Subgroups — #15774311

Order 3328.

394 conjugacy classes of subgroups.

The subgroup lattice has 224 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#13116635	C₂	1	1	yes	maximal, characteristic
2	#13113595	C₂	1	1	yes	maximal, characteristic
3	#13112885	C₂	1	4	yes	maximal, complemented
4	C₂×C₁₃×#494	C₂	1	1	yes	maximal, characteristic
5	C₂²×C₁₃×Q₁₆	—	2	1	no	—
6	C₂²×C₁₃×Q₁₆	C₄	1	2	yes	—
7	C₂×C₁₃×C₄⋊Q₈	—	2	1	no	—
8	C₁₃×#494	—	2	1	no	—
9	C₁₃×#494	—	2	1	no	—
10	C₁₃×#494	C₄	1	2	yes	—
11	C₁₃×#494	C₂²	1	2	yes	—
12	C₂×C₄×C₈×C₁₃	C₂²	1	1	yes	characteristic
13	#11718585	—	2	1	no	—
14	C₂²⋊#6090	—	2	1	no	—
15	C₄⋊#6090	C₂²	1	2	yes	—
16	C₄×#6090	—	2	1	no	—
17	#11718050	—	2	1	no	—
18	#11718050	C₂²	1	2	yes	—
19	C₂²×Q₈×C₁₃	—	2	2	no	—
20	C₂×C₁₃×Q₁₆	—	2	2	no	—
21	C₂×C₁₃×Q₁₆	—	2	2	no	—
22	C₂×C₁₃×Q₁₆	—	2	2	no	—
23	C₂×C₁₃×Q₁₆	—	4	4	no	—
24	C₂×C₁₃×Q₁₆	—	4	4	no	—
25	C₂×C₁₃×Q₁₆	C₂×C₄	1	4	yes	—
26	C₂²×C₈×C₁₃	C₂×C₄	1	1	yes	characteristic
27	C₂²×C₈×C₁₃	D₄	1	1	yes	characteristic
28	C₁₃×C₄⋊Q₈	—	2	2	no	—
29	C₁₃×C₄⋊Q₈	—	2	2	no	—
30	C₁₃×C₄⋊Q₈	—	4	1	no	—
31	C₁₃×C₄⋊Q₈	—	4	1	no	—
32	C₂×C₁₃×C₄⋊C₄	—	4	1	no	—
33	C₂×C₁₃×C₄⋊C₄	—	4	1	no	—
34	C₂×C₄²×C₁₃	D₄	1	1	yes	characteristic
35	C₂×#6099	—	4	1	no	—
36	C₄×C₈×C₁₃	C₂×C₄	1	1	yes	characteristic
37	C₄×C₈×C₁₃	D₄	1	1	yes	characteristic
38	C₄×C₈×C₁₃	D₄	1	1	yes	characteristic
39	C₄×C₈×C₁₃	C₂³	1	1	yes	characteristic
40	C₂×#6090	—	2	2	no	—
41	C₄⋊#1055	—	2	2	no	—
42	C₄⋊#1055	—	4	1	no	—
43	#11962	—	13	1	no	maximal, Sylow
44	C₂×Q₈×C₁₃	—	2	4	no	—
45	C₂×Q₈×C₁₃	—	4	4	no	—
46	C₂×Q₈×C₁₃	—	4	4	no	—
47	C₂×Q₈×C₁₃	—	4	2	no	—
48	C₂²×C₄×C₁₃	—	4	2	no	—
49	C₂²×C₄×C₁₃	C₂²⋊C₄	1	1	yes	characteristic
50	C₂²×C₄×C₁₃	D₈	1	1	yes	characteristic
51	C₂²×C₄×C₁₃	SD₁₆	1	1	yes	characteristic
52	C₁₃×Q₁₆	—	4	4	no	—
53	C₁₃×Q₁₆	—	4	4	no	—
54	C₁₃×Q₁₆	—	4	4	no	—
55	C₁₃×Q₁₆	—	4	4	no	—
56	C₂×C₈×C₁₃	—	2	2	no	—
57	C₂×C₈×C₁₃	—	2	2	no	—
58	C₂×C₈×C₁₃	C₂²⋊C₄	1	1	yes	characteristic
59	C₂×C₈×C₁₃	C₂²⋊C₄	1	1	yes	characteristic
60	C₂×C₈×C₁₃	C₂²×C₄	1	1	yes	characteristic
61	C₂×C₈×C₁₃	C₂×D₄	1	1	yes	characteristic
62	C₁₃×C₄⋊C₄	—	4	2	no	—
63	C₁₃×C₄⋊C₄	—	4	2	no	—
64	C₁₃×C₄⋊C₄	—	4	2	no	—
65	C₁₃×C₄⋊C₄	—	4	2	no	—
66	C₄²×C₁₃	C₂²⋊C₄	1	1	yes	characteristic
67	C₄²×C₁₃	D₈	1	1	yes	characteristic
68	C₄²×C₁₃	SD₁₆	1	1	yes	characteristic
69	C₄²×C₁₃	C₂×D₄	1	1	yes	characteristic
70	#6099	—	8	2	no	—
71	C₂×#1055	—	4	2	no	—
72	#6090	—	4	2	no	—
73	C₂×#494	D₁₃	1	1	yes	characteristic
74	#2162	—	13	1	no	—
75	#1567	—	13	1	no	—
76	#1362	—	13	4	no	—
77	C₂³×C₁₃	#102	1	1	yes	characteristic
78	Q₈×C₁₃	—	4	4	no	—
79	Q₈×C₁₃	—	4	4	no	—
80	Q₈×C₁₃	—	8	4	no	—
81	C₂×C₄×C₁₃	—	2	2	no	—
82	C₂×C₄×C₁₃	—	2	1	no	—
83	C₂×C₄×C₁₃	—	2	1	no	—
84	C₂×C₄×C₁₃	—	4	4	no	—
85	C₂×C₄×C₁₃	—	8	4	no	—
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	SD₃₂	1	2	yes	—
90	C₂×C₄×C₁₃	Q₃₂	1	2	yes	—
91	C₂×C₄×C₁₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
92	C₂×C₄×C₁₃	C₂×D₈	1	1	yes	characteristic
93	C₂×C₄×C₁₃	C₂×SD₁₆	1	1	yes	characteristic
94	C₈×C₁₃	—	2	1	no	—
95	C₈×C₁₃	—	2	1	no	—
96	C₈×C₁₃	#100	1	2	yes	—
97	C₈×C₁₃	AGL(1,ℤ/8)	1	2	yes	—
98	#1055	—	8	2	no	—
99	C₂²×Q₁₆	—	2	1	no	—
100	C₂²×Q₁₆	Dic₁₃	1	2	yes	—
101	C₂×C₄⋊Q₈	—	2	1	no	—
102	#494	—	2	1	no	—
103	#494	—	2	1	no	—
104	#494	Dic₁₃	1	2	yes	—
105	#494	C₂×D₁₃	1	2	yes	—
106	#423	—	26	1	no	—
107	C₂×C₄×C₈	C₂×D₁₃	1	1	yes	characteristic
108	C₄⋊C₁₆	—	13	2	no	—
109	C₂²⋊C₁₆	—	26	1	no	—
110	C₄×C₁₆	—	26	1	no	—
111	#335	—	13	2	no	—
112	#335	—	26	1	no	—
113	C₂²×C₁₃	—	2	1	no	—
114	C₂²×C₁₃	—	2	1	no	—
115	C₂²×C₁₃	#358	1	2	yes	—
116	C₂²×C₁₃	C₂×#102	1	1	yes	characteristic
117	C₄×C₁₃	—	2	1	no	—
118	C₄×C₁₃	—	2	1	no	—
119	C₄×C₁₃	—	8	4	no	—
120	C₄×C₁₃	#358	1	2	yes	—
121	C₄×C₁₃	#362	1	2	yes	—
122	C₄×C₁₃	C₂×#100	1	1	yes	characteristic
123	C₄×C₁₃	#418	1	1	yes	characteristic
124	C₄×C₁₃	C₂×SD₃₂	1	1	yes	characteristic
125	C₄×C₁₃	C₂×Q₃₂	1	1	yes	characteristic
126	C₂²×Q₈	—	2	2	no	—
127	C₂×Q₁₆	—	2	2	no	—
128	C₂×Q₁₆	—	2	2	no	—
129	C₂×Q₁₆	—	2	2	no	—
130	C₂×Q₁₆	—	4	4	no	—
131	C₂×Q₁₆	—	4	4	no	—
132	C₂×Q₁₆	C₂×Dic₁₃	1	4	yes	—
133	C₂×M₁₆	—	52	1	no	—
134	C₂²×C₈	C₂×Dic₁₃	1	1	yes	characteristic
135	C₂²×C₈	C₂²⋊D₁₃	1	1	yes	characteristic
136	C₄⋊Q₈	—	2	2	no	—
137	C₄⋊Q₈	—	2	2	no	—
138	C₄⋊Q₈	—	4	1	no	—
139	C₄⋊Q₈	—	4	1	no	—
140	C₂×C₄⋊C₄	—	4	1	no	—
141	C₂×C₄⋊C₄	—	4	1	no	—
142	C₂×C₄²	C₂²⋊D₁₃	1	1	yes	characteristic
143	C₂×C₁₆	—	26	2	no	—
144	C₄⋊C₈	—	26	2	no	—
145	C₄⋊C₈	—	52	1	no	—
146	C₄×C₈	C₂×Dic₁₃	1	1	yes	characteristic
147	C₄×C₈	C₂²⋊D₁₃	1	1	yes	characteristic
148	C₄×C₈	C₂²⋊D₁₃	1	1	yes	characteristic
149	C₄×C₈	C₂²×D₁₃	1	1	yes	characteristic
150	C₂×C₁₃	—	2	2	no	—
151	C₂×C₁₃	#1545	1	1	yes	characteristic
152	C₂×C₁₃	C₂×#358	1	1	yes	characteristic
153	C₂×C₁₃	C₂×#362	1	1	yes	characteristic
154	C₂×Q₈	—	2	4	no	—
155	C₂×Q₈	—	4	4	no	—
156	C₂×Q₈	—	4	4	no	—
157	C₂×Q₈	—	4	2	no	—
158	C₂²×C₄	—	4	2	no	—
159	C₂²×C₄	#6104	1	1	yes	characteristic
160	C₂²×C₄	#6105	1	1	yes	characteristic
161	C₂²×C₄	C₂²⋊Dic₁₃	1	1	yes	characteristic
162	Q₁₆	—	4	4	no	—
163	Q₁₆	—	4	4	no	—
164	Q₁₆	—	4	4	no	—
165	Q₁₆	—	4	4	no	—
166	M₁₆	—	104	2	no	—
167	C₂×C₈	—	2	2	no	—
168	C₂×C₈	—	2	2	no	—
169	C₂×C₈	—	52	2	no	—
170	C₂×C₈	C₂²⋊Dic₁₃	1	1	yes	characteristic
171	C₂×C₈	C₂²⋊Dic₁₃	1	1	yes	characteristic
172	C₂×C₈	C₂²×Dic₁₃	1	1	yes	characteristic
173	C₂×C₈	C₂×C₂²⋊D₁₃	1	1	yes	characteristic
174	C₄⋊C₄	—	4	2	no	—
175	C₄⋊C₄	—	4	2	no	—
176	C₄⋊C₄	—	4	2	no	—
177	C₄⋊C₄	—	4	2	no	—
178	C₄²	#6104	1	1	yes	characteristic
179	C₄²	#6105	1	1	yes	characteristic
180	C₄²	C₂²⋊Dic₁₃	1	1	yes	characteristic
181	C₄²	C₂×C₂²⋊D₁₃	1	1	yes	characteristic
182	C₁₆	—	52	2	no	—
183	C₁₃	#11962	1	1	yes	characteristic, complemented, Sylow
184	C₂³	#88072	1	1	yes	characteristic
185	Q₈	—	4	4	no	—
186	Q₈	—	4	4	no	—
187	Q₈	—	8	4	no	—
188	C₂×C₄	—	2	2	no	—
189	C₂×C₄	—	2	1	no	—
190	C₂×C₄	—	2	1	no	—
191	C₂×C₄	—	4	4	no	—
192	C₂×C₄	—	8	4	no	—
193	C₂×C₄	#88068	1	2	yes	—
194	C₂×C₄	#88069	1	2	yes	—
195	C₂×C₄	#88072	1	1	yes	characteristic
196	C₂×C₄	#88072	1	1	yes	characteristic
197	C₂×C₄	#88072	1	1	yes	characteristic
198	C₂×C₄	C₂×#6104	1	1	yes	characteristic
199	C₂×C₄	C₂×#6105	1	1	yes	characteristic
200	C₂×C₄	C₂×C₂²⋊Dic₁₃	1	1	yes	characteristic
201	C₈	—	2	1	no	—
202	C₈	—	2	1	no	—
203	C₈	—	104	2	no	—
204	C₈	#88073	1	2	yes	—
205	C₈	#88186	1	2	yes	—
206	C₂²	—	2	1	no	—
207	C₂²	—	2	1	no	—
208	C₂²	#11718156	1	2	yes	—
209	C₂²	C₂×#88072	1	1	yes	characteristic
210	C₄	—	2	1	no	—
211	C₄	—	2	1	no	—
212	C₄	—	8	4	no	—
213	C₄	#11718156	1	2	yes	—
214	C₄	#11718157	1	2	yes	—
215	C₄	C₂×#88068	1	1	yes	characteristic
216	C₄	C₂×#88069	1	1	yes	characteristic
217	C₄	#11718876	1	1	yes	characteristic
218	C₄	C₂×#88073	1	1	yes	characteristic
219	C₂	—	2	2	no	—
220	C₂	#13113504	1	1	yes	characteristic
221	C₂	C₂×#11718156	1	1	yes	characteristic
222	C₂	C₂×#11718157	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×#88072
commutator	C₂×C₈×C₁₃	C₂²×C₄
Frattini	C₄×C₈	C₂²×D₁₃
Fitting	C₂×C₁₃×#494	C₂
socle	C₂²×C₁₃	C₂×#102
prime radical	C₂³×C₁₃	#102
cosocle	C₄×C₈×C₁₃	C₂³
hypercenter	C₂×#494	D₁₃
hypocenter	C₁₃	#11962
LCS	G ▷ C₂×C₈×C₁₃ ▷ C₂×C₄×C₁₃ ▷ C₂×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ C₂×#494

Complements

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

N	H ≅ G/N	type
3. #13112885	C₂	semidirect
183. C₁₃	#11962	semidirect

Maximal subgroups

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

subgroup	index
#13116635	2
#13113595	2
#13112885	2
C₂×C₁₃×#494	2
#11962	13