Subgroups — #12253880

Order 1280.

642 conjugacy classes of subgroups.

The subgroup lattice has 368 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#65193	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×#65190	C₂	1	1	yes	maximal, characteristic, complemented
3	#10610987	C₂	1	2	yes	maximal, complemented
4	C₂×C₅×#368	C₂	1	1	yes	maximal, characteristic, complemented
5	#10601480	C₂	1	2	yes	maximal, complemented
6	#10600156	C₂	1	4	yes	maximal, complemented
7	#10600155	C₂	1	2	yes	maximal, complemented
8	#10600154	C₂	1	2	yes	maximal, complemented
9	#66082	—	2	2	no	—
10	C₂×#4050	C₂²	1	2	yes	—
11	C₂×#4047	C₂²	1	1	yes	characteristic
12	C₂²×C₅×C₁₆	C₂²	1	1	yes	characteristic
13	C₂×C₅×#108	C₂²	1	2	yes	—
14	#65421	—	2	2	no	—
15	C₂×#3944	C₂²	1	1	yes	characteristic
16	C₂×Dic₄₀	C₂²	1	1	yes	characteristic, complemented
17	#65193	—	2	1	no	—
18	#65193	—	2	1	no	—
19	#65193	—	2	1	no	—
20	#65193	—	2	1	no	—
21	C₂×#3932	C₂²	1	2	yes	complemented
22	C₂×D₈₀	C₂²	1	1	yes	characteristic, complemented
23	#65190	—	2	1	no	—
24	#65190	—	2	1	no	—
25	#65190	C₂²	1	2	yes	—
26	#65190	C₂²	1	2	yes	—
27	C₂×#3930	—	2	1	no	—
28	C₂×D₅×C₁₆	—	2	1	no	—
29	#65187	—	2	4	no	—
30	#65186	C₂²	1	4	yes	—
31	#65185	C₂²	1	4	yes	—
32	C₅×#368	C₂²	1	2	yes	complemented
33	C₅×#368	C₂²	1	2	yes	complemented
34	#64716	C₂²	1	4	yes	complemented
35	#64715	C₂²	1	4	yes	complemented
36	#33617	—	5	1	no	maximal, Sylow
37	C₂×#709	—	2	2	no	—
38	C₂×#709	D₄	1	1	yes	characteristic
39	C₂×C₅×M₁₆	—	2	2	no	—
40	C₂²×C₅×C₈	C₂³	1	1	yes	characteristic
41	C₂×#681	D₄	1	1	yes	characteristic
42	#4055	—	2	4	no	—
43	#4055	—	4	2	no	—
44	#4054	—	2	4	no	—
45	#4054	—	4	2	no	—
46	C₂×Dic₂₀	C₂³	1	2	yes	—
47	#4050	—	2	2	no	—
48	#4050	—	2	2	no	—
49	#4050	—	2	2	no	—
50	#4050	—	2	2	no	—
51	#4050	—	4	4	no	—
52	C₂×D₄₀	C₂³	1	2	yes	—
53	C₂×#677	—	2	2	no	—
54	#4047	—	2	1	no	—
55	#4047	—	2	1	no	—
56	#4047	D₄	1	2	yes	—
57	#4047	C₂³	1	2	yes	—
58	C₂×#676	—	2	1	no	—
59	C₂×C₈×D₅	—	2	1	no	—
60	C₂×C₅×C₁₆	D₄	1	1	yes	characteristic
61	C₂×C₅×C₁₆	D₄	1	1	yes	characteristic
62	C₂×C₅×C₁₆	D₄	1	1	yes	characteristic
63	C₂×C₅×C₁₆	D₄	1	1	yes	characteristic
64	C₂×C₅×C₁₆	C₂³	1	1	yes	characteristic
65	C₂×C₅×C₁₆	C₂³	1	1	yes	characteristic
66	C₅×#108	—	2	2	no	—
67	C₅×#108	C₂³	1	4	yes	—
68	#3956	—	2	4	no	—
69	#3955	—	2	4	no	—
70	#3944	—	2	1	no	—
71	#3944	C₂³	1	2	yes	—
72	C₂×#672	—	2	1	no	—
73	Dic₄₀	—	2	1	no	—
74	Dic₄₀	—	2	1	no	—
75	#3932	—	2	2	no	—
76	#3932	—	2	2	no	—
77	D₈₀	—	2	1	no	—
78	D₈₀	—	2	1	no	—
79	#3930	—	2	2	no	—
80	#3930	—	2	2	no	—
81	D₅×C₁₆	—	2	2	no	—
82	D₅×C₁₆	—	2	2	no	—
83	C₂×#508	—	5	1	no	—
84	C₂×#504	—	5	1	no	—
85	#3067	—	5	2	no	—
86	#2223	—	5	4	no	—
87	#2222	—	5	2	no	—
88	#2221	—	5	2	no	—
89	C₂×#368	D₅	1	1	yes	characteristic, complemented
90	#2148	—	5	2	no	—
91	C₂²×C₄×C₅	C₂×D₄	1	1	yes	characteristic
92	C₂×C₂²⋊D₅	—	2	1	no	—
93	C₂×C₂²⋊D₅	—	4	2	no	—
94	#709	—	2	1	no	—
95	#709	—	2	1	no	—
96	#709	—	4	2	no	—
97	#709	—	4	2	no	—
98	#709	—	4	2	no	—
99	#709	—	4	2	no	—
100	#709	D₈	1	2	yes	—
101	#709	C₂×D₄	1	2	yes	—
102	C₂×D₂₀	—	2	2	no	—
103	C₂×D₂₀	D₈	1	1	yes	characteristic
104	C₂×C₄×D₅	—	2	1	no	—
105	C₂×C₄×D₅	—	4	2	no	—
106	C₂×Dic₁₀	—	2	2	no	—
107	C₂×Dic₁₀	D₈	1	1	yes	characteristic
108	C₅×M₁₆	—	2	4	no	—
109	C₅×M₁₆	—	4	2	no	—
110	C₂×C₅×C₈	—	4	2	no	—
111	C₂×C₅×C₈	C₂×D₄	1	1	yes	characteristic
112	C₂×C₅×C₈	C₂×D₄	1	1	yes	characteristic
113	C₂×C₅×C₈	C₂×D₄	1	1	yes	characteristic
114	C₂×C₅×C₈	#41	1	1	yes	characteristic
115	C₂×C₅×C₈	#41	1	1	yes	characteristic
116	C₂×C₅×C₈	C₂⁴	1	1	yes	characteristic
117	#681	—	2	1	no	—
118	#681	C₂×D₄	1	2	yes	—
119	C₂×#168	—	2	1	no	—
120	Dic₂₀	—	2	2	no	—
121	Dic₂₀	—	2	2	no	—
122	Dic₂₀	—	4	4	no	—
123	D₄₀	—	2	2	no	—
124	D₄₀	—	2	2	no	—
125	D₄₀	—	4	4	no	—
126	#677	—	4	4	no	—
127	#677	—	4	4	no	—
128	#677	—	4	2	no	—
129	#677	—	4	2	no	—
130	#676	—	2	2	no	—
131	#676	—	2	2	no	—
132	C₈×D₅	—	2	2	no	—
133	C₈×D₅	—	2	2	no	—
134	C₅×C₁₆	C₂×D₄	1	1	yes	characteristic
135	C₅×C₁₆	C₂×D₄	1	1	yes	characteristic
136	C₅×C₁₆	C₂×D₄	1	1	yes	characteristic
137	C₅×C₁₆	C₂×D₄	1	1	yes	characteristic
138	#672	—	2	2	no	—
139	#575	—	10	2	no	—
140	C₂×#135	—	5	2	no	—
141	C₂×#131	—	5	1	no	—
142	#508	—	10	1	no	—
143	#508	—	10	1	no	—
144	#508	—	10	1	no	—
145	#508	—	10	1	no	—
146	C₂×Q₃₂	—	5	1	no	—
147	C₂×SD₃₂	—	5	2	no	—
148	C₂×D₁₆	—	5	1	no	—
149	#504	—	5	2	no	—
150	#504	—	5	2	no	—
151	#504	—	10	1	no	—
152	#504	—	10	1	no	—
153	C₂×M₃₂	—	5	1	no	—
154	C₂×M₃₂	—	10	1	no	—
155	C₂²×C₁₆	—	10	1	no	—
156	C₂²×C₁₆	C₂×D₅	1	1	yes	characteristic
157	#473	—	5	4	no	—
158	#472	—	5	4	no	—
159	#471	—	10	4	no	—
160	C₂×#108	C₂×D₅	1	2	yes	—
161	#413	—	10	2	no	—
162	#368	C₂×D₅	1	2	yes	complemented
163	#368	C₂×D₅	1	2	yes	complemented
164	#362	—	5	4	no	—
165	#361	—	5	4	no	—
166	C₂³×C₅	#135	1	1	yes	characteristic
167	C₂²×D₅	—	2	1	no	—
168	C₂²×D₅	—	4	2	no	—
169	C₂×C₄×C₅	#121	1	1	yes	characteristic
170	C₂×C₄×C₅	#121	1	1	yes	characteristic
171	C₂×C₄×C₅	C₂×D₈	1	1	yes	characteristic
172	C₂×C₄×C₅	C₂×D₈	1	1	yes	characteristic
173	C₂×C₄×C₅	#135	1	1	yes	characteristic
174	C₂×C₄×C₅	C₂²×D₄	1	1	yes	characteristic
175	C₂²⋊D₅	—	2	2	no	—
176	C₂²⋊D₅	—	2	2	no	—
177	C₂²⋊D₅	—	8	2	no	—
178	C₂²⋊D₅	—	8	2	no	—
179	C₂×Dic₅	—	2	1	no	—
180	C₂×Dic₅	—	4	2	no	—
181	D₂₀	—	2	1	no	—
182	D₂₀	—	4	2	no	—
183	D₂₀	—	4	2	no	—
184	D₂₀	C₂×D₈	1	2	yes	—
185	C₄×D₅	—	2	2	no	—
186	C₄×D₅	—	2	2	no	—
187	C₄×D₅	—	8	2	no	—
188	C₄×D₅	—	8	2	no	—
189	Dic₁₀	—	2	1	no	—
190	Dic₁₀	—	4	2	no	—
191	Dic₁₀	—	4	2	no	—
192	Dic₁₀	C₂×D₈	1	2	yes	—
193	C₅×C₈	—	4	4	no	—
194	C₅×C₈	#121	1	1	yes	characteristic
195	C₅×C₈	#121	1	1	yes	characteristic
196	C₅×C₈	C₂²×D₄	1	1	yes	characteristic
197	C₅×C₈	C₂×#41	1	1	yes	characteristic
198	#168	—	2	2	no	—
199	C₂×#41	—	5	1	no	—
200	C₂×#41	—	10	2	no	—
201	#137	—	10	4	no	—
202	#137	—	20	2	no	—
203	AGL(1,ℤ/8)	—	10	4	no	—
204	AGL(1,ℤ/8)	—	20	2	no	—
205	#135	—	10	2	no	—
206	#135	—	10	2	no	—
207	#135	—	10	2	no	—
208	#135	—	10	2	no	—
209	#135	—	20	4	no	—
210	C₂×Q₁₆	—	5	2	no	—
211	C₂×SD₁₆	—	10	2	no	—
212	C₂×D₈	—	5	2	no	—
213	#131	—	5	2	no	—
214	#131	—	5	2	no	—
215	#131	—	10	1	no	—
216	#131	—	10	1	no	—
217	C₂×M₁₆	—	2	2	no	—
218	C₂×M₁₆	—	5	1	no	—
219	C₂×M₁₆	—	10	1	no	—
220	C₂²×C₈	—	10	1	no	—
221	C₂²×C₈	C₂²×D₅	1	1	yes	characteristic
222	Q₃₂	—	10	1	no	—
223	Q₃₂	—	10	1	no	—
224	SD₃₂	—	10	2	no	—
225	SD₃₂	—	10	2	no	—
226	D₁₆	—	10	1	no	—
227	D₁₆	—	10	1	no	—
228	M₃₂	—	5	2	no	—
229	M₃₂	—	10	1	no	—
230	M₃₂	—	10	2	no	—
231	M₃₂	—	10	2	no	—
232	C₂×C₁₆	—	10	1	no	—
233	C₂×C₁₆	—	10	2	no	—
234	C₂×C₁₆	—	10	2	no	—
235	C₂×C₁₆	D₂₀	1	1	yes	characteristic
236	C₂×C₁₆	D₂₀	1	1	yes	characteristic
237	C₂×C₁₆	D₂₀	1	1	yes	characteristic
238	C₂×C₁₆	D₂₀	1	1	yes	characteristic
239	C₂×C₁₆	C₂²×D₅	1	1	yes	characteristic
240	C₂×C₁₆	C₂²×D₅	1	1	yes	characteristic
241	#108	—	2	2	no	—
242	#108	C₂²×D₅	1	4	yes	—
243	#101	—	10	4	no	—
244	#100	—	10	4	no	—
245	C₂²×C₅	—	4	1	no	—
246	C₂²×C₅	#466	1	1	yes	characteristic
247	C₂²×C₅	#466	1	1	yes	characteristic
248	C₂²×C₅	C₂×#135	1	1	yes	characteristic
249	C₂×D₅	—	2	2	no	—
250	C₂×D₅	—	4	2	no	—
251	C₂×D₅	—	8	2	no	—
252	C₂×D₅	—	16	2	no	—
253	C₄×C₅	#466	1	1	yes	characteristic
254	C₄×C₅	#466	1	1	yes	characteristic
255	C₄×C₅	C₂×#121	1	1	yes	characteristic
256	C₄×C₅	C₂²×D₈	1	1	yes	characteristic
257	Dic₅	—	2	2	no	—
258	Dic₅	—	8	2	no	—
259	#41	—	5	2	no	—
260	#41	—	5	2	no	—
261	#41	—	10	1	no	—
262	#41	—	10	1	no	—
263	#41	—	20	2	no	—
264	#41	—	20	2	no	—
265	#41	—	20	2	no	—
266	#41	—	20	2	no	—
267	C₂×Q₈	—	5	1	no	—
268	C₂×Q₈	—	10	2	no	—
269	C₂×D₄	—	5	1	no	—
270	C₂×D₄	—	10	1	no	—
271	C₂×D₄	—	10	2	no	—
272	C₂×D₄	—	20	2	no	—
273	C₂²×C₄	—	10	1	no	—
274	C₂²×C₄	—	20	2	no	—
275	C₂²×C₄	D₄×D₅	1	1	yes	characteristic
276	Q₁₆	—	10	2	no	—
277	Q₁₆	—	10	2	no	—
278	Q₁₆	—	20	4	no	—
279	SD₁₆	—	20	4	no	—
280	SD₁₆	—	20	4	no	—
281	SD₁₆	—	20	2	no	—
282	SD₁₆	—	20	2	no	—
283	D₈	—	10	2	no	—
284	D₈	—	10	2	no	—
285	D₈	—	20	4	no	—
286	M₁₆	—	2	4	no	—
287	M₁₆	—	4	2	no	—
288	M₁₆	—	5	2	no	—
289	M₁₆	—	10	1	no	—
290	M₁₆	—	10	2	no	—
291	M₁₆	—	10	2	no	—
292	C₂×C₈	—	4	2	no	—
293	C₂×C₈	—	10	1	no	—
294	C₂×C₈	—	10	2	no	—
295	C₂×C₈	—	10	2	no	—
296	C₂×C₈	C₂×D₂₀	1	1	yes	characteristic
297	C₂×C₈	C₂×D₂₀	1	1	yes	characteristic
298	C₂×C₈	D₄×D₅	1	1	yes	characteristic
299	C₂×C₈	#713	1	1	yes	characteristic
300	C₂×C₈	#713	1	1	yes	characteristic
301	C₂×C₈	C₂³×D₅	1	1	yes	characteristic
302	C₁₆	—	10	2	no	—
303	C₁₆	C₂×D₂₀	1	1	yes	characteristic
304	C₁₆	C₂×D₂₀	1	1	yes	characteristic
305	C₁₆	C₂×D₂₀	1	1	yes	characteristic
306	C₁₆	C₂×D₂₀	1	1	yes	characteristic
307	C₂×C₅	—	2	1	no	—
308	C₂×C₅	—	2	1	no	—
309	C₂×C₅	—	2	1	no	—
310	C₂×C₅	C₂×#466	1	1	yes	characteristic
311	D₅	—	4	2	no	—
312	D₅	—	16	2	no	—
313	C₂³	—	10	1	no	—
314	C₂³	—	20	2	no	—
315	C₂³	#4065	1	1	yes	characteristic
316	Q₈	—	5	2	no	—
317	Q₈	—	10	1	no	—
318	Q₈	—	20	2	no	—
319	Q₈	—	20	2	no	—
320	D₄	—	5	2	no	—
321	D₄	—	10	2	no	—
322	D₄	—	10	2	no	—
323	D₄	—	10	1	no	—
324	D₄	—	20	2	no	—
325	D₄	—	20	2	no	—
326	D₄	—	40	2	no	—
327	D₄	—	40	2	no	—
328	C₂×C₄	—	10	1	no	—
329	C₂×C₄	—	10	2	no	—
330	C₂×C₄	—	10	2	no	—
331	C₂×C₄	—	20	2	no	—
332	C₂×C₄	—	40	2	no	—
333	C₂×C₄	—	40	2	no	—
334	C₂×C₄	#4041	1	1	yes	characteristic
335	C₂×C₄	#4041	1	1	yes	characteristic
336	C₂×C₄	D₅×D₈	1	1	yes	characteristic
337	C₂×C₄	D₅×D₈	1	1	yes	characteristic
338	C₂×C₄	#4065	1	1	yes	characteristic
339	C₂×C₄	C₂×D₄×D₅	1	1	yes	characteristic
340	C₈	—	4	4	no	—
341	C₈	—	10	2	no	—
342	C₈	#4041	1	1	yes	characteristic
343	C₈	#4041	1	1	yes	characteristic
344	C₈	C₂²×D₂₀	1	1	yes	characteristic
345	C₈	C₂×#713	1	1	yes	characteristic
346	C₅	#33617	1	1	yes	characteristic, complemented, Sylow
347	C₂²	—	4	1	no	—
348	C₂²	—	10	2	no	—
349	C₂²	—	20	2	no	—
350	C₂²	—	40	2	no	—
351	C₂²	—	80	2	no	—
352	C₂²	#65172	1	1	yes	characteristic
353	C₂²	#65172	1	1	yes	characteristic
354	C₂²	C₂×#4065	1	1	yes	characteristic
355	C₄	—	10	2	no	—
356	C₄	—	40	2	no	—
357	C₄	#65172	1	1	yes	characteristic
358	C₄	#65172	1	1	yes	characteristic
359	C₄	C₂×#4041	1	1	yes	characteristic
360	C₄	C₂×D₅×D₈	1	1	yes	characteristic
361	C₂	—	2	1	no	—
362	C₂	—	2	1	no	—
363	C₂	—	2	1	no	—
364	C₂	—	20	2	no	—
365	C₂	—	80	2	no	—
366	C₂	C₂×#65172	1	1	yes	characteristic

	subgroup	G/H
center	C₄	#65172
commutator, cosocle	C₂×C₅×C₈	C₂⁴
Frattini	C₂×C₈	C₂³×D₅
Fitting	C₂×C₅×#368	C₂
socle	C₂×C₅	C₂×#466
hypercenter	C₂×#368	D₅
hypocenter	C₅	#33617
LCS	G ▷ C₂×C₅×C₈ ▷ C₄×C₅ ▷ C₂×C₅ ▷ C₅
UCS	C₁ ◁ C₄ ◁ C₂²×C₄ ◁ C₂²×C₈ ◁ C₂×#368

Complements

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

N	H ≅ G/N	type
1. C₂×#65193	C₂	semidirect
2. C₂×#65190	C₂	semidirect
3. #10610987	C₂	semidirect
4. C₂×C₅×#368	C₂	semidirect
5. #10601480	C₂	semidirect
6. #10600156	C₂	semidirect
7. #10600155	C₂	semidirect
8. #10600154	C₂	semidirect
16. C₂×Dic₄₀	C₂²	semidirect
21. C₂×#3932	C₂²	semidirect
22. C₂×D₈₀	C₂²	semidirect
32. C₅×#368	C₂²	semidirect
33. C₅×#368	C₂²	semidirect
34. #64716	C₂²	semidirect
35. #64715	C₂²	semidirect
89. C₂×#368	D₅	semidirect
162. #368	C₂×D₅	semidirect
163. #368	C₂×D₅	semidirect
346. C₅	#33617	semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
C₂×#65193	2
C₂×#65190	2
#10610987	2
C₂×C₅×#368	2
#10601480	2
#10600156	2
#10600155	2
#10600154	2
#33617	5