Subgroups — C₂²×C₅₃×#556 #27270218

Order 13568.

7846 conjugacy classes of subgroups.

The subgroup lattice has 318 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₅₃×#556	C₂	1	48	yes	maximal, direct product factor
2	C₂²×C₄⋊Q₈×C₅₃	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂²×#126×C₅₃	C₂	1	4	yes	maximal, complemented
4	C₂²×#124×C₅₃	C₂	1	2	yes	maximal
5	C₂²×C₂²⋊Q₈×C₅₃	C₂	1	2	yes	maximal, complemented
6	C₂²×C₄×Q₈×C₅₃	C₂	1	2	yes	maximal, complemented
7	C₂²×C₄×Q₈×C₅₃	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂²×C₄×D₄×C₅₃	C₂	1	1	yes	maximal, characteristic
9	C₂²×#117×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	C₅₃×#556	C₂²	1	256	yes	direct product factor
15	C₂×C₄⋊Q₈×C₅₃	C₂²	1	24	yes	complemented
16	C₂×#126×C₅₃	C₂²	1	96	yes	complemented
17	C₂×#124×C₅₃	C₂²	1	48	yes	—
18	C₂×C₂²⋊Q₈×C₅₃	C₂²	1	48	yes	complemented
19	C₂×C₄×Q₈×C₅₃	C₂²	1	48	yes	complemented
20	C₂×C₄×Q₈×C₅₃	C₂²	1	24	yes	complemented
21	C₂×C₄×D₄×C₅₃	C₂²	1	24	yes	—
22	C₂×#117×C₅₃	C₂²	1	48	yes	complemented
23	C₂²×C₄⋊C₄×C₅₃	C₂²	1	1	yes	characteristic
24	C₂²×C₄⋊C₄×C₅₃	C₂²	1	2	yes	—
25	C₂²×C₄⋊C₄×C₅₃	C₂²	1	1	yes	characteristic
26	C₂²×C₄⋊C₄×C₅₃	C₂²	1	4	yes	—
27	C₂²×C₄⋊C₄×C₅₃	C₂²	1	2	yes	—
28	C₂²×C₄⋊C₄×C₅₃	C₂²	1	2	yes	—
29	C₂²×C₂²⋊C₄×C₅₃	C₂²	1	4	yes	—
30	C₂²×C₂²⋊C₄×C₅₃	C₂²	1	2	yes	—
31	C₂²×C₂²⋊C₄×C₅₃	C₂²	1	4	yes	—
32	C₂²×C₄²×C₅₃	C₂²	1	4	yes	—
33	C₂²×C₄²×C₅₃	C₂²	1	1	yes	characteristic
34	C₂²×C₄²×C₅₃	C₂²	1	2	yes	—
35	C₂⁵×C₅₃	C₂³	1	2	yes	—
36	C₂²×Q₈×C₅₃	—	2	8	no	—
37	C₂²×Q₈×C₅₃	—	2	16	no	—
38	C₂²×Q₈×C₅₃	C₂³	1	12	yes	—
39	C₂²×Q₈×C₅₃	C₂³	1	24	yes	—
40	C₂²×D₄×C₅₃	—	2	8	no	—
41	C₂²×D₄×C₅₃	C₂³	1	12	yes	—
42	C₂³×C₄×C₅₃	—	2	16	no	—
43	C₂³×C₄×C₅₃	C₂³	1	24	yes	—
44	C₂³×C₄×C₅₃	C₂³	1	2	yes	—
45	C₂³×C₄×C₅₃	C₂³	1	2	yes	—
46	C₂³×C₄×C₅₃	C₂³	1	1	yes	characteristic
47	C₂³×C₄×C₅₃	C₂³	1	4	yes	—
48	C₂³×C₄×C₅₃	C₂³	1	2	yes	—
49	C₂³×C₄×C₅₃	C₂³	1	1	yes	characteristic
50	C₂³×C₄×C₅₃	C₂³	1	1	yes	characteristic
51	C₄⋊Q₈×C₅₃	C₂³	1	64	yes	complemented
52	#126×C₅₃	C₂³	1	256	yes	complemented
53	#124×C₅₃	C₂³	1	128	yes	—
54	C₂²⋊Q₈×C₅₃	C₂³	1	128	yes	complemented
55	C₄×Q₈×C₅₃	C₂³	1	64	yes	complemented
56	C₄×Q₈×C₅₃	C₂³	1	128	yes	complemented
57	C₄×D₄×C₅₃	C₂³	1	64	yes	—
58	#117×C₅₃	C₂³	1	128	yes	complemented
59	C₂×C₄⋊C₄×C₅₃	C₂³	1	12	yes	—
60	C₂×C₄⋊C₄×C₅₃	C₂³	1	24	yes	—
61	C₂×C₄⋊C₄×C₅₃	C₂³	1	12	yes	—
62	C₂×C₄⋊C₄×C₅₃	C₂³	1	48	yes	—
63	C₂×C₄⋊C₄×C₅₃	C₂³	1	24	yes	—
64	C₂×C₄⋊C₄×C₅₃	C₂³	1	24	yes	—
65	C₂×C₂²⋊C₄×C₅₃	C₂³	1	48	yes	—
66	C₂×C₂²⋊C₄×C₅₃	C₂³	1	24	yes	—
67	C₂×C₂²⋊C₄×C₅₃	C₂³	1	48	yes	—
68	C₂×C₄²×C₅₃	C₂³	1	48	yes	—
69	C₂×C₄²×C₅₃	C₂³	1	12	yes	—
70	C₂×C₄²×C₅₃	C₂³	1	24	yes	—
71	C₂⁴×C₅₃	—	2	8	no	—
72	C₂⁴×C₅₃	—	2	8	no	—
73	C₂⁴×C₅₃	—	2	8	no	—
74	C₂⁴×C₅₃	C₂⁴	1	12	yes	—
75	C₂⁴×C₅₃	C₂⁴	1	1	yes	characteristic
76	C₂×Q₈×C₅₃	—	2	48	no	—
77	C₂×Q₈×C₅₃	—	2	96	no	—
78	C₂×Q₈×C₅₃	C₂⁴	1	16	yes	—
79	C₂×Q₈×C₅₃	C₂⁴	1	32	yes	—
80	C₂×D₄×C₅₃	—	2	48	no	—
81	C₂×D₄×C₅₃	C₂⁴	1	16	yes	—
82	C₂²×C₄×C₅₃	—	2	8	no	—
83	C₂²×C₄×C₅₃	—	2	8	no	—
84	C₂²×C₄×C₅₃	—	2	16	no	—
85	C₂²×C₄×C₅₃	—	2	8	no	—
86	C₂²×C₄×C₅₃	—	2	4	no	—
87	C₂²×C₄×C₅₃	—	2	96	no	—
88	C₂²×C₄×C₅₃	#41	1	8	yes	—
89	C₂²×C₄×C₅₃	#41	1	8	yes	—
90	C₂²×C₄×C₅₃	C₂⁴	1	12	yes	—
91	C₂²×C₄×C₅₃	C₂⁴	1	12	yes	—
92	C₂²×C₄×C₅₃	C₂⁴	1	6	yes	—
93	C₂²×C₄×C₅₃	C₂⁴	1	24	yes	—
94	C₂²×C₄×C₅₃	C₂⁴	1	12	yes	—
95	C₂²×C₄×C₅₃	C₂⁴	1	32	yes	—
96	C₂²×C₄×C₅₃	C₂⁴	1	6	yes	—
97	C₂²×C₄×C₅₃	C₂⁴	1	6	yes	—
98	C₄⋊C₄×C₅₃	C₂⁴	1	16	yes	—
99	C₄⋊C₄×C₅₃	C₂⁴	1	32	yes	—
100	C₄⋊C₄×C₅₃	C₂⁴	1	16	yes	—
101	C₄⋊C₄×C₅₃	C₂⁴	1	64	yes	—
102	C₄⋊C₄×C₅₃	C₂⁴	1	32	yes	—
103	C₄⋊C₄×C₅₃	C₂⁴	1	32	yes	—
104	C₂²⋊C₄×C₅₃	C₂⁴	1	64	yes	—
105	C₂²⋊C₄×C₅₃	C₂⁴	1	32	yes	—
106	C₂²⋊C₄×C₅₃	C₂⁴	1	64	yes	—
107	C₄²×C₅₃	C₂⁴	1	64	yes	—
108	C₄²×C₅₃	C₂⁴	1	16	yes	—
109	C₄²×C₅₃	C₂⁴	1	32	yes	—
110	C₂³×C₅₃	—	2	24	no	—
111	C₂³×C₅₃	—	2	24	no	—
112	C₂³×C₅₃	—	2	24	no	—
113	C₂³×C₅₃	—	4	32	no	—
114	C₂³×C₅₃	C₂×#41	1	4	yes	—
115	C₂³×C₅₃	C₂×#41	1	4	yes	—
116	C₂³×C₅₃	2₋¹⁺⁴	1	4	yes	—
117	C₂³×C₅₃	C₂⁵	1	8	yes	—
118	C₂³×C₅₃	C₂⁵	1	3	yes	—
119	Q₈×C₅₃	—	2	32	no	—
120	Q₈×C₅₃	—	2	64	no	—
121	D₄×C₅₃	—	2	32	no	—
122	C₂×C₄×C₅₃	—	2	24	no	—
123	C₂×C₄×C₅₃	—	2	24	no	—
124	C₂×C₄×C₅₃	—	2	48	no	—
125	C₂×C₄×C₅₃	—	2	24	no	—
126	C₂×C₄×C₅₃	—	2	12	no	—
127	C₂×C₄×C₅₃	—	2	64	no	—
128	C₂×C₄×C₅₃	C₂×#41	1	24	yes	—
129	C₂×C₄×C₅₃	C₂×#41	1	24	yes	—
130	C₂×C₄×C₅₃	C₂⁵	1	8	yes	—
131	C₂×C₄×C₅₃	C₂⁵	1	8	yes	—
132	C₂×C₄×C₅₃	C₂⁵	1	16	yes	—
133	C₂×C₄×C₅₃	C₂⁵	1	8	yes	—
134	C₂×C₄×C₅₃	C₂⁵	1	4	yes	—
135	C₂×C₄×C₅₃	C₂⁵	1	4	yes	—
136	C₂×C₄×C₅₃	C₂⁵	1	4	yes	—
137	C₂²×#556	C₅₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
138	C₂²×C₅₃	—	2	8	no	—
139	C₂²×C₅₃	—	2	8	no	—
140	C₂²×C₅₃	—	2	8	no	—
141	C₂²×C₅₃	—	4	48	no	—
142	C₂²×C₅₃	#556	1	16	yes	direct product factor
143	C₂²×C₅₃	C₂²×#41	1	6	yes	—
144	C₂²×C₅₃	C₂²×#41	1	6	yes	—
145	C₂²×C₅₃	C₂×2₋¹⁺⁴	1	6	yes	—
146	C₂²×C₅₃	C₂⁶	1	1	yes	characteristic
147	C₄×C₅₃	—	2	8	no	—
148	C₄×C₅₃	—	2	8	no	—
149	C₄×C₅₃	—	2	16	no	—
150	C₄×C₅₃	—	2	8	no	—
151	C₄×C₅₃	—	2	4	no	—
152	C₄×C₅₃	C₂²×#41	1	8	yes	—
153	C₄×C₅₃	C₂²×#41	1	8	yes	—
154	C₂×#556	C₂×C₅₃	1	48	yes	direct product factor
155	C₂²×C₄⋊Q₈	C₂×C₅₃	1	1	yes	characteristic, complemented
156	C₂²×#126	C₂×C₅₃	1	4	yes	complemented
157	C₂²×#124	C₂×C₅₃	1	2	yes	—
158	C₂²×C₂²⋊Q₈	C₂×C₅₃	1	2	yes	complemented
159	C₂²×C₄×Q₈	C₂×C₅₃	1	1	yes	characteristic, complemented
160	C₂²×C₄×Q₈	C₂×C₅₃	1	2	yes	complemented
161	C₂²×C₄×D₄	C₂×C₅₃	1	1	yes	characteristic
162	C₂²×#117	C₂×C₅₃	1	2	yes	complemented
163	C₂×C₅₃	—	4	8	no	—
164	C₂×C₅₃	C₂×#556	1	12	yes	direct product factor
165	C₂×C₅₃	C₂³×#41	1	1	yes	characteristic
166	C₂×C₅₃	C₂³×#41	1	1	yes	characteristic
167	C₂×C₅₃	C₂²×2₋¹⁺⁴	1	1	yes	characteristic
168	C₂³×Q₈	C₂²×C₅₃	1	1	yes	characteristic
169	C₂³×Q₈	C₂²×C₅₃	1	2	yes	—
170	C₂³×D₄	C₂²×C₅₃	1	1	yes	characteristic
171	C₂⁴×C₄	C₂²×C₅₃	1	2	yes	—
172	#556	C₂²×C₅₃	1	256	yes	direct product factor
173	C₂×C₄⋊Q₈	C₂²×C₅₃	1	24	yes	complemented
174	C₂×#126	C₂²×C₅₃	1	96	yes	complemented
175	C₂×#124	C₂²×C₅₃	1	48	yes	—
176	C₂×C₂²⋊Q₈	C₂²×C₅₃	1	48	yes	complemented
177	C₂×C₄×Q₈	C₂²×C₅₃	1	48	yes	complemented
178	C₂×C₄×Q₈	C₂²×C₅₃	1	24	yes	complemented
179	C₂×C₄×D₄	C₂²×C₅₃	1	24	yes	—
180	C₂×#117	C₂²×C₅₃	1	48	yes	complemented
181	C₂²×C₄⋊C₄	C₂²×C₅₃	1	1	yes	characteristic
182	C₂²×C₄⋊C₄	C₂²×C₅₃	1	2	yes	—
183	C₂²×C₄⋊C₄	C₂²×C₅₃	1	1	yes	characteristic
184	C₂²×C₄⋊C₄	C₂²×C₅₃	1	4	yes	—
185	C₂²×C₄⋊C₄	C₂²×C₅₃	1	2	yes	—
186	C₂²×C₄⋊C₄	C₂²×C₅₃	1	2	yes	—
187	C₂²×C₂²⋊C₄	C₂²×C₅₃	1	4	yes	—
188	C₂²×C₂²⋊C₄	C₂²×C₅₃	1	2	yes	—
189	C₂²×C₂²⋊C₄	C₂²×C₅₃	1	4	yes	—
190	C₂²×C₄²	C₂²×C₅₃	1	4	yes	—
191	C₂²×C₄²	C₂²×C₅₃	1	1	yes	characteristic
192	C₂²×C₄²	C₂²×C₅₃	1	2	yes	—
193	C₅₃	C₂²×#556	1	1	yes	characteristic, direct product factor, Sylow
194	C₂⁵	C₂³×C₅₃	1	2	yes	—
195	C₂²×Q₈	—	2	8	no	—
196	C₂²×Q₈	—	2	16	no	—
197	C₂²×Q₈	C₂³×C₅₃	1	12	yes	—
198	C₂²×Q₈	C₂³×C₅₃	1	24	yes	—
199	C₂²×D₄	—	2	8	no	—
200	C₂²×D₄	C₂³×C₅₃	1	12	yes	—
201	C₂³×C₄	—	2	16	no	—
202	C₂³×C₄	C₂³×C₅₃	1	1	yes	characteristic
203	C₂³×C₄	C₂³×C₅₃	1	4	yes	—
204	C₂³×C₄	C₂³×C₅₃	1	2	yes	—
205	C₂³×C₄	C₂³×C₅₃	1	2	yes	—
206	C₂³×C₄	C₂³×C₅₃	1	2	yes	—
207	C₂³×C₄	C₂³×C₅₃	1	24	yes	—
208	C₂³×C₄	C₂³×C₅₃	1	1	yes	characteristic
209	C₂³×C₄	C₂³×C₅₃	1	1	yes	characteristic
210	C₄⋊Q₈	C₂³×C₅₃	1	64	yes	complemented
211	#126	C₂³×C₅₃	1	256	yes	complemented
212	#124	C₂³×C₅₃	1	128	yes	—
213	C₂²⋊Q₈	C₂³×C₅₃	1	128	yes	complemented
214	C₄×Q₈	C₂³×C₅₃	1	128	yes	complemented
215	C₄×Q₈	C₂³×C₅₃	1	64	yes	complemented
216	C₄×D₄	C₂³×C₅₃	1	64	yes	—
217	#117	C₂³×C₅₃	1	128	yes	complemented
218	C₂×C₄⋊C₄	C₂³×C₅₃	1	12	yes	—
219	C₂×C₄⋊C₄	C₂³×C₅₃	1	24	yes	—
220	C₂×C₄⋊C₄	C₂³×C₅₃	1	12	yes	—
221	C₂×C₄⋊C₄	C₂³×C₅₃	1	48	yes	—
222	C₂×C₄⋊C₄	C₂³×C₅₃	1	24	yes	—
223	C₂×C₄⋊C₄	C₂³×C₅₃	1	24	yes	—
224	C₂×C₂²⋊C₄	C₂³×C₅₃	1	48	yes	—
225	C₂×C₂²⋊C₄	C₂³×C₅₃	1	24	yes	—
226	C₂×C₂²⋊C₄	C₂³×C₅₃	1	48	yes	—
227	C₂×C₄²	C₂³×C₅₃	1	48	yes	—
228	C₂×C₄²	C₂³×C₅₃	1	12	yes	—
229	C₂×C₄²	C₂³×C₅₃	1	24	yes	—
230	C₂⁴	—	2	8	no	—
231	C₂⁴	—	2	8	no	—
232	C₂⁴	—	2	8	no	—
233	C₂⁴	C₂⁴×C₅₃	1	12	yes	—
234	C₂⁴	C₂⁴×C₅₃	1	1	yes	characteristic
235	C₂×Q₈	—	2	48	no	—
236	C₂×Q₈	—	2	96	no	—
237	C₂×Q₈	C₂⁴×C₅₃	1	16	yes	—
238	C₂×Q₈	C₂⁴×C₅₃	1	32	yes	—
239	C₂×D₄	—	2	48	no	—
240	C₂×D₄	C₂⁴×C₅₃	1	16	yes	—
241	C₂²×C₄	—	2	8	no	—
242	C₂²×C₄	—	2	8	no	—
243	C₂²×C₄	—	2	16	no	—
244	C₂²×C₄	—	2	4	no	—
245	C₂²×C₄	—	2	8	no	—
246	C₂²×C₄	—	2	96	no	—
247	C₂²×C₄	#41×C₅₃	1	8	yes	—
248	C₂²×C₄	#41×C₅₃	1	8	yes	—
249	C₂²×C₄	C₂⁴×C₅₃	1	12	yes	—
250	C₂²×C₄	C₂⁴×C₅₃	1	12	yes	—
251	C₂²×C₄	C₂⁴×C₅₃	1	6	yes	—
252	C₂²×C₄	C₂⁴×C₅₃	1	24	yes	—
253	C₂²×C₄	C₂⁴×C₅₃	1	12	yes	—
254	C₂²×C₄	C₂⁴×C₅₃	1	32	yes	—
255	C₂²×C₄	C₂⁴×C₅₃	1	6	yes	—
256	C₂²×C₄	C₂⁴×C₅₃	1	6	yes	—
257	C₄⋊C₄	C₂⁴×C₅₃	1	16	yes	—
258	C₄⋊C₄	C₂⁴×C₅₃	1	32	yes	—
259	C₄⋊C₄	C₂⁴×C₅₃	1	16	yes	—
260	C₄⋊C₄	C₂⁴×C₅₃	1	64	yes	—
261	C₄⋊C₄	C₂⁴×C₅₃	1	32	yes	—
262	C₄⋊C₄	C₂⁴×C₅₃	1	32	yes	—
263	C₂²⋊C₄	C₂⁴×C₅₃	1	64	yes	—
264	C₂²⋊C₄	C₂⁴×C₅₃	1	32	yes	—
265	C₂²⋊C₄	C₂⁴×C₅₃	1	64	yes	—
266	C₄²	C₂⁴×C₅₃	1	64	yes	—
267	C₄²	C₂⁴×C₅₃	1	16	yes	—
268	C₄²	C₂⁴×C₅₃	1	32	yes	—
269	C₂³	—	2	24	no	—
270	C₂³	—	2	24	no	—
271	C₂³	—	2	24	no	—
272	C₂³	—	4	32	no	—
273	C₂³	C₂×#41×C₅₃	1	4	yes	—
274	C₂³	C₂×#41×C₅₃	1	4	yes	—
275	C₂³	2₋¹⁺⁴×C₅₃	1	4	yes	—
276	C₂³	C₂⁵×C₅₃	1	8	yes	—
277	C₂³	C₂⁵×C₅₃	1	3	yes	—
278	Q₈	—	2	32	no	—
279	Q₈	—	2	64	no	—
280	D₄	—	2	32	no	—
281	C₂×C₄	—	2	24	no	—
282	C₂×C₄	—	2	24	no	—
283	C₂×C₄	—	2	48	no	—
284	C₂×C₄	—	2	24	no	—
285	C₂×C₄	—	2	12	no	—
286	C₂×C₄	—	2	64	no	—
287	C₂×C₄	C₂×#41×C₅₃	1	24	yes	—
288	C₂×C₄	C₂×#41×C₅₃	1	24	yes	—
289	C₂×C₄	C₂⁵×C₅₃	1	8	yes	—
290	C₂×C₄	C₂⁵×C₅₃	1	8	yes	—
291	C₂×C₄	C₂⁵×C₅₃	1	4	yes	—
292	C₂×C₄	C₂⁵×C₅₃	1	16	yes	—
293	C₂×C₄	C₂⁵×C₅₃	1	8	yes	—
294	C₂×C₄	C₂⁵×C₅₃	1	4	yes	—
295	C₂×C₄	C₂⁵×C₅₃	1	4	yes	—
296	C₂²	—	2	8	no	—
297	C₂²	—	2	8	no	—
298	C₂²	—	2	8	no	—
299	C₂²	—	4	48	no	—
300	C₂²	C₅₃×#556	1	16	yes	direct product factor
301	C₂²	C₂²×#41×C₅₃	1	6	yes	—
302	C₂²	C₂²×#41×C₅₃	1	6	yes	—
303	C₂²	C₂×2₋¹⁺⁴×C₅₃	1	6	yes	—
304	C₂²	C₂⁶×C₅₃	1	1	yes	characteristic
305	C₄	—	2	8	no	—
306	C₄	—	2	8	no	—
307	C₄	—	2	16	no	—
308	C₄	—	2	8	no	—
309	C₄	—	2	4	no	—
310	C₄	C₂²×#41×C₅₃	1	8	yes	—
311	C₄	C₂²×#41×C₅₃	1	8	yes	—
312	C₂	—	4	8	no	—
313	C₂	C₂×C₅₃×#556	1	12	yes	direct product factor
314	C₂	C₂³×#41×C₅₃	1	1	yes	characteristic
315	C₂	C₂³×#41×C₅₃	1	1	yes	characteristic
316	C₂	C₂²×2₋¹⁺⁴×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×#556	C₂	direct (N × H)
2. C₂²×C₄⋊Q₈×C₅₃	C₂	semidirect
3. C₂²×#126×C₅₃	C₂	semidirect
5. C₂²×C₂²⋊Q₈×C₅₃	C₂	semidirect
6. C₂²×C₄×Q₈×C₅₃	C₂	semidirect
7. C₂²×C₄×Q₈×C₅₃	C₂	semidirect
9. C₂²×#117×C₅₃	C₂	semidirect
14. C₅₃×#556	C₂²	direct (N × H)
15. C₂×C₄⋊Q₈×C₅₃	C₂²	semidirect
16. C₂×#126×C₅₃	C₂²	semidirect
18. C₂×C₂²⋊Q₈×C₅₃	C₂²	semidirect
19. C₂×C₄×Q₈×C₅₃	C₂²	semidirect
20. C₂×C₄×Q₈×C₅₃	C₂²	semidirect
22. C₂×#117×C₅₃	C₂²	semidirect
51. C₄⋊Q₈×C₅₃	C₂³	semidirect
52. #126×C₅₃	C₂³	semidirect
54. C₂²⋊Q₈×C₅₃	C₂³	semidirect
55. C₄×Q₈×C₅₃	C₂³	semidirect
56. C₄×Q₈×C₅₃	C₂³	semidirect
58. #117×C₅₃	C₂³	semidirect
137. C₂²×#556	C₅₃	direct (N × H)
142. C₂²×C₅₃	#556	direct (N × H)
154. C₂×#556	C₂×C₅₃	direct (N × H)
155. C₂²×C₄⋊Q₈	C₂×C₅₃	semidirect
156. C₂²×#126	C₂×C₅₃	semidirect
158. C₂²×C₂²⋊Q₈	C₂×C₅₃	semidirect
159. C₂²×C₄×Q₈	C₂×C₅₃	semidirect
160. C₂²×C₄×Q₈	C₂×C₅₃	semidirect
162. C₂²×#117	C₂×C₅₃	semidirect
164. C₂×C₅₃	C₂×#556	direct (N × H)
172. #556	C₂²×C₅₃	direct (N × H)
173. C₂×C₄⋊Q₈	C₂²×C₅₃	semidirect
174. C₂×#126	C₂²×C₅₃	semidirect
176. C₂×C₂²⋊Q₈	C₂²×C₅₃	semidirect
177. C₂×C₄×Q₈	C₂²×C₅₃	semidirect
178. C₂×C₄×Q₈	C₂²×C₅₃	semidirect
180. C₂×#117	C₂²×C₅₃	semidirect
193. C₅₃	C₂²×#556	direct (N × H)
210. C₄⋊Q₈	C₂³×C₅₃	semidirect
211. #126	C₂³×C₅₃	semidirect
213. C₂²⋊Q₈	C₂³×C₅₃	semidirect
214. C₄×Q₈	C₂³×C₅₃	semidirect
215. C₄×Q₈	C₂³×C₅₃	semidirect
217. #117	C₂³×C₅₃	semidirect
300. C₂²	C₅₃×#556	direct (N × H)
313. C₂	C₂×C₅₃×#556	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×#556	2
C₂²×C₄⋊Q₈×C₅₃	2
C₂²×#126×C₅₃	2
C₂²×#124×C₅₃	2
C₂²×C₂²⋊Q₈×C₅₃	2
C₂²×C₄×Q₈×C₅₃	2
C₂²×C₄×D₄×C₅₃	2
C₂²×#117×C₅₃	2
C₂²×#556	53

Subgroups — C22×C53×#556 #27270218

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂²×C₅₃×#556 #27270218