Subgroups — #16807255

Order 3328.

506 conjugacy classes of subgroups.

The subgroup lattice has 244 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#13131253	C₂	1	4	yes	maximal, complemented
2	#13131250	C₂	1	4	yes	maximal, complemented
3	#13131173	C₂	1	1	yes	maximal, characteristic
4	#13131171	C₂	1	1	yes	maximal, characteristic
5	#13131170	C₂	1	1	yes	maximal, characteristic
6	#13131047	C₂	1	1	yes	maximal, characteristic
7	#13131045	C₂	1	1	yes	maximal, characteristic
8	C₂×#11718257	C₂	1	1	yes	maximal, characteristic
9	#13123769	C₂	1	1	yes	maximal, characteristic
10	#11719222	C₄	1	1	yes	characteristic
11	C₂×#88106	C₂²	1	1	yes	characteristic
12	C₂×#88106	C₂²	1	1	yes	characteristic
13	C₂²×#6092	C₂²	1	1	yes	characteristic
14	C₂²×#6092	C₂²	1	1	yes	characteristic
15	C₁₃×#424	C₄	1	1	yes	characteristic
16	#11718783	C₂²	1	1	yes	characteristic
17	#11718779	C₄	1	1	yes	characteristic
18	#11718763	C₄	1	1	yes	characteristic
19	C₂×C₈×Dic₁₃	C₂²	1	1	yes	characteristic
20	#11718658	C₂²	1	1	yes	characteristic
21	#11718491	C₂²	1	4	yes	—
22	#11718489	C₄	1	4	yes	—
23	#11718260	C₂²	1	4	yes	—
24	#11718260	C₂²	1	4	yes	—
25	#11718258	C₂²	1	2	yes	—
26	#11718257	C₂²	1	2	yes	—
27	#11718257	C₂²	1	2	yes	—
28	#11718257	C₂²	1	2	yes	—
29	#11718231	C₂²	1	2	yes	—
30	C₄⋊#6092	C₂²	1	4	yes	—
31	C₄×#6092	C₂²	1	2	yes	—
32	C₂×C₁₃×M₁₆	C₂×C₄	1	1	yes	characteristic
33	C₂²×C₈×C₁₃	C₂×C₄	1	1	yes	characteristic
34	C₁₃×#117	C₂×C₄	1	1	yes	characteristic
35	#88180	C₂×C₄	1	1	yes	characteristic
36	C₂×C₄×Dic₁₃	C₂×C₄	1	1	yes	characteristic
37	C₂×#6099	C₂×C₄	1	1	yes	characteristic
38	C₂²×#1055	C₂³	1	1	yes	characteristic
39	#88144	C₈	1	2	yes	—
40	#88142	C₈	1	2	yes	—
41	#88132	C₈	1	4	yes	—
42	C₄×Dic₂₆	C₂×C₄	1	4	yes	—
43	#88106	—	2	2	no	—
44	#88106	—	2	2	no	—
45	C₂×#6092	—	2	2	no	—
46	C₂×#6092	Q₈	1	4	yes	complemented
47	C₂×#6092	C₂³	1	2	yes	—
48	C₂×#6092	C₂³	1	2	yes	—
49	C₂×#6092	C₂³	1	2	yes	—
50	C₂×#6092	C₂³	1	1	yes	characteristic
51	C₂×#6092	C₂³	1	1	yes	characteristic
52	C₁₃×C₄⋊C₈	C₂×C₄	1	4	yes	—
53	#88055	C₂×C₄	1	2	yes	—
54	#88054	C₂×C₄	1	4	yes	—
55	#88054	C₂×C₄	1	4	yes	—
56	C₈×Dic₁₃	C₂×C₄	1	2	yes	—
57	C₈×Dic₁₃	C₂³	1	2	yes	—
58	C₈×Dic₁₃	C₂³	1	2	yes	—
59	#88043	C₂×C₄	1	2	yes	—
60	C₄×#1055	C₂³	1	2	yes	—
61	#33474	—	13	1	no	maximal, Sylow
62	C₂²×C₄×C₁₃	C₂²×C₄	1	1	yes	characteristic
63	C₂²×Dic₁₃	C₂×C₈	1	2	yes	—
64	C₂×Dic₂₆	C₂×C₈	1	4	yes	—
65	C₁₃×M₁₆	—	2	2	no	—
66	C₂×C₈×C₁₃	—	2	2	no	—
67	C₂×C₈×C₁₃	C₂²×C₄	1	2	yes	—
68	C₂×C₈×C₁₃	C₂²×C₄	1	2	yes	—
69	C₁₃×C₄⋊C₄	C₂×C₈	1	2	yes	—
70	C₁₃×C₂²⋊C₄	C₂×C₈	1	2	yes	—
71	C₄²×C₁₃	C₂²×C₄	1	2	yes	—
72	C₂²⋊Dic₁₃	C₂×C₈	1	2	yes	—
73	C₄⋊Dic₁₃	C₂×C₈	1	4	yes	—
74	C₄⋊Dic₁₃	C₂×C₈	1	2	yes	—
75	#6101	C₂×C₈	1	4	yes	—
76	#6101	C₂×C₈	1	4	yes	—
77	C₄×Dic₁₃	C₂×C₈	1	1	yes	characteristic
78	C₄×Dic₁₃	C₂×C₈	1	1	yes	characteristic
79	C₄×Dic₁₃	C₂²×C₄	1	2	yes	—
80	C₄×Dic₁₃	C₂²×C₄	1	2	yes	—
81	#6099	—	2	2	no	—
82	C₂×#1055	C₂²×C₄	1	2	yes	—
83	C₂×#1055	C₂²×C₄	1	1	yes	characteristic
84	C₂×#1055	C₂×Q₈	1	2	yes	—
85	C₂×#1055	#41	1	2	yes	—
86	C₂×#1055	C₂⁴	1	1	yes	characteristic
87	#6092	—	2	2	no	—
88	#6092	—	2	2	no	—
89	#6092	—	2	2	no	—
90	#6092	C₂×Q₈	1	2	yes	—
91	#6092	C₂×Q₈	1	2	yes	—
92	#2960	—	13	1	no	—
93	#2183	—	13	4	no	—
94	Q₈×C₁₆	—	13	4	no	—
95	#2151	—	13	1	no	—
96	#2151	—	13	1	no	—
97	#2150	—	13	1	no	—
98	#2108	—	13	1	no	—
99	#2108	—	13	1	no	—
100	C₂×C₄×C₁₆	—	13	1	no	—
101	C₂³×C₁₃	C₂²×C₈	1	1	yes	characteristic
102	C₂×C₄×C₁₃	C₄×Q₈	1	2	yes	—
103	C₂×C₄×C₁₃	C₂²×C₈	1	4	yes	—
104	C₂×C₄×C₁₃	C₂²×C₈	1	1	yes	characteristic
105	C₂×C₄×C₁₃	#131	1	2	yes	—
106	C₂×C₄×C₁₃	C₂³×C₄	1	1	yes	characteristic
107	C₂×Dic₁₃	—	2	4	no	—
108	C₂×Dic₁₃	C₂²×C₈	1	2	yes	—
109	C₂×Dic₁₃	C₂²×C₈	1	4	yes	—
110	C₂×Dic₁₃	C₂²×C₈	1	2	yes	—
111	Dic₂₆	—	2	8	no	—
112	C₈×C₁₃	—	2	2	no	—
113	C₈×C₁₃	—	2	2	no	—
114	#1055	—	2	2	no	—
115	#1055	C₄×Q₈	1	2	yes	—
116	#1055	C₂²×Q₈	1	1	yes	characteristic
117	#1055	C₂×#41	1	1	yes	characteristic
118	#533	—	13	1	no	—
119	C₂×M₃₂	—	13	1	no	—
120	C₂×M₃₂	—	13	1	no	—
121	C₂²×C₁₆	—	13	1	no	—
122	C₂²×C₁₆	—	13	1	no	—
123	#446	—	13	4	no	—
124	C₈×Q₈	—	13	4	no	—
125	#424	—	13	1	no	—
126	#424	#261	1	1	yes	characteristic
127	#423	—	13	1	no	—
128	#405	—	13	1	no	—
129	#405	—	13	1	no	—
130	C₂×C₄×C₈	—	13	1	no	—
131	C₄⋊C₁₆	—	13	4	no	—
132	C₄⋊C₁₆	—	13	4	no	—
133	C₄⋊C₁₆	—	13	4	no	—
134	#346	—	13	2	no	—
135	#346	—	13	2	no	—
136	C₄×C₁₆	—	13	2	no	—
137	C₄×C₁₆	—	13	2	no	—
138	C₄×C₁₆	—	13	2	no	—
139	C₄×C₁₆	—	13	2	no	—
140	C₂²×C₁₃	—	2	2	no	—
141	C₂²×C₁₃	C₈×Q₈	1	2	yes	—
142	C₂²×C₁₃	C₂³×C₈	1	1	yes	characteristic
143	C₄×C₁₃	—	2	4	no	—
144	C₄×C₁₃	C₈×Q₈	1	2	yes	—
145	C₄×C₁₃	C₂×C₄×Q₈	1	1	yes	characteristic
146	C₄×C₁₃	C₂×#131	1	1	yes	characteristic
147	Dic₁₃	—	2	4	no	—
148	Dic₁₃	—	2	2	no	—
149	Dic₁₃	#504	1	4	yes	—
150	C₂×M₁₆	—	13	1	no	—
151	C₂×M₁₆	C₂×#261	1	1	yes	characteristic
152	C₂²×C₈	—	13	1	no	—
153	C₂²×C₈	C₂×#261	1	1	yes	characteristic
154	C₄⋊Q₈	—	13	2	no	—
155	#125	—	13	2	no	—
156	C₂²⋊Q₈	—	13	4	no	—
157	C₄×Q₈	—	13	4	no	—
158	#117	—	13	1	no	—
159	#117	C₂×#261	1	1	yes	characteristic
160	C₂×C₄²	—	13	1	no	—
161	M₃₂	—	26	2	no	—
162	M₃₂	—	26	2	no	—
163	C₂×C₁₆	—	13	4	no	—
164	C₂×C₁₆	—	13	1	no	—
165	C₂×C₁₆	—	13	1	no	—
166	C₂×C₁₆	—	13	2	no	—
167	C₂×C₁₆	—	13	2	no	—
168	C₂×C₁₆	—	13	2	no	—
169	C₂×C₁₆	—	26	2	no	—
170	C₄⋊C₈	—	13	4	no	—
171	C₄⋊C₈	—	13	4	no	—
172	C₄⋊C₈	C₂×#261	1	4	yes	—
173	#97	—	13	2	no	—
174	#97	—	13	2	no	—
175	C₄×C₈	—	13	2	no	—
176	C₄×C₈	—	13	2	no	—
177	C₄×C₈	—	13	2	no	—
178	C₄×C₈	—	13	2	no	—
179	C₂×C₁₃	—	2	2	no	—
180	C₂×C₁₃	C₂×C₈×Q₈	1	1	yes	characteristic
181	C₂×C₁₃	C₂×#504	1	2	yes	—
182	C₂×Q₈	—	13	4	no	—
183	C₂²×C₄	—	13	2	no	—
184	C₂²×C₄	C₂²×#261	1	1	yes	characteristic
185	M₁₆	—	2	2	no	—
186	M₁₆	—	26	2	no	—
187	C₂×C₈	—	2	2	no	—
188	C₂×C₈	—	13	2	no	—
189	C₂×C₈	—	13	2	no	—
190	C₂×C₈	—	13	1	no	—
191	C₂×C₈	—	13	1	no	—
192	C₂×C₈	—	13	2	no	—
193	C₂×C₈	C₂²×#261	1	2	yes	—
194	C₂×C₈	C₂²×#261	1	2	yes	—
195	C₄⋊C₄	—	13	2	no	—
196	C₄⋊C₄	—	13	4	no	—
197	C₄⋊C₄	—	13	4	no	—
198	C₄⋊C₄	—	13	4	no	—
199	C₄⋊C₄	D₁₃⋊C₈	1	2	yes	—
200	C₂²⋊C₄	—	13	2	no	—
201	C₂²⋊C₄	D₁₃⋊C₈	1	2	yes	—
202	C₄²	—	13	1	no	—
203	C₄²	—	13	1	no	—
204	C₄²	—	13	2	no	—
205	C₄²	—	13	2	no	—
206	C₄²	C₂²×#261	1	2	yes	—
207	C₁₆	—	13	2	no	—
208	C₁₆	—	13	2	no	—
209	C₁₆	—	26	2	no	—
210	C₁₆	—	26	2	no	—
211	C₁₆	—	26	2	no	—
212	C₁₃	#33474	1	1	yes	characteristic, complemented, Sylow
213	C₂³	C₂×D₁₃⋊C₈	1	1	yes	characteristic
214	Q₈	—	26	8	no	—
215	C₂×C₄	—	13	2	no	—
216	C₂×C₄	—	13	2	no	—
217	C₂×C₄	—	13	4	no	—
218	C₂×C₄	—	26	4	no	—
219	C₂×C₄	C₂×D₁₃⋊C₈	1	4	yes	—
220	C₂×C₄	C₂×D₁₃⋊C₈	1	1	yes	characteristic
221	C₂×C₄	#88238	1	2	yes	—
222	C₂×C₄	Q₈×#261	1	2	yes	—
223	C₂×C₄	C₂³×#261	1	1	yes	characteristic
224	C₈	—	2	2	no	—
225	C₈	—	2	2	no	—
226	C₈	—	13	1	no	—
227	C₈	—	13	1	no	—
228	C₈	—	13	2	no	—
229	C₈	—	26	2	no	—
230	C₂²	—	2	2	no	—
231	C₂²	#11719074	1	2	yes	—
232	C₂²	C₂²×D₁₃⋊C₈	1	1	yes	characteristic
233	C₄	—	2	4	no	—
234	C₄	—	13	4	no	—
235	C₄	—	26	2	no	—
236	C₄	—	26	4	no	—
237	C₄	#11719074	1	2	yes	—
238	C₄	C₂×#88238	1	1	yes	characteristic
239	C₄	C₂×Q₈×#261	1	1	yes	characteristic
240	C₂	—	2	2	no	—
241	C₂	C₂×#11719074	1	1	yes	characteristic
242	C₂	D₁₃⋊#504	1	2	yes	—

	subgroup	G/H
center, Frattini	C₂×C₄	C₂³×#261
commutator, socle	C₂²×C₁₃	C₂³×C₈
Fitting	C₁₃×#424	C₄
prime radical	C₂³×C₁₃	C₂²×C₈
cosocle	C₂×#1055	C₂⁴
hypercenter	#424	#261
hypocenter	C₁₃	#33474
LCS	G ▷ C₂²×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂×C₄ ◁ #424

Complements

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

N	H ≅ G/N	type
1. #13131253	C₂	semidirect
2. #13131250	C₂	semidirect
46. C₂×#6092	Q₈	semidirect
212. C₁₃	#33474	semidirect

Maximal subgroups

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

subgroup	index
#13131253	2
#13131250	2
#13131173	2
#13131171	2
#13131170	2
#13131047	2
#13131045	2
C₂×#11718257	2
#13123769	2
#33474	13