Subgroups — C₂×#13123687 #16727046

Order 3328.

3086 conjugacy classes of subgroups.

The subgroup lattice has 340 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂³×D₁₀₄	C₂	1	1	yes	maximal, characteristic
2	C₂×#11719226	C₂	1	2	yes	maximal, complemented
3	#13123687	C₂	1	16	yes	maximal, direct product factor
4	C₂×#11718530	C₂	1	4	yes	maximal, complemented
5	C₂×#11718533	C₂	1	4	yes	maximal, complemented
6	C₂×#11718662	C₂	1	2	yes	maximal, complemented
7	C₂×#11718762	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×C₁₃×#427	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂³×D₅₂	C₂²	1	2	yes	—
10	C₂³×C₈×C₁₃	C₂²	1	1	yes	characteristic
11	C₂×C₁₃×#117	C₂²	1	2	yes	—
12	C₂×#88180	C₂²	1	1	yes	characteristic
13	C₂²×#6099	C₂²	1	1	yes	characteristic
14	C₂²×D₁₀₄	C₄	1	16	yes	complemented
15	C₂²×D₁₀₄	C₂²	1	2	yes	—
16	C₂²×D₁₀₄	C₂²	1	2	yes	—
17	C₂²×D₁₀₄	C₂²	1	8	yes	—
18	#11719226	—	2	8	no	—
19	#11719226	C₂²	1	16	yes	complemented
20	C₂×#88147	C₂²	1	8	yes	complemented
21	C₂×C₂²⋊C₄×D₁₃	—	2	4	no	—
22	C₂×C₄×D₅₂	—	2	4	no	—
23	C₁₃×#427	C₂²	1	8	yes	complemented
24	C₂×C₁₃×#107	C₂²	1	2	yes	complemented
25	C₂×C₁₃×#106	C₂²	1	2	yes	complemented
26	#11718762	C₂²	1	8	yes	complemented
27	C₂×#88055	C₂²	1	2	yes	complemented
28	C₂×C₈×Dic₁₃	C₂²	1	2	yes	complemented
29	#11718662	C₂²	1	16	yes	complemented
30	C₂×#88049	C₂²	1	8	yes	complemented
31	#11718533	C₂²	1	32	yes	complemented
32	#11718530	C₂²	1	32	yes	complemented
33	C₂⁴×D₁₃	—	2	2	no	—
34	C₂³×C₄×C₁₃	C₂³	1	1	yes	characteristic
35	C₂²×D₅₂	—	2	2	no	—
36	C₂²×D₅₂	—	2	4	no	—
37	C₂²×D₅₂	—	2	8	no	—
38	C₂²×D₅₂	C₂×C₄	1	16	yes	complemented
39	C₂²×D₅₂	C₂³	1	4	yes	—
40	C₂²×D₅₂	C₂³	1	8	yes	—
41	C₂²×C₄×D₁₃	—	2	8	no	—
42	C₂²×C₈×C₁₃	C₂×C₄	1	8	yes	complemented
43	C₂²×C₈×C₁₃	C₂³	1	1	yes	characteristic
44	C₂²×C₈×C₁₃	C₂³	1	1	yes	characteristic
45	C₂²×C₈×C₁₃	C₂³	1	4	yes	—
46	C₁₃×#117	—	2	4	no	—
47	C₁₃×#117	C₂³	1	8	yes	—
48	C₂×C₁₃×C₄⋊C₄	C₂³	1	4	yes	—
49	C₂×C₁₃×C₂²⋊C₄	—	2	2	no	—
50	C₂×C₄²×C₁₃	—	2	2	no	—
51	C₂×C₂²⋊Dic₁₃	D₄	1	2	yes	—
52	C₂×#6103	—	2	8	no	—
53	#88180	D₄	1	4	yes	—
54	#88180	C₂³	1	4	yes	—
55	C₂×C₄⋊Dic₁₃	D₄	1	2	yes	—
56	C₂×C₄×Dic₁₃	C₂³	1	2	yes	—
57	C₂×#6099	—	2	4	no	—
58	C₂×#6099	C₂³	1	4	yes	—
59	C₂²×#1055	C₂³	1	2	yes	—
60	C₂×D₁₀₄	C₂×C₄	1	16	yes	complemented
61	C₂×D₁₀₄	C₂×C₄	1	16	yes	complemented
62	C₂×D₁₀₄	C₂×C₄	1	64	yes	complemented
63	C₂×D₁₀₄	C₂³	1	8	yes	—
64	C₂×D₁₀₄	C₂³	1	8	yes	—
65	#88147	—	2	16	no	—
66	#88147	C₂³	1	32	yes	complemented
67	C₂²⋊C₄×D₁₃	—	2	32	no	—
68	C₄×D₅₂	—	2	32	no	—
69	C₁₃×#107	C₂³	1	8	yes	complemented
70	C₁₃×#106	C₂³	1	8	yes	complemented
71	#88055	C₂³	1	8	yes	complemented
72	C₈×Dic₁₃	C₂³	1	8	yes	complemented
73	#88049	C₂³	1	32	yes	complemented
74	C₂×#2948	—	13	1	no	maximal, Sylow
75	C₂⁴×C₁₃	C₂×D₄	1	1	yes	characteristic
76	C₂³×D₁₃	—	2	4	no	—
77	C₂³×D₁₃	—	2	16	no	—
78	C₂³×D₁₃	—	2	8	no	—
79	C₂³×D₁₃	—	8	8	no	—
80	C₂²×C₄×C₁₃	—	2	4	no	—
81	C₂²×C₄×C₁₃	C₂²×C₄	1	4	yes	—
82	C₂²×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
83	C₂²×C₄×C₁₃	C₂×D₄	1	2	yes	—
84	C₂²×C₄×C₁₃	#41	1	4	yes	—
85	C₂²×C₄×C₁₃	C₂⁴	1	1	yes	characteristic
86	C₂²×C₄×C₁₃	C₂⁴	1	2	yes	—
87	C₂²×Dic₁₃	C₂×D₄	1	4	yes	—
88	C₂×D₅₂	—	2	32	no	—
89	C₂×D₅₂	—	2	16	no	—
90	C₂×D₅₂	—	2	8	no	—
91	C₂×D₅₂	—	2	4	no	—
92	C₂×D₅₂	—	2	8	no	—
93	C₂×D₅₂	—	2	16	no	—
94	C₂×D₅₂	C₂²×C₄	1	16	yes	—
95	C₂×D₅₂	C₂²×C₄	1	32	yes	complemented
96	C₂×D₅₂	C₂⁴	1	8	yes	—
97	C₂×C₄×D₁₃	—	2	32	no	—
98	C₂×C₄×D₁₃	—	8	16	no	—
99	C₂×C₈×C₁₃	C₂²×C₄	1	4	yes	—
100	C₂×C₈×C₁₃	C₂²×C₄	1	4	yes	—
101	C₂×C₈×C₁₃	C₂²×C₄	1	16	yes	complemented
102	C₂×C₈×C₁₃	C₂⁴	1	2	yes	—
103	C₂×C₈×C₁₃	C₂⁴	1	2	yes	—
104	C₁₃×C₄⋊C₄	—	2	4	no	—
105	C₁₃×C₄⋊C₄	C₂⁴	1	8	yes	—
106	C₁₃×C₂²⋊C₄	—	2	8	no	—
107	C₄²×C₁₃	—	2	8	no	—
108	C₂²⋊Dic₁₃	C₂×D₄	1	8	yes	—
109	#6103	—	2	32	no	—
110	C₄⋊Dic₁₃	C₂×D₄	1	8	yes	—
111	C₄×Dic₁₃	C₂×D₄	1	4	yes	—
112	C₄×Dic₁₃	C₂⁴	1	4	yes	—
113	#6099	—	2	8	no	—
114	C₂×#1055	—	2	4	no	—
115	C₂×#1055	C₂⁴	1	4	yes	—
116	D₁₀₄	C₂²×C₄	1	32	yes	complemented
117	D₁₀₄	C₂²×C₄	1	32	yes	complemented
118	C₂³×D₈	—	13	1	no	—
119	C₂×#518	—	13	2	no	—
120	#2948	—	13	16	no	—
121	C₂×#442	—	13	4	no	—
122	C₂×C₄×D₈	—	13	4	no	—
123	C₂×#427	D₁₃	1	1	yes	characteristic, complemented
124	C₂×#418	—	13	2	no	—
125	C₂×#405	—	13	1	no	—
126	C₂³×C₁₃	—	2	4	no	—
127	C₂³×C₁₃	C₄×D₄	1	4	yes	—
128	C₂³×C₁₃	C₂²×D₄	1	2	yes	—
129	C₂³×C₁₃	C₂²×D₄	1	1	yes	characteristic
130	C₂²×D₁₃	—	2	16	no	—
131	C₂²×D₁₃	—	2	8	no	—
132	C₂²×D₁₃	—	2	32	no	—
133	C₂²×D₁₃	—	4	16	no	—
134	C₂²×D₁₃	—	8	32	no	—
135	C₂²×D₁₃	—	8	16	no	—
136	C₂×C₄×C₁₃	—	2	8	no	—
137	C₂×C₄×C₁₃	—	4	8	no	—
138	C₂×C₄×C₁₃	C₄×D₄	1	8	yes	—
139	C₂×C₄×C₁₃	C₄×D₄	1	4	yes	—
140	C₂×C₄×C₁₃	C₂³×C₄	1	2	yes	—
141	C₂×C₄×C₁₃	C₂³×C₄	1	4	yes	—
142	C₂×C₄×C₁₃	C₂²×D₄	1	1	yes	characteristic
143	C₂×C₄×C₁₃	C₂²×D₄	1	2	yes	—
144	C₂×C₄×C₁₃	C₂×#41	1	2	yes	—
145	C₂×C₄×C₁₃	C₂×#41	1	4	yes	—
146	C₂×C₄×C₁₃	C₂⁵	1	1	yes	characteristic
147	C₂×Dic₁₃	—	2	8	no	—
148	C₂×Dic₁₃	C₂²×D₄	1	8	yes	—
149	D₅₂	—	2	16	no	—
150	D₅₂	—	2	32	no	—
151	D₅₂	—	2	8	no	—
152	D₅₂	C₂³×C₄	1	16	yes	—
153	C₄×D₁₃	—	8	32	no	—
154	C₈×C₁₃	C₂³×C₄	1	4	yes	—
155	C₈×C₁₃	C₂³×C₄	1	4	yes	—
156	#1055	—	2	4	no	—
157	C₂³×D₄	—	13	2	no	—
158	C₂²×D₈	—	13	16	no	—
159	C₂²×D₈	—	13	2	no	—
160	C₂²×D₈	—	13	2	no	—
161	C₂²×D₈	—	13	8	no	—
162	C₂²×M₁₆	—	13	1	no	—
163	C₂³×C₈	C₂×D₁₃	1	1	yes	characteristic
164	#518	—	13	16	no	—
165	#518	—	26	8	no	—
166	C₂×C₄×D₄	—	13	8	no	—
167	C₂×C₄×D₄	—	26	4	no	—
168	C₂×#117	—	13	1	no	—
169	C₂×#117	C₂×D₁₃	1	2	yes	—
170	C₂²×C₂²⋊C₄	—	26	4	no	—
171	#442	—	13	32	no	—
172	C₄×D₈	—	13	32	no	—
173	#427	C₂×D₁₃	1	8	yes	complemented
174	C₂×#107	C₂×D₁₃	1	2	yes	complemented
175	C₂×#106	C₂×D₁₃	1	2	yes	complemented
176	#418	—	13	16	no	—
177	C₂×#102	—	13	8	no	—
178	#405	—	13	8	no	—
179	C₂×#97	—	13	2	no	—
180	C₂×C₄×C₈	—	13	2	no	—
181	C₂²×C₁₃	—	2	8	no	—
182	C₂²×C₁₃	—	2	4	no	—
183	C₂²×C₁₃	C₂×C₄×D₄	1	4	yes	—
184	C₂²×C₁₃	C₂×C₄×D₄	1	2	yes	—
185	C₂²×C₁₃	#576	1	4	yes	—
186	C₂²×C₁₃	C₂³×D₄	1	1	yes	characteristic
187	C₂×D₁₃	—	2	16	no	—
188	C₂×D₁₃	—	4	32	no	—
189	C₂×D₁₃	—	4	16	no	—
190	C₂×D₁₃	—	8	32	no	—
191	C₄×C₁₃	—	4	8	no	—
192	C₄×C₁₃	C₂×C₄×D₄	1	2	yes	—
193	C₄×C₁₃	C₂×C₄×D₄	1	4	yes	—
194	C₄×C₁₃	C₂⁴×C₄	1	1	yes	characteristic
195	C₄×C₁₃	C₂²×#41	1	1	yes	characteristic
196	Dic₁₃	—	2	8	no	—
197	C₂⁵	—	26	2	no	—
198	C₂²×D₄	—	13	16	no	—
199	C₂²×D₄	—	13	4	no	—
200	C₂²×D₄	—	13	8	no	—
201	C₂²×D₄	—	26	2	no	—
202	C₂²×D₄	—	26	4	no	—
203	C₂²×D₄	—	26	8	no	—
204	C₂³×C₄	—	26	8	no	—
205	C₂³×C₄	C₂²×D₁₃	1	1	yes	characteristic
206	C₂×D₈	—	13	16	no	—
207	C₂×D₈	—	13	16	no	—
208	C₂×D₈	—	13	64	no	—
209	C₂×D₈	—	13	8	no	—
210	C₂×D₈	—	13	8	no	—
211	C₂×M₁₆	—	13	4	no	—
212	C₂×M₁₆	—	26	4	no	—
213	C₂²×C₈	—	13	2	no	—
214	C₂²×C₈	C₄×D₁₃	1	8	yes	complemented
215	C₂²×C₈	C₂²×D₁₃	1	1	yes	characteristic
216	C₂²×C₈	C₂²×D₁₃	1	1	yes	characteristic
217	C₂²×C₈	C₂²×D₁₃	1	4	yes	—
218	C₄×D₄	—	13	32	no	—
219	C₄×D₄	—	26	32	no	—
220	C₄×D₄	—	26	16	no	—
221	#117	—	2	4	no	—
222	#117	—	13	4	no	—
223	#117	—	13	4	no	—
224	#117	C₂²×D₁₃	1	8	yes	—
225	C₂×C₄⋊C₄	—	13	2	no	—
226	C₂×C₄⋊C₄	C₂²×D₁₃	1	4	yes	—
227	C₂×C₂²⋊C₄	—	2	2	no	—
228	C₂×C₂²⋊C₄	—	13	2	no	—
229	C₂×C₂²⋊C₄	—	26	32	no	—
230	C₂×C₂²⋊C₄	—	26	8	no	—
231	C₂×C₄²	—	2	2	no	—
232	C₂×C₄²	—	13	2	no	—
233	#107	C₂²×D₁₃	1	8	yes	complemented
234	#106	C₂²×D₁₃	1	8	yes	complemented
235	#102	—	13	32	no	—
236	#97	—	13	8	no	—
237	C₄×C₈	—	13	8	no	—
238	C₂×C₁₃	—	2	4	no	—
239	C₂×C₁₃	#2948	1	4	yes	complemented
240	C₂×C₁₃	C₂²×C₄×D₄	1	1	yes	characteristic
241	C₂×C₁₃	C₂×#576	1	2	yes	—
242	D₁₃	—	4	16	no	—
243	C₂⁴	—	26	4	no	—
244	C₂⁴	—	26	16	no	—
245	C₂⁴	—	26	8	no	—
246	C₂⁴	—	104	8	no	—
247	C₂⁴	D₄×D₁₃	1	1	yes	characteristic
248	C₂×D₄	—	13	16	no	—
249	C₂×D₄	—	13	32	no	—
250	C₂×D₄	—	13	8	no	—
251	C₂×D₄	—	26	16	no	—
252	C₂×D₄	—	26	32	no	—
253	C₂×D₄	—	26	8	no	—
254	C₂×D₄	—	26	4	no	—
255	C₂×D₄	—	26	8	no	—
256	C₂×D₄	—	26	16	no	—
257	C₂²×C₄	—	2	4	no	—
258	C₂²×C₄	—	13	4	no	—
259	C₂²×C₄	—	26	32	no	—
260	C₂²×C₄	—	104	16	no	—
261	C₂²×C₄	C₂×C₄×D₁₃	1	4	yes	—
262	C₂²×C₄	D₄×D₁₃	1	2	yes	—
263	C₂²×C₄	D₄×D₁₃	1	1	yes	characteristic
264	C₂²×C₄	#6131	1	4	yes	—
265	C₂²×C₄	C₂³×D₁₃	1	2	yes	—
266	C₂²×C₄	C₂³×D₁₃	1	1	yes	characteristic
267	D₈	—	13	32	no	—
268	D₈	—	13	32	no	—
269	M₁₆	—	26	8	no	—
270	C₂×C₈	—	13	4	no	—
271	C₂×C₈	—	26	4	no	—
272	C₂×C₈	C₂×C₄×D₁₃	1	4	yes	—
273	C₂×C₈	C₂×C₄×D₁₃	1	4	yes	—
274	C₂×C₈	C₂×C₄×D₁₃	1	16	yes	complemented
275	C₂×C₈	C₂³×D₁₃	1	2	yes	—
276	C₂×C₈	C₂³×D₁₃	1	2	yes	—
277	C₄⋊C₄	—	2	4	no	—
278	C₄⋊C₄	—	13	8	no	—
279	C₄⋊C₄	C₂³×D₁₃	1	8	yes	—
280	C₂²⋊C₄	—	2	8	no	—
281	C₂²⋊C₄	—	13	8	no	—
282	C₂²⋊C₄	—	26	32	no	—
283	C₄²	—	2	8	no	—
284	C₄²	—	13	4	no	—
285	C₄²	—	13	4	no	—
286	C₁₃	C₂×#2948	1	1	yes	characteristic, complemented, Sylow
287	C₂³	—	2	4	no	—
288	C₂³	—	26	16	no	—
289	C₂³	—	26	8	no	—
290	C₂³	—	26	32	no	—
291	C₂³	—	52	16	no	—
292	C₂³	—	104	32	no	—
293	C₂³	—	104	16	no	—
294	C₂³	#88147	1	4	yes	—
295	C₂³	C₂×D₄×D₁₃	1	2	yes	—
296	C₂³	C₂×D₄×D₁₃	1	1	yes	characteristic
297	D₄	—	13	16	no	—
298	D₄	—	26	16	no	—
299	D₄	—	26	32	no	—
300	D₄	—	26	8	no	—
301	C₂×C₄	—	2	8	no	—
302	C₂×C₄	—	4	8	no	—
303	C₂×C₄	—	13	8	no	—
304	C₂×C₄	—	26	8	no	—
305	C₂×C₄	—	104	32	no	—
306	C₂×C₄	#88147	1	8	yes	—
307	C₂×C₄	#88147	1	4	yes	—
308	C₂×C₄	C₂²×C₄×D₁₃	1	2	yes	—
309	C₂×C₄	C₂²×C₄×D₁₃	1	4	yes	—
310	C₂×C₄	C₂×D₄×D₁₃	1	2	yes	—
311	C₂×C₄	C₂×D₄×D₁₃	1	1	yes	characteristic
312	C₂×C₄	C₂×#6131	1	2	yes	—
313	C₂×C₄	C₂×#6131	1	4	yes	—
314	C₂×C₄	C₂⁴×D₁₃	1	1	yes	characteristic
315	C₈	—	26	4	no	—
316	C₈	C₂²×C₄×D₁₃	1	4	yes	—
317	C₈	C₂²×C₄×D₁₃	1	4	yes	—
318	C₂²	—	2	8	no	—
319	C₂²	—	2	4	no	—
320	C₂²	—	26	16	no	—
321	C₂²	—	52	32	no	—
322	C₂²	—	52	16	no	—
323	C₂²	—	104	32	no	—
324	C₂²	C₂×#88147	1	4	yes	—
325	C₂²	C₂×#88147	1	2	yes	—
326	C₂²	#11719474	1	4	yes	—
327	C₂²	C₂²×D₄×D₁₃	1	1	yes	characteristic
328	C₄	—	4	8	no	—
329	C₄	—	26	8	no	—
330	C₄	C₂×#88147	1	2	yes	—
331	C₄	C₂×#88147	1	4	yes	—
332	C₄	C₂³×C₄×D₁₃	1	1	yes	characteristic
333	C₄	C₂²×#6131	1	1	yes	characteristic
334	C₂	—	2	4	no	—
335	C₂	—	52	16	no	—
336	C₂	#13123687	1	4	yes	direct product factor
337	C₂	C₂²×#88147	1	1	yes	characteristic
338	C₂	C₂×#11719474	1	2	yes	—

	subgroup	G/H
center	C₂³	C₂×D₄×D₁₃
commutator	C₄×C₁₃	C₂⁴×C₄
Frattini	C₂×C₄	C₂⁴×D₁₃
Fitting	C₂×C₁₃×#427	C₂
socle	C₂³×C₁₃	C₂²×D₄
prime radical	C₂³×D₁₀₄	C₂
cosocle	C₂×C₄×C₁₃	C₂⁵
hypercenter	C₂×#427	D₁₃
hypocenter	C₁₃	C₂×#2948
LCS	G ▷ C₄×C₁₃ ▷ C₂×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂³ ◁ C₂³×C₄ ◁ C₂×#427

Complements

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

N	H ≅ G/N	type
2. C₂×#11719226	C₂	semidirect
3. #13123687	C₂	direct (N × H)
4. C₂×#11718530	C₂	semidirect
5. C₂×#11718533	C₂	semidirect
6. C₂×#11718662	C₂	semidirect
7. C₂×#11718762	C₂	semidirect
8. C₂×C₁₃×#427	C₂	semidirect
14. C₂²×D₁₀₄	C₄	semidirect
19. #11719226	C₂²	semidirect
20. C₂×#88147	C₂²	semidirect
23. C₁₃×#427	C₂²	semidirect
24. C₂×C₁₃×#107	C₂²	semidirect
25. C₂×C₁₃×#106	C₂²	semidirect
26. #11718762	C₂²	semidirect
27. C₂×#88055	C₂²	semidirect
28. C₂×C₈×Dic₁₃	C₂²	semidirect
29. #11718662	C₂²	semidirect
30. C₂×#88049	C₂²	semidirect
31. #11718533	C₂²	semidirect
32. #11718530	C₂²	semidirect
38. C₂²×D₅₂	C₂×C₄	semidirect
42. C₂²×C₈×C₁₃	C₂×C₄	semidirect
60. C₂×D₁₀₄	C₂×C₄	semidirect
61. C₂×D₁₀₄	C₂×C₄	semidirect
62. C₂×D₁₀₄	C₂×C₄	semidirect
66. #88147	C₂³	semidirect
69. C₁₃×#107	C₂³	semidirect
70. C₁₃×#106	C₂³	semidirect
71. #88055	C₂³	semidirect
72. C₈×Dic₁₃	C₂³	semidirect
73. #88049	C₂³	semidirect
95. C₂×D₅₂	C₂²×C₄	semidirect
101. C₂×C₈×C₁₃	C₂²×C₄	semidirect
116. D₁₀₄	C₂²×C₄	semidirect
117. D₁₀₄	C₂²×C₄	semidirect
123. C₂×#427	D₁₃	semidirect
173. #427	C₂×D₁₃	semidirect
174. C₂×#107	C₂×D₁₃	semidirect
175. C₂×#106	C₂×D₁₃	semidirect
214. C₂²×C₈	C₄×D₁₃	semidirect
233. #107	C₂²×D₁₃	semidirect
234. #106	C₂²×D₁₃	semidirect
239. C₂×C₁₃	#2948	semidirect
274. C₂×C₈	C₂×C₄×D₁₃	semidirect
286. C₁₃	C₂×#2948	semidirect
336. C₂	#13123687	direct (N × H)

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
C₂³×D₁₀₄	2
C₂×#11719226	2
#13123687	2
C₂×#11718530	2
C₂×#11718533	2
C₂×#11718662	2
C₂×#11718762	2
C₂×C₁₃×#427	2
C₂×#2948	13

Subgroups — C2×#13123687 #16727046

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#13123687 #16727046