Subgroups — #13118675

Order 1664.

484 conjugacy classes of subgroups.

The subgroup lattice has 258 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#88123	C₂	1	1	yes	maximal, characteristic
2	C₂×C₄×Dic₂₆	C₂	1	1	yes	maximal, characteristic
3	C₄×C₁₃×C₄⋊C₄	C₂	1	1	yes	maximal, characteristic
4	#11718642	C₂	1	2	yes	maximal
5	#11718640	C₂	1	2	yes	maximal
6	#11718590	C₂	1	1	yes	maximal, characteristic
7	#11718590	C₂	1	1	yes	maximal, characteristic
8	#11718589	C₂	1	1	yes	maximal, characteristic
9	#11718589	C₂	1	1	yes	maximal, characteristic
10	#11718318	C₂	1	2	yes	maximal
11	#11718307	C₂	1	2	yes	maximal
12	C₂²×Dic₂₆	C₂²	1	2	yes	—
13	C₂²×Dic₂₆	C₂²	1	1	yes	characteristic
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×C₄⋊Dic₁₃	C₂²	1	2	yes	—
20	C₂×C₄⋊Dic₁₃	C₂²	1	1	yes	characteristic
21	C₂×C₄⋊Dic₁₃	C₂²	1	1	yes	characteristic
22	C₂×C₄⋊Dic₁₃	C₂²	1	1	yes	characteristic
23	C₂×#6101	C₂²	1	2	yes	—
24	C₂×#6101	C₂²	1	2	yes	—
25	C₂×#6101	C₂²	1	2	yes	—
26	C₂×C₄×Dic₁₃	C₂²	1	2	yes	—
27	#88123	—	2	4	no	—
28	C₄×Dic₂₆	—	2	4	no	—
29	C₁₃×#95	C₂²	1	1	yes	characteristic
30	C₁₃×#95	C₂²	1	1	yes	characteristic
31	#88071	C₂²	1	2	yes	—
32	#88071	C₂²	1	2	yes	—
33	#88071	C₂²	1	2	yes	—
34	#88071	C₂²	1	2	yes	—
35	#88071	C₂²	1	2	yes	—
36	#88071	C₂²	1	2	yes	—
37	C₂²×C₄×C₁₃	C₂³	1	1	yes	characteristic
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×Dic₁₃	C₂³	1	2	yes	—
45	C₂²×Dic₁₃	C₂³	1	2	yes	—
46	C₂²×Dic₁₃	C₂³	1	2	yes	—
47	C₂²×Dic₁₃	C₂³	1	2	yes	—
48	C₂×Dic₂₆	—	2	2	no	—
49	C₂×Dic₂₆	—	2	2	no	—
50	C₂×Dic₂₆	—	2	4	no	—
51	C₂×Dic₂₆	—	2	4	no	—
52	C₂×Dic₂₆	—	4	2	no	—
53	C₂×Dic₂₆	Q₈	1	4	yes	complemented
54	C₁₃×C₄⋊C₄	—	2	2	no	—
55	C₁₃×C₄⋊C₄	Q₈	1	4	yes	complemented
56	C₄²×C₁₃	—	2	2	no	—
57	C₄²×C₁₃	—	2	2	no	—
58	C₄²×C₁₃	—	2	2	no	—
59	C₄⋊Dic₁₃	—	2	2	no	—
60	C₄⋊Dic₁₃	—	2	2	no	—
61	C₄⋊Dic₁₃	—	2	4	no	—
62	C₄⋊Dic₁₃	D₄	1	4	yes	—
63	#6101	—	2	4	no	—
64	#6101	—	4	2	no	—
65	#6101	—	4	2	no	—
66	C₄×Dic₁₃	—	2	4	no	—
67	#2451	—	13	1	no	maximal, Sylow
68	C₂³×C₁₃	C₂⁴	1	1	yes	characteristic
69	C₂×C₄×C₁₃	—	2	1	no	—
70	C₂×C₄×C₁₃	—	2	1	no	—
71	C₂×C₄×C₁₃	—	2	1	no	—
72	C₂×C₄×C₁₃	—	2	1	no	—
73	C₂×C₄×C₁₃	—	2	1	no	—
74	C₂×C₄×C₁₃	—	2	1	no	—
75	C₂×C₄×C₁₃	—	2	1	no	—
76	C₂×C₄×C₁₃	—	2	1	no	—
77	C₂×C₄×C₁₃	—	2	1	no	—
78	C₂×C₄×C₁₃	—	2	1	no	—
79	C₂×C₄×C₁₃	—	2	1	no	—
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	C₂×D₄	1	2	yes	—
84	C₂×C₄×C₁₃	C₂×Q₈	1	2	yes	—
85	C₂×C₄×C₁₃	C₂×Q₈	1	2	yes	—
86	C₂×C₄×C₁₃	C₂×Q₈	1	2	yes	—
87	C₂×C₄×C₁₃	C₂×Q₈	1	2	yes	—
88	C₂×C₄×C₁₃	#41	1	2	yes	—
89	C₂×C₄×C₁₃	#41	1	2	yes	—
90	C₂×Dic₁₃	—	2	2	no	—
91	C₂×Dic₁₃	—	2	2	no	—
92	C₂×Dic₁₃	—	2	2	no	—
93	C₂×Dic₁₃	—	2	2	no	—
94	C₂×Dic₁₃	—	2	2	no	—
95	C₂×Dic₁₃	—	2	2	no	—
96	C₂×Dic₁₃	—	2	2	no	—
97	C₂×Dic₁₃	—	2	2	no	—
98	C₂×Dic₁₃	—	2	2	no	—
99	C₂×Dic₁₃	—	2	2	no	—
100	C₂×Dic₁₃	C₂×D₄	1	4	yes	—
101	C₂×Dic₁₃	C₂×Q₈	1	4	yes	—
102	Dic₂₆	—	2	8	no	—
103	Dic₂₆	—	4	8	no	—
104	C₂×C₄⋊Q₈	—	13	1	no	—
105	C₂×C₄×Q₈	—	13	1	no	—
106	#398	—	13	2	no	—
107	#395	—	13	2	no	—
108	#391	—	13	1	no	—
109	#391	—	13	1	no	—
110	#391	—	13	2	no	—
111	#389	—	13	1	no	—
112	#389	—	13	1	no	—
113	#387	—	13	2	no	—
114	C₄×C₄⋊C₄	D₁₃	1	1	yes	characteristic
115	C₂²×C₁₃	C₂²×D₄	1	1	yes	characteristic
116	C₂²×C₁₃	C₂²×Q₈	1	1	yes	characteristic
117	C₂²×C₁₃	C₂²×Q₈	1	1	yes	characteristic
118	C₂²×C₁₃	C₂×#41	1	1	yes	characteristic
119	C₂²×C₁₃	C₂×#41	1	1	yes	characteristic
120	C₂²×C₁₃	H(5,𝔽₂)	1	1	yes	characteristic
121	C₂²×C₁₃	2₋¹⁺⁴	1	1	yes	characteristic
122	C₄×C₁₃	—	2	2	no	—
123	C₄×C₁₃	—	2	2	no	—
124	C₄×C₁₃	—	2	2	no	—
125	C₄×C₁₃	—	2	2	no	—
126	C₄×C₁₃	—	4	1	no	—
127	C₄×C₁₃	—	4	1	no	—
128	C₄×C₁₃	C₂²⋊Q₈	1	4	yes	—
129	Dic₁₃	—	2	4	no	—
130	Dic₁₃	—	2	4	no	—
131	Dic₁₃	—	4	2	no	—
132	Dic₁₃	—	4	2	no	—
133	C₂²×Q₈	—	13	1	no	—
134	C₂²×Q₈	—	13	2	no	—
135	C₄⋊Q₈	—	26	4	no	—
136	C₄×Q₈	—	26	4	no	—
137	C₂×C₄⋊C₄	—	13	1	no	—
138	C₂×C₄⋊C₄	—	13	2	no	—
139	C₂×C₄⋊C₄	—	13	2	no	—
140	C₂×C₄⋊C₄	—	13	1	no	—
141	C₂×C₄⋊C₄	—	13	1	no	—
142	C₂×C₄⋊C₄	—	13	2	no	—
143	C₂×C₄⋊C₄	—	13	2	no	—
144	C₂×C₄⋊C₄	C₂×D₁₃	1	1	yes	characteristic
145	C₂×C₄⋊C₄	C₂×D₁₃	1	1	yes	characteristic
146	C₂×C₄²	—	13	2	no	—
147	C₂×C₄²	C₂×D₁₃	1	1	yes	characteristic
148	C₂×C₄²	C₂×D₁₃	1	1	yes	characteristic
149	C₂×C₄²	C₂×D₁₃	1	1	yes	characteristic
150	#95	—	13	2	no	—
151	#95	—	13	2	no	—
152	#95	—	13	2	no	—
153	#95	—	13	2	no	—
154	#95	—	13	2	no	—
155	#95	—	13	2	no	—
156	#95	C₂×D₁₃	1	1	yes	characteristic
157	#95	C₂×D₁₃	1	1	yes	characteristic
158	C₂×C₁₃	C₂×C₂²⋊Q₈	1	1	yes	characteristic
159	C₂×C₁₃	#533	1	1	yes	characteristic
160	C₂×C₁₃	#542	1	1	yes	characteristic
161	C₂×C₁₃	#546	1	1	yes	characteristic
162	C₂×C₁₃	D₄×Q₈	1	1	yes	characteristic
163	C₂×C₁₃	#557	1	1	yes	characteristic
164	C₂×C₁₃	Q₈²	1	1	yes	characteristic
165	C₂×Q₈	—	13	4	no	—
166	C₂×Q₈	—	26	4	no	—
167	C₂×Q₈	—	26	2	no	—
168	C₂×Q₈	—	26	4	no	—
169	C₂×Q₈	—	26	2	no	—
170	C₂×Q₈	—	52	2	no	—
171	C₂²×C₄	—	13	2	no	—
172	C₂²×C₄	—	13	2	no	—
173	C₂²×C₄	—	13	2	no	—
174	C₂²×C₄	—	13	2	no	—
175	C₂²×C₄	C₂²×D₁₃	1	1	yes	characteristic
176	C₂²×C₄	C₂²×D₁₃	1	1	yes	characteristic
177	C₂²×C₄	C₂²×D₁₃	1	1	yes	characteristic
178	C₂²×C₄	C₂²×D₁₃	1	1	yes	characteristic
179	C₂²×C₄	C₂²×D₁₃	1	1	yes	characteristic
180	C₂²×C₄	C₂²×D₁₃	1	1	yes	characteristic
181	C₂²×C₄	C₂²×D₁₃	1	1	yes	characteristic
182	C₄⋊C₄	—	2	2	no	—
183	C₄⋊C₄	—	13	4	no	—
184	C₄⋊C₄	—	26	4	no	—
185	C₄⋊C₄	—	26	4	no	—
186	C₄⋊C₄	—	26	2	no	—
187	C₄⋊C₄	—	26	2	no	—
188	C₄⋊C₄	—	52	2	no	—
189	C₄⋊C₄	—	52	2	no	—
190	C₄⋊C₄	Dic₂₆	1	4	yes	complemented
191	C₄²	—	2	2	no	—
192	C₄²	—	2	2	no	—
193	C₄²	—	2	2	no	—
194	C₄²	—	26	4	no	—
195	C₁₃	#2451	1	1	yes	characteristic, complemented, Sylow
196	C₂³	C₂³×D₁₃	1	1	yes	characteristic
197	Q₈	—	26	8	no	—
198	Q₈	—	52	8	no	—
199	C₂×C₄	—	2	1	no	—
200	C₂×C₄	—	2	1	no	—
201	C₂×C₄	—	2	1	no	—
202	C₂×C₄	—	2	1	no	—
203	C₂×C₄	—	2	1	no	—
204	C₂×C₄	—	2	1	no	—
205	C₂×C₄	—	2	1	no	—
206	C₂×C₄	—	2	1	no	—
207	C₂×C₄	—	2	1	no	—
208	C₂×C₄	—	2	1	no	—
209	C₂×C₄	—	2	1	no	—
210	C₂×C₄	—	2	1	no	—
211	C₂×C₄	—	2	1	no	—
212	C₂×C₄	—	2	1	no	—
213	C₂×C₄	—	13	4	no	—
214	C₂×C₄	—	13	4	no	—
215	C₂×C₄	—	26	2	no	—
216	C₂×C₄	—	26	2	no	—
217	C₂×C₄	—	26	2	no	—
218	C₂×C₄	—	26	2	no	—
219	C₂×C₄	—	26	2	no	—
220	C₂×C₄	—	26	2	no	—
221	C₂×C₄	—	26	2	no	—
222	C₂×C₄	—	26	2	no	—
223	C₂×C₄	—	26	2	no	—
224	C₂×C₄	—	26	2	no	—
225	C₂×C₄	C₂×Dic₂₆	1	2	yes	—
226	C₂×C₄	C₂×Dic₂₆	1	2	yes	—
227	C₂×C₄	C₂×Dic₂₆	1	2	yes	—
228	C₂×C₄	#6127	1	2	yes	—
229	C₂×C₄	#6127	1	2	yes	—
230	C₂×C₄	D₄×D₁₃	1	2	yes	—
231	C₂×C₄	Q₈×D₁₃	1	2	yes	—
232	C₂²	C₂²×Dic₂₆	1	1	yes	characteristic
233	C₂²	C₂×#6127	1	1	yes	characteristic
234	C₂²	C₂×#6127	1	1	yes	characteristic
235	C₂²	C₂×D₄×D₁₃	1	1	yes	characteristic
236	C₂²	C₂×Q₈×D₁₃	1	1	yes	characteristic
237	C₂²	#88256	1	1	yes	characteristic
238	C₂²	#88257	1	1	yes	characteristic
239	C₄	—	2	2	no	—
240	C₄	—	2	2	no	—
241	C₄	—	2	2	no	—
242	C₄	—	2	2	no	—
243	C₄	—	4	1	no	—
244	C₄	—	4	1	no	—
245	C₄	—	26	4	no	—
246	C₄	—	26	4	no	—
247	C₄	—	52	2	no	—
248	C₄	—	52	2	no	—
249	C₄	#88150	1	4	yes	—
250	C₂	#11719175	1	1	yes	characteristic
251	C₂	C₂×#88150	1	1	yes	characteristic
252	C₂	#11719239	1	1	yes	characteristic
253	C₂	D₄×Dic₂₆	1	1	yes	characteristic
254	C₂	#11719255	1	1	yes	characteristic
255	C₂	Q₈×Dic₂₆	1	1	yes	characteristic
256	C₂	#11719272	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂³	C₂³×D₁₃
commutator, socle, prime radical, cosocle	C₂³×C₁₃	C₂⁴
Fitting	C₄×C₁₃×C₄⋊C₄	C₂
hypercenter	C₄×C₄⋊C₄	D₁₃
hypocenter	C₁₃	#2451
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
53. C₂×Dic₂₆	Q₈	semidirect
55. C₁₃×C₄⋊C₄	Q₈	semidirect
190. C₄⋊C₄	Dic₂₆	semidirect
195. C₁₃	#2451	semidirect

Maximal subgroups

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

subgroup	index
C₂×#88123	2
C₂×C₄×Dic₂₆	2
C₄×C₁₃×C₄⋊C₄	2
#11718642	2
#11718640	2
#11718590	2
#11718589	2
#11718318	2
#11718307	2
#2451	13