Subgroups — C₂×#19128498 #24922207

Order 10624.

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×D₈₃	C₂	1	2	yes	maximal, complemented
2	#19128498	C₂	1	16	yes	maximal, direct product factor
3	C₂×#14581824	C₂	1	2	yes	maximal, complemented
4	C₂×#14581821	C₂	1	2	yes	maximal, complemented
5	C₂×#14581819	C₂	1	2	yes	maximal, complemented
6	C₂×#14581816	C₂	1	2	yes	maximal, complemented
7	C₂×#14581814	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×#14581812	C₂	1	2	yes	maximal, complemented
9	C₂²×#13050573	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂×#135×C₈₃	C₂	1	1	yes	maximal, characteristic, complemented
11	C₂×#13050591	C₂²	1	2	yes	—
12	C₂×#13050590	—	2	2	no	—
13	C₂×#13050590	C₂²	1	2	yes	complemented
14	C₂×#13050589	—	2	2	no	—
15	C₂×#13050589	C₂²	1	2	yes	—
16	#41×D₈₃	—	2	8	no	—
17	#41×D₈₃	C₂²	1	16	yes	complemented
18	C₂×Q₈×D₈₃	C₂²	1	2	yes	complemented
19	C₂×D₄×D₈₃	—	2	2	no	—
20	C₂×D₄×D₈₃	C₂²	1	2	yes	complemented
21	C₂²×C₄×D₈₃	—	2	2	no	—
22	C₂²×C₄×D₈₃	C₂²	1	1	yes	characteristic
23	#14581824	C₂²	1	16	yes	complemented
24	#14581821	C₂²	1	16	yes	complemented
25	#14581819	C₂²	1	16	yes	complemented
26	#14581816	C₂²	1	16	yes	complemented
27	#14581814	C₂²	1	8	yes	complemented
28	#14581812	C₂²	1	16	yes	complemented
29	C₂×Dic₃₃₂	C₂²	1	1	yes	characteristic, complemented
30	C₂×#13050580	C₂²	1	2	yes	complemented
31	C₂×#13050579	C₂²	1	2	yes	complemented
32	C₂×#13050578	C₂²	1	2	yes	complemented
33	C₂×#13050577	C₂²	1	2	yes	complemented
34	C₂×D₆₆₄	C₂²	1	1	yes	characteristic, complemented
35	C₂×#13050575	C₂²	1	2	yes	complemented
36	C₂×#13050573	C₂²	1	2	yes	complemented
37	C₂×#13050573	C₂²	1	2	yes	complemented
38	C₂×#13050573	C₂²	1	8	yes	complemented
39	C₂²×C₈₃⋊C₈	C₂²	1	1	yes	characteristic, complemented
40	C₂×#41×C₈₃	C₂²	1	2	yes	complemented
41	#135×C₈₃	C₂²	1	8	yes	complemented
42	C₂×Q₁₆×C₈₃	C₂²	1	1	yes	characteristic, complemented
43	C₂×SD₁₆×C₈₃	C₂²	1	2	yes	complemented
44	C₂×D₈×C₈₃	C₂²	1	1	yes	characteristic, complemented
45	C₂²×C₈×C₈₃	C₂²	1	1	yes	characteristic, complemented
46	C₂³×D₈₃	—	2	2	no	—
47	C₂³×D₈₃	D₄	1	1	yes	characteristic
48	#13050591	—	2	4	no	—
49	#13050591	C₂³	1	8	yes	—
50	#13050590	—	2	4	no	—
51	#13050590	—	2	16	no	—
52	#13050590	C₂³	1	8	yes	complemented
53	#13050589	—	2	16	no	—
54	#13050589	—	2	4	no	—
55	#13050589	C₂³	1	8	yes	—
56	C₂×Dic₁₆₆	—	2	2	no	—
57	C₂×Dic₁₆₆	C₂³	1	2	yes	—
58	Q₈×D₈₃	—	2	4	no	—
59	Q₈×D₈₃	C₂³	1	8	yes	complemented
60	C₂×D₃₃₂	—	2	2	no	—
61	C₂×D₃₃₂	C₂³	1	2	yes	—
62	C₂×C₂²⋊D₈₃	—	2	2	no	—
63	C₂×C₂²⋊D₈₃	—	2	2	no	—
64	C₂×C₂²⋊D₈₃	—	2	2	no	—
65	D₄×D₈₃	—	2	16	no	—
66	D₄×D₈₃	—	2	4	no	—
67	D₄×D₈₃	C₂³	1	8	yes	complemented
68	C₂²×Dic₈₃	—	2	2	no	—
69	C₂²×Dic₈₃	D₄	1	1	yes	characteristic
70	C₂×C₄×D₈₃	—	2	2	no	—
71	C₂×C₄×D₈₃	—	2	2	no	—
72	C₂×C₄×D₈₃	—	2	2	no	—
73	C₂×C₄×D₈₃	—	2	16	no	—
74	C₂×C₄×D₈₃	D₄	1	2	yes	—
75	C₂×C₄×D₈₃	D₄	1	4	yes	—
76	C₂×C₄×D₈₃	C₂³	1	2	yes	—
77	C₂×C₄×D₈₃	C₂³	1	4	yes	—
78	Dic₃₃₂	C₂³	1	4	yes	complemented
79	#13050580	C₂³	1	8	yes	complemented
80	#13050579	C₂³	1	8	yes	complemented
81	#13050578	C₂³	1	8	yes	complemented
82	#13050577	C₂³	1	8	yes	complemented
83	D₆₆₄	C₂³	1	4	yes	complemented
84	#13050575	C₂³	1	8	yes	complemented
85	#13050573	C₂³	1	8	yes	complemented
86	#13050573	C₂³	1	8	yes	complemented
87	C₂×C₈₃⋊C₈	C₂³	1	2	yes	—
88	C₂×C₈₃⋊C₈	C₂³	1	4	yes	complemented
89	#41×C₈₃	—	2	4	no	—
90	#41×C₈₃	C₂³	1	8	yes	complemented
91	C₂×Q₈×C₈₃	C₂³	1	2	yes	—
92	C₂×D₄×C₈₃	—	2	2	no	—
93	C₂×D₄×C₈₃	C₂³	1	2	yes	—
94	C₂²×C₄×C₈₃	—	2	2	no	—
95	C₂²×C₄×C₈₃	C₂³	1	1	yes	characteristic
96	Q₁₆×C₈₃	C₂³	1	4	yes	complemented
97	SD₁₆×C₈₃	C₂³	1	8	yes	complemented
98	D₈×C₈₃	C₂³	1	4	yes	complemented
99	C₂×C₈×C₈₃	C₂³	1	1	yes	characteristic
100	C₂×C₈×C₈₃	C₂³	1	1	yes	characteristic
101	C₂×C₈×C₈₃	C₂³	1	4	yes	complemented
102	C₂³×C₈₃	—	2	2	no	—
103	C₂³×C₈₃	C₂×D₄	1	1	yes	characteristic
104	C₂²×D₈₃	—	2	2	no	—
105	C₂²×D₈₃	—	2	8	no	—
106	C₂²×D₈₃	—	4	2	no	—
107	C₂²×D₈₃	—	4	8	no	—
108	C₂²×D₈₃	C₂×D₄	1	2	yes	—
109	C₂²×D₈₃	C₂×D₄	1	4	yes	—
110	Q₈×C₈₃	—	2	2	no	—
111	Q₈×C₈₃	C₂⁴	1	4	yes	—
112	D₄×C₈₃	—	2	8	no	—
113	D₄×C₈₃	—	2	2	no	—
114	D₄×C₈₃	C₂⁴	1	4	yes	—
115	C₂×C₄×C₈₃	—	2	2	no	—
116	C₂×C₄×C₈₃	—	2	8	no	—
117	C₂×C₄×C₈₃	C₂×D₄	1	2	yes	—
118	C₂×C₄×C₈₃	C₂×D₄	1	1	yes	characteristic
119	C₂×C₄×C₈₃	C₂⁴	1	1	yes	characteristic
120	C₂×C₄×C₈₃	C₂⁴	1	2	yes	—
121	C₂²⋊D₈₃	—	2	8	no	—
122	C₂²⋊D₈₃	—	2	8	no	—
123	C₂²⋊D₈₃	—	2	8	no	—
124	C₂×Dic₈₃	—	2	2	no	—
125	C₂×Dic₈₃	—	2	8	no	—
126	C₂×Dic₈₃	C₂×D₄	1	2	yes	—
127	C₂×Dic₈₃	C₂×D₄	1	4	yes	—
128	D₃₃₂	—	2	8	no	—
129	D₃₃₂	—	2	2	no	—
130	D₃₃₂	C₂⁴	1	4	yes	—
131	C₄×D₈₃	—	2	8	no	—
132	C₄×D₈₃	—	2	8	no	—
133	C₄×D₈₃	—	2	8	no	—
134	C₄×D₈₃	C₂×D₄	1	8	yes	—
135	C₄×D₈₃	C₂×D₄	1	4	yes	—
136	C₄×D₈₃	C₂⁴	1	4	yes	—
137	Dic₁₆₆	—	2	2	no	—
138	Dic₁₆₆	—	2	8	no	—
139	Dic₁₆₆	C₂⁴	1	4	yes	—
140	C₈×C₈₃	C₂⁴	1	2	yes	—
141	C₈×C₈₃	C₂⁴	1	2	yes	—
142	C₈₃⋊C₈	C₂⁴	1	4	yes	—
143	C₂²×C₈₃	—	2	2	no	—
144	C₂²×C₈₃	—	2	4	no	—
145	C₂²×C₈₃	—	4	4	no	—
146	C₂²×C₈₃	C₂²×D₄	1	2	yes	—
147	C₂²×C₈₃	C₂²×D₄	1	1	yes	characteristic
148	C₂×D₈₃	—	2	4	no	—
149	C₂×D₈₃	—	2	4	no	—
150	C₂×D₈₃	—	4	4	no	—
151	C₂×D₈₃	—	4	16	no	—
152	C₂×D₈₃	—	4	4	no	—
153	C₂×D₈₃	C₂²×D₄	1	4	yes	—
154	C₄×C₈₃	—	2	4	no	—
155	C₄×C₈₃	C₂²×D₄	1	1	yes	characteristic
156	C₄×C₈₃	C₂²×D₄	1	2	yes	—
157	C₄×C₈₃	C₂⁵	1	1	yes	characteristic
158	Dic₈₃	—	2	4	no	—
159	Dic₈₃	C₂²×D₄	1	4	yes	—
160	C₂×C₈₃	—	2	2	no	—
161	C₂×C₈₃	—	4	4	no	—
162	C₂×C₈₃	#575	1	2	yes	complemented
163	C₂×C₈₃	C₂³×D₄	1	1	yes	characteristic
164	D₈₃	—	2	4	no	—
165	D₈₃	—	4	4	no	—
166	C₂×#575	—	83	1	no	maximal, Sylow
167	C₈₃	C₂×#575	1	1	yes	characteristic, complemented, Sylow
168	C₂²×#41	—	83	2	no	—
169	#575	—	83	16	no	—
170	C₂×#137	—	83	2	no	—
171	C₂×#137	—	83	2	no	—
172	C₂×AGL(1,ℤ/8)	—	83	2	no	—
173	C₂×AGL(1,ℤ/8)	—	83	2	no	—
174	C₂×#135	—	83	2	no	—
175	C₂×#135	—	83	1	no	—
176	C₂×#135	D₈₃	1	1	yes	characteristic, complemented
177	C₂²×M₁₆	—	83	1	no	—
178	C₂×#41	—	83	16	no	—
179	C₂×#41	—	83	2	no	—
180	C₂×#41	—	83	2	no	—
181	C₂×#41	—	83	2	no	—
182	C₂×#41	—	166	2	no	—
183	C₂×#41	—	166	2	no	—
184	C₂×#41	—	166	8	no	—
185	C₂×#41	C₂×D₈₃	1	2	yes	complemented
186	C₂²×Q₈	—	83	2	no	—
187	C₂²×D₄	—	83	2	no	—
188	C₂²×D₄	—	166	2	no	—
189	C₂³×C₄	—	83	1	no	—
190	C₂³×C₄	—	166	2	no	—
191	#137	—	83	16	no	—
192	#137	—	83	16	no	—
193	AGL(1,ℤ/8)	—	83	16	no	—
194	AGL(1,ℤ/8)	—	83	16	no	—
195	#135	—	83	16	no	—
196	#135	—	83	8	no	—
197	#135	C₂×D₈₃	1	8	yes	complemented
198	C₂×Q₁₆	—	83	2	no	—
199	C₂×Q₁₆	—	83	1	no	—
200	C₂×Q₁₆	C₂×D₈₃	1	1	yes	characteristic, complemented
201	C₂×SD₁₆	—	83	2	no	—
202	C₂×SD₁₆	—	83	2	no	—
203	C₂×SD₁₆	—	83	2	no	—
204	C₂×SD₁₆	C₂×D₈₃	1	2	yes	complemented
205	C₂×D₈	—	83	2	no	—
206	C₂×D₈	—	83	1	no	—
207	C₂×D₈	C₂×D₈₃	1	1	yes	characteristic, complemented
208	C₂×M₁₆	—	83	2	no	—
209	C₂×M₁₆	—	83	2	no	—
210	C₂×M₁₆	—	83	8	no	—
211	C₂²×C₈	—	83	1	no	—
212	C₂²×C₈	C₂×D₈₃	1	1	yes	characteristic, complemented
213	C₂⁴	—	83	1	no	—
214	C₂⁴	—	166	2	no	—
215	#41	—	2	4	no	—
216	#41	—	83	8	no	—
217	#41	—	83	8	no	—
218	#41	—	83	8	no	—
219	#41	—	166	16	no	—
220	#41	—	166	16	no	—
221	#41	—	166	4	no	—
222	#41	—	166	4	no	—
223	#41	—	166	4	no	—
224	#41	C₂²×D₈₃	1	8	yes	complemented
225	C₂×Q₈	—	83	8	no	—
226	C₂×Q₈	—	83	2	no	—
227	C₂×Q₈	—	166	2	no	—
228	C₂×Q₈	—	166	4	no	—
229	C₂×Q₈	C₂²×D₈₃	1	2	yes	—
230	C₂×D₄	—	2	2	no	—
231	C₂×D₄	—	83	8	no	—
232	C₂×D₄	—	83	2	no	—
233	C₂×D₄	—	166	2	no	—
234	C₂×D₄	—	166	4	no	—
235	C₂×D₄	—	166	2	no	—
236	C₂×D₄	—	166	16	no	—
237	C₂×D₄	—	166	2	no	—
238	C₂×D₄	—	166	2	no	—
239	C₂×D₄	C₂²×D₈₃	1	2	yes	—
240	C₂²×C₄	—	2	2	no	—
241	C₂²×C₄	—	83	1	no	—
242	C₂²×C₄	—	83	4	no	—
243	C₂²×C₄	—	83	2	no	—
244	C₂²×C₄	—	83	2	no	—
245	C₂²×C₄	—	83	4	no	—
246	C₂²×C₄	—	166	2	no	—
247	C₂²×C₄	—	166	16	no	—
248	C₂²×C₄	—	166	2	no	—
249	C₂²×C₄	—	166	2	no	—
250	C₂²×C₄	—	166	2	no	—
251	C₂²×C₄	C₂²×D₈₃	1	1	yes	characteristic
252	Q₁₆	—	83	8	no	—
253	Q₁₆	—	83	4	no	—
254	Q₁₆	C₂²×D₈₃	1	4	yes	complemented
255	SD₁₆	—	83	8	no	—
256	SD₁₆	—	83	8	no	—
257	SD₁₆	—	83	8	no	—
258	SD₁₆	C₂²×D₈₃	1	8	yes	complemented
259	D₈	—	83	8	no	—
260	D₈	—	83	4	no	—
261	D₈	C₂²×D₈₃	1	4	yes	complemented
262	M₁₆	—	83	8	no	—
263	M₁₆	—	83	8	no	—
264	C₂×C₈	—	83	2	no	—
265	C₂×C₈	—	83	4	no	—
266	C₂×C₈	C₂²×D₈₃	1	1	yes	characteristic
267	C₂×C₈	C₂²×D₈₃	1	1	yes	characteristic
268	C₂×C₈	C₂²×D₈₃	1	4	yes	complemented
269	C₂³	—	2	2	no	—
270	C₂³	—	332	2	no	—
271	C₂³	—	332	8	no	—
272	C₂³	—	83	4	no	—
273	C₂³	—	83	2	no	—
274	C₂³	—	166	8	no	—
275	C₂³	—	166	2	no	—
276	C₂³	D₄×D₈₃	1	1	yes	characteristic
277	Q₈	—	2	2	no	—
278	Q₈	—	83	4	no	—
279	Q₈	—	166	8	no	—
280	Q₈	—	166	2	no	—
281	Q₈	C₂³×D₈₃	1	4	yes	—
282	D₄	—	2	8	no	—
283	D₄	—	2	2	no	—
284	D₄	—	83	4	no	—
285	D₄	—	166	8	no	—
286	D₄	—	166	8	no	—
287	D₄	—	166	8	no	—
288	D₄	—	166	8	no	—
289	D₄	—	166	2	no	—
290	D₄	C₂³×D₈₃	1	4	yes	—
291	C₂×C₄	—	2	2	no	—
292	C₂×C₄	—	2	8	no	—
293	C₂×C₄	—	83	2	no	—
294	C₂×C₄	—	83	4	no	—
295	C₂×C₄	—	83	8	no	—
296	C₂×C₄	—	83	4	no	—
297	C₂×C₄	—	83	4	no	—
298	C₂×C₄	—	166	8	no	—
299	C₂×C₄	—	166	8	no	—
300	C₂×C₄	—	166	8	no	—
301	C₂×C₄	—	166	2	no	—
302	C₂×C₄	—	166	8	no	—
303	C₂×C₄	D₄×D₈₃	1	2	yes	—
304	C₂×C₄	D₄×D₈₃	1	1	yes	characteristic
305	C₂×C₄	C₂³×D₈₃	1	1	yes	characteristic
306	C₂×C₄	C₂³×D₈₃	1	2	yes	—
307	C₈	—	83	4	no	—
308	C₈	C₂³×D₈₃	1	2	yes	—
309	C₈	C₂³×D₈₃	1	2	yes	—
310	C₂²	—	2	2	no	—
311	C₂²	—	2	4	no	—
312	C₂²	—	4	4	no	—
313	C₂²	—	332	4	no	—
314	C₂²	—	332	16	no	—
315	C₂²	—	332	4	no	—
316	C₂²	—	83	4	no	—
317	C₂²	—	166	4	no	—
318	C₂²	—	166	4	no	—
319	C₂²	C₂×D₄×D₈₃	1	2	yes	—
320	C₂²	C₂×D₄×D₈₃	1	1	yes	characteristic
321	C₄	—	2	4	no	—
322	C₄	—	83	4	no	—
323	C₄	—	166	4	no	—
324	C₄	C₂×D₄×D₈₃	1	2	yes	—
325	C₄	C₂×D₄×D₈₃	1	1	yes	characteristic
326	C₄	C₂⁴×D₈₃	1	1	yes	characteristic
327	C₂	—	2	2	no	—
328	C₂	—	4	4	no	—
329	C₂	—	332	4	no	—
330	C₂	—	166	4	no	—
331	C₂	#19128498	1	2	yes	direct product factor
332	C₂	C₂²×D₄×D₈₃	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	D₄×D₈₃
commutator, cosocle	C₄×C₈₃	C₂⁵
Frattini	C₄	C₂⁴×D₈₃
Fitting	C₂×#135×C₈₃	C₂
socle	C₂²×C₈₃	C₂²×D₄
hypercenter	C₂×#135	D₈₃
hypocenter	C₈₃	C₂×#575
LCS	G ▷ C₄×C₈₃ ▷ C₂×C₈₃ ▷ C₈₃
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×D₈₃	C₂	semidirect
2. #19128498	C₂	direct (N × H)
3. C₂×#14581824	C₂	semidirect
4. C₂×#14581821	C₂	semidirect
5. C₂×#14581819	C₂	semidirect
6. C₂×#14581816	C₂	semidirect
7. C₂×#14581814	C₂	semidirect
8. C₂×#14581812	C₂	semidirect
9. C₂²×#13050573	C₂	semidirect
10. C₂×#135×C₈₃	C₂	semidirect
13. C₂×#13050590	C₂²	semidirect
17. #41×D₈₃	C₂²	semidirect
18. C₂×Q₈×D₈₃	C₂²	semidirect
20. C₂×D₄×D₈₃	C₂²	semidirect
23. #14581824	C₂²	semidirect
24. #14581821	C₂²	semidirect
25. #14581819	C₂²	semidirect
26. #14581816	C₂²	semidirect
27. #14581814	C₂²	semidirect
28. #14581812	C₂²	semidirect
29. C₂×Dic₃₃₂	C₂²	semidirect
30. C₂×#13050580	C₂²	semidirect
31. C₂×#13050579	C₂²	semidirect
32. C₂×#13050578	C₂²	semidirect
33. C₂×#13050577	C₂²	semidirect
34. C₂×D₆₆₄	C₂²	semidirect
35. C₂×#13050575	C₂²	semidirect
36. C₂×#13050573	C₂²	semidirect
37. C₂×#13050573	C₂²	semidirect
38. C₂×#13050573	C₂²	semidirect
39. C₂²×C₈₃⋊C₈	C₂²	semidirect
40. C₂×#41×C₈₃	C₂²	semidirect
41. #135×C₈₃	C₂²	semidirect
42. C₂×Q₁₆×C₈₃	C₂²	semidirect
43. C₂×SD₁₆×C₈₃	C₂²	semidirect
44. C₂×D₈×C₈₃	C₂²	semidirect
45. C₂²×C₈×C₈₃	C₂²	semidirect
52. #13050590	C₂³	semidirect
59. Q₈×D₈₃	C₂³	semidirect
67. D₄×D₈₃	C₂³	semidirect
78. Dic₃₃₂	C₂³	semidirect
79. #13050580	C₂³	semidirect
80. #13050579	C₂³	semidirect
81. #13050578	C₂³	semidirect
82. #13050577	C₂³	semidirect
83. D₆₆₄	C₂³	semidirect
84. #13050575	C₂³	semidirect
85. #13050573	C₂³	semidirect
86. #13050573	C₂³	semidirect
88. C₂×C₈₃⋊C₈	C₂³	semidirect
90. #41×C₈₃	C₂³	semidirect
96. Q₁₆×C₈₃	C₂³	semidirect
97. SD₁₆×C₈₃	C₂³	semidirect
98. D₈×C₈₃	C₂³	semidirect
101. C₂×C₈×C₈₃	C₂³	semidirect
162. C₂×C₈₃	#575	semidirect
167. C₈₃	C₂×#575	semidirect
176. C₂×#135	D₈₃	semidirect
185. C₂×#41	C₂×D₈₃	semidirect
197. #135	C₂×D₈₃	semidirect
200. C₂×Q₁₆	C₂×D₈₃	semidirect
204. C₂×SD₁₆	C₂×D₈₃	semidirect
207. C₂×D₈	C₂×D₈₃	semidirect
212. C₂²×C₈	C₂×D₈₃	semidirect
224. #41	C₂²×D₈₃	semidirect
254. Q₁₆	C₂²×D₈₃	semidirect
258. SD₁₆	C₂²×D₈₃	semidirect
261. D₈	C₂²×D₈₃	semidirect
268. C₂×C₈	C₂²×D₈₃	semidirect
331. C₂	#19128498	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×D₈₃	2
#19128498	2
C₂×#14581824	2
C₂×#14581821	2
C₂×#14581819	2
C₂×#14581816	2
C₂×#14581814	2
C₂×#14581812	2
C₂²×#13050573	2
C₂×#135×C₈₃	2
C₂×#575	83

Subgroups — C2×#19128498 #24922207

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#19128498 #24922207