Subgroups — C₂²×D₄×#88324 #28329418

Order 13568.

3694 conjugacy classes of subgroups.

The subgroup lattice has 264 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂⁴×#88324	C₂	1	2	yes	maximal, complemented
2	C₂×D₄×#88324	C₂	1	24	yes	maximal, direct product factor
3	C₂²×C₄⋊#88324	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂²×C₂²⋊#88324	C₂	1	2	yes	maximal, complemented
5	C₂²×C₄×#88324	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂²×D₄×Dic₅₃	C₂	1	1	yes	maximal, characteristic
7	C₂³×#88324	—	2	16	no	—
8	C₂³×#88324	C₂²	1	2	yes	—
9	C₂³×#88324	C₂²	1	1	yes	characteristic
10	C₂³×#88324	C₂²	1	1	yes	characteristic
11	C₂³×#88324	C₂²	1	24	yes	complemented
12	D₄×#88324	C₂²	1	64	yes	direct product factor
13	C₂×C₄⋊#88324	C₂²	1	12	yes	complemented
14	C₂×C₂²⋊#88324	C₂²	1	24	yes	complemented
15	C₂×C₄×#88324	C₂²	1	12	yes	complemented
16	C₂⁴×Dic₅₃	C₂²	1	2	yes	—
17	C₂×D₄×Dic₅₃	C₄	1	12	yes	—
18	C₂×D₄×Dic₅₃	C₂²	1	12	yes	—
19	C₂²×C₄⋊Dic₅₃	C₄	1	1	yes	characteristic
20	C₂²×C₂²⋊Dic₅₃	C₄	1	2	yes	—
21	C₂²×C₄×Dic₅₃	C₂²	1	1	yes	characteristic
22	C₂³×D₄×C₅₃	C₄	1	1	yes	characteristic
23	C₂²×#88324	—	2	8	no	—
24	C₂²×#88324	—	2	4	no	—
25	C₂²×#88324	—	2	96	no	—
26	C₂²×#88324	D₄	1	8	yes	direct product factor
27	C₂²×#88324	C₂³	1	6	yes	—
28	C₂²×#88324	C₂³	1	12	yes	—
29	C₂²×#88324	C₂³	1	6	yes	—
30	C₂²×#88324	C₂³	1	32	yes	complemented
31	C₄⋊#88324	C₂³	1	16	yes	complemented
32	C₂²⋊#88324	C₂³	1	32	yes	complemented
33	C₄×#88324	C₂³	1	16	yes	complemented
34	C₂³×Dic₅₃	—	2	8	no	—
35	C₂³×Dic₅₃	—	2	8	no	—
36	C₂³×Dic₅₃	C₂×C₄	1	1	yes	characteristic
37	C₂³×Dic₅₃	C₂×C₄	1	2	yes	—
38	C₂³×Dic₅₃	C₂×C₄	1	12	yes	—
39	C₂³×Dic₅₃	C₂³	1	12	yes	—
40	C₂³×Dic₅₃	C₂³	1	1	yes	characteristic
41	D₄×Dic₅₃	C₂×C₄	1	48	yes	—
42	D₄×Dic₅₃	C₂³	1	16	yes	—
43	C₂×C₄⋊Dic₅₃	C₂×C₄	1	12	yes	—
44	C₂×C₂²⋊Dic₅₃	C₂×C₄	1	24	yes	—
45	C₂×C₄×Dic₅₃	C₂×C₄	1	6	yes	—
46	C₂×C₄×Dic₅₃	C₂³	1	6	yes	—
47	C₂⁵×C₅₃	C₂×C₄	1	2	yes	—
48	C₂²×D₄×C₅₃	C₈	1	16	yes	complemented
49	C₂²×D₄×C₅₃	C₂×C₄	1	12	yes	—
50	C₂³×C₄×C₅₃	C₂×C₄	1	1	yes	characteristic
51	C₂⁴×C₅₃	—	2	8	no	—
52	C₂⁴×C₅₃	—	2	8	no	—
53	C₂⁴×C₅₃	C₂×C₈	1	16	yes	complemented
54	C₂⁴×C₅₃	C₂²×C₄	1	12	yes	—
55	C₂⁴×C₅₃	C₂²×C₄	1	1	yes	characteristic
56	C₂×D₄×C₅₃	C₂×C₈	1	96	yes	complemented
57	C₂×D₄×C₅₃	C₂²×C₄	1	16	yes	—
58	C₂²×C₄×C₅₃	C₂×C₈	1	8	yes	complemented
59	C₂²×C₄×C₅₃	C₂²×C₄	1	6	yes	—
60	C₂²×Dic₅₃	—	2	8	no	—
61	C₂²×Dic₅₃	—	2	4	no	—
62	C₂²×Dic₅₃	—	2	48	no	—
63	C₂²×Dic₅₃	—	2	24	no	—
64	C₂²×Dic₅₃	—	2	24	no	—
65	C₂²×Dic₅₃	C₂²×C₄	1	6	yes	—
66	C₂²×Dic₅₃	C₂²×C₄	1	12	yes	—
67	C₂²×Dic₅₃	C₂²×C₄	1	24	yes	—
68	C₂²×Dic₅₃	C₂²×C₄	1	3	yes	—
69	C₂²×Dic₅₃	C₂×D₄	1	4	yes	—
70	C₂²×Dic₅₃	#41	1	4	yes	—
71	C₂²×Dic₅₃	C₂⁴	1	8	yes	—
72	C₂²×Dic₅₃	C₂⁴	1	3	yes	—
73	C₂×#88324	—	2	24	no	—
74	C₂×#88324	—	2	12	no	—
75	C₂×#88324	—	2	64	no	—
76	C₂×#88324	C₂×D₄	1	24	yes	direct product factor
77	C₂×#88324	C₂⁴	1	4	yes	—
78	C₂×#88324	C₂⁴	1	8	yes	—
79	C₂×#88324	C₂⁴	1	4	yes	—
80	C₂²⋊Dic₅₃	C₂²×C₄	1	32	yes	—
81	C₄⋊Dic₅₃	C₂²×C₄	1	16	yes	—
82	C₄×Dic₅₃	C₂²×C₄	1	12	yes	—
83	C₄×Dic₅₃	C₂⁴	1	4	yes	—
84	C₂³×C₅₃	—	2	24	no	—
85	C₂³×C₅₃	—	2	24	no	—
86	C₂³×C₅₃	—	2	64	no	—
87	C₂³×C₅₃	C₄×D₄	1	4	yes	—
88	C₂³×C₅₃	C₂²×C₈	1	4	yes	—
89	C₂³×C₅₃	C₂²×C₈	1	48	yes	complemented
90	C₂³×C₅₃	#131	1	4	yes	—
91	C₂³×C₅₃	C₂³×C₄	1	8	yes	—
92	C₂³×C₅₃	C₂³×C₄	1	3	yes	—
93	D₄×C₅₃	C₂²×C₈	1	64	yes	complemented
94	C₂×C₄×C₅₃	C₂²×C₈	1	24	yes	complemented
95	C₂×C₄×C₅₃	C₂³×C₄	1	4	yes	—
96	C₂×Dic₅₃	—	2	24	no	—
97	C₂×Dic₅₃	—	2	12	no	—
98	C₂×Dic₅₃	—	2	48	no	—
99	C₂×Dic₅₃	—	2	8	no	—
100	C₂×Dic₅₃	—	2	8	no	—
101	C₂×Dic₅₃	C₄×D₄	1	12	yes	—
102	C₂×Dic₅₃	C₂³×C₄	1	8	yes	—
103	C₂×Dic₅₃	C₂³×C₄	1	4	yes	—
104	C₂×Dic₅₃	C₂³×C₄	1	3	yes	—
105	C₂×Dic₅₃	C₂²×D₄	1	6	yes	—
106	C₂×Dic₅₃	C₂×#41	1	6	yes	—
107	C₂×Dic₅₃	C₂⁵	1	1	yes	characteristic
108	#88324	—	2	8	no	—
109	#88324	—	2	4	no	—
110	#88324	C₂²×D₄	1	8	yes	direct product factor
111	C₂²×C₈×D₄	—	53	1	no	maximal, Sylow
112	C₂²×C₅₃	—	2	8	no	—
113	C₂²×C₅₃	—	2	96	no	—
114	C₂²×C₅₃	—	2	8	no	—
115	C₂²×C₅₃	C₈×D₄	1	16	yes	complemented
116	C₂²×C₅₃	C₂×C₄×D₄	1	6	yes	—
117	C₂²×C₅₃	C₂³×C₈	1	6	yes	—
118	C₂²×C₅₃	C₂³×C₈	1	16	yes	complemented
119	C₂²×C₅₃	C₂×#131	1	6	yes	—
120	C₂²×C₅₃	C₂⁴×C₄	1	1	yes	characteristic
121	C₄×C₅₃	C₂³×C₈	1	8	yes	complemented
122	Dic₅₃	—	2	8	no	—
123	Dic₅₃	—	2	4	no	—
124	Dic₅₃	C₂×C₄×D₄	1	6	yes	—
125	Dic₅₃	C₂³×D₄	1	1	yes	characteristic
126	Dic₅₃	C₂²×#41	1	1	yes	characteristic
127	C₂⁴×C₈	—	53	2	no	—
128	C₂²×C₄×D₄	—	53	1	no	—
129	C₂×C₈×D₄	—	53	24	no	—
130	C₂²×C₄⋊C₈	—	53	1	no	—
131	C₂²×C₂²⋊C₈	—	53	2	no	—
132	C₂²×C₄×C₈	—	53	1	no	—
133	C₂×C₅₃	—	2	16	no	—
134	C₂×C₅₃	C₂×C₈×D₄	1	12	yes	complemented
135	C₂×C₅₃	C₂²×C₄×D₄	1	1	yes	characteristic
136	C₂×C₅₃	C₂⁴×C₈	1	1	yes	characteristic
137	C₂×C₅₃	C₂²×#131	1	1	yes	characteristic
138	C₂³×D₄	#6157	1	1	yes	characteristic
139	C₂⁴×C₄	—	53	2	no	—
140	C₂³×C₈	—	53	1	no	—
141	C₂³×C₈	—	53	1	no	—
142	C₂³×C₈	—	53	2	no	—
143	C₂³×C₈	—	53	24	no	—
144	C₂³×C₈	—	106	16	no	—
145	C₂×C₄×D₄	—	53	12	no	—
146	C₂×C₄×D₄	—	53	12	no	—
147	C₂²×C₄⋊C₄	—	53	1	no	—
148	C₂²×C₂²⋊C₄	—	53	2	no	—
149	C₂²×C₄²	—	53	1	no	—
150	C₈×D₄	—	53	64	no	—
151	C₂×C₄⋊C₈	—	53	12	no	—
152	C₂×C₂²⋊C₈	—	53	24	no	—
153	C₂×C₄×C₈	—	53	12	no	—
154	C₅₃	C₂²×C₈×D₄	1	1	yes	characteristic, complemented, Sylow
155	C₂⁵	C₂×#6157	1	2	yes	—
156	C₂²×D₄	#88324	1	16	yes	direct product factor
157	C₂²×D₄	C₂×#6157	1	12	yes	—
158	C₂³×C₄	—	53	1	no	—
159	C₂³×C₄	—	53	12	no	—
160	C₂³×C₄	—	53	12	no	—
161	C₂³×C₄	—	53	1	no	—
162	C₂³×C₄	—	53	2	no	—
163	C₂³×C₄	—	106	8	no	—
164	C₂³×C₄	—	106	8	no	—
165	C₂³×C₄	C₂×#6157	1	1	yes	characteristic
166	C₂²×C₈	—	53	6	no	—
167	C₂²×C₈	—	53	8	no	—
168	C₂²×C₈	—	53	12	no	—
169	C₂²×C₈	—	53	32	no	—
170	C₂²×C₈	—	53	6	no	—
171	C₂²×C₈	—	106	96	no	—
172	C₂²×C₈	—	106	8	no	—
173	C₂²×C₈	—	106	4	no	—
174	C₄×D₄	—	53	48	no	—
175	C₄×D₄	—	53	16	no	—
176	C₂×C₄⋊C₄	—	53	12	no	—
177	C₂×C₂²⋊C₄	—	53	24	no	—
178	C₂×C₄²	—	53	6	no	—
179	C₂×C₄²	—	53	6	no	—
180	C₄⋊C₈	—	53	16	no	—
181	C₂²⋊C₈	—	53	32	no	—
182	C₄×C₈	—	53	16	no	—
183	C₂⁴	—	2	8	no	—
184	C₂⁴	—	2	8	no	—
185	C₂⁴	C₂×#88324	1	16	yes	complemented
186	C₂⁴	C₂²×#6157	1	12	yes	—
187	C₂⁴	C₂²×#6157	1	1	yes	characteristic
188	C₂×D₄	C₂×#88324	1	96	yes	direct product factor
189	C₂×D₄	C₂²×#6157	1	16	yes	—
190	C₂²×C₄	—	53	4	no	—
191	C₂²×C₄	—	53	4	no	—
192	C₂²×C₄	—	53	3	no	—
193	C₂²×C₄	—	53	3	no	—
194	C₂²×C₄	—	53	6	no	—
195	C₂²×C₄	—	53	12	no	—
196	C₂²×C₄	—	53	24	no	—
197	C₂²×C₄	—	53	8	no	—
198	C₂²×C₄	—	106	24	no	—
199	C₂²×C₄	—	106	48	no	—
200	C₂²×C₄	—	106	24	no	—
201	C₂²×C₄	—	106	8	no	—
202	C₂²×C₄	—	106	4	no	—
203	C₂²×C₄	C₂×#88324	1	8	yes	complemented
204	C₂²×C₄	C₂²×#6157	1	6	yes	—
205	C₂×C₈	—	53	24	no	—
206	C₂×C₈	—	53	4	no	—
207	C₂×C₈	—	53	4	no	—
208	C₂×C₈	—	53	8	no	—
209	C₂×C₈	—	106	64	no	—
210	C₂×C₈	—	106	12	no	—
211	C₂×C₈	—	106	24	no	—
212	C₄⋊C₄	—	53	16	no	—
213	C₂²⋊C₄	—	53	32	no	—
214	C₄²	—	53	4	no	—
215	C₄²	—	53	12	no	—
216	C₂³	—	2	64	no	—
217	C₂³	—	2	24	no	—
218	C₂³	—	2	24	no	—
219	C₂³	D₄×#6157	1	4	yes	—
220	C₂³	C₂²×#88324	1	4	yes	—
221	C₂³	C₂²×#88324	1	48	yes	complemented
222	C₂³	#13133304	1	4	yes	—
223	C₂³	C₂³×#6157	1	8	yes	—
224	C₂³	C₂³×#6157	1	3	yes	—
225	D₄	C₂²×#88324	1	64	yes	direct product factor
226	C₂×C₄	—	53	12	no	—
227	C₂×C₄	—	53	6	no	—
228	C₂×C₄	—	53	6	no	—
229	C₂×C₄	—	53	3	no	—
230	C₂×C₄	—	53	1	no	—
231	C₂×C₄	—	53	4	no	—
232	C₂×C₄	—	53	8	no	—
233	C₂×C₄	—	106	8	no	—
234	C₂×C₄	—	106	8	no	—
235	C₂×C₄	—	106	48	no	—
236	C₂×C₄	—	106	24	no	—
237	C₂×C₄	—	106	12	no	—
238	C₂×C₄	C₂²×#88324	1	24	yes	complemented
239	C₂×C₄	C₂³×#6157	1	4	yes	—
240	C₈	—	53	8	no	—
241	C₈	—	106	4	no	—
242	C₈	—	106	8	no	—
243	C₂²	—	2	96	no	—
244	C₂²	—	2	8	no	—
245	C₂²	—	2	8	no	—
246	C₂²	D₄×#88324	1	16	yes	direct product factor
247	C₂²	C₂×D₄×#6157	1	6	yes	—
248	C₂²	C₂³×#88324	1	6	yes	—
249	C₂²	C₂³×#88324	1	16	yes	complemented
250	C₂²	C₂×#13133304	1	6	yes	—
251	C₂²	C₂⁴×#6157	1	1	yes	characteristic
252	C₄	—	53	1	no	—
253	C₄	—	53	1	no	—
254	C₄	—	53	6	no	—
255	C₄	—	106	8	no	—
256	C₄	—	106	4	no	—
257	C₄	C₂³×#88324	1	8	yes	complemented
258	C₂	—	2	16	no	—
259	C₂	C₂×D₄×#88324	1	12	yes	direct product factor
260	C₂	C₂²×D₄×#6157	1	1	yes	characteristic
261	C₂	C₂⁴×#88324	1	1	yes	characteristic
262	C₂	C₂²×#13133304	1	1	yes	characteristic

	subgroup	G/H
center	C₂⁴	C₂²×#6157
commutator	C₂×C₅₃	C₂⁴×C₈
Frattini	C₂²	C₂⁴×#6157
Fitting, prime radical	C₂³×D₄×C₅₃	C₄
socle	C₂⁴×C₅₃	C₂²×C₄
cosocle	C₂×Dic₅₃	C₂⁵
hypercenter	C₂³×D₄	#6157
hypocenter	C₅₃	C₂²×C₈×D₄
LCS	G ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂⁴ ◁ C₂³×D₄

Complements

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

N	H ≅ G/N	type
1. C₂⁴×#88324	C₂	semidirect
2. C₂×D₄×#88324	C₂	direct (N × H)
3. C₂²×C₄⋊#88324	C₂	semidirect
4. C₂²×C₂²⋊#88324	C₂	semidirect
5. C₂²×C₄×#88324	C₂	semidirect
11. C₂³×#88324	C₂²	semidirect
12. D₄×#88324	C₂²	direct (N × H)
13. C₂×C₄⋊#88324	C₂²	semidirect
14. C₂×C₂²⋊#88324	C₂²	semidirect
15. C₂×C₄×#88324	C₂²	semidirect
26. C₂²×#88324	D₄	direct (N × H)
30. C₂²×#88324	C₂³	semidirect
31. C₄⋊#88324	C₂³	semidirect
32. C₂²⋊#88324	C₂³	semidirect
33. C₄×#88324	C₂³	semidirect
48. C₂²×D₄×C₅₃	C₈	semidirect
53. C₂⁴×C₅₃	C₂×C₈	semidirect
56. C₂×D₄×C₅₃	C₂×C₈	semidirect
58. C₂²×C₄×C₅₃	C₂×C₈	semidirect
76. C₂×#88324	C₂×D₄	direct (N × H)
89. C₂³×C₅₃	C₂²×C₈	semidirect
93. D₄×C₅₃	C₂²×C₈	semidirect
94. C₂×C₄×C₅₃	C₂²×C₈	semidirect
110. #88324	C₂²×D₄	direct (N × H)
115. C₂²×C₅₃	C₈×D₄	semidirect
118. C₂²×C₅₃	C₂³×C₈	semidirect
121. C₄×C₅₃	C₂³×C₈	semidirect
134. C₂×C₅₃	C₂×C₈×D₄	semidirect
154. C₅₃	C₂²×C₈×D₄	semidirect
156. C₂²×D₄	#88324	direct (N × H)
185. C₂⁴	C₂×#88324	semidirect
188. C₂×D₄	C₂×#88324	direct (N × H)
203. C₂²×C₄	C₂×#88324	semidirect
221. C₂³	C₂²×#88324	semidirect
225. D₄	C₂²×#88324	direct (N × H)
238. C₂×C₄	C₂²×#88324	semidirect
246. C₂²	D₄×#88324	direct (N × H)
249. C₂²	C₂³×#88324	semidirect
257. C₄	C₂³×#88324	semidirect
259. C₂	C₂×D₄×#88324	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂⁴×#88324	2
C₂×D₄×#88324	2
C₂²×C₄⋊#88324	2
C₂²×C₂²⋊#88324	2
C₂²×C₄×#88324	2
C₂²×D₄×Dic₅₃	2
C₂²×C₈×D₄	53

Subgroups — C22×D4×#88324 #28329418

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂²×D₄×#88324 #28329418