Subgroups — #106×Dic₂₆ #15881745

Order 3328.

838 conjugacy classes of subgroups.

The subgroup lattice has 294 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₈×Dic₂₆	C₂	1	1	yes	maximal, characteristic
2	C₄⋊C₄×Dic₂₆	C₂	1	2	yes	maximal
3	#13115599	C₂	1	2	yes	maximal
4	#13115598	C₂	1	4	yes	maximal
5	#13114993	C₂	1	1	yes	maximal, characteristic
6	#13114991	C₂	1	2	yes	maximal
7	#106×Dic₁₃	C₂	1	2	yes	maximal
8	C₄×C₁₃×#106	C₂	1	1	yes	maximal, characteristic
9	C₂×C₄×Dic₂₆	—	2	2	no	—
10	C₂×C₄×Dic₂₆	C₂²	1	1	yes	characteristic
11	C₂×C₁₃×#106	C₂²	1	1	yes	characteristic
12	C₂×C₁₃×#106	C₂²	1	1	yes	characteristic
13	C₂×C₄×C₈×C₁₃	C₂²	1	1	yes	characteristic
14	C₄×C₁₃×C₄⋊C₄	C₂²	1	2	yes	—
15	C₂×#88057	C₂²	1	2	yes	—
16	C₂×#88057	C₂²	1	2	yes	—
17	C₂×#88054	C₂²	1	2	yes	—
18	C₂×C₈×Dic₁₃	C₂²	1	2	yes	—
19	#11718637	C₂²	1	4	yes	—
20	C₄⋊C₄×Dic₁₃	C₂²	1	4	yes	—
21	C₂×#88048	C₂²	1	4	yes	—
22	#11718596	C₂²	1	2	yes	—
23	#11718589	—	2	2	no	—
24	C₂×C₄⋊#1055	C₂²	1	1	yes	characteristic
25	C₈×Dic₂₆	C₄	1	8	yes	complemented
26	#11718308	—	2	4	no	—
27	C₁₃×#340	C₂²	1	2	yes	—
28	#11718121	C₂²	1	4	yes	—
29	C₂²×Dic₂₆	D₄	1	1	yes	characteristic
30	C₂²×C₈×C₁₃	C₂³	1	1	yes	characteristic
31	C₂²×C₈×C₁₃	C₂³	1	1	yes	characteristic
32	C₂×C₁₃×C₄⋊C₄	C₂³	1	2	yes	—
33	C₂×C₁₃×C₄⋊C₄	C₂³	1	2	yes	—
34	C₂×C₄²×C₁₃	—	2	2	no	—
35	C₂×C₄²×C₁₃	C₂³	1	1	yes	characteristic
36	C₂×C₄⋊Dic₁₃	—	2	2	no	—
37	C₂×C₄⋊Dic₁₃	—	2	2	no	—
38	C₂×C₄⋊Dic₁₃	D₄	1	1	yes	characteristic
39	C₂×C₄⋊Dic₁₃	C₂³	1	4	yes	—
40	C₂×#6101	—	2	4	no	—
41	C₂×#6101	—	2	4	no	—
42	C₂×#6101	D₄	1	2	yes	—
43	C₂×C₄×Dic₁₃	—	2	4	no	—
44	C₂×C₄×Dic₁₃	C₂³	1	2	yes	—
45	C₂²×#1055	C₂³	1	2	yes	—
46	C₄×Dic₂₆	—	4	8	no	—
47	C₄×Dic₂₆	C₂×C₄	1	4	yes	—
48	C₄×Dic₂₆	Q₈	1	4	yes	—
49	C₁₃×#106	—	2	2	no	—
50	C₁₃×#106	Q₈	1	4	yes	complemented
51	C₄×C₈×C₁₃	C₂×C₄	1	4	yes	—
52	C₁₃×#95	—	2	2	no	—
53	#88071	—	2	4	no	—
54	#88057	—	2	4	no	—
55	#88057	—	4	2	no	—
56	#88054	C₂×C₄	1	8	yes	—
57	C₈×Dic₁₃	C₂×C₄	1	8	yes	—
58	#88048	—	4	4	no	—
59	C₄⋊#1055	C₂×C₄	1	4	yes	—
60	Q₈×#106	—	13	1	no	maximal, Sylow
61	C₂²×C₄×C₁₃	—	2	2	no	—
62	C₂²×C₄×C₁₃	—	2	2	no	—
63	C₂²×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
64	C₂²×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
65	C₂²×C₄×C₁₃	C₂⁴	1	1	yes	characteristic
66	C₂²×Dic₁₃	—	2	4	no	—
67	C₂²×Dic₁₃	C₂×D₄	1	2	yes	—
68	C₂²×Dic₁₃	C₂×D₄	1	2	yes	—
69	C₂×Dic₂₆	C₄⋊C₄	1	4	yes	—
70	C₂×Dic₂₆	SD₁₆	1	8	yes	—
71	C₂×C₈×C₁₃	—	2	1	no	—
72	C₂×C₈×C₁₃	—	2	1	no	—
73	C₂×C₈×C₁₃	C₂²×C₄	1	2	yes	—
74	C₂×C₈×C₁₃	C₂²×C₄	1	2	yes	—
75	C₂×C₈×C₁₃	C₂×Q₈	1	2	yes	—
76	C₂×C₈×C₁₃	#41	1	2	yes	—
77	C₁₃×C₄⋊C₄	—	2	2	no	—
78	C₁₃×C₄⋊C₄	—	2	2	no	—
79	C₁₃×C₄⋊C₄	—	2	2	no	—
80	C₁₃×C₄⋊C₄	C₂×Q₈	1	4	yes	—
81	C₄²×C₁₃	—	4	4	no	—
82	C₄²×C₁₃	C₂²×C₄	1	2	yes	—
83	C₄²×C₁₃	C₂×Q₈	1	2	yes	—
84	C₄⋊Dic₁₃	—	2	4	no	—
85	C₄⋊Dic₁₃	—	2	4	no	—
86	C₄⋊Dic₁₃	—	4	4	no	—
87	C₄⋊Dic₁₃	—	8	2	no	—
88	C₄⋊Dic₁₃	C₄⋊C₄	1	4	yes	—
89	#6101	—	4	8	no	—
90	#6101	—	8	4	no	—
91	#6101	C₄⋊C₄	1	8	yes	—
92	C₄×Dic₁₃	—	4	8	no	—
93	C₄×Dic₁₃	C₂²×C₄	1	4	yes	—
94	C₄×Dic₁₃	C₂×Q₈	1	4	yes	—
95	C₂×#1055	—	2	2	no	—
96	C₂×#1055	—	2	2	no	—
97	C₂×#1055	C₂²×C₄	1	4	yes	—
98	C₂×C₈×Q₈	—	13	1	no	—
99	Q₈×C₄⋊C₄	—	13	2	no	—
100	#1948	—	13	2	no	—
101	#1948	—	13	4	no	—
102	#1844	—	13	2	no	—
103	#1844	—	13	1	no	—
104	C₄×#106	—	13	2	no	—
105	C₄×#106	D₁₃	1	1	yes	characteristic
106	C₂³×C₁₃	C₂²×D₄	1	1	yes	characteristic
107	C₂×C₄×C₁₃	—	2	2	no	—
108	C₂×C₄×C₁₃	—	2	4	no	—
109	C₂×C₄×C₁₃	—	4	2	no	—
110	C₂×C₄×C₁₃	—	4	2	no	—
111	C₂×C₄×C₁₃	—	4	2	no	—
112	C₂×C₄×C₁₃	—	4	2	no	—
113	C₂×C₄×C₁₃	—	4	2	no	—
114	C₂×C₄×C₁₃	C₂×C₄⋊C₄	1	2	yes	—
115	C₂×C₄×C₁₃	C₂×C₄⋊C₄	1	2	yes	—
116	C₂×C₄×C₁₃	C₂×SD₁₆	1	4	yes	—
117	C₂×C₄×C₁₃	C₂³×C₄	1	1	yes	characteristic
118	C₂×C₄×C₁₃	C₂²×Q₈	1	1	yes	characteristic
119	C₂×C₄×C₁₃	C₂²×Q₈	1	1	yes	characteristic
120	C₂×C₄×C₁₃	C₂×#41	1	1	yes	characteristic
121	C₂×C₄×C₁₃	H(5,𝔽₂)	1	1	yes	characteristic
122	C₂×C₄×C₁₃	2₋¹⁺⁴	1	1	yes	characteristic
123	C₂×Dic₁₃	—	2	4	no	—
124	C₂×Dic₁₃	—	4	4	no	—
125	C₂×Dic₁₃	—	4	4	no	—
126	C₂×Dic₁₃	—	4	4	no	—
127	C₂×Dic₁₃	C₂×C₄⋊C₄	1	4	yes	—
128	C₂×Dic₁₃	C₂×C₄⋊C₄	1	4	yes	—
129	C₂×Dic₁₃	C₂×SD₁₆	1	8	yes	—
130	Dic₂₆	#106	1	16	yes	direct product factor
131	C₈×C₁₃	—	2	2	no	—
132	C₈×C₁₃	C₄×Q₈	1	4	yes	complemented
133	#1055	—	2	4	no	—
134	C₂×C₄×Q₈	—	13	1	no	—
135	C₂×C₄×Q₈	—	26	2	no	—
136	C₈×Q₈	—	13	8	no	—
137	C₂×#106	—	13	2	no	—
138	C₂×#106	—	13	2	no	—
139	C₂×#106	—	13	4	no	—
140	C₂×#106	C₂×D₁₃	1	1	yes	characteristic
141	C₂×#106	C₂×D₁₃	1	1	yes	characteristic
142	C₂×C₄⋊C₈	—	13	2	no	—
143	C₂×C₄⋊C₈	—	13	1	no	—
144	C₂×C₄×C₈	—	13	2	no	—
145	C₂×C₄×C₈	C₂×D₁₃	1	1	yes	characteristic
146	#389	—	26	4	no	—
147	#389	—	26	2	no	—
148	#384	—	13	4	no	—
149	#384	—	13	2	no	—
150	C₄×C₄⋊C₄	—	13	4	no	—
151	C₄×C₄⋊C₄	C₂×D₁₃	1	2	yes	—
152	#340	—	13	4	no	—
153	#340	C₂×D₁₃	1	2	yes	—
154	C₂²×C₁₃	C₂²×C₄⋊C₄	1	1	yes	characteristic
155	C₂²×C₁₃	D₄×Q₈	1	1	yes	characteristic
156	C₂²×C₁₃	#550	1	1	yes	characteristic
157	C₂²×C₁₃	C₂²×SD₁₆	1	2	yes	—
158	C₂²×C₁₃	#577	1	2	yes	—
159	C₄×C₁₃	—	2	2	no	—
160	C₄×C₁₃	—	4	4	no	—
161	C₄×C₁₃	—	8	2	no	—
162	C₄×C₁₃	C₂×#106	1	4	yes	—
163	C₄×C₁₃	C₂×C₄×Q₈	1	1	yes	characteristic
164	C₄×C₁₃	#520	1	1	yes	characteristic
165	C₄×C₁₃	#557	1	1	yes	characteristic
166	C₄×C₁₃	Q₈²	1	1	yes	characteristic
167	Dic₁₃	—	2	4	no	—
168	Dic₁₃	—	8	4	no	—
169	Dic₁₃	C₂×#106	1	8	yes	—
170	C₂²×Q₈	—	13	1	no	—
171	C₂²×C₈	—	13	2	no	—
172	C₂²×C₈	C₂²×D₁₃	1	1	yes	characteristic
173	C₂²×C₈	C₂²×D₁₃	1	1	yes	characteristic
174	C₄×Q₈	—	13	4	no	—
175	C₄×Q₈	—	13	4	no	—
176	C₄×Q₈	—	52	8	no	—
177	C₂×C₄⋊C₄	—	13	2	no	—
178	C₂×C₄⋊C₄	—	13	1	no	—
179	C₂×C₄⋊C₄	—	13	4	no	—
180	C₂×C₄⋊C₄	—	26	4	no	—
181	C₂×C₄⋊C₄	—	26	2	no	—
182	C₂×C₄⋊C₄	—	26	4	no	—
183	C₂×C₄⋊C₄	—	26	2	no	—
184	C₂×C₄⋊C₄	C₂²×D₁₃	1	2	yes	—
185	C₂×C₄⋊C₄	C₂²×D₁₃	1	2	yes	—
186	C₂×C₄²	—	2	2	no	—
187	C₂×C₄²	—	13	2	no	—
188	C₂×C₄²	—	26	4	no	—
189	C₂×C₄²	C₂²×D₁₃	1	1	yes	characteristic
190	#106	—	2	2	no	—
191	#106	—	26	4	no	—
192	#106	—	52	4	no	—
193	#106	—	52	2	no	—
194	#106	Dic₂₆	1	4	yes	direct product factor
195	C₄⋊C₈	—	13	8	no	—
196	C₄⋊C₈	—	13	4	no	—
197	C₄×C₈	—	13	8	no	—
198	C₄×C₈	C₄×D₁₃	1	4	yes	—
199	#95	—	2	2	no	—
200	#95	—	26	4	no	—
201	C₂×C₁₃	Q₈×C₄⋊C₄	1	1	yes	characteristic
202	C₂×C₁₃	C₂²×#106	1	1	yes	characteristic
203	C₂×C₁₃	#2912	1	1	yes	characteristic
204	C₂×C₁₃	Q₈×SD₁₆	1	2	yes	—
205	C₂×C₁₃	#3392	1	2	yes	—
206	C₂×Q₈	—	13	4	no	—
207	C₂×Q₈	—	13	8	no	—
208	C₂²×C₄	—	2	2	no	—
209	C₂²×C₄	—	2	2	no	—
210	C₂²×C₄	—	13	2	no	—
211	C₂²×C₄	—	13	2	no	—
212	C₂²×C₄	—	26	4	no	—
213	C₂²×C₄	D₄×D₁₃	1	1	yes	characteristic
214	C₂²×C₄	D₄×D₁₃	1	1	yes	characteristic
215	C₂²×C₄	C₂³×D₁₃	1	1	yes	characteristic
216	C₂×C₈	—	2	1	no	—
217	C₂×C₈	—	2	1	no	—
218	C₂×C₈	—	13	4	no	—
219	C₂×C₈	—	26	2	no	—
220	C₂×C₈	—	26	2	no	—
221	C₂×C₈	C₂×Dic₂₆	1	2	yes	—
222	C₂×C₈	C₂×C₄×D₁₃	1	2	yes	—
223	C₂×C₈	C₂×C₄×D₁₃	1	2	yes	—
224	C₂×C₈	#6127	1	2	yes	—
225	C₄⋊C₄	—	2	2	no	—
226	C₄⋊C₄	—	2	2	no	—
227	C₄⋊C₄	—	2	2	no	—
228	C₄⋊C₄	—	13	8	no	—
229	C₄⋊C₄	—	13	4	no	—
230	C₄⋊C₄	—	26	4	no	—
231	C₄⋊C₄	—	26	4	no	—
232	C₄⋊C₄	—	52	8	no	—
233	C₄⋊C₄	—	52	4	no	—
234	C₄⋊C₄	—	104	2	no	—
235	C₄⋊C₄	—	104	4	no	—
236	C₄⋊C₄	C₂×Dic₂₆	1	4	yes	—
237	C₄²	—	4	4	no	—
238	C₄²	—	13	4	no	—
239	C₄²	—	13	4	no	—
240	C₄²	—	52	8	no	—
241	C₄²	C₂×C₄×D₁₃	1	2	yes	—
242	C₄²	Q₈×D₁₃	1	2	yes	—
243	C₁₃	Q₈×#106	1	1	yes	characteristic, complemented, Sylow
244	C₂³	C₂×D₄×D₁₃	1	1	yes	characteristic
245	Q₈	—	13	16	no	—
246	C₂×C₄	—	2	2	no	—
247	C₂×C₄	—	2	4	no	—
248	C₂×C₄	—	4	2	no	—
249	C₂×C₄	—	4	2	no	—
250	C₂×C₄	—	4	2	no	—
251	C₂×C₄	—	4	2	no	—
252	C₂×C₄	—	4	2	no	—
253	C₂×C₄	—	13	4	no	—
254	C₂×C₄	—	13	8	no	—
255	C₂×C₄	—	13	4	no	—
256	C₂×C₄	—	26	4	no	—
257	C₂×C₄	—	52	4	no	—
258	C₂×C₄	—	52	4	no	—
259	C₂×C₄	—	52	4	no	—
260	C₂×C₄	C₄⋊C₄×D₁₃	1	2	yes	—
261	C₂×C₄	C₄⋊C₄×D₁₃	1	2	yes	—
262	C₂×C₄	SD₁₆×D₁₃	1	4	yes	—
263	C₂×C₄	C₂²×Dic₂₆	1	1	yes	characteristic
264	C₂×C₄	C₂²×C₄×D₁₃	1	1	yes	characteristic
265	C₂×C₄	C₂×#6127	1	1	yes	characteristic
266	C₂×C₄	C₂×Q₈×D₁₃	1	1	yes	characteristic
267	C₂×C₄	#88256	1	1	yes	characteristic
268	C₂×C₄	#88257	1	1	yes	characteristic
269	C₈	—	2	2	no	—
270	C₈	—	26	4	no	—
271	C₈	C₄×Dic₂₆	1	4	yes	complemented
272	C₂²	C₂×C₄⋊C₄×D₁₃	1	1	yes	characteristic
273	C₂²	D₄×Dic₂₆	1	1	yes	characteristic
274	C₂²	#11719258	1	1	yes	characteristic
275	C₂²	C₂×SD₁₆×D₁₃	1	2	yes	—
276	C₂²	#11719475	1	2	yes	—
277	C₄	—	2	2	no	—
278	C₄	—	4	4	no	—
279	C₄	—	8	2	no	—
280	C₄	—	13	8	no	—
281	C₄	—	26	4	no	—
282	C₄	—	104	4	no	—
283	C₄	D₁₃×#106	1	4	yes	—
284	C₄	C₂×C₄×Dic₂₆	1	1	yes	characteristic
285	C₄	#11719228	1	1	yes	characteristic
286	C₄	Q₈×Dic₂₆	1	1	yes	characteristic
287	C₄	#11719272	1	1	yes	characteristic
288	C₂	C₄⋊C₄×Dic₂₆	1	1	yes	characteristic
289	C₂	C₂×D₁₃×#106	1	1	yes	characteristic
290	C₂	#13123428	1	1	yes	characteristic
291	C₂	SD₁₆×Dic₂₆	1	2	yes	—
292	C₂	#13127815	1	2	yes	—

	subgroup	G/H
center	C₂³	C₂×D₄×D₁₃
commutator	C₂×C₄×C₁₃	C₂³×C₄
Frattini	C₂²×C₄	C₂³×D₁₃
Fitting	C₄×C₁₃×#106	C₂
socle, prime radical	C₂³×C₁₃	C₂²×D₄
cosocle	C₂²×C₄×C₁₃	C₂⁴
hypercenter	C₄×#106	D₁₃
hypocenter	C₁₃	Q₈×#106
LCS	G ▷ C₂×C₄×C₁₃ ▷ C₂×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₄×#106

Complements

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

N	H ≅ G/N	type
25. C₈×Dic₂₆	C₄	semidirect
50. C₁₃×#106	Q₈	semidirect
130. Dic₂₆	#106	direct (N × H)
132. C₈×C₁₃	C₄×Q₈	semidirect
194. #106	Dic₂₆	direct (N × H)
243. C₁₃	Q₈×#106	semidirect
271. C₈	C₄×Dic₂₆	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₈×Dic₂₆	2
C₄⋊C₄×Dic₂₆	2
#13115599	2
#13115598	2
#13114993	2
#13114991	2
#106×Dic₁₃	2
C₄×C₁₃×#106	2
Q₈×#106	13

Subgroups — #106×Dic26 #15881745

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — #106×Dic₂₆ #15881745