Subgroups — #15778997

Order 3328.

410 conjugacy classes of subgroups.

The subgroup lattice has 290 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₄×Dic₅₂	C₂	1	1	yes	maximal, characteristic
2	#13116592	C₂	1	1	yes	maximal, characteristic
3	C₂×#11718088	C₂	1	2	yes	maximal
4	#13112946	C₂	1	2	yes	maximal
5	C₁₃×#1848	C₂	1	1	yes	maximal, characteristic
6	C₂²×Dic₅₂	C₂²	1	1	yes	characteristic
7	C₂×C₄×Dic₂₆	—	2	1	no	—
8	C₂×C₁₃×#108	—	2	1	no	—
9	C₂×C₁₃×#108	C₂²	1	2	yes	—
10	C₁₃×#423	—	2	1	no	—
11	C₂×C₄×C₈×C₁₃	C₂²	1	1	yes	characteristic
12	C₂×#88058	C₂²	1	1	yes	characteristic
13	C₂×#88056	—	2	1	no	—
14	C₂×#88052	C₂²	1	2	yes	—
15	C₄×Dic₅₂	—	2	1	no	—
16	C₄×Dic₅₂	—	2	1	no	—
17	C₄×Dic₅₂	C₄	1	2	yes	—
18	C₄×Dic₅₂	C₄	1	2	yes	—
19	#11718088	—	2	4	no	—
20	C₄⋊#6090	—	2	1	no	—
21	C₄⋊#6090	—	2	1	no	—
22	C₂²×Dic₂₆	—	2	1	no	—
23	C₂×C₁₃×M₁₆	—	2	2	no	—
24	C₂²×C₈×C₁₃	D₄	1	1	yes	characteristic
25	C₂²×C₈×C₁₃	C₂³	1	1	yes	characteristic
26	C₂×C₄²×C₁₃	D₄	1	1	yes	characteristic
27	C₂×C₄⋊Dic₁₃	—	2	1	no	—
28	C₂×#6101	—	4	1	no	—
29	C₂×C₄×Dic₁₃	—	4	1	no	—
30	C₂×Dic₅₂	—	2	1	no	—
31	C₂×Dic₅₂	—	2	1	no	—
32	C₂×Dic₅₂	—	2	1	no	—
33	C₂×Dic₅₂	C₂×C₄	1	2	yes	—
34	C₂×Dic₅₂	D₄	1	2	yes	—
35	C₂×Dic₅₂	Q₈	1	2	yes	—
36	C₄×Dic₂₆	—	2	2	no	—
37	C₄×Dic₂₆	—	2	2	no	—
38	C₄×Dic₂₆	—	4	1	no	—
39	C₄×Dic₂₆	—	4	1	no	—
40	C₁₃×#108	—	2	2	no	—
41	C₁₃×#108	—	2	2	no	—
42	C₁₃×#108	—	4	2	no	—
43	C₁₃×C₄⋊C₈	—	4	1	no	—
44	C₁₃×C₄⋊C₈	—	4	1	no	—
45	C₄×C₈×C₁₃	C₂×C₄	1	1	yes	characteristic
46	C₄×C₈×C₁₃	C₂×C₄	1	1	yes	characteristic
47	C₄×C₈×C₁₃	D₄	1	1	yes	characteristic
48	C₄×C₈×C₁₃	D₄	1	1	yes	characteristic
49	#88058	—	2	1	no	—
50	#88058	C₂×C₄	1	2	yes	—
51	#88056	—	2	2	no	—
52	#88056	—	2	2	no	—
53	#88052	—	2	2	no	—
54	#88052	—	2	2	no	—
55	C₂×#6090	—	2	2	no	—
56	#12819	—	13	1	no	maximal, Sylow
57	C₂²×C₄×C₁₃	SD₁₆	1	1	yes	characteristic
58	C₂²×C₄×C₁₃	SD₁₆	1	1	yes	characteristic
59	C₂²×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
60	C₂²×Dic₁₃	—	4	1	no	—
61	C₂²×Dic₁₃	—	4	1	no	—
62	C₂×Dic₂₆	—	2	2	no	—
63	C₂×Dic₂₆	—	2	2	no	—
64	C₂×Dic₂₆	—	2	2	no	—
65	C₂×Dic₂₆	—	4	1	no	—
66	C₂×Dic₂₆	—	4	1	no	—
67	C₂×Dic₂₆	—	4	1	no	—
68	C₁₃×M₁₆	—	4	2	no	—
69	C₁₃×M₁₆	—	4	2	no	—
70	C₂×C₈×C₁₃	—	2	1	no	—
71	C₂×C₈×C₁₃	—	2	1	no	—
72	C₂×C₈×C₁₃	—	4	2	no	—
73	C₂×C₈×C₁₃	C₂²⋊C₄	1	1	yes	characteristic
74	C₂×C₈×C₁₃	C₂²⋊C₄	1	1	yes	characteristic
75	C₂×C₈×C₁₃	C₂²×C₄	1	1	yes	characteristic
76	C₂×C₈×C₁₃	C₂×D₄	1	1	yes	characteristic
77	C₂×C₈×C₁₃	C₂×D₄	1	1	yes	characteristic
78	C₂×C₈×C₁₃	C₂×Q₈	1	1	yes	characteristic
79	C₂×C₈×C₁₃	#41	1	1	yes	characteristic
80	C₂×C₈×C₁₃	#41	1	1	yes	characteristic
81	C₄²×C₁₃	C₂²⋊C₄	1	1	yes	characteristic
82	C₄²×C₁₃	C₂²⋊C₄	1	1	yes	characteristic
83	C₄²×C₁₃	D₈	1	1	yes	characteristic
84	C₄²×C₁₃	D₈	1	1	yes	characteristic
85	C₄⋊Dic₁₃	—	2	2	no	—
86	C₄⋊Dic₁₃	—	4	1	no	—
87	#6101	—	4	2	no	—
88	#6101	—	4	2	no	—
89	C₄×Dic₁₃	—	4	2	no	—
90	C₄×Dic₁₃	—	4	2	no	—
91	Dic₅₂	—	2	2	no	—
92	Dic₅₂	—	2	2	no	—
93	Dic₅₂	—	2	2	no	—
94	Dic₅₂	C₄⋊C₄	1	4	yes	—
95	#6090	—	4	2	no	—
96	C₂×C₄×Q₁₆	—	13	1	no	—
97	#2150	—	13	1	no	—
98	C₂×#362	—	13	2	no	—
99	#1848	D₁₃	1	1	yes	characteristic
100	#1387	—	13	2	no	—
101	C₂³×C₁₃	C₂×SD₁₆	1	1	yes	characteristic
102	C₂×C₄×C₁₃	—	2	2	no	—
103	C₂×C₄×C₁₃	—	4	1	no	—
104	C₂×C₄×C₁₃	#102	1	1	yes	characteristic
105	C₂×C₄×C₁₃	#102	1	1	yes	characteristic
106	C₂×C₄×C₁₃	#102	1	1	yes	characteristic
107	C₂×C₄×C₁₃	#102	1	1	yes	characteristic
108	C₂×C₄×C₁₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
109	C₂×C₄×C₁₃	#121	1	1	yes	characteristic
110	C₂×C₄×C₁₃	C₂²⋊Q₈	1	1	yes	characteristic
111	C₂×C₄×C₁₃	C₂×D₈	1	1	yes	characteristic
112	C₂×C₄×C₁₃	AGL(1,ℤ/8)	1	1	yes	characteristic
113	C₂×C₄×C₁₃	#137	1	1	yes	characteristic
114	C₂×Dic₁₃	—	4	2	no	—
115	C₂×Dic₁₃	—	4	2	no	—
116	C₂×Dic₁₃	—	4	2	no	—
117	C₂×Dic₁₃	—	4	2	no	—
118	C₂×Dic₁₃	—	8	2	no	—
119	Dic₂₆	—	2	4	no	—
120	Dic₂₆	—	4	2	no	—
121	Dic₂₆	—	4	2	no	—
122	Dic₂₆	—	4	2	no	—
123	C₈×C₁₃	—	2	1	no	—
124	C₈×C₁₃	—	2	1	no	—
125	C₈×C₁₃	—	8	2	no	—
126	C₈×C₁₃	C₂×C₄⋊C₄	1	1	yes	characteristic
127	C₈×C₁₃	#117	1	1	yes	characteristic
128	C₈×C₁₃	#121	1	1	yes	characteristic
129	C₈×C₁₃	C₂²⋊Q₈	1	1	yes	characteristic
130	C₂²×Q₁₆	—	13	1	no	—
131	C₂×C₄×Q₈	—	26	1	no	—
132	C₂×M₃₂	—	13	2	no	—
133	C₄×Q₁₆	—	13	2	no	—
134	C₄×Q₁₆	—	13	2	no	—
135	C₄×Q₁₆	—	26	1	no	—
136	C₄×Q₁₆	—	26	1	no	—
137	C₂×#108	—	2	1	no	—
138	C₂×#108	C₂×D₁₃	1	2	yes	—
139	C₂×#107	—	13	1	no	—
140	#423	—	2	1	no	—
141	C₂×#103	—	26	1	no	—
142	C₂×C₄×C₈	C₂×D₁₃	1	1	yes	characteristic
143	C₄⋊C₁₆	—	26	1	no	—
144	C₄⋊C₁₆	—	26	1	no	—
145	#362	—	26	4	no	—
146	C₂²×C₁₃	—	2	1	no	—
147	C₂²×C₁₃	—	2	1	no	—
148	C₂²×C₁₃	C₂×#102	1	1	yes	characteristic
149	C₂²×C₁₃	#461	1	1	yes	characteristic
150	C₂²×C₁₃	#476	1	1	yes	characteristic
151	C₄×C₁₃	—	2	1	no	—
152	C₄×C₁₃	—	2	1	no	—
153	C₄×C₁₃	—	2	1	no	—
154	C₄×C₁₃	#362	1	2	yes	complemented
155	C₄×C₁₃	#380	1	1	yes	characteristic
156	C₄×C₁₃	#417	1	1	yes	characteristic
157	C₄×C₁₃	#459	1	1	yes	characteristic
158	C₄×C₁₃	#474	1	1	yes	characteristic
159	Dic₁₃	—	4	4	no	—
160	Dic₁₃	—	8	2	no	—
161	C₂²×Q₈	—	26	1	no	—
162	C₂×Q₁₆	—	13	2	no	—
163	C₂×Q₁₆	—	13	2	no	—
164	C₂×Q₁₆	—	13	2	no	—
165	C₂×Q₁₆	—	26	1	no	—
166	C₂×Q₁₆	—	26	1	no	—
167	C₂×Q₁₆	—	26	1	no	—
168	C₂×M₁₆	—	2	2	no	—
169	C₂²×C₈	C₂²⋊D₁₃	1	1	yes	characteristic
170	C₂²×C₈	C₂²×D₁₃	1	1	yes	characteristic
171	C₄×Q₈	—	26	2	no	—
172	C₄×Q₈	—	26	2	no	—
173	C₄×Q₈	—	52	1	no	—
174	C₄×Q₈	—	52	1	no	—
175	C₂×C₄⋊C₄	—	26	1	no	—
176	C₂×C₄⋊C₄	—	52	1	no	—
177	C₂×C₄²	—	52	1	no	—
178	C₂×C₄²	C₂²⋊D₁₃	1	1	yes	characteristic
179	M₃₂	—	26	2	no	—
180	M₃₂	—	26	2	no	—
181	C₂×C₁₆	—	26	2	no	—
182	#108	—	2	2	no	—
183	#108	—	2	2	no	—
184	#108	—	4	2	no	—
185	#107	—	13	2	no	—
186	#107	—	26	1	no	—
187	C₄⋊C₈	—	4	1	no	—
188	C₄⋊C₈	—	4	1	no	—
189	#103	—	26	2	no	—
190	#103	—	26	2	no	—
191	C₄×C₈	C₄×D₁₃	1	1	yes	characteristic
192	C₄×C₈	C₄×D₁₃	1	1	yes	characteristic
193	C₄×C₈	D₅₂	1	1	yes	characteristic
194	C₄×C₈	D₅₂	1	1	yes	characteristic
195	C₂×C₁₃	—	2	1	no	—
196	C₂×C₁₃	—	2	1	no	—
197	C₂×C₁₃	#1802	1	1	yes	characteristic
198	C₂×C₁₃	C₂×#362	1	1	yes	characteristic
199	C₂×C₁₃	#2148	1	1	yes	characteristic
200	C₂×Q₈	—	26	2	no	—
201	C₂×Q₈	—	26	2	no	—
202	C₂×Q₈	—	26	2	no	—
203	C₂×Q₈	—	52	1	no	—
204	C₂×Q₈	—	52	1	no	—
205	C₂×Q₈	—	52	1	no	—
206	C₂²×C₄	—	52	1	no	—
207	C₂²×C₄	—	52	1	no	—
208	C₂²×C₄	#6106	1	1	yes	characteristic
209	C₂²×C₄	#6106	1	1	yes	characteristic
210	C₂²×C₄	C₂×C₂²⋊D₁₃	1	1	yes	characteristic
211	Q₁₆	—	13	4	no	—
212	Q₁₆	—	26	2	no	—
213	Q₁₆	—	26	2	no	—
214	Q₁₆	—	26	2	no	—
215	M₁₆	—	4	2	no	—
216	M₁₆	—	4	2	no	—
217	C₂×C₈	—	2	1	no	—
218	C₂×C₈	—	2	1	no	—
219	C₂×C₈	—	4	2	no	—
220	C₂×C₈	#6103	1	1	yes	characteristic
221	C₂×C₈	#6103	1	1	yes	characteristic
222	C₂×C₈	C₂×C₄×D₁₃	1	1	yes	characteristic
223	C₂×C₈	C₂×D₅₂	1	1	yes	characteristic
224	C₂×C₈	D₄×D₁₃	1	1	yes	characteristic
225	C₂×C₈	#6129	1	1	yes	characteristic
226	C₂×C₈	Q₈×D₁₃	1	1	yes	characteristic
227	C₂×C₈	#6131	1	1	yes	characteristic
228	C₄⋊C₄	—	26	2	no	—
229	C₄⋊C₄	—	52	1	no	—
230	C₄⋊C₄	—	52	2	no	—
231	C₄⋊C₄	—	52	2	no	—
232	C₄²	—	52	2	no	—
233	C₄²	—	52	2	no	—
234	C₄²	#6103	1	1	yes	characteristic
235	C₄²	#6103	1	1	yes	characteristic
236	C₄²	#6104	1	1	yes	characteristic
237	C₄²	#6104	1	1	yes	characteristic
238	C₁₆	—	52	2	no	—
239	C₁₃	#12819	1	1	yes	characteristic, complemented, Sylow
240	C₂³	C₂×#6106	1	1	yes	characteristic
241	Q₈	—	26	4	no	—
242	Q₈	—	52	2	no	—
243	Q₈	—	52	2	no	—
244	Q₈	—	52	2	no	—
245	C₂×C₄	—	2	2	no	—
246	C₂×C₄	—	4	1	no	—
247	C₂×C₄	—	52	2	no	—
248	C₂×C₄	—	52	2	no	—
249	C₂×C₄	—	52	2	no	—
250	C₂×C₄	—	52	2	no	—
251	C₂×C₄	—	104	2	no	—
252	C₂×C₄	#88049	1	1	yes	characteristic
253	C₂×C₄	#88049	1	1	yes	characteristic
254	C₂×C₄	#88049	1	1	yes	characteristic
255	C₂×C₄	#88049	1	1	yes	characteristic
256	C₂×C₄	C₂×#6103	1	1	yes	characteristic
257	C₂×C₄	C₂×#6104	1	1	yes	characteristic
258	C₂×C₄	#88186	1	1	yes	characteristic
259	C₂×C₄	#88192	1	1	yes	characteristic
260	C₂×C₄	#88196	1	1	yes	characteristic
261	C₂×C₄	#88200	1	1	yes	characteristic
262	C₈	—	2	1	no	—
263	C₈	—	2	1	no	—
264	C₈	—	8	2	no	—
265	C₈	C₄⋊C₄×D₁₃	1	1	yes	characteristic
266	C₈	#88146	1	1	yes	characteristic
267	C₈	#88149	1	1	yes	characteristic
268	C₈	#88151	1	1	yes	characteristic
269	C₂²	—	2	1	no	—
270	C₂²	—	2	1	no	—
271	C₂²	C₂×#88049	1	1	yes	characteristic
272	C₂²	#11718744	1	1	yes	characteristic
273	C₂²	#11718745	1	1	yes	characteristic
274	C₄	—	2	1	no	—
275	C₄	—	2	1	no	—
276	C₄	—	2	1	no	—
277	C₄	—	52	4	no	—
278	C₄	—	104	2	no	—
279	C₄	#11718088	1	2	yes	complemented
280	C₄	#11718627	1	1	yes	characteristic
281	C₄	#11718648	1	1	yes	characteristic
282	C₄	#11718733	1	1	yes	characteristic
283	C₄	#11718748	1	1	yes	characteristic
284	C₂	—	2	1	no	—
285	C₂	—	2	1	no	—
286	C₂	#13114833	1	1	yes	characteristic
287	C₂	C₂×#11718088	1	1	yes	characteristic
288	C₂	#13116583	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×#88049
commutator	C₂×C₈×C₁₃	C₂²×C₄
Frattini	C₂²×C₈	C₂²×D₁₃
Fitting	C₁₃×#1848	C₂
socle	C₂²×C₁₃	C₂×#102
prime radical	C₂³×C₁₃	C₂×SD₁₆
cosocle	C₂²×C₈×C₁₃	C₂³
hypercenter	#1848	D₁₃
hypocenter	C₁₃	#12819
LCS	G ▷ C₂×C₈×C₁₃ ▷ C₄×C₁₃ ▷ C₂×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ #1848

Complements

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

N	H ≅ G/N	type
154. C₄×C₁₃	#362	semidirect
239. C₁₃	#12819	semidirect
279. C₄	#11718088	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₄×Dic₅₂	2
#13116592	2
C₂×#11718088	2
#13112946	2
C₁₃×#1848	2
#12819	13