Subgroups — #15873405

Order 3328.

870 conjugacy classes of subgroups.

The subgroup lattice has 308 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#11718587	C₂	1	1	yes	maximal, characteristic
2	C₂×#11718585	C₂	1	1	yes	maximal, characteristic
3	#13115560	C₂	1	4	yes	maximal, complemented
4	#13115021	C₂	1	4	yes	maximal
5	#13114988	C₂	1	2	yes	maximal, complemented
6	#13114953	C₂	1	2	yes	maximal, complemented
7	C₄×C₁₃×#117	C₂	1	1	yes	maximal, characteristic
8	C₂×C₁₃×#117	C₄	1	2	yes	—
9	C₂²×C₄²×C₁₃	C₂²	1	1	yes	characteristic
10	C₂²×#6099	C₂²	1	2	yes	—
11	C₄×C₁₃×C₄⋊C₄	C₄	1	2	yes	—
12	C₄×C₁₃×C₄⋊C₄	C₂²	1	2	yes	—
13	C₄×C₁₃×C₂²⋊C₄	C₄	1	2	yes	—
14	C₄×C₁₃×C₂²⋊C₄	C₂²	1	2	yes	—
15	C₁₃×#376	C₄	1	1	yes	characteristic
16	C₁₃×#376	C₄	1	1	yes	characteristic
17	C₄³×C₁₃	C₂²	1	2	yes	—
18	C₂×C₂²⋊#1055	C₂²	1	2	yes	—
19	#11718587	—	2	4	no	—
20	#11718585	—	2	4	no	—
21	C₂×C₄⋊#1055	C₂²	1	2	yes	—
22	C₂×C₄⋊#1055	C₂²	1	2	yes	—
23	C₂×C₄⋊#1055	C₂²	1	1	yes	characteristic
24	C₂×C₄⋊#1055	C₂²	1	1	yes	characteristic
25	C₂×C₄⋊#1055	C₂²	1	8	yes	—
26	C₂×#88043	C₂²	1	2	yes	—
27	#11718116	C₂²	1	8	yes	—
28	C₂³×C₄×C₁₃	C₂×C₄	1	1	yes	characteristic
29	C₂³×C₄×C₁₃	C₂×C₄	1	1	yes	characteristic
30	C₂³×C₄×C₁₃	C₂³	1	1	yes	characteristic
31	C₁₃×#117	—	2	8	no	—
32	C₂×C₁₃×C₄⋊C₄	C₂×C₄	1	4	yes	—
33	C₂×C₁₃×C₂²⋊C₄	C₂×C₄	1	4	yes	—
34	C₂×C₄²×C₁₃	C₂×C₄	1	2	yes	—
35	C₂×C₄²×C₁₃	C₂×C₄	1	4	yes	—
36	C₂×C₄²×C₁₃	C₂×C₄	1	1	yes	characteristic
37	C₂×C₄²×C₁₃	C₂×C₄	1	1	yes	characteristic
38	C₂×C₄²×C₁₃	C₂×C₄	1	2	yes	—
39	C₂×C₄²×C₁₃	D₄	1	4	yes	—
40	C₂×C₄²×C₁₃	Q₈	1	4	yes	—
41	C₂×C₄²×C₁₃	C₂³	1	4	yes	—
42	C₂×C₄²×C₁₃	C₂³	1	1	yes	characteristic
43	C₂×C₄²×C₁₃	C₂³	1	1	yes	characteristic
44	C₂×#6099	—	2	4	no	—
45	C₂×#6099	—	4	4	no	—
46	C₂²×#1055	C₂³	1	4	yes	—
47	C₂²×#1055	C₂³	1	2	yes	—
48	C₂²×#1055	C₂³	1	2	yes	—
49	C₁₃×#95	C₂×C₄	1	2	yes	—
50	C₁₃×#95	C₂×C₄	1	2	yes	—
51	C₁₃×#95	C₂×C₄	1	2	yes	—
52	C₁₃×#95	C₂×C₄	1	2	yes	—
53	C₂²⋊#1055	—	2	4	no	—
54	C₄⋊#1055	—	2	4	no	—
55	C₄⋊#1055	—	2	4	no	—
56	C₄⋊#1055	—	2	2	no	—
57	C₄⋊#1055	—	2	2	no	—
58	C₄⋊#1055	—	4	8	no	—
59	#88043	—	2	4	no	—
60	#21978	—	13	1	no	maximal, Sylow
61	C₂⁴×C₁₃	C₂²×C₄	1	1	yes	characteristic
62	C₂²×C₄×C₁₃	—	2	4	no	—
63	C₂²×C₄×C₁₃	—	2	2	no	—
64	C₂²×C₄×C₁₃	—	2	2	no	—
65	C₂²×C₄×C₁₃	—	4	2	no	—
66	C₂²×C₄×C₁₃	C₄⋊C₄	1	4	yes	—
67	C₂²×C₄×C₁₃	C₄⋊C₄	1	4	yes	—
68	C₂²×C₄×C₁₃	C₂²×C₄	1	4	yes	—
69	C₂²×C₄×C₁₃	C₂²×C₄	1	1	yes	characteristic
70	C₂²×C₄×C₁₃	C₂²×C₄	1	1	yes	characteristic
71	C₂²×C₄×C₁₃	C₂²×C₄	1	1	yes	characteristic
72	C₂²×C₄×C₁₃	C₂²×C₄	1	1	yes	characteristic
73	C₂²×C₄×C₁₃	C₂²×C₄	1	4	yes	—
74	C₂²×C₄×C₁₃	C₂²×C₄	1	1	yes	characteristic
75	C₂²×C₄×C₁₃	C₂×D₄	1	2	yes	—
76	C₂²×C₄×C₁₃	C₂×Q₈	1	2	yes	—
77	C₂²×C₄×C₁₃	C₂⁴	1	1	yes	characteristic
78	C₁₃×C₄⋊C₄	—	2	8	no	—
79	C₁₃×C₂²⋊C₄	—	2	8	no	—
80	C₄²×C₁₃	—	2	4	no	—
81	C₄²×C₁₃	—	2	4	no	—
82	C₄²×C₁₃	—	2	4	no	—
83	C₄²×C₁₃	—	4	4	no	—
84	C₄²×C₁₃	C₄⋊C₄	1	4	yes	—
85	C₄²×C₁₃	C₄⋊C₄	1	4	yes	—
86	C₄²×C₁₃	M₁₆	1	8	yes	complemented
87	C₄²×C₁₃	C₂×D₄	1	2	yes	—
88	C₄²×C₁₃	C₂×D₄	1	2	yes	—
89	C₄²×C₁₃	C₂×Q₈	1	2	yes	—
90	C₄²×C₁₃	C₂×Q₈	1	2	yes	—
91	#6099	—	4	8	no	—
92	C₂×#1055	—	2	2	no	—
93	C₂×#1055	—	2	2	no	—
94	C₂×#1055	—	2	4	no	—
95	C₂×#1055	—	2	8	no	—
96	C₂×#1055	—	2	4	no	—
97	C₂×#1055	—	2	4	no	—
98	C₂×#432	—	13	1	no	—
99	C₂×#423	—	13	1	no	—
100	C₄×#117	D₁₃	1	1	yes	characteristic
101	#1940	—	13	4	no	—
102	#1851	—	13	4	no	—
103	#1842	—	13	2	no	—
104	#1831	—	13	2	no	—
105	C₂³×C₁₃	—	2	2	no	—
106	C₂³×C₁₃	—	2	2	no	—
107	C₂³×C₁₃	—	2	2	no	—
108	C₂³×C₁₃	C₂×C₄⋊C₄	1	2	yes	—
109	C₂³×C₁₃	C₂³×C₄	1	1	yes	characteristic
110	C₂×C₄×C₁₃	—	2	4	no	—
111	C₂×C₄×C₁₃	—	2	2	no	—
112	C₂×C₄×C₁₃	—	2	2	no	—
113	C₂×C₄×C₁₃	—	2	2	no	—
114	C₂×C₄×C₁₃	—	2	4	no	—
115	C₂×C₄×C₁₃	—	2	2	no	—
116	C₂×C₄×C₁₃	—	2	2	no	—
117	C₂×C₄×C₁₃	—	2	2	no	—
118	C₂×C₄×C₁₃	—	2	4	no	—
119	C₂×C₄×C₁₃	—	2	8	no	—
120	C₂×C₄×C₁₃	—	2	4	no	—
121	C₂×C₄×C₁₃	—	2	4	no	—
122	C₂×C₄×C₁₃	—	2	4	no	—
123	C₂×C₄×C₁₃	—	4	4	no	—
124	C₂×C₄×C₁₃	C₂×C₄⋊C₄	1	2	yes	—
125	C₂×C₄×C₁₃	C₂×C₄⋊C₄	1	2	yes	—
126	C₂×C₄×C₁₃	C₂×C₄⋊C₄	1	2	yes	—
127	C₂×C₄×C₁₃	C₂×C₄⋊C₄	1	2	yes	—
128	C₂×C₄×C₁₃	C₂×C₄⋊C₄	1	2	yes	—
129	C₂×C₄×C₁₃	C₂×M₁₆	1	8	yes	—
130	C₂×C₄×C₁₃	C₂×M₁₆	1	4	yes	—
131	C₂×C₄×C₁₃	C₂²×D₄	1	1	yes	characteristic
132	C₂×C₄×C₁₃	C₂²×Q₈	1	1	yes	characteristic
133	C₂×C₄×C₁₃	H(5,𝔽₂)	1	2	yes	—
134	C₂×C₄×C₁₃	2₋¹⁺⁴	1	2	yes	—
135	#1055	—	4	2	no	—
136	#1055	—	4	4	no	—
137	#1055	—	4	2	no	—
138	C₂²×M₁₆	—	13	2	no	—
139	C₂×#117	Dic₁₃	1	2	yes	—
140	C₂²×C₄²	C₂×D₁₃	1	1	yes	characteristic
141	#432	—	26	4	no	—
142	#423	—	26	4	no	—
143	C₂×C₄⋊C₈	—	13	1	no	—
144	C₂×C₄⋊C₈	—	13	2	no	—
145	C₂×C₄⋊C₈	—	13	1	no	—
146	C₂×C₄⋊C₈	—	13	2	no	—
147	C₂×C₄⋊C₈	—	13	8	no	—
148	C₂×C₂²⋊C₈	—	13	2	no	—
149	C₂×#97	—	13	2	no	—
150	C₄×C₄⋊C₄	Dic₁₃	1	2	yes	—
151	C₄×C₄⋊C₄	C₂×D₁₃	1	2	yes	—
152	C₄×C₂²⋊C₄	Dic₁₃	1	2	yes	—
153	C₄×C₂²⋊C₄	C₂×D₁₃	1	2	yes	—
154	#376	Dic₁₃	1	1	yes	characteristic
155	#376	Dic₁₃	1	1	yes	characteristic
156	C₄³	C₂×D₁₃	1	2	yes	—
157	#336	—	13	8	no	—
158	C₂²×C₁₃	—	2	2	no	—
159	C₂²×C₁₃	—	2	2	no	—
160	C₂²×C₁₃	—	2	2	no	—
161	C₂²×C₁₃	—	4	4	no	—
162	C₂²×C₁₃	C₂²×C₄⋊C₄	1	1	yes	characteristic
163	C₂²×C₁₃	#520	1	2	yes	—
164	C₂²×C₁₃	C₂²×M₁₆	1	2	yes	—
165	C₂²×C₁₃	#568	1	2	yes	—
166	C₄×C₁₃	—	2	2	no	—
167	C₄×C₁₃	—	2	2	no	—
168	C₄×C₁₃	—	2	2	no	—
169	C₄×C₁₃	—	2	2	no	—
170	C₄×C₁₃	—	2	8	no	—
171	C₄×C₁₃	—	4	4	no	—
172	C₄×C₁₃	#423	1	4	yes	—
173	C₄×C₁₃	#535	1	1	yes	characteristic
174	C₄×C₁₃	#536	1	1	yes	characteristic
175	C₄×C₁₃	#544	1	2	yes	—
176	C₂³×C₄	C₂×Dic₁₃	1	1	yes	characteristic
177	C₂³×C₄	C₂×Dic₁₃	1	1	yes	characteristic
178	C₂³×C₄	C₂²×D₁₃	1	1	yes	characteristic
179	C₂×M₁₆	—	26	4	no	—
180	C₂×M₁₆	—	52	4	no	—
181	C₂²×C₈	—	13	4	no	—
182	C₂²×C₈	—	13	2	no	—
183	C₂²×C₈	—	13	2	no	—
184	#117	—	2	8	no	—
185	C₂×C₄⋊C₄	C₂×Dic₁₃	1	4	yes	—
186	C₂×C₂²⋊C₄	C₂×Dic₁₃	1	4	yes	—
187	C₂×C₄²	Dic₂₆	1	4	yes	—
188	C₂×C₄²	D₅₂	1	4	yes	—
189	C₂×C₄²	C₂×Dic₁₃	1	2	yes	—
190	C₂×C₄²	C₂×Dic₁₃	1	4	yes	—
191	C₂×C₄²	C₂×Dic₁₃	1	2	yes	—
192	C₂×C₄²	C₂×Dic₁₃	1	1	yes	characteristic
193	C₂×C₄²	C₂×Dic₁₃	1	1	yes	characteristic
194	C₂×C₄²	C₂²×D₁₃	1	4	yes	—
195	C₂×C₄²	C₂²×D₁₃	1	1	yes	characteristic
196	C₂×C₄²	C₂²×D₁₃	1	1	yes	characteristic
197	C₄⋊C₈	—	26	4	no	—
198	C₄⋊C₈	—	26	4	no	—
199	C₄⋊C₈	—	26	2	no	—
200	C₄⋊C₈	—	26	2	no	—
201	C₄⋊C₈	—	52	8	no	—
202	C₂²⋊C₈	—	26	4	no	—
203	#97	—	26	4	no	—
204	#95	C₂×Dic₁₃	1	2	yes	—
205	#95	C₂×Dic₁₃	1	2	yes	—
206	#95	C₂×Dic₁₃	1	2	yes	—
207	#95	C₂×Dic₁₃	1	2	yes	—
208	C₂×C₁₃	—	4	2	no	—
209	C₂×C₁₃	#2317	1	1	yes	characteristic
210	C₂×C₁₃	C₂×#423	1	1	yes	characteristic
211	C₂×C₁₃	#2905	1	1	yes	characteristic
212	C₂×C₁₃	#2989	1	4	yes	—
213	C₂⁴	C₂²×Dic₁₃	1	1	yes	characteristic
214	C₂²×C₄	—	2	4	no	—
215	C₂²×C₄	—	2	2	no	—
216	C₂²×C₄	—	2	2	no	—
217	C₂²×C₄	—	4	2	no	—
218	C₂²×C₄	C₄⋊Dic₁₃	1	4	yes	—
219	C₂²×C₄	C₄⋊Dic₁₃	1	4	yes	—
220	C₂²×C₄	C₂×Dic₂₆	1	2	yes	—
221	C₂²×C₄	C₂×D₅₂	1	2	yes	—
222	C₂²×C₄	C₂²×Dic₁₃	1	4	yes	—
223	C₂²×C₄	C₂²×Dic₁₃	1	4	yes	—
224	C₂²×C₄	C₂²×Dic₁₃	1	1	yes	characteristic
225	C₂²×C₄	C₂²×Dic₁₃	1	1	yes	characteristic
226	C₂²×C₄	C₂²×Dic₁₃	1	1	yes	characteristic
227	C₂²×C₄	C₂²×Dic₁₃	1	1	yes	characteristic
228	C₂²×C₄	C₂²×Dic₁₃	1	1	yes	characteristic
229	C₂²×C₄	C₂³×D₁₃	1	1	yes	characteristic
230	M₁₆	—	52	8	no	—
231	C₂×C₈	—	26	2	no	—
232	C₂×C₈	—	26	2	no	—
233	C₂×C₈	—	26	4	no	—
234	C₂×C₈	—	26	4	no	—
235	C₂×C₈	—	26	8	no	—
236	C₂×C₈	—	26	4	no	—
237	C₄⋊C₄	—	2	8	no	—
238	C₂²⋊C₄	—	2	8	no	—
239	C₄²	—	2	4	no	—
240	C₄²	—	2	4	no	—
241	C₄²	—	2	4	no	—
242	C₄²	—	4	4	no	—
243	C₄²	#6099	1	8	yes	complemented
244	C₄²	C₄⋊Dic₁₃	1	4	yes	—
245	C₄²	C₄⋊Dic₁₃	1	4	yes	—
246	C₄²	C₂×Dic₂₆	1	2	yes	—
247	C₄²	C₂×Dic₂₆	1	2	yes	—
248	C₄²	C₂×D₅₂	1	2	yes	—
249	C₄²	C₂×D₅₂	1	2	yes	—
250	C₁₃	#21978	1	1	yes	characteristic, complemented, Sylow
251	C₂³	—	2	2	no	—
252	C₂³	—	2	2	no	—
253	C₂³	—	2	2	no	—
254	C₂³	C₂×C₄⋊Dic₁₃	1	2	yes	—
255	C₂³	C₂³×Dic₁₃	1	1	yes	characteristic
256	C₂×C₄	—	2	2	no	—
257	C₂×C₄	—	2	2	no	—
258	C₂×C₄	—	2	4	no	—
259	C₂×C₄	—	2	2	no	—
260	C₂×C₄	—	2	4	no	—
261	C₂×C₄	—	2	2	no	—
262	C₂×C₄	—	2	4	no	—
263	C₂×C₄	—	2	2	no	—
264	C₂×C₄	—	2	2	no	—
265	C₂×C₄	—	2	8	no	—
266	C₂×C₄	—	2	4	no	—
267	C₂×C₄	—	2	4	no	—
268	C₂×C₄	—	2	4	no	—
269	C₂×C₄	—	4	4	no	—
270	C₂×C₄	C₂×#6099	1	8	yes	—
271	C₂×C₄	C₂×#6099	1	4	yes	—
272	C₂×C₄	C₂×C₄⋊Dic₁₃	1	2	yes	—
273	C₂×C₄	C₂×C₄⋊Dic₁₃	1	2	yes	—
274	C₂×C₄	C₂×C₄⋊Dic₁₃	1	2	yes	—
275	C₂×C₄	C₂×C₄⋊Dic₁₃	1	2	yes	—
276	C₂×C₄	C₂×C₄⋊Dic₁₃	1	2	yes	—
277	C₂×C₄	C₂²×Dic₂₆	1	1	yes	characteristic
278	C₂×C₄	C₂²×D₅₂	1	1	yes	characteristic
279	C₂×C₄	#88256	1	2	yes	—
280	C₂×C₄	#88257	1	2	yes	—
281	C₈	—	52	2	no	—
282	C₈	—	52	4	no	—
283	C₈	—	52	2	no	—
284	C₂²	—	2	2	no	—
285	C₂²	—	2	2	no	—
286	C₂²	—	2	2	no	—
287	C₂²	—	4	4	no	—
288	C₂²	C₂²×#6099	1	2	yes	—
289	C₂²	C₂²×C₄⋊Dic₁₃	1	1	yes	characteristic
290	C₂²	#11719524	1	2	yes	—
291	C₂²	#11719532	1	2	yes	—
292	C₄	—	2	2	no	—
293	C₄	—	2	2	no	—
294	C₄	—	2	2	no	—
295	C₄	—	2	2	no	—
296	C₄	—	2	8	no	—
297	C₄	—	4	4	no	—
298	C₄	#11718585	1	4	yes	—
299	C₄	#11719224	1	2	yes	—
300	C₄	#11719230	1	1	yes	characteristic
301	C₄	#11719231	1	1	yes	characteristic
302	C₂	—	4	2	no	—
303	C₂	#13117459	1	1	yes	characteristic
304	C₂	C₂×#11718585	1	1	yes	characteristic
305	C₂	#13123400	1	1	yes	characteristic
306	C₂	#13123953	1	4	yes	—

	subgroup	G/H
center, Frattini	C₂²×C₄	C₂³×D₁₃
commutator, socle	C₂³×C₁₃	C₂³×C₄
Fitting	C₄×C₁₃×#117	C₂
prime radical	C₂⁴×C₁₃	C₂²×C₄
cosocle	C₂²×C₄×C₁₃	C₂⁴
hypercenter	C₄×#117	D₁₃
hypocenter	C₁₃	#21978
LCS	G ▷ C₂³×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂²×C₄ ◁ C₄×#117

Complements

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

N	H ≅ G/N	type
3. #13115560	C₂	semidirect
5. #13114988	C₂	semidirect
6. #13114953	C₂	semidirect
86. C₄²×C₁₃	M₁₆	semidirect
243. C₄²	#6099	semidirect
250. C₁₃	#21978	semidirect

Maximal subgroups

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

subgroup	index
C₂×#11718587	2
C₂×#11718585	2
#13115560	2
#13115021	2
#13114988	2
#13114953	2
C₄×C₁₃×#117	2
#21978	13