Subgroups — #16789294

Order 3328.

514 conjugacy classes of subgroups.

The subgroup lattice has 266 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₂²⋊#6092	C₂	1	2	yes	maximal
2	C₂×#11718231	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×C₄×#6092	C₂	1	1	yes	maximal, characteristic, complemented
4	#13129874	C₂	1	2	yes	maximal, complemented
5	C₂×#11718658	C₂	1	1	yes	maximal, characteristic
6	C₂×C₁₃×#117	C₄	1	1	yes	characteristic
7	C₂²×#6099	C₄	1	1	yes	characteristic
8	C₂³×#1055	C₂²	1	1	yes	characteristic
9	C₂²×#6092	C₂²	1	2	yes	—
10	C₂²×#6092	C₂²	1	2	yes	—
11	#11718658	—	2	2	no	—
12	#11718658	—	2	2	no	—
13	C₂×#88043	C₄	1	1	yes	characteristic
14	C₂×#88043	C₄	1	1	yes	characteristic
15	C₂×C₄×#1055	C₂²	1	1	yes	characteristic
16	C₂×C₄×#1055	C₂²	1	1	yes	characteristic
17	C₂²⋊#6092	C₄	1	8	yes	complemented
18	#11718231	—	2	2	no	—
19	C₄×#6092	—	2	2	no	—
20	C₂³×C₄×C₁₃	C₂×C₄	1	1	yes	characteristic
21	C₁₃×#117	—	2	2	no	—
22	C₁₃×#117	—	2	2	no	—
23	C₂×C₁₃×C₄⋊C₄	C₈	1	2	yes	—
24	C₂×C₁₃×C₂²⋊C₄	C₈	1	2	yes	—
25	C₂×C₄²×C₁₃	C₂×C₄	1	1	yes	characteristic
26	C₂×C₄²×C₁₃	C₂×C₄	1	1	yes	characteristic
27	C₂×#6099	—	2	2	no	—
28	C₂×#6099	—	2	2	no	—
29	C₂×#6099	C₈	1	4	yes	—
30	C₂²×#1055	—	2	1	no	—
31	C₂²×#1055	—	2	1	no	—
32	C₂²×#1055	—	2	2	no	—
33	C₂²×#1055	C₂×C₄	1	1	yes	characteristic
34	C₂²×#1055	C₂×C₄	1	4	yes	—
35	C₂²×#1055	C₂×C₄	1	1	yes	characteristic
36	C₂²×#1055	C₂×C₄	1	1	yes	characteristic
37	C₂²×#1055	C₂³	1	1	yes	characteristic
38	C₂×#6092	—	2	2	no	—
39	C₂×#6092	—	2	2	no	—
40	C₂×#6092	—	2	2	no	—
41	C₂×#6092	—	2	2	no	—
42	C₂×#6092	C₂×C₄	1	4	yes	—
43	C₂×#6092	C₂×C₄	1	4	yes	—
44	#88043	—	2	2	no	—
45	#88043	—	2	2	no	—
46	C₄×#1055	—	2	1	no	—
47	C₄×#1055	—	2	1	no	—
48	C₄×#1055	D₄	1	2	yes	—
49	C₄×#1055	D₄	1	2	yes	—
50	#12600	—	13	1	no	maximal, Sylow
51	C₂⁴×C₁₃	C₂×C₈	1	1	yes	characteristic
52	C₂²×C₄×C₁₃	—	2	1	no	—
53	C₂²×C₄×C₁₃	—	2	1	no	—
54	C₂²×C₄×C₁₃	—	2	1	no	—
55	C₂²×C₄×C₁₃	—	2	1	no	—
56	C₂²×C₄×C₁₃	C₄²	1	2	yes	—
57	C₂²×C₄×C₁₃	C₂×C₈	1	2	yes	—
58	C₂²×C₄×C₁₃	C₂×C₈	1	2	yes	—
59	C₂²×C₄×C₁₃	C₂×C₈	1	1	yes	characteristic
60	C₂²×C₄×C₁₃	C₂×C₈	1	2	yes	—
61	C₂²×C₄×C₁₃	C₂²×C₄	1	1	yes	characteristic
62	C₁₃×C₄⋊C₄	—	2	4	no	—
63	C₁₃×C₂²⋊C₄	—	2	4	no	—
64	C₄²×C₁₃	—	2	1	no	—
65	C₄²×C₁₃	—	2	1	no	—
66	C₄²×C₁₃	C₂²⋊C₄	1	2	yes	—
67	C₄²×C₁₃	M₁₆	1	2	yes	—
68	#6099	—	2	8	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	—	2	4	no	—
74	C₂×#1055	—	2	4	no	—
75	C₂×#1055	C₄²	1	2	yes	—
76	C₂×#1055	C₂²⋊C₄	1	2	yes	—
77	C₂×#1055	C₂×C₈	1	2	yes	—
78	C₂×#1055	C₂×C₈	1	2	yes	—
79	C₂×#1055	M₁₆	1	2	yes	—
80	C₂×#1055	C₂²×C₄	1	2	yes	—
81	C₂×#1055	C₂×D₄	1	1	yes	characteristic
82	C₂×#1055	C₂×D₄	1	1	yes	characteristic
83	C₂×#1055	#41	1	2	yes	—
84	#6092	—	2	4	no	—
85	#6092	—	2	4	no	—
86	C₂×#405	—	13	1	no	—
87	C₂×C₂²⋊C₁₆	—	13	2	no	—
88	C₂×#346	—	13	1	no	—
89	C₂×C₄×C₁₆	—	13	1	no	—
90	#1374	—	13	2	no	—
91	C₂³×C₁₃	—	2	1	no	—
92	C₂³×C₁₃	—	2	1	no	—
93	C₂³×C₁₃	—	2	4	no	—
94	C₂³×C₁₃	C₄×C₈	1	2	yes	—
95	C₂³×C₁₃	C₂²×C₈	1	1	yes	characteristic
96	C₂×C₄×C₁₃	—	2	1	no	—
97	C₂×C₄×C₁₃	—	2	1	no	—
98	C₂×C₄×C₁₃	—	2	4	no	—
99	C₂×C₄×C₁₃	—	2	2	no	—
100	C₂×C₄×C₁₃	—	2	2	no	—
101	C₂×C₄×C₁₃	—	2	2	no	—
102	C₂×C₄×C₁₃	—	2	2	no	—
103	C₂×C₄×C₁₃	—	2	2	no	—
104	C₂×C₄×C₁₃	—	2	4	no	—
105	C₂×C₄×C₁₃	C₄×C₈	1	2	yes	—
106	C₂×C₄×C₁₃	C₂²⋊C₈	1	4	yes	—
107	C₂×C₄×C₁₃	C₂×C₄²	1	1	yes	characteristic
108	C₂×C₄×C₁₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
109	C₂×C₄×C₁₃	#117	1	1	yes	characteristic
110	C₂×C₄×C₁₃	C₂²×C₈	1	1	yes	characteristic
111	C₂×C₄×C₁₃	C₂×M₁₆	1	1	yes	characteristic
112	C₂×C₄×C₁₃	C₂×M₁₆	1	1	yes	characteristic
113	#1055	—	2	2	no	—
114	#1055	—	2	2	no	—
115	#1055	C₂²⋊C₈	1	4	yes	—
116	#1055	C₄×D₄	1	2	yes	—
117	#1055	C₄×D₄	1	2	yes	—
118	C₂²×M₁₆	—	13	1	no	—
119	C₂³×C₈	—	13	1	no	—
120	C₂×#117	#261	1	1	yes	characteristic
121	C₂²×C₁₆	—	13	2	no	—
122	C₂²×C₁₆	—	13	2	no	—
123	#405	—	26	2	no	—
124	#405	—	26	2	no	—
125	C₂×#97	—	13	1	no	—
126	C₂×#97	—	13	1	no	—
127	C₂×C₄×C₈	—	13	1	no	—
128	C₂×C₄×C₈	—	13	1	no	—
129	C₂²⋊C₁₆	—	13	8	no	—
130	#346	—	26	2	no	—
131	C₄×C₁₆	—	26	2	no	—
132	C₂²×C₁₃	—	2	4	no	—
133	C₂²×C₁₃	—	2	2	no	—
134	C₂²×C₁₃	—	4	2	no	—
135	C₂²×C₁₃	—	4	2	no	—
136	C₂²×C₁₃	C₂×C₄×C₈	1	1	yes	characteristic
137	C₂²×C₁₃	C₂×C₂²⋊C₈	1	1	yes	characteristic
138	C₂²×C₁₃	#431	1	1	yes	characteristic
139	C₂²×C₁₃	#504	1	4	yes	—
140	C₄×C₁₃	—	2	4	no	—
141	C₄×C₁₃	—	2	2	no	—
142	C₄×C₁₃	—	4	2	no	—
143	C₄×C₁₃	C₄×C₂²⋊C₄	1	1	yes	characteristic
144	C₄×C₁₃	C₄×M₁₆	1	1	yes	characteristic
145	C₄×C₁₃	C₂×C₂²⋊C₈	1	1	yes	characteristic
146	C₄×C₁₃	#431	1	1	yes	characteristic
147	C₂³×C₄	C₂×#261	1	1	yes	characteristic
148	C₂×M₁₆	—	13	4	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	4	no	—
154	C₂²×C₈	—	13	1	no	—
155	C₂²×C₈	—	13	1	no	—
156	C₂²×C₈	—	26	2	no	—
157	C₂²×C₈	—	26	1	no	—
158	C₂²×C₈	—	26	1	no	—
159	#117	—	2	2	no	—
160	#117	—	2	2	no	—
161	C₂×C₄⋊C₄	#1057	1	2	yes	—
162	C₂×C₂²⋊C₄	#1057	1	2	yes	—
163	C₂×C₄²	C₂×#261	1	1	yes	characteristic
164	C₂×C₄²	C₂×#261	1	1	yes	characteristic
165	C₂×C₁₆	—	13	4	no	—
166	C₂×C₁₆	—	13	4	no	—
167	C₂×C₁₆	—	26	2	no	—
168	C₂×C₁₆	—	26	2	no	—
169	C₂×C₁₆	—	26	2	no	—
170	C₂×C₁₆	—	26	2	no	—
171	#97	—	26	2	no	—
172	#97	—	26	2	no	—
173	C₄×C₈	—	13	2	no	—
174	C₄×C₈	—	13	2	no	—
175	C₄×C₈	—	26	1	no	—
176	C₄×C₈	—	26	1	no	—
177	C₂×C₁₃	—	4	2	no	—
178	C₂×C₁₃	C₄×C₂²⋊C₈	1	1	yes	characteristic
179	C₂×C₁₃	#2108	1	2	yes	—
180	C₂×C₁₃	#2113	1	2	yes	—
181	C₂×C₁₃	#2164	1	2	yes	—
182	C₂⁴	C₂×#1057	1	1	yes	characteristic
183	C₂²×C₄	—	2	1	no	—
184	C₂²×C₄	—	2	1	no	—
185	C₂²×C₄	—	2	1	no	—
186	C₂²×C₄	—	2	1	no	—
187	C₂²×C₄	D₁₃⋊C₈	1	2	yes	—
188	C₂²×C₄	C₄×#261	1	2	yes	—
189	C₂²×C₄	C₂×#1057	1	2	yes	—
190	C₂²×C₄	C₂×#1057	1	1	yes	characteristic
191	C₂²×C₄	C₂×#1057	1	2	yes	—
192	C₂²×C₄	C₂²×#261	1	1	yes	characteristic
193	M₁₆	—	26	8	no	—
194	C₂×C₈	—	13	2	no	—
195	C₂×C₈	—	13	2	no	—
196	C₂×C₈	—	13	2	no	—
197	C₂×C₈	—	13	2	no	—
198	C₂×C₈	—	13	1	no	—
199	C₂×C₈	—	13	1	no	—
200	C₂×C₈	—	13	2	no	—
201	C₂×C₈	—	13	2	no	—
202	C₂×C₈	—	13	2	no	—
203	C₂×C₈	—	26	4	no	—
204	C₂×C₈	—	26	4	no	—
205	C₂×C₈	—	26	1	no	—
206	C₂×C₈	—	26	1	no	—
207	C₂×C₈	—	26	1	no	—
208	C₂×C₈	—	26	1	no	—
209	C₄⋊C₄	—	2	4	no	—
210	C₂²⋊C₄	—	2	4	no	—
211	C₄²	—	2	1	no	—
212	C₄²	—	2	1	no	—
213	C₄²	#6122	1	2	yes	—
214	C₄²	C₂²⋊#261	1	2	yes	—
215	C₁₆	—	26	4	no	—
216	C₁₆	—	26	4	no	—
217	C₁₃	#12600	1	1	yes	characteristic, complemented, Sylow
218	C₂³	—	2	1	no	—
219	C₂³	—	2	1	no	—
220	C₂³	—	2	4	no	—
221	C₂³	C₄×#1057	1	2	yes	—
222	C₂³	C₂²×#1057	1	1	yes	characteristic
223	C₂×C₄	—	2	1	no	—
224	C₂×C₄	—	2	1	no	—
225	C₂×C₄	—	2	4	no	—
226	C₂×C₄	—	2	2	no	—
227	C₂×C₄	—	2	2	no	—
228	C₂×C₄	—	2	2	no	—
229	C₂×C₄	—	2	2	no	—
230	C₂×C₄	—	2	2	no	—
231	C₂×C₄	—	2	4	no	—
232	C₂×C₄	C₄×#1057	1	2	yes	—
233	C₂×C₄	C₂²⋊#1057	1	4	yes	—
234	C₂×C₄	C₂×D₁₃⋊C₈	1	1	yes	characteristic
235	C₂×C₄	D₁₃⋊M₁₆	1	1	yes	characteristic
236	C₂×C₄	C₂×C₄×#261	1	1	yes	characteristic
237	C₂×C₄	#88237	1	1	yes	characteristic
238	C₂×C₄	C₂×#6122	1	1	yes	characteristic
239	C₂×C₄	C₂×C₂²⋊#261	1	1	yes	characteristic
240	C₈	—	13	4	no	—
241	C₈	—	13	2	no	—
242	C₈	—	13	2	no	—
243	C₈	—	26	2	no	—
244	C₈	—	26	2	no	—
245	C₂²	—	2	4	no	—
246	C₂²	—	2	2	no	—
247	C₂²	—	4	2	no	—
248	C₂²	—	4	2	no	—
249	C₂²	C₂×C₄×#1057	1	1	yes	characteristic
250	C₂²	#11719119	1	1	yes	characteristic
251	C₂²	#11719162	1	4	yes	—
252	C₂²	C₂×C₂²⋊#1057	1	1	yes	characteristic
253	C₄	—	2	4	no	—
254	C₄	—	2	2	no	—
255	C₄	—	4	2	no	—
256	C₄	C₄×#6122	1	1	yes	characteristic
257	C₄	D₁₃⋊(C₂²⋊C₈)	1	1	yes	characteristic
258	C₄	#11719125	1	1	yes	characteristic
259	C₄	C₄×C₂²⋊#261	1	1	yes	characteristic
260	C₂	—	4	2	no	—
261	C₂	C₄×C₂²⋊#1057	1	1	yes	characteristic
262	C₂	#13131045	1	2	yes	—
263	C₂	#13131066	1	2	yes	—
264	C₂	#13131201	1	2	yes	—

	subgroup	G/H
center, Frattini	C₂²×C₄	C₂²×#261
commutator	C₂²×C₁₃	C₂×C₄×C₈
Fitting	C₂×C₁₃×#117	C₄
socle	C₂³×C₁₃	C₂²×C₈
prime radical	C₂⁴×C₁₃	C₂×C₈
cosocle	C₂²×#1055	C₂³
hypercenter	C₂×#117	#261
hypocenter	C₁₃	#12600
LCS	G ▷ C₂²×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂²×C₄ ◁ C₂×#117

Complements

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

N	H ≅ G/N	type
2. C₂×#11718231	C₂	semidirect
3. C₂×C₄×#6092	C₂	semidirect
4. #13129874	C₂	semidirect
17. C₂²⋊#6092	C₄	semidirect
217. C₁₃	#12600	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₂²⋊#6092	2
C₂×#11718231	2
C₂×C₄×#6092	2
#13129874	2
C₂×#11718658	2
#12600	13