Subgroups — C₄×C₈×Dic₁₀₆ #27382698

Order 13568.

1070 conjugacy classes of subgroups.

The subgroup lattice has 276 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₈×Dic₁₀₆	C₂	1	2	yes	maximal
2	C₄²×Dic₁₀₆	C₂	1	1	yes	maximal, characteristic
3	C₈×#11719919	C₂	1	4	yes	maximal
4	C₈×C₄⋊Dic₅₃	C₂	1	2	yes	maximal
5	C₄×#13133146	C₂	1	2	yes	maximal
6	C₄×C₄⋊#88322	C₂	1	1	yes	maximal, characteristic
7	C₄×C₈×Dic₅₃	C₂	1	2	yes	maximal
8	C₄²×C₈×C₅₃	C₂	1	1	yes	maximal, characteristic
9	C₂×C₄×Dic₁₀₆	C₄	1	2	yes	—
10	C₂×C₄×Dic₁₀₆	C₂²	1	1	yes	characteristic
11	C₈×Dic₁₀₆	C₄	1	16	yes	direct product factor
12	C₂×C₄⋊#88322	C₂²	1	2	yes	—
13	C₂×#13133146	C₂²	1	4	yes	—
14	C₂×C₄×#88322	C₂²	1	2	yes	—
15	C₂×C₈×Dic₅₃	C₂²	1	4	yes	—
16	C₂×C₈×Dic₅₃	C₂²	1	4	yes	—
17	C₄×#11719919	C₄	1	4	yes	—
18	C₄×#11719919	C₂²	1	2	yes	—
19	C₄×C₄⋊Dic₅₃	C₄	1	2	yes	—
20	C₄×C₄⋊Dic₅₃	C₂²	1	1	yes	characteristic
21	C₄²×Dic₅₃	C₂²	1	2	yes	—
22	#16810378	C₂²	1	2	yes	—
23	#16810377	C₂²	1	4	yes	—
24	C₂×C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
25	C₂×C₄×C₈×C₅₃	C₂²	1	2	yes	—
26	C₂×C₄×C₈×C₅₃	C₂²	1	2	yes	—
27	C₂×C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
28	C₄³×C₅₃	C₂²	1	1	yes	characteristic
29	C₂²×Dic₁₀₆	C₂×C₄	1	1	yes	characteristic
30	C₂²×#88322	C₂³	1	4	yes	—
31	C₄×Dic₁₀₆	C₈	1	16	yes	direct product factor
32	C₄×Dic₁₀₆	C₂×C₄	1	8	yes	—
33	C₂×C₄⋊Dic₅₃	C₂×C₄	1	1	yes	characteristic
34	C₂×C₄⋊Dic₅₃	C₂×C₄	1	2	yes	—
35	C₂×#11719919	C₂×C₄	1	2	yes	—
36	C₂×#11719919	C₂×C₄	1	4	yes	—
37	C₂×C₄×Dic₅₃	C₂×C₄	1	4	yes	—
38	C₂×C₄×Dic₅₃	C₂×C₄	1	4	yes	—
39	C₂×C₄×Dic₅₃	C₂³	1	2	yes	—
40	C₂×C₄×Dic₅₃	C₂³	1	2	yes	—
41	C₄⋊#88322	C₂×C₄	1	8	yes	—
42	#13133146	C₂×C₄	1	16	yes	—
43	C₄×#88322	—	2	4	no	—
44	C₈×Dic₅₃	—	2	8	no	—
45	C₈×Dic₅₃	C₂×C₄	1	16	yes	—
46	#13133124	C₂×C₄	1	4	yes	—
47	#13133124	C₂×C₄	1	2	yes	—
48	C₂²×C₈×C₅₃	C₂³	1	2	yes	—
49	C₂²×C₈×C₅₃	C₂³	1	2	yes	—
50	C₂×C₄²×C₅₃	C₂×C₄	1	2	yes	—
51	C₂×C₄²×C₅₃	C₂×C₄	1	2	yes	—
52	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
53	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
54	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
55	C₄×C₈×C₅₃	—	2	4	no	—
56	C₄×C₈×C₅₃	—	2	2	no	—
57	C₄×C₈×C₅₃	C₂×C₄	1	8	yes	—
58	C₄×C₈×C₅₃	Q₈	1	4	yes	complemented
59	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
60	C₂²×C₄×C₅₃	C₂²×C₄	1	2	yes	—
61	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
62	C₂²×C₄×C₅₃	C₂²×C₄	1	2	yes	—
63	C₂²×C₄×C₅₃	C₂⁴	1	1	yes	characteristic
64	C₂²×Dic₅₃	C₂²×C₄	1	2	yes	—
65	C₂²×Dic₅₃	C₂²×C₄	1	4	yes	—
66	C₂²×Dic₅₃	C₂²×C₄	1	2	yes	—
67	C₂×Dic₁₀₆	C₄²	1	4	yes	—
68	C₂×Dic₁₀₆	C₂×C₈	1	8	yes	—
69	C₂×C₈×C₅₃	—	2	2	no	—
70	C₂×C₈×C₅₃	—	2	2	no	—
71	C₂×C₈×C₅₃	C₂²×C₄	1	4	yes	—
72	C₂×C₈×C₅₃	C₂²×C₄	1	4	yes	—
73	C₂×C₈×C₅₃	C₂×Q₈	1	4	yes	—
74	C₂×C₈×C₅₃	#41	1	4	yes	—
75	C₄²×C₅₃	—	2	4	no	—
76	C₄²×C₅₃	—	2	2	no	—
77	C₄²×C₅₃	C₂×C₈	1	8	yes	—
78	C₄²×C₅₃	C₂²×C₄	1	4	yes	—
79	C₄²×C₅₃	C₂×Q₈	1	2	yes	—
80	C₄²×C₅₃	#41	1	2	yes	—
81	C₄⋊Dic₅₃	C₄²	1	4	yes	—
82	C₄⋊Dic₅₃	C₂×C₈	1	8	yes	—
83	#11719919	C₄²	1	8	yes	—
84	#11719919	C₂×C₈	1	16	yes	—
85	C₄×Dic₅₃	—	2	8	no	—
86	C₄×Dic₅₃	—	2	4	no	—
87	C₄×Dic₅₃	C₂×C₈	1	16	yes	—
88	C₄×Dic₅₃	C₂²×C₄	1	8	yes	—
89	C₂×#88322	—	2	4	no	—
90	C₂×#88322	—	2	4	no	—
91	C₂×#88322	C₂²×C₄	1	8	yes	—
92	C₂³×C₅₃	C₂³×C₄	1	1	yes	characteristic
93	C₂×C₄×C₅₃	—	2	2	no	—
94	C₂×C₄×C₅₃	—	2	2	no	—
95	C₂×C₄×C₅₃	—	2	2	no	—
96	C₂×C₄×C₅₃	C₂×C₄²	1	2	yes	—
97	C₂×C₄×C₅₃	C₂×C₄²	1	2	yes	—
98	C₂×C₄×C₅₃	C₄×Q₈	1	4	yes	—
99	C₂×C₄×C₅₃	C₂²×C₈	1	4	yes	—
100	C₂×C₄×C₅₃	C₂²×C₈	1	4	yes	—
101	C₂×C₄×C₅₃	C₂²×C₈	1	4	yes	—
102	C₂×C₄×C₅₃	#131	1	4	yes	—
103	C₂×C₄×C₅₃	C₂³×C₄	1	2	yes	—
104	C₂×C₄×C₅₃	C₂²×Q₈	1	1	yes	characteristic
105	C₂×C₄×C₅₃	C₂×#41	1	2	yes	—
106	C₂×C₄×C₅₃	C₂×#41	1	1	yes	characteristic
107	C₂×Dic₅₃	—	2	4	no	—
108	C₂×Dic₅₃	—	2	4	no	—
109	C₂×Dic₅₃	—	2	4	no	—
110	C₂×Dic₅₃	C₂×C₄²	1	4	yes	—
111	C₂×Dic₅₃	C₂×C₄²	1	4	yes	—
112	C₂×Dic₅₃	C₂²×C₈	1	8	yes	—
113	C₂×Dic₅₃	C₂²×C₈	1	8	yes	—
114	Dic₁₀₆	C₄×C₈	1	16	yes	direct product factor
115	C₈×C₅₃	—	2	4	no	—
116	C₈×C₅₃	C₄×Q₈	1	8	yes	complemented
117	#88322	—	2	8	no	—
118	C₄×C₈×Q₈	—	53	1	no	maximal, Sylow
119	C₂²×C₅₃	C₂²×C₄²	1	1	yes	characteristic
120	C₂²×C₅₃	C₂×C₄×Q₈	1	1	yes	characteristic
121	C₂²×C₅₃	C₄×#41	1	1	yes	characteristic
122	C₂²×C₅₃	C₂³×C₈	1	2	yes	—
123	C₂²×C₅₃	C₂×#131	1	2	yes	—
124	C₄×C₅₃	—	2	2	no	—
125	C₄×C₅₃	—	2	4	no	—
126	C₄×C₅₃	C₂×C₄×C₈	1	4	yes	—
127	C₄×C₅₃	C₈×Q₈	1	8	yes	complemented
128	C₄×C₅₃	C₂×C₄×Q₈	1	2	yes	—
129	C₄×C₅₃	C₄×#41	1	2	yes	—
130	Dic₅₃	—	2	4	no	—
131	Dic₅₃	—	2	8	no	—
132	Dic₅₃	C₂×C₄×C₈	1	8	yes	—
133	C₂×C₈×Q₈	—	53	2	no	—
134	C₄²×Q₈	—	53	1	no	—
135	C₈×C₄⋊C₄	—	53	2	no	—
136	C₈×C₄⋊C₄	—	53	4	no	—
137	C₄×C₄⋊C₈	—	53	2	no	—
138	C₄×C₄⋊C₈	—	53	1	no	—
139	C₄²×C₈	—	53	2	no	—
140	C₄²×C₈	D₅₃	1	1	yes	characteristic
141	C₂×C₅₃	C₄²×Q₈	1	1	yes	characteristic
142	C₂×C₅₃	C₂²×C₄×C₈	1	1	yes	characteristic
143	C₂×C₅₃	C₄×#131	1	1	yes	characteristic
144	C₂×C₅₃	C₂×C₈×Q₈	1	2	yes	—
145	C₂×C₅₃	C₈×#41	1	2	yes	—
146	C₂×C₄×Q₈	—	53	2	no	—
147	C₂×C₄×Q₈	—	53	1	no	—
148	C₈×Q₈	—	53	16	no	—
149	C₂×C₄⋊C₈	—	53	4	no	—
150	C₂×C₄⋊C₈	—	53	2	no	—
151	C₂×C₄×C₈	—	53	4	no	—
152	C₂×C₄×C₈	—	53	4	no	—
153	C₂×C₄×C₈	—	53	2	no	—
154	C₂×C₄×C₈	C₂×D₅₃	1	1	yes	characteristic
155	C₂×C₄×C₈	C₂×D₅₃	1	2	yes	—
156	C₂×C₄×C₈	C₂×D₅₃	1	2	yes	—
157	C₂×C₄×C₈	C₂×D₅₃	1	1	yes	characteristic
158	C₄×C₄⋊C₄	—	53	2	no	—
159	C₄×C₄⋊C₄	—	53	4	no	—
160	C₄×C₄⋊C₄	—	53	1	no	—
161	C₄×C₄⋊C₄	—	53	2	no	—
162	C₄³	—	53	2	no	—
163	C₄³	C₂×D₅₃	1	1	yes	characteristic
164	#336	—	53	4	no	—
165	#336	—	53	2	no	—
166	C₅₃	C₄×C₈×Q₈	1	1	yes	characteristic, complemented, Sylow
167	C₂²×Q₈	—	53	1	no	—
168	C₂²×C₈	—	53	4	no	—
169	C₂²×C₈	C₂²×D₅₃	1	2	yes	—
170	C₂²×C₈	C₂²×D₅₃	1	2	yes	—
171	C₄×Q₈	—	53	8	no	—
172	C₄×Q₈	—	53	16	no	—
173	C₂×C₄⋊C₄	—	53	2	no	—
174	C₂×C₄⋊C₄	—	53	4	no	—
175	C₂×C₄⋊C₄	—	53	1	no	—
176	C₂×C₄⋊C₄	—	53	2	no	—
177	C₂×C₄²	—	53	4	no	—
178	C₂×C₄²	—	53	2	no	—
179	C₂×C₄²	—	53	4	no	—
180	C₂×C₄²	—	53	2	no	—
181	C₂×C₄²	C₄×D₅₃	1	2	yes	—
182	C₂×C₄²	C₄×D₅₃	1	2	yes	—
183	C₂×C₄²	C₂²×D₅₃	1	1	yes	characteristic
184	C₂×C₄²	C₂²×D₅₃	1	1	yes	characteristic
185	C₂×C₄²	C₂²×D₅₃	1	1	yes	characteristic
186	C₄⋊C₈	—	53	16	no	—
187	C₄⋊C₈	—	53	8	no	—
188	C₄×C₈	—	2	4	no	—
189	C₄×C₈	—	2	2	no	—
190	C₄×C₈	—	53	16	no	—
191	C₄×C₈	—	106	8	no	—
192	C₄×C₈	—	106	4	no	—
193	C₄×C₈	Dic₁₀₆	1	4	yes	direct product factor
194	C₄×C₈	C₄×D₅₃	1	8	yes	—
195	#95	—	53	2	no	—
196	#95	—	53	4	no	—
197	C₂×Q₈	—	53	4	no	—
198	C₂×Q₈	—	53	8	no	—
199	C₂²×C₄	—	53	2	no	—
200	C₂²×C₄	—	53	2	no	—
201	C₂²×C₄	—	53	4	no	—
202	C₂²×C₄	C₂×C₄×D₅₃	1	1	yes	characteristic
203	C₂²×C₄	C₂×C₄×D₅₃	1	2	yes	—
204	C₂²×C₄	C₂×C₄×D₅₃	1	1	yes	characteristic
205	C₂²×C₄	C₂×C₄×D₅₃	1	2	yes	—
206	C₂²×C₄	C₂³×D₅₃	1	1	yes	characteristic
207	C₂×C₈	—	2	2	no	—
208	C₂×C₈	—	2	2	no	—
209	C₂×C₈	—	53	8	no	—
210	C₂×C₈	—	106	4	no	—
211	C₂×C₈	—	106	4	no	—
212	C₂×C₈	C₂×Dic₁₀₆	1	4	yes	—
213	C₂×C₈	C₂×C₄×D₅₃	1	4	yes	—
214	C₂×C₈	C₂×C₄×D₅₃	1	4	yes	—
215	C₂×C₈	#11719945	1	4	yes	—
216	C₄⋊C₄	—	53	8	no	—
217	C₄⋊C₄	—	53	4	no	—
218	C₄⋊C₄	—	53	16	no	—
219	C₄⋊C₄	—	53	8	no	—
220	C₄²	—	2	4	no	—
221	C₄²	—	2	2	no	—
222	C₄²	—	53	16	no	—
223	C₄²	—	53	8	no	—
224	C₄²	—	106	8	no	—
225	C₄²	—	106	4	no	—
226	C₄²	C₈×D₅₃	1	8	yes	—
227	C₄²	C₂×Dic₁₀₆	1	2	yes	—
228	C₄²	C₂×C₄×D₅₃	1	4	yes	—
229	C₄²	#11719945	1	2	yes	—
230	C₂³	C₂²×C₄×D₅₃	1	1	yes	characteristic
231	Q₈	—	53	16	no	—
232	C₂×C₄	—	2	2	no	—
233	C₂×C₄	—	2	2	no	—
234	C₂×C₄	—	2	2	no	—
235	C₂×C₄	—	53	4	no	—
236	C₂×C₄	—	53	8	no	—
237	C₂×C₄	—	53	4	no	—
238	C₂×C₄	—	53	8	no	—
239	C₂×C₄	—	106	4	no	—
240	C₂×C₄	—	106	4	no	—
241	C₂×C₄	—	106	4	no	—
242	C₂×C₄	C₄²×D₅₃	1	2	yes	—
243	C₂×C₄	C₄²×D₅₃	1	2	yes	—
244	C₂×C₄	C₄×Dic₁₀₆	1	4	yes	—
245	C₂×C₄	C₂×C₈×D₅₃	1	4	yes	—
246	C₂×C₄	C₂×C₈×D₅₃	1	4	yes	—
247	C₂×C₄	C₂×C₈×D₅₃	1	4	yes	—
248	C₂×C₄	#13133223	1	4	yes	—
249	C₂×C₄	C₂²×C₄×D₅₃	1	2	yes	—
250	C₂×C₄	C₂²×Dic₁₀₆	1	1	yes	characteristic
251	C₂×C₄	C₂×#11719945	1	2	yes	—
252	C₂×C₄	C₂×#11719945	1	1	yes	characteristic
253	C₈	—	2	4	no	—
254	C₈	—	106	8	no	—
255	C₈	C₄×Dic₁₀₆	1	8	yes	direct product factor
256	C₂²	C₂×C₄²×D₅₃	1	1	yes	characteristic
257	C₂²	C₂×C₄×Dic₁₀₆	1	1	yes	characteristic
258	C₂²	C₄×#11719945	1	1	yes	characteristic
259	C₂²	C₂²×C₈×D₅₃	1	2	yes	—
260	C₂²	C₂×#13133223	1	2	yes	—
261	C₄	—	2	4	no	—
262	C₄	—	2	2	no	—
263	C₄	—	53	8	no	—
264	C₄	—	106	8	no	—
265	C₄	—	106	4	no	—
266	C₄	C₄×C₈×D₅₃	1	4	yes	—
267	C₄	C₈×Dic₁₀₆	1	8	yes	direct product factor
268	C₄	C₂×C₄×Dic₁₀₆	1	2	yes	—
269	C₄	C₄×#11719945	1	2	yes	—
270	C₂	C₄²×Dic₁₀₆	1	1	yes	characteristic
271	C₂	C₂×C₄×C₈×D₅₃	1	1	yes	characteristic
272	C₂	C₄×#13133223	1	1	yes	characteristic
273	C₂	C₂×C₈×Dic₁₀₆	1	2	yes	—
274	C₂	C₈×#11719945	1	2	yes	—

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

Complements

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

N	H ≅ G/N	type
11. C₈×Dic₁₀₆	C₄	direct (N × H)
31. C₄×Dic₁₀₆	C₈	direct (N × H)
58. C₄×C₈×C₅₃	Q₈	semidirect
114. Dic₁₀₆	C₄×C₈	direct (N × H)
116. C₈×C₅₃	C₄×Q₈	semidirect
127. C₄×C₅₃	C₈×Q₈	semidirect
166. C₅₃	C₄×C₈×Q₈	semidirect
193. C₄×C₈	Dic₁₀₆	direct (N × H)
255. C₈	C₄×Dic₁₀₆	direct (N × H)
267. C₄	C₈×Dic₁₀₆	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂×C₈×Dic₁₀₆	2
C₄²×Dic₁₀₆	2
C₈×#11719919	2
C₈×C₄⋊Dic₅₃	2
C₄×#13133146	2
C₄×C₄⋊#88322	2
C₄×C₈×Dic₅₃	2
C₄²×C₈×C₅₃	2
C₄×C₈×Q₈	53

Subgroups — C4×C8×Dic106 #27382698

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₄×C₈×Dic₁₀₆ #27382698