Subgroups — C₂×#13116962 #16017483

Order 3328.

638 conjugacy classes of subgroups.

The subgroup lattice has 300 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×Dic₁₀₄	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×#11718562	C₂	1	1	yes	maximal, characteristic
3	C₂×#11718557	C₂	1	2	yes	maximal
4	#13116962	C₂	1	8	yes	maximal, direct product factor
5	C₂×#11718088	C₂	1	2	yes	maximal, complemented
6	C₂×C₁₃×#368	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂×#88163	—	2	2	no	—
8	C₂²×Dic₅₂	C₂²	1	2	yes	—
9	C₂×#88155	C₂²	1	1	yes	characteristic
10	C₂²×C₁₃×C₁₆	C₂²	1	1	yes	characteristic
11	C₂×C₁₃×#108	C₂²	1	2	yes	—
12	C₂×#88064	—	2	2	no	—
13	C₂×#88052	C₂²	1	1	yes	characteristic
14	C₂×Dic₁₀₄	—	2	1	no	—
15	C₂×Dic₁₀₄	—	2	1	no	—
16	C₂×Dic₁₀₄	—	2	2	no	—
17	C₂×Dic₁₀₄	C₂²	1	4	yes	complemented
18	#11718562	—	2	2	no	—
19	#11718562	C₂²	1	4	yes	—
20	C₂×#88038	—	2	1	no	—
21	C₂×C₁₆×D₁₃	—	2	1	no	—
22	#11718557	—	2	4	no	—
23	#11718557	C₂²	1	8	yes	—
24	C₁₃×#368	C₂²	1	4	yes	complemented
25	#11718088	C₂²	1	8	yes	complemented
26	C₂×#6127	D₄	1	1	yes	characteristic
27	C₂²×Dic₂₆	—	2	2	no	—
28	C₂×C₁₃×M₁₆	—	2	2	no	—
29	C₂²×C₈×C₁₃	C₂³	1	1	yes	characteristic
30	C₂×#6099	D₄	1	1	yes	characteristic
31	#88163	—	2	8	no	—
32	#88163	—	4	4	no	—
33	C₂×Dic₅₂	—	2	2	no	—
34	C₂×Dic₅₂	—	2	2	no	—
35	C₂×Dic₅₂	—	2	2	no	—
36	C₂×Dic₅₂	—	2	2	no	—
37	C₂×Dic₅₂	—	2	2	no	—
38	C₂×Dic₅₂	—	4	2	no	—
39	C₂×Dic₅₂	C₂³	1	4	yes	—
40	C₂×#6095	—	4	2	no	—
41	#88155	—	2	1	no	—
42	#88155	—	2	1	no	—
43	#88155	D₄	1	2	yes	—
44	#88155	C₂³	1	2	yes	—
45	C₂×#6094	—	2	1	no	—
46	C₂×C₈×D₁₃	—	2	1	no	—
47	C₂×C₁₃×C₁₆	D₄	1	1	yes	characteristic
48	C₂×C₁₃×C₁₆	D₄	1	1	yes	characteristic
49	C₂×C₁₃×C₁₆	D₄	1	2	yes	—
50	C₂×C₁₃×C₁₆	C₂³	1	2	yes	—
51	C₁₃×#108	—	2	2	no	—
52	C₁₃×#108	C₂³	1	4	yes	—
53	#88064	—	2	8	no	—
54	#88052	—	2	1	no	—
55	#88052	C₂³	1	2	yes	—
56	C₂×#6090	—	2	1	no	—
57	Dic₁₀₄	—	2	4	no	—
58	Dic₁₀₄	—	2	4	no	—
59	#88038	—	2	4	no	—
60	C₁₆×D₁₃	—	2	4	no	—
61	C₂×#2222	—	13	1	no	maximal, Sylow
62	C₂²×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
63	C₂×C₂²⋊D₁₃	—	2	1	no	—
64	C₂²×Dic₁₃	—	4	2	no	—
65	#6127	—	2	1	no	—
66	#6127	—	2	1	no	—
67	#6127	D₈	1	2	yes	—
68	#6127	C₂×D₄	1	2	yes	—
69	C₂×D₅₂	D₈	1	1	yes	characteristic
70	C₂×C₄×D₁₃	—	2	1	no	—
71	C₂×Dic₂₆	—	2	4	no	—
72	C₂×Dic₂₆	—	4	2	no	—
73	C₂×Dic₂₆	—	4	2	no	—
74	C₂×Dic₂₆	—	4	2	no	—
75	C₂×Dic₂₆	—	4	2	no	—
76	C₂×Dic₂₆	—	4	2	no	—
77	C₂×Dic₂₆	D₈	1	1	yes	characteristic
78	C₁₃×M₁₆	—	2	4	no	—
79	C₁₃×M₁₆	—	4	2	no	—
80	C₂×C₈×C₁₃	—	4	2	no	—
81	C₂×C₈×C₁₃	C₂×D₄	1	1	yes	characteristic
82	C₂×C₈×C₁₃	C₂×D₄	1	1	yes	characteristic
83	C₂×C₈×C₁₃	C₂×D₄	1	1	yes	characteristic
84	C₂×C₈×C₁₃	#41	1	1	yes	characteristic
85	C₂×C₈×C₁₃	#41	1	1	yes	characteristic
86	C₂×C₈×C₁₃	C₂⁴	1	1	yes	characteristic
87	#6099	—	2	1	no	—
88	#6099	C₂×D₄	1	2	yes	—
89	C₂×#1055	—	2	1	no	—
90	Dic₅₂	—	2	4	no	—
91	Dic₅₂	—	2	4	no	—
92	Dic₅₂	—	4	8	no	—
93	Dic₅₂	—	4	2	no	—
94	Dic₅₂	—	4	2	no	—
95	#6095	—	4	8	no	—
96	#6094	—	2	2	no	—
97	#6094	—	2	2	no	—
98	C₈×D₁₃	—	2	2	no	—
99	C₈×D₁₃	—	2	2	no	—
100	C₁₃×C₁₆	C₂×D₄	1	2	yes	—
101	C₁₃×C₁₆	C₂×D₄	1	2	yes	—
102	#6090	—	2	2	no	—
103	C₂²×Q₃₂	—	13	1	no	—
104	C₂×#504	—	13	1	no	—
105	C₂×#473	—	13	2	no	—
106	#2222	—	13	8	no	—
107	C₂×#368	D₁₃	1	1	yes	characteristic, complemented
108	C₂×#362	—	13	2	no	—
109	C₂³×C₁₃	#135	1	1	yes	characteristic
110	C₂²×D₁₃	—	2	1	no	—
111	C₂×C₄×C₁₃	#121	1	1	yes	characteristic
112	C₂×C₄×C₁₃	#121	1	1	yes	characteristic
113	C₂×C₄×C₁₃	C₂×D₈	1	1	yes	characteristic
114	C₂×C₄×C₁₃	C₂×D₈	1	1	yes	characteristic
115	C₂×C₄×C₁₃	#135	1	1	yes	characteristic
116	C₂×C₄×C₁₃	C₂²×D₄	1	1	yes	characteristic
117	C₂²⋊D₁₃	—	2	2	no	—
118	C₂²⋊D₁₃	—	2	2	no	—
119	C₂×Dic₁₃	—	2	1	no	—
120	C₂×Dic₁₃	—	4	4	no	—
121	C₂×Dic₁₃	—	8	2	no	—
122	C₂×Dic₁₃	—	8	2	no	—
123	D₅₂	—	2	1	no	—
124	D₅₂	C₂×D₈	1	2	yes	—
125	C₄×D₁₃	—	2	2	no	—
126	C₄×D₁₃	—	2	2	no	—
127	Dic₂₆	—	2	1	no	—
128	Dic₂₆	—	4	4	no	—
129	Dic₂₆	—	4	4	no	—
130	Dic₂₆	—	8	2	no	—
131	Dic₂₆	—	8	2	no	—
132	Dic₂₆	C₂×D₈	1	2	yes	—
133	C₈×C₁₃	—	4	4	no	—
134	C₈×C₁₃	#121	1	1	yes	characteristic
135	C₈×C₁₃	#121	1	1	yes	characteristic
136	C₈×C₁₃	C₂²×D₄	1	1	yes	characteristic
137	C₈×C₁₃	C₂×#41	1	1	yes	characteristic
138	#1055	—	2	2	no	—
139	C₂×#137	—	26	2	no	—
140	C₂²×Q₁₆	—	13	2	no	—
141	C₂×#131	—	13	1	no	—
142	C₂×Q₃₂	—	13	4	no	—
143	C₂×Q₃₂	—	26	1	no	—
144	C₂×Q₃₂	—	26	1	no	—
145	C₂×Q₃₂	—	26	2	no	—
146	#504	—	13	4	no	—
147	#504	—	26	2	no	—
148	C₂×M₃₂	—	13	1	no	—
149	C₂×M₃₂	—	26	1	no	—
150	C₂²×C₁₆	—	26	1	no	—
151	C₂²×C₁₆	C₂×D₁₃	1	1	yes	characteristic
152	#473	—	13	8	no	—
153	#473	—	26	4	no	—
154	C₂×#108	C₂×D₁₃	1	2	yes	—
155	C₂×#101	—	26	2	no	—
156	#368	C₂×D₁₃	1	4	yes	complemented
157	#362	—	13	8	no	—
158	C₂²×C₁₃	—	2	2	no	—
159	C₂²×C₁₃	#466	1	1	yes	characteristic
160	C₂²×C₁₃	#466	1	1	yes	characteristic
161	C₂²×C₁₃	C₂×#135	1	1	yes	characteristic
162	C₂×D₁₃	—	2	2	no	—
163	C₂×D₁₃	—	4	2	no	—
164	C₄×C₁₃	#466	1	1	yes	characteristic
165	C₄×C₁₃	#466	1	1	yes	characteristic
166	C₄×C₁₃	C₂×#121	1	1	yes	characteristic
167	C₄×C₁₃	C₂²×D₈	1	1	yes	characteristic
168	Dic₁₃	—	2	2	no	—
169	Dic₁₃	—	8	4	no	—
170	C₂×#41	—	13	1	no	—
171	C₂²×Q₈	—	26	2	no	—
172	#137	—	26	8	no	—
173	#137	—	52	4	no	—
174	C₂×Q₁₆	—	13	4	no	—
175	C₂×Q₁₆	—	26	2	no	—
176	C₂×Q₁₆	—	26	2	no	—
177	C₂×Q₁₆	—	26	2	no	—
178	C₂×Q₁₆	—	26	2	no	—
179	C₂×Q₁₆	—	26	2	no	—
180	C₂×Q₁₆	—	52	2	no	—
181	C₂×SD₁₆	—	52	2	no	—
182	#131	—	13	2	no	—
183	#131	—	13	2	no	—
184	#131	—	26	1	no	—
185	#131	—	26	1	no	—
186	C₂×M₁₆	—	2	2	no	—
187	C₂×M₁₆	—	13	1	no	—
188	C₂×M₁₆	—	26	1	no	—
189	C₂²×C₈	—	26	1	no	—
190	C₂²×C₈	C₂²×D₁₃	1	1	yes	characteristic
191	Q₃₂	—	26	4	no	—
192	Q₃₂	—	26	4	no	—
193	M₃₂	—	13	2	no	—
194	M₃₂	—	26	1	no	—
195	M₃₂	—	26	4	no	—
196	C₂×C₁₆	—	26	1	no	—
197	C₂×C₁₆	—	26	4	no	—
198	C₂×C₁₆	D₅₂	1	1	yes	characteristic
199	C₂×C₁₆	D₅₂	1	1	yes	characteristic
200	C₂×C₁₆	D₅₂	1	2	yes	—
201	C₂×C₁₆	C₂²×D₁₃	1	2	yes	—
202	#108	—	2	2	no	—
203	#108	C₂²×D₁₃	1	4	yes	—
204	#101	—	26	8	no	—
205	C₂×C₁₃	—	2	1	no	—
206	C₂×C₁₃	—	2	1	no	—
207	C₂×C₁₃	#2222	1	2	yes	complemented
208	C₂×C₁₃	C₂×#466	1	1	yes	characteristic
209	D₁₃	—	4	2	no	—
210	#41	—	13	2	no	—
211	#41	—	13	2	no	—
212	#41	—	26	1	no	—
213	#41	—	26	1	no	—
214	C₂×Q₈	—	13	1	no	—
215	C₂×Q₈	—	26	4	no	—
216	C₂×Q₈	—	52	2	no	—
217	C₂×Q₈	—	52	2	no	—
218	C₂×Q₈	—	52	2	no	—
219	C₂×Q₈	—	52	2	no	—
220	C₂×Q₈	—	52	2	no	—
221	C₂×D₄	—	13	1	no	—
222	C₂×D₄	—	26	1	no	—
223	C₂²×C₄	—	26	1	no	—
224	C₂²×C₄	—	52	2	no	—
225	C₂²×C₄	D₄×D₁₃	1	1	yes	characteristic
226	Q₁₆	—	26	4	no	—
227	Q₁₆	—	26	4	no	—
228	Q₁₆	—	52	8	no	—
229	Q₁₆	—	52	2	no	—
230	Q₁₆	—	52	2	no	—
231	SD₁₆	—	52	8	no	—
232	M₁₆	—	2	4	no	—
233	M₁₆	—	4	2	no	—
234	M₁₆	—	13	2	no	—
235	M₁₆	—	26	1	no	—
236	M₁₆	—	26	2	no	—
237	M₁₆	—	26	2	no	—
238	C₂×C₈	—	4	2	no	—
239	C₂×C₈	—	26	1	no	—
240	C₂×C₈	—	26	2	no	—
241	C₂×C₈	—	26	2	no	—
242	C₂×C₈	C₂×D₅₂	1	1	yes	characteristic
243	C₂×C₈	C₂×D₅₂	1	1	yes	characteristic
244	C₂×C₈	D₄×D₁₃	1	1	yes	characteristic
245	C₂×C₈	#6131	1	1	yes	characteristic
246	C₂×C₈	#6131	1	1	yes	characteristic
247	C₂×C₈	C₂³×D₁₃	1	1	yes	characteristic
248	C₁₆	—	26	2	no	—
249	C₁₆	C₂×D₅₂	1	2	yes	—
250	C₁₆	C₂×D₅₂	1	2	yes	—
251	C₁₃	C₂×#2222	1	1	yes	characteristic, complemented, Sylow
252	C₂³	—	26	1	no	—
253	C₂³	#88173	1	1	yes	characteristic
254	Q₈	—	13	2	no	—
255	Q₈	—	26	1	no	—
256	Q₈	—	52	4	no	—
257	Q₈	—	52	4	no	—
258	Q₈	—	104	2	no	—
259	Q₈	—	104	2	no	—
260	D₄	—	13	2	no	—
261	D₄	—	26	2	no	—
262	D₄	—	26	2	no	—
263	D₄	—	26	1	no	—
264	C₂×C₄	—	26	1	no	—
265	C₂×C₄	—	26	2	no	—
266	C₂×C₄	—	26	2	no	—
267	C₂×C₄	—	52	4	no	—
268	C₂×C₄	—	104	2	no	—
269	C₂×C₄	—	104	2	no	—
270	C₂×C₄	#88149	1	1	yes	characteristic
271	C₂×C₄	#88149	1	1	yes	characteristic
272	C₂×C₄	D₈×D₁₃	1	1	yes	characteristic
273	C₂×C₄	D₈×D₁₃	1	1	yes	characteristic
274	C₂×C₄	#88173	1	1	yes	characteristic
275	C₂×C₄	C₂×D₄×D₁₃	1	1	yes	characteristic
276	C₈	—	4	4	no	—
277	C₈	—	26	2	no	—
278	C₈	#88149	1	1	yes	characteristic
279	C₈	#88149	1	1	yes	characteristic
280	C₈	C₂²×D₅₂	1	1	yes	characteristic
281	C₈	C₂×#6131	1	1	yes	characteristic
282	C₂²	—	2	2	no	—
283	C₂²	—	26	2	no	—
284	C₂²	—	52	2	no	—
285	C₂²	#11718544	1	1	yes	characteristic
286	C₂²	#11718544	1	1	yes	characteristic
287	C₂²	C₂×#88173	1	1	yes	characteristic
288	C₄	—	26	2	no	—
289	C₄	—	104	4	no	—
290	C₄	#11718544	1	1	yes	characteristic
291	C₄	#11718544	1	1	yes	characteristic
292	C₄	C₂×#88149	1	1	yes	characteristic
293	C₄	C₂×D₈×D₁₃	1	1	yes	characteristic
294	C₂	—	2	1	no	—
295	C₂	—	2	1	no	—
296	C₂	—	52	2	no	—
297	C₂	#13116962	1	2	yes	direct product factor
298	C₂	C₂×#11718544	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	#11718544
commutator, cosocle	C₂×C₈×C₁₃	C₂⁴
Frattini	C₂×C₈	C₂³×D₁₃
Fitting	C₂×C₁₃×#368	C₂
socle	C₂²×C₁₃	#466
prime radical	C₂×#6127	D₄
hypercenter	C₂×#368	D₁₃
hypocenter	C₁₃	C₂×#2222
LCS	G ▷ C₂×C₈×C₁₃ ▷ C₄×C₁₃ ▷ C₂×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂²×C₈ ◁ C₂×#368

Complements

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

N	H ≅ G/N	type
1. C₂²×Dic₁₀₄	C₂	semidirect
4. #13116962	C₂	direct (N × H)
5. C₂×#11718088	C₂	semidirect
6. C₂×C₁₃×#368	C₂	semidirect
17. C₂×Dic₁₀₄	C₂²	semidirect
24. C₁₃×#368	C₂²	semidirect
25. #11718088	C₂²	semidirect
107. C₂×#368	D₁₃	semidirect
156. #368	C₂×D₁₃	semidirect
207. C₂×C₁₃	#2222	semidirect
251. C₁₃	C₂×#2222	semidirect
297. C₂	#13116962	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂²×Dic₁₀₄	2
C₂×#11718562	2
C₂×#11718557	2
#13116962	2
C₂×#11718088	2
C₂×C₁₃×#368	2
C₂×#2222	13

Subgroups — C2×#13116962 #16017483

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#13116962 #16017483