Subgroups — C₂×#13116898 #16017175

Order 3328.

722 conjugacy classes of subgroups.

The subgroup lattice has 340 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×#88069	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂²×#88068	C₂	1	1	yes	maximal, characteristic
3	C₂×#11718357	C₂	1	1	yes	maximal, characteristic
4	#13116898	C₂	1	8	yes	maximal, direct product factor
5	C₂×C₄⋊#6090	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×#11718084	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂×#11718083	C₂	1	1	yes	maximal, characteristic
8	C₂×C₄×C₁₃×Q₁₆	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂²×C₁₃×Q₁₆	C₂²	1	1	yes	characteristic
10	C₂×C₄×Q₈×C₁₃	—	2	1	no	—
11	C₂²×Dic₅₂	C₂²	1	1	yes	characteristic
12	C₂²×D₁₀₄	C₂²	1	1	yes	characteristic
13	C₂²×#6095	—	2	1	no	—
14	C₂×#88129	—	2	1	no	—
15	C₄×C₁₃×Q₁₆	—	2	2	no	—
16	C₄×C₁₃×Q₁₆	C₂²	1	4	yes	complemented
17	C₂×C₁₃×#107	C₂²	1	1	yes	characteristic
18	C₂×C₁₃×#103	—	2	1	no	—
19	C₂×C₄×C₈×C₁₃	C₂²	1	1	yes	characteristic
20	C₂×#88069	—	2	4	no	—
21	C₂×#88069	C₂²	1	4	yes	complemented
22	C₂×#88068	—	2	4	no	—
23	C₂×#88068	C₂²	1	4	yes	—
24	C₂²×#6090	C₂²	1	1	yes	characteristic
25	C₂²×#6090	C₂²	1	1	yes	characteristic
26	#11718357	—	2	2	no	—
27	#11718357	C₂²	1	4	yes	—
28	#11718084	C₂²	1	4	yes	complemented
29	#11718083	C₂²	1	4	yes	—
30	C₄⋊#6090	C₂²	1	4	yes	complemented
31	C₂²×Q₈×C₁₃	—	2	1	no	—
32	C₂²×D₅₂	—	2	1	no	—
33	C₂²×Dic₂₆	—	2	1	no	—
34	C₂×C₁₃×Q₁₆	—	2	1	no	—
35	C₂×C₁₃×Q₁₆	—	2	2	no	—
36	C₂×C₁₃×Q₁₆	D₄	1	4	yes	—
37	C₂×C₁₃×Q₁₆	C₂³	1	2	yes	—
38	C₂²×C₈×C₁₃	D₄	1	1	yes	characteristic
39	C₂²×C₈×C₁₃	C₂³	1	1	yes	characteristic
40	C₄×Q₈×C₁₃	—	2	4	no	—
41	C₄×Q₈×C₁₃	—	4	2	no	—
42	C₂×C₁₃×C₄⋊C₄	—	2	1	no	—
43	C₂×C₁₃×C₄⋊C₄	—	4	1	no	—
44	C₂×C₄²×C₁₃	—	4	1	no	—
45	C₂×C₄²×C₁₃	D₄	1	1	yes	characteristic
46	C₂×#6103	—	4	1	no	—
47	C₂×#6103	—	4	1	no	—
48	C₂×Dic₅₂	—	2	2	no	—
49	C₂×Dic₅₂	—	2	2	no	—
50	C₂×Dic₅₂	—	2	1	no	—
51	C₂×Dic₅₂	C₂³	1	2	yes	—
52	C₂×D₁₀₄	—	2	2	no	—
53	C₂×D₁₀₄	—	2	2	no	—
54	C₂×D₁₀₄	—	2	1	no	—
55	C₂×D₁₀₄	C₂³	1	2	yes	—
56	C₂×#6095	—	2	4	no	—
57	C₂×#6095	—	4	4	no	—
58	#88129	—	2	4	no	—
59	#88129	—	4	2	no	—
60	C₁₃×#107	—	2	1	no	—
61	C₁₃×#107	C₂³	1	2	yes	—
62	C₁₃×#103	—	2	4	no	—
63	C₄×C₈×C₁₃	D₄	1	2	yes	—
64	C₄×C₈×C₁₃	C₂³	1	2	yes	—
65	#88069	—	2	8	no	—
66	#88068	—	2	8	no	—
67	C₂×#6090	—	2	2	no	—
68	C₂×#6090	—	2	2	no	—
69	C₂×#6090	C₂³	1	2	yes	—
70	C₂×#6090	C₂³	1	2	yes	—
71	C₂×#2211	—	13	1	no	maximal, Sylow
72	C₂³×D₁₃	—	4	1	no	—
73	C₂×Q₈×C₁₃	—	2	4	no	—
74	C₂×Q₈×C₁₃	—	2	2	no	—
75	C₂×Q₈×C₁₃	—	4	1	no	—
76	C₂×Q₈×C₁₃	—	4	2	no	—
77	C₂²×C₄×C₁₃	—	4	1	no	—
78	C₂²×C₄×C₁₃	—	4	1	no	—
79	C₂²×C₄×C₁₃	D₈	1	1	yes	characteristic
80	C₂²×C₄×C₁₃	D₈	1	1	yes	characteristic
81	C₂²×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
82	C₂²×Dic₁₃	—	4	1	no	—
83	C₂×D₅₂	—	2	2	no	—
84	C₂×D₅₂	—	4	2	no	—
85	C₂×D₅₂	—	4	2	no	—
86	C₂×D₅₂	—	4	1	no	—
87	C₂×Dic₂₆	—	2	2	no	—
88	C₂×Dic₂₆	—	4	2	no	—
89	C₂×Dic₂₆	—	4	2	no	—
90	C₂×Dic₂₆	—	4	1	no	—
91	C₁₃×Q₁₆	—	2	4	no	—
92	C₁₃×Q₁₆	—	2	2	no	—
93	C₁₃×Q₁₆	C₂×D₄	1	4	yes	—
94	C₂×C₈×C₁₃	—	2	2	no	—
95	C₂×C₈×C₁₃	C₂×D₄	1	2	yes	—
96	C₂×C₈×C₁₃	C₂×D₄	1	1	yes	characteristic
97	C₂×C₈×C₁₃	C₂×D₄	1	2	yes	—
98	C₂×C₈×C₁₃	#41	1	2	yes	—
99	C₂×C₈×C₁₃	C₂⁴	1	1	yes	characteristic
100	C₁₃×C₄⋊C₄	—	2	2	no	—
101	C₁₃×C₄⋊C₄	—	4	1	no	—
102	C₁₃×C₄⋊C₄	—	4	4	no	—
103	C₄²×C₁₃	—	4	4	no	—
104	C₄²×C₁₃	D₈	1	2	yes	—
105	C₄²×C₁₃	C₂×D₄	1	2	yes	—
106	#6103	—	4	4	no	—
107	#6103	—	4	4	no	—
108	Dic₅₂	—	2	2	no	—
109	Dic₅₂	—	2	2	no	—
110	Dic₅₂	—	4	2	no	—
111	D₁₀₄	—	2	2	no	—
112	D₁₀₄	—	2	2	no	—
113	D₁₀₄	—	4	2	no	—
114	#6095	—	4	8	no	—
115	#6090	—	2	2	no	—
116	#6090	—	2	2	no	—
117	C₂²×Q₃₂	—	13	1	no	—
118	C₂²×SD₃₂	—	13	1	no	—
119	C₂×#495	—	13	1	no	—
120	C₂×C₄×Q₁₆	D₁₃	1	1	yes	characteristic, complemented
121	#2211	—	13	8	no	—
122	C₂×C₄⋊C₁₆	—	13	1	no	—
123	C₂×#358	—	13	1	no	—
124	C₂×#357	—	13	1	no	—
125	C₂³×C₁₃	C₂×D₈	1	1	yes	characteristic
126	C₂²×D₁₃	—	4	2	no	—
127	C₂²×D₁₃	—	8	2	no	—
128	C₂²×D₁₃	—	8	2	no	—
129	C₂²×D₁₃	—	8	2	no	—
130	Q₈×C₁₃	—	2	4	no	—
131	Q₈×C₁₃	—	4	2	no	—
132	Q₈×C₁₃	—	4	4	no	—
133	C₂×C₄×C₁₃	—	2	2	no	—
134	C₂×C₄×C₁₃	—	2	2	no	—
135	C₂×C₄×C₁₃	—	4	2	no	—
136	C₂×C₄×C₁₃	—	4	4	no	—
137	C₂×C₄×C₁₃	—	4	2	no	—
138	C₂×C₄×C₁₃	—	8	2	no	—
139	C₂×C₄×C₁₃	#121	1	2	yes	—
140	C₂×C₄×C₁₃	C₂×D₈	1	2	yes	—
141	C₂×C₄×C₁₃	C₂×D₈	1	2	yes	—
142	C₂×C₄×C₁₃	C₂×D₈	1	1	yes	characteristic
143	C₂×C₄×C₁₃	AGL(1,ℤ/8)	1	2	yes	—
144	C₂×C₄×C₁₃	C₂²×D₄	1	1	yes	characteristic
145	C₂×Dic₁₃	—	4	2	no	—
146	C₂×Dic₁₃	—	8	2	no	—
147	D₅₂	—	4	2	no	—
148	D₅₂	—	4	2	no	—
149	D₅₂	—	8	2	no	—
150	Dic₂₆	—	4	2	no	—
151	Dic₂₆	—	4	2	no	—
152	Dic₂₆	—	8	2	no	—
153	C₈×C₁₃	—	2	2	no	—
154	C₈×C₁₃	#121	1	2	yes	—
155	C₈×C₁₃	C₂²×D₄	1	1	yes	characteristic
156	C₈×C₁₃	C₂×#41	1	1	yes	characteristic
157	C₂²×Q₁₆	—	13	1	no	—
158	C₂²×Q₁₆	C₂×D₁₃	1	1	yes	characteristic
159	C₂²×SD₁₆	—	26	1	no	—
160	C₂²×D₈	—	13	1	no	—
161	C₂×#124	—	26	1	no	—
162	C₂×C₄×Q₈	—	2	1	no	—
163	C₂×Q₃₂	—	13	4	no	—
164	C₂×Q₃₂	—	26	4	no	—
165	C₂×SD₃₂	—	13	4	no	—
166	C₂×SD₃₂	—	26	4	no	—
167	C₂²×C₁₆	—	13	1	no	—
168	C₂²×C₁₆	—	13	1	no	—
169	#495	—	13	4	no	—
170	#495	—	26	2	no	—
171	C₄×Q₁₆	—	2	2	no	—
172	C₄×Q₁₆	C₂×D₁₃	1	4	yes	complemented
173	C₂×#107	C₂×D₁₃	1	1	yes	characteristic
174	C₂×#103	—	2	1	no	—
175	C₂×C₄×C₈	C₂×D₁₃	1	1	yes	characteristic
176	C₄⋊C₁₆	—	13	4	no	—
177	#358	—	13	4	no	—
178	#357	—	13	4	no	—
179	C₂²×C₁₃	#459	1	2	yes	—
180	C₂²×C₁₃	#508	1	2	yes	—
181	C₂²×C₁₃	#510	1	2	yes	—
182	C₂²×C₁₃	C₂²×D₈	1	1	yes	characteristic
183	C₂×D₁₃	—	8	2	no	—
184	C₂×D₁₃	—	8	2	no	—
185	C₂×D₁₃	—	8	2	no	—
186	C₂×D₁₃	—	16	4	no	—
187	C₄×C₁₃	—	2	2	no	—
188	C₄×C₁₃	—	2	2	no	—
189	C₄×C₁₃	—	4	4	no	—
190	C₄×C₁₃	—	8	2	no	—
191	C₄×C₁₃	#459	1	2	yes	—
192	C₄×C₁₃	C₂×#121	1	1	yes	characteristic
193	C₄×C₁₃	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
194	Dic₁₃	—	8	2	no	—
195	C₂²×Q₈	—	2	1	no	—
196	C₂²×Q₈	—	26	1	no	—
197	C₂²×D₄	—	26	1	no	—
198	C₂×Q₁₆	—	2	2	no	—
199	C₂×Q₁₆	—	2	1	no	—
200	C₂×Q₁₆	—	13	2	no	—
201	C₂×Q₁₆	—	26	2	no	—
202	C₂×Q₁₆	—	26	2	no	—
203	C₂×Q₁₆	—	26	1	no	—
204	C₂×Q₁₆	D₅₂	1	4	yes	—
205	C₂×Q₁₆	C₂²×D₁₃	1	2	yes	—
206	C₂×SD₁₆	—	26	4	no	—
207	C₂×SD₁₆	—	52	4	no	—
208	C₂×D₈	—	13	2	no	—
209	C₂×D₈	—	26	2	no	—
210	C₂×D₈	—	26	2	no	—
211	C₂×D₈	—	26	1	no	—
212	C₂²×C₈	C₂²⋊D₁₃	1	1	yes	characteristic
213	C₂²×C₈	C₂²×D₁₃	1	1	yes	characteristic
214	#124	—	26	4	no	—
215	#124	—	52	2	no	—
216	C₄×Q₈	—	2	4	no	—
217	C₄×Q₈	—	4	2	no	—
218	C₂×C₄⋊C₄	—	2	1	no	—
219	C₂×C₄⋊C₄	—	4	1	no	—
220	C₂×C₂²⋊C₄	—	52	1	no	—
221	C₂×C₂²⋊C₄	—	52	1	no	—
222	C₂×C₄²	—	4	1	no	—
223	C₂×C₄²	C₂²⋊D₁₃	1	1	yes	characteristic
224	Q₃₂	—	26	8	no	—
225	SD₃₂	—	26	8	no	—
226	C₂×C₁₆	—	13	2	no	—
227	C₂×C₁₆	—	13	2	no	—
228	C₂×C₁₆	—	26	2	no	—
229	C₂×C₁₆	—	26	2	no	—
230	#107	—	2	1	no	—
231	#107	C₂²×D₁₃	1	2	yes	—
232	#103	—	2	4	no	—
233	C₄×C₈	C₂²⋊D₁₃	1	2	yes	—
234	C₄×C₈	C₂²×D₁₃	1	2	yes	—
235	C₂×C₁₃	#2211	1	4	yes	complemented
236	C₂×C₁₃	C₂×#459	1	1	yes	characteristic
237	C₂×C₁₃	C₂×#508	1	1	yes	characteristic
238	C₂×C₁₃	C₂×#510	1	1	yes	characteristic
239	D₁₃	—	16	2	no	—
240	C₂⁴	—	52	1	no	—
241	C₂×Q₈	—	2	4	no	—
242	C₂×Q₈	—	2	2	no	—
243	C₂×Q₈	—	4	1	no	—
244	C₂×Q₈	—	4	2	no	—
245	C₂×Q₈	—	26	2	no	—
246	C₂×Q₈	—	52	1	no	—
247	C₂×Q₈	—	52	2	no	—
248	C₂×Q₈	—	52	2	no	—
249	C₂×D₄	—	26	2	no	—
250	C₂×D₄	—	52	2	no	—
251	C₂×D₄	—	52	1	no	—
252	C₂×D₄	—	52	2	no	—
253	C₂²×C₄	—	4	1	no	—
254	C₂²×C₄	—	4	1	no	—
255	C₂²×C₄	—	52	1	no	—
256	C₂²×C₄	#6104	1	1	yes	characteristic
257	C₂²×C₄	#6104	1	1	yes	characteristic
258	C₂²×C₄	C₂×C₂²⋊D₁₃	1	1	yes	characteristic
259	Q₁₆	—	2	4	no	—
260	Q₁₆	—	2	2	no	—
261	Q₁₆	—	26	2	no	—
262	Q₁₆	—	26	2	no	—
263	Q₁₆	—	52	2	no	—
264	Q₁₆	C₂×D₅₂	1	4	yes	—
265	SD₁₆	—	52	8	no	—
266	D₈	—	26	2	no	—
267	D₈	—	26	2	no	—
268	D₈	—	52	2	no	—
269	C₂×C₈	—	2	2	no	—
270	C₂×C₈	C₂×D₅₂	1	2	yes	—
271	C₂×C₈	#6127	1	2	yes	—
272	C₂×C₈	C₂×C₂²⋊D₁₃	1	2	yes	—
273	C₂×C₈	C₂×C₂²⋊D₁₃	1	1	yes	characteristic
274	C₂×C₈	C₂³×D₁₃	1	1	yes	characteristic
275	C₄⋊C₄	—	2	2	no	—
276	C₄⋊C₄	—	4	1	no	—
277	C₄⋊C₄	—	4	4	no	—
278	C₂²⋊C₄	—	52	4	no	—
279	C₂²⋊C₄	—	52	4	no	—
280	C₄²	—	4	4	no	—
281	C₄²	#6104	1	2	yes	—
282	C₄²	C₂×C₂²⋊D₁₃	1	2	yes	—
283	C₁₆	—	26	2	no	—
284	C₁₆	—	26	2	no	—
285	C₁₃	C₂×#2211	1	1	yes	characteristic, complemented, Sylow
286	C₂³	—	52	2	no	—
287	C₂³	—	104	2	no	—
288	C₂³	—	104	2	no	—
289	C₂³	—	104	2	no	—
290	C₂³	C₂×#6104	1	1	yes	characteristic
291	Q₈	—	2	4	no	—
292	Q₈	—	4	4	no	—
293	Q₈	—	4	2	no	—
294	Q₈	—	52	2	no	—
295	Q₈	—	52	2	no	—
296	Q₈	—	104	2	no	—
297	D₄	—	52	2	no	—
298	D₄	—	52	2	no	—
299	D₄	—	104	2	no	—
300	C₂×C₄	—	2	2	no	—
301	C₂×C₄	—	2	2	no	—
302	C₂×C₄	—	4	4	no	—
303	C₂×C₄	—	4	2	no	—
304	C₂×C₄	—	4	2	no	—
305	C₂×C₄	—	8	2	no	—
306	C₂×C₄	—	52	2	no	—
307	C₂×C₄	—	104	2	no	—
308	C₂×C₄	#88184	1	2	yes	—
309	C₂×C₄	C₂×#6104	1	2	yes	—
310	C₂×C₄	C₂×#6104	1	2	yes	—
311	C₂×C₄	C₂×#6104	1	1	yes	characteristic
312	C₂×C₄	#88203	1	2	yes	—
313	C₂×C₄	C₂²×C₂²⋊D₁₃	1	1	yes	characteristic
314	C₈	—	2	2	no	—
315	C₈	#88184	1	2	yes	—
316	C₈	C₂²×D₅₂	1	1	yes	characteristic
317	C₈	C₂×#6127	1	1	yes	characteristic
318	C₂²	—	104	2	no	—
319	C₂²	—	104	2	no	—
320	C₂²	—	104	2	no	—
321	C₂²	—	208	4	no	—
322	C₂²	#11718676	1	2	yes	—
323	C₂²	#11718840	1	2	yes	—
324	C₂²	#11718855	1	2	yes	—
325	C₂²	C₂²×#6104	1	1	yes	characteristic
326	C₄	—	2	2	no	—
327	C₄	—	2	2	no	—
328	C₄	—	4	4	no	—
329	C₄	—	8	2	no	—
330	C₄	—	104	2	no	—
331	C₄	#11718676	1	2	yes	—
332	C₄	C₂×#88184	1	1	yes	characteristic
333	C₄	C₂×#88203	1	1	yes	characteristic
334	C₂	—	208	2	no	—
335	C₂	#13116898	1	4	yes	direct product factor
336	C₂	C₂×#11718676	1	1	yes	characteristic
337	C₂	C₂×#11718855	1	1	yes	characteristic
338	C₂	C₂×#11718840	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	C₂×#6104
commutator, cosocle	C₂×C₈×C₁₃	C₂⁴
Frattini	C₂×C₈	C₂³×D₁₃
Fitting	C₂×C₄×C₁₃×Q₁₆	C₂
socle	C₂³×C₁₃	C₂×D₈
prime radical	C₂²×D₁₀₄	C₂²
hypercenter	C₂×C₄×Q₁₆	D₁₃
hypocenter	C₁₃	C₂×#2211
LCS	G ▷ C₂×C₈×C₁₃ ▷ C₄×C₁₃ ▷ C₂×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ C₂×C₄×Q₁₆

Complements

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

N	H ≅ G/N	type
1. C₂²×#88069	C₂	semidirect
4. #13116898	C₂	direct (N × H)
5. C₂×C₄⋊#6090	C₂	semidirect
6. C₂×#11718084	C₂	semidirect
8. C₂×C₄×C₁₃×Q₁₆	C₂	semidirect
16. C₄×C₁₃×Q₁₆	C₂²	semidirect
21. C₂×#88069	C₂²	semidirect
28. #11718084	C₂²	semidirect
30. C₄⋊#6090	C₂²	semidirect
120. C₂×C₄×Q₁₆	D₁₃	semidirect
172. C₄×Q₁₆	C₂×D₁₃	semidirect
235. C₂×C₁₃	#2211	semidirect
285. C₁₃	C₂×#2211	semidirect
335. C₂	#13116898	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×#88069	2
C₂²×#88068	2
C₂×#11718357	2
#13116898	2
C₂×C₄⋊#6090	2
C₂×#11718084	2
C₂×#11718083	2
C₂×C₄×C₁₃×Q₁₆	2
C₂×#2211	13

Subgroups — C2×#13116898 #16017175

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#13116898 #16017175