Subgroups — C₂×#30631704 #44558907

Order 32128.

772 conjugacy classes of subgroups.

The subgroup lattice has 304 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×#16860794	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×#22548674	C₂	1	1	yes	maximal, characteristic, complemented
3	#30631704	C₂	1	8	yes	maximal, direct product factor
4	C₂×#22548642	C₂	1	4	yes	maximal, complemented
5	C₂²×#41×C₂₅₁	C₂	1	1	yes	maximal, characteristic
6	C₂×#16860794	—	2	1	no	—
7	C₂×#16860794	—	2	2	no	—
8	C₂×#16860794	C₂²	1	2	yes	complemented
9	C₂×#16860794	C₂²	1	4	yes	complemented
10	C₂²×C₂₅₁⋊C₈	—	2	1	no	—
11	#22548674	—	2	2	no	—
12	#22548674	C₂²	1	4	yes	complemented
13	C₂×C₄⋊Dic₂₅₁	—	2	1	no	—
14	C₂×C₂²⋊Dic₂₅₁	—	2	1	no	—
15	C₂×C₄×Dic₂₅₁	C₂²	1	2	yes	complemented
16	#22548642	C₂²	1	16	yes	complemented
17	C₂×#41×C₂₅₁	—	2	1	no	—
18	C₂×#41×C₂₅₁	—	2	2	no	—
19	C₂×#41×C₂₅₁	—	2	1	no	—
20	C₂×#41×C₂₅₁	—	2	2	no	—
21	C₂×#41×C₂₅₁	C₄	1	2	yes	complemented
22	C₂×#41×C₂₅₁	C₄	1	4	yes	complemented
23	C₂×#41×C₂₅₁	C₂²	1	2	yes	—
24	C₂×#41×C₂₅₁	C₂²	1	4	yes	—
25	C₂²×Q₈×C₂₅₁	C₄	1	1	yes	characteristic, complemented
26	C₂²×D₄×C₂₅₁	—	2	1	no	—
27	C₂²×D₄×C₂₅₁	C₄	1	1	yes	characteristic, complemented
28	C₂³×C₄×C₂₅₁	—	2	1	no	—
29	C₂³×C₄×C₂₅₁	C₂²	1	1	yes	characteristic
30	C₂²×Dic₂₅₁	—	2	2	no	—
31	#16860794	—	2	4	no	—
32	#16860794	—	2	2	no	—
33	#16860794	C₂³	1	4	yes	complemented
34	C₂×C₂₅₁⋊C₈	—	2	4	no	—
35	C₂×C₂₅₁⋊C₈	—	2	2	no	—
36	C₄⋊Dic₂₅₁	—	2	4	no	—
37	C₂²⋊Dic₂₅₁	—	2	4	no	—
38	C₄×Dic₂₅₁	—	2	2	no	—
39	C₄×Dic₂₅₁	C₂³	1	4	yes	complemented
40	C₂⁴×C₂₅₁	—	2	1	no	—
41	C₂⁴×C₂₅₁	D₄	1	1	yes	characteristic
42	#41×C₂₅₁	—	2	4	no	—
43	#41×C₂₅₁	—	2	4	no	—
44	#41×C₂₅₁	—	2	4	no	—
45	#41×C₂₅₁	—	2	2	no	—
46	#41×C₂₅₁	—	2	4	no	—
47	#41×C₂₅₁	—	2	4	no	—
48	#41×C₂₅₁	—	2	2	no	—
49	#41×C₂₅₁	C₂×C₄	1	4	yes	—
50	#41×C₂₅₁	C₂×C₄	1	8	yes	complemented
51	#41×C₂₅₁	C₂³	1	4	yes	—
52	C₂×Q₈×C₂₅₁	—	2	1	no	—
53	C₂×Q₈×C₂₅₁	—	2	2	no	—
54	C₂×Q₈×C₂₅₁	C₂×C₄	1	2	yes	—
55	C₂×Q₈×C₂₅₁	C₂×C₄	1	4	yes	complemented
56	C₂×D₄×C₂₅₁	—	2	2	no	—
57	C₂×D₄×C₂₅₁	—	2	4	no	—
58	C₂×D₄×C₂₅₁	—	2	2	no	—
59	C₂×D₄×C₂₅₁	—	2	4	no	—
60	C₂×D₄×C₂₅₁	—	2	2	no	—
61	C₂×D₄×C₂₅₁	—	2	1	no	—
62	C₂×D₄×C₂₅₁	C₂×C₄	1	2	yes	—
63	C₂×D₄×C₂₅₁	C₂×C₄	1	4	yes	complemented
64	C₂²×C₄×C₂₅₁	—	2	1	no	—
65	C₂²×C₄×C₂₅₁	—	2	2	no	—
66	C₂²×C₄×C₂₅₁	—	2	4	no	—
67	C₂²×C₄×C₂₅₁	—	2	4	no	—
68	C₂²×C₄×C₂₅₁	—	2	2	no	—
69	C₂²×C₄×C₂₅₁	C₂×C₄	1	1	yes	characteristic
70	C₂²×C₄×C₂₅₁	C₂×C₄	1	1	yes	characteristic
71	C₂²×C₄×C₂₅₁	C₂×C₄	1	2	yes	—
72	C₂²×C₄×C₂₅₁	D₄	1	1	yes	characteristic
73	C₂²×C₄×C₂₅₁	D₄	1	2	yes	—
74	C₂²×C₄×C₂₅₁	D₄	1	1	yes	characteristic
75	C₂²×C₄×C₂₅₁	D₄	1	1	yes	characteristic
76	C₂²×C₄×C₂₅₁	D₄	1	2	yes	—
77	C₂²×C₄×C₂₅₁	C₂³	1	1	yes	characteristic
78	C₂²×C₄×C₂₅₁	C₂³	1	2	yes	—
79	C₂×Dic₂₅₁	—	2	4	no	—
80	C₂×Dic₂₅₁	—	4	4	no	—
81	C₂₅₁⋊C₈	—	2	4	no	—
82	C₂³×C₂₅₁	—	2	2	no	—
83	C₂³×C₂₅₁	—	2	4	no	—
84	C₂³×C₂₅₁	—	4	2	no	—
85	C₂³×C₂₅₁	—	4	4	no	—
86	C₂³×C₂₅₁	C₂²⋊C₄	1	1	yes	characteristic
87	C₂³×C₂₅₁	C₂²⋊C₄	1	1	yes	characteristic
88	C₂³×C₂₅₁	C₂²⋊C₄	1	2	yes	—
89	C₂³×C₂₅₁	C₂×D₄	1	1	yes	characteristic
90	C₂³×C₂₅₁	C₂×D₄	1	2	yes	—
91	Q₈×C₂₅₁	—	2	2	no	—
92	Q₈×C₂₅₁	—	2	4	no	—
93	Q₈×C₂₅₁	C₂²×C₄	1	4	yes	—
94	D₄×C₂₅₁	—	2	4	no	—
95	D₄×C₂₅₁	—	2	8	no	—
96	D₄×C₂₅₁	—	2	4	no	—
97	D₄×C₂₅₁	—	2	2	no	—
98	D₄×C₂₅₁	—	2	4	no	—
99	D₄×C₂₅₁	C₂²×C₄	1	4	yes	—
100	C₂×C₄×C₂₅₁	—	2	2	no	—
101	C₂×C₄×C₂₅₁	—	2	4	no	—
102	C₂×C₄×C₂₅₁	—	2	4	no	—
103	C₂×C₄×C₂₅₁	—	2	8	no	—
104	C₂×C₄×C₂₅₁	—	2	4	no	—
105	C₂×C₄×C₂₅₁	C₂²⋊C₄	1	2	yes	—
106	C₂×C₄×C₂₅₁	C₂²⋊C₄	1	1	yes	characteristic
107	C₂×C₄×C₂₅₁	C₂²⋊C₄	1	2	yes	—
108	C₂×C₄×C₂₅₁	C₂²⋊C₄	1	1	yes	characteristic
109	C₂×C₄×C₂₅₁	C₂²⋊C₄	1	2	yes	—
110	C₂×C₄×C₂₅₁	C₂²⋊C₄	1	2	yes	—
111	C₂×C₄×C₂₅₁	C₂²⋊C₄	1	2	yes	—
112	C₂×C₄×C₂₅₁	C₂²×C₄	1	1	yes	characteristic
113	C₂×C₄×C₂₅₁	C₂²×C₄	1	2	yes	—
114	C₂×C₄×C₂₅₁	C₂²×C₄	1	1	yes	characteristic
115	C₂×C₄×C₂₅₁	C₂²×C₄	1	2	yes	—
116	C₂×C₄×C₂₅₁	C₂×D₄	1	1	yes	characteristic
117	C₂×C₄×C₂₅₁	C₂×D₄	1	2	yes	—
118	C₂×C₄×C₂₅₁	C₂×D₄	1	2	yes	—
119	C₂×C₄×C₂₅₁	C₂×D₄	1	2	yes	—
120	C₂×C₄×C₂₅₁	C₂×D₄	1	1	yes	characteristic
121	C₂×C₄×C₂₅₁	C₂×D₄	1	1	yes	characteristic
122	C₂×C₄×C₂₅₁	C₂⁴	1	1	yes	characteristic
123	C₂²×C₂₅₁	—	2	2	no	—
124	C₂²×C₂₅₁	—	2	2	no	—
125	C₂²×C₂₅₁	—	2	2	no	—
126	C₂²×C₂₅₁	—	2	4	no	—
127	C₂²×C₂₅₁	—	4	2	no	—
128	C₂²×C₂₅₁	—	4	2	no	—
129	C₂²×C₂₅₁	—	4	8	no	—
130	C₂²×C₂₅₁	—	4	4	no	—
131	C₂²×C₂₅₁	C₂×C₂²⋊C₄	1	2	yes	—
132	C₂²×C₂₅₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
133	C₂²×C₂₅₁	C₂×C₂²⋊C₄	1	2	yes	—
134	C₂²×C₂₅₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
135	C₂²×C₂₅₁	C₂²×D₄	1	1	yes	characteristic
136	C₄×C₂₅₁	—	2	4	no	—
137	C₄×C₂₅₁	C₂×C₂²⋊C₄	1	2	yes	—
138	C₄×C₂₅₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
139	C₄×C₂₅₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
140	C₄×C₂₅₁	C₂×C₂²⋊C₄	1	2	yes	—
141	C₄×C₂₅₁	C₂³×C₄	1	1	yes	characteristic
142	C₄×C₂₅₁	C₂²×D₄	1	1	yes	characteristic
143	Dic₂₅₁	—	4	4	no	—
144	C₂×C₂₅₁	—	2	1	no	—
145	C₂×C₂₅₁	—	2	2	no	—
146	C₂×C₂₅₁	—	2	2	no	—
147	C₂×C₂₅₁	—	2	1	no	—
148	C₂×C₂₅₁	—	4	4	no	—
149	C₂×C₂₅₁	#421	1	2	yes	complemented
150	C₂×C₂₅₁	C₂²×C₂²⋊C₄	1	1	yes	characteristic
151	C₂₅₁	C₂×#421	1	1	yes	characteristic, complemented, Sylow
152	C₂×#421	—	251	1	no	maximal, Sylow
153	C₂²×#41	D₂₅₁	1	1	yes	characteristic
154	C₂²×M₁₆	—	251	1	no	—
155	C₂×#117	—	251	1	no	—
156	#421	—	251	8	no	—
157	C₂×#104	—	251	4	no	—
158	C₂×#41	—	2	1	no	—
159	C₂×#41	—	2	2	no	—
160	C₂×#41	—	2	2	no	—
161	C₂×#41	—	2	1	no	—
162	C₂×#41	Dic₂₅₁	1	2	yes	complemented
163	C₂×#41	Dic₂₅₁	1	4	yes	complemented
164	C₂×#41	C₂×D₂₅₁	1	2	yes	—
165	C₂×#41	C₂×D₂₅₁	1	4	yes	—
166	C₂²×Q₈	Dic₂₅₁	1	1	yes	characteristic, complemented
167	C₂²×D₄	—	2	1	no	—
168	C₂²×D₄	Dic₂₅₁	1	1	yes	characteristic, complemented
169	C₂³×C₄	—	2	1	no	—
170	C₂³×C₄	C₂×D₂₅₁	1	1	yes	characteristic
171	C₂×M₁₆	—	502	2	no	—
172	C₂×M₁₆	—	502	1	no	—
173	C₂×M₁₆	—	251	2	no	—
174	C₂×M₁₆	—	251	4	no	—
175	C₂²×C₈	—	502	1	no	—
176	#117	—	502	2	no	—
177	#117	—	251	4	no	—
178	C₂×C₄⋊C₄	—	502	1	no	—
179	C₂×C₂²⋊C₄	—	502	1	no	—
180	C₂×C₄²	—	251	2	no	—
181	#104	—	251	16	no	—
182	C₂⁴	—	2	1	no	—
183	C₂⁴	C₂²⋊D₂₅₁	1	1	yes	characteristic
184	#41	—	2	4	no	—
185	#41	—	2	4	no	—
186	#41	—	2	4	no	—
187	#41	—	2	4	no	—
188	#41	—	2	2	no	—
189	#41	—	2	2	no	—
190	#41	—	2	4	no	—
191	#41	C₂×Dic₂₅₁	1	4	yes	—
192	#41	C₂×Dic₂₅₁	1	8	yes	complemented
193	#41	C₂²×D₂₅₁	1	4	yes	—
194	C₂×Q₈	—	2	1	no	—
195	C₂×Q₈	—	2	2	no	—
196	C₂×Q₈	C₂×Dic₂₅₁	1	2	yes	—
197	C₂×Q₈	C₂×Dic₂₅₁	1	4	yes	complemented
198	C₂×D₄	—	2	2	no	—
199	C₂×D₄	—	2	4	no	—
200	C₂×D₄	—	2	2	no	—
201	C₂×D₄	—	2	4	no	—
202	C₂×D₄	—	2	2	no	—
203	C₂×D₄	—	2	1	no	—
204	C₂×D₄	C₂×Dic₂₅₁	1	2	yes	—
205	C₂×D₄	C₂×Dic₂₅₁	1	4	yes	complemented
206	C₂²×C₄	—	2	1	no	—
207	C₂²×C₄	—	2	2	no	—
208	C₂²×C₄	—	2	4	no	—
209	C₂²×C₄	—	2	4	no	—
210	C₂²×C₄	—	2	2	no	—
211	C₂²×C₄	—	502	2	no	—
212	C₂²×C₄	C₂×Dic₂₅₁	1	1	yes	characteristic
213	C₂²×C₄	C₂×Dic₂₅₁	1	1	yes	characteristic
214	C₂²×C₄	C₂×Dic₂₅₁	1	2	yes	—
215	C₂²×C₄	C₂²⋊D₂₅₁	1	1	yes	characteristic
216	C₂²×C₄	C₂²⋊D₂₅₁	1	2	yes	—
217	C₂²×C₄	C₂²⋊D₂₅₁	1	2	yes	—
218	C₂²×C₄	C₂²⋊D₂₅₁	1	1	yes	characteristic
219	C₂²×C₄	C₂²⋊D₂₅₁	1	1	yes	characteristic
220	C₂²×C₄	C₂²×D₂₅₁	1	2	yes	—
221	C₂²×C₄	C₂²×D₂₅₁	1	1	yes	characteristic
222	M₁₆	—	502	4	no	—
223	M₁₆	—	502	2	no	—
224	M₁₆	—	251	4	no	—
225	C₂×C₈	—	502	4	no	—
226	C₂×C₈	—	502	2	no	—
227	C₄⋊C₄	—	502	4	no	—
228	C₂²⋊C₄	—	502	4	no	—
229	C₄²	—	502	2	no	—
230	C₄²	—	251	4	no	—
231	C₂³	—	2	2	no	—
232	C₂³	—	2	4	no	—
233	C₂³	—	4	2	no	—
234	C₂³	—	4	4	no	—
235	C₂³	C₂²⋊Dic₂₅₁	1	1	yes	characteristic
236	C₂³	C₂²⋊Dic₂₅₁	1	1	yes	characteristic
237	C₂³	C₂²⋊Dic₂₅₁	1	2	yes	—
238	C₂³	C₂×C₂²⋊D₂₅₁	1	1	yes	characteristic
239	C₂³	C₂×C₂²⋊D₂₅₁	1	2	yes	—
240	Q₈	—	2	2	no	—
241	Q₈	—	2	4	no	—
242	Q₈	C₂²×Dic₂₅₁	1	4	yes	—
243	D₄	—	2	4	no	—
244	D₄	—	2	4	no	—
245	D₄	—	2	8	no	—
246	D₄	—	2	2	no	—
247	D₄	—	2	4	no	—
248	D₄	C₂²×Dic₂₅₁	1	4	yes	—
249	C₂×C₄	—	2	2	no	—
250	C₂×C₄	—	2	4	no	—
251	C₂×C₄	—	2	4	no	—
252	C₂×C₄	—	2	4	no	—
253	C₂×C₄	—	2	8	no	—
254	C₂×C₄	—	1004	4	no	—
255	C₂×C₄	—	502	4	no	—
256	C₂×C₄	C₂²⋊Dic₂₅₁	1	1	yes	characteristic
257	C₂×C₄	C₂²⋊Dic₂₅₁	1	2	yes	—
258	C₂×C₄	C₂²⋊Dic₂₅₁	1	2	yes	—
259	C₂×C₄	C₂²⋊Dic₂₅₁	1	1	yes	characteristic
260	C₂×C₄	C₂²⋊Dic₂₅₁	1	2	yes	—
261	C₂×C₄	C₂²⋊Dic₂₅₁	1	2	yes	—
262	C₂×C₄	C₂²⋊Dic₂₅₁	1	2	yes	—
263	C₂×C₄	C₂²×Dic₂₅₁	1	1	yes	characteristic
264	C₂×C₄	C₂²×Dic₂₅₁	1	2	yes	—
265	C₂×C₄	C₂²×Dic₂₅₁	1	1	yes	characteristic
266	C₂×C₄	C₂²×Dic₂₅₁	1	2	yes	—
267	C₂×C₄	C₂×C₂²⋊D₂₅₁	1	2	yes	—
268	C₂×C₄	C₂×C₂²⋊D₂₅₁	1	2	yes	—
269	C₂×C₄	C₂×C₂²⋊D₂₅₁	1	1	yes	characteristic
270	C₂×C₄	C₂×C₂²⋊D₂₅₁	1	1	yes	characteristic
271	C₂×C₄	C₂×C₂²⋊D₂₅₁	1	2	yes	—
272	C₂×C₄	C₂×C₂²⋊D₂₅₁	1	1	yes	characteristic
273	C₂×C₄	C₂³×D₂₅₁	1	1	yes	characteristic
274	C₈	—	502	4	no	—
275	C₂²	—	2	2	no	—
276	C₂²	—	2	2	no	—
277	C₂²	—	2	2	no	—
278	C₂²	—	2	4	no	—
279	C₂²	—	4	2	no	—
280	C₂²	—	4	2	no	—
281	C₂²	—	4	8	no	—
282	C₂²	—	4	4	no	—
283	C₂²	C₂×C₂²⋊Dic₂₅₁	1	2	yes	—
284	C₂²	C₂×C₂²⋊Dic₂₅₁	1	1	yes	characteristic
285	C₂²	C₂×C₂²⋊Dic₂₅₁	1	2	yes	—
286	C₂²	C₂×C₂²⋊Dic₂₅₁	1	1	yes	characteristic
287	C₂²	C₂²×C₂²⋊D₂₅₁	1	1	yes	characteristic
288	C₄	—	2	4	no	—
289	C₄	—	1004	4	no	—
290	C₄	C₂×C₂²⋊Dic₂₅₁	1	2	yes	—
291	C₄	C₂×C₂²⋊Dic₂₅₁	1	1	yes	characteristic
292	C₄	C₂×C₂²⋊Dic₂₅₁	1	1	yes	characteristic
293	C₄	C₂×C₂²⋊Dic₂₅₁	1	2	yes	—
294	C₄	C₂³×Dic₂₅₁	1	1	yes	characteristic
295	C₄	C₂²×C₂²⋊D₂₅₁	1	1	yes	characteristic
296	C₂	—	2	1	no	—
297	C₂	—	2	2	no	—
298	C₂	—	2	1	no	—
299	C₂	—	2	2	no	—
300	C₂	—	4	4	no	—
301	C₂	#30631704	1	2	yes	direct product factor
302	C₂	C₂²×C₂²⋊Dic₂₅₁	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₂×C₂²⋊D₂₅₁
commutator	C₄×C₂₅₁	C₂³×C₄
Frattini	C₂×C₄	C₂³×D₂₅₁
Fitting, prime radical	C₂²×#41×C₂₅₁	C₂
socle	C₂²×C₂₅₁	C₂×C₂²⋊C₄
cosocle	C₂×C₄×C₂₅₁	C₂⁴
hypercenter	C₂²×#41	D₂₅₁
hypocenter	C₂₅₁	C₂×#421
LCS	G ▷ C₄×C₂₅₁ ▷ C₂×C₂₅₁ ▷ C₂₅₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₄ ◁ C₂²×#41

Complements

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

N	H ≅ G/N	type
1. C₂²×#16860794	C₂	semidirect
2. C₂×#22548674	C₂	semidirect
3. #30631704	C₂	direct (N × H)
4. C₂×#22548642	C₂	semidirect
8. C₂×#16860794	C₂²	semidirect
9. C₂×#16860794	C₂²	semidirect
12. #22548674	C₂²	semidirect
15. C₂×C₄×Dic₂₅₁	C₂²	semidirect
16. #22548642	C₂²	semidirect
21. C₂×#41×C₂₅₁	C₄	semidirect
22. C₂×#41×C₂₅₁	C₄	semidirect
25. C₂²×Q₈×C₂₅₁	C₄	semidirect
27. C₂²×D₄×C₂₅₁	C₄	semidirect
33. #16860794	C₂³	semidirect
39. C₄×Dic₂₅₁	C₂³	semidirect
50. #41×C₂₅₁	C₂×C₄	semidirect
55. C₂×Q₈×C₂₅₁	C₂×C₄	semidirect
63. C₂×D₄×C₂₅₁	C₂×C₄	semidirect
149. C₂×C₂₅₁	#421	semidirect
151. C₂₅₁	C₂×#421	semidirect
162. C₂×#41	Dic₂₅₁	semidirect
163. C₂×#41	Dic₂₅₁	semidirect
166. C₂²×Q₈	Dic₂₅₁	semidirect
168. C₂²×D₄	Dic₂₅₁	semidirect
192. #41	C₂×Dic₂₅₁	semidirect
197. C₂×Q₈	C₂×Dic₂₅₁	semidirect
205. C₂×D₄	C₂×Dic₂₅₁	semidirect
301. C₂	#30631704	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×#16860794	2
C₂×#22548674	2
#30631704	2
C₂×#22548642	2
C₂²×#41×C₂₅₁	2
C₂×#421	251

Subgroups — C2×#30631704 #44558907

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#30631704 #44558907