Subgroups — C₂²×C₈×Dic₁₄₂ #32861803

Order 18176.

2478 conjugacy classes of subgroups.

The subgroup lattice has 224 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₄×Dic₁₄₂	C₂	1	1	yes	maximal, characteristic
2	C₂×C₈×Dic₁₄₂	C₂	1	24	yes	maximal, direct product factor
3	C₂²×C₄⋊(C₇₁⋊C₈)	C₂	1	1	yes	maximal, characteristic
4	C₂²×Dic₇₁⋊C₈	C₂	1	2	yes	maximal
5	C₂²×C₈×Dic₇₁	C₂	1	2	yes	maximal
6	C₂²×C₄×C₈×C₇₁	C₂	1	1	yes	maximal, characteristic
7	C₂³×Dic₁₄₂	C₄	1	1	yes	characteristic
8	C₂³×C₇₁⋊C₈	C₂²	1	2	yes	—
9	C₂×C₄×Dic₁₄₂	C₄	1	12	yes	—
10	C₂×C₄×Dic₁₄₂	C₂²	1	12	yes	—
11	C₂²×C₄⋊Dic₇₁	C₄	1	1	yes	characteristic
12	C₂²×Dic₇₁⋊C₄	C₄	1	2	yes	—
13	C₂²×C₄×Dic₇₁	C₂²	1	2	yes	—
14	C₈×Dic₁₄₂	C₂²	1	64	yes	direct product factor
15	C₂×C₄⋊(C₇₁⋊C₈)	C₂²	1	12	yes	—
16	C₂×Dic₇₁⋊C₈	C₂²	1	24	yes	—
17	C₂×C₈×Dic₇₁	C₂²	1	24	yes	—
18	C₂³×C₈×C₇₁	C₂²	1	1	yes	characteristic
19	C₂³×C₈×C₇₁	C₂²	1	1	yes	characteristic
20	C₂²×C₄²×C₇₁	C₂²	1	1	yes	characteristic
21	C₂×C₄×C₈×C₇₁	C₂²	1	12	yes	—
22	C₂²×Dic₁₄₂	C₈	1	16	yes	direct product factor
23	C₂²×Dic₁₄₂	C₂×C₄	1	12	yes	—
24	C₂³×Dic₇₁	C₂×C₄	1	2	yes	—
25	C₂³×Dic₇₁	C₂×C₄	1	2	yes	—
26	C₂²×C₇₁⋊C₈	—	2	8	no	—
27	C₂²×C₇₁⋊C₈	C₂³	1	12	yes	—
28	C₄×Dic₁₄₂	C₂×C₄	1	48	yes	—
29	C₄×Dic₁₄₂	C₂³	1	16	yes	—
30	C₂×C₄⋊Dic₇₁	C₂×C₄	1	12	yes	—
31	C₂×Dic₇₁⋊C₄	C₂×C₄	1	24	yes	—
32	C₂×C₄×Dic₇₁	C₂×C₄	1	12	yes	—
33	C₂×C₄×Dic₇₁	C₂³	1	12	yes	—
34	C₄⋊(C₇₁⋊C₈)	C₂³	1	16	yes	—
35	Dic₇₁⋊C₈	C₂³	1	32	yes	—
36	C₈×Dic₇₁	C₂³	1	32	yes	—
37	C₂³×C₄×C₇₁	C₂×C₄	1	1	yes	characteristic
38	C₂³×C₄×C₇₁	C₂×C₄	1	1	yes	characteristic
39	C₂³×C₄×C₇₁	C₂³	1	1	yes	characteristic
40	C₂²×C₈×C₇₁	—	2	4	no	—
41	C₂²×C₈×C₇₁	Q₈	1	8	yes	complemented
42	C₂²×C₈×C₇₁	C₂³	1	6	yes	—
43	C₂²×C₈×C₇₁	C₂³	1	6	yes	—
44	C₂×C₄²×C₇₁	C₂×C₄	1	6	yes	—
45	C₂×C₄²×C₇₁	C₂³	1	6	yes	—
46	C₄×C₈×C₇₁	C₂³	1	16	yes	—
47	C₂×Dic₁₄₂	C₂×C₈	1	96	yes	direct product factor
48	C₂×Dic₁₄₂	C₂²×C₄	1	16	yes	—
49	C₂²×Dic₇₁	—	2	8	no	—
50	C₂²×Dic₇₁	C₂×C₈	1	16	yes	—
51	C₂²×Dic₇₁	C₂²×C₄	1	12	yes	—
52	C₂²×Dic₇₁	C₂²×C₄	1	12	yes	—
53	C₂×C₇₁⋊C₈	—	2	24	no	—
54	C₂×C₇₁⋊C₈	C₂⁴	1	8	yes	—
55	C₄⋊Dic₇₁	C₂²×C₄	1	16	yes	—
56	Dic₇₁⋊C₄	C₂²×C₄	1	32	yes	—
57	C₄×Dic₇₁	C₂²×C₄	1	24	yes	—
58	C₄×Dic₇₁	C₂⁴	1	8	yes	—
59	C₂⁴×C₇₁	C₂²×C₄	1	1	yes	characteristic
60	C₂²×C₄×C₇₁	—	2	4	no	—
61	C₂²×C₄×C₇₁	C₂×C₈	1	8	yes	—
62	C₂²×C₄×C₇₁	C₂²×C₄	1	6	yes	—
63	C₂²×C₄×C₇₁	C₂²×C₄	1	3	yes	—
64	C₂²×C₄×C₇₁	C₂²×C₄	1	6	yes	—
65	C₂²×C₄×C₇₁	C₂×Q₈	1	4	yes	—
66	C₂²×C₄×C₇₁	#41	1	4	yes	—
67	C₂²×C₄×C₇₁	C₂⁴	1	3	yes	—
68	C₂×C₈×C₇₁	—	2	12	no	—
69	C₂×C₈×C₇₁	C₂×Q₈	1	24	yes	complemented
70	C₂×C₈×C₇₁	C₂⁴	1	4	yes	—
71	C₂×C₈×C₇₁	C₂⁴	1	4	yes	—
72	C₄²×C₇₁	C₂²×C₄	1	12	yes	—
73	C₄²×C₇₁	C₂⁴	1	4	yes	—
74	C₂³×C₇₁	C₄×Q₈	1	4	yes	—
75	C₂³×C₇₁	C₂²×C₈	1	4	yes	—
76	C₂³×C₇₁	#131	1	4	yes	—
77	C₂³×C₇₁	C₂³×C₄	1	3	yes	—
78	C₂×C₄×C₇₁	—	2	12	no	—
79	C₂×C₄×C₇₁	C₄×Q₈	1	12	yes	—
80	C₂×C₄×C₇₁	C₂²×C₈	1	24	yes	—
81	C₂×C₄×C₇₁	C₂³×C₄	1	4	yes	—
82	C₂×C₄×C₇₁	C₂³×C₄	1	3	yes	—
83	C₂×C₄×C₇₁	C₂³×C₄	1	4	yes	—
84	C₂×C₄×C₇₁	C₂²×Q₈	1	6	yes	—
85	C₂×C₄×C₇₁	C₂×#41	1	6	yes	—
86	C₂×C₄×C₇₁	C₂⁵	1	1	yes	characteristic
87	C₂×Dic₇₁	—	2	24	no	—
88	C₂×Dic₇₁	C₂²×C₈	1	48	yes	—
89	C₂×Dic₇₁	C₂³×C₄	1	8	yes	—
90	C₂×Dic₇₁	C₂³×C₄	1	8	yes	—
91	Dic₁₄₂	C₂²×C₈	1	64	yes	direct product factor
92	C₈×C₇₁	—	2	4	no	—
93	C₈×C₇₁	C₂²×Q₈	1	8	yes	complemented
94	C₇₁⋊C₈	—	2	8	no	—
95	C₂²×C₇₁	C₈×Q₈	1	16	yes	complemented
96	C₂²×C₇₁	C₂×C₄×Q₈	1	6	yes	—
97	C₂²×C₇₁	C₂³×C₈	1	6	yes	—
98	C₂²×C₇₁	C₂×#131	1	6	yes	—
99	C₂²×C₇₁	C₂⁴×C₄	1	1	yes	characteristic
100	C₄×C₇₁	—	2	4	no	—
101	C₄×C₇₁	C₂×C₄×Q₈	1	6	yes	—
102	C₄×C₇₁	C₂³×C₈	1	8	yes	—
103	C₄×C₇₁	C₂³×Q₈	1	1	yes	characteristic
104	C₄×C₇₁	C₂²×#41	1	1	yes	characteristic
105	Dic₇₁	—	2	8	no	—
106	Dic₇₁	C₂³×C₈	1	16	yes	—
107	C₂²×C₈×Q₈	—	71	1	no	maximal, Sylow
108	C₂×C₇₁	C₂×C₈×Q₈	1	12	yes	complemented
109	C₂×C₇₁	C₂²×C₄×Q₈	1	1	yes	characteristic
110	C₂×C₇₁	C₂⁴×C₈	1	1	yes	characteristic
111	C₂×C₇₁	C₂²×#131	1	1	yes	characteristic
112	C₂²×C₄×Q₈	—	71	1	no	—
113	C₂×C₈×Q₈	—	71	24	no	—
114	C₂²×C₄⋊C₈	—	71	1	no	—
115	C₂²×C₄⋊C₈	—	71	2	no	—
116	C₂²×C₄×C₈	—	71	2	no	—
117	C₂²×C₄×C₈	D₇₁	1	1	yes	characteristic
118	C₇₁	C₂²×C₈×Q₈	1	1	yes	characteristic, complemented, Sylow
119	C₂³×Q₈	—	71	1	no	—
120	C₂³×C₈	—	71	2	no	—
121	C₂³×C₈	C₂×D₇₁	1	1	yes	characteristic
122	C₂³×C₈	C₂×D₇₁	1	1	yes	characteristic
123	C₂×C₄×Q₈	—	71	12	no	—
124	C₂×C₄×Q₈	—	71	12	no	—
125	C₂²×C₄⋊C₄	—	71	2	no	—
126	C₂²×C₄⋊C₄	—	71	1	no	—
127	C₂²×C₄²	—	71	2	no	—
128	C₂²×C₄²	C₂×D₇₁	1	1	yes	characteristic
129	C₈×Q₈	—	71	64	no	—
130	C₂×C₄⋊C₈	—	71	24	no	—
131	C₂×C₄⋊C₈	—	71	12	no	—
132	C₂×C₄×C₈	—	71	24	no	—
133	C₂×C₄×C₈	C₂×D₇₁	1	12	yes	—
134	C₂²×Q₈	—	71	16	no	—
135	C₂²×Q₈	—	71	12	no	—
136	C₂³×C₄	—	71	2	no	—
137	C₂³×C₄	—	71	2	no	—
138	C₂³×C₄	C₄×D₇₁	1	1	yes	characteristic
139	C₂³×C₄	C₄×D₇₁	1	1	yes	characteristic
140	C₂³×C₄	C₂²×D₇₁	1	1	yes	characteristic
141	C₂²×C₈	—	2	4	no	—
142	C₂²×C₈	—	71	12	no	—
143	C₂²×C₈	—	142	8	no	—
144	C₂²×C₈	Dic₁₄₂	1	8	yes	direct product factor
145	C₂²×C₈	C₂²×D₇₁	1	6	yes	—
146	C₂²×C₈	C₂²×D₇₁	1	6	yes	—
147	C₄×Q₈	—	71	48	no	—
148	C₄×Q₈	—	71	16	no	—
149	C₂×C₄⋊C₄	—	71	24	no	—
150	C₂×C₄⋊C₄	—	71	12	no	—
151	C₂×C₄²	—	71	12	no	—
152	C₂×C₄²	—	71	12	no	—
153	C₂×C₄²	C₄×D₇₁	1	6	yes	—
154	C₂×C₄²	C₂²×D₇₁	1	6	yes	—
155	C₄⋊C₈	—	71	32	no	—
156	C₄⋊C₈	—	71	16	no	—
157	C₄×C₈	—	71	32	no	—
158	C₄×C₈	C₂²×D₇₁	1	16	yes	—
159	C₂⁴	C₂×C₄×D₇₁	1	1	yes	characteristic
160	C₂×Q₈	—	71	16	no	—
161	C₂×Q₈	—	71	96	no	—
162	C₂²×C₄	—	2	4	no	—
163	C₂²×C₄	—	71	12	no	—
164	C₂²×C₄	—	71	16	no	—
165	C₂²×C₄	—	71	12	no	—
166	C₂²×C₄	—	142	8	no	—
167	C₂²×C₄	C₈×D₇₁	1	8	yes	—
168	C₂²×C₄	C₂×C₄×D₇₁	1	6	yes	—
169	C₂²×C₄	C₂×C₄×D₇₁	1	3	yes	—
170	C₂²×C₄	C₂×C₄×D₇₁	1	6	yes	—
171	C₂²×C₄	C₂×Dic₁₄₂	1	4	yes	—
172	C₂²×C₄	#11765865	1	4	yes	—
173	C₂²×C₄	C₂³×D₇₁	1	3	yes	—
174	C₂×C₈	—	2	12	no	—
175	C₂×C₈	—	71	8	no	—
176	C₂×C₈	—	142	24	no	—
177	C₂×C₈	C₂×Dic₁₄₂	1	24	yes	direct product factor
178	C₂×C₈	C₂³×D₇₁	1	4	yes	—
179	C₂×C₈	C₂³×D₇₁	1	4	yes	—
180	C₄⋊C₄	—	71	32	no	—
181	C₄⋊C₄	—	71	16	no	—
182	C₄²	—	71	24	no	—
183	C₄²	—	71	8	no	—
184	C₄²	C₂×C₄×D₇₁	1	12	yes	—
185	C₄²	C₂³×D₇₁	1	4	yes	—
186	C₂³	C₄×Dic₁₄₂	1	4	yes	—
187	C₂³	C₂×C₈×D₇₁	1	4	yes	—
188	C₂³	#14556134	1	4	yes	—
189	C₂³	C₂²×C₄×D₇₁	1	3	yes	—
190	Q₈	—	71	64	no	—
191	C₂×C₄	—	2	12	no	—
192	C₂×C₄	—	71	8	no	—
193	C₂×C₄	—	71	48	no	—
194	C₂×C₄	—	71	8	no	—
195	C₂×C₄	—	142	24	no	—
196	C₂×C₄	C₄×Dic₁₄₂	1	12	yes	—
197	C₂×C₄	C₂×C₈×D₇₁	1	24	yes	—
198	C₂×C₄	C₂²×C₄×D₇₁	1	4	yes	—
199	C₂×C₄	C₂²×C₄×D₇₁	1	3	yes	—
200	C₂×C₄	C₂²×C₄×D₇₁	1	4	yes	—
201	C₂×C₄	C₂²×Dic₁₄₂	1	6	yes	—
202	C₂×C₄	C₂×#11765865	1	6	yes	—
203	C₂×C₄	C₂⁴×D₇₁	1	1	yes	characteristic
204	C₈	—	2	4	no	—
205	C₈	—	142	8	no	—
206	C₈	C₂²×Dic₁₄₂	1	8	yes	direct product factor
207	C₂²	C₈×Dic₁₄₂	1	16	yes	direct product factor
208	C₂²	C₂×C₄×Dic₁₄₂	1	6	yes	—
209	C₂²	C₂²×C₈×D₇₁	1	6	yes	—
210	C₂²	C₂×#14556134	1	6	yes	—
211	C₂²	C₂³×C₄×D₇₁	1	1	yes	characteristic
212	C₄	—	2	4	no	—
213	C₄	—	71	16	no	—
214	C₄	—	142	8	no	—
215	C₄	C₂×C₄×Dic₁₄₂	1	6	yes	—
216	C₄	C₂²×C₈×D₇₁	1	8	yes	—
217	C₄	C₂³×Dic₁₄₂	1	1	yes	characteristic
218	C₄	C₂²×#11765865	1	1	yes	characteristic
219	C₂	C₂×C₈×Dic₁₄₂	1	12	yes	direct product factor
220	C₂	C₂²×C₄×Dic₁₄₂	1	1	yes	characteristic
221	C₂	C₂³×C₈×D₇₁	1	1	yes	characteristic
222	C₂	C₂²×#14556134	1	1	yes	characteristic

	subgroup	G/H
center	C₂³×C₈	C₂×D₇₁
commutator	C₂×C₇₁	C₂⁴×C₈
Frattini	C₂×C₄	C₂⁴×D₇₁
Fitting	C₂²×C₄×C₈×C₇₁	C₂
socle, prime radical	C₂⁴×C₇₁	C₂²×C₄
cosocle	C₂×C₄×C₇₁	C₂⁵
hypercenter	C₂²×C₄×C₈	D₇₁
hypocenter	C₇₁	C₂²×C₈×Q₈
LCS	G ▷ C₂×C₇₁ ▷ C₇₁
UCS	C₁ ◁ C₂³×C₈ ◁ C₂²×C₄×C₈

Complements

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

N	H ≅ G/N	type
2. C₂×C₈×Dic₁₄₂	C₂	direct (N × H)
14. C₈×Dic₁₄₂	C₂²	direct (N × H)
22. C₂²×Dic₁₄₂	C₈	direct (N × H)
41. C₂²×C₈×C₇₁	Q₈	semidirect
47. C₂×Dic₁₄₂	C₂×C₈	direct (N × H)
69. C₂×C₈×C₇₁	C₂×Q₈	semidirect
91. Dic₁₄₂	C₂²×C₈	direct (N × H)
93. C₈×C₇₁	C₂²×Q₈	semidirect
95. C₂²×C₇₁	C₈×Q₈	semidirect
108. C₂×C₇₁	C₂×C₈×Q₈	semidirect
118. C₇₁	C₂²×C₈×Q₈	semidirect
144. C₂²×C₈	Dic₁₄₂	direct (N × H)
177. C₂×C₈	C₂×Dic₁₄₂	direct (N × H)
206. C₈	C₂²×Dic₁₄₂	direct (N × H)
207. C₂²	C₈×Dic₁₄₂	direct (N × H)
219. C₂	C₂×C₈×Dic₁₄₂	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×Dic₁₄₂	2
C₂×C₈×Dic₁₄₂	2
C₂²×C₄⋊(C₇₁⋊C₈)	2
C₂²×Dic₇₁⋊C₈	2
C₂²×C₈×Dic₇₁	2
C₂²×C₄×C₈×C₇₁	2
C₂²×C₈×Q₈	71

Subgroups — C22×C8×Dic142 #32861803

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂²×C₈×Dic₁₄₂ #32861803