Subgroups — C₂²×D₄×#168 #12963132

Order 1280.

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	C₂²×C₄×C₅×D₄	C₂	1	1	yes	maximal, characteristic
2	C₂⁴×#168	C₂	1	2	yes	maximal, complemented
3	C₂²×C₂²⋊#168	C₂	1	2	yes	maximal, complemented
4	C₂×D₄×#168	C₂	1	24	yes	maximal, direct product factor
5	C₂²×C₄⋊#168	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂²×C₄×#168	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂³×C₅×D₄	C₄	1	1	yes	characteristic
8	C₂⁴×C₄×C₅	C₂²	1	2	yes	—
9	C₂×C₄×C₅×D₄	C₄	1	12	yes	—
10	C₂×C₄×C₅×D₄	C₂²	1	12	yes	—
11	C₂²×C₅×C₄⋊C₄	C₄	1	1	yes	characteristic
12	C₂²×C₅×C₂²⋊C₄	C₄	1	2	yes	—
13	C₂²×C₄²×C₅	C₂²	1	1	yes	characteristic
14	C₂³×#168	—	2	16	no	—
15	C₂³×#168	C₂²	1	1	yes	characteristic
16	C₂³×#168	C₂²	1	2	yes	—
17	C₂³×#168	C₂²	1	1	yes	characteristic
18	C₂³×#168	C₂²	1	24	yes	complemented
19	C₂×C₂²⋊#168	C₂²	1	24	yes	complemented
20	D₄×#168	C₂²	1	64	yes	direct product factor
21	C₂×C₄⋊#168	C₂²	1	12	yes	complemented
22	C₂×C₄×#168	C₂²	1	12	yes	complemented
23	C₂²×C₈×D₄	—	5	1	no	maximal, Sylow
24	C₂⁵×C₅	C₂×C₄	1	2	yes	—
25	C₂²×C₅×D₄	C₈	1	16	yes	complemented
26	C₂²×C₅×D₄	C₂×C₄	1	12	yes	—
27	C₂³×C₄×C₅	—	2	8	no	—
28	C₂³×C₄×C₅	—	2	8	no	—
29	C₂³×C₄×C₅	C₂×C₄	1	1	yes	characteristic
30	C₂³×C₄×C₅	C₂×C₄	1	2	yes	—
31	C₂³×C₄×C₅	C₂×C₄	1	12	yes	—
32	C₂³×C₄×C₅	C₂×C₄	1	1	yes	characteristic
33	C₂³×C₄×C₅	C₂³	1	12	yes	—
34	C₂³×C₄×C₅	C₂³	1	1	yes	characteristic
35	C₄×C₅×D₄	C₂×C₄	1	48	yes	—
36	C₄×C₅×D₄	C₂³	1	16	yes	—
37	C₂×C₅×C₄⋊C₄	C₂×C₄	1	12	yes	—
38	C₂×C₅×C₂²⋊C₄	C₂×C₄	1	24	yes	—
39	C₂×C₄²×C₅	C₂×C₄	1	6	yes	—
40	C₂×C₄²×C₅	C₂³	1	6	yes	—
41	C₂²×#168	—	2	8	no	—
42	C₂²×#168	—	2	4	no	—
43	C₂²×#168	—	2	96	no	—
44	C₂²×#168	D₄	1	8	yes	direct product factor
45	C₂²×#168	C₂³	1	12	yes	—
46	C₂²×#168	C₂³	1	6	yes	—
47	C₂²×#168	C₂³	1	6	yes	—
48	C₂²×#168	C₂³	1	32	yes	complemented
49	C₂²⋊#168	C₂³	1	32	yes	complemented
50	C₄⋊#168	C₂³	1	16	yes	complemented
51	C₄×#168	C₂³	1	16	yes	complemented
52	C₂⁴×C₈	—	5	2	no	—
53	C₂²×C₄×D₄	D₅	1	1	yes	characteristic
54	C₂×C₈×D₄	—	5	24	no	—
55	C₂²×C₄⋊C₈	—	5	1	no	—
56	C₂²×C₂²⋊C₈	—	5	2	no	—
57	C₂²×C₄×C₈	—	5	1	no	—
58	C₂⁴×C₅	—	2	8	no	—
59	C₂⁴×C₅	—	2	8	no	—
60	C₂⁴×C₅	C₂×C₈	1	16	yes	complemented
61	C₂⁴×C₅	C₂²×C₄	1	12	yes	—
62	C₂⁴×C₅	C₂²×C₄	1	1	yes	characteristic
63	C₂×C₅×D₄	C₂×C₈	1	96	yes	complemented
64	C₂×C₅×D₄	C₂²×C₄	1	16	yes	—
65	C₂²×C₄×C₅	—	2	8	no	—
66	C₂²×C₄×C₅	—	2	4	no	—
67	C₂²×C₄×C₅	—	2	48	no	—
68	C₂²×C₄×C₅	—	2	24	no	—
69	C₂²×C₄×C₅	—	2	24	no	—
70	C₂²×C₄×C₅	C₂×C₈	1	8	yes	complemented
71	C₂²×C₄×C₅	C₂²×C₄	1	6	yes	—
72	C₂²×C₄×C₅	C₂²×C₄	1	12	yes	—
73	C₂²×C₄×C₅	C₂²×C₄	1	24	yes	—
74	C₂²×C₄×C₅	C₂²×C₄	1	3	yes	—
75	C₂²×C₄×C₅	C₂²×C₄	1	6	yes	—
76	C₂²×C₄×C₅	C₂×D₄	1	4	yes	—
77	C₂²×C₄×C₅	#41	1	4	yes	—
78	C₂²×C₄×C₅	C₂⁴	1	8	yes	—
79	C₂²×C₄×C₅	C₂⁴	1	3	yes	—
80	C₅×C₄⋊C₄	C₂²×C₄	1	16	yes	—
81	C₅×C₂²⋊C₄	C₂²×C₄	1	32	yes	—
82	C₄²×C₅	C₂²×C₄	1	12	yes	—
83	C₄²×C₅	C₂⁴	1	4	yes	—
84	C₂×#168	—	2	24	no	—
85	C₂×#168	—	2	12	no	—
86	C₂×#168	—	2	64	no	—
87	C₂×#168	C₂×D₄	1	24	yes	direct product factor
88	C₂×#168	C₂⁴	1	4	yes	—
89	C₂×#168	C₂⁴	1	8	yes	—
90	C₂×#168	C₂⁴	1	4	yes	—
91	C₂³×D₄	Dic₅	1	1	yes	characteristic
92	C₂⁴×C₄	C₂×D₅	1	2	yes	—
93	C₂³×C₈	—	5	1	no	—
94	C₂³×C₈	—	5	24	no	—
95	C₂³×C₈	—	5	1	no	—
96	C₂³×C₈	—	5	2	no	—
97	C₂³×C₈	—	10	16	no	—
98	C₂×C₄×D₄	Dic₅	1	12	yes	—
99	C₂×C₄×D₄	C₂×D₅	1	12	yes	—
100	C₂²×C₄⋊C₄	Dic₅	1	1	yes	characteristic
101	C₂²×C₂²⋊C₄	Dic₅	1	2	yes	—
102	C₂²×C₄²	C₂×D₅	1	1	yes	characteristic
103	C₈×D₄	—	5	64	no	—
104	C₂×C₄⋊C₈	—	5	12	no	—
105	C₂×C₂²⋊C₈	—	5	24	no	—
106	C₂×C₄×C₈	—	5	12	no	—
107	C₂³×C₅	—	2	24	no	—
108	C₂³×C₅	—	2	24	no	—
109	C₂³×C₅	—	2	64	no	—
110	C₂³×C₅	C₄×D₄	1	4	yes	—
111	C₂³×C₅	C₂²×C₈	1	4	yes	—
112	C₂³×C₅	C₂²×C₈	1	48	yes	complemented
113	C₂³×C₅	#131	1	4	yes	—
114	C₂³×C₅	C₂³×C₄	1	8	yes	—
115	C₂³×C₅	C₂³×C₄	1	3	yes	—
116	C₅×D₄	C₂²×C₈	1	64	yes	complemented
117	C₂×C₄×C₅	—	2	24	no	—
118	C₂×C₄×C₅	—	2	12	no	—
119	C₂×C₄×C₅	—	2	48	no	—
120	C₂×C₄×C₅	—	2	8	no	—
121	C₂×C₄×C₅	—	2	8	no	—
122	C₂×C₄×C₅	C₄×D₄	1	12	yes	—
123	C₂×C₄×C₅	C₂²×C₈	1	24	yes	complemented
124	C₂×C₄×C₅	C₂³×C₄	1	4	yes	—
125	C₂×C₄×C₅	C₂³×C₄	1	8	yes	—
126	C₂×C₄×C₅	C₂³×C₄	1	3	yes	—
127	C₂×C₄×C₅	C₂³×C₄	1	4	yes	—
128	C₂×C₄×C₅	C₂²×D₄	1	6	yes	—
129	C₂×C₄×C₅	C₂×#41	1	6	yes	—
130	C₂×C₄×C₅	C₂⁵	1	1	yes	characteristic
131	#168	—	2	8	no	—
132	#168	—	2	4	no	—
133	#168	C₂²×D₄	1	8	yes	direct product factor
134	C₂⁵	C₂×Dic₅	1	2	yes	—
135	C₂²×D₄	#168	1	16	yes	direct product factor
136	C₂²×D₄	C₂×Dic₅	1	12	yes	—
137	C₂³×C₄	—	2	8	no	—
138	C₂³×C₄	—	2	8	no	—
139	C₂³×C₄	C₂×Dic₅	1	1	yes	characteristic
140	C₂³×C₄	C₂×Dic₅	1	2	yes	—
141	C₂³×C₄	C₂×Dic₅	1	12	yes	—
142	C₂³×C₄	C₂×Dic₅	1	1	yes	characteristic
143	C₂³×C₄	C₂²×D₅	1	12	yes	—
144	C₂³×C₄	C₂²×D₅	1	1	yes	characteristic
145	C₂²×C₈	—	5	8	no	—
146	C₂²×C₈	—	5	6	no	—
147	C₂²×C₈	—	5	12	no	—
148	C₂²×C₈	—	5	32	no	—
149	C₂²×C₈	—	5	6	no	—
150	C₂²×C₈	—	10	96	no	—
151	C₂²×C₈	—	10	8	no	—
152	C₂²×C₈	—	10	4	no	—
153	C₄×D₄	C₂×Dic₅	1	48	yes	—
154	C₄×D₄	C₂²×D₅	1	16	yes	—
155	C₂×C₄⋊C₄	C₂×Dic₅	1	12	yes	—
156	C₂×C₂²⋊C₄	C₂×Dic₅	1	24	yes	—
157	C₂×C₄²	C₂×Dic₅	1	6	yes	—
158	C₂×C₄²	C₂²×D₅	1	6	yes	—
159	C₄⋊C₈	—	5	16	no	—
160	C₂²⋊C₈	—	5	32	no	—
161	C₄×C₈	—	5	16	no	—
162	C₂²×C₅	—	2	8	no	—
163	C₂²×C₅	—	2	96	no	—
164	C₂²×C₅	—	2	8	no	—
165	C₂²×C₅	C₈×D₄	1	16	yes	complemented
166	C₂²×C₅	C₂×C₄×D₄	1	6	yes	—
167	C₂²×C₅	C₂³×C₈	1	6	yes	—
168	C₂²×C₅	C₂³×C₈	1	16	yes	complemented
169	C₂²×C₅	C₂×#131	1	6	yes	—
170	C₂²×C₅	C₂⁴×C₄	1	1	yes	characteristic
171	C₄×C₅	—	2	8	no	—
172	C₄×C₅	—	2	4	no	—
173	C₄×C₅	C₂×C₄×D₄	1	6	yes	—
174	C₄×C₅	C₂³×C₈	1	8	yes	complemented
175	C₄×C₅	C₂³×D₄	1	1	yes	characteristic
176	C₄×C₅	C₂²×#41	1	1	yes	characteristic
177	C₂⁴	—	2	8	no	—
178	C₂⁴	—	2	8	no	—
179	C₂⁴	C₂×#168	1	16	yes	complemented
180	C₂⁴	C₂²×Dic₅	1	12	yes	—
181	C₂⁴	C₂²×Dic₅	1	1	yes	characteristic
182	C₂×D₄	C₂×#168	1	96	yes	direct product factor
183	C₂×D₄	C₂²×Dic₅	1	16	yes	—
184	C₂²×C₄	—	2	8	no	—
185	C₂²×C₄	—	2	4	no	—
186	C₂²×C₄	—	2	24	no	—
187	C₂²×C₄	—	2	24	no	—
188	C₂²×C₄	—	2	48	no	—
189	C₂²×C₄	C₂×#168	1	8	yes	complemented
190	C₂²×C₄	D₄×D₅	1	4	yes	—
191	C₂²×C₄	#711	1	4	yes	—
192	C₂²×C₄	C₂²×Dic₅	1	6	yes	—
193	C₂²×C₄	C₂²×Dic₅	1	12	yes	—
194	C₂²×C₄	C₂²×Dic₅	1	24	yes	—
195	C₂²×C₄	C₂²×Dic₅	1	6	yes	—
196	C₂²×C₄	C₂²×Dic₅	1	3	yes	—
197	C₂²×C₄	C₂³×D₅	1	8	yes	—
198	C₂²×C₄	C₂³×D₅	1	3	yes	—
199	C₂×C₈	—	5	24	no	—
200	C₂×C₈	—	5	4	no	—
201	C₂×C₈	—	5	4	no	—
202	C₂×C₈	—	5	8	no	—
203	C₂×C₈	—	10	64	no	—
204	C₂×C₈	—	10	24	no	—
205	C₂×C₈	—	10	12	no	—
206	C₄⋊C₄	C₂²×Dic₅	1	16	yes	—
207	C₂²⋊C₄	C₂²×Dic₅	1	32	yes	—
208	C₄²	C₂²×Dic₅	1	12	yes	—
209	C₄²	C₂³×D₅	1	4	yes	—
210	C₂×C₅	—	2	16	no	—
211	C₂×C₅	C₂×C₈×D₄	1	12	yes	complemented
212	C₂×C₅	C₂²×C₄×D₄	1	1	yes	characteristic
213	C₂×C₅	C₂⁴×C₈	1	1	yes	characteristic
214	C₂×C₅	C₂²×#131	1	1	yes	characteristic
215	C₂³	—	2	64	no	—
216	C₂³	—	2	24	no	—
217	C₂³	—	2	24	no	—
218	C₂³	C₂²×#168	1	4	yes	—
219	C₂³	C₂²×#168	1	48	yes	complemented
220	C₂³	D₄×Dic₅	1	4	yes	—
221	C₂³	#4094	1	4	yes	—
222	C₂³	C₂³×Dic₅	1	8	yes	—
223	C₂³	C₂³×Dic₅	1	3	yes	—
224	D₄	C₂²×#168	1	64	yes	direct product factor
225	C₂×C₄	—	2	24	no	—
226	C₂×C₄	—	2	12	no	—
227	C₂×C₄	—	2	48	no	—
228	C₂×C₄	—	2	8	no	—
229	C₂×C₄	—	2	8	no	—
230	C₂×C₄	C₂²×#168	1	24	yes	complemented
231	C₂×C₄	D₄×Dic₅	1	12	yes	—
232	C₂×C₄	C₂×D₄×D₅	1	6	yes	—
233	C₂×C₄	C₂×#711	1	6	yes	—
234	C₂×C₄	C₂³×Dic₅	1	4	yes	—
235	C₂×C₄	C₂³×Dic₅	1	8	yes	—
236	C₂×C₄	C₂³×Dic₅	1	4	yes	—
237	C₂×C₄	C₂³×Dic₅	1	3	yes	—
238	C₂×C₄	C₂⁴×D₅	1	1	yes	characteristic
239	C₈	—	5	8	no	—
240	C₈	—	10	8	no	—
241	C₈	—	10	4	no	—
242	C₅	C₂²×C₈×D₄	1	1	yes	characteristic, complemented, Sylow
243	C₂²	—	2	96	no	—
244	C₂²	—	2	8	no	—
245	C₂²	—	2	8	no	—
246	C₂²	D₄×#168	1	16	yes	direct product factor
247	C₂²	C₂³×#168	1	6	yes	—
248	C₂²	C₂³×#168	1	16	yes	complemented
249	C₂²	C₂×D₄×Dic₅	1	6	yes	—
250	C₂²	C₂×#4094	1	6	yes	—
251	C₂²	C₂⁴×Dic₅	1	1	yes	characteristic
252	C₄	—	2	4	no	—
253	C₄	—	2	8	no	—
254	C₄	C₂³×#168	1	8	yes	complemented
255	C₄	C₂×D₄×Dic₅	1	6	yes	—
256	C₄	C₂²×D₄×D₅	1	1	yes	characteristic
257	C₄	C₂²×#711	1	1	yes	characteristic
258	C₂	—	2	16	no	—
259	C₂	C₂×D₄×#168	1	12	yes	direct product factor
260	C₂	C₂⁴×#168	1	1	yes	characteristic
261	C₂	C₂²×D₄×Dic₅	1	1	yes	characteristic
262	C₂	C₂²×#4094	1	1	yes	characteristic

	subgroup	G/H
center	C₂³×C₄	C₂²×D₅
commutator	C₂×C₅	C₂⁴×C₈
Frattini	C₂×C₄	C₂⁴×D₅
Fitting	C₂²×C₄×C₅×D₄	C₂
socle	C₂⁴×C₅	C₂²×C₄
prime radical	C₂³×C₅×D₄	C₄
cosocle	C₂×C₄×C₅	C₂⁵
hypercenter	C₂²×C₄×D₄	D₅
hypocenter	C₅	C₂²×C₈×D₄
LCS	G ▷ C₂×C₅ ▷ C₅
UCS	C₁ ◁ C₂³×C₄ ◁ C₂²×C₄×D₄

Complements

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

N	H ≅ G/N	type
2. C₂⁴×#168	C₂	semidirect
3. C₂²×C₂²⋊#168	C₂	semidirect
4. C₂×D₄×#168	C₂	direct (N × H)
5. C₂²×C₄⋊#168	C₂	semidirect
6. C₂²×C₄×#168	C₂	semidirect
18. C₂³×#168	C₂²	semidirect
19. C₂×C₂²⋊#168	C₂²	semidirect
20. D₄×#168	C₂²	direct (N × H)
21. C₂×C₄⋊#168	C₂²	semidirect
22. C₂×C₄×#168	C₂²	semidirect
25. C₂²×C₅×D₄	C₈	semidirect
44. C₂²×#168	D₄	direct (N × H)
48. C₂²×#168	C₂³	semidirect
49. C₂²⋊#168	C₂³	semidirect
50. C₄⋊#168	C₂³	semidirect
51. C₄×#168	C₂³	semidirect
60. C₂⁴×C₅	C₂×C₈	semidirect
63. C₂×C₅×D₄	C₂×C₈	semidirect
70. C₂²×C₄×C₅	C₂×C₈	semidirect
87. C₂×#168	C₂×D₄	direct (N × H)
112. C₂³×C₅	C₂²×C₈	semidirect
116. C₅×D₄	C₂²×C₈	semidirect
123. C₂×C₄×C₅	C₂²×C₈	semidirect
133. #168	C₂²×D₄	direct (N × H)
135. C₂²×D₄	#168	direct (N × H)
165. C₂²×C₅	C₈×D₄	semidirect
168. C₂²×C₅	C₂³×C₈	semidirect
174. C₄×C₅	C₂³×C₈	semidirect
179. C₂⁴	C₂×#168	semidirect
182. C₂×D₄	C₂×#168	direct (N × H)
189. C₂²×C₄	C₂×#168	semidirect
211. C₂×C₅	C₂×C₈×D₄	semidirect
219. C₂³	C₂²×#168	semidirect
224. D₄	C₂²×#168	direct (N × H)
230. C₂×C₄	C₂²×#168	semidirect
242. C₅	C₂²×C₈×D₄	semidirect
246. C₂²	D₄×#168	direct (N × H)
248. C₂²	C₂³×#168	semidirect
254. C₄	C₂³×#168	semidirect
259. C₂	C₂×D₄×#168	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×C₅×D₄	2
C₂⁴×#168	2
C₂²×C₂²⋊#168	2
C₂×D₄×#168	2
C₂²×C₄⋊#168	2
C₂²×C₄×#168	2
C₂²×C₈×D₄	5

Subgroups — C22×D4×#168 #12963132

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂²×D₄×#168 #12963132