Subgroups — C₂×C₁₃×#2222 #15719954

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

	subgroup	G/H
center, socle	C₂²×C₁₃	#466
commutator, Frattini, cosocle	C₂×C₈	C₂⁴×C₁₃
prime radical	C₂×C₁₃×#41	D₄
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₁₃ ◁ C₂²×C₄×C₁₃ ◁ C₂×C₁₃×#131 ◁ G

Complements

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

N	H ≅ G/N	type
1. C₂²×C₁₃×Q₃₂	C₂	semidirect
4. C₁₃×#2222	C₂	direct (N × H)
5. C₂×C₁₃×#368	C₂	semidirect
6. C₂×C₁₃×#362	C₂	semidirect
13. C₂×C₁₃×Q₃₂	C₂²	semidirect
24. C₁₃×#368	C₂²	semidirect
25. C₁₃×#362	C₂²	semidirect
61. C₂×#2222	C₁₃	direct (N × H)
103. C₂²×Q₃₂	C₂×C₁₃	semidirect
106. #2222	C₂×C₁₃	direct (N × H)
107. C₂×#368	C₂×C₁₃	semidirect
108. C₂×#362	C₂×C₁₃	semidirect
145. C₂×Q₃₂	C₂²×C₁₃	semidirect
156. #368	C₂²×C₁₃	semidirect
157. #362	C₂²×C₁₃	semidirect
208. C₂×C₁₃	#2222	direct (N × H)
251. C₁₃	C₂×#2222	direct (N × H)
297. C₂	C₁₃×#2222	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×C₁₃×Q₃₂	2
C₂×C₁₃×#504	2
C₂×C₁₃×#473	2
C₁₃×#2222	2
C₂×C₁₃×#368	2
C₂×C₁₃×#362	2
C₂×#2222	13

Subgroups — C2×C13×#2222 #15719954

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₁₃×#2222 #15719954