Subgroups — C₂×#13114377 #15845568

Order 3328.

702 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×#11718355	C₂	1	1	yes	maximal, characteristic
2	C₂×#11718511	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×#11718509	C₂	1	1	yes	maximal, characteristic
4	C₂×#11718340	C₂	1	1	yes	maximal, characteristic, complemented
5	#13114377	C₂	1	8	yes	maximal, direct product factor
6	C₂×#11718073	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂×#11718077	C₂	1	1	yes	maximal, characteristic
8	C₂×C₁₃×#335	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂²×#6095	—	2	1	no	—
10	C₂×#88128	C₂²	1	1	yes	characteristic
11	C₂×#88123	C₂²	1	1	yes	characteristic
12	C₂×C₄×Dic₂₆	C₂²	1	1	yes	characteristic
13	C₂×C₁₃×C₄⋊C₈	C₂²	1	1	yes	characteristic
14	C₂×C₁₃×C₄⋊C₈	C₂²	1	1	yes	characteristic
15	C₂×C₄×C₈×C₁₃	C₂²	1	1	yes	characteristic
16	C₂×#88061	—	2	1	no	—
17	C₂×#88061	—	2	1	no	—
18	C₂×#88061	—	2	1	no	—
19	C₂×#88058	—	2	1	no	—
20	C₂×#88056	—	2	1	no	—
21	C₂×#88055	—	2	1	no	—
22	C₂×#88054	—	2	1	no	—
23	C₂×C₄⋊#1055	C₂²	1	1	yes	characteristic
24	#11718511	—	2	2	no	—
25	#11718511	C₂²	1	4	yes	complemented
26	#11718509	—	2	2	no	—
27	#11718509	C₂²	1	4	yes	—
28	#11718355	—	2	2	no	—
29	#11718355	C₂²	1	4	yes	—
30	#11718340	—	2	2	no	—
31	#11718340	C₂²	1	4	yes	complemented
32	C₁₃×#335	C₂²	1	4	yes	complemented
33	#11718077	C₂²	1	4	yes	—
34	#11718073	C₂²	1	4	yes	complemented
35	C₂²×D₅₂	—	2	1	no	—
36	C₂²×D₅₂	—	2	1	no	—
37	C₂²×D₅₂	—	2	1	no	—
38	C₂²×Dic₂₆	—	2	1	no	—
39	C₂²×Dic₂₆	D₄	1	1	yes	characteristic
40	C₂²×C₈×C₁₃	—	2	1	no	—
41	C₂²×C₈×C₁₃	—	2	1	no	—
42	C₂²×C₈×C₁₃	D₄	1	1	yes	characteristic
43	C₂²×C₈×C₁₃	D₄	1	1	yes	characteristic
44	C₂×C₄²×C₁₃	C₂³	1	1	yes	characteristic
45	C₂×C₄⋊Dic₁₃	—	2	1	no	—
46	C₂×C₄⋊Dic₁₃	—	2	1	no	—
47	C₂×C₄⋊Dic₁₃	D₄	1	1	yes	characteristic
48	C₂×#6101	—	2	1	no	—
49	C₂×C₄×Dic₁₃	—	2	1	no	—
50	C₂²×#1055	—	2	1	no	—
51	C₂×#6095	—	2	4	no	—
52	C₂×#6095	—	4	4	no	—
53	#88128	—	2	1	no	—
54	#88128	—	2	1	no	—
55	#88128	—	2	1	no	—
56	#88128	C₂³	1	2	yes	—
57	#88123	—	2	1	no	—
58	#88123	—	2	1	no	—
59	#88123	—	2	1	no	—
60	#88123	C₂³	1	2	yes	—
61	C₄×Dic₂₆	—	2	1	no	—
62	C₄×Dic₂₆	—	2	1	no	—
63	C₄×Dic₂₆	D₄	1	2	yes	—
64	C₄×Dic₂₆	C₂³	1	2	yes	—
65	C₁₃×C₄⋊C₈	—	2	1	no	—
66	C₁₃×C₄⋊C₈	—	2	1	no	—
67	C₁₃×C₄⋊C₈	C₂³	1	2	yes	—
68	C₁₃×C₄⋊C₈	C₂³	1	2	yes	—
69	C₄×C₈×C₁₃	D₄	1	2	yes	—
70	C₄×C₈×C₁₃	C₂³	1	2	yes	—
71	#88061	—	2	4	no	—
72	#88061	—	2	4	no	—
73	#88061	—	2	4	no	—
74	#88058	—	2	4	no	—
75	#88056	—	2	4	no	—
76	#88055	—	2	4	no	—
77	#88054	—	2	4	no	—
78	C₄⋊#1055	—	2	1	no	—
79	C₄⋊#1055	C₂³	1	2	yes	—
80	C₂×#1698	—	13	1	no	maximal, Sylow
81	C₂³×D₁₃	—	4	1	no	—
82	C₂²×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
83	C₂²×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
84	C₂²×C₄×C₁₃	#41	1	1	yes	characteristic
85	C₂²×Dic₁₃	—	2	1	no	—
86	C₂²×Dic₁₃	—	2	1	no	—
87	C₂²×Dic₁₃	—	4	1	no	—
88	C₂×D₅₂	—	2	2	no	—
89	C₂×D₅₂	—	2	2	no	—
90	C₂×D₅₂	—	2	2	no	—
91	C₂×D₅₂	—	4	2	no	—
92	C₂×D₅₂	—	4	2	no	—
93	C₂×D₅₂	—	4	2	no	—
94	C₂×D₅₂	—	4	2	no	—
95	C₂×D₅₂	—	4	1	no	—
96	C₂×D₅₂	—	4	1	no	—
97	C₂×D₅₂	—	4	1	no	—
98	C₂×D₅₂	—	8	2	no	—
99	C₂×Dic₂₆	—	2	2	no	—
100	C₂×Dic₂₆	—	2	2	no	—
101	C₂×Dic₂₆	—	2	1	no	—
102	C₂×Dic₂₆	—	2	2	no	—
103	C₂×Dic₂₆	—	4	1	no	—
104	C₂×Dic₂₆	—	8	2	no	—
105	C₂×Dic₂₆	C₂×D₄	1	2	yes	—
106	C₂×C₈×C₁₃	—	2	2	no	—
107	C₂×C₈×C₁₃	—	2	2	no	—
108	C₂×C₈×C₁₃	—	2	2	no	—
109	C₂×C₈×C₁₃	—	2	2	no	—
110	C₂×C₈×C₁₃	—	4	2	no	—
111	C₂×C₈×C₁₃	—	4	2	no	—
112	C₂×C₈×C₁₃	C₂×D₄	1	2	yes	—
113	C₂×C₈×C₁₃	C₂×D₄	1	2	yes	—
114	C₄²×C₁₃	C₂×D₄	1	1	yes	characteristic
115	C₄²×C₁₃	C₂×D₄	1	1	yes	characteristic
116	C₄²×C₁₃	#41	1	1	yes	characteristic
117	C₄²×C₁₃	C₂⁴	1	1	yes	characteristic
118	C₄⋊Dic₁₃	—	2	1	no	—
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	C₄⋊Dic₁₃	C₂×D₄	1	2	yes	—
124	#6101	—	2	2	no	—
125	#6101	—	2	2	no	—
126	C₄×Dic₁₃	—	2	2	no	—
127	C₄×Dic₁₃	—	2	2	no	—
128	C₂×#1055	—	2	2	no	—
129	C₂×#1055	—	4	2	no	—
130	#6095	—	4	8	no	—
131	C₂×#486	—	13	1	no	—
132	C₂×#477	—	13	1	no	—
133	C₂×#460	—	13	1	no	—
134	C₂×#446	—	13	1	no	—
135	#1698	—	13	8	no	—
136	C₂×#335	D₁₃	1	1	yes	characteristic, complemented
137	C₂×#333	—	13	1	no	—
138	C₂×#331	—	13	1	no	—
139	C₂³×C₁₃	#121	1	1	yes	characteristic
140	C₂²×D₁₃	—	4	2	no	—
141	C₂²×D₁₃	—	8	2	no	—
142	C₂²×D₁₃	—	8	2	no	—
143	C₂²×D₁₃	—	8	2	no	—
144	C₂×C₄×C₁₃	—	2	2	no	—
145	C₂×C₄×C₁₃	#121	1	1	yes	characteristic
146	C₂×C₄×C₁₃	#121	1	1	yes	characteristic
147	C₂×C₄×C₁₃	#121	1	1	yes	characteristic
148	C₂×C₄×C₁₃	#135	1	2	yes	—
149	C₂×C₄×C₁₃	AGL(1,ℤ/8)	1	2	yes	—
150	C₂×C₄×C₁₃	#137	1	2	yes	—
151	C₂×C₄×C₁₃	#137	1	2	yes	—
152	C₂×C₄×C₁₃	C₂²×D₄	1	1	yes	characteristic
153	C₂×C₄×C₁₃	C₂²×D₄	1	1	yes	characteristic
154	C₂×C₄×C₁₃	C₂×#41	1	1	yes	characteristic
155	C₂×Dic₁₃	—	2	2	no	—
156	C₂×Dic₁₃	—	2	2	no	—
157	C₂×Dic₁₃	—	4	2	no	—
158	C₂×Dic₁₃	—	4	2	no	—
159	C₂×Dic₁₃	—	4	2	no	—
160	C₂×Dic₁₃	—	8	2	no	—
161	D₅₂	—	4	2	no	—
162	D₅₂	—	4	2	no	—
163	D₅₂	—	4	2	no	—
164	D₅₂	—	4	2	no	—
165	D₅₂	—	8	2	no	—
166	D₅₂	—	8	2	no	—
167	D₅₂	—	8	2	no	—
168	D₅₂	—	8	2	no	—
169	Dic₂₆	—	2	2	no	—
170	Dic₂₆	—	2	2	no	—
171	Dic₂₆	—	4	2	no	—
172	Dic₂₆	—	8	2	no	—
173	Dic₂₆	—	8	2	no	—
174	C₈×C₁₃	—	2	2	no	—
175	C₈×C₁₃	—	2	2	no	—
176	C₈×C₁₃	—	4	2	no	—
177	C₈×C₁₃	—	4	2	no	—
178	#1055	—	4	2	no	—
179	C₂²×SD₁₆	—	26	1	no	—
180	C₂×C₄⋊Q₈	—	13	1	no	—
181	C₂×#127	—	13	1	no	—
182	C₂×C₄×Q₈	—	13	1	no	—
183	#486	—	13	4	no	—
184	#486	—	26	2	no	—
185	#477	—	13	4	no	—
186	#477	—	26	2	no	—
187	#460	—	13	4	no	—
188	#460	—	26	2	no	—
189	#446	—	13	4	no	—
190	#446	—	26	2	no	—
191	C₂×#107	—	26	1	no	—
192	C₂×C₄⋊C₈	—	13	1	no	—
193	C₂×C₄⋊C₈	—	26	1	no	—
194	C₂×C₄⋊C₈	C₂×D₁₃	1	1	yes	characteristic
195	C₂×C₄⋊C₈	C₂×D₁₃	1	1	yes	characteristic
196	C₂×#103	—	26	1	no	—
197	C₂×#102	—	26	1	no	—
198	C₂×#102	—	26	1	no	—
199	C₂×#102	—	26	1	no	—
200	C₂×#97	—	26	1	no	—
201	C₂×C₄×C₈	C₂×D₁₃	1	1	yes	characteristic
202	#335	C₂×D₁₃	1	4	yes	complemented
203	#333	—	13	4	no	—
204	#331	—	13	4	no	—
205	C₂²×C₁₃	#464	1	2	yes	—
206	C₂²×C₁₃	#468	1	2	yes	—
207	C₂²×C₁₃	#472	1	2	yes	—
208	C₂²×C₁₃	C₂×#121	1	1	yes	characteristic
209	C₂×D₁₃	—	8	2	no	—
210	C₂×D₁₃	—	8	2	no	—
211	C₂×D₁₃	—	8	2	no	—
212	C₂×D₁₃	—	16	4	no	—
213	C₄×C₁₃	—	2	1	no	—
214	C₄×C₁₃	—	2	1	no	—
215	C₄×C₁₃	#464	1	2	yes	—
216	C₄×C₁₃	#468	1	2	yes	—
217	C₄×C₁₃	C₂×#135	1	1	yes	characteristic
218	C₄×C₁₃	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
219	C₄×C₁₃	C₂×#137	1	1	yes	characteristic
220	C₄×C₁₃	C₂×#137	1	1	yes	characteristic
221	Dic₁₃	—	4	2	no	—
222	Dic₁₃	—	4	2	no	—
223	Dic₁₃	—	8	2	no	—
224	C₂²×Q₈	—	13	1	no	—
225	C₂²×Q₈	—	26	1	no	—
226	C₂²×D₄	—	26	1	no	—
227	C₂²×D₄	—	26	1	no	—
228	C₂²×D₄	—	26	1	no	—
229	C₂×SD₁₆	—	26	4	no	—
230	C₂×SD₁₆	—	52	4	no	—
231	C₂²×C₈	—	2	1	no	—
232	C₂²×C₈	—	2	1	no	—
233	C₂²×C₈	—	26	1	no	—
234	C₂²×C₈	D₅₂	1	1	yes	characteristic
235	C₂²×C₈	D₅₂	1	1	yes	characteristic
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	—	13	2	no	—
241	#127	—	26	1	no	—
242	#127	—	26	1	no	—
243	#127	—	26	1	no	—
244	C₄×Q₈	—	13	2	no	—
245	C₄×Q₈	—	13	2	no	—
246	C₄×Q₈	—	26	1	no	—
247	C₄×Q₈	—	26	1	no	—
248	C₂×C₄⋊C₄	—	13	1	no	—
249	C₂×C₄⋊C₄	—	26	1	no	—
250	C₂×C₄⋊C₄	—	26	1	no	—
251	C₂×C₄⋊C₄	—	26	1	no	—
252	C₂×C₄²	—	26	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	—	2	1	no	—
257	C₄⋊C₈	—	13	2	no	—
258	C₄⋊C₈	—	26	4	no	—
259	C₄⋊C₈	—	26	1	no	—
260	C₄⋊C₈	C₂²×D₁₃	1	2	yes	—
261	C₄⋊C₈	C₂²×D₁₃	1	2	yes	—
262	#103	—	26	4	no	—
263	#102	—	26	4	no	—
264	#102	—	26	4	no	—
265	#102	—	26	4	no	—
266	#97	—	26	4	no	—
267	C₄×C₈	D₅₂	1	2	yes	—
268	C₄×C₈	C₂²×D₁₃	1	2	yes	—
269	C₂×C₁₃	#1698	1	4	yes	complemented
270	C₂×C₁₃	C₂×#464	1	1	yes	characteristic
271	C₂×C₁₃	C₂×#468	1	1	yes	characteristic
272	C₂×C₁₃	C₂×#472	1	1	yes	characteristic
273	D₁₃	—	16	2	no	—
274	C₂⁴	—	52	1	no	—
275	C₂×Q₈	—	13	2	no	—
276	C₂×Q₈	—	26	1	no	—
277	C₂×Q₈	—	26	2	no	—
278	C₂×Q₈	—	26	2	no	—
279	C₂×Q₈	—	26	2	no	—
280	C₂×Q₈	—	52	1	no	—
281	C₂×Q₈	—	104	2	no	—
282	C₂×D₄	—	26	2	no	—
283	C₂×D₄	—	26	2	no	—
284	C₂×D₄	—	26	2	no	—
285	C₂×D₄	—	52	1	no	—
286	C₂×D₄	—	52	1	no	—
287	C₂×D₄	—	52	1	no	—
288	C₂×D₄	—	52	2	no	—
289	C₂×D₄	—	52	2	no	—
290	C₂×D₄	—	52	2	no	—
291	C₂×D₄	—	52	2	no	—
292	C₂×D₄	—	104	2	no	—
293	C₂²×C₄	—	26	1	no	—
294	C₂²×C₄	—	26	1	no	—
295	C₂²×C₄	—	52	1	no	—
296	C₂²×C₄	C₂×D₅₂	1	1	yes	characteristic
297	C₂²×C₄	D₄×D₁₃	1	1	yes	characteristic
298	C₂²×C₄	#6131	1	1	yes	characteristic
299	SD₁₆	—	52	8	no	—
300	C₂×C₈	—	2	2	no	—
301	C₂×C₈	—	2	2	no	—
302	C₂×C₈	—	2	2	no	—
303	C₂×C₈	—	2	2	no	—
304	C₂×C₈	—	4	2	no	—
305	C₂×C₈	—	4	2	no	—
306	C₂×C₈	—	26	2	no	—
307	C₂×C₈	—	52	2	no	—
308	C₂×C₈	C₂×D₅₂	1	2	yes	—
309	C₂×C₈	C₂×D₅₂	1	2	yes	—
310	C₄⋊C₄	—	13	2	no	—
311	C₄⋊C₄	—	26	1	no	—
312	C₄⋊C₄	—	26	2	no	—
313	C₄⋊C₄	—	26	2	no	—
314	C₄⋊C₄	—	26	2	no	—
315	C₄⋊C₄	—	26	2	no	—
316	C₄⋊C₄	—	52	1	no	—
317	C₄⋊C₄	—	52	1	no	—
318	C₄²	—	26	2	no	—
319	C₄²	—	26	2	no	—
320	C₄²	C₂×D₅₂	1	1	yes	characteristic
321	C₄²	D₄×D₁₃	1	1	yes	characteristic
322	C₄²	#6131	1	1	yes	characteristic
323	C₄²	C₂³×D₁₃	1	1	yes	characteristic
324	C₁₃	C₂×#1698	1	1	yes	characteristic, complemented, Sylow
325	C₂³	—	52	2	no	—
326	C₂³	—	104	2	no	—
327	C₂³	—	104	2	no	—
328	C₂³	—	104	2	no	—
329	C₂³	#88149	1	1	yes	characteristic
330	Q₈	—	26	2	no	—
331	Q₈	—	26	2	no	—
332	Q₈	—	52	2	no	—
333	Q₈	—	104	2	no	—
334	Q₈	—	104	2	no	—
335	D₄	—	52	2	no	—
336	D₄	—	52	2	no	—
337	D₄	—	52	2	no	—
338	D₄	—	52	2	no	—
339	D₄	—	104	2	no	—
340	D₄	—	104	2	no	—
341	D₄	—	104	2	no	—
342	D₄	—	104	2	no	—
343	C₂×C₄	—	2	2	no	—
344	C₂×C₄	—	26	2	no	—
345	C₂×C₄	—	26	2	no	—
346	C₂×C₄	—	52	2	no	—
347	C₂×C₄	—	52	2	no	—
348	C₂×C₄	—	52	2	no	—
349	C₂×C₄	—	104	2	no	—
350	C₂×C₄	#88149	1	1	yes	characteristic
351	C₂×C₄	#88149	1	1	yes	characteristic
352	C₂×C₄	#88149	1	1	yes	characteristic
353	C₂×C₄	#88158	1	2	yes	—
354	C₂×C₄	#88163	1	2	yes	—
355	C₂×C₄	#88165	1	2	yes	—
356	C₂×C₄	#88172	1	2	yes	—
357	C₂×C₄	C₂²×D₅₂	1	1	yes	characteristic
358	C₂×C₄	C₂×D₄×D₁₃	1	1	yes	characteristic
359	C₂×C₄	C₂×#6131	1	1	yes	characteristic
360	C₈	—	2	2	no	—
361	C₈	—	2	2	no	—
362	C₈	—	4	2	no	—
363	C₈	—	4	2	no	—
364	C₈	—	52	2	no	—
365	C₂²	—	104	2	no	—
366	C₂²	—	104	2	no	—
367	C₂²	—	104	2	no	—
368	C₂²	—	208	4	no	—
369	C₂²	#11718506	1	2	yes	—
370	C₂²	#11718547	1	2	yes	—
371	C₂²	#11718558	1	2	yes	—
372	C₂²	C₂×#88149	1	1	yes	characteristic
373	C₄	—	2	1	no	—
374	C₄	—	2	1	no	—
375	C₄	—	52	2	no	—
376	C₄	—	52	2	no	—
377	C₄	—	104	2	no	—
378	C₄	#11718506	1	2	yes	—
379	C₄	#11718547	1	2	yes	—
380	C₄	C₂×#88158	1	1	yes	characteristic
381	C₄	C₂×#88163	1	1	yes	characteristic
382	C₄	C₂×#88165	1	1	yes	characteristic
383	C₄	C₂×#88172	1	1	yes	characteristic
384	C₂	—	208	2	no	—
385	C₂	#13114377	1	4	yes	direct product factor
386	C₂	C₂×#11718506	1	1	yes	characteristic
387	C₂	C₂×#11718547	1	1	yes	characteristic
388	C₂	C₂×#11718558	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	#88149
commutator, cosocle	C₄²×C₁₃	C₂⁴
Frattini	C₄²	C₂³×D₁₃
Fitting	C₂×C₁₃×#335	C₂
socle	C₂³×C₁₃	#121
prime radical	C₂×#88128	C₂²
hypercenter	C₂×#335	D₁₃
hypocenter	C₁₃	C₂×#1698
LCS	G ▷ C₄²×C₁₃ ▷ C₂²×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×#335

Complements

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

N	H ≅ G/N	type
2. C₂×#11718511	C₂	semidirect
4. C₂×#11718340	C₂	semidirect
5. #13114377	C₂	direct (N × H)
6. C₂×#11718073	C₂	semidirect
8. C₂×C₁₃×#335	C₂	semidirect
25. #11718511	C₂²	semidirect
31. #11718340	C₂²	semidirect
32. C₁₃×#335	C₂²	semidirect
34. #11718073	C₂²	semidirect
136. C₂×#335	D₁₃	semidirect
202. #335	C₂×D₁₃	semidirect
269. C₂×C₁₃	#1698	semidirect
324. C₁₃	C₂×#1698	semidirect
385. C₂	#13114377	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×#11718355	2
C₂×#11718511	2
C₂×#11718509	2
C₂×#11718340	2
#13114377	2
C₂×#11718073	2
C₂×#11718077	2
C₂×C₁₃×#335	2
C₂×#1698	13

Subgroups — C2×#13114377 #15845568

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#13114377 #15845568