Subgroups — #16789605

Order 3328.

374 conjugacy classes of subgroups.

The subgroup lattice has 200 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
2	C₂×#11718231	C₂	1	2	yes	maximal
3	C₂×C₄×#6092	C₂	1	1	yes	maximal, characteristic
4	#13129874	C₂	1	2	yes	maximal
5	C₄⋊C₄×#1055	C₂	1	1	yes	maximal, characteristic
6	C₂²×#6092	C₂²	1	1	yes	characteristic
7	C₂²×#6092	C₂²	1	2	yes	—
8	C₂²×#6092	C₂²	1	1	yes	characteristic
9	C₄×C₁₃×C₄⋊C₄	C₄	1	1	yes	characteristic
10	C₂×C₄⋊#1055	C₄	1	1	yes	characteristic
11	C₂×C₄×#1055	C₂²	1	2	yes	—
12	C₂×C₄×#1055	C₂²	1	1	yes	characteristic
13	#11718231	—	2	4	no	—
14	C₄⋊#6092	C₄	1	4	yes	complemented
15	C₄×#6092	C₄	1	4	yes	complemented
16	#11718116	C₄	1	2	yes	—
17	C₂×C₁₃×C₄⋊C₄	C₈	1	2	yes	—
18	C₂×C₄²×C₁₃	C₂×C₄	1	2	yes	—
19	C₂×C₄²×C₁₃	C₂×C₄	1	1	yes	characteristic
20	C₂²×#1055	C₂×C₄	1	1	yes	characteristic
21	C₂²×#1055	C₂×C₄	1	2	yes	—
22	C₂²×#1055	C₂³	1	1	yes	characteristic
23	C₂×#6092	—	2	2	no	—
24	C₂×#6092	—	2	1	no	—
25	C₂×#6092	—	2	1	no	—
26	C₂×#6092	—	2	2	no	—
27	C₂×#6092	C₂×C₄	1	2	yes	—
28	C₂×#6092	C₂×C₄	1	4	yes	—
29	C₂×#6092	C₂×C₄	1	2	yes	—
30	C₂×#6092	D₄	1	2	yes	—
31	C₂×#6092	Q₈	1	2	yes	—
32	C₁₃×#95	C₈	1	2	yes	—
33	C₄⋊#1055	C₈	1	4	yes	—
34	C₄×#1055	—	2	2	no	—
35	C₄×#1055	—	2	2	no	—
36	C₄×#1055	C₂×C₄	1	2	yes	—
37	C₄×#1055	C₂×C₄	1	2	yes	—
38	#12662	—	13	1	no	maximal, Sylow
39	C₂²×C₄×C₁₃	C₂×C₈	1	4	yes	—
40	C₂²×C₄×C₁₃	C₂×C₈	1	1	yes	characteristic
41	C₂²×C₄×C₁₃	C₂×C₈	1	1	yes	characteristic
42	C₂²×C₄×C₁₃	C₂²×C₄	1	1	yes	characteristic
43	C₁₃×C₄⋊C₄	—	2	4	no	—
44	C₄²×C₁₃	—	2	2	no	—
45	C₄²×C₁₃	—	2	2	no	—
46	C₄²×C₁₃	C₄²	1	2	yes	—
47	C₄²×C₁₃	C₂×C₈	1	2	yes	—
48	C₂×#1055	—	2	2	no	—
49	C₂×#1055	—	2	1	no	—
50	C₂×#1055	—	2	1	no	—
51	C₂×#1055	—	2	2	no	—
52	C₂×#1055	C₄²	1	2	yes	—
53	C₂×#1055	C₂×C₈	1	4	yes	—
54	C₂×#1055	C₂²×C₄	1	1	yes	characteristic
55	C₂×#1055	C₂²×C₄	1	1	yes	characteristic
56	C₂×#1055	C₂×D₄	1	1	yes	characteristic
57	C₂×#1055	C₂×Q₈	1	1	yes	characteristic
58	C₂×#1055	#41	1	1	yes	characteristic
59	C₂×#1055	#41	1	1	yes	characteristic
60	#6092	—	2	2	no	—
61	#6092	—	2	4	no	—
62	#6092	C₄⋊C₄	1	4	yes	complemented
63	C₂×C₄⋊C₁₆	—	13	1	no	—
64	C₂×#346	—	13	2	no	—
65	C₂×C₄×C₁₆	—	13	1	no	—
66	C₈×C₄⋊C₄	—	13	1	no	—
67	#1374	—	13	2	no	—
68	C₂³×C₁₃	C₂²×C₈	1	1	yes	characteristic
69	C₂×C₄×C₁₃	—	2	2	no	—
70	C₂×C₄×C₁₃	—	2	4	no	—
71	C₂×C₄×C₁₃	—	2	4	no	—
72	C₂×C₄×C₁₃	—	2	4	no	—
73	C₂×C₄×C₁₃	C₄×C₈	1	2	yes	—
74	C₂×C₄×C₁₃	C₄×C₈	1	2	yes	—
75	C₂×C₄×C₁₃	M₃₂	1	4	yes	—
76	C₂×C₄×C₁₃	C₂×C₄²	1	1	yes	characteristic
77	C₂×C₄×C₁₃	C₄×D₄	1	1	yes	characteristic
78	C₂×C₄×C₁₃	C₄×Q₈	1	1	yes	characteristic
79	C₂×C₄×C₁₃	C₂²×C₈	1	1	yes	characteristic
80	C₂×C₄×C₁₃	#131	1	1	yes	characteristic
81	C₂×C₄×C₁₃	#131	1	1	yes	characteristic
82	#1055	—	2	2	no	—
83	#1055	—	2	4	no	—
84	#1055	C₂×C₄⋊C₄	1	1	yes	characteristic
85	#1055	#117	1	1	yes	characteristic
86	#1055	C₄×D₄	1	1	yes	characteristic
87	#1055	C₄×Q₈	1	1	yes	characteristic
88	C₂²×C₁₆	—	13	1	no	—
89	C₂²×C₁₆	—	13	1	no	—
90	C₂²×C₁₆	—	13	2	no	—
91	C₂×C₄⋊C₈	—	13	1	no	—
92	C₂×C₄×C₈	—	13	1	no	—
93	C₂×C₄×C₈	—	13	2	no	—
94	C₄×C₄⋊C₄	#261	1	1	yes	characteristic
95	C₄⋊C₁₆	—	13	4	no	—
96	#346	—	26	4	no	—
97	C₄×C₁₆	—	13	4	no	—
98	#336	—	13	2	no	—
99	C₂²×C₁₃	C₂×C₄×C₈	1	1	yes	characteristic
100	C₂²×C₁₃	C₈×D₄	1	1	yes	characteristic
101	C₂²×C₁₃	C₈×Q₈	1	1	yes	characteristic
102	C₂²×C₁₃	C₂×M₃₂	1	2	yes	—
103	C₂²×C₁₃	#504	1	2	yes	—
104	C₄×C₁₃	—	2	2	no	—
105	C₄×C₁₃	—	4	4	no	—
106	C₄×C₁₃	#346	1	4	yes	complemented
107	C₄×C₁₃	C₄×C₄⋊C₄	1	1	yes	characteristic
108	C₄×C₁₃	#405	1	1	yes	characteristic
109	C₄×C₁₃	C₈×D₄	1	1	yes	characteristic
110	C₄×C₁₃	C₈×Q₈	1	1	yes	characteristic
111	C₂²×C₈	—	13	1	no	—
112	C₂²×C₈	—	13	1	no	—
113	C₂²×C₈	—	13	2	no	—
114	C₂×C₄⋊C₄	#1057	1	2	yes	—
115	C₂×C₄²	C₂×#261	1	2	yes	—
116	C₂×C₄²	C₂×#261	1	1	yes	characteristic
117	C₂×C₁₆	—	13	2	no	—
118	C₂×C₁₆	—	13	2	no	—
119	C₂×C₁₆	—	13	2	no	—
120	C₂×C₁₆	—	13	2	no	—
121	C₂×C₁₆	—	13	4	no	—
122	C₂×C₁₆	—	26	2	no	—
123	C₂×C₁₆	—	26	1	no	—
124	C₂×C₁₆	—	26	1	no	—
125	C₂×C₁₆	—	26	2	no	—
126	C₄⋊C₈	—	13	4	no	—
127	C₄×C₈	—	13	2	no	—
128	C₄×C₈	—	13	2	no	—
129	C₄×C₈	—	26	2	no	—
130	C₄×C₈	—	26	2	no	—
131	#95	#1057	1	2	yes	—
132	C₂×C₁₃	C₈×C₄⋊C₄	1	1	yes	characteristic
133	C₂×C₁₃	C₂×#346	1	1	yes	characteristic
134	C₂×C₁₃	#2108	1	1	yes	characteristic
135	C₂×C₁₃	#2169	1	2	yes	—
136	C₂×C₁₃	#2183	1	2	yes	—
137	C₂²×C₄	C₂×#1057	1	4	yes	—
138	C₂²×C₄	C₂×#1057	1	1	yes	characteristic
139	C₂²×C₄	C₂×#1057	1	1	yes	characteristic
140	C₂²×C₄	C₂²×#261	1	1	yes	characteristic
141	C₂×C₈	—	13	1	no	—
142	C₂×C₈	—	13	1	no	—
143	C₂×C₈	—	13	1	no	—
144	C₂×C₈	—	13	1	no	—
145	C₂×C₈	—	13	1	no	—
146	C₂×C₈	—	13	1	no	—
147	C₂×C₈	—	13	4	no	—
148	C₂×C₈	—	13	2	no	—
149	C₂×C₈	—	26	1	no	—
150	C₂×C₈	—	26	2	no	—
151	C₂×C₈	—	26	1	no	—
152	C₂×C₈	—	26	2	no	—
153	C₄⋊C₄	—	2	4	no	—
154	C₄²	—	2	2	no	—
155	C₄²	—	2	2	no	—
156	C₄²	D₁₃⋊C₈	1	2	yes	—
157	C₄²	C₄×#261	1	2	yes	—
158	C₁₆	—	13	4	no	—
159	C₁₆	—	26	4	no	—
160	C₁₆	—	26	2	no	—
161	C₁₃	#12662	1	1	yes	characteristic, complemented, Sylow
162	C₂³	C₂²×#1057	1	1	yes	characteristic
163	C₂×C₄	—	2	2	no	—
164	C₂×C₄	—	2	4	no	—
165	C₂×C₄	—	2	4	no	—
166	C₂×C₄	—	2	4	no	—
167	C₂×C₄	#88106	1	4	yes	—
168	C₂×C₄	C₄×#1057	1	2	yes	—
169	C₂×C₄	C₄×#1057	1	2	yes	—
170	C₂×C₄	C₂×D₁₃⋊C₈	1	1	yes	characteristic
171	C₂×C₄	C₂×C₄×#261	1	1	yes	characteristic
172	C₂×C₄	#88238	1	1	yes	characteristic
173	C₂×C₄	D₄×#261	1	1	yes	characteristic
174	C₂×C₄	#88240	1	1	yes	characteristic
175	C₂×C₄	Q₈×#261	1	1	yes	characteristic
176	C₈	—	13	1	no	—
177	C₈	—	13	1	no	—
178	C₈	—	13	1	no	—
179	C₈	—	13	1	no	—
180	C₈	—	26	2	no	—
181	C₈	—	26	4	no	—
182	C₂²	C₂×#88106	1	2	yes	—
183	C₂²	C₂×C₄×#1057	1	1	yes	characteristic
184	C₂²	D₄×#1057	1	1	yes	characteristic
185	C₂²	Q₈×#1057	1	1	yes	characteristic
186	C₂²	#11719162	1	2	yes	—
187	C₄	—	2	2	no	—
188	C₄	—	4	4	no	—
189	C₄	#11718231	1	4	yes	complemented
190	C₄	#11719072	1	1	yes	characteristic
191	C₄	#11719073	1	1	yes	characteristic
192	C₄	#11719074	1	1	yes	characteristic
193	C₄	C₄⋊C₄×#261	1	1	yes	characteristic
194	C₂	C₄⋊C₄×#1057	1	1	yes	characteristic
195	C₂	C₂×#11718231	1	1	yes	characteristic
196	C₂	#13131045	1	1	yes	characteristic
197	C₂	#13131220	1	2	yes	—
198	C₂	#13131251	1	2	yes	—

	subgroup	G/H
center, Frattini	C₂²×C₄	C₂²×#261
commutator	C₂²×C₁₃	C₂×C₄×C₈
Fitting	C₄×C₁₃×C₄⋊C₄	C₄
socle, prime radical	C₂³×C₁₃	C₂²×C₈
cosocle	C₂²×#1055	C₂³
hypercenter	C₄×C₄⋊C₄	#261
hypocenter	C₁₃	#12662
LCS	G ▷ C₂²×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂²×C₄ ◁ C₄×C₄⋊C₄

Complements

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

N	H ≅ G/N	type
14. C₄⋊#6092	C₄	semidirect
15. C₄×#6092	C₄	semidirect
62. #6092	C₄⋊C₄	semidirect
106. C₄×C₁₃	#346	semidirect
161. C₁₃	#12662	semidirect
189. C₄	#11718231	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×#11718231	2
C₂×C₄×#6092	2
#13129874	2
C₄⋊C₄×#1055	2
#12662	13