Subgroups — #16732599

Order 3328.

1834 conjugacy classes of subgroups.

The subgroup lattice has 250 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#13128655	C₂	1	2	yes	maximal
2	#13125382	C₂	1	6	yes	maximal, complemented
3	#13125367	C₂	1	12	yes	maximal, complemented
4	#13125350	C₂	1	3	yes	maximal, complemented
5	C₂×#11718341	C₂	1	3	yes	maximal, complemented
6	#13123431	C₂	1	4	yes	maximal
7	C₄×C₁₃×#131	C₂	1	1	yes	maximal, characteristic
8	C₂×C₁₃×#131	C₂²	1	2	yes	—
9	C₄×C₁₃×#41	C₂²	1	1	yes	characteristic
10	#11719532	C₂²	1	4	yes	—
11	#11719275	—	2	2	no	—
12	Q₈×Dic₂₆	—	2	2	no	—
13	#11719248	—	2	6	no	—
14	D₄×Dic₂₆	—	2	6	no	—
15	#11719224	C₂²	1	12	yes	—
16	#11719175	C₂²	1	6	yes	—
17	C₂×#88123	C₂²	1	6	yes	—
18	C₁₃×#405	C₂²	1	6	yes	—
19	C₄×C₁₃×M₁₆	C₂²	1	3	yes	—
20	C₂×C₄×C₈×C₁₃	C₂²	1	3	yes	—
21	#11718782	C₂²	1	24	yes	—
22	#11718767	C₂²	1	12	yes	—
23	C₂×#88057	C₂²	1	12	yes	—
24	#11718363	C₂²	1	24	yes	—
25	#11718344	C₂²	1	12	yes	—
26	#11718343	C₂²	1	6	yes	—
27	#11718341	C₂²	1	6	yes	—
28	#11718341	C₂²	1	3	yes	—
29	#11718341	C₂²	1	12	yes	—
30	#11718341	C₂²	1	1	yes	characteristic
31	C₂×C₁₃×#41	C₂³	1	1	yes	characteristic
32	C₂²×Dic₂₆	—	2	6	no	—
33	C₁₃×#131	Q₈	1	16	yes	complemented
34	C₂×C₁₃×M₁₆	C₂³	1	6	yes	—
35	C₂²×C₈×C₁₃	C₂³	1	6	yes	—
36	C₄×Q₈×C₁₃	D₄	1	2	yes	—
37	C₄×D₄×C₁₃	D₄	1	6	yes	—
38	C₁₃×#117	C₂³	1	3	yes	—
39	C₂×C₄²×C₁₃	C₂³	1	3	yes	—
40	Q₈×Dic₁₃	—	2	4	no	—
41	D₄×Dic₁₃	—	2	12	no	—
42	#88180	C₂³	1	12	yes	—
43	C₂×C₄⋊Dic₁₃	—	2	6	no	—
44	C₂×C₄⋊Dic₁₃	C₂³	1	12	yes	—
45	#88178	—	2	6	no	—
46	#88178	—	2	6	no	—
47	#88144	—	2	12	no	—
48	#88142	—	2	12	no	—
49	#88132	—	2	24	no	—
50	#88124	C₂³	1	12	yes	—
51	#88123	—	2	6	no	—
52	#88123	—	2	6	no	—
53	#88123	C₂³	1	6	yes	—
54	#88123	C₂³	1	12	yes	—
55	#88123	C₂³	1	2	yes	—
56	C₄×Dic₂₆	—	2	6	no	—
57	C₄×Dic₂₆	—	2	6	no	—
58	C₁₃×#97	C₂³	1	6	yes	—
59	C₄×C₈×C₁₃	C₂³	1	1	yes	characteristic
60	C₄×C₈×C₁₃	C₂³	1	3	yes	—
61	C₄×C₈×C₁₃	C₂³	1	6	yes	—
62	#88058	C₂³	1	24	yes	—
63	#88057	C₂³	1	12	yes	—
64	#88057	C₂³	1	24	yes	—
65	#88057	C₂³	1	4	yes	—
66	#60665	—	13	1	no	maximal, Sylow
67	C₁₃×#41	C₂×Q₈	1	8	yes	—
68	C₂×Q₈×C₁₃	C₂×D₄	1	1	yes	characteristic
69	C₂×D₄×C₁₃	C₂×D₄	1	3	yes	—
70	C₂²×C₄×C₁₃	C₂×D₄	1	6	yes	—
71	C₂²×C₄×C₁₃	C₂⁴	1	3	yes	—
72	C₂²×Dic₁₃	—	2	12	no	—
73	C₂×Dic₂₆	—	2	2	no	—
74	C₂×Dic₂₆	—	2	6	no	—
75	C₂×Dic₂₆	—	2	12	no	—
76	C₂×Dic₂₆	—	8	12	no	—
77	C₁₃×M₁₆	C₂×Q₈	1	24	yes	complemented
78	C₂×C₈×C₁₃	C₂×Q₈	1	24	yes	complemented
79	C₂×C₈×C₁₃	C₂⁴	1	2	yes	—
80	C₂×C₈×C₁₃	C₂⁴	1	6	yes	—
81	C₁₃×C₄⋊C₄	C₂×D₄	1	6	yes	—
82	C₁₃×C₂²⋊C₄	C₂×D₄	1	6	yes	—
83	C₄²×C₁₃	C₂×D₄	1	6	yes	—
84	C₄²×C₁₃	C₂⁴	1	1	yes	characteristic
85	C₄²×C₁₃	C₂⁴	1	3	yes	—
86	C₂²⋊Dic₁₃	—	2	12	no	—
87	C₄⋊Dic₁₃	—	2	2	no	—
88	C₄⋊Dic₁₃	—	2	6	no	—
89	C₄⋊Dic₁₃	—	2	12	no	—
90	C₄⋊Dic₁₃	—	2	12	no	—
91	C₄⋊Dic₁₃	C₂⁴	1	12	yes	—
92	C₄⋊Dic₁₃	C₂⁴	1	4	yes	—
93	#6101	—	2	12	no	—
94	#6101	—	2	12	no	—
95	C₄×Dic₁₃	—	2	12	no	—
96	#3503	—	13	2	no	—
97	#3165	—	13	6	no	—
98	#3163	—	13	12	no	—
99	#3160	—	13	3	no	—
100	C₂×#498	—	13	3	no	—
101	#2912	—	13	4	no	—
102	C₄×#131	D₁₃	1	1	yes	characteristic
103	C₂³×C₁₃	C₂²×D₄	1	3	yes	—
104	Q₈×C₁₃	C₄⋊Q₈	1	4	yes	—
105	D₄×C₁₃	C₄⋊Q₈	1	12	yes	—
106	C₂×C₄×C₁₃	—	2	12	no	—
107	C₂×C₄×C₁₃	C₂²×D₄	1	6	yes	—
108	C₂×C₄×C₁₃	C₂²×D₄	1	3	yes	—
109	C₂×C₄×C₁₃	C₂²×D₄	1	2	yes	—
110	C₂×C₄×C₁₃	C₂²×Q₈	1	12	yes	—
111	C₂×C₄×C₁₃	C₂⁵	1	1	yes	characteristic
112	C₂×Dic₁₃	—	2	4	no	—
113	C₂×Dic₁₃	—	2	12	no	—
114	C₂×Dic₁₃	—	8	12	no	—
115	Dic₂₆	—	4	4	no	—
116	Dic₂₆	—	4	12	no	—
117	Dic₂₆	—	8	12	no	—
118	C₈×C₁₃	C₂²×Q₈	1	4	yes	—
119	C₈×C₁₃	C₂²×Q₈	1	12	yes	—
120	C₂×#131	C₂×D₁₃	1	2	yes	—
121	Q₈²	—	26	2	no	—
122	#557	—	26	2	no	—
123	#554	—	26	6	no	—
124	D₄×Q₈	—	26	6	no	—
125	#544	—	13	12	no	—
126	#533	—	13	6	no	—
127	C₂×C₄⋊Q₈	—	13	6	no	—
128	#520	—	13	4	no	—
129	C₄×#41	C₂×D₁₃	1	1	yes	characteristic
130	#501	—	13	24	no	—
131	#500	—	13	6	no	—
132	#499	—	13	12	no	—
133	#498	—	13	3	no	—
134	#498	—	13	12	no	—
135	#498	—	13	6	no	—
136	#498	—	13	1	no	—
137	#428	—	13	24	no	—
138	#427	—	13	12	no	—
139	C₂×#106	—	13	12	no	—
140	#405	C₂×D₁₃	1	6	yes	—
141	C₄×M₁₆	C₂×D₁₃	1	3	yes	—
142	C₂×C₄×C₈	C₂×D₁₃	1	3	yes	—
143	C₂²×C₁₃	—	2	6	no	—
144	C₂²×C₁₃	C₂×C₄⋊Q₈	1	6	yes	—
145	C₂²×C₁₃	C₂³×D₄	1	1	yes	characteristic
146	C₄×C₁₃	—	2	6	no	—
147	C₄×C₁₃	C₂×C₄⋊Q₈	1	6	yes	—
148	C₄×C₁₃	#577	1	4	yes	—
149	C₄×C₁₃	C₂³×Q₈	1	2	yes	—
150	Dic₁₃	—	4	12	no	—
151	Dic₁₃	—	4	4	no	—
152	C₂×#41	C₂²×D₁₃	1	1	yes	characteristic
153	C₂²×Q₈	—	26	6	no	—
154	#131	Dic₂₆	1	16	yes	complemented
155	C₂×M₁₆	C₂²×D₁₃	1	6	yes	—
156	C₂²×C₈	C₂²×D₁₃	1	6	yes	—
157	C₄⋊Q₈	—	13	6	no	—
158	C₄⋊Q₈	—	13	12	no	—
159	C₄⋊Q₈	—	13	2	no	—
160	C₄⋊Q₈	—	26	12	no	—
161	C₄⋊Q₈	—	26	6	no	—
162	C₄⋊Q₈	—	26	6	no	—
163	#125	—	13	12	no	—
164	#125	—	26	12	no	—
165	C₂²⋊Q₈	—	26	6	no	—
166	C₂²⋊Q₈	—	26	6	no	—
167	C₂²⋊Q₈	—	26	24	no	—
168	C₄×Q₈	—	26	4	no	—
169	C₄×Q₈	—	26	6	no	—
170	C₄×Q₈	—	26	6	no	—
171	C₄×Q₈	D₅₂	1	2	yes	—
172	C₄×D₄	—	26	12	no	—
173	C₄×D₄	D₅₂	1	6	yes	—
174	#117	—	13	12	no	—
175	#117	C₂²×D₁₃	1	3	yes	—
176	C₂×C₄⋊C₄	—	13	12	no	—
177	C₂×C₄⋊C₄	—	26	6	no	—
178	C₂×C₄²	C₂²×D₁₃	1	3	yes	—
179	#107	—	13	24	no	—
180	#106	—	13	12	no	—
181	#106	—	13	24	no	—
182	#106	—	13	4	no	—
183	#97	C₂²×D₁₃	1	6	yes	—
184	C₄×C₈	C₂²×D₁₃	1	1	yes	characteristic
185	C₄×C₈	C₂²×D₁₃	1	3	yes	—
186	C₄×C₈	C₂²×D₁₃	1	6	yes	—
187	C₂×C₁₃	—	2	6	no	—
188	C₂×C₁₃	C₂²×C₄⋊Q₈	1	1	yes	characteristic
189	C₂×C₁₃	C₂×#577	1	2	yes	—
190	#41	C₂×Dic₂₆	1	8	yes	—
191	C₂×Q₈	—	26	2	no	—
192	C₂×Q₈	—	26	6	no	—
193	C₂×Q₈	—	26	12	no	—
194	C₂×Q₈	—	104	12	no	—
195	C₂×Q₈	C₂×D₅₂	1	1	yes	characteristic
196	C₂×D₄	C₂×D₅₂	1	3	yes	—
197	C₂²×C₄	—	26	12	no	—
198	C₂²×C₄	C₂×D₅₂	1	6	yes	—
199	C₂²×C₄	C₂³×D₁₃	1	3	yes	—
200	M₁₆	C₂×Dic₂₆	1	24	yes	complemented
201	C₂×C₈	C₂×Dic₂₆	1	24	yes	complemented
202	C₂×C₈	C₂³×D₁₃	1	2	yes	—
203	C₂×C₈	C₂³×D₁₃	1	6	yes	—
204	C₄⋊C₄	—	13	12	no	—
205	C₄⋊C₄	—	13	4	no	—
206	C₄⋊C₄	—	26	12	no	—
207	C₄⋊C₄	—	26	6	no	—
208	C₄⋊C₄	—	26	12	no	—
209	C₄⋊C₄	—	26	2	no	—
210	C₄⋊C₄	—	26	12	no	—
211	C₄⋊C₄	—	26	12	no	—
212	C₄⋊C₄	C₂×D₅₂	1	6	yes	—
213	C₂²⋊C₄	—	26	12	no	—
214	C₂²⋊C₄	C₂×D₅₂	1	6	yes	—
215	C₄²	—	26	12	no	—
216	C₄²	C₂×D₅₂	1	6	yes	—
217	C₄²	C₂³×D₁₃	1	1	yes	characteristic
218	C₄²	C₂³×D₁₃	1	3	yes	—
219	C₁₃	#60665	1	1	yes	characteristic, complemented, Sylow
220	C₂³	C₂²×D₅₂	1	3	yes	—
221	Q₈	—	52	4	no	—
222	Q₈	—	52	12	no	—
223	Q₈	—	104	12	no	—
224	Q₈	#88123	1	4	yes	—
225	D₄	#88123	1	12	yes	—
226	C₂×C₄	—	2	12	no	—
227	C₂×C₄	—	26	4	no	—
228	C₂×C₄	—	26	12	no	—
229	C₂×C₄	—	104	12	no	—
230	C₂×C₄	C₂²×Dic₂₆	1	12	yes	—
231	C₂×C₄	C₂²×D₅₂	1	6	yes	—
232	C₂×C₄	C₂²×D₅₂	1	3	yes	—
233	C₂×C₄	C₂²×D₅₂	1	2	yes	—
234	C₂×C₄	C₂⁴×D₁₃	1	1	yes	characteristic
235	C₈	C₂²×Dic₂₆	1	4	yes	—
236	C₈	C₂²×Dic₂₆	1	12	yes	—
237	C₂²	—	2	6	no	—
238	C₂²	C₂×#88123	1	6	yes	—
239	C₂²	C₂³×D₅₂	1	1	yes	characteristic
240	C₄	—	2	6	no	—
241	C₄	—	52	12	no	—
242	C₄	—	52	4	no	—
243	C₄	C₂×#88123	1	6	yes	—
244	C₄	#11719456	1	4	yes	—
245	C₄	C₂³×Dic₂₆	1	2	yes	—
246	C₂	—	2	6	no	—
247	C₂	C₂²×#88123	1	1	yes	characteristic
248	C₂	C₂×#11719456	1	2	yes	—

	subgroup	G/H
center	C₂²	C₂³×D₅₂
commutator, cosocle	C₂×C₄×C₁₃	C₂⁵
Frattini	C₂×C₄	C₂⁴×D₁₃
Fitting	C₄×C₁₃×#131	C₂
socle	C₂²×C₁₃	C₂³×D₄
prime radical	C₂×C₁₃×#41	C₂³
hypercenter	C₄×#131	D₁₃
hypocenter	C₁₃	#60665
LCS	G ▷ C₂×C₄×C₁₃ ▷ C₂×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂² ◁ C₄×#41 ◁ C₄×#131

Complements

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

N	H ≅ G/N	type
2. #13125382	C₂	semidirect
3. #13125367	C₂	semidirect
4. #13125350	C₂	semidirect
5. C₂×#11718341	C₂	semidirect
33. C₁₃×#131	Q₈	semidirect
77. C₁₃×M₁₆	C₂×Q₈	semidirect
78. C₂×C₈×C₁₃	C₂×Q₈	semidirect
154. #131	Dic₂₆	semidirect
200. M₁₆	C₂×Dic₂₆	semidirect
201. C₂×C₈	C₂×Dic₂₆	semidirect
219. C₁₃	#60665	semidirect

Maximal subgroups

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

subgroup	index
#13128655	2
#13125382	2
#13125367	2
#13125350	2
C₂×#11718341	2
#13123431	2
C₄×C₁₃×#131	2
#60665	13