Subgroups — C₂³×#88077 #16726507

Order 3328.

5758 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	C₂³×#6099	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂³×C₄×Dic₁₃	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂²×#88077	C₂	1	28	yes	maximal, direct product factor
4	C₂³×C₁₃×#41	C₂	1	1	yes	maximal, characteristic
5	C₂²×C₁₃×#41	—	2	7	no	—
6	C₂²×C₁₃×#41	—	2	7	no	—
7	C₂²×C₁₃×#41	C₄	1	14	yes	complemented
8	C₂²×C₁₃×#41	C₂²	1	14	yes	—
9	C₂³×Q₈×C₁₃	C₄	1	1	yes	characteristic, complemented
10	C₂³×D₄×C₁₃	—	2	1	no	—
11	C₂³×D₄×C₁₃	C₄	1	1	yes	characteristic, complemented
12	C₂⁴×C₄×C₁₃	—	2	1	no	—
13	C₂⁴×C₄×C₁₃	C₂²	1	1	yes	characteristic
14	C₂⁴×Dic₁₃	—	2	1	no	—
15	C₂²×C₄×Dic₁₃	—	2	7	no	—
16	C₂²×C₄×Dic₁₃	C₂²	1	14	yes	complemented
17	C₂²×#6099	—	2	7	no	—
18	C₂²×#6099	C₂²	1	14	yes	complemented
19	C₂³×#1055	—	2	1	no	—
20	C₂×#88077	C₂²	1	112	yes	direct product factor
21	C₂⁵×C₁₃	—	2	1	no	—
22	C₂⁵×C₁₃	D₄	1	1	yes	characteristic
23	C₂×C₁₃×#41	—	2	84	no	—
24	C₂×C₁₃×#41	—	2	42	no	—
25	C₂×C₁₃×#41	—	2	42	no	—
26	C₂×C₁₃×#41	C₂×C₄	1	84	yes	complemented
27	C₂×C₁₃×#41	C₂³	1	28	yes	—
28	C₂²×Q₈×C₁₃	—	2	7	no	—
29	C₂²×Q₈×C₁₃	C₂×C₄	1	14	yes	complemented
30	C₂²×D₄×C₁₃	—	2	14	no	—
31	C₂²×D₄×C₁₃	—	2	14	no	—
32	C₂²×D₄×C₁₃	—	2	7	no	—
33	C₂²×D₄×C₁₃	C₂×C₄	1	14	yes	complemented
34	C₂³×C₄×C₁₃	—	2	1	no	—
35	C₂³×C₄×C₁₃	—	2	14	no	—
36	C₂³×C₄×C₁₃	—	2	14	no	—
37	C₂³×C₄×C₁₃	C₂×C₄	1	1	yes	characteristic
38	C₂³×C₄×C₁₃	C₂×C₄	1	7	yes	—
39	C₂³×C₄×C₁₃	D₄	1	7	yes	—
40	C₂³×C₄×C₁₃	D₄	1	7	yes	—
41	C₂³×C₄×C₁₃	D₄	1	1	yes	characteristic
42	C₂³×C₄×C₁₃	C₂³	1	7	yes	—
43	C₂³×Dic₁₃	—	2	16	no	—
44	C₂³×Dic₁₃	—	2	14	no	—
45	C₂×C₄×Dic₁₃	—	2	42	no	—
46	C₂×C₄×Dic₁₃	C₂³	1	28	yes	complemented
47	C₂×#6099	—	2	42	no	—
48	C₂×#6099	C₂³	1	28	yes	complemented
49	C₂²×#1055	—	2	14	no	—
50	#88077	C₂³	1	64	yes	direct product factor
51	C₂³×#104	—	13	1	no	maximal, Sylow
52	C₂⁴×C₁₃	—	2	8	no	—
53	C₂⁴×C₁₃	—	2	14	no	—
54	C₂⁴×C₁₃	—	4	8	no	—
55	C₂⁴×C₁₃	C₂²⋊C₄	1	7	yes	—
56	C₂⁴×C₁₃	C₂²⋊C₄	1	1	yes	characteristic
57	C₂⁴×C₁₃	C₂×D₄	1	7	yes	—
58	C₁₃×#41	—	2	168	no	—
59	C₁₃×#41	—	2	28	no	—
60	C₁₃×#41	—	2	28	no	—
61	C₁₃×#41	C₂²×C₄	1	56	yes	complemented
62	C₁₃×#41	C₂⁴	1	8	yes	—
63	C₂×Q₈×C₁₃	—	2	42	no	—
64	C₂×Q₈×C₁₃	C₂²×C₄	1	28	yes	complemented
65	C₂×D₄×C₁₃	—	2	84	no	—
66	C₂×D₄×C₁₃	—	2	28	no	—
67	C₂×D₄×C₁₃	—	2	42	no	—
68	C₂×D₄×C₁₃	C₂²×C₄	1	28	yes	complemented
69	C₂²×C₄×C₁₃	—	2	14	no	—
70	C₂²×C₄×C₁₃	—	2	84	no	—
71	C₂²×C₄×C₁₃	—	2	28	no	—
72	C₂²×C₄×C₁₃	C₂²⋊C₄	1	7	yes	—
73	C₂²×C₄×C₁₃	C₂²⋊C₄	1	7	yes	—
74	C₂²×C₄×C₁₃	C₂²⋊C₄	1	42	yes	—
75	C₂²×C₄×C₁₃	C₂²×C₄	1	7	yes	—
76	C₂²×C₄×C₁₃	C₂²×C₄	1	21	yes	—
77	C₂²×C₄×C₁₃	C₂×D₄	1	21	yes	—
78	C₂²×C₄×C₁₃	C₂×D₄	1	7	yes	—
79	C₂²×C₄×C₁₃	C₂×D₄	1	21	yes	—
80	C₂²×C₄×C₁₃	C₂⁴	1	7	yes	—
81	C₂²×Dic₁₃	—	2	112	no	—
82	C₂²×Dic₁₃	—	2	28	no	—
83	C₄×Dic₁₃	—	2	28	no	—
84	C₄×Dic₁₃	C₂⁴	1	8	yes	complemented
85	#6099	—	2	28	no	—
86	#6099	C₂⁴	1	8	yes	complemented
87	C₂×#1055	—	2	28	no	—
88	C₂³×#41	D₁₃	1	1	yes	characteristic
89	C₂³×M₁₆	—	13	1	no	—
90	C₂³×C₄²	—	13	1	no	—
91	C₂²×#104	—	13	28	no	—
92	C₂³×C₁₃	—	2	28	no	—
93	C₂³×C₁₃	—	2	28	no	—
94	C₂³×C₁₃	—	2	28	no	—
95	C₂³×C₁₃	—	4	56	no	—
96	C₂³×C₁₃	#104	1	8	yes	complemented
97	C₂³×C₁₃	C₂×C₂²⋊C₄	1	21	yes	—
98	C₂³×C₁₃	C₂×C₂²⋊C₄	1	7	yes	—
99	C₂³×C₁₃	C₂²×D₄	1	7	yes	—
100	Q₈×C₁₃	—	2	28	no	—
101	Q₈×C₁₃	C₂³×C₄	1	8	yes	complemented
102	D₄×C₁₃	—	2	56	no	—
103	D₄×C₁₃	—	2	8	no	—
104	D₄×C₁₃	—	2	28	no	—
105	D₄×C₁₃	C₂³×C₄	1	8	yes	complemented
106	C₂×C₄×C₁₃	—	2	28	no	—
107	C₂×C₄×C₁₃	—	2	56	no	—
108	C₂×C₄×C₁₃	—	2	8	no	—
109	C₂×C₄×C₁₃	C₂×C₂²⋊C₄	1	21	yes	—
110	C₂×C₄×C₁₃	C₂×C₂²⋊C₄	1	21	yes	—
111	C₂×C₄×C₁₃	C₂×C₂²⋊C₄	1	42	yes	—
112	C₂×C₄×C₁₃	C₂³×C₄	1	7	yes	—
113	C₂×C₄×C₁₃	C₂³×C₄	1	7	yes	—
114	C₂×C₄×C₁₃	C₂²×D₄	1	7	yes	—
115	C₂×C₄×C₁₃	C₂²×D₄	1	7	yes	—
116	C₂×C₄×C₁₃	C₂²×D₄	1	7	yes	—
117	C₂×C₄×C₁₃	C₂⁵	1	1	yes	characteristic
118	C₂×Dic₁₃	—	2	112	no	—
119	C₂×Dic₁₃	—	2	8	no	—
120	#1055	—	2	8	no	—
121	C₂²×#41	—	2	7	no	—
122	C₂²×#41	—	2	7	no	—
123	C₂²×#41	Dic₁₃	1	14	yes	complemented
124	C₂²×#41	C₂×D₁₃	1	14	yes	—
125	C₂³×Q₈	Dic₁₃	1	1	yes	characteristic, complemented
126	C₂³×D₄	—	2	1	no	—
127	C₂³×D₄	Dic₁₃	1	1	yes	characteristic, complemented
128	C₂⁴×C₄	—	2	1	no	—
129	C₂⁴×C₄	—	26	1	no	—
130	C₂⁴×C₄	C₂×D₁₃	1	1	yes	characteristic
131	C₂²×M₁₆	—	13	14	no	—
132	C₂²×M₁₆	—	26	7	no	—
133	C₂³×C₈	—	26	1	no	—
134	C₂²×C₄²	—	13	14	no	—
135	C₂²×C₄²	—	26	7	no	—
136	C₂×#104	—	13	112	no	—
137	C₂²×C₁₃	—	2	14	no	—
138	C₂²×C₁₃	—	2	42	no	—
139	C₂²×C₁₃	—	2	8	no	—
140	C₂²×C₁₃	—	4	56	no	—
141	C₂²×C₁₃	C₂×#104	1	28	yes	complemented
142	C₂²×C₁₃	C₂²×C₂²⋊C₄	1	7	yes	—
143	C₂²×C₁₃	C₂²×C₂²⋊C₄	1	7	yes	—
144	C₂²×C₁₃	C₂³×D₄	1	1	yes	characteristic
145	C₄×C₁₃	—	2	8	no	—
146	C₄×C₁₃	C₂²×C₂²⋊C₄	1	7	yes	—
147	C₄×C₁₃	C₂²×C₂²⋊C₄	1	7	yes	—
148	C₄×C₁₃	C₂⁴×C₄	1	1	yes	characteristic
149	C₄×C₁₃	C₂³×D₄	1	1	yes	characteristic
150	Dic₁₃	—	2	16	no	—
151	C₂⁵	—	2	1	no	—
152	C₂⁵	C₂²⋊D₁₃	1	1	yes	characteristic
153	C₂×#41	—	2	84	no	—
154	C₂×#41	—	2	42	no	—
155	C₂×#41	—	2	42	no	—
156	C₂×#41	C₂×Dic₁₃	1	84	yes	complemented
157	C₂×#41	C₂²×D₁₃	1	28	yes	—
158	C₂²×Q₈	—	2	7	no	—
159	C₂²×Q₈	C₂×Dic₁₃	1	14	yes	complemented
160	C₂²×D₄	—	2	14	no	—
161	C₂²×D₄	—	2	14	no	—
162	C₂²×D₄	—	2	7	no	—
163	C₂²×D₄	C₂×Dic₁₃	1	14	yes	complemented
164	C₂³×C₄	—	2	1	no	—
165	C₂³×C₄	—	2	14	no	—
166	C₂³×C₄	—	2	14	no	—
167	C₂³×C₄	—	26	16	no	—
168	C₂³×C₄	—	26	14	no	—
169	C₂³×C₄	C₂×Dic₁₃	1	1	yes	characteristic
170	C₂³×C₄	C₂×Dic₁₃	1	7	yes	—
171	C₂³×C₄	C₂²⋊D₁₃	1	7	yes	—
172	C₂³×C₄	C₂²⋊D₁₃	1	7	yes	—
173	C₂³×C₄	C₂²⋊D₁₃	1	1	yes	characteristic
174	C₂³×C₄	C₂²×D₁₃	1	7	yes	—
175	C₂×M₁₆	—	13	28	no	—
176	C₂×M₁₆	—	26	42	no	—
177	C₂²×C₈	—	26	14	no	—
178	C₂×C₄²	—	13	28	no	—
179	C₂×C₄²	—	26	42	no	—
180	#104	—	13	64	no	—
181	C₂×C₁₃	—	2	1	no	—
182	C₂×C₁₃	—	2	7	no	—
183	C₂×C₁₃	—	4	8	no	—
184	C₂×C₁₃	C₂²×#104	1	14	yes	complemented
185	C₂×C₁₃	C₂³×C₂²⋊C₄	1	1	yes	characteristic
186	C₂⁴	—	2	8	no	—
187	C₂⁴	—	2	14	no	—
188	C₂⁴	—	4	8	no	—
189	C₂⁴	C₂²⋊Dic₁₃	1	7	yes	—
190	C₂⁴	C₂²⋊Dic₁₃	1	1	yes	characteristic
191	C₂⁴	C₂×C₂²⋊D₁₃	1	7	yes	—
192	#41	—	2	168	no	—
193	#41	—	2	28	no	—
194	#41	—	2	28	no	—
195	#41	C₂²×Dic₁₃	1	56	yes	complemented
196	#41	C₂³×D₁₃	1	8	yes	—
197	C₂×Q₈	—	2	42	no	—
198	C₂×Q₈	C₂²×Dic₁₃	1	28	yes	complemented
199	C₂×D₄	—	2	84	no	—
200	C₂×D₄	—	2	28	no	—
201	C₂×D₄	—	2	42	no	—
202	C₂×D₄	C₂²×Dic₁₃	1	28	yes	complemented
203	C₂²×C₄	—	2	14	no	—
204	C₂²×C₄	—	2	84	no	—
205	C₂²×C₄	—	2	28	no	—
206	C₂²×C₄	—	26	112	no	—
207	C₂²×C₄	—	26	28	no	—
208	C₂²×C₄	C₂²⋊Dic₁₃	1	7	yes	—
209	C₂²×C₄	C₂²⋊Dic₁₃	1	7	yes	—
210	C₂²×C₄	C₂²⋊Dic₁₃	1	42	yes	—
211	C₂²×C₄	C₂²×Dic₁₃	1	7	yes	—
212	C₂²×C₄	C₂²×Dic₁₃	1	21	yes	—
213	C₂²×C₄	C₂×C₂²⋊D₁₃	1	21	yes	—
214	C₂²×C₄	C₂×C₂²⋊D₁₃	1	7	yes	—
215	C₂²×C₄	C₂×C₂²⋊D₁₃	1	21	yes	—
216	C₂²×C₄	C₂³×D₁₃	1	7	yes	—
217	M₁₆	—	13	8	no	—
218	M₁₆	—	26	28	no	—
219	C₂×C₈	—	26	28	no	—
220	C₄²	—	13	8	no	—
221	C₄²	—	26	28	no	—
222	C₁₃	C₂³×#104	1	1	yes	characteristic, complemented, Sylow
223	C₂³	—	2	28	no	—
224	C₂³	—	2	28	no	—
225	C₂³	—	2	28	no	—
226	C₂³	—	4	56	no	—
227	C₂³	#88077	1	8	yes	direct product factor
228	C₂³	C₂×C₂²⋊Dic₁₃	1	21	yes	—
229	C₂³	C₂×C₂²⋊Dic₁₃	1	7	yes	—
230	C₂³	C₂²×C₂²⋊D₁₃	1	7	yes	—
231	Q₈	—	2	28	no	—
232	Q₈	C₂³×Dic₁₃	1	8	yes	complemented
233	D₄	—	2	56	no	—
234	D₄	—	2	8	no	—
235	D₄	—	2	28	no	—
236	D₄	C₂³×Dic₁₃	1	8	yes	complemented
237	C₂×C₄	—	2	28	no	—
238	C₂×C₄	—	2	56	no	—
239	C₂×C₄	—	2	8	no	—
240	C₂×C₄	—	26	112	no	—
241	C₂×C₄	—	26	8	no	—
242	C₂×C₄	C₂×C₂²⋊Dic₁₃	1	21	yes	—
243	C₂×C₄	C₂×C₂²⋊Dic₁₃	1	21	yes	—
244	C₂×C₄	C₂×C₂²⋊Dic₁₃	1	42	yes	—
245	C₂×C₄	C₂³×Dic₁₃	1	7	yes	—
246	C₂×C₄	C₂³×Dic₁₃	1	7	yes	—
247	C₂×C₄	C₂²×C₂²⋊D₁₃	1	7	yes	—
248	C₂×C₄	C₂²×C₂²⋊D₁₃	1	7	yes	—
249	C₂×C₄	C₂²×C₂²⋊D₁₃	1	7	yes	—
250	C₂×C₄	C₂⁴×D₁₃	1	1	yes	characteristic
251	C₈	—	26	8	no	—
252	C₂²	—	2	14	no	—
253	C₂²	—	2	42	no	—
254	C₂²	—	2	8	no	—
255	C₂²	—	4	56	no	—
256	C₂²	C₂×#88077	1	28	yes	direct product factor
257	C₂²	C₂²×C₂²⋊Dic₁₃	1	7	yes	—
258	C₂²	C₂²×C₂²⋊Dic₁₃	1	7	yes	—
259	C₂²	C₂³×C₂²⋊D₁₃	1	1	yes	characteristic
260	C₄	—	2	8	no	—
261	C₄	—	26	16	no	—
262	C₄	C₂²×C₂²⋊Dic₁₃	1	7	yes	—
263	C₄	C₂²×C₂²⋊Dic₁₃	1	7	yes	—
264	C₄	C₂⁴×Dic₁₃	1	1	yes	characteristic
265	C₄	C₂³×C₂²⋊D₁₃	1	1	yes	characteristic
266	C₂	—	2	1	no	—
267	C₂	—	2	7	no	—
268	C₂	—	4	8	no	—
269	C₂	C₂²×#88077	1	14	yes	direct product factor
270	C₂	C₂³×C₂²⋊Dic₁₃	1	1	yes	characteristic

	subgroup	G/H
center	C₂³×C₄	C₂²⋊D₁₃
commutator	C₄×C₁₃	C₂⁴×C₄
Frattini	C₂×C₄	C₂⁴×D₁₃
Fitting, prime radical	C₂³×C₁₃×#41	C₂
socle	C₂⁴×C₁₃	C₂²⋊C₄
cosocle	C₂×C₄×C₁₃	C₂⁵
hypercenter	C₂³×#41	D₁₃
hypocenter	C₁₃	C₂³×#104
LCS	G ▷ C₄×C₁₃ ▷ C₂×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂³×C₄ ◁ C₂⁴×C₄ ◁ C₂³×#41

Complements

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

N	H ≅ G/N	type
1. C₂³×#6099	C₂	semidirect
2. C₂³×C₄×Dic₁₃	C₂	semidirect
3. C₂²×#88077	C₂	direct (N × H)
7. C₂²×C₁₃×#41	C₄	semidirect
9. C₂³×Q₈×C₁₃	C₄	semidirect
11. C₂³×D₄×C₁₃	C₄	semidirect
16. C₂²×C₄×Dic₁₃	C₂²	semidirect
18. C₂²×#6099	C₂²	semidirect
20. C₂×#88077	C₂²	direct (N × H)
26. C₂×C₁₃×#41	C₂×C₄	semidirect
29. C₂²×Q₈×C₁₃	C₂×C₄	semidirect
33. C₂²×D₄×C₁₃	C₂×C₄	semidirect
46. C₂×C₄×Dic₁₃	C₂³	semidirect
48. C₂×#6099	C₂³	semidirect
50. #88077	C₂³	direct (N × H)
61. C₁₃×#41	C₂²×C₄	semidirect
64. C₂×Q₈×C₁₃	C₂²×C₄	semidirect
68. C₂×D₄×C₁₃	C₂²×C₄	semidirect
84. C₄×Dic₁₃	C₂⁴	semidirect
86. #6099	C₂⁴	semidirect
96. C₂³×C₁₃	#104	semidirect
101. Q₈×C₁₃	C₂³×C₄	semidirect
105. D₄×C₁₃	C₂³×C₄	semidirect
123. C₂²×#41	Dic₁₃	semidirect
125. C₂³×Q₈	Dic₁₃	semidirect
127. C₂³×D₄	Dic₁₃	semidirect
141. C₂²×C₁₃	C₂×#104	semidirect
156. C₂×#41	C₂×Dic₁₃	semidirect
159. C₂²×Q₈	C₂×Dic₁₃	semidirect
163. C₂²×D₄	C₂×Dic₁₃	semidirect
184. C₂×C₁₃	C₂²×#104	semidirect
195. #41	C₂²×Dic₁₃	semidirect
198. C₂×Q₈	C₂²×Dic₁₃	semidirect
202. C₂×D₄	C₂²×Dic₁₃	semidirect
222. C₁₃	C₂³×#104	semidirect
227. C₂³	#88077	direct (N × H)
232. Q₈	C₂³×Dic₁₃	semidirect
236. D₄	C₂³×Dic₁₃	semidirect
256. C₂²	C₂×#88077	direct (N × H)
269. C₂	C₂²×#88077	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂³×#6099	2
C₂³×C₄×Dic₁₃	2
C₂²×#88077	2
C₂³×C₁₃×#41	2
C₂³×#104	13

Subgroups — C23×#88077 #16726507

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂³×#88077 #16726507