Subgroups — #15779435

Order 3328.

404 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	C₂×#11718085	C₂	1	2	yes	maximal, complemented
2	C₂×C₂²⋊#6090	C₂	1	1	yes	maximal, characteristic
3	#13114874	C₂	1	1	yes	maximal, characteristic
4	#13112929	C₂	1	2	yes	maximal, complemented
5	C₂²×C₁₃×#108	C₂	1	1	yes	maximal, characteristic
6	C₂²×C₁₃×M₁₆	—	2	1	no	—
7	C₂³×C₈×C₁₃	C₂²	1	1	yes	characteristic
8	C₂×C₁₃×#108	—	2	1	no	—
9	C₂×C₁₃×#108	—	2	4	no	—
10	C₂×C₁₃×#108	C₂²	1	2	yes	—
11	#11718868	—	2	1	no	—
12	C₂×#88059	C₂²	1	1	yes	characteristic
13	C₂×#88059	C₂²	1	1	yes	characteristic
14	C₂²×#6090	C₂²	1	2	yes	—
15	#11718144	—	2	1	no	—
16	C₂²⋊#6090	C₄	1	2	yes	—
17	C₂²⋊#6090	C₄	1	2	yes	—
18	#11718085	—	2	4	no	—
19	C₂³×C₄×C₁₃	D₄	1	1	yes	characteristic
20	C₂×C₁₃×M₁₆	—	2	4	no	—
21	C₂×C₁₃×M₁₆	—	2	2	no	—
22	C₂×C₁₃×M₁₆	—	4	2	no	—
23	C₂×C₁₃×M₁₆	—	4	1	no	—
24	C₂²×C₈×C₁₃	—	2	2	no	—
25	C₂²×C₈×C₁₃	—	2	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	D₄	1	1	yes	characteristic
30	C₂²×C₈×C₁₃	Q₈	1	1	yes	characteristic
31	C₂²×C₈×C₁₃	Q₈	1	1	yes	characteristic
32	C₂²×C₈×C₁₃	C₂³	1	1	yes	characteristic
33	C₂×#6099	—	2	1	no	—
34	C₂×#6099	—	2	1	no	—
35	C₁₃×#108	—	2	4	no	—
36	C₁₃×#108	—	2	2	no	—
37	C₁₃×#108	D₄	1	4	yes	—
38	C₂²⋊#1055	—	4	1	no	—
39	C₂²⋊#1055	—	4	1	no	—
40	#88059	—	2	2	no	—
41	#88059	—	2	2	no	—
42	C₂×#6090	—	2	2	no	—
43	C₂×#6090	—	2	2	no	—
44	C₂×#6090	C₂×C₄	1	4	yes	—
45	#12922	—	13	1	no	maximal, Sylow
46	C₂⁴×C₁₃	Q₁₆	1	1	yes	characteristic
47	C₂²×C₄×C₁₃	—	2	1	no	—
48	C₂²×C₄×C₁₃	—	2	1	no	—
49	C₂²×C₄×C₁₃	—	2	1	no	—
50	C₂²×C₄×C₁₃	—	2	1	no	—
51	C₂²×C₄×C₁₃	C₄⋊C₄	1	1	yes	characteristic
52	C₂²×C₄×C₁₃	C₄⋊C₄	1	1	yes	characteristic
53	C₂²×C₄×C₁₃	D₈	1	1	yes	characteristic
54	C₂²×C₄×C₁₃	D₈	1	1	yes	characteristic
55	C₂²×C₄×C₁₃	Q₁₆	1	1	yes	characteristic
56	C₂²×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
57	C₁₃×M₁₆	—	2	4	no	—
58	C₁₃×M₁₆	—	4	4	no	—
59	C₁₃×M₁₆	—	4	2	no	—
60	C₂×C₈×C₁₃	—	2	1	no	—
61	C₂×C₈×C₁₃	—	2	1	no	—
62	C₂×C₈×C₁₃	—	2	2	no	—
63	C₂×C₈×C₁₃	—	2	2	no	—
64	C₂×C₈×C₁₃	—	2	2	no	—
65	C₂×C₈×C₁₃	—	2	2	no	—
66	C₂×C₈×C₁₃	—	4	4	no	—
67	C₂×C₈×C₁₃	—	4	2	no	—
68	C₂×C₈×C₁₃	C₄⋊C₄	1	1	yes	characteristic
69	C₂×C₈×C₁₃	C₄⋊C₄	1	1	yes	characteristic
70	C₂×C₈×C₁₃	C₂²×C₄	1	1	yes	characteristic
71	C₂×C₈×C₁₃	C₂×D₄	1	2	yes	—
72	C₂×C₈×C₁₃	C₂×Q₈	1	1	yes	characteristic
73	C₂×C₈×C₁₃	#41	1	2	yes	—
74	#6099	—	4	1	no	—
75	#6099	—	4	1	no	—
76	#6099	—	4	1	no	—
77	#6099	—	4	1	no	—
78	C₂×#1055	—	4	1	no	—
79	C₂×#1055	—	4	1	no	—
80	#6090	—	2	4	no	—
81	C₂²×#108	D₁₃	1	1	yes	characteristic
82	C₂×#368	—	13	2	no	—
83	C₂×C₂²⋊C₁₆	—	13	1	no	—
84	#1810	—	13	1	no	—
85	#1381	—	13	2	no	—
86	C₂³×C₁₃	—	2	2	no	—
87	C₂³×C₁₃	—	2	1	no	—
88	C₂³×C₁₃	—	2	1	no	—
89	C₂³×C₁₃	—	4	1	no	—
90	C₂³×C₁₃	#107	1	1	yes	characteristic
91	C₂³×C₁₃	#107	1	1	yes	characteristic
92	C₂³×C₁₃	C₂×Q₁₆	1	1	yes	characteristic
93	C₂×C₄×C₁₃	—	2	1	no	—
94	C₂×C₄×C₁₃	—	2	1	no	—
95	C₂×C₄×C₁₃	—	2	1	no	—
96	C₂×C₄×C₁₃	—	2	1	no	—
97	C₂×C₄×C₁₃	—	2	1	no	—
98	C₂×C₄×C₁₃	—	2	1	no	—
99	C₂×C₄×C₁₃	—	2	1	no	—
100	C₂×C₄×C₁₃	—	2	1	no	—
101	C₂×C₄×C₁₃	—	2	1	no	—
102	C₂×C₄×C₁₃	—	2	1	no	—
103	C₂×C₄×C₁₃	#107	1	1	yes	characteristic
104	C₂×C₄×C₁₃	#107	1	1	yes	characteristic
105	C₂×C₄×C₁₃	C₂×C₄⋊C₄	1	1	yes	characteristic
106	C₂×C₄×C₁₃	#120	1	1	yes	characteristic
107	C₂×C₄×C₁₃	#123	1	1	yes	characteristic
108	C₂×C₄×C₁₃	C₂×D₈	1	1	yes	characteristic
109	C₂×C₄×C₁₃	AGL(1,ℤ/8)	1	1	yes	characteristic
110	C₂×C₄×C₁₃	#137	1	1	yes	characteristic
111	C₈×C₁₃	—	2	1	no	—
112	C₈×C₁₃	—	2	1	no	—
113	C₈×C₁₃	—	4	4	no	—
114	C₈×C₁₃	C₄×D₄	1	2	yes	—
115	C₈×C₁₃	C₂²⋊Q₈	1	2	yes	—
116	#1055	—	8	1	no	—
117	#1055	—	8	1	no	—
118	C₂²×M₁₆	—	2	1	no	—
119	C₂³×C₈	C₂×D₁₃	1	1	yes	characteristic
120	C₂²×C₁₆	—	13	2	no	—
121	C₂×#108	—	2	1	no	—
122	C₂×#108	—	2	4	no	—
123	C₂×#108	—	13	1	no	—
124	C₂×#108	—	13	1	no	—
125	C₂×#108	C₂×D₁₃	1	2	yes	—
126	#407	—	26	1	no	—
127	#368	—	26	4	no	—
128	C₂²⋊C₁₆	—	13	2	no	—
129	C₂²⋊C₁₆	—	13	2	no	—
130	#341	—	26	1	no	—
131	C₂²×C₁₃	—	2	1	no	—
132	C₂²×C₁₃	—	2	1	no	—
133	C₂²×C₁₃	—	2	1	no	—
134	C₂²×C₁₃	—	2	1	no	—
135	C₂²×C₁₃	—	2	1	no	—
136	C₂²×C₁₃	—	2	1	no	—
137	C₂²×C₁₃	—	2	1	no	—
138	C₂²×C₁₃	—	4	2	no	—
139	C₂²×C₁₃	—	4	2	no	—
140	C₂²×C₁₃	#368	1	2	yes	complemented
141	C₂²×C₁₃	C₂×#107	1	1	yes	characteristic
142	C₂²×C₁₃	#451	1	1	yes	characteristic
143	C₂²×C₁₃	#484	1	1	yes	characteristic
144	C₄×C₁₃	—	2	1	no	—
145	C₄×C₁₃	—	2	1	no	—
146	C₄×C₁₃	#385	1	1	yes	characteristic
147	C₄×C₁₃	#428	1	1	yes	characteristic
148	C₄×C₁₃	#447	1	1	yes	characteristic
149	C₄×C₁₃	#480	1	1	yes	characteristic
150	C₂³×C₄	C₂²⋊D₁₃	1	1	yes	characteristic
151	C₂×M₁₆	—	2	4	no	—
152	C₂×M₁₆	—	2	2	no	—
153	C₂×M₁₆	—	4	2	no	—
154	C₂×M₁₆	—	4	1	no	—
155	C₂×M₁₆	—	26	1	no	—
156	C₂×M₁₆	—	26	1	no	—
157	C₂²×C₈	—	2	2	no	—
158	C₂²×C₈	—	2	2	no	—
159	C₂²×C₈	—	4	1	no	—
160	C₂²×C₈	Dic₂₆	1	1	yes	characteristic
161	C₂²×C₈	Dic₂₆	1	1	yes	characteristic
162	C₂²×C₈	C₄×D₁₃	1	1	yes	characteristic
163	C₂²×C₈	C₄×D₁₃	1	1	yes	characteristic
164	C₂²×C₈	C₂²⋊D₁₃	1	1	yes	characteristic
165	C₂²×C₈	C₂²×D₁₃	1	1	yes	characteristic
166	C₂×C₁₆	—	13	4	no	—
167	C₂×C₁₆	—	26	2	no	—
168	C₂×C₁₆	—	26	2	no	—
169	#108	—	2	4	no	—
170	#108	—	2	2	no	—
171	#108	—	26	2	no	—
172	#108	—	26	2	no	—
173	#108	C₂²⋊D₁₃	1	4	yes	—
174	C₂²⋊C₈	—	52	1	no	—
175	C₂²⋊C₈	—	52	1	no	—
176	C₂×C₁₃	—	2	2	no	—
177	C₂×C₁₃	—	2	1	no	—
178	C₂×C₁₃	—	2	1	no	—
179	C₂×C₁₃	—	4	1	no	—
180	C₂×C₁₃	#1852	1	1	yes	characteristic
181	C₂×C₁₃	C₂×#368	1	1	yes	characteristic
182	C₂×C₁₃	#2161	1	1	yes	characteristic
183	C₂⁴	#6107	1	1	yes	characteristic
184	C₂²×C₄	—	2	1	no	—
185	C₂²×C₄	—	2	1	no	—
186	C₂²×C₄	—	2	1	no	—
187	C₂²×C₄	—	2	1	no	—
188	C₂²×C₄	#6101	1	1	yes	characteristic
189	C₂²×C₄	#6101	1	1	yes	characteristic
190	C₂²×C₄	#6104	1	1	yes	characteristic
191	C₂²×C₄	#6104	1	1	yes	characteristic
192	C₂²×C₄	#6107	1	1	yes	characteristic
193	C₂²×C₄	C₂×C₂²⋊D₁₃	1	1	yes	characteristic
194	M₁₆	—	2	4	no	—
195	M₁₆	—	4	4	no	—
196	M₁₆	—	4	2	no	—
197	M₁₆	—	52	1	no	—
198	M₁₆	—	52	1	no	—
199	M₁₆	—	52	1	no	—
200	M₁₆	—	52	1	no	—
201	C₂×C₈	—	2	1	no	—
202	C₂×C₈	—	2	1	no	—
203	C₂×C₈	—	2	2	no	—
204	C₂×C₈	—	2	2	no	—
205	C₂×C₈	—	2	2	no	—
206	C₂×C₈	—	2	2	no	—
207	C₂×C₈	—	4	4	no	—
208	C₂×C₈	—	4	2	no	—
209	C₂×C₈	—	52	1	no	—
210	C₂×C₈	—	52	1	no	—
211	C₂×C₈	#6101	1	1	yes	characteristic
212	C₂×C₈	#6101	1	1	yes	characteristic
213	C₂×C₈	C₂×Dic₂₆	1	1	yes	characteristic
214	C₂×C₈	C₂×C₄×D₁₃	1	1	yes	characteristic
215	C₂×C₈	#6127	1	2	yes	—
216	C₂×C₈	C₂×C₂²⋊D₁₃	1	2	yes	—
217	C₁₆	—	26	4	no	—
218	C₁₃	#12922	1	1	yes	characteristic, complemented, Sylow
219	C₂³	—	2	2	no	—
220	C₂³	—	2	1	no	—
221	C₂³	—	2	1	no	—
222	C₂³	—	4	1	no	—
223	C₂³	#88047	1	1	yes	characteristic
224	C₂³	#88047	1	1	yes	characteristic
225	C₂³	C₂×#6107	1	1	yes	characteristic
226	C₂×C₄	—	2	1	no	—
227	C₂×C₄	—	2	1	no	—
228	C₂×C₄	—	2	1	no	—
229	C₂×C₄	—	2	1	no	—
230	C₂×C₄	—	2	1	no	—
231	C₂×C₄	—	2	1	no	—
232	C₂×C₄	—	2	1	no	—
233	C₂×C₄	—	2	1	no	—
234	C₂×C₄	—	2	1	no	—
235	C₂×C₄	—	2	1	no	—
236	C₂×C₄	#88047	1	1	yes	characteristic
237	C₂×C₄	#88047	1	1	yes	characteristic
238	C₂×C₄	C₂×#6101	1	1	yes	characteristic
239	C₂×C₄	#88183	1	1	yes	characteristic
240	C₂×C₄	C₂×#6104	1	1	yes	characteristic
241	C₂×C₄	#88186	1	1	yes	characteristic
242	C₂×C₄	#88196	1	1	yes	characteristic
243	C₂×C₄	#88207	1	1	yes	characteristic
244	C₈	—	2	1	no	—
245	C₈	—	2	1	no	—
246	C₈	—	4	4	no	—
247	C₈	—	104	1	no	—
248	C₈	—	104	1	no	—
249	C₈	#88178	1	2	yes	—
250	C₈	C₄×C₂²⋊D₁₃	1	2	yes	—
251	C₂²	—	2	1	no	—
252	C₂²	—	2	1	no	—
253	C₂²	—	2	1	no	—
254	C₂²	—	2	1	no	—
255	C₂²	—	2	1	no	—
256	C₂²	—	2	1	no	—
257	C₂²	—	2	1	no	—
258	C₂²	—	4	2	no	—
259	C₂²	—	4	2	no	—
260	C₂²	#11718085	1	2	yes	complemented
261	C₂²	C₂×#88047	1	1	yes	characteristic
262	C₂²	#11718631	1	1	yes	characteristic
263	C₂²	#11718889	1	1	yes	characteristic
264	C₄	—	2	1	no	—
265	C₄	—	2	1	no	—
266	C₄	#11718625	1	1	yes	characteristic
267	C₄	#11718628	1	1	yes	characteristic
268	C₄	#11718871	1	1	yes	characteristic
269	C₄	#11718878	1	1	yes	characteristic
270	C₂	—	2	2	no	—
271	C₂	—	2	1	no	—
272	C₂	—	2	1	no	—
273	C₂	—	4	1	no	—
274	C₂	#13115024	1	1	yes	characteristic
275	C₂	C₂×#11718085	1	1	yes	characteristic
276	C₂	#13116630	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₂×#6104
commutator	C₂×C₈×C₁₃	C₂²×C₄
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	#12922
LCS	G ▷ C₂×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
1. C₂×#11718085	C₂	semidirect
4. #13112929	C₂	semidirect
140. C₂²×C₁₃	#368	semidirect
218. C₁₃	#12922	semidirect
260. C₂²	#11718085	semidirect

Maximal subgroups

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

subgroup	index
C₂×#11718085	2
C₂×C₂²⋊#6090	2
#13114874	2
#13112929	2
C₂²×C₁₃×#108	2
#12922	13