Subgroups — #16789957

Order 3328.

486 conjugacy classes of subgroups.

The subgroup lattice has 264 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#13131168	C₂	1	2	yes	maximal, complemented
2	C₂×#11718284	C₂	1	2	yes	maximal
3	#13129883	C₂	1	2	yes	maximal, complemented
4	C₂×C₄×#6099	C₂	1	1	yes	maximal, characteristic
5	C₂²×C₄²×C₁₃	C₄	1	1	yes	characteristic
6	C₂²×#6099	C₄	1	1	yes	characteristic
7	C₂²×#6099	C₂²	1	1	yes	characteristic
8	C₂×#88106	C₂²	1	4	yes	—
9	C₄×#6099	—	2	4	no	—
10	C₂×#88043	C₄	1	1	yes	characteristic
11	C₂×#88043	C₂²	1	1	yes	characteristic
12	C₂×C₄×#1055	C₄	1	1	yes	characteristic
13	C₂×C₄×#1055	C₂²	1	1	yes	characteristic
14	#11718284	—	2	4	no	—
15	C₄⋊#6092	—	2	4	no	—
16	C₂³×C₄×C₁₃	C₈	1	2	yes	—
17	C₂³×C₄×C₁₃	C₂×C₄	1	1	yes	characteristic
18	C₂×C₄²×C₁₃	—	2	2	no	—
19	C₂×C₄²×C₁₃	—	2	1	no	—
20	C₂×C₄²×C₁₃	—	2	1	no	—
21	C₂×C₄²×C₁₃	C₈	1	2	yes	—
22	C₂×C₄²×C₁₃	C₂×C₄	1	1	yes	characteristic
23	C₂×C₄²×C₁₃	C₂×C₄	1	1	yes	characteristic
24	C₂×#6099	—	2	2	no	—
25	C₂×#6099	—	2	2	no	—
26	C₂×#6099	—	2	2	no	—
27	C₂×#6099	—	2	2	no	—
28	C₂×#6099	—	2	2	no	—
29	C₂×#6099	D₄	1	2	yes	—
30	C₂×#6099	Q₈	1	2	yes	—
31	C₂²×#1055	C₂×C₄	1	1	yes	characteristic
32	C₂²×#1055	C₂×C₄	1	1	yes	characteristic
33	C₂²×#1055	C₂×C₄	1	1	yes	characteristic
34	C₂²×#1055	C₂³	1	1	yes	characteristic
35	#88106	—	2	4	no	—
36	#88106	—	2	4	no	—
37	C₂×#6092	—	2	4	no	—
38	#88043	—	2	2	no	—
39	#88043	—	2	2	no	—
40	C₄×#1055	—	2	2	no	—
41	C₄×#1055	D₄	1	4	yes	—
42	#12785	—	13	1	no	maximal, Sylow
43	C₂⁴×C₁₃	C₂×C₈	1	1	yes	characteristic
44	C₂²×C₄×C₁₃	—	2	1	no	—
45	C₂²×C₄×C₁₃	—	2	1	no	—
46	C₂²×C₄×C₁₃	—	2	1	no	—
47	C₂²×C₄×C₁₃	—	2	1	no	—
48	C₂²×C₄×C₁₃	—	2	2	no	—
49	C₂²×C₄×C₁₃	—	2	2	no	—
50	C₂²×C₄×C₁₃	—	4	4	no	—
51	C₂²×C₄×C₁₃	C₄⋊C₄	1	2	yes	—
52	C₂²×C₄×C₁₃	C₂×C₈	1	2	yes	—
53	C₂²×C₄×C₁₃	C₂×C₈	1	1	yes	characteristic
54	C₂²×C₄×C₁₃	C₂×C₈	1	2	yes	—
55	C₂²×C₄×C₁₃	M₁₆	1	2	yes	—
56	C₂²×C₄×C₁₃	C₂²×C₄	1	1	yes	characteristic
57	C₄²×C₁₃	—	2	2	no	—
58	C₄²×C₁₃	—	2	2	no	—
59	C₄²×C₁₃	—	2	1	no	—
60	C₄²×C₁₃	—	2	1	no	—
61	C₄²×C₁₃	C₂²⋊C₄	1	2	yes	—
62	C₄²×C₁₃	M₁₆	1	1	yes	characteristic
63	C₄²×C₁₃	M₁₆	1	1	yes	characteristic
64	#6099	—	2	8	no	—
65	#6099	—	2	4	no	—
66	#6099	—	2	4	no	—
67	C₂×#1055	—	2	1	no	—
68	C₂×#1055	—	2	1	no	—
69	C₂×#1055	—	2	1	no	—
70	C₂×#1055	—	2	1	no	—
71	C₂×#1055	—	2	1	no	—
72	C₂×#1055	—	2	1	no	—
73	C₂×#1055	C₂²⋊C₄	1	2	yes	—
74	C₂×#1055	C₄⋊C₄	1	2	yes	—
75	C₂×#1055	M₁₆	1	2	yes	—
76	C₂×#1055	C₂×D₄	1	1	yes	characteristic
77	C₂×#1055	C₂×D₄	1	2	yes	—
78	C₂×#1055	C₂×Q₈	1	1	yes	characteristic
79	C₂×#1055	#41	1	2	yes	—
80	#6092	—	4	4	no	—
81	C₂×C₄×M₁₆	—	13	1	no	—
82	#2150	—	13	2	no	—
83	C₂×#349	—	13	2	no	—
84	#1377	—	13	2	no	—
85	C₂³×C₁₃	—	2	1	no	—
86	C₂³×C₁₃	—	2	1	no	—
87	C₂³×C₁₃	—	4	2	no	—
88	C₂³×C₁₃	C₄⋊C₈	1	2	yes	—
89	C₂³×C₁₃	C₂²×C₈	1	1	yes	characteristic
90	C₂×C₄×C₁₃	—	2	1	no	—
91	C₂×C₄×C₁₃	—	2	1	no	—
92	C₂×C₄×C₁₃	—	2	4	no	—
93	C₂×C₄×C₁₃	—	2	2	no	—
94	C₂×C₄×C₁₃	—	2	2	no	—
95	C₂×C₄×C₁₃	—	2	2	no	—
96	C₂×C₄×C₁₃	—	2	2	no	—
97	C₂×C₄×C₁₃	—	2	2	no	—
98	C₂×C₄×C₁₃	—	4	4	no	—
99	C₂×C₄×C₁₃	—	4	4	no	—
100	C₂×C₄×C₁₃	—	4	2	no	—
101	C₂×C₄×C₁₃	C₂²⋊C₈	1	2	yes	—
102	C₂×C₄×C₁₃	C₂²⋊C₈	1	2	yes	—
103	C₂×C₄×C₁₃	C₄⋊C₈	1	2	yes	—
104	C₂×C₄×C₁₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
105	C₂×C₄×C₁₃	C₂×C₄⋊C₄	1	1	yes	characteristic
106	C₂×C₄×C₁₃	#117	1	1	yes	characteristic
107	C₂×C₄×C₁₃	C₂×M₁₆	1	1	yes	characteristic
108	C₂×C₄×C₁₃	C₂×M₁₆	1	1	yes	characteristic
109	C₂×C₄×C₁₃	C₂×M₁₆	1	1	yes	characteristic
110	#1055	—	2	2	no	—
111	#1055	—	2	2	no	—
112	#1055	—	2	2	no	—
113	#1055	#121	1	2	yes	—
114	#1055	C₂²⋊Q₈	1	2	yes	—
115	C₂²×M₁₆	—	13	1	no	—
116	C₂²×M₁₆	—	13	1	no	—
117	C₂²×C₄²	#261	1	1	yes	characteristic
118	C₂×M₃₂	—	13	4	no	—
119	C₄×M₁₆	—	26	4	no	—
120	C₂×#97	—	13	1	no	—
121	C₂×#97	—	13	1	no	—
122	C₂×C₄×C₈	—	13	1	no	—
123	C₂×C₄×C₈	—	13	1	no	—
124	C₄⋊C₁₆	—	26	4	no	—
125	#349	—	26	4	no	—
126	C₂²×C₁₃	—	2	2	no	—
127	C₂²×C₁₃	—	2	2	no	—
128	C₂²×C₁₃	—	4	2	no	—
129	C₂²×C₁₃	—	4	2	no	—
130	C₂²×C₁₃	—	4	2	no	—
131	C₂²×C₁₃	C₂×C₂²⋊C₈	1	1	yes	characteristic
132	C₂²×C₁₃	C₂×C₄⋊C₈	1	1	yes	characteristic
133	C₂²×C₁₃	#431	1	1	yes	characteristic
134	C₄×C₁₃	—	2	2	no	—
135	C₄×C₁₃	—	2	2	no	—
136	C₄×C₁₃	—	4	2	no	—
137	C₄×C₁₃	#349	1	4	yes	complemented
138	C₄×C₁₃	#380	1	1	yes	characteristic
139	C₄×C₁₃	#407	1	1	yes	characteristic
140	C₄×C₁₃	#423	1	1	yes	characteristic
141	C₄×C₁₃	#432	1	1	yes	characteristic
142	C₂³×C₄	#1057	1	2	yes	—
143	C₂³×C₄	C₂×#261	1	1	yes	characteristic
144	C₂×M₁₆	—	13	2	no	—
145	C₂×M₁₆	—	13	2	no	—
146	C₂×M₁₆	—	26	2	no	—
147	C₂×M₁₆	—	26	2	no	—
148	C₂×M₁₆	—	26	2	no	—
149	C₂×M₁₆	—	26	2	no	—
150	C₂×M₁₆	—	26	2	no	—
151	C₂²×C₈	—	13	1	no	—
152	C₂²×C₈	—	13	1	no	—
153	C₂²×C₈	—	13	1	no	—
154	C₂²×C₈	—	13	1	no	—
155	C₂×C₄²	—	2	1	no	—
156	C₂×C₄²	—	2	1	no	—
157	C₂×C₄²	—	2	2	no	—
158	C₂×C₄²	#1057	1	2	yes	—
159	C₂×C₄²	C₂×#261	1	1	yes	characteristic
160	C₂×C₄²	C₂×#261	1	1	yes	characteristic
161	M₃₂	—	26	4	no	—
162	M₃₂	—	26	4	no	—
163	C₂×C₁₆	—	26	4	no	—
164	#97	—	26	2	no	—
165	#97	—	26	2	no	—
166	C₄×C₈	—	13	4	no	—
167	C₄×C₈	—	26	2	no	—
168	C₂×C₁₃	—	2	1	no	—
169	C₂×C₁₃	—	2	1	no	—
170	C₂×C₁₃	—	4	2	no	—
171	C₂×C₁₃	#1797	1	1	yes	characteristic
172	C₂×C₁₃	C₂×#349	1	2	yes	—
173	C₂⁴	C₂×#1057	1	1	yes	characteristic
174	C₂²×C₄	—	2	1	no	—
175	C₂²×C₄	—	2	1	no	—
176	C₂²×C₄	—	2	1	no	—
177	C₂²×C₄	—	2	1	no	—
178	C₂²×C₄	—	2	2	no	—
179	C₂²×C₄	—	2	2	no	—
180	C₂²×C₄	—	4	4	no	—
181	C₂²×C₄	#6118	1	2	yes	—
182	C₂²×C₄	C₄⋊#261	1	2	yes	—
183	C₂²×C₄	C₂×#1057	1	1	yes	characteristic
184	C₂²×C₄	C₂×#1057	1	2	yes	—
185	C₂²×C₄	C₂×#1057	1	2	yes	—
186	C₂²×C₄	C₂²×#261	1	1	yes	characteristic
187	M₁₆	—	26	4	no	—
188	M₁₆	—	26	4	no	—
189	M₁₆	—	26	8	no	—
190	C₂×C₈	—	13	1	no	—
191	C₂×C₈	—	13	1	no	—
192	C₂×C₈	—	13	2	no	—
193	C₂×C₈	—	13	2	no	—
194	C₂×C₈	—	13	2	no	—
195	C₂×C₈	—	13	2	no	—
196	C₂×C₈	—	13	2	no	—
197	C₂×C₈	—	26	1	no	—
198	C₂×C₈	—	26	1	no	—
199	C₂×C₈	—	26	1	no	—
200	C₂×C₈	—	26	1	no	—
201	C₂×C₈	—	26	1	no	—
202	C₂×C₈	—	26	1	no	—
203	C₄²	—	2	2	no	—
204	C₄²	—	2	2	no	—
205	C₄²	—	2	1	no	—
206	C₄²	—	2	1	no	—
207	C₄²	#6122	1	1	yes	characteristic
208	C₄²	#6122	1	1	yes	characteristic
209	C₄²	C₂²⋊#261	1	2	yes	—
210	C₁₆	—	52	4	no	—
211	C₁₃	#12785	1	1	yes	characteristic, complemented, Sylow
212	C₂³	—	2	1	no	—
213	C₂³	—	2	1	no	—
214	C₂³	—	4	2	no	—
215	C₂³	C₄⋊#1057	1	2	yes	—
216	C₂³	C₂²×#1057	1	1	yes	characteristic
217	C₂×C₄	—	2	1	no	—
218	C₂×C₄	—	2	1	no	—
219	C₂×C₄	—	2	4	no	—
220	C₂×C₄	—	2	2	no	—
221	C₂×C₄	—	2	2	no	—
222	C₂×C₄	—	2	2	no	—
223	C₂×C₄	—	2	2	no	—
224	C₂×C₄	—	2	2	no	—
225	C₂×C₄	—	4	4	no	—
226	C₂×C₄	—	4	4	no	—
227	C₂×C₄	—	4	2	no	—
228	C₂×C₄	C₄⋊#1057	1	2	yes	—
229	C₂×C₄	C₂²⋊#1057	1	2	yes	—
230	C₂×C₄	C₂²⋊#1057	1	2	yes	—
231	C₂×C₄	C₂×#6118	1	1	yes	characteristic
232	C₂×C₄	D₁₃⋊M₁₆	1	1	yes	characteristic
233	C₂×C₄	C₂×C₄⋊#261	1	1	yes	characteristic
234	C₂×C₄	#88237	1	1	yes	characteristic
235	C₂×C₄	C₂×#6122	1	1	yes	characteristic
236	C₂×C₄	C₂×C₂²⋊#261	1	1	yes	characteristic
237	C₈	—	13	2	no	—
238	C₈	—	13	2	no	—
239	C₈	—	26	2	no	—
240	C₈	—	26	2	no	—
241	C₈	—	26	2	no	—
242	C₂²	—	2	2	no	—
243	C₂²	—	2	2	no	—
244	C₂²	—	4	2	no	—
245	C₂²	—	4	2	no	—
246	C₂²	—	4	2	no	—
247	C₂²	C₂×C₄⋊#1057	1	1	yes	characteristic
248	C₂²	#11719119	1	1	yes	characteristic
249	C₂²	C₂×C₂²⋊#1057	1	1	yes	characteristic
250	C₄	—	2	2	no	—
251	C₄	—	2	2	no	—
252	C₄	—	4	2	no	—
253	C₄	#11718284	1	4	yes	complemented
254	C₄	#11719127	1	1	yes	characteristic
255	C₄	#11719129	1	1	yes	characteristic
256	C₄	#11719130	1	1	yes	characteristic
257	C₄	#11719136	1	1	yes	characteristic
258	C₂	—	2	1	no	—
259	C₂	—	2	1	no	—
260	C₂	—	4	2	no	—
261	C₂	#13130690	1	1	yes	characteristic
262	C₂	C₂×#11718284	1	2	yes	—

	subgroup	G/H
center	C₂×C₄	C₂×C₂²⋊#261
commutator	C₂³×C₁₃	C₂²×C₈
Frattini	C₂²×C₄	C₂²×#261
Fitting	C₂²×C₄²×C₁₃	C₄
socle	C₂²×C₁₃	C₂×C₂²⋊C₈
prime radical	C₂⁴×C₁₃	C₂×C₈
cosocle	C₂²×#1055	C₂³
hypercenter	C₂²×C₄²	#261
hypocenter	C₁₃	#12785
LCS	G ▷ C₂³×C₁₃ ▷ C₂×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ C₂²×C₄²

Complements

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

N	H ≅ G/N	type
1. #13131168	C₂	semidirect
3. #13129883	C₂	semidirect
137. C₄×C₁₃	#349	semidirect
211. C₁₃	#12785	semidirect
253. C₄	#11718284	semidirect

Maximal subgroups

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

subgroup	index
#13131168	2
C₂×#11718284	2
#13129883	2
C₂×C₄×#6099	2
#12785	13