Subgroups — #16807142

Order 3328.

690 conjugacy classes of subgroups.

The subgroup lattice has 328 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#11719096	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×#11719162	C₂	1	1	yes	maximal, characteristic, complemented
3	#13131221	C₂	1	4	yes	maximal, complemented
4	#13131214	C₂	1	4	yes	maximal, complemented
5	#13131170	C₂	1	1	yes	maximal, characteristic, complemented
6	#13131068	C₂	1	1	yes	maximal, characteristic, complemented
7	#13131066	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×#11718257	C₂	1	1	yes	maximal, characteristic, complemented
9	#13123544	C₂	1	1	yes	maximal, characteristic
10	#11719537	C₄	1	1	yes	characteristic
11	C₂×#88202	C₂²	1	1	yes	characteristic
12	C₂×#88161	C₂²	1	1	yes	characteristic
13	#11719162	—	2	4	no	—
14	C₂×#88106	C₂²	1	1	yes	characteristic
15	C₂×#88106	C₂²	1	2	yes	—
16	C₂×#88106	C₂²	1	1	yes	characteristic
17	C₂²×#6092	C₂²	1	2	yes	—
18	C₂²×#6092	C₂²	1	1	yes	characteristic
19	C₂²×#6092	C₂²	1	1	yes	characteristic
20	#11719096	—	2	4	no	—
21	C₂×#88098	C₂²	1	2	yes	—
22	C₂×D₁₃⋊C₁₆	C₂²	1	2	yes	—
23	C₁₃×#408	C₄	1	1	yes	characteristic
24	#11718779	C₄	1	1	yes	characteristic
25	#11718771	C₄	1	1	yes	characteristic
26	C₂×C₈×Dic₁₃	C₂²	1	1	yes	characteristic
27	#11718390	C₄	1	4	yes	—
28	#11718386	C₂²	1	4	yes	—
29	C₂²⋊#6092	C₂²	1	4	yes	—
30	#11718260	C₂²	1	4	yes	—
31	#11718259	C₂²	1	4	yes	—
32	#11718257	C₂²	1	2	yes	—
33	#11718257	C₂²	1	2	yes	—
34	C₂×C₁₃×#41	C₂×C₄	1	1	yes	characteristic
35	C₂×#6127	C₂×C₄	1	1	yes	characteristic
36	C₂×C₁₃×M₁₆	C₂×C₄	1	1	yes	characteristic
37	C₂²×C₈×C₁₃	C₂×C₄	1	1	yes	characteristic
38	#88202	—	2	2	no	—
39	#88202	—	2	2	no	—
40	#88201	C₈	1	1	yes	characteristic
41	#88198	C₈	1	1	yes	characteristic
42	#88194	C₈	1	1	yes	characteristic
43	#88193	C₈	1	4	yes	—
44	#88190	C₈	1	1	yes	characteristic
45	C₄×C₂²⋊D₁₃	C₂×C₄	1	4	yes	—
46	C₂×C₄×Dic₁₃	C₂×C₄	1	1	yes	characteristic
47	C₂×#6099	C₂×C₄	1	1	yes	characteristic
48	C₂×#6099	C₂×C₄	1	2	yes	—
49	C₂²×#1055	C₂³	1	2	yes	—
50	C₂²×#1055	C₂³	1	1	yes	characteristic
51	#88161	—	2	2	no	—
52	#88161	—	2	2	no	—
53	C₂×#6094	C₂×C₄	1	2	yes	—
54	C₂×C₈×D₁₃	C₂³	1	2	yes	—
55	#88106	—	2	2	no	—
56	#88106	—	2	2	no	—
57	#88106	—	2	4	no	—
58	C₂×#6092	—	2	2	no	—
59	C₂×#6092	—	2	4	no	—
60	C₂×#6092	D₄	1	4	yes	complemented
61	C₂×#6092	C₂³	1	2	yes	—
62	C₂×#6092	C₂³	1	2	yes	—
63	C₂×#6092	C₂³	1	2	yes	—
64	C₂×#6092	C₂³	1	1	yes	characteristic
65	C₂×#6092	C₂³	1	1	yes	characteristic
66	#88098	—	2	4	no	—
67	D₁₃⋊C₁₆	—	2	4	no	—
68	C₁₃×C₂²⋊C₈	C₂×C₄	1	4	yes	—
69	#88060	C₂×C₄	1	4	yes	—
70	#88054	C₂×C₄	1	4	yes	—
71	C₈×Dic₁₃	C₂×C₄	1	2	yes	—
72	C₈×Dic₁₃	C₂³	1	2	yes	—
73	#33434	—	13	1	no	maximal, Sylow
74	C₁₃×#41	—	2	2	no	—
75	C₁₃×#41	—	2	2	no	—
76	C₂×Q₈×C₁₃	C₂×C₈	1	1	yes	characteristic
77	C₂×D₄×C₁₃	C₂×C₈	1	1	yes	characteristic
78	C₂×D₄×C₁₃	C₂×C₈	1	2	yes	—
79	C₂²×C₄×C₁₃	C₂²×C₄	1	2	yes	—
80	C₂²×C₄×C₁₃	C₂²×C₄	1	1	yes	characteristic
81	C₂×C₂²⋊D₁₃	C₂×C₈	1	4	yes	—
82	C₂×C₂²⋊D₁₃	C₂×C₈	1	2	yes	—
83	C₂²×Dic₁₃	C₂×C₈	1	2	yes	—
84	#6127	—	2	2	no	—
85	#6127	—	2	2	no	—
86	C₂×D₅₂	C₂×C₈	1	1	yes	characteristic
87	C₂×C₄×D₁₃	C₂²×C₄	1	2	yes	—
88	C₂×Dic₂₆	C₂×C₈	1	1	yes	characteristic
89	C₁₃×M₁₆	—	2	2	no	—
90	C₂×C₈×C₁₃	—	2	2	no	—
91	C₂×C₈×C₁₃	C₂²×C₄	1	2	yes	—
92	C₂×C₈×C₁₃	C₂²×C₄	1	2	yes	—
93	C₂²⋊Dic₁₃	C₂×C₈	1	4	yes	—
94	#6103	C₂×C₈	1	4	yes	—
95	#6101	C₂×C₈	1	4	yes	—
96	C₄×Dic₁₃	C₂×C₈	1	1	yes	characteristic
97	C₄×Dic₁₃	C₂×C₈	1	1	yes	characteristic
98	C₄×Dic₁₃	C₂²×C₄	1	2	yes	—
99	#6099	—	2	2	no	—
100	#6099	—	2	4	no	—
101	C₂×#1055	—	2	2	no	—
102	C₂×#1055	—	2	2	no	—
103	C₂×#1055	C₂²×C₄	1	2	yes	—
104	C₂×#1055	C₂²×C₄	1	1	yes	characteristic
105	C₂×#1055	C₂×D₄	1	2	yes	—
106	C₂×#1055	#41	1	2	yes	—
107	C₂×#1055	C₂⁴	1	1	yes	characteristic
108	#6094	—	2	4	no	—
109	C₈×D₁₃	—	2	2	no	—
110	C₈×D₁₃	—	2	2	no	—
111	#6092	—	2	2	no	—
112	#6092	—	2	2	no	—
113	#6092	—	2	2	no	—
114	#6092	C₂×D₄	1	2	yes	—
115	#6092	C₂×D₄	1	2	yes	—
116	C₂×#504	—	13	1	no	—
117	C₂×#504	—	13	1	no	—
118	#2931	—	13	1	no	—
119	#2169	—	13	4	no	—
120	D₄×C₁₆	—	13	4	no	—
121	#2150	—	13	1	no	—
122	#2113	—	13	1	no	—
123	#2113	—	13	1	no	—
124	C₂×C₄×C₁₆	—	13	1	no	—
125	C₂³×C₁₃	C₂²×C₈	1	2	yes	—
126	C₂³×C₁₃	C₂²×C₈	1	1	yes	characteristic
127	C₂²×D₁₃	C₂²×C₈	1	2	yes	—
128	Q₈×C₁₃	—	2	2	no	—
129	D₄×C₁₃	—	2	4	no	—
130	D₄×C₁₃	—	2	2	no	—
131	C₂×C₄×C₁₃	—	2	2	no	—
132	C₂×C₄×C₁₃	—	2	2	no	—
133	C₂×C₄×C₁₃	C₄×D₄	1	2	yes	—
134	C₂×C₄×C₁₃	C₂²×C₈	1	2	yes	—
135	C₂×C₄×C₁₃	C₂²×C₈	1	1	yes	characteristic
136	C₂×C₄×C₁₃	#131	1	2	yes	—
137	C₂×C₄×C₁₃	C₂³×C₄	1	1	yes	characteristic
138	C₂²⋊D₁₃	—	2	4	no	—
139	C₂²⋊D₁₃	—	2	8	no	—
140	C₂×Dic₁₃	—	2	4	no	—
141	C₂×Dic₁₃	C₂²×C₈	1	2	yes	—
142	C₂×Dic₁₃	C₂²×C₈	1	2	yes	—
143	C₂×Dic₁₃	C₂²×C₈	1	2	yes	—
144	D₅₂	—	2	2	no	—
145	C₄×D₁₃	—	2	2	no	—
146	C₄×D₁₃	—	2	2	no	—
147	Dic₂₆	—	2	2	no	—
148	C₈×C₁₃	—	2	2	no	—
149	C₈×C₁₃	—	2	2	no	—
150	#1055	—	2	2	no	—
151	#1055	C₄×D₄	1	2	yes	—
152	#1055	C₂²×D₄	1	1	yes	characteristic
153	#1055	C₂×#41	1	1	yes	characteristic
154	C₂×#131	—	13	1	no	—
155	C₂×#131	—	13	1	no	—
156	#532	—	13	1	no	—
157	#504	—	26	4	no	—
158	#504	—	26	4	no	—
159	C₂×M₃₂	—	13	1	no	—
160	C₂×M₃₂	—	13	1	no	—
161	C₂×M₃₂	—	13	2	no	—
162	C₂×M₃₂	—	13	2	no	—
163	C₂²×C₁₆	—	13	1	no	—
164	C₂²×C₁₆	—	13	1	no	—
165	C₂²×C₁₆	—	13	2	no	—
166	C₂²×C₁₆	—	13	2	no	—
167	#436	—	13	4	no	—
168	C₈×D₄	—	13	4	no	—
169	#423	—	13	1	no	—
170	#408	—	13	1	no	—
171	#408	#261	1	1	yes	characteristic
172	C₂×C₄×C₈	—	13	1	no	—
173	C₄⋊C₁₆	—	13	4	no	—
174	C₂²⋊C₁₆	—	13	4	no	—
175	C₂²⋊C₁₆	—	13	4	no	—
176	C₄×C₁₆	—	13	2	no	—
177	C₄×C₁₆	—	13	2	no	—
178	C₂²×C₁₃	—	2	2	no	—
179	C₂²×C₁₃	—	2	4	no	—
180	C₂²×C₁₃	—	2	2	no	—
181	C₂²×C₁₃	C₈×D₄	1	2	yes	—
182	C₂²×C₁₃	C₂³×C₈	1	1	yes	characteristic
183	C₂×D₁₃	—	2	4	no	—
184	C₂×D₁₃	—	2	2	no	—
185	C₄×C₁₃	—	2	2	no	—
186	C₄×C₁₃	C₈×D₄	1	2	yes	—
187	C₄×C₁₃	C₂×C₄×D₄	1	1	yes	characteristic
188	C₄×C₁₃	C₂×#131	1	1	yes	characteristic
189	Dic₁₃	—	2	2	no	—
190	Dic₁₃	—	2	2	no	—
191	Dic₁₃	#504	1	4	yes	complemented
192	C₂×#41	—	13	1	no	—
193	C₂×#41	C₂×#261	1	1	yes	characteristic
194	#131	—	26	2	no	—
195	#131	—	26	2	no	—
196	#131	—	26	2	no	—
197	#131	—	26	2	no	—
198	C₂×M₁₆	—	13	1	no	—
199	C₂×M₁₆	—	13	2	no	—
200	C₂×M₁₆	—	13	2	no	—
201	C₂×M₁₆	C₂×#261	1	1	yes	characteristic
202	C₂²×C₈	—	13	1	no	—
203	C₂²×C₈	—	13	2	no	—
204	C₂²×C₈	—	13	2	no	—
205	C₂²×C₈	C₂×#261	1	1	yes	characteristic
206	C₄⋊Q₈	—	13	1	no	—
207	#127	—	13	1	no	—
208	#124	—	13	1	no	—
209	#124	—	13	1	no	—
210	#121	—	13	4	no	—
211	C₄×D₄	—	13	4	no	—
212	C₂×C₄²	—	13	1	no	—
213	M₃₂	—	26	4	no	—
214	M₃₂	—	26	4	no	—
215	M₃₂	—	26	2	no	—
216	M₃₂	—	26	2	no	—
217	C₂×C₁₆	—	13	4	no	—
218	C₂×C₁₆	—	13	1	no	—
219	C₂×C₁₆	—	13	1	no	—
220	C₂×C₁₆	—	13	2	no	—
221	C₂×C₁₆	—	13	2	no	—
222	C₂×C₁₆	—	13	2	no	—
223	C₂×C₁₆	—	26	4	no	—
224	C₂×C₁₆	—	26	4	no	—
225	C₂×C₁₆	—	26	2	no	—
226	C₄⋊C₈	—	13	4	no	—
227	C₂²⋊C₈	—	13	4	no	—
228	C₂²⋊C₈	C₂×#261	1	4	yes	—
229	C₄×C₈	—	13	2	no	—
230	C₄×C₈	—	13	2	no	—
231	C₂×C₁₃	—	2	2	no	—
232	C₂×C₁₃	—	4	2	no	—
233	C₂×C₁₃	C₂×C₈×D₄	1	1	yes	characteristic
234	C₂×C₁₃	C₂×#504	1	2	yes	—
235	D₁₃	—	4	2	no	—
236	#41	—	2	2	no	—
237	#41	—	2	2	no	—
238	#41	—	26	2	no	—
239	#41	—	26	2	no	—
240	C₂×Q₈	—	13	1	no	—
241	C₂×Q₈	D₁₃⋊C₈	1	1	yes	characteristic
242	C₂×D₄	—	13	4	no	—
243	C₂×D₄	—	13	2	no	—
244	C₂×D₄	—	13	1	no	—
245	C₂×D₄	D₁₃⋊C₈	1	2	yes	—
246	C₂×D₄	D₁₃⋊C₈	1	1	yes	characteristic
247	C₂²×C₄	—	13	2	no	—
248	C₂²×C₄	—	13	2	no	—
249	C₂²×C₄	C₂²×#261	1	2	yes	—
250	C₂²×C₄	C₂²×#261	1	1	yes	characteristic
251	M₁₆	—	2	2	no	—
252	M₁₆	—	26	4	no	—
253	M₁₆	—	26	4	no	—
254	M₁₆	—	26	2	no	—
255	C₂×C₈	—	2	2	no	—
256	C₂×C₈	—	13	2	no	—
257	C₂×C₈	—	13	2	no	—
258	C₂×C₈	—	13	1	no	—
259	C₂×C₈	—	13	1	no	—
260	C₂×C₈	—	13	2	no	—
261	C₂×C₈	—	26	2	no	—
262	C₂×C₈	—	26	2	no	—
263	C₂×C₈	—	26	2	no	—
264	C₂×C₈	—	26	2	no	—
265	C₂×C₈	C₂²×#261	1	2	yes	—
266	C₂×C₈	C₂²×#261	1	2	yes	—
267	C₄⋊C₄	—	13	4	no	—
268	C₂²⋊C₄	—	13	4	no	—
269	C₂²⋊C₄	—	13	4	no	—
270	C₄²	—	13	1	no	—
271	C₄²	—	13	1	no	—
272	C₄²	—	13	2	no	—
273	C₁₆	—	13	2	no	—
274	C₁₆	—	13	2	no	—
275	C₁₆	—	26	2	no	—
276	C₁₆	—	26	2	no	—
277	C₁₆	—	26	2	no	—
278	C₁₃	#33434	1	1	yes	characteristic, complemented, Sylow
279	C₂³	—	13	2	no	—
280	C₂³	C₂×D₁₃⋊C₈	1	2	yes	—
281	C₂³	C₂×D₁₃⋊C₈	1	1	yes	characteristic
282	Q₈	—	2	2	no	—
283	Q₈	—	26	2	no	—
284	D₄	—	2	4	no	—
285	D₄	—	2	2	no	—
286	D₄	—	26	8	no	—
287	D₄	—	26	4	no	—
288	D₄	—	26	2	no	—
289	C₂×C₄	—	2	2	no	—
290	C₂×C₄	—	2	2	no	—
291	C₂×C₄	—	13	2	no	—
292	C₂×C₄	—	13	2	no	—
293	C₂×C₄	—	13	2	no	—
294	C₂×C₄	—	26	4	no	—
295	C₂×C₄	—	26	2	no	—
296	C₂×C₄	—	26	2	no	—
297	C₂×C₄	C₂×D₁₃⋊C₈	1	2	yes	—
298	C₂×C₄	C₂×D₁₃⋊C₈	1	1	yes	characteristic
299	C₂×C₄	#88238	1	2	yes	—
300	C₂×C₄	D₄×#261	1	2	yes	—
301	C₂×C₄	C₂³×#261	1	1	yes	characteristic
302	C₈	—	2	2	no	—
303	C₈	—	2	2	no	—
304	C₈	—	13	1	no	—
305	C₈	—	13	1	no	—
306	C₈	—	13	2	no	—
307	C₈	—	26	2	no	—
308	C₂²	—	2	2	no	—
309	C₂²	—	2	2	no	—
310	C₂²	—	2	4	no	—
311	C₂²	—	26	2	no	—
312	C₂²	—	26	4	no	—
313	C₂²	#11719063	1	2	yes	—
314	C₂²	C₂²×D₁₃⋊C₈	1	1	yes	characteristic
315	C₄	—	2	2	no	—
316	C₄	—	13	4	no	—
317	C₄	—	26	2	no	—
318	C₄	—	26	2	no	—
319	C₄	#11719063	1	2	yes	—
320	C₄	C₂×#88238	1	1	yes	characteristic
321	C₄	C₂×D₄×#261	1	1	yes	characteristic
322	C₂	—	2	2	no	—
323	C₂	—	4	2	no	—
324	C₂	—	52	2	no	—
325	C₂	C₂×#11719063	1	1	yes	characteristic
326	C₂	D₁₃⋊#504	1	2	yes	—

	subgroup	G/H
center, Frattini	C₂×C₄	C₂³×#261
commutator, socle	C₂²×C₁₃	C₂³×C₈
Fitting	C₁₃×#408	C₄
prime radical	#11719537	C₄
cosocle	C₂×#1055	C₂⁴
hypercenter	#408	#261
hypocenter	C₁₃	#33434
LCS	G ▷ C₂²×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂×C₄ ◁ #408

Complements

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

N	H ≅ G/N	type
1. C₂×#11719096	C₂	semidirect
2. C₂×#11719162	C₂	semidirect
3. #13131221	C₂	semidirect
4. #13131214	C₂	semidirect
5. #13131170	C₂	semidirect
6. #13131068	C₂	semidirect
7. #13131066	C₂	semidirect
8. C₂×#11718257	C₂	semidirect
60. C₂×#6092	D₄	semidirect
191. Dic₁₃	#504	semidirect
278. C₁₃	#33434	semidirect

Maximal subgroups

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

subgroup	index
C₂×#11719096	2
C₂×#11719162	2
#13131221	2
#13131214	2
#13131170	2
#13131068	2
#13131066	2
C₂×#11718257	2
#13123544	2
#33434	13