Subgroups — C₂²×D₄×D₂₁₂ #28291003

Order 13568.

20038 conjugacy classes of subgroups.

The subgroup lattice has 420 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂⁴×D₂₁₂	C₂	1	2	yes	maximal, complemented
2	C₂³×D₄×D₅₃	C₂	1	2	yes	maximal, complemented
3	C₂×D₄×D₂₁₂	C₂	1	48	yes	maximal, direct product factor
4	C₂²×#13133214	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂²×#13133192	C₂	1	2	yes	maximal, complemented
6	C₂²×#13133191	C₂	1	2	yes	maximal, complemented
7	C₂²×#13133187	C₂	1	4	yes	maximal, complemented
8	C₂²×C₄×D₂₁₂	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂²×C₄×D₄×C₅₃	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂⁵×D₅₃	C₂²	1	4	yes	complemented
11	C₂³×D₂₁₂	—	2	32	no	—
12	C₂³×D₂₁₂	C₂²	1	1	yes	characteristic
13	C₂³×D₂₁₂	C₂²	1	4	yes	complemented
14	C₂³×D₂₁₂	C₂²	1	2	yes	complemented
15	C₂³×D₂₁₂	C₂²	1	2	yes	complemented
16	C₂³×D₂₁₂	C₂²	1	2	yes	complemented
17	C₂³×D₂₁₂	C₂²	1	48	yes	complemented
18	C₂³×C₂²⋊D₅₃	C₂²	1	4	yes	complemented
19	C₂²×D₄×D₅₃	—	2	32	no	—
20	C₂²×D₄×D₅₃	C₂²	1	48	yes	complemented
21	C₂³×C₄×D₅₃	C₂²	1	2	yes	complemented
22	D₄×D₂₁₂	C₂²	1	256	yes	direct product factor
23	C₂×#13133214	C₂²	1	24	yes	complemented
24	C₂×#13133192	C₂²	1	48	yes	complemented
25	C₂×#13133191	C₂²	1	48	yes	complemented
26	C₂×#13133187	C₂²	1	96	yes	complemented
27	C₂×C₄×D₂₁₂	C₂²	1	24	yes	complemented
28	C₂²×C₄⋊Dic₅₃	C₂²	1	1	yes	characteristic, complemented
29	C₂²×#11719921	C₂²	1	2	yes	complemented
30	C₂²×#11719921	C₂²	1	4	yes	complemented
31	C₂³×D₄×C₅₃	C₂²	1	1	yes	characteristic
32	C₂⁴×C₄×C₅₃	C₂²	1	2	yes	complemented
33	C₂×C₄×D₄×C₅₃	C₂²	1	24	yes	complemented
34	C₂²×C₄⋊C₄×C₅₃	C₂²	1	1	yes	characteristic, complemented
35	C₂²×C₂²⋊C₄×C₅₃	C₂²	1	2	yes	complemented
36	C₂²×C₄²×C₅₃	C₂²	1	1	yes	characteristic, complemented
37	C₂⁴×D₅₃	—	2	32	no	—
38	C₂⁴×D₅₃	—	2	32	no	—
39	C₂⁴×D₅₃	—	4	16	no	—
40	C₂⁴×D₅₃	C₂³	1	4	yes	—
41	C₂⁴×D₅₃	C₂³	1	2	yes	—
42	C₂⁴×D₅₃	C₂³	1	48	yes	complemented
43	C₂²×D₂₁₂	—	2	32	no	—
44	C₂²×D₂₁₂	—	2	16	no	—
45	C₂²×D₂₁₂	—	2	16	no	—
46	C₂²×D₂₁₂	—	2	384	no	—
47	C₂²×D₂₁₂	—	4	8	no	—
48	C₂²×D₂₁₂	D₄	1	16	yes	direct product factor
49	C₂²×D₂₁₂	C₂³	1	12	yes	—
50	C₂²×D₂₁₂	C₂³	1	128	yes	complemented
51	C₂²×D₂₁₂	C₂³	1	48	yes	complemented
52	C₂²×D₂₁₂	C₂³	1	24	yes	complemented
53	C₂²×D₂₁₂	C₂³	1	24	yes	complemented
54	C₂²×D₂₁₂	C₂³	1	24	yes	complemented
55	C₂²×C₂²⋊D₅₃	—	2	32	no	—
56	C₂²×C₂²⋊D₅₃	C₂³	1	48	yes	complemented
57	C₂×D₄×D₅₃	—	2	384	no	—
58	C₂×D₄×D₅₃	C₂³	1	128	yes	complemented
59	C₂³×Dic₅₃	C₂³	1	2	yes	—
60	C₂²×C₄×D₅₃	—	2	16	no	—
61	C₂²×C₄×D₅₃	C₂³	1	24	yes	complemented
62	#13133214	C₂³	1	64	yes	complemented
63	#13133192	C₂³	1	128	yes	complemented
64	#13133191	C₂³	1	128	yes	complemented
65	#13133187	C₂³	1	256	yes	complemented
66	C₄×D₂₁₂	C₂³	1	64	yes	complemented
67	C₂×C₄⋊Dic₅₃	C₂³	1	12	yes	complemented
68	C₂×#11719921	C₂³	1	24	yes	complemented
69	C₂×#11719921	C₂³	1	48	yes	complemented
70	C₂⁵×C₅₃	C₂³	1	2	yes	—
71	C₂²×D₄×C₅₃	D₄	1	16	yes	complemented
72	C₂²×D₄×C₅₃	C₂³	1	12	yes	—
73	C₂³×C₄×C₅₃	—	2	16	no	—
74	C₂³×C₄×C₅₃	C₂³	1	1	yes	characteristic
75	C₂³×C₄×C₅₃	C₂³	1	2	yes	—
76	C₂³×C₄×C₅₃	C₂³	1	1	yes	characteristic
77	C₂³×C₄×C₅₃	C₂³	1	1	yes	characteristic
78	C₂³×C₄×C₅₃	C₂³	1	24	yes	complemented
79	C₄×D₄×C₅₃	C₂³	1	64	yes	complemented
80	C₂×C₄⋊C₄×C₅₃	C₂³	1	12	yes	complemented
81	C₂×C₂²⋊C₄×C₅₃	C₂³	1	24	yes	complemented
82	C₂×C₄²×C₅₃	C₂³	1	12	yes	complemented
83	C₂⁴×C₅₃	—	2	8	no	—
84	C₂⁴×C₅₃	—	2	8	no	—
85	C₂⁴×C₅₃	C₂×D₄	1	16	yes	complemented
86	C₂⁴×C₅₃	C₂⁴	1	12	yes	—
87	C₂⁴×C₅₃	C₂⁴	1	1	yes	characteristic
88	C₂³×D₅₃	—	2	8	no	—
89	C₂³×D₅₃	—	2	8	no	—
90	C₂³×D₅₃	—	2	16	no	—
91	C₂³×D₅₃	—	2	16	no	—
92	C₂³×D₅₃	—	2	192	no	—
93	C₂³×D₅₃	—	2	16	no	—
94	C₂³×D₅₃	—	2	192	no	—
95	C₂³×D₅₃	—	4	96	no	—
96	C₂³×D₅₃	—	4	256	no	—
97	C₂³×D₅₃	C₂×D₄	1	16	yes	complemented
98	C₂³×D₅₃	C₂⁴	1	24	yes	—
99	C₂³×D₅₃	C₂⁴	1	12	yes	—
100	C₂³×D₅₃	C₂⁴	1	64	yes	complemented
101	C₂×D₄×C₅₃	C₂×D₄	1	96	yes	complemented
102	C₂×D₄×C₅₃	C₂⁴	1	16	yes	—
103	C₂²×C₄×C₅₃	—	2	8	no	—
104	C₂²×C₄×C₅₃	—	2	4	no	—
105	C₂²×C₄×C₅₃	—	2	96	no	—
106	C₂²×C₄×C₅₃	C₂×D₄	1	8	yes	complemented
107	C₂²×C₄×C₅₃	C₂×D₄	1	8	yes	complemented
108	C₂²×C₄×C₅₃	C₂⁴	1	6	yes	—
109	C₂²×C₄×C₅₃	C₂⁴	1	12	yes	—
110	C₂²×C₄×C₅₃	C₂⁴	1	6	yes	—
111	C₂²×C₄×C₅₃	C₂⁴	1	6	yes	—
112	C₂²×C₄×C₅₃	C₂⁴	1	32	yes	complemented
113	C₂×C₂²⋊D₅₃	—	2	192	no	—
114	C₂×C₂²⋊D₅₃	C₂⁴	1	64	yes	complemented
115	C₂²×Dic₅₃	—	2	8	no	—
116	C₂²×Dic₅₃	C₂⁴	1	12	yes	—
117	D₄×D₅₃	—	2	512	no	—
118	C₂×D₂₁₂	—	2	192	no	—
119	C₂×D₂₁₂	—	2	96	no	—
120	C₂×D₂₁₂	—	2	96	no	—
121	C₂×D₂₁₂	—	2	512	no	—
122	C₂×D₂₁₂	—	4	48	no	—
123	C₂×D₂₁₂	C₂×D₄	1	96	yes	direct product factor
124	C₂×D₂₁₂	C₂⁴	1	16	yes	—
125	C₂×D₂₁₂	C₂⁴	1	64	yes	complemented
126	C₂×D₂₁₂	C₂⁴	1	32	yes	complemented
127	C₂×D₂₁₂	C₂⁴	1	32	yes	complemented
128	C₂×D₂₁₂	C₂⁴	1	32	yes	complemented
129	C₂×C₄×D₅₃	—	2	96	no	—
130	C₂×C₄×D₅₃	C₂⁴	1	32	yes	complemented
131	C₄⋊C₄×C₅₃	C₂⁴	1	16	yes	complemented
132	C₂²⋊C₄×C₅₃	C₂⁴	1	32	yes	complemented
133	C₄²×C₅₃	C₂⁴	1	16	yes	complemented
134	#11719921	C₂⁴	1	32	yes	complemented
135	#11719921	C₂⁴	1	64	yes	complemented
136	C₄⋊Dic₅₃	C₂⁴	1	16	yes	complemented
137	C₂³×C₅₃	—	2	64	no	—
138	C₂³×C₅₃	—	2	24	no	—
139	C₂³×C₅₃	—	2	24	no	—
140	C₂³×C₅₃	C₂²×D₄	1	4	yes	—
141	C₂³×C₅₃	C₂²×D₄	1	4	yes	—
142	C₂³×C₅₃	C₂²×D₄	1	48	yes	complemented
143	C₂³×C₅₃	H(5,𝔽₂)	1	4	yes	—
144	C₂³×C₅₃	C₂⁵	1	8	yes	—
145	C₂³×C₅₃	C₂⁵	1	3	yes	—
146	C₂²×D₅₃	—	2	64	no	—
147	C₂²×D₅₃	—	2	24	no	—
148	C₂²×D₅₃	—	2	24	no	—
149	C₂²×D₅₃	—	2	48	no	—
150	C₂²×D₅₃	—	2	48	no	—
151	C₂²×D₅₃	—	2	128	no	—
152	C₂²×D₅₃	—	2	48	no	—
153	C₂²×D₅₃	—	2	128	no	—
154	C₂²×D₅₃	—	4	64	no	—
155	C₂²×D₅₃	—	4	768	no	—
156	C₂²×D₅₃	—	4	64	no	—
157	C₂²×D₅₃	C₂²×D₄	1	48	yes	complemented
158	C₂²×D₅₃	C₂⁵	1	16	yes	—
159	C₂²×D₅₃	C₂⁵	1	8	yes	—
160	D₄×C₅₃	C₂²×D₄	1	64	yes	complemented
161	C₂×C₄×C₅₃	—	2	24	no	—
162	C₂×C₄×C₅₃	—	2	12	no	—
163	C₂×C₄×C₅₃	—	2	64	no	—
164	C₂×C₄×C₅₃	C₂²×D₄	1	24	yes	complemented
165	C₂×C₄×C₅₃	C₂²×D₄	1	24	yes	complemented
166	C₂×C₄×C₅₃	C₂⁵	1	8	yes	—
167	C₂×C₄×C₅₃	C₂⁵	1	4	yes	—
168	C₂×C₄×C₅₃	C₂⁵	1	4	yes	—
169	C₂×C₄×C₅₃	C₂⁵	1	4	yes	—
170	C₂²⋊D₅₃	—	2	128	no	—
171	C₂×Dic₅₃	—	2	24	no	—
172	C₂×Dic₅₃	C₂⁵	1	8	yes	—
173	D₂₁₂	—	2	128	no	—
174	D₂₁₂	—	2	64	no	—
175	D₂₁₂	—	2	64	no	—
176	D₂₁₂	—	4	32	no	—
177	D₂₁₂	C₂²×D₄	1	64	yes	direct product factor
178	C₄×D₅₃	—	2	64	no	—
179	C₂²×D₄²	—	53	1	no	maximal, Sylow
180	C₂²×C₅₃	—	2	8	no	—
181	C₂²×C₅₃	—	2	8	no	—
182	C₂²×C₅₃	—	2	96	no	—
183	C₂²×C₅₃	D₄²	1	16	yes	complemented
184	C₂²×C₅₃	C₂³×D₄	1	6	yes	—
185	C₂²×C₅₃	C₂³×D₄	1	6	yes	—
186	C₂²×C₅₃	C₂³×D₄	1	16	yes	complemented
187	C₂²×C₅₃	C₂×H(5,𝔽₂)	1	6	yes	—
188	C₂²×C₅₃	C₂⁶	1	1	yes	characteristic
189	C₂×D₅₃	—	2	8	no	—
190	C₂×D₅₃	—	2	8	no	—
191	C₂×D₅₃	—	2	96	no	—
192	C₂×D₅₃	—	2	16	no	—
193	C₂×D₅₃	—	2	16	no	—
194	C₂×D₅₃	—	2	16	no	—
195	C₂×D₅₃	—	4	256	no	—
196	C₂×D₅₃	—	4	96	no	—
197	C₂×D₅₃	C₂³×D₄	1	16	yes	complemented
198	C₄×C₅₃	—	2	8	no	—
199	C₄×C₅₃	—	2	4	no	—
200	C₄×C₅₃	C₂³×D₄	1	8	yes	complemented
201	C₄×C₅₃	C₂³×D₄	1	8	yes	complemented
202	Dic₅₃	—	2	8	no	—
203	C₂⁴×D₄	—	53	2	no	—
204	C₂⁴×D₄	—	53	2	no	—
205	C₂×D₄²	—	53	48	no	—
206	C₂²×#127	—	53	1	no	—
207	C₂²×#121	—	53	2	no	—
208	C₂²×#121	—	53	2	no	—
209	C₂²×#120	—	53	4	no	—
210	C₂²×C₄×D₄	—	53	1	no	—
211	C₂²×C₄×D₄	D₅₃	1	1	yes	characteristic, complemented
212	C₂×C₅₃	—	2	16	no	—
213	C₂×C₅₃	C₂×D₄²	1	12	yes	complemented
214	C₂×C₅₃	C₂⁴×D₄	1	1	yes	characteristic
215	C₂×C₅₃	C₂⁴×D₄	1	1	yes	characteristic
216	C₂×C₅₃	C₂²×H(5,𝔽₂)	1	1	yes	characteristic
217	D₅₃	—	2	16	no	—
218	D₅₃	—	4	16	no	—
219	C₂⁶	—	53	4	no	—
220	C₂³×D₄	—	53	4	no	—
221	C₂³×D₄	—	53	48	no	—
222	C₂³×D₄	—	53	1	no	—
223	C₂³×D₄	—	53	4	no	—
224	C₂³×D₄	—	53	2	no	—
225	C₂³×D₄	—	53	2	no	—
226	C₂³×D₄	—	53	2	no	—
227	C₂³×D₄	—	53	48	no	—
228	C₂³×D₄	—	106	32	no	—
229	C₂³×D₄	—	106	32	no	—
230	C₂³×D₄	C₂×D₅₃	1	1	yes	characteristic
231	C₂⁴×C₄	—	53	2	no	—
232	C₂⁴×C₄	C₂×D₅₃	1	2	yes	complemented
233	D₄²	—	53	256	no	—
234	C₂×#127	—	53	24	no	—
235	C₂×#121	—	53	48	no	—
236	C₂×#121	—	53	48	no	—
237	C₂×#120	—	53	96	no	—
238	C₂×C₄×D₄	—	53	24	no	—
239	C₂×C₄×D₄	C₂×D₅₃	1	24	yes	complemented
240	C₂²×C₄⋊C₄	—	53	1	no	—
241	C₂²×C₄⋊C₄	C₂×D₅₃	1	1	yes	characteristic, complemented
242	C₂²×C₂²⋊C₄	—	53	2	no	—
243	C₂²×C₂²⋊C₄	—	53	4	no	—
244	C₂²×C₂²⋊C₄	C₂×D₅₃	1	2	yes	complemented
245	C₂²×C₄²	C₂×D₅₃	1	1	yes	characteristic, complemented
246	C₅₃	C₂²×D₄²	1	1	yes	characteristic, complemented, Sylow
247	C₂⁵	—	53	2	no	—
248	C₂⁵	—	53	4	no	—
249	C₂⁵	—	53	48	no	—
250	C₂⁵	—	106	32	no	—
251	C₂⁵	—	106	32	no	—
252	C₂⁵	—	212	16	no	—
253	C₂⁵	C₂²×D₅₃	1	2	yes	—
254	C₂²×D₄	—	53	48	no	—
255	C₂²×D₄	—	53	128	no	—
256	C₂²×D₄	—	53	16	no	—
257	C₂²×D₄	—	53	12	no	—
258	C₂²×D₄	—	53	48	no	—
259	C₂²×D₄	—	53	24	no	—
260	C₂²×D₄	—	53	24	no	—
261	C₂²×D₄	—	53	24	no	—
262	C₂²×D₄	—	53	128	no	—
263	C₂²×D₄	—	106	384	no	—
264	C₂²×D₄	—	106	32	no	—
265	C₂²×D₄	—	106	32	no	—
266	C₂²×D₄	—	106	384	no	—
267	C₂²×D₄	—	106	16	no	—
268	C₂²×D₄	—	106	16	no	—
269	C₂²×D₄	—	212	8	no	—
270	C₂²×D₄	D₂₁₂	1	16	yes	direct product factor
271	C₂²×D₄	C₂²×D₅₃	1	12	yes	—
272	C₂³×C₄	—	2	16	no	—
273	C₂³×C₄	—	53	2	no	—
274	C₂³×C₄	—	53	24	no	—
275	C₂³×C₄	—	106	16	no	—
276	C₂³×C₄	C₂²×D₅₃	1	1	yes	characteristic
277	C₂³×C₄	C₂²×D₅₃	1	2	yes	—
278	C₂³×C₄	C₂²×D₅₃	1	1	yes	characteristic
279	C₂³×C₄	C₂²×D₅₃	1	1	yes	characteristic
280	C₂³×C₄	C₂²×D₅₃	1	24	yes	complemented
281	#127	—	53	64	no	—
282	#121	—	53	128	no	—
283	#121	—	53	128	no	—
284	#120	—	53	256	no	—
285	C₄×D₄	—	53	64	no	—
286	C₄×D₄	C₂²×D₅₃	1	64	yes	complemented
287	C₂×C₄⋊C₄	—	53	12	no	—
288	C₂×C₄⋊C₄	C₂²×D₅₃	1	12	yes	complemented
289	C₂×C₂²⋊C₄	—	53	24	no	—
290	C₂×C₂²⋊C₄	—	53	48	no	—
291	C₂×C₂²⋊C₄	C₂²×D₅₃	1	24	yes	complemented
292	C₂×C₄²	C₂²×D₅₃	1	12	yes	complemented
293	C₂⁴	—	2	8	no	—
294	C₂⁴	—	2	8	no	—
295	C₂⁴	—	53	16	no	—
296	C₂⁴	—	53	12	no	—
297	C₂⁴	—	53	24	no	—
298	C₂⁴	—	53	64	no	—
299	C₂⁴	—	106	192	no	—
300	C₂⁴	—	106	8	no	—
301	C₂⁴	—	106	8	no	—
302	C₂⁴	—	106	16	no	—
303	C₂⁴	—	106	16	no	—
304	C₂⁴	—	106	192	no	—
305	C₂⁴	—	106	16	no	—
306	C₂⁴	—	212	96	no	—
307	C₂⁴	—	212	256	no	—
308	C₂⁴	C₂×D₂₁₂	1	16	yes	complemented
309	C₂⁴	C₂³×D₅₃	1	12	yes	—
310	C₂⁴	C₂³×D₅₃	1	1	yes	characteristic
311	C₂×D₄	—	53	64	no	—
312	C₂×D₄	—	53	96	no	—
313	C₂×D₄	—	53	16	no	—
314	C₂×D₄	—	53	64	no	—
315	C₂×D₄	—	53	32	no	—
316	C₂×D₄	—	53	32	no	—
317	C₂×D₄	—	53	32	no	—
318	C₂×D₄	—	106	512	no	—
319	C₂×D₄	—	106	192	no	—
320	C₂×D₄	—	106	192	no	—
321	C₂×D₄	—	106	512	no	—
322	C₂×D₄	—	106	96	no	—
323	C₂×D₄	—	106	96	no	—
324	C₂×D₄	—	212	48	no	—
325	C₂×D₄	C₂×D₂₁₂	1	96	yes	direct product factor
326	C₂×D₄	C₂³×D₅₃	1	16	yes	—
327	C₂²×C₄	—	2	8	no	—
328	C₂²×C₄	—	2	4	no	—
329	C₂²×C₄	—	2	96	no	—
330	C₂²×C₄	—	53	12	no	—
331	C₂²×C₄	—	53	32	no	—
332	C₂²×C₄	—	106	8	no	—
333	C₂²×C₄	—	106	96	no	—
334	C₂²×C₄	C₂×D₂₁₂	1	8	yes	complemented
335	C₂²×C₄	D₄×D₅₃	1	8	yes	complemented
336	C₂²×C₄	C₂³×D₅₃	1	6	yes	—
337	C₂²×C₄	C₂³×D₅₃	1	12	yes	—
338	C₂²×C₄	C₂³×D₅₃	1	6	yes	—
339	C₂²×C₄	C₂³×D₅₃	1	6	yes	—
340	C₂²×C₄	C₂³×D₅₃	1	32	yes	complemented
341	C₄⋊C₄	—	53	16	no	—
342	C₄⋊C₄	C₂³×D₅₃	1	16	yes	complemented
343	C₂²⋊C₄	—	53	32	no	—
344	C₂²⋊C₄	—	53	64	no	—
345	C₂²⋊C₄	C₂³×D₅₃	1	32	yes	complemented
346	C₄²	C₂³×D₅₃	1	16	yes	complemented
347	C₂³	—	2	24	no	—
348	C₂³	—	2	24	no	—
349	C₂³	—	2	64	no	—
350	C₂³	—	53	48	no	—
351	C₂³	—	53	8	no	—
352	C₂³	—	53	16	no	—
353	C₂³	—	106	128	no	—
354	C₂³	—	106	24	no	—
355	C₂³	—	106	24	no	—
356	C₂³	—	106	64	no	—
357	C₂³	—	106	48	no	—
358	C₂³	—	106	48	no	—
359	C₂³	—	106	128	no	—
360	C₂³	—	106	48	no	—
361	C₂³	—	212	64	no	—
362	C₂³	—	212	768	no	—
363	C₂³	—	212	64	no	—
364	C₂³	C₂×D₄×D₅₃	1	4	yes	—
365	C₂³	C₂²×D₂₁₂	1	4	yes	—
366	C₂³	C₂²×D₂₁₂	1	48	yes	complemented
367	C₂³	#13133263	1	4	yes	—
368	C₂³	C₂⁴×D₅₃	1	8	yes	—
369	C₂³	C₂⁴×D₅₃	1	3	yes	—
370	D₄	—	53	64	no	—
371	D₄	—	106	128	no	—
372	D₄	—	106	128	no	—
373	D₄	—	106	64	no	—
374	D₄	—	106	64	no	—
375	D₄	—	212	32	no	—
376	D₄	C₂²×D₂₁₂	1	64	yes	direct product factor
377	C₂×C₄	—	2	24	no	—
378	C₂×C₄	—	2	12	no	—
379	C₂×C₄	—	2	64	no	—
380	C₂×C₄	—	53	8	no	—
381	C₂×C₄	—	106	24	no	—
382	C₂×C₄	—	106	64	no	—
383	C₂×C₄	C₂×D₄×D₅₃	1	24	yes	complemented
384	C₂×C₄	C₂²×D₂₁₂	1	24	yes	complemented
385	C₂×C₄	C₂⁴×D₅₃	1	4	yes	—
386	C₂×C₄	C₂⁴×D₅₃	1	8	yes	—
387	C₂×C₄	C₂⁴×D₅₃	1	4	yes	—
388	C₂×C₄	C₂⁴×D₅₃	1	4	yes	—
389	C₂²	—	2	96	no	—
390	C₂²	—	2	8	no	—
391	C₂²	—	2	8	no	—
392	C₂²	—	53	16	no	—
393	C₂²	—	106	96	no	—
394	C₂²	—	106	8	no	—
395	C₂²	—	106	8	no	—
396	C₂²	—	106	16	no	—
397	C₂²	—	106	16	no	—
398	C₂²	—	106	16	no	—
399	C₂²	—	212	256	no	—
400	C₂²	—	212	96	no	—
401	C₂²	D₄×D₂₁₂	1	16	yes	direct product factor
402	C₂²	C₂²×D₄×D₅₃	1	6	yes	—
403	C₂²	C₂³×D₂₁₂	1	6	yes	—
404	C₂²	C₂³×D₂₁₂	1	16	yes	complemented
405	C₂²	C₂×#13133263	1	6	yes	—
406	C₂²	C₂⁵×D₅₃	1	1	yes	characteristic
407	C₄	—	2	8	no	—
408	C₄	—	2	4	no	—
409	C₄	—	106	8	no	—
410	C₄	C₂²×D₄×D₅₃	1	8	yes	complemented
411	C₄	C₂³×D₂₁₂	1	8	yes	complemented
412	C₂	—	2	16	no	—
413	C₂	—	106	16	no	—
414	C₂	—	212	16	no	—
415	C₂	C₂×D₄×D₂₁₂	1	12	yes	direct product factor
416	C₂	C₂³×D₄×D₅₃	1	1	yes	characteristic
417	C₂	C₂⁴×D₂₁₂	1	1	yes	characteristic
418	C₂	C₂²×#13133263	1	1	yes	characteristic

	subgroup	G/H
center	C₂⁴	C₂³×D₅₃
commutator, cosocle	C₂²×C₅₃	C₂⁶
Frattini	C₂²	C₂⁵×D₅₃
Fitting	C₂²×C₄×D₄×C₅₃	C₂
socle	C₂⁴×C₅₃	C₂⁴
hypercenter	C₂²×C₄×D₄	D₅₃
hypocenter	C₅₃	C₂²×D₄²
LCS	G ▷ C₂²×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂⁴ ◁ C₂²×C₄×D₄

Complements

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

N	H ≅ G/N	type
1. C₂⁴×D₂₁₂	C₂	semidirect
2. C₂³×D₄×D₅₃	C₂	semidirect
3. C₂×D₄×D₂₁₂	C₂	direct (N × H)
4. C₂²×#13133214	C₂	semidirect
5. C₂²×#13133192	C₂	semidirect
6. C₂²×#13133191	C₂	semidirect
7. C₂²×#13133187	C₂	semidirect
8. C₂²×C₄×D₂₁₂	C₂	semidirect
9. C₂²×C₄×D₄×C₅₃	C₂	semidirect
10. C₂⁵×D₅₃	C₂²	semidirect
13. C₂³×D₂₁₂	C₂²	semidirect
14. C₂³×D₂₁₂	C₂²	semidirect
15. C₂³×D₂₁₂	C₂²	semidirect
16. C₂³×D₂₁₂	C₂²	semidirect
17. C₂³×D₂₁₂	C₂²	semidirect
18. C₂³×C₂²⋊D₅₃	C₂²	semidirect
20. C₂²×D₄×D₅₃	C₂²	semidirect
21. C₂³×C₄×D₅₃	C₂²	semidirect
22. D₄×D₂₁₂	C₂²	direct (N × H)
23. C₂×#13133214	C₂²	semidirect
24. C₂×#13133192	C₂²	semidirect
25. C₂×#13133191	C₂²	semidirect
26. C₂×#13133187	C₂²	semidirect
27. C₂×C₄×D₂₁₂	C₂²	semidirect
28. C₂²×C₄⋊Dic₅₃	C₂²	semidirect
29. C₂²×#11719921	C₂²	semidirect
30. C₂²×#11719921	C₂²	semidirect
32. C₂⁴×C₄×C₅₃	C₂²	semidirect
33. C₂×C₄×D₄×C₅₃	C₂²	semidirect
34. C₂²×C₄⋊C₄×C₅₃	C₂²	semidirect
35. C₂²×C₂²⋊C₄×C₅₃	C₂²	semidirect
36. C₂²×C₄²×C₅₃	C₂²	semidirect
42. C₂⁴×D₅₃	C₂³	semidirect
48. C₂²×D₂₁₂	D₄	direct (N × H)
50. C₂²×D₂₁₂	C₂³	semidirect
51. C₂²×D₂₁₂	C₂³	semidirect
52. C₂²×D₂₁₂	C₂³	semidirect
53. C₂²×D₂₁₂	C₂³	semidirect
54. C₂²×D₂₁₂	C₂³	semidirect
56. C₂²×C₂²⋊D₅₃	C₂³	semidirect
58. C₂×D₄×D₅₃	C₂³	semidirect
61. C₂²×C₄×D₅₃	C₂³	semidirect
62. #13133214	C₂³	semidirect
63. #13133192	C₂³	semidirect
64. #13133191	C₂³	semidirect
65. #13133187	C₂³	semidirect
66. C₄×D₂₁₂	C₂³	semidirect
67. C₂×C₄⋊Dic₅₃	C₂³	semidirect
68. C₂×#11719921	C₂³	semidirect
69. C₂×#11719921	C₂³	semidirect
71. C₂²×D₄×C₅₃	D₄	semidirect
78. C₂³×C₄×C₅₃	C₂³	semidirect
79. C₄×D₄×C₅₃	C₂³	semidirect
80. C₂×C₄⋊C₄×C₅₃	C₂³	semidirect
81. C₂×C₂²⋊C₄×C₅₃	C₂³	semidirect
82. C₂×C₄²×C₅₃	C₂³	semidirect
85. C₂⁴×C₅₃	C₂×D₄	semidirect
97. C₂³×D₅₃	C₂×D₄	semidirect
100. C₂³×D₅₃	C₂⁴	semidirect
101. C₂×D₄×C₅₃	C₂×D₄	semidirect
106. C₂²×C₄×C₅₃	C₂×D₄	semidirect
107. C₂²×C₄×C₅₃	C₂×D₄	semidirect
112. C₂²×C₄×C₅₃	C₂⁴	semidirect
114. C₂×C₂²⋊D₅₃	C₂⁴	semidirect
123. C₂×D₂₁₂	C₂×D₄	direct (N × H)
125. C₂×D₂₁₂	C₂⁴	semidirect
126. C₂×D₂₁₂	C₂⁴	semidirect
127. C₂×D₂₁₂	C₂⁴	semidirect
128. C₂×D₂₁₂	C₂⁴	semidirect
130. C₂×C₄×D₅₃	C₂⁴	semidirect
131. C₄⋊C₄×C₅₃	C₂⁴	semidirect
132. C₂²⋊C₄×C₅₃	C₂⁴	semidirect
133. C₄²×C₅₃	C₂⁴	semidirect
134. #11719921	C₂⁴	semidirect
135. #11719921	C₂⁴	semidirect
136. C₄⋊Dic₅₃	C₂⁴	semidirect
142. C₂³×C₅₃	C₂²×D₄	semidirect
157. C₂²×D₅₃	C₂²×D₄	semidirect
160. D₄×C₅₃	C₂²×D₄	semidirect
164. C₂×C₄×C₅₃	C₂²×D₄	semidirect
165. C₂×C₄×C₅₃	C₂²×D₄	semidirect
177. D₂₁₂	C₂²×D₄	direct (N × H)
183. C₂²×C₅₃	D₄²	semidirect
186. C₂²×C₅₃	C₂³×D₄	semidirect
197. C₂×D₅₃	C₂³×D₄	semidirect
200. C₄×C₅₃	C₂³×D₄	semidirect
201. C₄×C₅₃	C₂³×D₄	semidirect
211. C₂²×C₄×D₄	D₅₃	semidirect
213. C₂×C₅₃	C₂×D₄²	semidirect
232. C₂⁴×C₄	C₂×D₅₃	semidirect
239. C₂×C₄×D₄	C₂×D₅₃	semidirect
241. C₂²×C₄⋊C₄	C₂×D₅₃	semidirect
244. C₂²×C₂²⋊C₄	C₂×D₅₃	semidirect
245. C₂²×C₄²	C₂×D₅₃	semidirect
246. C₅₃	C₂²×D₄²	semidirect
270. C₂²×D₄	D₂₁₂	direct (N × H)
280. C₂³×C₄	C₂²×D₅₃	semidirect
286. C₄×D₄	C₂²×D₅₃	semidirect
288. C₂×C₄⋊C₄	C₂²×D₅₃	semidirect
291. C₂×C₂²⋊C₄	C₂²×D₅₃	semidirect
292. C₂×C₄²	C₂²×D₅₃	semidirect
308. C₂⁴	C₂×D₂₁₂	semidirect
325. C₂×D₄	C₂×D₂₁₂	direct (N × H)
334. C₂²×C₄	C₂×D₂₁₂	semidirect
335. C₂²×C₄	D₄×D₅₃	semidirect
340. C₂²×C₄	C₂³×D₅₃	semidirect
342. C₄⋊C₄	C₂³×D₅₃	semidirect
345. C₂²⋊C₄	C₂³×D₅₃	semidirect
346. C₄²	C₂³×D₅₃	semidirect
366. C₂³	C₂²×D₂₁₂	semidirect
376. D₄	C₂²×D₂₁₂	direct (N × H)
383. C₂×C₄	C₂×D₄×D₅₃	semidirect
384. C₂×C₄	C₂²×D₂₁₂	semidirect
401. C₂²	D₄×D₂₁₂	direct (N × H)
404. C₂²	C₂³×D₂₁₂	semidirect
410. C₄	C₂²×D₄×D₅₃	semidirect
411. C₄	C₂³×D₂₁₂	semidirect
415. C₂	C₂×D₄×D₂₁₂	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂⁴×D₂₁₂	2
C₂³×D₄×D₅₃	2
C₂×D₄×D₂₁₂	2
C₂²×#13133214	2
C₂²×#13133192	2
C₂²×#13133191	2
C₂²×#13133187	2
C₂²×C₄×D₂₁₂	2
C₂²×C₄×D₄×C₅₃	2
C₂²×D₄²	53

Subgroups — C22×D4×D212 #28291003

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂²×D₄×D₂₁₂ #28291003