Subgroups — #16807273

Order 3328.

558 conjugacy classes of subgroups.

The subgroup lattice has 306 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#13131251	C₂	1	2	yes	maximal, complemented
2	#13131220	C₂	1	2	yes	maximal, complemented
3	#13131216	C₂	1	4	yes	maximal, complemented
4	#13131193	C₂	1	2	yes	maximal, complemented
5	#13131193	C₂	1	1	yes	maximal, characteristic, complemented
6	C₄×#88106	C₂	1	2	yes	maximal, complemented
7	C₂×C₄×#6092	C₂	1	1	yes	maximal, characteristic, complemented
8	#41×#1055	C₂	1	1	yes	maximal, characteristic
9	C₄×C₁₃×#41	C₄	1	1	yes	characteristic
10	C₂×#88106	C₂²	1	4	yes	—
11	C₂²×#6092	C₂²	1	2	yes	—
12	Q₈×#1055	C₄	1	1	yes	characteristic
13	Q₈×#1055	C₂²	1	1	yes	characteristic
14	D₄×#1055	C₄	1	1	yes	characteristic
15	D₄×#1055	C₄	1	2	yes	—
16	D₄×#1055	C₂²	1	2	yes	—
17	D₄×#1055	C₂²	1	1	yes	characteristic
18	#11718586	C₄	1	1	yes	characteristic
19	#11718586	C₄	1	2	yes	—
20	C₂×C₄×#1055	C₂²	1	1	yes	characteristic
21	C₂×C₄×#1055	C₂²	1	2	yes	—
22	C₂²⋊#6092	C₂²	1	2	yes	—
23	C₂²⋊#6092	C₂²	1	4	yes	—
24	#11718231	C₂²	1	4	yes	—
25	C₄⋊#6092	C₂²	1	4	yes	—
26	C₄⋊#6092	C₂²	1	2	yes	—
27	C₄×#6092	C₂²	1	1	yes	characteristic
28	C₄×#6092	C₂²	1	1	yes	characteristic
29	C₄×#6092	C₂²	1	2	yes	—
30	C₄×#6092	C₂²	1	2	yes	—
31	C₂×C₁₃×#41	C₈	1	1	yes	characteristic
32	C₄×Q₈×C₁₃	C₈	1	1	yes	characteristic
33	C₄×Q₈×C₁₃	C₂×C₄	1	1	yes	characteristic
34	C₄×D₄×C₁₃	C₈	1	2	yes	—
35	C₄×D₄×C₁₃	C₈	1	1	yes	characteristic
36	C₄×D₄×C₁₃	C₂×C₄	1	2	yes	—
37	C₄×D₄×C₁₃	C₂×C₄	1	1	yes	characteristic
38	C₁₃×#117	C₈	1	2	yes	—
39	C₁₃×#117	C₈	1	1	yes	characteristic
40	C₂×C₄²×C₁₃	C₂×C₄	1	2	yes	—
41	C₂×C₄²×C₁₃	C₂×C₄	1	1	yes	characteristic
42	C₂²×#1055	C₂×C₄	1	1	yes	characteristic
43	C₂²×#1055	C₂×C₄	1	2	yes	—
44	C₂²×#1055	C₂³	1	1	yes	characteristic
45	C₂²×#1055	C₂³	1	2	yes	—
46	#88106	—	2	8	no	—
47	C₂×#6092	—	2	4	no	—
48	C₂×#6092	C₂³	1	2	yes	—
49	C₂×#6092	C₂³	1	4	yes	—
50	C₂×#6092	C₂³	1	2	yes	—
51	C₂²⋊#1055	C₂×C₄	1	2	yes	—
52	C₂²⋊#1055	C₂×C₄	1	2	yes	—
53	C₂²⋊#1055	C₂×C₄	1	2	yes	—
54	C₄⋊#1055	C₂×C₄	1	2	yes	—
55	C₄⋊#1055	C₂×C₄	1	1	yes	characteristic
56	C₄⋊#1055	C₂×C₄	1	2	yes	—
57	C₄⋊#1055	C₂×C₄	1	1	yes	characteristic
58	C₄×#1055	C₂×C₄	1	1	yes	characteristic
59	C₄×#1055	C₂×C₄	1	2	yes	—
60	C₄×#1055	C₂×C₄	1	1	yes	characteristic
61	C₄×#1055	C₂×C₄	1	2	yes	—
62	C₄×#1055	C₂³	1	1	yes	characteristic
63	C₄×#1055	C₂³	1	2	yes	—
64	C₄×#1055	C₂³	1	1	yes	characteristic
65	#33479	—	13	1	no	maximal, Sylow
66	C₁₃×#41	—	2	4	no	—
67	C₂×Q₈×C₁₃	C₂×C₈	1	1	yes	characteristic
68	C₂×D₄×C₁₃	C₂×C₈	1	2	yes	—
69	C₂×D₄×C₁₃	C₂×C₈	1	1	yes	characteristic
70	C₂²×C₄×C₁₃	C₂×C₈	1	2	yes	—
71	C₂²×C₄×C₁₃	C₂×C₈	1	2	yes	—
72	C₂²×C₄×C₁₃	C₂×C₈	1	1	yes	characteristic
73	C₂²×C₄×C₁₃	C₂×C₈	1	1	yes	characteristic
74	C₂²×C₄×C₁₃	C₂²×C₄	1	2	yes	—
75	C₂²×C₄×C₁₃	C₂²×C₄	1	1	yes	characteristic
76	C₁₃×C₄⋊C₄	C₂×C₈	1	2	yes	—
77	C₁₃×C₄⋊C₄	C₂×C₈	1	1	yes	characteristic
78	C₁₃×C₄⋊C₄	C₂×C₈	1	2	yes	—
79	C₁₃×C₄⋊C₄	C₂×C₈	1	1	yes	characteristic
80	C₁₃×C₂²⋊C₄	C₂×C₈	1	2	yes	—
81	C₁₃×C₂²⋊C₄	C₂×C₈	1	2	yes	—
82	C₁₃×C₂²⋊C₄	C₂×C₈	1	2	yes	—
83	C₄²×C₁₃	C₂×C₈	1	2	yes	—
84	C₄²×C₁₃	C₂×C₈	1	1	yes	characteristic
85	C₄²×C₁₃	C₂×C₈	1	2	yes	—
86	C₄²×C₁₃	C₂×C₈	1	1	yes	characteristic
87	C₄²×C₁₃	C₂²×C₄	1	2	yes	—
88	C₄²×C₁₃	C₂²×C₄	1	1	yes	characteristic
89	C₄²×C₁₃	C₂²×C₄	1	1	yes	characteristic
90	C₂×#1055	—	2	2	no	—
91	C₂×#1055	—	2	4	no	—
92	C₂×#1055	—	2	2	no	—
93	C₂×#1055	—	2	4	no	—
94	C₂×#1055	C₂²×C₄	1	1	yes	characteristic
95	C₂×#1055	C₂²×C₄	1	2	yes	—
96	C₂×#1055	C₂²×C₄	1	1	yes	characteristic
97	C₂×#1055	C₂²×C₄	1	2	yes	—
98	C₂×#1055	C₂²×C₄	1	1	yes	characteristic
99	C₂×#1055	C₂⁴	1	1	yes	characteristic
100	#6092	—	2	2	no	—
101	#6092	—	2	4	no	—
102	#6092	#41	1	4	yes	complemented
103	C₈×#41	—	13	1	no	—
104	#2183	—	13	2	no	—
105	#2169	—	13	2	no	—
106	#2168	—	13	4	no	—
107	#2162	—	13	2	no	—
108	#2162	—	13	1	no	—
109	C₄×M₃₂	—	13	2	no	—
110	C₂×C₄×C₁₆	—	13	1	no	—
111	C₂³×C₁₃	C₂²×C₈	1	2	yes	—
112	C₂³×C₁₃	C₂²×C₈	1	1	yes	characteristic
113	Q₈×C₁₃	—	2	2	no	—
114	D₄×C₁₃	—	2	4	no	—
115	D₄×C₁₃	—	2	2	no	—
116	C₂×C₄×C₁₃	—	2	1	no	—
117	C₂×C₄×C₁₃	—	2	2	no	—
118	C₂×C₄×C₁₃	—	2	1	no	—
119	C₂×C₄×C₁₃	—	2	2	no	—
120	C₂×C₄×C₁₃	—	2	4	no	—
121	C₂×C₄×C₁₃	—	2	2	no	—
122	C₂×C₄×C₁₃	—	2	4	no	—
123	C₂×C₄×C₁₃	M₃₂	1	4	yes	complemented
124	C₂×C₄×C₁₃	C₂²×C₈	1	2	yes	—
125	C₂×C₄×C₁₃	C₂²×C₈	1	1	yes	characteristic
126	C₂×C₄×C₁₃	C₂²×C₈	1	2	yes	—
127	C₂×C₄×C₁₃	C₂²×C₈	1	1	yes	characteristic
128	C₂×C₄×C₁₃	C₂²×C₈	1	2	yes	—
129	C₂×C₄×C₁₃	C₂²×C₈	1	1	yes	characteristic
130	C₂×C₄×C₁₃	C₂²×C₈	1	1	yes	characteristic
131	C₂×C₄×C₁₃	C₂²×C₈	1	1	yes	characteristic
132	C₂×C₄×C₁₃	C₂³×C₄	1	1	yes	characteristic
133	#1055	—	2	1	no	—
134	#1055	—	2	2	no	—
135	#1055	—	2	1	no	—
136	#1055	—	2	2	no	—
137	#1055	#131	1	2	yes	—
138	#1055	C₂×#41	1	2	yes	—
139	C₄×#41	#261	1	1	yes	characteristic
140	C₂×M₃₂	—	13	4	no	—
141	C₂²×C₁₆	—	13	2	no	—
142	C₈×Q₈	—	13	1	no	—
143	C₈×Q₈	—	13	1	no	—
144	C₈×D₄	—	13	1	no	—
145	C₈×D₄	—	13	2	no	—
146	C₈×D₄	—	13	2	no	—
147	C₈×D₄	—	13	1	no	—
148	#431	—	13	2	no	—
149	#431	—	13	1	no	—
150	C₂×C₄×C₈	—	13	1	no	—
151	C₂×C₄×C₈	—	13	2	no	—
152	C₄⋊C₁₆	—	13	4	no	—
153	C₄⋊C₁₆	—	13	2	no	—
154	C₂²⋊C₁₆	—	13	2	no	—
155	C₂²⋊C₁₆	—	13	4	no	—
156	#346	—	13	4	no	—
157	C₄×C₁₆	—	13	1	no	—
158	C₄×C₁₆	—	13	1	no	—
159	C₄×C₁₆	—	13	2	no	—
160	C₄×C₁₆	—	13	2	no	—
161	C₂²×C₁₃	—	2	1	no	—
162	C₂²×C₁₃	—	2	1	no	—
163	C₂²×C₁₃	—	2	2	no	—
164	C₂²×C₁₃	—	2	2	no	—
165	C₂²×C₁₃	—	2	2	no	—
166	C₂²×C₁₃	C₂×M₃₂	1	2	yes	—
167	C₂²×C₁₃	C₂³×C₈	1	1	yes	characteristic
168	C₄×C₁₃	—	2	2	no	—
169	C₄×C₁₃	—	2	1	no	—
170	C₄×C₁₃	—	2	2	no	—
171	C₄×C₁₃	—	2	1	no	—
172	C₄×C₁₃	—	2	2	no	—
173	C₄×C₁₃	C₂×M₃₂	1	2	yes	—
174	C₄×C₁₃	C₂×M₃₂	1	2	yes	—
175	C₄×C₁₃	#504	1	2	yes	—
176	C₄×C₁₃	C₄×#41	1	1	yes	characteristic
177	C₄×C₁₃	C₂×#131	1	1	yes	characteristic
178	C₂×#41	#1057	1	1	yes	characteristic
179	C₂²×C₈	—	13	1	no	—
180	C₂²×C₈	—	13	2	no	—
181	C₂²×C₈	—	13	1	no	—
182	C₂²×C₈	—	13	2	no	—
183	C₄×Q₈	#1057	1	1	yes	characteristic
184	C₄×Q₈	C₂×#261	1	1	yes	characteristic
185	C₄×D₄	#1057	1	2	yes	—
186	C₄×D₄	#1057	1	1	yes	characteristic
187	C₄×D₄	C₂×#261	1	2	yes	—
188	C₄×D₄	C₂×#261	1	1	yes	characteristic
189	#117	#1057	1	2	yes	—
190	#117	#1057	1	1	yes	characteristic
191	C₂×C₄²	C₂×#261	1	2	yes	—
192	C₂×C₄²	C₂×#261	1	1	yes	characteristic
193	M₃₂	—	26	8	no	—
194	C₂×C₁₆	—	13	2	no	—
195	C₂×C₁₆	—	13	2	no	—
196	C₂×C₁₆	—	13	4	no	—
197	C₂×C₁₆	—	26	4	no	—
198	C₄⋊C₈	—	13	2	no	—
199	C₄⋊C₈	—	13	1	no	—
200	C₄⋊C₈	—	13	2	no	—
201	C₄⋊C₈	—	13	1	no	—
202	C₂²⋊C₈	—	13	2	no	—
203	C₂²⋊C₈	—	13	2	no	—
204	C₂²⋊C₈	—	13	2	no	—
205	C₄×C₈	—	13	1	no	—
206	C₄×C₈	—	13	2	no	—
207	C₄×C₈	—	13	1	no	—
208	C₄×C₈	—	13	2	no	—
209	C₄×C₈	—	13	1	no	—
210	C₄×C₈	—	13	1	no	—
211	C₄×C₈	—	13	2	no	—
212	C₂×C₁₃	—	2	2	no	—
213	C₂×C₁₃	—	4	2	no	—
214	C₂×C₁₃	C₈×#41	1	1	yes	characteristic
215	C₂×C₁₃	C₂²×M₃₂	1	1	yes	characteristic
216	C₂×C₁₃	C₂×#504	1	1	yes	characteristic
217	#41	—	2	4	no	—
218	C₂×Q₈	C₂×#1057	1	1	yes	characteristic
219	C₂×D₄	C₂×#1057	1	2	yes	—
220	C₂×D₄	C₂×#1057	1	1	yes	characteristic
221	C₂²×C₄	C₂×#1057	1	2	yes	—
222	C₂²×C₄	C₂×#1057	1	2	yes	—
223	C₂²×C₄	C₂×#1057	1	1	yes	characteristic
224	C₂²×C₄	C₂×#1057	1	1	yes	characteristic
225	C₂²×C₄	C₂²×#261	1	2	yes	—
226	C₂²×C₄	C₂²×#261	1	1	yes	characteristic
227	C₂×C₈	—	13	1	no	—
228	C₂×C₈	—	13	1	no	—
229	C₂×C₈	—	13	1	no	—
230	C₂×C₈	—	13	2	no	—
231	C₂×C₈	—	13	1	no	—
232	C₂×C₈	—	13	2	no	—
233	C₂×C₈	—	26	2	no	—
234	C₂×C₈	—	26	4	no	—
235	C₂×C₈	—	26	2	no	—
236	C₂×C₈	—	26	4	no	—
237	C₄⋊C₄	C₂×#1057	1	2	yes	—
238	C₄⋊C₄	C₂×#1057	1	1	yes	characteristic
239	C₄⋊C₄	C₂×#1057	1	2	yes	—
240	C₄⋊C₄	C₂×#1057	1	1	yes	characteristic
241	C₂²⋊C₄	C₂×#1057	1	2	yes	—
242	C₂²⋊C₄	C₂×#1057	1	2	yes	—
243	C₂²⋊C₄	C₂×#1057	1	2	yes	—
244	C₄²	C₂×#1057	1	2	yes	—
245	C₄²	C₂×#1057	1	1	yes	characteristic
246	C₄²	C₂×#1057	1	2	yes	—
247	C₄²	C₂×#1057	1	1	yes	characteristic
248	C₄²	C₂²×#261	1	2	yes	—
249	C₄²	C₂²×#261	1	1	yes	characteristic
250	C₄²	C₂²×#261	1	1	yes	characteristic
251	C₁₆	—	13	4	no	—
252	C₁₆	—	26	2	no	—
253	C₁₆	—	26	4	no	—
254	C₁₃	#33479	1	1	yes	characteristic, complemented, Sylow
255	C₂³	C₂²×#1057	1	2	yes	—
256	C₂³	C₂²×#1057	1	1	yes	characteristic
257	Q₈	—	2	2	no	—
258	D₄	—	2	4	no	—
259	D₄	—	2	2	no	—
260	C₂×C₄	—	2	4	no	—
261	C₂×C₄	—	2	1	no	—
262	C₂×C₄	—	2	2	no	—
263	C₂×C₄	—	2	1	no	—
264	C₂×C₄	—	2	2	no	—
265	C₂×C₄	—	2	2	no	—
266	C₂×C₄	—	2	4	no	—
267	C₂×C₄	#88106	1	4	yes	complemented
268	C₂×C₄	C₂²×#1057	1	2	yes	—
269	C₂×C₄	C₂²×#1057	1	1	yes	characteristic
270	C₂×C₄	C₂²×#1057	1	2	yes	—
271	C₂×C₄	C₂²×#1057	1	1	yes	characteristic
272	C₂×C₄	C₂²×#1057	1	2	yes	—
273	C₂×C₄	C₂²×#1057	1	1	yes	characteristic
274	C₂×C₄	C₂²×#1057	1	1	yes	characteristic
275	C₂×C₄	C₂²×#1057	1	1	yes	characteristic
276	C₂×C₄	C₂³×#261	1	1	yes	characteristic
277	C₈	—	13	2	no	—
278	C₈	—	13	2	no	—
279	C₈	—	26	1	no	—
280	C₈	—	26	2	no	—
281	C₈	—	26	1	no	—
282	C₈	—	26	2	no	—
283	C₂²	—	2	1	no	—
284	C₂²	—	2	1	no	—
285	C₂²	—	2	2	no	—
286	C₂²	—	2	2	no	—
287	C₂²	—	2	2	no	—
288	C₂²	C₂×#88106	1	2	yes	—
289	C₂²	C₂³×#1057	1	1	yes	characteristic
290	C₄	—	2	2	no	—
291	C₄	—	2	1	no	—
292	C₄	—	2	2	no	—
293	C₄	—	2	1	no	—
294	C₄	—	2	2	no	—
295	C₄	C₂×#88106	1	2	yes	—
296	C₄	C₂×#88106	1	2	yes	—
297	C₄	#11719162	1	2	yes	—
298	C₄	D₁₃⋊#131	1	1	yes	characteristic
299	C₄	#41×#261	1	1	yes	characteristic
300	C₂	—	2	2	no	—
301	C₂	—	4	2	no	—
302	C₂	#41×#1057	1	1	yes	characteristic
303	C₂	C₂²×#88106	1	1	yes	characteristic
304	C₂	C₂×#11719162	1	1	yes	characteristic

	subgroup	G/H
center	C₄²	C₂²×#261
commutator, socle	C₂²×C₁₃	C₂³×C₈
Frattini	C₂×C₄	C₂³×#261
Fitting	C₄×C₁₃×#41	C₄
prime radical	C₂×C₁₃×#41	C₈
cosocle	C₂×#1055	C₂⁴
hypercenter	C₄×#41	#261
hypocenter	C₁₃	#33479
LCS	G ▷ C₂²×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₄² ◁ C₄×#41

Complements

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

N	H ≅ G/N	type
1. #13131251	C₂	semidirect
2. #13131220	C₂	semidirect
3. #13131216	C₂	semidirect
4. #13131193	C₂	semidirect
5. #13131193	C₂	semidirect
6. C₄×#88106	C₂	semidirect
7. C₂×C₄×#6092	C₂	semidirect
102. #6092	#41	semidirect
123. C₂×C₄×C₁₃	M₃₂	semidirect
254. C₁₃	#33479	semidirect
267. C₂×C₄	#88106	semidirect

Maximal subgroups

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

subgroup	index
#13131251	2
#13131220	2
#13131216	2
#13131193	2
C₄×#88106	2
C₂×C₄×#6092	2
#41×#1055	2
#33479	13