Subgroups — #16789067

Order 3328.

586 conjugacy classes of subgroups.

The subgroup lattice has 222 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×#88106	C₂	1	2	yes	maximal
2	#13129880	C₂	1	4	yes	maximal, complemented
3	C₂×#11718658	C₂	1	1	yes	maximal, characteristic
4	C₂×C₁₃×#117	C₄	1	1	yes	characteristic
5	C₂²×#6099	C₄	1	1	yes	characteristic
6	C₂³×#1055	C₂²	1	1	yes	characteristic
7	C₂×#88106	—	2	4	no	—
8	C₂×#88106	—	2	2	no	—
9	C₂×#88106	C₄	1	8	yes	complemented
10	C₂×#88106	C₂²	1	4	yes	—
11	C₂²×#6092	—	2	2	no	—
12	#11718658	—	2	4	no	—
13	C₂×#88043	C₄	1	2	yes	—
14	C₂×C₄×#1055	C₂²	1	2	yes	—
15	C₂³×C₄×C₁₃	C₂×C₄	1	1	yes	characteristic
16	C₁₃×#117	—	2	2	no	—
17	C₁₃×#117	C₈	1	4	yes	—
18	C₂×C₁₃×C₄⋊C₄	—	2	1	no	—
19	C₂×C₁₃×C₂²⋊C₄	—	2	1	no	—
20	C₂×C₄²×C₁₃	C₂×C₄	1	2	yes	—
21	C₂×#6099	—	2	2	no	—
22	C₂×#6099	—	2	2	no	—
23	C₂×#6099	C₈	1	4	yes	—
24	C₂²×#1055	C₂×C₄	1	2	yes	—
25	C₂²×#1055	C₂×C₄	1	1	yes	characteristic
26	C₂²×#1055	C₂×C₄	1	4	yes	—
27	C₂²×#1055	D₄	1	2	yes	—
28	C₂²×#1055	D₄	1	4	yes	—
29	C₂²×#1055	Q₈	1	2	yes	—
30	C₂²×#1055	C₂³	1	1	yes	characteristic
31	#88106	—	2	8	no	—
32	#88106	—	2	4	no	—
33	#88106	C₂×C₄	1	8	yes	—
34	C₂×#6092	—	2	8	no	—
35	C₂×#6092	—	2	4	no	—
36	#88043	—	2	4	no	—
37	C₄×#1055	—	2	4	no	—
38	#12549	—	13	1	no	maximal, Sylow
39	C₂⁴×C₁₃	M₁₆	1	1	yes	characteristic
40	C₂²×C₄×C₁₃	—	2	2	no	—
41	C₂²×C₄×C₁₃	C₄²	1	2	yes	—
42	C₂²×C₄×C₁₃	C₂²⋊C₄	1	2	yes	—
43	C₂²×C₄×C₁₃	C₄⋊C₄	1	2	yes	—
44	C₂²×C₄×C₁₃	C₂×C₈	1	2	yes	—
45	C₂²×C₄×C₁₃	C₂×C₈	1	2	yes	—
46	C₂²×C₄×C₁₃	M₁₆	1	2	yes	—
47	C₂²×C₄×C₁₃	M₁₆	1	1	yes	characteristic
48	C₂²×C₄×C₁₃	C₂²×C₄	1	1	yes	characteristic
49	C₁₃×C₄⋊C₄	—	2	4	no	—
50	C₁₃×C₂²⋊C₄	—	2	4	no	—
51	C₄²×C₁₃	—	2	2	no	—
52	C₄²×C₁₃	C₂×C₈	1	4	yes	—
53	#6099	—	2	8	no	—
54	C₂×#1055	—	2	2	no	—
55	C₂×#1055	—	2	2	no	—
56	C₂×#1055	C₄²	1	2	yes	—
57	C₂×#1055	C₂²⋊C₄	1	2	yes	—
58	C₂×#1055	C₂²⋊C₄	1	4	yes	—
59	C₂×#1055	C₂²⋊C₄	1	4	yes	—
60	C₂×#1055	C₄⋊C₄	1	2	yes	—
61	C₂×#1055	C₄⋊C₄	1	8	yes	—
62	C₂×#1055	C₂×C₈	1	4	yes	—
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×Q₈	1	1	yes	characteristic
67	#6092	—	2	8	no	—
68	C₂²×M₃₂	—	13	2	no	—
69	C₂×#405	—	13	1	no	—
70	#1376	—	13	4	no	—
71	C₂³×C₁₃	—	4	2	no	—
72	C₂³×C₁₃	#97	1	2	yes	—
73	C₂³×C₁₃	C₂²⋊C₈	1	2	yes	—
74	C₂³×C₁₃	C₄⋊C₈	1	2	yes	—
75	C₂³×C₁₃	C₂×M₁₆	1	1	yes	characteristic
76	C₂×C₄×C₁₃	—	2	4	no	—
77	C₂×C₄×C₁₃	—	4	4	no	—
78	C₂×C₄×C₁₃	#95	1	4	yes	—
79	C₂×C₄×C₁₃	C₄×C₈	1	4	yes	—
80	C₂×C₄×C₁₃	#97	1	2	yes	—
81	C₂×C₄×C₁₃	C₂²⋊C₈	1	2	yes	—
82	C₂×C₄×C₁₃	C₂²⋊C₈	1	4	yes	—
83	C₂×C₄×C₁₃	C₄⋊C₈	1	2	yes	—
84	C₂×C₄×C₁₃	C₄⋊C₈	1	4	yes	—
85	C₂×C₄×C₁₃	C₂×C₄²	1	1	yes	characteristic
86	C₂×C₄×C₁₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
87	C₂×C₄×C₁₃	C₂×C₄⋊C₄	1	1	yes	characteristic
88	C₂×C₄×C₁₃	C₂²×C₈	1	1	yes	characteristic
89	C₂×C₄×C₁₃	C₂²×C₈	1	1	yes	characteristic
90	C₂×C₄×C₁₃	C₂×M₁₆	1	1	yes	characteristic
91	#1055	—	2	4	no	—
92	#1055	#95	1	4	yes	—
93	#1055	C₂×C₂²⋊C₄	1	2	yes	—
94	#1055	C₂×C₄⋊C₄	1	2	yes	—
95	C₂²×M₁₆	—	13	1	no	—
96	C₂³×C₈	—	13	1	no	—
97	C₂×#117	#261	1	1	yes	characteristic
98	C₂×M₃₂	—	13	4	no	—
99	C₂×M₃₂	—	13	8	no	—
100	C₂×M₃₂	—	26	2	no	—
101	C₂×M₃₂	—	26	4	no	—
102	C₂²×C₁₆	—	26	2	no	—
103	#405	—	26	4	no	—
104	C₂×#97	—	13	2	no	—
105	C₂×C₄×C₈	—	13	2	no	—
106	C₂²×C₁₃	—	2	2	no	—
107	C₂²×C₁₃	—	2	4	no	—
108	C₂²×C₁₃	—	4	2	no	—
109	C₂²×C₁₃	—	4	2	no	—
110	C₂²×C₁₃	#336	1	4	yes	—
111	C₂²×C₁₃	C₂×#97	1	1	yes	characteristic
112	C₂²×C₁₃	C₂×C₂²⋊C₈	1	1	yes	characteristic
113	C₂²×C₁₃	C₂×C₄⋊C₈	1	1	yes	characteristic
114	C₄×C₁₃	—	4	4	no	—
115	C₄×C₁₃	#336	1	4	yes	—
116	C₄×C₁₃	C₂×#95	1	1	yes	characteristic
117	C₄×C₁₃	C₂×C₄×C₈	1	1	yes	characteristic
118	C₄×C₁₃	C₂×C₂²⋊C₈	1	1	yes	characteristic
119	C₄×C₁₃	C₂×C₄⋊C₈	1	1	yes	characteristic
120	C₂³×C₄	C₂×#261	1	1	yes	characteristic
121	C₂×M₁₆	—	13	4	no	—
122	C₂×M₁₆	—	26	2	no	—
123	C₂×M₁₆	—	26	2	no	—
124	C₂²×C₈	—	13	4	no	—
125	C₂²×C₈	—	13	2	no	—
126	C₂²×C₈	—	13	2	no	—
127	C₂²×C₈	—	13	4	no	—
128	C₂²×C₈	—	13	1	no	—
129	C₂²×C₈	—	13	1	no	—
130	C₂²×C₈	—	13	2	no	—
131	#117	—	2	2	no	—
132	#117	#1057	1	4	yes	—
133	C₂×C₄⋊C₄	—	2	1	no	—
134	C₂×C₂²⋊C₄	—	2	1	no	—
135	C₂×C₄²	C₂×#261	1	2	yes	—
136	M₃₂	—	13	8	no	—
137	M₃₂	—	26	8	no	—
138	M₃₂	—	26	4	no	—
139	C₂×C₁₆	—	26	8	no	—
140	C₂×C₁₆	—	26	4	no	—
141	#97	—	26	4	no	—
142	C₄×C₈	—	26	4	no	—
143	C₂×C₁₃	—	2	1	no	—
144	C₂×C₁₃	—	2	4	no	—
145	C₂×C₁₃	—	2	1	no	—
146	C₂×C₁₃	C₂×#336	1	1	yes	characteristic
147	C₂×C₁₃	#2153	1	2	yes	—
148	C₂⁴	#6122	1	1	yes	characteristic
149	C₂²×C₄	—	2	2	no	—
150	C₂²×C₄	D₁₃⋊C₈	1	2	yes	—
151	C₂²×C₄	#6118	1	2	yes	—
152	C₂²×C₄	C₄×#261	1	2	yes	—
153	C₂²×C₄	C₄⋊#261	1	2	yes	—
154	C₂²×C₄	C₂×#1057	1	2	yes	—
155	C₂²×C₄	#6122	1	1	yes	characteristic
156	C₂²×C₄	C₂²⋊#261	1	2	yes	—
157	C₂²×C₄	C₂²×#261	1	1	yes	characteristic
158	M₁₆	—	26	8	no	—
159	C₂×C₈	—	13	4	no	—
160	C₂×C₈	—	13	4	no	—
161	C₂×C₈	—	13	8	no	—
162	C₂×C₈	—	13	2	no	—
163	C₂×C₈	—	13	2	no	—
164	C₂×C₈	—	13	1	no	—
165	C₂×C₈	—	13	2	no	—
166	C₂×C₈	—	13	1	no	—
167	C₂×C₈	—	13	2	no	—
168	C₂×C₈	—	13	2	no	—
169	C₂×C₈	—	13	4	no	—
170	C₂×C₈	—	26	2	no	—
171	C₂×C₈	—	26	2	no	—
172	C₄⋊C₄	—	2	4	no	—
173	C₂²⋊C₄	—	2	4	no	—
174	C₄²	—	2	2	no	—
175	C₄²	C₂×#1057	1	4	yes	—
176	C₁₆	—	26	8	no	—
177	C₁₃	#12549	1	1	yes	characteristic, complemented, Sylow
178	C₂³	—	4	2	no	—
179	C₂³	#88110	1	2	yes	—
180	C₂³	#88112	1	2	yes	—
181	C₂³	C₂²⋊#1057	1	2	yes	—
182	C₂³	C₂×#6122	1	1	yes	characteristic
183	C₂×C₄	—	2	4	no	—
184	C₂×C₄	—	4	4	no	—
185	C₂×C₄	C₄×#1057	1	4	yes	—
186	C₂×C₄	C₄⋊#1057	1	4	yes	—
187	C₂×C₄	#88110	1	2	yes	—
188	C₂×C₄	#88111	1	4	yes	—
189	C₂×C₄	#88112	1	2	yes	—
190	C₂×C₄	(C₂×#261)⋊C₄	1	4	yes	—
191	C₂×C₄	C₂²⋊#1057	1	2	yes	—
192	C₂×C₄	C₂×D₁₃⋊C₈	1	1	yes	characteristic
193	C₂×C₄	C₂×#6118	1	1	yes	characteristic
194	C₂×C₄	C₂×C₄×#261	1	1	yes	characteristic
195	C₂×C₄	C₂×C₄⋊#261	1	1	yes	characteristic
196	C₂×C₄	C₂²×#1057	1	1	yes	characteristic
197	C₂×C₄	C₂×C₂²⋊#261	1	1	yes	characteristic
198	C₈	—	13	2	no	—
199	C₈	—	13	4	no	—
200	C₈	—	13	2	no	—
201	C₈	—	26	4	no	—
202	C₂²	—	2	2	no	—
203	C₂²	—	2	4	no	—
204	C₂²	—	4	2	no	—
205	C₂²	—	4	2	no	—
206	C₂²	#11718290	1	4	yes	—
207	C₂²	C₂×#88110	1	1	yes	characteristic
208	C₂²	C₂×#88112	1	1	yes	characteristic
209	C₂²	C₂×C₂²⋊#1057	1	1	yes	characteristic
210	C₄	—	4	4	no	—
211	C₄	#11718290	1	4	yes	—
212	C₄	C₂×C₄×#1057	1	1	yes	characteristic
213	C₄	C₂×C₄⋊#1057	1	1	yes	characteristic
214	C₄	C₂×#88111	1	1	yes	characteristic
215	C₄	C₂×(C₂×#261)⋊C₄	1	1	yes	characteristic
216	C₂	—	2	1	no	—
217	C₂	—	2	4	no	—
218	C₂	—	2	1	no	—
219	C₂	C₂×#11718290	1	1	yes	characteristic
220	C₂	#13131179	1	2	yes	—

	subgroup	G/H
center	C₂×C₄	C₂×C₄⋊#261
commutator	C₄×C₁₃	C₂×C₄×C₈
Frattini	C₂²×C₄	C₂²×#261
Fitting	C₂×C₁₃×#117	C₄
socle	C₂²×C₁₃	C₂×C₄⋊C₈
prime radical	C₂⁴×C₁₃	M₁₆
cosocle	C₂²×#1055	C₂³
hypercenter	C₂×#117	#261
hypocenter	C₁₃	#12549
LCS	G ▷ C₄×C₁₃ ▷ C₂×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₄ ◁ C₂×#117

Complements

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

N	H ≅ G/N	type
2. #13129880	C₂	semidirect
9. C₂×#88106	C₄	semidirect
177. C₁₃	#12549	semidirect

Maximal subgroups

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

subgroup	index
C₂²×#88106	2
#13129880	2
C₂×#11718658	2
#12549	13