Subgroups — C₂×#10604413 #13043503

Order 1280.

754 conjugacy classes of subgroups.

The subgroup lattice has 336 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×#3998	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂³×#674	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×D₄×#168	C₂	1	1	yes	maximal, characteristic
4	C₂×C₂²⋊#674	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂×C₂²⋊#674	C₂	1	1	yes	maximal, characteristic, complemented
6	#10604413	C₂	1	8	yes	maximal, direct product factor
7	C₂×C₄⋊#674	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×#64859	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂×C₄×C₅×D₄	C₄	1	1	yes	characteristic
10	C₂³×#168	C₂²	1	1	yes	characteristic
11	C₂³×#168	C₂²	1	1	yes	characteristic
12	C₂×#3998	—	2	4	no	—
13	C₂×#3998	C₂²	1	4	yes	complemented
14	C₂²×#674	—	2	4	no	—
15	C₂²×#674	C₂²	1	1	yes	characteristic
16	C₂²×#674	C₂²	1	1	yes	characteristic
17	C₂²×#674	C₂²	1	1	yes	characteristic
18	C₂²×#674	C₂²	1	1	yes	characteristic
19	C₂²×#674	C₂²	1	4	yes	complemented
20	C₂×C₂²⋊#168	C₄	1	1	yes	characteristic
21	C₂×C₂²⋊#168	C₄	1	1	yes	characteristic
22	D₄×#168	C₄	1	4	yes	—
23	D₄×#168	C₂²	1	4	yes	—
24	C₂×C₄⋊#168	C₄	1	1	yes	characteristic
25	C₂×C₄×#168	C₂²	1	1	yes	characteristic
26	C₂²⋊#674	C₂²	1	4	yes	complemented
27	C₂²⋊#674	C₂²	1	4	yes	complemented
28	#64859	C₂²	1	4	yes	complemented
29	C₄⋊#674	C₂²	1	4	yes	complemented
30	C₂×#2168	—	5	1	no	maximal, Sylow
31	C₂²×C₅×D₄	C₈	1	1	yes	characteristic
32	C₂³×C₄×C₅	C₂×C₄	1	1	yes	characteristic
33	C₂³×C₄×C₅	C₂×C₄	1	1	yes	characteristic
34	C₄×C₅×D₄	C₈	1	4	yes	—
35	C₄×C₅×D₄	C₂×C₄	1	4	yes	—
36	C₂×C₅×C₄⋊C₄	C₈	1	1	yes	characteristic
37	C₂×C₅×C₂²⋊C₄	C₈	1	1	yes	characteristic
38	C₂×C₅×C₂²⋊C₄	C₈	1	1	yes	characteristic
39	C₂×C₄²×C₅	C₂×C₄	1	1	yes	characteristic
40	C₂²×#168	—	2	2	no	—
41	C₂²×#168	—	2	2	no	—
42	C₂²×#168	—	2	2	no	—
43	C₂²×#168	—	2	2	no	—
44	C₂²×#168	C₂×C₄	1	2	yes	—
45	C₂²×#168	C₂×C₄	1	2	yes	—
46	C₂²×#168	C₂×C₄	1	1	yes	characteristic
47	C₂²×#168	C₂×C₄	1	1	yes	characteristic
48	C₂²×#168	C₂×C₄	1	1	yes	characteristic
49	C₂²×#168	C₂³	1	2	yes	—
50	C₂²×#168	C₂³	1	2	yes	—
51	C₂²×#168	C₂³	1	1	yes	characteristic
52	#3998	—	2	8	no	—
53	C₂×#674	—	2	2	no	—
54	C₂×#674	—	2	2	no	—
55	C₂×#674	—	2	2	no	—
56	C₂×#674	—	2	8	no	—
57	C₂×#674	D₄	1	4	yes	complemented
58	C₂×#674	C₂³	1	2	yes	—
59	C₂×#674	C₂³	1	2	yes	—
60	C₂×#674	C₂³	1	2	yes	—
61	C₂×#674	C₂³	1	2	yes	—
62	C₂²⋊#168	C₂×C₄	1	4	yes	—
63	C₂²⋊#168	C₂×C₄	1	4	yes	—
64	C₄⋊#168	C₂×C₄	1	4	yes	—
65	C₄×#168	C₂×C₄	1	2	yes	—
66	C₄×#168	C₂³	1	2	yes	—
67	C₂²×M₃₂	—	5	1	no	—
68	C₂³×C₁₆	—	5	1	no	—
69	C₂×C₈×D₄	—	5	1	no	—
70	#2168	—	5	8	no	—
71	C₂×C₄⋊C₁₆	—	5	1	no	—
72	C₂×C₂²⋊C₁₆	—	5	1	no	—
73	C₂×C₂²⋊C₁₆	—	5	1	no	—
74	C₂×#346	—	5	1	no	—
75	C₂⁴×C₅	C₂×C₈	1	1	yes	characteristic
76	C₂⁴×C₅	C₂×C₈	1	1	yes	characteristic
77	C₂×C₅×D₄	—	2	4	no	—
78	C₂×C₅×D₄	C₂×C₈	1	4	yes	—
79	C₂²×C₄×C₅	—	2	2	no	—
80	C₂²×C₄×C₅	—	2	2	no	—
81	C₂²×C₄×C₅	—	2	2	no	—
82	C₂²×C₄×C₅	—	2	2	no	—
83	C₂²×C₄×C₅	C₂×C₈	1	1	yes	characteristic
84	C₂²×C₄×C₅	C₂×C₈	1	1	yes	characteristic
85	C₂²×C₄×C₅	C₂×C₈	1	1	yes	characteristic
86	C₂²×C₄×C₅	C₂×C₈	1	1	yes	characteristic
87	C₂²×C₄×C₅	C₂×C₈	1	2	yes	—
88	C₂²×C₄×C₅	C₂×C₈	1	2	yes	—
89	C₂²×C₄×C₅	C₂²×C₄	1	2	yes	—
90	C₂²×C₄×C₅	C₂²×C₄	1	2	yes	—
91	C₂²×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
92	C₅×C₄⋊C₄	C₂×C₈	1	4	yes	—
93	C₅×C₂²⋊C₄	C₂×C₈	1	4	yes	—
94	C₅×C₂²⋊C₄	C₂×C₈	1	4	yes	—
95	C₄²×C₅	C₂×C₈	1	2	yes	—
96	C₄²×C₅	C₂²×C₄	1	2	yes	—
97	C₂×#168	—	2	2	no	—
98	C₂×#168	—	2	2	no	—
99	C₂×#168	—	2	2	no	—
100	C₂×#168	—	2	4	no	—
101	C₂×#168	—	2	4	no	—
102	C₂×#168	—	2	2	no	—
103	C₂×#168	—	2	2	no	—
104	C₂×#168	—	2	2	no	—
105	C₂×#168	—	2	2	no	—
106	C₂×#168	C₂²×C₄	1	2	yes	—
107	C₂×#168	C₂²×C₄	1	2	yes	—
108	C₂×#168	C₂²×C₄	1	2	yes	—
109	C₂×#168	C₂²×C₄	1	1	yes	characteristic
110	C₂×#168	C₂×D₄	1	2	yes	—
111	C₂×#168	#41	1	2	yes	—
112	C₂×#168	C₂⁴	1	1	yes	characteristic
113	#674	—	2	2	no	—
114	#674	—	2	2	no	—
115	#674	—	2	2	no	—
116	#674	C₂×D₄	1	4	yes	complemented
117	C₂³×C₈	—	5	1	no	—
118	C₂³×C₈	—	5	1	no	—
119	C₂×C₄×D₄	AGL(1,𝔽₅)	1	1	yes	characteristic
120	C₂×M₃₂	—	5	4	no	—
121	C₂×M₃₂	—	10	4	no	—
122	C₂²×C₁₆	—	5	1	no	—
123	C₂²×C₁₆	—	5	1	no	—
124	C₂²×C₁₆	—	5	1	no	—
125	C₂²×C₁₆	—	5	1	no	—
126	C₂²×C₁₆	—	5	4	no	—
127	C₂²×C₁₆	—	10	4	no	—
128	C₈×D₄	—	5	4	no	—
129	C₈×D₄	—	5	4	no	—
130	C₂×C₄⋊C₈	—	5	1	no	—
131	C₂×C₂²⋊C₈	—	5	1	no	—
132	C₂×C₂²⋊C₈	—	5	1	no	—
133	C₂×C₄×C₈	—	5	1	no	—
134	C₄⋊C₁₆	—	5	4	no	—
135	C₂²⋊C₁₆	—	5	4	no	—
136	C₂²⋊C₁₆	—	5	4	no	—
137	#346	—	5	4	no	—
138	C₂³×C₅	—	2	2	no	—
139	C₂³×C₅	—	2	2	no	—
140	C₂³×C₅	—	2	2	no	—
141	C₂³×C₅	—	2	2	no	—
142	C₂³×C₅	—	2	2	no	—
143	C₂³×C₅	M₃₂	1	4	yes	complemented
144	C₂³×C₅	C₂²×C₈	1	2	yes	—
145	C₂³×C₅	C₂²×C₈	1	2	yes	—
146	C₂³×C₅	C₂²×C₈	1	1	yes	characteristic
147	C₅×D₄	—	2	8	no	—
148	C₂×C₄×C₅	—	2	2	no	—
149	C₂×C₄×C₅	—	2	2	no	—
150	C₂×C₄×C₅	—	2	2	no	—
151	C₂×C₄×C₅	—	2	2	no	—
152	C₂×C₄×C₅	—	2	2	no	—
153	C₂×C₄×C₅	—	2	2	no	—
154	C₂×C₄×C₅	—	2	2	no	—
155	C₂×C₄×C₅	—	2	2	no	—
156	C₂×C₄×C₅	—	2	4	no	—
157	C₂×C₄×C₅	—	2	4	no	—
158	C₂×C₄×C₅	C₄×D₄	1	2	yes	—
159	C₂×C₄×C₅	C₂²×C₈	1	2	yes	—
160	C₂×C₄×C₅	C₂²×C₈	1	2	yes	—
161	C₂×C₄×C₅	C₂²×C₈	1	2	yes	—
162	C₂×C₄×C₅	C₂²×C₈	1	2	yes	—
163	C₂×C₄×C₅	C₂²×C₈	1	1	yes	characteristic
164	C₂×C₄×C₅	#131	1	2	yes	—
165	C₂×C₄×C₅	C₂³×C₄	1	1	yes	characteristic
166	#168	—	2	2	no	—
167	#168	—	2	2	no	—
168	#168	—	2	2	no	—
169	#168	C₄×D₄	1	2	yes	—
170	#168	C₂²×D₄	1	1	yes	characteristic
171	#168	C₂×#41	1	1	yes	characteristic
172	C₂²×D₄	#170	1	1	yes	characteristic
173	C₂³×C₄	C₂×AGL(1,𝔽₅)	1	1	yes	characteristic
174	C₂³×C₄	C₂×AGL(1,𝔽₅)	1	1	yes	characteristic
175	C₂²×C₈	—	5	1	no	—
176	C₂²×C₈	—	5	2	no	—
177	C₂²×C₈	—	5	2	no	—
178	C₂²×C₈	—	5	2	no	—
179	C₂²×C₈	—	5	2	no	—
180	C₂²×C₈	—	5	1	no	—
181	C₂²×C₈	—	5	1	no	—
182	C₂²×C₈	—	5	1	no	—
183	C₂²×C₈	—	10	2	no	—
184	C₂²×C₈	—	10	2	no	—
185	C₂²×C₈	—	10	2	no	—
186	C₂²×C₈	—	10	2	no	—
187	C₄×D₄	#170	1	4	yes	—
188	C₄×D₄	C₂×AGL(1,𝔽₅)	1	4	yes	—
189	C₂×C₄⋊C₄	#170	1	1	yes	characteristic
190	C₂×C₂²⋊C₄	#170	1	1	yes	characteristic
191	C₂×C₂²⋊C₄	#170	1	1	yes	characteristic
192	C₂×C₄²	C₂×AGL(1,𝔽₅)	1	1	yes	characteristic
193	M₃₂	—	10	8	no	—
194	C₂×C₁₆	—	5	2	no	—
195	C₂×C₁₆	—	5	4	no	—
196	C₂×C₁₆	—	5	2	no	—
197	C₂×C₁₆	—	5	2	no	—
198	C₂×C₁₆	—	5	2	no	—
199	C₂×C₁₆	—	10	8	no	—
200	C₂×C₁₆	—	10	2	no	—
201	C₂×C₁₆	—	10	2	no	—
202	C₂×C₁₆	—	10	2	no	—
203	C₄⋊C₈	—	5	4	no	—
204	C₂²⋊C₈	—	5	4	no	—
205	C₂²⋊C₈	—	5	4	no	—
206	C₄×C₈	—	5	2	no	—
207	C₄×C₈	—	5	2	no	—
208	C₂²×C₅	—	2	2	no	—
209	C₂²×C₅	—	2	2	no	—
210	C₂²×C₅	—	2	8	no	—
211	C₂²×C₅	—	2	2	no	—
212	C₂²×C₅	—	2	2	no	—
213	C₂²×C₅	—	2	2	no	—
214	C₂²×C₅	—	4	4	no	—
215	C₂²×C₅	C₈×D₄	1	2	yes	—
216	C₂²×C₅	C₂×M₃₂	1	2	yes	—
217	C₂²×C₅	C₂×M₃₂	1	4	yes	complemented
218	C₂²×C₅	#504	1	2	yes	—
219	C₂²×C₅	C₂³×C₈	1	1	yes	characteristic
220	C₄×C₅	—	2	2	no	—
221	C₄×C₅	—	2	2	no	—
222	C₄×C₅	—	2	2	no	—
223	C₄×C₅	—	2	2	no	—
224	C₄×C₅	C₈×D₄	1	2	yes	—
225	C₄×C₅	C₂×C₄×D₄	1	1	yes	characteristic
226	C₄×C₅	C₂×#131	1	1	yes	characteristic
227	C₂⁴	C₂×#170	1	1	yes	characteristic
228	C₂⁴	C₂×#170	1	1	yes	characteristic
229	C₂×D₄	—	2	4	no	—
230	C₂×D₄	C₂×#170	1	4	yes	—
231	C₂²×C₄	—	2	2	no	—
232	C₂²×C₄	—	2	2	no	—
233	C₂²×C₄	—	2	2	no	—
234	C₂²×C₄	—	2	2	no	—
235	C₂²×C₄	C₂×#170	1	1	yes	characteristic
236	C₂²×C₄	C₂×#170	1	1	yes	characteristic
237	C₂²×C₄	C₂×#170	1	1	yes	characteristic
238	C₂²×C₄	C₂×#170	1	1	yes	characteristic
239	C₂²×C₄	C₂×#170	1	2	yes	—
240	C₂²×C₄	C₂×#170	1	2	yes	—
241	C₂²×C₄	C₂²×AGL(1,𝔽₅)	1	2	yes	—
242	C₂²×C₄	C₂²×AGL(1,𝔽₅)	1	2	yes	—
243	C₂²×C₄	C₂²×AGL(1,𝔽₅)	1	1	yes	characteristic
244	C₂×C₈	—	5	2	no	—
245	C₂×C₈	—	5	2	no	—
246	C₂×C₈	—	5	1	no	—
247	C₂×C₈	—	5	1	no	—
248	C₂×C₈	—	5	2	no	—
249	C₂×C₈	—	5	2	no	—
250	C₂×C₈	—	5	2	no	—
251	C₂×C₈	—	10	2	no	—
252	C₂×C₈	—	10	2	no	—
253	C₂×C₈	—	10	2	no	—
254	C₂×C₈	—	10	2	no	—
255	C₂×C₈	—	10	4	no	—
256	C₂×C₈	—	10	4	no	—
257	C₂×C₈	—	10	2	no	—
258	C₂×C₈	—	10	2	no	—
259	C₂×C₈	—	10	2	no	—
260	C₄⋊C₄	C₂×#170	1	4	yes	—
261	C₂²⋊C₄	C₂×#170	1	4	yes	—
262	C₂²⋊C₄	C₂×#170	1	4	yes	—
263	C₄²	C₂×#170	1	2	yes	—
264	C₄²	C₂²×AGL(1,𝔽₅)	1	2	yes	—
265	C₁₆	—	5	4	no	—
266	C₁₆	—	10	2	no	—
267	C₁₆	—	10	2	no	—
268	C₁₆	—	10	2	no	—
269	C₂×C₅	—	2	4	no	—
270	C₂×C₅	—	4	2	no	—
271	C₂×C₅	#2168	1	4	yes	complemented
272	C₂×C₅	C₂×C₈×D₄	1	1	yes	characteristic
273	C₂×C₅	C₂²×M₃₂	1	1	yes	characteristic
274	C₂×C₅	C₂×#504	1	1	yes	characteristic
275	C₂³	—	2	2	no	—
276	C₂³	—	2	2	no	—
277	C₂³	—	2	2	no	—
278	C₂³	—	2	2	no	—
279	C₂³	—	2	2	no	—
280	C₂³	#3998	1	4	yes	complemented
281	C₂³	C₂²×#170	1	2	yes	—
282	C₂³	C₂²×#170	1	2	yes	—
283	C₂³	C₂²×#170	1	1	yes	characteristic
284	D₄	—	2	8	no	—
285	C₂×C₄	—	2	2	no	—
286	C₂×C₄	—	2	2	no	—
287	C₂×C₄	—	2	2	no	—
288	C₂×C₄	—	2	2	no	—
289	C₂×C₄	—	2	2	no	—
290	C₂×C₄	—	2	2	no	—
291	C₂×C₄	—	2	2	no	—
292	C₂×C₄	—	2	2	no	—
293	C₂×C₄	—	2	4	no	—
294	C₂×C₄	—	2	4	no	—
295	C₂×C₄	#4131	1	2	yes	—
296	C₂×C₄	D₄×AGL(1,𝔽₅)	1	2	yes	—
297	C₂×C₄	C₂²×#170	1	2	yes	—
298	C₂×C₄	C₂²×#170	1	2	yes	—
299	C₂×C₄	C₂²×#170	1	2	yes	—
300	C₂×C₄	C₂²×#170	1	2	yes	—
301	C₂×C₄	C₂²×#170	1	1	yes	characteristic
302	C₂×C₄	C₂³×AGL(1,𝔽₅)	1	1	yes	characteristic
303	C₈	—	5	1	no	—
304	C₈	—	5	1	no	—
305	C₈	—	5	2	no	—
306	C₈	—	10	2	no	—
307	C₈	—	10	2	no	—
308	C₈	—	10	2	no	—
309	C₅	C₂×#2168	1	1	yes	characteristic, complemented, Sylow
310	C₂²	—	2	8	no	—
311	C₂²	—	2	2	no	—
312	C₂²	—	2	2	no	—
313	C₂²	—	2	2	no	—
314	C₂²	—	2	2	no	—
315	C₂²	—	2	2	no	—
316	C₂²	—	4	4	no	—
317	C₂²	C₂×#3998	1	2	yes	—
318	C₂²	C₂×#3998	1	4	yes	complemented
319	C₂²	D₄×#170	1	2	yes	—
320	C₂²	#65791	1	2	yes	—
321	C₂²	C₂³×#170	1	1	yes	characteristic
322	C₄	—	2	2	no	—
323	C₄	—	2	2	no	—
324	C₄	—	2	2	no	—
325	C₄	—	2	2	no	—
326	C₄	D₄×#170	1	2	yes	—
327	C₄	C₂×#4131	1	1	yes	characteristic
328	C₄	C₂×D₄×AGL(1,𝔽₅)	1	1	yes	characteristic
329	C₂	—	2	4	no	—
330	C₂	—	4	2	no	—
331	C₂	#10604413	1	4	yes	direct product factor
332	C₂	C₂²×#3998	1	1	yes	characteristic
333	C₂	C₂×D₄×#170	1	1	yes	characteristic
334	C₂	C₂×#65791	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄	C₂²×AGL(1,𝔽₅)
commutator	C₂²×C₅	C₂³×C₈
Frattini	C₂×C₄	C₂³×AGL(1,𝔽₅)
Fitting	C₂×C₄×C₅×D₄	C₄
socle	C₂³×C₅	C₂²×C₈
prime radical	C₂²×C₅×D₄	C₈
cosocle	C₂×#168	C₂⁴
hypercenter	C₂×C₄×D₄	AGL(1,𝔽₅)
hypocenter	C₅	C₂×#2168
LCS	G ▷ C₂²×C₅ ▷ C₅
UCS	C₁ ◁ C₂²×C₄ ◁ C₂×C₄×D₄

Complements

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

N	H ≅ G/N	type
1. C₂²×#3998	C₂	semidirect
2. C₂³×#674	C₂	semidirect
4. C₂×C₂²⋊#674	C₂	semidirect
5. C₂×C₂²⋊#674	C₂	semidirect
6. #10604413	C₂	direct (N × H)
7. C₂×C₄⋊#674	C₂	semidirect
8. C₂×#64859	C₂	semidirect
13. C₂×#3998	C₂²	semidirect
19. C₂²×#674	C₂²	semidirect
26. C₂²⋊#674	C₂²	semidirect
27. C₂²⋊#674	C₂²	semidirect
28. #64859	C₂²	semidirect
29. C₄⋊#674	C₂²	semidirect
57. C₂×#674	D₄	semidirect
116. #674	C₂×D₄	semidirect
143. C₂³×C₅	M₃₂	semidirect
217. C₂²×C₅	C₂×M₃₂	semidirect
271. C₂×C₅	#2168	semidirect
280. C₂³	#3998	semidirect
309. C₅	C₂×#2168	semidirect
318. C₂²	C₂×#3998	semidirect
331. C₂	#10604413	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×#3998	2
C₂³×#674	2
C₂×D₄×#168	2
C₂×C₂²⋊#674	2
#10604413	2
C₂×C₄⋊#674	2
C₂×#64859	2
C₂×#2168	5

Subgroups — C2×#10604413 #13043503

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#10604413 #13043503