Subgroups — #13124907

Order 1664.

390 conjugacy classes of subgroups.

The subgroup lattice has 274 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄×C₁₃×#41	C₂	1	1	yes	maximal, characteristic
2	#11719175	C₂	1	1	yes	maximal, characteristic, complemented
3	#11718900	C₂	1	2	yes	maximal
4	#11718692	C₂	1	1	yes	maximal, characteristic, complemented
5	C₄×#6107	C₂	1	2	yes	maximal, complemented
6	#11718681	C₂	1	1	yes	maximal, characteristic, complemented
7	#11718680	C₂	1	1	yes	maximal, characteristic, complemented
8	C₄×#6105	C₂	1	2	yes	maximal, complemented
9	#11718668	C₂	1	1	yes	maximal, characteristic, complemented
10	#11718657	C₂	1	2	yes	maximal, complemented
11	#11718586	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×C₁₃×#41	C₂²	1	1	yes	characteristic
13	C₄×Q₈×C₁₃	C₂²	1	1	yes	characteristic
14	C₄×Q₈×C₁₃	C₂²	1	1	yes	characteristic
15	C₄×D₄×C₁₃	—	2	1	no	—
16	C₄×D₄×C₁₃	—	2	1	no	—
17	C₄×D₄×C₁₃	C₂²	1	1	yes	characteristic
18	C₄×D₄×C₁₃	C₂²	1	1	yes	characteristic
19	C₁₃×#117	—	2	1	no	—
20	C₁₃×#117	C₂²	1	1	yes	characteristic
21	C₂×C₄²×C₁₃	—	2	1	no	—
22	C₂×C₄²×C₁₃	C₂²	1	1	yes	characteristic
23	C₂×#6107	C₂²	1	2	yes	—
24	C₂×#6105	C₂²	1	2	yes	—
25	#88180	—	2	1	no	—
26	#88178	—	2	1	no	—
27	#88178	C₂²	1	2	yes	—
28	#88124	—	2	1	no	—
29	#88123	C₂²	1	1	yes	characteristic
30	#88123	C₂²	1	1	yes	characteristic
31	C₄×Dic₂₆	—	2	1	no	—
32	C₄×Dic₂₆	C₂²	1	2	yes	—
33	#88075	C₂²	1	2	yes	—
34	#88072	C₂²	1	2	yes	—
35	C₂²⋊#1055	C₂²	1	2	yes	—
36	#88050	C₂²	1	2	yes	—
37	#88050	C₂²	1	2	yes	—
38	#88048	C₂²	1	2	yes	—
39	#88047	C₂²	1	2	yes	—
40	C₄⋊#1055	C₂²	1	1	yes	characteristic
41	C₄⋊#1055	C₂²	1	1	yes	characteristic
42	C₄×#1055	C₂²	1	2	yes	—
43	C₁₃×#41	—	2	2	no	—
44	C₁₃×#41	—	2	2	no	—
45	C₂×Q₈×C₁₃	C₂³	1	1	yes	characteristic
46	C₂×D₄×C₁₃	—	2	1	no	—
47	C₂×D₄×C₁₃	C₂³	1	1	yes	characteristic
48	C₂²×C₄×C₁₃	—	2	1	no	—
49	C₂²×C₄×C₁₃	—	2	1	no	—
50	C₂²×C₄×C₁₃	—	2	1	no	—
51	C₂²×C₄×C₁₃	D₄	1	1	yes	characteristic
52	C₂²×C₄×C₁₃	D₄	1	1	yes	characteristic
53	C₂²×C₄×C₁₃	C₂³	1	1	yes	characteristic
54	C₂×Dic₂₆	—	2	1	no	—
55	C₂×Dic₂₆	C₂³	1	2	yes	—
56	C₁₃×C₄⋊C₄	—	2	1	no	—
57	C₁₃×C₄⋊C₄	—	2	1	no	—
58	C₁₃×C₄⋊C₄	C₂³	1	1	yes	characteristic
59	C₁₃×C₄⋊C₄	C₂³	1	1	yes	characteristic
60	C₁₃×C₂²⋊C₄	—	2	1	no	—
61	C₁₃×C₂²⋊C₄	—	2	1	no	—
62	C₁₃×C₂²⋊C₄	—	2	1	no	—
63	C₄²×C₁₃	—	2	1	no	—
64	C₄²×C₁₃	—	2	1	no	—
65	C₄²×C₁₃	—	2	1	no	—
66	C₄²×C₁₃	D₄	1	1	yes	characteristic
67	C₄²×C₁₃	D₄	1	1	yes	characteristic
68	C₄²×C₁₃	C₂³	1	1	yes	characteristic
69	C₄²×C₁₃	C₂³	1	1	yes	characteristic
70	C₂²⋊Dic₁₃	—	2	2	no	—
71	#6107	—	2	4	no	—
72	#6105	—	2	4	no	—
73	C₄⋊Dic₁₃	—	2	1	no	—
74	C₄⋊Dic₁₃	—	2	1	no	—
75	C₄⋊Dic₁₃	—	2	1	no	—
76	C₄⋊Dic₁₃	C₂³	1	2	yes	—
77	#6101	—	2	2	no	—
78	#6101	—	2	2	no	—
79	C₄×Dic₁₃	—	2	2	no	—
80	C₂×#1055	C₂³	1	2	yes	—
81	C₂×#1055	C₂³	1	2	yes	—
82	#3107	—	13	1	no	maximal, Sylow
83	C₂³×C₁₃	—	2	1	no	—
84	C₂³×C₁₃	C₂×D₄	1	1	yes	characteristic
85	Q₈×C₁₃	—	2	1	no	—
86	Q₈×C₁₃	#41	1	2	yes	—
87	D₄×C₁₃	—	2	4	no	—
88	D₄×C₁₃	—	2	1	no	—
89	D₄×C₁₃	#41	1	2	yes	—
90	C₂×C₄×C₁₃	—	2	4	no	—
91	C₂×C₄×C₁₃	—	2	1	no	—
92	C₂×C₄×C₁₃	—	2	2	no	—
93	C₂×C₄×C₁₃	—	2	2	no	—
94	C₂×C₄×C₁₃	—	2	1	no	—
95	C₂×C₄×C₁₃	—	2	1	no	—
96	C₂×C₄×C₁₃	—	2	1	no	—
97	C₂×C₄×C₁₃	—	2	1	no	—
98	C₂×C₄×C₁₃	—	4	2	no	—
99	C₂×C₄×C₁₃	—	4	2	no	—
100	C₂×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
101	C₂×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
102	C₂×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
103	C₂×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
104	C₂×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
105	C₂×C₄×C₁₃	C₂⁴	1	1	yes	characteristic
106	C₂×Dic₁₃	—	2	2	no	—
107	C₂×Dic₁₃	—	2	2	no	—
108	Dic₂₆	—	2	2	no	—
109	Dic₂₆	—	2	2	no	—
110	Dic₂₆	—	4	2	no	—
111	#1055	—	2	2	no	—
112	#1055	—	2	2	no	—
113	#533	—	13	1	no	—
114	C₄×#41	D₁₃	1	1	yes	characteristic
115	#485	—	13	2	no	—
116	#475	—	13	1	no	—
117	#474	—	13	1	no	—
118	#462	—	13	1	no	—
119	#461	—	13	1	no	—
120	#452	—	13	2	no	—
121	C₄×Q₁₆	—	13	2	no	—
122	C₄×SD₁₆	—	13	2	no	—
123	#431	—	13	1	no	—
124	C₂²×C₁₃	—	2	1	no	—
125	C₂²×C₁₃	—	2	2	no	—
126	C₂²×C₁₃	—	2	1	no	—
127	C₂²×C₁₃	—	2	1	no	—
128	C₂²×C₁₃	—	4	1	no	—
129	C₂²×C₁₃	—	4	1	no	—
130	C₂²×C₁₃	C₂²×D₄	1	1	yes	characteristic
131	C₄×C₁₃	—	2	1	no	—
132	C₄×C₁₃	—	2	2	no	—
133	C₄×C₁₃	—	2	1	no	—
134	C₄×C₁₃	—	2	1	no	—
135	C₄×C₁₃	—	4	1	no	—
136	C₄×C₁₃	—	4	1	no	—
137	C₄×C₁₃	#135	1	2	yes	—
138	C₄×C₁₃	#137	1	2	yes	—
139	C₄×C₁₃	C₂×#41	1	1	yes	characteristic
140	C₄×C₁₃	C₂×#41	1	1	yes	characteristic
141	Dic₁₃	—	4	2	no	—
142	Dic₁₃	—	4	2	no	—
143	C₂×#41	C₂×D₁₃	1	1	yes	characteristic
144	C₂×Q₁₆	—	13	2	no	—
145	C₂×SD₁₆	—	13	2	no	—
146	C₄⋊Q₈	—	13	1	no	—
147	C₄⋊Q₈	—	13	1	no	—
148	#125	—	26	1	no	—
149	C₂²⋊Q₈	—	13	2	no	—
150	C₂²⋊Q₈	—	26	1	no	—
151	C₄×Q₈	—	13	2	no	—
152	C₄×Q₈	—	26	1	no	—
153	C₄×Q₈	C₂×D₁₃	1	1	yes	characteristic
154	C₄×Q₈	C₂×D₁₃	1	1	yes	characteristic
155	C₄×D₄	—	2	1	no	—
156	C₄×D₄	—	2	1	no	—
157	C₄×D₄	C₂×D₁₃	1	1	yes	characteristic
158	C₄×D₄	C₂×D₁₃	1	1	yes	characteristic
159	#117	—	2	1	no	—
160	#117	—	26	1	no	—
161	#117	C₂×D₁₃	1	1	yes	characteristic
162	C₂×C₄²	—	2	1	no	—
163	C₂×C₄²	C₂×D₁₃	1	1	yes	characteristic
164	#107	—	13	2	no	—
165	#106	—	13	2	no	—
166	C₄⋊C₈	—	13	1	no	—
167	C₄⋊C₈	—	13	1	no	—
168	#103	—	13	2	no	—
169	#103	—	13	2	no	—
170	#103	—	13	2	no	—
171	#102	—	13	2	no	—
172	C₂²⋊C₈	—	13	2	no	—
173	C₄×C₈	—	13	2	no	—
174	C₂×C₁₃	—	4	1	no	—
175	C₂×C₁₃	—	4	2	no	—
176	C₂×C₁₃	#525	1	1	yes	characteristic
177	C₂×C₁₃	C₂×#135	1	1	yes	characteristic
178	C₂×C₁₃	C₂×#137	1	1	yes	characteristic
179	#41	—	2	2	no	—
180	#41	—	2	2	no	—
181	C₂×Q₈	—	13	2	no	—
182	C₂×Q₈	—	26	1	no	—
183	C₂×Q₈	C₂²×D₁₃	1	1	yes	characteristic
184	C₂×D₄	—	2	1	no	—
185	C₂×D₄	C₂²×D₁₃	1	1	yes	characteristic
186	C₂²×C₄	—	2	1	no	—
187	C₂²×C₄	—	2	1	no	—
188	C₂²×C₄	—	2	1	no	—
189	C₂²×C₄	C₂²⋊D₁₃	1	1	yes	characteristic
190	C₂²×C₄	C₂²⋊D₁₃	1	1	yes	characteristic
191	C₂²×C₄	C₂²×D₁₃	1	1	yes	characteristic
192	Q₁₆	—	26	4	no	—
193	SD₁₆	—	26	4	no	—
194	C₂×C₈	—	13	2	no	—
195	C₂×C₈	—	13	2	no	—
196	C₄⋊C₄	—	2	1	no	—
197	C₄⋊C₄	—	2	1	no	—
198	C₄⋊C₄	—	13	2	no	—
199	C₄⋊C₄	—	26	1	no	—
200	C₄⋊C₄	—	26	1	no	—
201	C₄⋊C₄	—	26	2	no	—
202	C₄⋊C₄	—	26	2	no	—
203	C₄⋊C₄	—	26	1	no	—
204	C₄⋊C₄	C₂²×D₁₃	1	1	yes	characteristic
205	C₄⋊C₄	C₂²×D₁₃	1	1	yes	characteristic
206	C₂²⋊C₄	—	2	1	no	—
207	C₂²⋊C₄	—	2	1	no	—
208	C₂²⋊C₄	—	2	1	no	—
209	C₂²⋊C₄	—	26	2	no	—
210	C₄²	—	2	1	no	—
211	C₄²	—	2	1	no	—
212	C₄²	—	2	1	no	—
213	C₄²	—	26	2	no	—
214	C₄²	C₂²⋊D₁₃	1	1	yes	characteristic
215	C₄²	C₂²⋊D₁₃	1	1	yes	characteristic
216	C₄²	C₂²×D₁₃	1	1	yes	characteristic
217	C₄²	C₂²×D₁₃	1	1	yes	characteristic
218	C₁₃	#3107	1	1	yes	characteristic, complemented, Sylow
219	C₂³	—	2	1	no	—
220	C₂³	C₂×C₂²⋊D₁₃	1	1	yes	characteristic
221	Q₈	—	2	1	no	—
222	Q₈	—	26	2	no	—
223	Q₈	—	26	2	no	—
224	Q₈	—	52	2	no	—
225	Q₈	#6127	1	2	yes	—
226	D₄	—	2	4	no	—
227	D₄	—	2	1	no	—
228	D₄	#6127	1	2	yes	—
229	C₂×C₄	—	2	4	no	—
230	C₂×C₄	—	2	1	no	—
231	C₂×C₄	—	2	2	no	—
232	C₂×C₄	—	2	2	no	—
233	C₂×C₄	—	2	1	no	—
234	C₂×C₄	—	2	1	no	—
235	C₂×C₄	—	2	1	no	—
236	C₂×C₄	—	2	1	no	—
237	C₂×C₄	—	4	2	no	—
238	C₂×C₄	—	4	2	no	—
239	C₂×C₄	—	26	2	no	—
240	C₂×C₄	—	26	2	no	—
241	C₂×C₄	C₂×C₂²⋊D₁₃	1	1	yes	characteristic
242	C₂×C₄	C₂×C₂²⋊D₁₃	1	1	yes	characteristic
243	C₂×C₄	C₂×C₂²⋊D₁₃	1	1	yes	characteristic
244	C₂×C₄	C₂×C₂²⋊D₁₃	1	1	yes	characteristic
245	C₂×C₄	C₂×C₂²⋊D₁₃	1	1	yes	characteristic
246	C₂×C₄	C₂³×D₁₃	1	1	yes	characteristic
247	C₈	—	26	2	no	—
248	C₈	—	26	2	no	—
249	C₂²	—	2	1	no	—
250	C₂²	—	2	2	no	—
251	C₂²	—	2	1	no	—
252	C₂²	—	2	1	no	—
253	C₂²	—	4	1	no	—
254	C₂²	—	4	1	no	—
255	C₂²	C₂²×C₂²⋊D₁₃	1	1	yes	characteristic
256	C₄	—	2	1	no	—
257	C₄	—	2	2	no	—
258	C₄	—	2	1	no	—
259	C₄	—	2	1	no	—
260	C₄	—	4	1	no	—
261	C₄	—	4	1	no	—
262	C₄	—	52	2	no	—
263	C₄	—	52	2	no	—
264	C₄	#88204	1	2	yes	—
265	C₄	#88205	1	2	yes	—
266	C₄	C₂×#6127	1	1	yes	characteristic
267	C₄	C₂×#6127	1	1	yes	characteristic
268	C₂	—	4	1	no	—
269	C₂	—	4	2	no	—
270	C₂	#11719496	1	1	yes	characteristic
271	C₂	C₂×#88204	1	1	yes	characteristic
272	C₂	C₂×#88205	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₂×C₂²⋊D₁₃
commutator, cosocle	C₂×C₄×C₁₃	C₂⁴
Frattini	C₂×C₄	C₂³×D₁₃
Fitting	C₄×C₁₃×#41	C₂
socle	C₂²×C₁₃	C₂²×D₄
prime radical	C₂×C₁₃×#41	C₂²
hypercenter	C₄×#41	D₁₃
hypocenter	C₁₃	#3107
LCS	G ▷ C₂×C₄×C₁₃ ▷ C₂×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ C₄×#41

Complements

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

N	H ≅ G/N	type
2. #11719175	C₂	semidirect
4. #11718692	C₂	semidirect
5. C₄×#6107	C₂	semidirect
6. #11718681	C₂	semidirect
7. #11718680	C₂	semidirect
8. C₄×#6105	C₂	semidirect
9. #11718668	C₂	semidirect
10. #11718657	C₂	semidirect
11. #11718586	C₂	semidirect
218. C₁₃	#3107	semidirect

Maximal subgroups

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

subgroup	index
C₄×C₁₃×#41	2
#11719175	2
#11718900	2
#11718692	2
C₄×#6107	2
#11718681	2
#11718680	2
C₄×#6105	2
#11718668	2
#11718657	2
#11718586	2
#3107	13