Subgroups — #15777808

Order 3328.

554 conjugacy classes of subgroups.

The subgroup lattice has 296 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×#88052	C₂	1	1	yes	maximal, characteristic
2	C₂²×#88059	C₂	1	1	yes	maximal, characteristic
3	#13112929	C₂	1	4	yes	maximal, complemented
4	C₂²×C₁₃×#108	C₂	1	1	yes	maximal, characteristic
5	C₂²×C₁₃×M₁₆	—	2	1	no	—
6	C₂³×C₈×C₁₃	C₂²	1	1	yes	characteristic
7	C₂²×#6099	—	2	1	no	—
8	C₂×C₁₃×#108	—	2	1	no	—
9	C₂×C₁₃×#108	—	2	2	no	—
10	C₂×C₁₃×#108	C₄	1	4	yes	—
11	C₂×C₁₃×#108	C₂²	1	2	yes	—
12	C₂×#88059	—	2	2	no	—
13	C₂×#88059	—	2	1	no	—
14	C₂×#88059	C₄	1	4	yes	—
15	C₂×#88059	C₂²	1	2	yes	—
16	C₂×#88052	—	2	2	no	—
17	C₂×#88052	—	2	2	no	—
18	C₂×#88052	C₄	1	2	yes	—
19	C₂×#88052	C₄	1	2	yes	—
20	C₂²×#6090	C₂²	1	2	yes	—
21	C₂³×C₄×C₁₃	D₄	1	1	yes	characteristic
22	C₂×C₁₃×M₁₆	—	2	4	no	—
23	C₂×C₁₃×M₁₆	—	2	2	no	—
24	C₂×C₁₃×M₁₆	—	4	1	no	—
25	C₂×C₁₃×M₁₆	—	4	2	no	—
26	C₂²×C₈×C₁₃	—	4	1	no	—
27	C₂²×C₈×C₁₃	C₂×C₄	1	1	yes	characteristic
28	C₂²×C₈×C₁₃	C₂×C₄	1	1	yes	characteristic
29	C₂²×C₈×C₁₃	C₂×C₄	1	2	yes	—
30	C₂²×C₈×C₁₃	C₂×C₄	1	2	yes	—
31	C₂²×C₈×C₁₃	D₄	1	2	yes	—
32	C₂²×C₈×C₁₃	D₄	1	1	yes	characteristic
33	C₂²×C₈×C₁₃	D₄	1	2	yes	—
34	C₂²×C₈×C₁₃	Q₈	1	1	yes	characteristic
35	C₂²×C₈×C₁₃	Q₈	1	1	yes	characteristic
36	C₂²×C₈×C₁₃	C₂³	1	1	yes	characteristic
37	C₂×#6099	—	2	4	no	—
38	C₂×#6099	—	2	2	no	—
39	C₂×#6099	—	4	1	no	—
40	C₂×#6099	—	4	2	no	—
41	C₂²×#1055	—	4	1	no	—
42	C₁₃×#108	—	2	4	no	—
43	C₁₃×#108	—	2	2	no	—
44	C₁₃×#108	C₂×C₄	1	4	yes	—
45	#88059	—	2	4	no	—
46	#88059	—	2	2	no	—
47	#88059	C₂×C₄	1	4	yes	—
48	#88052	—	2	4	no	—
49	#88052	—	2	4	no	—
50	C₂×#6090	—	2	2	no	—
51	C₂×#6090	—	2	2	no	—
52	C₂×#6090	C₂×C₄	1	4	yes	—
53	#12559	—	13	1	no	maximal, Sylow
54	C₂⁴×C₁₃	Q₁₆	1	1	yes	characteristic
55	C₂²×C₄×C₁₃	C₂²⋊C₄	1	2	yes	—
56	C₂²×C₄×C₁₃	C₂²⋊C₄	1	2	yes	—
57	C₂²×C₄×C₁₃	C₄⋊C₄	1	1	yes	characteristic
58	C₂²×C₄×C₁₃	C₄⋊C₄	1	1	yes	characteristic
59	C₂²×C₄×C₁₃	D₈	1	1	yes	characteristic
60	C₂²×C₄×C₁₃	D₈	1	1	yes	characteristic
61	C₂²×C₄×C₁₃	SD₁₆	1	2	yes	—
62	C₂²×C₄×C₁₃	SD₁₆	1	2	yes	—
63	C₂²×C₄×C₁₃	Q₁₆	1	1	yes	characteristic
64	C₂²×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
65	C₁₃×M₁₆	—	2	4	no	—
66	C₁₃×M₁₆	—	4	4	no	—
67	C₁₃×M₁₆	—	4	2	no	—
68	C₂×C₈×C₁₃	—	4	4	no	—
69	C₂×C₈×C₁₃	—	4	2	no	—
70	C₂×C₈×C₁₃	C₄²	1	2	yes	—
71	C₂×C₈×C₁₃	C₄²	1	2	yes	—
72	C₂×C₈×C₁₃	C₂²⋊C₄	1	2	yes	—
73	C₂×C₈×C₁₃	C₂²⋊C₄	1	2	yes	—
74	C₂×C₈×C₁₃	C₂²⋊C₄	1	2	yes	—
75	C₂×C₈×C₁₃	C₂²⋊C₄	1	2	yes	—
76	C₂×C₈×C₁₃	C₄⋊C₄	1	1	yes	characteristic
77	C₂×C₈×C₁₃	C₄⋊C₄	1	1	yes	characteristic
78	C₂×C₈×C₁₃	C₄⋊C₄	1	2	yes	—
79	C₂×C₈×C₁₃	C₄⋊C₄	1	2	yes	—
80	C₂×C₈×C₁₃	C₄⋊C₄	1	2	yes	—
81	C₂×C₈×C₁₃	C₄⋊C₄	1	2	yes	—
82	C₂×C₈×C₁₃	C₂²×C₄	1	1	yes	characteristic
83	C₂×C₈×C₁₃	C₂²×C₄	1	1	yes	characteristic
84	C₂×C₈×C₁₃	C₂²×C₄	1	1	yes	characteristic
85	C₂×C₈×C₁₃	C₂×D₄	1	1	yes	characteristic
86	C₂×C₈×C₁₃	C₂×D₄	1	1	yes	characteristic
87	C₂×C₈×C₁₃	C₂×Q₈	1	1	yes	characteristic
88	#6099	—	2	4	no	—
89	#6099	—	4	4	no	—
90	#6099	—	4	2	no	—
91	C₂×#1055	—	4	4	no	—
92	C₂×#1055	—	4	2	no	—
93	#6090	—	2	4	no	—
94	C₂²×M₃₂	—	13	1	no	—
95	C₂²×#108	—	13	1	no	—
96	C₂²×#108	D₁₃	1	1	yes	characteristic
97	#1381	—	13	4	no	—
98	C₂³×C₁₃	—	4	2	no	—
99	C₂³×C₁₃	#103	1	2	yes	—
100	C₂³×C₁₃	#103	1	2	yes	—
101	C₂³×C₁₃	#107	1	1	yes	characteristic
102	C₂³×C₁₃	#107	1	1	yes	characteristic
103	C₂³×C₁₃	C₂×Q₁₆	1	1	yes	characteristic
104	C₂×C₄×C₁₃	#95	1	2	yes	—
105	C₂×C₄×C₁₃	#95	1	2	yes	—
106	C₂×C₄×C₁₃	#102	1	2	yes	—
107	C₂×C₄×C₁₃	#102	1	2	yes	—
108	C₂×C₄×C₁₃	#102	1	2	yes	—
109	C₂×C₄×C₁₃	#102	1	2	yes	—
110	C₂×C₄×C₁₃	#103	1	2	yes	—
111	C₂×C₄×C₁₃	#103	1	2	yes	—
112	C₂×C₄×C₁₃	#106	1	2	yes	—
113	C₂×C₄×C₁₃	#106	1	2	yes	—
114	C₂×C₄×C₁₃	#107	1	1	yes	characteristic
115	C₂×C₄×C₁₃	#107	1	1	yes	characteristic
116	C₂×C₄×C₁₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
117	C₂×C₄×C₁₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
118	C₂×C₄×C₁₃	C₂×C₄⋊C₄	1	1	yes	characteristic
119	C₂×C₄×C₁₃	C₂×D₈	1	1	yes	characteristic
120	C₂×C₄×C₁₃	C₂×SD₁₆	1	1	yes	characteristic
121	C₂×C₄×C₁₃	C₂×SD₁₆	1	1	yes	characteristic
122	C₈×C₁₃	—	4	4	no	—
123	C₈×C₁₃	#95	1	2	yes	—
124	C₈×C₁₃	#95	1	2	yes	—
125	C₈×C₁₃	C₂×C₄²	1	1	yes	characteristic
126	C₈×C₁₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
127	C₈×C₁₃	C₂×C₄⋊C₄	1	1	yes	characteristic
128	C₈×C₁₃	C₂×C₄⋊C₄	1	1	yes	characteristic
129	#1055	—	4	4	no	—
130	C₂²×M₁₆	—	2	1	no	—
131	C₂²×M₁₆	—	26	1	no	—
132	C₂³×C₈	C₂×D₁₃	1	1	yes	characteristic
133	C₂×M₃₂	—	13	2	no	—
134	C₂×M₃₂	—	13	2	no	—
135	C₂×M₃₂	—	26	2	no	—
136	C₂×M₃₂	—	26	2	no	—
137	C₂²×C₁₆	—	13	2	no	—
138	C₂×#108	—	2	1	no	—
139	C₂×#108	—	2	2	no	—
140	C₂×#108	—	13	4	no	—
141	C₂×#108	—	13	2	no	—
142	C₂×#108	—	26	2	no	—
143	C₂×#108	—	26	1	no	—
144	C₂×#108	Dic₁₃	1	4	yes	—
145	C₂×#108	C₂×D₁₃	1	2	yes	—
146	C₂²×C₁₃	—	2	2	no	—
147	C₂²×C₁₃	—	2	2	no	—
148	C₂²×C₁₃	—	2	2	no	—
149	C₂²×C₁₃	—	4	2	no	—
150	C₂²×C₁₃	—	4	2	no	—
151	C₂²×C₁₃	#340	1	2	yes	—
152	C₂²×C₁₃	#340	1	2	yes	—
153	C₂²×C₁₃	C₂×#103	1	1	yes	characteristic
154	C₂²×C₁₃	C₂×#103	1	1	yes	characteristic
155	C₂²×C₁₃	C₂×#107	1	1	yes	characteristic
156	C₄×C₁₃	#340	1	2	yes	—
157	C₄×C₁₃	#340	1	2	yes	—
158	C₄×C₁₃	C₂×#95	1	1	yes	characteristic
159	C₄×C₁₃	C₂×#102	1	1	yes	characteristic
160	C₄×C₁₃	C₂×#102	1	1	yes	characteristic
161	C₄×C₁₃	C₂×#106	1	1	yes	characteristic
162	C₂³×C₄	C₂²⋊D₁₃	1	1	yes	characteristic
163	C₂×M₁₆	—	2	4	no	—
164	C₂×M₁₆	—	2	2	no	—
165	C₂×M₁₆	—	4	1	no	—
166	C₂×M₁₆	—	4	2	no	—
167	C₂×M₁₆	—	26	4	no	—
168	C₂×M₁₆	—	26	2	no	—
169	C₂×M₁₆	—	52	1	no	—
170	C₂×M₁₆	—	52	2	no	—
171	C₂²×C₈	—	4	1	no	—
172	C₂²×C₈	—	52	1	no	—
173	C₂²×C₈	Dic₂₆	1	1	yes	characteristic
174	C₂²×C₈	Dic₂₆	1	1	yes	characteristic
175	C₂²×C₈	C₄×D₁₃	1	2	yes	—
176	C₂²×C₈	C₄×D₁₃	1	1	yes	characteristic
177	C₂²×C₈	C₄×D₁₃	1	1	yes	characteristic
178	C₂²×C₈	D₅₂	1	2	yes	—
179	C₂²×C₈	C₂×Dic₁₃	1	2	yes	—
180	C₂²×C₈	C₂²⋊D₁₃	1	1	yes	characteristic
181	C₂²×C₈	C₂²⋊D₁₃	1	2	yes	—
182	C₂²×C₈	C₂²×D₁₃	1	1	yes	characteristic
183	M₃₂	—	26	4	no	—
184	M₃₂	—	26	4	no	—
185	C₂×C₁₆	—	13	4	no	—
186	C₂×C₁₆	—	26	2	no	—
187	C₂×C₁₆	—	26	2	no	—
188	#108	—	2	4	no	—
189	#108	—	2	2	no	—
190	#108	—	13	4	no	—
191	#108	—	26	4	no	—
192	#108	—	26	2	no	—
193	#108	C₂×Dic₁₃	1	4	yes	—
194	C₂×C₁₃	—	2	2	no	—
195	C₂×C₁₃	—	2	2	no	—
196	C₂×C₁₃	—	2	1	no	—
197	C₂×C₁₃	—	2	1	no	—
198	C₂×C₁₃	C₂×#340	1	1	yes	characteristic
199	C₂×C₁₃	#2161	1	2	yes	—
200	C₂⁴	#6107	1	1	yes	characteristic
201	C₂²×C₄	#6101	1	1	yes	characteristic
202	C₂²×C₄	#6101	1	1	yes	characteristic
203	C₂²×C₄	#6103	1	2	yes	—
204	C₂²×C₄	#6104	1	1	yes	characteristic
205	C₂²×C₄	#6104	1	1	yes	characteristic
206	C₂²×C₄	#6105	1	2	yes	—
207	C₂²×C₄	#6106	1	2	yes	—
208	C₂²×C₄	#6107	1	1	yes	characteristic
209	C₂²×C₄	C₂²⋊Dic₁₃	1	2	yes	—
210	C₂²×C₄	C₂×C₂²⋊D₁₃	1	1	yes	characteristic
211	M₁₆	—	2	4	no	—
212	M₁₆	—	4	4	no	—
213	M₁₆	—	4	2	no	—
214	M₁₆	—	26	4	no	—
215	M₁₆	—	52	2	no	—
216	M₁₆	—	52	4	no	—
217	C₂×C₈	—	4	4	no	—
218	C₂×C₈	—	4	2	no	—
219	C₂×C₈	—	52	2	no	—
220	C₂×C₈	—	52	4	no	—
221	C₂×C₈	C₄×Dic₁₃	1	2	yes	—
222	C₂×C₈	C₄×Dic₁₃	1	2	yes	—
223	C₂×C₈	#6101	1	1	yes	characteristic
224	C₂×C₈	#6101	1	1	yes	characteristic
225	C₂×C₈	#6101	1	2	yes	—
226	C₂×C₈	#6101	1	2	yes	—
227	C₂×C₈	C₄⋊Dic₁₃	1	2	yes	—
228	C₂×C₈	C₄⋊Dic₁₃	1	2	yes	—
229	C₂×C₈	#6103	1	2	yes	—
230	C₂×C₈	#6103	1	2	yes	—
231	C₂×C₈	#6103	1	2	yes	—
232	C₂×C₈	C₂²⋊Dic₁₃	1	2	yes	—
233	C₂×C₈	C₂×Dic₂₆	1	1	yes	characteristic
234	C₂×C₈	C₂×C₄×D₁₃	1	1	yes	characteristic
235	C₂×C₈	C₂×C₄×D₁₃	1	1	yes	characteristic
236	C₂×C₈	C₂×D₅₂	1	1	yes	characteristic
237	C₂×C₈	C₂²×Dic₁₃	1	1	yes	characteristic
238	C₂×C₈	C₂×C₂²⋊D₁₃	1	1	yes	characteristic
239	C₁₆	—	26	4	no	—
240	C₁₃	#12559	1	1	yes	characteristic, complemented, Sylow
241	C₂³	—	4	2	no	—
242	C₂³	#88047	1	1	yes	characteristic
243	C₂³	#88047	1	1	yes	characteristic
244	C₂³	#88050	1	2	yes	—
245	C₂³	#88075	1	2	yes	—
246	C₂³	C₂×#6107	1	1	yes	characteristic
247	C₂×C₄	#88047	1	1	yes	characteristic
248	C₂×C₄	#88047	1	1	yes	characteristic
249	C₂×C₄	#88048	1	2	yes	—
250	C₂×C₄	#88048	1	2	yes	—
251	C₂×C₄	#88049	1	2	yes	—
252	C₂×C₄	#88049	1	2	yes	—
253	C₂×C₄	#88050	1	2	yes	—
254	C₂×C₄	#88071	1	2	yes	—
255	C₂×C₄	#88071	1	2	yes	—
256	C₂×C₄	#88072	1	2	yes	—
257	C₂×C₄	#88072	1	2	yes	—
258	C₂×C₄	#88075	1	2	yes	—
259	C₂×C₄	C₂×#6101	1	1	yes	characteristic
260	C₂×C₄	C₂×#6103	1	1	yes	characteristic
261	C₂×C₄	C₂×#6104	1	1	yes	characteristic
262	C₂×C₄	C₂×#6105	1	1	yes	characteristic
263	C₂×C₄	C₂×#6106	1	1	yes	characteristic
264	C₂×C₄	C₂×C₂²⋊Dic₁₃	1	1	yes	characteristic
265	C₈	—	4	4	no	—
266	C₈	—	52	4	no	—
267	C₈	#88071	1	2	yes	—
268	C₈	#88071	1	2	yes	—
269	C₈	C₂×C₄×Dic₁₃	1	1	yes	characteristic
270	C₈	C₂×#6101	1	1	yes	characteristic
271	C₈	C₂×C₄⋊Dic₁₃	1	1	yes	characteristic
272	C₈	C₂×#6103	1	1	yes	characteristic
273	C₂²	—	2	2	no	—
274	C₂²	—	2	2	no	—
275	C₂²	—	2	2	no	—
276	C₂²	—	4	2	no	—
277	C₂²	—	4	2	no	—
278	C₂²	#11718121	1	2	yes	—
279	C₂²	#11718121	1	2	yes	—
280	C₂²	C₂×#88047	1	1	yes	characteristic
281	C₂²	C₂×#88050	1	1	yes	characteristic
282	C₂²	C₂×#88075	1	1	yes	characteristic
283	C₄	#11718121	1	2	yes	—
284	C₄	#11718121	1	2	yes	—
285	C₄	C₂×#88048	1	1	yes	characteristic
286	C₄	C₂×#88049	1	1	yes	characteristic
287	C₄	C₂×#88071	1	1	yes	characteristic
288	C₄	C₂×#88072	1	1	yes	characteristic
289	C₂	—	2	1	no	—
290	C₂	—	2	1	no	—
291	C₂	—	2	2	no	—
292	C₂	—	2	2	no	—
293	C₂	C₂×#11718121	1	1	yes	characteristic
294	C₂	#13116630	1	2	yes	—

	subgroup	G/H
center	C₂×C₄	C₂×#6104
commutator	C₈×C₁₃	C₂×C₄²
Frattini	C₂²×C₈	C₂²×D₁₃
Fitting	C₂²×C₁₃×#108	C₂
socle	C₂²×C₁₃	C₂×#107
prime radical	C₂⁴×C₁₃	Q₁₆
cosocle	C₂²×C₈×C₁₃	C₂³
hypercenter	C₂²×#108	D₁₃
hypocenter	C₁₃	#12559
LCS	G ▷ C₈×C₁₃ ▷ C₄×C₁₃ ▷ C₂×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₄ ◁ C₂³×C₈ ◁ C₂²×#108

Complements

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

N	H ≅ G/N	type
3. #13112929	C₂	semidirect
240. C₁₃	#12559	semidirect

Maximal subgroups

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

subgroup	index
C₂²×#88052	2
C₂²×#88059	2
#13112929	2
C₂²×C₁₃×#108	2
#12559	13