Subgroups — C₂×#14521846 #16860455

Order 3968.

1484 conjugacy classes of subgroups.

The subgroup lattice has 334 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#41×D₃₁	C₂	1	2	yes	maximal, complemented
2	#14521846	C₂	1	16	yes	maximal, direct product factor
3	C₂×#11758495	C₂	1	2	yes	maximal, complemented
4	C₂×#11758492	C₂	1	2	yes	maximal, complemented
5	C₂×#11758491	C₂	1	2	yes	maximal, complemented
6	C₂×#11758488	C₂	1	2	yes	maximal, complemented
7	C₂×#11758481	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×#11758527	C₂	1	2	yes	maximal, complemented
9	C₂²×#92469	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂×C₃₁×#135	C₂	1	1	yes	maximal, characteristic, complemented
11	C₂×#41×C₃₁	C₂²	1	2	yes	complemented
12	#41×D₃₁	—	2	8	no	—
13	#41×D₃₁	C₂²	1	16	yes	complemented
14	C₂×#92499	C₂²	1	2	yes	—
15	C₂×Q₈×D₃₁	C₂²	1	2	yes	complemented
16	C₂×#92497	—	2	2	no	—
17	C₂×#92497	C₂²	1	2	yes	—
18	C₂×D₄×D₃₁	—	2	2	no	—
19	C₂×D₄×D₃₁	C₂²	1	2	yes	complemented
20	C₂×#92495	—	2	2	no	—
21	C₂×#92495	C₂²	1	2	yes	complemented
22	C₂²×C₄×D₃₁	—	2	2	no	—
23	C₂²×C₄×D₃₁	C₂²	1	1	yes	characteristic
24	C₃₁×#135	C₂²	1	8	yes	complemented
25	C₂×Q₁₆×C₃₁	C₂²	1	1	yes	characteristic, complemented
26	C₂×SD₁₆×C₃₁	C₂²	1	2	yes	complemented
27	C₂×D₈×C₃₁	C₂²	1	1	yes	characteristic, complemented
28	C₂²×C₈×C₃₁	C₂²	1	1	yes	characteristic, complemented
29	#11758527	C₂²	1	16	yes	complemented
30	C₂×#92482	C₂²	1	2	yes	complemented
31	C₂×#92481	C₂²	1	2	yes	complemented
32	C₂×#92480	C₂²	1	2	yes	complemented
33	C₂×#92479	C₂²	1	2	yes	complemented
34	C₂²×C₃₁⋊C₈	C₂²	1	1	yes	characteristic, complemented
35	#11758495	C₂²	1	16	yes	complemented
36	#11758492	C₂²	1	16	yes	complemented
37	#11758491	C₂²	1	16	yes	complemented
38	#11758488	C₂²	1	16	yes	complemented
39	C₂×Dic₁₂₄	C₂²	1	1	yes	characteristic, complemented
40	#11758481	C₂²	1	8	yes	complemented
41	C₂×D₂₄₈	C₂²	1	1	yes	characteristic, complemented
42	C₂×#92470	C₂²	1	2	yes	complemented
43	C₂×#92469	C₂²	1	2	yes	complemented
44	C₂×#92469	C₂²	1	2	yes	complemented
45	C₂×#92469	C₂²	1	8	yes	complemented
46	C₂³×D₃₁	—	2	2	no	—
47	C₂³×D₃₁	D₄	1	1	yes	characteristic
48	#41×C₃₁	—	2	4	no	—
49	#41×C₃₁	C₂³	1	8	yes	complemented
50	C₂×Q₈×C₃₁	C₂³	1	2	yes	—
51	C₂×D₄×C₃₁	—	2	2	no	—
52	C₂×D₄×C₃₁	C₂³	1	2	yes	—
53	C₂²×C₄×C₃₁	—	2	2	no	—
54	C₂²×C₄×C₃₁	C₂³	1	1	yes	characteristic
55	C₂×C₂²⋊D₃₁	—	2	2	no	—
56	C₂×C₂²⋊D₃₁	—	2	2	no	—
57	C₂×C₂²⋊D₃₁	—	2	2	no	—
58	C₂²×Dic₃₁	—	2	2	no	—
59	C₂²×Dic₃₁	D₄	1	1	yes	characteristic
60	#92499	—	2	4	no	—
61	#92499	C₂³	1	8	yes	—
62	Q₈×D₃₁	—	2	4	no	—
63	Q₈×D₃₁	C₂³	1	8	yes	complemented
64	#92497	—	2	16	no	—
65	#92497	—	2	4	no	—
66	#92497	C₂³	1	8	yes	—
67	D₄×D₃₁	—	2	16	no	—
68	D₄×D₃₁	—	2	4	no	—
69	D₄×D₃₁	C₂³	1	8	yes	complemented
70	#92495	—	2	4	no	—
71	#92495	—	2	16	no	—
72	#92495	C₂³	1	8	yes	complemented
73	C₂×D₁₂₄	—	2	2	no	—
74	C₂×D₁₂₄	C₂³	1	2	yes	—
75	C₂×C₄×D₃₁	—	2	2	no	—
76	C₂×C₄×D₃₁	—	2	2	no	—
77	C₂×C₄×D₃₁	—	2	2	no	—
78	C₂×C₄×D₃₁	—	2	16	no	—
79	C₂×C₄×D₃₁	D₄	1	2	yes	—
80	C₂×C₄×D₃₁	D₄	1	4	yes	—
81	C₂×C₄×D₃₁	C₂³	1	2	yes	—
82	C₂×C₄×D₃₁	C₂³	1	4	yes	—
83	C₂×Dic₆₂	—	2	2	no	—
84	C₂×Dic₆₂	C₂³	1	2	yes	—
85	Q₁₆×C₃₁	C₂³	1	4	yes	complemented
86	SD₁₆×C₃₁	C₂³	1	8	yes	complemented
87	D₈×C₃₁	C₂³	1	4	yes	complemented
88	C₂×C₈×C₃₁	C₂³	1	1	yes	characteristic
89	C₂×C₈×C₃₁	C₂³	1	1	yes	characteristic
90	C₂×C₈×C₃₁	C₂³	1	4	yes	complemented
91	#92482	C₂³	1	8	yes	complemented
92	#92481	C₂³	1	8	yes	complemented
93	#92480	C₂³	1	8	yes	complemented
94	#92479	C₂³	1	8	yes	complemented
95	C₂×C₃₁⋊C₈	C₂³	1	2	yes	—
96	C₂×C₃₁⋊C₈	C₂³	1	4	yes	complemented
97	Dic₁₂₄	C₂³	1	4	yes	complemented
98	D₂₄₈	C₂³	1	4	yes	complemented
99	#92470	C₂³	1	8	yes	complemented
100	#92469	C₂³	1	8	yes	complemented
101	#92469	C₂³	1	8	yes	complemented
102	C₂³×C₃₁	—	2	2	no	—
103	C₂³×C₃₁	C₂×D₄	1	1	yes	characteristic
104	C₂²×D₃₁	—	2	2	no	—
105	C₂²×D₃₁	—	2	8	no	—
106	C₂²×D₃₁	—	4	2	no	—
107	C₂²×D₃₁	—	4	8	no	—
108	C₂²×D₃₁	C₂×D₄	1	2	yes	—
109	C₂²×D₃₁	C₂×D₄	1	4	yes	—
110	Q₈×C₃₁	—	2	2	no	—
111	Q₈×C₃₁	C₂⁴	1	4	yes	—
112	D₄×C₃₁	—	2	8	no	—
113	D₄×C₃₁	—	2	2	no	—
114	D₄×C₃₁	C₂⁴	1	4	yes	—
115	C₂×C₄×C₃₁	—	2	2	no	—
116	C₂×C₄×C₃₁	—	2	8	no	—
117	C₂×C₄×C₃₁	C₂×D₄	1	2	yes	—
118	C₂×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
119	C₂×C₄×C₃₁	C₂⁴	1	1	yes	characteristic
120	C₂×C₄×C₃₁	C₂⁴	1	2	yes	—
121	C₂²⋊D₃₁	—	2	8	no	—
122	C₂²⋊D₃₁	—	2	8	no	—
123	C₂²⋊D₃₁	—	2	8	no	—
124	C₂×Dic₃₁	—	2	2	no	—
125	C₂×Dic₃₁	—	2	8	no	—
126	C₂×Dic₃₁	C₂×D₄	1	2	yes	—
127	C₂×Dic₃₁	C₂×D₄	1	4	yes	—
128	D₁₂₄	—	2	8	no	—
129	D₁₂₄	—	2	2	no	—
130	D₁₂₄	C₂⁴	1	4	yes	—
131	C₄×D₃₁	—	2	8	no	—
132	C₄×D₃₁	—	2	8	no	—
133	C₄×D₃₁	—	2	8	no	—
134	C₄×D₃₁	C₂×D₄	1	8	yes	—
135	C₄×D₃₁	C₂×D₄	1	4	yes	—
136	C₄×D₃₁	C₂⁴	1	4	yes	—
137	Dic₆₂	—	2	2	no	—
138	Dic₆₂	—	2	8	no	—
139	Dic₆₂	C₂⁴	1	4	yes	—
140	C₈×C₃₁	C₂⁴	1	2	yes	—
141	C₈×C₃₁	C₂⁴	1	2	yes	—
142	C₃₁⋊C₈	C₂⁴	1	4	yes	—
143	C₂×#575	—	31	1	no	maximal, Sylow
144	C₂²×C₃₁	—	2	2	no	—
145	C₂²×C₃₁	—	2	4	no	—
146	C₂²×C₃₁	—	4	4	no	—
147	C₂²×C₃₁	C₂²×D₄	1	2	yes	—
148	C₂²×C₃₁	C₂²×D₄	1	1	yes	characteristic
149	C₂×D₃₁	—	2	4	no	—
150	C₂×D₃₁	—	2	4	no	—
151	C₂×D₃₁	—	4	4	no	—
152	C₂×D₃₁	—	4	16	no	—
153	C₂×D₃₁	—	4	4	no	—
154	C₂×D₃₁	C₂²×D₄	1	4	yes	—
155	C₄×C₃₁	—	2	4	no	—
156	C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
157	C₄×C₃₁	C₂²×D₄	1	2	yes	—
158	C₄×C₃₁	C₂⁵	1	1	yes	characteristic
159	Dic₃₁	—	2	4	no	—
160	Dic₃₁	C₂²×D₄	1	4	yes	—
161	C₂²×#41	—	31	2	no	—
162	#575	—	31	16	no	—
163	C₂×#137	—	31	2	no	—
164	C₂×#137	—	31	2	no	—
165	C₂×AGL(1,ℤ/8)	—	31	2	no	—
166	C₂×AGL(1,ℤ/8)	—	31	2	no	—
167	C₂×#135	—	31	2	no	—
168	C₂×#135	—	31	1	no	—
169	C₂×#135	D₃₁	1	1	yes	characteristic, complemented
170	C₂²×M₁₆	—	31	1	no	—
171	C₂×C₃₁	—	2	2	no	—
172	C₂×C₃₁	—	4	4	no	—
173	C₂×C₃₁	#575	1	2	yes	complemented
174	C₂×C₃₁	C₂³×D₄	1	1	yes	characteristic
175	D₃₁	—	2	4	no	—
176	D₃₁	—	4	4	no	—
177	C₂×#41	—	31	16	no	—
178	C₂×#41	—	31	2	no	—
179	C₂×#41	—	31	2	no	—
180	C₂×#41	—	31	2	no	—
181	C₂×#41	—	62	2	no	—
182	C₂×#41	—	62	2	no	—
183	C₂×#41	—	62	8	no	—
184	C₂×#41	C₂×D₃₁	1	2	yes	complemented
185	C₂²×Q₈	—	31	2	no	—
186	C₂²×D₄	—	31	2	no	—
187	C₂²×D₄	—	62	2	no	—
188	C₂³×C₄	—	31	1	no	—
189	C₂³×C₄	—	62	2	no	—
190	#137	—	31	16	no	—
191	#137	—	31	16	no	—
192	AGL(1,ℤ/8)	—	31	16	no	—
193	AGL(1,ℤ/8)	—	31	16	no	—
194	#135	—	31	16	no	—
195	#135	—	31	8	no	—
196	#135	C₂×D₃₁	1	8	yes	complemented
197	C₂×Q₁₆	—	31	2	no	—
198	C₂×Q₁₆	—	31	1	no	—
199	C₂×Q₁₆	C₂×D₃₁	1	1	yes	characteristic, complemented
200	C₂×SD₁₆	—	31	2	no	—
201	C₂×SD₁₆	—	31	2	no	—
202	C₂×SD₁₆	—	31	2	no	—
203	C₂×SD₁₆	C₂×D₃₁	1	2	yes	complemented
204	C₂×D₈	—	31	2	no	—
205	C₂×D₈	—	31	1	no	—
206	C₂×D₈	C₂×D₃₁	1	1	yes	characteristic, complemented
207	C₂×M₁₆	—	31	2	no	—
208	C₂×M₁₆	—	31	2	no	—
209	C₂×M₁₆	—	31	8	no	—
210	C₂²×C₈	—	31	1	no	—
211	C₂²×C₈	C₂×D₃₁	1	1	yes	characteristic, complemented
212	C₃₁	C₂×#575	1	1	yes	characteristic, complemented, Sylow
213	C₂⁴	—	31	1	no	—
214	C₂⁴	—	62	2	no	—
215	#41	—	2	4	no	—
216	#41	—	31	8	no	—
217	#41	—	31	8	no	—
218	#41	—	31	8	no	—
219	#41	—	62	16	no	—
220	#41	—	62	16	no	—
221	#41	—	62	4	no	—
222	#41	—	62	4	no	—
223	#41	—	62	4	no	—
224	#41	C₂²×D₃₁	1	8	yes	complemented
225	C₂×Q₈	—	31	8	no	—
226	C₂×Q₈	—	31	2	no	—
227	C₂×Q₈	—	62	2	no	—
228	C₂×Q₈	—	62	4	no	—
229	C₂×Q₈	C₂²×D₃₁	1	2	yes	—
230	C₂×D₄	—	2	2	no	—
231	C₂×D₄	—	31	8	no	—
232	C₂×D₄	—	31	2	no	—
233	C₂×D₄	—	62	2	no	—
234	C₂×D₄	—	62	4	no	—
235	C₂×D₄	—	62	2	no	—
236	C₂×D₄	—	62	16	no	—
237	C₂×D₄	—	62	2	no	—
238	C₂×D₄	—	62	2	no	—
239	C₂×D₄	C₂²×D₃₁	1	2	yes	—
240	C₂²×C₄	—	2	2	no	—
241	C₂²×C₄	—	31	1	no	—
242	C₂²×C₄	—	31	4	no	—
243	C₂²×C₄	—	31	2	no	—
244	C₂²×C₄	—	31	2	no	—
245	C₂²×C₄	—	31	4	no	—
246	C₂²×C₄	—	62	2	no	—
247	C₂²×C₄	—	62	16	no	—
248	C₂²×C₄	—	62	2	no	—
249	C₂²×C₄	—	62	2	no	—
250	C₂²×C₄	—	62	2	no	—
251	C₂²×C₄	C₂²×D₃₁	1	1	yes	characteristic
252	Q₁₆	—	31	8	no	—
253	Q₁₆	—	31	4	no	—
254	Q₁₆	C₂²×D₃₁	1	4	yes	complemented
255	SD₁₆	—	31	8	no	—
256	SD₁₆	—	31	8	no	—
257	SD₁₆	—	31	8	no	—
258	SD₁₆	C₂²×D₃₁	1	8	yes	complemented
259	D₈	—	31	8	no	—
260	D₈	—	31	4	no	—
261	D₈	C₂²×D₃₁	1	4	yes	complemented
262	M₁₆	—	31	8	no	—
263	M₁₆	—	31	8	no	—
264	C₂×C₈	—	31	2	no	—
265	C₂×C₈	—	31	4	no	—
266	C₂×C₈	C₂²×D₃₁	1	1	yes	characteristic
267	C₂×C₈	C₂²×D₃₁	1	1	yes	characteristic
268	C₂×C₈	C₂²×D₃₁	1	4	yes	complemented
269	C₂³	—	2	2	no	—
270	C₂³	—	31	2	no	—
271	C₂³	—	31	4	no	—
272	C₂³	—	62	8	no	—
273	C₂³	—	62	2	no	—
274	C₂³	—	124	2	no	—
275	C₂³	—	124	8	no	—
276	C₂³	D₄×D₃₁	1	1	yes	characteristic
277	Q₈	—	2	2	no	—
278	Q₈	—	31	4	no	—
279	Q₈	—	62	8	no	—
280	Q₈	—	62	2	no	—
281	Q₈	C₂³×D₃₁	1	4	yes	—
282	D₄	—	2	8	no	—
283	D₄	—	2	2	no	—
284	D₄	—	31	4	no	—
285	D₄	—	62	8	no	—
286	D₄	—	62	8	no	—
287	D₄	—	62	8	no	—
288	D₄	—	62	8	no	—
289	D₄	—	62	2	no	—
290	D₄	C₂³×D₃₁	1	4	yes	—
291	C₂×C₄	—	2	2	no	—
292	C₂×C₄	—	2	8	no	—
293	C₂×C₄	—	31	2	no	—
294	C₂×C₄	—	31	4	no	—
295	C₂×C₄	—	31	8	no	—
296	C₂×C₄	—	31	4	no	—
297	C₂×C₄	—	31	4	no	—
298	C₂×C₄	—	62	8	no	—
299	C₂×C₄	—	62	8	no	—
300	C₂×C₄	—	62	8	no	—
301	C₂×C₄	—	62	2	no	—
302	C₂×C₄	—	62	8	no	—
303	C₂×C₄	D₄×D₃₁	1	2	yes	—
304	C₂×C₄	D₄×D₃₁	1	1	yes	characteristic
305	C₂×C₄	C₂³×D₃₁	1	1	yes	characteristic
306	C₂×C₄	C₂³×D₃₁	1	2	yes	—
307	C₈	—	31	4	no	—
308	C₈	C₂³×D₃₁	1	2	yes	—
309	C₈	C₂³×D₃₁	1	2	yes	—
310	C₂²	—	2	2	no	—
311	C₂²	—	2	4	no	—
312	C₂²	—	4	4	no	—
313	C₂²	—	31	4	no	—
314	C₂²	—	62	4	no	—
315	C₂²	—	62	4	no	—
316	C₂²	—	124	4	no	—
317	C₂²	—	124	16	no	—
318	C₂²	—	124	4	no	—
319	C₂²	C₂×D₄×D₃₁	1	2	yes	—
320	C₂²	C₂×D₄×D₃₁	1	1	yes	characteristic
321	C₄	—	2	4	no	—
322	C₄	—	31	4	no	—
323	C₄	—	62	4	no	—
324	C₄	C₂×D₄×D₃₁	1	2	yes	—
325	C₄	C₂×D₄×D₃₁	1	1	yes	characteristic
326	C₄	C₂⁴×D₃₁	1	1	yes	characteristic
327	C₂	—	2	2	no	—
328	C₂	—	4	4	no	—
329	C₂	—	62	4	no	—
330	C₂	—	124	4	no	—
331	C₂	#14521846	1	2	yes	direct product factor
332	C₂	C₂²×D₄×D₃₁	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	D₄×D₃₁
commutator, cosocle	C₄×C₃₁	C₂⁵
Frattini	C₄	C₂⁴×D₃₁
Fitting	C₂×C₃₁×#135	C₂
socle	C₂²×C₃₁	C₂²×D₄
hypercenter	C₂×#135	D₃₁
hypocenter	C₃₁	C₂×#575
LCS	G ▷ C₄×C₃₁ ▷ C₂×C₃₁ ▷ C₃₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄ ◁ C₂×#135

Complements

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

N	H ≅ G/N	type
1. C₂×#41×D₃₁	C₂	semidirect
2. #14521846	C₂	direct (N × H)
3. C₂×#11758495	C₂	semidirect
4. C₂×#11758492	C₂	semidirect
5. C₂×#11758491	C₂	semidirect
6. C₂×#11758488	C₂	semidirect
7. C₂×#11758481	C₂	semidirect
8. C₂×#11758527	C₂	semidirect
9. C₂²×#92469	C₂	semidirect
10. C₂×C₃₁×#135	C₂	semidirect
11. C₂×#41×C₃₁	C₂²	semidirect
13. #41×D₃₁	C₂²	semidirect
15. C₂×Q₈×D₃₁	C₂²	semidirect
19. C₂×D₄×D₃₁	C₂²	semidirect
21. C₂×#92495	C₂²	semidirect
24. C₃₁×#135	C₂²	semidirect
25. C₂×Q₁₆×C₃₁	C₂²	semidirect
26. C₂×SD₁₆×C₃₁	C₂²	semidirect
27. C₂×D₈×C₃₁	C₂²	semidirect
28. C₂²×C₈×C₃₁	C₂²	semidirect
29. #11758527	C₂²	semidirect
30. C₂×#92482	C₂²	semidirect
31. C₂×#92481	C₂²	semidirect
32. C₂×#92480	C₂²	semidirect
33. C₂×#92479	C₂²	semidirect
34. C₂²×C₃₁⋊C₈	C₂²	semidirect
35. #11758495	C₂²	semidirect
36. #11758492	C₂²	semidirect
37. #11758491	C₂²	semidirect
38. #11758488	C₂²	semidirect
39. C₂×Dic₁₂₄	C₂²	semidirect
40. #11758481	C₂²	semidirect
41. C₂×D₂₄₈	C₂²	semidirect
42. C₂×#92470	C₂²	semidirect
43. C₂×#92469	C₂²	semidirect
44. C₂×#92469	C₂²	semidirect
45. C₂×#92469	C₂²	semidirect
49. #41×C₃₁	C₂³	semidirect
63. Q₈×D₃₁	C₂³	semidirect
69. D₄×D₃₁	C₂³	semidirect
72. #92495	C₂³	semidirect
85. Q₁₆×C₃₁	C₂³	semidirect
86. SD₁₆×C₃₁	C₂³	semidirect
87. D₈×C₃₁	C₂³	semidirect
90. C₂×C₈×C₃₁	C₂³	semidirect
91. #92482	C₂³	semidirect
92. #92481	C₂³	semidirect
93. #92480	C₂³	semidirect
94. #92479	C₂³	semidirect
96. C₂×C₃₁⋊C₈	C₂³	semidirect
97. Dic₁₂₄	C₂³	semidirect
98. D₂₄₈	C₂³	semidirect
99. #92470	C₂³	semidirect
100. #92469	C₂³	semidirect
101. #92469	C₂³	semidirect
169. C₂×#135	D₃₁	semidirect
173. C₂×C₃₁	#575	semidirect
184. C₂×#41	C₂×D₃₁	semidirect
196. #135	C₂×D₃₁	semidirect
199. C₂×Q₁₆	C₂×D₃₁	semidirect
203. C₂×SD₁₆	C₂×D₃₁	semidirect
206. C₂×D₈	C₂×D₃₁	semidirect
211. C₂²×C₈	C₂×D₃₁	semidirect
212. C₃₁	C₂×#575	semidirect
224. #41	C₂²×D₃₁	semidirect
254. Q₁₆	C₂²×D₃₁	semidirect
258. SD₁₆	C₂²×D₃₁	semidirect
261. D₈	C₂²×D₃₁	semidirect
268. C₂×C₈	C₂²×D₃₁	semidirect
331. C₂	#14521846	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×#41×D₃₁	2
#14521846	2
C₂×#11758495	2
C₂×#11758492	2
C₂×#11758491	2
C₂×#11758488	2
C₂×#11758481	2
C₂×#11758527	2
C₂²×#92469	2
C₂×C₃₁×#135	2
C₂×#575	31

Subgroups — C2×#14521846 #16860455

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#14521846 #16860455