Subgroups — #16807148

Order 3328.

674 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	C₂×#11719096	C₂	1	2	yes	maximal, complemented
2	#13131219	C₂	1	8	yes	maximal, complemented
3	#13131168	C₂	1	1	yes	maximal, characteristic, complemented
4	#13131068	C₂	1	2	yes	maximal, complemented
5	C₂×#11718231	C₂	1	1	yes	maximal, characteristic, complemented
6	#13123547	C₂	1	1	yes	maximal, characteristic
7	C₂×#88161	C₂²	1	2	yes	—
8	#11719182	C₄	1	1	yes	characteristic
9	C₂×#88106	C₂²	1	2	yes	—
10	C₂²×#6092	C₂²	1	1	yes	characteristic
11	C₂²×#6092	C₂²	1	1	yes	characteristic
12	#11719096	—	2	8	no	—
13	C₂×#88098	C₂²	1	4	yes	—
14	C₂×D₁₃⋊C₁₆	C₂²	1	4	yes	—
15	C₁₃×#423	C₄	1	1	yes	characteristic
16	#11718790	C₄	1	2	yes	—
17	C₂×C₄×#1055	C₂²	1	1	yes	characteristic
18	#11718499	C₄	1	4	yes	—
19	#11718495	C₂²	1	4	yes	—
20	#11718259	C₂²	1	8	yes	—
21	#11718231	C₂²	1	4	yes	—
22	C₄⋊#6092	C₂²	1	4	yes	—
23	C₂×#6127	C₂×C₄	1	2	yes	—
24	C₂×C₁₃×M₁₆	C₂×C₄	1	2	yes	—
25	C₂×C₄²×C₁₃	C₂×C₄	1	1	yes	characteristic
26	#88184	C₈	1	4	yes	—
27	C₂²×#1055	C₂×C₄	1	1	yes	characteristic
28	C₂²×#1055	C₂³	1	1	yes	characteristic
29	#88161	—	2	4	no	—
30	#88161	—	2	4	no	—
31	C₂×#6094	C₂×C₄	1	4	yes	—
32	C₂×C₈×D₁₃	C₂³	1	4	yes	—
33	#88129	C₈	1	2	yes	—
34	#88128	C₈	1	1	yes	characteristic
35	C₄×D₅₂	C₂×C₄	1	4	yes	—
36	#88123	C₈	1	1	yes	characteristic
37	#88106	—	2	4	no	—
38	C₂×#6092	—	2	2	no	—
39	C₂×#6092	D₄	1	4	yes	complemented
40	C₂×#6092	C₂³	1	4	yes	—
41	C₂×#6092	C₂³	1	2	yes	—
42	C₂×#6092	C₂³	1	2	yes	—
43	#88098	—	2	8	no	—
44	D₁₃⋊C₁₆	—	2	8	no	—
45	C₁₃×C₄⋊C₈	C₂×C₄	1	4	yes	—
46	#88060	C₂×C₄	1	8	yes	—
47	C₄×#1055	C₂×C₄	1	2	yes	—
48	C₄×#1055	C₂³	1	2	yes	—
49	#33435	—	13	1	no	maximal, Sylow
50	C₂²×C₄×C₁₃	C₂×C₈	1	2	yes	—
51	C₂²×C₄×C₁₃	C₂²×C₄	1	1	yes	characteristic
52	C₂×C₂²⋊D₁₃	C₂×C₈	1	4	yes	—
53	#6127	—	2	4	no	—
54	#6127	—	2	4	no	—
55	C₂×D₅₂	C₂×C₈	1	2	yes	—
56	C₂×D₅₂	C₂×C₈	1	4	yes	—
57	C₂×C₄×D₁₃	C₂²×C₄	1	4	yes	—
58	C₂×Dic₂₆	C₂×C₈	1	2	yes	—
59	C₁₃×M₁₆	—	2	4	no	—
60	C₂×C₈×C₁₃	C₂²×C₄	1	4	yes	—
61	C₄²×C₁₃	C₂×C₈	1	1	yes	characteristic
62	C₄²×C₁₃	C₂×C₈	1	1	yes	characteristic
63	C₄²×C₁₃	C₂²×C₄	1	2	yes	—
64	#6103	C₂×C₈	1	8	yes	—
65	C₄⋊Dic₁₃	C₂×C₈	1	4	yes	—
66	C₂×#1055	—	2	2	no	—
67	C₂×#1055	C₂²×C₄	1	2	yes	—
68	C₂×#1055	C₂²×C₄	1	1	yes	characteristic
69	C₂×#1055	C₂×D₄	1	2	yes	—
70	C₂×#1055	#41	1	2	yes	—
71	C₂×#1055	C₂⁴	1	1	yes	characteristic
72	#6094	—	2	8	no	—
73	C₈×D₁₃	—	2	4	no	—
74	C₈×D₁₃	—	2	4	no	—
75	#6092	—	2	4	no	—
76	#6092	—	2	2	no	—
77	#6092	C₂×D₄	1	4	yes	—
78	C₂×#504	—	13	2	no	—
79	#2931	—	13	1	no	—
80	#2168	—	13	8	no	—
81	#2150	—	13	1	no	—
82	#2113	—	13	2	no	—
83	C₂×#346	—	13	1	no	—
84	C₂³×C₁₃	C₂²×C₈	1	1	yes	characteristic
85	C₂²×D₁₃	C₂²×C₈	1	4	yes	—
86	C₂×C₄×C₁₃	—	4	2	no	—
87	C₂×C₄×C₁₃	C₄×D₄	1	2	yes	—
88	C₂×C₄×C₁₃	C₂²×C₈	1	2	yes	—
89	C₂×C₄×C₁₃	C₂²×C₈	1	2	yes	—
90	C₂×C₄×C₁₃	C₂²×C₈	1	1	yes	characteristic
91	C₂×C₄×C₁₃	#131	1	2	yes	—
92	C₂×C₄×C₁₃	C₂³×C₄	1	1	yes	characteristic
93	C₂²⋊D₁₃	—	2	8	no	—
94	C₂×Dic₁₃	C₂²×C₈	1	4	yes	—
95	D₅₂	—	2	4	no	—
96	D₅₂	—	4	4	no	—
97	C₄×D₁₃	—	2	4	no	—
98	C₄×D₁₃	—	2	4	no	—
99	Dic₂₆	—	2	4	no	—
100	C₈×C₁₃	—	2	4	no	—
101	#1055	—	2	2	no	—
102	#1055	C₄×D₄	1	2	yes	—
103	#1055	C₂²×D₄	1	1	yes	characteristic
104	#1055	C₂×#41	1	1	yes	characteristic
105	C₂×#131	—	13	2	no	—
106	#532	—	13	1	no	—
107	#504	—	26	8	no	—
108	C₂×M₃₂	—	13	2	no	—
109	C₂×M₃₂	—	13	4	no	—
110	C₂²×C₁₆	—	13	1	no	—
111	C₂²×C₁₆	—	13	1	no	—
112	C₂²×C₁₆	—	13	4	no	—
113	#436	—	13	4	no	—
114	C₈×D₄	—	13	4	no	—
115	#423	#261	1	1	yes	characteristic
116	#408	—	13	2	no	—
117	C₂×C₄×C₈	—	13	1	no	—
118	C₄⋊C₁₆	—	13	4	no	—
119	C₂²⋊C₁₆	—	13	8	no	—
120	#346	—	13	4	no	—
121	C₂²×C₁₃	—	2	2	no	—
122	C₂²×C₁₃	C₈×D₄	1	2	yes	—
123	C₂²×C₁₃	C₂³×C₈	1	1	yes	characteristic
124	C₂×D₁₃	—	2	8	no	—
125	C₂×D₁₃	—	2	4	no	—
126	C₄×C₁₃	—	2	2	no	—
127	C₄×C₁₃	—	2	2	no	—
128	C₄×C₁₃	C₈×D₄	1	2	yes	—
129	C₄×C₁₃	C₂×C₄×D₄	1	1	yes	characteristic
130	C₄×C₁₃	C₂×#131	1	1	yes	characteristic
131	Dic₁₃	—	2	4	no	—
132	C₂×#41	—	13	2	no	—
133	#131	—	26	4	no	—
134	#131	—	26	4	no	—
135	C₂×M₁₆	—	13	4	no	—
136	C₂×M₁₆	C₂×#261	1	2	yes	—
137	C₂²×C₈	—	13	1	no	—
138	C₂²×C₈	—	13	1	no	—
139	C₂²×C₈	—	13	4	no	—
140	C₄⋊Q₈	—	13	1	no	—
141	#127	—	13	1	no	—
142	#124	—	13	2	no	—
143	#121	—	13	4	no	—
144	C₄×D₄	—	13	4	no	—
145	C₂×C₄²	C₂×#261	1	1	yes	characteristic
146	M₃₂	—	26	8	no	—
147	M₃₂	—	26	4	no	—
148	C₂×C₁₆	—	13	4	no	—
149	C₂×C₁₆	—	13	2	no	—
150	C₂×C₁₆	—	13	2	no	—
151	C₂×C₁₆	—	13	4	no	—
152	C₂×C₁₆	—	26	8	no	—
153	C₂×C₁₆	—	26	2	no	—
154	C₄⋊C₈	C₂×#261	1	4	yes	—
155	C₂²⋊C₈	—	13	8	no	—
156	C₄×C₈	—	13	2	no	—
157	C₄×C₈	—	13	2	no	—
158	C₂×C₁₃	—	2	2	no	—
159	C₂×C₁₃	C₂×C₈×D₄	1	1	yes	characteristic
160	C₂×C₁₃	#3407	1	2	yes	—
161	D₁₃	—	4	4	no	—
162	#41	—	26	4	no	—
163	#41	—	26	4	no	—
164	C₂×Q₈	—	13	2	no	—
165	C₂×D₄	—	13	4	no	—
166	C₂×D₄	—	13	2	no	—
167	C₂×D₄	—	13	4	no	—
168	C₂²×C₄	—	13	4	no	—
169	C₂²×C₄	D₁₃⋊C₈	1	2	yes	—
170	C₂²×C₄	C₂²×#261	1	1	yes	characteristic
171	M₁₆	—	2	4	no	—
172	M₁₆	—	26	8	no	—
173	C₂×C₈	—	13	2	no	—
174	C₂×C₈	—	13	2	no	—
175	C₂×C₈	—	13	1	no	—
176	C₂×C₈	—	13	1	no	—
177	C₂×C₈	—	13	2	no	—
178	C₂×C₈	—	26	4	no	—
179	C₂×C₈	—	26	4	no	—
180	C₂×C₈	—	26	2	no	—
181	C₂×C₈	C₂²×#261	1	4	yes	—
182	C₄⋊C₄	—	13	4	no	—
183	C₂²⋊C₄	—	13	8	no	—
184	C₄²	D₁₃⋊C₈	1	1	yes	characteristic
185	C₄²	D₁₃⋊C₈	1	1	yes	characteristic
186	C₄²	C₂²×#261	1	2	yes	—
187	C₁₆	—	13	4	no	—
188	C₁₆	—	26	4	no	—
189	C₁₆	—	26	2	no	—
190	C₁₃	#33435	1	1	yes	characteristic, complemented, Sylow
191	C₂³	—	13	4	no	—
192	C₂³	C₂×D₁₃⋊C₈	1	1	yes	characteristic
193	Q₈	—	26	4	no	—
194	D₄	—	26	8	no	—
195	D₄	—	26	4	no	—
196	D₄	—	52	4	no	—
197	C₂×C₄	—	4	2	no	—
198	C₂×C₄	—	13	4	no	—
199	C₂×C₄	—	26	4	no	—
200	C₂×C₄	—	26	4	no	—
201	C₂×C₄	C₂×D₁₃⋊C₈	1	2	yes	—
202	C₂×C₄	C₂×D₁₃⋊C₈	1	2	yes	—
203	C₂×C₄	C₂×D₁₃⋊C₈	1	1	yes	characteristic
204	C₂×C₄	D₄×#261	1	2	yes	—
205	C₂×C₄	#88240	1	2	yes	—
206	C₂×C₄	C₂³×#261	1	1	yes	characteristic
207	C₈	—	2	4	no	—
208	C₈	—	13	1	no	—
209	C₈	—	13	1	no	—
210	C₈	—	13	2	no	—
211	C₈	—	26	2	no	—
212	C₂²	—	2	2	no	—
213	C₂²	—	26	4	no	—
214	C₂²	—	26	8	no	—
215	C₂²	#11719073	1	2	yes	—
216	C₂²	C₂²×D₁₃⋊C₈	1	1	yes	characteristic
217	C₄	—	2	2	no	—
218	C₄	—	2	2	no	—
219	C₄	—	26	4	no	—
220	C₄	#11719073	1	2	yes	—
221	C₄	C₂×D₄×#261	1	1	yes	characteristic
222	C₄	C₂×#88240	1	1	yes	characteristic
223	C₂	—	2	2	no	—
224	C₂	—	52	4	no	—
225	C₂	C₂×#11719073	1	1	yes	characteristic
226	C₂	#13131805	1	2	yes	—

	subgroup	G/H
center, Frattini	C₂×C₄	C₂³×#261
commutator, socle	C₂²×C₁₃	C₂³×C₈
Fitting	C₁₃×#423	C₄
prime radical	#11719182	C₄
cosocle	C₂×#1055	C₂⁴
hypercenter	#423	#261
hypocenter	C₁₃	#33435
LCS	G ▷ C₂²×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂×C₄ ◁ #423

Complements

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

N	H ≅ G/N	type
1. C₂×#11719096	C₂	semidirect
2. #13131219	C₂	semidirect
3. #13131168	C₂	semidirect
4. #13131068	C₂	semidirect
5. C₂×#11718231	C₂	semidirect
39. C₂×#6092	D₄	semidirect
190. C₁₃	#33435	semidirect

Maximal subgroups

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

subgroup	index
C₂×#11719096	2
#13131219	2
#13131168	2
#13131068	2
C₂×#11718231	2
#13123547	2
#33435	13