Subgroups — D₄×#13133153 #27402345

Order 13568.

894 conjugacy classes of subgroups.

The subgroup lattice has 390 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	D₄×#11719917	C₂	1	2	yes	maximal
2	C₂²×#13133153	C₂	1	2	yes	maximal, complemented
3	#20275491	C₂	1	2	yes	maximal, complemented
4	#20274922	C₂	1	4	yes	maximal, complemented
5	#20274887	C₂	1	1	yes	maximal, characteristic, complemented
6	#20274753	C₂	1	2	yes	maximal, complemented
7	C₄×#13133153	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×C₈×D₄×C₅₃	C₂	1	1	yes	maximal, characteristic
9	C₂²×#11719917	C₂²	1	4	yes	—
10	#16810701	—	2	2	no	—
11	#16810695	—	2	4	no	—
12	D₄×#88322	—	2	2	no	—
13	C₂×#13133153	—	2	8	no	—
14	C₂×#13133153	C₂²	1	4	yes	—
15	C₂×#13133153	C₂²	1	1	yes	characteristic
16	C₂×#13133153	C₂²	1	2	yes	—
17	C₂×#13133153	C₂²	1	2	yes	—
18	C₂×#13133153	C₂²	1	1	yes	characteristic
19	#16810647	C₂²	1	2	yes	—
20	#16810587	C₂²	1	4	yes	—
21	C₄×#11719917	C₂²	1	2	yes	—
22	#16810388	C₂²	1	4	yes	—
23	#16810388	C₂²	1	2	yes	—
24	C₂³×C₈×C₅₃	C₂²	1	2	yes	—
25	C₂×C₄×D₄×C₅₃	C₂²	1	1	yes	characteristic
26	C₈×D₄×C₅₃	C₄	1	4	yes	—
27	C₈×D₄×C₅₃	C₄	1	2	yes	—
28	C₈×D₄×C₅₃	C₄	1	2	yes	—
29	C₂×C₄⋊C₈×C₅₃	C₂²	1	1	yes	characteristic
30	C₂×C₂²⋊C₈×C₅₃	C₂²	1	2	yes	—
31	C₂×C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
32	C₂×#11719917	—	2	8	no	—
33	C₂×#11719917	—	2	4	no	—
34	C₂×#11719917	—	4	4	no	—
35	C₂×#11719917	C₂³	1	2	yes	—
36	C₂×#11719917	C₂³	1	4	yes	—
37	C₂×#11719917	C₂³	1	2	yes	—
38	C₂²×#88322	—	2	4	no	—
39	#13133153	—	2	4	no	—
40	#13133153	—	2	2	no	—
41	#13133153	—	2	8	no	—
42	#13133153	—	4	2	no	—
43	#13133153	D₄	1	4	yes	direct product factor
44	C₄⋊#88322	—	2	2	no	—
45	C₄⋊#88322	—	2	2	no	—
46	C₂²⋊#88322	—	2	4	no	—
47	C₂²⋊#88322	—	2	4	no	—
48	#13133128	—	2	2	no	—
49	C₄×#88322	—	2	2	no	—
50	C₂²×D₄×C₅₃	Q₈	1	1	yes	characteristic
51	C₂³×C₄×C₅₃	C₂³	1	2	yes	—
52	C₂²×C₈×C₅₃	—	2	4	no	—
53	C₂²×C₈×C₅₃	—	2	4	no	—
54	C₂²×C₈×C₅₃	C₂×C₄	1	4	yes	—
55	C₂²×C₈×C₅₃	C₂×C₄	1	4	yes	—
56	C₂²×C₈×C₅₃	C₂³	1	2	yes	—
57	C₂²×C₈×C₅₃	C₂³	1	1	yes	characteristic
58	C₂²×C₈×C₅₃	C₂³	1	1	yes	characteristic
59	C₄×D₄×C₅₃	C₂×C₄	1	1	yes	characteristic
60	C₄×D₄×C₅₃	C₂×C₄	1	2	yes	—
61	C₄×D₄×C₅₃	C₂×C₄	1	1	yes	characteristic
62	C₄×D₄×C₅₃	D₄	1	1	yes	characteristic
63	C₄×D₄×C₅₃	D₄	1	2	yes	—
64	C₄×D₄×C₅₃	D₄	1	1	yes	characteristic
65	C₂×C₄⋊C₄×C₅₃	Q₈	1	1	yes	characteristic
66	C₂×C₂²⋊C₄×C₅₃	Q₈	1	2	yes	—
67	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
68	C₄⋊C₈×C₅₃	C₂×C₄	1	2	yes	—
69	C₄⋊C₈×C₅₃	C₂×C₄	1	2	yes	—
70	C₂²⋊C₈×C₅₃	C₂×C₄	1	4	yes	—
71	C₂²⋊C₈×C₅₃	C₂×C₄	1	4	yes	—
72	C₄×C₈×C₅₃	C₂×C₄	1	2	yes	—
73	C₄×C₈×C₅₃	C₂×C₄	1	2	yes	—
74	C₂⁴×C₅₃	C₂×Q₈	1	2	yes	—
75	C₂×D₄×C₅₃	—	2	4	no	—
76	C₂×D₄×C₅₃	C₄⋊C₄	1	2	yes	—
77	C₂×D₄×C₅₃	C₄⋊C₄	1	1	yes	characteristic
78	C₂×D₄×C₅₃	C₄⋊C₄	1	1	yes	characteristic
79	C₂²×C₄×C₅₃	—	2	2	no	—
80	C₂²×C₄×C₅₃	—	2	2	no	—
81	C₂²×C₄×C₅₃	—	2	2	no	—
82	C₂²×C₄×C₅₃	—	2	2	no	—
83	C₂²×C₄×C₅₃	C₂²×C₄	1	2	yes	—
84	C₂²×C₄×C₅₃	C₂²×C₄	1	2	yes	—
85	C₂²×C₄×C₅₃	C₂×D₄	1	2	yes	—
86	C₂²×C₄×C₅₃	C₂×D₄	1	2	yes	—
87	C₂²×C₄×C₅₃	C₂×Q₈	1	1	yes	characteristic
88	C₂²×C₄×C₅₃	C₂×Q₈	1	2	yes	—
89	C₂²×C₄×C₅₃	C₂×Q₈	1	1	yes	characteristic
90	C₂²×C₄×C₅₃	C₂⁴	1	1	yes	characteristic
91	C₂×C₈×C₅₃	—	2	2	no	—
92	C₂×C₈×C₅₃	—	2	2	no	—
93	C₂×C₈×C₅₃	—	2	1	no	—
94	C₂×C₈×C₅₃	—	2	1	no	—
95	C₂×C₈×C₅₃	—	2	8	no	—
96	C₂×C₈×C₅₃	—	2	8	no	—
97	C₂×C₈×C₅₃	C₂²×C₄	1	2	yes	—
98	C₂×C₈×C₅₃	C₂²×C₄	1	4	yes	—
99	C₂×C₈×C₅₃	C₂²×C₄	1	2	yes	—
100	C₂×C₈×C₅₃	C₂×D₄	1	2	yes	—
101	C₂×C₈×C₅₃	#41	1	2	yes	—
102	C₄⋊C₄×C₅₃	C₄⋊C₄	1	2	yes	—
103	C₄⋊C₄×C₅₃	C₄⋊C₄	1	1	yes	characteristic
104	C₄⋊C₄×C₅₃	C₄⋊C₄	1	1	yes	characteristic
105	C₂²⋊C₄×C₅₃	C₄⋊C₄	1	2	yes	—
106	C₂²⋊C₄×C₅₃	C₄⋊C₄	1	2	yes	—
107	C₂²⋊C₄×C₅₃	C₄⋊C₄	1	2	yes	—
108	C₂²⋊C₄×C₅₃	C₄⋊C₄	1	2	yes	—
109	C₄²×C₅₃	C₂²×C₄	1	1	yes	characteristic
110	C₄²×C₅₃	C₂²×C₄	1	1	yes	characteristic
111	C₄²×C₅₃	C₂×D₄	1	1	yes	characteristic
112	C₄²×C₅₃	C₂×D₄	1	1	yes	characteristic
113	#11719917	—	2	2	no	—
114	#11719917	—	2	4	no	—
115	#11719917	—	2	2	no	—
116	#11719917	—	2	4	no	—
117	#11719917	—	2	2	no	—
118	#11719917	—	4	8	no	—
119	#11719917	C₂×D₄	1	4	yes	—
120	C₂×#88322	—	2	2	no	—
121	C₂×#88322	—	2	2	no	—
122	C₂×#88322	—	2	4	no	—
123	C₂×#88322	—	4	8	no	—
124	C₂³×C₅₃	—	2	2	no	—
125	C₂³×C₅₃	—	2	2	no	—
126	C₂³×C₅₃	—	2	4	no	—
127	C₂³×C₅₃	—	4	2	no	—
128	C₂³×C₅₃	C₂×C₄⋊C₄	1	2	yes	—
129	C₂³×C₅₃	C₂×C₄⋊C₄	1	2	yes	—
130	C₂³×C₅₃	C₂²×Q₈	1	1	yes	characteristic
131	D₄×C₅₃	—	2	4	no	—
132	D₄×C₅₃	—	2	2	no	—
133	D₄×C₅₃	#108	1	4	yes	complemented
134	C₂×C₄×C₅₃	—	2	2	no	—
135	C₂×C₄×C₅₃	—	2	2	no	—
136	C₂×C₄×C₅₃	—	2	2	no	—
137	C₂×C₄×C₅₃	—	2	2	no	—
138	C₂×C₄×C₅₃	—	2	2	no	—
139	C₂×C₄×C₅₃	—	2	2	no	—
140	C₂×C₄×C₅₃	—	2	2	no	—
141	C₂×C₄×C₅₃	—	2	2	no	—
142	C₂×C₄×C₅₃	—	2	2	no	—
143	C₂×C₄×C₅₃	—	2	2	no	—
144	C₂×C₄×C₅₃	—	2	2	no	—
145	C₂×C₄×C₅₃	—	4	1	no	—
146	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	2	yes	—
147	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	2	yes	—
148	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
149	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
150	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
151	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
152	C₂×C₄×C₅₃	C₂³×C₄	1	1	yes	characteristic
153	C₂×C₄×C₅₃	C₂²×D₄	1	1	yes	characteristic
154	C₂×C₄×C₅₃	C₂²×D₄	1	1	yes	characteristic
155	C₂×C₄×C₅₃	C₂×#41	1	1	yes	characteristic
156	C₂×C₄×C₅₃	H(5,𝔽₂)	1	1	yes	characteristic
157	C₂×C₄×C₅₃	2₋¹⁺⁴	1	1	yes	characteristic
158	C₈×C₅₃	—	2	4	no	—
159	C₈×C₅₃	—	2	2	no	—
160	C₈×C₅₃	C₄×D₄	1	4	yes	—
161	#88322	—	2	4	no	—
162	#88322	—	4	4	no	—
163	#88322	—	4	2	no	—
164	D₄×#108	—	53	1	no	maximal, Sylow
165	C₂²×C₅₃	—	2	2	no	—
166	C₂²×C₅₃	—	2	1	no	—
167	C₂²×C₅₃	—	2	1	no	—
168	C₂²×C₅₃	—	2	2	no	—
169	C₂²×C₅₃	—	2	2	no	—
170	C₂²×C₅₃	—	2	2	no	—
171	C₂²×C₅₃	—	2	2	no	—
172	C₂²×C₅₃	—	4	4	no	—
173	C₂²×C₅₃	—	4	4	no	—
174	C₂²×C₅₃	C₂×#108	1	4	yes	complemented
175	C₂²×C₅₃	C₂²×C₄⋊C₄	1	1	yes	characteristic
176	C₂²×C₅₃	D₄×Q₈	1	1	yes	characteristic
177	C₂²×C₅₃	#554	1	1	yes	characteristic
178	C₄×C₅₃	—	2	2	no	—
179	C₄×C₅₃	—	2	2	no	—
180	C₄×C₅₃	—	2	1	no	—
181	C₄×C₅₃	—	2	1	no	—
182	C₄×C₅₃	—	2	1	no	—
183	C₄×C₅₃	C₂×#108	1	2	yes	complemented
184	C₄×C₅₃	C₂×C₄×D₄	1	1	yes	characteristic
185	C₄×C₅₃	#520	1	1	yes	characteristic
186	C₄×C₅₃	D₄²	1	1	yes	characteristic
187	C₄×C₅₃	#547	1	1	yes	characteristic
188	D₄×M₁₆	—	53	2	no	—
189	C₂×C₈×D₄	D₅₃	1	1	yes	characteristic
190	C₂²×#108	—	53	2	no	—
191	#1950	—	53	2	no	—
192	#1855	—	53	4	no	—
193	#1848	—	53	1	no	—
194	#1810	—	53	2	no	—
195	C₄×#108	—	53	1	no	—
196	C₂×C₅₃	—	2	1	no	—
197	C₂×C₅₃	—	2	1	no	—
198	C₂×C₅₃	—	2	4	no	—
199	C₂×C₅₃	—	4	2	no	—
200	C₂×C₅₃	D₄×C₄⋊C₄	1	1	yes	characteristic
201	C₂×C₅₃	C₂²×#108	1	1	yes	characteristic
202	C₂×C₅₃	#2916	1	1	yes	characteristic
203	C₂²×M₁₆	—	53	4	no	—
204	C₂³×C₈	C₂×D₅₃	1	2	yes	—
205	C₂×C₄×D₄	C₂×D₅₃	1	1	yes	characteristic
206	#436	—	106	2	no	—
207	#435	—	106	4	no	—
208	C₈×D₄	—	106	2	no	—
209	C₈×D₄	Dic₅₃	1	4	yes	—
210	C₈×D₄	Dic₅₃	1	2	yes	—
211	C₈×D₄	Dic₅₃	1	2	yes	—
212	C₂×#108	—	53	1	no	—
213	C₂×#108	—	53	1	no	—
214	C₂×#108	—	53	2	no	—
215	C₂×#108	—	53	4	no	—
216	C₂×#108	—	53	2	no	—
217	C₂×#108	—	106	8	no	—
218	#423	—	53	2	no	—
219	C₂×C₄⋊C₈	C₂×D₅₃	1	1	yes	characteristic
220	#407	—	53	4	no	—
221	C₂×C₂²⋊C₈	C₂×D₅₃	1	2	yes	—
222	C₄×M₁₆	—	53	2	no	—
223	C₂×C₄×C₈	C₂×D₅₃	1	1	yes	characteristic
224	#341	—	53	4	no	—
225	#341	—	53	2	no	—
226	C₅₃	D₄×#108	1	1	yes	characteristic, complemented, Sylow
227	C₂²×D₄	Dic₁₀₆	1	1	yes	characteristic
228	C₂³×C₄	C₂²×D₅₃	1	2	yes	—
229	C₂×M₁₆	—	53	2	no	—
230	C₂×M₁₆	—	53	4	no	—
231	C₂×M₁₆	—	53	2	no	—
232	C₂×M₁₆	—	106	4	no	—
233	C₂×M₁₆	—	106	8	no	—
234	C₂×M₁₆	—	212	4	no	—
235	C₂²×C₈	—	2	4	no	—
236	C₂²×C₈	—	2	4	no	—
237	C₂²×C₈	—	106	4	no	—
238	C₂²×C₈	C₂×Dic₅₃	1	4	yes	—
239	C₂²×C₈	C₂×Dic₅₃	1	4	yes	—
240	C₂²×C₈	C₂²×D₅₃	1	2	yes	—
241	C₂²×C₈	C₂²×D₅₃	1	1	yes	characteristic
242	C₂²×C₈	C₂²×D₅₃	1	1	yes	characteristic
243	C₄×D₄	D₂₁₂	1	1	yes	characteristic
244	C₄×D₄	D₂₁₂	1	2	yes	—
245	C₄×D₄	D₂₁₂	1	1	yes	characteristic
246	C₄×D₄	C₂×Dic₅₃	1	2	yes	—
247	C₄×D₄	C₂×Dic₅₃	1	1	yes	characteristic
248	C₄×D₄	C₂×Dic₅₃	1	1	yes	characteristic
249	C₂×C₄⋊C₄	Dic₁₀₆	1	1	yes	characteristic
250	C₂×C₂²⋊C₄	Dic₁₀₆	1	2	yes	—
251	C₂×C₄²	C₂²×D₅₃	1	1	yes	characteristic
252	#108	—	53	4	no	—
253	#108	—	106	8	no	—
254	#108	—	106	4	no	—
255	#108	—	106	2	no	—
256	#108	—	212	2	no	—
257	C₄⋊C₈	—	106	2	no	—
258	C₄⋊C₈	—	106	2	no	—
259	C₄⋊C₈	C₂×Dic₅₃	1	2	yes	—
260	C₄⋊C₈	C₂×Dic₅₃	1	2	yes	—
261	C₂²⋊C₈	—	106	4	no	—
262	C₂²⋊C₈	—	106	4	no	—
263	C₂²⋊C₈	C₂×Dic₅₃	1	4	yes	—
264	C₂²⋊C₈	C₂×Dic₅₃	1	4	yes	—
265	#97	—	106	2	no	—
266	C₄×C₈	—	106	2	no	—
267	C₄×C₈	C₂×Dic₅₃	1	2	yes	—
268	C₄×C₈	C₂×Dic₅₃	1	2	yes	—
269	C₂⁴	C₂×Dic₁₀₆	1	2	yes	—
270	C₂×D₄	—	2	4	no	—
271	C₂×D₄	C₄⋊Dic₅₃	1	2	yes	—
272	C₂×D₄	C₄⋊Dic₅₃	1	1	yes	characteristic
273	C₂×D₄	C₄⋊Dic₅₃	1	1	yes	characteristic
274	C₂²×C₄	—	2	2	no	—
275	C₂²×C₄	—	2	2	no	—
276	C₂²×C₄	—	2	2	no	—
277	C₂²×C₄	—	2	2	no	—
278	C₂²×C₄	C₂×Dic₁₀₆	1	1	yes	characteristic
279	C₂²×C₄	C₂×Dic₁₀₆	1	2	yes	—
280	C₂²×C₄	C₂×Dic₁₀₆	1	1	yes	characteristic
281	C₂²×C₄	C₂×D₂₁₂	1	2	yes	—
282	C₂²×C₄	C₂×D₂₁₂	1	2	yes	—
283	C₂²×C₄	C₂²×Dic₅₃	1	2	yes	—
284	C₂²×C₄	C₂²×Dic₅₃	1	2	yes	—
285	C₂²×C₄	C₂³×D₅₃	1	1	yes	characteristic
286	M₁₆	—	53	4	no	—
287	M₁₆	—	106	2	no	—
288	M₁₆	—	106	4	no	—
289	M₁₆	—	106	2	no	—
290	M₁₆	—	106	4	no	—
291	M₁₆	—	106	2	no	—
292	M₁₆	—	212	8	no	—
293	C₂×C₈	—	2	2	no	—
294	C₂×C₈	—	2	2	no	—
295	C₂×C₈	—	2	1	no	—
296	C₂×C₈	—	2	1	no	—
297	C₂×C₈	—	2	8	no	—
298	C₂×C₈	—	2	8	no	—
299	C₂×C₈	—	106	2	no	—
300	C₂×C₈	—	106	2	no	—
301	C₂×C₈	—	106	4	no	—
302	C₂×C₈	—	212	8	no	—
303	C₂×C₈	D₄×D₅₃	1	2	yes	—
304	C₂×C₈	#11719947	1	2	yes	—
305	C₂×C₈	C₂²×Dic₅₃	1	2	yes	—
306	C₂×C₈	C₂²×Dic₅₃	1	4	yes	—
307	C₂×C₈	C₂²×Dic₅₃	1	2	yes	—
308	C₄⋊C₄	C₄⋊Dic₅₃	1	2	yes	—
309	C₄⋊C₄	C₄⋊Dic₅₃	1	1	yes	characteristic
310	C₄⋊C₄	C₄⋊Dic₅₃	1	1	yes	characteristic
311	C₂²⋊C₄	C₄⋊Dic₅₃	1	2	yes	—
312	C₂²⋊C₄	C₄⋊Dic₅₃	1	2	yes	—
313	C₂²⋊C₄	C₄⋊Dic₅₃	1	2	yes	—
314	C₂²⋊C₄	C₄⋊Dic₅₃	1	2	yes	—
315	C₄²	C₂×D₂₁₂	1	1	yes	characteristic
316	C₄²	C₂×D₂₁₂	1	1	yes	characteristic
317	C₄²	C₂²×Dic₅₃	1	1	yes	characteristic
318	C₄²	C₂²×Dic₅₃	1	1	yes	characteristic
319	C₂³	—	2	2	no	—
320	C₂³	—	2	2	no	—
321	C₂³	—	2	4	no	—
322	C₂³	—	4	2	no	—
323	C₂³	C₂×C₄⋊Dic₅₃	1	2	yes	—
324	C₂³	C₂×C₄⋊Dic₅₃	1	2	yes	—
325	C₂³	C₂²×Dic₁₀₆	1	1	yes	characteristic
326	D₄	—	2	4	no	—
327	D₄	—	2	2	no	—
328	D₄	#13133153	1	4	yes	direct product factor
329	C₂×C₄	—	2	2	no	—
330	C₂×C₄	—	2	2	no	—
331	C₂×C₄	—	2	2	no	—
332	C₂×C₄	—	2	2	no	—
333	C₂×C₄	—	2	2	no	—
334	C₂×C₄	—	2	2	no	—
335	C₂×C₄	—	2	2	no	—
336	C₂×C₄	—	2	2	no	—
337	C₂×C₄	—	2	2	no	—
338	C₂×C₄	—	2	2	no	—
339	C₂×C₄	—	2	2	no	—
340	C₂×C₄	—	4	1	no	—
341	C₂×C₄	C₂×C₄⋊Dic₅₃	1	1	yes	characteristic
342	C₂×C₄	C₂×C₄⋊Dic₅₃	1	1	yes	characteristic
343	C₂×C₄	C₂×C₄⋊Dic₅₃	1	1	yes	characteristic
344	C₂×C₄	C₂×C₄⋊Dic₅₃	1	2	yes	—
345	C₂×C₄	C₂×C₄⋊Dic₅₃	1	2	yes	—
346	C₂×C₄	C₂×C₄⋊Dic₅₃	1	1	yes	characteristic
347	C₂×C₄	C₂³×Dic₅₃	1	1	yes	characteristic
348	C₂×C₄	C₂×D₄×D₅₃	1	1	yes	characteristic
349	C₂×C₄	C₂²×D₂₁₂	1	1	yes	characteristic
350	C₂×C₄	C₂×#11719947	1	1	yes	characteristic
351	C₂×C₄	#13133263	1	1	yes	characteristic
352	C₂×C₄	#13133264	1	1	yes	characteristic
353	C₈	—	2	4	no	—
354	C₈	—	2	2	no	—
355	C₈	—	106	4	no	—
356	C₈	—	212	2	no	—
357	C₈	—	212	4	no	—
358	C₈	D₄×Dic₅₃	1	4	yes	—
359	C₂²	—	2	2	no	—
360	C₂²	—	2	1	no	—
361	C₂²	—	2	1	no	—
362	C₂²	—	2	2	no	—
363	C₂²	—	2	2	no	—
364	C₂²	—	2	2	no	—
365	C₂²	—	2	2	no	—
366	C₂²	—	4	4	no	—
367	C₂²	—	4	4	no	—
368	C₂²	C₂×#13133153	1	4	yes	complemented
369	C₂²	C₂²×C₄⋊Dic₅₃	1	1	yes	characteristic
370	C₂²	D₄×Dic₁₀₆	1	1	yes	characteristic
371	C₂²	#16811330	1	1	yes	characteristic
372	C₄	—	2	2	no	—
373	C₄	—	2	2	no	—
374	C₄	—	2	1	no	—
375	C₄	—	2	1	no	—
376	C₄	—	2	1	no	—
377	C₄	C₂×#13133153	1	2	yes	complemented
378	C₄	C₂×D₄×Dic₅₃	1	1	yes	characteristic
379	C₄	#16811106	1	1	yes	characteristic
380	C₄	D₄×D₂₁₂	1	1	yes	characteristic
381	C₄	#16811277	1	1	yes	characteristic
382	C₂	—	2	1	no	—
383	C₂	—	2	1	no	—
384	C₂	—	2	4	no	—
385	C₂	—	4	2	no	—
386	C₂	D₄×C₄⋊Dic₅₃	1	1	yes	characteristic
387	C₂	C₂²×#13133153	1	1	yes	characteristic
388	C₂	#20283326	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₂²×D₂₁₂
commutator	C₂×C₄×C₅₃	C₂³×C₄
Frattini	C₂²×C₄	C₂³×D₅₃
Fitting	C₂×C₈×D₄×C₅₃	C₂
socle	C₂²×C₅₃	C₂²×C₄⋊C₄
prime radical	C₂²×D₄×C₅₃	Q₈
cosocle	C₂²×C₄×C₅₃	C₂⁴
hypercenter	C₂×C₈×D₄	D₅₃
hypocenter	C₅₃	D₄×#108
LCS	G ▷ C₂×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄×D₄ ◁ C₂×C₈×D₄

Complements

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

N	H ≅ G/N	type
2. C₂²×#13133153	C₂	semidirect
3. #20275491	C₂	semidirect
4. #20274922	C₂	semidirect
5. #20274887	C₂	semidirect
6. #20274753	C₂	semidirect
7. C₄×#13133153	C₂	semidirect
43. #13133153	D₄	direct (N × H)
133. D₄×C₅₃	#108	semidirect
174. C₂²×C₅₃	C₂×#108	semidirect
183. C₄×C₅₃	C₂×#108	semidirect
226. C₅₃	D₄×#108	semidirect
328. D₄	#13133153	direct (N × H)
368. C₂²	C₂×#13133153	semidirect
377. C₄	C₂×#13133153	semidirect

Maximal subgroups

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

subgroup	index
D₄×#11719917	2
C₂²×#13133153	2
#20275491	2
#20274922	2
#20274887	2
#20274753	2
C₄×#13133153	2
C₂×C₈×D₄×C₅₃	2
D₄×#108	53

Subgroups — D4×#13133153 #27402345

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — D₄×#13133153 #27402345