Subgroups — #15749486

Order 3328.

352 conjugacy classes of subgroups.

The subgroup lattice has 260 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄×C₄⋊#1055	C₂	1	1	yes	maximal, characteristic
2	C₄×C₄⋊#1055	C₂	1	1	yes	maximal, characteristic
3	C₁₃×#1842	C₂	1	1	yes	maximal, characteristic
4	C₂×C₁₃×C₄⋊C₈	—	2	1	no	—
5	C₂×C₁₃×C₄⋊C₈	—	2	1	no	—
6	C₂×C₁₃×C₄⋊C₈	C₄	1	2	yes	—
7	C₄³×C₁₃	C₂²	1	1	yes	characteristic
8	C₂×C₄⋊#1055	—	2	1	no	—
9	C₂×C₄⋊#1055	C₄	1	2	yes	—
10	C₂×C₄×#1055	—	2	1	no	—
11	C₂×C₄×#1055	—	2	1	no	—
12	#11718116	—	2	1	no	—
13	#11718116	C₄	1	2	yes	—
14	C₂²×C₈×C₁₃	—	2	2	no	—
15	C₂×C₄²×C₁₃	C₂×C₄	1	1	yes	characteristic
16	C₂×C₄²×C₁₃	C₂×C₄	1	1	yes	characteristic
17	C₂×C₄²×C₁₃	C₂×C₄	1	1	yes	characteristic
18	C₂×C₄²×C₁₃	D₄	1	1	yes	characteristic
19	C₂×C₄²×C₁₃	D₄	1	1	yes	characteristic
20	C₂×C₄²×C₁₃	D₄	1	1	yes	characteristic
21	C₂×C₄²×C₁₃	Q₈	1	1	yes	characteristic
22	C₂²×#1055	—	2	2	no	—
23	C₂²×#1055	—	2	2	no	—
24	C₁₃×C₄⋊C₈	—	4	2	no	—
25	C₁₃×C₄⋊C₈	—	4	1	no	—
26	C₁₃×C₄⋊C₈	—	4	1	no	—
27	C₁₃×C₄⋊C₈	—	4	1	no	—
28	C₁₃×C₄⋊C₈	—	4	1	no	—
29	C₄⋊#1055	—	2	4	no	—
30	C₄⋊#1055	—	2	2	no	—
31	C₄⋊#1055	C₈	1	4	yes	complemented
32	C₄×#1055	—	4	1	no	—
33	C₄×#1055	—	4	1	no	—
34	C₄×#1055	—	4	1	no	—
35	C₄×#1055	—	4	1	no	—
36	#7031	—	13	1	no	maximal, Sylow
37	C₂²×C₄×C₁₃	C₄²	1	1	yes	characteristic
38	C₂²×C₄×C₁₃	C₂²⋊C₄	1	1	yes	characteristic
39	C₂²×C₄×C₁₃	C₂²⋊C₄	1	1	yes	characteristic
40	C₂²×C₄×C₁₃	C₂²⋊C₄	1	1	yes	characteristic
41	C₂²×C₄×C₁₃	C₄⋊C₄	1	1	yes	characteristic
42	C₂²×C₄×C₁₃	C₄⋊C₄	1	1	yes	characteristic
43	C₂²×C₄×C₁₃	C₄⋊C₄	1	1	yes	characteristic
44	C₂×C₈×C₁₃	—	4	2	no	—
45	C₂×C₈×C₁₃	—	4	2	no	—
46	C₂×C₈×C₁₃	—	4	2	no	—
47	C₄²×C₁₃	—	2	1	no	—
48	C₄²×C₁₃	—	2	1	no	—
49	C₄²×C₁₃	—	2	2	no	—
50	C₄²×C₁₃	—	2	2	no	—
51	C₄²×C₁₃	—	4	1	no	—
52	C₄²×C₁₃	C₂×C₈	1	2	yes	—
53	C₄²×C₁₃	M₁₆	1	2	yes	—
54	C₄²×C₁₃	D₈	1	1	yes	characteristic
55	C₄²×C₁₃	D₈	1	1	yes	characteristic
56	C₄²×C₁₃	SD₁₆	1	1	yes	characteristic
57	C₄²×C₁₃	SD₁₆	1	1	yes	characteristic
58	C₄²×C₁₃	SD₁₆	1	1	yes	characteristic
59	C₄²×C₁₃	SD₁₆	1	1	yes	characteristic
60	C₄²×C₁₃	Q₁₆	1	1	yes	characteristic
61	C₄²×C₁₃	Q₁₆	1	1	yes	characteristic
62	C₂×#1055	—	2	4	no	—
63	C₂×#1055	—	2	4	no	—
64	C₂×#1055	—	4	2	no	—
65	C₂×#1055	—	4	2	no	—
66	C₂×#1055	—	4	2	no	—
67	C₂×#1055	—	4	2	no	—
68	#1842	D₁₃	1	1	yes	characteristic
69	C₄×C₄⋊C₈	—	13	1	no	—
70	C₄×C₄⋊C₈	—	13	1	no	—
71	C₂³×C₁₃	#95	1	1	yes	characteristic
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	C₄×C₈	1	1	yes	characteristic
82	C₂×C₄×C₁₃	#97	1	1	yes	characteristic
83	C₂×C₄×C₁₃	C₂²⋊C₈	1	2	yes	—
84	C₂×C₄×C₁₃	#100	1	1	yes	characteristic
85	C₂×C₄×C₁₃	#101	1	1	yes	characteristic
86	C₂×C₄×C₁₃	#102	1	1	yes	characteristic
87	C₂×C₄×C₁₃	#102	1	1	yes	characteristic
88	C₂×C₄×C₁₃	#102	1	1	yes	characteristic
89	C₂×C₄×C₁₃	#102	1	1	yes	characteristic
90	C₂×C₄×C₁₃	#103	1	1	yes	characteristic
91	C₂×C₄×C₁₃	#103	1	1	yes	characteristic
92	C₂×C₄×C₁₃	#103	1	1	yes	characteristic
93	C₂×C₄×C₁₃	#103	1	1	yes	characteristic
94	C₂×C₄×C₁₃	#104	1	2	yes	—
95	C₂×C₄×C₁₃	C₄⋊C₈	1	2	yes	—
96	C₂×C₄×C₁₃	#106	1	1	yes	characteristic
97	C₂×C₄×C₁₃	#106	1	1	yes	characteristic
98	C₂×C₄×C₁₃	#107	1	1	yes	characteristic
99	C₂×C₄×C₁₃	#107	1	1	yes	characteristic
100	C₂×C₄×C₁₃	#108	1	2	yes	—
101	C₈×C₁₃	—	8	2	no	—
102	#1055	—	4	4	no	—
103	#1055	—	4	4	no	—
104	C₂×C₄⋊C₈	—	2	1	no	—
105	C₂×C₄⋊C₈	—	2	1	no	—
106	C₂×C₄⋊C₈	—	13	2	no	—
107	C₂×C₄⋊C₈	—	26	1	no	—
108	C₂×C₄⋊C₈	Dic₁₃	1	2	yes	—
109	C₂×C₄×C₈	—	26	1	no	—
110	C₂×C₄×C₈	—	26	1	no	—
111	C₄³	C₂×D₁₃	1	1	yes	characteristic
112	#336	—	13	2	no	—
113	#336	—	26	1	no	—
114	C₂²×C₁₃	#336	1	1	yes	characteristic
115	C₂²×C₁₃	#339	1	1	yes	characteristic
116	C₂²×C₁₃	#340	1	1	yes	characteristic
117	C₂²×C₁₃	#340	1	1	yes	characteristic
118	C₂²×C₁₃	#341	1	1	yes	characteristic
119	C₂²×C₁₃	#343	1	1	yes	characteristic
120	C₂²×C₁₃	#344	1	1	yes	characteristic
121	C₄×C₁₃	—	2	1	no	—
122	C₄×C₁₃	—	2	1	no	—
123	C₄×C₁₃	—	2	1	no	—
124	C₄×C₁₃	—	2	1	no	—
125	C₄×C₁₃	—	2	1	no	—
126	C₄×C₁₃	—	2	1	no	—
127	C₄×C₁₃	—	4	1	no	—
128	C₄×C₁₃	D₄⋊C₈	1	2	yes	—
129	C₄×C₁₃	Q₈⋊C₈	1	2	yes	—
130	C₄×C₁₃	#331	1	1	yes	characteristic
131	C₄×C₁₃	#332	1	1	yes	characteristic
132	C₄×C₁₃	#332	1	1	yes	characteristic
133	C₄×C₁₃	#333	1	1	yes	characteristic
134	C₄×C₁₃	#334	1	2	yes	—
135	C₄×C₁₃	#335	1	2	yes	—
136	C₂²×C₈	—	2	2	no	—
137	C₂²×C₈	—	26	2	no	—
138	C₂²×C₈	—	26	2	no	—
139	C₂×C₄²	Dic₂₆	1	1	yes	characteristic
140	C₂×C₄²	C₄×D₁₃	1	1	yes	characteristic
141	C₂×C₄²	C₄×D₁₃	1	1	yes	characteristic
142	C₂×C₄²	D₅₂	1	1	yes	characteristic
143	C₂×C₄²	C₂×Dic₁₃	1	1	yes	characteristic
144	C₂×C₄²	C₂²⋊D₁₃	1	1	yes	characteristic
145	C₂×C₄²	C₂²⋊D₁₃	1	1	yes	characteristic
146	C₄⋊C₈	—	4	2	no	—
147	C₄⋊C₈	—	4	1	no	—
148	C₄⋊C₈	—	4	1	no	—
149	C₄⋊C₈	—	4	1	no	—
150	C₄⋊C₈	—	4	1	no	—
151	C₄⋊C₈	—	13	4	no	—
152	C₄⋊C₈	—	26	4	no	—
153	C₄⋊C₈	—	26	2	no	—
154	C₄×C₈	—	52	1	no	—
155	C₄×C₈	—	52	1	no	—
156	C₄×C₈	—	52	1	no	—
157	C₄×C₈	—	52	1	no	—
158	C₂×C₁₃	#1277	1	1	yes	characteristic
159	C₂×C₁₃	#1279	1	1	yes	characteristic
160	C₂×C₁₃	#1283	1	1	yes	characteristic
161	C₂×C₁₃	#1296	1	1	yes	characteristic
162	C₂×C₁₃	#1299	1	1	yes	characteristic
163	C₂×C₁₃	#1301	1	1	yes	characteristic
164	C₂×C₁₃	#1303	1	1	yes	characteristic
165	C₂²×C₄	C₄×Dic₁₃	1	1	yes	characteristic
166	C₂²×C₄	#6101	1	1	yes	characteristic
167	C₂²×C₄	#6101	1	1	yes	characteristic
168	C₂²×C₄	C₄⋊Dic₁₃	1	1	yes	characteristic
169	C₂²×C₄	#6103	1	1	yes	characteristic
170	C₂²×C₄	#6103	1	1	yes	characteristic
171	C₂²×C₄	C₂²⋊Dic₁₃	1	1	yes	characteristic
172	C₂×C₈	—	4	2	no	—
173	C₂×C₈	—	4	2	no	—
174	C₂×C₈	—	4	2	no	—
175	C₂×C₈	—	26	4	no	—
176	C₂×C₈	—	26	4	no	—
177	C₂×C₈	—	52	2	no	—
178	C₂×C₈	—	52	2	no	—
179	C₂×C₈	—	52	2	no	—
180	C₂×C₈	—	52	2	no	—
181	C₄²	—	2	2	no	—
182	C₄²	—	2	1	no	—
183	C₄²	—	2	1	no	—
184	C₄²	—	2	2	no	—
185	C₄²	—	4	1	no	—
186	C₄²	C₈×D₁₃	1	2	yes	—
187	C₄²	#6094	1	2	yes	—
188	C₄²	#6104	1	1	yes	characteristic
189	C₄²	#6104	1	1	yes	characteristic
190	C₄²	#6105	1	1	yes	characteristic
191	C₄²	#6105	1	1	yes	characteristic
192	C₄²	#6106	1	1	yes	characteristic
193	C₄²	#6106	1	1	yes	characteristic
194	C₄²	#6107	1	1	yes	characteristic
195	C₄²	#6107	1	1	yes	characteristic
196	C₁₃	#7031	1	1	yes	characteristic, complemented, Sylow
197	C₂³	#88071	1	1	yes	characteristic
198	C₂×C₄	—	2	1	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	#88047	1	1	yes	characteristic
208	C₂×C₄	#88047	1	1	yes	characteristic
209	C₂×C₄	#88048	1	1	yes	characteristic
210	C₂×C₄	#88048	1	1	yes	characteristic
211	C₂×C₄	#88049	1	1	yes	characteristic
212	C₂×C₄	#88049	1	1	yes	characteristic
213	C₂×C₄	#88050	1	1	yes	characteristic
214	C₂×C₄	#88050	1	1	yes	characteristic
215	C₂×C₄	C₈×Dic₁₃	1	1	yes	characteristic
216	C₂×C₄	#88054	1	2	yes	—
217	C₂×C₄	#88055	1	1	yes	characteristic
218	C₂×C₄	#88060	1	2	yes	—
219	C₂×C₄	#88062	1	2	yes	—
220	C₂×C₄	#88065	1	2	yes	—
221	C₂×C₄	#88072	1	1	yes	characteristic
222	C₂×C₄	#88072	1	1	yes	characteristic
223	C₂×C₄	#88073	1	1	yes	characteristic
224	C₂×C₄	#88075	1	1	yes	characteristic
225	C₂×C₄	#88075	1	1	yes	characteristic
226	C₂×C₄	#88076	1	1	yes	characteristic
227	C₈	—	8	2	no	—
228	C₈	—	52	4	no	—
229	C₈	—	52	4	no	—
230	C₂²	#11718121	1	1	yes	characteristic
231	C₂²	#11718121	1	1	yes	characteristic
232	C₂²	#11718122	1	1	yes	characteristic
233	C₂²	#11718142	1	1	yes	characteristic
234	C₂²	#11718147	1	1	yes	characteristic
235	C₂²	#11718148	1	1	yes	characteristic
236	C₂²	#11718150	1	1	yes	characteristic
237	C₄	—	2	1	no	—
238	C₄	—	2	1	no	—
239	C₄	—	2	1	no	—
240	C₄	—	2	1	no	—
241	C₄	—	2	1	no	—
242	C₄	—	2	1	no	—
243	C₄	—	4	1	no	—
244	C₄	#11718071	1	2	yes	—
245	C₄	#11718072	1	2	yes	—
246	C₄	#11718075	1	2	yes	—
247	C₄	#11718076	1	2	yes	—
248	C₄	#11718136	1	1	yes	characteristic
249	C₄	#11718138	1	1	yes	characteristic
250	C₄	#11718139	1	1	yes	characteristic
251	C₄	#11718139	1	1	yes	characteristic
252	C₂	#13112670	1	1	yes	characteristic
253	C₂	#13112675	1	1	yes	characteristic
254	C₂	#13112686	1	1	yes	characteristic
255	C₂	#13112719	1	1	yes	characteristic
256	C₂	#13112725	1	1	yes	characteristic
257	C₂	#13112731	1	1	yes	characteristic
258	C₂	#13112737	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄	C₂²⋊Dic₁₃
commutator	C₂×C₄×C₁₃	C₄×C₈
Frattini	C₄³	C₂×D₁₃
Fitting	C₁₃×#1842	C₂
socle, prime radical	C₂³×C₁₃	#95
cosocle	C₄³×C₁₃	C₂²
hypercenter	#1842	D₁₃
hypocenter	C₁₃	#7031
LCS	G ▷ C₂×C₄×C₁₃ ▷ C₂²×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂²×C₄ ◁ C₄³ ◁ #1842

Complements

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

N	H ≅ G/N	type
31. C₄⋊#1055	C₈	semidirect
196. C₁₃	#7031	semidirect

Maximal subgroups

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

subgroup	index
C₄×C₄⋊#1055	2
C₁₃×#1842	2
#7031	13