Subgroups — #16790855

Order 3328.

534 conjugacy classes of subgroups.

The subgroup lattice has 236 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₄⋊#6092	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×C₂²⋊#6092	C₂	1	4	yes	maximal, complemented
3	C₂×C₄×#6092	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂×D₄×#1055	C₂	1	1	yes	maximal, characteristic
5	C₂×C₄×D₄×C₁₃	C₄	1	1	yes	characteristic
6	C₂³×#1055	C₂²	1	2	yes	—
7	C₂²×#6092	C₂²	1	2	yes	—
8	C₂²×#6092	C₂²	1	2	yes	—
9	C₂×C₂²⋊#1055	C₄	1	2	yes	—
10	D₄×#1055	—	2	2	no	—
11	D₄×#1055	—	2	2	no	—
12	C₂×C₄⋊#1055	C₄	1	1	yes	characteristic
13	C₂×C₄×#1055	C₂²	1	1	yes	characteristic
14	C₂²⋊#6092	—	2	8	no	—
15	C₄⋊#6092	—	2	2	no	—
16	C₄×#6092	—	2	2	no	—
17	C₂²×D₄×C₁₃	C₈	1	1	yes	characteristic
18	C₂³×C₄×C₁₃	C₂×C₄	1	2	yes	—
19	C₄×D₄×C₁₃	—	2	2	no	—
20	C₄×D₄×C₁₃	—	2	2	no	—
21	C₂×C₁₃×C₄⋊C₄	C₈	1	1	yes	characteristic
22	C₂×C₁₃×C₂²⋊C₄	C₈	1	2	yes	—
23	C₂×C₄²×C₁₃	C₂×C₄	1	1	yes	characteristic
24	C₂²×#1055	—	2	2	no	—
25	C₂²×#1055	—	2	2	no	—
26	C₂²×#1055	—	2	4	no	—
27	C₂²×#1055	—	2	4	no	—
28	C₂²×#1055	C₂×C₄	1	2	yes	—
29	C₂²×#1055	C₂×C₄	1	1	yes	characteristic
30	C₂²×#1055	C₂³	1	1	yes	characteristic
31	C₂×#6092	—	2	2	no	—
32	C₂×#6092	—	2	2	no	—
33	C₂×#6092	—	2	2	no	—
34	C₂×#6092	—	2	4	no	—
35	C₂×#6092	D₄	1	4	yes	complemented
36	C₂²⋊#1055	—	2	4	no	—
37	C₄⋊#1055	—	2	2	no	—
38	C₄×#1055	D₄	1	2	yes	—
39	C₄×#1055	D₄	1	2	yes	—
40	#13189	—	13	1	no	maximal, Sylow
41	C₂⁴×C₁₃	C₂×C₈	1	2	yes	—
42	C₂×D₄×C₁₃	—	2	2	no	—
43	C₂×D₄×C₁₃	—	2	2	no	—
44	C₂×D₄×C₁₃	C₁₆	1	4	yes	complemented
45	C₂²×C₄×C₁₃	—	2	2	no	—
46	C₂²×C₄×C₁₃	—	2	4	no	—
47	C₂²×C₄×C₁₃	—	2	2	no	—
48	C₂²×C₄×C₁₃	—	2	4	no	—
49	C₂²×C₄×C₁₃	C₂×C₈	1	2	yes	—
50	C₂²×C₄×C₁₃	C₂×C₈	1	1	yes	characteristic
51	C₂²×C₄×C₁₃	C₂×C₈	1	1	yes	characteristic
52	C₂²×C₄×C₁₃	C₂²×C₄	1	1	yes	characteristic
53	C₁₃×C₄⋊C₄	—	2	2	no	—
54	C₁₃×C₂²⋊C₄	—	2	4	no	—
55	C₄²×C₁₃	C₂²⋊C₄	1	2	yes	—
56	C₄²×C₁₃	M₁₆	1	2	yes	—
57	C₂×#1055	—	2	2	no	—
58	C₂×#1055	—	2	2	no	—
59	C₂×#1055	—	2	4	no	—
60	C₂×#1055	—	4	4	no	—
61	C₂×#1055	—	4	2	no	—
62	C₂×#1055	—	4	2	no	—
63	C₂×#1055	C₂²⋊C₄	1	2	yes	—
64	C₂×#1055	C₂×D₄	1	1	yes	characteristic
65	C₂×#1055	C₂×D₄	1	2	yes	—
66	C₂×#1055	C₂×D₄	1	1	yes	characteristic
67	C₂×#1055	#41	1	2	yes	—
68	#6092	—	2	4	no	—
69	#6092	—	4	2	no	—
70	C₂×C₈×D₄	—	13	1	no	—
71	C₂×C₄⋊C₁₆	—	13	1	no	—
72	C₂×C₂²⋊C₁₆	—	13	4	no	—
73	C₂×C₄×C₁₆	—	13	1	no	—
74	C₂³×C₁₃	—	2	2	no	—
75	C₂³×C₁₃	—	2	4	no	—
76	C₂³×C₁₃	—	2	2	no	—
77	C₂³×C₁₃	—	2	4	no	—
78	C₂³×C₁₃	C₂×C₁₆	1	4	yes	—
79	C₂³×C₁₃	C₂²×C₈	1	1	yes	characteristic
80	D₄×C₁₃	—	2	8	no	—
81	C₂×C₄×C₁₃	—	2	2	no	—
82	C₂×C₄×C₁₃	—	2	2	no	—
83	C₂×C₄×C₁₃	—	2	4	no	—
84	C₂×C₄×C₁₃	—	4	2	no	—
85	C₂×C₄×C₁₃	—	4	2	no	—
86	C₂×C₄×C₁₃	—	4	4	no	—
87	C₂×C₄×C₁₃	C₂²⋊C₈	1	2	yes	—
88	C₂×C₄×C₁₃	C₂²⋊C₈	1	2	yes	—
89	C₂×C₄×C₁₃	C₂×C₁₆	1	2	yes	—
90	C₂×C₄×C₁₃	M₃₂	1	2	yes	—
91	C₂×C₄×C₁₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
92	C₂×C₄×C₁₃	C₄×D₄	1	2	yes	—
93	C₂×C₄×C₁₃	C₂×M₁₆	1	1	yes	characteristic
94	C₂×C₄×C₁₃	#131	1	2	yes	—
95	#1055	—	2	2	no	—
96	#1055	—	4	2	no	—
97	#1055	#121	1	2	yes	—
98	#1055	#124	1	1	yes	characteristic
99	#1055	#127	1	1	yes	characteristic
100	C₂³×C₈	—	13	2	no	—
101	C₂×C₄×D₄	#261	1	1	yes	characteristic
102	C₂²×C₁₆	—	13	2	no	—
103	C₂²×C₁₆	—	13	2	no	—
104	C₈×D₄	—	26	2	no	—
105	C₈×D₄	—	26	2	no	—
106	C₂×C₄⋊C₈	—	13	1	no	—
107	C₂×C₂²⋊C₈	—	13	2	no	—
108	C₂×C₄×C₈	—	13	1	no	—
109	C₄⋊C₁₆	—	26	2	no	—
110	C₂²⋊C₁₆	—	26	8	no	—
111	C₄×C₁₆	—	26	2	no	—
112	C₂²×C₁₃	—	2	4	no	—
113	C₂²×C₁₃	—	2	8	no	—
114	C₂²×C₁₃	—	4	2	no	—
115	C₂²×C₁₃	—	4	4	no	—
116	C₂²×C₁₃	—	4	2	no	—
117	C₂²×C₁₃	C₂×C₂²⋊C₈	1	1	yes	characteristic
118	C₂²×C₁₃	C₈×D₄	1	2	yes	—
119	C₂²×C₁₃	C₂²×C₁₆	1	1	yes	characteristic
120	C₂²×C₁₃	C₂×M₃₂	1	1	yes	characteristic
121	C₂²×C₁₃	#504	1	2	yes	—
122	C₄×C₁₃	—	2	2	no	—
123	C₄×C₁₃	—	4	2	no	—
124	C₄×C₁₃	C₂²⋊C₁₆	1	4	yes	complemented
125	C₄×C₁₃	#390	1	1	yes	characteristic
126	C₄×C₁₃	#408	1	1	yes	characteristic
127	C₄×C₁₃	#436	1	2	yes	—
128	C₂²×D₄	#1057	1	1	yes	characteristic
129	C₂³×C₄	C₂×#261	1	2	yes	—
130	C₂²×C₈	—	13	1	no	—
131	C₂²×C₈	—	13	1	no	—
132	C₂²×C₈	—	13	2	no	—
133	C₂²×C₈	—	26	4	no	—
134	C₂²×C₈	—	26	4	no	—
135	C₂²×C₈	—	26	2	no	—
136	C₂²×C₈	—	26	2	no	—
137	C₄×D₄	—	2	2	no	—
138	C₄×D₄	—	2	2	no	—
139	C₂×C₄⋊C₄	#1057	1	1	yes	characteristic
140	C₂×C₂²⋊C₄	#1057	1	2	yes	—
141	C₂×C₄²	C₂×#261	1	1	yes	characteristic
142	C₂×C₁₆	—	13	4	no	—
143	C₂×C₁₆	—	26	2	no	—
144	C₂×C₁₆	—	26	2	no	—
145	C₂×C₁₆	—	26	4	no	—
146	C₂×C₁₆	—	26	2	no	—
147	C₄⋊C₈	—	26	2	no	—
148	C₂²⋊C₈	—	26	4	no	—
149	C₄×C₈	—	13	2	no	—
150	C₄×C₈	—	13	2	no	—
151	C₂×C₁₃	—	4	4	no	—
152	C₂×C₁₃	#1954	1	1	yes	characteristic
153	C₂×C₁₃	C₂×C₂²⋊C₁₆	1	1	yes	characteristic
154	C₂×C₁₃	#2113	1	1	yes	characteristic
155	C₂×C₁₃	D₄×C₁₆	1	2	yes	—
156	C₂×C₁₃	#2169	1	2	yes	—
157	C₂⁴	C₂×#1057	1	2	yes	—
158	C₂×D₄	—	2	2	no	—
159	C₂×D₄	—	2	2	no	—
160	C₂×D₄	#6092	1	4	yes	complemented
161	C₂²×C₄	—	2	4	no	—
162	C₂²×C₄	—	2	4	no	—
163	C₂²×C₄	—	2	2	no	—
164	C₂²×C₄	—	2	2	no	—
165	C₂²×C₄	C₂×#1057	1	2	yes	—
166	C₂²×C₄	C₂×#1057	1	1	yes	characteristic
167	C₂²×C₄	C₂×#1057	1	1	yes	characteristic
168	C₂²×C₄	C₂²×#261	1	1	yes	characteristic
169	C₂×C₈	—	13	2	no	—
170	C₂×C₈	—	13	2	no	—
171	C₂×C₈	—	13	1	no	—
172	C₂×C₈	—	13	1	no	—
173	C₂×C₈	—	13	2	no	—
174	C₂×C₈	—	26	2	no	—
175	C₂×C₈	—	26	4	no	—
176	C₂×C₈	—	26	2	no	—
177	C₂×C₈	—	52	4	no	—
178	C₂×C₈	—	52	2	no	—
179	C₂×C₈	—	52	2	no	—
180	C₄⋊C₄	—	2	2	no	—
181	C₂²⋊C₄	—	2	4	no	—
182	C₄²	#6122	1	2	yes	—
183	C₄²	C₂²⋊#261	1	2	yes	—
184	C₁₆	—	26	4	no	—
185	C₁₆	—	52	2	no	—
186	C₁₃	#13189	1	1	yes	characteristic, complemented, Sylow
187	C₂³	—	2	2	no	—
188	C₂³	—	2	2	no	—
189	C₂³	—	2	4	no	—
190	C₂³	—	2	4	no	—
191	C₂³	C₂×#6092	1	4	yes	—
192	C₂³	C₂²×#1057	1	1	yes	characteristic
193	D₄	—	2	8	no	—
194	C₂×C₄	—	2	2	no	—
195	C₂×C₄	—	2	2	no	—
196	C₂×C₄	—	2	4	no	—
197	C₂×C₄	—	4	4	no	—
198	C₂×C₄	—	4	2	no	—
199	C₂×C₄	—	4	2	no	—
200	C₂×C₄	C₂×#6092	1	2	yes	—
201	C₂×C₄	#88106	1	2	yes	—
202	C₂×C₄	C₂²⋊#1057	1	2	yes	—
203	C₂×C₄	C₂²⋊#1057	1	2	yes	—
204	C₂×C₄	#88238	1	2	yes	—
205	C₂×C₄	D₄×#261	1	2	yes	—
206	C₂×C₄	C₂×#6122	1	1	yes	characteristic
207	C₂×C₄	C₂×C₂²⋊#261	1	1	yes	characteristic
208	C₈	—	13	2	no	—
209	C₈	—	13	1	no	—
210	C₈	—	13	1	no	—
211	C₈	—	26	2	no	—
212	C₈	—	52	2	no	—
213	C₂²	—	2	4	no	—
214	C₂²	—	2	8	no	—
215	C₂²	—	4	2	no	—
216	C₂²	—	4	2	no	—
217	C₂²	—	4	4	no	—
218	C₂²	C₂²×#6092	1	1	yes	characteristic
219	C₂²	C₂×#88106	1	1	yes	characteristic
220	C₂²	D₄×#1057	1	2	yes	—
221	C₂²	#11719162	1	2	yes	—
222	C₂²	C₂×C₂²⋊#1057	1	1	yes	characteristic
223	C₄	—	2	2	no	—
224	C₄	—	4	2	no	—
225	C₄	C₂²⋊#6092	1	4	yes	complemented
226	C₄	#11719145	1	2	yes	—
227	C₄	#11719147	1	1	yes	characteristic
228	C₄	#11719151	1	1	yes	characteristic
229	C₂	—	4	4	no	—
230	C₂	#13130957	1	1	yes	characteristic
231	C₂	C₂×C₂²⋊#6092	1	1	yes	characteristic
232	C₂	#13131066	1	1	yes	characteristic
233	C₂	D₄×#6092	1	2	yes	—
234	C₂	#13131220	1	2	yes	—

	subgroup	G/H
center, Frattini	C₂²×C₄	C₂²×#261
commutator	C₂²×C₁₃	C₂²×C₁₆
Fitting	C₂×C₄×D₄×C₁₃	C₄
socle	C₂³×C₁₃	C₂²×C₈
prime radical	C₂²×D₄×C₁₃	C₈
cosocle	C₂²×#1055	C₂³
hypercenter	C₂×C₄×D₄	#261
hypocenter	C₁₃	#13189
LCS	G ▷ C₂²×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂²×C₄ ◁ C₂×C₄×D₄

Complements

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

N	H ≅ G/N	type
1. C₂×C₄⋊#6092	C₂	semidirect
2. C₂×C₂²⋊#6092	C₂	semidirect
3. C₂×C₄×#6092	C₂	semidirect
35. C₂×#6092	D₄	semidirect
44. C₂×D₄×C₁₃	C₁₆	semidirect
124. C₄×C₁₃	C₂²⋊C₁₆	semidirect
160. C₂×D₄	#6092	semidirect
186. C₁₃	#13189	semidirect
225. C₄	C₂²⋊#6092	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₄⋊#6092	2
C₂×C₂²⋊#6092	2
C₂×C₄×#6092	2
C₂×D₄×#1055	2
#13189	13