Subgroups — C₂³×C₄×#117 #72339138

Order 1024.

185461 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	C₂⁴×#117	C₂	1	2	yes	maximal
2	C₂⁵×C₄²	C₂	1	1	yes	maximal, characteristic
3	C₂²×C₄×#117	C₂	1	112	yes	maximal, direct product factor
4	C₂³×C₄×C₄⋊C₄	C₂	1	4	yes	maximal, complemented
5	C₂³×C₄×C₂²⋊C₄	C₂	1	4	yes	maximal
6	C₂³×#376	C₂	1	2	yes	maximal, complemented
7	C₂³×C₄³	C₂	1	2	yes	maximal, complemented
8	C₂⁶×C₄	C₂²	1	2	yes	—
9	C₂⁶×C₄	C₂²	1	1	yes	characteristic
10	C₂³×#117	C₄	1	128	yes	direct product factor
11	C₂³×#117	C₂²	1	112	yes	—
12	C₂⁴×C₄⋊C₄	C₂²	1	4	yes	—
13	C₂⁴×C₂²⋊C₄	C₂²	1	4	yes	—
14	C₂⁴×C₄²	C₄	1	64	yes	complemented
15	C₂⁴×C₄²	C₂²	1	8	yes	—
16	C₂⁴×C₄²	C₂²	1	1	yes	characteristic
17	C₂⁴×C₄²	C₂²	1	2	yes	—
18	C₂⁴×C₄²	C₂²	1	56	yes	—
19	C₂⁴×C₄²	C₂²	1	4	yes	—
20	C₂⁴×C₄²	C₂²	1	1	yes	characteristic
21	C₂×C₄×#117	C₂²	1	1792	yes	direct product factor
22	C₂²×C₄×C₄⋊C₄	C₂²	1	224	yes	complemented
23	C₂²×C₄×C₂²⋊C₄	C₂²	1	224	yes	—
24	C₂²×#376	C₂²	1	112	yes	complemented
25	C₂³×#95	C₂²	1	8	yes	—
26	C₂²×C₄³	C₂²	1	112	yes	complemented
27	C₂⁷	C₂³	1	1	yes	characteristic
28	C₂⁵×C₄	—	2	64	no	—
29	C₂⁵×C₄	C₂×C₄	1	64	yes	—
30	C₂⁵×C₄	C₂×C₄	1	32	yes	—
31	C₂⁵×C₄	C₂×C₄	1	64	yes	—
32	C₂⁵×C₄	C₂³	1	2	yes	—
33	C₂⁵×C₄	C₂³	1	56	yes	—
34	C₂⁵×C₄	C₂³	1	8	yes	—
35	C₂⁵×C₄	C₂³	1	1	yes	characteristic
36	C₂⁵×C₄	C₂³	1	1	yes	characteristic
37	C₂⁵×C₄	C₂³	1	28	yes	—
38	C₂⁵×C₄	C₂³	1	2	yes	—
39	C₂²×#117	C₂×C₄	1	3584	yes	direct product factor
40	C₂²×#117	C₂³	1	896	yes	—
41	C₂³×C₄⋊C₄	C₂×C₄	1	128	yes	complemented
42	C₂³×C₄⋊C₄	C₂³	1	112	yes	—
43	C₂³×C₂²⋊C₄	C₂×C₄	1	128	yes	—
44	C₂³×C₂²⋊C₄	C₂³	1	112	yes	—
45	C₂³×C₄²	—	2	128	no	—
46	C₂³×C₄²	C₂×C₄	1	32	yes	—
47	C₂³×C₄²	C₂×C₄	1	64	yes	—
48	C₂³×C₄²	C₂×C₄	1	32	yes	—
49	C₂³×C₄²	C₂×C₄	1	1792	yes	complemented
50	C₂³×C₄²	C₂×C₄	1	128	yes	complemented
51	C₂³×C₄²	C₂³	1	224	yes	—
52	C₂³×C₄²	C₂³	1	28	yes	—
53	C₂³×C₄²	C₂³	1	56	yes	—
54	C₂³×C₄²	C₂³	1	28	yes	—
55	C₂³×C₄²	C₂³	1	448	yes	—
56	C₂³×C₄²	C₂³	1	112	yes	—
57	C₄×#117	C₂³	1	4096	yes	direct product factor
58	C₂×C₄×C₄⋊C₄	C₂³	1	1792	yes	complemented
59	C₂×C₄×C₂²⋊C₄	C₂³	1	1792	yes	—
60	C₂×#376	C₂³	1	896	yes	complemented
61	C₂²×#95	C₂³	1	224	yes	—
62	C₂×C₄³	C₂³	1	896	yes	complemented
63	C₂⁶	—	2	32	no	—
64	C₂⁶	C₂²×C₄	1	16	yes	—
65	C₂⁶	C₂²×C₄	1	32	yes	—
66	C₂⁶	C₂⁴	1	14	yes	—
67	C₂⁶	C₂⁴	1	1	yes	characteristic
68	C₂⁴×C₄	—	2	32	no	—
69	C₂⁴×C₄	—	2	128	no	—
70	C₂⁴×C₄	—	2	1024	no	—
71	C₂⁴×C₄	—	2	896	no	—
72	C₂⁴×C₄	C₄²	1	1024	yes	complemented
73	C₂⁴×C₄	C₂²×C₄	1	32	yes	—
74	C₂⁴×C₄	C₂²×C₄	1	896	yes	—
75	C₂⁴×C₄	C₂²×C₄	1	128	yes	—
76	C₂⁴×C₄	C₂²×C₄	1	16	yes	—
77	C₂⁴×C₄	C₂²×C₄	1	16	yes	—
78	C₂⁴×C₄	C₂²×C₄	1	32	yes	—
79	C₂⁴×C₄	C₂²×C₄	1	32	yes	—
80	C₂⁴×C₄	C₂²×C₄	1	448	yes	—
81	C₂⁴×C₄	C₂²×C₄	1	896	yes	—
82	C₂⁴×C₄	C₂²×C₄	1	32	yes	—
83	C₂⁴×C₄	#41	1	64	yes	—
84	C₂⁴×C₄	C₂⁴	1	28	yes	—
85	C₂⁴×C₄	C₂⁴	1	224	yes	—
86	C₂⁴×C₄	C₂⁴	1	112	yes	—
87	C₂⁴×C₄	C₂⁴	1	14	yes	—
88	C₂⁴×C₄	C₂⁴	1	14	yes	—
89	C₂⁴×C₄	C₂⁴	1	112	yes	—
90	C₂⁴×C₄	C₂⁴	1	28	yes	—
91	C₂×#117	C₂²×C₄	1	14336	yes	direct product factor
92	C₂×#117	C₂⁴	1	1024	yes	—
93	C₂²×C₄⋊C₄	C₂²×C₄	1	1792	yes	complemented
94	C₂²×C₄⋊C₄	C₂⁴	1	448	yes	—
95	C₂²×C₂²⋊C₄	C₂²×C₄	1	1792	yes	—
96	C₂²×C₂²⋊C₄	C₂⁴	1	448	yes	—
97	C₂²×C₄²	—	2	1792	no	—
98	C₂²×C₄²	C₂²×C₄	1	448	yes	—
99	C₂²×C₄²	C₂²×C₄	1	896	yes	—
100	C₂²×C₄²	C₂²×C₄	1	448	yes	—
101	C₂²×C₄²	C₂²×C₄	1	7168	yes	complemented
102	C₂²×C₄²	C₂²×C₄	1	1792	yes	complemented
103	C₂²×C₄²	C₂⁴	1	896	yes	—
104	C₂²×C₄²	C₂⁴	1	112	yes	—
105	C₂²×C₄²	C₂⁴	1	112	yes	—
106	C₂²×C₄²	C₂⁴	1	224	yes	—
107	C₂²×C₄²	C₂⁴	1	512	yes	—
108	C₂²×C₄²	C₂⁴	1	448	yes	—
109	C₄×C₄⋊C₄	C₂⁴	1	2048	yes	complemented
110	C₄×C₂²⋊C₄	C₂⁴	1	2048	yes	—
111	#376	C₂⁴	1	1024	yes	complemented
112	C₂×#95	C₂⁴	1	896	yes	—
113	C₄³	C₂⁴	1	1024	yes	complemented
114	C₂⁵	—	2	256	no	—
115	C₂⁵	—	2	512	no	—
116	C₂⁵	—	2	224	no	—
117	C₂⁵	C₂×C₄²	1	256	yes	—
118	C₂⁵	C₂³×C₄	1	224	yes	—
119	C₂⁵	C₂³×C₄	1	112	yes	—
120	C₂⁵	C₂³×C₄	1	8	yes	—
121	C₂⁵	C₂³×C₄	1	16	yes	—
122	C₂⁵	C₂×#41	1	32	yes	—
123	C₂⁵	C₂⁵	1	28	yes	—
124	C₂⁵	C₂⁵	1	7	yes	—
125	C₂³×C₄	—	2	256	no	—
126	C₂³×C₄	—	2	1024	no	—
127	C₂³×C₄	—	2	224	no	—
128	C₂³×C₄	—	2	896	no	—
129	C₂³×C₄	—	2	7168	no	—
130	C₂³×C₄	—	2	1792	no	—
131	C₂³×C₄	C₂×C₄²	1	256	yes	—
132	C₂³×C₄	C₂×C₄²	1	256	yes	—
133	C₂³×C₄	C₂×C₄²	1	7168	yes	complemented
134	C₂³×C₄	#117	1	512	yes	direct product factor
135	C₂³×C₄	C₂³×C₄	1	896	yes	—
136	C₂³×C₄	C₂³×C₄	1	224	yes	—
137	C₂³×C₄	C₂³×C₄	1	1792	yes	—
138	C₂³×C₄	C₂³×C₄	1	224	yes	—
139	C₂³×C₄	C₂³×C₄	1	224	yes	—
140	C₂³×C₄	C₂³×C₄	1	112	yes	—
141	C₂³×C₄	C₂³×C₄	1	112	yes	—
142	C₂³×C₄	C₂³×C₄	1	224	yes	—
143	C₂³×C₄	C₂³×C₄	1	1792	yes	—
144	C₂³×C₄	C₂³×C₄	1	896	yes	—
145	C₂³×C₄	C₂×#41	1	448	yes	—
146	C₂³×C₄	C₂⁵	1	224	yes	—
147	C₂³×C₄	C₂⁵	1	56	yes	—
148	C₂³×C₄	C₂⁵	1	128	yes	—
149	C₂³×C₄	C₂⁵	1	28	yes	—
150	C₂³×C₄	C₂⁵	1	64	yes	—
151	C₂³×C₄	C₂⁵	1	28	yes	—
152	C₂³×C₄	C₂⁵	1	56	yes	—
153	#117	C₂³×C₄	1	8192	yes	direct product factor
154	C₂×C₄⋊C₄	C₂³×C₄	1	3584	yes	complemented
155	C₂×C₄⋊C₄	C₂⁵	1	256	yes	—
156	C₂×C₂²⋊C₄	C₂³×C₄	1	3584	yes	—
157	C₂×C₂²⋊C₄	C₂⁵	1	256	yes	—
158	C₂×C₄²	—	2	3584	no	—
159	C₂×C₄²	C₂³×C₄	1	896	yes	—
160	C₂×C₄²	C₂³×C₄	1	1792	yes	—
161	C₂×C₄²	C₂³×C₄	1	896	yes	—
162	C₂×C₄²	C₂³×C₄	1	4096	yes	complemented
163	C₂×C₄²	C₂³×C₄	1	3584	yes	complemented
164	C₂×C₄²	C₂⁵	1	512	yes	—
165	C₂×C₄²	C₂⁵	1	64	yes	—
166	C₂×C₄²	C₂⁵	1	64	yes	—
167	C₂×C₄²	C₂⁵	1	128	yes	—
168	C₂×C₄²	C₂⁵	1	256	yes	—
169	#95	C₂⁵	1	512	yes	—
170	C₂⁴	—	2	224	no	—
171	C₂⁴	—	2	896	no	—
172	C₂⁴	—	2	1792	no	—
173	C₂⁴	—	2	2048	no	—
174	C₂⁴	C₂²×C₄²	1	896	yes	—
175	C₂⁴	C₂²×C₄²	1	64	yes	—
176	C₂⁴	C₂×#117	1	128	yes	—
177	C₂⁴	C₄×#41	1	256	yes	—
178	C₂⁴	C₂⁴×C₄	1	224	yes	—
179	C₂⁴	C₂⁴×C₄	1	112	yes	—
180	C₂⁴	C₂⁴×C₄	1	28	yes	—
181	C₂⁴	C₂⁴×C₄	1	56	yes	—
182	C₂⁴	C₂²×#41	1	112	yes	—
183	C₂⁴	C₂⁶	1	8	yes	—
184	C₂⁴	C₂⁶	1	7	yes	—
185	C₂²×C₄	—	2	896	no	—
186	C₂²×C₄	—	2	224	no	—
187	C₂²×C₄	—	2	896	no	—
188	C₂²×C₄	—	2	3584	no	—
189	C₂²×C₄	—	2	512	no	—
190	C₂²×C₄	—	2	7168	no	—
191	C₂²×C₄	C₂²×C₄²	1	896	yes	—
192	C₂²×C₄	C₂²×C₄²	1	896	yes	—
193	C₂²×C₄	C₂²×C₄²	1	7168	yes	complemented
194	C₂²×C₄	C₂×#117	1	1792	yes	direct product factor
195	C₂²×C₄	C₂⁴×C₄	1	224	yes	—
196	C₂²×C₄	C₂⁴×C₄	1	512	yes	—
197	C₂²×C₄	C₂⁴×C₄	1	896	yes	—
198	C₂²×C₄	C₂⁴×C₄	1	112	yes	—
199	C₂²×C₄	C₂⁴×C₄	1	112	yes	—
200	C₂²×C₄	C₂⁴×C₄	1	224	yes	—
201	C₂²×C₄	C₂⁴×C₄	1	224	yes	—
202	C₂²×C₄	C₂⁴×C₄	1	256	yes	—
203	C₂²×C₄	C₂⁴×C₄	1	512	yes	—
204	C₂²×C₄	C₂⁴×C₄	1	224	yes	—
205	C₂²×C₄	C₂²×#41	1	448	yes	—
206	C₂²×C₄	C₂⁶	1	16	yes	—
207	C₂²×C₄	C₂⁶	1	64	yes	—
208	C₂²×C₄	C₂⁶	1	8	yes	—
209	C₂²×C₄	C₂⁶	1	8	yes	—
210	C₂²×C₄	C₂⁶	1	16	yes	—
211	C₄⋊C₄	C₂⁴×C₄	1	1024	yes	complemented
212	C₂²⋊C₄	C₂⁴×C₄	1	1024	yes	—
213	C₄²	—	2	1024	no	—
214	C₄²	C₂⁴×C₄	1	256	yes	—
215	C₄²	C₂⁴×C₄	1	512	yes	—
216	C₄²	C₂⁴×C₄	1	256	yes	—
217	C₄²	C₂⁴×C₄	1	1024	yes	complemented
218	C₂³	—	2	32	no	—
219	C₂³	—	2	448	no	—
220	C₂³	—	2	896	no	—
221	C₂³	—	2	3584	no	—
222	C₂³	C₄×#117	1	512	yes	direct product factor
223	C₂³	C₂³×C₄²	1	448	yes	—
224	C₂³	C₂³×C₄²	1	112	yes	—
225	C₂³	C₂²×#117	1	224	yes	—
226	C₂³	C₂×C₄×#41	1	448	yes	—
227	C₂³	C₂⁵×C₄	1	32	yes	—
228	C₂³	C₂⁵×C₄	1	16	yes	—
229	C₂³	C₂⁵×C₄	1	14	yes	—
230	C₂³	C₂⁵×C₄	1	28	yes	—
231	C₂³	C₂³×#41	1	56	yes	—
232	C₂³	C₂⁷	1	1	yes	characteristic
233	C₂×C₄	—	2	128	no	—
234	C₂×C₄	—	2	32	no	—
235	C₂×C₄	—	2	448	no	—
236	C₂×C₄	—	2	1792	no	—
237	C₂×C₄	—	2	1024	no	—
238	C₂×C₄	C₂³×C₄²	1	448	yes	—
239	C₂×C₄	C₂³×C₄²	1	448	yes	—
240	C₂×C₄	C₂³×C₄²	1	1024	yes	complemented
241	C₂×C₄	C₂²×#117	1	896	yes	direct product factor
242	C₂×C₄	C₂⁵×C₄	1	32	yes	—
243	C₂×C₄	C₂⁵×C₄	1	128	yes	—
244	C₂×C₄	C₂⁵×C₄	1	16	yes	—
245	C₂×C₄	C₂⁵×C₄	1	16	yes	—
246	C₂×C₄	C₂⁵×C₄	1	32	yes	—
247	C₂×C₄	C₂⁵×C₄	1	32	yes	—
248	C₂×C₄	C₂⁵×C₄	1	32	yes	—
249	C₂×C₄	C₂³×#41	1	64	yes	—
250	C₂²	—	2	32	no	—
251	C₂²	—	2	64	no	—
252	C₂²	—	2	896	no	—
253	C₂²	C₂×C₄×#117	1	448	yes	direct product factor
254	C₂²	C₂⁴×C₄²	1	32	yes	—
255	C₂²	C₂⁴×C₄²	1	28	yes	—
256	C₂²	C₂³×#117	1	56	yes	—
257	C₂²	C₂²×C₄×#41	1	112	yes	—
258	C₂²	C₂⁶×C₄	1	1	yes	characteristic
259	C₂²	C₂⁶×C₄	1	2	yes	—
260	C₂²	C₂⁴×#41	1	4	yes	—
261	C₄	—	2	32	no	—
262	C₄	—	2	128	no	—
263	C₄	C₂⁴×C₄²	1	32	yes	—
264	C₄	C₂⁴×C₄²	1	32	yes	—
265	C₄	C₂³×#117	1	64	yes	direct product factor
266	C₂	—	2	32	no	—
267	C₂	C₂²×C₄×#117	1	56	yes	direct product factor
268	C₂	C₂⁵×C₄²	1	1	yes	characteristic
269	C₂	C₂⁴×#117	1	2	yes	—
270	C₂	C₂³×C₄×#41	1	4	yes	—

	subgroup	G/H
center, socle	C₂⁴×C₄²	C₂²
commutator	C₂	C₂⁵×C₄²
Frattini, cosocle	C₂³	C₂⁷
prime radical	C₂⁷	C₂³
LCS	G ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂⁴×C₄² ◁ G
LECS	G ▷ C₂³ ▷ C₁

Complements

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

N	H ≅ G/N	type
3. C₂²×C₄×#117	C₂	direct (N × H)
4. C₂³×C₄×C₄⋊C₄	C₂	semidirect
6. C₂³×#376	C₂	semidirect
7. C₂³×C₄³	C₂	semidirect
10. C₂³×#117	C₄	direct (N × H)
14. C₂⁴×C₄²	C₄	semidirect
21. C₂×C₄×#117	C₂²	direct (N × H)
22. C₂²×C₄×C₄⋊C₄	C₂²	semidirect
24. C₂²×#376	C₂²	semidirect
26. C₂²×C₄³	C₂²	semidirect
39. C₂²×#117	C₂×C₄	direct (N × H)
41. C₂³×C₄⋊C₄	C₂×C₄	semidirect
49. C₂³×C₄²	C₂×C₄	semidirect
50. C₂³×C₄²	C₂×C₄	semidirect
57. C₄×#117	C₂³	direct (N × H)
58. C₂×C₄×C₄⋊C₄	C₂³	semidirect
60. C₂×#376	C₂³	semidirect
62. C₂×C₄³	C₂³	semidirect
72. C₂⁴×C₄	C₄²	semidirect
91. C₂×#117	C₂²×C₄	direct (N × H)
93. C₂²×C₄⋊C₄	C₂²×C₄	semidirect
101. C₂²×C₄²	C₂²×C₄	semidirect
102. C₂²×C₄²	C₂²×C₄	semidirect
109. C₄×C₄⋊C₄	C₂⁴	semidirect
111. #376	C₂⁴	semidirect
113. C₄³	C₂⁴	semidirect
133. C₂³×C₄	C₂×C₄²	semidirect
134. C₂³×C₄	#117	direct (N × H)
153. #117	C₂³×C₄	direct (N × H)
154. C₂×C₄⋊C₄	C₂³×C₄	semidirect
162. C₂×C₄²	C₂³×C₄	semidirect
163. C₂×C₄²	C₂³×C₄	semidirect
193. C₂²×C₄	C₂²×C₄²	semidirect
194. C₂²×C₄	C₂×#117	direct (N × H)
211. C₄⋊C₄	C₂⁴×C₄	semidirect
217. C₄²	C₂⁴×C₄	semidirect
222. C₂³	C₄×#117	direct (N × H)
240. C₂×C₄	C₂³×C₄²	semidirect
241. C₂×C₄	C₂²×#117	direct (N × H)
253. C₂²	C₂×C₄×#117	direct (N × H)
265. C₄	C₂³×#117	direct (N × H)
267. C₂	C₂²×C₄×#117	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂⁴×#117	2
C₂⁵×C₄²	2
C₂²×C₄×#117	2
C₂³×C₄×C₄⋊C₄	2
C₂³×C₄×C₂²⋊C₄	2
C₂³×#376	2
C₂³×C₄³	2

Subgroups — C23×C4×#117 #72339138

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂³×C₄×#117 #72339138