Subgroups — C₂×#20291095 #28328295

Order 13568.

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×#13133291	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂³×#11719910	C₂	1	1	yes	maximal, characteristic, complemented
3	#20291095	C₂	1	8	yes	maximal, direct product factor
4	C₂×C₄⋊#11719910	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂×C₂²⋊#11719910	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×C₂²⋊#11719910	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂×#16811511	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×D₄×#88322	C₂	1	1	yes	maximal, characteristic
9	C₂×#13133291	—	2	4	no	—
10	C₂×#13133291	C₂²	1	4	yes	complemented
11	C₂²×#11719910	—	2	4	no	—
12	C₂²×#11719910	C₂²	1	1	yes	characteristic
13	C₂²×#11719910	C₂²	1	1	yes	characteristic
14	C₂²×#11719910	C₂²	1	1	yes	characteristic
15	C₂²×#11719910	C₂²	1	1	yes	characteristic
16	C₂²×#11719910	C₂²	1	4	yes	complemented
17	C₄⋊#11719910	C₂²	1	4	yes	complemented
18	C₂²⋊#11719910	C₂²	1	4	yes	complemented
19	C₂²⋊#11719910	C₂²	1	4	yes	complemented
20	#16811511	C₂²	1	4	yes	complemented
21	C₂³×#88322	C₂²	1	1	yes	characteristic
22	C₂³×#88322	C₂²	1	1	yes	characteristic
23	D₄×#88322	C₄	1	4	yes	—
24	D₄×#88322	C₂²	1	4	yes	—
25	C₂×C₄⋊#88322	C₄	1	1	yes	characteristic
26	C₂×C₂²⋊#88322	C₄	1	1	yes	characteristic
27	C₂×C₂²⋊#88322	C₄	1	1	yes	characteristic
28	C₂×C₄×#88322	C₂²	1	1	yes	characteristic
29	C₂×C₄×D₄×C₅₃	C₄	1	1	yes	characteristic
30	#13133291	—	2	8	no	—
31	C₂×#11719910	—	2	2	no	—
32	C₂×#11719910	—	2	2	no	—
33	C₂×#11719910	—	2	2	no	—
34	C₂×#11719910	—	2	8	no	—
35	C₂×#11719910	D₄	1	4	yes	complemented
36	C₂×#11719910	C₂³	1	2	yes	—
37	C₂×#11719910	C₂³	1	2	yes	—
38	C₂×#11719910	C₂³	1	2	yes	—
39	C₂×#11719910	C₂³	1	2	yes	—
40	C₂²×#88322	—	2	2	no	—
41	C₂²×#88322	—	2	2	no	—
42	C₂²×#88322	—	2	2	no	—
43	C₂²×#88322	—	2	2	no	—
44	C₂²×#88322	C₂×C₄	1	1	yes	characteristic
45	C₂²×#88322	C₂×C₄	1	1	yes	characteristic
46	C₂²×#88322	C₂×C₄	1	1	yes	characteristic
47	C₂²×#88322	C₂×C₄	1	2	yes	—
48	C₂²×#88322	C₂×C₄	1	2	yes	—
49	C₂²×#88322	C₂³	1	2	yes	—
50	C₂²×#88322	C₂³	1	2	yes	—
51	C₂²×#88322	C₂³	1	1	yes	characteristic
52	C₄⋊#88322	C₂×C₄	1	4	yes	—
53	C₂²⋊#88322	C₂×C₄	1	4	yes	—
54	C₂²⋊#88322	C₂×C₄	1	4	yes	—
55	C₄×#88322	C₂×C₄	1	2	yes	—
56	C₄×#88322	C₂³	1	2	yes	—
57	C₂²×D₄×C₅₃	C₈	1	1	yes	characteristic
58	C₂³×C₄×C₅₃	C₂×C₄	1	1	yes	characteristic
59	C₂³×C₄×C₅₃	C₂×C₄	1	1	yes	characteristic
60	C₄×D₄×C₅₃	C₈	1	4	yes	—
61	C₄×D₄×C₅₃	C₂×C₄	1	4	yes	—
62	C₂×C₄⋊C₄×C₅₃	C₈	1	1	yes	characteristic
63	C₂×C₂²⋊C₄×C₅₃	C₈	1	1	yes	characteristic
64	C₂×C₂²⋊C₄×C₅₃	C₈	1	1	yes	characteristic
65	C₂×C₄²×C₅₃	C₂×C₄	1	1	yes	characteristic
66	C₂⁴×C₅₃	C₂×C₈	1	1	yes	characteristic
67	C₂⁴×C₅₃	C₂×C₈	1	1	yes	characteristic
68	C₂×D₄×C₅₃	—	2	4	no	—
69	C₂×D₄×C₅₃	C₂×C₈	1	4	yes	—
70	C₂²×C₄×C₅₃	—	2	2	no	—
71	C₂²×C₄×C₅₃	—	2	2	no	—
72	C₂²×C₄×C₅₃	—	2	2	no	—
73	C₂²×C₄×C₅₃	—	2	2	no	—
74	C₂²×C₄×C₅₃	C₂×C₈	1	1	yes	characteristic
75	C₂²×C₄×C₅₃	C₂×C₈	1	1	yes	characteristic
76	C₂²×C₄×C₅₃	C₂×C₈	1	1	yes	characteristic
77	C₂²×C₄×C₅₃	C₂×C₈	1	1	yes	characteristic
78	C₂²×C₄×C₅₃	C₂×C₈	1	2	yes	—
79	C₂²×C₄×C₅₃	C₂×C₈	1	2	yes	—
80	C₂²×C₄×C₅₃	C₂²×C₄	1	2	yes	—
81	C₂²×C₄×C₅₃	C₂²×C₄	1	2	yes	—
82	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
83	C₄⋊C₄×C₅₃	C₂×C₈	1	4	yes	—
84	C₂²⋊C₄×C₅₃	C₂×C₈	1	4	yes	—
85	C₂²⋊C₄×C₅₃	C₂×C₈	1	4	yes	—
86	C₄²×C₅₃	C₂×C₈	1	2	yes	—
87	C₄²×C₅₃	C₂²×C₄	1	2	yes	—
88	C₂×#88322	—	2	2	no	—
89	C₂×#88322	—	2	2	no	—
90	C₂×#88322	—	2	2	no	—
91	C₂×#88322	—	2	4	no	—
92	C₂×#88322	—	2	4	no	—
93	C₂×#88322	—	2	2	no	—
94	C₂×#88322	—	2	2	no	—
95	C₂×#88322	—	2	2	no	—
96	C₂×#88322	—	2	2	no	—
97	C₂×#88322	C₂²×C₄	1	2	yes	—
98	C₂×#88322	C₂²×C₄	1	2	yes	—
99	C₂×#88322	C₂²×C₄	1	2	yes	—
100	C₂×#88322	C₂²×C₄	1	1	yes	characteristic
101	C₂×#88322	C₂×D₄	1	2	yes	—
102	C₂×#88322	#41	1	2	yes	—
103	C₂×#88322	C₂⁴	1	1	yes	characteristic
104	#11719910	—	2	2	no	—
105	#11719910	—	2	2	no	—
106	#11719910	—	2	2	no	—
107	#11719910	C₂×D₄	1	4	yes	complemented
108	C₂³×C₅₃	—	2	2	no	—
109	C₂³×C₅₃	—	2	2	no	—
110	C₂³×C₅₃	—	2	2	no	—
111	C₂³×C₅₃	—	2	2	no	—
112	C₂³×C₅₃	—	2	2	no	—
113	C₂³×C₅₃	M₃₂	1	4	yes	complemented
114	C₂³×C₅₃	C₂²×C₈	1	2	yes	—
115	C₂³×C₅₃	C₂²×C₈	1	2	yes	—
116	C₂³×C₅₃	C₂²×C₈	1	1	yes	characteristic
117	D₄×C₅₃	—	2	8	no	—
118	C₂×C₄×C₅₃	—	2	2	no	—
119	C₂×C₄×C₅₃	—	2	2	no	—
120	C₂×C₄×C₅₃	—	2	2	no	—
121	C₂×C₄×C₅₃	—	2	2	no	—
122	C₂×C₄×C₅₃	—	2	2	no	—
123	C₂×C₄×C₅₃	—	2	2	no	—
124	C₂×C₄×C₅₃	—	2	2	no	—
125	C₂×C₄×C₅₃	—	2	2	no	—
126	C₂×C₄×C₅₃	—	2	4	no	—
127	C₂×C₄×C₅₃	—	2	4	no	—
128	C₂×C₄×C₅₃	C₄×D₄	1	2	yes	—
129	C₂×C₄×C₅₃	C₂²×C₈	1	2	yes	—
130	C₂×C₄×C₅₃	C₂²×C₈	1	2	yes	—
131	C₂×C₄×C₅₃	C₂²×C₈	1	2	yes	—
132	C₂×C₄×C₅₃	C₂²×C₈	1	2	yes	—
133	C₂×C₄×C₅₃	C₂²×C₈	1	1	yes	characteristic
134	C₂×C₄×C₅₃	#131	1	2	yes	—
135	C₂×C₄×C₅₃	C₂³×C₄	1	1	yes	characteristic
136	#88322	—	2	2	no	—
137	#88322	—	2	2	no	—
138	#88322	—	2	2	no	—
139	#88322	C₄×D₄	1	2	yes	—
140	#88322	C₂²×D₄	1	1	yes	characteristic
141	#88322	C₂×#41	1	1	yes	characteristic
142	C₂×#2168	—	53	1	no	maximal, Sylow
143	C₂²×C₅₃	—	2	2	no	—
144	C₂²×C₅₃	—	2	2	no	—
145	C₂²×C₅₃	—	2	8	no	—
146	C₂²×C₅₃	—	2	2	no	—
147	C₂²×C₅₃	—	2	2	no	—
148	C₂²×C₅₃	—	2	2	no	—
149	C₂²×C₅₃	—	4	4	no	—
150	C₂²×C₅₃	C₈×D₄	1	2	yes	—
151	C₂²×C₅₃	C₂×M₃₂	1	2	yes	—
152	C₂²×C₅₃	C₂×M₃₂	1	4	yes	complemented
153	C₂²×C₅₃	#504	1	2	yes	—
154	C₂²×C₅₃	C₂³×C₈	1	1	yes	characteristic
155	C₄×C₅₃	—	2	2	no	—
156	C₄×C₅₃	—	2	2	no	—
157	C₄×C₅₃	—	2	2	no	—
158	C₄×C₅₃	—	2	2	no	—
159	C₄×C₅₃	C₈×D₄	1	2	yes	—
160	C₄×C₅₃	C₂×C₄×D₄	1	1	yes	characteristic
161	C₄×C₅₃	C₂×#131	1	1	yes	characteristic
162	C₂²×M₃₂	—	53	1	no	—
163	C₂³×C₁₆	—	53	1	no	—
164	C₂×C₈×D₄	—	53	1	no	—
165	#2168	—	53	8	no	—
166	C₂×C₄⋊C₁₆	—	53	1	no	—
167	C₂×C₂²⋊C₁₆	—	53	1	no	—
168	C₂×C₂²⋊C₁₆	—	53	1	no	—
169	C₂×#346	—	53	1	no	—
170	C₂×C₅₃	—	2	4	no	—
171	C₂×C₅₃	—	4	2	no	—
172	C₂×C₅₃	#2168	1	4	yes	complemented
173	C₂×C₅₃	C₂×C₈×D₄	1	1	yes	characteristic
174	C₂×C₅₃	C₂²×M₃₂	1	1	yes	characteristic
175	C₂×C₅₃	C₂×#504	1	1	yes	characteristic
176	C₂³×C₈	—	53	1	no	—
177	C₂³×C₈	—	53	1	no	—
178	C₂×C₄×D₄	#6157	1	1	yes	characteristic
179	C₂×M₃₂	—	53	4	no	—
180	C₂×M₃₂	—	106	4	no	—
181	C₂²×C₁₆	—	53	1	no	—
182	C₂²×C₁₆	—	53	1	no	—
183	C₂²×C₁₆	—	53	1	no	—
184	C₂²×C₁₆	—	53	1	no	—
185	C₂²×C₁₆	—	53	4	no	—
186	C₂²×C₁₆	—	106	4	no	—
187	C₈×D₄	—	53	4	no	—
188	C₈×D₄	—	53	4	no	—
189	C₂×C₄⋊C₈	—	53	1	no	—
190	C₂×C₂²⋊C₈	—	53	1	no	—
191	C₂×C₂²⋊C₈	—	53	1	no	—
192	C₂×C₄×C₈	—	53	1	no	—
193	C₄⋊C₁₆	—	53	4	no	—
194	C₂²⋊C₁₆	—	53	4	no	—
195	C₂²⋊C₁₆	—	53	4	no	—
196	#346	—	53	4	no	—
197	C₅₃	C₂×#2168	1	1	yes	characteristic, complemented, Sylow
198	C₂²×D₄	#88324	1	1	yes	characteristic
199	C₂³×C₄	C₂×#6157	1	1	yes	characteristic
200	C₂³×C₄	C₂×#6157	1	1	yes	characteristic
201	C₂²×C₈	—	53	1	no	—
202	C₂²×C₈	—	53	2	no	—
203	C₂²×C₈	—	53	2	no	—
204	C₂²×C₈	—	53	2	no	—
205	C₂²×C₈	—	53	2	no	—
206	C₂²×C₈	—	53	1	no	—
207	C₂²×C₈	—	53	1	no	—
208	C₂²×C₈	—	53	1	no	—
209	C₂²×C₈	—	106	2	no	—
210	C₂²×C₈	—	106	2	no	—
211	C₂²×C₈	—	106	2	no	—
212	C₂²×C₈	—	106	2	no	—
213	C₄×D₄	#88324	1	4	yes	—
214	C₄×D₄	C₂×#6157	1	4	yes	—
215	C₂×C₄⋊C₄	#88324	1	1	yes	characteristic
216	C₂×C₂²⋊C₄	#88324	1	1	yes	characteristic
217	C₂×C₂²⋊C₄	#88324	1	1	yes	characteristic
218	C₂×C₄²	C₂×#6157	1	1	yes	characteristic
219	M₃₂	—	106	8	no	—
220	C₂×C₁₆	—	53	2	no	—
221	C₂×C₁₆	—	53	4	no	—
222	C₂×C₁₆	—	53	2	no	—
223	C₂×C₁₆	—	53	2	no	—
224	C₂×C₁₆	—	53	2	no	—
225	C₂×C₁₆	—	106	8	no	—
226	C₂×C₁₆	—	106	2	no	—
227	C₂×C₁₆	—	106	2	no	—
228	C₂×C₁₆	—	106	2	no	—
229	C₄⋊C₈	—	53	4	no	—
230	C₂²⋊C₈	—	53	4	no	—
231	C₂²⋊C₈	—	53	4	no	—
232	C₄×C₈	—	53	2	no	—
233	C₄×C₈	—	53	2	no	—
234	C₂⁴	C₂×#88324	1	1	yes	characteristic
235	C₂⁴	C₂×#88324	1	1	yes	characteristic
236	C₂×D₄	—	2	4	no	—
237	C₂×D₄	C₂×#88324	1	4	yes	—
238	C₂²×C₄	—	2	2	no	—
239	C₂²×C₄	—	2	2	no	—
240	C₂²×C₄	—	2	2	no	—
241	C₂²×C₄	—	2	2	no	—
242	C₂²×C₄	C₂×#88324	1	1	yes	characteristic
243	C₂²×C₄	C₂×#88324	1	1	yes	characteristic
244	C₂²×C₄	C₂×#88324	1	1	yes	characteristic
245	C₂²×C₄	C₂×#88324	1	1	yes	characteristic
246	C₂²×C₄	C₂×#88324	1	2	yes	—
247	C₂²×C₄	C₂×#88324	1	2	yes	—
248	C₂²×C₄	C₂²×#6157	1	2	yes	—
249	C₂²×C₄	C₂²×#6157	1	2	yes	—
250	C₂²×C₄	C₂²×#6157	1	1	yes	characteristic
251	C₂×C₈	—	53	2	no	—
252	C₂×C₈	—	53	2	no	—
253	C₂×C₈	—	53	1	no	—
254	C₂×C₈	—	53	1	no	—
255	C₂×C₈	—	53	2	no	—
256	C₂×C₈	—	53	2	no	—
257	C₂×C₈	—	53	2	no	—
258	C₂×C₈	—	106	2	no	—
259	C₂×C₈	—	106	2	no	—
260	C₂×C₈	—	106	4	no	—
261	C₂×C₈	—	106	4	no	—
262	C₂×C₈	—	106	2	no	—
263	C₂×C₈	—	106	2	no	—
264	C₂×C₈	—	106	2	no	—
265	C₂×C₈	—	106	2	no	—
266	C₂×C₈	—	106	2	no	—
267	C₄⋊C₄	C₂×#88324	1	4	yes	—
268	C₂²⋊C₄	C₂×#88324	1	4	yes	—
269	C₂²⋊C₄	C₂×#88324	1	4	yes	—
270	C₄²	C₂×#88324	1	2	yes	—
271	C₄²	C₂²×#6157	1	2	yes	—
272	C₁₆	—	53	4	no	—
273	C₁₆	—	106	2	no	—
274	C₁₆	—	106	2	no	—
275	C₁₆	—	106	2	no	—
276	C₂³	—	2	2	no	—
277	C₂³	—	2	2	no	—
278	C₂³	—	2	2	no	—
279	C₂³	—	2	2	no	—
280	C₂³	—	2	2	no	—
281	C₂³	#13133291	1	4	yes	complemented
282	C₂³	C₂²×#88324	1	2	yes	—
283	C₂³	C₂²×#88324	1	2	yes	—
284	C₂³	C₂²×#88324	1	1	yes	characteristic
285	D₄	—	2	8	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	4	no	—
291	C₂×C₄	—	2	4	no	—
292	C₂×C₄	—	2	2	no	—
293	C₂×C₄	—	2	2	no	—
294	C₂×C₄	—	2	2	no	—
295	C₂×C₄	—	2	2	no	—
296	C₂×C₄	D₄×#6157	1	2	yes	—
297	C₂×C₄	C₂²×#88324	1	2	yes	—
298	C₂×C₄	C₂²×#88324	1	2	yes	—
299	C₂×C₄	C₂²×#88324	1	2	yes	—
300	C₂×C₄	C₂²×#88324	1	2	yes	—
301	C₂×C₄	C₂²×#88324	1	1	yes	characteristic
302	C₂×C₄	#13133304	1	2	yes	—
303	C₂×C₄	C₂³×#6157	1	1	yes	characteristic
304	C₈	—	53	2	no	—
305	C₈	—	53	1	no	—
306	C₈	—	53	1	no	—
307	C₈	—	106	2	no	—
308	C₈	—	106	2	no	—
309	C₈	—	106	2	no	—
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	D₄×#88324	1	2	yes	—
318	C₂²	C₂×#13133291	1	2	yes	—
319	C₂²	C₂×#13133291	1	4	yes	complemented
320	C₂²	#16811681	1	2	yes	—
321	C₂²	C₂³×#88324	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×#88324	1	2	yes	—
327	C₄	C₂×D₄×#6157	1	1	yes	characteristic
328	C₄	C₂×#13133304	1	1	yes	characteristic
329	C₂	—	2	4	no	—
330	C₂	—	4	2	no	—
331	C₂	#20291095	1	4	yes	direct product factor
332	C₂	C₂×D₄×#88324	1	1	yes	characteristic
333	C₂	C₂²×#13133291	1	1	yes	characteristic
334	C₂	C₂×#16811681	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄	C₂²×#6157
commutator	C₂²×C₅₃	C₂³×C₈
Frattini	C₂×C₄	C₂³×#6157
Fitting	C₂×C₄×D₄×C₅₃	C₄
socle	C₂³×C₅₃	C₂²×C₈
prime radical	C₂²×D₄×C₅₃	C₈
cosocle	C₂×#88322	C₂⁴
hypercenter	C₂×C₄×D₄	#6157
hypocenter	C₅₃	C₂×#2168
LCS	G ▷ C₂²×C₅₃ ▷ C₅₃
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×#13133291	C₂	semidirect
2. C₂³×#11719910	C₂	semidirect
3. #20291095	C₂	direct (N × H)
4. C₂×C₄⋊#11719910	C₂	semidirect
5. C₂×C₂²⋊#11719910	C₂	semidirect
6. C₂×C₂²⋊#11719910	C₂	semidirect
7. C₂×#16811511	C₂	semidirect
10. C₂×#13133291	C₂²	semidirect
16. C₂²×#11719910	C₂²	semidirect
17. C₄⋊#11719910	C₂²	semidirect
18. C₂²⋊#11719910	C₂²	semidirect
19. C₂²⋊#11719910	C₂²	semidirect
20. #16811511	C₂²	semidirect
35. C₂×#11719910	D₄	semidirect
107. #11719910	C₂×D₄	semidirect
113. C₂³×C₅₃	M₃₂	semidirect
152. C₂²×C₅₃	C₂×M₃₂	semidirect
172. C₂×C₅₃	#2168	semidirect
197. C₅₃	C₂×#2168	semidirect
281. C₂³	#13133291	semidirect
319. C₂²	C₂×#13133291	semidirect
331. C₂	#20291095	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×#13133291	2
C₂³×#11719910	2
#20291095	2
C₂×C₄⋊#11719910	2
C₂×C₂²⋊#11719910	2
C₂×#16811511	2
C₂×D₄×#88322	2
C₂×#2168	53

Subgroups — C2×#20291095 #28328295

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#20291095 #28328295