Subgroups — #107×Dic₂₆ #15881758

Order 3328.

838 conjugacy classes of subgroups.

The subgroup lattice has 312 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	#13115594	C₂	1	2	yes	maximal
4	#13115593	C₂	1	4	yes	maximal
5	#13114996	C₂	1	1	yes	maximal, characteristic
6	#13114994	C₂	1	2	yes	maximal
7	#107×Dic₁₃	C₂	1	2	yes	maximal
8	C₄×C₁₃×#107	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×#107	C₂²	1	1	yes	characteristic
12	C₂×C₁₃×#107	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×#88058	C₂²	1	2	yes	—
16	C₂×#88058	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×#88047	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×#107	—	2	2	no	—
50	C₁₃×#107	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	#88058	—	2	4	no	—
55	#88058	—	4	2	no	—
56	#88054	C₂×C₄	1	8	yes	—
57	C₈×Dic₁₃	C₂×C₄	1	8	yes	—
58	#88047	—	4	4	no	—
59	C₄⋊#1055	C₂×C₄	1	4	yes	—
60	Q₈×#107	—	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	D₈	1	4	yes	—
71	C₂×Dic₂₆	Q₁₆	1	4	yes	—
72	C₂×C₈×C₁₃	—	2	1	no	—
73	C₂×C₈×C₁₃	—	2	1	no	—
74	C₂×C₈×C₁₃	C₂²×C₄	1	2	yes	—
75	C₂×C₈×C₁₃	C₂²×C₄	1	2	yes	—
76	C₂×C₈×C₁₃	C₂×Q₈	1	2	yes	—
77	C₂×C₈×C₁₃	#41	1	2	yes	—
78	C₁₃×C₄⋊C₄	—	2	2	no	—
79	C₁₃×C₄⋊C₄	—	2	2	no	—
80	C₁₃×C₄⋊C₄	—	2	2	no	—
81	C₁₃×C₄⋊C₄	C₂×Q₈	1	4	yes	—
82	C₄²×C₁₃	—	4	4	no	—
83	C₄²×C₁₃	C₂²×C₄	1	2	yes	—
84	C₄²×C₁₃	C₂×Q₈	1	2	yes	—
85	C₄⋊Dic₁₃	—	2	4	no	—
86	C₄⋊Dic₁₃	—	2	4	no	—
87	C₄⋊Dic₁₃	—	4	4	no	—
88	C₄⋊Dic₁₃	—	8	2	no	—
89	C₄⋊Dic₁₃	C₄⋊C₄	1	4	yes	—
90	#6101	—	4	8	no	—
91	#6101	—	8	4	no	—
92	#6101	C₄⋊C₄	1	8	yes	—
93	C₄×Dic₁₃	—	4	8	no	—
94	C₄×Dic₁₃	C₂²×C₄	1	4	yes	—
95	C₄×Dic₁₃	C₂×Q₈	1	4	yes	—
96	C₂×#1055	—	2	2	no	—
97	C₂×#1055	—	2	2	no	—
98	C₂×#1055	C₂²×C₄	1	4	yes	—
99	C₂×C₈×Q₈	—	13	1	no	—
100	Q₈×C₄⋊C₄	—	13	2	no	—
101	#1947	—	13	2	no	—
102	#1947	—	13	4	no	—
103	#1845	—	13	2	no	—
104	#1845	—	13	1	no	—
105	C₄×#107	—	13	2	no	—
106	C₄×#107	D₁₃	1	1	yes	characteristic
107	C₂³×C₁₃	C₂²×D₄	1	1	yes	characteristic
108	C₂×C₄×C₁₃	—	2	1	no	—
109	C₂×C₄×C₁₃	—	2	1	no	—
110	C₂×C₄×C₁₃	—	2	4	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	—	4	2	no	—
115	C₂×C₄×C₁₃	—	4	2	no	—
116	C₂×C₄×C₁₃	C₂×C₄⋊C₄	1	2	yes	—
117	C₂×C₄×C₁₃	C₂×C₄⋊C₄	1	2	yes	—
118	C₂×C₄×C₁₃	C₂×D₈	1	2	yes	—
119	C₂×C₄×C₁₃	C₂×Q₁₆	1	2	yes	—
120	C₂×C₄×C₁₃	C₂³×C₄	1	1	yes	characteristic
121	C₂×C₄×C₁₃	C₂²×Q₈	1	1	yes	characteristic
122	C₂×C₄×C₁₃	C₂²×Q₈	1	1	yes	characteristic
123	C₂×C₄×C₁₃	C₂×#41	1	1	yes	characteristic
124	C₂×C₄×C₁₃	H(5,𝔽₂)	1	1	yes	characteristic
125	C₂×C₄×C₁₃	2₋¹⁺⁴	1	1	yes	characteristic
126	C₂×Dic₁₃	—	2	2	no	—
127	C₂×Dic₁₃	—	2	2	no	—
128	C₂×Dic₁₃	—	4	4	no	—
129	C₂×Dic₁₃	—	4	4	no	—
130	C₂×Dic₁₃	—	4	4	no	—
131	C₂×Dic₁₃	C₂×C₄⋊C₄	1	4	yes	—
132	C₂×Dic₁₃	C₂×C₄⋊C₄	1	4	yes	—
133	C₂×Dic₁₃	C₂×D₈	1	4	yes	—
134	C₂×Dic₁₃	C₂×Q₁₆	1	4	yes	—
135	Dic₂₆	#107	1	16	yes	direct product factor
136	C₈×C₁₃	—	2	2	no	—
137	C₈×C₁₃	C₄×Q₈	1	4	yes	complemented
138	#1055	—	2	4	no	—
139	C₂×C₄×Q₈	—	13	1	no	—
140	C₂×C₄×Q₈	—	26	2	no	—
141	C₈×Q₈	—	13	8	no	—
142	C₂×#107	—	13	2	no	—
143	C₂×#107	—	13	2	no	—
144	C₂×#107	—	13	4	no	—
145	C₂×#107	C₂×D₁₃	1	1	yes	characteristic
146	C₂×#107	C₂×D₁₃	1	1	yes	characteristic
147	C₂×C₄⋊C₈	—	13	2	no	—
148	C₂×C₄⋊C₈	—	13	1	no	—
149	C₂×C₄×C₈	—	13	2	no	—
150	C₂×C₄×C₈	C₂×D₁₃	1	1	yes	characteristic
151	#389	—	26	4	no	—
152	#389	—	26	2	no	—
153	#384	—	13	4	no	—
154	#384	—	13	2	no	—
155	C₄×C₄⋊C₄	—	13	4	no	—
156	C₄×C₄⋊C₄	C₂×D₁₃	1	2	yes	—
157	#340	—	13	4	no	—
158	#340	C₂×D₁₃	1	2	yes	—
159	C₂²×C₁₃	C₂²×C₄⋊C₄	1	1	yes	characteristic
160	C₂²×C₁₃	D₄×Q₈	1	1	yes	characteristic
161	C₂²×C₁₃	#550	1	1	yes	characteristic
162	C₂²×C₁₃	C₂²×D₈	1	1	yes	characteristic
163	C₂²×C₁₃	C₂²×Q₁₆	1	1	yes	characteristic
164	C₂²×C₁₃	#576	1	1	yes	characteristic
165	C₂²×C₁₃	#578	1	1	yes	characteristic
166	C₄×C₁₃	—	2	2	no	—
167	C₄×C₁₃	—	4	4	no	—
168	C₄×C₁₃	—	8	2	no	—
169	C₄×C₁₃	C₂×#107	1	4	yes	—
170	C₄×C₁₃	C₂×C₄×Q₈	1	1	yes	characteristic
171	C₄×C₁₃	#520	1	1	yes	characteristic
172	C₄×C₁₃	#557	1	1	yes	characteristic
173	C₄×C₁₃	Q₈²	1	1	yes	characteristic
174	Dic₁₃	—	2	4	no	—
175	Dic₁₃	—	8	4	no	—
176	Dic₁₃	C₂×#107	1	8	yes	—
177	C₂²×Q₈	—	13	1	no	—
178	C₂²×C₈	—	13	2	no	—
179	C₂²×C₈	C₂²×D₁₃	1	1	yes	characteristic
180	C₂²×C₈	C₂²×D₁₃	1	1	yes	characteristic
181	C₄×Q₈	—	13	4	no	—
182	C₄×Q₈	—	13	4	no	—
183	C₄×Q₈	—	52	8	no	—
184	C₂×C₄⋊C₄	—	13	2	no	—
185	C₂×C₄⋊C₄	—	13	1	no	—
186	C₂×C₄⋊C₄	—	13	4	no	—
187	C₂×C₄⋊C₄	—	26	4	no	—
188	C₂×C₄⋊C₄	—	26	2	no	—
189	C₂×C₄⋊C₄	—	26	4	no	—
190	C₂×C₄⋊C₄	—	26	2	no	—
191	C₂×C₄⋊C₄	C₂²×D₁₃	1	2	yes	—
192	C₂×C₄⋊C₄	C₂²×D₁₃	1	2	yes	—
193	C₂×C₄²	—	2	2	no	—
194	C₂×C₄²	—	13	2	no	—
195	C₂×C₄²	—	26	4	no	—
196	C₂×C₄²	C₂²×D₁₃	1	1	yes	characteristic
197	#107	—	2	2	no	—
198	#107	—	26	4	no	—
199	#107	—	52	4	no	—
200	#107	—	52	2	no	—
201	#107	Dic₂₆	1	4	yes	direct product factor
202	C₄⋊C₈	—	13	8	no	—
203	C₄⋊C₈	—	13	4	no	—
204	C₄×C₈	—	13	8	no	—
205	C₄×C₈	C₄×D₁₃	1	4	yes	—
206	#95	—	2	2	no	—
207	#95	—	26	4	no	—
208	C₂×C₁₃	Q₈×C₄⋊C₄	1	1	yes	characteristic
209	C₂×C₁₃	C₂²×#107	1	1	yes	characteristic
210	C₂×C₁₃	#2913	1	1	yes	characteristic
211	C₂×C₁₃	Q₈×D₈	1	1	yes	characteristic
212	C₂×C₁₃	Q₈×Q₁₆	1	1	yes	characteristic
213	C₂×C₁₃	#3391	1	1	yes	characteristic
214	C₂×C₁₃	#3395	1	1	yes	characteristic
215	C₂×Q₈	—	13	4	no	—
216	C₂×Q₈	—	13	4	no	—
217	C₂×Q₈	—	13	4	no	—
218	C₂²×C₄	—	2	2	no	—
219	C₂²×C₄	—	2	2	no	—
220	C₂²×C₄	—	13	2	no	—
221	C₂²×C₄	—	13	2	no	—
222	C₂²×C₄	—	26	4	no	—
223	C₂²×C₄	D₄×D₁₃	1	1	yes	characteristic
224	C₂²×C₄	D₄×D₁₃	1	1	yes	characteristic
225	C₂²×C₄	C₂³×D₁₃	1	1	yes	characteristic
226	C₂×C₈	—	2	1	no	—
227	C₂×C₈	—	2	1	no	—
228	C₂×C₈	—	13	4	no	—
229	C₂×C₈	—	26	2	no	—
230	C₂×C₈	—	26	2	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	—	2	2	no	—
236	C₄⋊C₄	—	2	2	no	—
237	C₄⋊C₄	—	2	2	no	—
238	C₄⋊C₄	—	13	8	no	—
239	C₄⋊C₄	—	13	4	no	—
240	C₄⋊C₄	—	26	4	no	—
241	C₄⋊C₄	—	26	4	no	—
242	C₄⋊C₄	—	52	8	no	—
243	C₄⋊C₄	—	52	4	no	—
244	C₄⋊C₄	—	104	2	no	—
245	C₄⋊C₄	—	104	4	no	—
246	C₄⋊C₄	C₂×Dic₂₆	1	4	yes	—
247	C₄²	—	4	4	no	—
248	C₄²	—	13	4	no	—
249	C₄²	—	13	4	no	—
250	C₄²	—	52	8	no	—
251	C₄²	C₂×C₄×D₁₃	1	2	yes	—
252	C₄²	Q₈×D₁₃	1	2	yes	—
253	C₁₃	Q₈×#107	1	1	yes	characteristic, complemented, Sylow
254	C₂³	C₂×D₄×D₁₃	1	1	yes	characteristic
255	Q₈	—	13	16	no	—
256	C₂×C₄	—	2	1	no	—
257	C₂×C₄	—	2	1	no	—
258	C₂×C₄	—	2	4	no	—
259	C₂×C₄	—	4	2	no	—
260	C₂×C₄	—	4	2	no	—
261	C₂×C₄	—	4	2	no	—
262	C₂×C₄	—	4	2	no	—
263	C₂×C₄	—	4	2	no	—
264	C₂×C₄	—	13	4	no	—
265	C₂×C₄	—	13	4	no	—
266	C₂×C₄	—	13	4	no	—
267	C₂×C₄	—	13	4	no	—
268	C₂×C₄	—	26	2	no	—
269	C₂×C₄	—	26	2	no	—
270	C₂×C₄	—	52	4	no	—
271	C₂×C₄	—	52	4	no	—
272	C₂×C₄	—	52	4	no	—
273	C₂×C₄	C₄⋊C₄×D₁₃	1	2	yes	—
274	C₂×C₄	C₄⋊C₄×D₁₃	1	2	yes	—
275	C₂×C₄	D₈×D₁₃	1	2	yes	—
276	C₂×C₄	Q₁₆×D₁₃	1	2	yes	—
277	C₂×C₄	C₂²×Dic₂₆	1	1	yes	characteristic
278	C₂×C₄	C₂²×C₄×D₁₃	1	1	yes	characteristic
279	C₂×C₄	C₂×#6127	1	1	yes	characteristic
280	C₂×C₄	C₂×Q₈×D₁₃	1	1	yes	characteristic
281	C₂×C₄	#88256	1	1	yes	characteristic
282	C₂×C₄	#88257	1	1	yes	characteristic
283	C₈	—	2	2	no	—
284	C₈	—	26	4	no	—
285	C₈	C₄×Dic₂₆	1	4	yes	complemented
286	C₂²	C₂×C₄⋊C₄×D₁₃	1	1	yes	characteristic
287	C₂²	D₄×Dic₂₆	1	1	yes	characteristic
288	C₂²	#11719258	1	1	yes	characteristic
289	C₂²	C₂×D₈×D₁₃	1	1	yes	characteristic
290	C₂²	C₂×Q₁₆×D₁₃	1	1	yes	characteristic
291	C₂²	#11719474	1	1	yes	characteristic
292	C₂²	#11719476	1	1	yes	characteristic
293	C₄	—	2	2	no	—
294	C₄	—	4	4	no	—
295	C₄	—	8	2	no	—
296	C₄	—	13	8	no	—
297	C₄	—	26	4	no	—
298	C₄	—	104	4	no	—
299	C₄	D₁₃×#107	1	4	yes	—
300	C₄	C₂×C₄×Dic₂₆	1	1	yes	characteristic
301	C₄	#11719228	1	1	yes	characteristic
302	C₄	Q₈×Dic₂₆	1	1	yes	characteristic
303	C₄	#11719272	1	1	yes	characteristic
304	C₂	C₄⋊C₄×Dic₂₆	1	1	yes	characteristic
305	C₂	C₂×D₁₃×#107	1	1	yes	characteristic
306	C₂	#13123433	1	1	yes	characteristic
307	C₂	D₈×Dic₂₆	1	1	yes	characteristic
308	C₂	Q₁₆×Dic₂₆	1	1	yes	characteristic
309	C₂	#13127810	1	1	yes	characteristic
310	C₂	#13127841	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	C₂×D₄×D₁₃
commutator	C₂×C₄×C₁₃	C₂³×C₄
Frattini	C₂²×C₄	C₂³×D₁₃
Fitting	C₄×C₁₃×#107	C₂
socle, prime radical	C₂³×C₁₃	C₂²×D₄
cosocle	C₂²×C₄×C₁₃	C₂⁴
hypercenter	C₄×#107	D₁₃
hypocenter	C₁₃	Q₈×#107
LCS	G ▷ C₂×C₄×C₁₃ ▷ C₂×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₄×#107

Complements

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

N	H ≅ G/N	type
25. C₈×Dic₂₆	C₄	semidirect
50. C₁₃×#107	Q₈	semidirect
135. Dic₂₆	#107	direct (N × H)
137. C₈×C₁₃	C₄×Q₈	semidirect
201. #107	Dic₂₆	direct (N × H)
253. C₁₃	Q₈×#107	semidirect
285. 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
#13115594	2
#13115593	2
#13114996	2
#13114994	2
#107×Dic₁₃	2
C₄×C₁₃×#107	2
Q₈×#107	13

Subgroups — #107×Dic26 #15881758

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — #107×Dic₂₆ #15881758