Subgroups — #15768654

Order 3328.

718 conjugacy classes of subgroups.

The subgroup lattice has 290 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₄×#1055	C₂	1	1	yes	maximal, characteristic
2	C₂×#11718150	C₂	1	2	yes	maximal
3	C₂×#11718150	C₂	1	2	yes	maximal
4	#13114530	C₂	1	1	yes	maximal, characteristic
5	C₁₃×#1728	C₂	1	1	yes	maximal, characteristic
6	C₂²×C₁₃×M₁₆	C₂²	1	2	yes	—
7	C₂²×C₄²×C₁₃	C₂²	1	1	yes	characteristic
8	C₂²×#6099	C₂²	1	2	yes	—
9	C₂³×#1055	C₂²	1	1	yes	characteristic
10	C₂³×#1055	C₂²	1	1	yes	characteristic
11	C₂×C₄×#1055	—	2	4	no	—
12	C₂×C₄×#1055	C₄	1	4	yes	—
13	C₁₃×#336	—	2	2	no	—
14	#11718150	—	2	4	no	—
15	#11718150	C₄	1	8	yes	complemented
16	#11718116	—	2	2	no	—
17	C₂³×C₄×C₁₃	Q₈	1	2	yes	—
18	C₂³×C₄×C₁₃	C₂³	1	1	yes	characteristic
19	C₂×C₁₃×M₁₆	—	2	2	no	—
20	C₂×C₁₃×M₁₆	—	2	2	no	—
21	C₂×C₁₃×M₁₆	—	2	2	no	—
22	C₂×C₁₃×M₁₆	—	2	2	no	—
23	C₂×C₁₃×M₁₆	—	2	2	no	—
24	C₂×C₁₃×M₁₆	C₂×C₄	1	4	yes	—
25	C₂²×C₈×C₁₃	—	2	2	no	—
26	C₂×C₄²×C₁₃	—	2	2	no	—
27	C₂×C₄²×C₁₃	—	2	2	no	—
28	C₂×C₄²×C₁₃	C₂×C₄	1	2	yes	—
29	C₂×C₄²×C₁₃	D₄	1	2	yes	—
30	C₂×#6099	—	2	2	no	—
31	C₂×#6099	—	2	2	no	—
32	C₂×#6099	—	2	2	no	—
33	C₂×#6099	—	2	2	no	—
34	C₂×#6099	—	2	2	no	—
35	C₂×#6099	C₂×C₄	1	4	yes	—
36	C₂²×#1055	—	2	2	no	—
37	C₂²×#1055	—	2	2	no	—
38	C₂²×#1055	—	2	1	no	—
39	C₂²×#1055	—	2	1	no	—
40	C₂²×#1055	—	2	1	no	—
41	C₂²×#1055	—	2	1	no	—
42	C₂²×#1055	—	2	2	no	—
43	C₂²×#1055	C₂×C₄	1	4	yes	—
44	C₂²×#1055	C₂×C₄	1	2	yes	—
45	C₂²×#1055	C₂×C₄	1	2	yes	—
46	C₂²×#1055	D₄	1	4	yes	—
47	C₄×#1055	—	2	8	no	—
48	#10822	—	13	1	no	maximal, Sylow
49	C₂⁴×C₁₃	C₂×Q₈	1	1	yes	characteristic
50	C₂²×C₄×C₁₃	—	2	4	no	—
51	C₂²×C₄×C₁₃	—	2	8	no	—
52	C₂²×C₄×C₁₃	C₄⋊C₄	1	4	yes	—
53	C₂²×C₄×C₁₃	C₂²×C₄	1	1	yes	characteristic
54	C₂²×C₄×C₁₃	C₂²×C₄	1	2	yes	—
55	C₂²×C₄×C₁₃	C₂×D₄	1	2	yes	—
56	C₂²×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
57	C₂²×C₄×C₁₃	#41	1	2	yes	—
58	C₂²×C₄×C₁₃	#41	1	2	yes	—
59	C₂²×C₄×C₁₃	#41	1	1	yes	characteristic
60	C₂²×C₄×C₁₃	#41	1	1	yes	characteristic
61	C₂²×C₄×C₁₃	#41	1	1	yes	characteristic
62	C₂²×C₄×C₁₃	#41	1	1	yes	characteristic
63	C₁₃×M₁₆	—	2	4	no	—
64	C₁₃×M₁₆	—	2	4	no	—
65	C₁₃×M₁₆	—	4	2	no	—
66	C₁₃×M₁₆	—	4	2	no	—
67	C₂×C₈×C₁₃	—	2	4	no	—
68	C₂×C₈×C₁₃	—	4	2	no	—
69	C₂×C₈×C₁₃	—	4	2	no	—
70	C₄²×C₁₃	—	2	2	no	—
71	C₄²×C₁₃	—	2	2	no	—
72	C₄²×C₁₃	—	2	4	no	—
73	#6099	—	2	4	no	—
74	#6099	—	2	4	no	—
75	#6099	—	4	2	no	—
76	#6099	—	4	2	no	—
77	C₂×#1055	—	2	2	no	—
78	C₂×#1055	—	2	2	no	—
79	C₂×#1055	—	2	2	no	—
80	C₂×#1055	—	2	2	no	—
81	C₂×#1055	—	2	4	no	—
82	C₂×#1055	—	2	4	no	—
83	C₂×#1055	—	2	4	no	—
84	C₂×#1055	—	2	4	no	—
85	C₂×#1055	—	2	4	no	—
86	C₂×#1055	—	4	2	no	—
87	C₂×#1055	—	4	2	no	—
88	C₂×#1055	C₄²	1	4	yes	—
89	C₂×#1055	C₂²⋊C₄	1	4	yes	—
90	C₂²×C₄×C₈	—	13	1	no	—
91	C₂×#341	—	13	2	no	—
92	C₂×#341	—	13	2	no	—
93	#1728	—	13	1	no	—
94	#1728	D₁₃	1	1	yes	characteristic
95	C₂³×C₁₃	—	2	4	no	—
96	C₂³×C₁₃	C₂×C₄⋊C₄	1	1	yes	characteristic
97	C₂³×C₁₃	C₄×Q₈	1	2	yes	—
98	C₂³×C₁₃	C₂²⋊Q₈	1	2	yes	—
99	C₂³×C₁₃	#125	1	1	yes	characteristic
100	C₂³×C₁₃	#125	1	1	yes	characteristic
101	C₂×C₄×C₁₃	—	2	16	no	—
102	C₂×C₄×C₁₃	—	2	4	no	—
103	C₂×C₄×C₁₃	—	2	4	no	—
104	C₂×C₄×C₁₃	#108	1	8	yes	—
105	C₂×C₄×C₁₃	C₂×C₄²	1	1	yes	characteristic
106	C₂×C₄×C₁₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
107	C₂×C₄×C₁₃	#117	1	1	yes	characteristic
108	C₂×C₄×C₁₃	#117	1	1	yes	characteristic
109	C₂×C₄×C₁₃	#117	1	2	yes	—
110	C₂×C₄×C₁₃	#117	1	2	yes	—
111	C₂×C₄×C₁₃	#117	1	1	yes	characteristic
112	C₂×C₄×C₁₃	#117	1	1	yes	characteristic
113	C₂×C₄×C₁₃	C₄×D₄	1	2	yes	—
114	C₂×C₄×C₁₃	C₄×D₄	1	2	yes	—
115	C₂×C₄×C₁₃	C₄×D₄	1	2	yes	—
116	C₂×C₄×C₁₃	#121	1	2	yes	—
117	C₂×C₄×C₁₃	#123	1	2	yes	—
118	C₂×C₄×C₁₃	#123	1	2	yes	—
119	C₂×C₄×C₁₃	#124	1	1	yes	characteristic
120	C₂×C₄×C₁₃	#124	1	1	yes	characteristic
121	C₂×C₄×C₁₃	#126	1	2	yes	—
122	C₂×C₄×C₁₃	#126	1	2	yes	—
123	C₈×C₁₃	—	4	4	no	—
124	#1055	—	2	4	no	—
125	#1055	—	2	4	no	—
126	#1055	—	4	4	no	—
127	C₂²×M₁₆	—	13	2	no	—
128	C₂²×M₁₆	C₂×D₁₃	1	2	yes	—
129	C₂³×C₈	—	13	1	no	—
130	C₂³×C₈	—	13	1	no	—
131	C₂²×C₄²	C₂×D₁₃	1	1	yes	characteristic
132	C₂×C₄×C₈	—	13	4	no	—
133	C₂×C₄×C₈	—	26	4	no	—
134	#341	—	13	8	no	—
135	#341	—	26	4	no	—
136	#336	—	2	2	no	—
137	#336	—	26	2	no	—
138	C₂²×C₁₃	—	2	4	no	—
139	C₂²×C₁₃	—	2	2	no	—
140	C₂²×C₁₃	—	2	2	no	—
141	C₂²×C₁₃	—	2	4	no	—
142	C₂²×C₁₃	C₄×C₄⋊C₄	1	1	yes	characteristic
143	C₂²×C₁₃	#380	1	1	yes	characteristic
144	C₂²×C₁₃	#382	1	1	yes	characteristic
145	C₂²×C₁₃	#387	1	2	yes	—
146	C₂²×C₁₃	#387	1	2	yes	—
147	C₂²×C₁₃	C₂×#108	1	4	yes	—
148	C₄×C₁₃	—	2	8	no	—
149	C₄×C₁₃	#376	1	1	yes	characteristic
150	C₄×C₁₃	C₄×C₂²⋊C₄	1	1	yes	characteristic
151	C₄×C₁₃	#381	1	1	yes	characteristic
152	C₄×C₁₃	#383	1	1	yes	characteristic
153	C₄×C₁₃	#388	1	2	yes	—
154	C₄×C₁₃	#388	1	2	yes	—
155	C₂³×C₄	Dic₂₆	1	2	yes	—
156	C₂³×C₄	C₂²×D₁₃	1	1	yes	characteristic
157	C₂×M₁₆	—	2	2	no	—
158	C₂×M₁₆	—	2	2	no	—
159	C₂×M₁₆	—	2	2	no	—
160	C₂×M₁₆	—	2	2	no	—
161	C₂×M₁₆	—	2	2	no	—
162	C₂×M₁₆	—	13	4	no	—
163	C₂×M₁₆	—	26	2	no	—
164	C₂×M₁₆	—	26	2	no	—
165	C₂×M₁₆	—	26	2	no	—
166	C₂×M₁₆	—	26	2	no	—
167	C₂×M₁₆	—	26	2	no	—
168	C₂×M₁₆	C₄×D₁₃	1	4	yes	—
169	C₂²×C₈	—	2	2	no	—
170	C₂²×C₈	—	13	4	no	—
171	C₂²×C₈	—	13	2	no	—
172	C₂²×C₈	—	13	2	no	—
173	C₂²×C₈	—	13	4	no	—
174	C₂²×C₈	—	26	2	no	—
175	C₂²×C₈	—	26	2	no	—
176	C₂²×C₈	—	26	1	no	—
177	C₂²×C₈	—	26	1	no	—
178	C₂²×C₈	—	26	1	no	—
179	C₂²×C₈	—	26	1	no	—
180	C₂²×C₈	—	26	2	no	—
181	C₂×C₄²	—	2	2	no	—
182	C₂×C₄²	—	2	2	no	—
183	C₂×C₄²	C₄×D₁₃	1	2	yes	—
184	C₂×C₄²	C₂²⋊D₁₃	1	2	yes	—
185	C₄×C₈	—	26	8	no	—
186	C₂×C₁₃	—	2	2	no	—
187	C₂×C₁₃	—	2	2	no	—
188	C₂×C₁₃	#1434	1	1	yes	characteristic
189	C₂×C₁₃	C₄×#108	1	2	yes	—
190	C₂×C₁₃	#1810	1	2	yes	—
191	C₂×C₁₃	#1837	1	2	yes	—
192	C₂⁴	C₂×Dic₂₆	1	1	yes	characteristic
193	C₂²×C₄	—	2	4	no	—
194	C₂²×C₄	—	2	8	no	—
195	C₂²×C₄	#6101	1	4	yes	—
196	C₂²×C₄	C₂×C₄×D₁₃	1	1	yes	characteristic
197	C₂²×C₄	C₂×C₄×D₁₃	1	2	yes	—
198	C₂²×C₄	#6127	1	2	yes	—
199	C₂²×C₄	D₄×D₁₃	1	2	yes	—
200	C₂²×C₄	#6129	1	2	yes	—
201	C₂²×C₄	#6129	1	1	yes	characteristic
202	C₂²×C₄	#6129	1	1	yes	characteristic
203	C₂²×C₄	#6131	1	1	yes	characteristic
204	C₂²×C₄	#6131	1	1	yes	characteristic
205	C₂²×C₄	C₂×C₂²⋊D₁₃	1	1	yes	characteristic
206	M₁₆	—	2	4	no	—
207	M₁₆	—	2	4	no	—
208	M₁₆	—	4	2	no	—
209	M₁₆	—	4	2	no	—
210	M₁₆	—	26	4	no	—
211	M₁₆	—	26	4	no	—
212	M₁₆	—	52	2	no	—
213	M₁₆	—	52	2	no	—
214	C₂×C₈	—	2	4	no	—
215	C₂×C₈	—	4	2	no	—
216	C₂×C₈	—	4	2	no	—
217	C₂×C₈	—	13	4	no	—
218	C₂×C₈	—	13	4	no	—
219	C₂×C₈	—	26	2	no	—
220	C₂×C₈	—	26	2	no	—
221	C₂×C₈	—	26	4	no	—
222	C₂×C₈	—	26	4	no	—
223	C₂×C₈	—	26	2	no	—
224	C₂×C₈	—	26	2	no	—
225	C₂×C₈	—	26	4	no	—
226	C₂×C₈	—	26	4	no	—
227	C₂×C₈	—	26	4	no	—
228	C₂×C₈	—	52	2	no	—
229	C₂×C₈	—	52	2	no	—
230	C₄²	—	2	2	no	—
231	C₄²	—	2	2	no	—
232	C₄²	—	2	4	no	—
233	C₁₃	#10822	1	1	yes	characteristic, complemented, Sylow
234	C₂³	—	2	4	no	—
235	C₂³	C₄×Dic₂₆	1	2	yes	—
236	C₂³	#88132	1	2	yes	—
237	C₂³	#88144	1	1	yes	characteristic
238	C₂³	#88144	1	1	yes	characteristic
239	C₂³	C₂×#6101	1	1	yes	characteristic
240	C₂×C₄	—	2	16	no	—
241	C₂×C₄	—	2	4	no	—
242	C₂×C₄	—	2	4	no	—
243	C₂×C₄	#88062	1	8	yes	—
244	C₂×C₄	C₄²×D₁₃	1	1	yes	characteristic
245	C₂×C₄	#88126	1	1	yes	characteristic
246	C₂×C₄	#88131	1	1	yes	characteristic
247	C₂×C₄	#88131	1	2	yes	—
248	C₂×C₄	#88133	1	2	yes	—
249	C₂×C₄	C₂²⋊C₄×D₁₃	1	1	yes	characteristic
250	C₂×C₄	#88135	1	2	yes	—
251	C₂×C₄	#88137	1	2	yes	—
252	C₂×C₄	#88146	1	1	yes	characteristic
253	C₂×C₄	#88146	1	2	yes	—
254	C₂×C₄	#88146	1	1	yes	characteristic
255	C₂×C₄	#88147	1	2	yes	—
256	C₂×C₄	#88148	1	2	yes	—
257	C₂×C₄	#88152	1	2	yes	—
258	C₂×C₄	C₄×C₂²⋊D₁₃	1	2	yes	—
259	C₂×C₄	#88190	1	1	yes	characteristic
260	C₂×C₄	#88193	1	2	yes	—
261	C₂×C₄	#88201	1	1	yes	characteristic
262	C₈	—	4	4	no	—
263	C₈	—	26	4	no	—
264	C₈	—	26	4	no	—
265	C₈	—	52	4	no	—
266	C₂²	—	2	2	no	—
267	C₂²	—	2	2	no	—
268	C₂²	—	2	4	no	—
269	C₂²	—	2	4	no	—
270	C₂²	#11718315	1	2	yes	—
271	C₂²	#11718316	1	2	yes	—
272	C₂²	C₄×#6101	1	1	yes	characteristic
273	C₂²	#11718603	1	1	yes	characteristic
274	C₂²	#11718638	1	1	yes	characteristic
275	C₂²	C₂×#88062	1	4	yes	—
276	C₄	—	2	8	no	—
277	C₄	#11718309	1	1	yes	characteristic
278	C₄	#11718312	1	1	yes	characteristic
279	C₄	#11718325	1	1	yes	characteristic
280	C₄	#11718327	1	1	yes	characteristic
281	C₄	#11718606	1	2	yes	—
282	C₄	#11718651	1	2	yes	—
283	C₂	—	2	2	no	—
284	C₂	—	2	2	no	—
285	C₂	#13113062	1	1	yes	characteristic
286	C₂	C₄×#88062	1	2	yes	—
287	C₂	#13114872	1	2	yes	—
288	C₂	#13114973	1	2	yes	—

	subgroup	G/H
center	C₂²×C₄	C₂×C₂²⋊D₁₃
commutator	C₂×C₄×C₁₃	C₂×C₄²
Frattini	C₂³×C₄	C₂²×D₁₃
Fitting	C₁₃×#1728	C₂
socle	C₂³×C₁₃	C₂×C₄⋊C₄
prime radical	C₂⁴×C₁₃	C₂×Q₈
cosocle	C₂³×C₄×C₁₃	C₂³
hypercenter	#1728	D₁₃
hypocenter	C₁₃	#10822
LCS	G ▷ C₂×C₄×C₁₃ ▷ C₂×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂²×C₄ ◁ C₂²×C₄² ◁ #1728

Complements

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

N	H ≅ G/N	type
15. #11718150	C₄	semidirect
233. C₁₃	#10822	semidirect

Maximal subgroups

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

subgroup	index
C₂²×C₄×#1055	2
C₂×#11718150	2
#13114530	2
C₁₃×#1728	2
#10822	13