Subgroups — #15881816

Order 3328.

638 conjugacy classes of subgroups.

The subgroup lattice has 292 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	M₁₆×Dic₂₆	C₂	1	2	yes	maximal
2	C₂×#11718340	C₂	1	1	yes	maximal, characteristic
3	#13115611	C₂	1	2	yes	maximal
4	#13115610	C₂	1	4	yes	maximal
5	#13115009	C₂	1	2	yes	maximal
6	#13115008	C₂	1	1	yes	maximal, characteristic
7	#13114744	C₂	1	2	yes	maximal
8	C₄×C₁₃×#108	C₂	1	1	yes	maximal, characteristic
9	C₂×C₄×Dic₂₆	C₂²	1	1	yes	characteristic
10	C₂×C₁₃×#108	C₂²	1	1	yes	characteristic
11	C₂×C₁₃×#108	C₂²	1	1	yes	characteristic
12	C₄×C₁₃×M₁₆	C₂²	1	2	yes	—
13	C₂×C₄×C₈×C₁₃	C₂²	1	1	yes	characteristic
14	C₂×#88062	C₂²	1	4	yes	—
15	#11718779	C₂²	1	4	yes	—
16	M₁₆×Dic₁₃	C₂²	1	4	yes	—
17	C₂×#88059	C₂²	1	2	yes	—
18	C₂×#88059	C₂²	1	2	yes	—
19	C₂×#88055	C₂²	1	2	yes	—
20	C₂×#88054	C₂²	1	2	yes	—
21	#11718585	C₂²	1	2	yes	—
22	C₂×C₄⋊#1055	C₂²	1	1	yes	characteristic
23	#11718362	—	2	4	no	—
24	#11718340	—	2	2	no	—
25	#11718340	C₄	1	4	yes	—
26	#11718340	C₄	1	4	yes	—
27	C₈×Dic₂₆	—	2	2	no	—
28	C₁₃×#341	C₂²	1	2	yes	—
29	#11718150	C₂²	1	4	yes	—
30	C₂²×Dic₂₆	Q₈	1	1	yes	characteristic
31	C₂×C₁₃×M₁₆	C₂³	1	2	yes	—
32	C₂×C₁₃×M₁₆	C₂³	1	2	yes	—
33	C₂²×C₈×C₁₃	C₂³	1	1	yes	characteristic
34	C₂²×C₈×C₁₃	C₂³	1	1	yes	characteristic
35	C₂×C₄²×C₁₃	C₂³	1	1	yes	characteristic
36	C₂×C₄⋊Dic₁₃	Q₈	1	1	yes	characteristic
37	C₂×#6101	Q₈	1	2	yes	—
38	C₂×C₄×Dic₁₃	C₂³	1	2	yes	—
39	C₂×#6099	C₂³	1	4	yes	—
40	C₂²×#1055	C₂³	1	2	yes	—
41	C₄×Dic₂₆	C₂×C₄	1	2	yes	—
42	C₄×Dic₂₆	C₂×C₄	1	2	yes	—
43	C₄×Dic₂₆	D₄	1	2	yes	—
44	C₄×Dic₂₆	D₄	1	2	yes	—
45	C₁₃×#108	—	2	2	no	—
46	C₁₃×#108	Q₈	1	4	yes	complemented
47	C₁₃×#97	—	2	2	no	—
48	C₄×C₈×C₁₃	—	2	2	no	—
49	C₄×C₈×C₁₃	C₂×C₄	1	2	yes	—
50	C₄×C₈×C₁₃	C₂×C₄	1	2	yes	—
51	#88062	—	4	4	no	—
52	#88059	—	2	4	no	—
53	#88059	—	4	2	no	—
54	#88055	—	2	4	no	—
55	#88055	C₂×C₄	1	8	yes	—
56	#88054	—	2	4	no	—
57	#88054	—	2	4	no	—
58	#88054	C₂×C₄	1	8	yes	—
59	C₈×Dic₁₃	—	2	4	no	—
60	C₄⋊#1055	—	2	2	no	—
61	C₄⋊#1055	—	2	2	no	—
62	C₄⋊#1055	C₂×C₄	1	2	yes	—
63	C₄⋊#1055	C₂×C₄	1	2	yes	—
64	#22733	—	13	1	no	maximal, Sylow
65	C₂²×C₄×C₁₃	C₂×Q₈	1	1	yes	characteristic
66	C₂²×C₄×C₁₃	C₂×Q₈	1	1	yes	characteristic
67	C₂²×C₄×C₁₃	C₂⁴	1	1	yes	characteristic
68	C₂²×Dic₁₃	C₂×Q₈	1	2	yes	—
69	C₂²×Dic₁₃	C₂×Q₈	1	2	yes	—
70	C₂×Dic₂₆	—	4	2	no	—
71	C₂×Dic₂₆	C₄⋊C₄	1	2	yes	—
72	C₂×Dic₂₆	C₄⋊C₄	1	2	yes	—
73	C₁₃×M₁₆	—	2	2	no	—
74	C₁₃×M₁₆	—	2	2	no	—
75	C₁₃×M₁₆	—	2	2	no	—
76	C₁₃×M₁₆	C₂×Q₈	1	4	yes	—
77	C₂×C₈×C₁₃	—	2	2	no	—
78	C₂×C₈×C₁₃	—	2	1	no	—
79	C₂×C₈×C₁₃	—	2	1	no	—
80	C₂×C₈×C₁₃	—	2	2	no	—
81	C₂×C₈×C₁₃	C₂²×C₄	1	2	yes	—
82	C₂×C₈×C₁₃	C₂²×C₄	1	2	yes	—
83	C₂×C₈×C₁₃	C₂×Q₈	1	2	yes	—
84	C₂×C₈×C₁₃	#41	1	2	yes	—
85	C₄²×C₁₃	C₂²×C₄	1	1	yes	characteristic
86	C₄²×C₁₃	C₂²×C₄	1	1	yes	characteristic
87	C₄²×C₁₃	C₂×D₄	1	1	yes	characteristic
88	C₄²×C₁₃	C₂×D₄	1	1	yes	characteristic
89	C₄⋊Dic₁₃	C₄⋊C₄	1	2	yes	—
90	C₄⋊Dic₁₃	C₄⋊C₄	1	2	yes	—
91	#6101	C₄⋊C₄	1	4	yes	—
92	#6101	C₄⋊C₄	1	4	yes	—
93	C₄×Dic₁₃	C₂²×C₄	1	4	yes	—
94	C₄×Dic₁₃	C₂×D₄	1	4	yes	—
95	#6099	—	2	4	no	—
96	#6099	—	2	4	no	—
97	C₂×#1055	—	2	2	no	—
98	C₂×#1055	—	2	2	no	—
99	C₂×#1055	—	2	4	no	—
100	C₂×#1055	C₂²×C₄	1	4	yes	—
101	Q₈×M₁₆	—	13	2	no	—
102	C₂×#446	—	13	1	no	—
103	#1950	—	13	2	no	—
104	#1950	—	13	4	no	—
105	#1849	—	13	2	no	—
106	#1848	—	13	1	no	—
107	#1778	—	13	2	no	—
108	C₄×#108	D₁₃	1	1	yes	characteristic
109	C₂³×C₁₃	C₂²×Q₈	1	1	yes	characteristic
110	C₂×C₄×C₁₃	—	2	2	no	—
111	C₂×C₄×C₁₃	—	4	1	no	—
112	C₂×C₄×C₁₃	C₂×C₄⋊C₄	1	1	yes	characteristic
113	C₂×C₄×C₁₃	C₂×C₄⋊C₄	1	1	yes	characteristic
114	C₂×C₄×C₁₃	C₂×C₄⋊C₄	1	1	yes	characteristic
115	C₂×C₄×C₁₃	C₂×C₄⋊C₄	1	1	yes	characteristic
116	C₂×C₄×C₁₃	C₂³×C₄	1	1	yes	characteristic
117	C₂×C₄×C₁₃	C₂²×D₄	1	1	yes	characteristic
118	C₂×C₄×C₁₃	C₂²×Q₈	1	1	yes	characteristic
119	C₂×C₄×C₁₃	C₂×#41	1	1	yes	characteristic
120	C₂×C₄×C₁₃	H(5,𝔽₂)	1	1	yes	characteristic
121	C₂×C₄×C₁₃	2₋¹⁺⁴	1	1	yes	characteristic
122	C₂×Dic₁₃	—	4	2	no	—
123	C₂×Dic₁₃	—	4	2	no	—
124	C₂×Dic₁₃	C₂×C₄⋊C₄	1	4	yes	—
125	C₂×Dic₁₃	C₂×C₄⋊C₄	1	4	yes	—
126	Dic₂₆	—	2	4	no	—
127	Dic₂₆	—	4	2	no	—
128	C₈×C₁₃	—	2	2	no	—
129	C₈×C₁₃	—	2	4	no	—
130	C₈×C₁₃	—	4	2	no	—
131	C₈×C₁₃	C₄×Q₈	1	4	yes	—
132	#1055	—	2	4	no	—
133	#1055	—	4	4	no	—
134	C₂×C₄×Q₈	—	13	1	no	—
135	#446	—	13	4	no	—
136	#446	—	13	4	no	—
137	#446	—	26	2	no	—
138	#446	—	26	4	no	—
139	C₈×Q₈	—	26	2	no	—
140	C₂×#108	—	13	2	no	—
141	C₂×#108	—	13	4	no	—
142	C₂×#108	—	13	2	no	—
143	C₂×#108	C₂×D₁₃	1	1	yes	characteristic
144	C₂×#108	C₂×D₁₃	1	1	yes	characteristic
145	#423	—	13	4	no	—
146	#423	—	13	2	no	—
147	C₂×C₄⋊C₈	—	13	2	no	—
148	C₂×C₄⋊C₈	—	13	1	no	—
149	C₄×M₁₆	—	13	4	no	—
150	C₄×M₁₆	C₂×D₁₃	1	2	yes	—
151	C₂×#97	—	13	2	no	—
152	C₂×C₄×C₈	C₂×D₁₃	1	1	yes	characteristic
153	#341	—	13	4	no	—
154	#341	C₂×D₁₃	1	2	yes	—
155	C₂²×C₁₃	—	2	1	no	—
156	C₂²×C₁₃	—	2	1	no	—
157	C₂²×C₁₃	C₂²×C₄⋊C₄	1	1	yes	characteristic
158	C₂²×C₁₃	#557	1	1	yes	characteristic
159	C₂²×C₁₃	Q₈²	1	1	yes	characteristic
160	C₄×C₁₃	—	2	1	no	—
161	C₄×C₁₃	—	2	1	no	—
162	C₄×C₁₃	—	2	1	no	—
163	C₄×C₁₃	#430	1	2	yes	—
164	C₄×C₁₃	C₂×C₄×Q₈	1	1	yes	characteristic
165	C₄×C₁₃	#520	1	1	yes	characteristic
166	C₄×C₁₃	D₄×Q₈	1	1	yes	characteristic
167	C₄×C₁₃	#550	1	1	yes	characteristic
168	Dic₁₃	—	2	4	no	—
169	Dic₁₃	—	2	4	no	—
170	C₂²×Q₈	—	13	1	no	—
171	C₂×M₁₆	—	13	4	no	—
172	C₂×M₁₆	C₂²×D₁₃	1	2	yes	—
173	C₂×M₁₆	C₂²×D₁₃	1	2	yes	—
174	C₂²×C₈	—	13	2	no	—
175	C₂²×C₈	C₂²×D₁₃	1	1	yes	characteristic
176	C₂²×C₈	C₂²×D₁₃	1	1	yes	characteristic
177	C₄×Q₈	—	13	2	no	—
178	C₄×Q₈	—	13	2	no	—
179	C₄×Q₈	—	13	2	no	—
180	C₄×Q₈	—	13	2	no	—
181	C₂×C₄⋊C₄	—	13	2	no	—
182	C₂×C₄⋊C₄	—	13	1	no	—
183	C₂×C₄²	—	13	2	no	—
184	C₂×C₄²	C₂²×D₁₃	1	1	yes	characteristic
185	#108	—	2	2	no	—
186	#108	—	26	4	no	—
187	#108	—	52	2	no	—
188	#108	—	52	4	no	—
189	#108	Dic₂₆	1	4	yes	complemented
190	C₄⋊C₈	—	13	8	no	—
191	C₄⋊C₈	—	13	2	no	—
192	C₄⋊C₈	—	13	2	no	—
193	C₄⋊C₈	—	26	4	no	—
194	C₄⋊C₈	—	26	4	no	—
195	C₄⋊C₈	—	26	2	no	—
196	C₄⋊C₈	—	26	2	no	—
197	#97	—	2	2	no	—
198	#97	—	13	8	no	—
199	#97	—	26	4	no	—
200	C₄×C₈	—	2	2	no	—
201	C₄×C₈	—	26	4	no	—
202	C₄×C₈	C₄×D₁₃	1	2	yes	—
203	C₄×C₈	C₄×D₁₃	1	2	yes	—
204	C₂×C₁₃	—	2	1	no	—
205	C₂×C₁₃	—	2	1	no	—
206	C₂×C₁₃	Q₈×C₄⋊C₄	1	1	yes	characteristic
207	C₂×C₁₃	C₂×#430	1	1	yes	characteristic
208	C₂×C₁₃	#2916	1	1	yes	characteristic
209	C₂×Q₈	—	13	2	no	—
210	C₂×Q₈	—	13	2	no	—
211	C₂×Q₈	—	52	2	no	—
212	C₂²×C₄	—	13	2	no	—
213	C₂²×C₄	—	13	2	no	—
214	C₂²×C₄	Q₈×D₁₃	1	1	yes	characteristic
215	C₂²×C₄	Q₈×D₁₃	1	1	yes	characteristic
216	C₂²×C₄	C₂³×D₁₃	1	1	yes	characteristic
217	M₁₆	—	2	2	no	—
218	M₁₆	—	2	2	no	—
219	M₁₆	—	2	2	no	—
220	M₁₆	—	26	4	no	—
221	M₁₆	—	26	4	no	—
222	M₁₆	C₂×Dic₂₆	1	4	yes	—
223	C₂×C₈	—	2	2	no	—
224	C₂×C₈	—	2	1	no	—
225	C₂×C₈	—	2	1	no	—
226	C₂×C₈	—	2	2	no	—
227	C₂×C₈	—	13	4	no	—
228	C₂×C₈	—	26	2	no	—
229	C₂×C₈	—	26	2	no	—
230	C₂×C₈	—	26	4	no	—
231	C₂×C₈	C₂×Dic₂₆	1	2	yes	—
232	C₂×C₈	C₂×C₄×D₁₃	1	2	yes	—
233	C₂×C₈	C₂×C₄×D₁₃	1	2	yes	—
234	C₂×C₈	#6127	1	2	yes	—
235	C₄⋊C₄	—	13	4	no	—
236	C₄⋊C₄	—	13	4	no	—
237	C₄⋊C₄	—	13	2	no	—
238	C₄⋊C₄	—	13	2	no	—
239	C₄²	—	13	4	no	—
240	C₄²	—	13	4	no	—
241	C₄²	C₂×C₄×D₁₃	1	1	yes	characteristic
242	C₄²	C₂×C₄×D₁₃	1	1	yes	characteristic
243	C₄²	D₄×D₁₃	1	1	yes	characteristic
244	C₄²	D₄×D₁₃	1	1	yes	characteristic
245	C₁₃	#22733	1	1	yes	characteristic, complemented, Sylow
246	C₂³	C₂×Q₈×D₁₃	1	1	yes	characteristic
247	Q₈	—	26	4	no	—
248	Q₈	—	52	2	no	—
249	C₂×C₄	—	2	2	no	—
250	C₂×C₄	—	4	1	no	—
251	C₂×C₄	—	13	4	no	—
252	C₂×C₄	—	13	4	no	—
253	C₂×C₄	—	52	2	no	—
254	C₂×C₄	—	52	2	no	—
255	C₂×C₄	C₄⋊C₄×D₁₃	1	1	yes	characteristic
256	C₂×C₄	C₄⋊C₄×D₁₃	1	1	yes	characteristic
257	C₂×C₄	C₄⋊C₄×D₁₃	1	1	yes	characteristic
258	C₂×C₄	C₄⋊C₄×D₁₃	1	1	yes	characteristic
259	C₂×C₄	C₂²×Dic₂₆	1	1	yes	characteristic
260	C₂×C₄	C₂²×C₄×D₁₃	1	1	yes	characteristic
261	C₂×C₄	C₂×#6127	1	1	yes	characteristic
262	C₂×C₄	C₂×D₄×D₁₃	1	1	yes	characteristic
263	C₂×C₄	#88256	1	1	yes	characteristic
264	C₂×C₄	#88257	1	1	yes	characteristic
265	C₈	—	2	2	no	—
266	C₈	—	2	4	no	—
267	C₈	—	4	2	no	—
268	C₈	—	26	4	no	—
269	C₈	—	52	4	no	—
270	C₈	C₄×Dic₂₆	1	4	yes	—
271	C₂²	—	2	1	no	—
272	C₂²	—	2	1	no	—
273	C₂²	C₂×C₄⋊C₄×D₁₃	1	1	yes	characteristic
274	C₂²	Q₈×Dic₂₆	1	1	yes	characteristic
275	C₂²	#11719272	1	1	yes	characteristic
276	C₄	—	2	1	no	—
277	C₄	—	2	1	no	—
278	C₄	—	2	1	no	—
279	C₄	—	26	4	no	—
280	C₄	—	26	4	no	—
281	C₄	#11718554	1	2	yes	—
282	C₄	C₂×C₄×Dic₂₆	1	1	yes	characteristic
283	C₄	#11719228	1	1	yes	characteristic
284	C₄	D₄×Dic₂₆	1	1	yes	characteristic
285	C₄	#11719258	1	1	yes	characteristic
286	C₂	—	2	1	no	—
287	C₂	—	2	1	no	—
288	C₂	C₄⋊C₄×Dic₂₆	1	1	yes	characteristic
289	C₂	C₂×#11718554	1	1	yes	characteristic
290	C₂	#13123444	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₂×D₄×D₁₃
commutator	C₂×C₄×C₁₃	C₂³×C₄
Frattini	C₂²×C₄	C₂³×D₁₃
Fitting	C₄×C₁₃×#108	C₂
socle	C₂²×C₁₃	C₂²×C₄⋊C₄
prime radical	C₂³×C₁₃	C₂²×Q₈
cosocle	C₂²×C₄×C₁₃	C₂⁴
hypercenter	C₄×#108	D₁₃
hypocenter	C₁₃	#22733
LCS	G ▷ C₂×C₄×C₁₃ ▷ C₂×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ C₄×#108

Complements

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

N	H ≅ G/N	type
46. C₁₃×#108	Q₈	semidirect
189. #108	Dic₂₆	semidirect
245. C₁₃	#22733	semidirect

Maximal subgroups

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

subgroup	index
M₁₆×Dic₂₆	2
C₂×#11718340	2
#13115611	2
#13115610	2
#13115009	2
#13115008	2
#13114744	2
C₄×C₁₃×#108	2
#22733	13