Subgroups — #13118708

Order 1664.

548 conjugacy classes of subgroups.

The subgroup lattice has 288 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₄×D₅₂	C₂	1	1	yes	maximal, characteristic
2	C₂×#88124	C₂	1	1	yes	maximal, characteristic, complemented
3	C₄×C₁₃×C₄⋊C₄	C₂	1	1	yes	maximal, characteristic, complemented
4	#11718653	C₂	1	2	yes	maximal, complemented
5	#11718650	C₂	1	2	yes	maximal, complemented
6	#11718600	C₂	1	1	yes	maximal, characteristic
7	#11718600	C₂	1	1	yes	maximal, characteristic
8	#11718589	C₂	1	1	yes	maximal, characteristic, complemented
9	#11718589	C₂	1	1	yes	maximal, characteristic, complemented
10	#11718337	C₂	1	2	yes	maximal, complemented
11	#11718328	C₂	1	2	yes	maximal, complemented
12	C₂²×D₅₂	C₂²	1	1	yes	characteristic
13	C₂²×C₄×D₁₃	C₂²	1	2	yes	—
14	C₂×C₁₃×C₄⋊C₄	C₂²	1	1	yes	characteristic
15	C₂×C₁₃×C₄⋊C₄	C₂²	1	1	yes	characteristic
16	C₂×C₄²×C₁₃	C₂²	1	1	yes	characteristic
17	C₂×C₄²×C₁₃	C₂²	1	1	yes	characteristic
18	C₂×C₄²×C₁₃	C₂²	1	1	yes	characteristic
19	C₂×#6103	C₂²	1	2	yes	—
20	C₂×#6103	C₂²	1	2	yes	—
21	C₂×#6103	C₂²	1	2	yes	—
22	C₂×#6103	C₂²	1	2	yes	—
23	C₂×#6103	C₂²	1	2	yes	—
24	C₂×C₄⋊Dic₁₃	C₂²	1	1	yes	characteristic
25	C₂×C₄⋊Dic₁₃	C₂²	1	1	yes	characteristic
26	C₂×C₄⋊Dic₁₃	C₂²	1	1	yes	characteristic
27	C₂×#6101	C₂²	1	2	yes	—
28	C₂×#6101	C₂²	1	2	yes	—
29	C₂×#6101	C₂²	1	2	yes	—
30	C₂×#6101	C₂²	1	2	yes	—
31	C₄×D₅₂	—	2	4	no	—
32	#88124	—	2	4	no	—
33	C₁₃×#95	C₂²	1	1	yes	characteristic
34	C₁₃×#95	C₂²	1	1	yes	characteristic
35	#88071	C₂²	1	2	yes	—
36	#88071	C₂²	1	2	yes	—
37	C₂³×D₁₃	C₂³	1	2	yes	—
38	C₂²×C₄×C₁₃	C₂³	1	1	yes	characteristic
39	C₂²×C₄×C₁₃	C₂³	1	1	yes	characteristic
40	C₂²×C₄×C₁₃	C₂³	1	1	yes	characteristic
41	C₂²×C₄×C₁₃	C₂³	1	1	yes	characteristic
42	C₂²×C₄×C₁₃	C₂³	1	1	yes	characteristic
43	C₂²×C₄×C₁₃	C₂³	1	1	yes	characteristic
44	C₂²×C₄×C₁₃	C₂³	1	1	yes	characteristic
45	C₂²×Dic₁₃	C₂³	1	2	yes	—
46	C₂²×Dic₁₃	C₂³	1	2	yes	—
47	C₂²×Dic₁₃	C₂³	1	2	yes	—
48	C₂×D₅₂	—	2	2	no	—
49	C₂×D₅₂	—	2	2	no	—
50	C₂×D₅₂	Q₈	1	4	yes	—
51	C₂×C₄×D₁₃	—	2	4	no	—
52	C₂×C₄×D₁₃	—	2	4	no	—
53	C₂×C₄×D₁₃	—	4	2	no	—
54	C₁₃×C₄⋊C₄	—	2	2	no	—
55	C₁₃×C₄⋊C₄	—	2	2	no	—
56	C₄²×C₁₃	—	2	2	no	—
57	C₄²×C₁₃	—	2	2	no	—
58	C₄²×C₁₃	—	2	2	no	—
59	#6103	—	2	4	no	—
60	#6103	—	2	4	no	—
61	#6103	—	4	2	no	—
62	#6103	—	4	2	no	—
63	#6103	—	4	2	no	—
64	C₄⋊Dic₁₃	—	2	2	no	—
65	C₄⋊Dic₁₃	—	2	2	no	—
66	C₄⋊Dic₁₃	D₄	1	4	yes	—
67	#6101	—	2	4	no	—
68	#6101	—	2	4	no	—
69	#6101	—	4	2	no	—
70	#6101	—	4	2	no	—
71	#2454	—	13	1	no	maximal, Sylow
72	C₂³×C₁₃	C₂⁴	1	1	yes	characteristic
73	C₂²×D₁₃	—	2	2	no	—
74	C₂²×D₁₃	—	2	2	no	—
75	C₂²×D₁₃	—	2	2	no	—
76	C₂²×D₁₃	—	2	2	no	—
77	C₂²×D₁₃	—	2	2	no	—
78	C₂²×D₁₃	—	2	2	no	—
79	C₂²×D₁₃	C₂×Q₈	1	4	yes	—
80	C₂×C₄×C₁₃	—	2	1	no	—
81	C₂×C₄×C₁₃	—	2	1	no	—
82	C₂×C₄×C₁₃	—	2	1	no	—
83	C₂×C₄×C₁₃	—	2	1	no	—
84	C₂×C₄×C₁₃	—	2	1	no	—
85	C₂×C₄×C₁₃	—	2	1	no	—
86	C₂×C₄×C₁₃	—	2	1	no	—
87	C₂×C₄×C₁₃	—	2	1	no	—
88	C₂×C₄×C₁₃	—	2	1	no	—
89	C₂×C₄×C₁₃	—	2	1	no	—
90	C₂×C₄×C₁₃	—	2	1	no	—
91	C₂×C₄×C₁₃	—	2	1	no	—
92	C₂×C₄×C₁₃	—	2	1	no	—
93	C₂×C₄×C₁₃	—	2	1	no	—
94	C₂×C₄×C₁₃	—	2	1	no	—
95	C₂×C₄×C₁₃	—	2	1	no	—
96	C₂×C₄×C₁₃	C₂×D₄	1	2	yes	—
97	C₂×C₄×C₁₃	C₂×Q₈	1	2	yes	—
98	C₂×C₄×C₁₃	#41	1	2	yes	—
99	C₂×C₄×C₁₃	#41	1	2	yes	—
100	C₂×C₄×C₁₃	#41	1	2	yes	—
101	C₂×Dic₁₃	—	2	2	no	—
102	C₂×Dic₁₃	—	2	2	no	—
103	C₂×Dic₁₃	—	2	2	no	—
104	C₂×Dic₁₃	—	2	2	no	—
105	C₂×Dic₁₃	—	2	2	no	—
106	C₂×Dic₁₃	—	2	2	no	—
107	C₂×Dic₁₃	—	2	2	no	—
108	C₂×Dic₁₃	—	2	2	no	—
109	C₂×Dic₁₃	C₂×D₄	1	4	yes	—
110	D₅₂	—	2	8	no	—
111	C₄×D₁₃	—	4	8	no	—
112	C₂×#125	—	13	1	no	—
113	C₂×C₄×D₄	—	13	1	no	—
114	#399	—	13	2	no	—
115	#396	—	13	2	no	—
116	#390	—	13	1	no	—
117	#390	—	13	1	no	—
118	#389	—	13	1	no	—
119	#389	—	13	1	no	—
120	#385	—	13	2	no	—
121	#380	—	13	2	no	—
122	C₄×C₄⋊C₄	D₁₃	1	1	yes	characteristic, complemented
123	C₂²×C₁₃	C₂²×D₄	1	1	yes	characteristic
124	C₂²×C₁₃	C₂²×Q₈	1	1	yes	characteristic
125	C₂²×C₁₃	C₂×#41	1	1	yes	characteristic
126	C₂²×C₁₃	C₂×#41	1	1	yes	characteristic
127	C₂²×C₁₃	C₂×#41	1	1	yes	characteristic
128	C₂²×C₁₃	H(5,𝔽₂)	1	1	yes	characteristic
129	C₂²×C₁₃	H(5,𝔽₂)	1	1	yes	characteristic
130	C₂×D₁₃	—	2	4	no	—
131	C₂×D₁₃	—	2	4	no	—
132	C₂×D₁₃	—	2	4	no	—
133	C₂×D₁₃	—	4	2	no	—
134	C₂×D₁₃	—	4	2	no	—
135	C₂×D₁₃	—	4	2	no	—
136	C₂×D₁₃	—	4	2	no	—
137	C₄×C₁₃	—	2	2	no	—
138	C₄×C₁₃	—	2	2	no	—
139	C₄×C₁₃	—	4	1	no	—
140	C₄×C₁₃	—	4	1	no	—
141	C₄×C₁₃	—	4	1	no	—
142	C₄×C₁₃	—	4	1	no	—
143	C₄×C₁₃	C₂²⋊Q₈	1	4	yes	—
144	Dic₁₃	—	2	4	no	—
145	Dic₁₃	—	4	2	no	—
146	Dic₁₃	—	4	2	no	—
147	C₂²×D₄	—	13	1	no	—
148	C₂³×C₄	—	13	2	no	—
149	#125	—	26	4	no	—
150	C₄×D₄	—	26	4	no	—
151	C₂×C₄⋊C₄	—	13	1	no	—
152	C₂×C₄⋊C₄	—	13	2	no	—
153	C₂×C₄⋊C₄	—	13	2	no	—
154	C₂×C₄⋊C₄	—	13	1	no	—
155	C₂×C₄⋊C₄	—	13	1	no	—
156	C₂×C₄⋊C₄	—	13	2	no	—
157	C₂×C₄⋊C₄	—	13	2	no	—
158	C₂×C₄⋊C₄	C₂×D₁₃	1	1	yes	characteristic
159	C₂×C₄⋊C₄	C₂×D₁₃	1	1	yes	characteristic
160	C₂×C₂²⋊C₄	—	13	2	no	—
161	C₂×C₂²⋊C₄	—	13	2	no	—
162	C₂×C₂²⋊C₄	—	13	2	no	—
163	C₂×C₂²⋊C₄	—	13	2	no	—
164	C₂×C₂²⋊C₄	—	13	2	no	—
165	C₂×C₄²	C₂×D₁₃	1	1	yes	characteristic
166	C₂×C₄²	C₂×D₁₃	1	1	yes	characteristic
167	C₂×C₄²	C₂×D₁₃	1	1	yes	characteristic
168	#95	—	13	2	no	—
169	#95	—	13	2	no	—
170	#95	C₂×D₁₃	1	1	yes	characteristic
171	#95	C₂×D₁₃	1	1	yes	characteristic
172	C₂×C₁₃	C₂×C₂²⋊Q₈	1	1	yes	characteristic
173	C₂×C₁₃	#529	1	1	yes	characteristic
174	C₂×C₁₃	#540	1	1	yes	characteristic
175	C₂×C₁₃	#546	1	1	yes	characteristic
176	C₂×C₁₃	#550	1	1	yes	characteristic
177	C₂×C₁₃	#554	1	1	yes	characteristic
178	C₂×C₁₃	#554	1	1	yes	characteristic
179	D₁₃	—	4	4	no	—
180	C₂⁴	—	13	2	no	—
181	C₂×D₄	—	13	4	no	—
182	C₂×D₄	—	26	2	no	—
183	C₂×D₄	—	26	2	no	—
184	C₂²×C₄	—	13	2	no	—
185	C₂²×C₄	—	13	2	no	—
186	C₂²×C₄	—	13	2	no	—
187	C₂²×C₄	—	26	4	no	—
188	C₂²×C₄	—	26	4	no	—
189	C₂²×C₄	—	52	2	no	—
190	C₂²×C₄	C₂²×D₁₃	1	1	yes	characteristic
191	C₂²×C₄	C₂²×D₁₃	1	1	yes	characteristic
192	C₂²×C₄	C₂²×D₁₃	1	1	yes	characteristic
193	C₂²×C₄	C₂²×D₁₃	1	1	yes	characteristic
194	C₂²×C₄	C₂²×D₁₃	1	1	yes	characteristic
195	C₂²×C₄	C₂²×D₁₃	1	1	yes	characteristic
196	C₂²×C₄	C₂²×D₁₃	1	1	yes	characteristic
197	C₄⋊C₄	—	2	2	no	—
198	C₄⋊C₄	—	2	2	no	—
199	C₄⋊C₄	—	13	4	no	—
200	C₄⋊C₄	—	26	4	no	—
201	C₄⋊C₄	—	26	4	no	—
202	C₄⋊C₄	—	26	2	no	—
203	C₄⋊C₄	—	26	2	no	—
204	C₄⋊C₄	—	52	2	no	—
205	C₄⋊C₄	—	52	2	no	—
206	C₂²⋊C₄	—	26	4	no	—
207	C₂²⋊C₄	—	26	4	no	—
208	C₂²⋊C₄	—	52	2	no	—
209	C₂²⋊C₄	—	52	2	no	—
210	C₂²⋊C₄	—	52	2	no	—
211	C₄²	—	2	2	no	—
212	C₄²	—	2	2	no	—
213	C₄²	—	2	2	no	—
214	C₁₃	#2454	1	1	yes	characteristic, complemented, Sylow
215	C₂³	—	13	4	no	—
216	C₂³	—	26	2	no	—
217	C₂³	—	26	2	no	—
218	C₂³	—	26	2	no	—
219	C₂³	—	26	2	no	—
220	C₂³	—	26	2	no	—
221	C₂³	—	26	2	no	—
222	C₂³	C₂³×D₁₃	1	1	yes	characteristic
223	D₄	—	26	8	no	—
224	C₂×C₄	—	2	1	no	—
225	C₂×C₄	—	2	1	no	—
226	C₂×C₄	—	2	1	no	—
227	C₂×C₄	—	2	1	no	—
228	C₂×C₄	—	2	1	no	—
229	C₂×C₄	—	2	1	no	—
230	C₂×C₄	—	2	1	no	—
231	C₂×C₄	—	2	1	no	—
232	C₂×C₄	—	2	1	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	—	2	1	no	—
238	C₂×C₄	—	2	1	no	—
239	C₂×C₄	—	2	1	no	—
240	C₂×C₄	—	13	4	no	—
241	C₂×C₄	—	26	2	no	—
242	C₂×C₄	—	26	2	no	—
243	C₂×C₄	—	26	2	no	—
244	C₂×C₄	—	26	2	no	—
245	C₂×C₄	—	26	2	no	—
246	C₂×C₄	—	26	2	no	—
247	C₂×C₄	—	26	2	no	—
248	C₂×C₄	—	26	2	no	—
249	C₂×C₄	—	52	8	no	—
250	C₂×C₄	#6127	1	2	yes	—
251	C₂×C₄	#6127	1	2	yes	—
252	C₂×C₄	#6127	1	2	yes	—
253	C₂×C₄	D₄×D₁₃	1	2	yes	—
254	C₂×C₄	Q₈×D₁₃	1	2	yes	—
255	C₂²	—	26	4	no	—
256	C₂²	—	26	4	no	—
257	C₂²	—	26	4	no	—
258	C₂²	—	52	2	no	—
259	C₂²	—	52	2	no	—
260	C₂²	—	52	2	no	—
261	C₂²	—	52	2	no	—
262	C₂²	C₂×#6127	1	1	yes	characteristic
263	C₂²	C₂×#6127	1	1	yes	characteristic
264	C₂²	C₂×#6127	1	1	yes	characteristic
265	C₂²	C₂×D₄×D₁₃	1	1	yes	characteristic
266	C₂²	C₂×Q₈×D₁₃	1	1	yes	characteristic
267	C₂²	#88256	1	1	yes	characteristic
268	C₂²	#88256	1	1	yes	characteristic
269	C₄	—	2	2	no	—
270	C₄	—	2	2	no	—
271	C₄	—	4	1	no	—
272	C₄	—	4	1	no	—
273	C₄	—	4	1	no	—
274	C₄	—	4	1	no	—
275	C₄	—	26	4	no	—
276	C₄	—	52	2	no	—
277	C₄	—	52	2	no	—
278	C₄	#88150	1	4	yes	—
279	C₂	—	52	4	no	—
280	C₂	#11719184	1	1	yes	characteristic
281	C₂	C₂×#88150	1	1	yes	characteristic
282	C₂	#11719240	1	1	yes	characteristic
283	C₂	#11719255	1	1	yes	characteristic
284	C₂	#11719258	1	1	yes	characteristic
285	C₂	#11719284	1	1	yes	characteristic
286	C₂	#11719284	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂³	C₂³×D₁₃
commutator, socle, cosocle	C₂³×C₁₃	C₂⁴
Fitting	C₄×C₁₃×C₄⋊C₄	C₂
prime radical	C₂²×D₅₂	C₂²
hypercenter	C₄×C₄⋊C₄	D₁₃
hypocenter	C₁₃	#2454
LCS	G ▷ C₂³×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂³ ◁ C₄×C₄⋊C₄

Complements

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

N	H ≅ G/N	type
2. C₂×#88124	C₂	semidirect
3. C₄×C₁₃×C₄⋊C₄	C₂	semidirect
4. #11718653	C₂	semidirect
5. #11718650	C₂	semidirect
8. #11718589	C₂	semidirect
9. #11718589	C₂	semidirect
10. #11718337	C₂	semidirect
11. #11718328	C₂	semidirect
122. C₄×C₄⋊C₄	D₁₃	semidirect
214. C₁₃	#2454	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₄×D₅₂	2
C₂×#88124	2
C₄×C₁₃×C₄⋊C₄	2
#11718653	2
#11718650	2
#11718600	2
#11718589	2
#11718337	2
#11718328	2
#2454	13