Subgroups — #12199142

Order 1280.

810 conjugacy classes of subgroups.

The subgroup lattice has 332 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄×C₅×C₄⋊Q₈	C₂	1	1	yes	maximal, characteristic
2	#10605042	C₂	1	1	yes	maximal, characteristic
3	#10602578	C₂	1	2	yes	maximal
4	#10602419	C₂	1	4	yes	maximal
5	#10600461	C₂	1	4	yes	maximal
6	#10600367	C₂	1	2	yes	maximal
7	C₄×C₄⋊#168	C₂	1	1	yes	maximal, characteristic
8	C₂×C₅×C₄⋊Q₈	C₂²	1	1	yes	characteristic
9	C₂×C₄×C₅×Q₈	C₂²	1	2	yes	—
10	C₂×#4015	—	2	1	no	—
11	C₂×#4015	—	2	1	no	—
12	C₂×#4015	C₂²	1	2	yes	—
13	C₅×#391	C₂²	1	2	yes	—
14	C₅×#389	—	2	1	no	—
15	C₅×#389	—	2	1	no	—
16	C₅×#384	C₂²	1	1	yes	characteristic
17	C₄×C₅×C₄⋊C₄	—	2	1	no	—
18	C₄×C₅×C₄⋊C₄	—	2	1	no	—
19	C₄³×C₅	C₂²	1	1	yes	characteristic
20	C₂×#3967	C₂²	1	8	yes	—
21	#65224	—	2	1	no	—
22	#65224	—	2	1	no	—
23	#65224	C₂²	1	2	yes	—
24	#65224	C₂²	1	2	yes	—
25	C₂×C₄⋊#168	C₂²	1	2	yes	—
26	C₂×C₄×#168	C₂²	1	2	yes	—
27	#64749	C₂²	1	8	yes	—
28	#64744	C₂²	1	2	yes	—
29	#29141	—	5	1	no	maximal, Sylow
30	C₂²×C₅×Q₈	C₂³	1	2	yes	—
31	C₂²×Dic₁₀	—	2	2	no	—
32	C₅×C₄⋊Q₈	—	2	2	no	—
33	C₅×C₄⋊Q₈	Q₈	1	4	yes	complemented
34	C₄×C₅×Q₈	—	2	8	no	—
35	C₂×C₅×C₄⋊C₄	—	2	1	no	—
36	C₂×C₅×C₄⋊C₄	—	2	1	no	—
37	C₂×C₅×C₄⋊C₄	—	2	1	no	—
38	C₂×C₅×C₄⋊C₄	—	2	1	no	—
39	C₂×C₅×C₄⋊C₄	—	2	2	no	—
40	C₂×C₅×C₄⋊C₄	C₂³	1	2	yes	—
41	C₂×C₄²×C₅	—	2	2	no	—
42	C₂×C₄²×C₅	D₄	1	1	yes	characteristic
43	C₂×C₄²×C₅	D₄	1	1	yes	characteristic
44	C₂×C₄²×C₅	D₄	1	1	yes	characteristic
45	C₂×C₄²×C₅	D₄	1	1	yes	characteristic
46	C₂×C₄²×C₅	C₂³	1	1	yes	characteristic
47	C₂×C₄²×C₅	C₂³	1	1	yes	characteristic
48	C₂×C₄²×C₅	C₂³	1	1	yes	characteristic
49	C₂×C₄⋊Dic₅	—	2	2	no	—
50	C₂×C₄⋊Dic₅	—	2	2	no	—
51	C₂×C₄⋊Dic₅	—	2	2	no	—
52	C₂×C₄⋊Dic₅	—	2	2	no	—
53	C₂×C₄⋊Dic₅	—	2	2	no	—
54	C₂×C₄⋊Dic₅	C₂³	1	4	yes	—
55	C₂²×#168	C₂³	1	4	yes	—
56	#4015	—	4	4	no	—
57	#4015	—	4	4	no	—
58	#4015	—	4	4	no	—
59	C₅×#95	—	2	2	no	—
60	C₅×#95	—	2	2	no	—
61	#3967	—	4	8	no	—
62	C₄⋊#168	—	2	4	no	—
63	C₄×#168	—	2	4	no	—
64	#2868	—	5	1	no	—
65	C₄×C₄⋊Q₈	D₅	1	1	yes	characteristic
66	#2024	—	5	4	no	—
67	#1956	—	5	2	no	—
68	#1834	—	5	2	no	—
69	#1803	—	5	4	no	—
70	C₄×C₄⋊C₈	—	5	1	no	—
71	C₂×C₅×Q₈	—	2	2	no	—
72	C₂×C₅×Q₈	—	2	4	no	—
73	C₂×C₅×Q₈	—	2	4	no	—
74	C₂×C₅×Q₈	C₂×Q₈	1	4	yes	—
75	C₂²×C₄×C₅	—	2	2	no	—
76	C₂²×C₄×C₅	—	2	2	no	—
77	C₂²×C₄×C₅	C₂×D₄	1	1	yes	characteristic
78	C₂²×C₄×C₅	C₂×D₄	1	1	yes	characteristic
79	C₂²×C₄×C₅	C₂×D₄	1	1	yes	characteristic
80	C₂²×C₄×C₅	C₂×D₄	1	1	yes	characteristic
81	C₂²×C₄×C₅	C₂×D₄	1	1	yes	characteristic
82	C₂²×C₄×C₅	C₂×D₄	1	1	yes	characteristic
83	C₂²×C₄×C₅	C₂⁴	1	1	yes	characteristic
84	C₂²×Dic₅	—	2	4	no	—
85	C₂×Dic₁₀	—	4	4	no	—
86	C₂×Dic₁₀	—	4	4	no	—
87	C₂×Dic₁₀	—	4	4	no	—
88	C₅×C₄⋊C₄	—	2	4	no	—
89	C₅×C₄⋊C₄	—	2	4	no	—
90	C₅×C₄⋊C₄	—	2	4	no	—
91	C₅×C₄⋊C₄	—	4	4	no	—
92	C₅×C₄⋊C₄	—	4	2	no	—
93	C₅×C₄⋊C₄	—	4	2	no	—
94	C₄²×C₅	—	2	2	no	—
95	C₄²×C₅	—	2	2	no	—
96	C₄²×C₅	—	2	2	no	—
97	C₄²×C₅	—	4	4	no	—
98	C₄²×C₅	SD₁₆	1	4	yes	—
99	C₄²×C₅	Q₁₆	1	4	yes	—
100	C₄²×C₅	C₂×Q₈	1	2	yes	—
101	C₄²×C₅	#41	1	2	yes	—
102	C₄²×C₅	#41	1	4	yes	—
103	C₄⋊Dic₅	—	2	8	no	—
104	C₄⋊Dic₅	—	4	4	no	—
105	C₄⋊Dic₅	—	4	4	no	—
106	C₄⋊Dic₅	—	4	4	no	—
107	C₄⋊Dic₅	—	4	4	no	—
108	C₄⋊Dic₅	—	4	4	no	—
109	C₂×#168	—	2	4	no	—
110	C₂×#168	—	2	4	no	—
111	C₂×#168	—	2	4	no	—
112	C₂×C₄⋊Q₈	—	5	2	no	—
113	C₂×C₄⋊Q₈	—	10	1	no	—
114	C₂×C₄⋊Q₈	—	10	1	no	—
115	C₂×C₄⋊Q₈	C₂×D₅	1	1	yes	characteristic
116	C₂×C₄×Q₈	C₂×D₅	1	2	yes	—
117	C₂×C₄⋊C₈	—	5	2	no	—
118	C₂×#103	—	5	8	no	—
119	C₂×C₄×C₈	—	5	2	no	—
120	#391	C₂×D₅	1	2	yes	—
121	#389	—	2	1	no	—
122	#389	—	2	1	no	—
123	#384	—	5	2	no	—
124	#384	—	5	2	no	—
125	#384	—	10	1	no	—
126	#384	—	10	1	no	—
127	#384	C₂×D₅	1	1	yes	characteristic
128	C₄×C₄⋊C₄	—	2	1	no	—
129	C₄×C₄⋊C₄	—	2	1	no	—
130	C₄³	C₂×D₅	1	1	yes	characteristic
131	#340	—	5	8	no	—
132	#336	—	5	2	no	—
133	C₂³×C₅	C₂²×D₄	1	1	yes	characteristic
134	C₅×Q₈	—	2	8	no	—
135	C₅×Q₈	—	4	4	no	—
136	C₂×C₄×C₅	—	2	1	no	—
137	C₂×C₄×C₅	—	2	1	no	—
138	C₂×C₄×C₅	—	2	1	no	—
139	C₂×C₄×C₅	—	2	1	no	—
140	C₂×C₄×C₅	—	2	1	no	—
141	C₂×C₄×C₅	—	2	1	no	—
142	C₂×C₄×C₅	—	2	4	no	—
143	C₂×C₄×C₅	—	4	2	no	—
144	C₂×C₄×C₅	—	4	2	no	—
145	C₂×C₄×C₅	—	4	2	no	—
146	C₂×C₄×C₅	—	4	2	no	—
147	C₂×C₄×C₅	—	4	2	no	—
148	C₂×C₄×C₅	C₂²⋊Q₈	1	2	yes	—
149	C₂×C₄×C₅	C₂²⋊Q₈	1	2	yes	—
150	C₂×C₄×C₅	#124	1	2	yes	—
151	C₂×C₄×C₅	#124	1	2	yes	—
152	C₂×C₄×C₅	C₂×SD₁₆	1	2	yes	—
153	C₂×C₄×C₅	C₂×SD₁₆	1	2	yes	—
154	C₂×C₄×C₅	C₂×SD₁₆	1	2	yes	—
155	C₂×C₄×C₅	C₂×Q₁₆	1	2	yes	—
156	C₂×C₄×C₅	C₂×Q₁₆	1	2	yes	—
157	C₂×C₄×C₅	C₂×Q₁₆	1	2	yes	—
158	C₂×C₄×C₅	AGL(1,ℤ/8)	1	2	yes	—
159	C₂×C₄×C₅	#137	1	2	yes	—
160	C₂×C₄×C₅	C₂²×Q₈	1	1	yes	characteristic
161	C₂×C₄×C₅	C₂×#41	1	1	yes	characteristic
162	C₂×C₄×C₅	C₂×#41	1	2	yes	—
163	C₂×C₄×C₅	H(5,𝔽₂)	1	2	yes	—
164	C₂×Dic₅	—	4	4	no	—
165	C₂×Dic₅	—	4	4	no	—
166	C₂×Dic₅	—	4	4	no	—
167	Dic₁₀	—	8	8	no	—
168	#168	—	4	4	no	—
169	C₂²×Q₈	—	10	2	no	—
170	C₂²×Q₈	C₂²×D₅	1	2	yes	—
171	C₂²×C₈	—	5	4	no	—
172	C₄⋊Q₈	—	2	2	no	—
173	C₄⋊Q₈	—	20	4	no	—
174	C₄⋊Q₈	—	20	4	no	—
175	C₄⋊Q₈	—	20	4	no	—
176	C₄⋊Q₈	Dic₁₀	1	4	yes	complemented
177	C₄×Q₈	—	2	8	no	—
178	C₂×C₄⋊C₄	—	2	1	no	—
179	C₂×C₄⋊C₄	—	2	1	no	—
180	C₂×C₄⋊C₄	—	2	2	no	—
181	C₂×C₄⋊C₄	—	2	1	no	—
182	C₂×C₄⋊C₄	—	2	1	no	—
183	C₂×C₄⋊C₄	—	5	4	no	—
184	C₂×C₄⋊C₄	—	10	2	no	—
185	C₂×C₄⋊C₄	—	10	2	no	—
186	C₂×C₄⋊C₄	—	10	2	no	—
187	C₂×C₄⋊C₄	—	10	2	no	—
188	C₂×C₄⋊C₄	—	10	2	no	—
189	C₂×C₄⋊C₄	C₂²×D₅	1	2	yes	—
190	C₂×C₄²	—	2	2	no	—
191	C₂×C₄²	C₂²⋊D₅	1	1	yes	characteristic
192	C₂×C₄²	C₂²⋊D₅	1	1	yes	characteristic
193	C₂×C₄²	C₂²⋊D₅	1	1	yes	characteristic
194	C₂×C₄²	C₂²⋊D₅	1	1	yes	characteristic
195	C₂×C₄²	C₂²×D₅	1	1	yes	characteristic
196	C₂×C₄²	C₂²×D₅	1	1	yes	characteristic
197	C₂×C₄²	C₂²×D₅	1	1	yes	characteristic
198	C₄⋊C₈	—	10	4	no	—
199	#103	—	20	8	no	—
200	C₄×C₈	—	10	4	no	—
201	#95	—	2	2	no	—
202	#95	—	2	2	no	—
203	C₂²×C₅	C₂×C₂²⋊Q₈	1	1	yes	characteristic
204	C₂²×C₅	C₂×#124	1	1	yes	characteristic
205	C₂²×C₅	#535	1	1	yes	characteristic
206	C₂²×C₅	C₂²×SD₁₆	1	1	yes	characteristic
207	C₂²×C₅	C₂²×Q₁₆	1	1	yes	characteristic
208	C₂²×C₅	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
209	C₂²×C₅	C₂×#137	1	1	yes	characteristic
210	C₄×C₅	—	2	2	no	—
211	C₄×C₅	—	2	2	no	—
212	C₄×C₅	—	2	2	no	—
213	C₄×C₅	—	4	4	no	—
214	C₄×C₅	—	8	2	no	—
215	C₄×C₅	#475	1	4	yes	—
216	C₄×C₅	#477	1	4	yes	—
217	C₄×C₅	#487	1	4	yes	—
218	C₄×C₅	#554	1	2	yes	—
219	C₄×C₅	#555	1	2	yes	—
220	Dic₅	—	8	4	no	—
221	C₂×Q₈	—	2	2	no	—
222	C₂×Q₈	—	2	4	no	—
223	C₂×Q₈	—	2	4	no	—
224	C₂×Q₈	—	20	4	no	—
225	C₂×Q₈	—	20	4	no	—
226	C₂×Q₈	—	20	4	no	—
227	C₂×Q₈	C₂×Dic₁₀	1	4	yes	—
228	C₂²×C₄	—	2	2	no	—
229	C₂²×C₄	—	2	2	no	—
230	C₂²×C₄	—	10	4	no	—
231	C₂²×C₄	C₂×C₂²⋊D₅	1	1	yes	characteristic
232	C₂²×C₄	C₂×C₂²⋊D₅	1	1	yes	characteristic
233	C₂²×C₄	C₂×C₂²⋊D₅	1	1	yes	characteristic
234	C₂²×C₄	C₂×C₂²⋊D₅	1	1	yes	characteristic
235	C₂²×C₄	C₂×C₂²⋊D₅	1	1	yes	characteristic
236	C₂²×C₄	C₂×C₂²⋊D₅	1	1	yes	characteristic
237	C₂²×C₄	C₂³×D₅	1	1	yes	characteristic
238	C₂×C₈	—	10	4	no	—
239	C₂×C₈	—	10	4	no	—
240	C₂×C₈	—	10	4	no	—
241	C₄⋊C₄	—	2	4	no	—
242	C₄⋊C₄	—	2	4	no	—
243	C₄⋊C₄	—	2	4	no	—
244	C₄⋊C₄	—	4	2	no	—
245	C₄⋊C₄	—	4	4	no	—
246	C₄⋊C₄	—	4	2	no	—
247	C₄⋊C₄	—	10	8	no	—
248	C₄⋊C₄	—	20	4	no	—
249	C₄⋊C₄	—	20	4	no	—
250	C₄⋊C₄	—	20	4	no	—
251	C₄⋊C₄	—	20	4	no	—
252	C₄⋊C₄	—	20	4	no	—
253	C₄²	—	2	2	no	—
254	C₄²	—	2	2	no	—
255	C₄²	—	2	2	no	—
256	C₄²	—	4	4	no	—
257	C₄²	#688	1	4	yes	—
258	C₄²	#689	1	4	yes	—
259	C₄²	C₂×Dic₁₀	1	2	yes	—
260	C₄²	#709	1	2	yes	—
261	C₄²	#711	1	4	yes	—
262	C₂×C₅	#2563	1	1	yes	characteristic
263	C₂×C₅	C₂×#475	1	1	yes	characteristic
264	C₂×C₅	C₂×#477	1	1	yes	characteristic
265	C₂×C₅	C₂×#487	1	1	yes	characteristic
266	C₂×C₅	#3141	1	1	yes	characteristic
267	C₂×C₅	#3208	1	1	yes	characteristic
268	C₂×C₅	#3209	1	1	yes	characteristic
269	C₂³	C₂²×C₂²⋊D₅	1	1	yes	characteristic
270	Q₈	—	2	8	no	—
271	Q₈	—	4	4	no	—
272	Q₈	—	40	8	no	—
273	C₂×C₄	—	2	1	no	—
274	C₂×C₄	—	2	1	no	—
275	C₂×C₄	—	2	1	no	—
276	C₂×C₄	—	2	1	no	—
277	C₂×C₄	—	2	1	no	—
278	C₂×C₄	—	2	1	no	—
279	C₂×C₄	—	2	4	no	—
280	C₂×C₄	—	4	2	no	—
281	C₂×C₄	—	4	2	no	—
282	C₂×C₄	—	4	2	no	—
283	C₂×C₄	—	4	2	no	—
284	C₂×C₄	—	4	2	no	—
285	C₂×C₄	—	20	4	no	—
286	C₂×C₄	—	20	4	no	—
287	C₂×C₄	—	20	4	no	—
288	C₂×C₄	#4070	1	2	yes	—
289	C₂×C₄	#4070	1	2	yes	—
290	C₂×C₄	#4082	1	2	yes	—
291	C₂×C₄	#4082	1	2	yes	—
292	C₂×C₄	C₂×#688	1	2	yes	—
293	C₂×C₄	C₂×#688	1	2	yes	—
294	C₂×C₄	C₂×#688	1	2	yes	—
295	C₂×C₄	C₂×#689	1	2	yes	—
296	C₂×C₄	C₂×#689	1	2	yes	—
297	C₂×C₄	C₂×#689	1	2	yes	—
298	C₂×C₄	#4095	1	2	yes	—
299	C₂×C₄	#4097	1	2	yes	—
300	C₂×C₄	C₂²×Dic₁₀	1	1	yes	characteristic
301	C₂×C₄	C₂×#709	1	1	yes	characteristic
302	C₂×C₄	C₂×#711	1	2	yes	—
303	C₂×C₄	#4149	1	2	yes	—
304	C₈	—	20	4	no	—
305	C₅	#29141	1	1	yes	characteristic, complemented, Sylow
306	C₂²	C₂×#4070	1	1	yes	characteristic
307	C₂²	C₂×#4082	1	1	yes	characteristic
308	C₂²	C₂²×#688	1	1	yes	characteristic
309	C₂²	C₂²×#689	1	1	yes	characteristic
310	C₂²	C₂×#4095	1	1	yes	characteristic
311	C₂²	C₂×#4097	1	1	yes	characteristic
312	C₂²	#66164	1	1	yes	characteristic
313	C₄	—	2	2	no	—
314	C₄	—	2	2	no	—
315	C₄	—	2	2	no	—
316	C₄	—	4	4	no	—
317	C₄	—	8	2	no	—
318	C₄	—	40	4	no	—
319	C₄	#65309	1	4	yes	—
320	C₄	#65310	1	4	yes	—
321	C₄	#65370	1	4	yes	—
322	C₄	#65877	1	2	yes	—
323	C₄	#65897	1	2	yes	—
324	C₂	#10614115	1	1	yes	characteristic
325	C₂	C₂×#65309	1	1	yes	characteristic
326	C₂	C₂×#65310	1	1	yes	characteristic
327	C₂	C₂×#65370	1	1	yes	characteristic
328	C₂	#10615014	1	1	yes	characteristic
329	C₂	#10615086	1	1	yes	characteristic
330	C₂	#10615101	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	C₂²×C₂²⋊D₅
commutator, cosocle	C₂²×C₄×C₅	C₂⁴
Frattini	C₂²×C₄	C₂³×D₅
Fitting	C₄×C₅×C₄⋊Q₈	C₂
socle, prime radical	C₂³×C₅	C₂²×D₄
hypercenter	C₄×C₄⋊Q₈	D₅
hypocenter	C₅	#29141
LCS	G ▷ C₂²×C₄×C₅ ▷ C₂×C₅ ▷ C₅
UCS	C₁ ◁ C₂³ ◁ C₄³ ◁ C₄×C₄⋊Q₈

Complements

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

N	H ≅ G/N	type
33. C₅×C₄⋊Q₈	Q₈	semidirect
176. C₄⋊Q₈	Dic₁₀	semidirect
305. C₅	#29141	semidirect

Maximal subgroups

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

subgroup	index
C₄×C₅×C₄⋊Q₈	2
#10605042	2
#10602578	2
#10602419	2
#10600461	2
#10600367	2
C₄×C₄⋊#168	2
#29141	5