Subgroups — #15867662

Order 3328.

746 conjugacy classes of subgroups.

The subgroup lattice has 302 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#13115777	C₂	1	2	yes	maximal
2	#13115560	C₂	1	2	yes	maximal
3	#13115395	C₂	1	4	yes	maximal
4	#13114689	C₂	1	2	yes	maximal
5	#13114689	C₂	1	1	yes	maximal, characteristic
6	#13114585	C₂	1	2	yes	maximal
7	C₄×#88043	C₂	1	1	yes	maximal, characteristic
8	C₄²×Q₈×C₁₃	C₂	1	1	yes	maximal, characteristic
9	C₂×C₄×Q₈×C₁₃	C₄	1	2	yes	—
10	C₂×C₄×Q₈×C₁₃	C₂²	1	1	yes	characteristic
11	C₄×C₁₃×C₄⋊C₄	C₄	1	2	yes	—
12	C₄×C₁₃×C₄⋊C₄	C₄	1	2	yes	—
13	C₄×C₁₃×C₄⋊C₄	C₄	1	2	yes	—
14	C₄×C₁₃×C₄⋊C₄	C₂²	1	2	yes	—
15	C₄×C₁₃×C₄⋊C₄	C₂²	1	1	yes	characteristic
16	C₄³×C₁₃	C₂²	1	1	yes	characteristic
17	C₄³×C₁₃	C₂²	1	2	yes	—
18	C₂×C₄⋊#1055	C₂²	1	4	yes	—
19	C₂×C₄⋊#1055	C₂²	1	2	yes	—
20	C₂×#88043	C₂²	1	1	yes	characteristic
21	C₂×#88043	C₂²	1	2	yes	—
22	C₂×#88043	C₂²	1	2	yes	—
23	C₂×#88043	C₂²	1	1	yes	characteristic
24	C₂×C₄×#1055	C₂²	1	2	yes	—
25	#11718116	C₂²	1	2	yes	—
26	#11718116	C₂²	1	4	yes	—
27	#11718116	C₂²	1	4	yes	—
28	#11718116	C₂²	1	4	yes	—
29	C₂²×Q₈×C₁₃	C₂×C₄	1	1	yes	characteristic
30	C₄×Q₈×C₁₃	—	2	8	no	—
31	C₄×Q₈×C₁₃	—	2	4	no	—
32	C₂×C₁₃×C₄⋊C₄	C₂×C₄	1	1	yes	characteristic
33	C₂×C₁₃×C₄⋊C₄	C₂×C₄	1	2	yes	—
34	C₂×C₁₃×C₄⋊C₄	C₂×C₄	1	2	yes	—
35	C₂×C₁₃×C₄⋊C₄	C₂×C₄	1	2	yes	—
36	C₂×C₁₃×C₄⋊C₄	C₂×C₄	1	2	yes	—
37	C₂×C₄²×C₁₃	C₂×C₄	1	2	yes	—
38	C₂×C₄²×C₁₃	C₂×C₄	1	2	yes	—
39	C₂×C₄²×C₁₃	C₂×C₄	1	2	yes	—
40	C₂×C₄²×C₁₃	C₂×C₄	1	2	yes	—
41	C₂×C₄²×C₁₃	C₂×C₄	1	2	yes	—
42	C₂×C₄²×C₁₃	C₂×C₄	1	2	yes	—
43	C₂×C₄²×C₁₃	C₂³	1	1	yes	characteristic
44	C₂×C₄²×C₁₃	C₂³	1	2	yes	—
45	C₂×C₄²×C₁₃	C₂³	1	2	yes	—
46	C₂×C₄²×C₁₃	C₂³	1	1	yes	characteristic
47	C₂×C₄²×C₁₃	C₂³	1	1	yes	characteristic
48	C₂²×#1055	C₂³	1	4	yes	—
49	C₂²×#1055	C₂³	1	2	yes	—
50	C₂²×#1055	C₂³	1	2	yes	—
51	C₁₃×#95	C₂×C₄	1	1	yes	characteristic
52	C₁₃×#95	C₂×C₄	1	2	yes	—
53	C₁₃×#95	C₂×C₄	1	2	yes	—
54	C₁₃×#95	C₂×C₄	1	1	yes	characteristic
55	C₄⋊#1055	—	2	4	no	—
56	C₄⋊#1055	—	2	8	no	—
57	#88043	—	2	4	no	—
58	#88043	—	2	2	no	—
59	#88043	—	2	4	no	—
60	#88043	Q₈	1	4	yes	complemented
61	C₄×#1055	—	2	4	no	—
62	#21346	—	13	1	no	maximal, Sylow
63	C₂×Q₈×C₁₃	—	2	2	no	—
64	C₂×Q₈×C₁₃	—	2	4	no	—
65	C₂²×C₄×C₁₃	C₂²×C₄	1	2	yes	—
66	C₂²×C₄×C₁₃	C₂²×C₄	1	2	yes	—
67	C₂²×C₄×C₁₃	C₂²×C₄	1	1	yes	characteristic
68	C₂²×C₄×C₁₃	C₂²×C₄	1	1	yes	characteristic
69	C₂²×C₄×C₁₃	C₂²×C₄	1	2	yes	—
70	C₂²×C₄×C₁₃	C₂²×C₄	1	2	yes	—
71	C₂²×C₄×C₁₃	C₂²×C₄	1	2	yes	—
72	C₂²×C₄×C₁₃	C₂²×C₄	1	2	yes	—
73	C₂²×C₄×C₁₃	C₂⁴	1	1	yes	characteristic
74	C₁₃×C₄⋊C₄	—	2	4	no	—
75	C₁₃×C₄⋊C₄	—	2	2	no	—
76	C₁₃×C₄⋊C₄	—	2	4	no	—
77	C₁₃×C₄⋊C₄	—	2	4	no	—
78	C₁₃×C₄⋊C₄	M₁₆	1	8	yes	—
79	C₄²×C₁₃	—	2	4	no	—
80	C₄²×C₁₃	—	2	4	no	—
81	C₄²×C₁₃	—	2	4	no	—
82	C₄²×C₁₃	—	2	4	no	—
83	C₄²×C₁₃	—	2	2	no	—
84	C₄²×C₁₃	—	2	4	no	—
85	C₄²×C₁₃	—	4	2	no	—
86	C₄²×C₁₃	—	4	2	no	—
87	C₄²×C₁₃	—	4	2	no	—
88	C₄²×C₁₃	C₂×Q₈	1	2	yes	—
89	C₄²×C₁₃	#41	1	4	yes	—
90	C₄²×C₁₃	#41	1	2	yes	—
91	C₂×#1055	—	2	2	no	—
92	C₂×#1055	—	2	2	no	—
93	C₂×#1055	—	2	2	no	—
94	C₂×#1055	—	2	2	no	—
95	C₂×#1055	—	2	4	no	—
96	C₂×#1055	—	2	4	no	—
97	C₂×#1055	—	2	4	no	—
98	C₂×#1055	C₂×Q₈	1	4	yes	—
99	C₂×#1055	#41	1	4	yes	—
100	C₄²×Q₈	D₁₃	1	1	yes	characteristic
101	#1985	—	13	2	no	—
102	#1940	—	13	2	no	—
103	#1916	—	13	4	no	—
104	#1767	—	13	1	no	—
105	#1767	—	13	2	no	—
106	#1745	—	13	2	no	—
107	C₄×#97	—	13	1	no	—
108	C₂³×C₁₃	C₂³×C₄	1	1	yes	characteristic
109	Q₈×C₁₃	—	2	8	no	—
110	C₂×C₄×C₁₃	—	2	2	no	—
111	C₂×C₄×C₁₃	—	2	2	no	—
112	C₂×C₄×C₁₃	—	2	2	no	—
113	C₂×C₄×C₁₃	—	2	2	no	—
114	C₂×C₄×C₁₃	—	2	2	no	—
115	C₂×C₄×C₁₃	—	2	2	no	—
116	C₂×C₄×C₁₃	—	2	2	no	—
117	C₂×C₄×C₁₃	—	2	2	no	—
118	C₂×C₄×C₁₃	—	2	2	no	—
119	C₂×C₄×C₁₃	—	2	2	no	—
120	C₂×C₄×C₁₃	—	2	2	no	—
121	C₂×C₄×C₁₃	—	2	2	no	—
122	C₂×C₄×C₁₃	—	2	2	no	—
123	C₂×C₄×C₁₃	—	2	2	no	—
124	C₂×C₄×C₁₃	—	2	2	no	—
125	C₂×C₄×C₁₃	—	2	2	no	—
126	C₂×C₄×C₁₃	#117	1	2	yes	—
127	C₂×C₄×C₁₃	#117	1	2	yes	—
128	C₂×C₄×C₁₃	C₄×Q₈	1	4	yes	—
129	C₂×C₄×C₁₃	C₂×M₁₆	1	4	yes	—
130	C₂×C₄×C₁₃	C₂×M₁₆	1	4	yes	—
131	C₂×C₄×C₁₃	C₂×M₁₆	1	4	yes	—
132	C₂×C₄×C₁₃	C₂²×Q₈	1	1	yes	characteristic
133	C₂×C₄×C₁₃	C₂×#41	1	1	yes	characteristic
134	C₂×C₄×C₁₃	C₂×#41	1	2	yes	—
135	C₂×C₄×C₁₃	C₂×#41	1	2	yes	—
136	#1055	—	2	4	no	—
137	#1055	—	2	4	no	—
138	#1055	—	4	4	no	—
139	C₂×C₄×Q₈	Dic₁₃	1	2	yes	—
140	C₂×C₄×Q₈	C₂×D₁₃	1	1	yes	characteristic
141	C₂×C₄⋊C₈	—	13	4	no	—
142	C₂×C₄⋊C₈	—	13	2	no	—
143	C₂×#97	—	13	1	no	—
144	C₂×#97	—	13	2	no	—
145	C₂×#97	—	13	1	no	—
146	C₂×#97	—	13	2	no	—
147	C₂×C₄×C₈	—	13	2	no	—
148	C₄×C₄⋊C₄	Dic₁₃	1	2	yes	—
149	C₄×C₄⋊C₄	Dic₁₃	1	2	yes	—
150	C₄×C₄⋊C₄	Dic₁₃	1	2	yes	—
151	C₄×C₄⋊C₄	C₂×D₁₃	1	2	yes	—
152	C₄×C₄⋊C₄	C₂×D₁₃	1	1	yes	characteristic
153	C₄³	C₂×D₁₃	1	1	yes	characteristic
154	C₄³	C₂×D₁₃	1	2	yes	—
155	#336	—	13	2	no	—
156	#336	—	13	4	no	—
157	#336	—	13	4	no	—
158	#336	—	13	4	no	—
159	C₂²×C₁₃	C₂×#117	1	1	yes	characteristic
160	C₂²×C₁₃	C₂×C₄×Q₈	1	1	yes	characteristic
161	C₂²×C₁₃	C₄×#41	1	1	yes	characteristic
162	C₂²×C₁₃	C₂²×M₁₆	1	2	yes	—
163	C₂²×C₁₃	#568	1	2	yes	—
164	C₄×C₁₃	—	2	2	no	—
165	C₄×C₁₃	—	2	4	no	—
166	C₄×C₁₃	—	2	2	no	—
167	C₄×C₁₃	—	2	4	no	—
168	C₄×C₁₃	—	2	4	no	—
169	C₄×C₁₃	—	2	4	no	—
170	C₄×C₁₃	—	4	2	no	—
171	C₄×C₁₃	—	4	2	no	—
172	C₄×C₁₃	#529	1	2	yes	—
173	C₄×C₁₃	#533	1	2	yes	—
174	C₂²×Q₈	C₂×Dic₁₃	1	1	yes	characteristic
175	C₂²×C₈	—	13	2	no	—
176	C₂²×C₈	—	13	2	no	—
177	C₂²×C₈	—	13	4	no	—
178	C₄×Q₈	—	2	8	no	—
179	C₄×Q₈	—	2	4	no	—
180	C₂×C₄⋊C₄	C₂×Dic₁₃	1	2	yes	—
181	C₂×C₄⋊C₄	C₂×Dic₁₃	1	2	yes	—
182	C₂×C₄⋊C₄	C₂×Dic₁₃	1	1	yes	characteristic
183	C₂×C₄⋊C₄	C₂×Dic₁₃	1	2	yes	—
184	C₂×C₄⋊C₄	C₂×Dic₁₃	1	2	yes	—
185	C₂×C₄²	C₂×Dic₁₃	1	2	yes	—
186	C₂×C₄²	C₂×Dic₁₃	1	2	yes	—
187	C₂×C₄²	C₂×Dic₁₃	1	2	yes	—
188	C₂×C₄²	C₂×Dic₁₃	1	2	yes	—
189	C₂×C₄²	C₂×Dic₁₃	1	2	yes	—
190	C₂×C₄²	C₂×Dic₁₃	1	2	yes	—
191	C₂×C₄²	C₂²×D₁₃	1	2	yes	—
192	C₂×C₄²	C₂²×D₁₃	1	1	yes	characteristic
193	C₂×C₄²	C₂²×D₁₃	1	2	yes	—
194	C₂×C₄²	C₂²×D₁₃	1	1	yes	characteristic
195	C₂×C₄²	C₂²×D₁₃	1	1	yes	characteristic
196	C₄⋊C₈	—	26	8	no	—
197	C₄⋊C₈	—	26	4	no	—
198	#97	—	13	4	no	—
199	#97	—	26	4	no	—
200	#97	—	26	2	no	—
201	#97	—	26	4	no	—
202	C₄×C₈	—	26	4	no	—
203	#95	C₂×Dic₁₃	1	2	yes	—
204	#95	C₂×Dic₁₃	1	1	yes	characteristic
205	#95	C₂×Dic₁₃	1	2	yes	—
206	#95	C₂×Dic₁₃	1	1	yes	characteristic
207	C₂×C₁₃	#2295	1	1	yes	characteristic
208	C₂×C₁₃	#2920	1	1	yes	characteristic
209	C₂×C₁₃	#2923	1	1	yes	characteristic
210	C₂×C₁₃	Q₈×M₁₆	1	2	yes	—
211	C₂×C₁₃	#2970	1	2	yes	—
212	C₂×Q₈	—	2	4	no	—
213	C₂×Q₈	—	2	2	no	—
214	C₂²×C₄	C₂²×Dic₁₃	1	2	yes	—
215	C₂²×C₄	C₂²×Dic₁₃	1	2	yes	—
216	C₂²×C₄	C₂²×Dic₁₃	1	2	yes	—
217	C₂²×C₄	C₂²×Dic₁₃	1	2	yes	—
218	C₂²×C₄	C₂²×Dic₁₃	1	2	yes	—
219	C₂²×C₄	C₂²×Dic₁₃	1	1	yes	characteristic
220	C₂²×C₄	C₂²×Dic₁₃	1	1	yes	characteristic
221	C₂²×C₄	C₂²×Dic₁₃	1	2	yes	—
222	C₂²×C₄	C₂³×D₁₃	1	1	yes	characteristic
223	C₂×C₈	—	13	4	no	—
224	C₂×C₈	—	13	4	no	—
225	C₂×C₈	—	26	2	no	—
226	C₂×C₈	—	26	2	no	—
227	C₂×C₈	—	26	2	no	—
228	C₂×C₈	—	26	2	no	—
229	C₂×C₈	—	26	4	no	—
230	C₂×C₈	—	26	4	no	—
231	C₂×C₈	—	26	4	no	—
232	C₄⋊C₄	—	2	4	no	—
233	C₄⋊C₄	—	2	4	no	—
234	C₄⋊C₄	—	2	4	no	—
235	C₄⋊C₄	—	2	2	no	—
236	C₄⋊C₄	#6099	1	8	yes	—
237	C₄²	—	2	4	no	—
238	C₄²	—	2	4	no	—
239	C₄²	—	2	4	no	—
240	C₄²	—	2	4	no	—
241	C₄²	—	2	2	no	—
242	C₄²	—	2	4	no	—
243	C₄²	—	4	2	no	—
244	C₄²	—	4	2	no	—
245	C₄²	—	4	2	no	—
246	C₄²	#6127	1	4	yes	—
247	C₄²	Q₈×D₁₃	1	2	yes	—
248	C₄²	#6131	1	2	yes	—
249	C₁₃	#21346	1	1	yes	characteristic, complemented, Sylow
250	C₂³	C₂³×Dic₁₃	1	1	yes	characteristic
251	Q₈	—	2	8	no	—
252	C₂×C₄	—	2	2	no	—
253	C₂×C₄	—	2	2	no	—
254	C₂×C₄	—	2	2	no	—
255	C₂×C₄	—	2	2	no	—
256	C₂×C₄	—	2	2	no	—
257	C₂×C₄	—	2	2	no	—
258	C₂×C₄	—	2	2	no	—
259	C₂×C₄	—	2	2	no	—
260	C₂×C₄	—	2	2	no	—
261	C₂×C₄	—	2	2	no	—
262	C₂×C₄	—	2	2	no	—
263	C₂×C₄	—	2	2	no	—
264	C₂×C₄	—	2	2	no	—
265	C₂×C₄	—	2	2	no	—
266	C₂×C₄	—	2	2	no	—
267	C₂×C₄	—	2	2	no	—
268	C₂×C₄	C₂×#6099	1	4	yes	—
269	C₂×C₄	C₂×#6099	1	4	yes	—
270	C₂×C₄	C₂×#6099	1	4	yes	—
271	C₂×C₄	#88180	1	2	yes	—
272	C₂×C₄	#88180	1	2	yes	—
273	C₂×C₄	Q₈×Dic₁₃	1	4	yes	—
274	C₂×C₄	C₂×#6127	1	2	yes	—
275	C₂×C₄	C₂×Q₈×D₁₃	1	1	yes	characteristic
276	C₂×C₄	C₂×#6131	1	1	yes	characteristic
277	C₂×C₄	#41×D₁₃	1	2	yes	—
278	C₈	—	26	4	no	—
279	C₈	—	26	4	no	—
280	C₈	—	52	4	no	—
281	C₂²	C₂²×#6099	1	2	yes	—
282	C₂²	C₂×#88180	1	1	yes	characteristic
283	C₂²	C₂×Q₈×Dic₁₃	1	1	yes	characteristic
284	C₂²	#11719524	1	2	yes	—
285	C₂²	#41×Dic₁₃	1	1	yes	characteristic
286	C₄	—	2	4	no	—
287	C₄	—	2	4	no	—
288	C₄	—	2	4	no	—
289	C₄	—	2	2	no	—
290	C₄	—	2	2	no	—
291	C₄	—	2	4	no	—
292	C₄	—	4	2	no	—
293	C₄	—	4	2	no	—
294	C₄	#11719283	1	2	yes	—
295	C₄	#11719285	1	2	yes	—
296	C₂	#13117314	1	1	yes	characteristic
297	C₂	#13123469	1	1	yes	characteristic
298	C₂	#13123487	1	1	yes	characteristic
299	C₂	Q₈×#6099	1	2	yes	—
300	C₂	#13123850	1	2	yes	—

	subgroup	G/H
center, Frattini	C₂²×C₄	C₂³×D₁₃
commutator, socle, prime radical	C₂³×C₁₃	C₂³×C₄
Fitting	C₄²×Q₈×C₁₃	C₂
cosocle	C₂²×C₄×C₁₃	C₂⁴
hypercenter	C₄²×Q₈	D₁₃
hypocenter	C₁₃	#21346
LCS	G ▷ C₂³×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂²×C₄ ◁ C₄²×Q₈

Complements

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

N	H ≅ G/N	type
60. #88043	Q₈	semidirect
249. C₁₃	#21346	semidirect

Maximal subgroups

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

subgroup	index
#13115777	2
#13115560	2
#13115395	2
#13114689	2
#13114585	2
C₄×#88043	2
C₄²×Q₈×C₁₃	2
#21346	13