Subgroups — C₂²×#16811425 #28297901

Order 13568.

11462 conjugacy classes of subgroups.

The subgroup lattice has 426 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×#13133263	C₂	1	2	yes	maximal, complemented
2	C₂×#16811425	C₂	1	48	yes	maximal, direct product factor
3	C₂²×#13133244	C₂	1	4	yes	maximal, complemented
4	C₂²×#13133242	C₂	1	2	yes	maximal, complemented
5	C₂²×#13133240	C₂	1	2	yes	maximal, complemented
6	C₂³×D₄₂₄	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂²×D₈×D₅₃	C₂	1	2	yes	maximal, complemented
8	C₂²×#13133223	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂²×#135×C₅₃	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂×#13133263	—	2	24	no	—
11	C₂×#13133263	C₂²	1	48	yes	complemented
12	C₂²×#11719949	C₂²	1	4	yes	complemented
13	C₂²×#11719945	—	2	2	no	—
14	C₂²×#11719945	C₂²	1	1	yes	characteristic
15	C₂³×D₂₁₂	—	2	2	no	—
16	C₂³×D₂₁₂	C₂²	1	2	yes	complemented
17	C₂²×D₄×D₅₃	—	2	4	no	—
18	C₂²×D₄×D₅₃	C₂²	1	4	yes	complemented
19	#16811425	C₂²	1	256	yes	direct product factor
20	C₂×#13133244	C₂²	1	96	yes	complemented
21	C₂×#13133242	C₂²	1	48	yes	complemented
22	C₂×#13133240	C₂²	1	48	yes	complemented
23	C₂²×#11719924	C₂²	1	4	yes	complemented
24	C₂²×D₄₂₄	C₂²	1	2	yes	complemented
25	C₂²×D₄₂₄	C₂²	1	2	yes	complemented
26	C₂²×D₄₂₄	C₂²	1	24	yes	complemented
27	C₂²×#11719922	C₂²	1	4	yes	complemented
28	C₂×D₈×D₅₃	C₂²	1	48	yes	complemented
29	C₂×#13133223	C₂²	1	24	yes	complemented
30	C₂²×#11719917	C₂²	1	1	yes	characteristic, complemented
31	C₂²×#11719912	C₂²	1	2	yes	complemented
32	C₂²×C₈×D₅₃	C₂²	1	2	yes	complemented
33	C₂²×#41×C₅₃	C₂²	1	2	yes	complemented
34	C₂×#135×C₅₃	C₂²	1	24	yes	complemented
35	C₂²×Q₁₆×C₅₃	C₂²	1	1	yes	characteristic, complemented
36	C₂²×SD₁₆×C₅₃	C₂²	1	2	yes	complemented
37	C₂²×D₈×C₅₃	C₂²	1	1	yes	characteristic, complemented
38	C₂³×C₈×C₅₃	C₂²	1	1	yes	characteristic, complemented
39	C₂⁴×D₅₃	—	2	2	no	—
40	C₂⁴×D₅₃	—	2	4	no	—
41	#13133263	—	2	192	no	—
42	#13133263	C₂³	1	128	yes	complemented
43	C₂×#11719949	—	2	24	no	—
44	C₂×#11719949	C₂³	1	48	yes	complemented
45	C₂×#11719945	—	2	48	no	—
46	C₂×#11719945	D₄	1	12	yes	—
47	C₂×#11719945	C₂³	1	12	yes	—
48	C₂²×Dic₁₀₆	D₄	1	1	yes	characteristic
49	C₂²×D₂₁₂	—	2	4	no	—
50	C₂²×D₂₁₂	—	2	2	no	—
51	C₂²×D₂₁₂	—	2	12	no	—
52	C₂²×D₂₁₂	—	2	48	no	—
53	C₂²×D₂₁₂	D₄	1	1	yes	characteristic
54	C₂²×D₂₁₂	C₂³	1	4	yes	—
55	C₂²×D₂₁₂	C₂³	1	24	yes	complemented
56	C₂²×C₂²⋊D₅₃	—	2	4	no	—
57	C₂²×C₂²⋊D₅₃	D₄	1	2	yes	—
58	C₂×D₄×D₅₃	—	2	96	no	—
59	C₂×D₄×D₅₃	—	2	24	no	—
60	C₂×D₄×D₅₃	C₂³	1	48	yes	complemented
61	C₂²×C₄×D₅₃	—	2	4	no	—
62	C₂²×C₄×D₅₃	C₂³	1	2	yes	—
63	#13133244	C₂³	1	256	yes	complemented
64	#13133242	C₂³	1	128	yes	complemented
65	#13133240	C₂³	1	128	yes	complemented
66	C₂×#11719924	C₂³	1	48	yes	complemented
67	C₂×D₄₂₄	C₂³	1	24	yes	complemented
68	C₂×D₄₂₄	C₂³	1	24	yes	complemented
69	C₂×D₄₂₄	C₂³	1	64	yes	complemented
70	C₂×#11719922	C₂³	1	48	yes	complemented
71	D₈×D₅₃	C₂³	1	128	yes	complemented
72	#13133223	C₂³	1	64	yes	complemented
73	C₂×#11719917	C₂³	1	12	yes	complemented
74	C₂×#11719912	C₂³	1	24	yes	complemented
75	C₂²×#88322	C₂³	1	2	yes	—
76	C₂×C₈×D₅₃	C₂³	1	24	yes	complemented
77	C₂×#41×C₅₃	—	2	12	no	—
78	C₂×#41×C₅₃	C₂³	1	24	yes	complemented
79	C₂²×Q₈×C₅₃	C₂³	1	2	yes	—
80	C₂²×D₄×C₅₃	—	2	2	no	—
81	C₂²×D₄×C₅₃	C₂³	1	2	yes	—
82	C₂³×C₄×C₅₃	—	2	2	no	—
83	C₂³×C₄×C₅₃	C₂³	1	1	yes	characteristic
84	#135×C₅₃	C₂³	1	64	yes	complemented
85	C₂×Q₁₆×C₅₃	C₂³	1	12	yes	complemented
86	C₂×SD₁₆×C₅₃	C₂³	1	24	yes	complemented
87	C₂×D₈×C₅₃	C₂³	1	12	yes	complemented
88	C₂²×C₈×C₅₃	C₂³	1	1	yes	characteristic
89	C₂²×C₈×C₅₃	C₂³	1	1	yes	characteristic
90	C₂²×C₈×C₅₃	C₂³	1	12	yes	complemented
91	C₂⁴×C₅₃	—	2	2	no	—
92	C₂⁴×C₅₃	C₂×D₄	1	1	yes	characteristic
93	C₂³×D₅₃	—	2	4	no	—
94	C₂³×D₅₃	—	2	24	no	—
95	C₂³×D₅₃	—	2	48	no	—
96	C₂³×D₅₃	—	8	16	no	—
97	C₂³×D₅₃	—	8	8	no	—
98	C₂³×D₅₃	C₂×D₄	1	2	yes	—
99	#41×C₅₃	—	2	48	no	—
100	#41×C₅₃	C₂⁴	1	32	yes	complemented
101	C₂×Q₈×C₅₃	—	2	6	no	—
102	C₂×Q₈×C₅₃	C₂⁴	1	12	yes	—
103	C₂×D₄×C₅₃	—	2	24	no	—
104	C₂×D₄×C₅₃	—	2	6	no	—
105	C₂×D₄×C₅₃	C₂⁴	1	12	yes	—
106	C₂²×C₄×C₅₃	—	2	2	no	—
107	C₂²×C₄×C₅₃	—	2	24	no	—
108	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
109	C₂²×C₄×C₅₃	C₂×D₄	1	6	yes	—
110	C₂²×C₄×C₅₃	C₂⁴	1	1	yes	characteristic
111	C₂²×C₄×C₅₃	C₂⁴	1	6	yes	—
112	C₂×C₂²⋊D₅₃	—	2	48	no	—
113	C₂×C₂²⋊D₅₃	C₂×D₄	1	24	yes	—
114	C₂²×Dic₅₃	C₂×D₄	1	2	yes	—
115	#11719949	—	2	96	no	—
116	#11719949	C₂⁴	1	64	yes	complemented
117	D₄×D₅₃	—	2	256	no	—
118	D₄×D₅₃	—	2	96	no	—
119	D₄×D₅₃	C₂⁴	1	64	yes	complemented
120	#11719945	—	2	128	no	—
121	#11719945	C₂×D₄	1	48	yes	—
122	#11719945	C₂⁴	1	16	yes	—
123	C₂×D₂₁₂	—	2	48	no	—
124	C₂×D₂₁₂	—	2	12	no	—
125	C₂×D₂₁₂	—	2	24	no	—
126	C₂×D₂₁₂	—	2	48	no	—
127	C₂×D₂₁₂	—	2	128	no	—
128	C₂×D₂₁₂	C₂×D₄	1	12	yes	—
129	C₂×D₂₁₂	C₂⁴	1	24	yes	—
130	C₂×D₂₁₂	C₂⁴	1	32	yes	complemented
131	C₂×C₄×D₅₃	—	2	48	no	—
132	C₂×C₄×D₅₃	C₂×D₄	1	12	yes	—
133	C₂×C₄×D₅₃	C₂⁴	1	12	yes	—
134	C₂×Dic₁₀₆	C₂×D₄	1	12	yes	—
135	Q₁₆×C₅₃	C₂⁴	1	16	yes	complemented
136	SD₁₆×C₅₃	C₂⁴	1	32	yes	complemented
137	D₈×C₅₃	C₂⁴	1	16	yes	complemented
138	C₂×C₈×C₅₃	C₂⁴	1	6	yes	—
139	C₂×C₈×C₅₃	C₂⁴	1	6	yes	—
140	C₂×C₈×C₅₃	C₂⁴	1	16	yes	complemented
141	#11719924	C₂⁴	1	64	yes	complemented
142	#11719922	C₂⁴	1	64	yes	complemented
143	#11719917	C₂⁴	1	16	yes	complemented
144	C₂×#88322	C₂⁴	1	12	yes	—
145	D₄₂₄	C₂⁴	1	32	yes	complemented
146	D₄₂₄	C₂⁴	1	32	yes	complemented
147	#11719912	C₂⁴	1	32	yes	complemented
148	C₈×D₅₃	C₂⁴	1	32	yes	complemented
149	C₂³×C₅₃	—	2	4	no	—
150	C₂³×C₅₃	—	2	12	no	—
151	C₂³×C₅₃	—	4	8	no	—
152	C₂³×C₅₃	C₂²×D₄	1	6	yes	—
153	C₂³×C₅₃	C₂²×D₄	1	1	yes	characteristic
154	C₂²×D₅₃	—	2	8	no	—
155	C₂²×D₅₃	—	2	24	no	—
156	C₂²×D₅₃	—	2	32	no	—
157	C₂²×D₅₃	—	2	64	no	—
158	C₂²×D₅₃	—	4	16	no	—
159	C₂²×D₅₃	—	8	96	no	—
160	C₂²×D₅₃	—	8	48	no	—
161	C₂²×D₅₃	C₂²×D₄	1	12	yes	—
162	Q₈×C₅₃	—	2	12	no	—
163	Q₈×C₅₃	C₂⁵	1	8	yes	—
164	D₄×C₅₃	—	2	32	no	—
165	D₄×C₅₃	—	2	12	no	—
166	D₄×C₅₃	C₂⁵	1	8	yes	—
167	C₂×C₄×C₅₃	—	2	12	no	—
168	C₂×C₄×C₅₃	—	2	32	no	—
169	C₂×C₄×C₅₃	C₂²×D₄	1	6	yes	—
170	C₂×C₄×C₅₃	C₂²×D₄	1	3	yes	—
171	C₂×C₄×C₅₃	C₂²×D₄	1	12	yes	—
172	C₂×C₄×C₅₃	C₂⁵	1	3	yes	—
173	C₂×C₄×C₅₃	C₂⁵	1	4	yes	—
174	C₂²⋊D₅₃	—	2	64	no	—
175	C₂²⋊D₅₃	C₂²×D₄	1	32	yes	—
176	C₂×Dic₅₃	C₂²×D₄	1	12	yes	—
177	D₂₁₂	—	2	64	no	—
178	D₂₁₂	—	2	32	no	—
179	D₂₁₂	—	2	24	no	—
180	D₂₁₂	C₂²×D₄	1	16	yes	—
181	D₂₁₂	C₂⁵	1	16	yes	—
182	C₄×D₅₃	—	2	64	no	—
183	C₄×D₅₃	C₂²×D₄	1	24	yes	—
184	C₄×D₅₃	C₂⁵	1	8	yes	—
185	Dic₁₀₆	C₂²×D₄	1	16	yes	—
186	C₈×C₅₃	C₂⁵	1	4	yes	—
187	C₈×C₅₃	C₂⁵	1	4	yes	—
188	#88322	C₂⁵	1	8	yes	—
189	C₂²×#576	—	53	1	no	maximal, Sylow
190	C₂²×C₅₃	—	2	12	no	—
191	C₂²×C₅₃	—	2	8	no	—
192	C₂²×C₅₃	—	4	24	no	—
193	C₂²×C₅₃	#576	1	4	yes	complemented
194	C₂²×C₅₃	C₂³×D₄	1	4	yes	—
195	C₂²×C₅₃	C₂³×D₄	1	3	yes	—
196	C₂×D₅₃	—	2	24	no	—
197	C₂×D₅₃	—	2	16	no	—
198	C₂×D₅₃	—	4	48	no	—
199	C₂×D₅₃	—	8	64	no	—
200	C₂×D₅₃	—	8	32	no	—
201	C₂×D₅₃	C₂³×D₄	1	8	yes	—
202	C₄×C₅₃	—	2	8	no	—
203	C₄×C₅₃	C₂³×D₄	1	4	yes	—
204	C₄×C₅₃	C₂³×D₄	1	3	yes	—
205	C₄×C₅₃	C₂⁶	1	1	yes	characteristic
206	Dic₅₃	C₂³×D₄	1	8	yes	—
207	C₂²×H(5,𝔽₂)	—	53	2	no	—
208	C₂×#576	—	53	48	no	—
209	C₂²×AGL(1,ℤ/8)	—	53	2	no	—
210	C₂²×AGL(1,ℤ/8)	—	53	4	no	—
211	C₂²×#135	—	53	2	no	—
212	C₂²×#135	D₅₃	1	1	yes	characteristic, complemented
213	C₂³×D₈	—	53	2	no	—
214	C₂³×D₈	—	53	1	no	—
215	C₂²×#131	—	53	1	no	—
216	C₂×C₅₃	—	2	4	no	—
217	C₂×C₅₃	—	4	8	no	—
218	C₂×C₅₃	C₂×#576	1	6	yes	complemented
219	C₂×C₅₃	C₂⁴×D₄	1	1	yes	characteristic
220	D₅₃	—	2	8	no	—
221	D₅₃	—	4	16	no	—
222	C₂×H(5,𝔽₂)	—	53	48	no	—
223	C₂×H(5,𝔽₂)	—	106	24	no	—
224	C₂²×#41	—	53	1	no	—
225	C₂²×#41	—	53	4	no	—
226	C₂²×#41	—	106	2	no	—
227	C₂²×#41	C₂×D₅₃	1	2	yes	complemented
228	C₂³×D₄	—	53	4	no	—
229	C₂³×D₄	—	53	2	no	—
230	C₂³×D₄	—	106	2	no	—
231	C₂³×D₄	—	106	4	no	—
232	#576	—	53	256	no	—
233	C₂×AGL(1,ℤ/8)	—	53	48	no	—
234	C₂×AGL(1,ℤ/8)	—	53	96	no	—
235	C₂×#135	—	53	48	no	—
236	C₂×#135	C₂×D₅₃	1	24	yes	complemented
237	C₂²×Q₁₆	C₂×D₅₃	1	1	yes	characteristic, complemented
238	C₂²×SD₁₆	—	53	4	no	—
239	C₂²×SD₁₆	C₂×D₅₃	1	2	yes	complemented
240	C₂²×D₈	—	53	4	no	—
241	C₂²×D₈	—	53	48	no	—
242	C₂²×D₈	—	53	2	no	—
243	C₂²×D₈	—	53	2	no	—
244	C₂²×D₈	—	53	24	no	—
245	C₂²×D₈	C₂×D₅₃	1	1	yes	characteristic, complemented
246	C₂×#131	—	53	24	no	—
247	C₂²×M₁₆	—	53	1	no	—
248	C₂²×M₁₆	—	53	2	no	—
249	C₂³×C₈	—	53	2	no	—
250	C₂³×C₈	C₂×D₅₃	1	1	yes	characteristic, complemented
251	C₅₃	C₂²×#576	1	1	yes	characteristic, complemented, Sylow
252	C₂⁵	—	106	4	no	—
253	C₂⁵	—	106	2	no	—
254	H(5,𝔽₂)	—	53	128	no	—
255	H(5,𝔽₂)	—	106	192	no	—
256	C₂×#41	—	2	12	no	—
257	C₂×#41	—	53	12	no	—
258	C₂×#41	—	53	12	no	—
259	C₂×#41	—	53	48	no	—
260	C₂×#41	—	106	48	no	—
261	C₂×#41	—	106	24	no	—
262	C₂×#41	C₂²×D₅₃	1	24	yes	complemented
263	C₂²×Q₈	—	53	1	no	—
264	C₂²×Q₈	C₂²×D₅₃	1	2	yes	—
265	C₂²×D₄	—	2	2	no	—
266	C₂²×D₄	—	53	2	no	—
267	C₂²×D₄	—	53	48	no	—
268	C₂²×D₄	—	53	1	no	—
269	C₂²×D₄	—	53	4	no	—
270	C₂²×D₄	—	53	24	no	—
271	C₂²×D₄	—	106	48	no	—
272	C₂²×D₄	—	106	4	no	—
273	C₂²×D₄	—	106	24	no	—
274	C₂²×D₄	—	106	4	no	—
275	C₂²×D₄	—	106	96	no	—
276	C₂²×D₄	—	106	2	no	—
277	C₂²×D₄	—	106	12	no	—
278	C₂²×D₄	C₂²×D₅₃	1	2	yes	—
279	C₂³×C₄	—	2	2	no	—
280	C₂³×C₄	—	53	2	no	—
281	C₂³×C₄	—	106	4	no	—
282	C₂³×C₄	C₂²×D₅₃	1	1	yes	characteristic
283	AGL(1,ℤ/8)	—	53	128	no	—
284	AGL(1,ℤ/8)	—	53	256	no	—
285	#135	—	53	128	no	—
286	#135	C₂²×D₅₃	1	64	yes	complemented
287	C₂×Q₁₆	C₂²×D₅₃	1	12	yes	complemented
288	C₂×SD₁₆	—	53	48	no	—
289	C₂×SD₁₆	C₂²×D₅₃	1	24	yes	complemented
290	C₂×D₈	—	53	48	no	—
291	C₂×D₈	—	53	128	no	—
292	C₂×D₈	—	53	24	no	—
293	C₂×D₈	—	53	24	no	—
294	C₂×D₈	—	53	64	no	—
295	C₂×D₈	C₂²×D₅₃	1	12	yes	complemented
296	#131	—	53	64	no	—
297	C₂×M₁₆	—	53	12	no	—
298	C₂×M₁₆	—	53	24	no	—
299	C₂²×C₈	—	53	2	no	—
300	C₂²×C₈	—	53	24	no	—
301	C₂²×C₈	C₂²×D₅₃	1	1	yes	characteristic
302	C₂²×C₈	C₂²×D₅₃	1	1	yes	characteristic
303	C₂²×C₈	C₂²×D₅₃	1	12	yes	complemented
304	C₂⁴	—	2	2	no	—
305	C₂⁴	—	53	2	no	—
306	C₂⁴	—	106	48	no	—
307	C₂⁴	—	106	24	no	—
308	C₂⁴	—	106	4	no	—
309	C₂⁴	—	424	8	no	—
310	C₂⁴	—	424	16	no	—
311	C₂⁴	D₄×D₅₃	1	1	yes	characteristic
312	#41	—	2	48	no	—
313	#41	—	53	48	no	—
314	#41	—	53	16	no	—
315	#41	—	53	64	no	—
316	#41	—	106	128	no	—
317	#41	—	106	96	no	—
318	#41	C₂³×D₅₃	1	32	yes	complemented
319	C₂×Q₈	—	2	6	no	—
320	C₂×Q₈	—	53	12	no	—
321	C₂×Q₈	C₂³×D₅₃	1	12	yes	—
322	C₂×D₄	—	2	24	no	—
323	C₂×D₄	—	2	6	no	—
324	C₂×D₄	—	53	24	no	—
325	C₂×D₄	—	53	64	no	—
326	C₂×D₄	—	53	12	no	—
327	C₂×D₄	—	53	24	no	—
328	C₂×D₄	—	53	32	no	—
329	C₂×D₄	—	106	128	no	—
330	C₂×D₄	—	106	48	no	—
331	C₂×D₄	—	106	96	no	—
332	C₂×D₄	—	106	48	no	—
333	C₂×D₄	—	106	256	no	—
334	C₂×D₄	—	106	24	no	—
335	C₂×D₄	—	106	12	no	—
336	C₂×D₄	—	106	48	no	—
337	C₂×D₄	C₂³×D₅₃	1	12	yes	—
338	C₂²×C₄	—	2	2	no	—
339	C₂²×C₄	—	2	24	no	—
340	C₂²×C₄	—	53	2	no	—
341	C₂²×C₄	—	53	12	no	—
342	C₂²×C₄	—	53	12	no	—
343	C₂²×C₄	—	106	48	no	—
344	C₂²×C₄	D₄×D₅₃	1	6	yes	—
345	C₂²×C₄	D₄×D₅₃	1	1	yes	characteristic
346	C₂²×C₄	C₂³×D₅₃	1	6	yes	—
347	C₂²×C₄	C₂³×D₅₃	1	1	yes	characteristic
348	Q₁₆	C₂³×D₅₃	1	16	yes	complemented
349	SD₁₆	—	53	64	no	—
350	SD₁₆	C₂³×D₅₃	1	32	yes	complemented
351	D₈	—	53	64	no	—
352	D₈	—	53	32	no	—
353	D₈	—	53	32	no	—
354	D₈	C₂³×D₅₃	1	16	yes	complemented
355	M₁₆	—	53	16	no	—
356	M₁₆	—	53	32	no	—
357	C₂×C₈	—	53	12	no	—
358	C₂×C₈	—	53	32	no	—
359	C₂×C₈	C₂³×D₅₃	1	6	yes	—
360	C₂×C₈	C₂³×D₅₃	1	6	yes	—
361	C₂×C₈	C₂³×D₅₃	1	16	yes	complemented
362	C₂³	—	2	4	no	—
363	C₂³	—	2	12	no	—
364	C₂³	—	4	8	no	—
365	C₂³	—	53	12	no	—
366	C₂³	—	106	64	no	—
367	C₂³	—	106	8	no	—
368	C₂³	—	106	32	no	—
369	C₂³	—	106	24	no	—
370	C₂³	—	424	48	no	—
371	C₂³	—	424	96	no	—
372	C₂³	—	212	16	no	—
373	C₂³	C₂×D₄×D₅₃	1	6	yes	—
374	C₂³	C₂×D₄×D₅₃	1	1	yes	characteristic
375	Q₈	—	2	12	no	—
376	Q₈	—	53	16	no	—
377	Q₈	C₂⁴×D₅₃	1	8	yes	—
378	D₄	—	2	32	no	—
379	D₄	—	2	12	no	—
380	D₄	—	53	32	no	—
381	D₄	—	53	16	no	—
382	D₄	—	53	16	no	—
383	D₄	—	106	64	no	—
384	D₄	—	106	64	no	—
385	D₄	—	106	32	no	—
386	D₄	—	106	24	no	—
387	D₄	C₂⁴×D₅₃	1	8	yes	—
388	C₂×C₄	—	2	12	no	—
389	C₂×C₄	—	2	32	no	—
390	C₂×C₄	—	53	12	no	—
391	C₂×C₄	—	53	24	no	—
392	C₂×C₄	—	53	8	no	—
393	C₂×C₄	—	106	64	no	—
394	C₂×C₄	C₂×D₄×D₅₃	1	6	yes	—
395	C₂×C₄	C₂×D₄×D₅₃	1	3	yes	—
396	C₂×C₄	C₂×D₄×D₅₃	1	12	yes	—
397	C₂×C₄	C₂⁴×D₅₃	1	3	yes	—
398	C₂×C₄	C₂⁴×D₅₃	1	4	yes	—
399	C₈	—	53	8	no	—
400	C₈	C₂⁴×D₅₃	1	4	yes	—
401	C₈	C₂⁴×D₅₃	1	4	yes	—
402	C₂²	—	2	12	no	—
403	C₂²	—	2	8	no	—
404	C₂²	—	4	24	no	—
405	C₂²	—	53	8	no	—
406	C₂²	—	106	24	no	—
407	C₂²	—	106	16	no	—
408	C₂²	—	424	32	no	—
409	C₂²	—	424	64	no	—
410	C₂²	—	212	48	no	—
411	C₂²	#16811425	1	4	yes	direct product factor
412	C₂²	C₂²×D₄×D₅₃	1	4	yes	—
413	C₂²	C₂²×D₄×D₅₃	1	3	yes	—
414	C₄	—	2	8	no	—
415	C₄	—	53	8	no	—
416	C₄	C₂²×D₄×D₅₃	1	4	yes	—
417	C₄	C₂²×D₄×D₅₃	1	3	yes	—
418	C₄	C₂⁵×D₅₃	1	1	yes	characteristic
419	C₂	—	2	4	no	—
420	C₂	—	4	8	no	—
421	C₂	—	106	8	no	—
422	C₂	—	212	16	no	—
423	C₂	C₂×#16811425	1	6	yes	direct product factor
424	C₂	C₂³×D₄×D₅₃	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	C₂×D₄×D₅₃
commutator, cosocle	C₄×C₅₃	C₂⁶
Frattini	C₄	C₂⁵×D₅₃
Fitting	C₂²×#135×C₅₃	C₂
socle	C₂³×C₅₃	C₂²×D₄
hypercenter	C₂²×#135	D₅₃
hypocenter	C₅₃	C₂²×#576
LCS	G ▷ C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂³ ◁ C₂³×C₄ ◁ C₂²×#135

Complements

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

N	H ≅ G/N	type
1. C₂²×#13133263	C₂	semidirect
2. C₂×#16811425	C₂	direct (N × H)
3. C₂²×#13133244	C₂	semidirect
4. C₂²×#13133242	C₂	semidirect
5. C₂²×#13133240	C₂	semidirect
6. C₂³×D₄₂₄	C₂	semidirect
7. C₂²×D₈×D₅₃	C₂	semidirect
8. C₂²×#13133223	C₂	semidirect
9. C₂²×#135×C₅₃	C₂	semidirect
11. C₂×#13133263	C₂²	semidirect
12. C₂²×#11719949	C₂²	semidirect
16. C₂³×D₂₁₂	C₂²	semidirect
18. C₂²×D₄×D₅₃	C₂²	semidirect
19. #16811425	C₂²	direct (N × H)
20. C₂×#13133244	C₂²	semidirect
21. C₂×#13133242	C₂²	semidirect
22. C₂×#13133240	C₂²	semidirect
23. C₂²×#11719924	C₂²	semidirect
24. C₂²×D₄₂₄	C₂²	semidirect
25. C₂²×D₄₂₄	C₂²	semidirect
26. C₂²×D₄₂₄	C₂²	semidirect
27. C₂²×#11719922	C₂²	semidirect
28. C₂×D₈×D₅₃	C₂²	semidirect
29. C₂×#13133223	C₂²	semidirect
30. C₂²×#11719917	C₂²	semidirect
31. C₂²×#11719912	C₂²	semidirect
32. C₂²×C₈×D₅₃	C₂²	semidirect
33. C₂²×#41×C₅₃	C₂²	semidirect
34. C₂×#135×C₅₃	C₂²	semidirect
35. C₂²×Q₁₆×C₅₃	C₂²	semidirect
36. C₂²×SD₁₆×C₅₃	C₂²	semidirect
37. C₂²×D₈×C₅₃	C₂²	semidirect
38. C₂³×C₈×C₅₃	C₂²	semidirect
42. #13133263	C₂³	semidirect
44. C₂×#11719949	C₂³	semidirect
55. C₂²×D₂₁₂	C₂³	semidirect
60. C₂×D₄×D₅₃	C₂³	semidirect
63. #13133244	C₂³	semidirect
64. #13133242	C₂³	semidirect
65. #13133240	C₂³	semidirect
66. C₂×#11719924	C₂³	semidirect
67. C₂×D₄₂₄	C₂³	semidirect
68. C₂×D₄₂₄	C₂³	semidirect
69. C₂×D₄₂₄	C₂³	semidirect
70. C₂×#11719922	C₂³	semidirect
71. D₈×D₅₃	C₂³	semidirect
72. #13133223	C₂³	semidirect
73. C₂×#11719917	C₂³	semidirect
74. C₂×#11719912	C₂³	semidirect
76. C₂×C₈×D₅₃	C₂³	semidirect
78. C₂×#41×C₅₃	C₂³	semidirect
84. #135×C₅₃	C₂³	semidirect
85. C₂×Q₁₆×C₅₃	C₂³	semidirect
86. C₂×SD₁₆×C₅₃	C₂³	semidirect
87. C₂×D₈×C₅₃	C₂³	semidirect
90. C₂²×C₈×C₅₃	C₂³	semidirect
100. #41×C₅₃	C₂⁴	semidirect
116. #11719949	C₂⁴	semidirect
119. D₄×D₅₃	C₂⁴	semidirect
130. C₂×D₂₁₂	C₂⁴	semidirect
135. Q₁₆×C₅₃	C₂⁴	semidirect
136. SD₁₆×C₅₃	C₂⁴	semidirect
137. D₈×C₅₃	C₂⁴	semidirect
140. C₂×C₈×C₅₃	C₂⁴	semidirect
141. #11719924	C₂⁴	semidirect
142. #11719922	C₂⁴	semidirect
143. #11719917	C₂⁴	semidirect
145. D₄₂₄	C₂⁴	semidirect
146. D₄₂₄	C₂⁴	semidirect
147. #11719912	C₂⁴	semidirect
148. C₈×D₅₃	C₂⁴	semidirect
193. C₂²×C₅₃	#576	semidirect
212. C₂²×#135	D₅₃	semidirect
218. C₂×C₅₃	C₂×#576	semidirect
227. C₂²×#41	C₂×D₅₃	semidirect
236. C₂×#135	C₂×D₅₃	semidirect
237. C₂²×Q₁₆	C₂×D₅₃	semidirect
239. C₂²×SD₁₆	C₂×D₅₃	semidirect
245. C₂²×D₈	C₂×D₅₃	semidirect
250. C₂³×C₈	C₂×D₅₃	semidirect
251. C₅₃	C₂²×#576	semidirect
262. C₂×#41	C₂²×D₅₃	semidirect
286. #135	C₂²×D₅₃	semidirect
287. C₂×Q₁₆	C₂²×D₅₃	semidirect
289. C₂×SD₁₆	C₂²×D₅₃	semidirect
295. C₂×D₈	C₂²×D₅₃	semidirect
303. C₂²×C₈	C₂²×D₅₃	semidirect
318. #41	C₂³×D₅₃	semidirect
348. Q₁₆	C₂³×D₅₃	semidirect
350. SD₁₆	C₂³×D₅₃	semidirect
354. D₈	C₂³×D₅₃	semidirect
361. C₂×C₈	C₂³×D₅₃	semidirect
411. C₂²	#16811425	direct (N × H)
423. C₂	C₂×#16811425	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×#13133263	2
C₂×#16811425	2
C₂²×#13133244	2
C₂²×#13133242	2
C₂²×#13133240	2
C₂³×D₄₂₄	2
C₂²×D₈×D₅₃	2
C₂²×#13133223	2
C₂²×#135×C₅₃	2
C₂²×#576	53

Subgroups — C22×#16811425 #28297901

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂²×#16811425 #28297901