Subgroups — C₂²×C₁₃×#324 #15703683

Order 3328.

1806 conjugacy classes of subgroups.

The subgroup lattice has 268 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂³×C₁₃×C₄⋊C₄	C₂	1	1	yes	maximal, characteristic
2	C₂²×C₁₃×C₂²⋊C₈	C₂	1	2	yes	maximal
3	C₂×C₁₃×#324	C₂	1	12	yes	maximal, direct product factor
4	C₂⁴×C₄×C₁₃	C₄	1	1	yes	characteristic
5	C₂⁴×C₄×C₁₃	C₄	1	1	yes	characteristic
6	C₂⁴×C₄×C₁₃	C₂²	1	1	yes	characteristic
7	C₂³×C₈×C₁₃	—	2	2	no	—
8	C₂²×C₁₃×C₄⋊C₄	—	2	6	no	—
9	C₂²×C₁₃×C₄⋊C₄	—	2	2	no	—
10	C₂²×C₁₃×C₄⋊C₄	C₄	1	6	yes	—
11	C₂²×C₁₃×C₄⋊C₄	C₂²	1	6	yes	—
12	C₂×C₁₃×C₂²⋊C₈	—	2	6	no	—
13	C₂×C₁₃×C₂²⋊C₈	C₂²	1	12	yes	—
14	C₁₃×#324	C₂²	1	16	yes	direct product factor
15	C₂⁵×C₁₃	C₂×C₄	1	1	yes	characteristic
16	C₂³×C₄×C₁₃	—	2	3	no	—
17	C₂³×C₄×C₁₃	—	2	1	no	—
18	C₂³×C₄×C₁₃	—	2	4	no	—
19	C₂³×C₄×C₁₃	—	2	3	no	—
20	C₂³×C₄×C₁₃	—	2	1	no	—
21	C₂³×C₄×C₁₃	—	4	4	no	—
22	C₂³×C₄×C₁₃	—	4	4	no	—
23	C₂³×C₄×C₁₃	C₈	1	8	yes	complemented
24	C₂³×C₄×C₁₃	C₂×C₄	1	6	yes	—
25	C₂³×C₄×C₁₃	C₂×C₄	1	6	yes	—
26	C₂³×C₄×C₁₃	C₂×C₄	1	3	yes	—
27	C₂³×C₄×C₁₃	D₄	1	2	yes	—
28	C₂³×C₄×C₁₃	D₄	1	6	yes	—
29	C₂³×C₄×C₁₃	C₂³	1	3	yes	—
30	C₂²×C₈×C₁₃	—	2	12	no	—
31	C₂²×C₈×C₁₃	—	4	8	no	—
32	C₂×C₁₃×C₄⋊C₄	—	2	12	no	—
33	C₂×C₁₃×C₄⋊C₄	—	2	12	no	—
34	C₂×C₁₃×C₄⋊C₄	—	2	12	no	—
35	C₂×C₁₃×C₄⋊C₄	—	2	12	no	—
36	C₂×C₁₃×C₄⋊C₄	C₂×C₄	1	12	yes	—
37	C₂×C₁₃×C₄⋊C₄	C₂³	1	4	yes	—
38	C₁₃×C₂²⋊C₈	—	2	12	no	—
39	C₁₃×C₂²⋊C₈	C₂³	1	8	yes	—
40	C₂²×#324	C₁₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
41	C₂⁴×C₁₃	—	2	4	no	—
42	C₂⁴×C₁₃	—	2	4	no	—
43	C₂⁴×C₁₃	C₂²⋊C₄	1	3	yes	—
44	C₂⁴×C₁₃	C₂²⋊C₄	1	1	yes	characteristic
45	C₂⁴×C₁₃	C₂×C₈	1	4	yes	—
46	C₂⁴×C₁₃	M₁₆	1	4	yes	—
47	C₂⁴×C₁₃	C₂²×C₄	1	3	yes	—
48	C₂²×C₄×C₁₃	—	2	24	no	—
49	C₂²×C₄×C₁₃	—	2	8	no	—
50	C₂²×C₄×C₁₃	—	2	8	no	—
51	C₂²×C₄×C₁₃	—	2	6	no	—
52	C₂²×C₄×C₁₃	—	2	6	no	—
53	C₂²×C₄×C₁₃	—	2	12	no	—
54	C₂²×C₄×C₁₃	—	2	6	no	—
55	C₂²×C₄×C₁₃	—	2	6	no	—
56	C₂²×C₄×C₁₃	—	4	6	no	—
57	C₂²×C₄×C₁₃	—	4	12	no	—
58	C₂²×C₄×C₁₃	—	4	6	no	—
59	C₂²×C₄×C₁₃	—	4	12	no	—
60	C₂²×C₄×C₁₃	—	4	4	no	—
61	C₂²×C₄×C₁₃	—	4	12	no	—
62	C₂²×C₄×C₁₃	C₂²⋊C₄	1	6	yes	—
63	C₂²×C₄×C₁₃	C₂²⋊C₄	1	6	yes	—
64	C₂²×C₄×C₁₃	C₂×C₈	1	24	yes	complemented
65	C₂²×C₄×C₁₃	C₂²×C₄	1	4	yes	—
66	C₂²×C₄×C₁₃	C₂²×C₄	1	4	yes	—
67	C₂²×C₄×C₁₃	C₂²×C₄	1	3	yes	—
68	C₂²×C₄×C₁₃	C₂×D₄	1	6	yes	—
69	C₂²×C₄×C₁₃	C₂×D₄	1	6	yes	—
70	C₂²×C₄×C₁₃	C₂⁴	1	1	yes	characteristic
71	C₂×C₈×C₁₃	—	2	8	no	—
72	C₂×C₈×C₁₃	—	4	24	no	—
73	C₁₃×C₄⋊C₄	—	2	24	no	—
74	C₁₃×C₄⋊C₄	—	2	8	no	—
75	C₂³×C₄⋊C₄	C₂×C₁₃	1	1	yes	characteristic
76	C₂²×C₂²⋊C₈	C₂×C₁₃	1	2	yes	—
77	C₂×#324	C₂×C₁₃	1	12	yes	direct product factor
78	C₂³×C₁₃	—	2	16	no	—
79	C₂³×C₁₃	—	2	16	no	—
80	C₂³×C₁₃	—	2	6	no	—
81	C₂³×C₁₃	—	2	6	no	—
82	C₂³×C₁₃	—	2	6	no	—
83	C₂³×C₁₃	—	2	6	no	—
84	C₂³×C₁₃	C₂²⋊C₈	1	12	yes	—
85	C₂³×C₁₃	C₂²⋊C₈	1	4	yes	—
86	C₂³×C₁₃	#99	1	4	yes	—
87	C₂³×C₁₃	#101	1	4	yes	—
88	C₂³×C₁₃	C₂×C₂²⋊C₄	1	3	yes	—
89	C₂³×C₁₃	C₂×C₂²⋊C₄	1	3	yes	—
90	C₂³×C₁₃	C₂²×C₈	1	6	yes	—
91	C₂³×C₁₃	C₂×M₁₆	1	6	yes	—
92	C₂³×C₁₃	C₂³×C₄	1	1	yes	characteristic
93	C₂×C₄×C₁₃	—	2	4	no	—
94	C₂×C₄×C₁₃	—	2	4	no	—
95	C₂×C₄×C₁₃	—	2	24	no	—
96	C₂×C₄×C₁₃	—	2	24	no	—
97	C₂×C₄×C₁₃	—	2	12	no	—
98	C₂×C₄×C₁₃	—	2	12	no	—
99	C₂×C₄×C₁₃	—	2	4	no	—
100	C₂×C₄×C₁₃	—	4	6	no	—
101	C₂×C₄×C₁₃	—	4	3	no	—
102	C₂×C₄×C₁₃	—	4	6	no	—
103	C₂×C₄×C₁₃	—	4	1	no	—
104	C₂×C₄×C₁₃	—	4	12	no	—
105	C₂×C₄×C₁₃	—	4	12	no	—
106	C₂×C₄×C₁₃	—	4	4	no	—
107	C₂×C₄×C₁₃	C₂×C₂²⋊C₄	1	6	yes	—
108	C₂×C₄×C₁₃	C₂²×C₈	1	8	yes	complemented
109	C₂×C₄×C₁₃	C₂²×D₄	1	2	yes	—
110	C₈×C₁₃	—	4	8	no	—
111	C₂⁴×C₄	C₄×C₁₃	1	1	yes	characteristic
112	C₂⁴×C₄	C₄×C₁₃	1	1	yes	characteristic
113	C₂⁴×C₄	C₂²×C₁₃	1	1	yes	characteristic
114	C₂³×C₈	—	2	2	no	—
115	C₂²×C₄⋊C₄	—	2	2	no	—
116	C₂²×C₄⋊C₄	—	2	6	no	—
117	C₂²×C₄⋊C₄	C₄×C₁₃	1	6	yes	—
118	C₂²×C₄⋊C₄	C₂²×C₁₃	1	6	yes	—
119	C₂×C₂²⋊C₈	—	2	6	no	—
120	C₂×C₂²⋊C₈	C₂²×C₁₃	1	12	yes	—
121	#324	C₂²×C₁₃	1	16	yes	direct product factor
122	C₂²×C₁₃	—	2	3	no	—
123	C₂²×C₁₃	—	2	1	no	—
124	C₂²×C₁₃	—	2	3	no	—
125	C₂²×C₁₃	—	2	1	no	—
126	C₂²×C₁₃	—	2	24	no	—
127	C₂²×C₁₃	—	2	12	no	—
128	C₂²×C₁₃	—	2	12	no	—
129	C₂²×C₁₃	#324	1	16	yes	direct product factor
130	C₂²×C₁₃	C₂×C₂²⋊C₈	1	6	yes	—
131	C₂²×C₁₃	C₂×C₂²⋊C₈	1	6	yes	—
132	C₂²×C₁₃	C₂×#99	1	6	yes	—
133	C₂²×C₁₃	C₂×#101	1	6	yes	—
134	C₂²×C₁₃	C₂²×C₂²⋊C₄	1	1	yes	characteristic
135	C₂²×C₁₃	C₂³×C₈	1	1	yes	characteristic
136	C₂²×C₁₃	C₂²×M₁₆	1	1	yes	characteristic
137	C₄×C₁₃	—	2	8	no	—
138	C₄×C₁₃	—	2	2	no	—
139	C₄×C₁₃	—	2	6	no	—
140	C₄×C₁₃	—	4	4	no	—
141	C₂⁵	C₂×C₄×C₁₃	1	1	yes	characteristic
142	C₂³×C₄	—	2	3	no	—
143	C₂³×C₄	—	2	1	no	—
144	C₂³×C₄	—	2	4	no	—
145	C₂³×C₄	—	2	3	no	—
146	C₂³×C₄	—	2	1	no	—
147	C₂³×C₄	—	4	4	no	—
148	C₂³×C₄	—	4	4	no	—
149	C₂³×C₄	C₈×C₁₃	1	8	yes	complemented
150	C₂³×C₄	C₂×C₄×C₁₃	1	6	yes	—
151	C₂³×C₄	C₂×C₄×C₁₃	1	6	yes	—
152	C₂³×C₄	C₂×C₄×C₁₃	1	3	yes	—
153	C₂³×C₄	D₄×C₁₃	1	2	yes	—
154	C₂³×C₄	D₄×C₁₃	1	6	yes	—
155	C₂³×C₄	C₂³×C₁₃	1	3	yes	—
156	C₂²×C₈	—	2	12	no	—
157	C₂²×C₈	—	4	8	no	—
158	C₂×C₄⋊C₄	—	2	12	no	—
159	C₂×C₄⋊C₄	—	2	12	no	—
160	C₂×C₄⋊C₄	—	2	12	no	—
161	C₂×C₄⋊C₄	—	2	12	no	—
162	C₂×C₄⋊C₄	C₂×C₄×C₁₃	1	12	yes	—
163	C₂×C₄⋊C₄	C₂³×C₁₃	1	4	yes	—
164	C₂²⋊C₈	—	2	12	no	—
165	C₂²⋊C₈	C₂³×C₁₃	1	8	yes	—
166	C₂×C₁₃	—	2	1	no	—
167	C₂×C₁₃	—	2	1	no	—
168	C₂×C₁₃	—	2	6	no	—
169	C₂×C₁₃	C₂×#324	1	12	yes	direct product factor
170	C₂×C₁₃	C₂²×C₂²⋊C₈	1	1	yes	characteristic
171	C₂×C₁₃	C₂²×#99	1	1	yes	characteristic
172	C₂×C₁₃	C₂²×#101	1	1	yes	characteristic
173	C₂⁴	—	2	4	no	—
174	C₂⁴	—	2	4	no	—
175	C₂⁴	C₁₃×C₂²⋊C₄	1	3	yes	—
176	C₂⁴	C₁₃×C₂²⋊C₄	1	1	yes	characteristic
177	C₂⁴	C₂×C₈×C₁₃	1	4	yes	—
178	C₂⁴	C₁₃×M₁₆	1	4	yes	—
179	C₂⁴	C₂²×C₄×C₁₃	1	3	yes	—
180	C₂²×C₄	—	2	24	no	—
181	C₂²×C₄	—	2	8	no	—
182	C₂²×C₄	—	2	8	no	—
183	C₂²×C₄	—	2	6	no	—
184	C₂²×C₄	—	2	6	no	—
185	C₂²×C₄	—	2	12	no	—
186	C₂²×C₄	—	2	6	no	—
187	C₂²×C₄	—	2	6	no	—
188	C₂²×C₄	—	4	6	no	—
189	C₂²×C₄	—	4	12	no	—
190	C₂²×C₄	—	4	6	no	—
191	C₂²×C₄	—	4	12	no	—
192	C₂²×C₄	—	4	4	no	—
193	C₂²×C₄	—	4	12	no	—
194	C₂²×C₄	C₁₃×C₂²⋊C₄	1	6	yes	—
195	C₂²×C₄	C₁₃×C₂²⋊C₄	1	6	yes	—
196	C₂²×C₄	C₂×C₈×C₁₃	1	24	yes	complemented
197	C₂²×C₄	C₂²×C₄×C₁₃	1	4	yes	—
198	C₂²×C₄	C₂²×C₄×C₁₃	1	4	yes	—
199	C₂²×C₄	C₂²×C₄×C₁₃	1	3	yes	—
200	C₂²×C₄	C₂×D₄×C₁₃	1	6	yes	—
201	C₂²×C₄	C₂×D₄×C₁₃	1	6	yes	—
202	C₂²×C₄	C₂⁴×C₁₃	1	1	yes	characteristic
203	C₂×C₈	—	2	8	no	—
204	C₂×C₈	—	4	24	no	—
205	C₄⋊C₄	—	2	24	no	—
206	C₄⋊C₄	—	2	8	no	—
207	C₁₃	C₂²×#324	1	1	yes	characteristic, direct product factor, Sylow
208	C₂³	—	2	16	no	—
209	C₂³	—	2	16	no	—
210	C₂³	—	2	6	no	—
211	C₂³	—	2	6	no	—
212	C₂³	—	2	6	no	—
213	C₂³	—	2	6	no	—
214	C₂³	C₁₃×C₂²⋊C₈	1	12	yes	—
215	C₂³	C₁₃×C₂²⋊C₈	1	4	yes	—
216	C₂³	C₁₃×#99	1	4	yes	—
217	C₂³	C₁₃×#101	1	4	yes	—
218	C₂³	C₂×C₁₃×C₂²⋊C₄	1	3	yes	—
219	C₂³	C₂×C₁₃×C₂²⋊C₄	1	3	yes	—
220	C₂³	C₂²×C₈×C₁₃	1	6	yes	—
221	C₂³	C₂×C₁₃×M₁₆	1	6	yes	—
222	C₂³	C₂³×C₄×C₁₃	1	1	yes	characteristic
223	C₂×C₄	—	2	24	no	—
224	C₂×C₄	—	2	24	no	—
225	C₂×C₄	—	2	12	no	—
226	C₂×C₄	—	2	12	no	—
227	C₂×C₄	—	2	4	no	—
228	C₂×C₄	—	2	4	no	—
229	C₂×C₄	—	2	4	no	—
230	C₂×C₄	—	4	6	no	—
231	C₂×C₄	—	4	3	no	—
232	C₂×C₄	—	4	6	no	—
233	C₂×C₄	—	4	1	no	—
234	C₂×C₄	—	4	12	no	—
235	C₂×C₄	—	4	12	no	—
236	C₂×C₄	—	4	4	no	—
237	C₂×C₄	C₂×C₁₃×C₂²⋊C₄	1	6	yes	—
238	C₂×C₄	C₂²×C₈×C₁₃	1	8	yes	complemented
239	C₂×C₄	C₂²×D₄×C₁₃	1	2	yes	—
240	C₈	—	4	8	no	—
241	C₂²	—	2	3	no	—
242	C₂²	—	2	1	no	—
243	C₂²	—	2	3	no	—
244	C₂²	—	2	1	no	—
245	C₂²	—	2	24	no	—
246	C₂²	—	2	12	no	—
247	C₂²	—	2	12	no	—
248	C₂²	C₁₃×#324	1	16	yes	direct product factor
249	C₂²	C₂×C₁₃×C₂²⋊C₈	1	6	yes	—
250	C₂²	C₂×C₁₃×C₂²⋊C₈	1	6	yes	—
251	C₂²	C₂×C₁₃×#99	1	6	yes	—
252	C₂²	C₂×C₁₃×#101	1	6	yes	—
253	C₂²	C₂²×C₁₃×C₂²⋊C₄	1	1	yes	characteristic
254	C₂²	C₂³×C₈×C₁₃	1	1	yes	characteristic
255	C₂²	C₂²×C₁₃×M₁₆	1	1	yes	characteristic
256	C₄	—	2	8	no	—
257	C₄	—	2	2	no	—
258	C₄	—	2	6	no	—
259	C₄	—	4	4	no	—
260	C₂	—	2	6	no	—
261	C₂	—	2	1	no	—
262	C₂	—	2	1	no	—
263	C₂	C₂×C₁₃×#324	1	12	yes	direct product factor
264	C₂	C₂²×C₁₃×C₂²⋊C₈	1	1	yes	characteristic
265	C₂	C₂²×C₁₃×#99	1	1	yes	characteristic
266	C₂	C₂²×C₁₃×#101	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂⁴×C₁₃	C₂²⋊C₄
commutator	C₂²	C₂³×C₈×C₁₃
Frattini, cosocle	C₂²×C₄	C₂⁴×C₁₃
prime radical	C₂⁵×C₁₃	C₂×C₄
LCS	G ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂⁴×C₁₃ ◁ C₂⁴×C₄×C₁₃ ◁ G

Complements

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

N	H ≅ G/N	type
3. C₂×C₁₃×#324	C₂	direct (N × H)
14. C₁₃×#324	C₂²	direct (N × H)
23. C₂³×C₄×C₁₃	C₈	semidirect
40. C₂²×#324	C₁₃	direct (N × H)
64. C₂²×C₄×C₁₃	C₂×C₈	semidirect
77. C₂×#324	C₂×C₁₃	direct (N × H)
108. C₂×C₄×C₁₃	C₂²×C₈	semidirect
121. #324	C₂²×C₁₃	direct (N × H)
129. C₂²×C₁₃	#324	direct (N × H)
149. C₂³×C₄	C₈×C₁₃	semidirect
169. C₂×C₁₃	C₂×#324	direct (N × H)
196. C₂²×C₄	C₂×C₈×C₁₃	semidirect
207. C₁₃	C₂²×#324	direct (N × H)
238. C₂×C₄	C₂²×C₈×C₁₃	semidirect
248. C₂²	C₁₃×#324	direct (N × H)
263. C₂	C₂×C₁₃×#324	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂³×C₁₃×C₄⋊C₄	2
C₂²×C₁₃×C₂²⋊C₈	2
C₂×C₁₃×#324	2
C₂²×#324	13

Subgroups — C22×C13×#324 #15703683

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂²×C₁₃×#324 #15703683