Subgroups — C₂×#13114744 #15861022

Order 3328.

1046 conjugacy classes of subgroups.

The subgroup lattice has 304 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×#88059	C₂	1	1	yes	maximal, characteristic
2	C₂×M₁₆×Dic₁₃	C₂	1	2	yes	maximal
3	C₂²×#88055	C₂	1	1	yes	maximal, characteristic
4	#13114744	C₂	1	8	yes	maximal, direct product factor
5	C₂×#11718150	C₂	1	2	yes	maximal
6	C₂²×C₁₃×#108	C₂	1	1	yes	maximal, characteristic
7	C₂²×C₁₃×M₁₆	C₂²	1	2	yes	—
8	C₂³×C₈×C₁₃	C₂²	1	1	yes	characteristic
9	C₂²×C₄×Dic₁₃	C₂²	1	1	yes	characteristic
10	C₂²×#6099	C₂²	1	2	yes	—
11	C₂³×#1055	C₂²	1	1	yes	characteristic
12	C₂×C₁₃×#108	C₄	1	8	yes	complemented
13	C₂×C₁₃×#108	C₂²	1	4	yes	—
14	M₁₆×Dic₁₃	—	2	4	no	—
15	M₁₆×Dic₁₃	C₂²	1	8	yes	—
16	C₂×#88059	C₄	1	8	yes	complemented
17	C₂×#88059	C₂²	1	4	yes	—
18	C₂×#88055	—	2	2	no	—
19	C₂×#88055	C₄	1	4	yes	—
20	C₂×#88055	C₄	1	4	yes	—
21	C₂×#88055	C₂²	1	4	yes	—
22	C₂×C₈×Dic₁₃	—	2	2	no	—
23	#11718150	C₂²	1	8	yes	—
24	C₂³×C₄×C₁₃	C₂³	1	1	yes	characteristic
25	C₂³×Dic₁₃	Q₈	1	2	yes	—
26	C₂×C₁₃×M₁₆	—	2	2	no	—
27	C₂×C₁₃×M₁₆	—	2	4	no	—
28	C₂×C₁₃×M₁₆	C₂×C₄	1	8	yes	—
29	C₂×C₁₃×M₁₆	C₂³	1	4	yes	—
30	C₂²×C₈×C₁₃	—	2	2	no	—
31	C₂²×C₈×C₁₃	C₂×C₄	1	2	yes	—
32	C₂²×C₈×C₁₃	C₂×C₄	1	2	yes	—
33	C₂²×C₈×C₁₃	C₂×C₄	1	4	yes	—
34	C₂²×C₈×C₁₃	C₂×C₄	1	4	yes	—
35	C₂²×C₈×C₁₃	C₂³	1	2	yes	—
36	C₂×C₄×Dic₁₃	C₂×C₄	1	2	yes	—
37	C₂×C₄×Dic₁₃	C₂×C₄	1	2	yes	—
38	C₂×C₄×Dic₁₃	D₄	1	2	yes	—
39	C₂×C₄×Dic₁₃	D₄	1	2	yes	—
40	C₂×C₄×Dic₁₃	Q₈	1	2	yes	—
41	C₂×C₄×Dic₁₃	C₂³	1	2	yes	—
42	C₂×#6099	—	2	2	no	—
43	C₂×#6099	—	2	4	no	—
44	C₂×#6099	C₂×C₄	1	8	yes	—
45	C₂×#6099	C₂³	1	4	yes	—
46	C₂²×#1055	—	2	2	no	—
47	C₂²×#1055	—	2	2	no	—
48	C₂²×#1055	—	2	2	no	—
49	C₂²×#1055	C₂×C₄	1	2	yes	—
50	C₂²×#1055	C₂×C₄	1	2	yes	—
51	C₂²×#1055	C₂³	1	2	yes	—
52	C₁₃×#108	C₂×C₄	1	16	yes	complemented
53	#88059	C₂×C₄	1	16	yes	complemented
54	#88055	—	2	8	no	—
55	#88055	C₂×C₄	1	16	yes	—
56	C₈×Dic₁₃	—	2	8	no	—
57	C₂×#1778	—	13	1	no	maximal, Sylow
58	C₂⁴×C₁₃	C₂×Q₈	1	1	yes	characteristic
59	C₂²×C₄×C₁₃	C₂²×C₄	1	1	yes	characteristic
60	C₂²×C₄×C₁₃	C₂²×C₄	1	1	yes	characteristic
61	C₂²×C₄×C₁₃	C₂²×C₄	1	2	yes	—
62	C₂²×C₄×C₁₃	C₂²×C₄	1	2	yes	—
63	C₂²×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
64	C₂²×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
65	C₂²×C₄×C₁₃	C₂×Q₈	1	1	yes	characteristic
66	C₂²×C₄×C₁₃	#41	1	2	yes	—
67	C₂²×C₄×C₁₃	#41	1	2	yes	—
68	C₂²×C₄×C₁₃	C₂⁴	1	1	yes	characteristic
69	C₂²×Dic₁₃	—	4	4	no	—
70	C₂²×Dic₁₃	C₄⋊C₄	1	4	yes	—
71	C₂²×Dic₁₃	C₄⋊C₄	1	4	yes	—
72	C₂²×Dic₁₃	C₂×Q₈	1	4	yes	—
73	C₁₃×M₁₆	—	2	8	no	—
74	C₁₃×M₁₆	—	2	4	no	—
75	C₁₃×M₁₆	C₂²×C₄	1	8	yes	—
76	C₂×C₈×C₁₃	—	2	8	no	—
77	C₂×C₈×C₁₃	—	2	4	no	—
78	C₂×C₈×C₁₃	C₄²	1	8	yes	—
79	C₂×C₈×C₁₃	C₄²	1	8	yes	—
80	C₂×C₈×C₁₃	C₂²×C₄	1	4	yes	—
81	C₂×C₈×C₁₃	C₂²×C₄	1	4	yes	—
82	C₂×C₈×C₁₃	C₂²×C₄	1	4	yes	—
83	C₄×Dic₁₃	C₄⋊C₄	1	4	yes	—
84	C₄×Dic₁₃	C₄⋊C₄	1	4	yes	—
85	C₄×Dic₁₃	C₂²×C₄	1	4	yes	—
86	C₄×Dic₁₃	C₂×D₄	1	4	yes	—
87	#6099	—	2	8	no	—
88	#6099	—	2	4	no	—
89	#6099	C₂²×C₄	1	8	yes	—
90	C₂×#1055	—	2	4	no	—
91	C₂×#1055	—	2	4	no	—
92	C₂×#1055	—	2	8	no	—
93	C₂×#1055	—	2	4	no	—
94	C₂×#1055	—	2	2	no	—
95	C₂×#1055	—	2	2	no	—
96	C₂×#1055	C₂²×C₄	1	4	yes	—
97	C₂²×#108	—	13	1	no	—
98	C₂²×#108	D₁₃	1	1	yes	characteristic
99	C₂×C₄×M₁₆	—	13	2	no	—
100	C₂²×#97	—	13	1	no	—
101	#1778	—	13	8	no	—
102	C₂×#341	—	13	2	no	—
103	C₂³×C₁₃	—	2	4	no	—
104	C₂³×C₁₃	C₂×C₄⋊C₄	1	1	yes	characteristic
105	C₂³×C₁₃	C₂×C₄⋊C₄	1	1	yes	characteristic
106	C₂³×C₁₃	C₄×Q₈	1	2	yes	—
107	C₂³×C₁₃	C₄×Q₈	1	2	yes	—
108	C₂³×C₁₃	C₂²×Q₈	1	1	yes	characteristic
109	C₂×C₄×C₁₃	C₂×C₄²	1	2	yes	—
110	C₂×C₄×C₁₃	C₂×C₄²	1	2	yes	—
111	C₂×C₄×C₁₃	C₂×C₄⋊C₄	1	1	yes	characteristic
112	C₂×C₄×C₁₃	C₂×C₄⋊C₄	1	1	yes	characteristic
113	C₂×C₄×C₁₃	#117	1	2	yes	—
114	C₂×C₄×C₁₃	#117	1	2	yes	—
115	C₂×C₄×C₁₃	C₄×D₄	1	2	yes	—
116	C₂×C₄×C₁₃	C₄×D₄	1	2	yes	—
117	C₂×C₄×C₁₃	C₄×D₄	1	2	yes	—
118	C₂×C₄×C₁₃	C₄×D₄	1	2	yes	—
119	C₂×C₄×C₁₃	C₄×Q₈	1	2	yes	—
120	C₂×C₄×C₁₃	C₄×Q₈	1	2	yes	—
121	C₂×C₄×C₁₃	C₂³×C₄	1	1	yes	characteristic
122	C₂×C₄×C₁₃	C₂³×C₄	1	1	yes	characteristic
123	C₂×C₄×C₁₃	C₂³×C₄	1	1	yes	characteristic
124	C₂×C₄×C₁₃	C₂²×D₄	1	1	yes	characteristic
125	C₂×C₄×C₁₃	C₂×#41	1	1	yes	characteristic
126	C₂×C₄×C₁₃	C₂×#41	1	1	yes	characteristic
127	C₂×Dic₁₃	—	2	8	no	—
128	C₂×Dic₁₃	—	4	4	no	—
129	C₂×Dic₁₃	—	4	4	no	—
130	C₂×Dic₁₃	C₂×C₄⋊C₄	1	8	yes	—
131	C₈×C₁₃	—	2	8	no	—
132	C₈×C₁₃	C₂×C₄²	1	8	yes	—
133	#1055	—	2	4	no	—
134	#1055	—	2	8	no	—
135	C₂²×M₁₆	—	13	2	no	—
136	C₂²×M₁₆	C₂×D₁₃	1	2	yes	—
137	C₂³×C₈	—	13	1	no	—
138	C₂³×C₈	C₂×D₁₃	1	1	yes	characteristic
139	C₂²×C₄²	—	13	1	no	—
140	C₂×#108	—	13	8	no	—
141	C₂×#108	—	13	4	no	—
142	C₂×#108	Dic₁₃	1	8	yes	complemented
143	C₂×#108	C₂×D₁₃	1	4	yes	—
144	C₄×M₁₆	—	13	8	no	—
145	C₄×M₁₆	—	26	4	no	—
146	C₂×#97	—	13	4	no	—
147	C₂×#97	—	13	4	no	—
148	C₂×#97	—	13	4	no	—
149	C₂×#97	—	26	2	no	—
150	C₂×C₄×C₈	—	26	2	no	—
151	#341	—	13	8	no	—
152	C₂²×C₁₃	—	2	2	no	—
153	C₂²×C₁₃	—	2	2	no	—
154	C₂²×C₁₃	—	2	4	no	—
155	C₂²×C₁₃	—	2	4	no	—
156	C₂²×C₁₃	C₄×C₄⋊C₄	1	2	yes	—
157	C₂²×C₁₃	C₄×C₄⋊C₄	1	2	yes	—
158	C₂²×C₁₃	#430	1	4	yes	—
159	C₂²×C₁₃	C₂²×C₄⋊C₄	1	1	yes	characteristic
160	C₂²×C₁₃	C₂×C₄×Q₈	1	1	yes	characteristic
161	C₂²×C₁₃	C₂×C₄×Q₈	1	1	yes	characteristic
162	C₄×C₁₃	C₄×C₄⋊C₄	1	2	yes	—
163	C₄×C₁₃	C₄×C₄⋊C₄	1	2	yes	—
164	C₄×C₁₃	C₂²×C₄²	1	1	yes	characteristic
165	C₄×C₁₃	C₂×#117	1	1	yes	characteristic
166	C₄×C₁₃	C₂×C₄×D₄	1	1	yes	characteristic
167	C₄×C₁₃	C₂×C₄×D₄	1	1	yes	characteristic
168	Dic₁₃	—	2	8	no	—
169	C₂³×C₄	—	13	2	no	—
170	C₂³×C₄	C₂²×D₁₃	1	1	yes	characteristic
171	C₂×M₁₆	—	2	2	no	—
172	C₂×M₁₆	—	2	4	no	—
173	C₂×M₁₆	—	13	8	no	—
174	C₂×M₁₆	—	13	4	no	—
175	C₂×M₁₆	—	26	2	no	—
176	C₂×M₁₆	—	26	4	no	—
177	C₂×M₁₆	C₂×Dic₁₃	1	8	yes	—
178	C₂×M₁₆	C₂²×D₁₃	1	4	yes	—
179	C₂²×C₈	—	2	2	no	—
180	C₂²×C₈	—	13	2	no	—
181	C₂²×C₈	—	13	2	no	—
182	C₂²×C₈	—	13	2	no	—
183	C₂²×C₈	—	26	2	no	—
184	C₂²×C₈	—	26	2	no	—
185	C₂²×C₈	—	26	2	no	—
186	C₂²×C₈	C₄×D₁₃	1	2	yes	—
187	C₂²×C₈	C₄×D₁₃	1	2	yes	—
188	C₂²×C₈	C₄×D₁₃	1	4	yes	—
189	C₂²×C₈	C₂×Dic₁₃	1	4	yes	—
190	C₂²×C₈	C₂²×D₁₃	1	2	yes	—
191	C₂×C₄²	—	13	2	no	—
192	C₂×C₄²	—	13	2	no	—
193	C₂×C₄²	—	13	2	no	—
194	C₂×C₄²	—	13	2	no	—
195	C₂×C₄²	—	13	2	no	—
196	C₂×C₄²	—	13	2	no	—
197	#108	—	13	16	no	—
198	#108	C₂×Dic₁₃	1	16	yes	complemented
199	#97	—	13	16	no	—
200	#97	—	26	8	no	—
201	C₄×C₈	—	26	8	no	—
202	C₂×C₁₃	—	2	1	no	—
203	C₂×C₁₃	—	2	1	no	—
204	C₂×C₁₃	—	2	2	no	—
205	C₂×C₁₃	#1778	1	4	yes	complemented
206	C₂×C₁₃	C₂×C₄×C₄⋊C₄	1	1	yes	characteristic
207	C₂×C₁₃	C₂×#430	1	2	yes	—
208	C₂⁴	Q₈×D₁₃	1	1	yes	characteristic
209	C₂²×C₄	—	13	4	no	—
210	C₂²×C₄	—	13	4	no	—
211	C₂²×C₄	—	13	4	no	—
212	C₂²×C₄	—	52	4	no	—
213	C₂²×C₄	C₂×C₄×D₁₃	1	2	yes	—
214	C₂²×C₄	C₂×C₄×D₁₃	1	1	yes	characteristic
215	C₂²×C₄	C₂×C₄×D₁₃	1	1	yes	characteristic
216	C₂²×C₄	D₄×D₁₃	1	1	yes	characteristic
217	C₂²×C₄	D₄×D₁₃	1	1	yes	characteristic
218	C₂²×C₄	#6129	1	2	yes	—
219	C₂²×C₄	Q₈×D₁₃	1	1	yes	characteristic
220	C₂²×C₄	#6131	1	2	yes	—
221	C₂²×C₄	C₂²×Dic₁₃	1	2	yes	—
222	C₂²×C₄	C₂³×D₁₃	1	1	yes	characteristic
223	M₁₆	—	2	8	no	—
224	M₁₆	—	2	4	no	—
225	M₁₆	—	13	8	no	—
226	M₁₆	—	26	8	no	—
227	M₁₆	—	26	4	no	—
228	M₁₆	C₂²×Dic₁₃	1	8	yes	—
229	C₂×C₈	—	2	8	no	—
230	C₂×C₈	—	2	4	no	—
231	C₂×C₈	—	13	4	no	—
232	C₂×C₈	—	26	4	no	—
233	C₂×C₈	—	26	4	no	—
234	C₂×C₈	—	26	8	no	—
235	C₂×C₈	—	26	4	no	—
236	C₂×C₈	—	26	2	no	—
237	C₂×C₈	—	26	2	no	—
238	C₂×C₈	C₄×Dic₁₃	1	8	yes	—
239	C₂×C₈	C₄×Dic₁₃	1	8	yes	—
240	C₂×C₈	C₂×C₄×D₁₃	1	4	yes	—
241	C₂×C₈	C₂×C₄×D₁₃	1	4	yes	—
242	C₂×C₈	C₂²×Dic₁₃	1	4	yes	—
243	C₄²	—	13	4	no	—
244	C₄²	—	13	4	no	—
245	C₄²	—	13	4	no	—
246	C₄²	—	13	4	no	—
247	C₁₃	C₂×#1778	1	1	yes	characteristic, complemented, Sylow
248	C₂³	—	2	4	no	—
249	C₂³	#88141	1	2	yes	—
250	C₂³	C₄⋊C₄×D₁₃	1	1	yes	characteristic
251	C₂³	C₄⋊C₄×D₁₃	1	1	yes	characteristic
252	C₂³	Q₈×Dic₁₃	1	2	yes	—
253	C₂³	C₂×Q₈×D₁₃	1	1	yes	characteristic
254	C₂×C₄	—	13	8	no	—
255	C₂×C₄	—	26	8	no	—
256	C₂×C₄	—	52	4	no	—
257	C₂×C₄	—	52	4	no	—
258	C₂×C₄	#88141	1	2	yes	—
259	C₂×C₄	C₄⋊C₄×D₁₃	1	1	yes	characteristic
260	C₂×C₄	C₄⋊C₄×D₁₃	1	1	yes	characteristic
261	C₂×C₄	#88146	1	2	yes	—
262	C₂×C₄	#88146	1	2	yes	—
263	C₂×C₄	#88147	1	2	yes	—
264	C₂×C₄	#88147	1	2	yes	—
265	C₂×C₄	C₂×C₄×Dic₁₃	1	2	yes	—
266	C₂×C₄	C₂×C₄×Dic₁₃	1	2	yes	—
267	C₂×C₄	D₄×Dic₁₃	1	2	yes	—
268	C₂×C₄	D₄×Dic₁₃	1	2	yes	—
269	C₂×C₄	Q₈×Dic₁₃	1	2	yes	—
270	C₂×C₄	C₂²×C₄×D₁₃	1	1	yes	characteristic
271	C₂×C₄	C₂²×C₄×D₁₃	1	1	yes	characteristic
272	C₂×C₄	C₂×D₄×D₁₃	1	1	yes	characteristic
273	C₂×C₄	C₂×#6129	1	1	yes	characteristic
274	C₂×C₄	C₂×#6131	1	1	yes	characteristic
275	C₂×C₄	C₂³×Dic₁₃	1	1	yes	characteristic
276	C₈	—	2	8	no	—
277	C₈	—	26	4	no	—
278	C₈	—	26	8	no	—
279	C₈	C₂×C₄×Dic₁₃	1	8	yes	—
280	C₂²	—	2	2	no	—
281	C₂²	—	2	2	no	—
282	C₂²	—	2	4	no	—
283	C₂²	—	2	4	no	—
284	C₂²	#11718554	1	4	yes	—
285	C₂²	C₄⋊C₄×Dic₁₃	1	2	yes	—
286	C₂²	C₄⋊C₄×Dic₁₃	1	2	yes	—
287	C₂²	C₂×#88141	1	1	yes	characteristic
288	C₂²	C₂×C₄⋊C₄×D₁₃	1	1	yes	characteristic
289	C₂²	C₂×Q₈×Dic₁₃	1	1	yes	characteristic
290	C₄	—	26	8	no	—
291	C₄	C₄⋊C₄×Dic₁₃	1	2	yes	—
292	C₄	C₄⋊C₄×Dic₁₃	1	2	yes	—
293	C₄	C₂×#88146	1	1	yes	characteristic
294	C₄	C₂×#88147	1	1	yes	characteristic
295	C₄	C₂²×C₄×Dic₁₃	1	1	yes	characteristic
296	C₄	C₂×D₄×Dic₁₃	1	1	yes	characteristic
297	C₂	—	2	1	no	—
298	C₂	—	2	1	no	—
299	C₂	—	2	2	no	—
300	C₂	#13114744	1	4	yes	direct product factor
301	C₂	C₂×C₄⋊C₄×Dic₁₃	1	1	yes	characteristic
302	C₂	C₂×#11718554	1	2	yes	—

	subgroup	G/H
center	C₂²×C₄	D₄×D₁₃
commutator	C₄×C₁₃	C₂²×C₄²
Frattini	C₂²×C₄	C₂³×D₁₃
Fitting	C₂²×C₁₃×#108	C₂
socle	C₂³×C₁₃	C₂×C₄⋊C₄
prime radical	C₂⁴×C₁₃	C₂×Q₈
cosocle	C₂²×C₄×C₁₃	C₂⁴
hypercenter	C₂²×#108	D₁₃
hypocenter	C₁₃	C₂×#1778
LCS	G ▷ C₄×C₁₃ ▷ C₂×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂²×C₄ ◁ C₂³×C₄ ◁ C₂²×#108

Complements

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

N	H ≅ G/N	type
4. #13114744	C₂	direct (N × H)
12. C₂×C₁₃×#108	C₄	semidirect
16. C₂×#88059	C₄	semidirect
52. C₁₃×#108	C₂×C₄	semidirect
53. #88059	C₂×C₄	semidirect
142. C₂×#108	Dic₁₃	semidirect
198. #108	C₂×Dic₁₃	semidirect
205. C₂×C₁₃	#1778	semidirect
247. C₁₃	C₂×#1778	semidirect
300. C₂	#13114744	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×#88059	2
C₂×M₁₆×Dic₁₃	2
C₂²×#88055	2
#13114744	2
C₂×#11718150	2
C₂²×C₁₃×#108	2
C₂×#1778	13

Subgroups — C2×#13114744 #15861022

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#13114744 #15861022