Subgroups — C₂³×#88150 #16769561

Order 3328.

14078 conjugacy classes of subgroups.

The subgroup lattice has 256 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂⁴×Dic₂₆	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂⁴×C₄×D₁₃	C₂	1	1	yes	maximal, characteristic
3	C₂²×#88150	C₂	1	56	yes	maximal, direct product factor
4	C₂³×#6101	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂³×#6101	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂³×#6103	C₂	1	1	yes	maximal, characteristic
7	C₂³×#6103	C₂	1	1	yes	maximal, characteristic
8	C₂³×C₁₃×C₄⋊C₄	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂⁵×D₁₃	C₂²	1	1	yes	characteristic
10	C₂⁴×C₄×C₁₃	C₂²	1	1	yes	characteristic
11	C₂⁴×C₄×C₁₃	C₂²	1	1	yes	characteristic
12	C₂⁴×C₄×C₁₃	C₂²	1	1	yes	characteristic
13	C₂⁴×Dic₁₃	C₂²	1	1	yes	characteristic
14	C₂⁴×Dic₁₃	C₂²	1	1	yes	characteristic
15	C₂⁴×Dic₁₃	C₂²	1	1	yes	characteristic
16	C₂³×C₄×D₁₃	—	2	16	no	—
17	C₂³×C₄×D₁₃	C₂²	1	28	yes	—
18	C₂³×Dic₂₆	—	2	16	no	—
19	C₂³×Dic₂₆	C₂²	1	28	yes	complemented
20	C₂²×C₁₃×C₄⋊C₄	C₂²	1	28	yes	complemented
21	C₂²×#6103	C₂²	1	28	yes	—
22	C₂²×#6103	C₂²	1	28	yes	—
23	C₂²×#6101	C₂²	1	28	yes	complemented
24	C₂²×#6101	C₂²	1	28	yes	complemented
25	C₂×#88150	C₂²	1	448	yes	direct product factor
26	C₂⁵×C₁₃	C₂³	1	1	yes	characteristic
27	C₂⁴×D₁₃	—	2	8	no	—
28	C₂⁴×D₁₃	—	2	8	no	—
29	C₂⁴×D₁₃	Q₈	1	16	yes	complemented
30	C₂⁴×D₁₃	C₂³	1	14	yes	—
31	C₂³×C₄×C₁₃	—	2	8	no	—
32	C₂³×C₄×C₁₃	—	2	8	no	—
33	C₂³×C₄×C₁₃	—	2	8	no	—
34	C₂³×C₄×C₁₃	C₂³	1	14	yes	—
35	C₂³×C₄×C₁₃	C₂³	1	14	yes	—
36	C₂³×C₄×C₁₃	C₂³	1	14	yes	—
37	C₂³×Dic₁₃	—	2	8	no	—
38	C₂³×Dic₁₃	—	2	8	no	—
39	C₂³×Dic₁₃	D₄	1	16	yes	—
40	C₂³×Dic₁₃	C₂³	1	14	yes	—
41	C₂³×Dic₁₃	C₂³	1	14	yes	—
42	C₂³×Dic₁₃	C₂³	1	14	yes	—
43	C₂²×C₄×D₁₃	—	2	224	no	—
44	C₂²×C₄×D₁₃	C₂³	1	112	yes	—
45	C₂²×Dic₂₆	—	2	224	no	—
46	C₂²×Dic₂₆	C₂³	1	112	yes	complemented
47	C₂×C₁₃×C₄⋊C₄	C₂³	1	112	yes	complemented
48	C₂×#6103	C₂³	1	112	yes	—
49	C₂×#6103	C₂³	1	112	yes	—
50	C₂×#6101	C₂³	1	112	yes	complemented
51	C₂×#6101	C₂³	1	112	yes	complemented
52	#88150	C₂³	1	512	yes	direct product factor
53	C₂³×C₂²⋊Q₈	—	13	1	no	maximal, Sylow
54	C₂⁴×C₁₃	C₂×D₄	1	8	yes	—
55	C₂⁴×C₁₃	C₂×Q₈	1	8	yes	—
56	C₂⁴×C₁₃	#41	1	8	yes	—
57	C₂⁴×C₁₃	C₂⁴	1	7	yes	—
58	C₂³×D₁₃	—	2	128	no	—
59	C₂³×D₁₃	—	2	56	no	—
60	C₂³×D₁₃	—	2	56	no	—
61	C₂³×D₁₃	C₂×Q₈	1	112	yes	complemented
62	C₂³×D₁₃	C₂⁴	1	28	yes	—
63	C₂²×C₄×C₁₃	—	2	56	no	—
64	C₂²×C₄×C₁₃	—	2	56	no	—
65	C₂²×C₄×C₁₃	—	2	56	no	—
66	C₂²×C₄×C₁₃	C₂⁴	1	28	yes	—
67	C₂²×C₄×C₁₃	C₂⁴	1	28	yes	—
68	C₂²×C₄×C₁₃	C₂⁴	1	28	yes	—
69	C₂²×Dic₁₃	—	2	56	no	—
70	C₂²×Dic₁₃	—	2	56	no	—
71	C₂²×Dic₁₃	C₂×D₄	1	112	yes	—
72	C₂²×Dic₁₃	C₂⁴	1	28	yes	—
73	C₂²×Dic₁₃	C₂⁴	1	28	yes	—
74	C₂²×Dic₁₃	C₂⁴	1	28	yes	—
75	C₂×C₄×D₁₃	—	2	448	no	—
76	C₂×C₄×D₁₃	C₂⁴	1	64	yes	—
77	C₂×Dic₂₆	—	2	448	no	—
78	C₂×Dic₂₆	C₂⁴	1	64	yes	complemented
79	C₁₃×C₄⋊C₄	C₂⁴	1	64	yes	complemented
80	#6103	C₂⁴	1	64	yes	—
81	#6103	C₂⁴	1	64	yes	—
82	#6101	C₂⁴	1	64	yes	complemented
83	#6101	C₂⁴	1	64	yes	complemented
84	C₂⁴×Q₈	—	13	1	no	—
85	C₂⁵×C₄	—	13	1	no	—
86	C₂²×C₂²⋊Q₈	—	13	56	no	—
87	C₂³×C₄⋊C₄	—	13	1	no	—
88	C₂³×C₄⋊C₄	—	13	1	no	—
89	C₂³×C₄⋊C₄	D₁₃	1	1	yes	characteristic, complemented
90	C₂³×C₂²⋊C₄	—	13	1	no	—
91	C₂³×C₂²⋊C₄	—	13	1	no	—
92	C₂³×C₁₃	C₂²⋊Q₈	1	64	yes	complemented
93	C₂³×C₁₃	C₂²×D₄	1	28	yes	—
94	C₂³×C₁₃	C₂²×Q₈	1	28	yes	—
95	C₂³×C₁₃	C₂×#41	1	28	yes	—
96	C₂³×C₁₃	C₂⁵	1	7	yes	—
97	C₂²×D₁₃	—	2	448	no	—
98	C₂²×D₁₃	—	2	56	no	—
99	C₂²×D₁₃	—	2	56	no	—
100	C₂²×D₁₃	C₂²×Q₈	1	112	yes	complemented
101	C₂²×D₁₃	C₂⁵	1	8	yes	—
102	C₂×C₄×C₁₃	—	2	56	no	—
103	C₂×C₄×C₁₃	—	2	56	no	—
104	C₂×C₄×C₁₃	—	2	56	no	—
105	C₂×C₄×C₁₃	C₂⁵	1	8	yes	—
106	C₂×C₄×C₁₃	C₂⁵	1	8	yes	—
107	C₂×C₄×C₁₃	C₂⁵	1	8	yes	—
108	C₂×Dic₁₃	—	2	56	no	—
109	C₂×Dic₁₃	—	2	56	no	—
110	C₂×Dic₁₃	C₂²×D₄	1	112	yes	—
111	C₂×Dic₁₃	C₂⁵	1	8	yes	—
112	C₂×Dic₁₃	C₂⁵	1	8	yes	—
113	C₂×Dic₁₃	C₂⁵	1	8	yes	—
114	C₄×D₁₃	—	2	128	no	—
115	Dic₂₆	—	2	128	no	—
116	C₂⁶	—	13	1	no	—
117	C₂³×Q₈	—	13	28	no	—
118	C₂³×Q₈	—	26	16	no	—
119	C₂⁴×C₄	—	13	1	no	—
120	C₂⁴×C₄	—	13	1	no	—
121	C₂⁴×C₄	—	13	1	no	—
122	C₂⁴×C₄	—	13	28	no	—
123	C₂⁴×C₄	—	26	16	no	—
124	C₂⁴×C₄	C₂×D₁₃	1	1	yes	characteristic
125	C₂⁴×C₄	C₂×D₁₃	1	1	yes	characteristic
126	C₂⁴×C₄	C₂×D₁₃	1	1	yes	characteristic
127	C₂×C₂²⋊Q₈	—	13	448	no	—
128	C₂²×C₄⋊C₄	—	13	28	no	—
129	C₂²×C₄⋊C₄	—	13	28	no	—
130	C₂²×C₄⋊C₄	C₂×D₁₃	1	28	yes	complemented
131	C₂²×C₂²⋊C₄	—	13	28	no	—
132	C₂²×C₂²⋊C₄	—	13	28	no	—
133	C₂²×C₁₃	C₂×C₂²⋊Q₈	1	112	yes	complemented
134	C₂²×C₁₃	C₂³×D₄	1	14	yes	—
135	C₂²×C₁₃	C₂³×Q₈	1	14	yes	—
136	C₂²×C₁₃	C₂²×#41	1	14	yes	—
137	C₂²×C₁₃	C₂⁶	1	1	yes	characteristic
138	C₂×D₁₃	—	2	8	no	—
139	C₂×D₁₃	—	2	8	no	—
140	C₂×D₁₃	—	2	224	no	—
141	C₂×D₁₃	C₂³×Q₈	1	16	yes	complemented
142	C₄×C₁₃	—	2	8	no	—
143	C₄×C₁₃	—	2	8	no	—
144	C₄×C₁₃	—	2	8	no	—
145	Dic₁₃	—	2	8	no	—
146	Dic₁₃	—	2	8	no	—
147	Dic₁₃	C₂³×D₄	1	16	yes	—
148	C₂⁵	—	13	16	no	—
149	C₂⁵	—	13	14	no	—
150	C₂⁵	—	26	8	no	—
151	C₂⁵	—	26	8	no	—
152	C₂⁵	C₂²×D₁₃	1	1	yes	characteristic
153	C₂²×Q₈	—	13	112	no	—
154	C₂²×Q₈	—	26	224	no	—
155	C₂³×C₄	—	2	8	no	—
156	C₂³×C₄	—	2	8	no	—
157	C₂³×C₄	—	2	8	no	—
158	C₂³×C₄	—	13	16	no	—
159	C₂³×C₄	—	13	14	no	—
160	C₂³×C₄	—	13	14	no	—
161	C₂³×C₄	—	13	14	no	—
162	C₂³×C₄	—	13	112	no	—
163	C₂³×C₄	—	26	224	no	—
164	C₂³×C₄	—	26	8	no	—
165	C₂³×C₄	—	26	8	no	—
166	C₂³×C₄	C₂²×D₁₃	1	14	yes	—
167	C₂³×C₄	C₂²×D₁₃	1	14	yes	—
168	C₂³×C₄	C₂²×D₁₃	1	14	yes	—
169	C₂²⋊Q₈	—	13	512	no	—
170	C₂×C₄⋊C₄	—	13	112	no	—
171	C₂×C₄⋊C₄	—	13	112	no	—
172	C₂×C₄⋊C₄	C₂²×D₁₃	1	112	yes	complemented
173	C₂×C₂²⋊C₄	—	13	112	no	—
174	C₂×C₂²⋊C₄	—	13	112	no	—
175	C₂×C₁₃	C₂²×C₂²⋊Q₈	1	28	yes	complemented
176	C₂×C₁₃	C₂⁴×D₄	1	1	yes	characteristic
177	C₂×C₁₃	C₂⁴×Q₈	1	1	yes	characteristic
178	C₂×C₁₃	C₂³×#41	1	1	yes	characteristic
179	D₁₃	—	2	16	no	—
180	C₂⁴	—	13	112	no	—
181	C₂⁴	—	13	28	no	—
182	C₂⁴	—	26	128	no	—
183	C₂⁴	—	26	56	no	—
184	C₂⁴	—	26	56	no	—
185	C₂⁴	#6127	1	8	yes	—
186	C₂⁴	D₄×D₁₃	1	8	yes	—
187	C₂⁴	Q₈×D₁₃	1	8	yes	—
188	C₂⁴	C₂³×D₁₃	1	7	yes	—
189	C₂×Q₈	—	13	64	no	—
190	C₂×Q₈	—	26	448	no	—
191	C₂²×C₄	—	2	56	no	—
192	C₂²×C₄	—	2	56	no	—
193	C₂²×C₄	—	2	56	no	—
194	C₂²×C₄	—	13	112	no	—
195	C₂²×C₄	—	13	28	no	—
196	C₂²×C₄	—	13	28	no	—
197	C₂²×C₄	—	13	28	no	—
198	C₂²×C₄	—	13	64	no	—
199	C₂²×C₄	—	26	448	no	—
200	C₂²×C₄	—	26	56	no	—
201	C₂²×C₄	—	26	56	no	—
202	C₂²×C₄	C₂³×D₁₃	1	28	yes	—
203	C₂²×C₄	C₂³×D₁₃	1	28	yes	—
204	C₂²×C₄	C₂³×D₁₃	1	28	yes	—
205	C₄⋊C₄	—	13	64	no	—
206	C₄⋊C₄	—	13	64	no	—
207	C₄⋊C₄	C₂³×D₁₃	1	64	yes	complemented
208	C₂²⋊C₄	—	13	64	no	—
209	C₂²⋊C₄	—	13	64	no	—
210	C₁₃	C₂³×C₂²⋊Q₈	1	1	yes	characteristic, complemented, Sylow
211	C₂³	—	13	112	no	—
212	C₂³	—	13	8	no	—
213	C₂³	—	26	448	no	—
214	C₂³	—	26	56	no	—
215	C₂³	—	26	56	no	—
216	C₂³	#88150	1	64	yes	direct product factor
217	C₂³	C₂×#6127	1	28	yes	—
218	C₂³	C₂×D₄×D₁₃	1	28	yes	—
219	C₂³	C₂×Q₈×D₁₃	1	28	yes	—
220	C₂³	C₂⁴×D₁₃	1	7	yes	—
221	Q₈	—	26	128	no	—
222	C₂×C₄	—	2	56	no	—
223	C₂×C₄	—	2	56	no	—
224	C₂×C₄	—	2	56	no	—
225	C₂×C₄	—	13	112	no	—
226	C₂×C₄	—	13	8	no	—
227	C₂×C₄	—	13	8	no	—
228	C₂×C₄	—	13	8	no	—
229	C₂×C₄	—	26	128	no	—
230	C₂×C₄	—	26	56	no	—
231	C₂×C₄	—	26	56	no	—
232	C₂×C₄	C₂⁴×D₁₃	1	8	yes	—
233	C₂×C₄	C₂⁴×D₁₃	1	8	yes	—
234	C₂×C₄	C₂⁴×D₁₃	1	8	yes	—
235	C₂²	—	13	16	no	—
236	C₂²	—	26	224	no	—
237	C₂²	—	26	8	no	—
238	C₂²	—	26	8	no	—
239	C₂²	C₂×#88150	1	112	yes	direct product factor
240	C₂²	C₂²×#6127	1	14	yes	—
241	C₂²	C₂²×D₄×D₁₃	1	14	yes	—
242	C₂²	C₂²×Q₈×D₁₃	1	14	yes	—
243	C₂²	C₂⁵×D₁₃	1	1	yes	characteristic
244	C₄	—	2	8	no	—
245	C₄	—	2	8	no	—
246	C₄	—	2	8	no	—
247	C₄	—	13	16	no	—
248	C₄	—	26	8	no	—
249	C₄	—	26	8	no	—
250	C₂	—	26	16	no	—
251	C₂	C₂²×#88150	1	28	yes	direct product factor
252	C₂	C₂³×D₄×D₁₃	1	1	yes	characteristic
253	C₂	C₂³×Q₈×D₁₃	1	1	yes	characteristic
254	C₂	C₂³×#6127	1	1	yes	characteristic

	subgroup	G/H
center	C₂⁵	C₂²×D₁₃
commutator, cosocle	C₂²×C₁₃	C₂⁶
Frattini	C₂²	C₂⁵×D₁₃
Fitting	C₂³×C₁₃×C₄⋊C₄	C₂
socle	C₂⁵×C₁₃	C₂³
prime radical	C₂⁵×D₁₃	C₂²
hypercenter	C₂³×C₄⋊C₄	D₁₃
hypocenter	C₁₃	C₂³×C₂²⋊Q₈
LCS	G ▷ C₂²×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂⁵ ◁ C₂³×C₄⋊C₄

Complements

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

N	H ≅ G/N	type
1. C₂⁴×Dic₂₆	C₂	semidirect
3. C₂²×#88150	C₂	direct (N × H)
4. C₂³×#6101	C₂	semidirect
5. C₂³×#6101	C₂	semidirect
8. C₂³×C₁₃×C₄⋊C₄	C₂	semidirect
19. C₂³×Dic₂₆	C₂²	semidirect
20. C₂²×C₁₃×C₄⋊C₄	C₂²	semidirect
23. C₂²×#6101	C₂²	semidirect
24. C₂²×#6101	C₂²	semidirect
25. C₂×#88150	C₂²	direct (N × H)
29. C₂⁴×D₁₃	Q₈	semidirect
46. C₂²×Dic₂₆	C₂³	semidirect
47. C₂×C₁₃×C₄⋊C₄	C₂³	semidirect
50. C₂×#6101	C₂³	semidirect
51. C₂×#6101	C₂³	semidirect
52. #88150	C₂³	direct (N × H)
61. C₂³×D₁₃	C₂×Q₈	semidirect
78. C₂×Dic₂₆	C₂⁴	semidirect
79. C₁₃×C₄⋊C₄	C₂⁴	semidirect
82. #6101	C₂⁴	semidirect
83. #6101	C₂⁴	semidirect
89. C₂³×C₄⋊C₄	D₁₃	semidirect
92. C₂³×C₁₃	C₂²⋊Q₈	semidirect
100. C₂²×D₁₃	C₂²×Q₈	semidirect
130. C₂²×C₄⋊C₄	C₂×D₁₃	semidirect
133. C₂²×C₁₃	C₂×C₂²⋊Q₈	semidirect
141. C₂×D₁₃	C₂³×Q₈	semidirect
172. C₂×C₄⋊C₄	C₂²×D₁₃	semidirect
175. C₂×C₁₃	C₂²×C₂²⋊Q₈	semidirect
207. C₄⋊C₄	C₂³×D₁₃	semidirect
210. C₁₃	C₂³×C₂²⋊Q₈	semidirect
216. C₂³	#88150	direct (N × H)
239. C₂²	C₂×#88150	direct (N × H)
251. C₂	C₂²×#88150	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×Dic₂₆	2
C₂⁴×C₄×D₁₃	2
C₂²×#88150	2
C₂³×#6101	2
C₂³×#6103	2
C₂³×C₁₃×C₄⋊C₄	2
C₂³×C₂²⋊Q₈	13

Subgroups — C23×#88150 #16769561

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂³×#88150 #16769561