Subgroups — C₂×#13130400 #16794945

Order 3328.

758 conjugacy classes of subgroups.

The subgroup lattice has 372 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#11719130	C₂	1	1	yes	maximal, characteristic
2	C₂×#11719129	C₂	1	1	yes	maximal, characteristic
3	#13130400	C₂	1	8	yes	maximal, direct product factor
4	C₂×#11718256	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂×#11718254	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×#11718255	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂×#11718253	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×M₁₆×Dic₁₃	C₂	1	1	yes	maximal, characteristic
9	C₂²×#6122	—	2	1	no	—
10	C₂²×C₁₃×M₁₆	C₄	1	1	yes	characteristic
11	C₂²×C₄×Dic₁₃	C₂²	1	1	yes	characteristic
12	C₂²×#6099	C₄	1	1	yes	characteristic
13	C₂×C₂²⋊#1057	—	2	1	no	—
14	#11719130	—	2	2	no	—
15	#11719130	C₂²	1	4	yes	—
16	#11719129	—	2	2	no	—
17	#11719129	C₂²	1	4	yes	—
18	C₂×#88112	—	2	1	no	—
19	C₂×#88110	—	2	1	no	—
20	C₂×C₄⋊#1057	C₂²	1	1	yes	characteristic
21	C₂×C₄⋊#1057	C₂²	1	1	yes	characteristic
22	C₂×C₄⋊#1057	C₂²	1	1	yes	characteristic
23	C₂×C₄⋊#1057	C₂²	1	1	yes	characteristic
24	M₁₆×Dic₁₃	C₄	1	4	yes	—
25	M₁₆×Dic₁₃	C₂²	1	4	yes	—
26	C₂×#88055	C₄	1	1	yes	characteristic
27	C₂×#88055	C₄	1	1	yes	characteristic
28	C₂×C₈×Dic₁₃	C₂²	1	1	yes	characteristic
29	C₂×C₈×Dic₁₃	C₂²	1	1	yes	characteristic
30	#11718256	C₂²	1	4	yes	complemented
31	#11718255	C₂²	1	4	yes	complemented
32	#11718254	C₂²	1	4	yes	complemented
33	#11718253	C₂²	1	4	yes	complemented
34	C₂³×C₄×C₁₃	C₂×C₄	1	1	yes	characteristic
35	C₂³×Dic₁₃	D₄	1	1	yes	characteristic
36	C₂³×Dic₁₃	Q₈	1	1	yes	characteristic
37	C₂×#6122	—	2	4	no	—
38	C₂×#6122	—	4	4	no	—
39	C₂²×#1057	—	2	1	no	—
40	C₂²×#1057	—	2	1	no	—
41	C₂²×#1057	—	2	1	no	—
42	C₂²×#1057	—	2	1	no	—
43	C₂×C₁₃×M₁₆	—	2	4	no	—
44	C₂×C₁₃×M₁₆	C₂×C₄	1	4	yes	—
45	C₂²×C₈×C₁₃	C₂×C₄	1	1	yes	characteristic
46	C₂²×C₈×C₁₃	C₂×C₄	1	1	yes	characteristic
47	C₂×C₄×Dic₁₃	C₂×C₄	1	1	yes	characteristic
48	C₂×C₄×Dic₁₃	C₂×C₄	1	2	yes	—
49	C₂×C₄×Dic₁₃	D₄	1	1	yes	characteristic
50	C₂×C₄×Dic₁₃	D₄	1	2	yes	—
51	C₂×C₄×Dic₁₃	Q₈	1	1	yes	characteristic
52	C₂×C₄×Dic₁₃	Q₈	1	2	yes	—
53	C₂×C₄×Dic₁₃	C₂³	1	1	yes	characteristic
54	C₂×C₄×Dic₁₃	C₂³	1	2	yes	—
55	C₂×#6099	—	2	4	no	—
56	C₂×#6099	C₂×C₄	1	4	yes	—
57	C₂²×#1055	C₂×C₄	1	1	yes	characteristic
58	C₂²×#1055	C₂×C₄	1	1	yes	characteristic
59	C₂²⋊#1057	—	2	4	no	—
60	#88112	—	2	4	no	—
61	#88110	—	2	4	no	—
62	C₄⋊#1057	—	2	1	no	—
63	C₄⋊#1057	—	2	1	no	—
64	C₄⋊#1057	—	2	1	no	—
65	C₄⋊#1057	—	2	1	no	—
66	C₄⋊#1057	C₂³	1	2	yes	—
67	C₄⋊#1057	C₂³	1	2	yes	—
68	C₄⋊#1057	C₂³	1	2	yes	—
69	C₄⋊#1057	C₂³	1	2	yes	—
70	#88055	C₂×C₄	1	4	yes	—
71	#88055	C₂×C₄	1	4	yes	—
72	C₈×Dic₁₃	C₂×C₄	1	2	yes	—
73	C₈×Dic₁₃	C₂×C₄	1	2	yes	—
74	C₈×Dic₁₃	C₂³	1	2	yes	—
75	C₈×Dic₁₃	C₂³	1	2	yes	—
76	C₂×#1569	—	13	1	no	maximal, Sylow
77	C₂⁴×C₁₃	C₄⋊C₄	1	1	yes	characteristic
78	C₂²×C₄×C₁₃	—	2	2	no	—
79	C₂²×C₄×C₁₃	—	2	2	no	—
80	C₂²×C₄×C₁₃	C₄⋊C₄	1	2	yes	—
81	C₂²×C₄×C₁₃	C₄⋊C₄	1	1	yes	characteristic
82	C₂²×C₄×C₁₃	C₂²×C₄	1	2	yes	—
83	C₂²×C₄×C₁₃	C₂²×C₄	1	1	yes	characteristic
84	C₂²×Dic₁₃	—	2	2	no	—
85	C₂²×Dic₁₃	—	2	2	no	—
86	C₂²×Dic₁₃	—	4	2	no	—
87	C₂²×Dic₁₃	C₄⋊C₄	1	1	yes	characteristic
88	C₂²×Dic₁₃	C₄⋊C₄	1	1	yes	characteristic
89	C₂²×Dic₁₃	C₄⋊C₄	1	2	yes	—
90	C₂²×Dic₁₃	C₄⋊C₄	1	2	yes	—
91	C₂²×Dic₁₃	C₂×D₄	1	2	yes	—
92	C₂²×Dic₁₃	C₂×D₄	1	1	yes	characteristic
93	C₂²×Dic₁₃	C₂×Q₈	1	1	yes	characteristic
94	C₂²×Dic₁₃	C₂×Q₈	1	2	yes	—
95	#6122	—	4	8	no	—
96	C₂×#1057	—	2	2	no	—
97	C₂×#1057	—	2	2	no	—
98	C₂×#1057	—	2	2	no	—
99	C₂×#1057	—	2	2	no	—
100	C₂×#1057	—	4	2	no	—
101	C₂×#1057	—	4	2	no	—
102	C₂×#1057	—	4	2	no	—
103	C₂×#1057	—	4	2	no	—
104	C₁₃×M₁₆	—	2	8	no	—
105	C₂×C₈×C₁₃	—	2	2	no	—
106	C₂×C₈×C₁₃	M₁₆	1	4	yes	complemented
107	C₂×C₈×C₁₃	C₂²×C₄	1	2	yes	—
108	C₂×C₈×C₁₃	C₂²×C₄	1	2	yes	—
109	C₄×Dic₁₃	C₄⋊C₄	1	2	yes	—
110	C₄×Dic₁₃	C₄⋊C₄	1	2	yes	—
111	C₄×Dic₁₃	C₄⋊C₄	1	2	yes	—
112	C₄×Dic₁₃	C₂²×C₄	1	1	yes	characteristic
113	C₄×Dic₁₃	C₂²×C₄	1	2	yes	—
114	C₄×Dic₁₃	C₂×D₄	1	1	yes	characteristic
115	C₄×Dic₁₃	C₂×D₄	1	2	yes	—
116	C₄×Dic₁₃	C₂×Q₈	1	2	yes	—
117	C₄×Dic₁₃	C₂×Q₈	1	1	yes	characteristic
118	C₄×Dic₁₃	C₂⁴	1	1	yes	characteristic
119	#6099	—	2	8	no	—
120	C₂×#1055	—	2	2	no	—
121	C₂×#1055	M₁₆	1	4	yes	complemented
122	C₂×#1055	C₂²×C₄	1	2	yes	—
123	C₂×#1055	C₂²×C₄	1	2	yes	—
124	C₂×#432	—	13	1	no	—
125	C₂×#423	—	13	1	no	—
126	C₂×C₄×M₁₆	—	13	1	no	—
127	#1569	—	13	8	no	—
128	C₂×#335	—	13	1	no	—
129	C₂×#335	—	13	1	no	—
130	C₂×#334	—	13	1	no	—
131	C₂×#334	—	13	1	no	—
132	C₂³×C₁₃	—	2	2	no	—
133	C₂³×C₁₃	—	2	2	no	—
134	C₂³×C₁₃	—	2	2	no	—
135	C₂³×C₁₃	C₂×C₄⋊C₄	1	2	yes	—
136	C₂³×C₁₃	C₂×C₄⋊C₄	1	1	yes	characteristic
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	C₂×C₄⋊C₄	1	2	yes	—
142	C₂×C₄×C₁₃	C₂×C₄⋊C₄	1	1	yes	characteristic
143	C₂×C₄×C₁₃	C₂×M₁₆	1	2	yes	—
144	C₂×C₄×C₁₃	C₂×M₁₆	1	2	yes	—
145	C₂×C₄×C₁₃	C₂³×C₄	1	1	yes	characteristic
146	C₂×Dic₁₃	—	2	2	no	—
147	C₂×Dic₁₃	—	2	2	no	—
148	C₂×Dic₁₃	—	2	2	no	—
149	C₂×Dic₁₃	—	2	2	no	—
150	C₂×Dic₁₃	—	2	2	no	—
151	C₂×Dic₁₃	—	2	4	no	—
152	C₂×Dic₁₃	—	4	2	no	—
153	C₂×Dic₁₃	—	4	2	no	—
154	C₂×Dic₁₃	C₂×C₄⋊C₄	1	1	yes	characteristic
155	C₂×Dic₁₃	C₂×C₄⋊C₄	1	2	yes	—
156	C₂×Dic₁₃	C₂×C₄⋊C₄	1	2	yes	—
157	C₂×Dic₁₃	C₂×C₄⋊C₄	1	1	yes	characteristic
158	C₂×Dic₁₃	AGL(1,ℤ/8)	1	2	yes	—
159	C₂×Dic₁₃	#137	1	2	yes	—
160	C₂×Dic₁₃	C₂²×D₄	1	1	yes	characteristic
161	C₂×Dic₁₃	C₂²×Q₈	1	1	yes	characteristic
162	#1057	—	4	2	no	—
163	#1057	—	4	2	no	—
164	#1057	—	4	2	no	—
165	#1057	—	4	2	no	—
166	C₈×C₁₃	—	2	2	no	—
167	C₈×C₁₃	C₂×M₁₆	1	4	yes	complemented
168	#1055	—	2	2	no	—
169	#1055	C₂×M₁₆	1	4	yes	complemented
170	C₂²×M₁₆	—	13	1	no	—
171	C₂²×M₁₆	—	26	1	no	—
172	C₂²×M₁₆	#261	1	1	yes	characteristic
173	C₂²×C₄²	—	13	1	no	—
174	#432	—	13	4	no	—
175	#432	—	26	2	no	—
176	#423	—	13	4	no	—
177	#423	—	26	2	no	—
178	C₂×C₄⋊C₈	—	13	1	no	—
179	C₂×C₄⋊C₈	—	13	1	no	—
180	C₂×C₄⋊C₈	—	13	1	no	—
181	C₂×C₄⋊C₈	—	13	1	no	—
182	C₂×C₄⋊C₈	—	26	1	no	—
183	C₂×C₂²⋊C₈	—	26	1	no	—
184	C₄×M₁₆	—	13	4	no	—
185	C₄×M₁₆	—	13	4	no	—
186	C₂×#97	—	13	1	no	—
187	C₂×#97	—	13	1	no	—
188	C₂×#97	—	26	1	no	—
189	C₂×C₄×C₈	—	13	1	no	—
190	C₂×C₄×C₈	—	13	1	no	—
191	#335	—	13	4	no	—
192	#335	—	13	4	no	—
193	#334	—	13	4	no	—
194	#334	—	13	4	no	—
195	C₂²×C₁₃	—	2	2	no	—
196	C₂²×C₁₃	—	2	2	no	—
197	C₂²×C₁₃	—	2	2	no	—
198	C₂²×C₁₃	—	4	4	no	—
199	C₂²×C₁₃	#423	1	2	yes	—
200	C₂²×C₁₃	#428	1	2	yes	—
201	C₂²×C₁₃	#430	1	2	yes	—
202	C₂²×C₁₃	C₂²×C₄⋊C₄	1	1	yes	characteristic
203	C₄×C₁₃	—	2	2	no	—
204	C₄×C₁₃	#423	1	2	yes	—
205	C₄×C₁₃	C₂²×M₁₆	1	1	yes	characteristic
206	C₄×C₁₃	C₂²×M₁₆	1	1	yes	characteristic
207	Dic₁₃	—	2	2	no	—
208	Dic₁₃	—	2	2	no	—
209	Dic₁₃	—	2	1	no	—
210	Dic₁₃	—	2	1	no	—
211	Dic₁₃	#428	1	2	yes	—
212	Dic₁₃	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
213	Dic₁₃	C₂×#137	1	1	yes	characteristic
214	C₂³×C₄	—	13	1	no	—
215	C₂³×C₄	—	13	1	no	—
216	C₂³×C₄	C₂×#261	1	1	yes	characteristic
217	C₂×M₁₆	—	2	4	no	—
218	C₂×M₁₆	—	13	4	no	—
219	C₂×M₁₆	—	26	4	no	—
220	C₂×M₁₆	—	26	4	no	—
221	C₂×M₁₆	—	52	4	no	—
222	C₂×M₁₆	C₂×#261	1	4	yes	—
223	C₂²×C₈	—	13	1	no	—
224	C₂²×C₈	—	13	1	no	—
225	C₂²×C₈	—	26	1	no	—
226	C₂²×C₈	—	26	1	no	—
227	C₂²×C₈	—	26	1	no	—
228	C₂²×C₈	—	26	1	no	—
229	C₂²×C₈	C₂×#261	1	1	yes	characteristic
230	C₂²×C₈	C₂×#261	1	1	yes	characteristic
231	C₂×C₄²	—	13	1	no	—
232	C₂×C₄²	—	13	2	no	—
233	C₂×C₄²	—	13	2	no	—
234	C₂×C₄²	—	13	1	no	—
235	C₂×C₄²	—	13	1	no	—
236	C₂×C₄²	—	13	2	no	—
237	C₂×C₄²	—	13	1	no	—
238	C₂×C₄²	—	13	2	no	—
239	C₄⋊C₈	—	13	2	no	—
240	C₄⋊C₈	—	13	2	no	—
241	C₄⋊C₈	—	13	2	no	—
242	C₄⋊C₈	—	13	2	no	—
243	C₄⋊C₈	—	26	1	no	—
244	C₄⋊C₈	—	26	1	no	—
245	C₄⋊C₈	—	26	1	no	—
246	C₄⋊C₈	—	26	1	no	—
247	C₄⋊C₈	—	26	4	no	—
248	C₂²⋊C₈	—	26	4	no	—
249	#97	—	13	4	no	—
250	#97	—	13	4	no	—
251	#97	—	26	4	no	—
252	C₄×C₈	—	13	2	no	—
253	C₄×C₈	—	13	2	no	—
254	C₄×C₈	—	13	2	no	—
255	C₄×C₈	—	13	2	no	—
256	C₂×C₁₃	—	4	2	no	—
257	C₂×C₁₃	#1569	1	4	yes	complemented
258	C₂×C₁₃	C₂×#423	1	1	yes	characteristic
259	C₂×C₁₃	C₂×#428	1	1	yes	characteristic
260	C₂×C₁₃	C₂×#430	1	1	yes	characteristic
261	C₂⁴	C₄⋊#261	1	1	yes	characteristic
262	C₂²×C₄	—	2	2	no	—
263	C₂²×C₄	—	2	2	no	—
264	C₂²×C₄	—	13	1	no	—
265	C₂²×C₄	—	13	2	no	—
266	C₂²×C₄	—	13	2	no	—
267	C₂²×C₄	—	13	1	no	—
268	C₂²×C₄	—	13	1	no	—
269	C₂²×C₄	—	13	1	no	—
270	C₂²×C₄	—	13	2	no	—
271	C₂²×C₄	—	13	2	no	—
272	C₂²×C₄	—	26	2	no	—
273	C₂²×C₄	—	26	2	no	—
274	C₂²×C₄	—	52	2	no	—
275	C₂²×C₄	C₄⋊#261	1	2	yes	—
276	C₂²×C₄	C₄⋊#261	1	1	yes	characteristic
277	C₂²×C₄	C₂²×#261	1	2	yes	—
278	C₂²×C₄	C₂²×#261	1	1	yes	characteristic
279	M₁₆	—	2	8	no	—
280	M₁₆	—	26	8	no	—
281	M₁₆	—	52	8	no	—
282	C₂×C₈	—	2	2	no	—
283	C₂×C₈	—	13	4	no	—
284	C₂×C₈	—	13	2	no	—
285	C₂×C₈	—	13	2	no	—
286	C₂×C₈	—	26	2	no	—
287	C₂×C₈	—	26	2	no	—
288	C₂×C₈	—	26	2	no	—
289	C₂×C₈	—	26	2	no	—
290	C₂×C₈	—	26	2	no	—
291	C₂×C₈	—	52	2	no	—
292	C₂×C₈	—	52	2	no	—
293	C₂×C₈	—	52	2	no	—
294	C₂×C₈	—	52	2	no	—
295	C₂×C₈	#6122	1	4	yes	complemented
296	C₂×C₈	C₂²×#261	1	2	yes	—
297	C₂×C₈	C₂²×#261	1	2	yes	—
298	C₄²	—	13	2	no	—
299	C₄²	—	13	1	no	—
300	C₄²	—	13	2	no	—
301	C₄²	—	13	2	no	—
302	C₄²	—	13	2	no	—
303	C₄²	—	13	1	no	—
304	C₄²	—	13	2	no	—
305	C₄²	—	13	1	no	—
306	C₄²	—	13	1	no	—
307	C₄²	—	13	2	no	—
308	C₁₃	C₂×#1569	1	1	yes	characteristic, complemented, Sylow
309	C₂³	—	2	2	no	—
310	C₂³	—	2	2	no	—
311	C₂³	—	2	2	no	—
312	C₂³	C₂×C₄⋊#261	1	2	yes	—
313	C₂³	C₂×C₄⋊#261	1	1	yes	characteristic
314	C₂×C₄	—	2	2	no	—
315	C₂×C₄	—	2	2	no	—
316	C₂×C₄	—	2	2	no	—
317	C₂×C₄	—	2	4	no	—
318	C₂×C₄	—	13	2	no	—
319	C₂×C₄	—	13	2	no	—
320	C₂×C₄	—	13	1	no	—
321	C₂×C₄	—	13	1	no	—
322	C₂×C₄	—	13	2	no	—
323	C₂×C₄	—	13	2	no	—
324	C₂×C₄	—	13	1	no	—
325	C₂×C₄	—	13	1	no	—
326	C₂×C₄	—	26	2	no	—
327	C₂×C₄	—	26	2	no	—
328	C₂×C₄	—	26	4	no	—
329	C₂×C₄	—	26	2	no	—
330	C₂×C₄	—	26	2	no	—
331	C₂×C₄	—	26	2	no	—
332	C₂×C₄	—	52	2	no	—
333	C₂×C₄	—	52	2	no	—
334	C₂×C₄	D₁₃⋊M₁₆	1	2	yes	—
335	C₂×C₄	C₂×C₄⋊#261	1	2	yes	—
336	C₂×C₄	C₂×C₄⋊#261	1	1	yes	characteristic
337	C₂×C₄	C₂×#6122	1	2	yes	—
338	C₂×C₄	C₂³×#261	1	1	yes	characteristic
339	C₈	—	2	2	no	—
340	C₈	—	13	4	no	—
341	C₈	—	26	2	no	—
342	C₈	—	52	2	no	—
343	C₈	—	52	2	no	—
344	C₈	—	52	2	no	—
345	C₈	—	52	2	no	—
346	C₈	C₂×#6122	1	4	yes	complemented
347	C₂²	—	2	2	no	—
348	C₂²	—	2	2	no	—
349	C₂²	—	2	2	no	—
350	C₂²	—	4	4	no	—
351	C₂²	#11719098	1	2	yes	—
352	C₂²	#11719100	1	2	yes	—
353	C₂²	#11719129	1	2	yes	—
354	C₂²	C₂²×C₄⋊#261	1	1	yes	characteristic
355	C₄	—	2	2	no	—
356	C₄	—	13	1	no	—
357	C₄	—	13	1	no	—
358	C₄	—	13	2	no	—
359	C₄	—	26	2	no	—
360	C₄	—	26	2	no	—
361	C₄	—	26	1	no	—
362	C₄	—	26	1	no	—
363	C₄	#11719129	1	2	yes	—
364	C₄	C₂×D₁₃⋊M₁₆	1	1	yes	characteristic
365	C₄	C₂²×#6122	1	1	yes	characteristic
366	C₂	—	4	2	no	—
367	C₂	#13130400	1	4	yes	direct product factor
368	C₂	C₂×#11719129	1	1	yes	characteristic
369	C₂	C₂×#11719098	1	1	yes	characteristic
370	C₂	C₂×#11719100	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	C₂×C₄⋊#261
commutator	C₂×C₄×C₁₃	C₂³×C₄
Frattini	C₂×C₄	C₂³×#261
Fitting	C₂²×C₁₃×M₁₆	C₄
socle	C₂³×C₁₃	C₂×C₄⋊C₄
prime radical	C₂⁴×C₁₃	C₄⋊C₄
cosocle	C₄×Dic₁₃	C₂⁴
hypercenter	C₂²×M₁₆	#261
hypocenter	C₁₃	C₂×#1569
LCS	G ▷ C₂×C₄×C₁₃ ▷ C₂×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂³ ◁ C₂³×C₄ ◁ C₂²×M₁₆

Complements

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

N	H ≅ G/N	type
3. #13130400	C₂	direct (N × H)
4. C₂×#11718256	C₂	semidirect
5. C₂×#11718254	C₂	semidirect
6. C₂×#11718255	C₂	semidirect
7. C₂×#11718253	C₂	semidirect
30. #11718256	C₂²	semidirect
31. #11718255	C₂²	semidirect
32. #11718254	C₂²	semidirect
33. #11718253	C₂²	semidirect
106. C₂×C₈×C₁₃	M₁₆	semidirect
121. C₂×#1055	M₁₆	semidirect
167. C₈×C₁₃	C₂×M₁₆	semidirect
169. #1055	C₂×M₁₆	semidirect
257. C₂×C₁₃	#1569	semidirect
295. C₂×C₈	#6122	semidirect
308. C₁₃	C₂×#1569	semidirect
346. C₈	C₂×#6122	semidirect
367. C₂	#13130400	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×#11719130	2
C₂×#11719129	2
#13130400	2
C₂×#11718256	2
C₂×#11718254	2
C₂×#11718255	2
C₂×#11718253	2
C₂×M₁₆×Dic₁₃	2
C₂×#1569	13

Subgroups — C2×#13130400 #16794945

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#13130400 #16794945