Subgroups — C₂×#23762760 #34270597

Order 20224.

870 conjugacy classes of subgroups.

The subgroup lattice has 324 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#19098940	C₂	1	1	yes	maximal, characteristic
2	C₂×#19098763	C₂	1	2	yes	maximal, complemented
3	#23762760	C₂	1	8	yes	maximal, direct product factor
4	C₂×#19098316	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂×#19098306	C₂	1	2	yes	maximal, complemented
6	C₂×C₈×Q₈×C₇₉	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂²×#11929611	—	2	2	no	—
8	C₂²×D₆₃₂	—	2	1	no	—
9	C₂²×D₆₃₂	—	2	1	no	—
10	C₂×#14579158	C₂²	1	2	yes	—
11	#19098940	—	2	2	no	—
12	#19098940	C₂²	1	4	yes	—
13	#19098763	—	2	4	no	—
14	#19098763	C₂²	1	8	yes	complemented
15	C₂×C₄⋊(C₇₉⋊C₈)	C₂²	1	2	yes	—
16	C₂×#14579081	—	2	2	no	—
17	#19098316	C₂²	1	4	yes	complemented
18	#19098306	C₂²	1	8	yes	complemented
19	C₂×C₄×Q₈×C₇₉	C₂²	1	1	yes	characteristic
20	C₈×Q₈×C₇₉	—	2	2	no	—
21	C₈×Q₈×C₇₉	C₂²	1	4	yes	complemented
22	C₂×C₄⋊C₈×C₇₉	—	2	1	no	—
23	C₂×C₄⋊C₈×C₇₉	C₂²	1	1	yes	characteristic
24	C₂×C₄×C₈×C₇₉	—	2	1	no	—
25	C₂×C₄×C₈×C₇₉	C₂²	1	1	yes	characteristic
26	C₂²×D₃₁₆	—	2	2	no	—
27	C₂²×D₃₁₆	—	2	2	no	—
28	C₂²×D₃₁₆	—	2	2	no	—
29	C₂×#11929611	—	2	8	no	—
30	C₂×#11929611	—	4	8	no	—
31	C₂×D₆₃₂	—	2	4	no	—
32	C₂×D₆₃₂	—	2	4	no	—
33	C₂×D₆₃₂	—	4	4	no	—
34	C₂×D₆₃₂	—	4	4	no	—
35	C₂²×C₇₉⋊C₈	—	2	2	no	—
36	#14579158	—	2	2	no	—
37	#14579158	—	2	2	no	—
38	#14579158	—	2	2	no	—
39	#14579158	C₂³	1	4	yes	—
40	C₄⋊(C₇₉⋊C₈)	—	2	2	no	—
41	C₄⋊(C₇₉⋊C₈)	C₂³	1	4	yes	—
42	#14579081	—	2	8	no	—
43	C₂²×Q₈×C₇₉	D₄	1	1	yes	characteristic
44	C₂²×C₈×C₇₉	—	2	1	no	—
45	C₂²×C₈×C₇₉	D₄	1	1	yes	characteristic
46	C₂²×C₈×C₇₉	D₄	1	1	yes	characteristic
47	C₄×Q₈×C₇₉	—	2	1	no	—
48	C₄×Q₈×C₇₉	—	2	1	no	—
49	C₄×Q₈×C₇₉	D₄	1	2	yes	—
50	C₄×Q₈×C₇₉	C₂³	1	2	yes	—
51	C₂×C₄⋊C₄×C₇₉	—	2	1	no	—
52	C₂×C₄⋊C₄×C₇₉	D₄	1	1	yes	characteristic
53	C₂×C₄²×C₇₉	—	2	1	no	—
54	C₂×C₄²×C₇₉	C₂³	1	1	yes	characteristic
55	C₄⋊C₈×C₇₉	—	2	4	no	—
56	C₄⋊C₈×C₇₉	—	2	1	no	—
57	C₄⋊C₈×C₇₉	C₂³	1	2	yes	—
58	C₄×C₈×C₇₉	—	2	4	no	—
59	C₄×C₈×C₇₉	D₄	1	2	yes	—
60	C₄×C₈×C₇₉	C₂³	1	2	yes	—
61	C₂³×D₇₉	—	4	2	no	—
62	C₂×D₃₁₆	—	2	4	no	—
63	C₂×D₃₁₆	—	2	4	no	—
64	C₂×D₃₁₆	—	2	4	no	—
65	C₂×D₃₁₆	—	4	2	no	—
66	C₂×D₃₁₆	—	4	2	no	—
67	C₂×D₃₁₆	—	4	2	no	—
68	C₂×D₃₁₆	—	4	4	no	—
69	C₂×D₃₁₆	—	4	4	no	—
70	C₂×D₃₁₆	—	4	4	no	—
71	C₂×D₃₁₆	—	4	4	no	—
72	C₂×D₃₁₆	—	8	4	no	—
73	#11929611	—	4	16	no	—
74	D₆₃₂	—	4	8	no	—
75	D₆₃₂	—	4	8	no	—
76	C₂×C₇₉⋊C₈	—	2	4	no	—
77	C₂×C₇₉⋊C₈	—	4	4	no	—
78	C₂×Q₈×C₇₉	—	2	1	no	—
79	C₂×Q₈×C₇₉	—	2	2	no	—
80	C₂×Q₈×C₇₉	D₈	1	4	yes	complemented
81	C₂×Q₈×C₇₉	C₂×D₄	1	2	yes	—
82	C₂²×C₄×C₇₉	—	2	1	no	—
83	C₂²×C₄×C₇₉	—	2	1	no	—
84	C₂²×C₄×C₇₉	C₂×D₄	1	1	yes	characteristic
85	C₂²×C₄×C₇₉	C₂×D₄	1	1	yes	characteristic
86	C₂²×C₄×C₇₉	#41	1	1	yes	characteristic
87	C₂×C₈×C₇₉	—	2	2	no	—
88	C₂×C₈×C₇₉	—	2	2	no	—
89	C₂×C₈×C₇₉	—	4	2	no	—
90	C₂×C₈×C₇₉	SD₁₆	1	4	yes	complemented
91	C₂×C₈×C₇₉	C₂×D₄	1	2	yes	—
92	C₂×C₈×C₇₉	C₂×D₄	1	2	yes	—
93	C₄⋊C₄×C₇₉	—	2	2	no	—
94	C₄⋊C₄×C₇₉	—	2	1	no	—
95	C₄⋊C₄×C₇₉	—	2	2	no	—
96	C₄⋊C₄×C₇₉	C₂×D₄	1	2	yes	—
97	C₄²×C₇₉	—	2	2	no	—
98	C₄²×C₇₉	—	2	2	no	—
99	C₄²×C₇₉	C₂×D₄	1	1	yes	characteristic
100	C₄²×C₇₉	C₂×D₄	1	1	yes	characteristic
101	C₄²×C₇₉	#41	1	1	yes	characteristic
102	C₄²×C₇₉	C₂⁴	1	1	yes	characteristic
103	C₂³×C₇₉	#121	1	1	yes	characteristic
104	C₂²×D₇₉	—	4	4	no	—
105	C₂²×D₇₉	—	8	4	no	—
106	C₂²×D₇₉	—	8	4	no	—
107	C₂²×D₇₉	—	8	4	no	—
108	Q₈×C₇₉	—	2	2	no	—
109	Q₈×C₇₉	—	2	4	no	—
110	Q₈×C₇₉	C₂×D₈	1	4	yes	complemented
111	C₂×C₄×C₇₉	—	2	2	no	—
112	C₂×C₄×C₇₉	—	2	4	no	—
113	C₂×C₄×C₇₉	—	2	2	no	—
114	C₂×C₄×C₇₉	—	2	2	no	—
115	C₂×C₄×C₇₉	—	4	2	no	—
116	C₂×C₄×C₇₉	#121	1	1	yes	characteristic
117	C₂×C₄×C₇₉	#121	1	1	yes	characteristic
118	C₂×C₄×C₇₉	#121	1	1	yes	characteristic
119	C₂×C₄×C₇₉	C₂×D₈	1	2	yes	—
120	C₂×C₄×C₇₉	C₂×SD₁₆	1	2	yes	—
121	C₂×C₄×C₇₉	#135	1	2	yes	—
122	C₂×C₄×C₇₉	AGL(1,ℤ/8)	1	2	yes	—
123	C₂×C₄×C₇₉	C₂²×D₄	1	1	yes	characteristic
124	C₂×C₄×C₇₉	C₂²×D₄	1	1	yes	characteristic
125	C₂×C₄×C₇₉	C₂×#41	1	1	yes	characteristic
126	D₃₁₆	—	4	4	no	—
127	D₃₁₆	—	4	4	no	—
128	D₃₁₆	—	4	4	no	—
129	D₃₁₆	—	4	4	no	—
130	D₃₁₆	—	8	4	no	—
131	D₃₁₆	—	8	4	no	—
132	D₃₁₆	—	8	4	no	—
133	D₃₁₆	—	8	4	no	—
134	C₈×C₇₉	—	2	2	no	—
135	C₈×C₇₉	—	4	2	no	—
136	C₈×C₇₉	C₂×SD₁₆	1	4	yes	complemented
137	C₇₉⋊C₈	—	4	4	no	—
138	C₂²×C₇₉	#460	1	2	yes	—
139	C₂²×C₇₉	#466	1	2	yes	—
140	C₂²×C₇₉	#472	1	2	yes	—
141	C₂²×C₇₉	C₂×#121	1	1	yes	characteristic
142	C₂×D₇₉	—	8	4	no	—
143	C₂×D₇₉	—	8	4	no	—
144	C₂×D₇₉	—	8	4	no	—
145	C₂×D₇₉	—	16	8	no	—
146	C₄×C₇₉	—	2	4	no	—
147	C₄×C₇₉	—	2	1	no	—
148	C₄×C₇₉	—	2	1	no	—
149	C₄×C₇₉	—	4	2	no	—
150	C₄×C₇₉	#460	1	2	yes	—
151	C₄×C₇₉	#466	1	2	yes	—
152	C₄×C₇₉	C₂²×D₈	1	1	yes	characteristic
153	C₄×C₇₉	C₂²×SD₁₆	1	1	yes	characteristic
154	C₄×C₇₉	C₂×#135	1	1	yes	characteristic
155	C₄×C₇₉	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
156	C₂×#1668	—	79	1	no	maximal, Sylow
157	C₂×C₇₉	#1668	1	4	yes	complemented
158	C₂×C₇₉	C₂×#460	1	1	yes	characteristic
159	C₂×C₇₉	C₂×#466	1	1	yes	characteristic
160	C₂×C₇₉	C₂×#472	1	1	yes	characteristic
161	D₇₉	—	16	4	no	—
162	C₂×#493	—	79	1	no	—
163	C₂×#460	—	79	2	no	—
164	C₂×C₈×Q₈	D₇₉	1	1	yes	characteristic, complemented
165	#1668	—	79	8	no	—
166	C₂×#335	—	79	1	no	—
167	C₂×#331	—	79	2	no	—
168	C₇₉	C₂×#1668	1	1	yes	characteristic, complemented, Sylow
169	C₂²×SD₁₆	—	158	2	no	—
170	C₂²×D₈	—	158	1	no	—
171	C₂²×D₈	—	158	1	no	—
172	C₂×#127	—	79	2	no	—
173	C₂×C₄×Q₈	C₂×D₇₉	1	1	yes	characteristic
174	#493	—	79	4	no	—
175	#493	—	158	2	no	—
176	#460	—	79	8	no	—
177	#460	—	158	4	no	—
178	C₈×Q₈	—	2	2	no	—
179	C₈×Q₈	C₂×D₇₉	1	4	yes	complemented
180	C₂×C₄⋊C₈	—	2	1	no	—
181	C₂×C₄⋊C₈	—	79	2	no	—
182	C₂×C₄⋊C₈	C₂×D₇₉	1	1	yes	characteristic
183	C₂×#102	—	158	2	no	—
184	C₂×C₄×C₈	—	2	1	no	—
185	C₂×C₄×C₈	C₂×D₇₉	1	1	yes	characteristic
186	#335	—	79	4	no	—
187	#331	—	79	8	no	—
188	C₂²×Q₈	D₃₁₆	1	1	yes	characteristic
189	C₂²×D₄	—	158	2	no	—
190	C₂²×D₄	—	158	2	no	—
191	C₂²×D₄	—	158	2	no	—
192	C₂×SD₁₆	—	316	8	no	—
193	C₂×SD₁₆	—	158	8	no	—
194	C₂×D₈	—	316	4	no	—
195	C₂×D₈	—	316	4	no	—
196	C₂×D₈	—	158	4	no	—
197	C₂×D₈	—	158	4	no	—
198	C₂²×C₈	—	2	1	no	—
199	C₂²×C₈	—	158	2	no	—
200	C₂²×C₈	C₂²⋊D₇₉	1	1	yes	characteristic
201	C₂²×C₈	C₂²⋊D₇₉	1	1	yes	characteristic
202	#127	—	79	4	no	—
203	#127	—	158	2	no	—
204	#127	—	158	2	no	—
205	#127	—	158	2	no	—
206	C₄×Q₈	—	2	1	no	—
207	C₄×Q₈	—	2	1	no	—
208	C₄×Q₈	D₃₁₆	1	2	yes	—
209	C₄×Q₈	C₂²×D₇₉	1	2	yes	—
210	C₂×C₄⋊C₄	—	2	1	no	—
211	C₂×C₄⋊C₄	D₃₁₆	1	1	yes	characteristic
212	C₂×C₄²	—	2	1	no	—
213	C₂×C₄²	C₂²×D₇₉	1	1	yes	characteristic
214	C₄⋊C₈	—	2	4	no	—
215	C₄⋊C₈	—	2	1	no	—
216	C₄⋊C₈	—	79	4	no	—
217	C₄⋊C₈	—	158	2	no	—
218	C₄⋊C₈	C₂²×D₇₉	1	2	yes	—
219	#102	—	158	8	no	—
220	C₄×C₈	—	2	4	no	—
221	C₄×C₈	C₂²⋊D₇₉	1	2	yes	—
222	C₄×C₈	C₂²×D₇₉	1	2	yes	—
223	C₂⁴	—	316	2	no	—
224	C₂×Q₈	—	2	1	no	—
225	C₂×Q₈	—	2	2	no	—
226	C₂×Q₈	D₆₃₂	1	4	yes	complemented
227	C₂×Q₈	C₂×D₃₁₆	1	2	yes	—
228	C₂×D₄	—	316	2	no	—
229	C₂×D₄	—	316	2	no	—
230	C₂×D₄	—	316	2	no	—
231	C₂×D₄	—	316	4	no	—
232	C₂×D₄	—	316	4	no	—
233	C₂×D₄	—	316	4	no	—
234	C₂×D₄	—	316	4	no	—
235	C₂×D₄	—	632	4	no	—
236	C₂×D₄	—	158	4	no	—
237	C₂×D₄	—	158	4	no	—
238	C₂×D₄	—	158	4	no	—
239	C₂²×C₄	—	2	1	no	—
240	C₂²×C₄	—	2	1	no	—
241	C₂²×C₄	C₂×C₂²⋊D₇₉	1	1	yes	characteristic
242	C₂²×C₄	C₂×D₃₁₆	1	1	yes	characteristic
243	C₂²×C₄	#11929624	1	1	yes	characteristic
244	SD₁₆	—	316	16	no	—
245	D₈	—	316	8	no	—
246	D₈	—	316	8	no	—
247	C₂×C₈	—	2	2	no	—
248	C₂×C₈	—	2	2	no	—
249	C₂×C₈	—	4	2	no	—
250	C₂×C₈	—	316	4	no	—
251	C₂×C₈	—	158	4	no	—
252	C₂×C₈	#11929611	1	4	yes	complemented
253	C₂×C₈	C₂×C₂²⋊D₇₉	1	2	yes	—
254	C₂×C₈	C₂×C₂²⋊D₇₉	1	2	yes	—
255	C₄⋊C₄	—	2	2	no	—
256	C₄⋊C₄	—	2	2	no	—
257	C₄⋊C₄	—	2	1	no	—
258	C₄⋊C₄	C₂×D₃₁₆	1	2	yes	—
259	C₄²	—	2	2	no	—
260	C₄²	—	2	2	no	—
261	C₄²	C₂×C₂²⋊D₇₉	1	1	yes	characteristic
262	C₄²	C₂×D₃₁₆	1	1	yes	characteristic
263	C₄²	#11929624	1	1	yes	characteristic
264	C₄²	C₂³×D₇₉	1	1	yes	characteristic
265	C₂³	—	316	4	no	—
266	C₂³	—	632	4	no	—
267	C₂³	—	632	4	no	—
268	C₂³	—	632	4	no	—
269	C₂³	#14579135	1	1	yes	characteristic
270	Q₈	—	2	4	no	—
271	Q₈	—	2	2	no	—
272	Q₈	C₂×D₆₃₂	1	4	yes	complemented
273	D₄	—	316	4	no	—
274	D₄	—	316	4	no	—
275	D₄	—	316	4	no	—
276	D₄	—	316	4	no	—
277	D₄	—	632	4	no	—
278	D₄	—	632	4	no	—
279	D₄	—	632	4	no	—
280	D₄	—	632	4	no	—
281	C₂×C₄	—	2	4	no	—
282	C₂×C₄	—	2	2	no	—
283	C₂×C₄	—	2	2	no	—
284	C₂×C₄	—	2	2	no	—
285	C₂×C₄	—	4	2	no	—
286	C₂×C₄	#14579135	1	1	yes	characteristic
287	C₂×C₄	#14579135	1	1	yes	characteristic
288	C₂×C₄	#14579135	1	1	yes	characteristic
289	C₂×C₄	C₂×D₆₃₂	1	2	yes	—
290	C₂×C₄	C₂×#11929611	1	2	yes	—
291	C₂×C₄	#14579183	1	2	yes	—
292	C₂×C₄	#14579186	1	2	yes	—
293	C₂×C₄	C₂²×C₂²⋊D₇₉	1	1	yes	characteristic
294	C₂×C₄	C₂²×D₃₁₆	1	1	yes	characteristic
295	C₂×C₄	C₂×#11929624	1	1	yes	characteristic
296	C₈	—	2	2	no	—
297	C₈	—	4	2	no	—
298	C₈	—	316	4	no	—
299	C₈	C₂×#11929611	1	4	yes	complemented
300	C₂²	—	632	4	no	—
301	C₂²	—	632	4	no	—
302	C₂²	—	632	4	no	—
303	C₂²	—	1264	8	no	—
304	C₂²	#19098763	1	2	yes	—
305	C₂²	#19098799	1	2	yes	—
306	C₂²	#19098833	1	2	yes	—
307	C₂²	C₂×#14579135	1	1	yes	characteristic
308	C₄	—	2	4	no	—
309	C₄	—	2	1	no	—
310	C₄	—	2	1	no	—
311	C₄	—	4	2	no	—
312	C₄	#19098763	1	2	yes	—
313	C₄	#19098799	1	2	yes	—
314	C₄	C₂²×D₆₃₂	1	1	yes	characteristic
315	C₄	C₂²×#11929611	1	1	yes	characteristic
316	C₄	C₂×#14579183	1	1	yes	characteristic
317	C₄	C₂×#14579186	1	1	yes	characteristic
318	C₂	—	1264	4	no	—
319	C₂	#23762760	1	4	yes	direct product factor
320	C₂	C₂×#19098763	1	1	yes	characteristic
321	C₂	C₂×#19098799	1	1	yes	characteristic
322	C₂	C₂×#19098833	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	#14579135
commutator, cosocle	C₄²×C₇₉	C₂⁴
Frattini	C₄²	C₂³×D₇₉
Fitting	C₂×C₈×Q₈×C₇₉	C₂
socle	C₂³×C₇₉	#121
prime radical	C₂×#19098940	C₂
hypercenter	C₂×C₈×Q₈	D₇₉
hypocenter	C₇₉	C₂×#1668
LCS	G ▷ C₄²×C₇₉ ▷ C₂²×C₇₉ ▷ C₇₉
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×C₈×Q₈

Complements

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

N	H ≅ G/N	type
2. C₂×#19098763	C₂	semidirect
3. #23762760	C₂	direct (N × H)
4. C₂×#19098316	C₂	semidirect
5. C₂×#19098306	C₂	semidirect
6. C₂×C₈×Q₈×C₇₉	C₂	semidirect
14. #19098763	C₂²	semidirect
17. #19098316	C₂²	semidirect
18. #19098306	C₂²	semidirect
21. C₈×Q₈×C₇₉	C₂²	semidirect
80. C₂×Q₈×C₇₉	D₈	semidirect
90. C₂×C₈×C₇₉	SD₁₆	semidirect
110. Q₈×C₇₉	C₂×D₈	semidirect
136. C₈×C₇₉	C₂×SD₁₆	semidirect
157. C₂×C₇₉	#1668	semidirect
164. C₂×C₈×Q₈	D₇₉	semidirect
168. C₇₉	C₂×#1668	semidirect
179. C₈×Q₈	C₂×D₇₉	semidirect
226. C₂×Q₈	D₆₃₂	semidirect
252. C₂×C₈	#11929611	semidirect
272. Q₈	C₂×D₆₃₂	semidirect
299. C₈	C₂×#11929611	semidirect
319. C₂	#23762760	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×#19098940	2
C₂×#19098763	2
#23762760	2
C₂×#19098316	2
C₂×#19098306	2
C₂×C₈×Q₈×C₇₉	2
C₂×#1668	79

Subgroups — C2×#23762760 #34270597

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#23762760 #34270597