Subgroups — C₂×#23762801 #34270628

Order 20224.

638 conjugacy classes of subgroups.

The subgroup lattice has 280 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#19098855	C₂	1	2	yes	maximal
2	C₂×C₈×Dic₁₅₈	C₂	1	1	yes	maximal, characteristic
3	#23762801	C₂	1	8	yes	maximal, direct product factor
4	C₂×#19098316	C₂	1	1	yes	maximal, characteristic
5	C₂×#19098312	C₂	1	2	yes	maximal
6	C₂×#500×C₇₉	C₂	1	1	yes	maximal, characteristic
7	C₂×C₄×Dic₁₅₈	C₂²	1	1	yes	characteristic
8	#19098855	—	2	4	no	—
9	#19098855	C₂²	1	8	yes	—
10	C₈×Dic₁₅₈	—	2	2	no	—
11	C₈×Dic₁₅₈	C₂²	1	4	yes	—
12	C₂×#14579094	—	2	2	no	—
13	C₂×C₄⋊(C₇₉⋊C₈)	C₂²	1	1	yes	characteristic
14	C₂×C₄⋊(C₇₉⋊C₈)	C₂²	1	2	yes	—
15	C₂×Dic₇₉⋊C₈	—	2	1	no	—
16	C₂×#14579084	—	2	2	no	—
17	C₂×C₈×Dic₇₉	—	2	1	no	—
18	#19098316	C₂²	1	4	yes	—
19	#19098312	C₂²	1	8	yes	—
20	C₂×C₄⋊Q₈×C₇₉	C₂²	1	2	yes	—
21	#500×C₇₉	—	2	2	no	—
22	#500×C₇₉	C₂²	1	4	yes	—
23	C₂×#107×C₇₉	—	2	1	no	—
24	C₂×#107×C₇₉	—	2	1	no	—
25	C₂×C₄×C₈×C₇₉	C₂²	1	1	yes	characteristic
26	C₂²×Dic₁₅₈	D₄	1	1	yes	characteristic
27	C₂²×C₇₉⋊C₈	—	2	1	no	—
28	C₂²×C₇₉⋊C₈	—	2	2	no	—
29	C₄×Dic₁₅₈	—	2	1	no	—
30	C₄×Dic₁₅₈	—	2	1	no	—
31	C₄×Dic₁₅₈	D₄	1	2	yes	—
32	C₄×Dic₁₅₈	C₂³	1	2	yes	—
33	C₂×C₄⋊Dic₇₉	D₄	1	1	yes	characteristic
34	C₂×Dic₇₉⋊C₄	—	2	1	no	—
35	C₂×C₄×Dic₇₉	—	2	1	no	—
36	#14579094	—	2	8	no	—
37	C₄⋊(C₇₉⋊C₈)	—	2	1	no	—
38	C₄⋊(C₇₉⋊C₈)	—	2	2	no	—
39	C₄⋊(C₇₉⋊C₈)	C₂³	1	2	yes	—
40	C₄⋊(C₇₉⋊C₈)	C₂³	1	4	yes	—
41	Dic₇₉⋊C₈	—	2	4	no	—
42	#14579084	—	2	8	no	—
43	C₈×Dic₇₉	—	2	4	no	—
44	C₂²×Q₈×C₇₉	—	2	2	no	—
45	C₂²×C₈×C₇₉	D₄	1	1	yes	characteristic
46	C₂²×C₈×C₇₉	D₄	1	1	yes	characteristic
47	C₄⋊Q₈×C₇₉	—	2	2	no	—
48	C₄⋊Q₈×C₇₉	—	2	2	no	—
49	C₄⋊Q₈×C₇₉	—	2	2	no	—
50	C₄⋊Q₈×C₇₉	C₂³	1	4	yes	—
51	C₂×C₄⋊C₄×C₇₉	—	2	2	no	—
52	C₂×C₄⋊C₄×C₇₉	—	2	2	no	—
53	C₂×C₄²×C₇₉	C₂³	1	1	yes	characteristic
54	#107×C₇₉	—	2	4	no	—
55	#107×C₇₉	—	2	4	no	—
56	C₄×C₈×C₇₉	D₄	1	2	yes	—
57	C₄×C₈×C₇₉	C₂³	1	2	yes	—
58	C₂×Dic₁₅₈	—	2	1	no	—
59	C₂×Dic₁₅₈	—	2	2	no	—
60	C₂×Dic₁₅₈	Q₁₆	1	4	yes	—
61	C₂×Dic₁₅₈	C₂×D₄	1	2	yes	—
62	C₂²×Dic₇₉	—	2	1	no	—
63	C₂²×Dic₇₉	—	2	1	no	—
64	C₂×C₇₉⋊C₈	—	2	2	no	—
65	C₂×C₇₉⋊C₈	—	2	4	no	—
66	C₂×C₇₉⋊C₈	—	4	4	no	—
67	C₂×C₇₉⋊C₈	—	4	2	no	—
68	C₄⋊Dic₇₉	—	2	1	no	—
69	C₄⋊Dic₇₉	C₂×D₄	1	2	yes	—
70	Dic₇₉⋊C₄	—	2	2	no	—
71	Dic₇₉⋊C₄	—	2	2	no	—
72	C₄×Dic₇₉	—	2	2	no	—
73	C₄×Dic₇₉	—	2	2	no	—
74	C₂×Q₈×C₇₉	—	2	4	no	—
75	C₂×Q₈×C₇₉	—	4	2	no	—
76	C₂×Q₈×C₇₉	—	8	4	no	—
77	C₂²×C₄×C₇₉	—	4	2	no	—
78	C₂²×C₄×C₇₉	C₂×D₄	1	1	yes	characteristic
79	C₂²×C₄×C₇₉	C₂×D₄	1	1	yes	characteristic
80	C₂²×C₄×C₇₉	#41	1	1	yes	characteristic
81	C₂×C₈×C₇₉	—	2	2	no	—
82	C₂×C₈×C₇₉	Q₁₆	1	4	yes	—
83	C₂×C₈×C₇₉	C₂×D₄	1	2	yes	—
84	C₂×C₈×C₇₉	C₂×D₄	1	2	yes	—
85	C₄⋊C₄×C₇₉	—	2	4	no	—
86	C₄⋊C₄×C₇₉	—	2	4	no	—
87	C₄⋊C₄×C₇₉	—	4	2	no	—
88	C₄⋊C₄×C₇₉	—	4	2	no	—
89	C₄²×C₇₉	C₂×D₄	1	1	yes	characteristic
90	C₄²×C₇₉	C₂×D₄	1	1	yes	characteristic
91	C₄²×C₇₉	#41	1	1	yes	characteristic
92	C₄²×C₇₉	C₂⁴	1	1	yes	characteristic
93	C₂³×C₇₉	#121	1	1	yes	characteristic
94	Q₈×C₇₉	—	8	4	no	—
95	Q₈×C₇₉	—	8	4	no	—
96	C₂×C₄×C₇₉	—	2	2	no	—
97	C₂×C₄×C₇₉	—	4	4	no	—
98	C₂×C₄×C₇₉	—	8	4	no	—
99	C₂×C₄×C₇₉	#121	1	1	yes	characteristic
100	C₂×C₄×C₇₉	#121	1	1	yes	characteristic
101	C₂×C₄×C₇₉	#121	1	1	yes	characteristic
102	C₂×C₄×C₇₉	C₂×Q₁₆	1	2	yes	—
103	C₂×C₄×C₇₉	C₂×Q₁₆	1	2	yes	—
104	C₂×C₄×C₇₉	#135	1	2	yes	—
105	C₂×C₄×C₇₉	#137	1	2	yes	—
106	C₂×C₄×C₇₉	C₂²×D₄	1	1	yes	characteristic
107	C₂×C₄×C₇₉	C₂²×D₄	1	1	yes	characteristic
108	C₂×C₄×C₇₉	C₂×#41	1	1	yes	characteristic
109	C₂×Dic₇₉	—	2	2	no	—
110	C₂×Dic₇₉	—	2	4	no	—
111	C₂×Dic₇₉	—	2	2	no	—
112	C₂×Dic₇₉	—	4	2	no	—
113	Dic₁₅₈	—	2	2	no	—
114	Dic₁₅₈	—	2	4	no	—
115	Dic₁₅₈	C₂×Q₁₆	1	4	yes	—
116	C₈×C₇₉	—	2	2	no	—
117	C₈×C₇₉	C₂×Q₁₆	1	4	yes	—
118	C₇₉⋊C₈	—	4	4	no	—
119	C₇₉⋊C₈	—	4	2	no	—
120	C₂²×C₇₉	#462	1	2	yes	—
121	C₂²×C₇₉	#467	1	2	yes	—
122	C₂²×C₇₉	#472	1	2	yes	—
123	C₂²×C₇₉	C₂×#121	1	1	yes	characteristic
124	C₄×C₇₉	—	2	1	no	—
125	C₄×C₇₉	—	2	1	no	—
126	C₄×C₇₉	—	8	4	no	—
127	C₄×C₇₉	#462	1	2	yes	—
128	C₄×C₇₉	#467	1	2	yes	—
129	C₄×C₇₉	C₂²×Q₁₆	1	1	yes	characteristic
130	C₄×C₇₉	C₂²×Q₁₆	1	1	yes	characteristic
131	C₄×C₇₉	C₂×#135	1	1	yes	characteristic
132	C₄×C₇₉	C₂×#137	1	1	yes	characteristic
133	Dic₇₉	—	2	4	no	—
134	Dic₇₉	—	4	2	no	—
135	C₂×#1676	—	79	1	no	maximal, Sylow
136	C₂×C₇₉	#1676	1	4	yes	complemented
137	C₂×C₇₉	C₂×#462	1	1	yes	characteristic
138	C₂×C₇₉	C₂×#467	1	1	yes	characteristic
139	C₂×C₇₉	C₂×#472	1	1	yes	characteristic
140	C₂×#500	D₇₉	1	1	yes	characteristic
141	C₂×#477	—	79	2	no	—
142	C₂×C₈×Q₈	—	79	1	no	—
143	#1676	—	79	8	no	—
144	C₂×#335	—	79	1	no	—
145	C₂×#333	—	79	2	no	—
146	C₇₉	C₂×#1676	1	1	yes	characteristic, complemented, Sylow
147	C₂×C₄⋊Q₈	C₂×D₇₉	1	2	yes	—
148	C₂×C₄×Q₈	—	79	1	no	—
149	#500	—	2	2	no	—
150	#500	C₂×D₇₉	1	4	yes	—
151	#477	—	79	8	no	—
152	#477	—	158	4	no	—
153	C₈×Q₈	—	79	4	no	—
154	C₈×Q₈	—	158	2	no	—
155	C₂×#107	—	2	1	no	—
156	C₂×#107	—	2	1	no	—
157	C₂×#107	—	158	2	no	—
158	C₂×C₄⋊C₈	—	79	2	no	—
159	C₂×C₄⋊C₈	—	79	1	no	—
160	C₂×C₄⋊C₈	—	158	1	no	—
161	C₂×#103	—	158	2	no	—
162	C₂×C₄×C₈	—	158	1	no	—
163	C₂×C₄×C₈	C₂×D₇₉	1	1	yes	characteristic
164	#335	—	79	4	no	—
165	#333	—	79	8	no	—
166	C₂²×Q₈	—	2	2	no	—
167	C₂²×Q₈	—	79	1	no	—
168	C₂²×C₈	—	158	2	no	—
169	C₂²×C₈	—	158	1	no	—
170	C₂²×C₈	C₂²⋊D₇₉	1	1	yes	characteristic
171	C₂²×C₈	C₂²⋊D₇₉	1	1	yes	characteristic
172	C₄⋊Q₈	—	2	2	no	—
173	C₄⋊Q₈	—	2	2	no	—
174	C₄⋊Q₈	—	2	2	no	—
175	C₄⋊Q₈	C₂²×D₇₉	1	4	yes	—
176	C₄×Q₈	—	79	2	no	—
177	C₄×Q₈	—	79	2	no	—
178	C₄×Q₈	—	158	1	no	—
179	C₄×Q₈	—	158	1	no	—
180	C₂×C₄⋊C₄	—	2	2	no	—
181	C₂×C₄⋊C₄	—	2	2	no	—
182	C₂×C₄⋊C₄	—	79	1	no	—
183	C₂×C₄⋊C₄	—	158	1	no	—
184	C₂×C₄²	—	158	1	no	—
185	C₂×C₄²	C₂²×D₇₉	1	1	yes	characteristic
186	#107	—	2	4	no	—
187	#107	—	2	4	no	—
188	#107	—	158	8	no	—
189	C₄⋊C₈	—	79	2	no	—
190	C₄⋊C₈	—	79	4	no	—
191	C₄⋊C₈	—	158	4	no	—
192	C₄⋊C₈	—	158	2	no	—
193	C₄⋊C₈	—	158	1	no	—
194	#103	—	158	8	no	—
195	C₄×C₈	—	158	4	no	—
196	C₄×C₈	C₂²⋊D₇₉	1	2	yes	—
197	C₄×C₈	C₂²×D₇₉	1	2	yes	—
198	C₂×Q₈	—	2	4	no	—
199	C₂×Q₈	—	4	2	no	—
200	C₂×Q₈	—	8	4	no	—
201	C₂×Q₈	—	79	4	no	—
202	C₂×Q₈	—	79	2	no	—
203	C₂×Q₈	—	158	1	no	—
204	C₂×Q₈	—	158	2	no	—
205	C₂²×C₄	—	4	2	no	—
206	C₂²×C₄	—	158	1	no	—
207	C₂²×C₄	—	158	1	no	—
208	C₂²×C₄	D₄×D₇₉	1	1	yes	characteristic
209	C₂²×C₄	C₂×C₂²⋊D₇₉	1	1	yes	characteristic
210	C₂²×C₄	#11929623	1	1	yes	characteristic
211	C₂×C₈	—	2	2	no	—
212	C₂×C₈	—	316	4	no	—
213	C₂×C₈	—	316	2	no	—
214	C₂×C₈	—	158	4	no	—
215	C₂×C₈	—	158	2	no	—
216	C₂×C₈	#11929614	1	4	yes	—
217	C₂×C₈	C₂×C₂²⋊D₇₉	1	2	yes	—
218	C₂×C₈	C₂×C₂²⋊D₇₉	1	2	yes	—
219	C₄⋊C₄	—	2	4	no	—
220	C₄⋊C₄	—	2	4	no	—
221	C₄⋊C₄	—	4	2	no	—
222	C₄⋊C₄	—	4	2	no	—
223	C₄⋊C₄	—	79	2	no	—
224	C₄⋊C₄	—	158	2	no	—
225	C₄⋊C₄	—	158	2	no	—
226	C₄⋊C₄	—	158	1	no	—
227	C₄²	—	158	2	no	—
228	C₄²	—	158	2	no	—
229	C₄²	D₄×D₇₉	1	1	yes	characteristic
230	C₄²	C₂×C₂²⋊D₇₉	1	1	yes	characteristic
231	C₄²	#11929623	1	1	yes	characteristic
232	C₄²	C₂³×D₇₉	1	1	yes	characteristic
233	C₂³	#14579132	1	1	yes	characteristic
234	Q₈	—	8	4	no	—
235	Q₈	—	8	4	no	—
236	Q₈	—	79	4	no	—
237	Q₈	—	158	4	no	—
238	Q₈	—	158	2	no	—
239	C₂×C₄	—	2	2	no	—
240	C₂×C₄	—	4	4	no	—
241	C₂×C₄	—	8	4	no	—
242	C₂×C₄	—	316	2	no	—
243	C₂×C₄	—	158	2	no	—
244	C₂×C₄	—	158	2	no	—
245	C₂×C₄	—	158	4	no	—
246	C₂×C₄	#14579132	1	1	yes	characteristic
247	C₂×C₄	#14579132	1	1	yes	characteristic
248	C₂×C₄	#14579132	1	1	yes	characteristic
249	C₂×C₄	Q₁₆×D₇₉	1	2	yes	—
250	C₂×C₄	C₂×#11929614	1	2	yes	—
251	C₂×C₄	#14579184	1	2	yes	—
252	C₂×C₄	#14579192	1	2	yes	—
253	C₂×C₄	C₂×D₄×D₇₉	1	1	yes	characteristic
254	C₂×C₄	C₂²×C₂²⋊D₇₉	1	1	yes	characteristic
255	C₂×C₄	C₂×#11929623	1	1	yes	characteristic
256	C₈	—	2	2	no	—
257	C₈	—	316	4	no	—
258	C₈	—	316	2	no	—
259	C₈	C₂×#11929614	1	4	yes	—
260	C₂²	#19098770	1	2	yes	—
261	C₂²	#19098801	1	2	yes	—
262	C₂²	#19098830	1	2	yes	—
263	C₂²	C₂×#14579132	1	1	yes	characteristic
264	C₄	—	2	1	no	—
265	C₄	—	2	1	no	—
266	C₄	—	8	4	no	—
267	C₄	—	316	2	no	—
268	C₄	—	158	4	no	—
269	C₄	#19098770	1	2	yes	—
270	C₄	#19098801	1	2	yes	—
271	C₄	C₂×Q₁₆×D₇₉	1	1	yes	characteristic
272	C₄	C₂²×#11929614	1	1	yes	characteristic
273	C₄	C₂×#14579184	1	1	yes	characteristic
274	C₄	C₂×#14579192	1	1	yes	characteristic
275	C₂	#23762801	1	4	yes	direct product factor
276	C₂	C₂×#19098770	1	1	yes	characteristic
277	C₂	C₂×#19098801	1	1	yes	characteristic
278	C₂	C₂×#19098830	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	#14579132
commutator, cosocle	C₄²×C₇₉	C₂⁴
Frattini	C₄²	C₂³×D₇₉
Fitting	C₂×#500×C₇₉	C₂
socle, prime radical	C₂³×C₇₉	#121
hypercenter	C₂×#500	D₇₉
hypocenter	C₇₉	C₂×#1676
LCS	G ▷ C₄²×C₇₉ ▷ C₂²×C₇₉ ▷ C₇₉
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×#500

Complements

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

N	H ≅ G/N	type
3. #23762801	C₂	direct (N × H)
136. C₂×C₇₉	#1676	semidirect
146. C₇₉	C₂×#1676	semidirect
275. C₂	#23762801	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×#19098855	2
C₂×C₈×Dic₁₅₈	2
#23762801	2
C₂×#19098316	2
C₂×#19098312	2
C₂×#500×C₇₉	2
C₂×#1676	79

Subgroups — C2×#23762801 #34270628

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#23762801 #34270628