Subgroups — C₂×#23765520 #34442826

Order 20224.

678 conjugacy classes of subgroups.

The subgroup lattice has 310 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×D₁₂₆₄	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×#19098833	C₂	1	2	yes	maximal, complemented
3	#23765520	C₂	1	8	yes	maximal, direct product factor
4	C₂×#19098393	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂×#19098377	C₂	1	2	yes	maximal, complemented
6	C₂×#504×C₇₉	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂×#14579186	—	2	2	no	—
8	C₂²×D₆₃₂	C₂²	1	2	yes	—
9	C₂×D₁₂₆₄	—	2	1	no	—
10	C₂×D₁₂₆₄	—	2	1	no	—
11	C₂×D₁₂₆₄	—	2	2	no	—
12	C₂×D₁₂₆₄	C₂²	1	4	yes	complemented
13	#19098833	—	2	4	no	—
14	#19098833	C₂²	1	8	yes	complemented
15	C₂×#14579097	C₂²	1	2	yes	—
16	C₂×#14579077	—	2	2	no	—
17	#19098393	C₂²	1	4	yes	complemented
18	#19098377	C₂²	1	8	yes	complemented
19	C₂×#131×C₇₉	C₂²	1	1	yes	characteristic
20	#504×C₇₉	—	2	2	no	—
21	#504×C₇₉	C₂²	1	4	yes	complemented
22	C₂×M₃₂×C₇₉	—	2	1	no	—
23	C₂×M₃₂×C₇₉	C₂²	1	1	yes	characteristic
24	C₂²×C₁₆×C₇₉	—	2	1	no	—
25	C₂²×C₁₆×C₇₉	C₂²	1	1	yes	characteristic
26	C₂²×D₃₁₆	—	2	2	no	—
27	#14579186	—	2	8	no	—
28	#14579186	—	4	4	no	—
29	C₂×#11929611	—	4	2	no	—
30	C₂×D₆₃₂	—	2	2	no	—
31	C₂×D₆₃₂	—	2	2	no	—
32	C₂×D₆₃₂	—	2	2	no	—
33	C₂×D₆₃₂	—	2	2	no	—
34	C₂×D₆₃₂	—	2	2	no	—
35	C₂×D₆₃₂	C₂³	1	4	yes	—
36	C₂×#11929609	—	4	2	no	—
37	C₂×#11929608	—	2	2	no	—
38	D₁₂₆₄	—	2	4	no	—
39	D₁₂₆₄	—	2	4	no	—
40	#14579097	—	2	2	no	—
41	#14579097	C₂³	1	4	yes	—
42	#14579077	—	2	8	no	—
43	C₂×#41×C₇₉	D₄	1	1	yes	characteristic
44	#131×C₇₉	—	2	1	no	—
45	#131×C₇₉	—	2	1	no	—
46	#131×C₇₉	D₄	1	2	yes	—
47	#131×C₇₉	C₂³	1	2	yes	—
48	C₂×M₁₆×C₇₉	—	2	1	no	—
49	C₂×M₁₆×C₇₉	D₄	1	1	yes	characteristic
50	C₂²×C₈×C₇₉	—	2	1	no	—
51	C₂²×C₈×C₇₉	C₂³	1	1	yes	characteristic
52	M₃₂×C₇₉	—	2	1	no	—
53	M₃₂×C₇₉	—	2	4	no	—
54	M₃₂×C₇₉	C₂³	1	2	yes	—
55	C₂×C₁₆×C₇₉	—	2	1	no	—
56	C₂×C₁₆×C₇₉	—	2	4	no	—
57	C₂×C₁₆×C₇₉	D₄	1	1	yes	characteristic
58	C₂×C₁₆×C₇₉	D₄	1	1	yes	characteristic
59	C₂×C₁₆×C₇₉	D₄	1	2	yes	—
60	C₂×C₁₆×C₇₉	C₂³	1	2	yes	—
61	C₂³×D₇₉	—	4	2	no	—
62	C₂×D₃₁₆	—	2	4	no	—
63	C₂×D₃₁₆	—	4	2	no	—
64	C₂×D₃₁₆	—	4	2	no	—
65	C₂×D₃₁₆	—	4	2	no	—
66	C₂×D₃₁₆	—	4	2	no	—
67	C₂×D₃₁₆	—	4	2	no	—
68	#11929611	—	4	8	no	—
69	D₆₃₂	—	2	4	no	—
70	D₆₃₂	—	2	4	no	—
71	D₆₃₂	—	4	2	no	—
72	D₆₃₂	—	4	2	no	—
73	#11929609	—	4	8	no	—
74	#11929608	—	2	4	no	—
75	#11929608	—	4	2	no	—
76	C₂×C₇₉⋊C₈	—	4	2	no	—
77	#41×C₇₉	—	2	1	no	—
78	#41×C₇₉	—	2	1	no	—
79	#41×C₇₉	D₈	1	2	yes	—
80	#41×C₇₉	C₂×D₄	1	2	yes	—
81	C₂×Q₈×C₇₉	D₈	1	1	yes	characteristic
82	C₂×D₄×C₇₉	—	2	1	no	—
83	C₂×D₄×C₇₉	D₈	1	1	yes	characteristic
84	C₂²×C₄×C₇₉	—	2	1	no	—
85	C₂²×C₄×C₇₉	C₂×D₄	1	1	yes	characteristic
86	M₁₆×C₇₉	—	2	2	no	—
87	M₁₆×C₇₉	—	2	2	no	—
88	M₁₆×C₇₉	—	2	1	no	—
89	M₁₆×C₇₉	C₂×D₄	1	2	yes	—
90	C₂×C₈×C₇₉	—	2	2	no	—
91	C₂×C₈×C₇₉	—	2	2	no	—
92	C₂×C₈×C₇₉	—	2	1	no	—
93	C₂×C₈×C₇₉	C₂×D₄	1	1	yes	characteristic
94	C₂×C₈×C₇₉	C₂×D₄	1	1	yes	characteristic
95	C₂×C₈×C₇₉	C₂×D₄	1	1	yes	characteristic
96	C₂×C₈×C₇₉	#41	1	1	yes	characteristic
97	C₂×C₈×C₇₉	#41	1	1	yes	characteristic
98	C₂×C₈×C₇₉	C₂⁴	1	1	yes	characteristic
99	C₁₆×C₇₉	—	2	2	no	—
100	C₁₆×C₇₉	C₂×D₄	1	2	yes	—
101	C₁₆×C₇₉	C₂×D₄	1	2	yes	—
102	C₂³×C₇₉	—	2	1	no	—
103	C₂³×C₇₉	#135	1	1	yes	characteristic
104	C₂²×D₇₉	—	4	4	no	—
105	C₂²×D₇₉	—	8	2	no	—
106	C₂²×D₇₉	—	8	2	no	—
107	C₂²×D₇₉	—	16	4	no	—
108	Q₈×C₇₉	—	2	1	no	—
109	Q₈×C₇₉	C₂×D₈	1	2	yes	—
110	D₄×C₇₉	—	2	2	no	—
111	D₄×C₇₉	—	2	2	no	—
112	D₄×C₇₉	—	2	1	no	—
113	D₄×C₇₉	C₂×D₈	1	2	yes	—
114	C₂×C₄×C₇₉	—	2	1	no	—
115	C₂×C₄×C₇₉	—	2	2	no	—
116	C₂×C₄×C₇₉	—	2	2	no	—
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	1	yes	characteristic
120	C₂×C₄×C₇₉	C₂×D₈	1	1	yes	characteristic
121	C₂×C₄×C₇₉	#135	1	1	yes	characteristic
122	C₂×C₄×C₇₉	C₂²×D₄	1	1	yes	characteristic
123	D₃₁₆	—	4	4	no	—
124	D₃₁₆	—	4	4	no	—
125	D₃₁₆	—	8	2	no	—
126	D₃₁₆	—	8	2	no	—
127	C₈×C₇₉	—	2	2	no	—
128	C₈×C₇₉	#121	1	1	yes	characteristic
129	C₈×C₇₉	#121	1	1	yes	characteristic
130	C₈×C₇₉	C₂²×D₄	1	1	yes	characteristic
131	C₈×C₇₉	C₂×#41	1	1	yes	characteristic
132	C₇₉⋊C₈	—	4	4	no	—
133	C₂²×C₇₉	—	2	2	no	—
134	C₂²×C₇₉	—	2	2	no	—
135	C₂²×C₇₉	—	4	2	no	—
136	C₂²×C₇₉	#466	1	1	yes	characteristic
137	C₂²×C₇₉	#466	1	1	yes	characteristic
138	C₂²×C₇₉	C₂×#135	1	1	yes	characteristic
139	C₂×D₇₉	—	8	4	no	—
140	C₂×D₇₉	—	16	4	no	—
141	C₂×D₇₉	—	16	4	no	—
142	C₂×D₇₉	—	16	4	no	—
143	C₄×C₇₉	—	2	2	no	—
144	C₄×C₇₉	#466	1	1	yes	characteristic
145	C₄×C₇₉	#466	1	1	yes	characteristic
146	C₄×C₇₉	C₂×#121	1	1	yes	characteristic
147	C₄×C₇₉	C₂²×D₈	1	1	yes	characteristic
148	C₂×#2221	—	79	1	no	maximal, Sylow
149	C₂×C₇₉	—	2	1	no	—
150	C₂×C₇₉	—	2	1	no	—
151	C₂×C₇₉	—	4	2	no	—
152	C₂×C₇₉	#2221	1	2	yes	complemented
153	C₂×C₇₉	C₂×#466	1	1	yes	characteristic
154	D₇₉	—	16	4	no	—
155	C₂²×D₁₆	—	79	1	no	—
156	C₂×#504	D₇₉	1	1	yes	characteristic, complemented
157	C₂×#472	—	79	2	no	—
158	#2221	—	79	8	no	—
159	C₂×#368	—	79	1	no	—
160	C₂×#361	—	79	2	no	—
161	C₇₉	C₂×#2221	1	1	yes	characteristic, complemented, Sylow
162	C₂×AGL(1,ℤ/8)	—	158	2	no	—
163	C₂²×D₈	—	79	2	no	—
164	C₂×#131	C₂×D₇₉	1	1	yes	characteristic
165	C₂×D₁₆	—	79	4	no	—
166	C₂×D₁₆	—	158	1	no	—
167	C₂×D₁₆	—	158	1	no	—
168	C₂×D₁₆	—	158	2	no	—
169	#504	—	2	2	no	—
170	#504	C₂×D₇₉	1	4	yes	complemented
171	C₂×M₃₂	—	2	1	no	—
172	C₂×M₃₂	C₂×D₇₉	1	1	yes	characteristic
173	C₂²×C₁₆	—	2	1	no	—
174	C₂²×C₁₆	C₂×D₇₉	1	1	yes	characteristic
175	#472	—	79	8	no	—
176	#472	—	158	4	no	—
177	C₂×#108	—	79	2	no	—
178	C₂×#100	—	158	2	no	—
179	#368	—	79	4	no	—
180	#361	—	79	8	no	—
181	C₂×#41	D₃₁₆	1	1	yes	characteristic
182	C₂²×D₄	—	158	2	no	—
183	AGL(1,ℤ/8)	—	316	4	no	—
184	AGL(1,ℤ/8)	—	158	8	no	—
185	C₂×SD₁₆	—	316	2	no	—
186	C₂×D₈	—	316	2	no	—
187	C₂×D₈	—	79	4	no	—
188	C₂×D₈	—	158	2	no	—
189	C₂×D₈	—	158	2	no	—
190	C₂×D₈	—	158	2	no	—
191	C₂×D₈	—	158	2	no	—
192	C₂×D₈	—	158	2	no	—
193	#131	—	2	1	no	—
194	#131	—	2	1	no	—
195	#131	D₃₁₆	1	2	yes	—
196	#131	C₂²×D₇₉	1	2	yes	—
197	C₂×M₁₆	—	2	1	no	—
198	C₂×M₁₆	—	158	2	no	—
199	C₂×M₁₆	D₃₁₆	1	1	yes	characteristic
200	C₂²×C₈	—	2	1	no	—
201	C₂²×C₈	C₂²×D₇₉	1	1	yes	characteristic
202	D₁₆	—	158	4	no	—
203	D₁₆	—	158	4	no	—
204	M₃₂	—	2	1	no	—
205	M₃₂	—	2	4	no	—
206	M₃₂	C₂²×D₇₉	1	2	yes	—
207	C₂×C₁₆	—	2	1	no	—
208	C₂×C₁₆	—	2	4	no	—
209	C₂×C₁₆	C₂²⋊D₇₉	1	1	yes	characteristic
210	C₂×C₁₆	C₂²⋊D₇₉	1	1	yes	characteristic
211	C₂×C₁₆	C₂²⋊D₇₉	1	2	yes	—
212	C₂×C₁₆	C₂²×D₇₉	1	2	yes	—
213	#108	—	79	4	no	—
214	#108	—	158	2	no	—
215	#100	—	158	8	no	—
216	C₂⁴	—	316	2	no	—
217	#41	—	2	1	no	—
218	#41	—	2	1	no	—
219	#41	D₆₃₂	1	2	yes	—
220	#41	C₂×D₃₁₆	1	2	yes	—
221	C₂×Q₈	D₆₃₂	1	1	yes	characteristic
222	C₂×D₄	—	2	1	no	—
223	C₂×D₄	—	316	2	no	—
224	C₂×D₄	—	316	2	no	—
225	C₂×D₄	—	316	2	no	—
226	C₂×D₄	—	316	2	no	—
227	C₂×D₄	—	316	2	no	—
228	C₂×D₄	—	158	4	no	—
229	C₂×D₄	D₆₃₂	1	1	yes	characteristic
230	C₂²×C₄	—	2	1	no	—
231	C₂²×C₄	C₂×D₃₁₆	1	1	yes	characteristic
232	SD₁₆	—	316	8	no	—
233	D₈	—	316	8	no	—
234	D₈	—	316	2	no	—
235	D₈	—	316	2	no	—
236	D₈	—	158	4	no	—
237	D₈	—	158	4	no	—
238	M₁₆	—	2	1	no	—
239	M₁₆	—	2	2	no	—
240	M₁₆	—	2	2	no	—
241	M₁₆	—	316	2	no	—
242	M₁₆	—	158	4	no	—
243	M₁₆	C₂×D₃₁₆	1	2	yes	—
244	C₂×C₈	—	2	1	no	—
245	C₂×C₈	—	2	2	no	—
246	C₂×C₈	—	2	2	no	—
247	C₂×C₈	—	316	2	no	—
248	C₂×C₈	C₂×C₂²⋊D₇₉	1	1	yes	characteristic
249	C₂×C₈	C₂×C₂²⋊D₇₉	1	1	yes	characteristic
250	C₂×C₈	C₂×D₃₁₆	1	1	yes	characteristic
251	C₂×C₈	#11929624	1	1	yes	characteristic
252	C₂×C₈	#11929624	1	1	yes	characteristic
253	C₂×C₈	C₂³×D₇₉	1	1	yes	characteristic
254	C₁₆	—	2	2	no	—
255	C₁₆	C₂×C₂²⋊D₇₉	1	2	yes	—
256	C₁₆	C₂×C₂²⋊D₇₉	1	2	yes	—
257	C₂³	—	2	1	no	—
258	C₂³	—	316	4	no	—
259	C₂³	—	632	2	no	—
260	C₂³	—	632	2	no	—
261	C₂³	—	1264	4	no	—
262	C₂³	#14579183	1	1	yes	characteristic
263	Q₈	—	2	1	no	—
264	Q₈	C₂×D₆₃₂	1	2	yes	—
265	D₄	—	2	2	no	—
266	D₄	—	2	2	no	—
267	D₄	—	2	1	no	—
268	D₄	—	316	4	no	—
269	D₄	—	316	4	no	—
270	D₄	—	632	2	no	—
271	D₄	—	632	2	no	—
272	D₄	C₂×D₆₃₂	1	2	yes	—
273	C₂×C₄	—	2	1	no	—
274	C₂×C₄	—	2	2	no	—
275	C₂×C₄	—	2	2	no	—
276	C₂×C₄	#14579135	1	1	yes	characteristic
277	C₂×C₄	#14579135	1	1	yes	characteristic
278	C₂×C₄	C₂×D₆₃₂	1	1	yes	characteristic
279	C₂×C₄	C₂×D₆₃₂	1	1	yes	characteristic
280	C₂×C₄	#14579183	1	1	yes	characteristic
281	C₂×C₄	C₂²×D₃₁₆	1	1	yes	characteristic
282	C₈	—	2	2	no	—
283	C₈	—	316	4	no	—
284	C₈	#14579135	1	1	yes	characteristic
285	C₈	#14579135	1	1	yes	characteristic
286	C₈	C₂²×C₂²⋊D₇₉	1	1	yes	characteristic
287	C₈	C₂×#11929624	1	1	yes	characteristic
288	C₂²	—	2	2	no	—
289	C₂²	—	2	2	no	—
290	C₂²	—	4	2	no	—
291	C₂²	—	632	4	no	—
292	C₂²	—	1264	4	no	—
293	C₂²	—	1264	4	no	—
294	C₂²	—	1264	4	no	—
295	C₂²	#19098799	1	1	yes	characteristic
296	C₂²	#19098799	1	1	yes	characteristic
297	C₂²	C₂×#14579183	1	1	yes	characteristic
298	C₄	—	2	2	no	—
299	C₄	#19098799	1	1	yes	characteristic
300	C₄	#19098799	1	1	yes	characteristic
301	C₄	C₂×#14579135	1	1	yes	characteristic
302	C₄	C₂²×D₆₃₂	1	1	yes	characteristic
303	C₂	—	2	1	no	—
304	C₂	—	2	1	no	—
305	C₂	—	4	2	no	—
306	C₂	—	1264	4	no	—
307	C₂	#23765520	1	2	yes	direct product factor
308	C₂	C₂×#19098799	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	#19098799
commutator, cosocle	C₂×C₈×C₇₉	C₂⁴
Frattini	C₂×C₈	C₂³×D₇₉
Fitting	C₂×#504×C₇₉	C₂
socle	C₂²×C₇₉	#466
hypercenter	C₂×#504	D₇₉
hypocenter	C₇₉	C₂×#2221
LCS	G ▷ C₂×C₈×C₇₉ ▷ C₄×C₇₉ ▷ C₂×C₇₉ ▷ C₇₉
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×#131 ◁ C₂×#504

Complements

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

N	H ≅ G/N	type
1. C₂²×D₁₂₆₄	C₂	semidirect
2. C₂×#19098833	C₂	semidirect
3. #23765520	C₂	direct (N × H)
4. C₂×#19098393	C₂	semidirect
5. C₂×#19098377	C₂	semidirect
6. C₂×#504×C₇₉	C₂	semidirect
12. C₂×D₁₂₆₄	C₂²	semidirect
14. #19098833	C₂²	semidirect
17. #19098393	C₂²	semidirect
18. #19098377	C₂²	semidirect
21. #504×C₇₉	C₂²	semidirect
152. C₂×C₇₉	#2221	semidirect
156. C₂×#504	D₇₉	semidirect
161. C₇₉	C₂×#2221	semidirect
170. #504	C₂×D₇₉	semidirect
307. C₂	#23765520	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×D₁₂₆₄	2
C₂×#19098833	2
#23765520	2
C₂×#19098393	2
C₂×#19098377	2
C₂×#504×C₇₉	2
C₂×#2221	79

Subgroups — C2×#23765520 #34442826

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#23765520 #34442826