Subgroups — #15881923

Order 3328.

642 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	M₁₆×Dic₂₆	C₂	1	2	yes	maximal
2	C₂×C₈×Dic₂₆	C₂	1	1	yes	maximal, characteristic
3	#13115611	C₂	1	2	yes	maximal
4	#13115608	C₂	1	4	yes	maximal
5	#13114974	C₂	1	1	yes	maximal, characteristic
6	#13114972	C₂	1	2	yes	maximal
7	#13114737	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	2	yes	—
11	C₄×C₁₃×M₁₆	C₂²	1	2	yes	—
12	C₂×C₄×C₈×C₁₃	C₂²	1	1	yes	characteristic
13	C₂×#88062	C₂²	1	4	yes	—
14	#11718779	C₂²	1	4	yes	—
15	M₁₆×Dic₁₃	C₂²	1	4	yes	—
16	C₂×#88054	C₂²	1	2	yes	—
17	C₂×C₈×Dic₁₃	C₂²	1	2	yes	—
18	#11718585	C₂²	1	2	yes	—
19	C₂×C₄⋊#1055	C₂²	1	1	yes	characteristic
20	#11718362	—	2	4	no	—
21	#11718340	—	2	2	no	—
22	C₈×Dic₂₆	—	2	2	no	—
23	C₈×Dic₂₆	C₄	1	4	yes	—
24	C₈×Dic₂₆	C₄	1	2	yes	—
25	C₈×Dic₂₆	C₄	1	2	yes	—
26	C₁₃×#341	C₂²	1	1	yes	characteristic
27	C₁₃×#341	C₂²	1	1	yes	characteristic
28	#11718150	C₂²	1	4	yes	—
29	#11718144	C₂²	1	2	yes	—
30	#11718144	C₂²	1	2	yes	—
31	C₂²×Dic₂₆	Q₈	1	1	yes	characteristic
32	C₂×C₁₃×M₁₆	C₂³	1	2	yes	—
33	C₂×C₁₃×M₁₆	C₂³	1	2	yes	—
34	C₂²×C₈×C₁₃	C₂³	1	1	yes	characteristic
35	C₂²×C₈×C₁₃	C₂³	1	1	yes	characteristic
36	C₂×C₄²×C₁₃	C₂³	1	1	yes	characteristic
37	C₂×C₄⋊Dic₁₃	Q₈	1	1	yes	characteristic
38	C₂×#6101	Q₈	1	2	yes	—
39	C₂×C₄×Dic₁₃	C₂³	1	2	yes	—
40	C₂×#6099	C₂³	1	4	yes	—
41	C₂²×#1055	C₂³	1	2	yes	—
42	C₄×Dic₂₆	C₂×C₄	1	1	yes	characteristic
43	C₄×Dic₂₆	C₂×C₄	1	2	yes	—
44	C₄×Dic₂₆	C₂×C₄	1	1	yes	characteristic
45	C₄×Dic₂₆	D₄	1	1	yes	characteristic
46	C₄×Dic₂₆	D₄	1	1	yes	characteristic
47	C₄×Dic₂₆	D₄	1	2	yes	—
48	C₁₃×#108	—	2	4	no	—
49	C₁₃×#97	—	2	2	no	—
50	C₄×C₈×C₁₃	—	2	2	no	—
51	C₄×C₈×C₁₃	C₂×C₄	1	2	yes	—
52	C₄×C₈×C₁₃	C₂×C₄	1	2	yes	—
53	#88062	—	2	8	no	—
54	#88055	—	2	4	no	—
55	#88054	—	2	4	no	—
56	#88054	—	2	4	no	—
57	#88054	C₂×C₄	1	4	yes	—
58	#88054	C₂×C₄	1	4	yes	—
59	C₈×Dic₁₃	—	2	4	no	—
60	C₈×Dic₁₃	C₂×C₄	1	4	yes	—
61	C₈×Dic₁₃	C₂×C₄	1	4	yes	—
62	C₄⋊#1055	—	2	2	no	—
63	C₄⋊#1055	—	2	2	no	—
64	C₄⋊#1055	C₂×C₄	1	2	yes	—
65	C₄⋊#1055	C₂×C₄	1	2	yes	—
66	#22746	—	13	1	no	maximal, Sylow
67	C₂²×C₄×C₁₃	C₂×Q₈	1	1	yes	characteristic
68	C₂²×C₄×C₁₃	C₂×Q₈	1	1	yes	characteristic
69	C₂²×C₄×C₁₃	C₂⁴	1	1	yes	characteristic
70	C₂²×Dic₁₃	C₂×Q₈	1	2	yes	—
71	C₂²×Dic₁₃	C₂×Q₈	1	2	yes	—
72	C₂×Dic₂₆	—	2	4	no	—
73	C₂×Dic₂₆	C₄⋊C₄	1	1	yes	characteristic
74	C₂×Dic₂₆	C₄⋊C₄	1	2	yes	—
75	C₂×Dic₂₆	C₄⋊C₄	1	1	yes	characteristic
76	C₁₃×M₁₆	—	2	2	no	—
77	C₁₃×M₁₆	—	2	2	no	—
78	C₁₃×M₁₆	—	2	2	no	—
79	C₁₃×M₁₆	#41	1	4	yes	—
80	C₂×C₈×C₁₃	—	2	2	no	—
81	C₂×C₈×C₁₃	—	2	2	no	—
82	C₂×C₈×C₁₃	—	2	2	no	—
83	C₂×C₈×C₁₃	—	2	2	no	—
84	C₂×C₈×C₁₃	C₂²×C₄	1	2	yes	—
85	C₂×C₈×C₁₃	C₂²×C₄	1	2	yes	—
86	C₄²×C₁₃	C₂²×C₄	1	1	yes	characteristic
87	C₄²×C₁₃	C₂²×C₄	1	1	yes	characteristic
88	C₄²×C₁₃	C₂×D₄	1	1	yes	characteristic
89	C₄²×C₁₃	C₂×D₄	1	1	yes	characteristic
90	C₄⋊Dic₁₃	C₄⋊C₄	1	2	yes	—
91	C₄⋊Dic₁₃	C₄⋊C₄	1	1	yes	characteristic
92	C₄⋊Dic₁₃	C₄⋊C₄	1	1	yes	characteristic
93	#6101	C₄⋊C₄	1	2	yes	—
94	#6101	C₄⋊C₄	1	2	yes	—
95	#6101	C₄⋊C₄	1	2	yes	—
96	#6101	C₄⋊C₄	1	2	yes	—
97	C₄×Dic₁₃	C₂²×C₄	1	2	yes	—
98	C₄×Dic₁₃	C₂²×C₄	1	2	yes	—
99	C₄×Dic₁₃	C₂×D₄	1	2	yes	—
100	C₄×Dic₁₃	C₂×D₄	1	2	yes	—
101	#6099	—	2	4	no	—
102	#6099	—	2	4	no	—
103	C₂×#1055	—	2	4	no	—
104	C₂×#1055	—	2	4	no	—
105	C₂×#1055	C₂²×C₄	1	4	yes	—
106	Q₈×M₁₆	—	13	2	no	—
107	C₂×C₈×Q₈	—	13	1	no	—
108	#1950	—	13	4	no	—
109	#1950	—	13	2	no	—
110	#1837	—	13	1	no	—
111	#1837	—	13	2	no	—
112	C₄×#108	—	13	2	no	—
113	C₄×#108	D₁₃	1	1	yes	characteristic
114	C₂³×C₁₃	C₂²×Q₈	1	1	yes	characteristic
115	C₂×C₄×C₁₃	—	2	2	no	—
116	C₂×C₄×C₁₃	—	4	1	no	—
117	C₂×C₄×C₁₃	C₂×C₄⋊C₄	1	1	yes	characteristic
118	C₂×C₄×C₁₃	C₂×C₄⋊C₄	1	1	yes	characteristic
119	C₂×C₄×C₁₃	C₂×C₄⋊C₄	1	1	yes	characteristic
120	C₂×C₄×C₁₃	C₂×C₄⋊C₄	1	1	yes	characteristic
121	C₂×C₄×C₁₃	C₂³×C₄	1	1	yes	characteristic
122	C₂×C₄×C₁₃	C₂²×D₄	1	1	yes	characteristic
123	C₂×C₄×C₁₃	C₂×#41	1	2	yes	—
124	C₂×C₄×C₁₃	2₋¹⁺⁴	1	2	yes	—
125	C₂×Dic₁₃	—	2	4	no	—
126	C₂×Dic₁₃	—	4	2	no	—
127	C₂×Dic₁₃	C₂×C₄⋊C₄	1	2	yes	—
128	C₂×Dic₁₃	C₂×C₄⋊C₄	1	2	yes	—
129	C₂×Dic₁₃	C₂×C₄⋊C₄	1	2	yes	—
130	C₂×Dic₁₃	C₂×C₄⋊C₄	1	2	yes	—
131	Dic₂₆	—	2	2	no	—
132	Dic₂₆	—	2	4	no	—
133	Dic₂₆	#108	1	4	yes	complemented
134	C₈×C₁₃	—	2	2	no	—
135	C₈×C₁₃	—	2	2	no	—
136	C₈×C₁₃	—	2	4	no	—
137	C₈×C₁₃	—	4	2	no	—
138	#1055	—	2	4	no	—
139	#1055	—	4	4	no	—
140	C₂×C₄×Q₈	—	13	1	no	—
141	#446	—	26	2	no	—
142	#446	—	26	4	no	—
143	C₈×Q₈	—	13	2	no	—
144	C₈×Q₈	—	13	4	no	—
145	C₈×Q₈	—	13	2	no	—
146	C₈×Q₈	—	26	2	no	—
147	C₂×#108	—	13	4	no	—
148	C₂×#108	C₂×D₁₃	1	2	yes	—
149	#423	—	13	4	no	—
150	#423	—	13	2	no	—
151	C₂×C₄⋊C₈	—	13	2	no	—
152	C₂×C₄⋊C₈	—	13	1	no	—
153	C₄×M₁₆	—	13	4	no	—
154	C₄×M₁₆	C₂×D₁₃	1	2	yes	—
155	C₂×C₄×C₈	—	13	2	no	—
156	C₂×C₄×C₈	C₂×D₁₃	1	1	yes	characteristic
157	#341	—	13	4	no	—
158	#341	—	13	2	no	—
159	#341	—	13	2	no	—
160	#341	C₂×D₁₃	1	1	yes	characteristic
161	#341	C₂×D₁₃	1	1	yes	characteristic
162	C₂²×C₁₃	—	2	1	no	—
163	C₂²×C₁₃	—	2	1	no	—
164	C₂²×C₁₃	C₂²×C₄⋊C₄	1	1	yes	characteristic
165	C₂²×C₁₃	#557	1	2	yes	—
166	C₄×C₁₃	—	2	1	no	—
167	C₄×C₁₃	—	2	1	no	—
168	C₄×C₁₃	—	2	1	no	—
169	C₄×C₁₃	C₂×#108	1	2	yes	—
170	C₄×C₁₃	C₄×#41	1	1	yes	characteristic
171	C₄×C₁₃	#519	1	1	yes	characteristic
172	C₄×C₁₃	#548	1	2	yes	—
173	Dic₁₃	—	2	2	no	—
174	Dic₁₃	—	2	2	no	—
175	Dic₁₃	—	2	2	no	—
176	Dic₁₃	C₂×#108	1	4	yes	—
177	C₂²×Q₈	—	13	1	no	—
178	C₂×M₁₆	—	13	4	no	—
179	C₂×M₁₆	C₂²×D₁₃	1	2	yes	—
180	C₂×M₁₆	C₂²×D₁₃	1	2	yes	—
181	C₂²×C₈	—	13	2	no	—
182	C₂²×C₈	C₂²×D₁₃	1	1	yes	characteristic
183	C₂²×C₈	C₂²×D₁₃	1	1	yes	characteristic
184	C₄×Q₈	—	13	1	no	—
185	C₄×Q₈	—	13	2	no	—
186	C₄×Q₈	—	13	1	no	—
187	C₄×Q₈	—	13	1	no	—
188	C₄×Q₈	—	13	2	no	—
189	C₄×Q₈	—	13	1	no	—
190	C₂×C₄⋊C₄	—	13	2	no	—
191	C₂×C₄⋊C₄	—	13	1	no	—
192	C₂×C₄²	—	13	2	no	—
193	C₂×C₄²	C₂²×D₁₃	1	1	yes	characteristic
194	#108	—	2	4	no	—
195	#108	—	26	8	no	—
196	C₄⋊C₈	—	13	4	no	—
197	C₄⋊C₈	—	13	4	no	—
198	C₄⋊C₈	—	13	2	no	—
199	C₄⋊C₈	—	13	2	no	—
200	C₄⋊C₈	—	26	4	no	—
201	C₄⋊C₈	—	26	4	no	—
202	C₄⋊C₈	—	26	2	no	—
203	C₄⋊C₈	—	26	2	no	—
204	#97	—	2	2	no	—
205	#97	—	26	4	no	—
206	C₄×C₈	—	2	2	no	—
207	C₄×C₈	—	13	4	no	—
208	C₄×C₈	—	13	4	no	—
209	C₄×C₈	—	26	4	no	—
210	C₄×C₈	C₄×D₁₃	1	2	yes	—
211	C₄×C₈	C₄×D₁₃	1	2	yes	—
212	C₂×C₁₃	—	2	2	no	—
213	C₂×C₁₃	#2351	1	1	yes	characteristic
214	C₂×C₁₃	C₂²×#108	1	1	yes	characteristic
215	C₂×C₁₃	#2916	1	1	yes	characteristic
216	C₂×Q₈	—	13	1	no	—
217	C₂×Q₈	—	13	2	no	—
218	C₂×Q₈	—	13	1	no	—
219	C₂×Q₈	—	26	4	no	—
220	C₂²×C₄	—	13	2	no	—
221	C₂²×C₄	—	13	2	no	—
222	C₂²×C₄	Q₈×D₁₃	1	1	yes	characteristic
223	C₂²×C₄	Q₈×D₁₃	1	1	yes	characteristic
224	C₂²×C₄	C₂³×D₁₃	1	1	yes	characteristic
225	M₁₆	—	2	2	no	—
226	M₁₆	—	2	2	no	—
227	M₁₆	—	2	2	no	—
228	M₁₆	—	26	4	no	—
229	M₁₆	—	26	4	no	—
230	M₁₆	#6127	1	4	yes	—
231	C₂×C₈	—	2	2	no	—
232	C₂×C₈	—	2	2	no	—
233	C₂×C₈	—	2	2	no	—
234	C₂×C₈	—	2	2	no	—
235	C₂×C₈	—	13	4	no	—
236	C₂×C₈	—	26	4	no	—
237	C₂×C₈	—	26	4	no	—
238	C₂×C₈	C₂×C₄×D₁₃	1	2	yes	—
239	C₂×C₈	C₂×C₄×D₁₃	1	2	yes	—
240	C₄⋊C₄	—	13	2	no	—
241	C₄⋊C₄	—	13	2	no	—
242	C₄⋊C₄	—	13	2	no	—
243	C₄⋊C₄	—	13	2	no	—
244	C₄⋊C₄	—	13	2	no	—
245	C₄⋊C₄	—	13	1	no	—
246	C₄⋊C₄	—	13	1	no	—
247	C₄²	—	13	2	no	—
248	C₄²	—	13	2	no	—
249	C₄²	—	13	2	no	—
250	C₄²	—	13	2	no	—
251	C₄²	C₂×C₄×D₁₃	1	1	yes	characteristic
252	C₄²	C₂×C₄×D₁₃	1	1	yes	characteristic
253	C₄²	D₄×D₁₃	1	1	yes	characteristic
254	C₄²	D₄×D₁₃	1	1	yes	characteristic
255	C₁₃	#22746	1	1	yes	characteristic, complemented, Sylow
256	C₂³	C₂×Q₈×D₁₃	1	1	yes	characteristic
257	Q₈	—	13	4	no	—
258	Q₈	—	26	4	no	—
259	Q₈	—	26	2	no	—
260	C₂×C₄	—	2	2	no	—
261	C₂×C₄	—	4	1	no	—
262	C₂×C₄	—	13	2	no	—
263	C₂×C₄	—	13	2	no	—
264	C₂×C₄	—	13	2	no	—
265	C₂×C₄	—	13	2	no	—
266	C₂×C₄	—	26	4	no	—
267	C₂×C₄	—	52	2	no	—
268	C₂×C₄	C₄⋊C₄×D₁₃	1	1	yes	characteristic
269	C₂×C₄	C₄⋊C₄×D₁₃	1	1	yes	characteristic
270	C₂×C₄	C₄⋊C₄×D₁₃	1	1	yes	characteristic
271	C₂×C₄	C₄⋊C₄×D₁₃	1	1	yes	characteristic
272	C₂×C₄	C₂²×C₄×D₁₃	1	1	yes	characteristic
273	C₂×C₄	C₂×#6127	1	2	yes	—
274	C₂×C₄	C₂×D₄×D₁₃	1	1	yes	characteristic
275	C₂×C₄	#88257	1	2	yes	—
276	C₈	—	2	2	no	—
277	C₈	—	2	2	no	—
278	C₈	—	2	4	no	—
279	C₈	—	4	2	no	—
280	C₈	—	26	4	no	—
281	C₈	—	52	4	no	—
282	C₂²	—	2	1	no	—
283	C₂²	—	2	1	no	—
284	C₂²	C₂×C₄⋊C₄×D₁₃	1	1	yes	characteristic
285	C₂²	#11719272	1	2	yes	—
286	C₄	—	2	1	no	—
287	C₄	—	2	1	no	—
288	C₄	—	2	1	no	—
289	C₄	—	13	4	no	—
290	C₄	—	26	2	no	—
291	C₄	—	26	2	no	—
292	C₄	—	26	2	no	—
293	C₄	D₁₃×#108	1	2	yes	—
294	C₄	C₄×#6127	1	1	yes	characteristic
295	C₄	#11719221	1	1	yes	characteristic
296	C₄	#11719257	1	2	yes	—
297	C₂	—	2	2	no	—
298	C₂	#13117756	1	1	yes	characteristic
299	C₂	C₂×D₁₃×#108	1	1	yes	characteristic
300	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	#22746
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
133. Dic₂₆	#108	semidirect
255. C₁₃	#22746	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×C₈×Dic₂₆	2
#13115611	2
#13115608	2
#13114974	2
#13114972	2
#13114737	2
C₄×C₁₃×#108	2
#22746	13