Subgroups — C₂×#13116714 #16016316

Order 3328.

662 conjugacy classes of subgroups.

The subgroup lattice has 392 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#11718855	C₂	1	1	yes	maximal, characteristic
2	C₂×#11718857	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×#11718556	C₂	1	1	yes	maximal, characteristic, complemented
4	#13116714	C₂	1	8	yes	maximal, direct product factor
5	C₂×#11718085	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×#11718102	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂×C₁₆×Dic₁₃	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×C₁₃×#359	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂×C₁₃×#135	C₂²	1	1	yes	characteristic
10	C₂×#88202	—	2	1	no	—
11	C₂×#88161	—	2	1	no	—
12	C₂×#88158	C₂²	1	1	yes	characteristic
13	C₂²×C₁₃×C₁₆	C₂²	1	1	yes	characteristic
14	C₂×C₁₃×#108	C₂²	1	1	yes	characteristic
15	C₂×#88077	—	2	1	no	—
16	#11718857	—	2	2	no	—
17	#11718857	C₂²	1	4	yes	complemented
18	#11718855	C₄	1	4	yes	complemented
19	#11718855	C₂²	1	4	yes	—
20	C₂×#88069	C₄	1	1	yes	characteristic, complemented
21	C₂×#88068	C₄	1	1	yes	characteristic, complemented
22	C₂×#88067	C₄	1	1	yes	characteristic, complemented
23	C₂×#88066	C₄	1	1	yes	characteristic, complemented
24	C₂×#88065	—	2	1	no	—
25	C₂×#88059	C₂²	1	1	yes	characteristic
26	C₂×C₈×Dic₁₃	C₂²	1	1	yes	characteristic
27	C₂²×#6090	C₂²	1	1	yes	characteristic
28	#11718556	—	2	2	no	—
29	#11718556	C₂²	1	4	yes	complemented
30	C₁₃×#359	C₂²	1	4	yes	complemented
31	#11718102	C₂²	1	4	yes	complemented
32	C₁₆×Dic₁₃	C₂²	1	4	yes	complemented
33	#11718085	C₂²	1	4	yes	complemented
34	C₂×C₁₃×#41	—	2	1	no	—
35	C₂×#6127	—	2	1	no	—
36	C₁₃×#135	—	2	1	no	—
37	C₁₃×#135	—	2	1	no	—
38	C₁₃×#135	C₂×C₄	1	2	yes	—
39	C₁₃×#135	C₂³	1	2	yes	—
40	C₂×C₁₃×Q₁₆	C₂×C₄	1	1	yes	characteristic
41	C₂×C₁₃×SD₁₆	—	2	1	no	—
42	C₂×C₁₃×D₈	C₂×C₄	1	1	yes	characteristic
43	C₂×C₁₃×M₁₆	—	2	1	no	—
44	C₂²×C₈×C₁₃	C₂³	1	1	yes	characteristic
45	#88202	—	2	4	no	—
46	#88202	—	4	2	no	—
47	C₂×C₄×Dic₁₃	D₄	1	1	yes	characteristic
48	C₂×#6099	—	2	1	no	—
49	C₂×#6099	—	4	1	no	—
50	C₂²×#1055	—	4	1	no	—
51	C₂²×#1055	D₄	1	1	yes	characteristic
52	#88161	—	2	4	no	—
53	#88161	—	4	2	no	—
54	C₂×Dic₅₂	C₂×C₄	1	1	yes	characteristic
55	#88158	—	2	1	no	—
56	#88158	—	2	1	no	—
57	#88158	C₂×C₄	1	2	yes	—
58	#88158	C₂³	1	2	yes	—
59	C₂×D₁₀₄	C₂×C₄	1	1	yes	characteristic
60	C₂×#6095	—	2	1	no	—
61	C₂×#6094	—	4	1	no	—
62	C₂×C₈×D₁₃	—	4	1	no	—
63	C₂×C₁₃×C₁₆	—	2	1	no	—
64	C₂×C₁₃×C₁₆	—	2	1	no	—
65	C₂×C₁₃×C₁₆	C₂³	1	2	yes	—
66	C₁₃×#108	—	2	1	no	—
67	C₁₃×#108	C₂³	1	2	yes	—
68	#88077	—	2	4	no	—
69	#88069	C₂×C₄	1	4	yes	complemented
70	#88068	C₂×C₄	1	4	yes	complemented
71	#88067	C₂×C₄	1	4	yes	complemented
72	#88066	C₂×C₄	1	4	yes	complemented
73	#88065	—	2	4	no	—
74	#88059	—	2	1	no	—
75	#88059	C₂³	1	2	yes	—
76	C₈×Dic₁₃	D₄	1	2	yes	—
77	C₈×Dic₁₃	C₂³	1	2	yes	—
78	C₂×#6090	C₂×C₄	1	1	yes	characteristic
79	C₂×#6090	C₂×C₄	1	1	yes	characteristic
80	C₂×#6090	C₂×C₄	1	2	yes	—
81	C₂×#6090	C₂³	1	2	yes	—
82	C₂×#2178	—	13	1	no	maximal, Sylow
83	C₁₃×#41	—	2	2	no	—
84	C₁₃×#41	—	2	2	no	—
85	C₁₃×#41	—	4	1	no	—
86	C₁₃×#41	—	4	1	no	—
87	C₂×Q₈×C₁₃	—	2	1	no	—
88	C₂×D₄×C₁₃	—	2	1	no	—
89	C₂×D₄×C₁₃	—	4	1	no	—
90	C₂²×C₄×C₁₃	—	4	1	no	—
91	C₂²×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
92	C₂×C₂²⋊D₁₃	—	4	1	no	—
93	C₂²×Dic₁₃	—	2	1	no	—
94	#6127	—	2	2	no	—
95	#6127	—	2	2	no	—
96	#6127	—	4	1	no	—
97	#6127	—	4	1	no	—
98	C₂×D₅₂	—	2	1	no	—
99	C₂×C₄×D₁₃	—	4	1	no	—
100	C₂×Dic₂₆	—	2	1	no	—
101	C₁₃×Q₁₆	—	2	1	no	—
102	C₁₃×Q₁₆	C₂²×C₄	1	2	yes	—
103	C₁₃×SD₁₆	—	2	2	no	—
104	C₁₃×SD₁₆	—	2	2	no	—
105	C₁₃×D₈	—	2	1	no	—
106	C₁₃×D₈	C₂²×C₄	1	2	yes	—
107	C₁₃×M₁₆	—	2	2	no	—
108	C₁₃×M₁₆	—	4	1	no	—
109	C₂×C₈×C₁₃	—	4	1	no	—
110	C₂×C₈×C₁₃	C₂²×C₄	1	1	yes	characteristic
111	C₂×C₈×C₁₃	C₂²×C₄	1	1	yes	characteristic
112	C₂×C₈×C₁₃	C₂×D₄	1	1	yes	characteristic
113	C₂×C₈×C₁₃	#41	1	1	yes	characteristic
114	C₂×C₈×C₁₃	#41	1	1	yes	characteristic
115	C₂×C₈×C₁₃	C₂⁴	1	1	yes	characteristic
116	C₄×Dic₁₃	—	2	1	no	—
117	C₄×Dic₁₃	C₂×D₄	1	2	yes	—
118	#6099	—	2	2	no	—
119	#6099	—	4	1	no	—
120	#6099	—	4	4	no	—
121	C₂×#1055	—	4	1	no	—
122	C₂×#1055	—	4	4	no	—
123	C₂×#1055	D₈	1	1	yes	characteristic
124	C₂×#1055	D₈	1	1	yes	characteristic
125	C₂×#1055	D₈	1	2	yes	—
126	C₂×#1055	C₂×D₄	1	2	yes	—
127	Dic₅₂	—	2	1	no	—
128	Dic₅₂	C₂²×C₄	1	2	yes	—
129	D₁₀₄	—	2	1	no	—
130	D₁₀₄	C₂²×C₄	1	2	yes	—
131	#6095	—	2	2	no	—
132	#6095	—	2	2	no	—
133	#6094	—	4	4	no	—
134	C₈×D₁₃	—	4	4	no	—
135	C₁₃×C₁₆	—	2	2	no	—
136	#6090	C₂²×C₄	1	2	yes	—
137	#6090	C₂²×C₄	1	2	yes	—
138	C₂×#508	—	13	1	no	—
139	C₂×#443	—	13	1	no	—
140	C₂×#443	—	13	1	no	—
141	#2178	—	13	8	no	—
142	C₂×#368	—	13	1	no	—
143	C₂×#359	—	13	1	no	—
144	C₂×#359	D₁₃	1	1	yes	characteristic, complemented
145	C₂×C₄×C₁₆	—	13	1	no	—
146	C₂³×C₁₃	—	4	1	no	—
147	C₂³×C₁₃	#135	1	1	yes	characteristic
148	C₂²×D₁₃	—	4	1	no	—
149	Q₈×C₁₃	—	2	2	no	—
150	Q₈×C₁₃	—	4	1	no	—
151	D₄×C₁₃	—	2	2	no	—
152	D₄×C₁₃	—	4	2	no	—
153	D₄×C₁₃	—	4	2	no	—
154	D₄×C₁₃	—	4	1	no	—
155	C₂×C₄×C₁₃	—	4	1	no	—
156	C₂×C₄×C₁₃	—	4	2	no	—
157	C₂×C₄×C₁₃	—	4	2	no	—
158	C₂×C₄×C₁₃	C₄×D₄	1	1	yes	characteristic
159	C₂×C₄×C₁₃	C₄×D₄	1	1	yes	characteristic
160	C₂×C₄×C₁₃	C₂×D₈	1	1	yes	characteristic
161	C₂×C₄×C₁₃	C₂×D₈	1	1	yes	characteristic
162	C₂×C₄×C₁₃	#135	1	1	yes	characteristic
163	C₂×C₄×C₁₃	C₂²×D₄	1	1	yes	characteristic
164	C₂²⋊D₁₃	—	4	2	no	—
165	C₂²⋊D₁₃	—	4	2	no	—
166	C₂×Dic₁₃	—	2	4	no	—
167	C₂×Dic₁₃	—	2	2	no	—
168	C₂×Dic₁₃	—	4	1	no	—
169	D₅₂	—	2	2	no	—
170	D₅₂	—	4	1	no	—
171	C₄×D₁₃	—	4	2	no	—
172	C₄×D₁₃	—	4	2	no	—
173	Dic₂₆	—	2	2	no	—
174	Dic₂₆	—	4	1	no	—
175	C₈×C₁₃	—	4	2	no	—
176	C₈×C₁₃	C₄×D₄	1	1	yes	characteristic
177	C₈×C₁₃	C₄×D₄	1	1	yes	characteristic
178	C₈×C₁₃	C₂³×C₄	1	1	yes	characteristic
179	C₈×C₁₃	C₂×#41	1	1	yes	characteristic
180	#1055	—	4	2	no	—
181	#1055	C₂×D₈	1	2	yes	—
182	#1055	C₂×D₈	1	2	yes	—
183	C₂×#135	—	13	1	no	—
184	C₂×#135	C₂×D₁₃	1	1	yes	characteristic
185	C₂×#131	—	26	1	no	—
186	C₂×#131	—	26	1	no	—
187	#508	—	13	4	no	—
188	#508	—	13	4	no	—
189	C₂×Q₃₂	—	13	1	no	—
190	C₂×SD₃₂	—	13	1	no	—
191	C₂×SD₃₂	—	13	1	no	—
192	C₂×D₁₆	—	13	1	no	—
193	C₂²×C₁₆	—	13	1	no	—
194	C₂²×C₁₆	C₂×D₁₃	1	1	yes	characteristic
195	#443	—	13	4	no	—
196	#443	—	13	4	no	—
197	#443	—	26	2	no	—
198	#443	—	26	2	no	—
199	C₂×#108	—	13	1	no	—
200	C₂×#108	C₂×D₁₃	1	1	yes	characteristic
201	C₂×#104	—	26	1	no	—
202	C₂×#104	—	26	1	no	—
203	C₂×C₄×C₈	—	13	1	no	—
204	#368	—	13	4	no	—
205	#359	—	13	4	no	—
206	#359	C₂×D₁₃	1	4	yes	complemented
207	C₄×C₁₆	—	13	4	no	—
208	C₂²×C₁₃	—	2	2	no	—
209	C₂²×C₁₃	—	4	2	no	—
210	C₂²×C₁₃	—	8	2	no	—
211	C₂²×C₁₃	C₄×D₈	1	1	yes	characteristic
212	C₂²×C₁₃	C₄×D₈	1	1	yes	characteristic
213	C₂²×C₁₃	C₂×#135	1	1	yes	characteristic
214	C₂×D₁₃	—	4	2	no	—
215	C₂×D₁₃	—	8	2	no	—
216	C₄×C₁₃	—	4	2	no	—
217	C₄×C₁₃	C₄×D₈	1	1	yes	characteristic
218	C₄×C₁₃	C₄×D₈	1	1	yes	characteristic
219	C₄×C₁₃	C₂×C₄×D₄	1	1	yes	characteristic
220	C₄×C₁₃	C₂²×D₈	1	1	yes	characteristic
221	Dic₁₃	—	2	4	no	—
222	Dic₁₃	—	4	2	no	—
223	C₂×#41	—	2	1	no	—
224	C₂×#41	—	26	1	no	—
225	#135	—	2	1	no	—
226	#135	—	2	1	no	—
227	#135	—	13	2	no	—
228	#135	—	13	2	no	—
229	#135	—	26	1	no	—
230	#135	—	26	1	no	—
231	#135	C₄×D₁₃	1	2	yes	—
232	#135	C₂²×D₁₃	1	2	yes	—
233	C₂×Q₁₆	—	13	1	no	—
234	C₂×Q₁₆	C₄×D₁₃	1	1	yes	characteristic
235	C₂×SD₁₆	—	2	1	no	—
236	C₂×SD₁₆	—	26	1	no	—
237	C₂×D₈	—	13	1	no	—
238	C₂×D₈	C₄×D₁₃	1	1	yes	characteristic
239	#131	—	26	4	no	—
240	#131	—	26	4	no	—
241	#131	—	52	2	no	—
242	#131	—	52	2	no	—
243	C₂×M₁₆	—	2	1	no	—
244	C₂×M₁₆	—	26	1	no	—
245	C₂×M₁₆	—	52	1	no	—
246	C₂×M₁₆	—	52	1	no	—
247	C₂²×C₈	—	13	1	no	—
248	C₂²×C₈	—	52	1	no	—
249	C₂²×C₈	—	52	1	no	—
250	C₂²×C₈	C₂²×D₁₃	1	1	yes	characteristic
251	C₂×C₄²	—	13	1	no	—
252	Q₃₂	—	13	4	no	—
253	SD₃₂	—	13	4	no	—
254	SD₃₂	—	13	4	no	—
255	D₁₆	—	13	4	no	—
256	C₂×C₁₆	—	2	1	no	—
257	C₂×C₁₆	—	2	1	no	—
258	C₂×C₁₆	—	13	1	no	—
259	C₂×C₁₆	—	13	1	no	—
260	C₂×C₁₆	—	13	2	no	—
261	C₂×C₁₆	—	13	2	no	—
262	C₂×C₁₆	C₂²×D₁₃	1	2	yes	—
263	#108	—	2	1	no	—
264	#108	—	13	2	no	—
265	#108	—	26	1	no	—
266	#108	C₂²×D₁₃	1	2	yes	—
267	#104	—	26	4	no	—
268	#104	—	26	4	no	—
269	C₄×C₈	—	13	2	no	—
270	C₄×C₈	—	13	2	no	—
271	C₂×C₁₃	—	2	1	no	—
272	C₂×C₁₃	—	2	1	no	—
273	C₂×C₁₃	—	8	2	no	—
274	C₂×C₁₃	#2178	1	2	yes	complemented
275	C₂×C₁₃	C₂×C₄×D₈	1	1	yes	characteristic
276	D₁₃	—	8	2	no	—
277	#41	—	2	2	no	—
278	#41	—	2	2	no	—
279	#41	—	4	1	no	—
280	#41	—	4	1	no	—
281	#41	—	26	2	no	—
282	#41	—	26	2	no	—
283	#41	—	52	1	no	—
284	#41	—	52	1	no	—
285	C₂×Q₈	—	2	1	no	—
286	C₂×Q₈	—	26	1	no	—
287	C₂×D₄	—	2	1	no	—
288	C₂×D₄	—	4	1	no	—
289	C₂×D₄	—	26	1	no	—
290	C₂×D₄	—	52	1	no	—
291	C₂²×C₄	—	4	1	no	—
292	C₂²×C₄	—	26	1	no	—
293	C₂²×C₄	—	52	1	no	—
294	C₂²×C₄	D₄×D₁₃	1	1	yes	characteristic
295	Q₁₆	—	2	1	no	—
296	Q₁₆	—	13	2	no	—
297	Q₁₆	—	26	1	no	—
298	Q₁₆	C₂×C₄×D₁₃	1	2	yes	—
299	SD₁₆	—	2	2	no	—
300	SD₁₆	—	2	2	no	—
301	SD₁₆	—	26	2	no	—
302	SD₁₆	—	26	2	no	—
303	D₈	—	2	1	no	—
304	D₈	—	13	2	no	—
305	D₈	—	26	1	no	—
306	D₈	C₂×C₄×D₁₃	1	2	yes	—
307	M₁₆	—	2	2	no	—
308	M₁₆	—	4	1	no	—
309	M₁₆	—	26	2	no	—
310	M₁₆	—	52	4	no	—
311	M₁₆	—	52	4	no	—
312	M₁₆	—	52	1	no	—
313	C₂×C₈	—	4	1	no	—
314	C₂×C₈	—	13	2	no	—
315	C₂×C₈	—	13	1	no	—
316	C₂×C₈	—	13	1	no	—
317	C₂×C₈	—	13	2	no	—
318	C₂×C₈	—	52	4	no	—
319	C₂×C₈	—	52	4	no	—
320	C₂×C₈	—	52	1	no	—
321	C₂×C₈	C₂×C₄×D₁₃	1	1	yes	characteristic
322	C₂×C₈	C₂×C₄×D₁₃	1	1	yes	characteristic
323	C₂×C₈	D₄×D₁₃	1	1	yes	characteristic
324	C₂×C₈	#6129	1	1	yes	characteristic
325	C₂×C₈	#6129	1	1	yes	characteristic
326	C₂×C₈	C₂³×D₁₃	1	1	yes	characteristic
327	C₄²	—	13	2	no	—
328	C₄²	—	26	1	no	—
329	C₁₆	—	2	2	no	—
330	C₁₆	—	13	2	no	—
331	C₁₆	—	13	2	no	—
332	C₁₃	C₂×#2178	1	1	yes	characteristic, complemented, Sylow
333	C₂³	—	4	1	no	—
334	C₂³	—	52	1	no	—
335	C₂³	#88170	1	1	yes	characteristic
336	Q₈	—	2	2	no	—
337	Q₈	—	4	1	no	—
338	Q₈	—	26	2	no	—
339	Q₈	—	52	1	no	—
340	D₄	—	2	2	no	—
341	D₄	—	4	2	no	—
342	D₄	—	4	2	no	—
343	D₄	—	4	1	no	—
344	D₄	—	26	2	no	—
345	D₄	—	52	2	no	—
346	D₄	—	52	2	no	—
347	D₄	—	52	1	no	—
348	C₂×C₄	—	4	1	no	—
349	C₂×C₄	—	4	2	no	—
350	C₂×C₄	—	4	2	no	—
351	C₂×C₄	—	26	4	no	—
352	C₂×C₄	—	26	2	no	—
353	C₂×C₄	—	52	1	no	—
354	C₂×C₄	—	52	2	no	—
355	C₂×C₄	—	52	2	no	—
356	C₂×C₄	#88135	1	1	yes	characteristic
357	C₂×C₄	#88135	1	1	yes	characteristic
358	C₂×C₄	D₈×D₁₃	1	1	yes	characteristic
359	C₂×C₄	D₈×D₁₃	1	1	yes	characteristic
360	C₂×C₄	#88170	1	1	yes	characteristic
361	C₂×C₄	C₂×D₄×D₁₃	1	1	yes	characteristic
362	C₈	—	4	2	no	—
363	C₈	—	13	2	no	—
364	C₈	—	13	2	no	—
365	C₈	—	52	2	no	—
366	C₈	#88135	1	1	yes	characteristic
367	C₈	#88135	1	1	yes	characteristic
368	C₈	C₂²×C₄×D₁₃	1	1	yes	characteristic
369	C₈	C₂×#6129	1	1	yes	characteristic
370	C₂²	—	2	2	no	—
371	C₂²	—	4	2	no	—
372	C₂²	—	8	2	no	—
373	C₂²	—	52	2	no	—
374	C₂²	—	104	2	no	—
375	C₂²	#11718417	1	1	yes	characteristic
376	C₂²	#11718417	1	1	yes	characteristic
377	C₂²	C₂×#88170	1	1	yes	characteristic
378	C₄	—	4	2	no	—
379	C₄	—	26	4	no	—
380	C₄	—	52	2	no	—
381	C₄	#11718417	1	1	yes	characteristic
382	C₄	#11718417	1	1	yes	characteristic
383	C₄	C₂×#88135	1	1	yes	characteristic
384	C₄	C₂×D₈×D₁₃	1	1	yes	characteristic
385	C₂	—	2	1	no	—
386	C₂	—	2	1	no	—
387	C₂	—	8	2	no	—
388	C₂	—	104	2	no	—
389	C₂	#13116714	1	2	yes	direct product factor
390	C₂	C₂×#11718417	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	D₈×D₁₃
commutator	C₈×C₁₃	C₂³×C₄
Frattini	C₂×C₈	C₂³×D₁₃
Fitting	C₂×C₁₃×#359	C₂
socle	C₂²×C₁₃	C₄×D₈
prime radical	C₂×#11718855	C₂
cosocle	C₂×C₈×C₁₃	C₂⁴
hypercenter	C₂×#359	D₁₃
hypocenter	C₁₃	C₂×#2178
LCS	G ▷ C₈×C₁₃ ▷ C₄×C₁₃ ▷ C₂×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄ ◁ C₂²×C₈ ◁ C₂×#359

Complements

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

N	H ≅ G/N	type
2. C₂×#11718857	C₂	semidirect
3. C₂×#11718556	C₂	semidirect
4. #13116714	C₂	direct (N × H)
5. C₂×#11718085	C₂	semidirect
6. C₂×#11718102	C₂	semidirect
7. C₂×C₁₆×Dic₁₃	C₂	semidirect
8. C₂×C₁₃×#359	C₂	semidirect
17. #11718857	C₂²	semidirect
18. #11718855	C₄	semidirect
20. C₂×#88069	C₄	semidirect
21. C₂×#88068	C₄	semidirect
22. C₂×#88067	C₄	semidirect
23. C₂×#88066	C₄	semidirect
29. #11718556	C₂²	semidirect
30. C₁₃×#359	C₂²	semidirect
31. #11718102	C₂²	semidirect
32. C₁₆×Dic₁₃	C₂²	semidirect
33. #11718085	C₂²	semidirect
69. #88069	C₂×C₄	semidirect
70. #88068	C₂×C₄	semidirect
71. #88067	C₂×C₄	semidirect
72. #88066	C₂×C₄	semidirect
144. C₂×#359	D₁₃	semidirect
206. #359	C₂×D₁₃	semidirect
274. C₂×C₁₃	#2178	semidirect
332. C₁₃	C₂×#2178	semidirect
389. C₂	#13116714	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂×#11718855	2
C₂×#11718857	2
C₂×#11718556	2
#13116714	2
C₂×#11718085	2
C₂×#11718102	2
C₂×C₁₆×Dic₁₃	2
C₂×C₁₃×#359	2
C₂×#2178	13

Subgroups — C2×#13116714 #16016316

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#13116714 #16016316