₂²	—	976	4	no	—
328	C₂²	—	488	8	no	—
329	C₂²	—	488	4	no	—
330	C₂²	#16839957	1	1	yes	characteristic
331	C₂²	#16839957	1	1	yes	characteristic
332	C₂²	C₂×#14519802	1	1	yes	characteristic
333	C₄	—	2	4	no	—
334	C₄	—	4	2	no	—
335	C₄	#16839957	1	1	yes	characteristic
336	C₄	#16839957	1	1	yes	characteristic
337	C₄	C₂×#14519751	1	1	yes	characteristic
338	C₄	C₂²×#11758199	1	1	yes	characteristic
339	C₂	—	2	1	no	—
340	C₂	—	2	1	no	—
341	C₂	—	8	2	no	—
342	C₂	—	976	4	no	—
343	C₂	#21447895	1	2	yes	direct product factor
344	C₂	C₂×#16839957	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	#16839957
commutator, cosocle	C₂×C₈×C₆₁	C₂⁴
Frattini	C₂×C₈	C₂³×D₆₁
Fitting	C₂×C₆₁×#443	C₂
socle	C₂²×C₆₁	#459
hypercenter	C₂×#443	D₆₁
hypocenter	C₆₁	C₂×#2213
LCS	G ▷ C₂×C₈×C₆₁ ▷ C₄×C₆₁ ▷ C₂×C₆₁ ▷ C₆₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×C₄×C₈ ◁ C₂×#443

Complements

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

N	H ≅ G/N	type
1. C₂×#16840197	C₂	semidirect
2. C₂×#16840139	C₂	semidirect
3. #21447895	C₂	direct (N × H)
4. C₂×#16839583	C₂	semidirect
5. C₂×#16839574	C₂	semidirect
6. C₂×C₆₁×#443	C₂	semidirect
11. #16840197	C₂²	semidirect
16. #16840139	C₂²	semidirect
17. #16839583	C₂²	semidirect
18. #16839574	C₂²	semidirect
22. C₆₁×#443	C₂²	semidirect
48. #135×C₆₁	D₄	semidirect
50. C₂×Q₁₆×C₆₁	D₄	semidirect
52. C₂×D₈×C₆₁	D₄	semidirect
86. Q₁₆×C₆₁	C₂×D₄	semidirect
90. D₈×C₆₁	C₂×D₄	semidirect
169. C₂×#443	D₆₁	semidirect
176. C₂×C₆₁	#2213	semidirect
192. #443	C₂×D₆₁	semidirect
198. C₆₁	C₂×#2213	semidirect
204. #135	D₂₄₄	semidirect
206. C₂×Q₁₆	D₂₄₄	semidirect
218. C₂×D₈	D₂₄₄	semidirect
259. Q₁₆	C₂×D₂₄₄	semidirect
269. D₈	C₂×D₂₄₄	semidirect
343. C₂	#21447895	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×#16840197	2
C₂×#16840139	2
#21447895	2
C₂×#16839583	2
C₂×#16839574	2
C₂×C₆₁×#443	2
C₂×#2213	61

Subgroups — C2×#21447895 #29834393

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#21447895 #29834393