Subgroups — C₂×#30632491 #44565117

Order 32128.

1512 conjugacy classes of subgroups.

The subgroup lattice has 416 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#22548761	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×#22548760	C₂	1	1	yes	maximal, characteristic, complemented
3	#30632491	C₂	1	16	yes	maximal, direct product factor
4	C₂×#22548743	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂×#22548742	C₂	1	2	yes	maximal, complemented
6	C₂×#22548739	C₂	1	2	yes	maximal, complemented
7	C₂×#22548736	C₂	1	2	yes	maximal, complemented
8	C₂×#22548735	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂²×#16860795	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂×D₈×D₂₅₁	C₂	1	2	yes	maximal, complemented
11	C₂×#22548723	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×AGL(1,ℤ/8)×C₂₅₁	C₂	1	1	yes	maximal, characteristic, complemented
13	#22548761	—	2	4	no	—
14	#22548761	C₂²	1	8	yes	complemented
15	#22548760	—	2	4	no	—
16	#22548760	C₂²	1	8	yes	complemented
17	C₂×#16860811	C₂²	1	2	yes	complemented
18	C₂×#16860810	—	2	1	no	—
19	C₂×#16860810	C₂²	1	1	yes	characteristic
20	C₂×#16860810	C₂²	1	1	yes	characteristic, complemented
21	C₂×#16860809	—	2	1	no	—
22	C₂×#16860809	C₂²	1	2	yes	complemented
23	C₂²×D₁₀₀₄	—	2	1	no	—
24	C₂²×D₁₀₀₄	C₂²	1	1	yes	characteristic, complemented
25	C₂²×C₂²⋊D₂₅₁	—	2	2	no	—
26	C₂×D₄×D₂₅₁	—	2	1	no	—
27	C₂×D₄×D₂₅₁	—	2	2	no	—
28	C₂×D₄×D₂₅₁	C₂²	1	2	yes	complemented
29	C₂×D₄×D₂₅₁	C₂²	1	2	yes	complemented
30	#22548743	C₂²	1	8	yes	complemented
31	#22548742	C₂²	1	16	yes	complemented
32	#22548739	C₂²	1	16	yes	complemented
33	#22548736	C₂²	1	16	yes	complemented
34	#22548735	C₂²	1	8	yes	complemented
35	C₂×#16860800	C₂²	1	1	yes	characteristic, complemented
36	C₂×#16860799	C₂²	1	2	yes	complemented
37	C₂×#16860798	C₂²	1	1	yes	characteristic, complemented
38	C₂×#16860797	C₂²	1	1	yes	characteristic, complemented
39	C₂×D₂₀₀₈	C₂²	1	2	yes	complemented
40	C₂×#16860795	C₂²	1	1	yes	characteristic, complemented
41	C₂×#16860795	C₂²	1	2	yes	complemented
42	C₂×#16860795	C₂²	1	2	yes	complemented
43	C₂×#16860795	C₂²	1	8	yes	complemented
44	D₈×D₂₅₁	C₂²	1	16	yes	complemented
45	#22548723	C₂²	1	8	yes	complemented
46	C₂×#16860793	C₂²	1	2	yes	complemented
47	C₂²×C₂₅₁⋊C₈	C₂²	1	1	yes	characteristic, complemented
48	C₂×C₈×D₂₅₁	C₂²	1	2	yes	complemented
49	C₂×#41×C₂₅₁	C₂²	1	1	yes	characteristic, complemented
50	C₂²×D₄×C₂₅₁	C₂²	1	1	yes	characteristic, complemented
51	AGL(1,ℤ/8)×C₂₅₁	C₂²	1	8	yes	complemented
52	C₂×SD₁₆×C₂₅₁	C₂²	1	2	yes	complemented
53	C₂×D₈×C₂₅₁	C₂²	1	2	yes	complemented
54	C₂×M₁₆×C₂₅₁	C₂²	1	1	yes	characteristic, complemented
55	C₂³×D₂₅₁	—	2	2	no	—
56	C₂³×D₂₅₁	—	2	2	no	—
57	C₂³×D₂₅₁	—	2	1	no	—
58	#16860811	—	2	4	no	—
59	#16860811	C₂³	1	8	yes	complemented
60	#16860810	—	2	8	no	—
61	#16860810	—	2	2	no	—
62	#16860810	D₄	1	4	yes	—
63	#16860810	C₂³	1	4	yes	—
64	#16860810	C₂³	1	4	yes	complemented
65	#16860809	—	2	8	no	—
66	#16860809	—	2	4	no	—
67	#16860809	C₂³	1	8	yes	complemented
68	C₂×Dic₅₀₂	D₄	1	1	yes	characteristic
69	C₂×Dic₅₀₂	C₂³	1	1	yes	characteristic
70	C₂×D₁₀₀₄	—	2	2	no	—
71	C₂×D₁₀₀₄	—	2	8	no	—
72	C₂×D₁₀₀₄	—	2	1	no	—
73	C₂×D₁₀₀₄	—	2	2	no	—
74	C₂×D₁₀₀₄	D₄	1	1	yes	characteristic
75	C₂×D₁₀₀₄	C₂³	1	1	yes	characteristic
76	C₂×D₁₀₀₄	C₂³	1	2	yes	—
77	C₂×D₁₀₀₄	C₂³	1	4	yes	complemented
78	C₂×C₂²⋊D₂₅₁	—	2	2	no	—
79	C₂×C₂²⋊D₂₅₁	—	2	16	no	—
80	C₂×C₂²⋊D₂₅₁	—	2	2	no	—
81	C₂×C₂²⋊D₂₅₁	—	2	2	no	—
82	C₂×C₂²⋊D₂₅₁	—	2	1	no	—
83	C₂×C₂²⋊D₂₅₁	D₄	1	2	yes	—
84	D₄×D₂₅₁	—	2	8	no	—
85	D₄×D₂₅₁	—	2	16	no	—
86	D₄×D₂₅₁	—	2	4	no	—
87	D₄×D₂₅₁	—	2	4	no	—
88	D₄×D₂₅₁	C₂³	1	8	yes	complemented
89	D₄×D₂₅₁	C₂³	1	8	yes	complemented
90	C₂²×Dic₂₅₁	—	2	2	no	—
91	C₂×C₄×D₂₅₁	—	2	2	no	—
92	C₂×C₄×D₂₅₁	—	2	1	no	—
93	C₂×C₄×D₂₅₁	C₂³	1	2	yes	—
94	#16860800	C₂³	1	4	yes	complemented
95	#16860799	C₂³	1	8	yes	complemented
96	#16860798	C₂³	1	4	yes	complemented
97	#16860797	C₂³	1	4	yes	complemented
98	D₂₀₀₈	C₂³	1	8	yes	complemented
99	#16860795	C₂³	1	8	yes	complemented
100	#16860795	C₂³	1	8	yes	complemented
101	#16860795	C₂³	1	4	yes	complemented
102	#16860793	C₂³	1	8	yes	complemented
103	C₂×C₂₅₁⋊C₈	C₂³	1	1	yes	characteristic
104	C₂×C₂₅₁⋊C₈	C₂³	1	1	yes	characteristic
105	C₂×C₂₅₁⋊C₈	C₂³	1	4	yes	complemented
106	C₈×D₂₅₁	C₂³	1	8	yes	complemented
107	C₂⁴×C₂₅₁	—	2	1	no	—
108	#41×C₂₅₁	—	2	2	no	—
109	#41×C₂₅₁	C₂³	1	4	yes	complemented
110	C₂×Q₈×C₂₅₁	C₂³	1	1	yes	characteristic
111	C₂×D₄×C₂₅₁	—	2	1	no	—
112	C₂×D₄×C₂₅₁	—	2	1	no	—
113	C₂×D₄×C₂₅₁	—	2	2	no	—
114	C₂×D₄×C₂₅₁	C₂³	1	1	yes	characteristic
115	C₂×D₄×C₂₅₁	C₂³	1	2	yes	—
116	C₂×D₄×C₂₅₁	C₂³	1	4	yes	complemented
117	C₂²×C₄×C₂₅₁	—	2	1	no	—
118	C₂²×C₄×C₂₅₁	C₂³	1	1	yes	characteristic
119	SD₁₆×C₂₅₁	C₂³	1	8	yes	complemented
120	D₈×C₂₅₁	C₂³	1	8	yes	complemented
121	M₁₆×C₂₅₁	C₂³	1	4	yes	complemented
122	C₂×C₈×C₂₅₁	C₂³	1	2	yes	—
123	C₂²×D₂₅₁	—	2	1	no	—
124	C₂²×D₂₅₁	—	2	2	no	—
125	C₂²×D₂₅₁	—	2	4	no	—
126	C₂²×D₂₅₁	—	2	8	no	—
127	C₂²×D₂₅₁	—	2	8	no	—
128	C₂²×D₂₅₁	—	8	4	no	—
129	C₂²×D₂₅₁	—	8	4	no	—
130	C₂²×D₂₅₁	—	8	2	no	—
131	C₂²×D₂₅₁	C₂×D₄	1	2	yes	—
132	Dic₅₀₂	—	2	1	no	—
133	Dic₅₀₂	C₂×D₄	1	4	yes	—
134	Dic₅₀₂	C₂⁴	1	2	yes	—
135	D₁₀₀₄	—	2	8	no	—
136	D₁₀₀₄	—	2	1	no	—
137	D₁₀₀₄	—	2	2	no	—
138	D₁₀₀₄	—	2	4	no	—
139	D₁₀₀₄	C₂×D₄	1	4	yes	—
140	D₁₀₀₄	C₂⁴	1	4	yes	—
141	D₁₀₀₄	C₂⁴	1	2	yes	—
142	C₂²⋊D₂₅₁	—	2	4	no	—
143	C₂²⋊D₂₅₁	—	2	8	no	—
144	C₂²⋊D₂₅₁	—	2	8	no	—
145	C₂²⋊D₂₅₁	—	2	8	no	—
146	C₂²⋊D₂₅₁	C₂×D₄	1	8	yes	—
147	C₂×Dic₂₅₁	—	2	1	no	—
148	C₂×Dic₂₅₁	—	2	8	no	—
149	C₂×Dic₂₅₁	C₂×D₄	1	2	yes	—
150	C₄×D₂₅₁	—	2	4	no	—
151	C₄×D₂₅₁	—	2	8	no	—
152	C₄×D₂₅₁	C₂×D₄	1	4	yes	—
153	C₄×D₂₅₁	C₂⁴	1	4	yes	—
154	C₂₅₁⋊C₈	C₂⁴	1	2	yes	—
155	C₂₅₁⋊C₈	C₂⁴	1	2	yes	—
156	C₂³×C₂₅₁	—	2	1	no	—
157	C₂³×C₂₅₁	—	2	2	no	—
158	C₂³×C₂₅₁	—	2	4	no	—
159	C₂³×C₂₅₁	—	8	2	no	—
160	C₂³×C₂₅₁	C₂×D₄	1	1	yes	characteristic
161	Q₈×C₂₅₁	—	2	1	no	—
162	Q₈×C₂₅₁	C₂⁴	1	2	yes	—
163	D₄×C₂₅₁	—	2	4	no	—
164	D₄×C₂₅₁	—	2	2	no	—
165	D₄×C₂₅₁	—	2	4	no	—
166	D₄×C₂₅₁	—	2	1	no	—
167	D₄×C₂₅₁	C₂⁴	1	2	yes	—
168	D₄×C₂₅₁	C₂⁴	1	4	yes	—
169	C₂×C₄×C₂₅₁	—	2	1	no	—
170	C₂×C₄×C₂₅₁	—	2	4	no	—
171	C₂×C₄×C₂₅₁	C₂×D₄	1	1	yes	characteristic
172	C₂×C₄×C₂₅₁	C₂×D₄	1	2	yes	—
173	C₂×C₄×C₂₅₁	C₂⁴	1	1	yes	characteristic
174	C₂×C₄×C₂₅₁	C₂⁴	1	2	yes	—
175	C₈×C₂₅₁	C₂⁴	1	4	yes	—
176	C₂²×C₂₅₁	—	2	2	no	—
177	C₂²×C₂₅₁	—	2	2	no	—
178	C₂²×C₂₅₁	—	2	4	no	—
179	C₂²×C₂₅₁	—	4	4	no	—
180	C₂²×C₂₅₁	—	4	2	no	—
181	C₂²×C₂₅₁	—	8	4	no	—
182	C₂²×C₂₅₁	C₂²×D₄	1	2	yes	—
183	C₂²×C₂₅₁	C₂²×D₄	1	1	yes	characteristic
184	C₂×D₂₅₁	—	2	4	no	—
185	C₂×D₂₅₁	—	2	2	no	—
186	C₂×D₂₅₁	—	2	4	no	—
187	C₂×D₂₅₁	—	4	2	no	—
188	C₂×D₂₅₁	—	4	4	no	—
189	C₂×D₂₅₁	—	8	8	no	—
190	C₂×D₂₅₁	—	8	8	no	—
191	C₂×D₂₅₁	—	8	4	no	—
192	C₂×D₂₅₁	C₂²×D₄	1	4	yes	—
193	C₄×C₂₅₁	—	2	2	no	—
194	C₄×C₂₅₁	C₂²×D₄	1	2	yes	—
195	C₄×C₂₅₁	C₂²×D₄	1	1	yes	characteristic
196	C₄×C₂₅₁	C₂⁵	1	1	yes	characteristic
197	Dic₂₅₁	—	2	2	no	—
198	Dic₂₅₁	C₂²×D₄	1	4	yes	—
199	C₂×C₂₅₁	—	2	2	no	—
200	C₂×C₂₅₁	—	4	2	no	—
201	C₂×C₂₅₁	—	4	4	no	—
202	C₂×C₂₅₁	#576	1	2	yes	complemented
203	C₂×C₂₅₁	C₂³×D₄	1	1	yes	characteristic
204	D₂₅₁	—	2	4	no	—
205	D₂₅₁	—	4	2	no	—
206	D₂₅₁	—	4	4	no	—
207	C₂₅₁	C₂×#576	1	1	yes	characteristic, complemented, Sylow
208	C₂×#576	—	251	1	no	maximal, Sylow
209	C₂×H(5,𝔽₂)	—	251	1	no	—
210	C₂×H(5,𝔽₂)	—	251	1	no	—
211	#576	—	251	16	no	—
212	C₂×AGL(1,ℤ/8)	—	251	2	no	—
213	C₂×AGL(1,ℤ/8)	—	251	2	no	—
214	C₂×AGL(1,ℤ/8)	—	251	1	no	—
215	C₂×AGL(1,ℤ/8)	D₂₅₁	1	1	yes	characteristic, complemented
216	C₂×#135	—	251	1	no	—
217	C₂×#135	—	251	2	no	—
218	C₂²×D₈	—	251	1	no	—
219	C₂²×D₈	—	251	2	no	—
220	C₂×#131	—	251	1	no	—
221	H(5,𝔽₂)	—	502	4	no	—
222	H(5,𝔽₂)	—	502	4	no	—
223	H(5,𝔽₂)	—	251	8	no	—
224	H(5,𝔽₂)	—	251	8	no	—
225	C₂×#41	—	502	1	no	—
226	C₂×#41	—	502	1	no	—
227	C₂×#41	—	251	2	no	—
228	C₂×#41	—	251	2	no	—
229	C₂×#41	—	251	1	no	—
230	C₂×#41	—	251	1	no	—
231	C₂×#41	C₂×D₂₅₁	1	1	yes	characteristic, complemented
232	C₂²×D₄	—	502	1	no	—
233	C₂²×D₄	—	502	2	no	—
234	C₂²×D₄	—	502	1	no	—
235	C₂²×D₄	—	502	2	no	—
236	C₂²×D₄	—	251	2	no	—
237	C₂²×D₄	—	251	2	no	—
238	C₂²×D₄	—	251	1	no	—
239	C₂²×D₄	C₂×D₂₅₁	1	1	yes	characteristic, complemented
240	AGL(1,ℤ/8)	—	251	16	no	—
241	AGL(1,ℤ/8)	—	251	16	no	—
242	AGL(1,ℤ/8)	—	251	8	no	—
243	AGL(1,ℤ/8)	C₂×D₂₅₁	1	8	yes	complemented
244	#135	—	251	8	no	—
245	#135	—	251	16	no	—
246	C₂×Q₁₆	—	251	1	no	—
247	C₂×SD₁₆	—	251	1	no	—
248	C₂×SD₁₆	—	251	1	no	—
249	C₂×SD₁₆	—	251	2	no	—
250	C₂×SD₁₆	C₂×D₂₅₁	1	2	yes	complemented
251	C₂×D₈	—	251	2	no	—
252	C₂×D₈	—	251	1	no	—
253	C₂×D₈	—	251	2	no	—
254	C₂×D₈	—	251	8	no	—
255	C₂×D₈	—	251	16	no	—
256	C₂×D₈	—	251	2	no	—
257	C₂×D₈	C₂×D₂₅₁	1	2	yes	complemented
258	#131	—	251	8	no	—
259	C₂×M₁₆	—	251	2	no	—
260	C₂×M₁₆	C₂×D₂₅₁	1	1	yes	characteristic, complemented
261	C₂²×C₈	—	251	1	no	—
262	C₂²×C₈	—	251	2	no	—
263	C₂⁴	—	2	1	no	—
264	C₂⁴	—	502	2	no	—
265	C₂⁴	—	502	2	no	—
266	C₂⁴	—	502	1	no	—
267	#41	—	2	2	no	—
268	#41	—	502	8	no	—
269	#41	—	502	8	no	—
270	#41	—	502	4	no	—
271	#41	—	502	4	no	—
272	#41	—	502	2	no	—
273	#41	—	251	8	no	—
274	#41	—	251	8	no	—
275	#41	—	251	4	no	—
276	#41	—	251	4	no	—
277	#41	—	251	4	no	—
278	#41	C₂²×D₂₅₁	1	4	yes	complemented
279	C₂×Q₈	—	251	1	no	—
280	C₂×Q₈	—	251	1	no	—
281	C₂×Q₈	C₂²×D₂₅₁	1	1	yes	characteristic
282	C₂×D₄	—	2	1	no	—
283	C₂×D₄	—	2	1	no	—
284	C₂×D₄	—	2	2	no	—
285	C₂×D₄	—	502	2	no	—
286	C₂×D₄	—	502	8	no	—
287	C₂×D₄	—	502	16	no	—
288	C₂×D₄	—	502	8	no	—
289	C₂×D₄	—	502	2	no	—
290	C₂×D₄	—	502	1	no	—
291	C₂×D₄	—	502	16	no	—
292	C₂×D₄	—	502	4	no	—
293	C₂×D₄	—	502	4	no	—
294	C₂×D₄	—	502	2	no	—
295	C₂×D₄	—	502	2	no	—
296	C₂×D₄	—	502	1	no	—
297	C₂×D₄	—	502	2	no	—
298	C₂×D₄	—	251	8	no	—
299	C₂×D₄	—	251	8	no	—
300	C₂×D₄	—	251	2	no	—
301	C₂×D₄	—	251	1	no	—
302	C₂×D₄	—	251	2	no	—
303	C₂×D₄	—	251	1	no	—
304	C₂×D₄	—	251	4	no	—
305	C₂×D₄	C₂²×D₂₅₁	1	2	yes	—
306	C₂×D₄	C₂²×D₂₅₁	1	1	yes	characteristic
307	C₂×D₄	C₂²×D₂₅₁	1	4	yes	complemented
308	C₂²×C₄	—	2	1	no	—
309	C₂²×C₄	—	502	2	no	—
310	C₂²×C₄	—	502	2	no	—
311	C₂²×C₄	—	502	1	no	—
312	C₂²×C₄	—	251	2	no	—
313	C₂²×C₄	C₂²×D₂₅₁	1	1	yes	characteristic
314	Q₁₆	—	251	4	no	—
315	SD₁₆	—	251	4	no	—
316	SD₁₆	—	251	4	no	—
317	SD₁₆	—	251	8	no	—
318	SD₁₆	C₂²×D₂₅₁	1	8	yes	complemented
319	D₈	—	251	8	no	—
320	D₈	—	251	4	no	—
321	D₈	—	251	8	no	—
322	D₈	—	251	8	no	—
323	D₈	C₂²×D₂₅₁	1	8	yes	complemented
324	M₁₆	—	251	8	no	—
325	M₁₆	C₂²×D₂₅₁	1	4	yes	complemented
326	C₂×C₈	—	251	1	no	—
327	C₂×C₈	—	251	1	no	—
328	C₂×C₈	—	251	4	no	—
329	C₂×C₈	—	251	8	no	—
330	C₂×C₈	C₂²×D₂₅₁	1	2	yes	—
331	C₂³	—	2	1	no	—
332	C₂³	—	2	2	no	—
333	C₂³	—	2	4	no	—
334	C₂³	—	8	2	no	—
335	C₂³	—	2008	2	no	—
336	C₂³	—	2008	4	no	—
337	C₂³	—	2008	4	no	—
338	C₂³	—	502	8	no	—
339	C₂³	—	502	8	no	—
340	C₂³	—	502	4	no	—
341	C₂³	—	502	2	no	—
342	C₂³	—	502	1	no	—
343	C₂³	—	251	2	no	—
344	C₂³	D₄×D₂₅₁	1	1	yes	characteristic
345	Q₈	—	2	1	no	—
346	Q₈	—	502	1	no	—
347	Q₈	—	251	2	no	—
348	Q₈	—	251	4	no	—
349	Q₈	C₂³×D₂₅₁	1	2	yes	—
350	D₄	—	2	4	no	—
351	D₄	—	2	2	no	—
352	D₄	—	2	4	no	—
353	D₄	—	2	1	no	—
354	D₄	—	502	8	no	—
355	D₄	—	502	8	no	—
356	D₄	—	502	4	no	—
357	D₄	—	502	8	no	—
358	D₄	—	502	8	no	—
359	D₄	—	502	4	no	—
360	D₄	—	502	1	no	—
361	D₄	—	502	2	no	—
362	D₄	—	251	8	no	—
363	D₄	—	251	4	no	—
364	D₄	—	251	4	no	—
365	D₄	—	251	2	no	—
366	D₄	C₂³×D₂₅₁	1	4	yes	—
367	D₄	C₂³×D₂₅₁	1	2	yes	—
368	C₂×C₄	—	2	1	no	—
369	C₂×C₄	—	2	4	no	—
370	C₂×C₄	—	502	8	no	—
371	C₂×C₄	—	502	1	no	—
372	C₂×C₄	—	502	8	no	—
373	C₂×C₄	—	502	4	no	—
374	C₂×C₄	—	251	2	no	—
375	C₂×C₄	—	251	4	no	—
376	C₂×C₄	—	251	4	no	—
377	C₂×C₄	D₄×D₂₅₁	1	2	yes	—
378	C₂×C₄	D₄×D₂₅₁	1	1	yes	characteristic
379	C₂×C₄	C₂³×D₂₅₁	1	2	yes	—
380	C₂×C₄	C₂³×D₂₅₁	1	1	yes	characteristic
381	C₈	—	251	2	no	—
382	C₈	—	251	2	no	—
383	C₈	C₂³×D₂₅₁	1	4	yes	—
384	C₂²	—	2	2	no	—
385	C₂²	—	2	2	no	—
386	C₂²	—	2	4	no	—
387	C₂²	—	4	4	no	—
388	C₂²	—	4	2	no	—
389	C₂²	—	8	4	no	—
390	C₂²	—	2008	4	no	—
391	C₂²	—	2008	8	no	—
392	C₂²	—	2008	8	no	—
393	C₂²	—	1004	4	no	—
394	C₂²	—	1004	2	no	—
395	C₂²	—	502	4	no	—
396	C₂²	—	502	2	no	—
397	C₂²	—	502	4	no	—
398	C₂²	—	251	4	no	—
399	C₂²	C₂×D₄×D₂₅₁	1	2	yes	—
400	C₂²	C₂×D₄×D₂₅₁	1	1	yes	characteristic
401	C₄	—	2	2	no	—
402	C₄	—	502	2	no	—
403	C₄	—	251	4	no	—
404	C₄	C₂×D₄×D₂₅₁	1	2	yes	—
405	C₄	C₂×D₄×D₂₅₁	1	1	yes	characteristic
406	C₄	C₂⁴×D₂₅₁	1	1	yes	characteristic
407	C₂	—	2	2	no	—
408	C₂	—	4	4	no	—
409	C₂	—	4	2	no	—
410	C₂	—	1004	4	no	—
411	C₂	—	1004	2	no	—
412	C₂	—	502	4	no	—
413	C₂	#30632491	1	2	yes	direct product factor
414	C₂	C₂²×D₄×D₂₅₁	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×D₄×D₂₅₁
commutator, cosocle	C₄×C₂₅₁	C₂⁵
Frattini	C₄	C₂⁴×D₂₅₁
Fitting	C₂×AGL(1,ℤ/8)×C₂₅₁	C₂
socle	C₂²×C₂₅₁	C₂²×D₄
hypercenter	C₂×AGL(1,ℤ/8)	D₂₅₁
hypocenter	C₂₅₁	C₂×#576
LCS	G ▷ C₄×C₂₅₁ ▷ C₂×C₂₅₁ ▷ C₂₅₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×AGL(1,ℤ/8)

Complements

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

N	H ≅ G/N	type
1. C₂×#22548761	C₂	semidirect
2. C₂×#22548760	C₂	semidirect
3. #30632491	C₂	direct (N × H)
4. C₂×#22548743	C₂	semidirect
5. C₂×#22548742	C₂	semidirect
6. C₂×#22548739	C₂	semidirect
7. C₂×#22548736	C₂	semidirect
8. C₂×#22548735	C₂	semidirect
9. C₂²×#16860795	C₂	semidirect
10. C₂×D₈×D₂₅₁	C₂	semidirect
11. C₂×#22548723	C₂	semidirect
12. C₂×AGL(1,ℤ/8)×C₂₅₁	C₂	semidirect
14. #22548761	C₂²	semidirect
16. #22548760	C₂²	semidirect
17. C₂×#16860811	C₂²	semidirect
20. C₂×#16860810	C₂²	semidirect
22. C₂×#16860809	C₂²	semidirect
24. C₂²×D₁₀₀₄	C₂²	semidirect
28. C₂×D₄×D₂₅₁	C₂²	semidirect
29. C₂×D₄×D₂₅₁	C₂²	semidirect
30. #22548743	C₂²	semidirect
31. #22548742	C₂²	semidirect
32. #22548739	C₂²	semidirect
33. #22548736	C₂²	semidirect
34. #22548735	C₂²	semidirect
35. C₂×#16860800	C₂²	semidirect
36. C₂×#16860799	C₂²	semidirect
37. C₂×#16860798	C₂²	semidirect
38. C₂×#16860797	C₂²	semidirect
39. C₂×D₂₀₀₈	C₂²	semidirect
40. C₂×#16860795	C₂²	semidirect
41. C₂×#16860795	C₂²	semidirect
42. C₂×#16860795	C₂²	semidirect
43. C₂×#16860795	C₂²	semidirect
44. D₈×D₂₅₁	C₂²	semidirect
45. #22548723	C₂²	semidirect
46. C₂×#16860793	C₂²	semidirect
47. C₂²×C₂₅₁⋊C₈	C₂²	semidirect
48. C₂×C₈×D₂₅₁	C₂²	semidirect
49. C₂×#41×C₂₅₁	C₂²	semidirect
50. C₂²×D₄×C₂₅₁	C₂²	semidirect
51. AGL(1,ℤ/8)×C₂₅₁	C₂²	semidirect
52. C₂×SD₁₆×C₂₅₁	C₂²	semidirect
53. C₂×D₈×C₂₅₁	C₂²	semidirect
54. C₂×M₁₆×C₂₅₁	C₂²	semidirect
59. #16860811	C₂³	semidirect
64. #16860810	C₂³	semidirect
67. #16860809	C₂³	semidirect
77. C₂×D₁₀₀₄	C₂³	semidirect
88. D₄×D₂₅₁	C₂³	semidirect
89. D₄×D₂₅₁	C₂³	semidirect
94. #16860800	C₂³	semidirect
95. #16860799	C₂³	semidirect
96. #16860798	C₂³	semidirect
97. #16860797	C₂³	semidirect
98. D₂₀₀₈	C₂³	semidirect
99. #16860795	C₂³	semidirect
100. #16860795	C₂³	semidirect
101. #16860795	C₂³	semidirect
102. #16860793	C₂³	semidirect
105. C₂×C₂₅₁⋊C₈	C₂³	semidirect
106. C₈×D₂₅₁	C₂³	semidirect
109. #41×C₂₅₁	C₂³	semidirect
116. C₂×D₄×C₂₅₁	C₂³	semidirect
119. SD₁₆×C₂₅₁	C₂³	semidirect
120. D₈×C₂₅₁	C₂³	semidirect
121. M₁₆×C₂₅₁	C₂³	semidirect
202. C₂×C₂₅₁	#576	semidirect
207. C₂₅₁	C₂×#576	semidirect
215. C₂×AGL(1,ℤ/8)	D₂₅₁	semidirect
231. C₂×#41	C₂×D₂₅₁	semidirect
239. C₂²×D₄	C₂×D₂₅₁	semidirect
243. AGL(1,ℤ/8)	C₂×D₂₅₁	semidirect
250. C₂×SD₁₆	C₂×D₂₅₁	semidirect
257. C₂×D₈	C₂×D₂₅₁	semidirect
260. C₂×M₁₆	C₂×D₂₅₁	semidirect
278. #41	C₂²×D₂₅₁	semidirect
307. C₂×D₄	C₂²×D₂₅₁	semidirect
318. SD₁₆	C₂²×D₂₅₁	semidirect
323. D₈	C₂²×D₂₅₁	semidirect
325. M₁₆	C₂²×D₂₅₁	semidirect
413. C₂	#30632491	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂×#22548761	2
C₂×#22548760	2
#30632491	2
C₂×#22548743	2
C₂×#22548742	2
C₂×#22548739	2
C₂×#22548736	2
C₂×#22548735	2
C₂²×#16860795	2
C₂×D₈×D₂₅₁	2
C₂×#22548723	2
C₂×AGL(1,ℤ/8)×C₂₅₁	2
C₂×#576	251

Subgroups — C2×#30632491 #44565117

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#30632491 #44565117