Subgroups — #16018430

Order 3328.

474 conjugacy classes of subgroups.

The subgroup lattice has 278 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#13125369	C₂	1	2	yes	maximal
2	#13117121	C₂	1	2	yes	maximal, complemented
3	#13117119	C₂	1	2	yes	maximal, complemented
4	#13117117	C₂	1	1	yes	maximal, characteristic, complemented
5	#13117115	C₂	1	1	yes	maximal, characteristic, complemented
6	#13117114	C₂	1	1	yes	maximal, characteristic, complemented
7	#13116623	C₂	1	2	yes	maximal
8	#13116619	C₂	1	1	yes	maximal, characteristic
9	D₁₃×#367	C₂	1	1	yes	maximal, characteristic
10	#13116431	C₂	1	1	yes	maximal, characteristic
11	C₁₃×#2254	C₂	1	1	yes	maximal, characteristic, complemented
12	#11719434	—	2	2	no	—
13	C₁₃×#500	C₂²	1	2	yes	—
14	#11718741	—	2	2	no	—
15	C₂×#88038	C₂²	1	1	yes	characteristic
16	C₂×C₁₆×D₁₃	C₂²	1	1	yes	characteristic
17	#11718542	—	2	2	no	—
18	#11718541	C₂²	1	2	yes	—
19	D₁₃×#107	C₂²	1	2	yes	—
20	#11718539	C₂²	1	2	yes	—
21	#11718537	—	2	2	no	—
22	#11718535	C₂²	1	2	yes	—
23	#11718350	C₂²	1	1	yes	characteristic
24	#11718344	C₂²	1	2	yes	—
25	C₁₃×#367	C₂²	1	1	yes	characteristic
26	C₁₃×#367	C₂²	1	1	yes	characteristic
27	C₁₃×#367	C₂²	1	2	yes	—
28	C₄×C₁₃×C₁₆	C₂²	1	1	yes	characteristic
29	#11718097	C₂²	1	1	yes	characteristic
30	#11718097	C₂²	1	1	yes	characteristic
31	#11718096	C₂²	1	2	yes	—
32	#11718094	C₂²	1	1	yes	characteristic
33	C₁₆×Dic₁₃	C₂²	1	1	yes	characteristic
34	#11718082	C₂²	1	2	yes	—
35	#11718082	C₂²	1	2	yes	—
36	#11718081	C₂²	1	2	yes	—
37	#11718081	C₂²	1	2	yes	—
38	#11718053	C₂²	1	1	yes	characteristic
39	C₁₃×C₄⋊Q₈	—	2	2	no	—
40	#88200	—	4	2	no	—
41	#88198	—	4	2	no	—
42	C₂×#6094	D₄	1	1	yes	characteristic
43	C₂×C₈×D₁₃	C₂³	1	1	yes	characteristic
44	#88150	—	4	2	no	—
45	#88146	—	2	2	no	—
46	C₄⋊C₄×D₁₃	—	2	2	no	—
47	#88144	—	2	2	no	—
48	#88143	—	4	2	no	—
49	#88142	—	2	2	no	—
50	#88126	D₄	1	1	yes	characteristic
51	#88124	—	2	2	no	—
52	C₂×C₁₃×C₁₆	C₂³	1	1	yes	characteristic
53	C₂×C₁₃×C₁₆	C₂³	1	1	yes	characteristic
54	C₁₃×#107	—	2	2	no	—
55	C₁₃×#107	C₂³	1	2	yes	—
56	C₁₃×#107	C₂³	1	2	yes	—
57	C₄×C₈×C₁₃	C₂³	1	1	yes	characteristic
58	#88058	C₂³	1	2	yes	—
59	#88058	C₂³	1	2	yes	—
60	#88057	—	2	2	no	—
61	#88055	D₄	1	1	yes	characteristic
62	C₈×Dic₁₃	C₂³	1	1	yes	characteristic
63	C₂×#6090	C₂³	1	1	yes	characteristic
64	C₂×#6090	C₂³	1	1	yes	characteristic
65	#88048	—	2	2	no	—
66	#88048	—	2	2	no	—
67	#88047	—	2	2	no	—
68	#88047	—	2	2	no	—
69	#88043	D₄	1	1	yes	characteristic
70	#88038	Q₈	1	4	yes	complemented
71	C₁₆×D₁₃	Q₈	1	4	yes	complemented
72	#33730	—	13	1	no	maximal, Sylow
73	C₂×Q₈×C₁₃	—	4	2	no	—
74	C₂×C₄×D₁₃	—	4	2	no	—
75	C₂×C₄×D₁₃	C₂×D₄	1	1	yes	characteristic
76	C₂×Dic₂₆	—	4	2	no	—
77	C₂×C₈×C₁₃	C₂×D₄	1	1	yes	characteristic
78	C₂×C₈×C₁₃	C₂⁴	1	1	yes	characteristic
79	C₁₃×C₄⋊C₄	—	2	2	no	—
80	C₁₃×C₄⋊C₄	—	2	2	no	—
81	C₁₃×C₄⋊C₄	—	4	2	no	—
82	C₄²×C₁₃	C₂×D₄	1	1	yes	characteristic
83	#6103	—	4	2	no	—
84	#6103	D₈	1	2	yes	—
85	C₄⋊Dic₁₃	—	2	2	no	—
86	C₄⋊Dic₁₃	—	2	2	no	—
87	C₄⋊Dic₁₃	—	4	2	no	—
88	#6101	—	4	2	no	—
89	#6101	—	4	2	no	—
90	#6101	—	4	2	no	—
91	#6101	—	4	2	no	—
92	#6101	—	4	2	no	—
93	#6101	D₈	1	2	yes	—
94	C₄×Dic₁₃	—	4	2	no	—
95	C₄×Dic₁₃	C₂×D₄	1	1	yes	characteristic
96	C₂×#1055	C₂×D₄	1	1	yes	characteristic
97	C₂×#1055	C₂×D₄	1	1	yes	characteristic
98	#6094	Q₁₆	1	4	yes	—
99	C₈×D₁₃	C₂×Q₈	1	2	yes	—
100	C₈×D₁₃	C₂×Q₈	1	2	yes	—
101	C₁₃×C₁₆	C₂×Q₈	1	2	yes	—
102	C₁₃×C₁₆	C₂×Q₈	1	2	yes	—
103	#6090	C₂×Q₈	1	2	yes	—
104	#6090	C₂×Q₈	1	2	yes	—
105	#3164	—	13	2	no	—
106	#2255	—	13	2	no	—
107	#2255	—	13	2	no	—
108	#2254	—	13	1	no	—
109	#2254	D₁₃	1	1	yes	characteristic, complemented
110	#2253	—	13	1	no	—
111	#2253	—	13	1	no	—
112	#2159	—	13	2	no	—
113	#2158	—	13	1	no	—
114	C₂×#367	—	13	1	no	—
115	#2108	—	13	1	no	—
116	C₂²×D₁₃	C₂×D₈	1	1	yes	characteristic
117	Q₈×C₁₃	—	16	2	no	—
118	C₂×C₄×C₁₃	—	4	2	no	—
119	C₂×C₄×C₁₃	—	4	2	no	—
120	C₂×C₄×C₁₃	C₂×D₈	1	2	yes	—
121	C₂×C₄×C₁₃	C₂²×D₄	1	1	yes	characteristic
122	C₂×Dic₁₃	—	4	2	no	—
123	C₂×Dic₁₃	—	4	2	no	—
124	C₂×Dic₁₃	C₂×D₈	1	2	yes	—
125	C₂×Dic₁₃	C₂×D₈	1	1	yes	characteristic
126	C₄×D₁₃	—	16	2	no	—
127	C₄×D₁₃	C₄⋊Q₈	1	2	yes	—
128	C₄×D₁₃	C₂×Q₁₆	1	2	yes	—
129	Dic₂₆	—	16	2	no	—
130	C₈×C₁₃	C₂×Q₁₆	1	2	yes	—
131	C₈×C₁₃	C₂²×Q₈	1	1	yes	characteristic
132	C₈×C₁₃	C₂²×Q₈	1	1	yes	characteristic
133	#1055	C₄⋊Q₈	1	2	yes	—
134	#1055	C₂×Q₁₆	1	2	yes	—
135	#544	—	26	2	no	—
136	C₂×M₃₂	—	13	1	no	—
137	C₂²×C₁₆	—	13	1	no	—
138	#501	—	26	2	no	—
139	#501	—	26	2	no	—
140	#500	—	13	2	no	—
141	#500	C₂×D₁₃	1	2	yes	—
142	#499	—	13	2	no	—
143	#499	—	13	2	no	—
144	#428	—	26	2	no	—
145	#427	—	13	2	no	—
146	C₂×#107	—	13	2	no	—
147	#405	—	13	1	no	—
148	#367	—	13	1	no	—
149	#367	—	13	1	no	—
150	#367	—	13	2	no	—
151	#367	—	13	2	no	—
152	#367	C₂×D₁₃	1	1	yes	characteristic
153	#367	C₂×D₁₃	1	1	yes	characteristic
154	#367	C₂×D₁₃	1	2	yes	—
155	#366	—	13	2	no	—
156	#366	—	13	2	no	—
157	#366	—	13	2	no	—
158	#346	—	13	1	no	—
159	#346	—	13	1	no	—
160	C₄×C₁₆	—	13	1	no	—
161	C₄×C₁₆	C₂×D₁₃	1	1	yes	characteristic
162	C₂²×C₁₃	C₂²×D₈	1	1	yes	characteristic
163	C₂×D₁₃	—	2	2	no	—
164	C₂×D₁₃	#500	1	2	yes	—
165	C₄×C₁₃	—	2	2	no	—
166	C₄×C₁₃	—	8	2	no	—
167	C₄×C₁₃	—	8	2	no	—
168	C₄×C₁₃	C₂×C₄⋊Q₈	1	1	yes	characteristic
169	C₄×C₁₃	C₂²×Q₁₆	1	1	yes	characteristic
170	Dic₁₃	—	2	2	no	—
171	Dic₁₃	—	8	2	no	—
172	Dic₁₃	—	8	2	no	—
173	Dic₁₃	#500	1	2	yes	—
174	C₂×M₁₆	—	13	1	no	—
175	C₂²×C₈	—	13	1	no	—
176	C₄⋊Q₈	—	2	2	no	—
177	C₄⋊Q₈	—	26	2	no	—
178	C₄⋊Q₈	—	52	2	no	—
179	#125	—	26	2	no	—
180	#125	—	26	2	no	—
181	#125	—	52	2	no	—
182	C₂²⋊Q₈	—	52	2	no	—
183	C₂²⋊Q₈	—	52	2	no	—
184	#117	—	13	1	no	—
185	#117	—	26	2	no	—
186	C₂×C₄⋊C₄	—	26	2	no	—
187	M₃₂	—	13	4	no	—
188	C₂×C₁₆	—	13	1	no	—
189	C₂×C₁₆	—	13	1	no	—
190	C₂×C₁₆	—	13	4	no	—
191	C₂×C₁₆	C₂²×D₁₃	1	1	yes	characteristic
192	C₂×C₁₆	C₂²×D₁₃	1	1	yes	characteristic
193	#107	—	2	2	no	—
194	#107	—	13	2	no	—
195	#107	—	13	2	no	—
196	#107	—	26	2	no	—
197	#107	—	26	2	no	—
198	#107	C₂²×D₁₃	1	2	yes	—
199	#107	C₂²×D₁₃	1	2	yes	—
200	#106	—	26	2	no	—
201	#106	—	26	2	no	—
202	#106	—	26	2	no	—
203	#97	—	13	1	no	—
204	#97	—	13	1	no	—
205	C₄×C₈	—	13	1	no	—
206	C₄×C₈	C₂²×D₁₃	1	1	yes	characteristic
207	C₂×C₁₃	C₂×#500	1	1	yes	characteristic
208	C₂×C₁₃	#3416	1	2	yes	—
209	D₁₃	—	2	2	no	—
210	C₂×Q₈	—	4	2	no	—
211	C₂×Q₈	—	52	2	no	—
212	C₂²×C₄	—	13	1	no	—
213	C₂²×C₄	—	52	2	no	—
214	M₁₆	—	13	4	no	—
215	C₂×C₈	—	13	1	no	—
216	C₂×C₈	—	13	1	no	—
217	C₂×C₈	—	13	2	no	—
218	C₂×C₈	—	13	2	no	—
219	C₂×C₈	D₄×D₁₃	1	1	yes	characteristic
220	C₂×C₈	C₂³×D₁₃	1	1	yes	characteristic
221	C₄⋊C₄	—	2	2	no	—
222	C₄⋊C₄	—	2	2	no	—
223	C₄⋊C₄	—	4	2	no	—
224	C₄⋊C₄	—	13	2	no	—
225	C₄⋊C₄	—	26	2	no	—
226	C₄⋊C₄	—	26	2	no	—
227	C₄⋊C₄	—	52	2	no	—
228	C₄⋊C₄	—	52	2	no	—
229	C₄⋊C₄	—	52	2	no	—
230	C₄⋊C₄	—	52	2	no	—
231	C₄⋊C₄	—	52	2	no	—
232	C₄⋊C₄	—	52	2	no	—
233	C₂²⋊C₄	—	13	2	no	—
234	C₂²⋊C₄	—	52	2	no	—
235	C₄²	—	13	1	no	—
236	C₄²	—	52	2	no	—
237	C₄²	D₄×D₁₃	1	1	yes	characteristic
238	C₁₆	—	13	2	no	—
239	C₁₆	—	13	2	no	—
240	C₁₆	Q₈×D₁₃	1	2	yes	—
241	C₁₆	Q₈×D₁₃	1	2	yes	—
242	C₁₃	#33730	1	1	yes	characteristic, complemented, Sylow
243	C₂³	—	13	1	no	—
244	Q₈	—	16	2	no	—
245	Q₈	—	208	2	no	—
246	C₂×C₄	—	4	2	no	—
247	C₂×C₄	—	4	2	no	—
248	C₂×C₄	—	13	1	no	—
249	C₂×C₄	—	13	2	no	—
250	C₂×C₄	—	13	2	no	—
251	C₂×C₄	—	13	2	no	—
252	C₂×C₄	—	52	2	no	—
253	C₂×C₄	—	52	2	no	—
254	C₂×C₄	—	208	2	no	—
255	C₂×C₄	D₈×D₁₃	1	2	yes	—
256	C₂×C₄	C₂×D₄×D₁₃	1	1	yes	characteristic
257	C₈	—	13	2	no	—
258	C₈	—	13	2	no	—
259	C₈	Q₁₆×D₁₃	1	2	yes	—
260	C₈	C₂×Q₈×D₁₃	1	1	yes	characteristic
261	C₈	C₂×Q₈×D₁₃	1	1	yes	characteristic
262	C₂²	—	13	2	no	—
263	C₂²	—	26	2	no	—
264	C₂²	C₂×D₈×D₁₃	1	1	yes	characteristic
265	C₄	—	2	2	no	—
266	C₄	—	8	2	no	—
267	C₄	—	8	2	no	—
268	C₄	—	13	2	no	—
269	C₄	—	26	2	no	—
270	C₄	—	104	2	no	—
271	C₄	—	104	2	no	—
272	C₄	D₁₃×C₄⋊Q₈	1	1	yes	characteristic
273	C₄	C₂×Q₁₆×D₁₃	1	1	yes	characteristic
274	C₂	—	26	2	no	—
275	C₂	D₁₃×#500	1	1	yes	characteristic
276	C₂	#13127978	1	2	yes	—

	subgroup	G/H
center	C₂²	C₂×D₈×D₁₃
commutator, cosocle	C₂×C₈×C₁₃	C₂⁴
Frattini	C₂×C₈	C₂³×D₁₃
Fitting	C₁₃×#2254	C₂
socle	C₂²×C₁₃	C₂²×D₈
prime radical	C₂²×D₁₃	C₂×D₈
hypercenter	#2254	D₁₃
hypocenter	C₁₃	#33730
LCS	G ▷ C₂×C₈×C₁₃ ▷ C₄×C₁₃ ▷ C₂×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₄×C₈ ◁ #2254

Complements

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

N	H ≅ G/N	type
2. #13117121	C₂	semidirect
3. #13117119	C₂	semidirect
4. #13117117	C₂	semidirect
5. #13117115	C₂	semidirect
6. #13117114	C₂	semidirect
11. C₁₃×#2254	C₂	semidirect
70. #88038	Q₈	semidirect
71. C₁₆×D₁₃	Q₈	semidirect
109. #2254	D₁₃	semidirect
242. C₁₃	#33730	semidirect

Maximal subgroups

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

subgroup	index
#13125369	2
#13117121	2
#13117119	2
#13117117	2
#13117115	2
#13117114	2
#13116623	2
#13116619	2
D₁₃×#367	2
#13116431	2
C₁₃×#2254	2
#33730	13