Subgroups — C₂×#20288493 #28292414

Order 13568.

12982 conjugacy classes of subgroups.

The subgroup lattice has 356 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	#20288493	C₂	1	32	yes	maximal, direct product factor
3	C₂×#16811295	C₂	1	4	yes	maximal, complemented
4	C₂×D₄×D₂₁₂	C₂	1	12	yes	maximal, complemented
5	C₂×#16811192	C₂	1	6	yes	maximal, complemented
6	C₂×#16811173	C₂	1	3	yes	maximal, complemented
7	C₂²×#13133214	C₂	1	3	yes	maximal, complemented
8	C₂×C₄×#41×C₅₃	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂×#13133263	—	2	32	no	—
10	C₂×#13133263	C₂²	1	32	yes	complemented
11	C₂²×#11719949	C₂²	1	4	yes	complemented
12	C₂²×#11719945	C₂²	1	6	yes	complemented
13	C₂³×D₂₁₂	C₂²	1	6	yes	complemented
14	C₂³×D₂₁₂	C₂²	1	12	yes	complemented
15	C₂²×D₄×D₅₃	C₂²	1	12	yes	complemented
16	#16811295	C₂²	1	64	yes	complemented
17	D₄×D₂₁₂	C₂²	1	192	yes	complemented
18	#16811192	C₂²	1	96	yes	complemented
19	#16811173	C₂²	1	48	yes	complemented
20	C₂×#13133217	C₂²	1	1	yes	characteristic, complemented
21	C₂×#13133214	C₂²	1	6	yes	complemented
22	C₂×#13133214	C₂²	1	12	yes	complemented
23	C₂×#13133214	C₂²	1	3	yes	complemented
24	C₂×#13133214	C₂²	1	48	yes	complemented
25	C₂×#13133205	C₂²	1	6	yes	complemented
26	C₂×#13133192	C₂²	1	24	yes	complemented
27	C₂×#13133191	C₂²	1	12	yes	complemented
28	C₂×#13133187	C₂²	1	24	yes	complemented
29	C₂×C₄×D₂₁₂	C₂²	1	12	yes	complemented
30	C₂²×#41×C₅₃	C₂²	1	1	yes	characteristic
31	C₄×#41×C₅₃	C₂²	1	16	yes	complemented
32	C₂×C₄×Q₈×C₅₃	C₂²	1	2	yes	complemented
33	C₂×C₄×D₄×C₅₃	C₂²	1	6	yes	complemented
34	C₂×#117×C₅₃	C₂²	1	3	yes	complemented
35	C₂²×C₄²×C₅₃	C₂²	1	3	yes	complemented
36	C₂⁴×D₅₃	C₂³	1	12	yes	—
37	#13133263	—	2	256	no	—
38	C₂×#11719949	—	2	32	no	—
39	C₂×#11719949	C₂³	1	32	yes	complemented
40	C₂×#11719945	—	2	48	no	—
41	C₂×#11719945	C₂³	1	48	yes	complemented
42	C₂²×Dic₁₀₆	C₂³	1	2	yes	—
43	C₂²×D₂₁₂	—	2	48	no	—
44	C₂²×D₂₁₂	—	4	48	no	—
45	C₂²×D₂₁₂	C₂³	1	12	yes	—
46	C₂²×D₂₁₂	C₂³	1	12	yes	—
47	C₂²×D₂₁₂	C₂³	1	6	yes	—
48	C₂²×D₂₁₂	C₂³	1	24	yes	—
49	C₂²×D₂₁₂	C₂³	1	48	yes	complemented
50	C₂²×D₂₁₂	C₂³	1	96	yes	complemented
51	C₂²×C₂²⋊D₅₃	C₂³	1	12	yes	—
52	C₂×D₄×D₅₃	—	2	96	no	—
53	C₂×D₄×D₅₃	C₂³	1	96	yes	complemented
54	C₂²×C₄×D₅₃	C₂³	1	12	yes	—
55	#13133217	C₂³	1	8	yes	complemented
56	#13133214	C₂³	1	96	yes	complemented
57	#13133214	C₂³	1	24	yes	complemented
58	#13133214	C₂³	1	48	yes	complemented
59	#13133205	C₂³	1	48	yes	complemented
60	#13133192	C₂³	1	192	yes	complemented
61	#13133191	C₂³	1	96	yes	complemented
62	#13133187	C₂³	1	192	yes	complemented
63	C₄×D₂₁₂	C₂³	1	96	yes	complemented
64	C₂×C₄⋊Dic₅₃	C₂³	1	4	yes	—
65	C₂×#11719921	C₂³	1	24	yes	—
66	C₂×#41×C₅₃	D₄	1	16	yes	complemented
67	C₂×#41×C₅₃	C₂³	1	8	yes	—
68	C₂²×Q₈×C₅₃	C₂³	1	1	yes	characteristic
69	C₂²×D₄×C₅₃	C₂³	1	3	yes	—
70	C₂³×C₄×C₅₃	C₂³	1	6	yes	—
71	C₂³×C₄×C₅₃	C₂³	1	3	yes	—
72	C₄×Q₈×C₅₃	C₂³	1	16	yes	complemented
73	C₄×D₄×C₅₃	C₂³	1	48	yes	complemented
74	#117×C₅₃	C₂³	1	24	yes	complemented
75	C₂×C₄⋊C₄×C₅₃	C₂³	1	6	yes	—
76	C₂×C₂²⋊C₄×C₅₃	C₂³	1	6	yes	—
77	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
78	C₂×C₄²×C₅₃	C₂³	1	3	yes	—
79	C₂×C₄²×C₅₃	C₂³	1	6	yes	—
80	C₂×C₄²×C₅₃	C₂³	1	24	yes	complemented
81	C₂⁴×C₅₃	C₂⁴	1	3	yes	—
82	C₂³×D₅₃	—	2	48	no	—
83	C₂³×D₅₃	—	4	48	no	—
84	C₂³×D₅₃	C₂⁴	1	12	yes	—
85	C₂³×D₅₃	C₂⁴	1	48	yes	—
86	#41×C₅₃	C₂×D₄	1	64	yes	complemented
87	C₂×Q₈×C₅₃	C₂×D₄	1	8	yes	complemented
88	C₂×Q₈×C₅₃	C₂⁴	1	4	yes	—
89	C₂×D₄×C₅₃	C₂×D₄	1	24	yes	complemented
90	C₂×D₄×C₅₃	C₂⁴	1	12	yes	—
91	C₂²×C₄×C₅₃	—	2	24	no	—
92	C₂²×C₄×C₅₃	C₂×D₄	1	24	yes	complemented
93	C₂²×C₄×C₅₃	C₂⁴	1	6	yes	—
94	C₂²×C₄×C₅₃	C₂⁴	1	24	yes	—
95	C₂²×C₄×C₅₃	C₂⁴	1	1	yes	characteristic
96	C₂²×C₄×C₅₃	C₂⁴	1	3	yes	—
97	C₂²×C₄×C₅₃	C₂⁴	1	12	yes	—
98	C₂²×C₄×C₅₃	C₂⁴	1	2	yes	—
99	C₂×C₂²⋊D₅₃	—	2	48	no	—
100	C₂×C₂²⋊D₅₃	C₂⁴	1	48	yes	—
101	C₂²×Dic₅₃	C₂⁴	1	4	yes	—
102	#11719949	—	2	128	no	—
103	D₄×D₅₃	—	2	384	no	—
104	#11719945	—	2	192	no	—
105	C₂×D₂₁₂	—	2	48	no	—
106	C₂×D₂₁₂	—	2	24	no	—
107	C₂×D₂₁₂	—	2	192	no	—
108	C₂×D₂₁₂	—	2	48	no	—
109	C₂×D₂₁₂	—	4	192	no	—
110	C₂×D₂₁₂	—	4	48	no	—
111	C₂×D₂₁₂	C₂⁴	1	48	yes	—
112	C₂×D₂₁₂	C₂⁴	1	48	yes	—
113	C₂×D₂₁₂	C₂⁴	1	24	yes	—
114	C₂×D₂₁₂	C₂⁴	1	96	yes	—
115	C₂×C₄×D₅₃	—	2	48	no	—
116	C₂×C₄×D₅₃	C₂⁴	1	48	yes	—
117	C₂×Dic₁₀₆	—	2	8	no	—
118	C₂×Dic₁₀₆	C₂⁴	1	8	yes	—
119	C₄⋊C₄×C₅₃	C₂⁴	1	24	yes	—
120	C₂²⋊C₄×C₅₃	C₂⁴	1	24	yes	—
121	C₄²×C₅₃	C₂⁴	1	4	yes	—
122	C₄²×C₅₃	C₂⁴	1	12	yes	—
123	C₄²×C₅₃	C₂⁴	1	24	yes	—
124	#11719921	C₂⁴	1	96	yes	—
125	C₄⋊Dic₅₃	C₂⁴	1	16	yes	—
126	C₂³×C₅₃	—	2	12	no	—
127	C₂³×C₅₃	C₂²×D₄	1	12	yes	—
128	C₂³×C₅₃	C₂⁵	1	6	yes	—
129	C₂³×C₅₃	C₂⁵	1	1	yes	characteristic
130	C₂²×D₅₃	—	2	24	no	—
131	C₂²×D₅₃	—	2	48	no	—
132	C₂²×D₅₃	—	2	96	no	—
133	C₂²×D₅₃	—	4	96	no	—
134	C₂²×D₅₃	—	8	96	no	—
135	C₂²×D₅₃	C₂⁵	1	24	yes	—
136	Q₈×C₅₃	C₂²×D₄	1	16	yes	complemented
137	D₄×C₅₃	C₂²×D₄	1	48	yes	complemented
138	C₂×C₄×C₅₃	—	2	12	no	—
139	C₂×C₄×C₅₃	—	2	48	no	—
140	C₂×C₄×C₅₃	C₂²×D₄	1	4	yes	—
141	C₂×C₄×C₅₃	C₂²×D₄	1	12	yes	—
142	C₂×C₄×C₅₃	C₂²×D₄	1	48	yes	complemented
143	C₂×C₄×C₅₃	H(5,𝔽₂)	1	8	yes	complemented
144	C₂×C₄×C₅₃	C₂⁵	1	12	yes	—
145	C₂×C₄×C₅₃	C₂⁵	1	2	yes	—
146	C₂×C₄×C₅₃	C₂⁵	1	6	yes	—
147	C₂×C₄×C₅₃	C₂⁵	1	4	yes	—
148	C₂²⋊D₅₃	—	2	96	no	—
149	C₂×Dic₅₃	—	2	8	no	—
150	C₂×Dic₅₃	C₂⁵	1	8	yes	—
151	D₂₁₂	—	2	48	no	—
152	D₂₁₂	—	2	96	no	—
153	D₂₁₂	—	2	96	no	—
154	D₂₁₂	—	4	96	no	—
155	C₄×D₅₃	—	2	96	no	—
156	Dic₁₀₆	—	2	16	no	—
157	C₂×#3498	—	53	1	no	maximal, Sylow
158	C₂²×C₅₃	—	2	12	no	—
159	C₂²×C₅₃	—	2	24	no	—
160	C₂²×C₅₃	C₂³×D₄	1	12	yes	—
161	C₂²×C₅₃	C₂³×D₄	1	2	yes	—
162	C₂²×C₅₃	C₂×H(5,𝔽₂)	1	4	yes	—
163	C₂²×C₅₃	C₂⁶	1	1	yes	characteristic
164	C₂×D₅₃	—	2	24	no	—
165	C₂×D₅₃	—	2	48	no	—
166	C₂×D₅₃	—	4	48	no	—
167	C₂×D₅₃	—	8	96	no	—
168	C₄×C₅₃	—	2	12	no	—
169	C₄×C₅₃	C₂³×D₄	1	4	yes	—
170	C₄×C₅₃	C₂³×D₄	1	12	yes	—
171	C₄×C₅₃	C₂×H(5,𝔽₂)	1	8	yes	complemented
172	Dic₅₃	—	2	8	no	—
173	C₂²×H(5,𝔽₂)	—	53	2	no	—
174	#3498	—	53	32	no	—
175	C₂×#550	—	53	4	no	—
176	C₂×D₄²	—	53	12	no	—
177	C₂×#535	—	53	6	no	—
178	C₂×#532	—	53	3	no	—
179	C₂²×#127	—	53	3	no	—
180	C₂×C₄×#41	D₅₃	1	1	yes	characteristic, complemented
181	C₂×C₅₃	—	2	12	no	—
182	C₂×C₅₃	#3498	1	4	yes	complemented
183	C₂×C₅₃	C₂⁴×D₄	1	1	yes	characteristic
184	C₂×C₅₃	C₂²×H(5,𝔽₂)	1	2	yes	—
185	D₅₃	—	4	24	no	—
186	C₂×H(5,𝔽₂)	—	53	32	no	—
187	C₂×H(5,𝔽₂)	—	106	32	no	—
188	C₂²×#41	—	53	4	no	—
189	C₂²×#41	—	53	6	no	—
190	C₂²×#41	C₂×D₅₃	1	1	yes	characteristic
191	C₂³×D₄	—	53	12	no	—
192	C₂³×D₄	—	53	6	no	—
193	C₂³×D₄	—	53	12	no	—
194	#550	—	53	64	no	—
195	D₄²	—	53	192	no	—
196	#535	—	53	96	no	—
197	#532	—	53	48	no	—
198	C₂×C₄⋊Q₈	—	53	1	no	—
199	C₂×#127	—	53	3	no	—
200	C₂×#127	—	53	6	no	—
201	C₂×#127	—	53	12	no	—
202	C₂×#127	—	53	48	no	—
203	C₂×#124	—	53	6	no	—
204	C₂×#121	—	53	12	no	—
205	C₂×#121	—	53	24	no	—
206	C₂×#120	—	53	24	no	—
207	C₄×#41	C₂×D₅₃	1	16	yes	complemented
208	C₂×C₄×Q₈	C₂×D₅₃	1	2	yes	complemented
209	C₂×C₄×D₄	—	53	12	no	—
210	C₂×C₄×D₄	C₂×D₅₃	1	6	yes	complemented
211	C₂×#117	C₂×D₅₃	1	3	yes	complemented
212	C₂²×C₄²	C₂×D₅₃	1	3	yes	complemented
213	C₅₃	C₂×#3498	1	1	yes	characteristic, complemented, Sylow
214	C₂⁵	—	53	12	no	—
215	H(5,𝔽₂)	—	106	256	no	—
216	C₂×#41	—	53	32	no	—
217	C₂×#41	—	53	48	no	—
218	C₂×#41	—	106	32	no	—
219	C₂×#41	—	106	48	no	—
220	C₂×#41	D₂₁₂	1	16	yes	complemented
221	C₂×#41	C₂²×D₅₃	1	8	yes	—
222	C₂²×Q₈	—	53	2	no	—
223	C₂²×Q₈	C₂²×D₅₃	1	1	yes	characteristic
224	C₂²×D₄	—	53	12	no	—
225	C₂²×D₄	—	53	96	no	—
226	C₂²×D₄	—	53	12	no	—
227	C₂²×D₄	—	53	12	no	—
228	C₂²×D₄	—	53	6	no	—
229	C₂²×D₄	—	53	48	no	—
230	C₂²×D₄	—	53	24	no	—
231	C₂²×D₄	—	53	96	no	—
232	C₂²×D₄	—	106	96	no	—
233	C₂²×D₄	—	106	48	no	—
234	C₂²×D₄	—	212	48	no	—
235	C₂²×D₄	C₂²×D₅₃	1	3	yes	—
236	C₂³×C₄	—	53	12	no	—
237	C₂³×C₄	C₂²×D₅₃	1	6	yes	—
238	C₂³×C₄	C₂²×D₅₃	1	3	yes	—
239	C₄⋊Q₈	—	53	8	no	—
240	#127	—	53	24	no	—
241	#127	—	53	48	no	—
242	#127	—	53	96	no	—
243	#124	—	53	48	no	—
244	#121	—	53	96	no	—
245	#121	—	53	192	no	—
246	#120	—	53	192	no	—
247	C₄×Q₈	C₂²×D₅₃	1	16	yes	complemented
248	C₄×D₄	—	53	96	no	—
249	C₄×D₄	C₂²×D₅₃	1	48	yes	complemented
250	#117	C₂²×D₅₃	1	24	yes	complemented
251	C₂×C₄⋊C₄	—	53	4	no	—
252	C₂×C₄⋊C₄	C₂²×D₅₃	1	6	yes	—
253	C₂×C₂²⋊C₄	—	53	24	no	—
254	C₂×C₂²⋊C₄	C₂²×D₅₃	1	6	yes	—
255	C₂×C₄²	C₂²×D₅₃	1	3	yes	—
256	C₂×C₄²	C₂²×D₅₃	1	6	yes	—
257	C₂×C₄²	C₂²×D₅₃	1	1	yes	characteristic
258	C₂×C₄²	C₂²×D₅₃	1	24	yes	complemented
259	C₂⁴	—	53	12	no	—
260	C₂⁴	—	53	48	no	—
261	C₂⁴	—	106	48	no	—
262	C₂⁴	—	212	48	no	—
263	C₂⁴	C₂³×D₅₃	1	3	yes	—
264	#41	—	106	128	no	—
265	#41	—	106	192	no	—
266	#41	C₂×D₂₁₂	1	64	yes	complemented
267	C₂×Q₈	—	53	8	no	—
268	C₂×Q₈	—	106	8	no	—
269	C₂×Q₈	C₂×D₂₁₂	1	8	yes	complemented
270	C₂×Q₈	C₂³×D₅₃	1	4	yes	—
271	C₂×D₄	—	53	48	no	—
272	C₂×D₄	—	53	48	no	—
273	C₂×D₄	—	53	48	no	—
274	C₂×D₄	—	53	24	no	—
275	C₂×D₄	—	53	96	no	—
276	C₂×D₄	—	106	48	no	—
277	C₂×D₄	—	106	384	no	—
278	C₂×D₄	—	106	48	no	—
279	C₂×D₄	—	106	24	no	—
280	C₂×D₄	—	106	192	no	—
281	C₂×D₄	—	106	48	no	—
282	C₂×D₄	—	212	192	no	—
283	C₂×D₄	—	212	48	no	—
284	C₂×D₄	C₂×D₂₁₂	1	24	yes	complemented
285	C₂×D₄	C₂³×D₅₃	1	12	yes	—
286	C₂²×C₄	—	2	24	no	—
287	C₂²×C₄	—	53	4	no	—
288	C₂²×C₄	—	53	48	no	—
289	C₂²×C₄	—	106	48	no	—
290	C₂²×C₄	C₂×D₂₁₂	1	24	yes	complemented
291	C₂²×C₄	C₂³×D₅₃	1	24	yes	—
292	C₂²×C₄	C₂³×D₅₃	1	6	yes	—
293	C₂²×C₄	C₂³×D₅₃	1	1	yes	characteristic
294	C₂²×C₄	C₂³×D₅₃	1	3	yes	—
295	C₂²×C₄	C₂³×D₅₃	1	12	yes	—
296	C₂²×C₄	C₂³×D₅₃	1	2	yes	—
297	C₄⋊C₄	—	53	16	no	—
298	C₄⋊C₄	C₂³×D₅₃	1	24	yes	—
299	C₂²⋊C₄	—	53	96	no	—
300	C₂²⋊C₄	C₂³×D₅₃	1	24	yes	—
301	C₄²	C₂³×D₅₃	1	4	yes	—
302	C₄²	C₂³×D₅₃	1	12	yes	—
303	C₄²	C₂³×D₅₃	1	24	yes	—
304	C₂³	—	2	12	no	—
305	C₂³	—	53	24	no	—
306	C₂³	—	106	96	no	—
307	C₂³	—	106	24	no	—
308	C₂³	—	106	48	no	—
309	C₂³	—	424	96	no	—
310	C₂³	—	212	96	no	—
311	C₂³	C₂²×D₂₁₂	1	12	yes	—
312	C₂³	C₂⁴×D₅₃	1	6	yes	—
313	C₂³	C₂⁴×D₅₃	1	1	yes	characteristic
314	Q₈	—	106	16	no	—
315	Q₈	C₂²×D₂₁₂	1	16	yes	complemented
316	D₄	—	106	96	no	—
317	D₄	—	106	96	no	—
318	D₄	—	106	48	no	—
319	D₄	—	106	96	no	—
320	D₄	—	212	96	no	—
321	D₄	C₂²×D₂₁₂	1	48	yes	complemented
322	C₂×C₄	—	2	12	no	—
323	C₂×C₄	—	2	48	no	—
324	C₂×C₄	—	53	8	no	—
325	C₂×C₄	—	106	8	no	—
326	C₂×C₄	—	106	96	no	—
327	C₂×C₄	C₂²×D₂₁₂	1	12	yes	—
328	C₂×C₄	C₂²×D₂₁₂	1	4	yes	—
329	C₂×C₄	C₂²×D₂₁₂	1	48	yes	complemented
330	C₂×C₄	#13133263	1	8	yes	complemented
331	C₂×C₄	C₂⁴×D₅₃	1	12	yes	—
332	C₂×C₄	C₂⁴×D₅₃	1	2	yes	—
333	C₂×C₄	C₂⁴×D₅₃	1	6	yes	—
334	C₂×C₄	C₂⁴×D₅₃	1	4	yes	—
335	C₂²	—	2	24	no	—
336	C₂²	—	2	12	no	—
337	C₂²	—	106	24	no	—
338	C₂²	—	106	48	no	—
339	C₂²	—	424	96	no	—
340	C₂²	—	212	48	no	—
341	C₂²	C₂³×D₂₁₂	1	12	yes	—
342	C₂²	C₂³×D₂₁₂	1	2	yes	—
343	C₂²	C₂×#13133263	1	4	yes	—
344	C₂²	C₂⁵×D₅₃	1	1	yes	characteristic
345	C₄	—	2	12	no	—
346	C₄	—	106	8	no	—
347	C₄	C₂³×D₂₁₂	1	12	yes	—
348	C₄	C₂³×D₂₁₂	1	4	yes	—
349	C₄	C₂×#13133263	1	8	yes	complemented
350	C₂	—	2	12	no	—
351	C₂	—	212	24	no	—
352	C₂	#20288493	1	4	yes	direct product factor
353	C₂	C₂⁴×D₂₁₂	1	1	yes	characteristic
354	C₂	C₂²×#13133263	1	2	yes	—

	subgroup	G/H
center	C₂³	C₂⁴×D₅₃
commutator, cosocle	C₂²×C₅₃	C₂⁶
Frattini	C₂²	C₂⁵×D₅₃
Fitting	C₂×C₄×#41×C₅₃	C₂
socle	C₂³×C₅₃	C₂⁵
hypercenter	C₂×C₄×#41	D₅₃
hypocenter	C₅₃	C₂×#3498
LCS	G ▷ C₂²×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂³ ◁ C₂×C₄×#41

Complements

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

N	H ≅ G/N	type
1. C₂²×#13133263	C₂	semidirect
2. #20288493	C₂	direct (N × H)
3. C₂×#16811295	C₂	semidirect
4. C₂×D₄×D₂₁₂	C₂	semidirect
5. C₂×#16811192	C₂	semidirect
6. C₂×#16811173	C₂	semidirect
7. C₂²×#13133214	C₂	semidirect
8. C₂×C₄×#41×C₅₃	C₂	semidirect
10. C₂×#13133263	C₂²	semidirect
11. C₂²×#11719949	C₂²	semidirect
12. C₂²×#11719945	C₂²	semidirect
13. C₂³×D₂₁₂	C₂²	semidirect
14. C₂³×D₂₁₂	C₂²	semidirect
15. C₂²×D₄×D₅₃	C₂²	semidirect
16. #16811295	C₂²	semidirect
17. D₄×D₂₁₂	C₂²	semidirect
18. #16811192	C₂²	semidirect
19. #16811173	C₂²	semidirect
20. C₂×#13133217	C₂²	semidirect
21. C₂×#13133214	C₂²	semidirect
22. C₂×#13133214	C₂²	semidirect
23. C₂×#13133214	C₂²	semidirect
24. C₂×#13133214	C₂²	semidirect
25. C₂×#13133205	C₂²	semidirect
26. C₂×#13133192	C₂²	semidirect
27. C₂×#13133191	C₂²	semidirect
28. C₂×#13133187	C₂²	semidirect
29. C₂×C₄×D₂₁₂	C₂²	semidirect
31. C₄×#41×C₅₃	C₂²	semidirect
32. C₂×C₄×Q₈×C₅₃	C₂²	semidirect
33. C₂×C₄×D₄×C₅₃	C₂²	semidirect
34. C₂×#117×C₅₃	C₂²	semidirect
35. C₂²×C₄²×C₅₃	C₂²	semidirect
39. C₂×#11719949	C₂³	semidirect
41. C₂×#11719945	C₂³	semidirect
49. C₂²×D₂₁₂	C₂³	semidirect
50. C₂²×D₂₁₂	C₂³	semidirect
53. C₂×D₄×D₅₃	C₂³	semidirect
55. #13133217	C₂³	semidirect
56. #13133214	C₂³	semidirect
57. #13133214	C₂³	semidirect
58. #13133214	C₂³	semidirect
59. #13133205	C₂³	semidirect
60. #13133192	C₂³	semidirect
61. #13133191	C₂³	semidirect
62. #13133187	C₂³	semidirect
63. C₄×D₂₁₂	C₂³	semidirect
66. C₂×#41×C₅₃	D₄	semidirect
72. C₄×Q₈×C₅₃	C₂³	semidirect
73. C₄×D₄×C₅₃	C₂³	semidirect
74. #117×C₅₃	C₂³	semidirect
80. C₂×C₄²×C₅₃	C₂³	semidirect
86. #41×C₅₃	C₂×D₄	semidirect
87. C₂×Q₈×C₅₃	C₂×D₄	semidirect
89. C₂×D₄×C₅₃	C₂×D₄	semidirect
92. C₂²×C₄×C₅₃	C₂×D₄	semidirect
136. Q₈×C₅₃	C₂²×D₄	semidirect
137. D₄×C₅₃	C₂²×D₄	semidirect
142. C₂×C₄×C₅₃	C₂²×D₄	semidirect
143. C₂×C₄×C₅₃	H(5,𝔽₂)	semidirect
171. C₄×C₅₃	C₂×H(5,𝔽₂)	semidirect
180. C₂×C₄×#41	D₅₃	semidirect
182. C₂×C₅₃	#3498	semidirect
207. C₄×#41	C₂×D₅₃	semidirect
208. C₂×C₄×Q₈	C₂×D₅₃	semidirect
210. C₂×C₄×D₄	C₂×D₅₃	semidirect
211. C₂×#117	C₂×D₅₃	semidirect
212. C₂²×C₄²	C₂×D₅₃	semidirect
213. C₅₃	C₂×#3498	semidirect
220. C₂×#41	D₂₁₂	semidirect
247. C₄×Q₈	C₂²×D₅₃	semidirect
249. C₄×D₄	C₂²×D₅₃	semidirect
250. #117	C₂²×D₅₃	semidirect
258. C₂×C₄²	C₂²×D₅₃	semidirect
266. #41	C₂×D₂₁₂	semidirect
269. C₂×Q₈	C₂×D₂₁₂	semidirect
284. C₂×D₄	C₂×D₂₁₂	semidirect
290. C₂²×C₄	C₂×D₂₁₂	semidirect
315. Q₈	C₂²×D₂₁₂	semidirect
321. D₄	C₂²×D₂₁₂	semidirect
329. C₂×C₄	C₂²×D₂₁₂	semidirect
330. C₂×C₄	#13133263	semidirect
349. C₄	C₂×#13133263	semidirect
352. C₂	#20288493	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
#20288493	2
C₂×#16811295	2
C₂×D₄×D₂₁₂	2
C₂×#16811192	2
C₂×#16811173	2
C₂²×#13133214	2
C₂×C₄×#41×C₅₃	2
C₂×#3498	53

Subgroups — C2×#20288493 #28292414

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#20288493 #28292414