Subgroups — C₂×C₁₃×#1696 #15705488

Order 3328.

618 conjugacy classes of subgroups.

The subgroup lattice has 272 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₁₃×#500	C₂	1	1	yes	maximal, characteristic
2	C₂×C₁₃×#477	C₂	1	2	yes	maximal
3	C₂×C₁₃×#446	C₂	1	1	yes	maximal, characteristic
4	C₁₃×#1696	C₂	1	8	yes	maximal, direct product factor
5	C₂×C₁₃×#334	C₂	1	1	yes	maximal, characteristic
6	C₂×C₁₃×#333	C₂	1	2	yes	maximal
7	C₂×C₁₃×C₄⋊Q₈	C₂²	1	2	yes	—
8	C₂×C₄×Q₈×C₁₃	C₂²	1	1	yes	characteristic
9	C₁₃×#500	—	2	2	no	—
10	C₁₃×#500	C₂²	1	4	yes	—
11	C₁₃×#477	—	2	4	no	—
12	C₁₃×#477	C₂²	1	8	yes	—
13	C₁₃×#446	—	2	2	no	—
14	C₁₃×#446	C₂²	1	4	yes	—
15	C₂×C₁₃×#107	—	2	1	no	—
16	C₂×C₁₃×#107	—	2	1	no	—
17	C₂×C₁₃×#107	—	2	2	no	—
18	C₂×C₁₃×C₄⋊C₈	—	2	1	no	—
19	C₂×C₁₃×C₄⋊C₈	C₂²	1	2	yes	—
20	C₂×C₁₃×C₄⋊C₈	C₂²	1	1	yes	characteristic
21	C₂×C₁₃×#103	—	2	2	no	—
22	C₂×C₁₃×#97	—	2	1	no	—
23	C₂×C₄×C₈×C₁₃	C₂²	1	1	yes	characteristic
24	C₁₃×#334	C₂²	1	4	yes	—
25	C₁₃×#333	C₂²	1	8	yes	—
26	C₂²×Q₈×C₁₃	—	2	2	no	—
27	C₂²×Q₈×C₁₃	D₄	1	1	yes	characteristic
28	C₂²×C₈×C₁₃	—	2	1	no	—
29	C₂²×C₈×C₁₃	—	2	2	no	—
30	C₂²×C₈×C₁₃	D₄	1	1	yes	characteristic
31	C₂²×C₈×C₁₃	D₄	1	1	yes	characteristic
32	C₁₃×C₄⋊Q₈	—	2	2	no	—
33	C₁₃×C₄⋊Q₈	—	2	2	no	—
34	C₁₃×C₄⋊Q₈	—	2	2	no	—
35	C₁₃×C₄⋊Q₈	C₂³	1	4	yes	—
36	C₄×Q₈×C₁₃	—	2	1	no	—
37	C₄×Q₈×C₁₃	—	2	1	no	—
38	C₄×Q₈×C₁₃	D₄	1	2	yes	—
39	C₄×Q₈×C₁₃	C₂³	1	2	yes	—
40	C₂×C₁₃×C₄⋊C₄	—	2	1	no	—
41	C₂×C₁₃×C₄⋊C₄	—	2	2	no	—
42	C₂×C₁₃×C₄⋊C₄	—	2	2	no	—
43	C₂×C₁₃×C₄⋊C₄	D₄	1	1	yes	characteristic
44	C₂×C₄²×C₁₃	—	2	1	no	—
45	C₂×C₄²×C₁₃	C₂³	1	1	yes	characteristic
46	C₁₃×#107	—	2	4	no	—
47	C₁₃×#107	—	2	4	no	—
48	C₁₃×#107	—	2	8	no	—
49	C₁₃×C₄⋊C₈	—	2	4	no	—
50	C₁₃×C₄⋊C₈	—	2	1	no	—
51	C₁₃×C₄⋊C₈	—	2	2	no	—
52	C₁₃×C₄⋊C₈	C₂³	1	2	yes	—
53	C₁₃×C₄⋊C₈	C₂³	1	4	yes	—
54	C₁₃×#103	—	2	8	no	—
55	C₁₃×#97	—	2	4	no	—
56	C₄×C₈×C₁₃	D₄	1	2	yes	—
57	C₄×C₈×C₁₃	C₂³	1	2	yes	—
58	C₂×#1696	C₁₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
59	C₂×Q₈×C₁₃	—	2	1	no	—
60	C₂×Q₈×C₁₃	—	2	4	no	—
61	C₂×Q₈×C₁₃	—	2	4	no	—
62	C₂×Q₈×C₁₃	—	4	2	no	—
63	C₂×Q₈×C₁₃	—	8	4	no	—
64	C₂×Q₈×C₁₃	C₂×D₄	1	2	yes	—
65	C₂²×C₄×C₁₃	—	2	1	no	—
66	C₂²×C₄×C₁₃	—	2	1	no	—
67	C₂²×C₄×C₁₃	—	4	2	no	—
68	C₂²×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
69	C₂²×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
70	C₂²×C₄×C₁₃	#41	1	1	yes	characteristic
71	C₂×C₈×C₁₃	—	2	2	no	—
72	C₂×C₈×C₁₃	—	2	2	no	—
73	C₂×C₈×C₁₃	—	2	4	no	—
74	C₂×C₈×C₁₃	—	4	2	no	—
75	C₂×C₈×C₁₃	—	4	4	no	—
76	C₂×C₈×C₁₃	Q₁₆	1	4	yes	—
77	C₂×C₈×C₁₃	C₂×D₄	1	2	yes	—
78	C₂×C₈×C₁₃	C₂×D₄	1	2	yes	—
79	C₁₃×C₄⋊C₄	—	2	2	no	—
80	C₁₃×C₄⋊C₄	—	2	1	no	—
81	C₁₃×C₄⋊C₄	—	2	2	no	—
82	C₁₃×C₄⋊C₄	—	2	4	no	—
83	C₁₃×C₄⋊C₄	—	2	4	no	—
84	C₁₃×C₄⋊C₄	—	4	2	no	—
85	C₁₃×C₄⋊C₄	—	4	2	no	—
86	C₁₃×C₄⋊C₄	C₂×D₄	1	2	yes	—
87	C₄²×C₁₃	—	2	2	no	—
88	C₄²×C₁₃	—	2	2	no	—
89	C₄²×C₁₃	C₂×D₄	1	1	yes	characteristic
90	C₄²×C₁₃	C₂×D₄	1	1	yes	characteristic
91	C₄²×C₁₃	#41	1	1	yes	characteristic
92	C₄²×C₁₃	C₂⁴	1	1	yes	characteristic
93	C₂×#500	C₂×C₁₃	1	1	yes	characteristic
94	C₂×#477	C₂×C₁₃	1	2	yes	—
95	C₂×#446	C₂×C₁₃	1	1	yes	characteristic
96	#1696	C₂×C₁₃	1	8	yes	direct product factor
97	C₂×#334	C₂×C₁₃	1	1	yes	characteristic
98	C₂×#333	C₂×C₁₃	1	2	yes	—
99	C₂³×C₁₃	#121	1	1	yes	characteristic
100	Q₈×C₁₃	—	2	4	no	—
101	Q₈×C₁₃	—	4	2	no	—
102	Q₈×C₁₃	—	8	4	no	—
103	Q₈×C₁₃	—	8	4	no	—
104	C₂×C₄×C₁₃	—	2	2	no	—
105	C₂×C₄×C₁₃	—	2	2	no	—
106	C₂×C₄×C₁₃	—	2	2	no	—
107	C₂×C₄×C₁₃	—	4	2	no	—
108	C₂×C₄×C₁₃	—	4	2	no	—
109	C₂×C₄×C₁₃	—	4	4	no	—
110	C₂×C₄×C₁₃	—	8	4	no	—
111	C₂×C₄×C₁₃	#121	1	1	yes	characteristic
112	C₂×C₄×C₁₃	#121	1	1	yes	characteristic
113	C₂×C₄×C₁₃	#121	1	1	yes	characteristic
114	C₂×C₄×C₁₃	C₂×Q₁₆	1	2	yes	—
115	C₂×C₄×C₁₃	#137	1	2	yes	—
116	C₂×C₄×C₁₃	#137	1	4	yes	—
117	C₂×C₄×C₁₃	C₂²×D₄	1	1	yes	characteristic
118	C₂×C₄×C₁₃	C₂²×D₄	1	1	yes	characteristic
119	C₂×C₄×C₁₃	C₂×#41	1	1	yes	characteristic
120	C₈×C₁₃	—	2	2	no	—
121	C₈×C₁₃	—	4	2	no	—
122	C₈×C₁₃	—	4	4	no	—
123	C₈×C₁₃	C₂×Q₁₆	1	4	yes	—
124	C₂×C₄⋊Q₈	C₂²×C₁₃	1	2	yes	—
125	C₂×C₄×Q₈	C₂²×C₁₃	1	1	yes	characteristic
126	#500	—	2	2	no	—
127	#500	C₂²×C₁₃	1	4	yes	—
128	#477	—	2	4	no	—
129	#477	C₂²×C₁₃	1	8	yes	—
130	#446	—	2	2	no	—
131	#446	C₂²×C₁₃	1	4	yes	—
132	C₂×#107	—	2	1	no	—
133	C₂×#107	—	2	1	no	—
134	C₂×#107	—	2	2	no	—
135	C₂×C₄⋊C₈	—	2	1	no	—
136	C₂×C₄⋊C₈	C₂²×C₁₃	1	2	yes	—
137	C₂×C₄⋊C₈	C₂²×C₁₃	1	1	yes	characteristic
138	C₂×#103	—	2	2	no	—
139	C₂×#97	—	2	1	no	—
140	C₂×C₄×C₈	C₂²×C₁₃	1	1	yes	characteristic
141	#334	C₂²×C₁₃	1	4	yes	—
142	#333	C₂²×C₁₃	1	8	yes	—
143	C₂²×C₁₃	#462	1	2	yes	—
144	C₂²×C₁₃	#470	1	2	yes	—
145	C₂²×C₁₃	#472	1	2	yes	—
146	C₂²×C₁₃	C₂×#121	1	1	yes	characteristic
147	C₄×C₁₃	—	2	1	no	—
148	C₄×C₁₃	—	2	1	no	—
149	C₄×C₁₃	—	4	2	no	—
150	C₄×C₁₃	—	4	2	no	—
151	C₄×C₁₃	—	8	4	no	—
152	C₄×C₁₃	#462	1	2	yes	—
153	C₄×C₁₃	#470	1	2	yes	—
154	C₄×C₁₃	C₂²×Q₁₆	1	1	yes	characteristic
155	C₄×C₁₃	C₂×#137	1	1	yes	characteristic
156	C₄×C₁₃	C₂×#137	1	2	yes	—
157	C₂²×Q₈	—	2	2	no	—
158	C₂²×Q₈	D₄×C₁₃	1	1	yes	characteristic
159	C₂²×C₈	—	2	1	no	—
160	C₂²×C₈	—	2	2	no	—
161	C₂²×C₈	D₄×C₁₃	1	1	yes	characteristic
162	C₂²×C₈	D₄×C₁₃	1	1	yes	characteristic
163	C₄⋊Q₈	—	2	2	no	—
164	C₄⋊Q₈	—	2	2	no	—
165	C₄⋊Q₈	—	2	2	no	—
166	C₄⋊Q₈	C₂³×C₁₃	1	4	yes	—
167	C₄×Q₈	—	2	1	no	—
168	C₄×Q₈	—	2	1	no	—
169	C₄×Q₈	D₄×C₁₃	1	2	yes	—
170	C₄×Q₈	C₂³×C₁₃	1	2	yes	—
171	C₂×C₄⋊C₄	—	2	1	no	—
172	C₂×C₄⋊C₄	—	2	2	no	—
173	C₂×C₄⋊C₄	—	2	2	no	—
174	C₂×C₄⋊C₄	D₄×C₁₃	1	1	yes	characteristic
175	C₂×C₄²	—	2	1	no	—
176	C₂×C₄²	C₂³×C₁₃	1	1	yes	characteristic
177	#107	—	2	4	no	—
178	#107	—	2	4	no	—
179	#107	—	2	8	no	—
180	C₄⋊C₈	—	2	4	no	—
181	C₄⋊C₈	—	2	1	no	—
182	C₄⋊C₈	—	2	2	no	—
183	C₄⋊C₈	C₂³×C₁₃	1	2	yes	—
184	C₄⋊C₈	C₂³×C₁₃	1	4	yes	—
185	#103	—	2	8	no	—
186	#97	—	2	4	no	—
187	C₄×C₈	D₄×C₁₃	1	2	yes	—
188	C₄×C₈	C₂³×C₁₃	1	2	yes	—
189	C₂×C₁₃	#1696	1	4	yes	direct product factor
190	C₂×C₁₃	C₂×#462	1	1	yes	characteristic
191	C₂×C₁₃	C₂×#470	1	1	yes	characteristic
192	C₂×C₁₃	C₂×#472	1	1	yes	characteristic
193	C₂×Q₈	—	2	1	no	—
194	C₂×Q₈	—	2	4	no	—
195	C₂×Q₈	—	2	4	no	—
196	C₂×Q₈	—	4	2	no	—
197	C₂×Q₈	—	8	4	no	—
198	C₂×Q₈	C₂×D₄×C₁₃	1	2	yes	—
199	C₂²×C₄	—	2	1	no	—
200	C₂²×C₄	—	2	1	no	—
201	C₂²×C₄	—	4	2	no	—
202	C₂²×C₄	C₂×D₄×C₁₃	1	1	yes	characteristic
203	C₂²×C₄	C₂×D₄×C₁₃	1	1	yes	characteristic
204	C₂²×C₄	C₁₃×#41	1	1	yes	characteristic
205	C₂×C₈	—	2	2	no	—
206	C₂×C₈	—	2	2	no	—
207	C₂×C₈	—	2	4	no	—
208	C₂×C₈	—	4	2	no	—
209	C₂×C₈	—	4	4	no	—
210	C₂×C₈	C₁₃×Q₁₆	1	4	yes	—
211	C₂×C₈	C₂×D₄×C₁₃	1	2	yes	—
212	C₂×C₈	C₂×D₄×C₁₃	1	2	yes	—
213	C₄⋊C₄	—	2	2	no	—
214	C₄⋊C₄	—	2	1	no	—
215	C₄⋊C₄	—	2	2	no	—
216	C₄⋊C₄	—	2	4	no	—
217	C₄⋊C₄	—	2	4	no	—
218	C₄⋊C₄	—	4	2	no	—
219	C₄⋊C₄	—	4	2	no	—
220	C₄⋊C₄	C₂×D₄×C₁₃	1	2	yes	—
221	C₄²	—	2	2	no	—
222	C₄²	—	2	2	no	—
223	C₄²	C₂×D₄×C₁₃	1	1	yes	characteristic
224	C₄²	C₂×D₄×C₁₃	1	1	yes	characteristic
225	C₄²	C₁₃×#41	1	1	yes	characteristic
226	C₄²	C₂⁴×C₁₃	1	1	yes	characteristic
227	C₁₃	C₂×#1696	1	1	yes	characteristic, direct product factor, Sylow
228	C₂³	C₁₃×#121	1	1	yes	characteristic
229	Q₈	—	2	4	no	—
230	Q₈	—	4	2	no	—
231	Q₈	—	8	4	no	—
232	Q₈	—	8	4	no	—
233	C₂×C₄	—	2	2	no	—
234	C₂×C₄	—	2	2	no	—
235	C₂×C₄	—	2	2	no	—
236	C₂×C₄	—	4	2	no	—
237	C₂×C₄	—	4	2	no	—
238	C₂×C₄	—	4	4	no	—
239	C₂×C₄	—	8	4	no	—
240	C₂×C₄	C₁₃×#121	1	1	yes	characteristic
241	C₂×C₄	C₁₃×#121	1	1	yes	characteristic
242	C₂×C₄	C₁₃×#121	1	1	yes	characteristic
243	C₂×C₄	C₂×C₁₃×Q₁₆	1	2	yes	—
244	C₂×C₄	C₁₃×#137	1	2	yes	—
245	C₂×C₄	C₁₃×#137	1	4	yes	—
246	C₂×C₄	C₂²×D₄×C₁₃	1	1	yes	characteristic
247	C₂×C₄	C₂²×D₄×C₁₃	1	1	yes	characteristic
248	C₂×C₄	C₂×C₁₃×#41	1	1	yes	characteristic
249	C₈	—	2	2	no	—
250	C₈	—	4	2	no	—
251	C₈	—	4	4	no	—
252	C₈	C₂×C₁₃×Q₁₆	1	4	yes	—
253	C₂²	C₁₃×#462	1	2	yes	—
254	C₂²	C₁₃×#470	1	2	yes	—
255	C₂²	C₁₃×#472	1	2	yes	—
256	C₂²	C₂×C₁₃×#121	1	1	yes	characteristic
257	C₄	—	2	1	no	—
258	C₄	—	2	1	no	—
259	C₄	—	4	2	no	—
260	C₄	—	4	2	no	—
261	C₄	—	8	4	no	—
262	C₄	C₁₃×#462	1	2	yes	—
263	C₄	C₁₃×#470	1	2	yes	—
264	C₄	C₂²×C₁₃×Q₁₆	1	1	yes	characteristic
265	C₄	C₂×C₁₃×#137	1	1	yes	characteristic
266	C₄	C₂×C₁₃×#137	1	2	yes	—
267	C₂	C₁₃×#1696	1	4	yes	direct product factor
268	C₂	C₂×C₁₃×#462	1	1	yes	characteristic
269	C₂	C₂×C₁₃×#470	1	1	yes	characteristic
270	C₂	C₂×C₁₃×#472	1	1	yes	characteristic

	subgroup	G/H
center, socle, prime radical	C₂³×C₁₃	#121
commutator, Frattini, cosocle	C₄²	C₂⁴×C₁₃
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³×C₁₃ ◁ C₂×C₄²×C₁₃ ◁ G

Complements

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

N	H ≅ G/N	type
4. C₁₃×#1696	C₂	direct (N × H)
58. C₂×#1696	C₁₃	direct (N × H)
96. #1696	C₂×C₁₃	direct (N × H)
189. C₂×C₁₃	#1696	direct (N × H)
227. C₁₃	C₂×#1696	direct (N × H)
267. C₂	C₁₃×#1696	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂×C₁₃×#500	2
C₂×C₁₃×#477	2
C₂×C₁₃×#446	2
C₁₃×#1696	2
C₂×C₁₃×#334	2
C₂×C₁₃×#333	2
C₂×#1696	13

Subgroups — C2×C13×#1696 #15705488

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₁₃×#1696 #15705488