Subgroups — #15867740

Order 3328.

926 conjugacy classes of subgroups.

The subgroup lattice has 302 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×C₄×#6099	C₂	1	1	yes	maximal, characteristic
3	#13115555	C₂	1	4	yes	maximal, complemented
4	#13115021	C₂	1	4	yes	maximal
5	#13114689	C₂	1	4	yes	maximal, complemented
6	C₄×C₁₃×#117	C₂	1	1	yes	maximal, characteristic
7	C₂×C₁₃×#117	C₄	1	2	yes	—
8	C₂²×C₄²×C₁₃	C₂²	1	1	yes	characteristic
9	C₂²×#6099	C₂²	1	2	yes	—
10	C₄×C₁₃×C₄⋊C₄	C₄	1	1	yes	characteristic
11	C₄×C₁₃×C₄⋊C₄	C₄	1	1	yes	characteristic
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	2	yes	—
16	C₄³×C₁₃	C₂²	1	2	yes	—
17	C₂×C₂²⋊#1055	C₂²	1	2	yes	—
18	#11718587	—	2	4	no	—
19	C₂×C₄⋊#1055	C₂²	1	8	yes	—
20	C₂×C₄⋊#1055	C₂²	1	2	yes	—
21	C₄×#6099	—	2	4	no	—
22	C₂×#88043	C₂²	1	2	yes	—
23	C₂×#88043	C₂²	1	2	yes	—
24	C₂×C₄×#1055	C₂²	1	2	yes	—
25	#11718116	C₂²	1	8	yes	—
26	C₂³×C₄×C₁₃	C₂×C₄	1	1	yes	characteristic
27	C₂³×C₄×C₁₃	C₂×C₄	1	1	yes	characteristic
28	C₂³×C₄×C₁₃	C₂³	1	1	yes	characteristic
29	C₁₃×#117	—	2	8	no	—
30	C₂×C₁₃×C₄⋊C₄	C₂×C₄	1	2	yes	—
31	C₂×C₁₃×C₄⋊C₄	C₂×C₄	1	2	yes	—
32	C₂×C₁₃×C₂²⋊C₄	C₂×C₄	1	4	yes	—
33	C₂×C₄²×C₁₃	C₂×C₄	1	4	yes	—
34	C₂×C₄²×C₁₃	C₂×C₄	1	2	yes	—
35	C₂×C₄²×C₁₃	C₂×C₄	1	2	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	D₄	1	4	yes	—
39	C₂×C₄²×C₁₃	Q₈	1	4	yes	—
40	C₂×C₄²×C₁₃	C₂³	1	4	yes	—
41	C₂×C₄²×C₁₃	C₂³	1	1	yes	characteristic
42	C₂×C₄²×C₁₃	C₂³	1	1	yes	characteristic
43	C₂×#6099	—	2	4	no	—
44	C₂×#6099	—	2	8	no	—
45	C₂²×#1055	C₂³	1	4	yes	—
46	C₂²×#1055	C₂³	1	4	yes	—
47	C₁₃×#95	C₂×C₄	1	4	yes	—
48	C₁₃×#95	C₂×C₄	1	2	yes	—
49	C₁₃×#95	C₂×C₄	1	2	yes	—
50	C₂²⋊#1055	—	2	4	no	—
51	C₄⋊#1055	—	2	4	no	—
52	C₄⋊#1055	—	2	16	no	—
53	#88043	—	2	4	no	—
54	#88043	—	2	4	no	—
55	C₄×#1055	—	2	4	no	—
56	#21356	—	13	1	no	maximal, Sylow
57	C₂⁴×C₁₃	C₂²×C₄	1	1	yes	characteristic
58	C₂²×C₄×C₁₃	—	2	4	no	—
59	C₂²×C₄×C₁₃	—	2	4	no	—
60	C₂²×C₄×C₁₃	—	4	1	no	—
61	C₂²×C₄×C₁₃	—	4	1	no	—
62	C₂²×C₄×C₁₃	C₄⋊C₄	1	4	yes	—
63	C₂²×C₄×C₁₃	C₄⋊C₄	1	4	yes	—
64	C₂²×C₄×C₁₃	C₂²×C₄	1	4	yes	—
65	C₂²×C₄×C₁₃	C₂²×C₄	1	1	yes	characteristic
66	C₂²×C₄×C₁₃	C₂²×C₄	1	1	yes	characteristic
67	C₂²×C₄×C₁₃	C₂²×C₄	1	1	yes	characteristic
68	C₂²×C₄×C₁₃	C₂²×C₄	1	1	yes	characteristic
69	C₂²×C₄×C₁₃	C₂²×C₄	1	4	yes	—
70	C₂²×C₄×C₁₃	C₂²×C₄	1	1	yes	characteristic
71	C₂²×C₄×C₁₃	C₂×D₄	1	2	yes	—
72	C₂²×C₄×C₁₃	C₂×Q₈	1	2	yes	—
73	C₂²×C₄×C₁₃	C₂⁴	1	1	yes	characteristic
74	C₁₃×C₄⋊C₄	—	2	4	no	—
75	C₁₃×C₄⋊C₄	M₁₆	1	8	yes	complemented
76	C₁₃×C₂²⋊C₄	—	2	8	no	—
77	C₄²×C₁₃	—	2	8	no	—
78	C₄²×C₁₃	—	2	8	no	—
79	C₄²×C₁₃	—	4	2	no	—
80	C₄²×C₁₃	—	4	2	no	—
81	C₄²×C₁₃	C₄⋊C₄	1	4	yes	—
82	C₄²×C₁₃	C₄⋊C₄	1	4	yes	—
83	C₄²×C₁₃	C₂×D₄	1	2	yes	—
84	C₄²×C₁₃	C₂×D₄	1	2	yes	—
85	C₄²×C₁₃	C₂×Q₈	1	2	yes	—
86	C₄²×C₁₃	C₂×Q₈	1	2	yes	—
87	#6099	—	2	16	no	—
88	C₂×#1055	—	2	4	no	—
89	C₂×#1055	—	2	4	no	—
90	C₂×#1055	—	2	4	no	—
91	C₂×#1055	—	2	8	no	—
92	C₂×#1055	#41	1	8	yes	—
93	C₂×#432	—	13	1	no	—
94	C₂×C₄×M₁₆	—	13	1	no	—
95	C₄×#117	D₁₃	1	1	yes	characteristic
96	#1939	—	13	4	no	—
97	#1851	—	13	4	no	—
98	#1767	—	13	4	no	—
99	C₂³×C₁₃	—	2	2	no	—
100	C₂³×C₁₃	—	2	2	no	—
101	C₂³×C₁₃	—	2	1	no	—
102	C₂³×C₁₃	—	2	1	no	—
103	C₂³×C₁₃	C₂×C₄⋊C₄	1	2	yes	—
104	C₂³×C₁₃	C₂³×C₄	1	1	yes	characteristic
105	C₂×C₄×C₁₃	—	2	4	no	—
106	C₂×C₄×C₁₃	—	2	2	no	—
107	C₂×C₄×C₁₃	—	2	2	no	—
108	C₂×C₄×C₁₃	—	2	1	no	—
109	C₂×C₄×C₁₃	—	2	1	no	—
110	C₂×C₄×C₁₃	—	2	4	no	—
111	C₂×C₄×C₁₃	—	2	1	no	—
112	C₂×C₄×C₁₃	—	2	1	no	—
113	C₂×C₄×C₁₃	—	2	4	no	—
114	C₂×C₄×C₁₃	—	2	4	no	—
115	C₂×C₄×C₁₃	—	2	8	no	—
116	C₂×C₄×C₁₃	—	2	4	no	—
117	C₂×C₄×C₁₃	—	2	4	no	—
118	C₂×C₄×C₁₃	—	2	4	no	—
119	C₂×C₄×C₁₃	—	4	4	no	—
120	C₂×C₄×C₁₃	C₂×C₄⋊C₄	1	2	yes	—
121	C₂×C₄×C₁₃	C₂×C₄⋊C₄	1	2	yes	—
122	C₂×C₄×C₁₃	C₂×C₄⋊C₄	1	2	yes	—
123	C₂×C₄×C₁₃	C₂×C₄⋊C₄	1	2	yes	—
124	C₂×C₄×C₁₃	C₂×C₄⋊C₄	1	2	yes	—
125	C₂×C₄×C₁₃	C₂×M₁₆	1	8	yes	—
126	C₂×C₄×C₁₃	C₂×M₁₆	1	4	yes	—
127	C₂×C₄×C₁₃	C₂²×D₄	1	1	yes	characteristic
128	C₂×C₄×C₁₃	C₂²×Q₈	1	1	yes	characteristic
129	C₂×C₄×C₁₃	C₂×#41	1	4	yes	—
130	#1055	—	2	8	no	—
131	#1055	—	4	4	no	—
132	C₂²×M₁₆	—	13	2	no	—
133	C₂×#117	Dic₁₃	1	2	yes	—
134	C₂²×C₄²	C₂×D₁₃	1	1	yes	characteristic
135	#432	—	26	4	no	—
136	C₂×C₄⋊C₈	—	13	8	no	—
137	C₂×C₄⋊C₈	—	13	2	no	—
138	C₂×C₂²⋊C₈	—	13	2	no	—
139	C₄×M₁₆	—	26	4	no	—
140	C₂×#97	—	13	2	no	—
141	C₂×#97	—	13	2	no	—
142	C₂×C₄×C₈	—	13	2	no	—
143	C₄×C₄⋊C₄	Dic₁₃	1	1	yes	characteristic
144	C₄×C₄⋊C₄	Dic₁₃	1	1	yes	characteristic
145	C₄×C₄⋊C₄	C₂×D₁₃	1	2	yes	—
146	C₄×C₂²⋊C₄	Dic₁₃	1	2	yes	—
147	C₄×C₂²⋊C₄	C₂×D₁₃	1	2	yes	—
148	#376	Dic₁₃	1	2	yes	—
149	C₄³	C₂×D₁₃	1	2	yes	—
150	#336	—	13	8	no	—
151	C₂²×C₁₃	—	2	2	no	—
152	C₂²×C₁₃	—	2	2	no	—
153	C₂²×C₁₃	—	2	2	no	—
154	C₂²×C₁₃	—	4	2	no	—
155	C₂²×C₁₃	—	4	2	no	—
156	C₂²×C₁₃	C₂²×C₄⋊C₄	1	1	yes	characteristic
157	C₂²×C₁₃	C₄×#41	1	2	yes	—
158	C₂²×C₁₃	C₂²×M₁₆	1	2	yes	—
159	C₂²×C₁₃	#568	1	1	yes	characteristic
160	C₂²×C₁₃	#568	1	1	yes	characteristic
161	C₄×C₁₃	—	2	2	no	—
162	C₄×C₁₃	—	2	2	no	—
163	C₄×C₁₃	—	2	2	no	—
164	C₄×C₁₃	—	2	8	no	—
165	C₄×C₁₃	—	2	2	no	—
166	C₄×C₁₃	—	4	4	no	—
167	C₄×C₁₃	#423	1	4	yes	—
168	C₄×C₁₃	#525	1	2	yes	—
169	C₄×C₁₃	#533	1	2	yes	—
170	C₂³×C₄	C₂×Dic₁₃	1	1	yes	characteristic
171	C₂³×C₄	C₂×Dic₁₃	1	1	yes	characteristic
172	C₂³×C₄	C₂²×D₁₃	1	1	yes	characteristic
173	C₂×M₁₆	—	26	8	no	—
174	C₂×M₁₆	—	26	4	no	—
175	C₂²×C₈	—	13	4	no	—
176	C₂²×C₈	—	13	4	no	—
177	#117	—	2	8	no	—
178	C₂×C₄⋊C₄	C₂×Dic₁₃	1	2	yes	—
179	C₂×C₄⋊C₄	C₂×Dic₁₃	1	2	yes	—
180	C₂×C₂²⋊C₄	C₂×Dic₁₃	1	4	yes	—
181	C₂×C₄²	Dic₂₆	1	4	yes	—
182	C₂×C₄²	D₅₂	1	4	yes	—
183	C₂×C₄²	C₂×Dic₁₃	1	4	yes	—
184	C₂×C₄²	C₂×Dic₁₃	1	2	yes	—
185	C₂×C₄²	C₂×Dic₁₃	1	2	yes	—
186	C₂×C₄²	C₂×Dic₁₃	1	1	yes	characteristic
187	C₂×C₄²	C₂×Dic₁₃	1	1	yes	characteristic
188	C₂×C₄²	C₂²×D₁₃	1	4	yes	—
189	C₂×C₄²	C₂²×D₁₃	1	1	yes	characteristic
190	C₂×C₄²	C₂²×D₁₃	1	1	yes	characteristic
191	C₄⋊C₈	—	26	16	no	—
192	C₄⋊C₈	—	26	4	no	—
193	C₂²⋊C₈	—	26	4	no	—
194	#97	—	26	4	no	—
195	#97	—	26	4	no	—
196	C₄×C₈	—	26	4	no	—
197	#95	C₂×Dic₁₃	1	4	yes	—
198	#95	C₂×Dic₁₃	1	2	yes	—
199	#95	C₂×Dic₁₃	1	2	yes	—
200	C₂×C₁₃	—	4	2	no	—
201	C₂×C₁₃	#2296	1	1	yes	characteristic
202	C₂×C₁₃	C₂×#423	1	1	yes	characteristic
203	C₂×C₁₃	#2905	1	1	yes	characteristic
204	C₂×C₁₃	#2970	1	2	yes	—
205	C₂×C₁₃	#2971	1	2	yes	—
206	C₂⁴	C₂²×Dic₁₃	1	1	yes	characteristic
207	C₂²×C₄	—	2	4	no	—
208	C₂²×C₄	—	2	4	no	—
209	C₂²×C₄	—	4	1	no	—
210	C₂²×C₄	—	4	1	no	—
211	C₂²×C₄	C₄⋊Dic₁₃	1	4	yes	—
212	C₂²×C₄	C₄⋊Dic₁₃	1	4	yes	—
213	C₂²×C₄	C₂×Dic₂₆	1	2	yes	—
214	C₂²×C₄	C₂×D₅₂	1	2	yes	—
215	C₂²×C₄	C₂²×Dic₁₃	1	4	yes	—
216	C₂²×C₄	C₂²×Dic₁₃	1	4	yes	—
217	C₂²×C₄	C₂²×Dic₁₃	1	1	yes	characteristic
218	C₂²×C₄	C₂²×Dic₁₃	1	1	yes	characteristic
219	C₂²×C₄	C₂²×Dic₁₃	1	1	yes	characteristic
220	C₂²×C₄	C₂²×Dic₁₃	1	1	yes	characteristic
221	C₂²×C₄	C₂²×Dic₁₃	1	1	yes	characteristic
222	C₂²×C₄	C₂³×D₁₃	1	1	yes	characteristic
223	M₁₆	—	26	16	no	—
224	C₂×C₈	—	13	8	no	—
225	C₂×C₈	—	26	4	no	—
226	C₂×C₈	—	26	4	no	—
227	C₂×C₈	—	26	4	no	—
228	C₂×C₈	—	26	8	no	—
229	C₄⋊C₄	—	2	4	no	—
230	C₄⋊C₄	#6099	1	8	yes	complemented
231	C₂²⋊C₄	—	2	8	no	—
232	C₄²	—	2	8	no	—
233	C₄²	—	2	8	no	—
234	C₄²	—	4	2	no	—
235	C₄²	—	4	2	no	—
236	C₄²	C₄⋊Dic₁₃	1	4	yes	—
237	C₄²	C₄⋊Dic₁₃	1	4	yes	—
238	C₄²	C₂×Dic₂₆	1	2	yes	—
239	C₄²	C₂×Dic₂₆	1	2	yes	—
240	C₄²	C₂×D₅₂	1	2	yes	—
241	C₄²	C₂×D₅₂	1	2	yes	—
242	C₁₃	#21356	1	1	yes	characteristic, complemented, Sylow
243	C₂³	—	2	2	no	—
244	C₂³	—	2	1	no	—
245	C₂³	—	2	1	no	—
246	C₂³	—	2	2	no	—
247	C₂³	C₂×C₄⋊Dic₁₃	1	2	yes	—
248	C₂³	C₂³×Dic₁₃	1	1	yes	characteristic
249	C₂×C₄	—	2	2	no	—
250	C₂×C₄	—	2	1	no	—
251	C₂×C₄	—	2	1	no	—
252	C₂×C₄	—	2	4	no	—
253	C₂×C₄	—	2	1	no	—
254	C₂×C₄	—	2	1	no	—
255	C₂×C₄	—	2	4	no	—
256	C₂×C₄	—	2	2	no	—
257	C₂×C₄	—	2	4	no	—
258	C₂×C₄	—	2	4	no	—
259	C₂×C₄	—	2	8	no	—
260	C₂×C₄	—	2	4	no	—
261	C₂×C₄	—	2	4	no	—
262	C₂×C₄	—	2	4	no	—
263	C₂×C₄	—	4	4	no	—
264	C₂×C₄	C₂×#6099	1	8	yes	—
265	C₂×C₄	C₂×#6099	1	4	yes	—
266	C₂×C₄	C₂×C₄⋊Dic₁₃	1	2	yes	—
267	C₂×C₄	C₂×C₄⋊Dic₁₃	1	2	yes	—
268	C₂×C₄	C₂×C₄⋊Dic₁₃	1	2	yes	—
269	C₂×C₄	C₂×C₄⋊Dic₁₃	1	2	yes	—
270	C₂×C₄	C₂×C₄⋊Dic₁₃	1	2	yes	—
271	C₂×C₄	C₂²×Dic₂₆	1	1	yes	characteristic
272	C₂×C₄	C₂²×D₅₂	1	1	yes	characteristic
273	C₂×C₄	#41×D₁₃	1	4	yes	—
274	C₈	—	26	8	no	—
275	C₈	—	52	4	no	—
276	C₂²	—	2	2	no	—
277	C₂²	—	2	2	no	—
278	C₂²	—	2	2	no	—
279	C₂²	—	4	2	no	—
280	C₂²	—	4	2	no	—
281	C₂²	C₂²×#6099	1	2	yes	—
282	C₂²	C₂²×C₄⋊Dic₁₃	1	1	yes	characteristic
283	C₂²	#11719524	1	1	yes	characteristic
284	C₂²	#11719524	1	1	yes	characteristic
285	C₂²	#41×Dic₁₃	1	2	yes	—
286	C₄	—	2	8	no	—
287	C₄	—	2	2	no	—
288	C₄	—	2	2	no	—
289	C₄	—	2	2	no	—
290	C₄	—	2	2	no	—
291	C₄	—	4	4	no	—
292	C₄	#11718585	1	4	yes	—
293	C₄	#11719222	1	2	yes	—
294	C₄	#11719229	1	2	yes	—
295	C₂	—	4	2	no	—
296	C₂	#13117320	1	1	yes	characteristic
297	C₂	C₂×#11718585	1	1	yes	characteristic
298	C₂	#13123400	1	1	yes	characteristic
299	C₂	#13123850	1	2	yes	—
300	C₂	#13123856	1	2	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	#21356
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. #13115555	C₂	semidirect
5. #13114689	C₂	semidirect
75. C₁₃×C₄⋊C₄	M₁₆	semidirect
230. C₄⋊C₄	#6099	semidirect
242. C₁₃	#21356	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×C₄×#6099	2
#13115555	2
#13115021	2
#13114689	2
C₄×C₁₃×#117	2
#21356	13