Subgroups — C₂×#13117027 #16017689

Order 3328.

638 conjugacy classes of subgroups.

The subgroup lattice has 324 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#11718776	C₂	1	1	yes	maximal, characteristic
2	#13117027	C₂	1	8	yes	maximal, direct product factor
3	C₂×#11718082	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂×#11718081	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂×#11718083	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×#11718083	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂×C₂²⋊#6090	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×C₁₃×#427	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂³×C₈×C₁₃	C₂²	1	1	yes	characteristic
10	C₂×C₁₃×#117	—	2	1	no	—
11	C₂×#88184	—	2	1	no	—
12	C₂²×D₁₀₄	C₂²	1	1	yes	characteristic
13	C₁₃×#427	—	2	2	no	—
14	C₁₃×#427	C₂²	1	4	yes	complemented
15	C₂×C₁₃×#107	C₂²	1	1	yes	characteristic
16	C₂×C₁₃×#107	C₂²	1	1	yes	characteristic
17	C₂×C₁₃×#106	—	2	1	no	—
18	#11718776	—	2	2	no	—
19	#11718776	C₂²	1	4	yes	—
20	C₂×#88061	—	2	1	no	—
21	C₂×#88058	C₂²	1	1	yes	characteristic
22	C₂²×#6090	C₂²	1	1	yes	characteristic
23	C₂²×#6090	C₂²	1	1	yes	characteristic
24	C₂²⋊#6090	C₂²	1	4	yes	complemented
25	#11718083	C₂²	1	4	yes	complemented
26	#11718083	C₂²	1	4	yes	complemented
27	#11718082	C₂²	1	4	yes	complemented
28	#11718081	C₂²	1	4	yes	complemented
29	C₂³×C₄×C₁₃	D₄	1	1	yes	characteristic
30	C₂²×C₂²⋊D₁₃	—	4	1	no	—
31	C₂²×D₅₂	—	2	1	no	—
32	C₂²×C₈×C₁₃	—	2	2	no	—
33	C₂²×C₈×C₁₃	—	2	2	no	—
34	C₂²×C₈×C₁₃	D₄	1	1	yes	characteristic
35	C₂²×C₈×C₁₃	D₄	1	2	yes	—
36	C₂²×C₈×C₁₃	C₂³	1	2	yes	—
37	C₂²×C₈×C₁₃	C₂³	1	1	yes	characteristic
38	C₁₃×#117	—	2	4	no	—
39	C₁₃×#117	—	4	2	no	—
40	C₂×C₁₃×C₄⋊C₄	—	2	1	no	—
41	C₂×C₁₃×C₄⋊C₄	—	2	1	no	—
42	C₂×C₁₃×C₂²⋊C₄	—	4	1	no	—
43	C₂×C₄²×C₁₃	—	4	1	no	—
44	#88184	—	2	4	no	—
45	#88184	—	4	2	no	—
46	C₂×#6103	—	4	1	no	—
47	C₂×C₄⋊Dic₁₃	—	2	1	no	—
48	C₂×D₁₀₄	—	2	2	no	—
49	C₂×D₁₀₄	—	2	2	no	—
50	C₂×D₁₀₄	—	2	1	no	—
51	C₂×D₁₀₄	C₂³	1	2	yes	—
52	C₁₃×#107	—	2	1	no	—
53	C₁₃×#107	—	2	1	no	—
54	C₁₃×#107	C₂³	1	2	yes	—
55	C₁₃×#107	C₂³	1	2	yes	—
56	C₁₃×#106	—	2	4	no	—
57	#88061	—	2	4	no	—
58	#88058	—	2	1	no	—
59	#88058	C₂³	1	2	yes	—
60	C₂×#6090	—	2	2	no	—
61	C₂×#6090	—	2	2	no	—
62	C₂×#6090	C₂³	1	2	yes	—
63	C₂×#6090	C₂³	1	2	yes	—
64	C₂×#2234	—	13	1	no	maximal, Sylow
65	C₂⁴×C₁₃	D₈	1	1	yes	characteristic
66	C₂³×D₁₃	—	4	1	no	—
67	C₂²×C₄×C₁₃	—	2	2	no	—
68	C₂²×C₄×C₁₃	—	2	2	no	—
69	C₂²×C₄×C₁₃	—	4	1	no	—
70	C₂²×C₄×C₁₃	—	4	1	no	—
71	C₂²×C₄×C₁₃	D₈	1	1	yes	characteristic
72	C₂²×C₄×C₁₃	D₈	1	2	yes	—
73	C₂²×C₄×C₁₃	C₂×D₄	1	2	yes	—
74	C₂²×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
75	C₂×C₂²⋊D₁₃	—	4	4	no	—
76	C₂×C₂²⋊D₁₃	—	8	4	no	—
77	C₂²×Dic₁₃	—	4	1	no	—
78	C₂×D₅₂	—	2	2	no	—
79	C₂×D₅₂	—	4	2	no	—
80	C₂×D₅₂	—	4	2	no	—
81	C₂×D₅₂	—	4	1	no	—
82	C₂×C₈×C₁₃	—	2	2	no	—
83	C₂×C₈×C₁₃	—	2	4	no	—
84	C₂×C₈×C₁₃	—	2	2	no	—
85	C₂×C₈×C₁₃	—	2	2	no	—
86	C₂×C₈×C₁₃	C₂×D₄	1	2	yes	—
87	C₂×C₈×C₁₃	C₂×D₄	1	1	yes	characteristic
88	C₂×C₈×C₁₃	#41	1	2	yes	—
89	C₂×C₈×C₁₃	#41	1	2	yes	—
90	C₂×C₈×C₁₃	C₂⁴	1	1	yes	characteristic
91	C₁₃×C₄⋊C₄	—	2	2	no	—
92	C₁₃×C₄⋊C₄	—	2	2	no	—
93	C₁₃×C₄⋊C₄	—	4	1	no	—
94	C₁₃×C₄⋊C₄	—	4	1	no	—
95	C₁₃×C₂²⋊C₄	—	4	4	no	—
96	C₄²×C₁₃	—	4	4	no	—
97	#6103	—	4	4	no	—
98	C₄⋊Dic₁₃	—	2	2	no	—
99	C₄⋊Dic₁₃	—	4	1	no	—
100	D₁₀₄	—	2	2	no	—
101	D₁₀₄	—	2	2	no	—
102	D₁₀₄	—	4	2	no	—
103	#6090	—	2	2	no	—
104	#6090	—	2	2	no	—
105	C₂×#466	—	13	1	no	—
106	C₂×#427	D₁₃	1	1	yes	characteristic, complemented
107	#2234	—	13	8	no	—
108	C₂×#367	—	13	1	no	—
109	C₂×#366	—	13	1	no	—
110	C₂×#357	—	13	1	no	—
111	C₂×#357	—	13	1	no	—
112	C₂×C₂²⋊C₁₆	—	13	1	no	—
113	C₂³×C₁₃	—	2	2	no	—
114	C₂³×C₁₃	—	2	2	no	—
115	C₂³×C₁₃	—	2	2	no	—
116	C₂³×C₁₃	C₂×D₈	1	2	yes	—
117	C₂³×C₁₃	C₂×D₈	1	1	yes	characteristic
118	C₂²×D₁₃	—	4	2	no	—
119	C₂²×D₁₃	—	8	2	no	—
120	C₂²×D₁₃	—	8	2	no	—
121	C₂²×D₁₃	—	8	2	no	—
122	C₂×C₄×C₁₃	—	2	2	no	—
123	C₂×C₄×C₁₃	—	2	2	no	—
124	C₂×C₄×C₁₃	—	2	2	no	—
125	C₂×C₄×C₁₃	—	2	4	no	—
126	C₂×C₄×C₁₃	—	4	2	no	—
127	C₂×C₄×C₁₃	—	4	2	no	—
128	C₂×C₄×C₁₃	—	8	2	no	—
129	C₂×C₄×C₁₃	—	8	2	no	—
130	C₂×C₄×C₁₃	#123	1	2	yes	—
131	C₂×C₄×C₁₃	C₂×D₈	1	2	yes	—
132	C₂×C₄×C₁₃	C₂×D₈	1	1	yes	characteristic
133	C₂×C₄×C₁₃	#137	1	2	yes	—
134	C₂×C₄×C₁₃	C₂²×D₄	1	1	yes	characteristic
135	C₂²⋊D₁₃	—	8	8	no	—
136	C₂×Dic₁₃	—	4	2	no	—
137	C₂×Dic₁₃	—	8	2	no	—
138	D₅₂	—	4	2	no	—
139	D₅₂	—	4	2	no	—
140	D₅₂	—	8	2	no	—
141	C₈×C₁₃	—	2	2	no	—
142	C₈×C₁₃	#123	1	2	yes	—
143	C₈×C₁₃	C₂×#41	1	1	yes	characteristic
144	C₈×C₁₃	C₂×#41	1	1	yes	characteristic
145	C₂²×D₈	—	13	1	no	—
146	C₂³×C₈	C₂×D₁₃	1	1	yes	characteristic
147	C₂×#121	—	26	1	no	—
148	C₂×#117	—	2	1	no	—
149	C₂²×C₁₆	—	13	1	no	—
150	C₂²×C₁₆	—	13	1	no	—
151	#466	—	13	4	no	—
152	#466	—	26	2	no	—
153	#427	—	2	2	no	—
154	#427	C₂×D₁₃	1	4	yes	complemented
155	C₂×#107	—	13	1	no	—
156	C₂×#107	C₂×D₁₃	1	1	yes	characteristic
157	C₂×#107	C₂×D₁₃	1	1	yes	characteristic
158	C₂×#106	—	2	1	no	—
159	C₂×#102	—	26	1	no	—
160	#367	—	13	4	no	—
161	#366	—	13	4	no	—
162	#357	—	13	4	no	—
163	#357	—	13	4	no	—
164	C₂²⋊C₁₆	—	13	4	no	—
165	C₂²×C₁₃	—	2	2	no	—
166	C₂²×C₁₃	—	2	2	no	—
167	C₂²×C₁₃	—	2	2	no	—
168	C₂²×C₁₃	—	4	4	no	—
169	C₂²×C₁₃	#480	1	2	yes	—
170	C₂²×C₁₃	#508	1	2	yes	—
171	C₂²×C₁₃	#509	1	2	yes	—
172	C₂²×C₁₃	C₂²×D₈	1	1	yes	characteristic
173	C₂×D₁₃	—	8	2	no	—
174	C₂×D₁₃	—	8	2	no	—
175	C₂×D₁₃	—	8	2	no	—
176	C₂×D₁₃	—	16	4	no	—
177	C₄×C₁₃	—	2	2	no	—
178	C₄×C₁₃	—	8	2	no	—
179	C₄×C₁₃	—	8	2	no	—
180	C₄×C₁₃	#480	1	2	yes	—
181	C₄×C₁₃	C₂×#123	1	1	yes	characteristic
182	C₄×C₁₃	C₂×#137	1	1	yes	characteristic
183	Dic₁₃	—	8	2	no	—
184	C₂²×D₄	—	26	1	no	—
185	C₂²×D₄	—	52	1	no	—
186	C₂³×C₄	C₂²⋊D₁₃	1	1	yes	characteristic
187	C₂×D₈	—	13	2	no	—
188	C₂×D₈	—	26	2	no	—
189	C₂×D₈	—	26	2	no	—
190	C₂×D₈	—	26	1	no	—
191	C₂²×C₈	—	2	2	no	—
192	C₂²×C₈	—	2	2	no	—
193	C₂²×C₈	C₂²⋊D₁₃	1	1	yes	characteristic
194	C₂²×C₈	C₂²⋊D₁₃	1	2	yes	—
195	C₂²×C₈	C₂²×D₁₃	1	2	yes	—
196	C₂²×C₈	C₂²×D₁₃	1	1	yes	characteristic
197	#121	—	26	4	no	—
198	#121	—	52	2	no	—
199	#117	—	2	4	no	—
200	#117	—	4	2	no	—
201	C₂×C₄⋊C₄	—	2	1	no	—
202	C₂×C₄⋊C₄	—	2	1	no	—
203	C₂×C₄⋊C₄	—	26	1	no	—
204	C₂×C₂²⋊C₄	—	4	1	no	—
205	C₂×C₂²⋊C₄	—	52	1	no	—
206	C₂×C₄²	—	4	1	no	—
207	C₂×C₁₆	—	13	2	no	—
208	C₂×C₁₆	—	13	2	no	—
209	C₂×C₁₆	—	26	2	no	—
210	C₂×C₁₆	—	26	2	no	—
211	#107	—	2	1	no	—
212	#107	—	2	1	no	—
213	#107	—	13	2	no	—
214	#107	—	26	1	no	—
215	#107	C₂²×D₁₃	1	2	yes	—
216	#107	C₂²×D₁₃	1	2	yes	—
217	#106	—	2	4	no	—
218	#102	—	26	4	no	—
219	C₂×C₁₃	—	4	2	no	—
220	C₂×C₁₃	#2234	1	4	yes	complemented
221	C₂×C₁₃	C₂×#480	1	1	yes	characteristic
222	C₂×C₁₃	C₂×#508	1	1	yes	characteristic
223	C₂×C₁₃	C₂×#509	1	1	yes	characteristic
224	D₁₃	—	16	2	no	—
225	C₂⁴	—	52	1	no	—
226	C₂⁴	#6104	1	1	yes	characteristic
227	C₂×D₄	—	26	2	no	—
228	C₂×D₄	—	52	4	no	—
229	C₂×D₄	—	52	2	no	—
230	C₂×D₄	—	52	1	no	—
231	C₂×D₄	—	52	2	no	—
232	C₂×D₄	—	104	4	no	—
233	C₂²×C₄	—	2	2	no	—
234	C₂²×C₄	—	2	2	no	—
235	C₂²×C₄	—	4	1	no	—
236	C₂²×C₄	—	4	1	no	—
237	C₂²×C₄	—	52	1	no	—
238	C₂²×C₄	#6104	1	2	yes	—
239	C₂²×C₄	#6104	1	1	yes	characteristic
240	C₂²×C₄	C₂×C₂²⋊D₁₃	1	2	yes	—
241	C₂²×C₄	C₂×C₂²⋊D₁₃	1	1	yes	characteristic
242	D₈	—	26	2	no	—
243	D₈	—	26	2	no	—
244	D₈	—	52	2	no	—
245	C₂×C₈	—	2	2	no	—
246	C₂×C₈	—	2	4	no	—
247	C₂×C₈	—	2	2	no	—
248	C₂×C₈	—	2	2	no	—
249	C₂×C₈	#6127	1	2	yes	—
250	C₂×C₈	#6127	1	2	yes	—
251	C₂×C₈	C₂×C₂²⋊D₁₃	1	2	yes	—
252	C₂×C₈	C₂×C₂²⋊D₁₃	1	1	yes	characteristic
253	C₂×C₈	C₂³×D₁₃	1	1	yes	characteristic
254	C₄⋊C₄	—	2	2	no	—
255	C₄⋊C₄	—	2	2	no	—
256	C₄⋊C₄	—	4	1	no	—
257	C₄⋊C₄	—	4	1	no	—
258	C₄⋊C₄	—	26	2	no	—
259	C₄⋊C₄	—	52	1	no	—
260	C₂²⋊C₄	—	4	4	no	—
261	C₂²⋊C₄	—	52	4	no	—
262	C₄²	—	4	4	no	—
263	C₁₆	—	26	2	no	—
264	C₁₆	—	26	2	no	—
265	C₁₃	C₂×#2234	1	1	yes	characteristic, complemented, Sylow
266	C₂³	—	2	2	no	—
267	C₂³	—	2	2	no	—
268	C₂³	—	2	2	no	—
269	C₂³	—	52	2	no	—
270	C₂³	—	104	2	no	—
271	C₂³	—	104	2	no	—
272	C₂³	—	104	2	no	—
273	C₂³	C₂×#6104	1	2	yes	—
274	C₂³	C₂×#6104	1	1	yes	characteristic
275	D₄	—	52	2	no	—
276	D₄	—	52	2	no	—
277	D₄	—	104	8	no	—
278	D₄	—	104	2	no	—
279	C₂×C₄	—	2	2	no	—
280	C₂×C₄	—	2	2	no	—
281	C₂×C₄	—	2	2	no	—
282	C₂×C₄	—	2	4	no	—
283	C₂×C₄	—	4	2	no	—
284	C₂×C₄	—	4	2	no	—
285	C₂×C₄	—	8	2	no	—
286	C₂×C₄	—	8	2	no	—
287	C₂×C₄	—	52	2	no	—
288	C₂×C₄	—	104	2	no	—
289	C₂×C₄	#88183	1	2	yes	—
290	C₂×C₄	C₂×#6104	1	2	yes	—
291	C₂×C₄	C₂×#6104	1	1	yes	characteristic
292	C₂×C₄	#88196	1	2	yes	—
293	C₂×C₄	C₂²×C₂²⋊D₁₃	1	1	yes	characteristic
294	C₈	—	2	2	no	—
295	C₈	#88183	1	2	yes	—
296	C₈	C₂×#6127	1	1	yes	characteristic
297	C₈	C₂×#6127	1	1	yes	characteristic
298	C₂²	—	2	2	no	—
299	C₂²	—	2	2	no	—
300	C₂²	—	2	2	no	—
301	C₂²	—	4	4	no	—
302	C₂²	—	104	2	no	—
303	C₂²	—	104	2	no	—
304	C₂²	—	104	2	no	—
305	C₂²	—	208	4	no	—
306	C₂²	#11718628	1	2	yes	—
307	C₂²	#11718854	1	2	yes	—
308	C₂²	#11718855	1	2	yes	—
309	C₂²	C₂²×#6104	1	1	yes	characteristic
310	C₄	—	2	2	no	—
311	C₄	—	8	2	no	—
312	C₄	—	8	2	no	—
313	C₄	—	104	2	no	—
314	C₄	#11718628	1	2	yes	—
315	C₄	C₂×#88183	1	1	yes	characteristic
316	C₄	C₂×#88196	1	1	yes	characteristic
317	C₂	—	4	2	no	—
318	C₂	—	208	2	no	—
319	C₂	#13117027	1	4	yes	direct product factor
320	C₂	C₂×#11718628	1	1	yes	characteristic
321	C₂	C₂×#11718855	1	1	yes	characteristic
322	C₂	C₂×#11718854	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	C₂×#6104
commutator, cosocle	C₂×C₈×C₁₃	C₂⁴
Frattini	C₂×C₈	C₂³×D₁₃
Fitting	C₂×C₁₃×#427	C₂
socle	C₂³×C₁₃	C₂×D₈
prime radical	C₂×#11718776	C₂
hypercenter	C₂×#427	D₁₃
hypocenter	C₁₃	C₂×#2234
LCS	G ▷ C₂×C₈×C₁₃ ▷ C₄×C₁₃ ▷ C₂×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂³ ◁ C₂³×C₄ ◁ C₂³×C₈ ◁ C₂×#427

Complements

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

N	H ≅ G/N	type
2. #13117027	C₂	direct (N × H)
3. C₂×#11718082	C₂	semidirect
4. C₂×#11718081	C₂	semidirect
5. C₂×#11718083	C₂	semidirect
6. C₂×#11718083	C₂	semidirect
7. C₂×C₂²⋊#6090	C₂	semidirect
8. C₂×C₁₃×#427	C₂	semidirect
14. C₁₃×#427	C₂²	semidirect
24. C₂²⋊#6090	C₂²	semidirect
25. #11718083	C₂²	semidirect
26. #11718083	C₂²	semidirect
27. #11718082	C₂²	semidirect
28. #11718081	C₂²	semidirect
106. C₂×#427	D₁₃	semidirect
154. #427	C₂×D₁₃	semidirect
220. C₂×C₁₃	#2234	semidirect
265. C₁₃	C₂×#2234	semidirect
319. C₂	#13117027	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×#11718776	2
#13117027	2
C₂×#11718082	2
C₂×#11718081	2
C₂×#11718083	2
C₂×C₂²⋊#6090	2
C₂×C₁₃×#427	2
C₂×#2234	13

Subgroups — C2×#13117027 #16017689

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#13117027 #16017689