Subgroups — #15774630

Order 3328.

244 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	#13116668	C₂	1	1	yes	maximal, characteristic, complemented
2	D₄×#6090	C₂	1	1	yes	maximal, characteristic, complemented
3	#13112906	C₂	1	1	yes	maximal, characteristic, complemented
4	D₄⋊#6090	C₂	1	1	yes	maximal, characteristic, complemented
5	D₄⋊#6090	C₂	1	1	yes	maximal, characteristic, complemented
6	#13112753	C₂	1	1	yes	maximal, characteristic, complemented
7	C₈×C₁₃×D₈	C₂	1	1	yes	maximal, characteristic
8	C₄×C₁₃×D₈	C₄	1	1	yes	characteristic
9	C₈×D₄×C₁₃	C₂²	1	1	yes	characteristic
10	C₈×D₄×C₁₃	C₂²	1	1	yes	characteristic
11	C₂×#88052	—	2	1	no	—
12	C₂²×#6090	—	2	1	no	—
13	C₁₃×#335	C₄	1	1	yes	characteristic
14	C₁₃×D₄⋊C₈	C₄	1	1	yes	characteristic
15	C₁₃×D₄⋊C₈	C₄	1	1	yes	characteristic
16	C₈²×C₁₃	C₂²	1	1	yes	characteristic
17	C₂²⋊#6090	—	2	1	no	—
18	C₂²⋊#6090	—	2	1	no	—
19	C₄⋊#6090	C₂²	1	1	yes	characteristic
20	C₄⋊#6090	C₂²	1	1	yes	characteristic
21	C₄×#6090	C₂²	1	1	yes	characteristic
22	C₄×#6090	C₂²	1	1	yes	characteristic
23	C₂×C₁₃×D₈	C₈	1	1	yes	characteristic
24	C₂²×C₈×C₁₃	—	2	1	no	—
25	C₂²×C₈×C₁₃	—	2	1	no	—
26	C₄×D₄×C₁₃	C₂×C₄	1	1	yes	characteristic
27	C₄×D₄×C₁₃	C₂×C₄	1	1	yes	characteristic
28	C₁₃×#107	C₈	1	1	yes	characteristic
29	C₁₃×C₄⋊C₈	C₂×C₄	1	1	yes	characteristic
30	C₁₃×C₄⋊C₈	C₂×C₄	1	1	yes	characteristic
31	C₁₃×#102	C₈	1	1	yes	characteristic
32	C₁₃×#102	C₈	1	1	yes	characteristic
33	C₁₃×C₂²⋊C₈	—	2	1	no	—
34	C₁₃×C₂²⋊C₈	—	2	1	no	—
35	C₄×C₈×C₁₃	C₂×C₄	1	1	yes	characteristic
36	C₄×C₈×C₁₃	C₂×C₄	1	1	yes	characteristic
37	C₄×C₈×C₁₃	C₂³	1	1	yes	characteristic
38	#88052	—	4	2	no	—
39	C₂×#6090	—	2	1	no	—
40	C₂×#6090	—	2	1	no	—
41	C₂×#6090	—	2	1	no	—
42	C₂×#6090	—	4	2	no	—
43	C₂×#6090	D₄	1	1	yes	characteristic
44	C₂×#6090	D₄	1	1	yes	characteristic
45	#12038	—	13	1	no	maximal, Sylow
46	C₂×D₄×C₁₃	C₂×C₈	1	1	yes	characteristic
47	C₂×D₄×C₁₃	C₂×C₈	1	1	yes	characteristic
48	C₂²×C₄×C₁₃	—	2	1	no	—
49	C₂²×C₄×C₁₃	—	2	1	no	—
50	C₁₃×D₈	—	2	2	no	—
51	C₂×C₈×C₁₃	—	2	1	no	—
52	C₂×C₈×C₁₃	—	2	1	no	—
53	C₂×C₈×C₁₃	—	2	1	no	—
54	C₂×C₈×C₁₃	—	4	1	no	—
55	C₂×C₈×C₁₃	—	4	1	no	—
56	C₂×C₈×C₁₃	—	4	1	no	—
57	C₂×C₈×C₁₃	—	4	1	no	—
58	C₂×C₈×C₁₃	C₂×C₈	1	1	yes	characteristic
59	C₂×C₈×C₁₃	C₂×C₈	1	1	yes	characteristic
60	C₂×C₈×C₁₃	C₂×D₄	1	1	yes	characteristic
61	C₂×C₈×C₁₃	#41	1	1	yes	characteristic
62	C₁₃×C₄⋊C₄	C₂×C₈	1	1	yes	characteristic
63	C₁₃×C₄⋊C₄	C₂×C₈	1	1	yes	characteristic
64	C₁₃×C₂²⋊C₄	—	2	1	no	—
65	C₁₃×C₂²⋊C₄	—	2	1	no	—
66	C₄²×C₁₃	C₂²×C₄	1	1	yes	characteristic
67	#6090	—	2	2	no	—
68	#6090	—	2	1	no	—
69	#6090	—	4	1	no	—
70	#6090	—	4	1	no	—
71	#6090	D₈	1	2	yes	complemented
72	#2169	—	13	1	no	—
73	D₄×C₁₆	—	13	1	no	—
74	C₈×D₈	D₁₃	1	1	yes	characteristic
75	#1370	—	13	1	no	—
76	D₄⋊C₁₆	—	13	1	no	—
77	D₄⋊C₁₆	—	13	1	no	—
78	#1311	—	13	1	no	—
79	C₂³×C₁₃	—	2	1	no	—
80	C₂³×C₁₃	—	2	1	no	—
81	D₄×C₁₃	—	2	1	no	—
82	D₄×C₁₃	—	2	1	no	—
83	D₄×C₁₃	—	2	1	no	—
84	D₄×C₁₃	M₃₂	1	2	yes	complemented
85	C₂×C₄×C₁₃	—	2	1	no	—
86	C₂×C₄×C₁₃	—	2	1	no	—
87	C₂×C₄×C₁₃	—	4	1	no	—
88	C₂×C₄×C₁₃	—	4	1	no	—
89	C₂×C₄×C₁₃	—	4	1	no	—
90	C₂×C₄×C₁₃	—	4	1	no	—
91	C₂×C₄×C₁₃	C₄×D₄	1	1	yes	characteristic
92	C₂×C₄×C₁₃	C₂²×C₈	1	1	yes	characteristic
93	C₂×C₄×C₁₃	#131	1	1	yes	characteristic
94	C₈×C₁₃	—	2	1	no	—
95	C₈×C₁₃	—	2	1	no	—
96	C₈×C₁₃	—	2	1	no	—
97	C₈×C₁₃	—	4	1	no	—
98	C₈×C₁₃	—	4	1	no	—
99	C₈×C₁₃	—	4	1	no	—
100	C₈×C₁₃	C₂×D₈	1	1	yes	characteristic
101	C₈×C₁₃	#135	1	1	yes	characteristic
102	C₂×M₃₂	—	26	1	no	—
103	C₂²×C₁₆	—	26	1	no	—
104	C₄×D₈	Dic₁₃	1	1	yes	characteristic
105	C₈×D₄	C₂×D₁₃	1	1	yes	characteristic
106	C₈×D₄	C₂×D₁₃	1	1	yes	characteristic
107	C₄⋊C₁₆	—	13	1	no	—
108	C₄⋊C₁₆	—	13	1	no	—
109	C₂²⋊C₁₆	—	26	1	no	—
110	C₂²⋊C₁₆	—	26	1	no	—
111	C₄×C₁₆	—	13	1	no	—
112	C₄×C₁₆	—	13	1	no	—
113	#335	Dic₁₃	1	1	yes	characteristic
114	D₄⋊C₈	Dic₁₃	1	1	yes	characteristic
115	D₄⋊C₈	Dic₁₃	1	1	yes	characteristic
116	C₈²	C₂×D₁₃	1	1	yes	characteristic
117	C₂²×C₁₃	—	2	2	no	—
118	C₂²×C₁₃	—	4	1	no	—
119	C₂²×C₁₃	—	4	1	no	—
120	C₂²×C₁₃	—	4	1	no	—
121	C₂²×C₁₃	—	4	1	no	—
122	C₂²×C₁₃	—	4	1	no	—
123	C₂²×C₁₃	C₈×D₄	1	1	yes	characteristic
124	C₄×C₁₃	—	2	1	no	—
125	C₄×C₁₃	—	4	1	no	—
126	C₄×C₁₃	—	4	1	no	—
127	C₄×C₁₃	C₄×D₈	1	1	yes	characteristic
128	C₄×C₁₃	#443	1	1	yes	characteristic
129	C₄×C₁₃	C₂×M₃₂	1	1	yes	characteristic
130	C₄×C₁₃	#504	1	1	yes	characteristic
131	C₂×D₈	#1055	1	1	yes	characteristic
132	C₂²×C₈	—	2	1	no	—
133	C₂²×C₈	—	2	1	no	—
134	C₄×D₄	C₂×Dic₁₃	1	1	yes	characteristic
135	C₄×D₄	C₂×Dic₁₃	1	1	yes	characteristic
136	M₃₂	—	52	2	no	—
137	C₂×C₁₆	—	13	1	no	—
138	C₂×C₁₆	—	13	1	no	—
139	C₂×C₁₆	—	26	1	no	—
140	C₂×C₁₆	—	26	1	no	—
141	C₂×C₁₆	—	26	1	no	—
142	C₂×C₁₆	—	52	2	no	—
143	#107	#1055	1	1	yes	characteristic
144	C₄⋊C₈	C₂×Dic₁₃	1	1	yes	characteristic
145	C₄⋊C₈	C₂×Dic₁₃	1	1	yes	characteristic
146	#102	#1055	1	1	yes	characteristic
147	#102	#1055	1	1	yes	characteristic
148	C₂²⋊C₈	—	2	1	no	—
149	C₂²⋊C₈	—	2	1	no	—
150	C₄×C₈	C₂×Dic₁₃	1	1	yes	characteristic
151	C₄×C₈	C₂×Dic₁₃	1	1	yes	characteristic
152	C₄×C₈	C₂²×D₁₃	1	1	yes	characteristic
153	C₂×C₁₃	—	4	2	no	—
154	C₂×C₁₃	—	8	1	no	—
155	C₂×C₁₃	C₈×D₈	1	1	yes	characteristic
156	C₂×C₁₃	#2168	1	1	yes	characteristic
157	C₂×C₁₃	#2171	1	1	yes	characteristic
158	C₂×D₄	C₂×#1055	1	1	yes	characteristic
159	C₂×D₄	C₂×#1055	1	1	yes	characteristic
160	C₂²×C₄	—	2	1	no	—
161	C₂²×C₄	—	2	1	no	—
162	D₈	—	2	2	no	—
163	C₂×C₈	—	2	1	no	—
164	C₂×C₈	—	2	1	no	—
165	C₂×C₈	—	2	1	no	—
166	C₂×C₈	—	4	1	no	—
167	C₂×C₈	—	4	1	no	—
168	C₂×C₈	—	4	1	no	—
169	C₂×C₈	—	4	1	no	—
170	C₂×C₈	C₂×#1055	1	1	yes	characteristic
171	C₂×C₈	C₂×#1055	1	1	yes	characteristic
172	C₂×C₈	D₄×D₁₃	1	1	yes	characteristic
173	C₂×C₈	#6129	1	1	yes	characteristic
174	C₄⋊C₄	C₂×#1055	1	1	yes	characteristic
175	C₄⋊C₄	C₂×#1055	1	1	yes	characteristic
176	C₂²⋊C₄	—	2	1	no	—
177	C₂²⋊C₄	—	2	1	no	—
178	C₄²	C₂²×Dic₁₃	1	1	yes	characteristic
179	C₁₆	—	13	2	no	—
180	C₁₆	—	26	2	no	—
181	C₁₆	—	26	1	no	—
182	C₁₆	—	52	1	no	—
183	C₁₆	—	52	1	no	—
184	C₁₃	#12038	1	1	yes	characteristic, complemented, Sylow
185	C₂³	—	2	1	no	—
186	C₂³	—	2	1	no	—
187	D₄	—	2	1	no	—
188	D₄	—	2	1	no	—
189	D₄	—	2	1	no	—
190	D₄	#88052	1	2	yes	complemented
191	C₂×C₄	—	2	1	no	—
192	C₂×C₄	—	2	1	no	—
193	C₂×C₄	—	4	1	no	—
194	C₂×C₄	—	4	1	no	—
195	C₂×C₄	—	4	1	no	—
196	C₂×C₄	—	4	1	no	—
197	C₂×C₄	C₂²×#1055	1	1	yes	characteristic
198	C₂×C₄	D₄×Dic₁₃	1	1	yes	characteristic
199	C₂×C₄	#88202	1	1	yes	characteristic
200	C₈	—	2	1	no	—
201	C₈	—	2	1	no	—
202	C₈	—	2	1	no	—
203	C₈	—	4	1	no	—
204	C₈	—	4	1	no	—
205	C₈	—	4	1	no	—
206	C₈	D₈×D₁₃	1	1	yes	characteristic
207	C₈	#88166	1	1	yes	characteristic
208	C₂²	—	2	2	no	—
209	C₂²	—	4	1	no	—
210	C₂²	—	4	1	no	—
211	C₂²	—	4	1	no	—
212	C₂²	—	4	1	no	—
213	C₂²	—	4	1	no	—
214	C₂²	D₄×#1055	1	1	yes	characteristic
215	C₄	—	2	1	no	—
216	C₄	—	4	1	no	—
217	C₄	—	4	1	no	—
218	C₄	C₂×#88052	1	1	yes	characteristic
219	C₄	#11718801	1	1	yes	characteristic
220	C₄	D₈×Dic₁₃	1	1	yes	characteristic
221	C₄	#11718857	1	1	yes	characteristic
222	C₂	—	4	2	no	—
223	C₂	—	8	1	no	—
224	C₂	D₈×#1055	1	1	yes	characteristic
225	C₂	#13116662	1	1	yes	characteristic
226	C₂	#13116678	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₈	D₄×D₁₃
commutator	C₂×C₄×C₁₃	C₂²×C₈
Frattini	C₄×C₈	C₂²×D₁₃
Fitting	C₈×C₁₃×D₈	C₂
socle	C₂²×C₁₃	C₈×D₄
prime radical	C₂×C₁₃×D₈	C₈
cosocle	C₄×C₈×C₁₃	C₂³
hypercenter	C₈×D₈	D₁₃
hypocenter	C₁₃	#12038
LCS	G ▷ C₂×C₄×C₁₃ ▷ C₂×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂×C₈ ◁ C₄×C₈ ◁ C₈×D₈

Complements

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

N	H ≅ G/N	type
1. #13116668	C₂	semidirect
2. D₄×#6090	C₂	semidirect
3. #13112906	C₂	semidirect
4. D₄⋊#6090	C₂	semidirect
5. D₄⋊#6090	C₂	semidirect
6. #13112753	C₂	semidirect
71. #6090	D₈	semidirect
84. D₄×C₁₃	M₃₂	semidirect
184. C₁₃	#12038	semidirect
190. D₄	#88052	semidirect

Maximal subgroups

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

subgroup	index
#13116668	2
D₄×#6090	2
#13112906	2
D₄⋊#6090	2
#13112753	2
C₈×C₁₃×D₈	2
#12038	13