Subgroups — C₂×#13117006 #16017648

Order 3328.

558 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×#11718516	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×#11718556	C₂	1	1	yes	maximal, characteristic
3	#13117006	C₂	1	8	yes	maximal, direct product factor
4	C₂×#11718086	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂×#11718103	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×#11718110	C₂	1	1	yes	maximal, characteristic
7	C₂×#11718080	C₂	1	1	yes	maximal, characteristic
8	C₂×C₁₃×#365	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂×#88161	—	2	1	no	—
10	C₂×#88158	C₂²	1	1	yes	characteristic
11	C₂×#88142	—	2	1	no	—
12	C₂×C₁₃×M₃₂	C₂²	1	1	yes	characteristic
13	C₂×C₁₃×#108	C₂²	1	1	yes	characteristic
14	C₂×C₁₃×#106	C₂²	1	1	yes	characteristic
15	C₂×#88065	—	2	1	no	—
16	C₂×#88057	C₂²	1	1	yes	characteristic
17	C₂×C₈×Dic₁₃	C₂²	1	1	yes	characteristic
18	C₂×#88052	C₂²	1	1	yes	characteristic
19	C₂×#88048	—	2	1	no	—
20	#11718556	—	2	2	no	—
21	#11718556	C₂²	1	4	yes	—
22	#11718516	—	2	2	no	—
23	#11718516	C₂²	1	4	yes	complemented
24	C₁₃×#365	C₂²	1	4	yes	complemented
25	#11718110	C₂²	1	4	yes	—
26	#11718103	C₂²	1	4	yes	complemented
27	#11718086	C₂²	1	4	yes	complemented
28	#11718080	C₂²	1	4	yes	—
29	C₂×#6127	—	2	1	no	—
30	C₂²×Dic₂₆	—	4	1	no	—
31	C₂×C₁₃×M₁₆	—	2	1	no	—
32	C₂²×C₈×C₁₃	C₂³	1	1	yes	characteristic
33	C₂×C₁₃×C₄⋊C₄	—	2	1	no	—
34	C₂×C₄⋊Dic₁₃	—	2	1	no	—
35	C₂×#6101	—	4	1	no	—
36	C₂×C₄×Dic₁₃	D₄	1	1	yes	characteristic
37	C₂²×#1055	D₄	1	1	yes	characteristic
38	#88161	—	2	4	no	—
39	#88161	—	4	2	no	—
40	C₂×Dic₅₂	Q₈	1	1	yes	characteristic, complemented
41	#88158	—	2	1	no	—
42	#88158	—	2	1	no	—
43	#88158	Q₈	1	2	yes	complemented
44	#88158	C₂³	1	2	yes	—
45	C₂×D₁₀₄	Q₈	1	1	yes	characteristic, complemented
46	C₂×#6095	—	2	1	no	—
47	C₂×#6094	—	4	1	no	—
48	C₂×C₈×D₁₃	—	4	1	no	—
49	#88142	—	2	4	no	—
50	#88142	—	4	2	no	—
51	C₁₃×M₃₂	—	2	1	no	—
52	C₁₃×M₃₂	C₂³	1	2	yes	—
53	C₂×C₁₃×C₁₆	—	2	1	no	—
54	C₁₃×#108	—	2	1	no	—
55	C₁₃×#108	C₂³	1	2	yes	—
56	C₁₃×#106	—	2	1	no	—
57	C₁₃×#106	C₂³	1	2	yes	—
58	#88065	—	2	4	no	—
59	#88057	—	2	1	no	—
60	#88057	C₂³	1	2	yes	—
61	C₈×Dic₁₃	D₄	1	2	yes	—
62	C₈×Dic₁₃	C₂³	1	2	yes	—
63	#88052	—	2	1	no	—
64	#88052	C₂³	1	2	yes	—
65	C₂×#6090	—	2	1	no	—
66	#88048	—	2	4	no	—
67	C₂×#2230	—	13	1	no	maximal, Sylow
68	C₂²×C₄×C₁₃	—	4	1	no	—
69	C₂²×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
70	C₂×C₂²⋊D₁₃	—	4	1	no	—
71	C₂²×Dic₁₃	—	2	1	no	—
72	C₂²×Dic₁₃	—	4	1	no	—
73	#6127	—	2	2	no	—
74	#6127	—	2	2	no	—
75	#6127	—	4	1	no	—
76	#6127	—	4	1	no	—
77	C₂×D₅₂	—	2	1	no	—
78	C₂×C₄×D₁₃	—	4	1	no	—
79	C₂×Dic₂₆	—	2	1	no	—
80	C₂×Dic₂₆	—	4	4	no	—
81	C₂×Dic₂₆	—	8	4	no	—
82	C₁₃×M₁₆	—	2	2	no	—
83	C₁₃×M₁₆	—	4	1	no	—
84	C₂×C₈×C₁₃	—	4	1	no	—
85	C₂×C₈×C₁₃	C₂×D₄	1	1	yes	characteristic
86	C₂×C₈×C₁₃	C₂×Q₈	1	1	yes	characteristic
87	C₂×C₈×C₁₃	C₂×Q₈	1	1	yes	characteristic
88	C₂×C₈×C₁₃	#41	1	1	yes	characteristic
89	C₂×C₈×C₁₃	#41	1	1	yes	characteristic
90	C₂×C₈×C₁₃	C₂⁴	1	1	yes	characteristic
91	C₁₃×C₄⋊C₄	—	2	2	no	—
92	C₁₃×C₄⋊C₄	—	4	1	no	—
93	C₄⋊Dic₁₃	—	2	2	no	—
94	C₄⋊Dic₁₃	—	4	1	no	—
95	#6101	—	4	4	no	—
96	C₄×Dic₁₃	—	2	1	no	—
97	C₄×Dic₁₃	C₂×D₄	1	2	yes	—
98	C₂×#1055	SD₁₆	1	1	yes	characteristic
99	C₂×#1055	SD₁₆	1	1	yes	characteristic
100	C₂×#1055	SD₁₆	1	2	yes	—
101	C₂×#1055	C₂×D₄	1	2	yes	—
102	Dic₅₂	—	2	1	no	—
103	Dic₅₂	C₂×Q₈	1	2	yes	complemented
104	D₁₀₄	—	2	1	no	—
105	D₁₀₄	C₂×Q₈	1	2	yes	complemented
106	#6095	—	2	2	no	—
107	#6095	—	2	2	no	—
108	#6094	—	4	4	no	—
109	C₈×D₁₃	—	4	4	no	—
110	C₁₃×C₁₆	—	2	2	no	—
111	#6090	—	2	2	no	—
112	C₂×#498	—	13	1	no	—
113	C₂×#443	—	13	1	no	—
114	#2230	—	13	8	no	—
115	C₂×#365	—	13	1	no	—
116	C₂×#365	D₁₃	1	1	yes	characteristic, complemented
117	C₂×#364	—	13	1	no	—
118	C₂×#360	—	13	1	no	—
119	C₂×#360	—	13	1	no	—
120	C₂³×C₁₃	AGL(1,ℤ/8)	1	1	yes	characteristic
121	C₂²×D₁₃	—	4	1	no	—
122	C₂×C₄×C₁₃	—	4	2	no	—
123	C₂×C₄×C₁₃	—	8	2	no	—
124	C₂×C₄×C₁₃	C₂²⋊Q₈	1	1	yes	characteristic
125	C₂×C₄×C₁₃	C₂²⋊Q₈	1	1	yes	characteristic
126	C₂×C₄×C₁₃	C₂×SD₁₆	1	1	yes	characteristic
127	C₂×C₄×C₁₃	C₂×SD₁₆	1	1	yes	characteristic
128	C₂×C₄×C₁₃	AGL(1,ℤ/8)	1	1	yes	characteristic
129	C₂×C₄×C₁₃	C₂²×D₄	1	1	yes	characteristic
130	C₂²⋊D₁₃	—	4	2	no	—
131	C₂²⋊D₁₃	—	4	2	no	—
132	C₂×Dic₁₃	—	2	4	no	—
133	C₂×Dic₁₃	—	2	2	no	—
134	C₂×Dic₁₃	—	4	1	no	—
135	C₂×Dic₁₃	—	4	2	no	—
136	C₂×Dic₁₃	—	8	2	no	—
137	D₅₂	—	2	2	no	—
138	D₅₂	—	4	1	no	—
139	C₄×D₁₃	—	4	2	no	—
140	C₄×D₁₃	—	4	2	no	—
141	Dic₂₆	—	2	2	no	—
142	Dic₂₆	—	4	1	no	—
143	Dic₂₆	—	8	8	no	—
144	C₈×C₁₃	—	4	2	no	—
145	C₈×C₁₃	C₂²⋊Q₈	1	1	yes	characteristic
146	C₈×C₁₃	C₂²⋊Q₈	1	1	yes	characteristic
147	C₈×C₁₃	C₂²×Q₈	1	1	yes	characteristic
148	C₈×C₁₃	C₂×#41	1	1	yes	characteristic
149	#1055	C₂×SD₁₆	1	2	yes	—
150	#1055	C₂×SD₁₆	1	2	yes	—
151	C₂×#135	—	13	1	no	—
152	C₂×#131	—	26	1	no	—
153	C₂×C₄⋊Q₈	—	26	1	no	—
154	C₂×M₃₂	—	13	1	no	—
155	C₂×M₃₂	C₂×D₁₃	1	1	yes	characteristic
156	#498	—	13	4	no	—
157	#498	—	26	2	no	—
158	#443	—	13	4	no	—
159	#443	—	26	2	no	—
160	C₂×#108	C₂×D₁₃	1	1	yes	characteristic
161	C₂×#106	—	13	1	no	—
162	C₂×#106	—	26	1	no	—
163	C₂×#106	C₂×D₁₃	1	1	yes	characteristic
164	C₂×#104	—	26	1	no	—
165	C₂×C₄×C₈	—	13	1	no	—
166	#365	—	13	4	no	—
167	#365	C₂×D₁₃	1	4	yes	complemented
168	#364	—	13	4	no	—
169	#360	—	13	4	no	—
170	#360	—	13	4	no	—
171	C₂²×C₁₃	—	2	2	no	—
172	C₂²×C₁₃	#476	1	1	yes	characteristic
173	C₂²×C₁₃	#476	1	1	yes	characteristic
174	C₂²×C₁₃	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
175	C₂×D₁₃	—	4	2	no	—
176	C₂×D₁₃	—	8	2	no	—
177	C₄×C₁₃	—	8	2	no	—
178	C₄×C₁₃	#476	1	1	yes	characteristic
179	C₄×C₁₃	#476	1	1	yes	characteristic
180	C₄×C₁₃	C₂×C₂²⋊Q₈	1	1	yes	characteristic
181	C₄×C₁₃	C₂²×SD₁₆	1	1	yes	characteristic
182	Dic₁₃	—	2	4	no	—
183	Dic₁₃	—	4	2	no	—
184	Dic₁₃	—	8	2	no	—
185	C₂×#41	—	26	1	no	—
186	C₂²×Q₈	—	52	1	no	—
187	#135	—	13	2	no	—
188	#135	—	13	2	no	—
189	#135	—	26	1	no	—
190	#135	—	26	1	no	—
191	C₂×Q₁₆	—	13	1	no	—
192	C₂×SD₁₆	—	26	1	no	—
193	C₂×D₈	—	13	1	no	—
194	#131	—	26	4	no	—
195	#131	—	52	2	no	—
196	C₂×M₁₆	—	2	1	no	—
197	C₂×M₁₆	—	52	1	no	—
198	C₂²×C₈	—	13	1	no	—
199	C₂²×C₈	—	52	1	no	—
200	C₂²×C₈	C₂²×D₁₃	1	1	yes	characteristic
201	C₄⋊Q₈	—	26	4	no	—
202	C₄⋊Q₈	—	52	2	no	—
203	C₂×C₄⋊C₄	—	2	1	no	—
204	C₂×C₄⋊C₄	—	26	1	no	—
205	C₂×C₄⋊C₄	—	52	1	no	—
206	C₂×C₄²	—	13	1	no	—
207	M₃₂	—	2	1	no	—
208	M₃₂	—	13	2	no	—
209	M₃₂	—	26	1	no	—
210	M₃₂	C₂²×D₁₃	1	2	yes	—
211	C₂×C₁₆	—	2	1	no	—
212	C₂×C₁₆	—	26	1	no	—
213	#108	—	2	1	no	—
214	#108	C₂²×D₁₃	1	2	yes	—
215	#106	—	2	1	no	—
216	#106	—	13	2	no	—
217	#106	—	26	4	no	—
218	#106	—	26	1	no	—
219	#106	C₂²×D₁₃	1	2	yes	—
220	#104	—	26	4	no	—
221	C₄×C₈	—	13	2	no	—
222	C₄×C₈	—	13	2	no	—
223	C₂×C₁₃	—	2	1	no	—
224	C₂×C₁₃	—	2	1	no	—
225	C₂×C₁₃	#2230	1	2	yes	complemented
226	C₂×C₁₃	C₂×#476	1	1	yes	characteristic
227	D₁₃	—	8	2	no	—
228	#41	—	26	2	no	—
229	#41	—	26	2	no	—
230	#41	—	52	1	no	—
231	#41	—	52	1	no	—
232	C₂×Q₈	—	26	1	no	—
233	C₂×Q₈	—	52	4	no	—
234	C₂×Q₈	—	104	4	no	—
235	C₂×D₄	—	26	1	no	—
236	C₂×D₄	—	52	1	no	—
237	C₂²×C₄	—	4	1	no	—
238	C₂²×C₄	—	26	1	no	—
239	C₂²×C₄	—	52	1	no	—
240	C₂²×C₄	—	52	1	no	—
241	C₂²×C₄	D₄×D₁₃	1	1	yes	characteristic
242	Q₁₆	—	13	2	no	—
243	Q₁₆	—	26	1	no	—
244	SD₁₆	—	26	2	no	—
245	SD₁₆	—	26	2	no	—
246	D₈	—	13	2	no	—
247	D₈	—	26	1	no	—
248	M₁₆	—	2	2	no	—
249	M₁₆	—	4	1	no	—
250	M₁₆	—	52	4	no	—
251	C₂×C₈	—	4	1	no	—
252	C₂×C₈	—	13	1	no	—
253	C₂×C₈	—	13	1	no	—
254	C₂×C₈	—	13	2	no	—
255	C₂×C₈	—	13	2	no	—
256	C₂×C₈	—	52	4	no	—
257	C₂×C₈	#6127	1	1	yes	characteristic
258	C₂×C₈	#6127	1	1	yes	characteristic
259	C₂×C₈	D₄×D₁₃	1	1	yes	characteristic
260	C₂×C₈	Q₈×D₁₃	1	1	yes	characteristic
261	C₂×C₈	Q₈×D₁₃	1	1	yes	characteristic
262	C₂×C₈	C₂³×D₁₃	1	1	yes	characteristic
263	C₄⋊C₄	—	2	2	no	—
264	C₄⋊C₄	—	4	1	no	—
265	C₄⋊C₄	—	26	2	no	—
266	C₄⋊C₄	—	52	4	no	—
267	C₄⋊C₄	—	52	1	no	—
268	C₄²	—	13	2	no	—
269	C₄²	—	26	1	no	—
270	C₁₆	—	2	2	no	—
271	C₁₆	—	26	2	no	—
272	C₁₃	C₂×#2230	1	1	yes	characteristic, complemented, Sylow
273	C₂³	—	52	1	no	—
274	C₂³	#88168	1	1	yes	characteristic
275	Q₈	—	26	2	no	—
276	Q₈	—	52	1	no	—
277	Q₈	—	104	8	no	—
278	D₄	—	26	2	no	—
279	D₄	—	52	2	no	—
280	D₄	—	52	2	no	—
281	D₄	—	52	1	no	—
282	C₂×C₄	—	4	2	no	—
283	C₂×C₄	—	8	2	no	—
284	C₂×C₄	—	26	4	no	—
285	C₂×C₄	—	26	2	no	—
286	C₂×C₄	—	52	1	no	—
287	C₂×C₄	—	52	2	no	—
288	C₂×C₄	—	52	2	no	—
289	C₂×C₄	—	52	2	no	—
290	C₂×C₄	—	104	2	no	—
291	C₂×C₄	#88150	1	1	yes	characteristic
292	C₂×C₄	#88150	1	1	yes	characteristic
293	C₂×C₄	SD₁₆×D₁₃	1	1	yes	characteristic
294	C₂×C₄	SD₁₆×D₁₃	1	1	yes	characteristic
295	C₂×C₄	#88168	1	1	yes	characteristic
296	C₂×C₄	C₂×D₄×D₁₃	1	1	yes	characteristic
297	C₈	—	4	2	no	—
298	C₈	—	13	2	no	—
299	C₈	—	13	2	no	—
300	C₈	#88150	1	1	yes	characteristic
301	C₈	#88150	1	1	yes	characteristic
302	C₈	C₂×#6127	1	1	yes	characteristic
303	C₈	C₂×Q₈×D₁₃	1	1	yes	characteristic
304	C₂²	—	2	2	no	—
305	C₂²	—	52	2	no	—
306	C₂²	—	104	2	no	—
307	C₂²	#11718531	1	1	yes	characteristic
308	C₂²	#11718531	1	1	yes	characteristic
309	C₂²	C₂×#88168	1	1	yes	characteristic
310	C₄	—	8	2	no	—
311	C₄	—	26	4	no	—
312	C₄	—	52	2	no	—
313	C₄	—	104	2	no	—
314	C₄	#11718531	1	1	yes	characteristic
315	C₄	#11718531	1	1	yes	characteristic
316	C₄	C₂×#88150	1	1	yes	characteristic
317	C₄	C₂×SD₁₆×D₁₃	1	1	yes	characteristic
318	C₂	—	2	1	no	—
319	C₂	—	2	1	no	—
320	C₂	—	104	2	no	—
321	C₂	#13117006	1	2	yes	direct product factor
322	C₂	C₂×#11718531	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	#11718531
commutator, cosocle	C₂×C₈×C₁₃	C₂⁴
Frattini	C₂×C₈	C₂³×D₁₃
Fitting	C₂×C₁₃×#365	C₂
socle	C₂²×C₁₃	#476
prime radical	C₂×#88158	C₂²
hypercenter	C₂×#365	D₁₃
hypocenter	C₁₃	C₂×#2230
LCS	G ▷ C₂×C₈×C₁₃ ▷ C₄×C₁₃ ▷ C₂×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂²×C₈ ◁ C₂×#365

Complements

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

N	H ≅ G/N	type
1. C₂×#11718516	C₂	semidirect
3. #13117006	C₂	direct (N × H)
4. C₂×#11718086	C₂	semidirect
5. C₂×#11718103	C₂	semidirect
8. C₂×C₁₃×#365	C₂	semidirect
23. #11718516	C₂²	semidirect
24. C₁₃×#365	C₂²	semidirect
26. #11718103	C₂²	semidirect
27. #11718086	C₂²	semidirect
40. C₂×Dic₅₂	Q₈	semidirect
43. #88158	Q₈	semidirect
45. C₂×D₁₀₄	Q₈	semidirect
103. Dic₅₂	C₂×Q₈	semidirect
105. D₁₀₄	C₂×Q₈	semidirect
116. C₂×#365	D₁₃	semidirect
167. #365	C₂×D₁₃	semidirect
225. C₂×C₁₃	#2230	semidirect
272. C₁₃	C₂×#2230	semidirect
321. C₂	#13117006	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×#11718516	2
C₂×#11718556	2
#13117006	2
C₂×#11718086	2
C₂×#11718103	2
C₂×#11718110	2
C₂×#11718080	2
C₂×C₁₃×#365	2
C₂×#2230	13

Subgroups — C2×#13117006 #16017648

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#13117006 #16017648