Subgroups — #16015957

Order 3328.

674 conjugacy classes of subgroups.

The subgroup lattice has 296 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×#88052	C₂	1	1	yes	maximal, characteristic
2	C₂×#11718767	C₂	1	1	yes	maximal, characteristic
3	#13116625	C₂	1	4	yes	maximal, complemented
4	#13116622	C₂	1	4	yes	maximal, complemented
5	#13116618	C₂	1	4	yes	maximal, complemented
6	C₂×C₁₃×#427	C₂	1	1	yes	maximal, characteristic
7	C₂³×C₈×C₁₃	C₂²	1	1	yes	characteristic
8	C₂×C₁₃×#117	—	2	1	no	—
9	C₂×#88180	—	2	1	no	—
10	C₁₃×#427	—	2	1	no	—
11	C₁₃×#427	—	2	1	no	—
12	C₁₃×#427	C₂²	1	2	yes	—
13	C₁₃×#427	C₂²	1	2	yes	—
14	C₂×C₁₃×#107	C₂²	1	2	yes	—
15	C₂×C₁₃×#106	—	2	1	no	—
16	#11718767	—	2	1	no	—
17	#11718767	—	2	1	no	—
18	#11718767	C₂²	1	2	yes	—
19	#11718767	C₂²	1	2	yes	—
20	C₂×#88058	C₂²	1	2	yes	—
21	C₂×#88057	—	2	1	no	—
22	C₂×#88052	C₄	1	8	yes	complemented
23	C₂×#88052	C₂²	1	2	yes	—
24	C₂×#88052	C₂²	1	2	yes	—
25	C₂²×#6090	C₂²	1	2	yes	—
26	#11718082	C₂²	1	8	yes	complemented
27	#11718081	C₂²	1	8	yes	complemented
28	C₂³×C₄×C₁₃	D₄	1	1	yes	characteristic
29	C₂²×C₈×C₁₃	C₂×C₄	1	4	yes	—
30	C₂²×C₈×C₁₃	D₄	1	1	yes	characteristic
31	C₂²×C₈×C₁₃	D₄	1	1	yes	characteristic
32	C₂²×C₈×C₁₃	D₄	1	1	yes	characteristic
33	C₂²×C₈×C₁₃	Q₈	1	4	yes	—
34	C₂²×C₈×C₁₃	C₂³	1	1	yes	characteristic
35	C₂²×C₈×C₁₃	C₂³	1	1	yes	characteristic
36	C₂²×C₈×C₁₃	C₂³	1	1	yes	characteristic
37	C₁₃×#117	—	2	2	no	—
38	C₁₃×#117	—	2	2	no	—
39	C₁₃×#117	—	4	1	no	—
40	C₁₃×#117	—	4	1	no	—
41	C₂×C₁₃×C₄⋊C₄	—	2	2	no	—
42	C₂×C₁₃×C₂²⋊C₄	—	4	1	no	—
43	C₂×C₄²×C₁₃	—	4	1	no	—
44	C₂×C₂²⋊Dic₁₃	—	4	1	no	—
45	#88180	—	2	2	no	—
46	#88180	—	2	2	no	—
47	#88180	—	4	1	no	—
48	#88180	—	4	1	no	—
49	C₂×C₄⋊Dic₁₃	—	2	2	no	—
50	C₂×C₄×Dic₁₃	—	4	1	no	—
51	C₁₃×#107	—	2	2	no	—
52	C₁₃×#107	C₂³	1	4	yes	—
53	C₁₃×#106	—	2	2	no	—
54	C₁₃×#106	—	2	2	no	—
55	#88058	—	2	2	no	—
56	#88058	C₂³	1	4	yes	—
57	#88057	—	2	2	no	—
58	#88057	—	2	2	no	—
59	#88052	C₂×C₄	1	8	yes	complemented
60	#88052	C₂×C₄	1	8	yes	complemented
61	C₂×#6090	C₂×C₄	1	8	yes	complemented
62	C₂×#6090	C₂³	1	4	yes	—
63	#33409	—	13	1	no	maximal, Sylow
64	C₂⁴×C₁₃	D₈	1	1	yes	characteristic
65	C₂²×C₄×C₁₃	—	4	2	no	—
66	C₂²×C₄×C₁₃	C₄⋊C₄	1	4	yes	—
67	C₂²×C₄×C₁₃	D₈	1	1	yes	characteristic
68	C₂²×C₄×C₁₃	D₈	1	1	yes	characteristic
69	C₂²×C₄×C₁₃	D₈	1	1	yes	characteristic
70	C₂²×C₄×C₁₃	Q₁₆	1	4	yes	—
71	C₂²×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
72	C₂²×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
73	C₂²×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
74	C₂²×Dic₁₃	—	4	2	no	—
75	C₂×C₈×C₁₃	C₄⋊C₄	1	4	yes	—
76	C₂×C₈×C₁₃	C₄⋊C₄	1	4	yes	—
77	C₂×C₈×C₁₃	C₄⋊C₄	1	4	yes	—
78	C₂×C₈×C₁₃	C₂²×C₄	1	2	yes	—
79	C₂×C₈×C₁₃	C₂²×C₄	1	2	yes	—
80	C₂×C₈×C₁₃	C₂²×C₄	1	2	yes	—
81	C₂×C₈×C₁₃	C₂×D₄	1	1	yes	characteristic
82	C₂×C₈×C₁₃	C₂×D₄	1	1	yes	characteristic
83	C₂×C₈×C₁₃	C₂×D₄	1	1	yes	characteristic
84	C₂×C₈×C₁₃	C₂×Q₈	1	2	yes	—
85	C₂×C₈×C₁₃	C₂×Q₈	1	2	yes	—
86	C₂×C₈×C₁₃	C₂×Q₈	1	2	yes	—
87	C₂×C₈×C₁₃	C₂⁴	1	1	yes	characteristic
88	C₁₃×C₄⋊C₄	—	2	4	no	—
89	C₁₃×C₄⋊C₄	—	4	2	no	—
90	C₁₃×C₂²⋊C₄	—	4	2	no	—
91	C₁₃×C₂²⋊C₄	—	4	2	no	—
92	C₄²×C₁₃	—	4	2	no	—
93	C₄²×C₁₃	—	4	2	no	—
94	C₂²⋊Dic₁₃	—	4	2	no	—
95	C₂²⋊Dic₁₃	—	4	2	no	—
96	C₄⋊Dic₁₃	—	2	4	no	—
97	C₄⋊Dic₁₃	—	4	2	no	—
98	C₄×Dic₁₃	—	4	2	no	—
99	C₄×Dic₁₃	—	4	2	no	—
100	#6090	C₂²×C₄	1	8	yes	—
101	C₂²×M₃₂	—	13	1	no	—
102	C₂×#427	—	13	1	no	—
103	C₂×#427	D₁₃	1	1	yes	characteristic
104	#2159	—	13	4	no	—
105	#2159	—	13	4	no	—
106	#2158	—	13	4	no	—
107	C₂³×C₁₃	—	4	2	no	—
108	C₂³×C₁₃	#107	1	4	yes	—
109	C₂³×C₁₃	C₂×D₈	1	1	yes	characteristic
110	C₂³×C₁₃	C₂×D₈	1	1	yes	characteristic
111	C₂³×C₁₃	C₂×D₈	1	1	yes	characteristic
112	C₂×C₄×C₁₃	—	4	4	no	—
113	C₂×C₄×C₁₃	—	8	4	no	—
114	C₂×C₄×C₁₃	#107	1	4	yes	—
115	C₂×C₄×C₁₃	#107	1	4	yes	—
116	C₂×C₄×C₁₃	#107	1	4	yes	—
117	C₂×C₄×C₁₃	C₂×C₄⋊C₄	1	2	yes	—
118	C₂×C₄×C₁₃	C₂×C₄⋊C₄	1	2	yes	—
119	C₂×C₄×C₁₃	C₂×C₄⋊C₄	1	2	yes	—
120	C₂×C₄×C₁₃	C₂×D₈	1	1	yes	characteristic
121	C₂×C₄×C₁₃	C₂×D₈	1	1	yes	characteristic
122	C₂×C₄×C₁₃	C₂×D₈	1	1	yes	characteristic
123	C₂×C₄×C₁₃	C₂×Q₁₆	1	2	yes	—
124	C₂×C₄×C₁₃	C₂×Q₁₆	1	2	yes	—
125	C₂×C₄×C₁₃	C₂×Q₁₆	1	2	yes	—
126	C₂×C₄×C₁₃	C₂²×D₄	1	1	yes	characteristic
127	C₂×Dic₁₃	—	4	4	no	—
128	C₂×Dic₁₃	—	8	4	no	—
129	C₈×C₁₃	C₂×C₄⋊C₄	1	2	yes	—
130	C₈×C₁₃	C₂×C₄⋊C₄	1	2	yes	—
131	C₈×C₁₃	C₂×C₄⋊C₄	1	2	yes	—
132	C₈×C₁₃	C₂³×C₄	1	1	yes	characteristic
133	C₈×C₁₃	C₂²×Q₈	1	1	yes	characteristic
134	C₂³×C₈	C₂×D₁₃	1	1	yes	characteristic
135	C₂×#117	—	2	1	no	—
136	C₂×#117	—	26	1	no	—
137	C₂×M₃₂	—	13	8	no	—
138	C₂×M₃₂	—	13	2	no	—
139	C₂×M₃₂	—	13	2	no	—
140	C₂²×C₁₆	—	13	2	no	—
141	#427	—	2	1	no	—
142	#427	—	2	1	no	—
143	#427	—	13	2	no	—
144	#427	—	13	2	no	—
145	#427	—	26	1	no	—
146	#427	—	26	1	no	—
147	#427	C₂×D₁₃	1	2	yes	—
148	#427	C₂×D₁₃	1	2	yes	—
149	C₂×#107	—	13	2	no	—
150	C₂×#107	C₂×D₁₃	1	2	yes	—
151	C₂×#106	—	2	1	no	—
152	C₂×#106	—	26	1	no	—
153	#367	—	13	8	no	—
154	#366	—	13	8	no	—
155	C₂²×C₁₃	—	2	2	no	—
156	C₂²×C₁₃	—	2	2	no	—
157	C₂²×C₁₃	—	2	2	no	—
158	C₂²×C₁₃	—	4	4	no	—
159	C₂²×C₁₃	C₂×#107	1	2	yes	—
160	C₂²×C₁₃	C₂×#107	1	2	yes	—
161	C₂²×C₁₃	C₂×#107	1	2	yes	—
162	C₂²×C₁₃	C₂²×D₈	1	1	yes	characteristic
163	C₄×C₁₃	—	8	4	no	—
164	C₄×C₁₃	C₂×#107	1	2	yes	—
165	C₄×C₁₃	C₂×#107	1	2	yes	—
166	C₄×C₁₃	C₂×#107	1	2	yes	—
167	C₄×C₁₃	C₂²×C₄⋊C₄	1	1	yes	characteristic
168	C₄×C₁₃	C₂²×Q₁₆	1	1	yes	characteristic
169	Dic₁₃	—	8	4	no	—
170	C₂³×C₄	C₂²⋊D₁₃	1	1	yes	characteristic
171	C₂²×C₈	Dic₂₆	1	4	yes	—
172	C₂²×C₈	C₄×D₁₃	1	4	yes	—
173	C₂²×C₈	C₂²⋊D₁₃	1	1	yes	characteristic
174	C₂²×C₈	C₂²⋊D₁₃	1	1	yes	characteristic
175	C₂²×C₈	C₂²⋊D₁₃	1	1	yes	characteristic
176	C₂²×C₈	C₂²×D₁₃	1	1	yes	characteristic
177	C₂²×C₈	C₂²×D₁₃	1	1	yes	characteristic
178	C₂²×C₈	C₂²×D₁₃	1	1	yes	characteristic
179	#117	—	2	2	no	—
180	#117	—	2	2	no	—
181	#117	—	4	1	no	—
182	#117	—	4	1	no	—
183	#117	—	26	2	no	—
184	#117	—	26	2	no	—
185	#117	—	52	1	no	—
186	#117	—	52	1	no	—
187	C₂×C₄⋊C₄	—	2	2	no	—
188	C₂×C₄⋊C₄	—	26	2	no	—
189	C₂×C₂²⋊C₄	—	4	1	no	—
190	C₂×C₂²⋊C₄	—	52	1	no	—
191	C₂×C₄²	—	4	1	no	—
192	C₂×C₄²	—	52	1	no	—
193	M₃₂	—	13	8	no	—
194	M₃₂	—	13	8	no	—
195	C₂×C₁₆	—	13	8	no	—
196	C₂×C₁₆	—	13	4	no	—
197	#107	—	2	2	no	—
198	#107	—	13	4	no	—
199	#107	—	26	2	no	—
200	#107	C₂²×D₁₃	1	4	yes	—
201	#106	—	2	2	no	—
202	#106	—	2	2	no	—
203	#106	—	26	2	no	—
204	#106	—	26	2	no	—
205	C₂×C₁₃	—	2	2	no	—
206	C₂×C₁₃	—	2	2	no	—
207	C₂×C₁₃	—	2	2	no	—
208	C₂×C₁₃	C₂²×#107	1	1	yes	characteristic
209	C₂×C₁₃	#3414	1	2	yes	—
210	C₂⁴	#6104	1	1	yes	characteristic
211	C₂²×C₄	—	4	2	no	—
212	C₂²×C₄	—	52	2	no	—
213	C₂²×C₄	#6101	1	4	yes	—
214	C₂²×C₄	#6104	1	1	yes	characteristic
215	C₂²×C₄	#6104	1	1	yes	characteristic
216	C₂²×C₄	#6104	1	1	yes	characteristic
217	C₂²×C₄	#6107	1	4	yes	—
218	C₂²×C₄	C₂×C₂²⋊D₁₃	1	1	yes	characteristic
219	C₂²×C₄	C₂×C₂²⋊D₁₃	1	1	yes	characteristic
220	C₂²×C₄	C₂×C₂²⋊D₁₃	1	1	yes	characteristic
221	C₂×C₈	#6101	1	4	yes	—
222	C₂×C₈	#6101	1	4	yes	—
223	C₂×C₈	#6101	1	4	yes	—
224	C₂×C₈	C₂×Dic₂₆	1	2	yes	—
225	C₂×C₈	C₂×Dic₂₆	1	2	yes	—
226	C₂×C₈	C₂×Dic₂₆	1	2	yes	—
227	C₂×C₈	C₂×C₄×D₁₃	1	2	yes	—
228	C₂×C₈	C₂×C₄×D₁₃	1	2	yes	—
229	C₂×C₈	C₂×C₄×D₁₃	1	2	yes	—
230	C₂×C₈	C₂×C₂²⋊D₁₃	1	1	yes	characteristic
231	C₂×C₈	C₂×C₂²⋊D₁₃	1	1	yes	characteristic
232	C₂×C₈	C₂×C₂²⋊D₁₃	1	1	yes	characteristic
233	C₂×C₈	C₂³×D₁₃	1	1	yes	characteristic
234	C₄⋊C₄	—	2	4	no	—
235	C₄⋊C₄	—	4	2	no	—
236	C₄⋊C₄	—	26	4	no	—
237	C₄⋊C₄	—	52	2	no	—
238	C₂²⋊C₄	—	4	2	no	—
239	C₂²⋊C₄	—	4	2	no	—
240	C₂²⋊C₄	—	52	2	no	—
241	C₂²⋊C₄	—	52	2	no	—
242	C₄²	—	4	2	no	—
243	C₄²	—	4	2	no	—
244	C₄²	—	52	2	no	—
245	C₄²	—	52	2	no	—
246	C₁₆	—	13	8	no	—
247	C₁₃	#33409	1	1	yes	characteristic, complemented, Sylow
248	C₂³	—	4	2	no	—
249	C₂³	#88047	1	4	yes	—
250	C₂³	C₂×#6104	1	1	yes	characteristic
251	C₂³	C₂×#6104	1	1	yes	characteristic
252	C₂³	C₂×#6104	1	1	yes	characteristic
253	C₂×C₄	—	4	4	no	—
254	C₂×C₄	—	8	4	no	—
255	C₂×C₄	—	52	4	no	—
256	C₂×C₄	—	104	4	no	—
257	C₂×C₄	#88047	1	4	yes	—
258	C₂×C₄	#88047	1	4	yes	—
259	C₂×C₄	#88047	1	4	yes	—
260	C₂×C₄	C₂×#6101	1	2	yes	—
261	C₂×C₄	C₂×#6101	1	2	yes	—
262	C₂×C₄	C₂×#6101	1	2	yes	—
263	C₂×C₄	C₂×#6104	1	1	yes	characteristic
264	C₂×C₄	C₂×#6104	1	1	yes	characteristic
265	C₂×C₄	C₂×#6104	1	1	yes	characteristic
266	C₂×C₄	C₂×#6107	1	2	yes	—
267	C₂×C₄	C₂×#6107	1	2	yes	—
268	C₂×C₄	C₂×#6107	1	2	yes	—
269	C₂×C₄	C₂²×C₂²⋊D₁₃	1	1	yes	characteristic
270	C₈	C₂×#6101	1	2	yes	—
271	C₈	C₂×#6101	1	2	yes	—
272	C₈	C₂×#6101	1	2	yes	—
273	C₈	C₂²×Dic₂₆	1	1	yes	characteristic
274	C₈	C₂²×C₄×D₁₃	1	1	yes	characteristic
275	C₂²	—	2	2	no	—
276	C₂²	—	2	2	no	—
277	C₂²	—	2	2	no	—
278	C₂²	—	4	4	no	—
279	C₂²	C₂×#88047	1	2	yes	—
280	C₂²	C₂×#88047	1	2	yes	—
281	C₂²	C₂×#88047	1	2	yes	—
282	C₂²	C₂²×#6104	1	1	yes	characteristic
283	C₄	—	8	4	no	—
284	C₄	—	104	4	no	—
285	C₄	C₂×#88047	1	2	yes	—
286	C₄	C₂×#88047	1	2	yes	—
287	C₄	C₂×#88047	1	2	yes	—
288	C₄	C₂²×#6101	1	1	yes	characteristic
289	C₄	C₂²×#6107	1	1	yes	characteristic
290	C₂	—	2	2	no	—
291	C₂	—	2	2	no	—
292	C₂	—	2	2	no	—
293	C₂	C₂²×#88047	1	1	yes	characteristic
294	C₂	#13127970	1	2	yes	—

	subgroup	G/H
center	C₂×C₄	C₂×#6104
commutator	C₈×C₁₃	C₂³×C₄
Frattini	C₂×C₈	C₂³×D₁₃
Fitting	C₂×C₁₃×#427	C₂
socle	C₂²×C₁₃	C₂²×D₈
prime radical	C₂⁴×C₁₃	D₈
cosocle	C₂×C₈×C₁₃	C₂⁴
hypercenter	C₂×#427	D₁₃
hypocenter	C₁₃	#33409
LCS	G ▷ C₈×C₁₃ ▷ C₄×C₁₃ ▷ C₂×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₄ ◁ C₂³×C₈ ◁ C₂×#427

Complements

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

N	H ≅ G/N	type
3. #13116625	C₂	semidirect
4. #13116622	C₂	semidirect
5. #13116618	C₂	semidirect
22. C₂×#88052	C₄	semidirect
26. #11718082	C₂²	semidirect
27. #11718081	C₂²	semidirect
59. #88052	C₂×C₄	semidirect
60. #88052	C₂×C₄	semidirect
61. C₂×#6090	C₂×C₄	semidirect
247. C₁₃	#33409	semidirect

Maximal subgroups

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

subgroup	index
C₂²×#88052	2
C₂×#11718767	2
#13116625	2
#13116622	2
#13116618	2
C₂×C₁₃×#427	2
#33409	13