Subgroups — #15770988

Order 3328.

608 conjugacy classes of subgroups.

The subgroup lattice has 298 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#11718786	C₂	1	1	yes	maximal, characteristic
2	#13123399	C₂	1	1	yes	maximal, characteristic
3	C₂×#11718148	C₂	1	2	yes	maximal
4	C₂×#11718152	C₂	1	2	yes	maximal
5	C₂×C₁₃×#338	C₂	1	1	yes	maximal, characteristic
6	C₂²×C₁₃×M₁₆	C₂²	1	1	yes	characteristic
7	C₂²×C₁₃×M₁₆	C₂²	1	2	yes	—
8	C₂×#88180	C₂²	1	1	yes	characteristic
9	C₂²×#6099	C₂²	1	2	yes	—
10	C₂²×#6099	C₂²	1	1	yes	characteristic
11	#11718786	—	2	4	no	—
12	#11718780	C₄	1	4	yes	—
13	#11718779	—	2	2	no	—
14	C₂×#88059	—	2	2	no	—
15	#11718763	—	2	2	no	—
16	C₁₃×#338	—	2	2	no	—
17	#11718152	—	2	4	no	—
18	#11718148	—	2	4	no	—
19	C₂³×C₄×C₁₃	C₂³	1	1	yes	characteristic
20	C₂×C₁₃×M₁₆	—	2	2	no	—
21	C₂×C₁₃×M₁₆	—	2	2	no	—
22	C₂×C₁₃×M₁₆	—	2	2	no	—
23	C₂×C₁₃×M₁₆	—	2	2	no	—
24	C₂×C₁₃×M₁₆	—	2	2	no	—
25	C₂×C₁₃×M₁₆	—	2	2	no	—
26	C₂×C₁₃×M₁₆	—	2	2	no	—
27	C₂×C₁₃×M₁₆	—	2	2	no	—
28	C₂×C₁₃×M₁₆	C₂×C₄	1	2	yes	—
29	C₂×C₁₃×M₁₆	D₄	1	2	yes	—
30	C₂²×C₈×C₁₃	D₄	1	2	yes	—
31	C₂²×C₈×C₁₃	Q₈	1	4	yes	—
32	C₂×C₂²⋊Dic₁₃	Q₈	1	2	yes	—
33	#88180	—	2	1	no	—
34	#88180	—	2	1	no	—
35	#88180	C₂×C₄	1	2	yes	—
36	#88180	D₄	1	2	yes	—
37	C₂×C₄⋊Dic₁₃	—	2	1	no	—
38	C₂×C₄×Dic₁₃	—	2	1	no	—
39	C₂×#6099	—	2	2	no	—
40	C₂×#6099	—	2	1	no	—
41	C₂×#6099	—	2	1	no	—
42	C₂×#6099	—	2	2	no	—
43	C₂×#6099	—	2	2	no	—
44	C₂×#6099	—	2	2	no	—
45	C₂×#6099	—	2	2	no	—
46	C₂×#6099	—	2	2	no	—
47	C₂×#6099	—	2	2	no	—
48	C₂×#6099	D₄	1	4	yes	—
49	C₂²×#1055	—	2	2	no	—
50	C₂²×#1055	C₂×C₄	1	2	yes	—
51	#88059	—	2	8	no	—
52	#88054	—	2	8	no	—
53	#11235	—	13	1	no	maximal, Sylow
54	C₂⁴×C₁₃	C₂×Q₈	1	1	yes	characteristic
55	C₂²×C₄×C₁₃	C₂²×C₄	1	1	yes	characteristic
56	C₂²×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
57	C₂²×C₄×C₁₃	C₂×D₄	1	2	yes	—
58	C₂²×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
59	C₂²×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
60	C₂²×C₄×C₁₃	C₂×Q₈	1	2	yes	—
61	C₂²×C₄×C₁₃	#41	1	2	yes	—
62	C₂²×C₄×C₁₃	#41	1	1	yes	characteristic
63	C₂²×C₄×C₁₃	#41	1	2	yes	—
64	C₂²×C₄×C₁₃	#41	1	1	yes	characteristic
65	C₂²×Dic₁₃	—	2	2	no	—
66	C₁₃×M₁₆	—	2	4	no	—
67	C₁₃×M₁₆	—	2	4	no	—
68	C₁₃×M₁₆	—	4	2	no	—
69	C₁₃×M₁₆	—	4	2	no	—
70	C₁₃×M₁₆	—	4	2	no	—
71	C₁₃×M₁₆	—	4	2	no	—
72	C₂×C₈×C₁₃	—	2	2	no	—
73	C₂×C₈×C₁₃	—	2	2	no	—
74	C₂×C₈×C₁₃	—	2	4	no	—
75	C₂×C₈×C₁₃	—	2	4	no	—
76	C₂×C₈×C₁₃	—	2	4	no	—
77	C₂×C₈×C₁₃	C₂²⋊C₄	1	4	yes	—
78	C₂²⋊Dic₁₃	—	2	2	no	—
79	C₂²⋊Dic₁₃	C₄⋊C₄	1	4	yes	—
80	C₄⋊Dic₁₃	—	2	2	no	—
81	C₄⋊Dic₁₃	—	2	2	no	—
82	C₄×Dic₁₃	—	2	2	no	—
83	C₄×Dic₁₃	—	2	2	no	—
84	#6099	—	2	4	no	—
85	#6099	—	2	4	no	—
86	#6099	—	4	1	no	—
87	#6099	—	4	2	no	—
88	#6099	—	4	2	no	—
89	#6099	—	4	1	no	—
90	#6099	—	4	2	no	—
91	C₂×#1055	—	2	2	no	—
92	C₂×#1055	—	2	4	no	—
93	C₂×#1055	—	2	4	no	—
94	C₂×#1055	—	4	2	no	—
95	C₂×#1055	—	4	2	no	—
96	C₂×#430	—	13	1	no	—
97	#2905	—	13	1	no	—
98	C₂×#344	—	13	2	no	—
99	C₂×#338	—	13	2	no	—
100	C₂×#338	D₁₃	1	1	yes	characteristic
101	C₂³×C₁₃	—	8	1	no	—
102	C₂³×C₁₃	C₂×C₄⋊C₄	1	1	yes	characteristic
103	C₂³×C₁₃	C₂²⋊Q₈	1	1	yes	characteristic
104	C₂³×C₁₃	C₂²⋊Q₈	1	1	yes	characteristic
105	C₂³×C₁₃	#125	1	2	yes	—
106	C₂³×C₁₃	C₄⋊Q₈	1	2	yes	—
107	C₂×C₄×C₁₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
108	C₂×C₄×C₁₃	#117	1	1	yes	characteristic
109	C₂×C₄×C₁₃	C₄×D₄	1	2	yes	—
110	C₂×C₄×C₁₃	C₄×Q₈	1	2	yes	—
111	C₂×C₄×C₁₃	#120	1	1	yes	characteristic
112	C₂×C₄×C₁₃	#121	1	2	yes	—
113	C₂×C₄×C₁₃	#121	1	2	yes	—
114	C₂×C₄×C₁₃	#121	1	1	yes	characteristic
115	C₂×C₄×C₁₃	#121	1	1	yes	characteristic
116	C₂×C₄×C₁₃	C₂²⋊Q₈	1	2	yes	—
117	C₂×C₄×C₁₃	C₂²⋊Q₈	1	2	yes	—
118	C₂×C₄×C₁₃	C₂²⋊Q₈	1	2	yes	—
119	C₂×C₄×C₁₃	#123	1	1	yes	characteristic
120	C₂×C₄×C₁₃	#123	1	2	yes	—
121	C₂×C₄×C₁₃	#124	1	1	yes	characteristic
122	C₂×C₄×C₁₃	#125	1	2	yes	—
123	C₂×C₄×C₁₃	#126	1	2	yes	—
124	C₂×C₄×C₁₃	C₄⋊Q₈	1	1	yes	characteristic
125	C₂×Dic₁₃	—	2	4	no	—
126	C₂×Dic₁₃	—	8	2	no	—
127	C₈×C₁₃	—	2	4	no	—
128	C₈×C₁₃	—	4	4	no	—
129	#1055	—	4	2	no	—
130	#1055	—	4	4	no	—
131	C₂²×M₁₆	—	13	1	no	—
132	C₂²×M₁₆	—	13	2	no	—
133	C₂²×M₁₆	C₂×D₁₃	1	2	yes	—
134	C₂²×M₁₆	C₂×D₁₃	1	1	yes	characteristic
135	C₂×#117	—	13	1	no	—
136	#430	—	26	4	no	—
137	C₂×#108	—	26	2	no	—
138	#424	—	13	4	no	—
139	#424	—	26	2	no	—
140	#423	—	26	2	no	—
141	#344	—	26	4	no	—
142	#338	—	2	2	no	—
143	#338	—	26	4	no	—
144	C₂²×C₁₃	—	4	1	no	—
145	C₂²×C₁₃	—	4	1	no	—
146	C₂²×C₁₃	—	4	1	no	—
147	C₂²×C₁₃	—	4	2	no	—
148	C₂²×C₁₃	—	4	2	no	—
149	C₂²×C₁₃	#380	1	1	yes	characteristic
150	C₂²×C₁₃	#389	1	2	yes	—
151	C₂²×C₁₃	#395	1	1	yes	characteristic
152	C₂²×C₁₃	#398	1	2	yes	—
153	C₂²×C₁₃	#400	1	1	yes	characteristic
154	C₄×C₁₃	#386	1	1	yes	characteristic
155	C₄×C₁₃	#387	1	2	yes	—
156	C₄×C₁₃	#391	1	1	yes	characteristic
157	C₄×C₁₃	#394	1	1	yes	characteristic
158	C₄×C₁₃	#396	1	2	yes	—
159	C₄×C₁₃	#398	1	1	yes	characteristic
160	Dic₁₃	—	4	4	no	—
161	C₂³×C₄	C₂²×D₁₃	1	1	yes	characteristic
162	C₂×M₁₆	—	2	2	no	—
163	C₂×M₁₆	—	2	2	no	—
164	C₂×M₁₆	—	2	2	no	—
165	C₂×M₁₆	—	2	2	no	—
166	C₂×M₁₆	—	2	2	no	—
167	C₂×M₁₆	—	2	2	no	—
168	C₂×M₁₆	—	2	2	no	—
169	C₂×M₁₆	—	2	2	no	—
170	C₂×M₁₆	—	13	4	no	—
171	C₂×M₁₆	—	26	1	no	—
172	C₂×M₁₆	—	26	2	no	—
173	C₂×M₁₆	—	26	2	no	—
174	C₂×M₁₆	—	26	2	no	—
175	C₂×M₁₆	—	26	1	no	—
176	C₂×M₁₆	—	26	2	no	—
177	C₂×M₁₆	—	26	2	no	—
178	C₂×M₁₆	—	26	2	no	—
179	C₂×M₁₆	—	26	2	no	—
180	C₂×M₁₆	C₄×D₁₃	1	2	yes	—
181	C₂×M₁₆	C₂²⋊D₁₃	1	2	yes	—
182	C₂²×C₈	—	13	2	no	—
183	C₂²×C₈	—	26	2	no	—
184	C₂²×C₈	Dic₂₆	1	4	yes	—
185	C₂²×C₈	D₅₂	1	2	yes	—
186	#117	—	13	2	no	—
187	#117	—	13	2	no	—
188	#117	—	26	1	no	—
189	#117	—	26	1	no	—
190	C₂×C₄⋊C₄	—	26	1	no	—
191	C₂×C₂²⋊C₄	—	13	2	no	—
192	C₂×C₄²	—	26	1	no	—
193	#108	—	26	8	no	—
194	C₄⋊C₈	—	26	8	no	—
195	C₂×C₁₃	—	2	1	no	—
196	C₂×C₁₃	—	2	2	no	—
197	C₂×C₁₃	—	2	1	no	—
198	C₂×C₁₃	—	2	2	no	—
199	C₂×C₁₃	—	2	1	no	—
200	C₂×C₁₃	#1441	1	1	yes	characteristic
201	C₂⁴	Q₈×D₁₃	1	1	yes	characteristic
202	C₂²×C₄	—	26	2	no	—
203	C₂²×C₄	C₂×Dic₂₆	1	2	yes	—
204	C₂²×C₄	C₂×C₄×D₁₃	1	1	yes	characteristic
205	C₂²×C₄	C₂×D₅₂	1	1	yes	characteristic
206	C₂²×C₄	#6127	1	2	yes	—
207	C₂²×C₄	D₄×D₁₃	1	1	yes	characteristic
208	C₂²×C₄	D₄×D₁₃	1	2	yes	—
209	C₂²×C₄	#6129	1	1	yes	characteristic
210	C₂²×C₄	#6129	1	2	yes	—
211	C₂²×C₄	#6131	1	1	yes	characteristic
212	C₂²×C₄	C₂×C₂²⋊D₁₃	1	1	yes	characteristic
213	M₁₆	—	2	4	no	—
214	M₁₆	—	2	4	no	—
215	M₁₆	—	4	2	no	—
216	M₁₆	—	4	2	no	—
217	M₁₆	—	4	2	no	—
218	M₁₆	—	4	2	no	—
219	M₁₆	—	26	4	no	—
220	M₁₆	—	26	4	no	—
221	M₁₆	—	52	1	no	—
222	M₁₆	—	52	1	no	—
223	M₁₆	—	52	2	no	—
224	M₁₆	—	52	2	no	—
225	M₁₆	—	52	2	no	—
226	C₂×C₈	—	2	2	no	—
227	C₂×C₈	—	2	2	no	—
228	C₂×C₈	—	2	4	no	—
229	C₂×C₈	—	2	4	no	—
230	C₂×C₈	—	2	4	no	—
231	C₂×C₈	—	26	4	no	—
232	C₂×C₈	—	26	4	no	—
233	C₂×C₈	—	26	2	no	—
234	C₂×C₈	—	52	2	no	—
235	C₂×C₈	—	52	2	no	—
236	C₂×C₈	#6103	1	4	yes	—
237	C₄⋊C₄	—	26	2	no	—
238	C₄⋊C₄	—	26	2	no	—
239	C₂²⋊C₄	—	13	4	no	—
240	C₂²⋊C₄	—	26	2	no	—
241	C₄²	—	26	2	no	—
242	C₄²	—	26	2	no	—
243	C₁₃	#11235	1	1	yes	characteristic, complemented, Sylow
244	C₂³	—	8	1	no	—
245	C₂³	#88142	1	2	yes	—
246	C₂³	#88143	1	2	yes	—
247	C₂³	C₄⋊C₄×D₁₃	1	1	yes	characteristic
248	C₂³	#88151	1	1	yes	characteristic
249	C₂³	#88200	1	1	yes	characteristic
250	C₂×C₄	—	26	4	no	—
251	C₂×C₄	—	104	2	no	—
252	C₂×C₄	C₄×Dic₂₆	1	2	yes	—
253	C₂×C₄	#88123	1	1	yes	characteristic
254	C₂×C₄	#88129	1	1	yes	characteristic
255	C₂×C₄	#88132	1	2	yes	—
256	C₂×C₄	#88132	1	2	yes	—
257	C₂×C₄	#88133	1	2	yes	—
258	C₂×C₄	#88135	1	2	yes	—
259	C₂×C₄	#88136	1	1	yes	characteristic
260	C₂×C₄	#88138	1	2	yes	—
261	C₂×C₄	#88140	1	1	yes	characteristic
262	C₂×C₄	#88144	1	2	yes	—
263	C₂×C₄	#88146	1	1	yes	characteristic
264	C₂×C₄	#88148	1	2	yes	—
265	C₂×C₄	#88149	1	1	yes	characteristic
266	C₂×C₄	#88178	1	2	yes	—
267	C₂×C₄	C₂×#6103	1	1	yes	characteristic
268	C₂×C₄	#88192	1	1	yes	characteristic
269	C₂×C₄	#88193	1	2	yes	—
270	C₈	—	2	4	no	—
271	C₈	—	4	4	no	—
272	C₈	—	52	4	no	—
273	C₈	—	52	2	no	—
274	C₂²	—	4	1	no	—
275	C₂²	—	4	2	no	—
276	C₂²	—	4	2	no	—
277	C₂²	—	4	1	no	—
278	C₂²	—	4	1	no	—
279	C₂²	#11718308	1	2	yes	—
280	C₂²	#11718317	1	1	yes	characteristic
281	C₂²	#11718320	1	1	yes	characteristic
282	C₂²	#11718645	1	2	yes	—
283	C₂²	#11718648	1	1	yes	characteristic
284	C₄	—	52	4	no	—
285	C₄	#11718316	1	2	yes	—
286	C₄	#11718322	1	1	yes	characteristic
287	C₄	#11718333	1	1	yes	characteristic
288	C₄	#11718590	1	1	yes	characteristic
289	C₄	#11718609	1	2	yes	—
290	C₄	#11718619	1	1	yes	characteristic
291	C₂	—	2	1	no	—
292	C₂	—	2	2	no	—
293	C₂	—	2	1	no	—
294	C₂	—	2	1	no	—
295	C₂	—	2	2	no	—
296	C₂	#13113090	1	1	yes	characteristic

	subgroup	G/H
center	C₄	#11718590
commutator	C₂²×C₄×C₁₃	C₂²×C₄
Frattini	C₂³×C₄	C₂²×D₁₃
Fitting	C₂×C₁₃×#338	C₂
socle	C₂×C₁₃	#1441
prime radical	C₂⁴×C₁₃	C₂×Q₈
cosocle	C₂³×C₄×C₁₃	C₂³
hypercenter	C₂×#338	D₁₃
hypocenter	C₁₃	#11235
LCS	G ▷ C₂²×C₄×C₁₃ ▷ C₂×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₄ ◁ C₂³×C₄ ◁ C₂×#338

Complements

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

N	H ≅ G/N	type
243. C₁₃	#11235	semidirect

Maximal subgroups

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

subgroup	index
C₂×#11718786	2
#13123399	2
C₂×#11718148	2
C₂×#11718152	2
C₂×C₁₃×#338	2
#11235	13