Subgroups — C₂×#13116961 #16017482

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×#11718804	C₂	1	2	yes	maximal
3	#13116961	C₂	1	8	yes	maximal, direct product factor
4	C₂×#11718098	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂×#11718109	C₂	1	2	yes	maximal, complemented
6	C₂×C₁₃×#504	C₂	1	1	yes	maximal, characteristic
7	C₂×C₁₃×#131	C₂²	1	1	yes	characteristic
8	C₂×#88205	—	2	2	no	—
9	C₂²×Dic₅₂	C₂²	1	2	yes	—
10	C₁₃×#504	—	2	2	no	—
11	C₁₃×#504	C₂²	1	4	yes	—
12	C₂×C₁₃×M₃₂	—	2	1	no	—
13	C₂×C₁₃×M₃₂	C₂²	1	1	yes	characteristic
14	C₂²×C₁₃×C₁₆	—	2	1	no	—
15	C₂²×C₁₃×C₁₆	C₂²	1	1	yes	characteristic
16	#11718804	—	2	4	no	—
17	#11718804	C₂²	1	8	yes	—
18	C₂×#88064	—	2	2	no	—
19	C₂×#88059	C₂²	1	2	yes	—
20	C₂×Dic₁₀₄	—	2	1	no	—
21	C₂×Dic₁₀₄	—	2	1	no	—
22	C₂×Dic₁₀₄	—	2	2	no	—
23	C₂×Dic₁₀₄	C₂²	1	4	yes	complemented
24	#11718109	C₂²	1	8	yes	complemented
25	#11718098	C₂²	1	4	yes	complemented
26	C₂×C₁₃×#41	D₄	1	1	yes	characteristic
27	C₂²×Dic₂₆	—	2	2	no	—
28	C₁₃×#131	—	2	1	no	—
29	C₁₃×#131	—	2	1	no	—
30	C₁₃×#131	D₄	1	2	yes	—
31	C₁₃×#131	C₂³	1	2	yes	—
32	C₂×C₁₃×M₁₆	—	2	1	no	—
33	C₂×C₁₃×M₁₆	D₄	1	1	yes	characteristic
34	C₂²×C₈×C₁₃	—	2	1	no	—
35	C₂²×C₈×C₁₃	C₂³	1	1	yes	characteristic
36	#88205	—	2	8	no	—
37	#88205	—	4	4	no	—
38	C₂×#6107	—	4	2	no	—
39	C₂×#6105	—	4	2	no	—
40	C₂×#6099	—	2	2	no	—
41	C₂×Dic₅₂	—	2	2	no	—
42	C₂×Dic₅₂	—	2	2	no	—
43	C₂×Dic₅₂	—	2	2	no	—
44	C₂×Dic₅₂	—	2	2	no	—
45	C₂×Dic₅₂	—	2	2	no	—
46	C₂×Dic₅₂	C₂³	1	4	yes	—
47	C₁₃×M₃₂	—	2	1	no	—
48	C₁₃×M₃₂	—	2	4	no	—
49	C₁₃×M₃₂	C₂³	1	2	yes	—
50	C₂×C₁₃×C₁₆	—	2	1	no	—
51	C₂×C₁₃×C₁₆	—	2	4	no	—
52	C₂×C₁₃×C₁₆	D₄	1	1	yes	characteristic
53	C₂×C₁₃×C₁₆	D₄	1	1	yes	characteristic
54	C₂×C₁₃×C₁₆	D₄	1	2	yes	—
55	C₂×C₁₃×C₁₆	C₂³	1	2	yes	—
56	#88064	—	2	8	no	—
57	#88059	—	2	2	no	—
58	#88059	C₂³	1	4	yes	—
59	Dic₁₀₄	—	2	4	no	—
60	Dic₁₀₄	—	2	4	no	—
61	C₂×#2222	—	13	1	no	maximal, Sylow
62	C₁₃×#41	—	2	1	no	—
63	C₁₃×#41	—	2	1	no	—
64	C₁₃×#41	D₈	1	2	yes	—
65	C₁₃×#41	C₂×D₄	1	2	yes	—
66	C₂×Q₈×C₁₃	D₈	1	1	yes	characteristic
67	C₂×D₄×C₁₃	—	2	1	no	—
68	C₂×D₄×C₁₃	D₈	1	1	yes	characteristic
69	C₂²×C₄×C₁₃	—	2	1	no	—
70	C₂²×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
71	C₂²×Dic₁₃	—	4	2	no	—
72	C₂×Dic₂₆	—	2	4	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	—	4	2	no	—
78	C₁₃×M₁₆	—	2	1	no	—
79	C₁₃×M₁₆	—	2	2	no	—
80	C₁₃×M₁₆	—	2	2	no	—
81	C₁₃×M₁₆	C₂×D₄	1	2	yes	—
82	C₂×C₈×C₁₃	—	2	1	no	—
83	C₂×C₈×C₁₃	—	2	2	no	—
84	C₂×C₈×C₁₃	—	2	2	no	—
85	C₂×C₈×C₁₃	C₂×D₄	1	1	yes	characteristic
86	C₂×C₈×C₁₃	C₂×D₄	1	1	yes	characteristic
87	C₂×C₈×C₁₃	C₂×D₄	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	#6107	—	4	8	no	—
92	#6105	—	4	8	no	—
93	#6099	—	2	4	no	—
94	#6099	—	4	2	no	—
95	C₂×#1055	—	4	2	no	—
96	Dic₅₂	—	2	4	no	—
97	Dic₅₂	—	2	4	no	—
98	Dic₅₂	—	4	2	no	—
99	Dic₅₂	—	4	2	no	—
100	C₁₃×C₁₆	—	2	2	no	—
101	C₁₃×C₁₆	C₂×D₄	1	2	yes	—
102	C₁₃×C₁₆	C₂×D₄	1	2	yes	—
103	C₂²×Q₃₂	—	13	1	no	—
104	C₂×#504	D₁₃	1	1	yes	characteristic
105	C₂×#473	—	13	2	no	—
106	#2222	—	13	8	no	—
107	C₂×#368	—	13	1	no	—
108	C₂×#362	—	13	2	no	—
109	C₂³×C₁₃	—	2	1	no	—
110	C₂³×C₁₃	#135	1	1	yes	characteristic
111	Q₈×C₁₃	—	2	1	no	—
112	Q₈×C₁₃	C₂×D₈	1	2	yes	—
113	D₄×C₁₃	—	2	2	no	—
114	D₄×C₁₃	—	2	2	no	—
115	D₄×C₁₃	—	2	1	no	—
116	D₄×C₁₃	C₂×D₈	1	2	yes	—
117	C₂×C₄×C₁₃	—	2	1	no	—
118	C₂×C₄×C₁₃	—	2	2	no	—
119	C₂×C₄×C₁₃	—	2	2	no	—
120	C₂×C₄×C₁₃	#121	1	1	yes	characteristic
121	C₂×C₄×C₁₃	#121	1	1	yes	characteristic
122	C₂×C₄×C₁₃	C₂×D₈	1	1	yes	characteristic
123	C₂×C₄×C₁₃	C₂×D₈	1	1	yes	characteristic
124	C₂×C₄×C₁₃	#135	1	1	yes	characteristic
125	C₂×C₄×C₁₃	C₂²×D₄	1	1	yes	characteristic
126	C₂×Dic₁₃	—	4	4	no	—
127	C₂×Dic₁₃	—	8	2	no	—
128	C₂×Dic₁₃	—	8	2	no	—
129	Dic₂₆	—	4	4	no	—
130	Dic₂₆	—	4	4	no	—
131	Dic₂₆	—	8	2	no	—
132	Dic₂₆	—	8	2	no	—
133	C₈×C₁₃	—	2	2	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	—	4	4	no	—
139	C₂×#137	—	26	2	no	—
140	C₂²×Q₁₆	—	13	2	no	—
141	C₂×#131	C₂×D₁₃	1	1	yes	characteristic
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	—	2	2	no	—
147	#504	C₂×D₁₃	1	4	yes	—
148	C₂×M₃₂	—	2	1	no	—
149	C₂×M₃₂	C₂×D₁₃	1	1	yes	characteristic
150	C₂²×C₁₆	—	2	1	no	—
151	C₂²×C₁₆	C₂×D₁₃	1	1	yes	characteristic
152	#473	—	13	8	no	—
153	#473	—	26	4	no	—
154	C₂×#108	—	13	2	no	—
155	C₂×#101	—	26	2	no	—
156	#368	—	13	4	no	—
157	#362	—	13	8	no	—
158	C₂²×C₁₃	—	2	2	no	—
159	C₂²×C₁₃	—	2	2	no	—
160	C₂²×C₁₃	—	4	2	no	—
161	C₂²×C₁₃	#466	1	1	yes	characteristic
162	C₂²×C₁₃	#466	1	1	yes	characteristic
163	C₂²×C₁₃	C₂×#135	1	1	yes	characteristic
164	C₄×C₁₃	—	2	2	no	—
165	C₄×C₁₃	#466	1	1	yes	characteristic
166	C₄×C₁₃	#466	1	1	yes	characteristic
167	C₄×C₁₃	C₂×#121	1	1	yes	characteristic
168	C₄×C₁₃	C₂²×D₈	1	1	yes	characteristic
169	Dic₁₃	—	8	4	no	—
170	C₂×#41	D₅₂	1	1	yes	characteristic
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	—	2	1	no	—
183	#131	—	2	1	no	—
184	#131	D₅₂	1	2	yes	—
185	#131	C₂²×D₁₃	1	2	yes	—
186	C₂×M₁₆	—	2	1	no	—
187	C₂×M₁₆	—	26	2	no	—
188	C₂×M₁₆	D₅₂	1	1	yes	characteristic
189	C₂²×C₈	—	2	1	no	—
190	C₂²×C₈	C₂²×D₁₃	1	1	yes	characteristic
191	Q₃₂	—	26	4	no	—
192	Q₃₂	—	26	4	no	—
193	M₃₂	—	2	1	no	—
194	M₃₂	—	2	4	no	—
195	M₃₂	C₂²×D₁₃	1	2	yes	—
196	C₂×C₁₆	—	2	1	no	—
197	C₂×C₁₆	—	2	4	no	—
198	C₂×C₁₆	C₂²⋊D₁₃	1	1	yes	characteristic
199	C₂×C₁₆	C₂²⋊D₁₃	1	1	yes	characteristic
200	C₂×C₁₆	C₂²⋊D₁₃	1	2	yes	—
201	C₂×C₁₆	C₂²×D₁₃	1	2	yes	—
202	#108	—	13	4	no	—
203	#108	—	26	2	no	—
204	#101	—	26	8	no	—
205	C₂×C₁₃	—	2	1	no	—
206	C₂×C₁₃	—	2	1	no	—
207	C₂×C₁₃	—	4	2	no	—
208	C₂×C₁₃	#2222	1	2	yes	complemented
209	C₂×C₁₃	C₂×#466	1	1	yes	characteristic
210	#41	—	2	1	no	—
211	#41	—	2	1	no	—
212	#41	D₁₀₄	1	2	yes	—
213	#41	C₂×D₅₂	1	2	yes	—
214	C₂×Q₈	—	26	4	no	—
215	C₂×Q₈	—	52	2	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	D₁₀₄	1	1	yes	characteristic
221	C₂×D₄	—	2	1	no	—
222	C₂×D₄	D₁₀₄	1	1	yes	characteristic
223	C₂²×C₄	—	2	1	no	—
224	C₂²×C₄	—	52	2	no	—
225	C₂²×C₄	C₂×D₅₂	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	1	no	—
233	M₁₆	—	2	2	no	—
234	M₁₆	—	2	2	no	—
235	M₁₆	—	26	4	no	—
236	M₁₆	—	52	2	no	—
237	M₁₆	C₂×D₅₂	1	2	yes	—
238	C₂×C₈	—	2	1	no	—
239	C₂×C₈	—	2	2	no	—
240	C₂×C₈	—	2	2	no	—
241	C₂×C₈	—	52	2	no	—
242	C₂×C₈	C₂×D₅₂	1	1	yes	characteristic
243	C₂×C₈	#6127	1	1	yes	characteristic
244	C₂×C₈	#6127	1	1	yes	characteristic
245	C₂×C₈	C₂×C₂²⋊D₁₃	1	1	yes	characteristic
246	C₂×C₈	C₂×C₂²⋊D₁₃	1	1	yes	characteristic
247	C₂×C₈	C₂³×D₁₃	1	1	yes	characteristic
248	C₁₆	—	2	2	no	—
249	C₁₆	C₂×C₂²⋊D₁₃	1	2	yes	—
250	C₁₆	C₂×C₂²⋊D₁₃	1	2	yes	—
251	C₁₃	C₂×#2222	1	1	yes	characteristic, complemented, Sylow
252	C₂³	—	2	1	no	—
253	C₂³	#88158	1	1	yes	characteristic
254	Q₈	—	2	1	no	—
255	Q₈	—	52	4	no	—
256	Q₈	—	52	4	no	—
257	Q₈	—	104	2	no	—
258	Q₈	—	104	2	no	—
259	Q₈	C₂×D₁₀₄	1	2	yes	—
260	D₄	—	2	2	no	—
261	D₄	—	2	2	no	—
262	D₄	—	2	1	no	—
263	D₄	C₂×D₁₀₄	1	2	yes	—
264	C₂×C₄	—	2	1	no	—
265	C₂×C₄	—	2	2	no	—
266	C₂×C₄	—	2	2	no	—
267	C₂×C₄	—	52	4	no	—
268	C₂×C₄	—	104	2	no	—
269	C₂×C₄	—	104	2	no	—
270	C₂×C₄	C₂×D₁₀₄	1	1	yes	characteristic
271	C₂×C₄	C₂×D₁₀₄	1	1	yes	characteristic
272	C₂×C₄	#88158	1	1	yes	characteristic
273	C₂×C₄	#88184	1	1	yes	characteristic
274	C₂×C₄	#88184	1	1	yes	characteristic
275	C₂×C₄	C₂²×D₅₂	1	1	yes	characteristic
276	C₈	—	2	2	no	—
277	C₈	—	52	4	no	—
278	C₈	#88184	1	1	yes	characteristic
279	C₈	#88184	1	1	yes	characteristic
280	C₈	C₂×#6127	1	1	yes	characteristic
281	C₈	C₂²×C₂²⋊D₁₃	1	1	yes	characteristic
282	C₂²	—	2	2	no	—
283	C₂²	—	2	2	no	—
284	C₂²	—	4	2	no	—
285	C₂²	#11718776	1	1	yes	characteristic
286	C₂²	#11718776	1	1	yes	characteristic
287	C₂²	C₂×#88158	1	1	yes	characteristic
288	C₄	—	2	2	no	—
289	C₄	—	104	4	no	—
290	C₄	#11718776	1	1	yes	characteristic
291	C₄	#11718776	1	1	yes	characteristic
292	C₄	C₂²×D₁₀₄	1	1	yes	characteristic
293	C₄	C₂×#88184	1	1	yes	characteristic
294	C₂	—	2	1	no	—
295	C₂	—	2	1	no	—
296	C₂	—	4	2	no	—
297	C₂	#13116961	1	2	yes	direct product factor
298	C₂	C₂×#11718776	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	#11718776
commutator, cosocle	C₂×C₈×C₁₃	C₂⁴
Frattini	C₂×C₈	C₂³×D₁₃
Fitting	C₂×C₁₃×#504	C₂
socle	C₂²×C₁₃	#466
prime radical	C₂×C₁₃×#41	D₄
hypercenter	C₂×#504	D₁₃
hypocenter	C₁₃	C₂×#2222
LCS	G ▷ C₂×C₈×C₁₃ ▷ C₄×C₁₃ ▷ C₂×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×#131 ◁ C₂×#504

Complements

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

N	H ≅ G/N	type
1. C₂²×Dic₁₀₄	C₂	semidirect
3. #13116961	C₂	direct (N × H)
4. C₂×#11718098	C₂	semidirect
5. C₂×#11718109	C₂	semidirect
23. C₂×Dic₁₀₄	C₂²	semidirect
24. #11718109	C₂²	semidirect
25. #11718098	C₂²	semidirect
208. C₂×C₁₃	#2222	semidirect
251. C₁₃	C₂×#2222	semidirect
297. C₂	#13116961	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×#11718804	2
#13116961	2
C₂×#11718098	2
C₂×#11718109	2
C₂×C₁₃×#504	2
C₂×#2222	13

Subgroups — C2×#13116961 #16017482

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#13116961 #16017482