Subgroups — #12116103

Order 1280.

1086 conjugacy classes of subgroups.

The subgroup lattice has 306 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#65904	C₂	1	1	yes	maximal, characteristic
2	C₅×Q₈×C₄⋊C₄	C₂	1	1	yes	maximal, characteristic
3	C₂×Q₈×#168	C₂	1	1	yes	maximal, characteristic
4	#10600381	C₂	1	3	yes	maximal
5	#10600372	C₂	1	3	yes	maximal
6	#10600279	C₂	1	3	yes	maximal
7	C₄×#3942	C₂	1	3	yes	maximal
8	C₂×C₄×C₅×Q₈	—	2	1	no	—
9	C₂×C₄×C₅×Q₈	C₂²	1	1	yes	characteristic
10	C₂×Q₈×Dic₅	—	2	1	no	—
11	#65904	—	2	4	no	—
12	#65904	C₄	1	8	yes	complemented
13	C₂×#4036	—	2	3	no	—
14	C₂×#4015	C₂²	1	3	yes	—
15	C₂×C₄×Dic₁₀	C₂²	1	3	yes	—
16	C₅×#389	—	2	3	no	—
17	C₅×#384	C₂²	1	3	yes	—
18	C₄×C₅×C₄⋊C₄	C₂²	1	3	yes	—
19	Q₈×#168	—	2	4	no	—
20	C₂×#3942	C₂²	1	3	yes	—
21	C₂×#3942	C₂²	1	3	yes	—
22	C₂×#3942	C₂²	1	6	yes	—
23	C₂×C₄⋊#168	C₂²	1	3	yes	—
24	C₂×C₄×#168	C₂²	1	3	yes	—
25	#64749	C₂²	1	1	yes	characteristic
26	#64749	C₂²	1	3	yes	—
27	#22542	—	5	1	no	maximal, Sylow
28	C₂²×C₅×Q₈	D₄	1	1	yes	characteristic
29	C₂²×Dic₁₀	C₂³	1	3	yes	—
30	C₄×C₅×Q₈	—	4	4	no	—
31	C₄×C₅×Q₈	C₂×C₄	1	4	yes	—
32	C₄×C₅×Q₈	D₄	1	4	yes	—
33	C₂×C₅×C₄⋊C₄	—	2	3	no	—
34	C₂×C₅×C₄⋊C₄	—	2	3	no	—
35	C₂×C₅×C₄⋊C₄	D₄	1	3	yes	—
36	C₂×C₅×C₄⋊C₄	C₂³	1	3	yes	—
37	C₂×C₅×C₄⋊C₄	C₂³	1	1	yes	characteristic
38	C₂×C₄²×C₅	—	2	3	no	—
39	C₂×C₄²×C₅	C₂³	1	3	yes	—
40	Q₈×Dic₅	—	2	8	no	—
41	C₂×C₄⋊Dic₅	—	2	3	no	—
42	C₂×C₄⋊Dic₅	—	2	3	no	—
43	C₂×C₄⋊Dic₅	C₂³	1	1	yes	characteristic
44	C₂×#683	—	2	3	no	—
45	C₂×C₄×Dic₅	—	2	3	no	—
46	C₂²×#168	C₂³	1	3	yes	—
47	C₂²×#168	C₂³	1	1	yes	characteristic
48	#4036	—	2	24	no	—
49	#4015	—	2	6	no	—
50	#4015	C₂×C₄	1	12	yes	—
51	C₄×Dic₁₀	—	2	6	no	—
52	C₄×Dic₁₀	C₂×C₄	1	12	yes	—
53	C₅×#95	—	2	3	no	—
54	#3942	—	2	6	no	—
55	#3942	—	2	6	no	—
56	#3942	—	4	6	no	—
57	C₄⋊#168	—	2	6	no	—
58	C₄×#168	—	2	6	no	—
59	C₂×#557	—	5	1	no	—
60	C₂×C₈×Q₈	—	5	1	no	—
61	Q₈×C₄⋊C₄	D₅	1	1	yes	characteristic
62	#1981	—	5	3	no	—
63	#1956	—	5	3	no	—
64	#1803	—	5	3	no	—
65	C₄×#103	—	5	3	no	—
66	C₂×C₅×Q₈	C₂²⋊C₄	1	4	yes	—
67	C₂×C₅×Q₈	SD₁₆	1	4	yes	—
68	C₂×C₅×Q₈	Q₁₆	1	4	yes	—
69	C₂²×C₄×C₅	—	2	1	no	—
70	C₂²×C₄×C₅	—	2	3	no	—
71	C₂²×C₄×C₅	C₂×D₄	1	3	yes	—
72	C₂²×C₄×C₅	C₂×D₄	1	3	yes	—
73	C₂²×C₄×C₅	C₂⁴	1	1	yes	characteristic
74	C₂²×Dic₅	—	2	3	no	—
75	C₂²×Dic₅	—	2	1	no	—
76	C₂×Dic₁₀	—	2	3	no	—
77	C₂×Dic₁₀	—	2	3	no	—
78	C₂×Dic₁₀	—	2	3	no	—
79	C₂×Dic₁₀	—	2	3	no	—
80	C₂×Dic₁₀	—	2	3	no	—
81	C₂×Dic₁₀	C₂²×C₄	1	6	yes	—
82	C₅×C₄⋊C₄	—	2	1	no	—
83	C₅×C₄⋊C₄	—	2	3	no	—
84	C₅×C₄⋊C₄	—	2	3	no	—
85	C₅×C₄⋊C₄	—	4	6	no	—
86	C₅×C₄⋊C₄	—	8	3	no	—
87	C₅×C₄⋊C₄	C₂²⋊C₄	1	12	yes	—
88	C₅×C₄⋊C₄	#41	1	2	yes	—
89	C₄²×C₅	—	4	6	no	—
90	C₄²×C₅	C₂²×C₄	1	6	yes	—
91	C₄²×C₅	C₂×D₄	1	6	yes	—
92	C₄⋊Dic₅	—	2	12	no	—
93	C₄⋊Dic₅	—	2	12	no	—
94	C₄⋊Dic₅	—	2	1	no	—
95	C₄⋊Dic₅	C₂²×C₄	1	2	yes	—
96	#683	—	2	12	no	—
97	C₄×Dic₅	—	2	12	no	—
98	C₂×#168	—	2	1	no	—
99	C₂×#168	—	2	1	no	—
100	C₂×#168	—	2	3	no	—
101	C₂×#168	—	2	3	no	—
102	C₂×#168	—	2	3	no	—
103	C₂×#168	#41	1	2	yes	—
104	#557	—	5	8	no	—
105	#557	—	10	4	no	—
106	C₂×C₄⋊Q₈	—	5	3	no	—
107	C₂×#125	—	10	3	no	—
108	C₂×C₄×Q₈	—	2	1	no	—
109	C₂×C₄×Q₈	—	5	3	no	—
110	C₂×C₄×Q₈	—	10	1	no	—
111	C₂×C₄×Q₈	C₂×D₅	1	1	yes	characteristic
112	C₈×Q₈	—	10	4	no	—
113	C₂×C₄⋊C₈	—	5	3	no	—
114	C₂×#103	—	5	3	no	—
115	C₂×#103	—	5	3	no	—
116	C₂×#103	—	5	6	no	—
117	C₂×C₄×C₈	—	5	3	no	—
118	#389	—	2	3	no	—
119	#384	C₂×D₅	1	3	yes	—
120	C₄×C₄⋊C₄	C₂×D₅	1	3	yes	—
121	#340	—	5	3	no	—
122	#340	—	5	1	no	—
123	C₂³×C₅	C₂²×D₄	1	1	yes	characteristic
124	C₅×Q₈	#103	1	16	yes	complemented
125	C₂×C₄×C₅	—	2	3	no	—
126	C₂×C₄×C₅	—	2	3	no	—
127	C₂×C₄×C₅	—	2	2	no	—
128	C₂×C₄×C₅	—	4	1	no	—
129	C₂×C₄×C₅	—	4	1	no	—
130	C₂×C₄×C₅	—	4	3	no	—
131	C₂×C₄×C₅	—	4	3	no	—
132	C₂×C₄×C₅	—	4	3	no	—
133	C₂×C₄×C₅	C₂×C₂²⋊C₄	1	6	yes	—
134	C₂×C₄×C₅	C₂×C₂²⋊C₄	1	6	yes	—
135	C₂×C₄×C₅	C₂×SD₁₆	1	6	yes	—
136	C₂×C₄×C₅	C₂×Q₁₆	1	6	yes	—
137	C₂×C₄×C₅	C₂³×C₄	1	1	yes	characteristic
138	C₂×C₄×C₅	C₂²×D₄	1	1	yes	characteristic
139	C₂×C₄×C₅	C₂×#41	1	1	yes	characteristic
140	C₂×C₄×C₅	C₂×#41	1	1	yes	characteristic
141	C₂×C₄×C₅	H(5,𝔽₂)	1	1	yes	characteristic
142	C₂×C₄×C₅	2₋¹⁺⁴	1	1	yes	characteristic
143	C₂×Dic₅	—	2	2	no	—
144	C₂×Dic₅	—	2	6	no	—
145	C₂×Dic₅	—	4	3	no	—
146	C₂×Dic₅	—	4	3	no	—
147	C₂×Dic₅	—	4	1	no	—
148	C₂×Dic₅	—	4	1	no	—
149	Dic₁₀	—	2	6	no	—
150	Dic₁₀	—	2	6	no	—
151	Dic₁₀	—	4	3	no	—
152	Dic₁₀	—	4	3	no	—
153	#168	—	2	2	no	—
154	#168	—	4	3	no	—
155	C₂²×Q₈	—	5	3	no	—
156	C₂²×Q₈	C₂²⋊D₅	1	1	yes	characteristic
157	C₂²×C₈	—	5	1	no	—
158	C₂²×C₈	—	5	3	no	—
159	C₄⋊Q₈	—	5	12	no	—
160	C₄⋊Q₈	—	10	6	no	—
161	#125	—	10	24	no	—
162	C₄×Q₈	—	4	4	no	—
163	C₄×Q₈	—	5	12	no	—
164	C₄×Q₈	—	10	8	no	—
165	C₄×Q₈	—	10	6	no	—
166	C₄×Q₈	C₄×D₅	1	4	yes	—
167	C₄×Q₈	D₂₀	1	4	yes	—
168	C₂×C₄⋊C₄	—	2	3	no	—
169	C₂×C₄⋊C₄	—	2	3	no	—
170	C₂×C₄⋊C₄	—	5	1	no	—
171	C₂×C₄⋊C₄	—	10	3	no	—
172	C₂×C₄⋊C₄	—	10	3	no	—
173	C₂×C₄⋊C₄	—	10	3	no	—
174	C₂×C₄⋊C₄	C₂²⋊D₅	1	3	yes	—
175	C₂×C₄⋊C₄	C₂²×D₅	1	3	yes	—
176	C₂×C₄⋊C₄	C₂²×D₅	1	1	yes	characteristic
177	C₂×C₄²	—	2	3	no	—
178	C₂×C₄²	—	10	3	no	—
179	C₂×C₄²	C₂²×D₅	1	3	yes	—
180	C₄⋊C₈	—	10	6	no	—
181	#103	—	10	6	no	—
182	#103	—	10	6	no	—
183	#103	—	20	6	no	—
184	C₄×C₈	—	10	6	no	—
185	#95	—	2	3	no	—
186	C₂²×C₅	C₂²×C₂²⋊C₄	1	1	yes	characteristic
187	C₂²×C₅	#548	1	1	yes	characteristic
188	C₂²×C₅	#550	1	1	yes	characteristic
189	C₂²×C₅	C₂²×SD₁₆	1	1	yes	characteristic
190	C₂²×C₅	C₂²×Q₁₆	1	1	yes	characteristic
191	C₂²×C₅	#577	1	1	yes	characteristic
192	C₂²×C₅	#578	1	1	yes	characteristic
193	C₄×C₅	—	2	6	no	—
194	C₄×C₅	—	4	2	no	—
195	C₄×C₅	—	8	3	no	—
196	C₄×C₅	C₂×#103	1	12	yes	—
197	C₄×C₅	C₄×#41	1	1	yes	characteristic
198	C₄×C₅	#520	1	1	yes	characteristic
199	C₄×C₅	#548	1	1	yes	characteristic
200	C₄×C₅	#550	1	1	yes	characteristic
201	Dic₅	—	4	6	no	—
202	Dic₅	—	4	2	no	—
203	C₂×Q₈	—	5	6	no	—
204	C₂×Q₈	—	10	3	no	—
205	C₂×Q₈	—	10	3	no	—
206	C₂×Q₈	—	10	3	no	—
207	C₂×Q₈	—	10	3	no	—
208	C₂×Q₈	—	10	3	no	—
209	C₂×Q₈	#685	1	4	yes	—
210	C₂×Q₈	#687	1	4	yes	—
211	C₂×Q₈	#689	1	4	yes	—
212	C₂²×C₄	—	2	1	no	—
213	C₂²×C₄	—	2	3	no	—
214	C₂²×C₄	—	10	1	no	—
215	C₂²×C₄	—	10	3	no	—
216	C₂²×C₄	C₂×C₂²⋊D₅	1	3	yes	—
217	C₂²×C₄	C₂×C₂²⋊D₅	1	3	yes	—
218	C₂²×C₄	C₂³×D₅	1	1	yes	characteristic
219	C₂×C₈	—	5	2	no	—
220	C₂×C₈	—	10	1	no	—
221	C₂×C₈	—	10	1	no	—
222	C₂×C₈	—	10	3	no	—
223	C₂×C₈	—	10	3	no	—
224	C₂×C₈	—	10	3	no	—
225	C₄⋊C₄	—	2	1	no	—
226	C₄⋊C₄	—	2	3	no	—
227	C₄⋊C₄	—	2	3	no	—
228	C₄⋊C₄	—	4	6	no	—
229	C₄⋊C₄	—	5	2	no	—
230	C₄⋊C₄	—	8	3	no	—
231	C₄⋊C₄	—	10	1	no	—
232	C₄⋊C₄	—	10	12	no	—
233	C₄⋊C₄	—	10	12	no	—
234	C₄⋊C₄	—	10	12	no	—
235	C₄⋊C₄	#685	1	12	yes	—
236	C₄⋊C₄	#713	1	2	yes	—
237	C₄²	—	4	6	no	—
238	C₄²	—	10	12	no	—
239	C₄²	C₂×C₄×D₅	1	6	yes	—
240	C₄²	C₂×D₂₀	1	6	yes	—
241	C₂×C₅	#2341	1	1	yes	characteristic
242	C₂×C₅	C₂²×#103	1	1	yes	characteristic
243	C₂×C₅	#2898	1	1	yes	characteristic
244	C₂×C₅	#3359	1	1	yes	characteristic
245	C₂×C₅	#3363	1	1	yes	characteristic
246	C₂×C₅	#3392	1	1	yes	characteristic
247	C₂×C₅	#3395	1	1	yes	characteristic
248	C₂³	C₂²×C₂²⋊D₅	1	1	yes	characteristic
249	Q₈	—	10	6	no	—
250	Q₈	—	10	6	no	—
251	Q₈	—	20	3	no	—
252	Q₈	—	20	3	no	—
253	Q₈	#3942	1	16	yes	complemented
254	C₂×C₄	—	2	3	no	—
255	C₂×C₄	—	2	3	no	—
256	C₂×C₄	—	2	2	no	—
257	C₂×C₄	—	4	1	no	—
258	C₂×C₄	—	4	1	no	—
259	C₂×C₄	—	4	3	no	—
260	C₂×C₄	—	4	3	no	—
261	C₂×C₄	—	4	3	no	—
262	C₂×C₄	—	10	2	no	—
263	C₂×C₄	—	10	6	no	—
264	C₂×C₄	—	20	3	no	—
265	C₂×C₄	—	20	3	no	—
266	C₂×C₄	—	20	1	no	—
267	C₂×C₄	—	20	1	no	—
268	C₂×C₄	C₂×#685	1	6	yes	—
269	C₂×C₄	C₂×#685	1	6	yes	—
270	C₂×C₄	C₂×#687	1	6	yes	—
271	C₂×C₄	C₂×#689	1	6	yes	—
272	C₂×C₄	C₂²×C₄×D₅	1	1	yes	characteristic
273	C₂×C₄	C₂²×D₂₀	1	1	yes	characteristic
274	C₂×C₄	C₂×#713	1	1	yes	characteristic
275	C₂×C₄	#4147	1	1	yes	characteristic
276	C₂×C₄	D₅×#41	1	1	yes	characteristic
277	C₂×C₄	#4149	1	1	yes	characteristic
278	C₈	—	10	2	no	—
279	C₈	—	20	3	no	—
280	C₅	#22542	1	1	yes	characteristic, complemented, Sylow
281	C₂²	C₂²×#685	1	1	yes	characteristic
282	C₂²	C₂²×#687	1	1	yes	characteristic
283	C₂²	C₂²×#689	1	1	yes	characteristic
284	C₂²	#66151	1	1	yes	characteristic
285	C₂²	#66168	1	1	yes	characteristic
286	C₂²	#66170	1	1	yes	characteristic
287	C₂²	#66172	1	1	yes	characteristic
288	C₄	—	2	6	no	—
289	C₄	—	4	2	no	—
290	C₄	—	8	3	no	—
291	C₄	—	20	2	no	—
292	C₄	—	20	6	no	—
293	C₄	C₂×#3942	1	12	yes	—
294	C₄	C₄×#713	1	1	yes	characteristic
295	C₄	#65908	1	1	yes	characteristic
296	C₄	#65910	1	1	yes	characteristic
297	C₄	#65911	1	1	yes	characteristic
298	C₂	C₂²×#3942	1	1	yes	characteristic
299	C₂	#10614188	1	1	yes	characteristic
300	C₂	#10614301	1	1	yes	characteristic
301	C₂	#10615299	1	1	yes	characteristic
302	C₂	#10615319	1	1	yes	characteristic
303	C₂	#10615432	1	1	yes	characteristic
304	C₂	#10615438	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	C₂²×C₂²⋊D₅
commutator	C₂×C₄×C₅	C₂³×C₄
Frattini	C₂²×C₄	C₂³×D₅
Fitting	C₅×Q₈×C₄⋊C₄	C₂
socle, prime radical	C₂³×C₅	C₂²×D₄
cosocle	C₂²×C₄×C₅	C₂⁴
hypercenter	Q₈×C₄⋊C₄	D₅
hypocenter	C₅	#22542
LCS	G ▷ C₂×C₄×C₅ ▷ C₂×C₅ ▷ C₅
UCS	C₁ ◁ C₂³ ◁ C₂×C₄×Q₈ ◁ Q₈×C₄⋊C₄

Complements

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

N	H ≅ G/N	type
12. #65904	C₄	semidirect
124. C₅×Q₈	#103	semidirect
253. Q₈	#3942	semidirect
280. C₅	#22542	semidirect

Maximal subgroups

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

subgroup	index
C₂×#65904	2
C₅×Q₈×C₄⋊C₄	2
C₂×Q₈×#168	2
#10600381	2
#10600372	2
#10600279	2
C₄×#3942	2
#22542	5