Subgroups — #28328251

Order 13568.

600 conjugacy classes of subgroups.

The subgroup lattice has 338 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20291080	C₂	1	4	yes	maximal, complemented
2	#20291076	C₂	1	2	yes	maximal, complemented
3	#20291072	C₂	1	2	yes	maximal, complemented
4	#20291050	C₂	1	2	yes	maximal, complemented
5	#20290942	C₂	1	1	yes	maximal, characteristic
6	C₂×C₂²⋊#11719910	C₂	1	1	yes	maximal, characteristic
7	#20290924	C₂	1	2	yes	maximal, complemented
8	C₂×#16810579	C₂	1	1	yes	maximal, characteristic
9	C₂×#13133291	C₂²	1	2	yes	—
10	C₂²×#11719910	C₂²	1	2	yes	—
11	C₄⋊#11719910	C₂²	1	4	yes	complemented
12	C₄⋊#11719910	C₂²	1	4	yes	complemented
13	C₂²⋊#11719910	C₂²	1	1	yes	characteristic
14	C₂²⋊#11719910	C₂²	1	2	yes	—
15	C₂²⋊#11719910	C₂²	1	1	yes	characteristic
16	C₂²⋊#11719910	C₂²	1	2	yes	—
17	C₂²⋊#11719910	C₂²	1	2	yes	—
18	#16811511	C₂²	1	4	yes	complemented
19	C₄×#11719910	C₂²	1	4	yes	complemented
20	C₂²×#11719917	C₄	1	1	yes	characteristic
21	C₂³×#88322	C₂²	1	1	yes	characteristic
22	#16810579	C₄	1	2	yes	—
23	#16810579	C₄	1	1	yes	characteristic
24	#16810579	C₄	1	1	yes	characteristic
25	#16810579	C₂²	1	2	yes	—
26	#16810579	C₂²	1	1	yes	characteristic
27	#16810579	C₂²	1	1	yes	characteristic
28	C₂×#13133128	C₄	1	2	yes	—
29	C₂×C₄×#88322	C₂²	1	2	yes	—
30	C₂×#117×C₅₃	C₄	1	1	yes	characteristic
31	#13133291	—	2	4	no	—
32	C₂×#11719910	—	2	4	no	—
33	C₂×#11719910	C₂³	1	2	yes	—
34	C₂×#11719910	C₂³	1	2	yes	—
35	C₂×#11719910	C₂³	1	4	yes	—
36	C₂×#11719917	—	2	4	no	—
37	C₂×#11719917	C₂×C₄	1	2	yes	—
38	C₂×#11719917	C₂×C₄	1	1	yes	characteristic
39	C₂×#11719917	C₂×C₄	1	1	yes	characteristic
40	C₂²×#88322	—	2	4	no	—
41	C₂²×#88322	C₂×C₄	1	2	yes	—
42	C₂²×#88322	C₂×C₄	1	1	yes	characteristic
43	C₂²×#88322	C₂×C₄	1	1	yes	characteristic
44	C₂²×#88322	C₂×C₄	1	1	yes	characteristic
45	C₂²×#88322	C₂³	1	1	yes	characteristic
46	C₂²×#88322	C₂³	1	1	yes	characteristic
47	C₂²×#88322	C₂³	1	1	yes	characteristic
48	#13133128	C₂×C₄	1	2	yes	—
49	#13133128	C₂×C₄	1	2	yes	—
50	#13133128	C₂×C₄	1	2	yes	—
51	#13133128	C₂×C₄	1	2	yes	—
52	C₄×#88322	C₂×C₄	1	2	yes	—
53	C₄×#88322	C₂×C₄	1	2	yes	—
54	C₄×#88322	C₂³	1	2	yes	—
55	C₄×#88322	C₂³	1	2	yes	—
56	C₂³×C₄×C₅₃	C₂×C₄	1	1	yes	characteristic
57	#117×C₅₃	C₈	1	2	yes	—
58	#117×C₅₃	C₈	1	1	yes	characteristic
59	#117×C₅₃	C₈	1	1	yes	characteristic
60	#117×C₅₃	C₂×C₄	1	2	yes	—
61	#117×C₅₃	C₂×C₄	1	1	yes	characteristic
62	#117×C₅₃	C₂×C₄	1	1	yes	characteristic
63	C₂×C₄⋊C₄×C₅₃	C₈	1	1	yes	characteristic
64	C₂×C₄⋊C₄×C₅₃	C₈	1	1	yes	characteristic
65	C₂×C₂²⋊C₄×C₅₃	C₈	1	2	yes	—
66	C₂×C₄²×C₅₃	C₂×C₄	1	2	yes	—
67	C₂⁴×C₅₃	C₂×C₈	1	1	yes	characteristic
68	C₂²×C₄×C₅₃	—	2	2	no	—
69	C₂²×C₄×C₅₃	—	2	2	no	—
70	C₂²×C₄×C₅₃	C₂×C₈	1	2	yes	—
71	C₂²×C₄×C₅₃	C₂×C₈	1	1	yes	characteristic
72	C₂²×C₄×C₅₃	C₂×C₈	1	1	yes	characteristic
73	C₂²×C₄×C₅₃	C₂×C₈	1	1	yes	characteristic
74	C₂²×C₄×C₅₃	C₂×C₈	1	2	yes	—
75	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
76	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
77	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
78	C₄⋊C₄×C₅₃	C₂×C₈	1	2	yes	—
79	C₄⋊C₄×C₅₃	C₂×C₈	1	2	yes	—
80	C₄⋊C₄×C₅₃	C₂×C₈	1	1	yes	characteristic
81	C₄⋊C₄×C₅₃	C₂×C₈	1	1	yes	characteristic
82	C₄⋊C₄×C₅₃	C₂×C₈	1	1	yes	characteristic
83	C₄⋊C₄×C₅₃	C₂×C₈	1	1	yes	characteristic
84	C₂²⋊C₄×C₅₃	C₂×C₈	1	2	yes	—
85	C₂²⋊C₄×C₅₃	C₂×C₈	1	2	yes	—
86	C₂²⋊C₄×C₅₃	C₂×C₈	1	2	yes	—
87	C₂²⋊C₄×C₅₃	C₂×C₈	1	2	yes	—
88	C₄²×C₅₃	C₂×C₈	1	2	yes	—
89	C₄²×C₅₃	C₂×C₈	1	2	yes	—
90	C₄²×C₅₃	C₂²×C₄	1	2	yes	—
91	C₄²×C₅₃	C₂²×C₄	1	2	yes	—
92	#11719917	—	2	2	no	—
93	#11719917	—	2	4	no	—
94	#11719917	M₁₆	1	4	yes	—
95	C₂×#88322	—	2	2	no	—
96	C₂×#88322	—	2	2	no	—
97	C₂×#88322	—	2	2	no	—
98	C₂×#88322	—	2	4	no	—
99	C₂×#88322	—	2	2	no	—
100	C₂×#88322	—	2	2	no	—
101	C₂×#88322	C₂²×C₄	1	1	yes	characteristic
102	C₂×#88322	C₂²×C₄	1	2	yes	—
103	C₂×#88322	C₂²×C₄	1	1	yes	characteristic
104	C₂×#88322	C₂²×C₄	1	1	yes	characteristic
105	C₂×#88322	C₂²×C₄	1	2	yes	—
106	C₂×#88322	#41	1	4	yes	—
107	C₂×#88322	C₂⁴	1	1	yes	characteristic
108	#11719910	—	2	2	no	—
109	#11719910	—	2	2	no	—
110	#11719910	—	2	4	no	—
111	C₂³×C₅₃	—	2	2	no	—
112	C₂³×C₅₃	—	2	2	no	—
113	C₂³×C₅₃	—	4	1	no	—
114	C₂³×C₅₃	C₂²×C₈	1	1	yes	characteristic
115	C₂³×C₅₃	C₂²×C₈	1	1	yes	characteristic
116	C₂³×C₅₃	C₂²×C₈	1	1	yes	characteristic
117	C₂×C₄×C₅₃	—	2	2	no	—
118	C₂×C₄×C₅₃	—	2	4	no	—
119	C₂×C₄×C₅₃	—	2	1	no	—
120	C₂×C₄×C₅₃	—	2	1	no	—
121	C₂×C₄×C₅₃	—	2	1	no	—
122	C₂×C₄×C₅₃	—	2	1	no	—
123	C₂×C₄×C₅₃	—	2	2	no	—
124	C₂×C₄×C₅₃	—	2	2	no	—
125	C₂×C₄×C₅₃	—	4	2	no	—
126	C₂×C₄×C₅₃	#117	1	2	yes	—
127	C₂×C₄×C₅₃	C₂²×C₈	1	1	yes	characteristic
128	C₂×C₄×C₅₃	C₂²×C₈	1	1	yes	characteristic
129	C₂×C₄×C₅₃	C₂²×C₈	1	2	yes	—
130	C₂×C₄×C₅₃	C₂²×C₈	1	2	yes	—
131	C₂×C₄×C₅₃	C₂²×C₈	1	2	yes	—
132	C₂×C₄×C₅₃	C₂²×C₈	1	2	yes	—
133	C₂×C₄×C₅₃	C₂²×C₈	1	1	yes	characteristic
134	C₂×C₄×C₅₃	C₂×M₁₆	1	2	yes	—
135	C₂×C₄×C₅₃	C₂³×C₄	1	1	yes	characteristic
136	#88322	—	2	1	no	—
137	#88322	—	2	1	no	—
138	#88322	—	2	2	no	—
139	#88322	#117	1	2	yes	—
140	#88322	C₂×M₁₆	1	4	yes	—
141	#88322	C₂×#41	1	2	yes	—
142	#33422	—	53	1	no	maximal, Sylow
143	C₂²×C₅₃	—	2	1	no	—
144	C₂²×C₅₃	—	2	1	no	—
145	C₂²×C₅₃	—	2	1	no	—
146	C₂²×C₅₃	—	2	1	no	—
147	C₂²×C₅₃	—	2	1	no	—
148	C₂²×C₅₃	—	2	1	no	—
149	C₂²×C₅₃	—	2	1	no	—
150	C₂²×C₅₃	—	4	4	no	—
151	C₂²×C₅₃	#431	1	2	yes	—
152	C₂²×C₅₃	#504	1	2	yes	—
153	C₂²×C₅₃	C₂³×C₈	1	1	yes	characteristic
154	C₄×C₅₃	—	2	2	no	—
155	C₄×C₅₃	—	2	2	no	—
156	C₄×C₅₃	—	2	2	no	—
157	C₄×C₅₃	—	2	2	no	—
158	C₄×C₅₃	—	2	1	no	—
159	C₄×C₅₃	—	2	1	no	—
160	C₄×C₅₃	#431	1	2	yes	—
161	C₄×C₅₃	C₂×#117	1	1	yes	characteristic
162	C₄×C₅₃	C₂²×M₁₆	1	1	yes	characteristic
163	C₂×#405	—	53	1	no	—
164	#2164	—	53	4	no	—
165	#2163	—	53	2	no	—
166	#2162	—	53	2	no	—
167	#2151	—	53	2	no	—
168	#2112	—	53	1	no	—
169	C₂×C₂²⋊C₁₆	—	53	1	no	—
170	#2108	—	53	2	no	—
171	C₂×C₅₃	—	2	2	no	—
172	C₂×C₅₃	—	2	2	no	—
173	C₂×C₅₃	—	4	1	no	—
174	C₂×C₅₃	C₂×#431	1	1	yes	characteristic
175	C₂×C₅₃	C₂×#504	1	1	yes	characteristic
176	C₂×C₅₃	#3407	1	1	yes	characteristic
177	C₂²×M₁₆	—	53	1	no	—
178	C₂³×C₈	—	53	1	no	—
179	C₂×#117	#6157	1	1	yes	characteristic
180	C₂×M₃₂	—	53	2	no	—
181	C₂²×C₁₆	—	53	2	no	—
182	#405	—	53	1	no	—
183	#405	—	53	1	no	—
184	#405	—	53	2	no	—
185	#405	—	53	1	no	—
186	#405	—	53	1	no	—
187	#405	—	53	2	no	—
188	C₂×#97	—	53	2	no	—
189	C₂×C₄×C₈	—	53	2	no	—
190	C₄⋊C₁₆	—	53	4	no	—
191	C₄⋊C₁₆	—	53	4	no	—
192	C₂²⋊C₁₆	—	53	1	no	—
193	C₂²⋊C₁₆	—	53	1	no	—
194	C₂²⋊C₁₆	—	53	2	no	—
195	C₂²⋊C₁₆	—	53	2	no	—
196	C₂²⋊C₁₆	—	53	2	no	—
197	#346	—	53	4	no	—
198	C₄×C₁₆	—	53	4	no	—
199	C₅₃	#33422	1	1	yes	characteristic, complemented, Sylow
200	C₂³×C₄	C₂×#6157	1	1	yes	characteristic
201	C₂×M₁₆	—	53	1	no	—
202	C₂×M₁₆	—	53	2	no	—
203	C₂×M₁₆	—	53	1	no	—
204	C₂×M₁₆	—	106	4	no	—
205	C₂²×C₈	—	53	1	no	—
206	C₂²×C₈	—	53	1	no	—
207	C₂²×C₈	—	53	1	no	—
208	C₂²×C₈	—	53	1	no	—
209	C₂²×C₈	—	53	1	no	—
210	C₂²×C₈	—	53	1	no	—
211	C₂²×C₈	—	53	2	no	—
212	C₂²×C₈	—	106	4	no	—
213	#117	#88324	1	1	yes	characteristic
214	#117	#88324	1	1	yes	characteristic
215	#117	#88324	1	2	yes	—
216	#117	C₂×#6157	1	2	yes	—
217	#117	C₂×#6157	1	1	yes	characteristic
218	#117	C₂×#6157	1	1	yes	characteristic
219	C₂×C₄⋊C₄	#88324	1	1	yes	characteristic
220	C₂×C₄⋊C₄	#88324	1	1	yes	characteristic
221	C₂×C₂²⋊C₄	#88324	1	2	yes	—
222	C₂×C₄²	C₂×#6157	1	2	yes	—
223	M₃₂	—	106	4	no	—
224	C₂×C₁₆	—	53	2	no	—
225	C₂×C₁₆	—	53	4	no	—
226	C₂×C₁₆	—	53	2	no	—
227	C₂×C₁₆	—	106	4	no	—
228	#97	—	53	2	no	—
229	#97	—	53	2	no	—
230	#97	—	53	2	no	—
231	#97	—	53	2	no	—
232	C₄×C₈	—	53	2	no	—
233	C₄×C₈	—	53	2	no	—
234	C₄×C₈	—	53	2	no	—
235	C₄×C₈	—	53	2	no	—
236	C₂⁴	C₂×#88324	1	1	yes	characteristic
237	C₂²×C₄	—	2	2	no	—
238	C₂²×C₄	—	2	2	no	—
239	C₂²×C₄	C₂×#88324	1	2	yes	—
240	C₂²×C₄	C₂×#88324	1	1	yes	characteristic
241	C₂²×C₄	C₂×#88324	1	1	yes	characteristic
242	C₂²×C₄	C₂×#88324	1	1	yes	characteristic
243	C₂²×C₄	C₂×#88324	1	2	yes	—
244	C₂²×C₄	C₂²×#6157	1	1	yes	characteristic
245	C₂²×C₄	C₂²×#6157	1	1	yes	characteristic
246	C₂²×C₄	C₂²×#6157	1	1	yes	characteristic
247	M₁₆	—	53	4	no	—
248	M₁₆	—	106	4	no	—
249	M₁₆	—	106	2	no	—
250	C₂×C₈	—	53	4	no	—
251	C₂×C₈	—	53	1	no	—
252	C₂×C₈	—	53	2	no	—
253	C₂×C₈	—	53	1	no	—
254	C₂×C₈	—	53	1	no	—
255	C₂×C₈	—	53	2	no	—
256	C₂×C₈	—	53	1	no	—
257	C₂×C₈	—	106	2	no	—
258	C₂×C₈	—	106	2	no	—
259	C₂×C₈	—	106	2	no	—
260	C₂×C₈	—	106	2	no	—
261	C₂×C₈	—	106	4	no	—
262	C₂×C₈	—	106	2	no	—
263	C₄⋊C₄	C₂×#88324	1	2	yes	—
264	C₄⋊C₄	C₂×#88324	1	2	yes	—
265	C₄⋊C₄	C₂×#88324	1	1	yes	characteristic
266	C₄⋊C₄	C₂×#88324	1	1	yes	characteristic
267	C₄⋊C₄	C₂×#88324	1	1	yes	characteristic
268	C₄⋊C₄	C₂×#88324	1	1	yes	characteristic
269	C₂²⋊C₄	C₂×#88324	1	2	yes	—
270	C₂²⋊C₄	C₂×#88324	1	2	yes	—
271	C₂²⋊C₄	C₂×#88324	1	2	yes	—
272	C₂²⋊C₄	C₂×#88324	1	2	yes	—
273	C₄²	C₂×#88324	1	2	yes	—
274	C₄²	C₂×#88324	1	2	yes	—
275	C₄²	C₂²×#6157	1	2	yes	—
276	C₄²	C₂²×#6157	1	2	yes	—
277	C₁₆	—	106	2	no	—
278	C₁₆	—	106	2	no	—
279	C₁₆	—	106	4	no	—
280	C₂³	—	2	2	no	—
281	C₂³	—	2	2	no	—
282	C₂³	—	4	1	no	—
283	C₂³	C₂²×#88324	1	1	yes	characteristic
284	C₂³	C₂²×#88324	1	1	yes	characteristic
285	C₂³	C₂²×#88324	1	1	yes	characteristic
286	C₂×C₄	—	2	2	no	—
287	C₂×C₄	—	2	4	no	—
288	C₂×C₄	—	2	2	no	—
289	C₂×C₄	—	2	2	no	—
290	C₂×C₄	—	2	1	no	—
291	C₂×C₄	—	2	1	no	—
292	C₂×C₄	—	2	1	no	—
293	C₂×C₄	—	2	1	no	—
294	C₂×C₄	—	4	2	no	—
295	C₂×C₄	#13133296	1	2	yes	—
296	C₂×C₄	C₂²×#88324	1	1	yes	characteristic
297	C₂×C₄	C₂²×#88324	1	2	yes	—
298	C₂×C₄	C₂²×#88324	1	2	yes	—
299	C₂×C₄	C₂²×#88324	1	2	yes	—
300	C₂×C₄	C₂²×#88324	1	2	yes	—
301	C₂×C₄	C₂²×#88324	1	1	yes	characteristic
302	C₂×C₄	C₂²×#88324	1	1	yes	characteristic
303	C₂×C₄	D₅₃⋊M₁₆	1	2	yes	—
304	C₂×C₄	C₂³×#6157	1	1	yes	characteristic
305	C₈	—	53	2	no	—
306	C₈	—	53	2	no	—
307	C₈	—	53	4	no	—
308	C₈	—	106	1	no	—
309	C₈	—	106	1	no	—
310	C₈	—	106	2	no	—
311	C₂²	—	2	1	no	—
312	C₂²	—	2	1	no	—
313	C₂²	—	2	1	no	—
314	C₂²	—	2	1	no	—
315	C₂²	—	2	1	no	—
316	C₂²	—	2	1	no	—
317	C₂²	—	2	1	no	—
318	C₂²	—	4	4	no	—
319	C₂²	#16811639	1	2	yes	—
320	C₂²	#16811681	1	2	yes	—
321	C₂²	C₂³×#88324	1	1	yes	characteristic
322	C₄	—	2	1	no	—
323	C₄	—	2	1	no	—
324	C₄	—	2	2	no	—
325	C₄	—	2	2	no	—
326	C₄	—	2	2	no	—
327	C₄	—	2	2	no	—
328	C₄	#16811639	1	2	yes	—
329	C₄	C₂×#13133296	1	1	yes	characteristic
330	C₄	C₂×D₅₃⋊M₁₆	1	1	yes	characteristic
331	C₂	—	2	2	no	—
332	C₂	—	2	2	no	—
333	C₂	—	4	1	no	—
334	C₂	C₂×#16811639	1	1	yes	characteristic
335	C₂	C₂×#16811681	1	1	yes	characteristic
336	C₂	#20291682	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂×C₄	C₂³×#6157
commutator, socle	C₂²×C₅₃	C₂³×C₈
Fitting	C₂×#117×C₅₃	C₄
prime radical	C₂⁴×C₅₃	C₂×C₈
cosocle	C₂×#88322	C₂⁴
hypercenter	C₂×#117	#6157
hypocenter	C₅₃	#33422
LCS	G ▷ C₂²×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂×#117

Complements

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

N	H ≅ G/N	type
1. #20291080	C₂	semidirect
2. #20291076	C₂	semidirect
3. #20291072	C₂	semidirect
4. #20291050	C₂	semidirect
7. #20290924	C₂	semidirect
11. C₄⋊#11719910	C₂²	semidirect
12. C₄⋊#11719910	C₂²	semidirect
18. #16811511	C₂²	semidirect
19. C₄×#11719910	C₂²	semidirect
199. C₅₃	#33422	semidirect

Maximal subgroups

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

subgroup	index
#20291080	2
#20291076	2
#20291072	2
#20291050	2
#20290942	2
C₂×C₂²⋊#11719910	2
#20290924	2
C₂×#16810579	2
#33422	53