Subgroups — C₂×#20289119 #28291893

Order 13568.

8802 conjugacy classes of subgroups.

The subgroup lattice has 408 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20289119	C₂	1	32	yes	maximal, direct product factor
2	C₂×#16811365	C₂	1	3	yes	maximal, complemented
3	C₂×#16811335	C₂	1	6	yes	maximal, complemented
4	C₂×#16811255	C₂	1	6	yes	maximal, complemented
5	C₂×#16811191	C₂	1	6	yes	maximal, complemented
6	C₂×#16811188	C₂	1	6	yes	maximal, complemented
7	C₂×#16811172	C₂	1	3	yes	maximal, complemented
8	C₂×C₅₃×H(3,𝔽₄)	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂²×#11719949	C₂²	1	3	yes	complemented
10	C₂²×#11719947	C₂²	1	3	yes	complemented
11	C₂³×C₂²⋊D₅₃	C₂²	1	6	yes	complemented
12	C₂²×D₄×D₅₃	C₂²	1	6	yes	complemented
13	#16811365	C₂²	1	48	yes	complemented
14	#16811335	C₂²	1	96	yes	complemented
15	#16811255	C₂²	1	96	yes	complemented
16	#16811191	C₂²	1	96	yes	complemented
17	#16811188	C₂²	1	96	yes	complemented
18	#16811172	C₂²	1	48	yes	complemented
19	C₂×#13133215	C₂²	1	3	yes	complemented
20	C₂×#13133214	C₂²	1	2	yes	complemented
21	C₂×#13133213	C₂²	1	3	yes	—
22	C₂×#13133213	C₂²	1	6	yes	complemented
23	C₂×#13133206	C₂²	1	6	yes	complemented
24	C₂×#13133205	C₂²	1	3	yes	complemented
25	C₂×#13133203	C₂²	1	12	yes	complemented
26	C₂×#13133190	C₂²	1	12	yes	complemented
27	C₂×#13133190	C₂²	1	12	yes	complemented
28	C₂×#13133189	C₂²	1	12	yes	complemented
29	C₂×#13133187	C₂²	1	6	yes	complemented
30	C₂×#13133187	C₂²	1	12	yes	complemented
31	C₂×#13133186	C₂²	1	2	yes	complemented
32	C₂×#13133185	C₂²	1	6	yes	complemented
33	C₂×#13133177	C₂²	1	12	yes	complemented
34	C₂×#13133174	C₂²	1	6	yes	complemented
35	C₂×#13133173	C₂²	1	6	yes	complemented
36	C₂×C₄²×D₅₃	C₂²	1	1	yes	characteristic, complemented
37	C₅₃×H(3,𝔽₄)	C₂²	1	16	yes	complemented
38	C₂×#124×C₅₃	C₂²	1	6	yes	complemented
39	C₂×#124×C₅₃	C₂²	1	3	yes	complemented
40	C₂×#120×C₅₃	C₂²	1	6	yes	complemented
41	C₂⁴×D₅₃	C₂³	1	6	yes	—
42	C₂×#11719949	—	4	12	no	—
43	C₂×#11719949	C₂³	1	24	yes	complemented
44	C₂×#11719947	—	4	12	no	—
45	C₂×#11719947	C₂³	1	24	yes	complemented
46	C₂²×Dic₁₀₆	C₂³	1	3	yes	—
47	C₂²×D₂₁₂	C₂³	1	3	yes	—
48	C₂²×D₂₁₂	C₂³	1	6	yes	—
49	C₂²×D₂₁₂	C₂³	1	6	yes	—
50	C₂²×C₂²⋊D₅₃	—	4	24	no	—
51	C₂²×C₂²⋊D₅₃	C₂³	1	12	yes	—
52	C₂²×C₂²⋊D₅₃	C₂³	1	12	yes	—
53	C₂²×C₂²⋊D₅₃	C₂³	1	6	yes	—
54	C₂²×C₂²⋊D₅₃	C₂³	1	48	yes	complemented
55	C₂×D₄×D₅₃	—	4	24	no	—
56	C₂×D₄×D₅₃	C₂³	1	48	yes	complemented
57	C₂³×Dic₅₃	C₂³	1	6	yes	—
58	C₂²×C₄×D₅₃	C₂³	1	3	yes	—
59	C₂²×C₄×D₅₃	C₂³	1	6	yes	—
60	#13133215	C₂³	1	24	yes	complemented
61	#13133214	C₂³	1	16	yes	complemented
62	#13133213	C₂³	1	24	yes	—
63	#13133213	C₂³	1	48	yes	complemented
64	#13133206	C₂³	1	48	yes	complemented
65	#13133205	C₂³	1	24	yes	complemented
66	#13133203	C₂³	1	96	yes	complemented
67	#13133190	C₂³	1	96	yes	complemented
68	#13133190	C₂³	1	96	yes	complemented
69	#13133189	C₂³	1	96	yes	complemented
70	#13133187	C₂³	1	96	yes	complemented
71	#13133187	C₂³	1	48	yes	complemented
72	#13133186	C₂³	1	16	yes	complemented
73	#13133185	C₂³	1	48	yes	complemented
74	#13133177	C₂³	1	96	yes	complemented
75	#13133174	C₂³	1	48	yes	complemented
76	#13133173	C₂³	1	48	yes	complemented
77	C₂×#11719919	C₂³	1	12	yes	—
78	C₂×C₂²⋊Dic₅₃	C₂³	1	6	yes	—
79	C₂×#11719921	C₂³	1	12	yes	—
80	C₂×#11719921	C₂³	1	12	yes	—
81	C₂×C₄×Dic₅₃	C₂³	1	3	yes	—
82	C₂×C₄×Dic₅₃	C₂³	1	6	yes	—
83	C₄²×D₅₃	C₂³	1	8	yes	complemented
84	C₂⁵×C₅₃	C₂³	1	2	yes	—
85	C₂²×Q₈×C₅₃	C₂³	1	3	yes	—
86	C₂²×D₄×C₅₃	C₂³	1	3	yes	—
87	C₂²×D₄×C₅₃	C₂³	1	6	yes	—
88	#124×C₅₃	C₂³	1	48	yes	complemented
89	#124×C₅₃	C₂³	1	24	yes	complemented
90	#120×C₅₃	C₂³	1	48	yes	complemented
91	C₂×C₂²⋊C₄×C₅₃	C₂³	1	6	yes	—
92	C₂×C₂²⋊C₄×C₅₃	C₂³	1	12	yes	—
93	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
94	C₂×C₄²×C₅₃	C₂³	1	2	yes	—
95	C₂⁴×C₅₃	—	4	12	no	—
96	C₂⁴×C₅₃	C₂⁴	1	6	yes	—
97	C₂⁴×C₅₃	C₂⁴	1	8	yes	—
98	C₂³×D₅₃	—	4	24	no	—
99	C₂³×D₅₃	—	4	12	no	—
100	C₂³×D₅₃	C₂⁴	1	6	yes	—
101	C₂³×D₅₃	C₂⁴	1	1	yes	characteristic
102	C₂³×D₅₃	C₂⁴	1	24	yes	—
103	C₂×Q₈×C₅₃	—	4	6	no	—
104	C₂×Q₈×C₅₃	C₂⁴	1	12	yes	—
105	C₂×D₄×C₅₃	—	4	12	no	—
106	C₂×D₄×C₅₃	—	4	6	no	—
107	C₂×D₄×C₅₃	C₂⁴	1	12	yes	—
108	C₂×D₄×C₅₃	C₂⁴	1	24	yes	—
109	C₂²×C₄×C₅₃	C₂⁴	1	3	yes	—
110	C₂²×C₄×C₅₃	C₂⁴	1	6	yes	—
111	C₂×C₂²⋊D₅₃	—	2	48	no	—
112	C₂×C₂²⋊D₅₃	—	4	96	no	—
113	C₂×C₂²⋊D₅₃	—	4	24	no	—
114	C₂×C₂²⋊D₅₃	—	4	12	no	—
115	C₂×C₂²⋊D₅₃	C₂⁴	1	48	yes	—
116	C₂×C₂²⋊D₅₃	C₂⁴	1	48	yes	—
117	C₂×C₂²⋊D₅₃	C₂⁴	1	24	yes	—
118	C₂²×Dic₅₃	—	2	24	no	—
119	C₂²×Dic₅₃	C₂⁴	1	6	yes	—
120	C₂²×Dic₅₃	C₂⁴	1	24	yes	—
121	C₂²×Dic₅₃	C₂⁴	1	3	yes	—
122	#11719949	—	4	48	no	—
123	#11719947	—	4	48	no	—
124	D₄×D₅₃	—	4	96	no	—
125	C₂×D₂₁₂	—	4	12	no	—
126	C₂×D₂₁₂	—	4	6	no	—
127	C₂×D₂₁₂	—	4	12	no	—
128	C₂×D₂₁₂	C₂⁴	1	12	yes	—
129	C₂×D₂₁₂	C₂⁴	1	24	yes	—
130	C₂×D₂₁₂	C₂⁴	1	24	yes	—
131	C₂×C₄×D₅₃	—	2	24	no	—
132	C₂×C₄×D₅₃	—	2	12	no	—
133	C₂×C₄×D₅₃	C₂⁴	1	12	yes	—
134	C₂×C₄×D₅₃	C₂⁴	1	24	yes	—
135	C₂×Dic₁₀₆	—	4	6	no	—
136	C₂×Dic₁₀₆	C₂⁴	1	12	yes	—
137	C₂²⋊C₄×C₅₃	C₂⁴	1	24	yes	—
138	C₂²⋊C₄×C₅₃	C₂⁴	1	48	yes	—
139	C₄²×C₅₃	C₂⁴	1	4	yes	—
140	C₄²×C₅₃	C₂⁴	1	8	yes	—
141	C₂²⋊Dic₅₃	C₂⁴	1	24	yes	—
142	#11719921	C₂⁴	1	48	yes	—
143	#11719921	C₂⁴	1	48	yes	—
144	#11719919	C₂⁴	1	48	yes	—
145	C₄×Dic₅₃	C₂⁴	1	12	yes	—
146	C₄×Dic₅₃	C₂⁴	1	24	yes	—
147	C₂³×C₅₃	—	2	12	no	—
148	C₂³×C₅₃	—	2	24	no	—
149	C₂³×C₅₃	—	4	24	no	—
150	C₂³×C₅₃	—	8	16	no	—
151	C₂³×C₅₃	C₂⁵	1	12	yes	—
152	C₂³×C₅₃	C₂⁵	1	1	yes	characteristic
153	C₂²×D₅₃	—	2	6	no	—
154	C₂²×D₅₃	—	2	12	no	—
155	C₂²×D₅₃	—	2	24	no	—
156	C₂²×D₅₃	—	4	48	no	—
157	C₂²×D₅₃	—	4	24	no	—
158	C₂²×D₅₃	—	8	48	no	—
159	C₂²×D₅₃	C₂⁵	1	2	yes	—
160	C₂²×D₅₃	C₂⁵	1	12	yes	—
161	Q₈×C₅₃	—	4	12	no	—
162	D₄×C₅₃	—	4	24	no	—
163	D₄×C₅₃	—	4	12	no	—
164	C₂×C₄×C₅₃	—	2	6	no	—
165	C₂×C₄×C₅₃	—	2	12	no	—
166	C₂×C₄×C₅₃	C₂⁵	1	6	yes	—
167	C₂×C₄×C₅₃	C₂⁵	1	12	yes	—
168	C₂²⋊D₅₃	—	2	96	no	—
169	C₂²⋊D₅₃	—	4	24	no	—
170	C₂²⋊D₅₃	—	4	48	no	—
171	C₂×Dic₅₃	—	2	12	no	—
172	C₂×Dic₅₃	—	2	48	no	—
173	C₂×Dic₅₃	H(5,𝔽₂)	1	12	yes	—
174	C₂×Dic₅₃	C₂⁵	1	12	yes	—
175	C₂×Dic₅₃	C₂⁵	1	6	yes	—
176	D₂₁₂	—	4	24	no	—
177	D₂₁₂	—	4	12	no	—
178	D₂₁₂	—	4	24	no	—
179	C₄×D₅₃	—	2	48	no	—
180	C₄×D₅₃	—	2	24	no	—
181	Dic₁₀₆	—	4	12	no	—
182	C₂×#3549	—	53	1	no	maximal, Sylow
183	C₂²×C₅₃	—	2	12	no	—
184	C₂²×C₅₃	—	2	24	no	—
185	C₂²×C₅₃	—	4	24	no	—
186	C₂²×C₅₃	—	8	16	no	—
187	C₂²×C₅₃	C₂×H(5,𝔽₂)	1	6	yes	—
188	C₂²×C₅₃	C₂⁶	1	1	yes	characteristic
189	C₂×D₅₃	—	2	6	no	—
190	C₂×D₅₃	—	2	12	no	—
191	C₂×D₅₃	—	2	24	no	—
192	C₂×D₅₃	—	4	4	no	—
193	C₂×D₅₃	—	4	24	no	—
194	C₂×D₅₃	—	8	48	no	—
195	C₄×C₅₃	—	2	6	no	—
196	C₄×C₅₃	—	2	12	no	—
197	Dic₅₃	—	2	12	no	—
198	Dic₅₃	C₂×H(5,𝔽₂)	1	12	yes	—
199	#3549	—	53	32	no	—
200	C₂×H(3,𝔽₄)	—	53	3	no	—
201	C₂×H(3,𝔽₄)	D₅₃	1	1	yes	characteristic, complemented
202	C₂×#555	—	53	6	no	—
203	C₂×D₄²	—	53	6	no	—
204	C₂×#535	—	53	6	no	—
205	C₂×#535	—	53	6	no	—
206	C₂×#532	—	53	3	no	—
207	C₂×C₅₃	—	4	12	no	—
208	C₂×C₅₃	#3549	1	4	yes	complemented
209	C₂×C₅₃	C₂²×H(5,𝔽₂)	1	3	yes	—
210	D₅₃	—	4	2	no	—
211	D₅₃	—	4	12	no	—
212	C₂²×#41	—	53	3	no	—
213	C₂²×#41	—	53	3	no	—
214	C₂³×D₄	—	53	6	no	—
215	C₂³×D₄	—	53	6	no	—
216	H(3,𝔽₄)	—	53	48	no	—
217	H(3,𝔽₄)	C₂×D₅₃	1	16	yes	complemented
218	#555	—	53	96	no	—
219	D₄²	—	53	96	no	—
220	#535	—	53	96	no	—
221	#535	—	53	96	no	—
222	#532	—	53	48	no	—
223	C₂×C₄⋊Q₈	—	53	3	no	—
224	C₂×#127	—	53	3	no	—
225	C₂×#127	—	53	6	no	—
226	C₂×#127	—	53	2	no	—
227	C₂×#124	—	53	6	no	—
228	C₂×#124	—	53	12	no	—
229	C₂×#124	—	53	3	no	—
230	C₂×#124	C₂×D₅₃	1	3	yes	complemented
231	C₂×#124	C₂×D₅₃	1	6	yes	complemented
232	C₂×#121	—	53	12	no	—
233	C₂×#121	—	53	12	no	—
234	C₂×#121	—	53	12	no	—
235	C₂×#120	—	53	2	no	—
236	C₂×#120	—	53	6	no	—
237	C₂×#120	—	53	6	no	—
238	C₂×#120	—	53	12	no	—
239	C₂×#120	C₂×D₅₃	1	6	yes	complemented
240	C₂×C₄×D₄	—	53	12	no	—
241	C₂×#117	—	53	6	no	—
242	C₂×#117	—	53	6	no	—
243	C₂²×C₄²	—	53	1	no	—
244	C₅₃	C₂×#3549	1	1	yes	characteristic, complemented, Sylow
245	C₂⁵	—	53	6	no	—
246	C₂⁵	C₂²×D₅₃	1	2	yes	—
247	C₂×#41	—	53	24	no	—
248	C₂×#41	—	53	24	no	—
249	C₂×#41	—	212	12	no	—
250	C₂×#41	—	212	12	no	—
251	C₂²×Q₈	—	53	3	no	—
252	C₂²×Q₈	C₂²×D₅₃	1	3	yes	—
253	C₂²×D₄	—	53	12	no	—
254	C₂²×D₄	—	53	12	no	—
255	C₂²×D₄	—	53	6	no	—
256	C₂²×D₄	—	53	48	no	—
257	C₂²×D₄	—	53	48	no	—
258	C₂²×D₄	—	53	3	no	—
259	C₂²×D₄	—	53	6	no	—
260	C₂²×D₄	—	53	6	no	—
261	C₂²×D₄	—	212	24	no	—
262	C₂²×D₄	—	212	24	no	—
263	C₂²×D₄	C₂²×D₅₃	1	3	yes	—
264	C₂²×D₄	C₂²×D₅₃	1	6	yes	—
265	C₂³×C₄	—	53	6	no	—
266	C₂³×C₄	—	53	6	no	—
267	C₂³×C₄	—	53	3	no	—
268	C₄⋊Q₈	—	53	24	no	—
269	#127	—	53	24	no	—
270	#127	—	53	48	no	—
271	#127	—	53	16	no	—
272	#124	—	53	48	no	—
273	#124	—	53	96	no	—
274	#124	—	53	24	no	—
275	#124	C₂²×D₅₃	1	24	yes	complemented
276	#124	C₂²×D₅₃	1	48	yes	complemented
277	#121	—	53	96	no	—
278	#121	—	53	96	no	—
279	#121	—	53	96	no	—
280	#120	—	53	16	no	—
281	#120	—	53	48	no	—
282	#120	—	53	48	no	—
283	#120	—	53	96	no	—
284	#120	C₂²×D₅₃	1	48	yes	complemented
285	C₄×D₄	—	53	96	no	—
286	#117	—	53	48	no	—
287	#117	—	53	48	no	—
288	C₂×C₄⋊C₄	—	53	12	no	—
289	C₂×C₂²⋊C₄	—	53	6	no	—
290	C₂×C₂²⋊C₄	—	53	12	no	—
291	C₂×C₂²⋊C₄	—	53	12	no	—
292	C₂×C₂²⋊C₄	C₂²×D₅₃	1	6	yes	—
293	C₂×C₂²⋊C₄	C₂²×D₅₃	1	12	yes	—
294	C₂×C₄²	—	53	6	no	—
295	C₂×C₄²	—	53	3	no	—
296	C₂×C₄²	—	53	8	no	—
297	C₂×C₄²	C₂²×D₅₃	1	2	yes	—
298	C₂×C₄²	C₂²×D₅₃	1	1	yes	characteristic
299	C₂⁴	—	4	12	no	—
300	C₂⁴	—	53	6	no	—
301	C₂⁴	—	53	1	no	—
302	C₂⁴	—	53	24	no	—
303	C₂⁴	—	212	12	no	—
304	C₂⁴	—	212	24	no	—
305	C₂⁴	C₂³×D₅₃	1	6	yes	—
306	C₂⁴	C₂³×D₅₃	1	8	yes	—
307	#41	—	212	48	no	—
308	#41	—	212	48	no	—
309	C₂×Q₈	—	4	6	no	—
310	C₂×Q₈	—	53	12	no	—
311	C₂×Q₈	—	212	6	no	—
312	C₂×Q₈	C₂³×D₅₃	1	12	yes	—
313	C₂×D₄	—	4	12	no	—
314	C₂×D₄	—	4	6	no	—
315	C₂×D₄	—	53	48	no	—
316	C₂×D₄	—	53	48	no	—
317	C₂×D₄	—	53	24	no	—
318	C₂×D₄	—	53	12	no	—
319	C₂×D₄	—	53	24	no	—
320	C₂×D₄	—	53	24	no	—
321	C₂×D₄	—	106	48	no	—
322	C₂×D₄	—	212	12	no	—
323	C₂×D₄	—	212	12	no	—
324	C₂×D₄	—	212	6	no	—
325	C₂×D₄	—	212	96	no	—
326	C₂×D₄	—	212	96	no	—
327	C₂×D₄	—	212	24	no	—
328	C₂×D₄	—	212	12	no	—
329	C₂×D₄	C₂³×D₅₃	1	12	yes	—
330	C₂×D₄	C₂³×D₅₃	1	24	yes	—
331	C₂²×C₄	—	53	3	no	—
332	C₂²×C₄	—	53	24	no	—
333	C₂²×C₄	—	53	6	no	—
334	C₂²×C₄	—	53	12	no	—
335	C₂²×C₄	—	53	24	no	—
336	C₂²×C₄	—	106	24	no	—
337	C₂²×C₄	—	106	12	no	—
338	C₂²×C₄	—	106	24	no	—
339	C₂²×C₄	C₂³×D₅₃	1	3	yes	—
340	C₂²×C₄	C₂³×D₅₃	1	6	yes	—
341	C₄⋊C₄	—	53	48	no	—
342	C₂²⋊C₄	—	53	24	no	—
343	C₂²⋊C₄	—	53	48	no	—
344	C₂²⋊C₄	—	53	48	no	—
345	C₂²⋊C₄	C₂³×D₅₃	1	24	yes	—
346	C₂²⋊C₄	C₂³×D₅₃	1	48	yes	—
347	C₄²	—	53	12	no	—
348	C₄²	—	53	24	no	—
349	C₄²	C₂³×D₅₃	1	4	yes	—
350	C₄²	C₂³×D₅₃	1	8	yes	—
351	C₂³	—	2	12	no	—
352	C₂³	—	2	24	no	—
353	C₂³	—	4	24	no	—
354	C₂³	—	8	16	no	—
355	C₂³	—	53	12	no	—
356	C₂³	—	53	2	no	—
357	C₂³	—	106	6	no	—
358	C₂³	—	106	12	no	—
359	C₂³	—	106	24	no	—
360	C₂³	—	424	48	no	—
361	C₂³	—	212	24	no	—
362	C₂³	—	212	48	no	—
363	C₂³	C₂⁴×D₅₃	1	12	yes	—
364	C₂³	C₂⁴×D₅₃	1	1	yes	characteristic
365	Q₈	—	4	12	no	—
366	Q₈	—	212	12	no	—
367	D₄	—	4	24	no	—
368	D₄	—	4	12	no	—
369	D₄	—	106	96	no	—
370	D₄	—	212	24	no	—
371	D₄	—	212	24	no	—
372	D₄	—	212	12	no	—
373	D₄	—	212	48	no	—
374	D₄	—	212	24	no	—
375	C₂×C₄	—	2	12	no	—
376	C₂×C₄	—	2	6	no	—
377	C₂×C₄	—	53	12	no	—
378	C₂×C₄	—	53	6	no	—
379	C₂×C₄	—	53	12	no	—
380	C₂×C₄	—	106	48	no	—
381	C₂×C₄	—	106	24	no	—
382	C₂×C₄	—	106	48	no	—
383	C₂×C₄	—	106	12	no	—
384	C₂×C₄	C₂⁴×D₅₃	1	6	yes	—
385	C₂×C₄	C₂⁴×D₅₃	1	12	yes	—
386	C₂²	—	2	12	no	—
387	C₂²	—	2	24	no	—
388	C₂²	—	4	24	no	—
389	C₂²	—	8	16	no	—
390	C₂²	—	106	6	no	—
391	C₂²	—	106	12	no	—
392	C₂²	—	106	24	no	—
393	C₂²	—	424	48	no	—
394	C₂²	—	212	24	no	—
395	C₂²	—	212	4	no	—
396	C₂²	H(5,𝔽₂)×D₅₃	1	6	yes	—
397	C₂²	C₂⁵×D₅₃	1	1	yes	characteristic
398	C₄	—	2	12	no	—
399	C₄	—	2	6	no	—
400	C₄	—	53	12	no	—
401	C₄	—	106	12	no	—
402	C₂	—	4	12	no	—
403	C₂	—	212	12	no	—
404	C₂	—	212	2	no	—
405	C₂	#20289119	1	4	yes	direct product factor
406	C₂	C₂×H(5,𝔽₂)×D₅₃	1	3	yes	—

	subgroup	G/H
center	C₂³	C₂⁴×D₅₃
commutator, cosocle	C₂²×C₅₃	C₂⁶
Frattini	C₂²	C₂⁵×D₅₃
Fitting	C₂×C₅₃×H(3,𝔽₄)	C₂
socle	C₂³×C₅₃	C₂⁵
hypercenter	C₂×H(3,𝔽₄)	D₅₃
hypocenter	C₅₃	C₂×#3549
LCS	G ▷ C₂²×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂³ ◁ C₂×H(3,𝔽₄)

Complements

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

N	H ≅ G/N	type
1. #20289119	C₂	direct (N × H)
2. C₂×#16811365	C₂	semidirect
3. C₂×#16811335	C₂	semidirect
4. C₂×#16811255	C₂	semidirect
5. C₂×#16811191	C₂	semidirect
6. C₂×#16811188	C₂	semidirect
7. C₂×#16811172	C₂	semidirect
8. C₂×C₅₃×H(3,𝔽₄)	C₂	semidirect
9. C₂²×#11719949	C₂²	semidirect
10. C₂²×#11719947	C₂²	semidirect
11. C₂³×C₂²⋊D₅₃	C₂²	semidirect
12. C₂²×D₄×D₅₃	C₂²	semidirect
13. #16811365	C₂²	semidirect
14. #16811335	C₂²	semidirect
15. #16811255	C₂²	semidirect
16. #16811191	C₂²	semidirect
17. #16811188	C₂²	semidirect
18. #16811172	C₂²	semidirect
19. C₂×#13133215	C₂²	semidirect
20. C₂×#13133214	C₂²	semidirect
22. C₂×#13133213	C₂²	semidirect
23. C₂×#13133206	C₂²	semidirect
24. C₂×#13133205	C₂²	semidirect
25. C₂×#13133203	C₂²	semidirect
26. C₂×#13133190	C₂²	semidirect
27. C₂×#13133190	C₂²	semidirect
28. C₂×#13133189	C₂²	semidirect
29. C₂×#13133187	C₂²	semidirect
30. C₂×#13133187	C₂²	semidirect
31. C₂×#13133186	C₂²	semidirect
32. C₂×#13133185	C₂²	semidirect
33. C₂×#13133177	C₂²	semidirect
34. C₂×#13133174	C₂²	semidirect
35. C₂×#13133173	C₂²	semidirect
36. C₂×C₄²×D₅₃	C₂²	semidirect
37. C₅₃×H(3,𝔽₄)	C₂²	semidirect
38. C₂×#124×C₅₃	C₂²	semidirect
39. C₂×#124×C₅₃	C₂²	semidirect
40. C₂×#120×C₅₃	C₂²	semidirect
43. C₂×#11719949	C₂³	semidirect
45. C₂×#11719947	C₂³	semidirect
54. C₂²×C₂²⋊D₅₃	C₂³	semidirect
56. C₂×D₄×D₅₃	C₂³	semidirect
60. #13133215	C₂³	semidirect
61. #13133214	C₂³	semidirect
63. #13133213	C₂³	semidirect
64. #13133206	C₂³	semidirect
65. #13133205	C₂³	semidirect
66. #13133203	C₂³	semidirect
67. #13133190	C₂³	semidirect
68. #13133190	C₂³	semidirect
69. #13133189	C₂³	semidirect
70. #13133187	C₂³	semidirect
71. #13133187	C₂³	semidirect
72. #13133186	C₂³	semidirect
73. #13133185	C₂³	semidirect
74. #13133177	C₂³	semidirect
75. #13133174	C₂³	semidirect
76. #13133173	C₂³	semidirect
83. C₄²×D₅₃	C₂³	semidirect
88. #124×C₅₃	C₂³	semidirect
89. #124×C₅₃	C₂³	semidirect
90. #120×C₅₃	C₂³	semidirect
201. C₂×H(3,𝔽₄)	D₅₃	semidirect
208. C₂×C₅₃	#3549	semidirect
217. H(3,𝔽₄)	C₂×D₅₃	semidirect
230. C₂×#124	C₂×D₅₃	semidirect
231. C₂×#124	C₂×D₅₃	semidirect
239. C₂×#120	C₂×D₅₃	semidirect
244. C₅₃	C₂×#3549	semidirect
275. #124	C₂²×D₅₃	semidirect
276. #124	C₂²×D₅₃	semidirect
284. #120	C₂²×D₅₃	semidirect
405. C₂	#20289119	direct (N × H)

Maximal subgroups

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

subgroup	index
#20289119	2
C₂×#16811365	2
C₂×#16811335	2
C₂×#16811255	2
C₂×#16811191	2
C₂×#16811188	2
C₂×#16811172	2
C₂×C₅₃×H(3,𝔽₄)	2
C₂×#3549	53

Subgroups — C2×#20289119 #28291893

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#20289119 #28291893