Subgroups — C₂²×C₅₃×#554 #27270214

Order 13568.

8454 conjugacy classes of subgroups.

The subgroup lattice has 324 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₅₃×#554	C₂	1	48	yes	maximal, direct product factor
2	C₂²×C₄⋊Q₈×C₅₃	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂²×#125×C₅₃	C₂	1	2	yes	maximal, complemented
4	C₂²×C₂²⋊Q₈×C₅₃	C₂	1	4	yes	maximal, complemented
5	C₂²×C₂²⋊Q₈×C₅₃	C₂	1	2	yes	maximal, complemented
6	C₂²×C₄×Q₈×C₅₃	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂²×C₄×D₄×C₅₃	C₂	1	2	yes	maximal
8	C₂²×C₄×D₄×C₅₃	C₂	1	1	yes	maximal, characteristic
9	C₂³×C₄⋊C₄×C₅₃	C₂	1	2	yes	maximal, complemented
10	C₂³×Q₈×C₅₃	C₂²	1	1	yes	characteristic
11	C₂³×Q₈×C₅₃	C₂²	1	2	yes	—
12	C₂³×D₄×C₅₃	C₂²	1	1	yes	characteristic
13	C₂⁴×C₄×C₅₃	C₂²	1	2	yes	—
14	C₂⁴×C₄×C₅₃	C₂²	1	4	yes	—
15	C₅₃×#554	C₂²	1	256	yes	direct product factor
16	C₂×C₄⋊Q₈×C₅₃	C₂²	1	24	yes	complemented
17	C₂×#125×C₅₃	C₂²	1	48	yes	complemented
18	C₂×C₂²⋊Q₈×C₅₃	C₂²	1	96	yes	complemented
19	C₂×C₂²⋊Q₈×C₅₃	C₂²	1	48	yes	complemented
20	C₂×C₄×Q₈×C₅₃	C₂²	1	24	yes	complemented
21	C₂×C₄×D₄×C₅₃	C₂²	1	48	yes	—
22	C₂×C₄×D₄×C₅₃	C₂²	1	24	yes	—
23	C₂²×C₄⋊C₄×C₅₃	C₂²	1	1	yes	characteristic
24	C₂²×C₄⋊C₄×C₅₃	C₂²	1	4	yes	—
25	C₂²×C₄⋊C₄×C₅₃	C₂²	1	2	yes	—
26	C₂²×C₄⋊C₄×C₅₃	C₂²	1	1	yes	characteristic
27	C₂²×C₄⋊C₄×C₅₃	C₂²	1	2	yes	—
28	C₂²×C₄⋊C₄×C₅₃	C₂²	1	2	yes	—
29	C₂²×C₄⋊C₄×C₅₃	C₂²	1	4	yes	—
30	C₂²×C₄⋊C₄×C₅₃	C₂²	1	48	yes	complemented
31	C₂²×C₂²⋊C₄×C₅₃	C₂²	1	4	yes	—
32	C₂²×C₂²⋊C₄×C₅₃	C₂²	1	2	yes	—
33	C₂²×C₄²×C₅₃	C₂²	1	2	yes	—
34	C₂²×C₄²×C₅₃	C₂²	1	1	yes	characteristic
35	C₂⁵×C₅₃	C₂³	1	2	yes	—
36	C₂²×Q₈×C₅₃	—	2	8	no	—
37	C₂²×Q₈×C₅₃	—	4	8	no	—
38	C₂²×Q₈×C₅₃	C₂³	1	12	yes	—
39	C₂²×Q₈×C₅₃	C₂³	1	24	yes	—
40	C₂²×D₄×C₅₃	Q₈	1	16	yes	complemented
41	C₂²×D₄×C₅₃	C₂³	1	12	yes	—
42	C₂³×C₄×C₅₃	—	2	16	no	—
43	C₂³×C₄×C₅₃	—	4	16	no	—
44	C₂³×C₄×C₅₃	C₂³	1	24	yes	—
45	C₂³×C₄×C₅₃	C₂³	1	2	yes	—
46	C₂³×C₄×C₅₃	C₂³	1	1	yes	characteristic
47	C₂³×C₄×C₅₃	C₂³	1	2	yes	—
48	C₂³×C₄×C₅₃	C₂³	1	2	yes	—
49	C₂³×C₄×C₅₃	C₂³	1	4	yes	—
50	C₂³×C₄×C₅₃	C₂³	1	48	yes	—
51	C₂³×C₄×C₅₃	C₂³	1	1	yes	characteristic
52	C₂³×C₄×C₅₃	C₂³	1	1	yes	characteristic
53	C₄⋊Q₈×C₅₃	C₂³	1	64	yes	complemented
54	#125×C₅₃	C₂³	1	128	yes	complemented
55	C₂²⋊Q₈×C₅₃	C₂³	1	128	yes	complemented
56	C₂²⋊Q₈×C₅₃	C₂³	1	256	yes	complemented
57	C₄×Q₈×C₅₃	C₂³	1	64	yes	complemented
58	C₄×D₄×C₅₃	C₂³	1	64	yes	—
59	C₄×D₄×C₅₃	C₂³	1	128	yes	—
60	C₂×C₄⋊C₄×C₅₃	C₂³	1	12	yes	—
61	C₂×C₄⋊C₄×C₅₃	C₂³	1	48	yes	—
62	C₂×C₄⋊C₄×C₅₃	C₂³	1	24	yes	—
63	C₂×C₄⋊C₄×C₅₃	C₂³	1	12	yes	—
64	C₂×C₄⋊C₄×C₅₃	C₂³	1	24	yes	—
65	C₂×C₄⋊C₄×C₅₃	C₂³	1	24	yes	—
66	C₂×C₄⋊C₄×C₅₃	C₂³	1	48	yes	—
67	C₂×C₄⋊C₄×C₅₃	C₂³	1	128	yes	complemented
68	C₂×C₂²⋊C₄×C₅₃	C₂³	1	48	yes	—
69	C₂×C₂²⋊C₄×C₅₃	C₂³	1	24	yes	—
70	C₂×C₄²×C₅₃	C₂³	1	24	yes	—
71	C₂×C₄²×C₅₃	C₂³	1	12	yes	—
72	C₂⁴×C₅₃	—	2	8	no	—
73	C₂⁴×C₅₃	—	2	8	no	—
74	C₂⁴×C₅₃	C₂×Q₈	1	16	yes	—
75	C₂⁴×C₅₃	C₂⁴	1	12	yes	—
76	C₂⁴×C₅₃	C₂⁴	1	1	yes	characteristic
77	C₂×Q₈×C₅₃	—	2	48	no	—
78	C₂×Q₈×C₅₃	—	4	48	no	—
79	C₂×Q₈×C₅₃	C₂⁴	1	16	yes	—
80	C₂×Q₈×C₅₃	C₂⁴	1	32	yes	—
81	C₂×D₄×C₅₃	C₂×Q₈	1	96	yes	complemented
82	C₂×D₄×C₅₃	C₂⁴	1	16	yes	—
83	C₂²×C₄×C₅₃	—	2	8	no	—
84	C₂²×C₄×C₅₃	—	2	8	no	—
85	C₂²×C₄×C₅₃	—	2	8	no	—
86	C₂²×C₄×C₅₃	—	2	16	no	—
87	C₂²×C₄×C₅₃	—	2	4	no	—
88	C₂²×C₄×C₅₃	—	2	96	no	—
89	C₂²×C₄×C₅₃	—	4	96	no	—
90	C₂²×C₄×C₅₃	C₂×Q₈	1	8	yes	—
91	C₂²×C₄×C₅₃	#41	1	8	yes	—
92	C₂²×C₄×C₅₃	C₂⁴	1	12	yes	—
93	C₂²×C₄×C₅₃	C₂⁴	1	6	yes	—
94	C₂²×C₄×C₅₃	C₂⁴	1	12	yes	—
95	C₂²×C₄×C₅₃	C₂⁴	1	12	yes	—
96	C₂²×C₄×C₅₃	C₂⁴	1	24	yes	—
97	C₂²×C₄×C₅₃	C₂⁴	1	32	yes	—
98	C₂²×C₄×C₅₃	C₂⁴	1	64	yes	—
99	C₂²×C₄×C₅₃	C₂⁴	1	6	yes	—
100	C₂²×C₄×C₅₃	C₂⁴	1	6	yes	—
101	C₄⋊C₄×C₅₃	C₂⁴	1	16	yes	—
102	C₄⋊C₄×C₅₃	C₂⁴	1	64	yes	—
103	C₄⋊C₄×C₅₃	C₂⁴	1	32	yes	—
104	C₄⋊C₄×C₅₃	C₂⁴	1	16	yes	—
105	C₄⋊C₄×C₅₃	C₂⁴	1	32	yes	—
106	C₄⋊C₄×C₅₃	C₂⁴	1	32	yes	—
107	C₄⋊C₄×C₅₃	C₂⁴	1	64	yes	—
108	C₂²⋊C₄×C₅₃	C₂⁴	1	64	yes	—
109	C₂²⋊C₄×C₅₃	C₂⁴	1	32	yes	—
110	C₄²×C₅₃	C₂⁴	1	32	yes	—
111	C₄²×C₅₃	C₂⁴	1	16	yes	—
112	C₂³×C₅₃	—	2	64	no	—
113	C₂³×C₅₃	—	2	24	no	—
114	C₂³×C₅₃	—	2	24	no	—
115	C₂³×C₅₃	C₂²×Q₈	1	48	yes	—
116	C₂³×C₅₃	C₂²×Q₈	1	4	yes	—
117	C₂³×C₅₃	C₂×#41	1	4	yes	—
118	C₂³×C₅₃	H(5,𝔽₂)	1	4	yes	—
119	C₂³×C₅₃	C₂⁵	1	8	yes	—
120	C₂³×C₅₃	C₂⁵	1	3	yes	—
121	Q₈×C₅₃	—	2	32	no	—
122	Q₈×C₅₃	—	4	32	no	—
123	D₄×C₅₃	C₂²×Q₈	1	64	yes	complemented
124	C₂×C₄×C₅₃	—	2	24	no	—
125	C₂×C₄×C₅₃	—	2	24	no	—
126	C₂×C₄×C₅₃	—	2	24	no	—
127	C₂×C₄×C₅₃	—	2	48	no	—
128	C₂×C₄×C₅₃	—	2	12	no	—
129	C₂×C₄×C₅₃	—	2	64	no	—
130	C₂×C₄×C₅₃	—	4	64	no	—
131	C₂×C₄×C₅₃	C₂²×Q₈	1	24	yes	—
132	C₂×C₄×C₅₃	C₂×#41	1	24	yes	—
133	C₂×C₄×C₅₃	C₂⁵	1	8	yes	—
134	C₂×C₄×C₅₃	C₂⁵	1	16	yes	—
135	C₂×C₄×C₅₃	C₂⁵	1	8	yes	—
136	C₂×C₄×C₅₃	C₂⁵	1	4	yes	—
137	C₂×C₄×C₅₃	C₂⁵	1	8	yes	—
138	C₂×C₄×C₅₃	C₂⁵	1	4	yes	—
139	C₂×C₄×C₅₃	C₂⁵	1	4	yes	—
140	C₂²×#554	C₅₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
141	C₂²×C₅₃	—	2	8	no	—
142	C₂²×C₅₃	—	2	8	no	—
143	C₂²×C₅₃	—	2	96	no	—
144	C₂²×C₅₃	#554	1	16	yes	direct product factor
145	C₂²×C₅₃	C₂³×Q₈	1	16	yes	—
146	C₂²×C₅₃	C₂³×Q₈	1	6	yes	—
147	C₂²×C₅₃	C₂²×#41	1	6	yes	—
148	C₂²×C₅₃	C₂×H(5,𝔽₂)	1	6	yes	—
149	C₂²×C₅₃	C₂⁶	1	1	yes	characteristic
150	C₄×C₅₃	—	2	8	no	—
151	C₄×C₅₃	—	2	8	no	—
152	C₄×C₅₃	—	2	8	no	—
153	C₄×C₅₃	—	2	16	no	—
154	C₄×C₅₃	—	2	4	no	—
155	C₄×C₅₃	C₂³×Q₈	1	8	yes	—
156	C₄×C₅₃	C₂²×#41	1	8	yes	—
157	C₂×#554	C₂×C₅₃	1	48	yes	direct product factor
158	C₂²×C₄⋊Q₈	C₂×C₅₃	1	1	yes	characteristic, complemented
159	C₂²×#125	C₂×C₅₃	1	2	yes	complemented
160	C₂²×C₂²⋊Q₈	C₂×C₅₃	1	4	yes	complemented
161	C₂²×C₂²⋊Q₈	C₂×C₅₃	1	2	yes	complemented
162	C₂²×C₄×Q₈	C₂×C₅₃	1	1	yes	characteristic, complemented
163	C₂²×C₄×D₄	C₂×C₅₃	1	1	yes	characteristic
164	C₂²×C₄×D₄	C₂×C₅₃	1	2	yes	—
165	C₂³×C₄⋊C₄	C₂×C₅₃	1	2	yes	complemented
166	C₂×C₅₃	—	2	16	no	—
167	C₂×C₅₃	C₂×#554	1	12	yes	direct product factor
168	C₂×C₅₃	C₂⁴×Q₈	1	1	yes	characteristic
169	C₂×C₅₃	C₂³×#41	1	1	yes	characteristic
170	C₂×C₅₃	C₂²×H(5,𝔽₂)	1	1	yes	characteristic
171	C₂³×Q₈	C₂²×C₅₃	1	1	yes	characteristic
172	C₂³×Q₈	C₂²×C₅₃	1	2	yes	—
173	C₂³×D₄	C₂²×C₅₃	1	1	yes	characteristic
174	C₂⁴×C₄	C₂²×C₅₃	1	2	yes	—
175	C₂⁴×C₄	C₂²×C₅₃	1	4	yes	—
176	#554	C₂²×C₅₃	1	256	yes	direct product factor
177	C₂×C₄⋊Q₈	C₂²×C₅₃	1	24	yes	complemented
178	C₂×#125	C₂²×C₅₃	1	48	yes	complemented
179	C₂×C₂²⋊Q₈	C₂²×C₅₃	1	96	yes	complemented
180	C₂×C₂²⋊Q₈	C₂²×C₅₃	1	48	yes	complemented
181	C₂×C₄×Q₈	C₂²×C₅₃	1	24	yes	complemented
182	C₂×C₄×D₄	C₂²×C₅₃	1	48	yes	—
183	C₂×C₄×D₄	C₂²×C₅₃	1	24	yes	—
184	C₂²×C₄⋊C₄	C₂²×C₅₃	1	1	yes	characteristic
185	C₂²×C₄⋊C₄	C₂²×C₅₃	1	4	yes	—
186	C₂²×C₄⋊C₄	C₂²×C₅₃	1	2	yes	—
187	C₂²×C₄⋊C₄	C₂²×C₅₃	1	1	yes	characteristic
188	C₂²×C₄⋊C₄	C₂²×C₅₃	1	2	yes	—
189	C₂²×C₄⋊C₄	C₂²×C₅₃	1	2	yes	—
190	C₂²×C₄⋊C₄	C₂²×C₅₃	1	4	yes	—
191	C₂²×C₄⋊C₄	C₂²×C₅₃	1	48	yes	complemented
192	C₂²×C₂²⋊C₄	C₂²×C₅₃	1	4	yes	—
193	C₂²×C₂²⋊C₄	C₂²×C₅₃	1	2	yes	—
194	C₂²×C₄²	C₂²×C₅₃	1	2	yes	—
195	C₂²×C₄²	C₂²×C₅₃	1	1	yes	characteristic
196	C₅₃	C₂²×#554	1	1	yes	characteristic, direct product factor, Sylow
197	C₂⁵	C₂³×C₅₃	1	2	yes	—
198	C₂²×Q₈	—	2	8	no	—
199	C₂²×Q₈	—	4	8	no	—
200	C₂²×Q₈	C₂³×C₅₃	1	12	yes	—
201	C₂²×Q₈	C₂³×C₅₃	1	24	yes	—
202	C₂²×D₄	Q₈×C₅₃	1	16	yes	complemented
203	C₂²×D₄	C₂³×C₅₃	1	12	yes	—
204	C₂³×C₄	—	2	16	no	—
205	C₂³×C₄	—	4	16	no	—
206	C₂³×C₄	C₂³×C₅₃	1	2	yes	—
207	C₂³×C₄	C₂³×C₅₃	1	1	yes	characteristic
208	C₂³×C₄	C₂³×C₅₃	1	2	yes	—
209	C₂³×C₄	C₂³×C₅₃	1	4	yes	—
210	C₂³×C₄	C₂³×C₅₃	1	2	yes	—
211	C₂³×C₄	C₂³×C₅₃	1	24	yes	—
212	C₂³×C₄	C₂³×C₅₃	1	48	yes	—
213	C₂³×C₄	C₂³×C₅₃	1	1	yes	characteristic
214	C₂³×C₄	C₂³×C₅₃	1	1	yes	characteristic
215	C₄⋊Q₈	C₂³×C₅₃	1	64	yes	complemented
216	#125	C₂³×C₅₃	1	128	yes	complemented
217	C₂²⋊Q₈	C₂³×C₅₃	1	256	yes	complemented
218	C₂²⋊Q₈	C₂³×C₅₃	1	128	yes	complemented
219	C₄×Q₈	C₂³×C₅₃	1	64	yes	complemented
220	C₄×D₄	C₂³×C₅₃	1	128	yes	—
221	C₄×D₄	C₂³×C₅₃	1	64	yes	—
222	C₂×C₄⋊C₄	C₂³×C₅₃	1	12	yes	—
223	C₂×C₄⋊C₄	C₂³×C₅₃	1	48	yes	—
224	C₂×C₄⋊C₄	C₂³×C₅₃	1	24	yes	—
225	C₂×C₄⋊C₄	C₂³×C₅₃	1	12	yes	—
226	C₂×C₄⋊C₄	C₂³×C₅₃	1	24	yes	—
227	C₂×C₄⋊C₄	C₂³×C₅₃	1	24	yes	—
228	C₂×C₄⋊C₄	C₂³×C₅₃	1	48	yes	—
229	C₂×C₄⋊C₄	C₂³×C₅₃	1	128	yes	complemented
230	C₂×C₂²⋊C₄	C₂³×C₅₃	1	48	yes	—
231	C₂×C₂²⋊C₄	C₂³×C₅₃	1	24	yes	—
232	C₂×C₄²	C₂³×C₅₃	1	24	yes	—
233	C₂×C₄²	C₂³×C₅₃	1	12	yes	—
234	C₂⁴	—	2	8	no	—
235	C₂⁴	—	2	8	no	—
236	C₂⁴	C₂×Q₈×C₅₃	1	16	yes	—
237	C₂⁴	C₂⁴×C₅₃	1	12	yes	—
238	C₂⁴	C₂⁴×C₅₃	1	1	yes	characteristic
239	C₂×Q₈	—	2	48	no	—
240	C₂×Q₈	—	4	48	no	—
241	C₂×Q₈	C₂⁴×C₅₃	1	16	yes	—
242	C₂×Q₈	C₂⁴×C₅₃	1	32	yes	—
243	C₂×D₄	C₂×Q₈×C₅₃	1	96	yes	complemented
244	C₂×D₄	C₂⁴×C₅₃	1	16	yes	—
245	C₂²×C₄	—	2	8	no	—
246	C₂²×C₄	—	2	8	no	—
247	C₂²×C₄	—	2	8	no	—
248	C₂²×C₄	—	2	16	no	—
249	C₂²×C₄	—	2	4	no	—
250	C₂²×C₄	—	2	96	no	—
251	C₂²×C₄	—	4	96	no	—
252	C₂²×C₄	C₂×Q₈×C₅₃	1	8	yes	—
253	C₂²×C₄	#41×C₅₃	1	8	yes	—
254	C₂²×C₄	C₂⁴×C₅₃	1	12	yes	—
255	C₂²×C₄	C₂⁴×C₅₃	1	6	yes	—
256	C₂²×C₄	C₂⁴×C₅₃	1	12	yes	—
257	C₂²×C₄	C₂⁴×C₅₃	1	12	yes	—
258	C₂²×C₄	C₂⁴×C₅₃	1	24	yes	—
259	C₂²×C₄	C₂⁴×C₅₃	1	32	yes	—
260	C₂²×C₄	C₂⁴×C₅₃	1	64	yes	—
261	C₂²×C₄	C₂⁴×C₅₃	1	6	yes	—
262	C₂²×C₄	C₂⁴×C₅₃	1	6	yes	—
263	C₄⋊C₄	C₂⁴×C₅₃	1	16	yes	—
264	C₄⋊C₄	C₂⁴×C₅₃	1	64	yes	—
265	C₄⋊C₄	C₂⁴×C₅₃	1	32	yes	—
266	C₄⋊C₄	C₂⁴×C₅₃	1	16	yes	—
267	C₄⋊C₄	C₂⁴×C₅₃	1	32	yes	—
268	C₄⋊C₄	C₂⁴×C₅₃	1	32	yes	—
269	C₄⋊C₄	C₂⁴×C₅₃	1	64	yes	—
270	C₂²⋊C₄	C₂⁴×C₅₃	1	64	yes	—
271	C₂²⋊C₄	C₂⁴×C₅₃	1	32	yes	—
272	C₄²	C₂⁴×C₅₃	1	32	yes	—
273	C₄²	C₂⁴×C₅₃	1	16	yes	—
274	C₂³	—	2	64	no	—
275	C₂³	—	2	24	no	—
276	C₂³	—	2	24	no	—
277	C₂³	C₂²×Q₈×C₅₃	1	48	yes	—
278	C₂³	C₂²×Q₈×C₅₃	1	4	yes	—
279	C₂³	C₂×#41×C₅₃	1	4	yes	—
280	C₂³	H(5,𝔽₂)×C₅₃	1	4	yes	—
281	C₂³	C₂⁵×C₅₃	1	8	yes	—
282	C₂³	C₂⁵×C₅₃	1	3	yes	—
283	Q₈	—	2	32	no	—
284	Q₈	—	4	32	no	—
285	D₄	C₂²×Q₈×C₅₃	1	64	yes	complemented
286	C₂×C₄	—	2	24	no	—
287	C₂×C₄	—	2	24	no	—
288	C₂×C₄	—	2	24	no	—
289	C₂×C₄	—	2	48	no	—
290	C₂×C₄	—	2	12	no	—
291	C₂×C₄	—	2	64	no	—
292	C₂×C₄	—	4	64	no	—
293	C₂×C₄	C₂²×Q₈×C₅₃	1	24	yes	—
294	C₂×C₄	C₂×#41×C₅₃	1	24	yes	—
295	C₂×C₄	C₂⁵×C₅₃	1	8	yes	—
296	C₂×C₄	C₂⁵×C₅₃	1	4	yes	—
297	C₂×C₄	C₂⁵×C₅₃	1	8	yes	—
298	C₂×C₄	C₂⁵×C₅₃	1	8	yes	—
299	C₂×C₄	C₂⁵×C₅₃	1	16	yes	—
300	C₂×C₄	C₂⁵×C₅₃	1	4	yes	—
301	C₂×C₄	C₂⁵×C₅₃	1	4	yes	—
302	C₂²	—	2	96	no	—
303	C₂²	—	2	8	no	—
304	C₂²	—	2	8	no	—
305	C₂²	C₅₃×#554	1	16	yes	direct product factor
306	C₂²	C₂³×Q₈×C₅₃	1	16	yes	—
307	C₂²	C₂³×Q₈×C₅₃	1	6	yes	—
308	C₂²	C₂²×#41×C₅₃	1	6	yes	—
309	C₂²	C₂×H(5,𝔽₂)×C₅₃	1	6	yes	—
310	C₂²	C₂⁶×C₅₃	1	1	yes	characteristic
311	C₄	—	2	8	no	—
312	C₄	—	2	8	no	—
313	C₄	—	2	8	no	—
314	C₄	—	2	16	no	—
315	C₄	—	2	4	no	—
316	C₄	C₂³×Q₈×C₅₃	1	8	yes	—
317	C₄	C₂²×#41×C₅₃	1	8	yes	—
318	C₂	—	2	16	no	—
319	C₂	C₂×C₅₃×#554	1	12	yes	direct product factor
320	C₂	C₂⁴×Q₈×C₅₃	1	1	yes	characteristic
321	C₂	C₂³×#41×C₅₃	1	1	yes	characteristic
322	C₂	C₂²×H(5,𝔽₂)×C₅₃	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂⁴×C₅₃	C₂⁴
commutator, Frattini, cosocle	C₂²	C₂⁶×C₅₃
prime radical	C₂³×D₄×C₅₃	C₂²
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂⁴×C₅₃ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₂×C₅₃×#554	C₂	direct (N × H)
2. C₂²×C₄⋊Q₈×C₅₃	C₂	semidirect
3. C₂²×#125×C₅₃	C₂	semidirect
4. C₂²×C₂²⋊Q₈×C₅₃	C₂	semidirect
5. C₂²×C₂²⋊Q₈×C₅₃	C₂	semidirect
6. C₂²×C₄×Q₈×C₅₃	C₂	semidirect
9. C₂³×C₄⋊C₄×C₅₃	C₂	semidirect
15. C₅₃×#554	C₂²	direct (N × H)
16. C₂×C₄⋊Q₈×C₅₃	C₂²	semidirect
17. C₂×#125×C₅₃	C₂²	semidirect
18. C₂×C₂²⋊Q₈×C₅₃	C₂²	semidirect
19. C₂×C₂²⋊Q₈×C₅₃	C₂²	semidirect
20. C₂×C₄×Q₈×C₅₃	C₂²	semidirect
30. C₂²×C₄⋊C₄×C₅₃	C₂²	semidirect
40. C₂²×D₄×C₅₃	Q₈	semidirect
53. C₄⋊Q₈×C₅₃	C₂³	semidirect
54. #125×C₅₃	C₂³	semidirect
55. C₂²⋊Q₈×C₅₃	C₂³	semidirect
56. C₂²⋊Q₈×C₅₃	C₂³	semidirect
57. C₄×Q₈×C₅₃	C₂³	semidirect
67. C₂×C₄⋊C₄×C₅₃	C₂³	semidirect
81. C₂×D₄×C₅₃	C₂×Q₈	semidirect
123. D₄×C₅₃	C₂²×Q₈	semidirect
140. C₂²×#554	C₅₃	direct (N × H)
144. C₂²×C₅₃	#554	direct (N × H)
157. C₂×#554	C₂×C₅₃	direct (N × H)
158. C₂²×C₄⋊Q₈	C₂×C₅₃	semidirect
159. C₂²×#125	C₂×C₅₃	semidirect
160. C₂²×C₂²⋊Q₈	C₂×C₅₃	semidirect
161. C₂²×C₂²⋊Q₈	C₂×C₅₃	semidirect
162. C₂²×C₄×Q₈	C₂×C₅₃	semidirect
165. C₂³×C₄⋊C₄	C₂×C₅₃	semidirect
167. C₂×C₅₃	C₂×#554	direct (N × H)
176. #554	C₂²×C₅₃	direct (N × H)
177. C₂×C₄⋊Q₈	C₂²×C₅₃	semidirect
178. C₂×#125	C₂²×C₅₃	semidirect
179. C₂×C₂²⋊Q₈	C₂²×C₅₃	semidirect
180. C₂×C₂²⋊Q₈	C₂²×C₅₃	semidirect
181. C₂×C₄×Q₈	C₂²×C₅₃	semidirect
191. C₂²×C₄⋊C₄	C₂²×C₅₃	semidirect
196. C₅₃	C₂²×#554	direct (N × H)
202. C₂²×D₄	Q₈×C₅₃	semidirect
215. C₄⋊Q₈	C₂³×C₅₃	semidirect
216. #125	C₂³×C₅₃	semidirect
217. C₂²⋊Q₈	C₂³×C₅₃	semidirect
218. C₂²⋊Q₈	C₂³×C₅₃	semidirect
219. C₄×Q₈	C₂³×C₅₃	semidirect
229. C₂×C₄⋊C₄	C₂³×C₅₃	semidirect
243. C₂×D₄	C₂×Q₈×C₅₃	semidirect
285. D₄	C₂²×Q₈×C₅₃	semidirect
305. C₂²	C₅₃×#554	direct (N × H)
319. C₂	C₂×C₅₃×#554	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂×C₅₃×#554	2
C₂²×C₄⋊Q₈×C₅₃	2
C₂²×#125×C₅₃	2
C₂²×C₂²⋊Q₈×C₅₃	2
C₂²×C₄×Q₈×C₅₃	2
C₂²×C₄×D₄×C₅₃	2
C₂³×C₄⋊C₄×C₅₃	2
C₂²×#554	53

Subgroups — C22×C53×#554 #27270214

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂²×C₅₃×#554 #27270214