Subgroups — C₂×C₄×#14556060 #31995491

Order 18176.

1358 conjugacy classes of subgroups.

The subgroup lattice has 238 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×#14556060	C₂	1	2	yes	maximal
2	C₂×C₄×C₄⋊Dic₇₁	C₂	1	2	yes	maximal
3	C₄×#14556060	C₂	1	8	yes	maximal, direct product factor
4	C₂×#17985762	C₂	1	2	yes	maximal
5	C₂²×C₄×C₈×C₇₁	C₂	1	1	yes	maximal, characteristic
6	C₂²×C₄⋊Dic₇₁	C₂²	1	4	yes	—
7	C₂²×C₄×Dic₇₁	—	2	2	no	—
8	C₂×#14556060	C₄	1	16	yes	direct product factor
9	C₂×#14556060	C₂²	1	8	yes	—
10	C₄×C₄⋊Dic₇₁	—	2	4	no	—
11	C₄×C₄⋊Dic₇₁	C₂²	1	8	yes	—
12	C₂×C₇₁⋊#95	—	2	2	no	—
13	#17985762	C₂²	1	8	yes	—
14	C₂³×C₈×C₇₁	C₂²	1	1	yes	characteristic
15	C₂³×C₈×C₇₁	C₂²	1	1	yes	characteristic
16	C₂²×C₄²×C₇₁	C₂²	1	1	yes	characteristic
17	C₂×C₄×C₈×C₇₁	C₄	1	8	yes	complemented
18	C₂×C₄×C₈×C₇₁	C₂²	1	4	yes	—
19	C₂³×Dic₇₁	—	2	4	no	—
20	C₂×C₄⋊Dic₇₁	—	2	4	no	—
21	C₂×C₄⋊Dic₇₁	—	2	8	no	—
22	C₂×C₄⋊Dic₇₁	C₂×C₄	1	16	yes	—
23	C₂×C₄⋊Dic₇₁	C₂³	1	8	yes	—
24	C₂×C₄×Dic₇₁	—	2	8	no	—
25	C₂×C₄×Dic₇₁	—	4	8	no	—
26	#14556060	C₂×C₄	1	32	yes	direct product factor
27	C₇₁⋊#95	—	2	8	no	—
28	C₂³×C₄×C₇₁	D₄	1	1	yes	characteristic
29	C₂³×C₄×C₇₁	D₄	1	1	yes	characteristic
30	C₂³×C₄×C₇₁	C₂³	1	1	yes	characteristic
31	C₂²×C₈×C₇₁	—	2	4	no	—
32	C₂²×C₈×C₇₁	C₂×C₄	1	4	yes	—
33	C₂²×C₈×C₇₁	C₂×C₄	1	8	yes	—
34	C₂²×C₈×C₇₁	C₂×C₄	1	4	yes	—
35	C₂²×C₈×C₇₁	C₂³	1	2	yes	—
36	C₂²×C₈×C₇₁	C₂³	1	2	yes	—
37	C₂×C₄²×C₇₁	C₂×C₄	1	4	yes	—
38	C₂×C₄²×C₇₁	D₄	1	2	yes	—
39	C₂×C₄²×C₇₁	Q₈	1	4	yes	—
40	C₂×C₄²×C₇₁	C₂³	1	2	yes	—
41	C₄×C₈×C₇₁	C₂×C₄	1	16	yes	complemented
42	C₂²×Dic₇₁	—	2	16	no	—
43	C₂²×Dic₇₁	—	2	8	no	—
44	C₂²×Dic₇₁	—	4	8	no	—
45	C₂²×Dic₇₁	—	4	8	no	—
46	C₄⋊Dic₇₁	—	2	16	no	—
47	C₄⋊Dic₇₁	—	2	8	no	—
48	C₄⋊Dic₇₁	C₂²×C₄	1	16	yes	—
49	C₄×Dic₇₁	—	4	16	no	—
50	C₂⁴×C₇₁	C₂×D₄	1	1	yes	characteristic
51	C₂²×C₄×C₇₁	—	2	4	no	—
52	C₂²×C₄×C₇₁	C₄⋊C₄	1	4	yes	—
53	C₂²×C₄×C₇₁	C₄⋊C₄	1	4	yes	—
54	C₂²×C₄×C₇₁	SD₁₆	1	8	yes	—
55	C₂²×C₄×C₇₁	C₂²×C₄	1	4	yes	—
56	C₂²×C₄×C₇₁	C₂²×C₄	1	2	yes	—
57	C₂²×C₄×C₇₁	C₂×D₄	1	2	yes	—
58	C₂²×C₄×C₇₁	C₂×D₄	1	1	yes	characteristic
59	C₂²×C₄×C₇₁	C₂×D₄	1	2	yes	—
60	C₂²×C₄×C₇₁	C₂×Q₈	1	2	yes	—
61	C₂²×C₄×C₇₁	#41	1	4	yes	—
62	C₂²×C₄×C₇₁	C₂⁴	1	1	yes	characteristic
63	C₂×C₈×C₇₁	—	2	8	no	—
64	C₂×C₈×C₇₁	—	2	4	no	—
65	C₂×C₈×C₇₁	C₄²	1	16	yes	complemented
66	C₂×C₈×C₇₁	C₂²×C₄	1	8	yes	—
67	C₂×C₈×C₇₁	C₂²×C₄	1	4	yes	—
68	C₂×C₈×C₇₁	C₂²×C₄	1	4	yes	—
69	C₄²×C₇₁	C₄⋊C₄	1	8	yes	—
70	C₄²×C₇₁	C₂²×C₄	1	4	yes	—
71	C₄²×C₇₁	C₂×Q₈	1	4	yes	—
72	C₂³×C₇₁	C₂×C₄⋊C₄	1	2	yes	—
73	C₂³×C₇₁	C₄×D₄	1	4	yes	—
74	C₂³×C₇₁	C₂×SD₁₆	1	4	yes	—
75	C₂³×C₇₁	#135	1	4	yes	—
76	C₂³×C₇₁	C₂²×D₄	1	1	yes	characteristic
77	C₂×C₄×C₇₁	—	2	4	no	—
78	C₂×C₄×C₇₁	—	2	8	no	—
79	C₂×C₄×C₇₁	#106	1	16	yes	complemented
80	C₂×C₄×C₇₁	C₂×C₄²	1	4	yes	—
81	C₂×C₄×C₇₁	C₂×C₄⋊C₄	1	4	yes	—
82	C₂×C₄×C₇₁	C₂×C₄⋊C₄	1	2	yes	—
83	C₂×C₄×C₇₁	C₂×C₄⋊C₄	1	4	yes	—
84	C₂×C₄×C₇₁	#117	1	4	yes	—
85	C₂×C₄×C₇₁	C₄×D₄	1	4	yes	—
86	C₂×C₄×C₇₁	C₄×Q₈	1	8	yes	—
87	C₂×C₄×C₇₁	C₂×SD₁₆	1	8	yes	—
88	C₂×C₄×C₇₁	C₂³×C₄	1	2	yes	—
89	C₂×C₄×C₇₁	C₂³×C₄	1	1	yes	characteristic
90	C₂×C₄×C₇₁	C₂²×Q₈	1	1	yes	characteristic
91	C₂×C₄×C₇₁	C₂×#41	1	2	yes	—
92	C₂×Dic₇₁	—	2	16	no	—
93	C₂×Dic₇₁	—	4	8	no	—
94	C₂×Dic₇₁	—	4	8	no	—
95	C₂×Dic₇₁	—	4	32	no	—
96	C₈×C₇₁	—	2	4	no	—
97	C₈×C₇₁	C₂×C₄²	1	8	yes	complemented
98	C₂²×C₇₁	C₄×C₄⋊C₄	1	4	yes	—
99	C₂²×C₇₁	C₂×#106	1	4	yes	—
100	C₂²×C₇₁	#427	1	4	yes	—
101	C₂²×C₇₁	C₄×SD₁₆	1	16	yes	—
102	C₂²×C₇₁	C₂²×C₄⋊C₄	1	1	yes	characteristic
103	C₂²×C₇₁	C₂×C₄×D₄	1	2	yes	—
104	C₂²×C₇₁	C₂²×SD₁₆	1	2	yes	—
105	C₂²×C₇₁	C₂×#135	1	2	yes	—
106	C₄×C₇₁	—	2	4	no	—
107	C₄×C₇₁	C₄×C₄⋊C₄	1	4	yes	—
108	C₄×C₇₁	C₂×#106	1	8	yes	complemented
109	C₄×C₇₁	C₂²×C₄²	1	1	yes	characteristic
110	C₄×C₇₁	C₂×#117	1	1	yes	characteristic
111	C₄×C₇₁	C₂×C₄×Q₈	1	2	yes	—
112	Dic₇₁	—	4	16	no	—
113	C₂×C₄×#106	—	71	1	no	maximal, Sylow
114	C₂×C₇₁	C₄×#106	1	8	yes	complemented
115	C₂×C₇₁	C₂×C₄×C₄⋊C₄	1	1	yes	characteristic
116	C₂×C₇₁	C₂²×#106	1	1	yes	characteristic
117	C₂×C₇₁	C₂×#427	1	1	yes	characteristic
118	C₂×C₇₁	C₂×C₄×SD₁₆	1	4	yes	—
119	C₂²×#106	—	71	2	no	—
120	C₂²×C₄×C₈	D₇₁	1	1	yes	characteristic
121	C₂×C₄×C₄⋊C₄	—	71	2	no	—
122	C₄×#106	—	71	8	no	—
123	C₂×#340	—	71	2	no	—
124	C₇₁	C₂×C₄×#106	1	1	yes	characteristic, complemented, Sylow
125	C₂³×C₈	C₂×D₇₁	1	1	yes	characteristic
126	C₂³×C₈	C₂×D₇₁	1	1	yes	characteristic
127	C₂²×C₄⋊C₄	—	71	4	no	—
128	C₂²×C₄²	—	142	2	no	—
129	C₂²×C₄²	C₂×D₇₁	1	1	yes	characteristic
130	C₂×#106	—	71	16	no	—
131	C₂×#106	—	71	8	no	—
132	C₂×C₄×C₈	Dic₇₁	1	8	yes	complemented
133	C₂×C₄×C₈	C₂×D₇₁	1	4	yes	—
134	C₄×C₄⋊C₄	—	71	8	no	—
135	C₄×C₄⋊C₄	—	142	4	no	—
136	C₂×#95	—	142	2	no	—
137	#340	—	71	8	no	—
138	C₂³×C₄	—	142	4	no	—
139	C₂³×C₄	D₂₈₄	1	1	yes	characteristic
140	C₂³×C₄	D₂₈₄	1	1	yes	characteristic
141	C₂³×C₄	C₂²×D₇₁	1	1	yes	characteristic
142	C₂²×C₈	—	2	4	no	—
143	C₂²×C₈	C₄×D₇₁	1	8	yes	—
144	C₂²×C₈	C₂×Dic₇₁	1	4	yes	—
145	C₂²×C₈	C₂×Dic₇₁	1	4	yes	—
146	C₂²×C₈	C₂²×D₇₁	1	2	yes	—
147	C₂²×C₈	C₂²×D₇₁	1	2	yes	—
148	C₂×C₄⋊C₄	—	71	16	no	—
149	C₂×C₄⋊C₄	—	71	8	no	—
150	C₂×C₄⋊C₄	—	142	4	no	—
151	C₂×C₄⋊C₄	—	142	8	no	—
152	C₂×C₄²	—	284	8	no	—
153	C₂×C₄²	—	142	8	no	—
154	C₂×C₄²	Dic₁₄₂	1	4	yes	—
155	C₂×C₄²	D₂₈₄	1	2	yes	—
156	C₂×C₄²	C₂×Dic₇₁	1	4	yes	—
157	C₂×C₄²	C₂²×D₇₁	1	2	yes	—
158	#106	—	71	32	no	—
159	C₄×C₈	C₂×Dic₇₁	1	16	yes	complemented
160	#95	—	142	8	no	—
161	C₂⁴	C₂×D₂₈₄	1	1	yes	characteristic
162	C₂²×C₄	—	2	4	no	—
163	C₂²×C₄	—	284	8	no	—
164	C₂²×C₄	—	284	8	no	—
165	C₂²×C₄	—	142	8	no	—
166	C₂²×C₄	—	142	16	no	—
167	C₂²×C₄	C₄⋊Dic₇₁	1	4	yes	—
168	C₂²×C₄	C₄⋊Dic₇₁	1	4	yes	—
169	C₂²×C₄	#11765854	1	8	yes	—
170	C₂²×C₄	C₂×C₄×D₇₁	1	4	yes	—
171	C₂²×C₄	C₂²×Dic₇₁	1	2	yes	—
172	C₂²×C₄	C₂×D₂₈₄	1	2	yes	—
173	C₂²×C₄	C₂×D₂₈₄	1	1	yes	characteristic
174	C₂²×C₄	C₂×D₂₈₄	1	2	yes	—
175	C₂²×C₄	C₂×Dic₁₄₂	1	2	yes	—
176	C₂²×C₄	#11765865	1	4	yes	—
177	C₂²×C₄	C₂³×D₇₁	1	1	yes	characteristic
178	C₂×C₈	—	2	4	no	—
179	C₂×C₈	—	2	8	no	—
180	C₂×C₈	C₄×Dic₇₁	1	16	yes	complemented
181	C₂×C₈	C₂×C₄×D₇₁	1	8	yes	—
182	C₂×C₈	C₂²×Dic₇₁	1	4	yes	—
183	C₂×C₈	C₂²×Dic₇₁	1	4	yes	—
184	C₄⋊C₄	—	71	16	no	—
185	C₄⋊C₄	—	142	8	no	—
186	C₄⋊C₄	—	142	16	no	—
187	C₄²	—	284	16	no	—
188	C₄²	C₄⋊Dic₇₁	1	8	yes	—
189	C₄²	C₂²×Dic₇₁	1	4	yes	—
190	C₄²	C₂×Dic₁₄₂	1	4	yes	—
191	C₂³	C₂×C₄⋊Dic₇₁	1	2	yes	—
192	C₂³	C₄×D₂₈₄	1	4	yes	—
193	C₂³	C₂×#11765854	1	4	yes	—
194	C₂³	#14556150	1	4	yes	—
195	C₂³	C₂²×D₂₈₄	1	1	yes	characteristic
196	C₂×C₄	—	2	4	no	—
197	C₂×C₄	—	2	8	no	—
198	C₂×C₄	—	284	32	no	—
199	C₂×C₄	—	284	8	no	—
200	C₂×C₄	—	284	8	no	—
201	C₂×C₄	—	142	16	no	—
202	C₂×C₄	#14556060	1	16	yes	direct product factor
203	C₂×C₄	C₂×C₄×Dic₇₁	1	4	yes	—
204	C₂×C₄	C₂×C₄⋊Dic₇₁	1	4	yes	—
205	C₂×C₄	C₂×C₄⋊Dic₇₁	1	2	yes	—
206	C₂×C₄	C₂×C₄⋊Dic₇₁	1	4	yes	—
207	C₂×C₄	#14556087	1	4	yes	—
208	C₂×C₄	C₄×D₂₈₄	1	4	yes	—
209	C₂×C₄	C₄×Dic₁₄₂	1	8	yes	—
210	C₂×C₄	C₂×#11765854	1	8	yes	—
211	C₂×C₄	C₂²×C₄×D₇₁	1	2	yes	—
212	C₂×C₄	C₂³×Dic₇₁	1	1	yes	characteristic
213	C₂×C₄	C₂²×Dic₁₄₂	1	1	yes	characteristic
214	C₂×C₄	C₂×#11765865	1	2	yes	—
215	C₈	—	2	4	no	—
216	C₈	C₂×C₄×Dic₇₁	1	8	yes	complemented
217	C₂²	C₄×C₄⋊Dic₇₁	1	4	yes	—
218	C₂²	C₂×#14556060	1	4	yes	—
219	C₂²	#17986037	1	4	yes	—
220	C₂²	C₄×#11765854	1	16	yes	—
221	C₂²	C₂²×C₄⋊Dic₇₁	1	1	yes	characteristic
222	C₂²	C₂×C₄×D₂₈₄	1	2	yes	—
223	C₂²	C₂²×#11765854	1	2	yes	—
224	C₂²	C₂×#14556150	1	2	yes	—
225	C₄	—	2	4	no	—
226	C₄	—	284	16	no	—
227	C₄	C₄×C₄⋊Dic₇₁	1	4	yes	—
228	C₄	C₂×#14556060	1	8	yes	direct product factor
229	C₄	C₂²×C₄×Dic₇₁	1	1	yes	characteristic
230	C₄	C₂×#14556087	1	1	yes	characteristic
231	C₄	C₂×C₄×Dic₁₄₂	1	2	yes	—
232	C₂	C₄×#14556060	1	8	yes	direct product factor
233	C₂	C₂×C₄×C₄⋊Dic₇₁	1	1	yes	characteristic
234	C₂	C₂²×#14556060	1	1	yes	characteristic
235	C₂	C₂×#17986037	1	1	yes	characteristic
236	C₂	C₂×C₄×#11765854	1	4	yes	—

	subgroup	G/H
center	C₂³×C₄	D₂₈₄
commutator	C₄×C₇₁	C₂²×C₄²
Frattini	C₂²×C₄	C₂³×D₇₁
Fitting	C₂²×C₄×C₈×C₇₁	C₂
socle, prime radical	C₂⁴×C₇₁	C₂×D₄
cosocle	C₂²×C₄×C₇₁	C₂⁴
hypercenter	C₂²×C₄×C₈	D₇₁
hypocenter	C₇₁	C₂×C₄×#106
LCS	G ▷ C₄×C₇₁ ▷ C₂×C₇₁ ▷ C₇₁
UCS	C₁ ◁ C₂³×C₄ ◁ C₂²×C₄² ◁ C₂²×C₄×C₈

Complements

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

N	H ≅ G/N	type
3. C₄×#14556060	C₂	direct (N × H)
8. C₂×#14556060	C₄	direct (N × H)
17. C₂×C₄×C₈×C₇₁	C₄	semidirect
26. #14556060	C₂×C₄	direct (N × H)
41. C₄×C₈×C₇₁	C₂×C₄	semidirect
65. C₂×C₈×C₇₁	C₄²	semidirect
79. C₂×C₄×C₇₁	#106	semidirect
97. C₈×C₇₁	C₂×C₄²	semidirect
108. C₄×C₇₁	C₂×#106	semidirect
114. C₂×C₇₁	C₄×#106	semidirect
124. C₇₁	C₂×C₄×#106	semidirect
132. C₂×C₄×C₈	Dic₇₁	semidirect
159. C₄×C₈	C₂×Dic₇₁	semidirect
180. C₂×C₈	C₄×Dic₇₁	semidirect
202. C₂×C₄	#14556060	direct (N × H)
216. C₈	C₂×C₄×Dic₇₁	semidirect
228. C₄	C₂×#14556060	direct (N × H)
232. C₂	C₄×#14556060	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×#14556060	2
C₂×C₄×C₄⋊Dic₇₁	2
C₄×#14556060	2
C₂×#17985762	2
C₂²×C₄×C₈×C₇₁	2
C₂×C₄×#106	71

Subgroups — C2×C4×#14556060 #31995491

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₄×#14556060 #31995491