Subgroups — C₂×#13116967 #16017489

Order 3328.

658 conjugacy classes of subgroups.

The subgroup lattice has 408 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×#88040	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×#11718562	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×#11718557	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂×#11718558	C₂	1	1	yes	maximal, characteristic
5	#13116967	C₂	1	8	yes	maximal, direct product factor
6	C₂×#11718088	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂×#11718087	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×C₁₃×#368	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂×#88163	—	2	1	no	—
10	C₂×#88162	—	2	1	no	—
11	C₂²×Dic₅₂	C₂²	1	1	yes	characteristic
12	C₂²×D₁₀₄	C₂²	1	1	yes	characteristic
13	C₂×#88155	C₂²	1	1	yes	characteristic
14	C₂²×C₁₃×C₁₆	C₂²	1	1	yes	characteristic
15	C₂×C₁₃×#108	C₂²	1	1	yes	characteristic
16	C₂×C₁₃×#108	C₂²	1	1	yes	characteristic
17	C₂×#88064	—	2	1	no	—
18	C₂×#88063	—	2	1	no	—
19	C₂×#88052	C₂²	1	1	yes	characteristic
20	C₂×#88040	—	2	1	no	—
21	C₂×#88040	—	2	1	no	—
22	C₂×#88040	—	2	2	no	—
23	C₂×#88040	C₂²	1	4	yes	complemented
24	#11718562	—	2	2	no	—
25	#11718562	C₂²	1	4	yes	complemented
26	C₂×#88038	—	2	1	no	—
27	C₂×C₁₆×D₁₃	—	2	1	no	—
28	#11718558	—	2	2	no	—
29	#11718558	C₂²	1	4	yes	—
30	#11718557	—	2	2	no	—
31	#11718557	C₂²	1	4	yes	complemented
32	C₁₃×#368	C₂²	1	4	yes	complemented
33	#11718088	C₂²	1	4	yes	complemented
34	#11718087	C₂²	1	4	yes	complemented
35	C₂×#6127	D₄	1	1	yes	characteristic
36	C₂²×D₅₂	—	2	1	no	—
37	C₂²×Dic₂₆	—	2	1	no	—
38	C₂×C₁₃×M₁₆	—	2	1	no	—
39	C₂×C₁₃×M₁₆	—	2	1	no	—
40	C₂²×C₈×C₁₃	C₂³	1	1	yes	characteristic
41	C₂×#6099	D₄	1	1	yes	characteristic
42	#88163	—	2	4	no	—
43	#88163	—	4	2	no	—
44	#88162	—	2	4	no	—
45	#88162	—	4	2	no	—
46	C₂×Dic₅₂	—	2	1	no	—
47	C₂×Dic₅₂	—	2	1	no	—
48	C₂×Dic₅₂	—	2	1	no	—
49	C₂×Dic₅₂	—	2	1	no	—
50	C₂×Dic₅₂	—	2	1	no	—
51	C₂×Dic₅₂	—	4	1	no	—
52	C₂×Dic₅₂	C₂³	1	2	yes	—
53	C₂×D₁₀₄	—	2	1	no	—
54	C₂×D₁₀₄	—	2	1	no	—
55	C₂×D₁₀₄	—	2	1	no	—
56	C₂×D₁₀₄	—	2	1	no	—
57	C₂×D₁₀₄	—	2	1	no	—
58	C₂×D₁₀₄	—	4	1	no	—
59	C₂×D₁₀₄	C₂³	1	2	yes	—
60	C₂×#6095	—	4	1	no	—
61	C₂×#6095	—	4	1	no	—
62	#88155	—	2	1	no	—
63	#88155	—	2	1	no	—
64	#88155	D₄	1	2	yes	—
65	#88155	C₂³	1	2	yes	—
66	C₂×#6094	—	2	1	no	—
67	C₂×C₈×D₁₃	—	2	1	no	—
68	C₂×C₁₃×C₁₆	D₄	1	1	yes	characteristic
69	C₂×C₁₃×C₁₆	D₄	1	1	yes	characteristic
70	C₂×C₁₃×C₁₆	D₄	1	2	yes	—
71	C₂×C₁₃×C₁₆	C₂³	1	2	yes	—
72	C₁₃×#108	—	2	1	no	—
73	C₁₃×#108	—	2	1	no	—
74	C₁₃×#108	C₂³	1	2	yes	—
75	C₁₃×#108	C₂³	1	2	yes	—
76	#88064	—	2	4	no	—
77	#88063	—	2	4	no	—
78	#88052	—	2	1	no	—
79	#88052	C₂³	1	2	yes	—
80	C₂×#6090	—	2	1	no	—
81	#88040	—	2	4	no	—
82	#88040	—	2	4	no	—
83	#88038	—	2	4	no	—
84	C₁₆×D₁₃	—	2	4	no	—
85	C₂×#2223	—	13	1	no	maximal, Sylow
86	C₂³×D₁₃	—	4	1	no	—
87	C₂²×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
88	C₂×C₂²⋊D₁₃	—	2	1	no	—
89	C₂²×Dic₁₃	—	4	1	no	—
90	#6127	—	2	1	no	—
91	#6127	—	2	1	no	—
92	#6127	D₈	1	2	yes	—
93	#6127	C₂×D₄	1	2	yes	—
94	C₂×D₅₂	—	2	2	no	—
95	C₂×D₅₂	—	4	1	no	—
96	C₂×D₅₂	—	4	1	no	—
97	C₂×D₅₂	—	4	1	no	—
98	C₂×D₅₂	—	4	1	no	—
99	C₂×D₅₂	—	4	1	no	—
100	C₂×D₅₂	D₈	1	1	yes	characteristic
101	C₂×C₄×D₁₃	—	2	1	no	—
102	C₂×Dic₂₆	—	2	2	no	—
103	C₂×Dic₂₆	—	4	1	no	—
104	C₂×Dic₂₆	—	4	1	no	—
105	C₂×Dic₂₆	—	4	1	no	—
106	C₂×Dic₂₆	—	4	1	no	—
107	C₂×Dic₂₆	—	4	1	no	—
108	C₂×Dic₂₆	D₈	1	1	yes	characteristic
109	C₁₃×M₁₆	—	2	2	no	—
110	C₁₃×M₁₆	—	2	2	no	—
111	C₁₃×M₁₆	—	4	1	no	—
112	C₁₃×M₁₆	—	4	1	no	—
113	C₂×C₈×C₁₃	—	4	1	no	—
114	C₂×C₈×C₁₃	—	4	1	no	—
115	C₂×C₈×C₁₃	C₂×D₄	1	1	yes	characteristic
116	C₂×C₈×C₁₃	C₂×D₄	1	1	yes	characteristic
117	C₂×C₈×C₁₃	C₂×D₄	1	1	yes	characteristic
118	C₂×C₈×C₁₃	#41	1	1	yes	characteristic
119	C₂×C₈×C₁₃	#41	1	1	yes	characteristic
120	C₂×C₈×C₁₃	C₂⁴	1	1	yes	characteristic
121	#6099	—	2	1	no	—
122	#6099	C₂×D₄	1	2	yes	—
123	C₂×#1055	—	2	1	no	—
124	Dic₅₂	—	2	2	no	—
125	Dic₅₂	—	2	2	no	—
126	Dic₅₂	—	4	4	no	—
127	Dic₅₂	—	4	1	no	—
128	Dic₅₂	—	4	1	no	—
129	D₁₀₄	—	2	2	no	—
130	D₁₀₄	—	2	2	no	—
131	D₁₀₄	—	4	4	no	—
132	D₁₀₄	—	4	1	no	—
133	D₁₀₄	—	4	1	no	—
134	#6095	—	4	4	no	—
135	#6095	—	4	4	no	—
136	#6094	—	2	2	no	—
137	#6094	—	2	2	no	—
138	C₈×D₁₃	—	2	2	no	—
139	C₈×D₁₃	—	2	2	no	—
140	C₁₃×C₁₆	C₂×D₄	1	2	yes	—
141	C₁₃×C₁₆	C₂×D₄	1	2	yes	—
142	#6090	—	2	2	no	—
143	C₂²×SD₃₂	—	13	1	no	—
144	C₂×#504	—	13	1	no	—
145	C₂×#473	—	13	1	no	—
146	C₂×#472	—	13	1	no	—
147	#2223	—	13	8	no	—
148	C₂×#368	D₁₃	1	1	yes	characteristic, complemented
149	C₂×#362	—	13	1	no	—
150	C₂×#361	—	13	1	no	—
151	C₂³×C₁₃	#135	1	1	yes	characteristic
152	C₂²×D₁₃	—	2	1	no	—
153	C₂²×D₁₃	—	4	2	no	—
154	C₂²×D₁₃	—	8	1	no	—
155	C₂²×D₁₃	—	8	1	no	—
156	C₂²×D₁₃	—	16	2	no	—
157	C₂×C₄×C₁₃	#121	1	1	yes	characteristic
158	C₂×C₄×C₁₃	#121	1	1	yes	characteristic
159	C₂×C₄×C₁₃	C₂×D₈	1	1	yes	characteristic
160	C₂×C₄×C₁₃	C₂×D₈	1	1	yes	characteristic
161	C₂×C₄×C₁₃	#135	1	1	yes	characteristic
162	C₂×C₄×C₁₃	C₂²×D₄	1	1	yes	characteristic
163	C₂²⋊D₁₃	—	2	2	no	—
164	C₂²⋊D₁₃	—	2	2	no	—
165	C₂×Dic₁₃	—	2	1	no	—
166	C₂×Dic₁₃	—	4	2	no	—
167	C₂×Dic₁₃	—	8	1	no	—
168	C₂×Dic₁₃	—	8	1	no	—
169	D₅₂	—	2	1	no	—
170	D₅₂	—	4	2	no	—
171	D₅₂	—	4	2	no	—
172	D₅₂	—	8	1	no	—
173	D₅₂	—	8	1	no	—
174	D₅₂	C₂×D₈	1	2	yes	—
175	C₄×D₁₃	—	2	2	no	—
176	C₄×D₁₃	—	2	2	no	—
177	Dic₂₆	—	2	1	no	—
178	Dic₂₆	—	4	2	no	—
179	Dic₂₆	—	4	2	no	—
180	Dic₂₆	—	8	1	no	—
181	Dic₂₆	—	8	1	no	—
182	Dic₂₆	C₂×D₈	1	2	yes	—
183	C₈×C₁₃	—	4	2	no	—
184	C₈×C₁₃	—	4	2	no	—
185	C₈×C₁₃	#121	1	1	yes	characteristic
186	C₈×C₁₃	#121	1	1	yes	characteristic
187	C₈×C₁₃	C₂²×D₄	1	1	yes	characteristic
188	C₈×C₁₃	C₂×#41	1	1	yes	characteristic
189	#1055	—	2	2	no	—
190	C₂×#137	—	26	1	no	—
191	C₂×AGL(1,ℤ/8)	—	26	1	no	—
192	C₂²×Q₁₆	—	13	1	no	—
193	C₂²×D₈	—	13	1	no	—
194	C₂×#131	—	13	1	no	—
195	C₂×SD₃₂	—	13	4	no	—
196	C₂×SD₃₂	—	26	1	no	—
197	C₂×SD₃₂	—	26	1	no	—
198	C₂×SD₃₂	—	26	2	no	—
199	#504	—	13	4	no	—
200	#504	—	26	2	no	—
201	C₂×M₃₂	—	13	1	no	—
202	C₂×M₃₂	—	26	1	no	—
203	C₂²×C₁₆	—	26	1	no	—
204	C₂²×C₁₆	C₂×D₁₃	1	1	yes	characteristic
205	#473	—	13	4	no	—
206	#473	—	26	2	no	—
207	#472	—	13	4	no	—
208	#472	—	26	2	no	—
209	C₂×#108	C₂×D₁₃	1	1	yes	characteristic
210	C₂×#108	C₂×D₁₃	1	1	yes	characteristic
211	C₂×#101	—	26	1	no	—
212	C₂×#100	—	26	1	no	—
213	#368	C₂×D₁₃	1	4	yes	complemented
214	#362	—	13	4	no	—
215	#361	—	13	4	no	—
216	C₂²×C₁₃	—	2	2	no	—
217	C₂²×C₁₃	#466	1	1	yes	characteristic
218	C₂²×C₁₃	#466	1	1	yes	characteristic
219	C₂²×C₁₃	C₂×#135	1	1	yes	characteristic
220	C₂×D₁₃	—	2	2	no	—
221	C₂×D₁₃	—	4	2	no	—
222	C₂×D₁₃	—	8	2	no	—
223	C₂×D₁₃	—	16	2	no	—
224	C₂×D₁₃	—	16	2	no	—
225	C₂×D₁₃	—	16	2	no	—
226	C₄×C₁₃	#466	1	1	yes	characteristic
227	C₄×C₁₃	#466	1	1	yes	characteristic
228	C₄×C₁₃	C₂×#121	1	1	yes	characteristic
229	C₄×C₁₃	C₂²×D₈	1	1	yes	characteristic
230	Dic₁₃	—	2	2	no	—
231	Dic₁₃	—	8	2	no	—
232	C₂×#41	—	13	1	no	—
233	C₂²×Q₈	—	26	1	no	—
234	C₂²×D₄	—	26	1	no	—
235	#137	—	26	4	no	—
236	#137	—	52	2	no	—
237	AGL(1,ℤ/8)	—	26	4	no	—
238	AGL(1,ℤ/8)	—	52	2	no	—
239	C₂×Q₁₆	—	13	2	no	—
240	C₂×Q₁₆	—	26	1	no	—
241	C₂×Q₁₆	—	26	1	no	—
242	C₂×Q₁₆	—	26	1	no	—
243	C₂×Q₁₆	—	26	1	no	—
244	C₂×Q₁₆	—	26	1	no	—
245	C₂×Q₁₆	—	52	1	no	—
246	C₂×SD₁₆	—	52	1	no	—
247	C₂×SD₁₆	—	52	1	no	—
248	C₂×D₈	—	13	2	no	—
249	C₂×D₈	—	26	1	no	—
250	C₂×D₈	—	26	1	no	—
251	C₂×D₈	—	26	1	no	—
252	C₂×D₈	—	26	1	no	—
253	C₂×D₈	—	26	1	no	—
254	C₂×D₈	—	52	1	no	—
255	#131	—	13	2	no	—
256	#131	—	13	2	no	—
257	#131	—	26	1	no	—
258	#131	—	26	1	no	—
259	C₂×M₁₆	—	2	1	no	—
260	C₂×M₁₆	—	2	1	no	—
261	C₂×M₁₆	—	13	1	no	—
262	C₂×M₁₆	—	26	1	no	—
263	C₂²×C₈	—	26	1	no	—
264	C₂²×C₈	C₂²×D₁₃	1	1	yes	characteristic
265	SD₃₂	—	26	4	no	—
266	SD₃₂	—	26	4	no	—
267	M₃₂	—	13	2	no	—
268	M₃₂	—	26	1	no	—
269	M₃₂	—	26	4	no	—
270	C₂×C₁₆	—	26	1	no	—
271	C₂×C₁₆	—	26	4	no	—
272	C₂×C₁₆	D₅₂	1	1	yes	characteristic
273	C₂×C₁₆	D₅₂	1	1	yes	characteristic
274	C₂×C₁₆	D₅₂	1	2	yes	—
275	C₂×C₁₆	C₂²×D₁₃	1	2	yes	—
276	#108	—	2	1	no	—
277	#108	—	2	1	no	—
278	#108	C₂²×D₁₃	1	2	yes	—
279	#108	C₂²×D₁₃	1	2	yes	—
280	#101	—	26	4	no	—
281	#100	—	26	4	no	—
282	C₂×C₁₃	—	2	1	no	—
283	C₂×C₁₃	—	2	1	no	—
284	C₂×C₁₃	#2223	1	2	yes	complemented
285	C₂×C₁₃	C₂×#466	1	1	yes	characteristic
286	D₁₃	—	4	2	no	—
287	D₁₃	—	16	2	no	—
288	C₂⁴	—	52	1	no	—
289	#41	—	13	2	no	—
290	#41	—	13	2	no	—
291	#41	—	26	1	no	—
292	#41	—	26	1	no	—
293	C₂×Q₈	—	13	1	no	—
294	C₂×Q₈	—	26	2	no	—
295	C₂×Q₈	—	52	1	no	—
296	C₂×Q₈	—	52	1	no	—
297	C₂×Q₈	—	52	1	no	—
298	C₂×Q₈	—	52	1	no	—
299	C₂×Q₈	—	52	1	no	—
300	C₂×D₄	—	13	1	no	—
301	C₂×D₄	—	26	1	no	—
302	C₂×D₄	—	26	2	no	—
303	C₂×D₄	—	52	1	no	—
304	C₂×D₄	—	52	1	no	—
305	C₂×D₄	—	52	1	no	—
306	C₂×D₄	—	52	1	no	—
307	C₂×D₄	—	52	1	no	—
308	C₂²×C₄	—	26	1	no	—
309	C₂²×C₄	—	52	1	no	—
310	C₂²×C₄	D₄×D₁₃	1	1	yes	characteristic
311	Q₁₆	—	26	2	no	—
312	Q₁₆	—	26	2	no	—
313	Q₁₆	—	52	4	no	—
314	Q₁₆	—	52	1	no	—
315	Q₁₆	—	52	1	no	—
316	SD₁₆	—	52	4	no	—
317	SD₁₆	—	52	4	no	—
318	D₈	—	26	2	no	—
319	D₈	—	26	2	no	—
320	D₈	—	52	4	no	—
321	D₈	—	52	1	no	—
322	D₈	—	52	1	no	—
323	M₁₆	—	2	2	no	—
324	M₁₆	—	2	2	no	—
325	M₁₆	—	4	1	no	—
326	M₁₆	—	4	1	no	—
327	M₁₆	—	13	2	no	—
328	M₁₆	—	26	1	no	—
329	M₁₆	—	26	2	no	—
330	M₁₆	—	26	2	no	—
331	C₂×C₈	—	4	1	no	—
332	C₂×C₈	—	4	1	no	—
333	C₂×C₈	—	26	1	no	—
334	C₂×C₈	—	26	2	no	—
335	C₂×C₈	—	26	2	no	—
336	C₂×C₈	C₂×D₅₂	1	1	yes	characteristic
337	C₂×C₈	C₂×D₅₂	1	1	yes	characteristic
338	C₂×C₈	D₄×D₁₃	1	1	yes	characteristic
339	C₂×C₈	#6131	1	1	yes	characteristic
340	C₂×C₈	#6131	1	1	yes	characteristic
341	C₂×C₈	C₂³×D₁₃	1	1	yes	characteristic
342	C₁₆	—	26	2	no	—
343	C₁₆	C₂×D₅₂	1	2	yes	—
344	C₁₆	C₂×D₅₂	1	2	yes	—
345	C₁₃	C₂×#2223	1	1	yes	characteristic, complemented, Sylow
346	C₂³	—	26	1	no	—
347	C₂³	—	52	2	no	—
348	C₂³	—	104	1	no	—
349	C₂³	—	104	1	no	—
350	C₂³	—	208	2	no	—
351	C₂³	#88173	1	1	yes	characteristic
352	Q₈	—	13	2	no	—
353	Q₈	—	26	1	no	—
354	Q₈	—	52	2	no	—
355	Q₈	—	52	2	no	—
356	Q₈	—	104	1	no	—
357	Q₈	—	104	1	no	—
358	D₄	—	13	2	no	—
359	D₄	—	26	2	no	—
360	D₄	—	26	2	no	—
361	D₄	—	26	1	no	—
362	D₄	—	52	2	no	—
363	D₄	—	52	2	no	—
364	D₄	—	104	1	no	—
365	D₄	—	104	1	no	—
366	C₂×C₄	—	26	1	no	—
367	C₂×C₄	—	26	2	no	—
368	C₂×C₄	—	26	2	no	—
369	C₂×C₄	—	52	2	no	—
370	C₂×C₄	—	104	1	no	—
371	C₂×C₄	—	104	1	no	—
372	C₂×C₄	#88149	1	1	yes	characteristic
373	C₂×C₄	#88149	1	1	yes	characteristic
374	C₂×C₄	D₈×D₁₃	1	1	yes	characteristic
375	C₂×C₄	D₈×D₁₃	1	1	yes	characteristic
376	C₂×C₄	#88173	1	1	yes	characteristic
377	C₂×C₄	C₂×D₄×D₁₃	1	1	yes	characteristic
378	C₈	—	4	2	no	—
379	C₈	—	4	2	no	—
380	C₈	—	26	2	no	—
381	C₈	#88149	1	1	yes	characteristic
382	C₈	#88149	1	1	yes	characteristic
383	C₈	C₂²×D₅₂	1	1	yes	characteristic
384	C₈	C₂×#6131	1	1	yes	characteristic
385	C₂²	—	2	2	no	—
386	C₂²	—	26	2	no	—
387	C₂²	—	52	2	no	—
388	C₂²	—	104	2	no	—
389	C₂²	—	208	2	no	—
390	C₂²	—	208	2	no	—
391	C₂²	—	208	2	no	—
392	C₂²	#11718544	1	1	yes	characteristic
393	C₂²	#11718544	1	1	yes	characteristic
394	C₂²	C₂×#88173	1	1	yes	characteristic
395	C₄	—	26	2	no	—
396	C₄	—	104	2	no	—
397	C₄	#11718544	1	1	yes	characteristic
398	C₄	#11718544	1	1	yes	characteristic
399	C₄	C₂×#88149	1	1	yes	characteristic
400	C₄	C₂×D₈×D₁₃	1	1	yes	characteristic
401	C₂	—	2	1	no	—
402	C₂	—	2	1	no	—
403	C₂	—	52	2	no	—
404	C₂	—	208	2	no	—
405	C₂	#13116967	1	2	yes	direct product factor
406	C₂	C₂×#11718544	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	#11718544
commutator, cosocle	C₂×C₈×C₁₃	C₂⁴
Frattini	C₂×C₈	C₂³×D₁₃
Fitting	C₂×C₁₃×#368	C₂
socle	C₂²×C₁₃	#466
prime radical	C₂×#11718558	C₂
hypercenter	C₂×#368	D₁₃
hypocenter	C₁₃	C₂×#2223
LCS	G ▷ C₂×C₈×C₁₃ ▷ C₄×C₁₃ ▷ C₂×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂²×C₈ ◁ C₂×#368

Complements

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

N	H ≅ G/N	type
1. C₂²×#88040	C₂	semidirect
2. C₂×#11718562	C₂	semidirect
3. C₂×#11718557	C₂	semidirect
5. #13116967	C₂	direct (N × H)
6. C₂×#11718088	C₂	semidirect
7. C₂×#11718087	C₂	semidirect
8. C₂×C₁₃×#368	C₂	semidirect
23. C₂×#88040	C₂²	semidirect
25. #11718562	C₂²	semidirect
31. #11718557	C₂²	semidirect
32. C₁₃×#368	C₂²	semidirect
33. #11718088	C₂²	semidirect
34. #11718087	C₂²	semidirect
148. C₂×#368	D₁₃	semidirect
213. #368	C₂×D₁₃	semidirect
284. C₂×C₁₃	#2223	semidirect
345. C₁₃	C₂×#2223	semidirect
405. C₂	#13116967	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×#88040	2
C₂×#11718562	2
C₂×#11718557	2
C₂×#11718558	2
#13116967	2
C₂×#11718088	2
C₂×#11718087	2
C₂×C₁₃×#368	2
C₂×#2223	13

Subgroups — C2×#13116967 #16017489

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#13116967 #16017489