Subgroups — #16016295

Order 3328.

646 conjugacy classes of subgroups.

The subgroup lattice has 292 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×#88069	C₂	1	1	yes	maximal, characteristic
2	#13123705	C₂	1	1	yes	maximal, characteristic
3	#13116705	C₂	1	4	yes	maximal, complemented
4	C₄×#88069	C₂	1	4	yes	maximal, complemented
5	#13116618	C₂	1	1	yes	maximal, characteristic
6	#13116556	C₂	1	1	yes	maximal, characteristic
7	#13116555	C₂	1	1	yes	maximal, characteristic
8	#13116433	C₂	1	1	yes	maximal, characteristic
9	C₁₃×#2950	C₂	1	1	yes	maximal, characteristic
10	C₂²×C₁₃×Q₁₆	C₂²	1	1	yes	characteristic
11	C₁₃×#519	—	2	1	no	—
12	C₂²×Dic₅₂	C₂²	1	1	yes	characteristic
13	#11719221	—	2	1	no	—
14	C₁₃×#441	—	2	2	no	—
15	C₄×C₁₃×Q₁₆	C₂²	1	4	yes	—
16	C₁₃×#427	C₂²	1	1	yes	characteristic
17	C₁₃×#419	—	2	1	no	—
18	C₁₃×#405	C₂²	1	1	yes	characteristic
19	C₂×#88069	C₄	1	8	yes	complemented
20	C₂×#88069	C₂²	1	2	yes	—
21	C₂×#88069	C₂²	1	2	yes	—
22	#11718781	—	2	1	no	—
23	#11718767	C₂²	1	1	yes	characteristic
24	C₂×#88052	C₂²	1	1	yes	characteristic
25	C₂²×#6090	C₂²	1	1	yes	characteristic
26	#11718377	—	2	2	no	—
27	C₄×Dic₅₂	C₂²	1	4	yes	—
28	#11718156	C₂²	1	4	yes	—
29	#11718084	C₂²	1	4	yes	—
30	#11718082	C₂²	1	2	yes	—
31	#11718081	C₂²	1	2	yes	—
32	#11718053	C₂²	1	2	yes	—
33	C₄×#6090	C₂²	1	2	yes	—
34	C₂²×Q₈×C₁₃	—	2	1	no	—
35	C₂²×Dic₂₆	—	2	1	no	—
36	C₂×C₁₃×Q₁₆	—	2	1	no	—
37	C₂×C₁₃×Q₁₆	—	2	2	no	—
38	C₂×C₁₃×Q₁₆	C₂×C₄	1	4	yes	—
39	C₂×C₁₃×Q₁₆	C₂³	1	2	yes	—
40	C₂×C₁₃×M₁₆	D₄	1	1	yes	characteristic
41	C₂²×C₈×C₁₃	C₂³	1	1	yes	characteristic
42	C₄×Q₈×C₁₃	—	2	4	no	—
43	C₄×Q₈×C₁₃	—	4	2	no	—
44	C₁₃×#117	—	2	1	no	—
45	C₁₃×#117	—	4	2	no	—
46	C₁₃×#117	D₄	1	1	yes	characteristic
47	#88180	—	2	1	no	—
48	C₂×Dic₅₂	—	2	1	no	—
49	C₂×Dic₅₂	—	2	2	no	—
50	C₂×Dic₅₂	C₂×C₄	1	4	yes	—
51	C₂×Dic₅₂	C₂³	1	2	yes	—
52	#88141	—	4	2	no	—
53	#88131	—	4	2	no	—
54	C₄×Dic₂₆	—	2	4	no	—
55	C₁₃×#107	C₂³	1	2	yes	—
56	C₁₃×#106	—	2	1	no	—
57	C₁₃×#103	—	2	4	no	—
58	C₁₃×#97	D₄	1	2	yes	—
59	C₄×C₈×C₁₃	C₂³	1	2	yes	—
60	#88069	C₂×C₄	1	8	yes	—
61	#88069	C₂×C₄	1	8	yes	—
62	#88058	C₂³	1	2	yes	—
63	#88057	—	2	1	no	—
64	#88056	—	2	4	no	—
65	#88052	—	2	2	no	—
66	C₂×#6090	C₂×C₄	1	4	yes	—
67	C₂×#6090	C₂³	1	2	yes	—
68	C₂×#6090	C₂³	1	1	yes	characteristic
69	C₂×#6090	C₂³	1	1	yes	characteristic
70	#33458	—	13	1	no	maximal, Sylow
71	C₂×Q₈×C₁₃	—	2	4	no	—
72	C₂×Q₈×C₁₃	—	2	2	no	—
73	C₂×Q₈×C₁₃	—	4	1	no	—
74	C₂×Q₈×C₁₃	—	4	2	no	—
75	C₂²×C₄×C₁₃	—	4	1	no	—
76	C₂²×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
77	C₂²×Dic₁₃	—	4	1	no	—
78	C₂×Dic₂₆	—	2	4	no	—
79	C₂×Dic₂₆	—	2	2	no	—
80	C₂×Dic₂₆	—	4	1	no	—
81	C₂×Dic₂₆	—	4	2	no	—
82	C₁₃×Q₁₆	—	2	4	no	—
83	C₁₃×Q₁₆	—	2	2	no	—
84	C₁₃×Q₁₆	C₂²×C₄	1	4	yes	—
85	C₁₃×M₁₆	—	2	2	no	—
86	C₂×C₈×C₁₃	C₂²×C₄	1	2	yes	—
87	C₂×C₈×C₁₃	C₂×D₄	1	2	yes	—
88	C₂×C₈×C₁₃	C₂×D₄	1	1	yes	characteristic
89	C₂×C₈×C₁₃	#41	1	2	yes	—
90	C₂×C₈×C₁₃	C₂⁴	1	1	yes	characteristic
91	C₁₃×C₄⋊C₄	—	2	2	no	—
92	C₁₃×C₄⋊C₄	—	4	4	no	—
93	C₁₃×C₄⋊C₄	D₈	1	2	yes	—
94	C₁₃×C₂²⋊C₄	—	4	1	no	—
95	C₁₃×C₂²⋊C₄	D₈	1	2	yes	—
96	C₄²×C₁₃	—	4	1	no	—
97	C₄²×C₁₃	—	4	4	no	—
98	C₄²×C₁₃	C₂×D₄	1	2	yes	—
99	C₂²⋊Dic₁₃	—	4	1	no	—
100	C₄⋊Dic₁₃	—	2	2	no	—
101	#6101	—	4	4	no	—
102	C₄×Dic₁₃	—	4	4	no	—
103	C₄×Dic₁₃	—	4	1	no	—
104	Dic₅₂	—	2	4	no	—
105	Dic₅₂	—	2	2	no	—
106	Dic₅₂	C₂²×C₄	1	4	yes	—
107	#6090	—	2	2	no	—
108	#6090	C₂²×C₄	1	2	yes	—
109	#6090	C₂²×C₄	1	2	yes	—
110	C₂²×Q₃₂	—	13	1	no	—
111	#2950	—	13	1	no	—
112	#2950	D₁₃	1	1	yes	characteristic
113	#2176	—	13	4	no	—
114	C₄×Q₃₂	—	13	4	no	—
115	#2158	—	13	1	no	—
116	#2141	—	13	1	no	—
117	#2141	—	13	1	no	—
118	#2108	—	13	1	no	—
119	C₂³×C₁₃	C₂×D₈	1	1	yes	characteristic
120	Q₈×C₁₃	—	2	4	no	—
121	Q₈×C₁₃	—	4	2	no	—
122	Q₈×C₁₃	—	4	4	no	—
123	C₂×C₄×C₁₃	—	4	2	no	—
124	C₂×C₄×C₁₃	—	4	4	no	—
125	C₂×C₄×C₁₃	—	4	2	no	—
126	C₂×C₄×C₁₃	C₄×D₄	1	2	yes	—
127	C₂×C₄×C₁₃	C₂×D₈	1	4	yes	—
128	C₂×C₄×C₁₃	C₂×D₈	1	1	yes	characteristic
129	C₂×C₄×C₁₃	#135	1	2	yes	—
130	C₂×C₄×C₁₃	C₂²×D₄	1	1	yes	characteristic
131	C₂×Dic₁₃	—	4	2	no	—
132	C₂×Dic₁₃	—	4	4	no	—
133	C₂×Dic₁₃	—	4	2	no	—
134	Dic₂₆	—	2	4	no	—
135	Dic₂₆	—	4	2	no	—
136	Dic₂₆	—	4	4	no	—
137	C₈×C₁₃	—	2	2	no	—
138	C₈×C₁₃	C₄×D₄	1	2	yes	—
139	C₈×C₁₃	C₂³×C₄	1	1	yes	characteristic
140	C₈×C₁₃	C₂×#41	1	1	yes	characteristic
141	C₂²×Q₁₆	—	13	1	no	—
142	C₂²×Q₁₆	C₂×D₁₃	1	1	yes	characteristic
143	#519	—	2	1	no	—
144	#519	—	26	1	no	—
145	C₂×Q₃₂	—	13	8	no	—
146	C₂×Q₃₂	—	13	2	no	—
147	C₂×Q₃₂	—	13	2	no	—
148	C₂×M₃₂	—	13	1	no	—
149	C₂²×C₁₆	—	13	1	no	—
150	#441	—	2	2	no	—
151	#441	—	26	2	no	—
152	C₄×Q₁₆	—	13	4	no	—
153	C₄×Q₁₆	C₂×D₁₃	1	4	yes	—
154	#427	—	13	1	no	—
155	#427	C₂×D₁₃	1	1	yes	characteristic
156	#419	—	2	1	no	—
157	#419	—	26	1	no	—
158	#405	C₂×D₁₃	1	1	yes	characteristic
159	#367	—	13	2	no	—
160	#366	—	13	2	no	—
161	#358	—	13	4	no	—
162	#358	—	13	4	no	—
163	#346	—	13	2	no	—
164	C₄×C₁₆	—	13	2	no	—
165	C₂²×C₁₃	—	2	2	no	—
166	C₂²×C₁₃	C₄×D₈	1	2	yes	—
167	C₂²×C₁₃	C₂²×D₈	1	1	yes	characteristic
168	C₄×C₁₃	—	2	4	no	—
169	C₄×C₁₃	—	4	4	no	—
170	C₄×C₁₃	—	8	2	no	—
171	C₄×C₁₃	C₄×D₈	1	2	yes	—
172	C₄×C₁₃	C₂×C₄×D₄	1	1	yes	characteristic
173	C₄×C₁₃	C₂×#135	1	1	yes	characteristic
174	Dic₁₃	—	4	4	no	—
175	Dic₁₃	—	8	2	no	—
176	C₂²×Q₈	—	2	1	no	—
177	C₂²×Q₈	—	26	1	no	—
178	C₂×Q₁₆	—	2	2	no	—
179	C₂×Q₁₆	—	2	1	no	—
180	C₂×Q₁₆	—	13	4	no	—
181	C₂×Q₁₆	—	13	2	no	—
182	C₂×Q₁₆	—	26	2	no	—
183	C₂×Q₁₆	—	26	1	no	—
184	C₂×Q₁₆	C₄×D₁₃	1	4	yes	—
185	C₂×Q₁₆	C₂²×D₁₃	1	2	yes	—
186	C₂×M₁₆	C₂²⋊D₁₃	1	1	yes	characteristic
187	C₂²×C₈	C₂²×D₁₃	1	1	yes	characteristic
188	C₄×Q₈	—	2	4	no	—
189	C₄×Q₈	—	4	2	no	—
190	C₄×Q₈	—	26	4	no	—
191	C₄×Q₈	—	52	2	no	—
192	#117	—	2	1	no	—
193	#117	—	4	2	no	—
194	#117	—	26	1	no	—
195	#117	—	52	2	no	—
196	#117	C₂²⋊D₁₃	1	1	yes	characteristic
197	Q₃₂	—	13	8	no	—
198	Q₃₂	—	13	8	no	—
199	M₃₂	—	26	2	no	—
200	C₂×C₁₆	—	13	4	no	—
201	C₂×C₁₆	—	13	1	no	—
202	C₂×C₁₆	—	13	1	no	—
203	C₂×C₁₆	—	13	2	no	—
204	#107	—	13	2	no	—
205	#107	C₂²×D₁₃	1	2	yes	—
206	#106	—	2	1	no	—
207	#106	—	26	1	no	—
208	#103	—	2	4	no	—
209	#103	—	26	4	no	—
210	#97	C₂²⋊D₁₃	1	2	yes	—
211	C₄×C₈	C₂²×D₁₃	1	2	yes	—
212	C₂×C₁₃	—	2	2	no	—
213	C₂×C₁₃	C₂×C₄×D₈	1	1	yes	characteristic
214	C₂×C₁₃	#3417	1	2	yes	—
215	C₂×Q₈	—	2	4	no	—
216	C₂×Q₈	—	2	2	no	—
217	C₂×Q₈	—	4	1	no	—
218	C₂×Q₈	—	4	2	no	—
219	C₂×Q₈	—	26	4	no	—
220	C₂×Q₈	—	26	2	no	—
221	C₂×Q₈	—	52	1	no	—
222	C₂×Q₈	—	52	2	no	—
223	C₂²×C₄	—	4	1	no	—
224	C₂²×C₄	—	52	1	no	—
225	C₂²×C₄	C₂×C₂²⋊D₁₃	1	1	yes	characteristic
226	Q₁₆	—	2	4	no	—
227	Q₁₆	—	2	2	no	—
228	Q₁₆	—	13	4	no	—
229	Q₁₆	—	26	4	no	—
230	Q₁₆	—	26	2	no	—
231	Q₁₆	C₂×C₄×D₁₃	1	4	yes	—
232	M₁₆	—	2	2	no	—
233	C₂×C₈	C₂×C₄×D₁₃	1	2	yes	—
234	C₂×C₈	#6127	1	2	yes	—
235	C₂×C₈	C₂×C₂²⋊D₁₃	1	2	yes	—
236	C₂×C₈	C₂×C₂²⋊D₁₃	1	1	yes	characteristic
237	C₂×C₈	C₂³×D₁₃	1	1	yes	characteristic
238	C₄⋊C₄	—	2	2	no	—
239	C₄⋊C₄	—	4	4	no	—
240	C₄⋊C₄	—	26	2	no	—
241	C₄⋊C₄	—	52	4	no	—
242	C₄⋊C₄	#6104	1	2	yes	—
243	C₂²⋊C₄	—	4	1	no	—
244	C₂²⋊C₄	—	52	1	no	—
245	C₂²⋊C₄	#6104	1	2	yes	—
246	C₄²	—	4	1	no	—
247	C₄²	—	4	4	no	—
248	C₄²	—	52	1	no	—
249	C₄²	—	52	4	no	—
250	C₄²	C₂×C₂²⋊D₁₃	1	2	yes	—
251	C₁₆	—	13	2	no	—
252	C₁₆	—	13	2	no	—
253	C₁₆	—	26	2	no	—
254	C₁₃	#33458	1	1	yes	characteristic, complemented, Sylow
255	C₂³	C₂×#6104	1	1	yes	characteristic
256	Q₈	—	2	4	no	—
257	Q₈	—	4	4	no	—
258	Q₈	—	4	2	no	—
259	Q₈	—	26	4	no	—
260	Q₈	—	52	4	no	—
261	Q₈	—	52	2	no	—
262	C₂×C₄	—	4	4	no	—
263	C₂×C₄	—	4	2	no	—
264	C₂×C₄	—	4	2	no	—
265	C₂×C₄	—	52	4	no	—
266	C₂×C₄	—	52	2	no	—
267	C₂×C₄	—	52	2	no	—
268	C₂×C₄	C₄×C₂²⋊D₁₃	1	2	yes	—
269	C₂×C₄	C₂×#6104	1	4	yes	—
270	C₂×C₄	C₂×#6104	1	1	yes	characteristic
271	C₂×C₄	#88204	1	2	yes	—
272	C₂×C₄	C₂²×C₂²⋊D₁₃	1	1	yes	characteristic
273	C₈	—	2	2	no	—
274	C₈	C₄×C₂²⋊D₁₃	1	2	yes	—
275	C₈	C₂²×C₄×D₁₃	1	1	yes	characteristic
276	C₈	C₂×#6127	1	1	yes	characteristic
277	C₂²	—	2	2	no	—
278	C₂²	C₄×#6104	1	2	yes	—
279	C₂²	C₂²×#6104	1	1	yes	characteristic
280	C₄	—	2	4	no	—
281	C₄	—	4	4	no	—
282	C₄	—	8	2	no	—
283	C₄	—	52	4	no	—
284	C₄	—	104	2	no	—
285	C₄	C₄×#6104	1	2	yes	—
286	C₄	C₂×C₄×C₂²⋊D₁₃	1	1	yes	characteristic
287	C₄	C₂×#88204	1	1	yes	characteristic
288	C₂	—	2	2	no	—
289	C₂	C₂×C₄×#6104	1	1	yes	characteristic
290	C₂	#13127987	1	2	yes	—

	subgroup	G/H
center	C₂²	C₂²×#6104
commutator	C₈×C₁₃	C₂³×C₄
Frattini	C₂×C₈	C₂³×D₁₃
Fitting	C₁₃×#2950	C₂
socle	C₂²×C₁₃	C₂²×D₈
prime radical	C₂³×C₁₃	C₂×D₈
cosocle	C₂×C₈×C₁₃	C₂⁴
hypercenter	#2950	D₁₃
hypocenter	C₁₃	#33458
LCS	G ▷ C₈×C₁₃ ▷ C₄×C₁₃ ▷ C₂×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂² ◁ #117 ◁ #405 ◁ #2950

Complements

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

N	H ≅ G/N	type
3. #13116705	C₂	semidirect
4. C₄×#88069	C₂	semidirect
19. C₂×#88069	C₄	semidirect
254. C₁₃	#33458	semidirect

Maximal subgroups

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

subgroup	index
C₂²×#88069	2
#13123705	2
#13116705	2
C₄×#88069	2
#13116618	2
#13116556	2
#13116555	2
#13116433	2
C₁₃×#2950	2
#33458	13