Subgroups — #15881808

Order 3328.

658 conjugacy classes of subgroups.

The subgroup lattice has 324 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#13123787	C₂	1	2	yes	maximal
2	C₂×C₈×Dic₂₆	C₂	1	1	yes	maximal, characteristic
3	#13115609	C₂	1	2	yes	maximal
4	#13115608	C₂	1	4	yes	maximal
5	#13115006	C₂	1	2	yes	maximal
6	#108×Dic₁₃	C₂	1	2	yes	maximal
7	C₄×#88059	C₂	1	1	yes	maximal, characteristic
8	C₁₃×#1848	C₂	1	1	yes	maximal, characteristic
9	C₂×C₄×Dic₂₆	C₂²	1	1	yes	characteristic
10	C₂×C₁₃×#108	C₂²	1	2	yes	—
11	C₂×C₁₃×#108	C₂²	1	2	yes	—
12	C₁₃×#423	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	1	yes	characteristic
18	C₂×#88059	C₂²	1	1	yes	characteristic
19	C₂×#88054	C₂²	1	2	yes	—
20	C₂×C₈×Dic₁₃	C₂²	1	2	yes	—
21	C₂×C₄⋊#1055	C₂²	1	1	yes	characteristic
22	C₄×#6099	C₂²	1	2	yes	—
23	#11718492	—	2	2	no	—
24	#11718491	—	2	2	no	—
25	#11718489	—	2	4	no	—
26	C₈×Dic₂₆	C₄	1	4	yes	—
27	C₈×Dic₂₆	C₄	1	2	yes	—
28	C₈×Dic₂₆	C₄	1	2	yes	—
29	#11718150	C₂²	1	4	yes	—
30	#11718144	C₂²	1	2	yes	—
31	C₂²×Dic₂₆	Q₈	1	1	yes	characteristic
32	C₂×C₁₃×M₁₆	C₂³	1	4	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	2	yes	—
40	C₂×#6099	C₂³	1	2	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	2	yes	—
46	C₄×Dic₂₆	D₄	1	1	yes	characteristic
47	C₄×Dic₂₆	D₄	1	1	yes	characteristic
48	C₁₃×#108	—	2	4	no	—
49	C₁₃×#108	—	4	2	no	—
50	C₁₃×C₄⋊C₈	—	2	2	no	—
51	C₁₃×C₄⋊C₈	—	2	2	no	—
52	C₄×C₈×C₁₃	C₂×C₄	1	2	yes	—
53	C₄×C₈×C₁₃	C₂×C₄	1	2	yes	—
54	#88062	—	4	4	no	—
55	#88059	—	2	2	no	—
56	#88059	Q₈	1	4	yes	complemented
57	#88055	—	2	4	no	—
58	#88054	—	2	4	no	—
59	#88054	—	2	4	no	—
60	#88054	C₂×C₄	1	4	yes	—
61	#88054	C₂×C₄	1	4	yes	—
62	C₈×Dic₁₃	—	2	4	no	—
63	C₈×Dic₁₃	C₂×C₄	1	4	yes	—
64	C₈×Dic₁₃	C₂×C₄	1	4	yes	—
65	C₄⋊#1055	C₂×C₄	1	2	yes	—
66	C₄⋊#1055	C₂×C₄	1	2	yes	—
67	#88043	—	2	2	no	—
68	C₄×#1055	—	2	2	no	—
69	Q₈×#108	—	13	1	no	maximal, Sylow
70	C₂²×C₄×C₁₃	C₂×Q₈	1	1	yes	characteristic
71	C₂²×C₄×C₁₃	C₂×Q₈	1	1	yes	characteristic
72	C₂²×C₄×C₁₃	C₂⁴	1	1	yes	characteristic
73	C₂²×Dic₁₃	C₂×Q₈	1	2	yes	—
74	C₂²×Dic₁₃	C₂×Q₈	1	2	yes	—
75	C₂×Dic₂₆	—	2	4	no	—
76	C₂×Dic₂₆	C₄⋊C₄	1	1	yes	characteristic
77	C₂×Dic₂₆	C₄⋊C₄	1	2	yes	—
78	C₂×Dic₂₆	C₄⋊C₄	1	1	yes	characteristic
79	C₁₃×M₁₆	—	2	4	no	—
80	C₁₃×M₁₆	—	2	4	no	—
81	C₂×C₈×C₁₃	—	2	1	no	—
82	C₂×C₈×C₁₃	—	2	1	no	—
83	C₂×C₈×C₁₃	—	2	4	no	—
84	C₂×C₈×C₁₃	C₂²×C₄	1	2	yes	—
85	C₂×C₈×C₁₃	C₂²×C₄	1	2	yes	—
86	C₂×C₈×C₁₃	C₂×Q₈	1	2	yes	—
87	C₂×C₈×C₁₃	#41	1	2	yes	—
88	C₄²×C₁₃	C₂²×C₄	1	1	yes	characteristic
89	C₄²×C₁₃	C₂²×C₄	1	1	yes	characteristic
90	C₄²×C₁₃	C₂×D₄	1	1	yes	characteristic
91	C₄²×C₁₃	C₂×D₄	1	1	yes	characteristic
92	C₄⋊Dic₁₃	C₄⋊C₄	1	2	yes	—
93	C₄⋊Dic₁₃	C₄⋊C₄	1	1	yes	characteristic
94	C₄⋊Dic₁₃	C₄⋊C₄	1	1	yes	characteristic
95	#6101	C₄⋊C₄	1	2	yes	—
96	#6101	C₄⋊C₄	1	2	yes	—
97	#6101	C₄⋊C₄	1	2	yes	—
98	#6101	C₄⋊C₄	1	2	yes	—
99	C₄×Dic₁₃	C₂²×C₄	1	2	yes	—
100	C₄×Dic₁₃	C₂²×C₄	1	2	yes	—
101	C₄×Dic₁₃	C₂×D₄	1	2	yes	—
102	C₄×Dic₁₃	C₂×D₄	1	2	yes	—
103	#6099	—	2	2	no	—
104	#6099	—	2	2	no	—
105	#6099	—	2	2	no	—
106	#6099	C₂×Q₈	1	4	yes	—
107	C₂×#1055	—	2	2	no	—
108	C₂×#1055	—	2	2	no	—
109	C₂×#1055	—	2	2	no	—
110	C₂×#1055	—	2	2	no	—
111	C₂×#1055	C₂²×C₄	1	4	yes	—
112	Q₈×M₁₆	—	13	2	no	—
113	C₂×C₈×Q₈	—	13	1	no	—
114	#1950	—	13	4	no	—
115	#1950	—	13	2	no	—
116	#1848	—	13	2	no	—
117	#1848	D₁₃	1	1	yes	characteristic
118	C₄×#108	—	13	2	no	—
119	C₄×#108	—	13	1	no	—
120	C₂³×C₁₃	C₂²×Q₈	1	1	yes	characteristic
121	C₂×C₄×C₁₃	—	2	2	no	—
122	C₂×C₄×C₁₃	—	4	1	no	—
123	C₂×C₄×C₁₃	C₂×C₄⋊C₄	1	1	yes	characteristic
124	C₂×C₄×C₁₃	C₂×C₄⋊C₄	1	1	yes	characteristic
125	C₂×C₄×C₁₃	C₂×C₄⋊C₄	1	1	yes	characteristic
126	C₂×C₄×C₁₃	C₂×C₄⋊C₄	1	1	yes	characteristic
127	C₂×C₄×C₁₃	C₂³×C₄	1	1	yes	characteristic
128	C₂×C₄×C₁₃	C₂²×D₄	1	1	yes	characteristic
129	C₂×C₄×C₁₃	C₂²×Q₈	1	1	yes	characteristic
130	C₂×C₄×C₁₃	C₂×#41	1	1	yes	characteristic
131	C₂×C₄×C₁₃	H(5,𝔽₂)	1	1	yes	characteristic
132	C₂×C₄×C₁₃	2₋¹⁺⁴	1	1	yes	characteristic
133	C₂×Dic₁₃	—	2	4	no	—
134	C₂×Dic₁₃	—	4	2	no	—
135	C₂×Dic₁₃	C₂×C₄⋊C₄	1	2	yes	—
136	C₂×Dic₁₃	C₂×C₄⋊C₄	1	2	yes	—
137	C₂×Dic₁₃	C₂×C₄⋊C₄	1	2	yes	—
138	C₂×Dic₁₃	C₂×C₄⋊C₄	1	2	yes	—
139	Dic₂₆	—	2	2	no	—
140	Dic₂₆	—	2	4	no	—
141	Dic₂₆	#108	1	4	yes	complemented
142	C₈×C₁₃	—	2	2	no	—
143	C₈×C₁₃	—	4	4	no	—
144	C₈×C₁₃	C₄×Q₈	1	4	yes	—
145	#1055	—	2	4	no	—
146	#1055	—	2	4	no	—
147	#1055	—	4	2	no	—
148	C₂×C₄×Q₈	—	13	1	no	—
149	#446	—	26	4	no	—
150	#446	—	26	2	no	—
151	C₈×Q₈	—	13	2	no	—
152	C₈×Q₈	—	13	4	no	—
153	C₈×Q₈	—	13	2	no	—
154	C₈×Q₈	—	26	2	no	—
155	C₂×#108	—	13	1	no	—
156	C₂×#108	—	13	1	no	—
157	C₂×#108	—	13	4	no	—
158	C₂×#108	C₂×D₁₃	1	2	yes	—
159	C₂×#108	C₂×D₁₃	1	2	yes	—
160	#423	—	13	4	no	—
161	#423	C₂×D₁₃	1	2	yes	—
162	C₂×C₄⋊C₈	—	13	2	no	—
163	C₂×C₄⋊C₈	—	13	1	no	—
164	C₄×M₁₆	—	13	4	no	—
165	C₄×M₁₆	—	13	2	no	—
166	C₂×C₄×C₈	—	13	2	no	—
167	C₂×C₄×C₈	C₂×D₁₃	1	1	yes	characteristic
168	#341	—	13	4	no	—
169	#341	—	13	2	no	—
170	C₂²×C₁₃	—	2	1	no	—
171	C₂²×C₁₃	—	2	1	no	—
172	C₂²×C₁₃	C₂²×C₄⋊C₄	1	1	yes	characteristic
173	C₂²×C₁₃	#557	1	1	yes	characteristic
174	C₂²×C₁₃	Q₈²	1	1	yes	characteristic
175	C₄×C₁₃	—	2	1	no	—
176	C₄×C₁₃	—	2	1	no	—
177	C₄×C₁₃	—	2	1	no	—
178	C₄×C₁₃	C₂×#108	1	2	yes	—
179	C₄×C₁₃	C₂×C₄×Q₈	1	1	yes	characteristic
180	C₄×C₁₃	#520	1	1	yes	characteristic
181	C₄×C₁₃	D₄×Q₈	1	1	yes	characteristic
182	C₄×C₁₃	#550	1	1	yes	characteristic
183	Dic₁₃	—	2	2	no	—
184	Dic₁₃	—	2	2	no	—
185	Dic₁₃	—	2	2	no	—
186	Dic₁₃	C₂×#108	1	4	yes	—
187	C₂²×Q₈	—	13	1	no	—
188	C₂×M₁₆	—	13	2	no	—
189	C₂×M₁₆	—	13	2	no	—
190	C₂×M₁₆	C₂²×D₁₃	1	4	yes	—
191	C₂²×C₈	—	13	2	no	—
192	C₂²×C₈	C₂²×D₁₃	1	1	yes	characteristic
193	C₂²×C₈	C₂²×D₁₃	1	1	yes	characteristic
194	C₄×Q₈	—	13	1	no	—
195	C₄×Q₈	—	13	2	no	—
196	C₄×Q₈	—	13	1	no	—
197	C₄×Q₈	—	13	1	no	—
198	C₄×Q₈	—	13	2	no	—
199	C₄×Q₈	—	13	1	no	—
200	C₂×C₄⋊C₄	—	13	2	no	—
201	C₂×C₄⋊C₄	—	13	1	no	—
202	C₂×C₄²	—	13	2	no	—
203	C₂×C₄²	C₂²×D₁₃	1	1	yes	characteristic
204	#108	—	2	4	no	—
205	#108	—	4	2	no	—
206	#108	—	13	4	no	—
207	#108	—	26	2	no	—
208	#108	—	52	4	no	—
209	C₄⋊C₈	—	2	2	no	—
210	C₄⋊C₈	—	2	2	no	—
211	C₄⋊C₈	—	13	4	no	—
212	C₄⋊C₈	—	13	4	no	—
213	C₄⋊C₈	—	13	2	no	—
214	C₄⋊C₈	—	13	2	no	—
215	C₄⋊C₈	—	26	4	no	—
216	C₄⋊C₈	—	26	4	no	—
217	#97	—	26	4	no	—
218	#97	—	26	2	no	—
219	C₄×C₈	—	13	4	no	—
220	C₄×C₈	—	13	4	no	—
221	C₄×C₈	—	26	4	no	—
222	C₄×C₈	—	26	2	no	—
223	C₄×C₈	C₄×D₁₃	1	2	yes	—
224	C₄×C₈	C₄×D₁₃	1	2	yes	—
225	C₂×C₁₃	—	2	1	no	—
226	C₂×C₁₃	—	2	1	no	—
227	C₂×C₁₃	Q₈×C₄⋊C₄	1	1	yes	characteristic
228	C₂×C₁₃	C₂²×#108	1	1	yes	characteristic
229	C₂×C₁₃	#2916	1	1	yes	characteristic
230	C₂×Q₈	—	13	1	no	—
231	C₂×Q₈	—	13	2	no	—
232	C₂×Q₈	—	13	1	no	—
233	C₂×Q₈	—	26	4	no	—
234	C₂²×C₄	—	13	2	no	—
235	C₂²×C₄	—	13	2	no	—
236	C₂²×C₄	Q₈×D₁₃	1	1	yes	characteristic
237	C₂²×C₄	Q₈×D₁₃	1	1	yes	characteristic
238	C₂²×C₄	C₂³×D₁₃	1	1	yes	characteristic
239	M₁₆	—	2	4	no	—
240	M₁₆	—	2	4	no	—
241	M₁₆	—	13	4	no	—
242	M₁₆	—	26	2	no	—
243	M₁₆	—	26	2	no	—
244	M₁₆	—	26	2	no	—
245	C₂×C₈	—	2	1	no	—
246	C₂×C₈	—	2	1	no	—
247	C₂×C₈	—	2	4	no	—
248	C₂×C₈	—	13	4	no	—
249	C₂×C₈	—	26	2	no	—
250	C₂×C₈	—	26	2	no	—
251	C₂×C₈	—	26	2	no	—
252	C₂×C₈	—	26	2	no	—
253	C₂×C₈	C₂×C₄×D₁₃	1	2	yes	—
254	C₂×C₈	C₂×C₄×D₁₃	1	2	yes	—
255	C₂×C₈	#6129	1	2	yes	—
256	C₂×C₈	Q₈×D₁₃	1	2	yes	—
257	C₄⋊C₄	—	13	2	no	—
258	C₄⋊C₄	—	13	2	no	—
259	C₄⋊C₄	—	13	2	no	—
260	C₄⋊C₄	—	13	2	no	—
261	C₄⋊C₄	—	13	2	no	—
262	C₄⋊C₄	—	13	1	no	—
263	C₄⋊C₄	—	13	1	no	—
264	C₄²	—	13	2	no	—
265	C₄²	—	13	2	no	—
266	C₄²	—	13	2	no	—
267	C₄²	—	13	2	no	—
268	C₄²	C₂×C₄×D₁₃	1	1	yes	characteristic
269	C₄²	C₂×C₄×D₁₃	1	1	yes	characteristic
270	C₄²	D₄×D₁₃	1	1	yes	characteristic
271	C₄²	D₄×D₁₃	1	1	yes	characteristic
272	C₁₃	Q₈×#108	1	1	yes	characteristic, complemented, Sylow
273	C₂³	C₂×Q₈×D₁₃	1	1	yes	characteristic
274	Q₈	—	13	4	no	—
275	Q₈	—	26	4	no	—
276	Q₈	—	26	2	no	—
277	C₂×C₄	—	2	2	no	—
278	C₂×C₄	—	4	1	no	—
279	C₂×C₄	—	13	2	no	—
280	C₂×C₄	—	13	2	no	—
281	C₂×C₄	—	13	2	no	—
282	C₂×C₄	—	13	2	no	—
283	C₂×C₄	—	26	4	no	—
284	C₂×C₄	—	52	2	no	—
285	C₂×C₄	C₄⋊C₄×D₁₃	1	1	yes	characteristic
286	C₂×C₄	C₄⋊C₄×D₁₃	1	1	yes	characteristic
287	C₂×C₄	C₄⋊C₄×D₁₃	1	1	yes	characteristic
288	C₂×C₄	C₄⋊C₄×D₁₃	1	1	yes	characteristic
289	C₂×C₄	C₂²×C₄×D₁₃	1	1	yes	characteristic
290	C₂×C₄	C₂×D₄×D₁₃	1	1	yes	characteristic
291	C₂×C₄	C₂×#6129	1	1	yes	characteristic
292	C₂×C₄	#88251	1	1	yes	characteristic
293	C₂×C₄	C₂×Q₈×D₁₃	1	1	yes	characteristic
294	C₂×C₄	#88254	1	1	yes	characteristic
295	C₈	—	2	2	no	—
296	C₈	—	4	4	no	—
297	C₈	—	26	4	no	—
298	C₈	—	26	4	no	—
299	C₈	—	52	2	no	—
300	C₈	#88141	1	4	yes	—
301	C₂²	—	2	1	no	—
302	C₂²	—	2	1	no	—
303	C₂²	C₂×C₄⋊C₄×D₁₃	1	1	yes	characteristic
304	C₂²	#11719426	1	1	yes	characteristic
305	C₂²	#11719427	1	1	yes	characteristic
306	C₄	—	2	1	no	—
307	C₄	—	2	1	no	—
308	C₄	—	2	1	no	—
309	C₄	—	13	4	no	—
310	C₄	—	26	2	no	—
311	C₄	—	26	2	no	—
312	C₄	—	26	2	no	—
313	C₄	D₁₃×#108	1	2	yes	—
314	C₄	C₂×#88141	1	1	yes	characteristic
315	C₄	#11719209	1	1	yes	characteristic
316	C₄	#11719423	1	1	yes	characteristic
317	C₄	#11719432	1	1	yes	characteristic
318	C₂	—	2	1	no	—
319	C₂	—	2	1	no	—
320	C₂	#13117748	1	1	yes	characteristic
321	C₂	C₂×D₁₃×#108	1	1	yes	characteristic
322	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	C₁₃×#1848	C₂
socle	C₂²×C₁₃	C₂²×C₄⋊C₄
prime radical	C₂³×C₁₃	C₂²×Q₈
cosocle	C₂²×C₄×C₁₃	C₂⁴
hypercenter	#1848	D₁₃
hypocenter	C₁₃	Q₈×#108
LCS	G ▷ C₂×C₄×C₁₃ ▷ C₂×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #1848

Complements

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

N	H ≅ G/N	type
56. #88059	Q₈	semidirect
141. Dic₂₆	#108	semidirect
272. C₁₃	Q₈×#108	semidirect

Maximal subgroups

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

subgroup	index
#13123787	2
C₂×C₈×Dic₂₆	2
#13115609	2
#13115608	2
#13115006	2
#108×Dic₁₃	2
C₄×#88059	2
C₁₃×#1848	2
Q₈×#108	13