Subgroups — #16732615

Order 3328.

1834 conjugacy classes of subgroups.

The subgroup lattice has 274 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	#13125375	C₂	1	12	yes	maximal, complemented
4	#13125350	C₂	1	3	yes	maximal, complemented
5	C₂×#11718343	C₂	1	3	yes	maximal, complemented
6	#13123436	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×#88058	C₂²	1	12	yes	—
24	#11718363	C₂²	1	24	yes	—
25	#11718344	C₂²	1	12	yes	—
26	#11718343	C₂²	1	6	yes	—
27	#11718343	C₂²	1	3	yes	—
28	#11718343	C₂²	1	12	yes	—
29	#11718343	C₂²	1	1	yes	characteristic
30	#11718341	C₂²	1	6	yes	—
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	1	yes	characteristic
37	C₄×Q₈×C₁₃	D₄	1	1	yes	characteristic
38	C₄×D₄×C₁₃	D₄	1	3	yes	—
39	C₄×D₄×C₁₃	D₄	1	3	yes	—
40	C₁₃×#117	C₂³	1	3	yes	—
41	C₂×C₄²×C₁₃	C₂³	1	3	yes	—
42	Q₈×Dic₁₃	—	2	4	no	—
43	D₄×Dic₁₃	—	2	12	no	—
44	#88180	C₂³	1	12	yes	—
45	C₂×C₄⋊Dic₁₃	—	2	6	no	—
46	C₂×C₄⋊Dic₁₃	C₂³	1	12	yes	—
47	#88178	—	2	6	no	—
48	#88178	—	2	6	no	—
49	#88144	—	2	12	no	—
50	#88142	—	2	12	no	—
51	#88132	—	2	24	no	—
52	#88124	C₂³	1	12	yes	—
53	#88123	—	2	6	no	—
54	#88123	—	2	6	no	—
55	#88123	C₂³	1	6	yes	—
56	#88123	C₂³	1	12	yes	—
57	#88123	C₂³	1	2	yes	—
58	C₄×Dic₂₆	—	2	6	no	—
59	C₄×Dic₂₆	—	2	6	no	—
60	C₁₃×#97	C₂³	1	6	yes	—
61	C₄×C₈×C₁₃	C₂³	1	1	yes	characteristic
62	C₄×C₈×C₁₃	C₂³	1	3	yes	—
63	C₄×C₈×C₁₃	C₂³	1	6	yes	—
64	#88058	C₂³	1	12	yes	—
65	#88058	C₂³	1	24	yes	—
66	#88058	C₂³	1	4	yes	—
67	#88057	C₂³	1	24	yes	—
68	#60667	—	13	1	no	maximal, Sylow
69	C₁₃×#41	C₂×Q₈	1	8	yes	—
70	C₂×Q₈×C₁₃	C₂×D₄	1	1	yes	characteristic
71	C₂×D₄×C₁₃	C₂×D₄	1	3	yes	—
72	C₂²×C₄×C₁₃	C₂×D₄	1	3	yes	—
73	C₂²×C₄×C₁₃	C₂×D₄	1	3	yes	—
74	C₂²×C₄×C₁₃	C₂⁴	1	3	yes	—
75	C₂²×Dic₁₃	—	2	12	no	—
76	C₂×Dic₂₆	—	2	2	no	—
77	C₂×Dic₂₆	—	2	6	no	—
78	C₂×Dic₂₆	—	2	12	no	—
79	C₂×Dic₂₆	—	8	12	no	—
80	C₁₃×M₁₆	C₂×Q₈	1	24	yes	complemented
81	C₂×C₈×C₁₃	C₂×Q₈	1	24	yes	complemented
82	C₂×C₈×C₁₃	C₂⁴	1	2	yes	—
83	C₂×C₈×C₁₃	C₂⁴	1	6	yes	—
84	C₁₃×C₄⋊C₄	C₂×D₄	1	3	yes	—
85	C₁₃×C₄⋊C₄	C₂×D₄	1	3	yes	—
86	C₁₃×C₂²⋊C₄	C₂×D₄	1	6	yes	—
87	C₄²×C₁₃	C₂×D₄	1	3	yes	—
88	C₄²×C₁₃	C₂×D₄	1	3	yes	—
89	C₄²×C₁₃	C₂⁴	1	1	yes	characteristic
90	C₄²×C₁₃	C₂⁴	1	3	yes	—
91	C₂²⋊Dic₁₃	—	2	12	no	—
92	C₄⋊Dic₁₃	—	2	2	no	—
93	C₄⋊Dic₁₃	—	2	6	no	—
94	C₄⋊Dic₁₃	—	2	12	no	—
95	C₄⋊Dic₁₃	—	2	12	no	—
96	C₄⋊Dic₁₃	C₂⁴	1	12	yes	—
97	C₄⋊Dic₁₃	C₂⁴	1	4	yes	—
98	#6101	—	2	12	no	—
99	#6101	—	2	12	no	—
100	C₄×Dic₁₃	—	2	12	no	—
101	#3503	—	13	2	no	—
102	#3165	—	13	6	no	—
103	#3164	—	13	12	no	—
104	#3160	—	13	3	no	—
105	C₂×#500	—	13	3	no	—
106	#2913	—	13	4	no	—
107	C₄×#131	D₁₃	1	1	yes	characteristic
108	C₂³×C₁₃	C₂²×D₄	1	3	yes	—
109	Q₈×C₁₃	C₄⋊Q₈	1	4	yes	—
110	D₄×C₁₃	C₄⋊Q₈	1	12	yes	—
111	C₂×C₄×C₁₃	—	2	6	no	—
112	C₂×C₄×C₁₃	—	2	6	no	—
113	C₂×C₄×C₁₃	C₂²×D₄	1	3	yes	—
114	C₂×C₄×C₁₃	C₂²×D₄	1	3	yes	—
115	C₂×C₄×C₁₃	C₂²×D₄	1	3	yes	—
116	C₂×C₄×C₁₃	C₂²×D₄	1	1	yes	characteristic
117	C₂×C₄×C₁₃	C₂²×D₄	1	1	yes	characteristic
118	C₂×C₄×C₁₃	C₂²×Q₈	1	12	yes	—
119	C₂×C₄×C₁₃	C₂⁵	1	1	yes	characteristic
120	C₂×Dic₁₃	—	2	4	no	—
121	C₂×Dic₁₃	—	2	12	no	—
122	C₂×Dic₁₃	—	8	12	no	—
123	Dic₂₆	—	4	12	no	—
124	Dic₂₆	—	4	4	no	—
125	Dic₂₆	—	8	12	no	—
126	C₈×C₁₃	C₂²×Q₈	1	4	yes	—
127	C₈×C₁₃	C₂²×Q₈	1	12	yes	—
128	C₂×#131	C₂×D₁₃	1	2	yes	—
129	Q₈²	—	26	2	no	—
130	#557	—	26	2	no	—
131	#554	—	26	6	no	—
132	D₄×Q₈	—	26	6	no	—
133	#544	—	13	12	no	—
134	#533	—	13	6	no	—
135	C₂×C₄⋊Q₈	—	13	6	no	—
136	#520	—	13	4	no	—
137	C₄×#41	C₂×D₁₃	1	1	yes	characteristic
138	#501	—	13	24	no	—
139	#500	—	13	3	no	—
140	#500	—	13	12	no	—
141	#500	—	13	6	no	—
142	#500	—	13	1	no	—
143	#499	—	13	12	no	—
144	#498	—	13	6	no	—
145	#428	—	13	24	no	—
146	#427	—	13	12	no	—
147	C₂×#107	—	13	12	no	—
148	#405	C₂×D₁₃	1	6	yes	—
149	C₄×M₁₆	C₂×D₁₃	1	3	yes	—
150	C₂×C₄×C₈	C₂×D₁₃	1	3	yes	—
151	C₂²×C₁₃	—	2	3	no	—
152	C₂²×C₁₃	—	2	3	no	—
153	C₂²×C₁₃	C₂×C₄⋊Q₈	1	6	yes	—
154	C₂²×C₁₃	C₂³×D₄	1	1	yes	characteristic
155	C₄×C₁₃	—	2	3	no	—
156	C₄×C₁₃	—	2	3	no	—
157	C₄×C₁₃	C₂×C₄⋊Q₈	1	6	yes	—
158	C₄×C₁₃	#576	1	2	yes	—
159	C₄×C₁₃	#578	1	2	yes	—
160	C₄×C₁₃	C₂³×Q₈	1	2	yes	—
161	Dic₁₃	—	4	12	no	—
162	Dic₁₃	—	4	4	no	—
163	C₂×#41	C₂²×D₁₃	1	1	yes	characteristic
164	C₂²×Q₈	—	26	6	no	—
165	#131	Dic₂₆	1	16	yes	complemented
166	C₂×M₁₆	C₂²×D₁₃	1	6	yes	—
167	C₂²×C₈	C₂²×D₁₃	1	6	yes	—
168	C₄⋊Q₈	—	13	6	no	—
169	C₄⋊Q₈	—	13	12	no	—
170	C₄⋊Q₈	—	13	2	no	—
171	C₄⋊Q₈	—	26	12	no	—
172	C₄⋊Q₈	—	26	6	no	—
173	C₄⋊Q₈	—	26	6	no	—
174	#125	—	13	12	no	—
175	#125	—	26	12	no	—
176	C₂²⋊Q₈	—	26	6	no	—
177	C₂²⋊Q₈	—	26	6	no	—
178	C₂²⋊Q₈	—	26	24	no	—
179	C₄×Q₈	—	26	4	no	—
180	C₄×Q₈	—	26	6	no	—
181	C₄×Q₈	—	26	6	no	—
182	C₄×Q₈	D₅₂	1	1	yes	characteristic
183	C₄×Q₈	D₅₂	1	1	yes	characteristic
184	C₄×D₄	—	26	12	no	—
185	C₄×D₄	D₅₂	1	3	yes	—
186	C₄×D₄	D₅₂	1	3	yes	—
187	#117	—	13	12	no	—
188	#117	C₂²×D₁₃	1	3	yes	—
189	C₂×C₄⋊C₄	—	13	12	no	—
190	C₂×C₄⋊C₄	—	26	6	no	—
191	C₂×C₄²	C₂²×D₁₃	1	3	yes	—
192	#107	—	13	12	no	—
193	#107	—	13	24	no	—
194	#107	—	13	4	no	—
195	#106	—	13	24	no	—
196	#97	C₂²×D₁₃	1	6	yes	—
197	C₄×C₈	C₂²×D₁₃	1	1	yes	characteristic
198	C₄×C₈	C₂²×D₁₃	1	3	yes	—
199	C₄×C₈	C₂²×D₁₃	1	6	yes	—
200	C₂×C₁₃	—	2	6	no	—
201	C₂×C₁₃	C₂²×C₄⋊Q₈	1	1	yes	characteristic
202	C₂×C₁₃	C₂×#576	1	1	yes	characteristic
203	C₂×C₁₃	C₂×#578	1	1	yes	characteristic
204	#41	C₂×Dic₂₆	1	8	yes	—
205	C₂×Q₈	—	26	2	no	—
206	C₂×Q₈	—	26	6	no	—
207	C₂×Q₈	—	26	12	no	—
208	C₂×Q₈	—	104	12	no	—
209	C₂×Q₈	C₂×D₅₂	1	1	yes	characteristic
210	C₂×D₄	C₂×D₅₂	1	3	yes	—
211	C₂²×C₄	—	26	12	no	—
212	C₂²×C₄	C₂×D₅₂	1	3	yes	—
213	C₂²×C₄	C₂×D₅₂	1	3	yes	—
214	C₂²×C₄	C₂³×D₁₃	1	3	yes	—
215	M₁₆	C₂×Dic₂₆	1	24	yes	complemented
216	C₂×C₈	C₂×Dic₂₆	1	24	yes	complemented
217	C₂×C₈	C₂³×D₁₃	1	2	yes	—
218	C₂×C₈	C₂³×D₁₃	1	6	yes	—
219	C₄⋊C₄	—	13	4	no	—
220	C₄⋊C₄	—	13	12	no	—
221	C₄⋊C₄	—	26	12	no	—
222	C₄⋊C₄	—	26	6	no	—
223	C₄⋊C₄	—	26	12	no	—
224	C₄⋊C₄	—	26	2	no	—
225	C₄⋊C₄	—	26	12	no	—
226	C₄⋊C₄	—	26	12	no	—
227	C₄⋊C₄	C₂×D₅₂	1	3	yes	—
228	C₄⋊C₄	C₂×D₅₂	1	3	yes	—
229	C₂²⋊C₄	—	26	12	no	—
230	C₂²⋊C₄	C₂×D₅₂	1	6	yes	—
231	C₄²	—	26	12	no	—
232	C₄²	C₂×D₅₂	1	3	yes	—
233	C₄²	C₂×D₅₂	1	3	yes	—
234	C₄²	C₂³×D₁₃	1	1	yes	characteristic
235	C₄²	C₂³×D₁₃	1	3	yes	—
236	C₁₃	#60667	1	1	yes	characteristic, complemented, Sylow
237	C₂³	C₂²×D₅₂	1	3	yes	—
238	Q₈	—	52	12	no	—
239	Q₈	—	52	4	no	—
240	Q₈	—	104	12	no	—
241	Q₈	#88123	1	4	yes	—
242	D₄	#88123	1	12	yes	—
243	C₂×C₄	—	2	6	no	—
244	C₂×C₄	—	2	6	no	—
245	C₂×C₄	—	26	4	no	—
246	C₂×C₄	—	26	12	no	—
247	C₂×C₄	—	104	12	no	—
248	C₂×C₄	C₂²×Dic₂₆	1	12	yes	—
249	C₂×C₄	C₂²×D₅₂	1	3	yes	—
250	C₂×C₄	C₂²×D₅₂	1	3	yes	—
251	C₂×C₄	C₂²×D₅₂	1	3	yes	—
252	C₂×C₄	C₂²×D₅₂	1	1	yes	characteristic
253	C₂×C₄	C₂²×D₅₂	1	1	yes	characteristic
254	C₂×C₄	C₂⁴×D₁₃	1	1	yes	characteristic
255	C₈	C₂²×Dic₂₆	1	4	yes	—
256	C₈	C₂²×Dic₂₆	1	12	yes	—
257	C₂²	—	2	3	no	—
258	C₂²	—	2	3	no	—
259	C₂²	C₂×#88123	1	6	yes	—
260	C₂²	C₂³×D₅₂	1	1	yes	characteristic
261	C₄	—	2	3	no	—
262	C₄	—	2	3	no	—
263	C₄	—	52	12	no	—
264	C₄	—	52	4	no	—
265	C₄	C₂×#88123	1	6	yes	—
266	C₄	#11719457	1	2	yes	—
267	C₄	#11719458	1	2	yes	—
268	C₄	C₂³×Dic₂₆	1	2	yes	—
269	C₂	—	2	6	no	—
270	C₂	C₂²×#88123	1	1	yes	characteristic
271	C₂	C₂×#11719457	1	1	yes	characteristic
272	C₂	C₂×#11719458	1	1	yes	characteristic

	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	#60667
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. #13125375	C₂	semidirect
4. #13125350	C₂	semidirect
5. C₂×#11718343	C₂	semidirect
33. C₁₃×#131	Q₈	semidirect
80. C₁₃×M₁₆	C₂×Q₈	semidirect
81. C₂×C₈×C₁₃	C₂×Q₈	semidirect
165. #131	Dic₂₆	semidirect
215. M₁₆	C₂×Dic₂₆	semidirect
216. C₂×C₈	C₂×Dic₂₆	semidirect
236. C₁₃	#60667	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
#13125375	2
#13125350	2
C₂×#11718343	2
#13123436	2
C₄×C₁₃×#131	2
#60667	13