Subgroups — #15761492

Order 3328.

378 conjugacy classes of subgroups.

The subgroup lattice has 228 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#13115555	C₂	1	2	yes	maximal
2	C₂×#11718050	C₂	1	1	yes	maximal, characteristic
3	C₂×#11718046	C₂	1	1	yes	maximal, characteristic
4	#13112677	C₂	1	2	yes	maximal
5	C₄×C₁₃×#106	C₂	1	1	yes	maximal, characteristic
6	C₂×C₁₃×#106	C₄	1	2	yes	—
7	C₂×C₄×C₈×C₁₃	C₂²	1	1	yes	characteristic
8	C₄×C₁₃×C₄⋊C₄	C₂²	1	2	yes	—
9	C₂×C₄⋊#1055	—	2	2	no	—
10	C₂×C₄⋊#1055	C₂²	1	2	yes	—
11	C₂×C₄×#1055	C₂²	1	1	yes	characteristic
12	C₂×C₄×#1055	C₂²	1	1	yes	characteristic
13	C₁₃×#340	C₄	1	1	yes	characteristic
14	C₁₃×#340	C₄	1	1	yes	characteristic
15	#11718116	—	2	2	no	—
16	#11718050	—	2	2	no	—
17	#11718046	—	2	2	no	—
18	C₂²×C₈×C₁₃	C₂×C₄	1	1	yes	characteristic
19	C₂²×C₈×C₁₃	C₂×C₄	1	1	yes	characteristic
20	C₂×C₁₃×C₄⋊C₄	C₂×C₄	1	2	yes	—
21	C₂×C₁₃×C₄⋊C₄	C₂×C₄	1	2	yes	—
22	C₂×C₄²×C₁₃	—	2	2	no	—
23	C₂×C₄²×C₁₃	C₂³	1	1	yes	characteristic
24	C₂²×#1055	—	2	1	no	—
25	C₂²×#1055	—	2	2	no	—
26	C₂²×#1055	D₄	1	1	yes	characteristic
27	C₂²×#1055	D₄	1	1	yes	characteristic
28	C₁₃×#106	—	2	4	no	—
29	C₄×C₈×C₁₃	D₄	1	2	yes	—
30	C₄×C₈×C₁₃	Q₈	1	2	yes	—
31	C₁₃×#95	—	2	2	no	—
32	C₄⋊#1055	—	2	2	no	—
33	C₄⋊#1055	—	2	2	no	—
34	C₄⋊#1055	—	4	4	no	—
35	C₄×#1055	—	2	1	no	—
36	C₄×#1055	—	2	1	no	—
37	C₄×#1055	—	2	1	no	—
38	C₄×#1055	Q₈	1	2	yes	—
39	#9600	—	13	1	no	maximal, Sylow
40	C₂²×C₄×C₁₃	—	2	2	no	—
41	C₂²×C₄×C₁₃	—	2	2	no	—
42	C₂²×C₄×C₁₃	C₂²×C₄	1	1	yes	characteristic
43	C₂²×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
44	C₂²×C₄×C₁₃	#41	1	1	yes	characteristic
45	C₂×C₈×C₁₃	—	2	2	no	—
46	C₂×C₈×C₁₃	—	2	2	no	—
47	C₂×C₈×C₁₃	C₄⋊C₄	1	2	yes	—
48	C₂×C₈×C₁₃	C₄⋊C₄	1	2	yes	—
49	C₁₃×C₄⋊C₄	—	2	2	no	—
50	C₁₃×C₄⋊C₄	—	2	2	no	—
51	C₁₃×C₄⋊C₄	—	2	2	no	—
52	C₁₃×C₄⋊C₄	M₁₆	1	4	yes	—
53	C₄²×C₁₃	—	4	2	no	—
54	C₄²×C₁₃	—	4	2	no	—
55	C₄²×C₁₃	C₂×D₄	1	1	yes	characteristic
56	C₄²×C₁₃	C₂×Q₈	1	1	yes	characteristic
57	C₄²×C₁₃	C₂×Q₈	1	1	yes	characteristic
58	C₄²×C₁₃	#41	1	1	yes	characteristic
59	C₂×#1055	—	2	2	no	—
60	C₂×#1055	—	2	2	no	—
61	C₂×#1055	—	2	2	no	—
62	C₂×#1055	—	2	1	no	—
63	C₂×#1055	—	2	1	no	—
64	C₂×#1055	—	2	2	no	—
65	C₂×#1055	—	2	2	no	—
66	C₂×#1055	—	4	2	no	—
67	C₂×#1055	—	4	2	no	—
68	C₂×#1055	—	4	2	no	—
69	#1939	—	13	2	no	—
70	C₄×#106	D₁₃	1	1	yes	characteristic
71	C₂×#335	—	13	1	no	—
72	C₂×#322	—	13	1	no	—
73	#1279	—	13	2	no	—
74	C₂³×C₁₃	C₄×D₄	1	1	yes	characteristic
75	C₂×C₄×C₁₃	—	2	1	no	—
76	C₂×C₄×C₁₃	—	2	1	no	—
77	C₂×C₄×C₁₃	—	2	4	no	—
78	C₂×C₄×C₁₃	—	4	2	no	—
79	C₂×C₄×C₁₃	—	4	2	no	—
80	C₂×C₄×C₁₃	—	4	2	no	—
81	C₂×C₄×C₁₃	—	4	2	no	—
82	C₂×C₄×C₁₃	—	4	2	no	—
83	C₂×C₄×C₁₃	C₂×C₄⋊C₄	1	1	yes	characteristic
84	C₂×C₄×C₁₃	C₄×Q₈	1	1	yes	characteristic
85	C₂×C₄×C₁₃	#121	1	1	yes	characteristic
86	C₂×C₄×C₁₃	C₂²⋊Q₈	1	1	yes	characteristic
87	C₂×C₄×C₁₃	#125	1	1	yes	characteristic
88	C₂×C₄×C₁₃	C₄⋊Q₈	1	1	yes	characteristic
89	C₂×C₄×C₁₃	C₂×M₁₆	1	2	yes	—
90	C₂×C₄×C₁₃	#131	1	2	yes	—
91	C₂×C₄×C₁₃	AGL(1,ℤ/8)	1	2	yes	—
92	C₂×C₄×C₁₃	#137	1	2	yes	—
93	C₈×C₁₃	—	2	2	no	—
94	C₈×C₁₃	—	2	2	no	—
95	#1055	—	2	4	no	—
96	#1055	—	4	1	no	—
97	#1055	—	4	1	no	—
98	#1055	—	8	2	no	—
99	C₂×#106	Dic₁₃	1	2	yes	—
100	C₂×C₄⋊C₈	—	13	2	no	—
101	C₂×C₄⋊C₈	—	26	2	no	—
102	C₂×C₄×C₈	—	13	1	no	—
103	C₂×C₄×C₈	—	13	1	no	—
104	C₂×C₄×C₈	C₂×D₁₃	1	1	yes	characteristic
105	C₄×C₄⋊C₄	C₂×D₁₃	1	2	yes	—
106	#340	Dic₁₃	1	1	yes	characteristic
107	#340	Dic₁₃	1	1	yes	characteristic
108	#336	—	26	2	no	—
109	#335	—	26	2	no	—
110	#322	—	26	2	no	—
111	C₂²×C₁₃	#389	1	1	yes	characteristic
112	C₂²×C₁₃	#435	1	2	yes	—
113	C₂²×C₁₃	#440	1	2	yes	—
114	C₂²×C₁₃	#443	1	1	yes	characteristic
115	C₂²×C₁₃	#443	1	1	yes	characteristic
116	C₄×C₁₃	—	2	1	no	—
117	C₄×C₁₃	—	2	1	no	—
118	C₄×C₁₃	—	4	4	no	—
119	C₄×C₁₃	—	8	2	no	—
120	C₄×C₁₃	#423	1	1	yes	characteristic
121	C₄×C₁₃	#424	1	1	yes	characteristic
122	C₄×C₁₃	#446	1	2	yes	—
123	C₄×C₁₃	#469	1	1	yes	characteristic
124	C₄×C₁₃	#470	1	1	yes	characteristic
125	C₄×C₁₃	#501	1	2	yes	—
126	C₂²×C₈	—	13	1	no	—
127	C₂²×C₈	—	13	1	no	—
128	C₂²×C₈	—	26	1	no	—
129	C₂²×C₈	—	26	2	no	—
130	C₂²×C₈	C₂×Dic₁₃	1	1	yes	characteristic
131	C₂²×C₈	C₂×Dic₁₃	1	1	yes	characteristic
132	C₂×C₄⋊C₄	C₂×Dic₁₃	1	2	yes	—
133	C₂×C₄⋊C₄	C₂×Dic₁₃	1	2	yes	—
134	C₂×C₄²	—	2	2	no	—
135	C₂×C₄²	C₂²×D₁₃	1	1	yes	characteristic
136	#106	—	2	4	no	—
137	C₄⋊C₈	—	26	2	no	—
138	C₄⋊C₈	—	26	2	no	—
139	C₄⋊C₈	—	52	4	no	—
140	C₄×C₈	—	13	2	no	—
141	C₄×C₈	—	26	1	no	—
142	C₄×C₈	—	26	1	no	—
143	C₄×C₈	—	26	1	no	—
144	C₄×C₈	Dic₂₆	1	2	yes	—
145	C₄×C₈	D₅₂	1	2	yes	—
146	#95	—	2	2	no	—
147	C₂×C₁₃	#1585	1	2	yes	—
148	C₂×C₁₃	#1588	1	2	yes	—
149	C₂×C₁₃	#1940	1	1	yes	characteristic
150	C₂×C₁₃	#1944	1	1	yes	characteristic
151	C₂×C₁₃	#1945	1	1	yes	characteristic
152	C₂²×C₄	—	2	2	no	—
153	C₂²×C₄	—	2	2	no	—
154	C₂²×C₄	D₄×D₁₃	1	1	yes	characteristic
155	C₂²×C₄	#6129	1	1	yes	characteristic
156	C₂²×C₄	C₂²×Dic₁₃	1	1	yes	characteristic
157	C₂×C₈	—	2	2	no	—
158	C₂×C₈	—	2	2	no	—
159	C₂×C₈	—	26	2	no	—
160	C₂×C₈	—	26	2	no	—
161	C₂×C₈	—	26	2	no	—
162	C₂×C₈	—	26	1	no	—
163	C₂×C₈	—	26	1	no	—
164	C₂×C₈	—	26	2	no	—
165	C₂×C₈	—	26	2	no	—
166	C₂×C₈	—	52	2	no	—
167	C₂×C₈	—	52	2	no	—
168	C₂×C₈	—	52	2	no	—
169	C₂×C₈	C₄⋊Dic₁₃	1	2	yes	—
170	C₂×C₈	C₄⋊Dic₁₃	1	2	yes	—
171	C₄⋊C₄	—	2	2	no	—
172	C₄⋊C₄	—	2	2	no	—
173	C₄⋊C₄	—	2	2	no	—
174	C₄⋊C₄	#6099	1	4	yes	—
175	C₄²	—	4	2	no	—
176	C₄²	—	4	2	no	—
177	C₄²	C₂×Dic₂₆	1	1	yes	characteristic
178	C₄²	C₂×D₅₂	1	1	yes	characteristic
179	C₄²	Q₈×D₁₃	1	1	yes	characteristic
180	C₄²	#6131	1	1	yes	characteristic
181	C₁₃	#9600	1	1	yes	characteristic, complemented, Sylow
182	C₂³	D₄×Dic₁₃	1	1	yes	characteristic
183	C₂×C₄	—	2	1	no	—
184	C₂×C₄	—	2	1	no	—
185	C₂×C₄	—	2	4	no	—
186	C₂×C₄	—	4	2	no	—
187	C₂×C₄	—	4	2	no	—
188	C₂×C₄	—	4	2	no	—
189	C₂×C₄	—	4	2	no	—
190	C₂×C₄	—	4	2	no	—
191	C₂×C₄	#88142	1	1	yes	characteristic
192	C₂×C₄	#88144	1	1	yes	characteristic
193	C₂×C₄	#88149	1	1	yes	characteristic
194	C₂×C₄	#88151	1	1	yes	characteristic
195	C₂×C₄	#88168	1	2	yes	—
196	C₂×C₄	#88169	1	2	yes	—
197	C₂×C₄	C₂×#6099	1	2	yes	—
198	C₂×C₄	C₂×C₄⋊Dic₁₃	1	1	yes	characteristic
199	C₂×C₄	Q₈×Dic₁₃	1	1	yes	characteristic
200	C₂×C₄	#88202	1	2	yes	—
201	C₈	—	2	2	no	—
202	C₈	—	2	2	no	—
203	C₈	—	26	4	no	—
204	C₈	—	52	1	no	—
205	C₈	—	52	1	no	—
206	C₈	—	104	2	no	—
207	C₂²	#11718646	1	1	yes	characteristic
208	C₂²	#11718672	1	2	yes	—
209	C₂²	#11718825	1	2	yes	—
210	C₂²	#11718857	1	1	yes	characteristic
211	C₂²	#11718857	1	1	yes	characteristic
212	C₄	—	2	1	no	—
213	C₄	—	2	1	no	—
214	C₄	—	4	4	no	—
215	C₄	—	8	2	no	—
216	C₄	#11718517	1	2	yes	—
217	C₄	#11718526	1	1	yes	characteristic
218	C₄	#11718529	1	1	yes	characteristic
219	C₄	#11718585	1	1	yes	characteristic
220	C₄	#11718660	1	1	yes	characteristic
221	C₄	#11718685	1	2	yes	—
222	C₂	#13113694	1	2	yes	—
223	C₂	#13113715	1	2	yes	—
224	C₂	#13115560	1	1	yes	characteristic
225	C₂	#13115580	1	1	yes	characteristic
226	C₂	#13115585	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄	D₄×D₁₃
commutator	C₂²×C₄×C₁₃	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₁₃
Fitting	C₄×C₁₃×#106	C₂
socle, prime radical	C₂³×C₁₃	C₄×D₄
cosocle	C₂×C₄²×C₁₃	C₂³
hypercenter	C₄×#106	D₁₃
hypocenter	C₁₃	#9600
LCS	G ▷ C₂²×C₄×C₁₃ ▷ C₂×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂²×C₄ ◁ C₂×C₄² ◁ C₄×#106

Complements

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

N	H ≅ G/N	type
181. C₁₃	#9600	semidirect

Maximal subgroups

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

subgroup	index
#13115555	2
C₂×#11718050	2
C₂×#11718046	2
#13112677	2
C₄×C₁₃×#106	2
#9600	13