Subgroups — C₂×#13116947 #16017375

Order 3328.

734 conjugacy classes of subgroups.

The subgroup lattice has 352 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×#88067	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂²×#88052	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×#11718777	C₂	1	1	yes	maximal, characteristic, complemented
4	#13116947	C₂	1	8	yes	maximal, direct product factor
5	C₂×#11718081	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×#11718084	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂×#11718154	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×C₁₃×#466	C₂	1	1	yes	maximal, characteristic
9	C₂²×C₁₃×D₈	C₂²	1	1	yes	characteristic
10	C₂³×C₈×C₁₃	C₂²	1	1	yes	characteristic
11	C₂×C₁₃×#121	—	2	1	no	—
12	C₂×#88178	—	2	1	no	—
13	C₂²×Dic₅₂	C₂²	1	1	yes	characteristic
14	C₁₃×#466	—	2	2	no	—
15	C₁₃×#466	C₂²	1	4	yes	—
16	C₂×C₁₃×#107	C₂²	1	1	yes	characteristic
17	C₂×C₁₃×#102	—	2	1	no	—
18	C₂×#88067	—	2	4	no	—
19	C₂×#88067	C₂²	1	4	yes	complemented
20	#11718777	—	2	2	no	—
21	#11718777	C₂²	1	4	yes	complemented
22	C₂×#88058	C₂²	1	1	yes	characteristic
23	C₂×#88056	—	2	1	no	—
24	C₂×#88052	—	2	4	no	—
25	C₂×#88052	C₂²	1	4	yes	complemented
26	C₂²×#6090	C₂²	1	1	yes	characteristic
27	C₂²×#6090	C₂²	1	1	yes	characteristic
28	#11718154	C₂²	1	4	yes	complemented
29	#11718084	C₂²	1	4	yes	complemented
30	#11718081	C₂²	1	4	yes	complemented
31	C₂²×D₄×C₁₃	—	2	1	no	—
32	C₂²×D₄×C₁₃	—	4	1	no	—
33	C₂³×C₄×C₁₃	D₄	1	1	yes	characteristic
34	C₂²×Dic₂₆	—	2	1	no	—
35	C₂×C₁₃×D₈	—	2	2	no	—
36	C₂×C₁₃×D₈	—	2	1	no	—
37	C₂×C₁₃×D₈	—	2	2	no	—
38	C₂×C₁₃×D₈	C₂³	1	2	yes	—
39	C₂²×C₈×C₁₃	—	2	2	no	—
40	C₂²×C₈×C₁₃	—	2	2	no	—
41	C₂²×C₈×C₁₃	D₄	1	1	yes	characteristic
42	C₂²×C₈×C₁₃	D₄	1	2	yes	—
43	C₂²×C₈×C₁₃	C₂³	1	2	yes	—
44	C₂²×C₈×C₁₃	C₂³	1	1	yes	characteristic
45	C₁₃×#121	—	2	4	no	—
46	C₁₃×#121	—	4	2	no	—
47	C₂×C₁₃×C₄⋊C₄	—	2	1	no	—
48	C₂×C₁₃×C₂²⋊C₄	—	4	1	no	—
49	C₂×C₂²⋊Dic₁₃	—	4	1	no	—
50	C₂×C₄⋊Dic₁₃	—	2	1	no	—
51	#88178	—	2	4	no	—
52	#88178	—	4	2	no	—
53	C₂×#6101	—	4	1	no	—
54	C₂×Dic₅₂	—	2	2	no	—
55	C₂×Dic₅₂	—	2	2	no	—
56	C₂×Dic₅₂	—	2	1	no	—
57	C₂×Dic₅₂	C₂³	1	2	yes	—
58	C₁₃×#107	—	2	1	no	—
59	C₁₃×#107	C₂³	1	2	yes	—
60	C₁₃×#102	—	2	4	no	—
61	#88067	—	2	8	no	—
62	#88058	—	2	1	no	—
63	#88058	C₂³	1	2	yes	—
64	#88056	—	2	4	no	—
65	#88052	—	2	8	no	—
66	C₂×#6090	—	2	2	no	—
67	C₂×#6090	D₄	1	4	yes	complemented
68	C₂×#6090	C₂³	1	2	yes	—
69	C₂×#6090	C₂³	1	2	yes	—
70	C₂×#2220	—	13	1	no	maximal, Sylow
71	C₂⁴×C₁₃	—	4	1	no	—
72	C₂⁴×C₁₃	D₈	1	1	yes	characteristic
73	C₂×D₄×C₁₃	—	2	2	no	—
74	C₂×D₄×C₁₃	—	4	4	no	—
75	C₂×D₄×C₁₃	—	4	2	no	—
76	C₂×D₄×C₁₃	—	4	1	no	—
77	C₂×D₄×C₁₃	—	4	2	no	—
78	C₂×D₄×C₁₃	—	8	4	no	—
79	C₂²×C₄×C₁₃	—	2	2	no	—
80	C₂²×C₄×C₁₃	—	2	2	no	—
81	C₂²×C₄×C₁₃	—	4	1	no	—
82	C₂²×C₄×C₁₃	D₈	1	1	yes	characteristic
83	C₂²×C₄×C₁₃	D₈	1	2	yes	—
84	C₂²×C₄×C₁₃	C₂×D₄	1	2	yes	—
85	C₂²×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
86	C₂²×Dic₁₃	—	4	1	no	—
87	C₂²×Dic₁₃	—	4	1	no	—
88	C₂×Dic₂₆	—	2	2	no	—
89	C₂×Dic₂₆	—	4	2	no	—
90	C₂×Dic₂₆	—	4	2	no	—
91	C₂×Dic₂₆	—	4	1	no	—
92	C₁₃×D₈	—	2	2	no	—
93	C₁₃×D₈	—	2	2	no	—
94	C₁₃×D₈	—	4	2	no	—
95	C₂×C₈×C₁₃	—	2	2	no	—
96	C₂×C₈×C₁₃	—	2	4	no	—
97	C₂×C₈×C₁₃	—	2	2	no	—
98	C₂×C₈×C₁₃	—	2	2	no	—
99	C₂×C₈×C₁₃	C₂×D₄	1	2	yes	—
100	C₂×C₈×C₁₃	C₂×D₄	1	1	yes	characteristic
101	C₂×C₈×C₁₃	C₂×D₄	1	2	yes	—
102	C₂×C₈×C₁₃	#41	1	2	yes	—
103	C₂×C₈×C₁₃	C₂⁴	1	1	yes	characteristic
104	C₁₃×C₄⋊C₄	—	2	2	no	—
105	C₁₃×C₄⋊C₄	—	4	1	no	—
106	C₁₃×C₂²⋊C₄	—	4	4	no	—
107	C₂²⋊Dic₁₃	—	4	4	no	—
108	C₄⋊Dic₁₃	—	2	2	no	—
109	C₄⋊Dic₁₃	—	4	1	no	—
110	#6101	—	4	4	no	—
111	Dic₅₂	—	2	2	no	—
112	Dic₅₂	—	2	2	no	—
113	Dic₅₂	—	4	2	no	—
114	#6090	—	2	2	no	—
115	#6090	C₂×D₄	1	4	yes	complemented
116	C₂²×SD₃₂	—	13	1	no	—
117	C₂²×M₃₂	—	13	1	no	—
118	C₂×#467	—	13	1	no	—
119	C₂×#466	D₁₃	1	1	yes	characteristic
120	#2220	—	13	8	no	—
121	C₂×#366	—	13	1	no	—
122	C₂×#358	—	13	1	no	—
123	C₂×#357	—	13	1	no	—
124	C₂³×C₁₃	—	2	2	no	—
125	C₂³×C₁₃	—	2	2	no	—
126	C₂³×C₁₃	—	2	2	no	—
127	C₂³×C₁₃	—	4	2	no	—
128	C₂³×C₁₃	—	8	2	no	—
129	C₂³×C₁₃	—	8	2	no	—
130	C₂³×C₁₃	—	8	2	no	—
131	C₂³×C₁₃	C₂×D₈	1	2	yes	—
132	C₂³×C₁₃	C₂×D₈	1	1	yes	characteristic
133	D₄×C₁₃	—	4	2	no	—
134	D₄×C₁₃	—	4	2	no	—
135	D₄×C₁₃	—	8	8	no	—
136	D₄×C₁₃	—	8	2	no	—
137	C₂×C₄×C₁₃	—	2	2	no	—
138	C₂×C₄×C₁₃	—	2	2	no	—
139	C₂×C₄×C₁₃	—	2	2	no	—
140	C₂×C₄×C₁₃	—	2	4	no	—
141	C₂×C₄×C₁₃	—	4	2	no	—
142	C₂×C₄×C₁₃	—	8	2	no	—
143	C₂×C₄×C₁₃	#121	1	2	yes	—
144	C₂×C₄×C₁₃	C₂×D₈	1	2	yes	—
145	C₂×C₄×C₁₃	C₂×D₈	1	1	yes	characteristic
146	C₂×C₄×C₁₃	#135	1	2	yes	—
147	C₂×C₄×C₁₃	C₂²×D₄	1	1	yes	characteristic
148	C₂×Dic₁₃	—	4	2	no	—
149	C₂×Dic₁₃	—	4	2	no	—
150	C₂×Dic₁₃	—	8	2	no	—
151	C₂×Dic₁₃	—	8	2	no	—
152	Dic₂₆	—	4	2	no	—
153	Dic₂₆	—	4	2	no	—
154	Dic₂₆	—	8	2	no	—
155	C₈×C₁₃	—	2	2	no	—
156	C₈×C₁₃	#121	1	2	yes	—
157	C₈×C₁₃	C₂²×D₄	1	1	yes	characteristic
158	C₈×C₁₃	C₂×#41	1	1	yes	characteristic
159	C₂²×Q₁₆	—	13	1	no	—
160	C₂²×D₈	C₂×D₁₃	1	1	yes	characteristic
161	C₂³×C₈	C₂×D₁₃	1	1	yes	characteristic
162	C₂×C₂²⋊Q₈	—	26	1	no	—
163	C₂×#121	—	2	1	no	—
164	C₂×SD₃₂	—	13	4	no	—
165	C₂×SD₃₂	—	26	4	no	—
166	C₂×M₃₂	—	13	4	no	—
167	C₂×M₃₂	—	26	4	no	—
168	C₂²×C₁₆	—	13	1	no	—
169	C₂²×C₁₆	—	13	1	no	—
170	#467	—	13	4	no	—
171	#467	—	26	2	no	—
172	#466	—	2	2	no	—
173	#466	C₂×D₁₃	1	4	yes	—
174	C₂×#107	—	13	1	no	—
175	C₂×#107	C₂×D₁₃	1	1	yes	characteristic
176	C₂×#103	—	26	1	no	—
177	C₂×#102	—	2	1	no	—
178	#366	—	13	4	no	—
179	#358	—	13	4	no	—
180	#357	—	13	4	no	—
181	C₂²×C₁₃	—	2	2	no	—
182	C₂²×C₁₃	—	2	2	no	—
183	C₂²×C₁₃	—	2	2	no	—
184	C₂²×C₁₃	—	4	4	no	—
185	C₂²×C₁₃	—	8	2	no	—
186	C₂²×C₁₃	—	8	2	no	—
187	C₂²×C₁₃	—	8	2	no	—
188	C₂²×C₁₃	—	16	4	no	—
189	C₂²×C₁₃	#466	1	2	yes	—
190	C₂²×C₁₃	#509	1	2	yes	—
191	C₂²×C₁₃	#510	1	2	yes	—
192	C₂²×C₁₃	C₂²×D₈	1	1	yes	characteristic
193	C₄×C₁₃	—	2	2	no	—
194	C₄×C₁₃	—	8	2	no	—
195	C₄×C₁₃	#466	1	2	yes	—
196	C₄×C₁₃	C₂×#121	1	1	yes	characteristic
197	C₄×C₁₃	C₂×#135	1	1	yes	characteristic
198	Dic₁₃	—	8	2	no	—
199	Dic₁₃	—	8	2	no	—
200	C₂²×Q₈	—	26	1	no	—
201	C₂²×D₄	—	2	1	no	—
202	C₂²×D₄	—	4	1	no	—
203	C₂³×C₄	C₂²⋊D₁₃	1	1	yes	characteristic
204	C₂×Q₁₆	—	13	2	no	—
205	C₂×Q₁₆	—	26	2	no	—
206	C₂×Q₁₆	—	26	2	no	—
207	C₂×Q₁₆	—	26	1	no	—
208	C₂×D₈	—	2	2	no	—
209	C₂×D₈	—	2	2	no	—
210	C₂×D₈	—	2	1	no	—
211	C₂×D₈	C₂²×D₁₃	1	2	yes	—
212	C₂²×C₈	—	2	2	no	—
213	C₂²×C₈	—	2	2	no	—
214	C₂²×C₈	C₂²⋊D₁₃	1	1	yes	characteristic
215	C₂²×C₈	C₂²⋊D₁₃	1	2	yes	—
216	C₂²×C₈	C₂²×D₁₃	1	2	yes	—
217	C₂²×C₈	C₂²×D₁₃	1	1	yes	characteristic
218	C₂²⋊Q₈	—	26	4	no	—
219	C₂²⋊Q₈	—	52	2	no	—
220	#121	—	2	4	no	—
221	#121	—	4	2	no	—
222	C₂×C₄⋊C₄	—	2	1	no	—
223	C₂×C₄⋊C₄	—	26	1	no	—
224	C₂×C₄⋊C₄	—	52	1	no	—
225	C₂×C₂²⋊C₄	—	4	1	no	—
226	C₂×C₂²⋊C₄	—	52	1	no	—
227	SD₃₂	—	26	8	no	—
228	M₃₂	—	26	8	no	—
229	C₂×C₁₆	—	13	4	no	—
230	C₂×C₁₆	—	13	2	no	—
231	C₂×C₁₆	—	13	2	no	—
232	C₂×C₁₆	—	26	2	no	—
233	#107	—	2	1	no	—
234	#107	—	13	2	no	—
235	#107	—	26	1	no	—
236	#107	C₂²×D₁₃	1	2	yes	—
237	#103	—	26	4	no	—
238	#102	—	2	4	no	—
239	C₂×C₁₃	—	4	2	no	—
240	C₂×C₁₃	—	16	2	no	—
241	C₂×C₁₃	#2220	1	4	yes	complemented
242	C₂×C₁₃	C₂×#466	1	1	yes	characteristic
243	C₂×C₁₃	C₂×#509	1	1	yes	characteristic
244	C₂×C₁₃	C₂×#510	1	1	yes	characteristic
245	C₂⁴	—	4	1	no	—
246	C₂⁴	#6104	1	1	yes	characteristic
247	C₂×Q₈	—	26	2	no	—
248	C₂×Q₈	—	52	1	no	—
249	C₂×Q₈	—	52	2	no	—
250	C₂×Q₈	—	52	2	no	—
251	C₂×D₄	—	2	2	no	—
252	C₂×D₄	—	4	4	no	—
253	C₂×D₄	—	4	1	no	—
254	C₂×D₄	—	4	2	no	—
255	C₂×D₄	—	4	2	no	—
256	C₂×D₄	—	8	4	no	—
257	C₂²×C₄	—	2	2	no	—
258	C₂²×C₄	—	2	2	no	—
259	C₂²×C₄	—	4	1	no	—
260	C₂²×C₄	—	52	1	no	—
261	C₂²×C₄	—	52	1	no	—
262	C₂²×C₄	#6104	1	2	yes	—
263	C₂²×C₄	#6104	1	1	yes	characteristic
264	C₂²×C₄	C₂×C₂²⋊D₁₃	1	2	yes	—
265	C₂²×C₄	C₂×C₂²⋊D₁₃	1	1	yes	characteristic
266	Q₁₆	—	26	2	no	—
267	Q₁₆	—	26	2	no	—
268	Q₁₆	—	52	2	no	—
269	D₈	—	2	2	no	—
270	D₈	—	2	2	no	—
271	D₈	—	4	2	no	—
272	C₂×C₈	—	2	2	no	—
273	C₂×C₈	—	2	4	no	—
274	C₂×C₈	—	2	2	no	—
275	C₂×C₈	—	2	2	no	—
276	C₂×C₈	D₄×D₁₃	1	2	yes	—
277	C₂×C₈	#6129	1	2	yes	—
278	C₂×C₈	C₂×C₂²⋊D₁₃	1	2	yes	—
279	C₂×C₈	C₂×C₂²⋊D₁₃	1	1	yes	characteristic
280	C₂×C₈	C₂³×D₁₃	1	1	yes	characteristic
281	C₄⋊C₄	—	2	2	no	—
282	C₄⋊C₄	—	4	1	no	—
283	C₄⋊C₄	—	26	2	no	—
284	C₄⋊C₄	—	52	1	no	—
285	C₄⋊C₄	—	52	4	no	—
286	C₂²⋊C₄	—	4	4	no	—
287	C₂²⋊C₄	—	52	4	no	—
288	C₁₆	—	13	4	no	—
289	C₁₆	—	26	2	no	—
290	C₁₃	C₂×#2220	1	1	yes	characteristic, complemented, Sylow
291	C₂³	—	2	2	no	—
292	C₂³	—	2	2	no	—
293	C₂³	—	2	2	no	—
294	C₂³	—	4	2	no	—
295	C₂³	—	8	2	no	—
296	C₂³	—	8	2	no	—
297	C₂³	—	8	2	no	—
298	C₂³	C₂×#6104	1	2	yes	—
299	C₂³	C₂×#6104	1	1	yes	characteristic
300	Q₈	—	52	2	no	—
301	Q₈	—	52	2	no	—
302	Q₈	—	104	2	no	—
303	D₄	—	4	2	no	—
304	D₄	—	4	2	no	—
305	D₄	—	8	8	no	—
306	D₄	—	8	2	no	—
307	C₂×C₄	—	2	2	no	—
308	C₂×C₄	—	2	4	no	—
309	C₂×C₄	—	2	2	no	—
310	C₂×C₄	—	2	2	no	—
311	C₂×C₄	—	4	2	no	—
312	C₂×C₄	—	8	2	no	—
313	C₂×C₄	—	52	2	no	—
314	C₂×C₄	—	52	2	no	—
315	C₂×C₄	—	104	2	no	—
316	C₂×C₄	—	104	2	no	—
317	C₂×C₄	C₂×#6104	1	2	yes	—
318	C₂×C₄	C₂×#6104	1	1	yes	characteristic
319	C₂×C₄	#88193	1	2	yes	—
320	C₂×C₄	#88204	1	2	yes	—
321	C₂×C₄	C₂²×C₂²⋊D₁₃	1	1	yes	characteristic
322	C₈	—	2	2	no	—
323	C₈	#88193	1	2	yes	—
324	C₈	C₂×D₄×D₁₃	1	1	yes	characteristic
325	C₈	C₂×#6129	1	1	yes	characteristic
326	C₂²	—	2	2	no	—
327	C₂²	—	2	2	no	—
328	C₂²	—	2	2	no	—
329	C₂²	—	4	4	no	—
330	C₂²	—	8	2	no	—
331	C₂²	—	8	2	no	—
332	C₂²	—	8	2	no	—
333	C₂²	—	16	4	no	—
334	C₂²	#11718699	1	2	yes	—
335	C₂²	#11718808	1	2	yes	—
336	C₂²	#11718856	1	2	yes	—
337	C₂²	C₂²×#6104	1	1	yes	characteristic
338	C₄	—	2	2	no	—
339	C₄	—	8	2	no	—
340	C₄	—	104	2	no	—
341	C₄	—	104	2	no	—
342	C₄	#11718699	1	2	yes	—
343	C₄	C₂×#88193	1	1	yes	characteristic
344	C₄	C₂×#88204	1	1	yes	characteristic
345	C₂	—	4	2	no	—
346	C₂	—	16	2	no	—
347	C₂	#13116947	1	4	yes	direct product factor
348	C₂	C₂×#11718699	1	1	yes	characteristic
349	C₂	C₂×#11718808	1	1	yes	characteristic
350	C₂	C₂×#11718856	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	C₂×#6104
commutator, cosocle	C₂×C₈×C₁₃	C₂⁴
Frattini	C₂×C₈	C₂³×D₁₃
Fitting, prime radical	C₂×C₁₃×#466	C₂
socle	C₂³×C₁₃	C₂×D₈
hypercenter	C₂×#466	D₁₃
hypocenter	C₁₃	C₂×#2220
LCS	G ▷ C₂×C₈×C₁₃ ▷ C₄×C₁₃ ▷ C₂×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂³ ◁ C₂³×C₄ ◁ C₂³×C₈ ◁ C₂×#466

Complements

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

N	H ≅ G/N	type
1. C₂²×#88067	C₂	semidirect
2. C₂²×#88052	C₂	semidirect
3. C₂×#11718777	C₂	semidirect
4. #13116947	C₂	direct (N × H)
5. C₂×#11718081	C₂	semidirect
6. C₂×#11718084	C₂	semidirect
7. C₂×#11718154	C₂	semidirect
19. C₂×#88067	C₂²	semidirect
21. #11718777	C₂²	semidirect
25. C₂×#88052	C₂²	semidirect
28. #11718154	C₂²	semidirect
29. #11718084	C₂²	semidirect
30. #11718081	C₂²	semidirect
67. C₂×#6090	D₄	semidirect
115. #6090	C₂×D₄	semidirect
241. C₂×C₁₃	#2220	semidirect
290. C₁₃	C₂×#2220	semidirect
347. C₂	#13116947	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×#88067	2
C₂²×#88052	2
C₂×#11718777	2
#13116947	2
C₂×#11718081	2
C₂×#11718084	2
C₂×#11718154	2
C₂×C₁₃×#466	2
C₂×#2220	13

Subgroups — C2×#13116947 #16017375

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#13116947 #16017375