Subgroups — #15881763

Order 3328.

750 conjugacy classes of subgroups.

The subgroup lattice has 306 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	#13117791	C₂	1	2	yes	maximal
3	#13115607	C₂	1	4	yes	maximal
4	#13115599	C₂	1	2	yes	maximal
5	#13114999	C₂	1	2	yes	maximal
6	#13114996	C₂	1	1	yes	maximal, characteristic
7	#107×Dic₁₃	C₂	1	2	yes	maximal
8	C₄×C₁₃×#107	C₂	1	1	yes	maximal, characteristic
9	C₂×C₄×Dic₂₆	C₂²	1	1	yes	characteristic
10	C₂×C₁₃×#107	C₂²	1	1	yes	characteristic
11	C₂×C₁₃×#107	C₂²	1	1	yes	characteristic
12	C₂×C₄×C₈×C₁₃	C₂²	1	1	yes	characteristic
13	C₄×C₁₃×C₄⋊C₄	C₂²	1	2	yes	—
14	C₂×#88058	C₂²	1	2	yes	—
15	C₂×#88058	C₂²	1	2	yes	—
16	C₂×#88054	C₂²	1	2	yes	—
17	C₂×C₈×Dic₁₃	C₂²	1	2	yes	—
18	#11718638	C₂²	1	4	yes	—
19	C₄⋊C₄×Dic₁₃	C₂²	1	4	yes	—
20	C₂×#88048	C₂²	1	4	yes	—
21	#11718596	C₂²	1	2	yes	—
22	C₄×C₄⋊Dic₁₃	—	2	2	no	—
23	#11718590	—	2	2	no	—
24	C₂×C₄⋊#1055	C₂²	1	1	yes	characteristic
25	C₈×Dic₂₆	C₄	1	8	yes	complemented
26	#11718316	—	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	D₄	1	2	yes	—
41	C₂×C₄×Dic₁₃	—	2	4	no	—
42	C₂×C₄×Dic₁₃	C₂³	1	2	yes	—
43	C₂²×#1055	C₂³	1	2	yes	—
44	C₄×Dic₂₆	C₂×C₄	1	4	yes	—
45	C₄×Dic₂₆	Q₈	1	4	yes	—
46	C₁₃×#107	—	2	2	no	—
47	C₁₃×#107	Q₈	1	4	yes	complemented
48	C₄×C₈×C₁₃	C₂×C₄	1	4	yes	—
49	C₁₃×#95	—	2	2	no	—
50	#88071	—	2	4	no	—
51	#88071	—	2	4	no	—
52	#88071	—	2	4	no	—
53	#88058	—	2	4	no	—
54	#88058	—	4	2	no	—
55	#88054	C₂×C₄	1	8	yes	—
56	C₈×Dic₁₃	C₂×C₄	1	8	yes	—
57	#88048	—	4	4	no	—
58	C₄⋊#1055	C₂×C₄	1	4	yes	—
59	#22727	—	13	1	no	maximal, Sylow
60	C₂²×C₄×C₁₃	—	2	2	no	—
61	C₂²×C₄×C₁₃	—	2	2	no	—
62	C₂²×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
63	C₂²×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
64	C₂²×C₄×C₁₃	C₂⁴	1	1	yes	characteristic
65	C₂²×Dic₁₃	—	2	4	no	—
66	C₂²×Dic₁₃	C₂×D₄	1	2	yes	—
67	C₂²×Dic₁₃	C₂×D₄	1	2	yes	—
68	C₂×Dic₂₆	—	2	2	no	—
69	C₂×Dic₂₆	—	2	2	no	—
70	C₂×Dic₂₆	C₄⋊C₄	1	4	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	C₄⋊C₄	1	8	yes	—
90	C₄×Dic₁₃	—	8	4	no	—
91	C₄×Dic₁₃	C₂²×C₄	1	4	yes	—
92	C₄×Dic₁₃	C₂×Q₈	1	4	yes	—
93	C₂×#1055	—	2	2	no	—
94	C₂×#1055	—	2	2	no	—
95	C₂×#1055	C₂²×C₄	1	4	yes	—
96	C₂×C₈×Q₈	—	13	1	no	—
97	#2355	—	13	2	no	—
98	#1949	—	13	4	no	—
99	#1948	—	13	2	no	—
100	#1846	—	13	2	no	—
101	#1845	—	13	1	no	—
102	C₄×#107	—	13	2	no	—
103	C₄×#107	D₁₃	1	1	yes	characteristic
104	C₂³×C₁₃	C₂²×D₄	1	1	yes	characteristic
105	C₂×C₄×C₁₃	—	2	1	no	—
106	C₂×C₄×C₁₃	—	2	1	no	—
107	C₂×C₄×C₁₃	—	2	4	no	—
108	C₂×C₄×C₁₃	—	4	2	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	C₂×C₄⋊C₄	1	2	yes	—
114	C₂×C₄×C₁₃	C₂×C₄⋊C₄	1	2	yes	—
115	C₂×C₄×C₁₃	#135	1	2	yes	—
116	C₂×C₄×C₁₃	#135	1	2	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	2	no	—
124	C₂×Dic₁₃	—	2	2	no	—
125	C₂×Dic₁₃	—	2	2	no	—
126	C₂×Dic₁₃	—	2	2	no	—
127	C₂×Dic₁₃	—	4	4	no	—
128	C₂×Dic₁₃	—	4	4	no	—
129	C₂×Dic₁₃	—	4	4	no	—
130	C₂×Dic₁₃	C₂×C₄⋊C₄	1	4	yes	—
131	C₂×Dic₁₃	C₂×C₄⋊C₄	1	4	yes	—
132	Dic₂₆	—	2	8	no	—
133	C₈×C₁₃	—	2	2	no	—
134	C₈×C₁₃	C₄×Q₈	1	4	yes	—
135	#1055	—	2	4	no	—
136	C₂×C₄×Q₈	—	13	1	no	—
137	C₈×Q₈	—	13	8	no	—
138	C₂×#107	—	13	2	no	—
139	C₂×#107	—	13	2	no	—
140	C₂×#107	C₂×D₁₃	1	1	yes	characteristic
141	C₂×#107	C₂×D₁₃	1	1	yes	characteristic
142	C₂×#106	—	13	4	no	—
143	C₂×C₄⋊C₈	—	13	2	no	—
144	C₂×C₄⋊C₈	—	13	1	no	—
145	C₂×C₄×C₈	—	13	2	no	—
146	C₂×C₄×C₈	C₂×D₁₃	1	1	yes	characteristic
147	#391	—	26	2	no	—
148	#387	—	26	4	no	—
149	#384	—	13	2	no	—
150	#382	—	13	4	no	—
151	C₄×C₄⋊C₄	—	13	4	no	—
152	C₄×C₄⋊C₄	—	26	2	no	—
153	C₄×C₄⋊C₄	C₂×D₁₃	1	2	yes	—
154	#340	—	13	4	no	—
155	#340	C₂×D₁₃	1	2	yes	—
156	C₂²×C₁₃	C₂²×C₄⋊C₄	1	1	yes	characteristic
157	C₂²×C₁₃	D₄×Q₈	1	1	yes	characteristic
158	C₂²×C₁₃	#550	1	1	yes	characteristic
159	C₂²×C₁₃	C₂×#135	1	1	yes	characteristic
160	C₂²×C₁₃	C₂×#135	1	1	yes	characteristic
161	C₂²×C₁₃	#576	1	1	yes	characteristic
162	C₂²×C₁₃	#578	1	1	yes	characteristic
163	C₄×C₁₃	—	2	2	no	—
164	C₄×C₁₃	—	4	4	no	—
165	C₄×C₁₃	—	8	2	no	—
166	C₄×C₁₃	#427	1	4	yes	—
167	C₄×C₁₃	C₂×C₄×Q₈	1	1	yes	characteristic
168	C₄×C₁₃	#520	1	1	yes	characteristic
169	C₄×C₁₃	#557	1	1	yes	characteristic
170	C₄×C₁₃	Q₈²	1	1	yes	characteristic
171	Dic₁₃	—	2	4	no	—
172	Dic₁₃	—	2	4	no	—
173	Dic₁₃	—	8	4	no	—
174	C₂²×Q₈	—	13	1	no	—
175	C₂²×C₈	—	13	2	no	—
176	C₂²×C₈	C₂²×D₁₃	1	1	yes	characteristic
177	C₂²×C₈	C₂²×D₁₃	1	1	yes	characteristic
178	C₄×Q₈	—	13	4	no	—
179	C₄×Q₈	—	13	4	no	—
180	C₂×C₄⋊C₄	—	13	2	no	—
181	C₂×C₄⋊C₄	—	13	1	no	—
182	C₂×C₄⋊C₄	—	13	4	no	—
183	C₂×C₄⋊C₄	—	26	2	no	—
184	C₂×C₄⋊C₄	—	26	2	no	—
185	C₂×C₄⋊C₄	C₂²×D₁₃	1	2	yes	—
186	C₂×C₄⋊C₄	C₂²×D₁₃	1	2	yes	—
187	C₂×C₄²	—	2	2	no	—
188	C₂×C₄²	—	13	2	no	—
189	C₂×C₄²	—	26	4	no	—
190	C₂×C₄²	C₂²×D₁₃	1	1	yes	characteristic
191	#107	—	2	2	no	—
192	#107	—	26	4	no	—
193	#107	—	52	2	no	—
194	#107	Dic₂₆	1	4	yes	complemented
195	#106	—	52	4	no	—
196	C₄⋊C₈	—	13	8	no	—
197	C₄⋊C₈	—	13	4	no	—
198	C₄×C₈	—	13	8	no	—
199	C₄×C₈	C₄×D₁₃	1	4	yes	—
200	#95	—	2	2	no	—
201	#95	—	26	4	no	—
202	#95	—	26	4	no	—
203	#95	—	26	4	no	—
204	C₂×C₁₃	Q₈×C₄⋊C₄	1	1	yes	characteristic
205	C₂×C₁₃	C₂×#427	1	1	yes	characteristic
206	C₂×C₁₃	#2913	1	1	yes	characteristic
207	C₂×C₁₃	#3380	1	1	yes	characteristic
208	C₂×C₁₃	#3383	1	1	yes	characteristic
209	C₂×C₁₃	#3393	1	1	yes	characteristic
210	C₂×C₁₃	#3396	1	1	yes	characteristic
211	C₂×Q₈	—	13	4	no	—
212	C₂×Q₈	—	26	2	no	—
213	C₂×Q₈	—	26	2	no	—
214	C₂²×C₄	—	2	2	no	—
215	C₂²×C₄	—	2	2	no	—
216	C₂²×C₄	—	13	2	no	—
217	C₂²×C₄	—	13	2	no	—
218	C₂²×C₄	—	26	4	no	—
219	C₂²×C₄	D₄×D₁₃	1	1	yes	characteristic
220	C₂²×C₄	D₄×D₁₃	1	1	yes	characteristic
221	C₂²×C₄	C₂³×D₁₃	1	1	yes	characteristic
222	C₂×C₈	—	2	1	no	—
223	C₂×C₈	—	2	1	no	—
224	C₂×C₈	—	13	4	no	—
225	C₂×C₈	—	26	2	no	—
226	C₂×C₈	—	26	2	no	—
227	C₂×C₈	C₂×Dic₂₆	1	2	yes	—
228	C₂×C₈	C₂×C₄×D₁₃	1	2	yes	—
229	C₂×C₈	C₂×C₄×D₁₃	1	2	yes	—
230	C₂×C₈	#6127	1	2	yes	—
231	C₄⋊C₄	—	2	2	no	—
232	C₄⋊C₄	—	2	2	no	—
233	C₄⋊C₄	—	2	2	no	—
234	C₄⋊C₄	—	13	8	no	—
235	C₄⋊C₄	—	13	4	no	—
236	C₄⋊C₄	—	26	4	no	—
237	C₄⋊C₄	—	26	4	no	—
238	C₄⋊C₄	—	52	4	no	—
239	C₄⋊C₄	—	104	2	no	—
240	C₄⋊C₄	C₂×Dic₂₆	1	4	yes	—
241	C₄²	—	4	4	no	—
242	C₄²	—	13	4	no	—
243	C₄²	—	13	4	no	—
244	C₄²	—	104	4	no	—
245	C₄²	C₂×C₄×D₁₃	1	2	yes	—
246	C₄²	Q₈×D₁₃	1	2	yes	—
247	C₁₃	#22727	1	1	yes	characteristic, complemented, Sylow
248	C₂³	C₂×D₄×D₁₃	1	1	yes	characteristic
249	Q₈	—	26	8	no	—
250	C₂×C₄	—	2	1	no	—
251	C₂×C₄	—	2	1	no	—
252	C₂×C₄	—	2	4	no	—
253	C₂×C₄	—	4	2	no	—
254	C₂×C₄	—	4	2	no	—
255	C₂×C₄	—	4	2	no	—
256	C₂×C₄	—	4	2	no	—
257	C₂×C₄	—	4	2	no	—
258	C₂×C₄	—	13	4	no	—
259	C₂×C₄	—	13	4	no	—
260	C₂×C₄	—	26	2	no	—
261	C₂×C₄	—	26	2	no	—
262	C₂×C₄	—	26	2	no	—
263	C₂×C₄	—	26	2	no	—
264	C₂×C₄	—	52	4	no	—
265	C₂×C₄	—	52	4	no	—
266	C₂×C₄	—	52	4	no	—
267	C₂×C₄	C₄⋊C₄×D₁₃	1	2	yes	—
268	C₂×C₄	C₄⋊C₄×D₁₃	1	2	yes	—
269	C₂×C₄	#88166	1	2	yes	—
270	C₂×C₄	#88173	1	2	yes	—
271	C₂×C₄	C₂²×Dic₂₆	1	1	yes	characteristic
272	C₂×C₄	C₂²×C₄×D₁₃	1	1	yes	characteristic
273	C₂×C₄	C₂×#6127	1	1	yes	characteristic
274	C₂×C₄	C₂×Q₈×D₁₃	1	1	yes	characteristic
275	C₂×C₄	#88256	1	1	yes	characteristic
276	C₂×C₄	#88257	1	1	yes	characteristic
277	C₈	—	2	2	no	—
278	C₈	—	26	4	no	—
279	C₈	C₄×Dic₂₆	1	4	yes	—
280	C₂²	C₂×C₄⋊C₄×D₁₃	1	1	yes	characteristic
281	C₂²	D₄×Dic₂₆	1	1	yes	characteristic
282	C₂²	#11719258	1	1	yes	characteristic
283	C₂²	C₂×#88166	1	1	yes	characteristic
284	C₂²	C₂×#88173	1	1	yes	characteristic
285	C₂²	#11719474	1	1	yes	characteristic
286	C₂²	#11719476	1	1	yes	characteristic
287	C₄	—	2	2	no	—
288	C₄	—	4	4	no	—
289	C₄	—	8	2	no	—
290	C₄	—	26	4	no	—
291	C₄	—	26	4	no	—
292	C₄	—	104	4	no	—
293	C₄	#11718541	1	4	yes	—
294	C₄	C₂×C₄×Dic₂₆	1	1	yes	characteristic
295	C₄	#11719228	1	1	yes	characteristic
296	C₄	Q₈×Dic₂₆	1	1	yes	characteristic
297	C₄	#11719272	1	1	yes	characteristic
298	C₂	C₄⋊C₄×Dic₂₆	1	1	yes	characteristic
299	C₂	C₂×#11718541	1	1	yes	characteristic
300	C₂	#13123433	1	1	yes	characteristic
301	C₂	#13127701	1	1	yes	characteristic
302	C₂	#13127725	1	1	yes	characteristic
303	C₂	#13127822	1	1	yes	characteristic
304	C₂	#13127846	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	#22727
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
47. C₁₃×#107	Q₈	semidirect
194. #107	Dic₂₆	semidirect
247. C₁₃	#22727	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
#13117791	2
#13115607	2
#13115599	2
#13114999	2
#13114996	2
#107×Dic₁₃	2
C₄×C₁₃×#107	2
#22727	13