Subgroups — C₂×#13125486 #16732753

Order 3328.

2098 conjugacy classes of subgroups.

The subgroup lattice has 370 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#11719240	C₂	1	2	yes	maximal, complemented
2	#13125486	C₂	1	16	yes	maximal, direct product factor
3	C₂×#11718344	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂×#11718356	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂×#11718374	C₂	1	2	yes	maximal, complemented
6	C₂×#11718796	C₂	1	2	yes	maximal, complemented
7	C₂×#11718776	C₂	1	2	yes	maximal, complemented
8	C₂×#11718372	C₂	1	2	yes	maximal, complemented
9	C₂×C₄×D₁₀₄	C₂	1	2	yes	maximal, complemented
10	C₂×C₁₃×#405	C₂	1	1	yes	maximal, characteristic, complemented
11	C₂²×C₁₃×M₁₆	C₂²	1	1	yes	characteristic
12	C₂³×C₈×C₁₃	C₂²	1	1	yes	characteristic
13	C₂×C₁₃×#117	C₂²	1	1	yes	characteristic
14	C₂×#88184	C₂²	1	4	yes	—
15	C₂²×D₁₀₄	C₂²	1	2	yes	—
16	C₂²×D₁₀₄	C₂²	1	2	yes	—
17	#11719240	—	2	8	no	—
18	#11719240	C₂²	1	16	yes	complemented
19	C₂×#88148	—	2	4	no	—
20	C₂×#88138	—	2	4	no	—
21	C₂×#88128	C₂²	1	2	yes	—
22	C₂×C₄×D₅₂	C₂²	1	4	yes	—
23	C₂×#88124	C₂²	1	2	yes	—
24	C₁₃×#405	C₂²	1	8	yes	complemented
25	C₂×C₁₃×#97	C₂²	1	1	yes	characteristic
26	C₂×C₁₃×#97	C₂²	1	1	yes	characteristic
27	C₂×C₄×C₈×C₁₃	C₂²	1	1	yes	characteristic
28	C₂×C₄×C₈×C₁₃	C₂²	1	1	yes	characteristic
29	#11718796	C₂²	1	16	yes	complemented
30	#11718776	C₂²	1	16	yes	complemented
31	C₂×#88061	C₂²	1	4	yes	—
32	C₂×#88061	C₂²	1	4	yes	—
33	C₂×#88058	C₂²	1	2	yes	—
34	C₂×#88057	C₂²	1	2	yes	—
35	#11718374	C₂²	1	16	yes	complemented
36	#11718372	C₂²	1	16	yes	complemented
37	#11718356	C₂²	1	8	yes	complemented
38	C₄×D₁₀₄	C₂²	1	16	yes	complemented
39	#11718344	C₂²	1	8	yes	complemented
40	C₂³×C₄×C₁₃	C₂³	1	1	yes	characteristic
41	C₂²×C₂²⋊D₁₃	—	2	4	no	—
42	C₂²×D₅₂	—	2	4	no	—
43	C₂²×D₅₂	C₂³	1	4	yes	—
44	C₂²×C₄×D₁₃	—	2	4	no	—
45	C₂×C₁₃×M₁₆	—	2	4	no	—
46	C₂×C₁₃×M₁₆	C₂³	1	4	yes	—
47	C₂²×C₈×C₁₃	—	2	4	no	—
48	C₂²×C₈×C₁₃	C₂³	1	1	yes	characteristic
49	C₂²×C₈×C₁₃	C₂³	1	1	yes	characteristic
50	C₂²×C₈×C₁₃	C₂³	1	4	yes	—
51	C₂²×C₈×C₁₃	C₂³	1	1	yes	characteristic
52	C₂²×C₈×C₁₃	C₂³	1	1	yes	characteristic
53	C₁₃×#117	D₄	1	4	yes	—
54	C₁₃×#117	C₂³	1	4	yes	—
55	C₂×C₁₃×C₄⋊C₄	D₄	1	2	yes	—
56	C₂×C₁₃×C₂²⋊C₄	D₄	1	2	yes	—
57	C₂×C₄²×C₁₃	C₂³	1	1	yes	characteristic
58	C₂×C₄²×C₁₃	C₂³	1	1	yes	characteristic
59	#88184	—	2	8	no	—
60	#88184	C₂³	1	16	yes	—
61	C₂×#6103	—	2	4	no	—
62	C₂×#6103	—	2	4	no	—
63	C₂×C₄⋊Dic₁₃	C₂³	1	4	yes	—
64	C₂×#6101	—	2	4	no	—
65	C₂×D₁₀₄	—	2	8	no	—
66	C₂×D₁₀₄	—	2	8	no	—
67	C₂×D₁₀₄	C₂³	1	8	yes	—
68	C₂×D₁₀₄	C₂³	1	8	yes	—
69	#88148	—	2	32	no	—
70	#88138	—	2	32	no	—
71	#88128	—	2	4	no	—
72	#88128	C₂³	1	8	yes	—
73	C₄×D₅₂	—	2	8	no	—
74	C₄×D₅₂	C₂³	1	16	yes	—
75	#88124	—	2	4	no	—
76	#88124	C₂³	1	8	yes	—
77	C₁₃×#97	C₂³	1	4	yes	—
78	C₁₃×#97	C₂³	1	4	yes	—
79	C₄×C₈×C₁₃	C₂³	1	4	yes	—
80	C₄×C₈×C₁₃	C₂³	1	4	yes	—
81	#88061	C₂³	1	16	yes	—
82	#88061	C₂³	1	16	yes	—
83	#88058	C₂³	1	8	yes	—
84	#88057	C₂³	1	8	yes	—
85	C₂×#3176	—	13	1	no	maximal, Sylow
86	C₂⁴×C₁₃	C₂×D₄	1	1	yes	characteristic
87	C₂³×D₁₃	—	2	4	no	—
88	C₂²×C₄×C₁₃	—	2	2	no	—
89	C₂²×C₄×C₁₃	—	2	2	no	—
90	C₂²×C₄×C₁₃	C₂×D₄	1	2	yes	—
91	C₂²×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
92	C₂²×C₄×C₁₃	C₂×D₄	1	2	yes	—
93	C₂²×C₄×C₁₃	C₂×D₄	1	2	yes	—
94	C₂²×C₄×C₁₃	C₂⁴	1	2	yes	—
95	C₂²×C₄×C₁₃	C₂⁴	1	1	yes	characteristic
96	C₂×C₂²⋊D₁₃	—	2	16	no	—
97	C₂×C₂²⋊D₁₃	—	8	8	no	—
98	C₂²×Dic₁₃	—	2	4	no	—
99	C₂×D₅₂	—	2	8	no	—
100	C₂×D₅₂	—	2	8	no	—
101	C₂×D₅₂	—	2	4	no	—
102	C₂×D₅₂	—	2	16	no	—
103	C₂×D₅₂	—	8	8	no	—
104	C₂×D₅₂	C₂⁴	1	8	yes	—
105	C₂×C₄×D₁₃	—	2	16	no	—
106	C₂×C₄×D₁₃	—	8	8	no	—
107	C₁₃×M₁₆	—	2	8	no	—
108	C₂×C₈×C₁₃	—	2	2	no	—
109	C₂×C₈×C₁₃	—	2	2	no	—
110	C₂×C₈×C₁₃	—	2	8	no	—
111	C₂×C₈×C₁₃	#41	1	4	yes	—
112	C₂×C₈×C₁₃	#41	1	4	yes	—
113	C₂×C₈×C₁₃	C₂⁴	1	2	yes	—
114	C₂×C₈×C₁₃	C₂⁴	1	2	yes	—
115	C₂×C₈×C₁₃	C₂⁴	1	2	yes	—
116	C₂×C₈×C₁₃	C₂⁴	1	2	yes	—
117	C₁₃×C₄⋊C₄	C₂×D₄	1	8	yes	—
118	C₁₃×C₂²⋊C₄	C₂×D₄	1	8	yes	—
119	C₄²×C₁₃	C₂×D₄	1	2	yes	—
120	C₄²×C₁₃	C₂×D₄	1	2	yes	—
121	C₄²×C₁₃	C₂⁴	1	2	yes	—
122	C₄²×C₁₃	C₂⁴	1	2	yes	—
123	#6103	—	2	16	no	—
124	#6103	—	2	16	no	—
125	C₄⋊Dic₁₃	—	2	4	no	—
126	C₄⋊Dic₁₃	C₂⁴	1	8	yes	—
127	#6101	—	2	16	no	—
128	D₁₀₄	—	2	16	no	—
129	D₁₀₄	—	2	16	no	—
130	C₂×#540	—	13	2	no	—
131	#3176	—	13	16	no	—
132	C₂×#499	—	13	1	no	—
133	C₂×#493	—	13	1	no	—
134	C₂×#490	—	13	2	no	—
135	C₂×#469	—	13	2	no	—
136	C₂×#466	—	13	2	no	—
137	C₂×#442	—	13	2	no	—
138	C₂×C₄×D₈	—	13	2	no	—
139	C₂×#405	D₁₃	1	1	yes	characteristic, complemented
140	C₂³×C₁₃	—	2	2	no	—
141	C₂³×C₁₃	—	2	4	no	—
142	C₂³×C₁₃	C₂²×D₄	1	2	yes	—
143	C₂³×C₁₃	C₂²×D₄	1	1	yes	characteristic
144	C₂²×D₁₃	—	2	8	no	—
145	C₂²×D₁₃	—	4	8	no	—
146	C₂²×D₁₃	—	4	8	no	—
147	C₂²×D₁₃	—	4	8	no	—
148	C₂×C₄×C₁₃	—	2	2	no	—
149	C₂×C₄×C₁₃	—	2	4	no	—
150	C₂×C₄×C₁₃	—	2	4	no	—
151	C₂×C₄×C₁₃	—	2	4	no	—
152	C₂×C₄×C₁₃	—	2	2	no	—
153	C₂×C₄×C₁₃	—	2	2	no	—
154	C₂×C₄×C₁₃	C₂²×D₄	1	4	yes	—
155	C₂×C₄×C₁₃	C₂²×D₄	1	2	yes	—
156	C₂×C₄×C₁₃	C₂²×D₄	1	4	yes	—
157	C₂×C₄×C₁₃	C₂²×D₄	1	1	yes	characteristic
158	C₂×C₄×C₁₃	C₂×#41	1	2	yes	—
159	C₂×C₄×C₁₃	C₂×#41	1	2	yes	—
160	C₂×C₄×C₁₃	C₂⁵	1	1	yes	characteristic
161	C₂²⋊D₁₃	—	8	16	no	—
162	C₂×Dic₁₃	—	2	8	no	—
163	C₂×Dic₁₃	—	4	8	no	—
164	D₅₂	—	2	8	no	—
165	D₅₂	—	2	8	no	—
166	D₅₂	—	4	8	no	—
167	D₅₂	—	8	16	no	—
168	C₄×D₁₃	—	8	16	no	—
169	C₈×C₁₃	—	2	2	no	—
170	C₈×C₁₃	—	2	2	no	—
171	C₈×C₁₃	C₂×#41	1	4	yes	—
172	C₈×C₁₃	C₂×#41	1	4	yes	—
173	C₂²×D₈	—	13	2	no	—
174	C₂²×D₈	—	13	2	no	—
175	C₂²×M₁₆	C₂×D₁₃	1	1	yes	characteristic
176	C₂³×C₈	C₂×D₁₃	1	1	yes	characteristic
177	#540	—	13	16	no	—
178	#540	—	26	8	no	—
179	C₂×#127	—	13	2	no	—
180	C₂×#125	—	13	2	no	—
181	C₂×#123	—	26	4	no	—
182	C₂×#121	—	13	4	no	—
183	C₂×#121	—	26	4	no	—
184	C₂×C₄×D₄	—	13	4	no	—
185	C₂×#117	C₂×D₁₃	1	1	yes	characteristic
186	#499	—	13	8	no	—
187	#493	—	13	8	no	—
188	#490	—	13	16	no	—
189	#469	—	13	16	no	—
190	#466	—	13	16	no	—
191	#442	—	13	16	no	—
192	C₄×D₈	—	13	16	no	—
193	C₂×#107	—	13	2	no	—
194	C₂×#106	—	13	2	no	—
195	C₂×#102	—	13	4	no	—
196	C₂×#102	—	13	4	no	—
197	#405	C₂×D₁₃	1	8	yes	complemented
198	C₂×#97	C₂×D₁₃	1	1	yes	characteristic
199	C₂×#97	C₂×D₁₃	1	1	yes	characteristic
200	C₂×C₄×C₈	C₂×D₁₃	1	1	yes	characteristic
201	C₂×C₄×C₈	C₂×D₁₃	1	1	yes	characteristic
202	C₂²×C₁₃	—	2	2	no	—
203	C₂²×C₁₃	—	2	4	no	—
204	C₂²×C₁₃	—	4	4	no	—
205	C₂²×C₁₃	#532	1	2	yes	—
206	C₂²×C₁₃	#576	1	4	yes	—
207	C₂²×C₁₃	C₂³×D₄	1	1	yes	characteristic
208	C₂×D₁₃	—	4	8	no	—
209	C₂×D₁₃	—	4	8	no	—
210	C₂×D₁₃	—	4	8	no	—
211	C₂×D₁₃	—	8	16	no	—
212	C₄×C₁₃	—	2	2	no	—
213	C₄×C₁₃	—	2	4	no	—
214	C₄×C₁₃	—	2	4	no	—
215	C₄×C₁₃	#532	1	2	yes	—
216	C₄×C₁₃	C₂²×#41	1	1	yes	characteristic
217	C₄×C₁₃	C₂²×#41	1	1	yes	characteristic
218	Dic₁₃	—	4	8	no	—
219	C₂²×D₄	—	13	4	no	—
220	C₂²×D₄	—	26	4	no	—
221	C₂²×D₄	—	26	4	no	—
222	C₂³×C₄	—	26	4	no	—
223	C₂³×C₄	C₂²×D₁₃	1	1	yes	characteristic
224	C₂×D₈	—	13	8	no	—
225	C₂×D₈	—	13	8	no	—
226	C₂×D₈	—	26	8	no	—
227	C₂×D₈	—	26	8	no	—
228	C₂×M₁₆	—	2	4	no	—
229	C₂×M₁₆	C₂²×D₁₃	1	4	yes	—
230	C₂²×C₈	—	2	4	no	—
231	C₂²×C₈	C₂²×D₁₃	1	1	yes	characteristic
232	C₂²×C₈	C₂²×D₁₃	1	4	yes	—
233	C₂²×C₈	C₂²×D₁₃	1	1	yes	characteristic
234	C₂²×C₈	C₂²×D₁₃	1	1	yes	characteristic
235	C₂²×C₈	C₂²×D₁₃	1	1	yes	characteristic
236	#127	—	13	8	no	—
237	#127	—	26	4	no	—
238	#125	—	13	8	no	—
239	#125	—	26	4	no	—
240	#123	—	26	32	no	—
241	#121	—	13	16	no	—
242	#121	—	26	8	no	—
243	#121	—	26	32	no	—
244	C₄×D₄	—	13	16	no	—
245	C₄×D₄	—	26	8	no	—
246	#117	D₅₂	1	4	yes	—
247	#117	C₂²×D₁₃	1	4	yes	—
248	C₂×C₄⋊C₄	—	13	4	no	—
249	C₂×C₄⋊C₄	—	26	4	no	—
250	C₂×C₄⋊C₄	D₅₂	1	2	yes	—
251	C₂×C₂²⋊C₄	—	26	4	no	—
252	C₂×C₂²⋊C₄	—	26	4	no	—
253	C₂×C₂²⋊C₄	D₅₂	1	2	yes	—
254	C₂×C₄²	C₂²×D₁₃	1	1	yes	characteristic
255	C₂×C₄²	C₂²×D₁₃	1	1	yes	characteristic
256	#107	—	13	8	no	—
257	#106	—	13	8	no	—
258	#102	—	13	16	no	—
259	#102	—	13	16	no	—
260	#97	C₂²×D₁₃	1	4	yes	—
261	#97	C₂²×D₁₃	1	4	yes	—
262	C₄×C₈	C₂²×D₁₃	1	4	yes	—
263	C₄×C₈	C₂²×D₁₃	1	4	yes	—
264	C₂×C₁₃	—	4	2	no	—
265	C₂×C₁₃	#3176	1	4	yes	complemented
266	C₂×C₁₃	C₂×#532	1	1	yes	characteristic
267	C₂×C₁₃	C₂×#576	1	2	yes	—
268	D₁₃	—	8	8	no	—
269	C₂⁴	—	26	4	no	—
270	C₂⁴	C₂×D₅₂	1	1	yes	characteristic
271	C₂×D₄	—	13	8	no	—
272	C₂×D₄	—	26	16	no	—
273	C₂×D₄	—	26	4	no	—
274	C₂×D₄	—	26	16	no	—
275	C₂×D₄	—	26	8	no	—
276	C₂×D₄	—	26	8	no	—
277	C₂×D₄	—	104	8	no	—
278	C₂×D₄	—	104	8	no	—
279	C₂²×C₄	—	2	2	no	—
280	C₂²×C₄	—	2	2	no	—
281	C₂²×C₄	—	26	4	no	—
282	C₂²×C₄	—	26	16	no	—
283	C₂²×C₄	—	104	8	no	—
284	C₂²×C₄	C₂×D₅₂	1	2	yes	—
285	C₂²×C₄	C₂×D₅₂	1	2	yes	—
286	C₂²×C₄	C₂×D₅₂	1	2	yes	—
287	C₂²×C₄	C₂×D₅₂	1	1	yes	characteristic
288	C₂²×C₄	C₂³×D₁₃	1	2	yes	—
289	C₂²×C₄	C₂³×D₁₃	1	1	yes	characteristic
290	D₈	—	26	16	no	—
291	D₈	—	26	16	no	—
292	M₁₆	—	2	8	no	—
293	C₂×C₈	—	2	2	no	—
294	C₂×C₈	—	2	2	no	—
295	C₂×C₈	—	2	8	no	—
296	C₂×C₈	#6127	1	4	yes	—
297	C₂×C₈	#6127	1	4	yes	—
298	C₂×C₈	C₂³×D₁₃	1	2	yes	—
299	C₂×C₈	C₂³×D₁₃	1	2	yes	—
300	C₂×C₈	C₂³×D₁₃	1	2	yes	—
301	C₂×C₈	C₂³×D₁₃	1	2	yes	—
302	C₄⋊C₄	—	13	8	no	—
303	C₄⋊C₄	—	26	4	no	—
304	C₄⋊C₄	—	26	16	no	—
305	C₄⋊C₄	C₂×D₅₂	1	8	yes	—
306	C₂²⋊C₄	—	26	16	no	—
307	C₂²⋊C₄	—	26	16	no	—
308	C₂²⋊C₄	C₂×D₅₂	1	8	yes	—
309	C₄²	C₂×D₅₂	1	2	yes	—
310	C₄²	C₂×D₅₂	1	2	yes	—
311	C₄²	C₂³×D₁₃	1	2	yes	—
312	C₄²	C₂³×D₁₃	1	2	yes	—
313	C₁₃	C₂×#3176	1	1	yes	characteristic, complemented, Sylow
314	C₂³	—	2	2	no	—
315	C₂³	—	2	4	no	—
316	C₂³	—	26	8	no	—
317	C₂³	—	52	8	no	—
318	C₂³	—	52	8	no	—
319	C₂³	—	52	8	no	—
320	C₂³	C₂²×D₅₂	1	2	yes	—
321	C₂³	C₂²×D₅₂	1	1	yes	characteristic
322	D₄	—	26	8	no	—
323	D₄	—	26	8	no	—
324	D₄	—	52	8	no	—
325	D₄	—	104	16	no	—
326	D₄	—	104	16	no	—
327	C₂×C₄	—	2	2	no	—
328	C₂×C₄	—	2	4	no	—
329	C₂×C₄	—	2	4	no	—
330	C₂×C₄	—	2	4	no	—
331	C₂×C₄	—	2	2	no	—
332	C₂×C₄	—	2	2	no	—
333	C₂×C₄	—	26	8	no	—
334	C₂×C₄	—	52	8	no	—
335	C₂×C₄	—	104	16	no	—
336	C₂×C₄	C₂²×D₅₂	1	4	yes	—
337	C₂×C₄	C₂²×D₅₂	1	4	yes	—
338	C₂×C₄	C₂²×D₅₂	1	2	yes	—
339	C₂×C₄	C₂²×D₅₂	1	1	yes	characteristic
340	C₂×C₄	C₂×#6127	1	2	yes	—
341	C₂×C₄	C₂×#6127	1	2	yes	—
342	C₂×C₄	C₂⁴×D₁₃	1	1	yes	characteristic
343	C₈	—	2	2	no	—
344	C₈	—	2	2	no	—
345	C₈	C₂×#6127	1	4	yes	—
346	C₈	C₂×#6127	1	4	yes	—
347	C₂²	—	2	4	no	—
348	C₂²	—	2	2	no	—
349	C₂²	—	4	4	no	—
350	C₂²	—	52	8	no	—
351	C₂²	—	52	8	no	—
352	C₂²	—	52	8	no	—
353	C₂²	—	104	16	no	—
354	C₂²	#11719182	1	2	yes	—
355	C₂²	#11719457	1	4	yes	—
356	C₂²	C₂³×D₅₂	1	1	yes	characteristic
357	C₄	—	2	4	no	—
358	C₄	—	2	4	no	—
359	C₄	—	2	2	no	—
360	C₄	—	52	8	no	—
361	C₄	#11719182	1	2	yes	—
362	C₄	C₂²×#6127	1	1	yes	characteristic
363	C₄	C₂²×#6127	1	1	yes	characteristic
364	C₂	—	4	2	no	—
365	C₂	—	104	8	no	—
366	C₂	#13125486	1	4	yes	direct product factor
367	C₂	C₂×#11719182	1	1	yes	characteristic
368	C₂	C₂×#11719457	1	2	yes	—

	subgroup	G/H
center	C₂³	C₂²×D₅₂
commutator, cosocle	C₂×C₄×C₁₃	C₂⁵
Frattini	C₂×C₄	C₂⁴×D₁₃
Fitting	C₂×C₁₃×#405	C₂
socle	C₂³×C₁₃	C₂²×D₄
hypercenter	C₂×#405	D₁₃
hypocenter	C₁₃	C₂×#3176
LCS	G ▷ C₂×C₄×C₁₃ ▷ C₂×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂³ ◁ C₂×#117 ◁ C₂×#405

Complements

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

N	H ≅ G/N	type
1. C₂×#11719240	C₂	semidirect
2. #13125486	C₂	direct (N × H)
3. C₂×#11718344	C₂	semidirect
4. C₂×#11718356	C₂	semidirect
5. C₂×#11718374	C₂	semidirect
6. C₂×#11718796	C₂	semidirect
7. C₂×#11718776	C₂	semidirect
8. C₂×#11718372	C₂	semidirect
9. C₂×C₄×D₁₀₄	C₂	semidirect
10. C₂×C₁₃×#405	C₂	semidirect
18. #11719240	C₂²	semidirect
24. C₁₃×#405	C₂²	semidirect
29. #11718796	C₂²	semidirect
30. #11718776	C₂²	semidirect
35. #11718374	C₂²	semidirect
36. #11718372	C₂²	semidirect
37. #11718356	C₂²	semidirect
38. C₄×D₁₀₄	C₂²	semidirect
39. #11718344	C₂²	semidirect
139. C₂×#405	D₁₃	semidirect
197. #405	C₂×D₁₃	semidirect
265. C₂×C₁₃	#3176	semidirect
313. C₁₃	C₂×#3176	semidirect
366. C₂	#13125486	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×#11719240	2
#13125486	2
C₂×#11718344	2
C₂×#11718356	2
C₂×#11718374	2
C₂×#11718796	2
C₂×#11718776	2
C₂×#11718372	2
C₂×C₄×D₁₀₄	2
C₂×C₁₃×#405	2
C₂×#3176	13

Subgroups — C2×#13125486 #16732753

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#13125486 #16732753