Subgroups — C₂×#13116630 #16015974

Order 3328.

642 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×#88052	C₂	1	1	yes	maximal, characteristic
2	C₂²×#88059	C₂	1	1	yes	maximal, characteristic
3	#13116630	C₂	1	8	yes	maximal, direct product factor
4	C₂×#11718085	C₂	1	4	yes	maximal, complemented
5	C₂²×C₁₃×#108	C₂	1	1	yes	maximal, characteristic
6	C₂²×C₁₃×M₁₆	—	2	1	no	—
7	C₂³×C₈×C₁₃	C₂²	1	1	yes	characteristic
8	C₂²×#6099	—	2	1	no	—
9	C₂×C₁₃×#108	—	2	2	no	—
10	C₂×C₁₃×#108	—	2	1	no	—
11	C₂×C₁₃×#108	C₂²	1	2	yes	—
12	C₂×C₁₃×#108	C₂²	1	4	yes	—
13	C₂×#88059	—	2	1	no	—
14	C₂×#88059	—	2	2	no	—
15	C₂×#88059	C₂²	1	2	yes	—
16	C₂×#88059	C₂²	1	4	yes	—
17	C₂×#88052	C₄	1	2	yes	—
18	C₂×#88052	C₄	1	2	yes	—
19	C₂×#88052	C₄	1	4	yes	—
20	C₂×#88052	C₂²	1	4	yes	—
21	C₂²×#6090	C₂²	1	2	yes	—
22	#11718085	C₂²	1	16	yes	complemented
23	C₂³×C₄×C₁₃	D₄	1	1	yes	characteristic
24	C₂×C₁₃×M₁₆	—	2	2	no	—
25	C₂×C₁₃×M₁₆	—	2	4	no	—
26	C₂×C₁₃×M₁₆	—	4	1	no	—
27	C₂×C₁₃×M₁₆	—	4	2	no	—
28	C₂²×C₈×C₁₃	—	4	1	no	—
29	C₂²×C₈×C₁₃	C₂×C₄	1	1	yes	characteristic
30	C₂²×C₈×C₁₃	C₂×C₄	1	1	yes	characteristic
31	C₂²×C₈×C₁₃	C₂×C₄	1	2	yes	—
32	C₂²×C₈×C₁₃	D₄	1	2	yes	—
33	C₂²×C₈×C₁₃	D₄	1	1	yes	characteristic
34	C₂²×C₈×C₁₃	Q₈	1	1	yes	characteristic
35	C₂²×C₈×C₁₃	Q₈	1	1	yes	characteristic
36	C₂²×C₈×C₁₃	Q₈	1	2	yes	—
37	C₂²×C₈×C₁₃	C₂³	1	1	yes	characteristic
38	C₂²×C₈×C₁₃	C₂³	1	2	yes	—
39	C₂×#6099	—	2	2	no	—
40	C₂×#6099	—	2	4	no	—
41	C₂×#6099	—	4	1	no	—
42	C₂×#6099	—	4	2	no	—
43	C₂²×#1055	—	4	1	no	—
44	C₁₃×#108	—	2	2	no	—
45	C₁₃×#108	—	2	4	no	—
46	C₁₃×#108	C₂³	1	4	yes	—
47	#88059	—	2	2	no	—
48	#88059	—	2	4	no	—
49	#88059	C₂³	1	4	yes	—
50	#88052	C₂×C₄	1	8	yes	—
51	#88052	C₂×C₄	1	8	yes	—
52	C₂×#6090	C₂×C₄	1	4	yes	—
53	C₂×#6090	C₂×C₄	1	4	yes	—
54	C₂×#6090	C₂³	1	4	yes	—
55	C₂×#2161	—	13	1	no	maximal, Sylow
56	C₂⁴×C₁₃	Q₁₆	1	1	yes	characteristic
57	C₂²×C₄×C₁₃	C₄⋊C₄	1	1	yes	characteristic
58	C₂²×C₄×C₁₃	C₄⋊C₄	1	1	yes	characteristic
59	C₂²×C₄×C₁₃	C₄⋊C₄	1	2	yes	—
60	C₂²×C₄×C₁₃	D₈	1	1	yes	characteristic
61	C₂²×C₄×C₁₃	D₈	1	1	yes	characteristic
62	C₂²×C₄×C₁₃	D₈	1	2	yes	—
63	C₂²×C₄×C₁₃	Q₁₆	1	1	yes	characteristic
64	C₂²×C₄×C₁₃	Q₁₆	1	2	yes	—
65	C₂²×C₄×C₁₃	C₂×D₄	1	2	yes	—
66	C₂²×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
67	C₁₃×M₁₆	—	2	4	no	—
68	C₁₃×M₁₆	—	4	4	no	—
69	C₁₃×M₁₆	—	4	2	no	—
70	C₂×C₈×C₁₃	—	4	4	no	—
71	C₂×C₈×C₁₃	—	4	2	no	—
72	C₂×C₈×C₁₃	C₄⋊C₄	1	2	yes	—
73	C₂×C₈×C₁₃	C₄⋊C₄	1	2	yes	—
74	C₂×C₈×C₁₃	C₄⋊C₄	1	2	yes	—
75	C₂×C₈×C₁₃	C₄⋊C₄	1	2	yes	—
76	C₂×C₈×C₁₃	C₄⋊C₄	1	2	yes	—
77	C₂×C₈×C₁₃	C₄⋊C₄	1	1	yes	characteristic
78	C₂×C₈×C₁₃	C₄⋊C₄	1	1	yes	characteristic
79	C₂×C₈×C₁₃	C₂²×C₄	1	1	yes	characteristic
80	C₂×C₈×C₁₃	C₂²×C₄	1	1	yes	characteristic
81	C₂×C₈×C₁₃	C₂²×C₄	1	2	yes	—
82	C₂×C₈×C₁₃	C₂²×C₄	1	2	yes	—
83	C₂×C₈×C₁₃	C₂×D₄	1	1	yes	characteristic
84	C₂×C₈×C₁₃	C₂×D₄	1	2	yes	—
85	C₂×C₈×C₁₃	C₂×Q₈	1	1	yes	characteristic
86	C₂×C₈×C₁₃	C₂×Q₈	1	1	yes	characteristic
87	C₂×C₈×C₁₃	C₂×Q₈	1	2	yes	—
88	C₂×C₈×C₁₃	C₂×Q₈	1	2	yes	—
89	C₂×C₈×C₁₃	C₂⁴	1	1	yes	characteristic
90	#6099	—	2	4	no	—
91	#6099	—	4	4	no	—
92	#6099	—	4	2	no	—
93	C₂×#1055	—	4	4	no	—
94	C₂×#1055	—	4	2	no	—
95	#6090	C₂²×C₄	1	8	yes	—
96	C₂²×M₃₂	—	13	1	no	—
97	C₂²×#108	—	13	1	no	—
98	C₂²×#108	D₁₃	1	1	yes	characteristic
99	#2161	—	13	8	no	—
100	C₂×#368	—	13	4	no	—
101	C₂³×C₁₃	—	4	2	no	—
102	C₂³×C₁₃	#107	1	1	yes	characteristic
103	C₂³×C₁₃	#107	1	1	yes	characteristic
104	C₂³×C₁₃	#107	1	2	yes	—
105	C₂³×C₁₃	C₂×Q₁₆	1	1	yes	characteristic
106	C₂³×C₁₃	C₂×Q₁₆	1	2	yes	—
107	C₂×C₄×C₁₃	#107	1	2	yes	—
108	C₂×C₄×C₁₃	#107	1	2	yes	—
109	C₂×C₄×C₁₃	#107	1	2	yes	—
110	C₂×C₄×C₁₃	#107	1	2	yes	—
111	C₂×C₄×C₁₃	#107	1	2	yes	—
112	C₂×C₄×C₁₃	#107	1	1	yes	characteristic
113	C₂×C₄×C₁₃	#107	1	1	yes	characteristic
114	C₂×C₄×C₁₃	C₂×C₄⋊C₄	1	1	yes	characteristic
115	C₂×C₄×C₁₃	C₂×C₄⋊C₄	1	2	yes	—
116	C₂×C₄×C₁₃	C₂×C₄⋊C₄	1	2	yes	—
117	C₂×C₄×C₁₃	C₂×C₄⋊C₄	1	1	yes	characteristic
118	C₂×C₄×C₁₃	C₂×D₈	1	2	yes	—
119	C₂×C₄×C₁₃	C₂×D₈	1	2	yes	—
120	C₂×C₄×C₁₃	C₂×D₈	1	1	yes	characteristic
121	C₂×C₄×C₁₃	C₂×D₈	1	1	yes	characteristic
122	C₂×C₄×C₁₃	C₂×Q₁₆	1	2	yes	—
123	C₂×C₄×C₁₃	C₂×Q₁₆	1	1	yes	characteristic
124	C₂×C₄×C₁₃	C₂²×D₄	1	1	yes	characteristic
125	C₈×C₁₃	—	4	4	no	—
126	C₈×C₁₃	C₂×C₄⋊C₄	1	1	yes	characteristic
127	C₈×C₁₃	C₂×C₄⋊C₄	1	1	yes	characteristic
128	C₈×C₁₃	C₂×C₄⋊C₄	1	2	yes	—
129	C₈×C₁₃	C₂×C₄⋊C₄	1	2	yes	—
130	C₈×C₁₃	C₂³×C₄	1	1	yes	characteristic
131	C₈×C₁₃	C₂²×Q₈	1	1	yes	characteristic
132	#1055	—	4	4	no	—
133	C₂²×M₁₆	—	2	1	no	—
134	C₂²×M₁₆	—	26	1	no	—
135	C₂³×C₈	C₂×D₁₃	1	1	yes	characteristic
136	C₂×M₃₂	—	13	2	no	—
137	C₂×M₃₂	—	13	2	no	—
138	C₂×M₃₂	—	13	4	no	—
139	C₂×M₃₂	—	13	4	no	—
140	C₂²×C₁₆	—	13	2	no	—
141	C₂×#108	—	2	2	no	—
142	C₂×#108	—	2	1	no	—
143	C₂×#108	—	13	2	no	—
144	C₂×#108	—	13	4	no	—
145	C₂×#108	—	26	2	no	—
146	C₂×#108	—	26	1	no	—
147	C₂×#108	C₂×D₁₃	1	2	yes	—
148	C₂×#108	C₂×D₁₃	1	4	yes	—
149	#368	—	13	16	no	—
150	C₂²×C₁₃	—	2	2	no	—
151	C₂²×C₁₃	—	2	2	no	—
152	C₂²×C₁₃	—	2	2	no	—
153	C₂²×C₁₃	—	4	2	no	—
154	C₂²×C₁₃	—	4	2	no	—
155	C₂²×C₁₃	C₂×#107	1	2	yes	—
156	C₂²×C₁₃	C₂×#107	1	2	yes	—
157	C₂²×C₁₃	C₂×#107	1	1	yes	characteristic
158	C₂²×C₁₃	C₂×#107	1	1	yes	characteristic
159	C₂²×C₁₃	C₂²×Q₁₆	1	1	yes	characteristic
160	C₄×C₁₃	C₂×#107	1	2	yes	—
161	C₄×C₁₃	C₂×#107	1	2	yes	—
162	C₄×C₁₃	C₂×#107	1	1	yes	characteristic
163	C₄×C₁₃	C₂×#107	1	1	yes	characteristic
164	C₄×C₁₃	C₂²×C₄⋊C₄	1	1	yes	characteristic
165	C₄×C₁₃	C₂²×D₈	1	1	yes	characteristic
166	C₂³×C₄	C₂²⋊D₁₃	1	1	yes	characteristic
167	C₂×M₁₆	—	2	4	no	—
168	C₂×M₁₆	—	2	2	no	—
169	C₂×M₁₆	—	4	2	no	—
170	C₂×M₁₆	—	4	1	no	—
171	C₂×M₁₆	—	26	4	no	—
172	C₂×M₁₆	—	26	2	no	—
173	C₂×M₁₆	—	52	2	no	—
174	C₂×M₁₆	—	52	1	no	—
175	C₂²×C₈	—	4	1	no	—
176	C₂²×C₈	—	52	1	no	—
177	C₂²×C₈	Dic₂₆	1	1	yes	characteristic
178	C₂²×C₈	Dic₂₆	1	1	yes	characteristic
179	C₂²×C₈	Dic₂₆	1	2	yes	—
180	C₂²×C₈	C₄×D₁₃	1	1	yes	characteristic
181	C₂²×C₈	C₄×D₁₃	1	1	yes	characteristic
182	C₂²×C₈	C₄×D₁₃	1	2	yes	—
183	C₂²×C₈	C₂²⋊D₁₃	1	2	yes	—
184	C₂²×C₈	C₂²⋊D₁₃	1	1	yes	characteristic
185	C₂²×C₈	C₂²×D₁₃	1	1	yes	characteristic
186	C₂²×C₈	C₂²×D₁₃	1	2	yes	—
187	M₃₂	—	13	8	no	—
188	M₃₂	—	13	8	no	—
189	C₂×C₁₆	—	13	4	no	—
190	C₂×C₁₆	—	13	4	no	—
191	C₂×C₁₆	—	13	4	no	—
192	#108	—	2	2	no	—
193	#108	—	2	4	no	—
194	#108	—	13	4	no	—
195	#108	—	26	2	no	—
196	#108	—	26	4	no	—
197	#108	C₂²×D₁₃	1	4	yes	—
198	C₂×C₁₃	—	2	1	no	—
199	C₂×C₁₃	—	2	2	no	—
200	C₂×C₁₃	—	2	2	no	—
201	C₂×C₁₃	—	2	1	no	—
202	C₂×C₁₃	#2161	1	2	yes	complemented
203	C₂×C₁₃	C₂²×#107	1	1	yes	characteristic
204	C₂⁴	#6107	1	1	yes	characteristic
205	C₂²×C₄	#6101	1	1	yes	characteristic
206	C₂²×C₄	#6101	1	1	yes	characteristic
207	C₂²×C₄	#6101	1	2	yes	—
208	C₂²×C₄	#6104	1	2	yes	—
209	C₂²×C₄	#6104	1	1	yes	characteristic
210	C₂²×C₄	#6104	1	1	yes	characteristic
211	C₂²×C₄	#6107	1	1	yes	characteristic
212	C₂²×C₄	#6107	1	2	yes	—
213	C₂²×C₄	C₂×C₂²⋊D₁₃	1	2	yes	—
214	C₂²×C₄	C₂×C₂²⋊D₁₃	1	1	yes	characteristic
215	M₁₆	—	2	4	no	—
216	M₁₆	—	4	4	no	—
217	M₁₆	—	4	2	no	—
218	M₁₆	—	26	4	no	—
219	M₁₆	—	52	4	no	—
220	M₁₆	—	52	2	no	—
221	C₂×C₈	—	4	4	no	—
222	C₂×C₈	—	4	2	no	—
223	C₂×C₈	—	52	4	no	—
224	C₂×C₈	—	52	2	no	—
225	C₂×C₈	#6101	1	2	yes	—
226	C₂×C₈	#6101	1	2	yes	—
227	C₂×C₈	#6101	1	2	yes	—
228	C₂×C₈	#6101	1	2	yes	—
229	C₂×C₈	#6101	1	1	yes	characteristic
230	C₂×C₈	#6101	1	1	yes	characteristic
231	C₂×C₈	#6101	1	2	yes	—
232	C₂×C₈	C₂×Dic₂₆	1	1	yes	characteristic
233	C₂×C₈	C₂×Dic₂₆	1	1	yes	characteristic
234	C₂×C₈	C₂×Dic₂₆	1	2	yes	—
235	C₂×C₈	C₂×Dic₂₆	1	2	yes	—
236	C₂×C₈	C₂×C₄×D₁₃	1	1	yes	characteristic
237	C₂×C₈	C₂×C₄×D₁₃	1	1	yes	characteristic
238	C₂×C₈	C₂×C₄×D₁₃	1	2	yes	—
239	C₂×C₈	C₂×C₄×D₁₃	1	2	yes	—
240	C₂×C₈	C₂×C₂²⋊D₁₃	1	1	yes	characteristic
241	C₂×C₈	C₂×C₂²⋊D₁₃	1	2	yes	—
242	C₂×C₈	C₂³×D₁₃	1	1	yes	characteristic
243	C₁₆	—	13	8	no	—
244	C₁₃	C₂×#2161	1	1	yes	characteristic, complemented, Sylow
245	C₂³	—	4	2	no	—
246	C₂³	#88047	1	1	yes	characteristic
247	C₂³	#88047	1	1	yes	characteristic
248	C₂³	#88047	1	2	yes	—
249	C₂³	C₂×#6107	1	1	yes	characteristic
250	C₂³	C₂×#6107	1	2	yes	—
251	C₂×C₄	#88047	1	2	yes	—
252	C₂×C₄	#88047	1	2	yes	—
253	C₂×C₄	#88047	1	2	yes	—
254	C₂×C₄	#88047	1	1	yes	characteristic
255	C₂×C₄	#88047	1	1	yes	characteristic
256	C₂×C₄	#88047	1	2	yes	—
257	C₂×C₄	#88047	1	2	yes	—
258	C₂×C₄	C₂×#6101	1	2	yes	—
259	C₂×C₄	C₂×#6101	1	2	yes	—
260	C₂×C₄	C₂×#6101	1	1	yes	characteristic
261	C₂×C₄	C₂×#6101	1	1	yes	characteristic
262	C₂×C₄	C₂×#6104	1	2	yes	—
263	C₂×C₄	C₂×#6104	1	2	yes	—
264	C₂×C₄	C₂×#6104	1	1	yes	characteristic
265	C₂×C₄	C₂×#6104	1	1	yes	characteristic
266	C₂×C₄	C₂×#6107	1	2	yes	—
267	C₂×C₄	C₂×#6107	1	1	yes	characteristic
268	C₂×C₄	C₂²×C₂²⋊D₁₃	1	1	yes	characteristic
269	C₈	—	4	4	no	—
270	C₈	—	52	4	no	—
271	C₈	C₂×#6101	1	1	yes	characteristic
272	C₈	C₂×#6101	1	1	yes	characteristic
273	C₈	C₂×#6101	1	2	yes	—
274	C₈	C₂×#6101	1	2	yes	—
275	C₈	C₂²×Dic₂₆	1	1	yes	characteristic
276	C₈	C₂²×C₄×D₁₃	1	1	yes	characteristic
277	C₂²	—	2	2	no	—
278	C₂²	—	2	2	no	—
279	C₂²	—	2	2	no	—
280	C₂²	—	4	2	no	—
281	C₂²	—	4	2	no	—
282	C₂²	C₂×#88047	1	2	yes	—
283	C₂²	C₂×#88047	1	2	yes	—
284	C₂²	C₂×#88047	1	1	yes	characteristic
285	C₂²	C₂×#88047	1	1	yes	characteristic
286	C₂²	C₂²×#6107	1	1	yes	characteristic
287	C₄	C₂×#88047	1	2	yes	—
288	C₄	C₂×#88047	1	2	yes	—
289	C₄	C₂×#88047	1	1	yes	characteristic
290	C₄	C₂×#88047	1	1	yes	characteristic
291	C₄	C₂²×#6101	1	1	yes	characteristic
292	C₄	C₂²×#6104	1	1	yes	characteristic
293	C₂	—	2	1	no	—
294	C₂	—	2	1	no	—
295	C₂	—	2	2	no	—
296	C₂	—	2	2	no	—
297	C₂	#13116630	1	2	yes	direct product factor
298	C₂	C₂²×#88047	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₂×#6104
commutator	C₈×C₁₃	C₂³×C₄
Frattini	C₂×C₈	C₂³×D₁₃
Fitting	C₂²×C₁₃×#108	C₂
socle	C₂²×C₁₃	C₂×#107
prime radical	C₂⁴×C₁₃	Q₁₆
cosocle	C₂×C₈×C₁₃	C₂⁴
hypercenter	C₂²×#108	D₁₃
hypocenter	C₁₃	C₂×#2161
LCS	G ▷ C₈×C₁₃ ▷ C₄×C₁₃ ▷ C₂×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₄ ◁ C₂³×C₈ ◁ C₂²×#108

Complements

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

N	H ≅ G/N	type
3. #13116630	C₂	direct (N × H)
4. C₂×#11718085	C₂	semidirect
22. #11718085	C₂²	semidirect
202. C₂×C₁₃	#2161	semidirect
244. C₁₃	C₂×#2161	semidirect
297. C₂	#13116630	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×#88052	2
C₂²×#88059	2
#13116630	2
C₂×#11718085	2
C₂²×C₁₃×#108	2
C₂×#2161	13

Subgroups — C2×#13116630 #16015974

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#13116630 #16015974