Subgroups — C₂×#13131184 #16807023

Order 3328.

658 conjugacy classes of subgroups.

The subgroup lattice has 336 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×#88103	C₂	1	1	yes	maximal, characteristic
2	C₂×#11719092	C₂	1	1	yes	maximal, characteristic
3	#13131184	C₂	1	8	yes	maximal, direct product factor
4	C₂×#11718220	C₂	1	2	yes	maximal, complemented
5	C₂×#11718219	C₂	1	2	yes	maximal, complemented
6	C₂×D₁₃×M₃₂	C₂	1	1	yes	maximal, characteristic
7	C₂×#88237	—	2	1	no	—
8	C₂²×#6118	—	2	1	no	—
9	C₂²×C₈×D₁₃	C₂²	1	1	yes	characteristic
10	C₂×#88103	—	2	1	no	—
11	C₂×#88103	—	2	2	no	—
12	C₂×#88103	C₂²	1	2	yes	—
13	C₂×#88103	C₂²	1	4	yes	—
14	#11719092	—	2	2	no	—
15	#11719092	C₂²	1	4	yes	—
16	C₂×#88102	—	2	1	no	—
17	C₂×#88101	C₂²	1	1	yes	characteristic
18	C₂×#88101	C₂²	1	1	yes	characteristic
19	C₂×C₁₃×M₃₂	C₄	1	1	yes	characteristic, complemented
20	C₂×#88052	C₄	1	1	yes	characteristic, complemented
21	D₁₃×M₃₂	C₄	1	4	yes	complemented
22	D₁₃×M₃₂	C₂²	1	4	yes	—
23	C₂×#88038	C₂²	1	2	yes	—
24	C₂×C₁₆×D₁₃	C₄	1	2	yes	complemented
25	#11718220	C₂²	1	8	yes	complemented
26	#11718219	C₂²	1	8	yes	complemented
27	C₂²×C₄×D₁₃	D₄	1	1	yes	characteristic
28	C₂×C₂²⋊#261	—	4	1	no	—
29	C₂²×#1057	—	4	1	no	—
30	#88237	—	2	4	no	—
31	#88237	—	4	2	no	—
32	C₂×C₄⋊#261	—	2	1	no	—
33	C₂×C₄⋊#261	—	2	1	no	—
34	C₂×C₄×#261	—	4	1	no	—
35	C₂×#6118	—	2	2	no	—
36	C₂×#6118	—	2	4	no	—
37	C₂×#6118	—	4	1	no	—
38	C₂×#6118	—	4	2	no	—
39	C₂²×C₈×C₁₃	C₂×C₄	1	1	yes	characteristic
40	C₂²×#1055	Q₈	1	1	yes	characteristic
41	C₂×C₈×D₁₃	C₂×C₄	1	1	yes	characteristic
42	C₂×C₈×D₁₃	C₂×C₄	1	2	yes	—
43	C₂×C₈×D₁₃	D₄	1	1	yes	characteristic
44	C₂×C₈×D₁₃	D₄	1	2	yes	—
45	C₂×C₈×D₁₃	Q₈	1	2	yes	—
46	C₂×C₈×D₁₃	Q₈	1	1	yes	characteristic
47	C₂×C₈×D₁₃	C₂³	1	1	yes	characteristic
48	C₂×C₈×D₁₃	C₂³	1	2	yes	—
49	#88103	—	2	2	no	—
50	#88103	—	2	4	no	—
51	#88103	C₂³	1	4	yes	—
52	#88102	—	2	4	no	—
53	#88101	—	2	1	no	—
54	#88101	—	2	1	no	—
55	#88101	C₂³	1	2	yes	—
56	#88101	C₂³	1	2	yes	—
57	C₁₃×M₃₂	C₂×C₄	1	4	yes	complemented
58	C₂×C₁₃×C₁₆	C₂×C₄	1	2	yes	—
59	#88052	C₂×C₄	1	4	yes	complemented
60	C₂×#6090	C₂×C₄	1	2	yes	—
61	#88038	C₂×C₄	1	4	yes	—
62	#88038	C₂³	1	4	yes	—
63	C₁₆×D₁₃	C₂×C₄	1	8	yes	complemented
64	C₂×#2155	—	13	1	no	maximal, Sylow
65	C₂³×D₁₃	SD₁₆	1	1	yes	characteristic
66	C₂²×#261	—	4	1	no	—
67	C₂²×#261	—	4	1	no	—
68	C₂²×C₄×C₁₃	C₄⋊C₄	1	1	yes	characteristic
69	C₂²×Dic₁₃	SD₁₆	1	1	yes	characteristic
70	C₂×C₄×D₁₃	C₄⋊C₄	1	1	yes	characteristic
71	C₂×C₄×D₁₃	C₄⋊C₄	1	2	yes	—
72	C₂×C₄×D₁₃	SD₁₆	1	1	yes	characteristic
73	C₂×C₄×D₁₃	SD₁₆	1	2	yes	—
74	C₂×C₄×D₁₃	SD₁₆	1	1	yes	characteristic
75	C₂×C₄×D₁₃	SD₁₆	1	2	yes	—
76	C₂×C₄×D₁₃	C₂×D₄	1	1	yes	characteristic
77	C₂×C₄×D₁₃	C₂×D₄	1	2	yes	—
78	C₂²⋊#261	—	4	4	no	—
79	C₂×#1057	—	4	4	no	—
80	C₂×#1057	—	4	2	no	—
81	C₄⋊#261	—	2	2	no	—
82	C₄⋊#261	—	2	2	no	—
83	C₄⋊#261	—	4	1	no	—
84	C₄⋊#261	—	4	1	no	—
85	C₄×#261	—	4	4	no	—
86	#6118	—	2	4	no	—
87	#6118	—	4	2	no	—
88	#6118	—	4	4	no	—
89	C₂×C₈×C₁₃	C₄⋊C₄	1	1	yes	characteristic
90	C₂×C₈×C₁₃	C₄⋊C₄	1	2	yes	—
91	C₂×C₈×C₁₃	C₂²×C₄	1	1	yes	characteristic
92	C₂×C₈×C₁₃	C₂²×C₄	1	2	yes	—
93	C₂×#1055	C₄⋊C₄	1	1	yes	characteristic
94	C₂×#1055	C₄⋊C₄	1	2	yes	—
95	C₂×#1055	C₂×Q₈	1	1	yes	characteristic
96	C₂×#1055	C₂×Q₈	1	2	yes	—
97	C₈×D₁₃	C₄⋊C₄	1	2	yes	—
98	C₈×D₁₃	C₄⋊C₄	1	2	yes	—
99	C₈×D₁₃	C₄⋊C₄	1	2	yes	—
100	C₈×D₁₃	C₂²×C₄	1	2	yes	—
101	C₈×D₁₃	C₂²×C₄	1	1	yes	characteristic
102	C₈×D₁₃	C₂×D₄	1	1	yes	characteristic
103	C₈×D₁₃	C₂×D₄	1	2	yes	—
104	C₈×D₁₃	C₂×Q₈	1	1	yes	characteristic
105	C₈×D₁₃	C₂×Q₈	1	2	yes	—
106	C₈×D₁₃	C₂⁴	1	1	yes	characteristic
107	C₁₃×C₁₆	C₂²×C₄	1	4	yes	—
108	#6090	C₂²×C₄	1	4	yes	—
109	C₂²×M₃₂	—	13	1	no	—
110	C₂²×#108	—	13	1	no	—
111	C₂×#427	—	13	1	no	—
112	#2155	—	13	8	no	—
113	C₂×#365	—	13	2	no	—
114	C₂×#365	—	13	2	no	—
115	C₂³×C₁₃	#106	1	1	yes	characteristic
116	C₂²×D₁₃	—	4	2	no	—
117	C₂²×D₁₃	#106	1	1	yes	characteristic
118	C₂²×D₁₃	#106	1	2	yes	—
119	C₂²×D₁₃	C₂×SD₁₆	1	1	yes	characteristic
120	C₂²×D₁₃	C₂×SD₁₆	1	2	yes	—
121	C₂×#261	—	4	2	no	—
122	C₂×#261	—	4	2	no	—
123	C₂×#261	—	8	2	no	—
124	C₂×#261	—	8	2	no	—
125	C₂×C₄×C₁₃	#106	1	2	yes	—
126	C₂×C₄×C₁₃	#106	1	1	yes	characteristic
127	C₂×C₄×C₁₃	C₂×C₄⋊C₄	1	2	yes	—
128	C₂×C₄×C₁₃	C₂×C₄⋊C₄	1	1	yes	characteristic
129	C₂×Dic₁₃	#106	1	2	yes	—
130	C₂×Dic₁₃	#106	1	1	yes	characteristic
131	C₂×Dic₁₃	C₂×SD₁₆	1	2	yes	—
132	C₂×Dic₁₃	C₂×SD₁₆	1	1	yes	characteristic
133	C₄×D₁₃	#106	1	2	yes	—
134	C₄×D₁₃	#106	1	2	yes	—
135	C₄×D₁₃	#106	1	2	yes	—
136	C₄×D₁₃	C₂×C₄⋊C₄	1	2	yes	—
137	C₄×D₁₃	C₂×C₄⋊C₄	1	1	yes	characteristic
138	C₄×D₁₃	C₂×SD₁₆	1	1	yes	characteristic
139	C₄×D₁₃	C₂×SD₁₆	1	2	yes	—
140	C₄×D₁₃	C₂×SD₁₆	1	2	yes	—
141	C₄×D₁₃	C₂×SD₁₆	1	1	yes	characteristic
142	C₄×D₁₃	C₂²×D₄	1	1	yes	characteristic
143	#1057	—	4	4	no	—
144	C₈×C₁₃	C₂×C₄⋊C₄	1	1	yes	characteristic
145	C₈×C₁₃	C₂×C₄⋊C₄	1	2	yes	—
146	C₈×C₁₃	C₂³×C₄	1	1	yes	characteristic
147	#1055	C₂×C₄⋊C₄	1	2	yes	—
148	#1055	C₂×C₄⋊C₄	1	1	yes	characteristic
149	#1055	C₂²×Q₈	1	1	yes	characteristic
150	C₂²×M₁₆	—	26	1	no	—
151	C₂³×C₈	—	13	1	no	—
152	C₂×#117	—	26	1	no	—
153	C₂×M₃₂	—	13	1	no	—
154	C₂×M₃₂	—	13	4	no	—
155	C₂×M₃₂	—	13	4	no	—
156	C₂×M₃₂	—	13	2	no	—
157	C₂×M₃₂	#261	1	1	yes	characteristic, complemented
158	C₂²×C₁₆	—	13	2	no	—
159	C₂×#108	—	13	2	no	—
160	C₂×#108	—	13	4	no	—
161	C₂×#108	—	26	1	no	—
162	C₂×#108	—	26	2	no	—
163	#427	—	13	4	no	—
164	#427	—	26	2	no	—
165	C₂×#107	—	26	1	no	—
166	C₂×#106	—	13	1	no	—
167	C₂×#106	—	13	1	no	—
168	#365	—	13	8	no	—
169	#365	—	13	8	no	—
170	C₂²×C₁₃	—	2	2	no	—
171	C₂²×C₁₃	C₂×#106	1	2	yes	—
172	C₂²×C₁₃	C₂×#106	1	1	yes	characteristic
173	C₂×D₁₃	—	2	2	no	—
174	C₂×D₁₃	—	2	2	no	—
175	C₂×D₁₃	—	4	2	no	—
176	C₂×D₁₃	—	4	2	no	—
177	C₂×D₁₃	C₂×#106	1	2	yes	—
178	C₂×D₁₃	C₂×#106	1	1	yes	characteristic
179	C₂×D₁₃	C₂²×SD₁₆	1	1	yes	characteristic
180	#261	—	8	2	no	—
181	#261	—	8	2	no	—
182	C₄×C₁₃	C₂×#106	1	2	yes	—
183	C₄×C₁₃	C₂×#106	1	1	yes	characteristic
184	C₄×C₁₃	C₂²×C₄⋊C₄	1	1	yes	characteristic
185	Dic₁₃	C₂×#106	1	2	yes	—
186	Dic₁₃	C₂×#106	1	1	yes	characteristic
187	Dic₁₃	C₂²×SD₁₆	1	1	yes	characteristic
188	C₂³×C₄	—	13	1	no	—
189	C₂×M₁₆	—	26	2	no	—
190	C₂×M₁₆	—	26	4	no	—
191	C₂×M₁₆	—	52	2	no	—
192	C₂×M₁₆	—	52	1	no	—
193	C₂²×C₈	—	13	1	no	—
194	C₂²×C₈	—	13	1	no	—
195	C₂²×C₈	—	13	2	no	—
196	C₂²×C₈	—	13	2	no	—
197	C₂²×C₈	—	13	1	no	—
198	C₂²×C₈	—	13	1	no	—
199	C₂²×C₈	—	13	2	no	—
200	C₂²×C₈	—	13	1	no	—
201	C₂²×C₈	—	13	2	no	—
202	C₂²×C₈	—	52	1	no	—
203	C₂²×C₈	C₂×#261	1	1	yes	characteristic
204	#117	—	26	4	no	—
205	#117	—	52	2	no	—
206	C₂×C₄⋊C₄	—	26	1	no	—
207	C₂×C₄⋊C₄	—	26	1	no	—
208	C₂×C₂²⋊C₄	—	52	1	no	—
209	C₂×C₄²	—	52	1	no	—
210	M₃₂	—	13	4	no	—
211	M₃₂	—	13	4	no	—
212	M₃₂	—	13	4	no	—
213	M₃₂	C₂×#261	1	4	yes	complemented
214	C₂×C₁₆	—	13	2	no	—
215	C₂×C₁₆	—	13	8	no	—
216	C₂×C₁₆	C₂×#261	1	2	yes	—
217	#108	—	13	4	no	—
218	#108	—	26	2	no	—
219	#108	—	26	4	no	—
220	#107	—	26	4	no	—
221	#106	—	13	2	no	—
222	#106	—	13	2	no	—
223	#106	—	26	1	no	—
224	#106	—	26	1	no	—
225	C₂×C₁₃	—	2	2	no	—
226	C₂×C₁₃	#2155	1	2	yes	complemented
227	C₂×C₁₃	C₂²×#106	1	1	yes	characteristic
228	D₁₃	—	2	1	no	—
229	D₁₃	—	2	1	no	—
230	D₁₃	—	2	2	no	—
231	C₂⁴	—	13	1	no	—
232	C₂²×C₄	—	13	1	no	—
233	C₂²×C₄	—	13	1	no	—
234	C₂²×C₄	—	13	2	no	—
235	C₂²×C₄	—	13	2	no	—
236	C₂²×C₄	—	13	1	no	—
237	C₂²×C₄	—	13	1	no	—
238	C₂²×C₄	—	13	2	no	—
239	C₂²×C₄	—	13	2	no	—
240	C₂²×C₄	—	13	1	no	—
241	C₂²×C₄	—	52	1	no	—
242	C₂²×C₄	—	52	1	no	—
243	C₂²×C₄	C₄⋊#261	1	1	yes	characteristic
244	M₁₆	—	26	4	no	—
245	M₁₆	—	52	2	no	—
246	M₁₆	—	52	4	no	—
247	C₂×C₈	—	13	2	no	—
248	C₂×C₈	—	13	1	no	—
249	C₂×C₈	—	13	2	no	—
250	C₂×C₈	—	13	1	no	—
251	C₂×C₈	—	13	1	no	—
252	C₂×C₈	—	13	2	no	—
253	C₂×C₈	—	13	2	no	—
254	C₂×C₈	—	13	2	no	—
255	C₂×C₈	—	13	2	no	—
256	C₂×C₈	—	13	1	no	—
257	C₂×C₈	—	13	2	no	—
258	C₂×C₈	—	13	1	no	—
259	C₂×C₈	—	13	2	no	—
260	C₂×C₈	—	13	1	no	—
261	C₂×C₈	—	52	4	no	—
262	C₂×C₈	—	52	2	no	—
263	C₂×C₈	C₄⋊#261	1	2	yes	—
264	C₂×C₈	C₄⋊#261	1	1	yes	characteristic
265	C₂×C₈	C₂²×#261	1	1	yes	characteristic
266	C₂×C₈	C₂²×#261	1	2	yes	—
267	C₄⋊C₄	—	26	2	no	—
268	C₄⋊C₄	—	26	2	no	—
269	C₄⋊C₄	—	52	1	no	—
270	C₄⋊C₄	—	52	1	no	—
271	C₂²⋊C₄	—	52	4	no	—
272	C₄²	—	52	4	no	—
273	C₁₆	—	13	4	no	—
274	C₁₆	C₂²×#261	1	4	yes	—
275	C₁₃	C₂×#2155	1	1	yes	characteristic, complemented, Sylow
276	C₂³	—	13	1	no	—
277	C₂³	—	13	1	no	—
278	C₂³	—	13	2	no	—
279	C₂³	—	13	2	no	—
280	C₂³	—	52	2	no	—
281	C₂³	#88101	1	1	yes	characteristic
282	C₂×C₄	—	13	2	no	—
283	C₂×C₄	—	13	2	no	—
284	C₂×C₄	—	13	1	no	—
285	C₂×C₄	—	13	1	no	—
286	C₂×C₄	—	13	2	no	—
287	C₂×C₄	—	13	1	no	—
288	C₂×C₄	—	13	1	no	—
289	C₂×C₄	—	13	2	no	—
290	C₂×C₄	—	13	2	no	—
291	C₂×C₄	—	13	2	no	—
292	C₂×C₄	—	13	2	no	—
293	C₂×C₄	—	13	1	no	—
294	C₂×C₄	—	13	2	no	—
295	C₂×C₄	—	13	1	no	—
296	C₂×C₄	—	52	2	no	—
297	C₂×C₄	—	52	2	no	—
298	C₂×C₄	—	104	2	no	—
299	C₂×C₄	—	104	2	no	—
300	C₂×C₄	#88101	1	2	yes	—
301	C₂×C₄	#88101	1	1	yes	characteristic
302	C₂×C₄	C₂×C₄⋊#261	1	2	yes	—
303	C₂×C₄	C₂×C₄⋊#261	1	1	yes	characteristic
304	C₈	—	13	1	no	—
305	C₈	—	13	2	no	—
306	C₈	—	13	1	no	—
307	C₈	—	52	4	no	—
308	C₈	C₂×C₄⋊#261	1	1	yes	characteristic
309	C₈	C₂×C₄⋊#261	1	2	yes	—
310	C₈	C₂³×#261	1	1	yes	characteristic
311	C₂²	—	2	2	no	—
312	C₂²	—	13	1	no	—
313	C₂²	—	13	1	no	—
314	C₂²	—	13	2	no	—
315	C₂²	—	26	2	no	—
316	C₂²	—	26	2	no	—
317	C₂²	—	52	2	no	—
318	C₂²	—	52	2	no	—
319	C₂²	C₂×#88101	1	2	yes	—
320	C₂²	C₂×#88101	1	1	yes	characteristic
321	C₄	—	13	1	no	—
322	C₄	—	13	2	no	—
323	C₄	—	13	1	no	—
324	C₄	—	104	2	no	—
325	C₄	—	104	2	no	—
326	C₄	C₂×#88101	1	2	yes	—
327	C₄	C₂×#88101	1	1	yes	characteristic
328	C₄	C₂²×C₄⋊#261	1	1	yes	characteristic
329	C₂	—	2	2	no	—
330	C₂	—	26	1	no	—
331	C₂	—	26	1	no	—
332	C₂	—	26	2	no	—
333	C₂	#13131184	1	2	yes	direct product factor
334	C₂	C₂²×#88101	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×#88101
commutator	C₈×C₁₃	C₂³×C₄
Frattini	C₈	C₂³×#261
Fitting	C₂×C₁₃×M₃₂	C₄
socle	C₂²×C₁₃	C₂×#106
prime radical	C₂³×D₁₃	SD₁₆
cosocle	C₈×D₁₃	C₂⁴
hypercenter	C₂×M₃₂	#261
hypocenter	C₁₃	C₂×#2155
LCS	G ▷ C₈×C₁₃ ▷ C₄×C₁₃ ▷ C₂×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂²×C₈ ◁ C₂×M₃₂

Complements

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

N	H ≅ G/N	type
3. #13131184	C₂	direct (N × H)
4. C₂×#11718220	C₂	semidirect
5. C₂×#11718219	C₂	semidirect
19. C₂×C₁₃×M₃₂	C₄	semidirect
20. C₂×#88052	C₄	semidirect
21. D₁₃×M₃₂	C₄	semidirect
24. C₂×C₁₆×D₁₃	C₄	semidirect
25. #11718220	C₂²	semidirect
26. #11718219	C₂²	semidirect
57. C₁₃×M₃₂	C₂×C₄	semidirect
59. #88052	C₂×C₄	semidirect
63. C₁₆×D₁₃	C₂×C₄	semidirect
157. C₂×M₃₂	#261	semidirect
213. M₃₂	C₂×#261	semidirect
226. C₂×C₁₃	#2155	semidirect
275. C₁₃	C₂×#2155	semidirect
333. C₂	#13131184	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂²×#88103	2
C₂×#11719092	2
#13131184	2
C₂×#11718220	2
C₂×#11718219	2
C₂×D₁₃×M₃₂	2
C₂×#2155	13

Subgroups — C2×#13131184 #16807023

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#13131184 #16807023