Subgroups — C₂×C₄⋊C₄×#88322 #27382022

Order 13568.

1438 conjugacy classes of subgroups.

The subgroup lattice has 296 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₄⋊#88322	C₂	1	1	yes	maximal, characteristic
2	C₂²×C₄×#88322	C₂	1	2	yes	maximal
3	C₂²×C₄×#88322	C₂	1	1	yes	maximal, characteristic
4	C₄⋊C₄×#88322	C₂	1	8	yes	maximal, direct product factor
5	C₂×#16810378	C₂	1	2	yes	maximal
6	C₂×C₄×C₄⋊C₄×C₅₃	C₂	1	1	yes	maximal, characteristic
7	C₂³×#88322	C₂²	1	2	yes	—
8	C₂³×#88322	C₂²	1	1	yes	characteristic
9	C₂³×#88322	C₂²	1	1	yes	characteristic
10	C₂×C₄⋊#88322	C₄	1	8	yes	complemented
11	C₂×C₄⋊#88322	C₂²	1	4	yes	—
12	C₂×C₄×#88322	—	2	8	no	—
13	C₂×C₄×#88322	C₄	1	8	yes	complemented
14	C₂×C₄×#88322	C₂²	1	8	yes	—
15	C₂×C₄×#88322	C₂²	1	4	yes	—
16	#16810378	C₂²	1	8	yes	—
17	C₂²×C₄⋊C₄×C₅₃	C₄	1	2	yes	—
18	C₂²×C₄²×C₅₃	C₂²	1	2	yes	—
19	C₂²×C₄²×C₅₃	C₂²	1	1	yes	characteristic
20	C₄×C₄⋊C₄×C₅₃	C₄	1	4	yes	—
21	C₄×C₄⋊C₄×C₅₃	C₂²	1	4	yes	—
22	C₂×#95×C₅₃	C₄	1	2	yes	—
23	C₂²×#88322	—	2	4	no	—
24	C₂²×#88322	—	2	2	no	—
25	C₂²×#88322	—	2	4	no	—
26	C₂²×#88322	—	2	2	no	—
27	C₂²×#88322	C₂×C₄	1	4	yes	—
28	C₂²×#88322	C₂×C₄	1	8	yes	—
29	C₂²×#88322	C₂×C₄	1	4	yes	—
30	C₂²×#88322	D₄	1	4	yes	—
31	C₂²×#88322	Q₈	1	4	yes	—
32	C₂²×#88322	C₂³	1	2	yes	—
33	C₂²×#88322	C₂³	1	4	yes	—
34	C₂²×#88322	C₂³	1	2	yes	—
35	C₄⋊#88322	C₂×C₄	1	16	yes	complemented
36	C₄×#88322	—	2	16	no	—
37	C₄×#88322	C₂×C₄	1	16	yes	complemented
38	C₂³×C₄×C₅₃	C₂×C₄	1	1	yes	characteristic
39	C₂³×C₄×C₅₃	C₂×C₄	1	4	yes	—
40	C₂³×C₄×C₅₃	C₂×C₄	1	1	yes	characteristic
41	C₂³×C₄×C₅₃	C₂³	1	1	yes	characteristic
42	C₂×C₄⋊C₄×C₅₃	C₈	1	16	yes	complemented
43	C₂×C₄⋊C₄×C₅₃	C₂×C₄	1	8	yes	—
44	C₂×C₄²×C₅₃	—	2	4	no	—
45	C₂×C₄²×C₅₃	—	2	4	no	—
46	C₂×C₄²×C₅₃	C₂×C₄	1	4	yes	—
47	C₂×C₄²×C₅₃	C₂×C₄	1	4	yes	—
48	C₂×C₄²×C₅₃	C₂×C₄	1	4	yes	—
49	C₂×C₄²×C₅₃	C₂×C₄	1	2	yes	—
50	C₂×C₄²×C₅₃	C₂³	1	4	yes	—
51	C₂×C₄²×C₅₃	C₂³	1	2	yes	—
52	#95×C₅₃	C₂×C₄	1	8	yes	—
53	C₂⁴×C₅₃	C₂²×C₄	1	1	yes	characteristic
54	C₂²×C₄×C₅₃	—	2	8	no	—
55	C₂²×C₄×C₅₃	—	2	4	no	—
56	C₂²×C₄×C₅₃	—	2	8	no	—
57	C₂²×C₄×C₅₃	C₄²	1	4	yes	—
58	C₂²×C₄×C₅₃	C₄²	1	4	yes	—
59	C₂²×C₄×C₅₃	C₂×C₈	1	16	yes	—
60	C₂²×C₄×C₅₃	C₂×C₈	1	8	yes	—
61	C₂²×C₄×C₅₃	C₂²×C₄	1	2	yes	—
62	C₂²×C₄×C₅₃	C₂²×C₄	1	8	yes	—
63	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
64	C₂²×C₄×C₅₃	C₂²×C₄	1	2	yes	—
65	C₂²×C₄×C₅₃	C₂²×C₄	1	2	yes	—
66	C₂²×C₄×C₅₃	C₂²×C₄	1	2	yes	—
67	C₂²×C₄×C₅₃	C₂×D₄	1	2	yes	—
68	C₂²×C₄×C₅₃	C₂×Q₈	1	2	yes	—
69	C₂²×C₄×C₅₃	#41	1	2	yes	—
70	C₂²×C₄×C₅₃	#41	1	2	yes	—
71	C₂²×C₄×C₅₃	C₂⁴	1	1	yes	characteristic
72	C₄⋊C₄×C₅₃	C₂×C₈	1	32	yes	complemented
73	C₄²×C₅₃	—	2	8	no	—
74	C₄²×C₅₃	—	2	4	no	—
75	C₄²×C₅₃	—	2	4	no	—
76	C₄²×C₅₃	C₄²	1	8	yes	—
77	C₄²×C₅₃	C₂²×C₄	1	4	yes	—
78	C₄²×C₅₃	C₂²×C₄	1	4	yes	—
79	C₂×#88322	—	2	8	no	—
80	C₂×#88322	—	2	16	no	—
81	C₂×#88322	—	2	4	no	—
82	C₂×#88322	—	2	2	no	—
83	C₂×#88322	—	2	2	no	—
84	C₂×#88322	—	2	4	no	—
85	C₂×#88322	C₄⋊C₄	1	16	yes	direct product factor
86	C₂×#88322	C₂²×C₄	1	4	yes	—
87	C₂×#88322	C₂²×C₄	1	8	yes	—
88	C₂×#88322	C₂²×C₄	1	4	yes	—
89	C₂×#88322	C₂×D₄	1	4	yes	—
90	C₂×#88322	C₂×Q₈	1	4	yes	—
91	C₂³×C₅₃	C₂×C₄²	1	2	yes	—
92	C₂³×C₅₃	C₄×D₄	1	2	yes	—
93	C₂³×C₅₃	C₄×Q₈	1	2	yes	—
94	C₂³×C₅₃	C₂²×C₈	1	4	yes	—
95	C₂³×C₅₃	#131	1	4	yes	—
96	C₂³×C₅₃	C₂³×C₄	1	1	yes	characteristic
97	C₂×C₄×C₅₃	—	2	8	no	—
98	C₂×C₄×C₅₃	—	2	4	no	—
99	C₂×C₄×C₅₃	—	2	8	no	—
100	C₂×C₄×C₅₃	—	2	8	no	—
101	C₂×C₄×C₅₃	—	2	32	no	—
102	C₂×C₄×C₅₃	C₄×C₈	1	16	yes	complemented
103	C₂×C₄×C₅₃	C₂×C₄²	1	4	yes	—
104	C₂×C₄×C₅₃	C₂×C₄²	1	2	yes	—
105	C₂×C₄×C₅₃	C₂×C₄²	1	4	yes	—
106	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	4	yes	—
107	C₂×C₄×C₅₃	#117	1	4	yes	—
108	C₂×C₄×C₅₃	C₄×D₄	1	4	yes	—
109	C₂×C₄×C₅₃	C₄×D₄	1	2	yes	—
110	C₂×C₄×C₅₃	C₄×Q₈	1	4	yes	—
111	C₂×C₄×C₅₃	C₄×Q₈	1	2	yes	—
112	C₂×C₄×C₅₃	C₂²×C₈	1	16	yes	—
113	C₂×C₄×C₅₃	C₂²×C₈	1	8	yes	—
114	C₂×C₄×C₅₃	C₂³×C₄	1	1	yes	characteristic
115	C₂×C₄×C₅₃	C₂³×C₄	1	1	yes	characteristic
116	C₂×C₄×C₅₃	C₂²×D₄	1	1	yes	characteristic
117	C₂×C₄×C₅₃	C₂²×Q₈	1	1	yes	characteristic
118	C₂×C₄×C₅₃	C₂×#41	1	1	yes	characteristic
119	C₂×C₄×C₅₃	C₂×#41	1	1	yes	characteristic
120	#88322	—	2	4	no	—
121	#88322	—	2	8	no	—
122	#88322	C₂×C₄⋊C₄	1	8	yes	direct product factor
123	C₂×C₈×C₄⋊C₄	—	53	1	no	maximal, Sylow
124	C₂²×C₅₃	C₄×C₄⋊C₄	1	4	yes	—
125	C₂²×C₅₃	C₂×C₄×C₈	1	4	yes	—
126	C₂²×C₅₃	#405	1	4	yes	—
127	C₂²×C₅₃	C₈×D₄	1	8	yes	—
128	C₂²×C₅₃	C₈×Q₈	1	8	yes	—
129	C₂²×C₅₃	C₂²×C₄²	1	1	yes	characteristic
130	C₂²×C₅₃	C₂×C₄×D₄	1	1	yes	characteristic
131	C₂²×C₅₃	C₂×C₄×Q₈	1	1	yes	characteristic
132	C₂²×C₅₃	C₂³×C₈	1	2	yes	—
133	C₂²×C₅₃	C₂×#131	1	2	yes	—
134	C₄×C₅₃	—	2	4	no	—
135	C₄×C₅₃	—	2	16	no	—
136	C₄×C₅₃	C₄×C₄⋊C₄	1	4	yes	—
137	C₄×C₅₃	C₂×C₄×C₈	1	8	yes	complemented
138	C₄×C₅₃	C₂²×C₄⋊C₄	1	1	yes	characteristic
139	C₄×C₅₃	C₂×#117	1	1	yes	characteristic
140	C₄×C₅₃	C₂×C₄×D₄	1	1	yes	characteristic
141	C₄×C₅₃	C₂×C₄×Q₈	1	1	yes	characteristic
142	C₂²×C₄⋊C₈	—	53	1	no	—
143	C₂²×C₄×C₈	—	53	1	no	—
144	C₂²×C₄×C₈	—	53	2	no	—
145	C₂×C₄×C₄⋊C₄	D₅₃	1	1	yes	characteristic
146	C₈×C₄⋊C₄	—	53	8	no	—
147	C₂×#336	—	53	2	no	—
148	C₂×C₅₃	C₈×C₄⋊C₄	1	8	yes	complemented
149	C₂×C₅₃	C₂×C₄×C₄⋊C₄	1	1	yes	characteristic
150	C₂×C₅₃	C₂²×C₄×C₈	1	1	yes	characteristic
151	C₂×C₅₃	C₂×#405	1	1	yes	characteristic
152	C₂×C₅₃	C₂×C₈×D₄	1	2	yes	—
153	C₂×C₅₃	C₂×C₈×Q₈	1	2	yes	—
154	C₂³×C₈	—	53	1	no	—
155	C₂³×C₈	—	53	2	no	—
156	C₂³×C₈	—	53	1	no	—
157	C₂²×C₄⋊C₄	Dic₅₃	1	2	yes	—
158	C₂²×C₄²	C₂×D₅₃	1	2	yes	—
159	C₂²×C₄²	C₂×D₅₃	1	1	yes	characteristic
160	C₂×C₄⋊C₈	—	53	8	no	—
161	C₂×C₄⋊C₈	—	53	4	no	—
162	C₂×C₄×C₈	—	53	8	no	—
163	C₂×C₄×C₈	—	53	4	no	—
164	C₂×C₄×C₈	—	53	8	no	—
165	C₂×C₄×C₈	—	106	8	no	—
166	C₄×C₄⋊C₄	Dic₅₃	1	4	yes	—
167	C₄×C₄⋊C₄	C₂×D₅₃	1	4	yes	—
168	C₂×#95	Dic₅₃	1	2	yes	—
169	#336	—	53	8	no	—
170	C₅₃	C₂×C₈×C₄⋊C₄	1	1	yes	characteristic, complemented, Sylow
171	C₂³×C₄	C₂×Dic₅₃	1	4	yes	—
172	C₂³×C₄	C₂×Dic₅₃	1	1	yes	characteristic
173	C₂³×C₄	C₂×Dic₅₃	1	1	yes	characteristic
174	C₂³×C₄	C₂²×D₅₃	1	1	yes	characteristic
175	C₂²×C₈	—	53	4	no	—
176	C₂²×C₈	—	53	4	no	—
177	C₂²×C₈	—	53	4	no	—
178	C₂²×C₈	—	53	2	no	—
179	C₂²×C₈	—	53	4	no	—
180	C₂²×C₈	—	53	8	no	—
181	C₂²×C₈	—	53	2	no	—
182	C₂²×C₈	—	53	4	no	—
183	C₂²×C₈	—	106	4	no	—
184	C₂²×C₈	—	106	2	no	—
185	C₂²×C₈	—	106	2	no	—
186	C₂²×C₈	—	106	4	no	—
187	C₂×C₄⋊C₄	#88322	1	16	yes	direct product factor
188	C₂×C₄⋊C₄	C₂×Dic₅₃	1	8	yes	—
189	C₂×C₄²	—	2	4	no	—
190	C₂×C₄²	—	2	4	no	—
191	C₂×C₄²	C₄×D₅₃	1	4	yes	—
192	C₂×C₄²	C₄×D₅₃	1	4	yes	—
193	C₂×C₄²	C₂×Dic₅₃	1	4	yes	—
194	C₂×C₄²	C₂×Dic₅₃	1	2	yes	—
195	C₂×C₄²	C₂²×D₅₃	1	4	yes	—
196	C₂×C₄²	C₂²×D₅₃	1	2	yes	—
197	C₄⋊C₈	—	53	16	no	—
198	C₄×C₈	—	53	16	no	—
199	C₄×C₈	—	106	16	no	—
200	#95	C₂×Dic₅₃	1	8	yes	—
201	C₂⁴	C₂²×Dic₅₃	1	1	yes	characteristic
202	C₂²×C₄	—	2	8	no	—
203	C₂²×C₄	—	2	4	no	—
204	C₂²×C₄	—	2	8	no	—
205	C₂²×C₄	C₂×#88322	1	16	yes	—
206	C₂²×C₄	C₂×#88322	1	8	yes	—
207	C₂²×C₄	C₄×Dic₅₃	1	4	yes	—
208	C₂²×C₄	C₄×Dic₅₃	1	4	yes	—
209	C₂²×C₄	C₂×C₄×D₅₃	1	2	yes	—
210	C₂²×C₄	C₂×C₄×D₅₃	1	2	yes	—
211	C₂²×C₄	D₄×D₅₃	1	2	yes	—
212	C₂²×C₄	#11719947	1	2	yes	—
213	C₂²×C₄	Q₈×D₅₃	1	2	yes	—
214	C₂²×C₄	#11719949	1	2	yes	—
215	C₂²×C₄	C₂²×Dic₅₃	1	2	yes	—
216	C₂²×C₄	C₂²×Dic₅₃	1	8	yes	—
217	C₂²×C₄	C₂²×Dic₅₃	1	1	yes	characteristic
218	C₂²×C₄	C₂²×Dic₅₃	1	2	yes	—
219	C₂²×C₄	C₂³×D₅₃	1	1	yes	characteristic
220	C₂×C₈	—	53	16	no	—
221	C₂×C₈	—	53	4	no	—
222	C₂×C₈	—	53	4	no	—
223	C₂×C₈	—	53	4	no	—
224	C₂×C₈	—	53	4	no	—
225	C₂×C₈	—	53	8	no	—
226	C₂×C₈	—	106	8	no	—
227	C₂×C₈	—	106	16	no	—
228	C₂×C₈	—	106	4	no	—
229	C₂×C₈	—	106	2	no	—
230	C₂×C₈	—	106	2	no	—
231	C₂×C₈	—	106	4	no	—
232	C₄⋊C₄	C₂×#88322	1	32	yes	direct product factor
233	C₄²	—	2	8	no	—
234	C₄²	—	2	4	no	—
235	C₄²	—	2	4	no	—
236	C₄²	C₄×Dic₅₃	1	8	yes	—
237	C₄²	C₂×C₄×D₅₃	1	4	yes	—
238	C₄²	C₂×C₄×D₅₃	1	4	yes	—
239	C₂³	C₂×C₄×Dic₅₃	1	2	yes	—
240	C₂³	D₄×Dic₅₃	1	2	yes	—
241	C₂³	Q₈×Dic₅₃	1	2	yes	—
242	C₂³	C₂²×#88322	1	4	yes	—
243	C₂³	#13133224	1	4	yes	—
244	C₂³	C₂³×Dic₅₃	1	1	yes	characteristic
245	C₂×C₄	—	2	8	no	—
246	C₂×C₄	—	2	8	no	—
247	C₂×C₄	—	2	4	no	—
248	C₂×C₄	—	2	8	no	—
249	C₂×C₄	—	2	32	no	—
250	C₂×C₄	C₄×#88322	1	16	yes	complemented
251	C₂×C₄	C₂×C₄×Dic₅₃	1	4	yes	—
252	C₂×C₄	C₂×C₄×Dic₅₃	1	4	yes	—
253	C₂×C₄	C₂×C₄×Dic₅₃	1	2	yes	—
254	C₂×C₄	C₄⋊C₄×D₅₃	1	4	yes	—
255	C₂×C₄	#13133175	1	4	yes	—
256	C₂×C₄	#13133179	1	4	yes	—
257	C₂×C₄	D₄×Dic₅₃	1	2	yes	—
258	C₂×C₄	#13133182	1	4	yes	—
259	C₂×C₄	Q₈×Dic₅₃	1	2	yes	—
260	C₂×C₄	C₂²×#88322	1	16	yes	—
261	C₂×C₄	C₂²×#88322	1	8	yes	—
262	C₂×C₄	C₂²×C₄×D₅₃	1	1	yes	characteristic
263	C₂×C₄	C₂²×C₄×D₅₃	1	1	yes	characteristic
264	C₂×C₄	C₂×D₄×D₅₃	1	1	yes	characteristic
265	C₂×C₄	C₂×Q₈×D₅₃	1	1	yes	characteristic
266	C₂×C₄	C₂×#11719947	1	1	yes	characteristic
267	C₂×C₄	C₂×#11719949	1	1	yes	characteristic
268	C₈	—	53	8	no	—
269	C₈	—	106	4	no	—
270	C₈	—	106	8	no	—
271	C₂²	C₄⋊C₄×Dic₅₃	1	4	yes	—
272	C₂²	C₂×C₄×#88322	1	4	yes	—
273	C₂²	#16810579	1	4	yes	—
274	C₂²	D₄×#88322	1	8	yes	—
275	C₂²	Q₈×#88322	1	8	yes	—
276	C₂²	C₂²×C₄×Dic₅₃	1	1	yes	characteristic
277	C₂²	C₂×D₄×Dic₅₃	1	1	yes	characteristic
278	C₂²	C₂×Q₈×Dic₅₃	1	1	yes	characteristic
279	C₂²	C₂³×#88322	1	2	yes	—
280	C₂²	C₂×#13133224	1	2	yes	—
281	C₄	—	2	4	no	—
282	C₄	—	2	16	no	—
283	C₄	C₄⋊C₄×Dic₅₃	1	4	yes	—
284	C₄	C₂×C₄×#88322	1	8	yes	complemented
285	C₄	C₂×C₄⋊C₄×D₅₃	1	1	yes	characteristic
286	C₄	C₂×#13133175	1	1	yes	characteristic
287	C₄	C₂×#13133179	1	1	yes	characteristic
288	C₄	C₂×#13133182	1	1	yes	characteristic
289	C₂	C₄⋊C₄×#88322	1	8	yes	direct product factor
290	C₂	C₂×C₄⋊C₄×Dic₅₃	1	1	yes	characteristic
291	C₂	C₂²×C₄×#88322	1	1	yes	characteristic
292	C₂	C₂×#16810579	1	1	yes	characteristic
293	C₂	C₂×D₄×#88322	1	2	yes	—
294	C₂	C₂×Q₈×#88322	1	2	yes	—

	subgroup	G/H
center	C₂³×C₄	C₂²×D₅₃
commutator	C₂×C₅₃	C₂²×C₄×C₈
Frattini	C₂²×C₄	C₂³×D₅₃
Fitting	C₂×C₄×C₄⋊C₄×C₅₃	C₂
socle, prime radical	C₂⁴×C₅₃	C₂²×C₄
cosocle	C₂²×C₄×C₅₃	C₂⁴
hypercenter	C₂×C₄×C₄⋊C₄	D₅₃
hypocenter	C₅₃	C₂×C₈×C₄⋊C₄
LCS	G ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂³×C₄ ◁ C₂×C₄×C₄⋊C₄

Complements

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

N	H ≅ G/N	type
4. C₄⋊C₄×#88322	C₂	direct (N × H)
10. C₂×C₄⋊#88322	C₄	semidirect
13. C₂×C₄×#88322	C₄	semidirect
35. C₄⋊#88322	C₂×C₄	semidirect
37. C₄×#88322	C₂×C₄	semidirect
42. C₂×C₄⋊C₄×C₅₃	C₈	semidirect
72. C₄⋊C₄×C₅₃	C₂×C₈	semidirect
85. C₂×#88322	C₄⋊C₄	direct (N × H)
102. C₂×C₄×C₅₃	C₄×C₈	semidirect
122. #88322	C₂×C₄⋊C₄	direct (N × H)
137. C₄×C₅₃	C₂×C₄×C₈	semidirect
148. C₂×C₅₃	C₈×C₄⋊C₄	semidirect
170. C₅₃	C₂×C₈×C₄⋊C₄	semidirect
187. C₂×C₄⋊C₄	#88322	direct (N × H)
232. C₄⋊C₄	C₂×#88322	direct (N × H)
250. C₂×C₄	C₄×#88322	semidirect
284. C₄	C₂×C₄×#88322	semidirect
289. C₂	C₄⋊C₄×#88322	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×C₄⋊#88322	2
C₂²×C₄×#88322	2
C₄⋊C₄×#88322	2
C₂×#16810378	2
C₂×C₄×C₄⋊C₄×C₅₃	2
C₂×C₈×C₄⋊C₄	53

Subgroups — C2×C4⋊C4×#88322 #27382022

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₄⋊C₄×#88322 #27382022