Subgroups — C₂×Dic₅×#108 #12097398

Order 1280.

1086 conjugacy classes of subgroups.

The subgroup lattice has 318 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₅×#108	C₂	1	1	yes	maximal, characteristic
2	C₂×M₁₆×Dic₅	C₂	1	2	yes	maximal
3	C₂²×#3951	C₂	1	1	yes	maximal, characteristic
4	C₂²×C₈×Dic₅	C₂	1	1	yes	maximal, characteristic
5	C₂×#64778	C₂	1	2	yes	maximal
6	Dic₅×#108	C₂	1	8	yes	maximal, direct product factor
7	C₂²×C₅×M₁₆	C₂²	1	2	yes	—
8	C₂³×C₅×C₈	C₂²	1	1	yes	characteristic
9	C₂²×C₄×Dic₅	C₂²	1	1	yes	characteristic
10	C₂²×#681	C₂²	1	2	yes	—
11	C₂³×#168	C₂²	1	1	yes	characteristic
12	C₂×C₅×#108	C₄	1	8	yes	complemented
13	C₂×C₅×#108	C₂²	1	4	yes	—
14	M₁₆×Dic₅	—	2	4	no	—
15	M₁₆×Dic₅	C₂²	1	8	yes	—
16	C₂×#3951	C₄	1	8	yes	complemented
17	C₂×#3951	C₂²	1	4	yes	—
18	C₂×#3947	—	2	2	no	—
19	C₂×C₈×Dic₅	—	2	2	no	—
20	C₂×C₈×Dic₅	C₄	1	4	yes	—
21	C₂×C₈×Dic₅	C₄	1	4	yes	—
22	C₂×C₈×Dic₅	C₂²	1	4	yes	—
23	#64778	C₂²	1	8	yes	—
24	C₂×C₄×#108	—	5	1	no	maximal, Sylow
25	C₂³×C₄×C₅	C₂³	1	1	yes	characteristic
26	C₂³×Dic₅	Q₈	1	2	yes	—
27	C₂×C₅×M₁₆	—	2	2	no	—
28	C₂×C₅×M₁₆	—	2	4	no	—
29	C₂×C₅×M₁₆	C₂×C₄	1	8	yes	—
30	C₂×C₅×M₁₆	C₂³	1	4	yes	—
31	C₂²×C₅×C₈	—	2	2	no	—
32	C₂²×C₅×C₈	C₂×C₄	1	2	yes	—
33	C₂²×C₅×C₈	C₂×C₄	1	2	yes	—
34	C₂²×C₅×C₈	C₂×C₄	1	4	yes	—
35	C₂²×C₅×C₈	C₂×C₄	1	4	yes	—
36	C₂²×C₅×C₈	C₂³	1	2	yes	—
37	C₂×C₄×Dic₅	C₂×C₄	1	2	yes	—
38	C₂×C₄×Dic₅	C₂×C₄	1	2	yes	—
39	C₂×C₄×Dic₅	D₄	1	1	yes	characteristic
40	C₂×C₄×Dic₅	D₄	1	1	yes	characteristic
41	C₂×C₄×Dic₅	D₄	1	2	yes	—
42	C₂×C₄×Dic₅	Q₈	1	2	yes	—
43	C₂×C₄×Dic₅	C₂³	1	2	yes	—
44	C₂×#681	—	2	2	no	—
45	C₂×#681	—	2	4	no	—
46	C₂×#681	C₂×C₄	1	8	yes	—
47	C₂×#681	C₂³	1	4	yes	—
48	C₂²×#168	—	2	2	no	—
49	C₂²×#168	—	2	2	no	—
50	C₂²×#168	—	2	2	no	—
51	C₂²×#168	C₂×C₄	1	2	yes	—
52	C₂²×#168	C₂×C₄	1	2	yes	—
53	C₂²×#168	C₂³	1	2	yes	—
54	C₅×#108	C₂×C₄	1	16	yes	complemented
55	#3951	C₂×C₄	1	16	yes	complemented
56	#3947	—	2	8	no	—
57	C₈×Dic₅	—	2	8	no	—
58	C₈×Dic₅	C₂×C₄	1	16	yes	—
59	C₂²×#108	—	5	1	no	—
60	C₂²×#108	D₅	1	1	yes	characteristic
61	C₂×C₄×M₁₆	—	5	2	no	—
62	C₂²×C₄×C₈	—	5	1	no	—
63	C₄×#108	—	5	8	no	—
64	C₂×#341	—	5	2	no	—
65	C₂⁴×C₅	C₂×Q₈	1	1	yes	characteristic
66	C₂²×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
67	C₂²×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
68	C₂²×C₄×C₅	C₂²×C₄	1	2	yes	—
69	C₂²×C₄×C₅	C₂²×C₄	1	2	yes	—
70	C₂²×C₄×C₅	C₂×D₄	1	1	yes	characteristic
71	C₂²×C₄×C₅	C₂×D₄	1	1	yes	characteristic
72	C₂²×C₄×C₅	C₂×Q₈	1	1	yes	characteristic
73	C₂²×C₄×C₅	#41	1	2	yes	—
74	C₂²×C₄×C₅	#41	1	2	yes	—
75	C₂²×C₄×C₅	C₂⁴	1	1	yes	characteristic
76	C₂²×Dic₅	—	2	8	no	—
77	C₂²×Dic₅	C₄⋊C₄	1	2	yes	—
78	C₂²×Dic₅	C₄⋊C₄	1	4	yes	—
79	C₂²×Dic₅	C₄⋊C₄	1	2	yes	—
80	C₂²×Dic₅	C₂×Q₈	1	4	yes	—
81	C₅×M₁₆	—	2	8	no	—
82	C₅×M₁₆	—	2	4	no	—
83	C₅×M₁₆	C₂²×C₄	1	8	yes	—
84	C₂×C₅×C₈	—	2	8	no	—
85	C₂×C₅×C₈	—	2	4	no	—
86	C₂×C₅×C₈	C₄²	1	8	yes	—
87	C₂×C₅×C₈	C₄²	1	8	yes	—
88	C₂×C₅×C₈	C₂²×C₄	1	4	yes	—
89	C₂×C₅×C₈	C₂²×C₄	1	4	yes	—
90	C₂×C₅×C₈	C₂²×C₄	1	4	yes	—
91	C₄×Dic₅	C₄⋊C₄	1	2	yes	—
92	C₄×Dic₅	C₄⋊C₄	1	4	yes	—
93	C₄×Dic₅	C₄⋊C₄	1	2	yes	—
94	C₄×Dic₅	C₂²×C₄	1	4	yes	—
95	C₄×Dic₅	C₂×D₄	1	2	yes	—
96	C₄×Dic₅	C₂×D₄	1	2	yes	—
97	#681	—	2	8	no	—
98	#681	—	2	4	no	—
99	#681	C₂²×C₄	1	8	yes	—
100	C₂×#168	—	2	4	no	—
101	C₂×#168	—	2	4	no	—
102	C₂×#168	—	2	8	no	—
103	C₂×#168	—	2	4	no	—
104	C₂×#168	—	2	2	no	—
105	C₂×#168	—	2	2	no	—
106	C₂×#168	C₂²×C₄	1	4	yes	—
107	C₂²×M₁₆	—	5	2	no	—
108	C₂²×M₁₆	C₂×D₅	1	2	yes	—
109	C₂³×C₈	—	5	1	no	—
110	C₂³×C₈	C₂×D₅	1	1	yes	characteristic
111	C₂²×C₄²	—	5	1	no	—
112	C₂×#108	—	5	8	no	—
113	C₂×#108	—	5	4	no	—
114	C₂×#108	Dic₅	1	8	yes	direct product factor
115	C₂×#108	C₂×D₅	1	4	yes	—
116	C₄×M₁₆	—	5	8	no	—
117	C₄×M₁₆	—	10	4	no	—
118	C₂×#97	—	10	2	no	—
119	C₂×C₄×C₈	—	5	4	no	—
120	C₂×C₄×C₈	—	5	4	no	—
121	C₂×C₄×C₈	—	5	4	no	—
122	C₂×C₄×C₈	—	10	2	no	—
123	#341	—	5	8	no	—
124	C₂³×C₅	—	2	4	no	—
125	C₂³×C₅	C₂×C₄⋊C₄	1	1	yes	characteristic
126	C₂³×C₅	C₂×C₄⋊C₄	1	1	yes	characteristic
127	C₂³×C₅	C₄×Q₈	1	2	yes	—
128	C₂³×C₅	C₄×Q₈	1	2	yes	—
129	C₂³×C₅	C₂²×Q₈	1	1	yes	characteristic
130	C₂×C₄×C₅	C₂×C₄²	1	2	yes	—
131	C₂×C₄×C₅	C₂×C₄²	1	2	yes	—
132	C₂×C₄×C₅	C₂×C₄⋊C₄	1	1	yes	characteristic
133	C₂×C₄×C₅	C₂×C₄⋊C₄	1	1	yes	characteristic
134	C₂×C₄×C₅	#117	1	2	yes	—
135	C₂×C₄×C₅	#117	1	2	yes	—
136	C₂×C₄×C₅	C₄×D₄	1	2	yes	—
137	C₂×C₄×C₅	C₄×D₄	1	2	yes	—
138	C₂×C₄×C₅	C₄×D₄	1	2	yes	—
139	C₂×C₄×C₅	C₄×D₄	1	2	yes	—
140	C₂×C₄×C₅	C₄×Q₈	1	2	yes	—
141	C₂×C₄×C₅	C₄×Q₈	1	2	yes	—
142	C₂×C₄×C₅	C₂³×C₄	1	1	yes	characteristic
143	C₂×C₄×C₅	C₂³×C₄	1	1	yes	characteristic
144	C₂×C₄×C₅	C₂³×C₄	1	1	yes	characteristic
145	C₂×C₄×C₅	C₂²×D₄	1	1	yes	characteristic
146	C₂×C₄×C₅	C₂×#41	1	1	yes	characteristic
147	C₂×C₄×C₅	C₂×#41	1	1	yes	characteristic
148	C₂×Dic₅	—	2	4	no	—
149	C₂×Dic₅	—	2	8	no	—
150	C₂×Dic₅	—	2	8	no	—
151	C₂×Dic₅	#108	1	8	yes	direct product factor
152	C₂×Dic₅	C₂×C₄⋊C₄	1	4	yes	—
153	C₂×Dic₅	C₂×C₄⋊C₄	1	4	yes	—
154	C₅×C₈	—	2	8	no	—
155	C₅×C₈	C₂×C₄²	1	8	yes	—
156	#168	—	2	4	no	—
157	#168	—	2	8	no	—
158	C₂³×C₄	—	5	2	no	—
159	C₂³×C₄	C₂²×D₅	1	1	yes	characteristic
160	C₂×M₁₆	—	2	2	no	—
161	C₂×M₁₆	—	2	4	no	—
162	C₂×M₁₆	—	5	8	no	—
163	C₂×M₁₆	—	5	4	no	—
164	C₂×M₁₆	—	10	2	no	—
165	C₂×M₁₆	—	10	4	no	—
166	C₂×M₁₆	C₂×Dic₅	1	8	yes	—
167	C₂×M₁₆	C₂²×D₅	1	4	yes	—
168	C₂²×C₈	—	2	2	no	—
169	C₂²×C₈	—	5	2	no	—
170	C₂²×C₈	—	5	2	no	—
171	C₂²×C₈	—	5	2	no	—
172	C₂²×C₈	—	10	2	no	—
173	C₂²×C₈	—	10	2	no	—
174	C₂²×C₈	—	10	2	no	—
175	C₂²×C₈	C₄×D₅	1	2	yes	—
176	C₂²×C₈	C₄×D₅	1	2	yes	—
177	C₂²×C₈	C₄×D₅	1	4	yes	—
178	C₂²×C₈	C₂×Dic₅	1	4	yes	—
179	C₂²×C₈	C₂²×D₅	1	2	yes	—
180	C₂×C₄²	—	5	2	no	—
181	C₂×C₄²	—	5	2	no	—
182	C₂×C₄²	—	5	2	no	—
183	C₂×C₄²	—	5	1	no	—
184	C₂×C₄²	—	5	1	no	—
185	C₂×C₄²	—	5	2	no	—
186	C₂×C₄²	—	5	2	no	—
187	#108	—	5	16	no	—
188	#108	C₂×Dic₅	1	16	yes	direct product factor
189	#97	—	10	8	no	—
190	C₄×C₈	—	5	16	no	—
191	C₄×C₈	—	10	8	no	—
192	C₂²×C₅	—	2	2	no	—
193	C₂²×C₅	—	2	2	no	—
194	C₂²×C₅	—	2	4	no	—
195	C₂²×C₅	—	2	4	no	—
196	C₂²×C₅	C₄×C₄⋊C₄	1	2	yes	—
197	C₂²×C₅	C₄×C₄⋊C₄	1	2	yes	—
198	C₂²×C₅	C₂×#108	1	4	yes	—
199	C₂²×C₅	C₂²×C₄⋊C₄	1	1	yes	characteristic
200	C₂²×C₅	C₂×C₄×Q₈	1	1	yes	characteristic
201	C₂²×C₅	C₂×C₄×Q₈	1	1	yes	characteristic
202	C₄×C₅	C₄×C₄⋊C₄	1	2	yes	—
203	C₄×C₅	C₄×C₄⋊C₄	1	2	yes	—
204	C₄×C₅	C₂²×C₄²	1	1	yes	characteristic
205	C₄×C₅	C₂×#117	1	1	yes	characteristic
206	C₄×C₅	C₂×C₄×D₄	1	1	yes	characteristic
207	C₄×C₅	C₂×C₄×D₄	1	1	yes	characteristic
208	Dic₅	—	2	4	no	—
209	Dic₅	C₂×#108	1	8	yes	direct product factor
210	C₂⁴	Q₈×D₅	1	1	yes	characteristic
211	C₂²×C₄	—	5	2	no	—
212	C₂²×C₄	—	5	4	no	—
213	C₂²×C₄	—	5	4	no	—
214	C₂²×C₄	—	5	2	no	—
215	C₂²×C₄	—	10	8	no	—
216	C₂²×C₄	C₂×C₄×D₅	1	2	yes	—
217	C₂²×C₄	C₂×C₄×D₅	1	1	yes	characteristic
218	C₂²×C₄	C₂×C₄×D₅	1	1	yes	characteristic
219	C₂²×C₄	D₄×D₅	1	1	yes	characteristic
220	C₂²×C₄	D₄×D₅	1	1	yes	characteristic
221	C₂²×C₄	#711	1	2	yes	—
222	C₂²×C₄	Q₈×D₅	1	1	yes	characteristic
223	C₂²×C₄	#713	1	2	yes	—
224	C₂²×C₄	C₂²×Dic₅	1	2	yes	—
225	C₂²×C₄	C₂³×D₅	1	1	yes	characteristic
226	M₁₆	—	2	4	no	—
227	M₁₆	—	2	8	no	—
228	M₁₆	—	5	8	no	—
229	M₁₆	—	10	4	no	—
230	M₁₆	—	10	8	no	—
231	M₁₆	C₂²×Dic₅	1	8	yes	—
232	C₂×C₈	—	2	4	no	—
233	C₂×C₈	—	2	8	no	—
234	C₂×C₈	—	5	4	no	—
235	C₂×C₈	—	10	4	no	—
236	C₂×C₈	—	10	2	no	—
237	C₂×C₈	—	10	2	no	—
238	C₂×C₈	—	10	4	no	—
239	C₂×C₈	—	10	4	no	—
240	C₂×C₈	—	10	8	no	—
241	C₂×C₈	C₄×Dic₅	1	8	yes	—
242	C₂×C₈	C₄×Dic₅	1	8	yes	—
243	C₂×C₈	C₂×C₄×D₅	1	4	yes	—
244	C₂×C₈	C₂×C₄×D₅	1	4	yes	—
245	C₂×C₈	C₂²×Dic₅	1	4	yes	—
246	C₄²	—	5	4	no	—
247	C₄²	—	5	2	no	—
248	C₄²	—	5	2	no	—
249	C₄²	—	5	2	no	—
250	C₄²	—	5	4	no	—
251	C₄²	—	5	2	no	—
252	C₂×C₅	—	2	1	no	—
253	C₂×C₅	—	2	1	no	—
254	C₂×C₅	—	2	2	no	—
255	C₂×C₅	C₄×#108	1	4	yes	complemented
256	C₂×C₅	C₂×C₄×C₄⋊C₄	1	1	yes	characteristic
257	C₂×C₅	C₂²×#108	1	2	yes	—
258	C₂³	—	2	4	no	—
259	C₂³	#4033	1	2	yes	—
260	C₂³	D₅×C₄⋊C₄	1	1	yes	characteristic
261	C₂³	D₅×C₄⋊C₄	1	1	yes	characteristic
262	C₂³	Q₈×Dic₅	1	2	yes	—
263	C₂³	C₂×Q₈×D₅	1	1	yes	characteristic
264	C₂×C₄	—	5	4	no	—
265	C₂×C₄	—	5	8	no	—
266	C₂×C₄	—	5	4	no	—
267	C₂×C₄	—	10	8	no	—
268	C₂×C₄	—	10	8	no	—
269	C₂×C₄	—	10	4	no	—
270	C₂×C₄	#4033	1	2	yes	—
271	C₂×C₄	D₅×C₄⋊C₄	1	1	yes	characteristic
272	C₂×C₄	D₅×C₄⋊C₄	1	1	yes	characteristic
273	C₂×C₄	#4038	1	2	yes	—
274	C₂×C₄	#4038	1	2	yes	—
275	C₂×C₄	#4039	1	2	yes	—
276	C₂×C₄	#4039	1	2	yes	—
277	C₂×C₄	C₂×C₄×Dic₅	1	2	yes	—
278	C₂×C₄	C₂×C₄×Dic₅	1	2	yes	—
279	C₂×C₄	D₄×Dic₅	1	2	yes	—
280	C₂×C₄	D₄×Dic₅	1	2	yes	—
281	C₂×C₄	Q₈×Dic₅	1	2	yes	—
282	C₂×C₄	C₂²×C₄×D₅	1	1	yes	characteristic
283	C₂×C₄	C₂²×C₄×D₅	1	1	yes	characteristic
284	C₂×C₄	C₂×D₄×D₅	1	1	yes	characteristic
285	C₂×C₄	C₂×#711	1	1	yes	characteristic
286	C₂×C₄	C₂×#713	1	1	yes	characteristic
287	C₂×C₄	C₂³×Dic₅	1	1	yes	characteristic
288	C₈	—	2	8	no	—
289	C₈	—	10	4	no	—
290	C₈	—	10	8	no	—
291	C₈	C₂×C₄×Dic₅	1	8	yes	—
292	C₅	C₂×C₄×#108	1	1	yes	characteristic, complemented, Sylow
293	C₂²	—	2	4	no	—
294	C₂²	—	2	4	no	—
295	C₂²	—	2	2	no	—
296	C₂²	—	2	2	no	—
297	C₂²	D₅×#108	1	4	yes	—
298	C₂²	C₄⋊C₄×Dic₅	1	2	yes	—
299	C₂²	C₄⋊C₄×Dic₅	1	2	yes	—
300	C₂²	C₂×#4033	1	1	yes	characteristic
301	C₂²	C₂×D₅×C₄⋊C₄	1	1	yes	characteristic
302	C₂²	C₂×Q₈×Dic₅	1	1	yes	characteristic
303	C₄	—	5	8	no	—
304	C₄	—	10	4	no	—
305	C₄	C₄⋊C₄×Dic₅	1	2	yes	—
306	C₄	C₄⋊C₄×Dic₅	1	2	yes	—
307	C₄	C₂×#4038	1	1	yes	characteristic
308	C₄	C₂×#4039	1	1	yes	characteristic
309	C₄	C₂²×C₄×Dic₅	1	1	yes	characteristic
310	C₄	C₂×D₄×Dic₅	1	1	yes	characteristic
311	C₂	—	2	1	no	—
312	C₂	—	2	1	no	—
313	C₂	—	2	2	no	—
314	C₂	Dic₅×#108	1	4	yes	direct product factor
315	C₂	C₂×D₅×#108	1	2	yes	—
316	C₂	C₂×C₄⋊C₄×Dic₅	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄	D₄×D₅
commutator	C₄×C₅	C₂²×C₄²
Frattini	C₂²×C₄	C₂³×D₅
Fitting	C₂²×C₅×#108	C₂
socle	C₂³×C₅	C₂×C₄⋊C₄
prime radical	C₂⁴×C₅	C₂×Q₈
cosocle	C₂²×C₄×C₅	C₂⁴
hypercenter	C₂²×#108	D₅
hypocenter	C₅	C₂×C₄×#108
LCS	G ▷ C₄×C₅ ▷ C₂×C₅ ▷ C₅
UCS	C₁ ◁ C₂²×C₄ ◁ C₂³×C₄ ◁ C₂²×#108

Complements

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

N	H ≅ G/N	type
6. Dic₅×#108	C₂	direct (N × H)
12. C₂×C₅×#108	C₄	semidirect
16. C₂×#3951	C₄	semidirect
54. C₅×#108	C₂×C₄	semidirect
55. #3951	C₂×C₄	semidirect
114. C₂×#108	Dic₅	direct (N × H)
151. C₂×Dic₅	#108	direct (N × H)
188. #108	C₂×Dic₅	direct (N × H)
209. Dic₅	C₂×#108	direct (N × H)
255. C₂×C₅	C₄×#108	semidirect
292. C₅	C₂×C₄×#108	semidirect
314. C₂	Dic₅×#108	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂²×C₅×#108	2
C₂×M₁₆×Dic₅	2
C₂²×#3951	2
C₂²×C₈×Dic₅	2
C₂×#64778	2
Dic₅×#108	2
C₂×C₄×#108	5

Subgroups — C2×Dic5×#108 #12097398

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×Dic₅×#108 #12097398