Subgroups — C₁₃×C₄⋊#338 #15697177

Order 3328.

762 conjugacy classes of subgroups.

The subgroup lattice has 232 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₁₃×#423	C₂	1	2	yes	maximal
2	C₂×C₄×C₁₃×M₁₆	C₂	1	1	yes	maximal, characteristic
3	C₂×C₁₃×#338	C₂	1	4	yes	maximal
4	C₂²×C₁₃×M₁₆	C₂²	1	4	yes	—
5	C₂²×C₁₃×M₁₆	C₂²	1	2	yes	—
6	C₂²×C₄²×C₁₃	C₂²	1	1	yes	characteristic
7	C₁₃×#423	—	2	2	no	—
8	C₁₃×#423	—	2	2	no	—
9	C₁₃×#423	—	2	4	no	—
10	C₂×C₁₃×C₄⋊C₈	C₄	1	8	yes	—
11	C₄×C₁₃×M₁₆	—	2	2	no	—
12	C₄×C₁₃×M₁₆	—	2	2	no	—
13	C₂×C₁₃×#97	C₄	1	2	yes	—
14	C₂×C₄×C₈×C₁₃	C₄	1	2	yes	—
15	C₁₃×#338	—	2	8	no	—
16	C₂³×C₄×C₁₃	Q₈	1	2	yes	—
17	C₂³×C₄×C₁₃	C₂³	1	1	yes	characteristic
18	C₂×C₁₃×M₁₆	—	2	4	no	—
19	C₂×C₁₃×M₁₆	—	2	4	no	—
20	C₂×C₁₃×M₁₆	—	2	8	no	—
21	C₂×C₁₃×M₁₆	—	2	4	no	—
22	C₂×C₁₃×M₁₆	—	2	4	no	—
23	C₂×C₁₃×M₁₆	—	2	8	no	—
24	C₂×C₁₃×M₁₆	D₄	1	4	yes	—
25	C₂×C₁₃×M₁₆	Q₈	1	4	yes	—
26	C₂²×C₈×C₁₃	C₂×C₄	1	8	yes	—
27	C₂²×C₈×C₁₃	C₂×C₄	1	4	yes	—
28	C₂×C₄²×C₁₃	C₂×C₄	1	1	yes	characteristic
29	C₂×C₄²×C₁₃	C₂×C₄	1	1	yes	characteristic
30	C₂×C₄²×C₁₃	C₂×C₄	1	4	yes	—
31	C₂×C₄²×C₁₃	D₄	1	2	yes	—
32	C₂×C₄²×C₁₃	D₄	1	2	yes	—
33	C₂×C₄²×C₁₃	D₄	1	2	yes	—
34	C₁₃×C₄⋊C₈	—	2	8	no	—
35	C₁₃×C₄⋊C₈	—	2	8	no	—
36	C₁₃×#97	—	2	4	no	—
37	C₄×C₈×C₁₃	—	2	2	no	—
38	C₄×C₈×C₁₃	—	2	2	no	—
39	C₄⋊#338	C₁₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
40	C₂⁴×C₁₃	C₂×Q₈	1	1	yes	characteristic
41	C₂²×C₄×C₁₃	—	4	4	no	—
42	C₂²×C₄×C₁₃	C₄⋊C₄	1	2	yes	—
43	C₂²×C₄×C₁₃	C₄⋊C₄	1	4	yes	—
44	C₂²×C₄×C₁₃	C₄⋊C₄	1	2	yes	—
45	C₂²×C₄×C₁₃	C₄⋊C₄	1	4	yes	—
46	C₂²×C₄×C₁₃	C₂²×C₄	1	1	yes	characteristic
47	C₂²×C₄×C₁₃	C₂²×C₄	1	2	yes	—
48	C₂²×C₄×C₁₃	C₂×D₄	1	2	yes	—
49	C₂²×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
50	C₂²×C₄×C₁₃	C₂×D₄	1	2	yes	—
51	C₂²×C₄×C₁₃	C₂×Q₈	1	2	yes	—
52	C₂²×C₄×C₁₃	#41	1	4	yes	—
53	C₁₃×M₁₆	—	2	8	no	—
54	C₁₃×M₁₆	—	2	8	no	—
55	C₁₃×M₁₆	—	4	8	no	—
56	C₁₃×M₁₆	—	4	4	no	—
57	C₁₃×M₁₆	—	4	4	no	—
58	C₂×C₈×C₁₃	—	2	4	no	—
59	C₂×C₈×C₁₃	—	2	4	no	—
60	C₂×C₈×C₁₃	—	2	8	no	—
61	C₂×C₈×C₁₃	—	2	16	no	—
62	C₂×C₈×C₁₃	C₄⋊C₄	1	8	yes	—
63	C₄²×C₁₃	C₄²	1	2	yes	—
64	C₄²×C₁₃	C₄²	1	2	yes	—
65	C₄²×C₁₃	C₂²⋊C₄	1	1	yes	characteristic
66	C₄²×C₁₃	C₂²⋊C₄	1	1	yes	characteristic
67	C₄²×C₁₃	C₂²⋊C₄	1	2	yes	—
68	C₄²×C₁₃	C₂²⋊C₄	1	4	yes	—
69	C₄²×C₁₃	C₂²⋊C₄	1	4	yes	—
70	C₂×#423	C₂×C₁₃	1	2	yes	—
71	C₂×C₄×M₁₆	C₂×C₁₃	1	1	yes	characteristic
72	C₂×#338	C₂×C₁₃	1	4	yes	—
73	C₂³×C₁₃	—	4	2	no	—
74	C₂³×C₁₃	C₂×C₄⋊C₄	1	1	yes	characteristic
75	C₂³×C₁₃	C₂×C₄⋊C₄	1	2	yes	—
76	C₂³×C₁₃	#125	1	2	yes	—
77	C₂³×C₁₃	C₄⋊Q₈	1	2	yes	—
78	C₂×C₄×C₁₃	—	2	4	no	—
79	C₂×C₄×C₁₃	—	2	8	no	—
80	C₂×C₄×C₁₃	—	4	2	no	—
81	C₂×C₄×C₁₃	—	4	4	no	—
82	C₂×C₄×C₁₃	#95	1	2	yes	—
83	C₂×C₄×C₁₃	#95	1	4	yes	—
84	C₂×C₄×C₁₃	#95	1	2	yes	—
85	C₂×C₄×C₁₃	C₂×C₄²	1	1	yes	characteristic
86	C₂×C₄×C₁₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
87	C₂×C₄×C₁₃	C₂×C₂²⋊C₄	1	2	yes	—
88	C₂×C₄×C₁₃	C₂×C₄⋊C₄	1	2	yes	—
89	C₂×C₄×C₁₃	#117	1	2	yes	—
90	C₂×C₄×C₁₃	C₄×D₄	1	4	yes	—
91	C₂×C₄×C₁₃	C₄×Q₈	1	4	yes	—
92	C₂×C₄×C₁₃	#121	1	4	yes	—
93	C₂×C₄×C₁₃	C₂²⋊Q₈	1	4	yes	—
94	C₂×C₄×C₁₃	#124	1	2	yes	—
95	C₂×C₄×C₁₃	#127	1	1	yes	characteristic
96	C₂×C₄×C₁₃	C₄⋊Q₈	1	1	yes	characteristic
97	C₈×C₁₃	—	2	8	no	—
98	C₈×C₁₃	—	4	8	no	—
99	C₂²×M₁₆	C₂²×C₁₃	1	4	yes	—
100	C₂²×M₁₆	C₂²×C₁₃	1	2	yes	—
101	C₂²×C₄²	C₂²×C₁₃	1	1	yes	characteristic
102	#423	—	2	2	no	—
103	#423	—	2	2	no	—
104	#423	—	2	4	no	—
105	C₂×C₄⋊C₈	C₄×C₁₃	1	8	yes	—
106	C₄×M₁₆	—	2	2	no	—
107	C₄×M₁₆	—	2	2	no	—
108	C₂×#97	C₄×C₁₃	1	2	yes	—
109	C₂×C₄×C₈	C₄×C₁₃	1	2	yes	—
110	#338	—	2	8	no	—
111	C₂²×C₁₃	—	2	2	no	—
112	C₂²×C₁₃	—	2	4	no	—
113	C₂²×C₁₃	—	4	2	no	—
114	C₂²×C₁₃	—	4	2	no	—
115	C₂²×C₁₃	C₂×#95	1	1	yes	characteristic
116	C₂²×C₁₃	#382	1	1	yes	characteristic
117	C₂²×C₁₃	#384	1	1	yes	characteristic
118	C₂²×C₁₃	#389	1	4	yes	—
119	C₄×C₁₃	—	2	2	no	—
120	C₄×C₁₃	—	2	4	no	—
121	C₄×C₁₃	#338	1	4	yes	complemented
122	C₄×C₁₃	C₄×C₄⋊C₄	1	2	yes	—
123	C₄×C₁₃	#380	1	2	yes	—
124	C₄×C₁₃	#390	1	2	yes	—
125	C₄×C₁₃	#391	1	2	yes	—
126	C₂³×C₄	Q₈×C₁₃	1	2	yes	—
127	C₂³×C₄	C₂³×C₁₃	1	1	yes	characteristic
128	C₂×M₁₆	—	2	4	no	—
129	C₂×M₁₆	—	2	4	no	—
130	C₂×M₁₆	—	2	8	no	—
131	C₂×M₁₆	—	2	4	no	—
132	C₂×M₁₆	—	2	4	no	—
133	C₂×M₁₆	—	2	8	no	—
134	C₂×M₁₆	D₄×C₁₃	1	4	yes	—
135	C₂×M₁₆	Q₈×C₁₃	1	4	yes	—
136	C₂²×C₈	C₂×C₄×C₁₃	1	8	yes	—
137	C₂²×C₈	C₂×C₄×C₁₃	1	4	yes	—
138	C₂×C₄²	C₂×C₄×C₁₃	1	1	yes	characteristic
139	C₂×C₄²	C₂×C₄×C₁₃	1	1	yes	characteristic
140	C₂×C₄²	C₂×C₄×C₁₃	1	4	yes	—
141	C₂×C₄²	D₄×C₁₃	1	2	yes	—
142	C₂×C₄²	D₄×C₁₃	1	2	yes	—
143	C₂×C₄²	D₄×C₁₃	1	2	yes	—
144	C₄⋊C₈	—	2	8	no	—
145	C₄⋊C₈	—	2	8	no	—
146	#97	—	2	4	no	—
147	C₄×C₈	—	2	2	no	—
148	C₄×C₈	—	2	2	no	—
149	C₂×C₁₃	—	2	1	no	—
150	C₂×C₁₃	—	2	1	no	—
151	C₂×C₁₃	—	2	4	no	—
152	C₂×C₁₃	C₄⋊#95	1	1	yes	characteristic
153	C₂×C₁₃	C₂×#338	1	2	yes	—
154	C₂⁴	C₂×Q₈×C₁₃	1	1	yes	characteristic
155	C₂²×C₄	—	4	4	no	—
156	C₂²×C₄	C₁₃×C₄⋊C₄	1	2	yes	—
157	C₂²×C₄	C₁₃×C₄⋊C₄	1	4	yes	—
158	C₂²×C₄	C₁₃×C₄⋊C₄	1	2	yes	—
159	C₂²×C₄	C₁₃×C₄⋊C₄	1	4	yes	—
160	C₂²×C₄	C₂²×C₄×C₁₃	1	1	yes	characteristic
161	C₂²×C₄	C₂²×C₄×C₁₃	1	2	yes	—
162	C₂²×C₄	C₂×D₄×C₁₃	1	2	yes	—
163	C₂²×C₄	C₂×D₄×C₁₃	1	1	yes	characteristic
164	C₂²×C₄	C₂×D₄×C₁₃	1	2	yes	—
165	C₂²×C₄	C₂×Q₈×C₁₃	1	2	yes	—
166	C₂²×C₄	C₁₃×#41	1	4	yes	—
167	M₁₆	—	2	8	no	—
168	M₁₆	—	2	8	no	—
169	M₁₆	—	4	8	no	—
170	M₁₆	—	4	4	no	—
171	M₁₆	—	4	4	no	—
172	C₂×C₈	—	2	4	no	—
173	C₂×C₈	—	2	4	no	—
174	C₂×C₈	—	2	8	no	—
175	C₂×C₈	—	2	16	no	—
176	C₂×C₈	C₁₃×C₄⋊C₄	1	8	yes	—
177	C₄²	C₄²×C₁₃	1	2	yes	—
178	C₄²	C₄²×C₁₃	1	2	yes	—
179	C₄²	C₁₃×C₂²⋊C₄	1	1	yes	characteristic
180	C₄²	C₁₃×C₂²⋊C₄	1	1	yes	characteristic
181	C₄²	C₁₃×C₂²⋊C₄	1	2	yes	—
182	C₄²	C₁₃×C₂²⋊C₄	1	4	yes	—
183	C₄²	C₁₃×C₂²⋊C₄	1	4	yes	—
184	C₁₃	C₄⋊#338	1	1	yes	characteristic, direct product factor, Sylow
185	C₂³	—	4	2	no	—
186	C₂³	C₂×C₁₃×C₄⋊C₄	1	1	yes	characteristic
187	C₂³	C₂×C₁₃×C₄⋊C₄	1	2	yes	—
188	C₂³	C₁₃×#125	1	2	yes	—
189	C₂³	C₁₃×C₄⋊Q₈	1	2	yes	—
190	C₂×C₄	—	2	4	no	—
191	C₂×C₄	—	2	8	no	—
192	C₂×C₄	—	4	2	no	—
193	C₂×C₄	—	4	4	no	—
194	C₂×C₄	C₁₃×#95	1	2	yes	—
195	C₂×C₄	C₁₃×#95	1	4	yes	—
196	C₂×C₄	C₁₃×#95	1	2	yes	—
197	C₂×C₄	C₂×C₄²×C₁₃	1	1	yes	characteristic
198	C₂×C₄	C₂×C₁₃×C₂²⋊C₄	1	1	yes	characteristic
199	C₂×C₄	C₂×C₁₃×C₂²⋊C₄	1	2	yes	—
200	C₂×C₄	C₂×C₁₃×C₄⋊C₄	1	2	yes	—
201	C₂×C₄	C₁₃×#117	1	2	yes	—
202	C₂×C₄	C₄×D₄×C₁₃	1	4	yes	—
203	C₂×C₄	C₄×Q₈×C₁₃	1	4	yes	—
204	C₂×C₄	C₁₃×#121	1	4	yes	—
205	C₂×C₄	C₁₃×C₂²⋊Q₈	1	4	yes	—
206	C₂×C₄	C₁₃×#124	1	2	yes	—
207	C₂×C₄	C₁₃×#127	1	1	yes	characteristic
208	C₂×C₄	C₁₃×C₄⋊Q₈	1	1	yes	characteristic
209	C₈	—	2	8	no	—
210	C₈	—	4	8	no	—
211	C₂²	—	2	2	no	—
212	C₂²	—	2	4	no	—
213	C₂²	—	4	2	no	—
214	C₂²	—	4	2	no	—
215	C₂²	C₂×C₁₃×#95	1	1	yes	characteristic
216	C₂²	C₁₃×#382	1	1	yes	characteristic
217	C₂²	C₁₃×#384	1	1	yes	characteristic
218	C₂²	C₁₃×#389	1	4	yes	—
219	C₄	—	2	2	no	—
220	C₄	—	2	4	no	—
221	C₄	C₁₃×#338	1	4	yes	complemented
222	C₄	C₄×C₁₃×C₄⋊C₄	1	2	yes	—
223	C₄	C₁₃×#380	1	2	yes	—
224	C₄	C₁₃×#390	1	2	yes	—
225	C₄	C₁₃×#391	1	2	yes	—
226	C₂	—	2	1	no	—
227	C₂	—	2	1	no	—
228	C₂	—	2	4	no	—
229	C₂	C₁₃×C₄⋊#95	1	1	yes	characteristic
230	C₂	C₂×C₁₃×#338	1	2	yes	—

	subgroup	G/H
center	C₂×C₄×C₁₃	C₂×C₄²
commutator	C₂×C₄	C₂×C₄²×C₁₃
Frattini, cosocle	C₂³×C₄	C₂³×C₁₃
socle	C₂²×C₁₃	C₂×#95
prime radical	C₂⁴×C₁₃	C₂×Q₈
LCS	G ▷ C₂×C₄ ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₁₃ ◁ G

Complements

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

N	H ≅ G/N	type
39. C₄⋊#338	C₁₃	direct (N × H)
121. C₄×C₁₃	#338	semidirect
184. C₁₃	C₄⋊#338	direct (N × H)
221. C₄	C₁₃×#338	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₁₃×#423	2
C₂×C₄×C₁₃×M₁₆	2
C₂×C₁₃×#338	2
C₄⋊#338	13

Subgroups — C13×C4⋊#338 #15697177

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₃×C₄⋊#338 #15697177