Subgroups — C₂²×Q₈⋊#88322 #27350438

Order 13568.

1454 conjugacy classes of subgroups.

The subgroup lattice has 272 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₄⋊#88322	C₂	1	1	yes	maximal, characteristic
2	C₂²×C₄×#88322	C₂	1	1	yes	maximal, characteristic
3	C₂×Q₈⋊#88322	C₂	1	12	yes	maximal, direct product factor
4	C₂²×C₄×Q₈×C₅₃	C₂	1	1	yes	maximal, characteristic
5	C₂³×#88322	—	2	1	no	—
6	C₂³×#88322	—	2	1	no	—
7	C₂×C₄⋊#88322	—	2	3	no	—
8	C₂×C₄⋊#88322	C₂²	1	6	yes	—
9	C₂×C₄×#88322	—	2	3	no	—
10	C₂×C₄×#88322	C₂²	1	6	yes	—
11	Q₈⋊#88322	C₂²	1	16	yes	direct product factor
12	C₂³×Q₈×C₅₃	C₄	1	1	yes	characteristic
13	C₂×C₄×Q₈×C₅₃	—	2	3	no	—
14	C₂×C₄×Q₈×C₅₃	—	2	3	no	—
15	C₂×C₄×Q₈×C₅₃	C₄	1	6	yes	—
16	C₂×C₄×Q₈×C₅₃	C₂²	1	6	yes	—
17	C₂²×C₄⋊C₄×C₅₃	—	2	1	no	—
18	C₂²×C₄⋊C₄×C₅₃	C₄	1	1	yes	characteristic
19	C₂²×C₄²×C₅₃	—	2	1	no	—
20	C₂²×C₄²×C₅₃	C₂²	1	1	yes	characteristic
21	C₂²×#88322	—	2	8	no	—
22	C₂²×#88322	—	2	6	no	—
23	C₂²×#88322	—	2	6	no	—
24	C₂²×#88322	—	4	4	no	—
25	C₄⋊#88322	—	2	6	no	—
26	C₄⋊#88322	C₂³	1	4	yes	—
27	C₄×#88322	—	2	6	no	—
28	C₄×#88322	C₂³	1	4	yes	—
29	C₂²×Q₈×C₅₃	—	2	3	no	—
30	C₂²×Q₈×C₅₃	—	2	4	no	—
31	C₂²×Q₈×C₅₃	C₈	1	8	yes	complemented
32	C₂²×Q₈×C₅₃	C₂×C₄	1	6	yes	—
33	C₂³×C₄×C₅₃	—	2	1	no	—
34	C₂³×C₄×C₅₃	—	2	1	no	—
35	C₂³×C₄×C₅₃	C₂×C₄	1	1	yes	characteristic
36	C₂³×C₄×C₅₃	D₄	1	1	yes	characteristic
37	C₂³×C₄×C₅₃	D₄	1	1	yes	characteristic
38	C₄×Q₈×C₅₃	—	2	12	no	—
39	C₄×Q₈×C₅₃	—	2	6	no	—
40	C₄×Q₈×C₅₃	—	2	6	no	—
41	C₄×Q₈×C₅₃	C₂×C₄	1	12	yes	—
42	C₄×Q₈×C₅₃	C₂³	1	4	yes	—
43	C₂×C₄⋊C₄×C₅₃	—	2	6	no	—
44	C₂×C₄⋊C₄×C₅₃	—	2	3	no	—
45	C₂×C₄⋊C₄×C₅₃	—	2	6	no	—
46	C₂×C₄⋊C₄×C₅₃	C₂×C₄	1	6	yes	—
47	C₂×C₄²×C₅₃	—	2	6	no	—
48	C₂×C₄²×C₅₃	—	2	6	no	—
49	C₂×C₄²×C₅₃	C₂×C₄	1	3	yes	—
50	C₂×C₄²×C₅₃	D₄	1	3	yes	—
51	C₂×C₄²×C₅₃	D₄	1	3	yes	—
52	C₂×C₄²×C₅₃	C₂³	1	3	yes	—
53	C₂⁴×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
54	C₂×Q₈×C₅₃	—	2	24	no	—
55	C₂×Q₈×C₅₃	—	2	6	no	—
56	C₂×Q₈×C₅₃	—	2	12	no	—
57	C₂×Q₈×C₅₃	C₂×C₈	1	24	yes	complemented
58	C₂×Q₈×C₅₃	C₂²×C₄	1	4	yes	—
59	C₂²×C₄×C₅₃	—	2	8	no	—
60	C₂²×C₄×C₅₃	—	2	4	no	—
61	C₂²×C₄×C₅₃	—	2	6	no	—
62	C₂²×C₄×C₅₃	—	2	6	no	—
63	C₂²×C₄×C₅₃	—	4	4	no	—
64	C₂²×C₄×C₅₃	C₂²⋊C₄	1	3	yes	—
65	C₂²×C₄×C₅₃	C₂²⋊C₄	1	3	yes	—
66	C₂²×C₄×C₅₃	C₂²⋊C₄	1	3	yes	—
67	C₂²×C₄×C₅₃	C₂×C₈	1	4	yes	—
68	C₂²×C₄×C₅₃	M₁₆	1	4	yes	—
69	C₂²×C₄×C₅₃	SD₁₆	1	4	yes	—
70	C₂²×C₄×C₅₃	Q₁₆	1	4	yes	—
71	C₂²×C₄×C₅₃	C₂²×C₄	1	3	yes	—
72	C₂²×C₄×C₅₃	C₂×D₄	1	3	yes	—
73	C₂²×C₄×C₅₃	C₂×D₄	1	3	yes	—
74	C₄⋊C₄×C₅₃	—	2	12	no	—
75	C₄⋊C₄×C₅₃	—	2	6	no	—
76	C₄⋊C₄×C₅₃	—	2	4	no	—
77	C₄⋊C₄×C₅₃	C₂²×C₄	1	4	yes	—
78	C₄²×C₅₃	—	2	12	no	—
79	C₄²×C₅₃	—	2	4	no	—
80	C₄²×C₅₃	C₂²⋊C₄	1	6	yes	—
81	C₄²×C₅₃	C₂²×C₄	1	3	yes	—
82	C₄²×C₅₃	C₂×D₄	1	3	yes	—
83	C₄²×C₅₃	C₂×D₄	1	3	yes	—
84	C₄²×C₅₃	C₂⁴	1	1	yes	characteristic
85	C₂×#88322	—	2	24	no	—
86	C₂×#88322	—	2	4	no	—
87	C₂×#88322	—	2	4	no	—
88	C₂×#88322	—	4	12	no	—
89	C₂³×C₅₃	C₂²⋊C₈	1	4	yes	—
90	C₂³×C₅₃	#103	1	4	yes	—
91	C₂³×C₅₃	#104	1	4	yes	—
92	C₂³×C₅₃	C₂×C₂²⋊C₄	1	3	yes	—
93	Q₈×C₅₃	—	2	4	no	—
94	Q₈×C₅₃	—	2	24	no	—
95	Q₈×C₅₃	C₂²×C₈	1	8	yes	complemented
96	C₂×C₄×C₅₃	—	2	4	no	—
97	C₂×C₄×C₅₃	—	2	24	no	—
98	C₂×C₄×C₅₃	—	2	6	no	—
99	C₂×C₄×C₅₃	—	2	6	no	—
100	C₂×C₄×C₅₃	—	2	4	no	—
101	C₂×C₄×C₅₃	—	4	12	no	—
102	C₂×C₄×C₅₃	C₂²⋊C₈	1	12	yes	—
103	C₂×C₄×C₅₃	#103	1	12	yes	—
104	C₂×C₄×C₅₃	C₂×C₂²⋊C₄	1	3	yes	—
105	C₂×C₄×C₅₃	C₂×C₂²⋊C₄	1	3	yes	—
106	C₂×C₄×C₅₃	C₂×C₂²⋊C₄	1	3	yes	—
107	C₂×C₄×C₅₃	C₂²×C₈	1	6	yes	—
108	C₂×C₄×C₅₃	C₂×M₁₆	1	6	yes	—
109	C₂×C₄×C₅₃	C₂×SD₁₆	1	6	yes	—
110	C₂×C₄×C₅₃	C₂×Q₁₆	1	6	yes	—
111	C₂×C₄×C₅₃	C₂³×C₄	1	1	yes	characteristic
112	C₂×C₄×C₅₃	C₂²×D₄	1	1	yes	characteristic
113	C₂×C₄×C₅₃	C₂²×D₄	1	1	yes	characteristic
114	#88322	—	2	8	no	—
115	#88322	—	4	4	no	—
116	C₂²×Q₈⋊C₈	—	53	1	no	maximal, Sylow
117	C₂²×C₅₃	Q₈⋊C₈	1	16	yes	complemented
118	C₂²×C₅₃	C₂×C₂²⋊C₈	1	6	yes	—
119	C₂²×C₅₃	C₂×#103	1	6	yes	—
120	C₂²×C₅₃	C₂×#104	1	6	yes	—
121	C₂²×C₅₃	C₂²×C₂²⋊C₄	1	1	yes	characteristic
122	C₄×C₅₃	—	2	8	no	—
123	C₄×C₅₃	—	2	1	no	—
124	C₄×C₅₃	—	2	3	no	—
125	C₄×C₅₃	—	4	4	no	—
126	C₄×C₅₃	C₂×C₂²⋊C₈	1	6	yes	—
127	C₄×C₅₃	C₂×#103	1	6	yes	—
128	C₄×C₅₃	C₂³×C₈	1	1	yes	characteristic
129	C₄×C₅₃	C₂²×M₁₆	1	1	yes	characteristic
130	C₄×C₅₃	C₂²×SD₁₆	1	1	yes	characteristic
131	C₄×C₅₃	C₂²×Q₁₆	1	1	yes	characteristic
132	C₂²×C₄×Q₈	D₅₃	1	1	yes	characteristic
133	C₂²×C₄⋊C₈	—	53	1	no	—
134	C₂²×C₄×C₈	—	53	1	no	—
135	C₂×Q₈⋊C₈	—	53	12	no	—
136	C₂×C₅₃	C₂×Q₈⋊C₈	1	12	yes	complemented
137	C₂×C₅₃	C₂²×C₂²⋊C₈	1	1	yes	characteristic
138	C₂×C₅₃	C₂²×#103	1	1	yes	characteristic
139	C₂×C₅₃	C₂²×#104	1	1	yes	characteristic
140	C₂³×Q₈	Dic₅₃	1	1	yes	characteristic
141	C₂³×C₈	—	106	1	no	—
142	C₂³×C₈	—	106	1	no	—
143	C₂×C₄×Q₈	—	2	3	no	—
144	C₂×C₄×Q₈	—	2	3	no	—
145	C₂×C₄×Q₈	Dic₅₃	1	6	yes	—
146	C₂×C₄×Q₈	C₂×D₅₃	1	6	yes	—
147	C₂²×C₄⋊C₄	—	2	1	no	—
148	C₂²×C₄⋊C₄	Dic₅₃	1	1	yes	characteristic
149	C₂²×C₄²	—	2	1	no	—
150	C₂²×C₄²	C₂×D₅₃	1	1	yes	characteristic
151	C₂×C₄⋊C₈	—	53	6	no	—
152	C₂×C₄⋊C₈	—	106	3	no	—
153	C₂×C₄×C₈	—	53	6	no	—
154	C₂×C₄×C₈	—	106	3	no	—
155	Q₈⋊C₈	—	53	16	no	—
156	C₅₃	C₂²×Q₈⋊C₈	1	1	yes	characteristic, complemented, Sylow
157	C₂²×Q₈	—	2	4	no	—
158	C₂²×Q₈	—	2	3	no	—
159	C₂²×Q₈	#88322	1	8	yes	complemented
160	C₂²×Q₈	C₂×Dic₅₃	1	6	yes	—
161	C₂³×C₄	—	2	1	no	—
162	C₂³×C₄	—	2	1	no	—
163	C₂³×C₄	C₂×Dic₅₃	1	1	yes	characteristic
164	C₂³×C₄	C₂²⋊D₅₃	1	1	yes	characteristic
165	C₂³×C₄	C₂²⋊D₅₃	1	1	yes	characteristic
166	C₂²×C₈	—	106	8	no	—
167	C₂²×C₈	—	106	6	no	—
168	C₂²×C₈	—	106	6	no	—
169	C₂²×C₈	—	212	4	no	—
170	C₄×Q₈	—	2	6	no	—
171	C₄×Q₈	—	2	6	no	—
172	C₄×Q₈	—	2	12	no	—
173	C₄×Q₈	C₂×Dic₅₃	1	12	yes	—
174	C₄×Q₈	C₂²×D₅₃	1	4	yes	—
175	C₂×C₄⋊C₄	—	2	6	no	—
176	C₂×C₄⋊C₄	—	2	6	no	—
177	C₂×C₄⋊C₄	—	2	3	no	—
178	C₂×C₄⋊C₄	C₂×Dic₅₃	1	6	yes	—
179	C₂×C₄²	—	2	6	no	—
180	C₂×C₄²	—	2	6	no	—
181	C₂×C₄²	C₂×Dic₅₃	1	3	yes	—
182	C₂×C₄²	C₂²⋊D₅₃	1	3	yes	—
183	C₂×C₄²	C₂²⋊D₅₃	1	3	yes	—
184	C₂×C₄²	C₂²×D₅₃	1	3	yes	—
185	C₄⋊C₈	—	53	4	no	—
186	C₄⋊C₈	—	106	6	no	—
187	C₄×C₈	—	53	4	no	—
188	C₄×C₈	—	106	6	no	—
189	C₂⁴	C₂²⋊Dic₅₃	1	1	yes	characteristic
190	C₂×Q₈	—	2	24	no	—
191	C₂×Q₈	—	2	6	no	—
192	C₂×Q₈	—	2	12	no	—
193	C₂×Q₈	C₂×#88322	1	24	yes	complemented
194	C₂×Q₈	C₂²×Dic₅₃	1	4	yes	—
195	C₂²×C₄	—	2	8	no	—
196	C₂²×C₄	—	2	4	no	—
197	C₂²×C₄	—	2	6	no	—
198	C₂²×C₄	—	2	6	no	—
199	C₂²×C₄	—	4	4	no	—
200	C₂²×C₄	C₂×#88322	1	4	yes	—
201	C₂²×C₄	#11719917	1	4	yes	—
202	C₂²×C₄	#11719924	1	4	yes	—
203	C₂²×C₄	#11719925	1	4	yes	—
204	C₂²×C₄	C₂²⋊Dic₅₃	1	3	yes	—
205	C₂²×C₄	C₂²⋊Dic₅₃	1	3	yes	—
206	C₂²×C₄	C₂²⋊Dic₅₃	1	3	yes	—
207	C₂²×C₄	C₂²×Dic₅₃	1	3	yes	—
208	C₂²×C₄	C₂×C₂²⋊D₅₃	1	3	yes	—
209	C₂²×C₄	C₂×C₂²⋊D₅₃	1	3	yes	—
210	C₂×C₈	—	106	24	no	—
211	C₂×C₈	—	106	4	no	—
212	C₂×C₈	—	106	4	no	—
213	C₂×C₈	—	212	12	no	—
214	C₄⋊C₄	—	2	12	no	—
215	C₄⋊C₄	—	2	6	no	—
216	C₄⋊C₄	—	2	4	no	—
217	C₄⋊C₄	C₂²×Dic₅₃	1	4	yes	—
218	C₄²	—	2	12	no	—
219	C₄²	—	2	4	no	—
220	C₄²	C₂²⋊Dic₅₃	1	6	yes	—
221	C₄²	C₂²×Dic₅₃	1	3	yes	—
222	C₄²	C₂×C₂²⋊D₅₃	1	3	yes	—
223	C₄²	C₂×C₂²⋊D₅₃	1	3	yes	—
224	C₄²	C₂³×D₅₃	1	1	yes	characteristic
225	C₂³	C₂²⋊#88322	1	4	yes	—
226	C₂³	#13133141	1	4	yes	—
227	C₂³	#13133144	1	4	yes	—
228	C₂³	C₂×C₂²⋊Dic₅₃	1	3	yes	—
229	Q₈	—	2	24	no	—
230	Q₈	—	2	4	no	—
231	Q₈	C₂²×#88322	1	8	yes	complemented
232	C₂×C₄	—	2	4	no	—
233	C₂×C₄	—	2	24	no	—
234	C₂×C₄	—	2	6	no	—
235	C₂×C₄	—	2	6	no	—
236	C₂×C₄	—	2	4	no	—
237	C₂×C₄	—	4	12	no	—
238	C₂×C₄	C₂²⋊#88322	1	12	yes	—
239	C₂×C₄	#13133141	1	12	yes	—
240	C₂×C₄	C₂×C₂²⋊Dic₅₃	1	3	yes	—
241	C₂×C₄	C₂×C₂²⋊Dic₅₃	1	3	yes	—
242	C₂×C₄	C₂×C₂²⋊Dic₅₃	1	3	yes	—
243	C₂×C₄	C₂²×#88322	1	6	yes	—
244	C₂×C₄	C₂×#11719917	1	6	yes	—
245	C₂×C₄	C₂×#11719924	1	6	yes	—
246	C₂×C₄	C₂×#11719925	1	6	yes	—
247	C₂×C₄	C₂³×Dic₅₃	1	1	yes	characteristic
248	C₂×C₄	C₂²×C₂²⋊D₅₃	1	1	yes	characteristic
249	C₂×C₄	C₂²×C₂²⋊D₅₃	1	1	yes	characteristic
250	C₈	—	106	8	no	—
251	C₈	—	212	4	no	—
252	C₂²	Q₈⋊#88322	1	16	yes	direct product factor
253	C₂²	C₂×C₂²⋊#88322	1	6	yes	—
254	C₂²	C₂×#13133141	1	6	yes	—
255	C₂²	C₂×#13133144	1	6	yes	—
256	C₂²	C₂²×C₂²⋊Dic₅₃	1	1	yes	characteristic
257	C₄	—	2	8	no	—
258	C₄	—	2	1	no	—
259	C₄	—	2	3	no	—
260	C₄	—	4	4	no	—
261	C₄	C₂×C₂²⋊#88322	1	6	yes	—
262	C₄	C₂×#13133141	1	6	yes	—
263	C₄	C₂³×#88322	1	1	yes	characteristic
264	C₄	C₂²×#11719917	1	1	yes	characteristic
265	C₄	C₂²×#11719924	1	1	yes	characteristic
266	C₄	C₂²×#11719925	1	1	yes	characteristic
267	C₂	C₂×Q₈⋊#88322	1	12	yes	direct product factor
268	C₂	C₂²×C₂²⋊#88322	1	1	yes	characteristic
269	C₂	C₂²×#13133141	1	1	yes	characteristic
270	C₂	C₂²×#13133144	1	1	yes	characteristic

	subgroup	G/H
center	C₂³×C₄	C₂²⋊D₅₃
commutator	C₄×C₅₃	C₂³×C₈
Frattini	C₄²	C₂³×D₅₃
Fitting	C₂²×C₄×Q₈×C₅₃	C₂
socle, prime radical	C₂⁴×C₅₃	C₂²⋊C₄
cosocle	C₄²×C₅₃	C₂⁴
hypercenter	C₂²×C₄×Q₈	D₅₃
hypocenter	C₅₃	C₂²×Q₈⋊C₈
LCS	G ▷ C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂³×C₄ ◁ C₂²×C₄² ◁ C₂²×C₄×Q₈

Complements

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

N	H ≅ G/N	type
3. C₂×Q₈⋊#88322	C₂	direct (N × H)
11. Q₈⋊#88322	C₂²	direct (N × H)
31. C₂²×Q₈×C₅₃	C₈	semidirect
57. C₂×Q₈×C₅₃	C₂×C₈	semidirect
95. Q₈×C₅₃	C₂²×C₈	semidirect
117. C₂²×C₅₃	Q₈⋊C₈	semidirect
136. C₂×C₅₃	C₂×Q₈⋊C₈	semidirect
156. C₅₃	C₂²×Q₈⋊C₈	semidirect
159. C₂²×Q₈	#88322	semidirect
193. C₂×Q₈	C₂×#88322	semidirect
231. Q₈	C₂²×#88322	semidirect
252. C₂²	Q₈⋊#88322	direct (N × H)
267. C₂	C₂×Q₈⋊#88322	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×C₄⋊#88322	2
C₂²×C₄×#88322	2
C₂×Q₈⋊#88322	2
C₂²×C₄×Q₈×C₅₃	2
C₂²×Q₈⋊C₈	53

Subgroups — C22×Q8⋊#88322 #27350438

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂²×Q₈⋊#88322 #27350438