Subgroups — C₂²×#16811104 #28290681

Order 13568.

12270 conjugacy classes of subgroups.

The subgroup lattice has 360 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂³×#11719945	C₂	1	1	yes	maximal, characteristic
2	C₂×#16811104	C₂	1	48	yes	maximal, direct product factor
3	C₂²×#13133182	C₂	1	2	yes	maximal, complemented
4	C₂²×#13133179	C₂	1	2	yes	maximal, complemented
5	C₂²×C₄×C₂²⋊D₅₃	C₂	1	4	yes	maximal, complemented
6	C₂²×#13133176	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂²×#13133175	C₂	1	2	yes	maximal, complemented
8	C₂²×C₄⋊C₄×D₅₃	C₂	1	2	yes	maximal, complemented
9	C₂³×C₄⋊C₄×C₅₃	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂²×#11719945	C₄	1	32	yes	complemented
11	C₂²×#11719945	C₂²	1	24	yes	—
12	C₂³×Dic₁₀₆	C₂²	1	1	yes	characteristic
13	C₂³×D₂₁₂	C₂²	1	1	yes	characteristic
14	C₂³×C₂²⋊D₅₃	C₂²	1	2	yes	—
15	C₂³×C₄×D₅₃	C₂²	1	4	yes	—
16	C₂³×C₄×D₅₃	C₂²	1	2	yes	—
17	#16811104	C₂²	1	256	yes	direct product factor
18	C₂×#13133182	C₂²	1	48	yes	complemented
19	C₂×#13133179	C₂²	1	48	yes	complemented
20	C₂×C₄×C₂²⋊D₅₃	C₂²	1	96	yes	complemented
21	C₂×#13133176	C₂²	1	24	yes	complemented
22	C₂×#13133175	C₂²	1	48	yes	complemented
23	C₂²×C₄⋊Dic₅₃	C₂²	1	1	yes	characteristic
24	C₂²×C₄⋊Dic₅₃	C₂²	1	1	yes	characteristic
25	C₂²×#11719919	C₂²	1	4	yes	—
26	C₂×C₄⋊C₄×D₅₃	C₂²	1	48	yes	complemented
27	C₂²×C₂²⋊Dic₅₃	C₂²	1	2	yes	—
28	C₂²×#11719921	C₂²	1	4	yes	—
29	C₂²×C₄×Dic₅₃	C₂²	1	4	yes	—
30	C₂²×C₄×Dic₅₃	C₂²	1	2	yes	—
31	C₂⁴×C₄×C₅₃	C₂²	1	2	yes	—
32	C₂⁴×C₄×C₅₃	C₂²	1	1	yes	characteristic
33	C₂²×C₄⋊C₄×C₅₃	C₂²	1	2	yes	—
34	C₂²×C₄⋊C₄×C₅₃	C₂²	1	2	yes	—
35	C₂²×C₄⋊C₄×C₅₃	C₂²	1	24	yes	complemented
36	C₂⁴×D₅₃	C₂³	1	2	yes	—
37	C₂×#11719945	C₂×C₄	1	384	yes	complemented
38	C₂×#11719945	C₂³	1	64	yes	—
39	C₂²×Dic₁₀₆	C₂×C₄	1	16	yes	complemented
40	C₂²×Dic₁₀₆	C₂³	1	12	yes	—
41	C₂²×D₂₁₂	C₂×C₄	1	16	yes	complemented
42	C₂²×D₂₁₂	C₂³	1	12	yes	—
43	C₂²×C₂²⋊D₅₃	C₂×C₄	1	32	yes	complemented
44	C₂²×C₂²⋊D₅₃	C₂³	1	24	yes	—
45	C₂³×Dic₅₃	C₂³	1	2	yes	—
46	C₂³×Dic₅₃	C₂³	1	2	yes	—
47	C₂³×Dic₅₃	C₂³	1	2	yes	—
48	C₂²×C₄×D₅₃	—	4	16	no	—
49	C₂²×C₄×D₅₃	C₂×C₄	1	32	yes	complemented
50	C₂²×C₄×D₅₃	C₂³	1	48	yes	—
51	C₂²×C₄×D₅₃	C₂³	1	24	yes	—
52	#13133182	C₂³	1	128	yes	complemented
53	#13133179	C₂³	1	128	yes	complemented
54	C₄×C₂²⋊D₅₃	C₂³	1	256	yes	complemented
55	#13133176	C₂³	1	64	yes	complemented
56	#13133175	C₂³	1	128	yes	complemented
57	C₂×C₄⋊Dic₅₃	C₂³	1	12	yes	—
58	C₂×C₄⋊Dic₅₃	C₂³	1	12	yes	—
59	C₂×#11719919	C₂³	1	48	yes	—
60	C₄⋊C₄×D₅₃	C₂³	1	128	yes	complemented
61	C₂×C₂²⋊Dic₅₃	C₂³	1	24	yes	—
62	C₂×#11719921	C₂³	1	48	yes	—
63	C₂×C₄×Dic₅₃	C₂³	1	48	yes	—
64	C₂×C₄×Dic₅₃	C₂³	1	24	yes	—
65	C₂⁵×C₅₃	C₂³	1	1	yes	characteristic
66	C₂³×C₄×C₅₃	—	4	8	no	—
67	C₂³×C₄×C₅₃	C₂×C₄	1	16	yes	complemented
68	C₂³×C₄×C₅₃	C₂³	1	4	yes	—
69	C₂³×C₄×C₅₃	C₂³	1	24	yes	—
70	C₂³×C₄×C₅₃	C₂³	1	1	yes	characteristic
71	C₂³×C₄×C₅₃	C₂³	1	1	yes	characteristic
72	C₂³×C₄×C₅₃	C₂³	1	12	yes	—
73	C₂×C₄⋊C₄×C₅₃	C₂³	1	24	yes	—
74	C₂×C₄⋊C₄×C₅₃	C₂³	1	24	yes	—
75	C₂×C₄⋊C₄×C₅₃	C₂³	1	64	yes	complemented
76	C₂⁴×C₅₃	—	2	4	no	—
77	C₂⁴×C₅₃	—	2	4	no	—
78	C₂⁴×C₅₃	C₂²×C₄	1	8	yes	—
79	C₂⁴×C₅₃	C₂⁴	1	6	yes	—
80	C₂⁴×C₅₃	C₂⁴	1	1	yes	characteristic
81	C₂³×D₅₃	—	2	8	no	—
82	C₂³×D₅₃	—	2	8	no	—
83	C₂³×D₅₃	C₂²×C₄	1	16	yes	—
84	C₂³×D₅₃	C₂⁴	1	12	yes	—
85	C₂²×C₄×C₅₃	—	2	16	no	—
86	C₂²×C₄×C₅₃	—	4	48	no	—
87	C₂²×C₄×C₅₃	C₂²×C₄	1	8	yes	—
88	C₂²×C₄×C₅₃	C₂²×C₄	1	8	yes	—
89	C₂²×C₄×C₅₃	C₂²×C₄	1	96	yes	complemented
90	C₂²×C₄×C₅₃	C₂⁴	1	24	yes	—
91	C₂²×C₄×C₅₃	C₂⁴	1	32	yes	—
92	C₂²×C₄×C₅₃	C₂⁴	1	6	yes	—
93	C₂²×C₄×C₅₃	C₂⁴	1	6	yes	—
94	C₂²×C₄×C₅₃	C₂⁴	1	16	yes	—
95	C₂×C₂²⋊D₅₃	C₂²×C₄	1	192	yes	complemented
96	C₂×C₂²⋊D₅₃	C₂⁴	1	32	yes	—
97	C₂²×Dic₅₃	—	2	8	no	—
98	C₂²×Dic₅₃	—	2	8	no	—
99	C₂²×Dic₅₃	C₂²×C₄	1	16	yes	—
100	C₂²×Dic₅₃	C₂⁴	1	12	yes	—
101	C₂²×Dic₅₃	C₂⁴	1	12	yes	—
102	C₂²×Dic₅₃	C₂⁴	1	12	yes	—
103	#11719945	C₂²×C₄	1	512	yes	complemented
104	C₂×D₂₁₂	C₂²×C₄	1	96	yes	complemented
105	C₂×D₂₁₂	C₂⁴	1	16	yes	—
106	C₂×C₄×D₅₃	—	4	96	no	—
107	C₂×C₄×D₅₃	C₂²×C₄	1	192	yes	complemented
108	C₂×C₄×D₅₃	C₂⁴	1	64	yes	—
109	C₂×C₄×D₅₃	C₂⁴	1	32	yes	—
110	C₂×Dic₁₀₆	C₂²×C₄	1	96	yes	complemented
111	C₂×Dic₁₀₆	C₂⁴	1	16	yes	—
112	C₄⋊C₄×C₅₃	C₂⁴	1	32	yes	—
113	C₄⋊C₄×C₅₃	C₂⁴	1	32	yes	—
114	C₂²⋊Dic₅₃	C₂⁴	1	32	yes	—
115	#11719921	C₂⁴	1	64	yes	—
116	C₄⋊Dic₅₃	C₂⁴	1	16	yes	—
117	C₄⋊Dic₅₃	C₂⁴	1	16	yes	—
118	#11719919	C₂⁴	1	64	yes	—
119	C₄×Dic₅₃	C₂⁴	1	64	yes	—
120	C₄×Dic₅₃	C₂⁴	1	32	yes	—
121	C₂³×C₅₃	—	2	32	no	—
122	C₂³×C₅₃	—	2	12	no	—
123	C₂³×C₅₃	—	2	12	no	—
124	C₂³×C₅₃	C₂³×C₄	1	24	yes	—
125	C₂³×C₅₃	C₂³×C₄	1	4	yes	—
126	C₂³×C₅₃	H(5,𝔽₂)	1	4	yes	—
127	C₂³×C₅₃	2₋¹⁺⁴	1	4	yes	—
128	C₂³×C₅₃	C₂⁵	1	4	yes	—
129	C₂³×C₅₃	C₂⁵	1	3	yes	—
130	C₂²×D₅₃	—	2	64	no	—
131	C₂²×D₅₃	—	2	24	no	—
132	C₂²×D₅₃	—	2	24	no	—
133	C₂²×D₅₃	C₂³×C₄	1	48	yes	—
134	C₂²×D₅₃	C₂⁵	1	8	yes	—
135	C₂×C₄×C₅₃	—	2	48	no	—
136	C₂×C₄×C₅₃	—	4	32	no	—
137	C₂×C₄×C₅₃	C₂³×C₄	1	24	yes	—
138	C₂×C₄×C₅₃	C₂³×C₄	1	24	yes	—
139	C₂×C₄×C₅₃	C₂³×C₄	1	64	yes	complemented
140	C₂×C₄×C₅₃	C₂⁵	1	16	yes	—
141	C₂×C₄×C₅₃	C₂⁵	1	4	yes	—
142	C₂×C₄×C₅₃	C₂⁵	1	4	yes	—
143	C₂²⋊D₅₃	C₂³×C₄	1	128	yes	complemented
144	C₂×Dic₅₃	—	2	24	no	—
145	C₂×Dic₅₃	—	2	24	no	—
146	C₂×Dic₅₃	C₂³×C₄	1	48	yes	—
147	C₂×Dic₅₃	C₂⁵	1	8	yes	—
148	C₂×Dic₅₃	C₂⁵	1	8	yes	—
149	C₂×Dic₅₃	C₂⁵	1	8	yes	—
150	D₂₁₂	C₂³×C₄	1	64	yes	complemented
151	C₄×D₅₃	—	4	64	no	—
152	C₄×D₅₃	C₂³×C₄	1	128	yes	complemented
153	Dic₁₀₆	C₂³×C₄	1	64	yes	complemented
154	C₂²×#520	—	53	1	no	maximal, Sylow
155	C₂²×C₅₃	—	2	4	no	—
156	C₂²×C₅₃	—	2	4	no	—
157	C₂²×C₅₃	—	2	48	no	—
158	C₂²×C₅₃	#520	1	16	yes	complemented
159	C₂²×C₅₃	C₂⁴×C₄	1	8	yes	—
160	C₂²×C₅₃	C₂⁴×C₄	1	6	yes	—
161	C₂²×C₅₃	C₂×H(5,𝔽₂)	1	6	yes	—
162	C₂²×C₅₃	C₂×2₋¹⁺⁴	1	6	yes	—
163	C₂²×C₅₃	C₂⁶	1	1	yes	characteristic
164	C₂×D₅₃	—	2	8	no	—
165	C₂×D₅₃	—	2	8	no	—
166	C₂×D₅₃	—	2	96	no	—
167	C₂×D₅₃	C₂⁴×C₄	1	16	yes	—
168	C₄×C₅₃	—	2	16	no	—
169	C₄×C₅₃	C₂⁴×C₄	1	8	yes	—
170	C₄×C₅₃	C₂⁴×C₄	1	8	yes	—
171	Dic₅₃	—	2	8	no	—
172	Dic₅₃	—	2	8	no	—
173	Dic₅₃	C₂⁴×C₄	1	16	yes	—
174	C₂³×#41	—	53	1	no	—
175	C₂×#520	—	53	48	no	—
176	C₂²×C₄×Q₈	—	53	2	no	—
177	C₂²×C₄×D₄	—	53	4	no	—
178	C₂²×C₄×D₄	—	53	2	no	—
179	C₂²×#117	—	53	1	no	—
180	C₂²×#117	—	53	2	no	—
181	C₂³×C₄⋊C₄	—	53	2	no	—
182	C₂³×C₄⋊C₄	D₅₃	1	1	yes	characteristic, complemented
183	C₂×C₅₃	—	2	8	no	—
184	C₂×C₅₃	C₂×#520	1	12	yes	complemented
185	C₂×C₅₃	C₂⁵×C₄	1	1	yes	characteristic
186	C₂×C₅₃	C₂²×H(5,𝔽₂)	1	1	yes	characteristic
187	C₂×C₅₃	C₂²×2₋¹⁺⁴	1	1	yes	characteristic
188	D₅₃	—	2	16	no	—
189	C₂²×#41	—	53	32	no	—
190	C₂²×#41	—	53	24	no	—
191	C₂³×Q₈	—	53	1	no	—
192	C₂³×D₄	—	53	2	no	—
193	C₂³×D₄	—	53	1	no	—
194	C₂⁴×C₄	—	53	2	no	—
195	C₂⁴×C₄	—	53	4	no	—
196	C₂⁴×C₄	C₂×D₅₃	1	2	yes	—
197	C₂⁴×C₄	C₂×D₅₃	1	1	yes	characteristic
198	#520	—	53	256	no	—
199	C₂×C₄×Q₈	—	53	48	no	—
200	C₂×C₄×D₄	—	53	96	no	—
201	C₂×C₄×D₄	—	53	48	no	—
202	C₂×#117	—	53	24	no	—
203	C₂×#117	—	53	48	no	—
204	C₂²×C₄⋊C₄	—	53	4	no	—
205	C₂²×C₄⋊C₄	—	53	1	no	—
206	C₂²×C₄⋊C₄	—	53	1	no	—
207	C₂²×C₄⋊C₄	—	53	48	no	—
208	C₂²×C₄⋊C₄	C₂×D₅₃	1	2	yes	—
209	C₂²×C₄⋊C₄	C₂×D₅₃	1	2	yes	—
210	C₂²×C₄⋊C₄	C₂×D₅₃	1	24	yes	complemented
211	C₂²×C₂²⋊C₄	—	53	2	no	—
212	C₂²×C₂²⋊C₄	—	53	4	no	—
213	C₂²×C₄²	—	53	4	no	—
214	C₂²×C₄²	—	53	2	no	—
215	C₅₃	C₂²×#520	1	1	yes	characteristic, complemented, Sylow
216	C₂⁵	—	53	2	no	—
217	C₂⁵	C₂²×D₅₃	1	1	yes	characteristic
218	C₂×#41	—	53	384	no	—
219	C₂×#41	—	53	64	no	—
220	C₂²×Q₈	—	53	16	no	—
221	C₂²×Q₈	—	53	12	no	—
222	C₂²×D₄	—	53	32	no	—
223	C₂²×D₄	—	53	24	no	—
224	C₂²×D₄	—	53	16	no	—
225	C₂²×D₄	—	53	12	no	—
226	C₂³×C₄	—	4	8	no	—
227	C₂³×C₄	—	53	2	no	—
228	C₂³×C₄	—	53	32	no	—
229	C₂³×C₄	—	53	24	no	—
230	C₂³×C₄	—	53	2	no	—
231	C₂³×C₄	—	53	2	no	—
232	C₂³×C₄	—	53	48	no	—
233	C₂³×C₄	—	212	16	no	—
234	C₂³×C₄	C₄×D₅₃	1	16	yes	complemented
235	C₂³×C₄	C₂²×D₅₃	1	4	yes	—
236	C₂³×C₄	C₂²×D₅₃	1	24	yes	—
237	C₂³×C₄	C₂²×D₅₃	1	1	yes	characteristic
238	C₂³×C₄	C₂²×D₅₃	1	1	yes	characteristic
239	C₂³×C₄	C₂²×D₅₃	1	12	yes	—
240	C₄×Q₈	—	53	128	no	—
241	C₄×D₄	—	53	256	no	—
242	C₄×D₄	—	53	128	no	—
243	#117	—	53	64	no	—
244	#117	—	53	128	no	—
245	C₂×C₄⋊C₄	—	53	12	no	—
246	C₂×C₄⋊C₄	—	53	48	no	—
247	C₂×C₄⋊C₄	—	53	12	no	—
248	C₂×C₄⋊C₄	—	53	128	no	—
249	C₂×C₄⋊C₄	C₂²×D₅₃	1	24	yes	—
250	C₂×C₄⋊C₄	C₂²×D₅₃	1	24	yes	—
251	C₂×C₄⋊C₄	C₂²×D₅₃	1	64	yes	complemented
252	C₂×C₂²⋊C₄	—	53	24	no	—
253	C₂×C₂²⋊C₄	—	53	48	no	—
254	C₂×C₄²	—	53	48	no	—
255	C₂×C₄²	—	53	24	no	—
256	C₂⁴	—	2	4	no	—
257	C₂⁴	—	2	4	no	—
258	C₂⁴	—	53	16	no	—
259	C₂⁴	—	53	12	no	—
260	C₂⁴	—	106	8	no	—
261	C₂⁴	—	106	8	no	—
262	C₂⁴	C₂×C₄×D₅₃	1	8	yes	—
263	C₂⁴	C₂³×D₅₃	1	6	yes	—
264	C₂⁴	C₂³×D₅₃	1	1	yes	characteristic
265	#41	—	53	512	no	—
266	C₂×Q₈	—	53	96	no	—
267	C₂×Q₈	—	53	16	no	—
268	C₂×D₄	—	53	192	no	—
269	C₂×D₄	—	53	32	no	—
270	C₂×D₄	—	53	96	no	—
271	C₂×D₄	—	53	16	no	—
272	C₂²×C₄	—	2	16	no	—
273	C₂²×C₄	—	4	48	no	—
274	C₂²×C₄	—	53	16	no	—
275	C₂²×C₄	—	53	12	no	—
276	C₂²×C₄	—	53	192	no	—
277	C₂²×C₄	—	53	32	no	—
278	C₂²×C₄	—	53	12	no	—
279	C₂²×C₄	—	53	12	no	—
280	C₂²×C₄	—	53	64	no	—
281	C₂²×C₄	—	106	8	no	—
282	C₂²×C₄	—	106	8	no	—
283	C₂²×C₄	—	212	96	no	—
284	C₂²×C₄	C₂×C₄×D₅₃	1	8	yes	—
285	C₂²×C₄	C₂×C₄×D₅₃	1	8	yes	—
286	C₂²×C₄	C₂×C₄×D₅₃	1	96	yes	complemented
287	C₂²×C₄	C₂³×D₅₃	1	24	yes	—
288	C₂²×C₄	C₂³×D₅₃	1	32	yes	—
289	C₂²×C₄	C₂³×D₅₃	1	6	yes	—
290	C₂²×C₄	C₂³×D₅₃	1	6	yes	—
291	C₂²×C₄	C₂³×D₅₃	1	16	yes	—
292	C₄⋊C₄	—	53	16	no	—
293	C₄⋊C₄	—	53	64	no	—
294	C₄⋊C₄	—	53	16	no	—
295	C₄⋊C₄	C₂³×D₅₃	1	32	yes	—
296	C₄⋊C₄	C₂³×D₅₃	1	32	yes	—
297	C₂²⋊C₄	—	53	32	no	—
298	C₂²⋊C₄	—	53	64	no	—
299	C₄²	—	53	64	no	—
300	C₄²	—	53	32	no	—
301	C₂³	—	2	32	no	—
302	C₂³	—	2	12	no	—
303	C₂³	—	2	12	no	—
304	C₂³	—	53	48	no	—
305	C₂³	—	53	8	no	—
306	C₂³	—	106	64	no	—
307	C₂³	—	106	24	no	—
308	C₂³	—	106	24	no	—
309	C₂³	C₂²×C₄×D₅₃	1	24	yes	—
310	C₂³	C₂²×C₄×D₅₃	1	4	yes	—
311	C₂³	#13133262	1	4	yes	—
312	C₂³	#13133265	1	4	yes	—
313	C₂³	C₂⁴×D₅₃	1	4	yes	—
314	C₂³	C₂⁴×D₅₃	1	3	yes	—
315	Q₈	—	53	64	no	—
316	D₄	—	53	128	no	—
317	D₄	—	53	64	no	—
318	C₂×C₄	—	2	48	no	—
319	C₂×C₄	—	4	32	no	—
320	C₂×C₄	—	53	48	no	—
321	C₂×C₄	—	53	8	no	—
322	C₂×C₄	—	53	128	no	—
323	C₂×C₄	—	53	8	no	—
324	C₂×C₄	—	53	8	no	—
325	C₂×C₄	—	106	24	no	—
326	C₂×C₄	—	106	24	no	—
327	C₂×C₄	—	212	64	no	—
328	C₂×C₄	C₂²×C₄×D₅₃	1	24	yes	—
329	C₂×C₄	C₂²×C₄×D₅₃	1	24	yes	—
330	C₂×C₄	C₂²×C₄×D₅₃	1	64	yes	complemented
331	C₂×C₄	C₂⁴×D₅₃	1	16	yes	—
332	C₂×C₄	C₂⁴×D₅₃	1	4	yes	—
333	C₂×C₄	C₂⁴×D₅₃	1	4	yes	—
334	C₂²	—	2	48	no	—
335	C₂²	—	2	4	no	—
336	C₂²	—	2	4	no	—
337	C₂²	—	53	16	no	—
338	C₂²	—	106	96	no	—
339	C₂²	—	106	8	no	—
340	C₂²	—	106	8	no	—
341	C₂²	#16811104	1	16	yes	direct product factor
342	C₂²	C₂³×C₄×D₅₃	1	8	yes	—
343	C₂²	C₂³×C₄×D₅₃	1	6	yes	—
344	C₂²	C₂×#13133262	1	6	yes	—
345	C₂²	C₂×#13133265	1	6	yes	—
346	C₂²	C₂⁵×D₅₃	1	1	yes	characteristic
347	C₄	—	2	16	no	—
348	C₄	—	53	16	no	—
349	C₄	—	106	8	no	—
350	C₄	—	106	8	no	—
351	C₄	C₂³×C₄×D₅₃	1	8	yes	—
352	C₄	C₂³×C₄×D₅₃	1	8	yes	—
353	C₂	—	2	8	no	—
354	C₂	—	106	16	no	—
355	C₂	C₂×#16811104	1	12	yes	direct product factor
356	C₂	C₂⁴×C₄×D₅₃	1	1	yes	characteristic
357	C₂	C₂²×#13133262	1	1	yes	characteristic
358	C₂	C₂²×#13133265	1	1	yes	characteristic

	subgroup	G/H
center	C₂⁴	C₂³×D₅₃
commutator	C₂×C₅₃	C₂⁵×C₄
Frattini	C₂²	C₂⁵×D₅₃
Fitting	C₂³×C₄⋊C₄×C₅₃	C₂
socle	C₂⁴×C₅₃	C₂⁴
prime radical	C₂³×#11719945	C₂
cosocle	C₂²×C₅₃	C₂⁶
hypercenter	C₂³×C₄⋊C₄	D₅₃
hypocenter	C₅₃	C₂²×#520
LCS	G ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂⁴ ◁ C₂³×C₄⋊C₄

Complements

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

N	H ≅ G/N	type
2. C₂×#16811104	C₂	direct (N × H)
3. C₂²×#13133182	C₂	semidirect
4. C₂²×#13133179	C₂	semidirect
5. C₂²×C₄×C₂²⋊D₅₃	C₂	semidirect
6. C₂²×#13133176	C₂	semidirect
7. C₂²×#13133175	C₂	semidirect
8. C₂²×C₄⋊C₄×D₅₃	C₂	semidirect
9. C₂³×C₄⋊C₄×C₅₃	C₂	semidirect
10. C₂²×#11719945	C₄	semidirect
17. #16811104	C₂²	direct (N × H)
18. C₂×#13133182	C₂²	semidirect
19. C₂×#13133179	C₂²	semidirect
20. C₂×C₄×C₂²⋊D₅₃	C₂²	semidirect
21. C₂×#13133176	C₂²	semidirect
22. C₂×#13133175	C₂²	semidirect
26. C₂×C₄⋊C₄×D₅₃	C₂²	semidirect
35. C₂²×C₄⋊C₄×C₅₃	C₂²	semidirect
37. C₂×#11719945	C₂×C₄	semidirect
39. C₂²×Dic₁₀₆	C₂×C₄	semidirect
41. C₂²×D₂₁₂	C₂×C₄	semidirect
43. C₂²×C₂²⋊D₅₃	C₂×C₄	semidirect
49. C₂²×C₄×D₅₃	C₂×C₄	semidirect
52. #13133182	C₂³	semidirect
53. #13133179	C₂³	semidirect
54. C₄×C₂²⋊D₅₃	C₂³	semidirect
55. #13133176	C₂³	semidirect
56. #13133175	C₂³	semidirect
60. C₄⋊C₄×D₅₃	C₂³	semidirect
67. C₂³×C₄×C₅₃	C₂×C₄	semidirect
75. C₂×C₄⋊C₄×C₅₃	C₂³	semidirect
89. C₂²×C₄×C₅₃	C₂²×C₄	semidirect
95. C₂×C₂²⋊D₅₃	C₂²×C₄	semidirect
103. #11719945	C₂²×C₄	semidirect
104. C₂×D₂₁₂	C₂²×C₄	semidirect
107. C₂×C₄×D₅₃	C₂²×C₄	semidirect
110. C₂×Dic₁₀₆	C₂²×C₄	semidirect
139. C₂×C₄×C₅₃	C₂³×C₄	semidirect
143. C₂²⋊D₅₃	C₂³×C₄	semidirect
150. D₂₁₂	C₂³×C₄	semidirect
152. C₄×D₅₃	C₂³×C₄	semidirect
153. Dic₁₀₆	C₂³×C₄	semidirect
158. C₂²×C₅₃	#520	semidirect
182. C₂³×C₄⋊C₄	D₅₃	semidirect
184. C₂×C₅₃	C₂×#520	semidirect
210. C₂²×C₄⋊C₄	C₂×D₅₃	semidirect
215. C₅₃	C₂²×#520	semidirect
234. C₂³×C₄	C₄×D₅₃	semidirect
251. C₂×C₄⋊C₄	C₂²×D₅₃	semidirect
286. C₂²×C₄	C₂×C₄×D₅₃	semidirect
330. C₂×C₄	C₂²×C₄×D₅₃	semidirect
341. C₂²	#16811104	direct (N × H)
355. C₂	C₂×#16811104	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂³×#11719945	2
C₂×#16811104	2
C₂²×#13133182	2
C₂²×#13133179	2
C₂²×C₄×C₂²⋊D₅₃	2
C₂²×#13133176	2
C₂²×#13133175	2
C₂²×C₄⋊C₄×D₅₃	2
C₂³×C₄⋊C₄×C₅₃	2
C₂²×#520	53

Subgroups — C22×#16811104 #28290681

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂²×#16811104 #28290681