Subgroups — C₂×#13114250 #15845293

Order 3328.

722 conjugacy classes of subgroups.

The subgroup lattice has 390 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#11718731	C₂	1	1	yes	maximal, characteristic
2	C₂×#11718682	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×Q₈×#1055	C₂	1	1	yes	maximal, characteristic, complemented
4	#13114250	C₂	1	8	yes	maximal, direct product factor
5	C₂×#11718072	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×#11718073	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂×#11718136	C₂	1	1	yes	maximal, characteristic
8	C₂×C₁₃×#460	C₂	1	1	yes	maximal, characteristic
9	C₂²×C₁₃×SD₁₆	—	2	1	no	—
10	C₂×C₁₃×#127	C₂²	1	1	yes	characteristic
11	C₂×C₄×Q₈×C₁₃	C₂²	1	1	yes	characteristic
12	C₂×#88123	C₂²	1	1	yes	characteristic
13	C₁₃×#460	—	2	2	no	—
14	C₁₃×#460	C₂²	1	4	yes	—
15	C₂×C₁₃×C₄⋊C₈	C₂²	1	1	yes	characteristic
16	C₂×C₁₃×#102	—	2	1	no	—
17	C₂×#88075	—	2	1	no	—
18	C₂×#88072	—	2	1	no	—
19	C₂×#88072	—	2	1	no	—
20	#11718731	—	2	2	no	—
21	#11718731	C₂²	1	4	yes	—
22	Q₈×#1055	—	2	2	no	—
23	Q₈×#1055	C₂²	1	4	yes	complemented
24	#11718682	—	2	2	no	—
25	#11718682	C₂²	1	4	yes	complemented
26	C₂×#88047	—	2	1	no	—
27	C₂×C₄⋊#1055	—	2	1	no	—
28	C₂×C₄⋊#1055	C₂²	1	1	yes	characteristic
29	C₂×C₄⋊#1055	C₂²	1	1	yes	characteristic
30	C₂×C₄×#1055	—	2	1	no	—
31	C₂×C₄×#1055	C₂²	1	1	yes	characteristic
32	#11718136	C₂²	1	4	yes	—
33	#11718073	C₂²	1	4	yes	complemented
34	#11718072	C₂²	1	4	yes	complemented
35	C₂²×Q₈×C₁₃	D₄	1	1	yes	characteristic
36	C₂²×D₄×C₁₃	—	2	1	no	—
37	C₂²×D₄×C₁₃	—	2	1	no	—
38	C₂²×D₄×C₁₃	—	2	1	no	—
39	C₂²×Dic₂₆	—	2	1	no	—
40	C₂×C₁₃×SD₁₆	—	2	4	no	—
41	C₂×C₁₃×SD₁₆	—	4	4	no	—
42	C₂²×C₈×C₁₃	—	2	1	no	—
43	C₁₃×#127	—	2	1	no	—
44	C₁₃×#127	—	2	1	no	—
45	C₁₃×#127	—	2	1	no	—
46	C₁₃×#127	C₂³	1	2	yes	—
47	C₄×Q₈×C₁₃	—	2	1	no	—
48	C₄×Q₈×C₁₃	—	2	1	no	—
49	C₄×Q₈×C₁₃	D₄	1	2	yes	—
50	C₄×Q₈×C₁₃	C₂³	1	2	yes	—
51	C₂×C₁₃×C₄⋊C₄	—	2	1	no	—
52	C₂×C₁₃×C₄⋊C₄	D₄	1	1	yes	characteristic
53	C₂×C₄²×C₁₃	—	2	1	no	—
54	C₂×C₄²×C₁₃	C₂³	1	1	yes	characteristic
55	C₂×C₄⋊Dic₁₃	—	2	1	no	—
56	C₂×C₄⋊Dic₁₃	—	2	1	no	—
57	C₂²×#1055	—	2	1	no	—
58	C₂²×#1055	—	2	1	no	—
59	C₂²×#1055	D₄	1	1	yes	characteristic
60	C₂²×#1055	D₄	1	1	yes	characteristic
61	#88123	—	2	1	no	—
62	#88123	—	2	1	no	—
63	#88123	—	2	1	no	—
64	#88123	C₂³	1	2	yes	—
65	C₁₃×C₄⋊C₈	—	2	1	no	—
66	C₁₃×C₄⋊C₈	C₂³	1	2	yes	—
67	C₁₃×#102	—	2	4	no	—
68	#88075	—	2	4	no	—
69	#88072	—	2	4	no	—
70	#88072	—	2	4	no	—
71	#88047	—	2	4	no	—
72	C₄⋊#1055	—	2	1	no	—
73	C₄⋊#1055	—	2	1	no	—
74	C₄⋊#1055	—	2	4	no	—
75	C₄⋊#1055	C₂³	1	2	yes	—
76	C₄⋊#1055	C₂³	1	2	yes	—
77	C₄×#1055	—	2	4	no	—
78	C₄×#1055	D₄	1	2	yes	—
79	C₄×#1055	C₂³	1	2	yes	—
80	C₂×#1678	—	13	1	no	maximal, Sylow
81	C₂⁴×C₁₃	—	4	1	no	—
82	C₂×Q₈×C₁₃	—	2	1	no	—
83	C₂×Q₈×C₁₃	—	2	2	no	—
84	C₂×Q₈×C₁₃	SD₁₆	1	4	yes	—
85	C₂×Q₈×C₁₃	C₂×D₄	1	2	yes	—
86	C₂×D₄×C₁₃	—	2	2	no	—
87	C₂×D₄×C₁₃	—	2	2	no	—
88	C₂×D₄×C₁₃	—	2	2	no	—
89	C₂×D₄×C₁₃	—	4	1	no	—
90	C₂×D₄×C₁₃	—	4	1	no	—
91	C₂×D₄×C₁₃	—	4	1	no	—
92	C₂×D₄×C₁₃	—	4	2	no	—
93	C₂×D₄×C₁₃	—	4	2	no	—
94	C₂×D₄×C₁₃	—	4	2	no	—
95	C₂×D₄×C₁₃	—	4	2	no	—
96	C₂×D₄×C₁₃	—	8	2	no	—
97	C₂²×C₄×C₁₃	—	2	1	no	—
98	C₂²×C₄×C₁₃	—	2	1	no	—
99	C₂²×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
100	C₂²×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
101	C₂²×C₄×C₁₃	#41	1	1	yes	characteristic
102	C₂²×Dic₁₃	—	4	1	no	—
103	C₂×Dic₂₆	—	2	2	no	—
104	C₂×Dic₂₆	—	4	1	no	—
105	C₂×Dic₂₆	—	8	2	no	—
106	C₁₃×SD₁₆	—	4	8	no	—
107	C₂×C₈×C₁₃	—	2	2	no	—
108	C₂×C₈×C₁₃	—	4	2	no	—
109	C₁₃×C₄⋊C₄	—	2	2	no	—
110	C₁₃×C₄⋊C₄	—	2	1	no	—
111	C₁₃×C₄⋊C₄	—	2	2	no	—
112	C₁₃×C₄⋊C₄	C₂×D₄	1	2	yes	—
113	C₄²×C₁₃	—	2	2	no	—
114	C₄²×C₁₃	—	2	2	no	—
115	C₄²×C₁₃	C₂×D₄	1	1	yes	characteristic
116	C₄²×C₁₃	C₂×D₄	1	1	yes	characteristic
117	C₄²×C₁₃	#41	1	1	yes	characteristic
118	C₄²×C₁₃	C₂⁴	1	1	yes	characteristic
119	C₄⋊Dic₁₃	—	2	2	no	—
120	C₄⋊Dic₁₃	—	2	2	no	—
121	C₄⋊Dic₁₃	—	4	1	no	—
122	C₄⋊Dic₁₃	—	4	1	no	—
123	C₂×#1055	—	2	2	no	—
124	C₂×#1055	—	2	2	no	—
125	C₂×#1055	—	2	2	no	—
126	C₂×#1055	—	2	2	no	—
127	C₂×#1055	—	4	2	no	—
128	C₂×#1055	—	4	2	no	—
129	C₂×#1055	C₂×D₄	1	2	yes	—
130	C₂×#1055	C₂×D₄	1	2	yes	—
131	C₂×#486	—	13	1	no	—
132	C₂×#477	—	13	1	no	—
133	C₂×#460	D₁₃	1	1	yes	characteristic
134	C₂×C₈×Q₈	—	13	1	no	—
135	#1678	—	13	8	no	—
136	C₂×#334	—	13	1	no	—
137	C₂×#333	—	13	1	no	—
138	C₂×#331	—	13	1	no	—
139	C₂³×C₁₃	—	4	2	no	—
140	C₂³×C₁₃	—	8	2	no	—
141	C₂³×C₁₃	—	8	2	no	—
142	C₂³×C₁₃	—	8	2	no	—
143	C₂³×C₁₃	#121	1	1	yes	characteristic
144	Q₈×C₁₃	—	2	2	no	—
145	Q₈×C₁₃	—	2	4	no	—
146	Q₈×C₁₃	C₂×SD₁₆	1	4	yes	—
147	D₄×C₁₃	—	4	2	no	—
148	D₄×C₁₃	—	4	2	no	—
149	D₄×C₁₃	—	4	2	no	—
150	D₄×C₁₃	—	4	2	no	—
151	D₄×C₁₃	—	8	2	no	—
152	D₄×C₁₃	—	8	2	no	—
153	D₄×C₁₃	—	8	2	no	—
154	D₄×C₁₃	—	8	2	no	—
155	C₂×C₄×C₁₃	—	2	2	no	—
156	C₂×C₄×C₁₃	—	2	4	no	—
157	C₂×C₄×C₁₃	—	2	2	no	—
158	C₂×C₄×C₁₃	—	2	2	no	—
159	C₂×C₄×C₁₃	—	4	2	no	—
160	C₂×C₄×C₁₃	#121	1	1	yes	characteristic
161	C₂×C₄×C₁₃	#121	1	1	yes	characteristic
162	C₂×C₄×C₁₃	#121	1	1	yes	characteristic
163	C₂×C₄×C₁₃	C₂×SD₁₆	1	2	yes	—
164	C₂×C₄×C₁₃	#135	1	2	yes	—
165	C₂×C₄×C₁₃	#135	1	2	yes	—
166	C₂×C₄×C₁₃	#137	1	2	yes	—
167	C₂×C₄×C₁₃	C₂²×D₄	1	1	yes	characteristic
168	C₂×C₄×C₁₃	C₂²×D₄	1	1	yes	characteristic
169	C₂×C₄×C₁₃	C₂×#41	1	1	yes	characteristic
170	C₂×Dic₁₃	—	4	2	no	—
171	C₂×Dic₁₃	—	8	2	no	—
172	Dic₂₆	—	8	2	no	—
173	Dic₂₆	—	8	2	no	—
174	C₈×C₁₃	—	4	2	no	—
175	#1055	—	2	2	no	—
176	#1055	—	2	2	no	—
177	#1055	—	4	2	no	—
178	#1055	—	4	2	no	—
179	C₂²×SD₁₆	—	2	1	no	—
180	C₂×C₄⋊Q₈	—	13	1	no	—
181	C₂×#127	C₂×D₁₃	1	1	yes	characteristic
182	C₂×C₄×Q₈	C₂×D₁₃	1	1	yes	characteristic
183	#486	—	13	4	no	—
184	#486	—	26	2	no	—
185	#477	—	13	4	no	—
186	#477	—	26	2	no	—
187	#460	—	2	2	no	—
188	#460	C₂×D₁₃	1	4	yes	—
189	C₈×Q₈	—	13	4	no	—
190	C₈×Q₈	—	26	2	no	—
191	C₂×#107	—	26	1	no	—
192	C₂×C₄⋊C₈	—	13	1	no	—
193	C₂×C₄⋊C₈	—	13	1	no	—
194	C₂×C₄⋊C₈	—	26	1	no	—
195	C₂×C₄⋊C₈	C₂×D₁₃	1	1	yes	characteristic
196	C₂×#103	—	26	1	no	—
197	C₂×#102	—	2	1	no	—
198	C₂×#102	—	26	1	no	—
199	C₂×#102	—	26	1	no	—
200	C₂×C₄×C₈	—	13	1	no	—
201	C₂×C₄×C₈	—	26	1	no	—
202	#334	—	13	4	no	—
203	#333	—	13	4	no	—
204	#331	—	13	4	no	—
205	C₂²×C₁₃	—	8	2	no	—
206	C₂²×C₁₃	—	8	2	no	—
207	C₂²×C₁₃	—	8	2	no	—
208	C₂²×C₁₃	—	16	4	no	—
209	C₂²×C₁₃	#464	1	2	yes	—
210	C₂²×C₁₃	#465	1	2	yes	—
211	C₂²×C₁₃	#472	1	2	yes	—
212	C₂²×C₁₃	C₂×#121	1	1	yes	characteristic
213	C₄×C₁₃	—	2	4	no	—
214	C₄×C₁₃	—	2	1	no	—
215	C₄×C₁₃	—	2	1	no	—
216	C₄×C₁₃	—	4	2	no	—
217	C₄×C₁₃	#464	1	2	yes	—
218	C₄×C₁₃	#465	1	2	yes	—
219	C₄×C₁₃	C₂²×SD₁₆	1	1	yes	characteristic
220	C₄×C₁₃	C₂×#135	1	1	yes	characteristic
221	C₄×C₁₃	C₂×#135	1	1	yes	characteristic
222	C₄×C₁₃	C₂×#137	1	1	yes	characteristic
223	Dic₁₃	—	8	2	no	—
224	C₂²×Q₈	—	26	1	no	—
225	C₂²×Q₈	C₂²⋊D₁₃	1	1	yes	characteristic
226	C₂²×D₄	—	2	1	no	—
227	C₂²×D₄	—	2	1	no	—
228	C₂²×D₄	—	2	1	no	—
229	C₂×SD₁₆	—	2	4	no	—
230	C₂×SD₁₆	—	4	4	no	—
231	C₂²×C₈	—	2	1	no	—
232	C₂²×C₈	—	13	1	no	—
233	C₂²×C₈	—	13	1	no	—
234	C₂²×C₈	—	26	1	no	—
235	C₂²×C₈	—	26	1	no	—
236	C₄⋊Q₈	—	13	2	no	—
237	C₄⋊Q₈	—	26	1	no	—
238	C₄⋊Q₈	—	26	1	no	—
239	C₄⋊Q₈	—	26	1	no	—
240	#127	—	2	1	no	—
241	#127	—	2	1	no	—
242	#127	—	2	1	no	—
243	#127	C₂²×D₁₃	1	2	yes	—
244	C₄×Q₈	—	2	1	no	—
245	C₄×Q₈	—	2	1	no	—
246	C₄×Q₈	C₂²⋊D₁₃	1	2	yes	—
247	C₄×Q₈	C₂²×D₁₃	1	2	yes	—
248	C₂×C₄⋊C₄	—	2	1	no	—
249	C₂×C₄⋊C₄	—	26	1	no	—
250	C₂×C₄⋊C₄	—	26	1	no	—
251	C₂×C₄⋊C₄	C₂²⋊D₁₃	1	1	yes	characteristic
252	C₂×C₄²	—	2	1	no	—
253	C₂×C₄²	C₂²×D₁₃	1	1	yes	characteristic
254	#107	—	26	4	no	—
255	C₄⋊C₈	—	2	1	no	—
256	C₄⋊C₈	—	13	2	no	—
257	C₄⋊C₈	—	13	2	no	—
258	C₄⋊C₈	—	26	4	no	—
259	C₄⋊C₈	—	26	1	no	—
260	C₄⋊C₈	—	26	1	no	—
261	C₄⋊C₈	C₂²×D₁₃	1	2	yes	—
262	#103	—	26	4	no	—
263	#102	—	2	4	no	—
264	#102	—	26	4	no	—
265	#102	—	26	4	no	—
266	C₄×C₈	—	13	2	no	—
267	C₄×C₈	—	13	2	no	—
268	C₄×C₈	—	26	4	no	—
269	C₂×C₁₃	—	16	2	no	—
270	C₂×C₁₃	#1678	1	4	yes	complemented
271	C₂×C₁₃	C₂×#464	1	1	yes	characteristic
272	C₂×C₁₃	C₂×#465	1	1	yes	characteristic
273	C₂×C₁₃	C₂×#472	1	1	yes	characteristic
274	C₂⁴	—	4	1	no	—
275	C₂×Q₈	—	2	1	no	—
276	C₂×Q₈	—	2	2	no	—
277	C₂×Q₈	—	26	2	no	—
278	C₂×Q₈	—	52	1	no	—
279	C₂×Q₈	—	104	2	no	—
280	C₂×Q₈	#6105	1	4	yes	—
281	C₂×Q₈	C₂×C₂²⋊D₁₃	1	2	yes	—
282	C₂×D₄	—	2	2	no	—
283	C₂×D₄	—	2	2	no	—
284	C₂×D₄	—	2	2	no	—
285	C₂×D₄	—	4	2	no	—
286	C₂×D₄	—	4	2	no	—
287	C₂×D₄	—	4	2	no	—
288	C₂×D₄	—	4	1	no	—
289	C₂×D₄	—	4	1	no	—
290	C₂×D₄	—	4	1	no	—
291	C₂×D₄	—	4	2	no	—
292	C₂×D₄	—	8	2	no	—
293	C₂²×C₄	—	2	1	no	—
294	C₂²×C₄	—	2	1	no	—
295	C₂²×C₄	—	52	1	no	—
296	C₂²×C₄	D₄×D₁₃	1	1	yes	characteristic
297	C₂²×C₄	#6129	1	1	yes	characteristic
298	C₂²×C₄	C₂×C₂²⋊D₁₃	1	1	yes	characteristic
299	SD₁₆	—	4	8	no	—
300	C₂×C₈	—	2	2	no	—
301	C₂×C₈	—	4	2	no	—
302	C₂×C₈	—	13	2	no	—
303	C₂×C₈	—	13	2	no	—
304	C₂×C₈	—	26	2	no	—
305	C₂×C₈	—	26	2	no	—
306	C₂×C₈	—	26	2	no	—
307	C₂×C₈	—	26	2	no	—
308	C₂×C₈	—	52	2	no	—
309	C₂×C₈	—	52	2	no	—
310	C₄⋊C₄	—	2	2	no	—
311	C₄⋊C₄	—	2	1	no	—
312	C₄⋊C₄	—	2	2	no	—
313	C₄⋊C₄	—	26	2	no	—
314	C₄⋊C₄	—	26	2	no	—
315	C₄⋊C₄	—	52	1	no	—
316	C₄⋊C₄	—	52	1	no	—
317	C₄⋊C₄	C₂×C₂²⋊D₁₃	1	2	yes	—
318	C₄²	—	2	2	no	—
319	C₄²	—	2	2	no	—
320	C₄²	D₄×D₁₃	1	1	yes	characteristic
321	C₄²	#6129	1	1	yes	characteristic
322	C₄²	C₂×C₂²⋊D₁₃	1	1	yes	characteristic
323	C₄²	C₂³×D₁₃	1	1	yes	characteristic
324	C₁₃	C₂×#1678	1	1	yes	characteristic, complemented, Sylow
325	C₂³	—	4	2	no	—
326	C₂³	—	8	2	no	—
327	C₂³	—	8	2	no	—
328	C₂³	—	8	2	no	—
329	C₂³	#88193	1	1	yes	characteristic
330	Q₈	—	2	4	no	—
331	Q₈	—	2	2	no	—
332	Q₈	—	104	2	no	—
333	Q₈	—	104	2	no	—
334	Q₈	C₂×#6105	1	4	yes	—
335	D₄	—	4	2	no	—
336	D₄	—	4	2	no	—
337	D₄	—	4	2	no	—
338	D₄	—	4	2	no	—
339	D₄	—	8	2	no	—
340	D₄	—	8	2	no	—
341	D₄	—	8	2	no	—
342	D₄	—	8	2	no	—
343	C₂×C₄	—	2	4	no	—
344	C₂×C₄	—	2	2	no	—
345	C₂×C₄	—	2	2	no	—
346	C₂×C₄	—	2	2	no	—
347	C₂×C₄	—	4	2	no	—
348	C₂×C₄	—	52	2	no	—
349	C₂×C₄	—	104	2	no	—
350	C₂×C₄	#88169	1	2	yes	—
351	C₂×C₄	#88170	1	2	yes	—
352	C₂×C₄	C₂×#6105	1	2	yes	—
353	C₂×C₄	#88193	1	1	yes	characteristic
354	C₂×C₄	#88193	1	1	yes	characteristic
355	C₂×C₄	#88193	1	1	yes	characteristic
356	C₂×C₄	#88204	1	2	yes	—
357	C₂×C₄	C₂×D₄×D₁₃	1	1	yes	characteristic
358	C₂×C₄	C₂×#6129	1	1	yes	characteristic
359	C₂×C₄	C₂²×C₂²⋊D₁₃	1	1	yes	characteristic
360	C₈	—	4	2	no	—
361	C₈	—	26	2	no	—
362	C₈	—	26	2	no	—
363	C₈	—	52	2	no	—
364	C₈	—	52	2	no	—
365	C₂²	—	8	2	no	—
366	C₂²	—	8	2	no	—
367	C₂²	—	8	2	no	—
368	C₂²	—	16	4	no	—
369	C₂²	#11718702	1	2	yes	—
370	C₂²	#11718827	1	2	yes	—
371	C₂²	#11718860	1	2	yes	—
372	C₂²	C₂×#88193	1	1	yes	characteristic
373	C₄	—	2	4	no	—
374	C₄	—	2	1	no	—
375	C₄	—	2	1	no	—
376	C₄	—	4	2	no	—
377	C₄	—	104	2	no	—
378	C₄	#11718702	1	2	yes	—
379	C₄	#11718827	1	2	yes	—
380	C₄	C₂×#88169	1	1	yes	characteristic
381	C₄	C₂×#88170	1	1	yes	characteristic
382	C₄	C₂²×#6105	1	1	yes	characteristic
383	C₄	C₂×#88204	1	1	yes	characteristic
384	C₂	—	16	2	no	—
385	C₂	#13114250	1	4	yes	direct product factor
386	C₂	C₂×#11718827	1	1	yes	characteristic
387	C₂	C₂×#11718702	1	1	yes	characteristic
388	C₂	C₂×#11718860	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	#88193
commutator, cosocle	C₄²×C₁₃	C₂⁴
Frattini	C₄²	C₂³×D₁₃
Fitting	C₂×C₁₃×#460	C₂
socle	C₂³×C₁₃	#121
prime radical	C₂×C₁₃×#127	C₂²
hypercenter	C₂×#460	D₁₃
hypocenter	C₁₃	C₂×#1678
LCS	G ▷ C₄²×C₁₃ ▷ C₂²×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×#460

Complements

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

N	H ≅ G/N	type
2. C₂×#11718682	C₂	semidirect
3. C₂×Q₈×#1055	C₂	semidirect
4. #13114250	C₂	direct (N × H)
5. C₂×#11718072	C₂	semidirect
6. C₂×#11718073	C₂	semidirect
23. Q₈×#1055	C₂²	semidirect
25. #11718682	C₂²	semidirect
33. #11718073	C₂²	semidirect
34. #11718072	C₂²	semidirect
270. C₂×C₁₃	#1678	semidirect
324. C₁₃	C₂×#1678	semidirect
385. C₂	#13114250	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×#11718731	2
C₂×#11718682	2
C₂×Q₈×#1055	2
#13114250	2
C₂×#11718072	2
C₂×#11718073	2
C₂×#11718136	2
C₂×C₁₃×#460	2
C₂×#1678	13

Subgroups — C2×#13114250 #15845293

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#13114250 #15845293