Subgroups — C₂×#13116724 #16016349

Order 3328.

646 conjugacy classes of subgroups.

The subgroup lattice has 360 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#11718856	C₂	1	1	yes	maximal, characteristic
2	C₂×#11718858	C₂	1	1	yes	maximal, characteristic
3	C₂×#11718514	C₂	1	1	yes	maximal, characteristic, complemented
4	#13116724	C₂	1	8	yes	maximal, direct product factor
5	C₂×#11718079	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×#11718109	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂×#11718104	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×C₁₃×#360	C₂	1	1	yes	maximal, characteristic
9	C₂×C₁₃×#135	C₂²	1	1	yes	characteristic
10	C₂×#88202	—	2	1	no	—
11	C₂²×Dic₅₂	C₂²	1	1	yes	characteristic
12	C₂×#88141	—	2	1	no	—
13	C₂×C₁₃×M₃₂	C₂²	1	1	yes	characteristic
14	C₂×C₁₃×#106	C₂²	1	1	yes	characteristic
15	C₂×#88077	—	2	1	no	—
16	#11718858	—	2	2	no	—
17	#11718858	C₂²	1	4	yes	—
18	#11718856	C₄	1	4	yes	complemented
19	#11718856	C₂²	1	4	yes	—
20	C₂×#88069	C₄	1	2	yes	complemented
21	C₂×#88067	C₄	1	2	yes	complemented
22	C₂×#88059	C₂²	1	1	yes	characteristic
23	C₂×#88055	C₂²	1	1	yes	characteristic
24	C₂×#88052	C₂²	1	1	yes	characteristic
25	C₂×#88050	—	2	1	no	—
26	#11718514	—	2	2	no	—
27	#11718514	C₂²	1	4	yes	complemented
28	C₁₃×#360	C₂²	1	4	yes	—
29	#11718109	C₂²	1	4	yes	complemented
30	#11718104	C₂²	1	4	yes	complemented
31	#11718079	C₂²	1	4	yes	complemented
32	C₂×C₁₃×#41	—	2	1	no	—
33	C₂²×Dic₂₆	—	2	1	no	—
34	C₁₃×#135	—	2	1	no	—
35	C₁₃×#135	—	2	1	no	—
36	C₁₃×#135	C₂×C₄	1	2	yes	—
37	C₁₃×#135	C₂³	1	2	yes	—
38	C₂×C₁₃×Q₁₆	C₂×C₄	1	1	yes	characteristic
39	C₂×C₁₃×SD₁₆	—	2	1	no	—
40	C₂×C₁₃×D₈	C₂×C₄	1	1	yes	characteristic
41	C₂²×C₈×C₁₃	C₂³	1	1	yes	characteristic
42	C₂×C₁₃×C₄⋊C₄	—	2	1	no	—
43	#88202	—	2	4	no	—
44	#88202	—	4	2	no	—
45	C₂×#6101	—	4	1	no	—
46	C₂×C₄×Dic₁₃	—	4	1	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	C₂×Dic₅₂	—	2	1	no	—
53	C₂×Dic₅₂	—	2	1	no	—
54	C₂×Dic₅₂	—	2	1	no	—
55	C₂×Dic₅₂	C₂×C₄	1	2	yes	—
56	C₂×Dic₅₂	C₂×C₄	1	2	yes	—
57	C₂×Dic₅₂	C₂³	1	2	yes	—
58	#88141	—	2	4	no	—
59	#88141	—	4	2	no	—
60	C₁₃×M₃₂	—	2	1	no	—
61	C₁₃×M₃₂	C₂³	1	2	yes	—
62	C₂×C₁₃×C₁₆	—	2	1	no	—
63	C₁₃×#106	—	2	1	no	—
64	C₁₃×#106	C₂³	1	2	yes	—
65	#88077	—	2	4	no	—
66	#88069	C₂×C₄	1	8	yes	complemented
67	#88067	C₂×C₄	1	8	yes	complemented
68	#88059	—	2	1	no	—
69	#88059	C₂³	1	2	yes	—
70	#88055	D₄	1	2	yes	—
71	#88055	C₂³	1	2	yes	—
72	#88052	C₂×C₄	1	2	yes	—
73	#88052	C₂³	1	2	yes	—
74	C₂×#6090	C₂×C₄	1	2	yes	—
75	#88050	—	2	4	no	—
76	C₂×#2180	—	13	1	no	maximal, Sylow
77	C₁₃×#41	—	2	2	no	—
78	C₁₃×#41	—	2	2	no	—
79	C₁₃×#41	—	4	1	no	—
80	C₁₃×#41	—	4	1	no	—
81	C₂×Q₈×C₁₃	—	2	1	no	—
82	C₂×D₄×C₁₃	—	2	1	no	—
83	C₂×D₄×C₁₃	—	4	1	no	—
84	C₂²×C₄×C₁₃	—	4	1	no	—
85	C₂²×C₄×C₁₃	—	4	1	no	—
86	C₂²×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
87	C₂²×Dic₁₃	—	2	1	no	—
88	C₂²×Dic₁₃	—	4	1	no	—
89	C₂×Dic₂₆	—	2	2	no	—
90	C₂×Dic₂₆	—	2	2	no	—
91	C₂×Dic₂₆	—	2	2	no	—
92	C₂×Dic₂₆	—	4	1	no	—
93	C₂×Dic₂₆	—	4	1	no	—
94	C₂×Dic₂₆	—	4	1	no	—
95	C₁₃×Q₁₆	—	2	1	no	—
96	C₁₃×Q₁₆	C₂²×C₄	1	2	yes	—
97	C₁₃×SD₁₆	—	2	2	no	—
98	C₁₃×SD₁₆	—	2	2	no	—
99	C₁₃×D₈	—	2	1	no	—
100	C₁₃×D₈	C₂²×C₄	1	2	yes	—
101	C₂×C₈×C₁₃	C₂²×C₄	1	1	yes	characteristic
102	C₂×C₈×C₁₃	C₂²×C₄	1	1	yes	characteristic
103	C₂×C₈×C₁₃	C₂×D₄	1	1	yes	characteristic
104	C₂×C₈×C₁₃	#41	1	1	yes	characteristic
105	C₂×C₈×C₁₃	#41	1	1	yes	characteristic
106	C₂×C₈×C₁₃	C₂⁴	1	1	yes	characteristic
107	C₁₃×C₄⋊C₄	—	2	2	no	—
108	C₁₃×C₄⋊C₄	—	4	1	no	—
109	#6101	—	4	4	no	—
110	C₄×Dic₁₃	—	2	1	no	—
111	C₄×Dic₁₃	—	4	4	no	—
112	C₄×Dic₁₃	C₂×D₄	1	2	yes	—
113	#6099	—	2	2	no	—
114	#6099	—	4	4	no	—
115	#6099	—	4	1	no	—
116	C₂×#1055	—	2	1	no	—
117	C₂×#1055	—	2	1	no	—
118	C₂×#1055	—	4	4	no	—
119	C₂×#1055	—	4	1	no	—
120	C₂×#1055	C₂×D₄	1	2	yes	—
121	Dic₅₂	—	2	2	no	—
122	Dic₅₂	—	2	2	no	—
123	Dic₅₂	—	2	2	no	—
124	Dic₅₂	C₂²×C₄	1	4	yes	—
125	C₁₃×C₁₆	—	2	2	no	—
126	#6090	C₂²×C₄	1	4	yes	—
127	C₂×#510	—	13	1	no	—
128	C₂×#444	—	13	1	no	—
129	C₂×#441	—	13	1	no	—
130	#2180	—	13	8	no	—
131	C₂×#365	—	13	1	no	—
132	C₂×#362	—	13	1	no	—
133	C₂×#360	D₁₃	1	1	yes	characteristic
134	C₂×#347	—	13	1	no	—
135	C₂³×C₁₃	—	4	1	no	—
136	C₂³×C₁₃	AGL(1,ℤ/8)	1	1	yes	characteristic
137	Q₈×C₁₃	—	2	2	no	—
138	Q₈×C₁₃	—	4	1	no	—
139	D₄×C₁₃	—	2	2	no	—
140	D₄×C₁₃	—	4	2	no	—
141	D₄×C₁₃	—	4	2	no	—
142	D₄×C₁₃	—	4	1	no	—
143	C₂×C₄×C₁₃	—	4	1	no	—
144	C₂×C₄×C₁₃	—	4	2	no	—
145	C₂×C₄×C₁₃	—	4	2	no	—
146	C₂×C₄×C₁₃	—	4	2	no	—
147	C₂×C₄×C₁₃	—	8	2	no	—
148	C₂×C₄×C₁₃	C₄×D₄	1	1	yes	characteristic
149	C₂×C₄×C₁₃	C₄×D₄	1	1	yes	characteristic
150	C₂×C₄×C₁₃	AGL(1,ℤ/8)	1	1	yes	characteristic
151	C₂×C₄×C₁₃	AGL(1,ℤ/8)	1	1	yes	characteristic
152	C₂×C₄×C₁₃	AGL(1,ℤ/8)	1	1	yes	characteristic
153	C₂×C₄×C₁₃	C₂²×D₄	1	1	yes	characteristic
154	C₂×Dic₁₃	—	2	2	no	—
155	C₂×Dic₁₃	—	4	2	no	—
156	C₂×Dic₁₃	—	4	2	no	—
157	C₂×Dic₁₃	—	4	2	no	—
158	C₂×Dic₁₃	—	4	2	no	—
159	Dic₂₆	—	2	4	no	—
160	Dic₂₆	—	4	2	no	—
161	Dic₂₆	—	4	2	no	—
162	Dic₂₆	—	4	2	no	—
163	C₈×C₁₃	C₄×D₄	1	1	yes	characteristic
164	C₈×C₁₃	C₄×D₄	1	1	yes	characteristic
165	C₈×C₁₃	C₂³×C₄	1	1	yes	characteristic
166	C₈×C₁₃	C₂×#41	1	1	yes	characteristic
167	#1055	—	2	2	no	—
168	#1055	—	4	2	no	—
169	C₂×#135	C₂×D₁₃	1	1	yes	characteristic
170	C₂²×Q₁₆	—	13	1	no	—
171	C₂×#131	—	26	1	no	—
172	C₂×C₄×Q₈	—	26	1	no	—
173	#510	—	13	4	no	—
174	#510	—	13	4	no	—
175	C₂×Q₃₂	—	13	2	no	—
176	C₂×SD₃₂	—	13	2	no	—
177	C₂×M₃₂	—	13	1	no	—
178	C₂×M₃₂	C₂×D₁₃	1	1	yes	characteristic
179	#444	—	13	4	no	—
180	#444	—	26	2	no	—
181	#441	—	13	4	no	—
182	#441	—	26	2	no	—
183	C₂×#108	—	13	1	no	—
184	C₂×#106	C₂×D₁₃	1	1	yes	characteristic
185	C₂×#104	—	26	1	no	—
186	C₂×#103	—	26	1	no	—
187	C₂×#97	—	13	1	no	—
188	#365	—	13	4	no	—
189	#362	—	13	4	no	—
190	#360	C₂×D₁₃	1	4	yes	—
191	#347	—	13	4	no	—
192	C₂²×C₁₃	—	2	2	no	—
193	C₂²×C₁₃	—	4	2	no	—
194	C₂²×C₁₃	—	8	2	no	—
195	C₂²×C₁₃	#442	1	1	yes	characteristic
196	C₂²×C₁₃	#442	1	1	yes	characteristic
197	C₂²×C₁₃	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
198	C₄×C₁₃	—	4	2	no	—
199	C₄×C₁₃	—	8	2	no	—
200	C₄×C₁₃	#442	1	1	yes	characteristic
201	C₄×C₁₃	#442	1	1	yes	characteristic
202	C₄×C₁₃	C₂×C₄×D₄	1	1	yes	characteristic
203	C₄×C₁₃	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
204	Dic₁₃	—	4	2	no	—
205	Dic₁₃	—	4	4	no	—
206	C₂×#41	—	2	1	no	—
207	C₂²×Q₈	—	26	1	no	—
208	#135	—	2	1	no	—
209	#135	—	2	1	no	—
210	#135	C₄×D₁₃	1	2	yes	—
211	#135	C₂²×D₁₃	1	2	yes	—
212	C₂×Q₁₆	—	13	2	no	—
213	C₂×Q₁₆	—	13	2	no	—
214	C₂×Q₁₆	—	13	2	no	—
215	C₂×Q₁₆	—	26	1	no	—
216	C₂×Q₁₆	—	26	1	no	—
217	C₂×Q₁₆	—	26	1	no	—
218	C₂×Q₁₆	C₄×D₁₃	1	1	yes	characteristic
219	C₂×SD₁₆	—	2	1	no	—
220	C₂×D₈	C₄×D₁₃	1	1	yes	characteristic
221	#131	—	26	4	no	—
222	#131	—	52	2	no	—
223	C₂×M₁₆	—	26	1	no	—
224	C₂×M₁₆	—	52	1	no	—
225	C₂²×C₈	—	13	1	no	—
226	C₂²×C₈	—	52	1	no	—
227	C₂²×C₈	C₂²×D₁₃	1	1	yes	characteristic
228	C₄×Q₈	—	26	4	no	—
229	C₄×Q₈	—	52	2	no	—
230	C₂×C₄⋊C₄	—	2	1	no	—
231	C₂×C₄⋊C₄	—	52	1	no	—
232	C₂×C₄²	—	13	1	no	—
233	C₂×C₄²	—	52	1	no	—
234	Q₃₂	—	13	8	no	—
235	SD₃₂	—	13	8	no	—
236	M₃₂	—	2	1	no	—
237	M₃₂	—	13	2	no	—
238	M₃₂	—	13	2	no	—
239	M₃₂	C₂²×D₁₃	1	2	yes	—
240	C₂×C₁₆	—	2	1	no	—
241	C₂×C₁₆	—	13	2	no	—
242	#108	—	13	2	no	—
243	#108	—	26	1	no	—
244	#106	—	2	1	no	—
245	#106	C₂²×D₁₃	1	2	yes	—
246	#104	—	26	4	no	—
247	#103	—	26	4	no	—
248	#97	—	13	2	no	—
249	#97	—	13	2	no	—
250	C₂×C₁₃	—	2	1	no	—
251	C₂×C₁₃	—	2	1	no	—
252	C₂×C₁₃	—	8	2	no	—
253	C₂×C₁₃	#2180	1	2	yes	complemented
254	C₂×C₁₃	C₂×#442	1	1	yes	characteristic
255	#41	—	2	2	no	—
256	#41	—	2	2	no	—
257	#41	—	4	1	no	—
258	#41	—	4	1	no	—
259	C₂×Q₈	—	2	1	no	—
260	C₂×Q₈	—	26	2	no	—
261	C₂×Q₈	—	26	2	no	—
262	C₂×Q₈	—	26	2	no	—
263	C₂×Q₈	—	52	1	no	—
264	C₂×Q₈	—	52	1	no	—
265	C₂×Q₈	—	52	1	no	—
266	C₂×D₄	—	2	1	no	—
267	C₂×D₄	—	4	1	no	—
268	C₂²×C₄	—	4	1	no	—
269	C₂²×C₄	—	4	1	no	—
270	C₂²×C₄	—	26	1	no	—
271	C₂²×C₄	—	52	1	no	—
272	C₂²×C₄	D₄×D₁₃	1	1	yes	characteristic
273	Q₁₆	—	2	1	no	—
274	Q₁₆	—	13	4	no	—
275	Q₁₆	—	26	2	no	—
276	Q₁₆	—	26	2	no	—
277	Q₁₆	—	26	2	no	—
278	Q₁₆	C₂×C₄×D₁₃	1	2	yes	—
279	SD₁₆	—	2	2	no	—
280	SD₁₆	—	2	2	no	—
281	D₈	—	2	1	no	—
282	D₈	C₂×C₄×D₁₃	1	2	yes	—
283	M₁₆	—	26	2	no	—
284	M₁₆	—	52	4	no	—
285	M₁₆	—	52	1	no	—
286	C₂×C₈	—	13	2	no	—
287	C₂×C₈	—	26	1	no	—
288	C₂×C₈	—	26	1	no	—
289	C₂×C₈	—	52	4	no	—
290	C₂×C₈	—	52	1	no	—
291	C₂×C₈	C₂×C₄×D₁₃	1	1	yes	characteristic
292	C₂×C₈	C₂×C₄×D₁₃	1	1	yes	characteristic
293	C₂×C₈	D₄×D₁₃	1	1	yes	characteristic
294	C₂×C₈	#6129	1	1	yes	characteristic
295	C₂×C₈	#6129	1	1	yes	characteristic
296	C₂×C₈	C₂³×D₁₃	1	1	yes	characteristic
297	C₄⋊C₄	—	2	2	no	—
298	C₄⋊C₄	—	4	1	no	—
299	C₄⋊C₄	—	52	4	no	—
300	C₄²	—	13	2	no	—
301	C₄²	—	26	1	no	—
302	C₄²	—	52	4	no	—
303	C₁₆	—	2	2	no	—
304	C₁₆	—	13	4	no	—
305	C₁₃	C₂×#2180	1	1	yes	characteristic, complemented, Sylow
306	C₂³	—	4	1	no	—
307	C₂³	#88165	1	1	yes	characteristic
308	Q₈	—	2	2	no	—
309	Q₈	—	4	1	no	—
310	Q₈	—	26	4	no	—
311	Q₈	—	52	2	no	—
312	Q₈	—	52	2	no	—
313	Q₈	—	52	2	no	—
314	D₄	—	2	2	no	—
315	D₄	—	4	2	no	—
316	D₄	—	4	2	no	—
317	D₄	—	4	1	no	—
318	C₂×C₄	—	4	1	no	—
319	C₂×C₄	—	4	2	no	—
320	C₂×C₄	—	4	2	no	—
321	C₂×C₄	—	4	2	no	—
322	C₂×C₄	—	8	2	no	—
323	C₂×C₄	—	26	2	no	—
324	C₂×C₄	—	52	2	no	—
325	C₂×C₄	—	52	2	no	—
326	C₂×C₄	—	52	2	no	—
327	C₂×C₄	—	52	2	no	—
328	C₂×C₄	#88135	1	1	yes	characteristic
329	C₂×C₄	#88135	1	1	yes	characteristic
330	C₂×C₄	#88165	1	1	yes	characteristic
331	C₂×C₄	#88168	1	1	yes	characteristic
332	C₂×C₄	#88168	1	1	yes	characteristic
333	C₂×C₄	C₂×D₄×D₁₃	1	1	yes	characteristic
334	C₈	—	26	2	no	—
335	C₈	—	52	2	no	—
336	C₈	#88135	1	1	yes	characteristic
337	C₈	#88135	1	1	yes	characteristic
338	C₈	C₂²×C₄×D₁₃	1	1	yes	characteristic
339	C₈	C₂×#6129	1	1	yes	characteristic
340	C₂²	—	2	2	no	—
341	C₂²	—	4	2	no	—
342	C₂²	—	8	2	no	—
343	C₂²	#11718446	1	1	yes	characteristic
344	C₂²	#11718446	1	1	yes	characteristic
345	C₂²	C₂×#88165	1	1	yes	characteristic
346	C₄	—	4	2	no	—
347	C₄	—	8	2	no	—
348	C₄	—	52	2	no	—
349	C₄	—	52	4	no	—
350	C₄	#11718446	1	1	yes	characteristic
351	C₄	#11718446	1	1	yes	characteristic
352	C₄	C₂×#88135	1	1	yes	characteristic
353	C₄	C₂×#88168	1	1	yes	characteristic
354	C₂	—	2	1	no	—
355	C₂	—	2	1	no	—
356	C₂	—	8	2	no	—
357	C₂	#13116724	1	2	yes	direct product factor
358	C₂	C₂×#11718446	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	#11718446
commutator	C₈×C₁₃	C₂³×C₄
Frattini	C₂×C₈	C₂³×D₁₃
Fitting	C₂×C₁₃×#360	C₂
socle	C₂²×C₁₃	#442
prime radical	C₂×C₁₃×#135	C₂²
cosocle	C₂×C₈×C₁₃	C₂⁴
hypercenter	C₂×#360	D₁₃
hypocenter	C₁₃	C₂×#2180
LCS	G ▷ C₈×C₁₃ ▷ C₄×C₁₃ ▷ C₂×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂²×C₈ ◁ C₂×#360

Complements

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

N	H ≅ G/N	type
3. C₂×#11718514	C₂	semidirect
4. #13116724	C₂	direct (N × H)
5. C₂×#11718079	C₂	semidirect
6. C₂×#11718109	C₂	semidirect
7. C₂×#11718104	C₂	semidirect
18. #11718856	C₄	semidirect
20. C₂×#88069	C₄	semidirect
21. C₂×#88067	C₄	semidirect
27. #11718514	C₂²	semidirect
29. #11718109	C₂²	semidirect
30. #11718104	C₂²	semidirect
31. #11718079	C₂²	semidirect
66. #88069	C₂×C₄	semidirect
67. #88067	C₂×C₄	semidirect
253. C₂×C₁₃	#2180	semidirect
305. C₁₃	C₂×#2180	semidirect
357. C₂	#13116724	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×#11718856	2
C₂×#11718858	2
C₂×#11718514	2
#13116724	2
C₂×#11718079	2
C₂×#11718109	2
C₂×#11718104	2
C₂×C₁₃×#360	2
C₂×#2180	13

Subgroups — C2×#13116724 #16016349

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#13116724 #16016349