Subgroups — D₄×#13133291 #28328328

Order 13568.

734 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×#13133291	C₂	1	2	yes	maximal, complemented
2	#20291099	C₂	1	2	yes	maximal, complemented
3	#20291095	C₂	1	4	yes	maximal, complemented
4	D₄×#11719910	C₂	1	2	yes	maximal, complemented
5	#20291047	C₂	1	1	yes	maximal, characteristic, complemented
6	#20290942	C₂	1	2	yes	maximal, complemented
7	C₄×#13133291	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×D₄×#88322	C₂	1	1	yes	maximal, characteristic
9	C₂×#13133291	—	2	8	no	—
10	C₂×#13133291	C₂²	1	2	yes	—
11	C₂×#13133291	C₂²	1	1	yes	characteristic
12	C₂×#13133291	C₂²	1	1	yes	characteristic
13	C₂×#13133291	C₂²	1	4	yes	complemented
14	C₂²×#11719910	C₂²	1	4	yes	complemented
15	C₄⋊#11719910	C₂²	1	2	yes	complemented
16	C₄⋊#11719910	C₂²	1	2	yes	complemented
17	C₂²⋊#11719910	C₂²	1	4	yes	complemented
18	C₂²⋊#11719910	C₂²	1	4	yes	complemented
19	#16811511	C₂²	1	2	yes	complemented
20	C₄×#11719910	C₂²	1	2	yes	complemented
21	C₂³×#88322	C₂²	1	2	yes	—
22	D₄×#88322	C₄	1	2	yes	—
23	D₄×#88322	C₄	1	1	yes	characteristic
24	D₄×#88322	C₄	1	1	yes	characteristic
25	D₄×#88322	C₂²	1	1	yes	characteristic
26	D₄×#88322	C₂²	1	2	yes	—
27	D₄×#88322	C₂²	1	1	yes	characteristic
28	C₂×C₄⋊#88322	C₄	1	1	yes	characteristic
29	C₂×C₂²⋊#88322	C₄	1	2	yes	—
30	C₂×C₄×#88322	C₂²	1	1	yes	characteristic
31	C₂×C₄×D₄×C₅₃	C₄	1	1	yes	characteristic
32	#13133291	—	2	4	no	—
33	#13133291	—	2	2	no	—
34	#13133291	—	2	8	no	—
35	#13133291	D₄	1	4	yes	direct product factor
36	C₂×#11719910	—	2	8	no	—
37	C₂×#11719910	C₂³	1	4	yes	—
38	C₂×#11719910	C₂³	1	2	yes	—
39	C₂×#11719910	C₂³	1	2	yes	—
40	C₂²×#88322	—	2	4	no	—
41	C₂²×#88322	—	2	4	no	—
42	C₂²×#88322	C₂×C₄	1	1	yes	characteristic
43	C₂²×#88322	C₂×C₄	1	2	yes	—
44	C₂²×#88322	C₂×C₄	1	2	yes	—
45	C₂²×#88322	C₂×C₄	1	2	yes	—
46	C₂²×#88322	C₂³	1	2	yes	—
47	C₂²×#88322	C₂³	1	2	yes	—
48	C₂²×#88322	C₂³	1	1	yes	characteristic
49	C₄⋊#88322	C₂×C₄	1	1	yes	characteristic
50	C₄⋊#88322	C₂×C₄	1	2	yes	—
51	C₄⋊#88322	C₂×C₄	1	1	yes	characteristic
52	C₂²⋊#88322	C₂×C₄	1	2	yes	—
53	C₂²⋊#88322	C₂×C₄	1	2	yes	—
54	C₂²⋊#88322	C₂×C₄	1	2	yes	—
55	C₂²⋊#88322	C₂×C₄	1	2	yes	—
56	C₄×#88322	C₂×C₄	1	1	yes	characteristic
57	C₄×#88322	C₂×C₄	1	1	yes	characteristic
58	C₄×#88322	C₂³	1	1	yes	characteristic
59	C₄×#88322	C₂³	1	1	yes	characteristic
60	C₂²×D₄×C₅₃	C₈	1	1	yes	characteristic
61	C₂³×C₄×C₅₃	C₂×C₄	1	2	yes	—
62	C₄×D₄×C₅₃	C₈	1	1	yes	characteristic
63	C₄×D₄×C₅₃	C₈	1	2	yes	—
64	C₄×D₄×C₅₃	C₈	1	1	yes	characteristic
65	C₄×D₄×C₅₃	C₂×C₄	1	2	yes	—
66	C₄×D₄×C₅₃	C₂×C₄	1	1	yes	characteristic
67	C₄×D₄×C₅₃	C₂×C₄	1	1	yes	characteristic
68	C₂×C₄⋊C₄×C₅₃	C₈	1	1	yes	characteristic
69	C₂×C₂²⋊C₄×C₅₃	C₈	1	2	yes	—
70	C₂×C₄²×C₅₃	C₂×C₄	1	1	yes	characteristic
71	C₂⁴×C₅₃	C₂×C₈	1	2	yes	—
72	C₂×D₄×C₅₃	—	2	4	no	—
73	C₂×D₄×C₅₃	C₂×C₈	1	2	yes	—
74	C₂×D₄×C₅₃	C₂×C₈	1	1	yes	characteristic
75	C₂×D₄×C₅₃	C₂×C₈	1	1	yes	characteristic
76	C₂²×C₄×C₅₃	—	2	4	no	—
77	C₂²×C₄×C₅₃	—	2	4	no	—
78	C₂²×C₄×C₅₃	C₂×C₈	1	1	yes	characteristic
79	C₂²×C₄×C₅₃	C₂×C₈	1	2	yes	—
80	C₂²×C₄×C₅₃	C₂×C₈	1	2	yes	—
81	C₂²×C₄×C₅₃	C₂×C₈	1	2	yes	—
82	C₂²×C₄×C₅₃	C₂×C₈	1	1	yes	characteristic
83	C₂²×C₄×C₅₃	C₂²×C₄	1	2	yes	—
84	C₂²×C₄×C₅₃	C₂²×C₄	1	2	yes	—
85	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
86	C₄⋊C₄×C₅₃	C₂×C₈	1	2	yes	—
87	C₄⋊C₄×C₅₃	C₂×C₈	1	1	yes	characteristic
88	C₄⋊C₄×C₅₃	C₂×C₈	1	1	yes	characteristic
89	C₂²⋊C₄×C₅₃	C₂×C₈	1	2	yes	—
90	C₂²⋊C₄×C₅₃	C₂×C₈	1	2	yes	—
91	C₂²⋊C₄×C₅₃	C₂×C₈	1	2	yes	—
92	C₂²⋊C₄×C₅₃	C₂×C₈	1	2	yes	—
93	C₄²×C₅₃	C₂×C₈	1	1	yes	characteristic
94	C₄²×C₅₃	C₂×C₈	1	1	yes	characteristic
95	C₄²×C₅₃	C₂²×C₄	1	1	yes	characteristic
96	C₄²×C₅₃	C₂²×C₄	1	1	yes	characteristic
97	C₂×#88322	—	2	4	no	—
98	C₂×#88322	—	2	2	no	—
99	C₂×#88322	—	2	4	no	—
100	C₂×#88322	—	2	4	no	—
101	C₂×#88322	—	2	2	no	—
102	C₂×#88322	—	2	2	no	—
103	C₂×#88322	—	2	2	no	—
104	C₂×#88322	—	2	2	no	—
105	C₂×#88322	C₂²×C₄	1	1	yes	characteristic
106	C₂×#88322	C₂²×C₄	1	1	yes	characteristic
107	C₂×#88322	C₂²×C₄	1	2	yes	—
108	C₂×#88322	C₂²×C₄	1	2	yes	—
109	C₂×#88322	C₂²×C₄	1	1	yes	characteristic
110	C₂×#88322	C₂×D₄	1	2	yes	—
111	C₂×#88322	#41	1	2	yes	—
112	C₂×#88322	C₂⁴	1	1	yes	characteristic
113	#11719910	—	2	4	no	—
114	#11719910	—	2	2	no	—
115	#11719910	C₂×D₄	1	4	yes	complemented
116	C₂³×C₅₃	—	2	4	no	—
117	C₂³×C₅₃	—	2	4	no	—
118	C₂³×C₅₃	—	4	2	no	—
119	C₂³×C₅₃	C₂²×C₈	1	2	yes	—
120	C₂³×C₅₃	C₂²×C₈	1	1	yes	characteristic
121	C₂³×C₅₃	C₂²×C₈	1	2	yes	—
122	D₄×C₅₃	—	2	4	no	—
123	D₄×C₅₃	—	2	2	no	—
124	D₄×C₅₃	M₃₂	1	4	yes	complemented
125	C₂×C₄×C₅₃	—	2	4	no	—
126	C₂×C₄×C₅₃	—	2	2	no	—
127	C₂×C₄×C₅₃	—	2	2	no	—
128	C₂×C₄×C₅₃	—	2	2	no	—
129	C₂×C₄×C₅₃	—	2	2	no	—
130	C₂×C₄×C₅₃	—	2	4	no	—
131	C₂×C₄×C₅₃	—	2	4	no	—
132	C₂×C₄×C₅₃	—	2	2	no	—
133	C₂×C₄×C₅₃	—	4	1	no	—
134	C₂×C₄×C₅₃	C₄×D₄	1	2	yes	—
135	C₂×C₄×C₅₃	C₂²×C₈	1	2	yes	—
136	C₂×C₄×C₅₃	C₂²×C₈	1	2	yes	—
137	C₂×C₄×C₅₃	C₂²×C₈	1	1	yes	characteristic
138	C₂×C₄×C₅₃	C₂²×C₈	1	1	yes	characteristic
139	C₂×C₄×C₅₃	C₂²×C₈	1	1	yes	characteristic
140	C₂×C₄×C₅₃	C₂²×C₈	1	1	yes	characteristic
141	C₂×C₄×C₅₃	C₂²×C₈	1	1	yes	characteristic
142	C₂×C₄×C₅₃	#131	1	2	yes	—
143	C₂×C₄×C₅₃	C₂³×C₄	1	1	yes	characteristic
144	#88322	—	2	2	no	—
145	#88322	—	2	2	no	—
146	#88322	—	2	1	no	—
147	#88322	—	2	1	no	—
148	#88322	C₄×D₄	1	2	yes	—
149	#88322	C₂²×D₄	1	1	yes	characteristic
150	#88322	C₂×#41	1	1	yes	characteristic
151	D₄×M₃₂	—	53	1	no	maximal, Sylow
152	C₂²×C₅₃	—	2	1	no	—
153	C₂²×C₅₃	—	2	1	no	—
154	C₂²×C₅₃	—	2	2	no	—
155	C₂²×C₅₃	—	2	2	no	—
156	C₂²×C₅₃	—	2	2	no	—
157	C₂²×C₅₃	—	2	2	no	—
158	C₂²×C₅₃	—	2	2	no	—
159	C₂²×C₅₃	—	4	8	no	—
160	C₂²×C₅₃	C₈×D₄	1	2	yes	—
161	C₂²×C₅₃	C₂×M₃₂	1	4	yes	complemented
162	C₂²×C₅₃	C₂³×C₈	1	1	yes	characteristic
163	C₄×C₅₃	—	2	1	no	—
164	C₄×C₅₃	—	2	1	no	—
165	C₄×C₅₃	—	2	1	no	—
166	C₄×C₅₃	—	2	2	no	—
167	C₄×C₅₃	—	2	2	no	—
168	C₄×C₅₃	C₈×D₄	1	2	yes	—
169	C₄×C₅₃	C₂×M₃₂	1	2	yes	complemented
170	C₄×C₅₃	C₂×C₄×D₄	1	1	yes	characteristic
171	C₄×C₅₃	C₂×#131	1	1	yes	characteristic
172	C₂²×M₃₂	—	53	2	no	—
173	C₂×C₈×D₄	—	53	1	no	—
174	#2169	—	53	2	no	—
175	#2168	—	53	4	no	—
176	D₄×C₁₆	—	53	2	no	—
177	#2150	—	53	1	no	—
178	#2112	—	53	2	no	—
179	C₄×M₃₂	—	53	1	no	—
180	C₂×C₅₃	—	2	2	no	—
181	C₂×C₅₃	—	2	4	no	—
182	C₂×C₅₃	—	4	2	no	—
183	C₂×C₅₃	C₂×C₈×D₄	1	1	yes	characteristic
184	C₂×C₅₃	C₂²×M₃₂	1	1	yes	characteristic
185	C₂×C₅₃	#3407	1	1	yes	characteristic
186	C₂³×C₈	—	53	2	no	—
187	C₂×C₄×D₄	#6157	1	1	yes	characteristic
188	C₂×M₃₂	—	53	1	no	—
189	C₂×M₃₂	—	53	1	no	—
190	C₂×M₃₂	—	53	2	no	—
191	C₂×M₃₂	—	53	4	no	—
192	C₂×M₃₂	—	106	8	no	—
193	C₂²×C₁₆	—	53	4	no	—
194	C₈×D₄	—	53	1	no	—
195	C₈×D₄	—	53	2	no	—
196	C₈×D₄	—	53	1	no	—
197	C₈×D₄	—	53	2	no	—
198	C₈×D₄	—	53	1	no	—
199	C₈×D₄	—	53	1	no	—
200	C₂×C₄⋊C₈	—	53	1	no	—
201	C₂×C₂²⋊C₈	—	53	2	no	—
202	C₂×C₄×C₈	—	53	1	no	—
203	C₄⋊C₁₆	—	53	2	no	—
204	C₄⋊C₁₆	—	53	2	no	—
205	C₂²⋊C₁₆	—	53	4	no	—
206	C₂²⋊C₁₆	—	53	4	no	—
207	#346	—	53	2	no	—
208	C₄×C₁₆	—	53	2	no	—
209	C₅₃	D₄×M₃₂	1	1	yes	characteristic, complemented, Sylow
210	C₂²×D₄	#88324	1	1	yes	characteristic
211	C₂³×C₄	C₂×#6157	1	2	yes	—
212	C₂²×C₈	—	53	1	no	—
213	C₂²×C₈	—	53	2	no	—
214	C₂²×C₈	—	53	2	no	—
215	C₂²×C₈	—	53	2	no	—
216	C₂²×C₈	—	53	2	no	—
217	C₂²×C₈	—	53	2	no	—
218	C₂²×C₈	—	53	1	no	—
219	C₂²×C₈	—	106	4	no	—
220	C₂²×C₈	—	106	4	no	—
221	C₄×D₄	#88324	1	2	yes	—
222	C₄×D₄	#88324	1	1	yes	characteristic
223	C₄×D₄	#88324	1	1	yes	characteristic
224	C₄×D₄	C₂×#6157	1	2	yes	—
225	C₄×D₄	C₂×#6157	1	1	yes	characteristic
226	C₄×D₄	C₂×#6157	1	1	yes	characteristic
227	C₂×C₄⋊C₄	#88324	1	1	yes	characteristic
228	C₂×C₂²⋊C₄	#88324	1	2	yes	—
229	C₂×C₄²	C₂×#6157	1	1	yes	characteristic
230	M₃₂	—	53	4	no	—
231	M₃₂	—	106	8	no	—
232	M₃₂	—	106	2	no	—
233	M₃₂	—	106	4	no	—
234	C₂×C₁₆	—	53	2	no	—
235	C₂×C₁₆	—	53	2	no	—
236	C₂×C₁₆	—	53	4	no	—
237	C₂×C₁₆	—	106	8	no	—
238	C₄⋊C₈	—	53	1	no	—
239	C₄⋊C₈	—	53	2	no	—
240	C₄⋊C₈	—	53	1	no	—
241	C₂²⋊C₈	—	53	2	no	—
242	C₂²⋊C₈	—	53	2	no	—
243	C₂²⋊C₈	—	53	2	no	—
244	C₂²⋊C₈	—	53	2	no	—
245	C₄×C₈	—	53	1	no	—
246	C₄×C₈	—	53	1	no	—
247	C₄×C₈	—	53	1	no	—
248	C₄×C₈	—	53	1	no	—
249	C₂⁴	C₂×#88324	1	2	yes	—
250	C₂×D₄	—	2	4	no	—
251	C₂×D₄	C₂×#88324	1	2	yes	—
252	C₂×D₄	C₂×#88324	1	1	yes	characteristic
253	C₂×D₄	C₂×#88324	1	1	yes	characteristic
254	C₂²×C₄	—	2	4	no	—
255	C₂²×C₄	—	2	4	no	—
256	C₂²×C₄	C₂×#88324	1	1	yes	characteristic
257	C₂²×C₄	C₂×#88324	1	2	yes	—
258	C₂²×C₄	C₂×#88324	1	2	yes	—
259	C₂²×C₄	C₂×#88324	1	2	yes	—
260	C₂²×C₄	C₂×#88324	1	1	yes	characteristic
261	C₂²×C₄	C₂²×#6157	1	2	yes	—
262	C₂²×C₄	C₂²×#6157	1	2	yes	—
263	C₂²×C₄	C₂²×#6157	1	1	yes	characteristic
264	C₂×C₈	—	53	2	no	—
265	C₂×C₈	—	53	2	no	—
266	C₂×C₈	—	53	1	no	—
267	C₂×C₈	—	53	1	no	—
268	C₂×C₈	—	53	1	no	—
269	C₂×C₈	—	53	1	no	—
270	C₂×C₈	—	53	2	no	—
271	C₂×C₈	—	53	2	no	—
272	C₂×C₈	—	106	2	no	—
273	C₂×C₈	—	106	2	no	—
274	C₂×C₈	—	106	4	no	—
275	C₂×C₈	—	106	4	no	—
276	C₂×C₈	—	106	2	no	—
277	C₂×C₈	—	106	2	no	—
278	C₂×C₈	—	106	4	no	—
279	C₂×C₈	—	106	2	no	—
280	C₄⋊C₄	C₂×#88324	1	2	yes	—
281	C₄⋊C₄	C₂×#88324	1	1	yes	characteristic
282	C₄⋊C₄	C₂×#88324	1	1	yes	characteristic
283	C₂²⋊C₄	C₂×#88324	1	2	yes	—
284	C₂²⋊C₄	C₂×#88324	1	2	yes	—
285	C₂²⋊C₄	C₂×#88324	1	2	yes	—
286	C₂²⋊C₄	C₂×#88324	1	2	yes	—
287	C₄²	C₂×#88324	1	1	yes	characteristic
288	C₄²	C₂×#88324	1	1	yes	characteristic
289	C₄²	C₂²×#6157	1	1	yes	characteristic
290	C₄²	C₂²×#6157	1	1	yes	characteristic
291	C₁₆	—	53	4	no	—
292	C₁₆	—	106	4	no	—
293	C₁₆	—	106	2	no	—
294	C₂³	—	2	4	no	—
295	C₂³	—	2	4	no	—
296	C₂³	—	4	2	no	—
297	C₂³	C₂²×#88324	1	2	yes	—
298	C₂³	C₂²×#88324	1	1	yes	characteristic
299	C₂³	C₂²×#88324	1	2	yes	—
300	D₄	—	2	4	no	—
301	D₄	—	2	2	no	—
302	D₄	#13133291	1	4	yes	direct product factor
303	C₂×C₄	—	2	4	no	—
304	C₂×C₄	—	2	4	no	—
305	C₂×C₄	—	2	4	no	—
306	C₂×C₄	—	2	2	no	—
307	C₂×C₄	—	2	2	no	—
308	C₂×C₄	—	2	2	no	—
309	C₂×C₄	—	2	2	no	—
310	C₂×C₄	—	2	2	no	—
311	C₂×C₄	—	4	1	no	—
312	C₂×C₄	D₄×#6157	1	2	yes	—
313	C₂×C₄	C₂²×#88324	1	1	yes	characteristic
314	C₂×C₄	C₂²×#88324	1	1	yes	characteristic
315	C₂×C₄	C₂²×#88324	1	1	yes	characteristic
316	C₂×C₄	C₂²×#88324	1	2	yes	—
317	C₂×C₄	C₂²×#88324	1	2	yes	—
318	C₂×C₄	C₂²×#88324	1	1	yes	characteristic
319	C₂×C₄	C₂²×#88324	1	1	yes	characteristic
320	C₂×C₄	#13133304	1	2	yes	—
321	C₂×C₄	C₂³×#6157	1	1	yes	characteristic
322	C₈	—	53	2	no	—
323	C₈	—	53	1	no	—
324	C₈	—	53	1	no	—
325	C₈	—	106	2	no	—
326	C₈	—	106	2	no	—
327	C₈	—	106	1	no	—
328	C₈	—	106	1	no	—
329	C₂²	—	2	2	no	—
330	C₂²	—	2	1	no	—
331	C₂²	—	2	1	no	—
332	C₂²	—	2	2	no	—
333	C₂²	—	2	2	no	—
334	C₂²	—	2	2	no	—
335	C₂²	—	2	2	no	—
336	C₂²	—	4	8	no	—
337	C₂²	D₄×#88324	1	2	yes	—
338	C₂²	C₂×#13133291	1	4	yes	complemented
339	C₂²	C₂³×#88324	1	1	yes	characteristic
340	C₄	—	2	2	no	—
341	C₄	—	2	2	no	—
342	C₄	—	2	1	no	—
343	C₄	—	2	1	no	—
344	C₄	—	2	1	no	—
345	C₄	D₄×#88324	1	2	yes	—
346	C₄	C₂×#13133291	1	2	yes	complemented
347	C₄	C₂×D₄×#6157	1	1	yes	characteristic
348	C₄	C₂×#13133304	1	1	yes	characteristic
349	C₂	—	2	2	no	—
350	C₂	—	2	4	no	—
351	C₂	—	4	2	no	—
352	C₂	C₂×D₄×#88324	1	1	yes	characteristic
353	C₂	C₂²×#13133291	1	1	yes	characteristic
354	C₂	#20291682	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂×C₄	C₂³×#6157
commutator, socle	C₂²×C₅₃	C₂³×C₈
Fitting	C₂×C₄×D₄×C₅₃	C₄
prime radical	C₂²×D₄×C₅₃	C₈
cosocle	C₂×#88322	C₂⁴
hypercenter	C₂×C₄×D₄	#6157
hypocenter	C₅₃	D₄×M₃₂
LCS	G ▷ C₂²×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄×D₄

Complements

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

N	H ≅ G/N	type
1. C₂²×#13133291	C₂	semidirect
2. #20291099	C₂	semidirect
3. #20291095	C₂	semidirect
4. D₄×#11719910	C₂	semidirect
5. #20291047	C₂	semidirect
6. #20290942	C₂	semidirect
7. C₄×#13133291	C₂	semidirect
13. C₂×#13133291	C₂²	semidirect
14. C₂²×#11719910	C₂²	semidirect
15. C₄⋊#11719910	C₂²	semidirect
16. C₄⋊#11719910	C₂²	semidirect
17. C₂²⋊#11719910	C₂²	semidirect
18. C₂²⋊#11719910	C₂²	semidirect
19. #16811511	C₂²	semidirect
20. C₄×#11719910	C₂²	semidirect
35. #13133291	D₄	direct (N × H)
115. #11719910	C₂×D₄	semidirect
124. D₄×C₅₃	M₃₂	semidirect
161. C₂²×C₅₃	C₂×M₃₂	semidirect
169. C₄×C₅₃	C₂×M₃₂	semidirect
209. C₅₃	D₄×M₃₂	semidirect
302. D₄	#13133291	direct (N × H)
338. C₂²	C₂×#13133291	semidirect
346. C₄	C₂×#13133291	semidirect

Maximal subgroups

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

subgroup	index
C₂²×#13133291	2
#20291099	2
#20291095	2
D₄×#11719910	2
#20291047	2
#20290942	2
C₄×#13133291	2
C₂×D₄×#88322	2
D₄×M₃₂	53

Subgroups — D4×#13133291 #28328328

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — D₄×#13133291 #28328328