Subgroups — C₂×#23761415 #34198514

Order 20224.

498 conjugacy classes of subgroups.

The subgroup lattice has 308 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₂²⋊(C₇₉⋊C₁₆)	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×#19098328	C₂	1	1	yes	maximal, characteristic, complemented
3	#23761415	C₂	1	4	yes	maximal, direct product factor
4	C₂×#466×C₇₉	C₂	1	1	yes	maximal, characteristic
5	C₂²×#11929608	—	2	1	no	—
6	C₂²×C₇₉⋊C₁₆	—	2	1	no	—
7	C₂²⋊(C₇₉⋊C₁₆)	—	2	1	no	—
8	C₂²⋊(C₇₉⋊C₁₆)	C₂²	1	2	yes	complemented
9	#19098328	—	2	1	no	—
10	#19098328	C₂²	1	2	yes	complemented
11	C₂²×D₈×C₇₉	C₄	1	1	yes	characteristic
12	C₂³×C₈×C₇₉	C₂²	1	1	yes	characteristic
13	C₂×#121×C₇₉	—	2	1	no	—
14	#466×C₇₉	—	2	1	no	—
15	#466×C₇₉	—	2	1	no	—
16	#466×C₇₉	C₄	1	2	yes	—
17	#466×C₇₉	C₂²	1	2	yes	—
18	C₂×#107×C₇₉	C₄	1	1	yes	characteristic
19	C₂×#102×C₇₉	—	2	1	no	—
20	C₂×#11929608	—	2	4	no	—
21	C₂×#11929608	—	2	2	no	—
22	C₂×#11929608	—	4	2	no	—
23	C₂×#11929608	—	4	1	no	—
24	C₂²×C₇₉⋊C₈	—	4	1	no	—
25	C₂×C₇₉⋊C₁₆	—	2	2	no	—
26	C₂×C₇₉⋊C₁₆	—	4	2	no	—
27	C₂²×D₄×C₇₉	—	2	1	no	—
28	C₂²×D₄×C₇₉	—	4	1	no	—
29	C₂³×C₄×C₇₉	D₄	1	1	yes	characteristic
30	C₂×D₈×C₇₉	—	2	1	no	—
31	C₂×D₈×C₇₉	—	4	2	no	—
32	C₂×D₈×C₇₉	C₂×C₄	1	2	yes	—
33	C₂²×C₈×C₇₉	—	4	2	no	—
34	C₂²×C₈×C₇₉	C₂×C₄	1	1	yes	characteristic
35	C₂²×C₈×C₇₉	C₂×C₄	1	1	yes	characteristic
36	C₂²×C₈×C₇₉	D₄	1	1	yes	characteristic
37	C₂²×C₈×C₇₉	D₄	1	1	yes	characteristic
38	C₂²×C₈×C₇₉	D₄	1	1	yes	characteristic
39	C₂²×C₈×C₇₉	C₂³	1	1	yes	characteristic
40	#121×C₇₉	—	2	2	no	—
41	#121×C₇₉	—	2	2	no	—
42	#121×C₇₉	—	4	1	no	—
43	#121×C₇₉	—	4	1	no	—
44	C₂×C₄⋊C₄×C₇₉	—	2	1	no	—
45	C₂×C₂²⋊C₄×C₇₉	—	4	1	no	—
46	#107×C₇₉	—	2	1	no	—
47	#107×C₇₉	C₂×C₄	1	2	yes	—
48	#102×C₇₉	—	2	2	no	—
49	#102×C₇₉	—	2	2	no	—
50	#11929608	—	2	4	no	—
51	#11929608	—	4	4	no	—
52	#11929608	—	4	2	no	—
53	C₂×C₇₉⋊C₈	—	4	4	no	—
54	C₂×C₇₉⋊C₈	—	4	2	no	—
55	C₇₉⋊C₁₆	—	4	2	no	—
56	C₂⁴×C₇₉	—	4	1	no	—
57	C₂⁴×C₇₉	SD₁₆	1	1	yes	characteristic
58	C₂×D₄×C₇₉	—	2	2	no	—
59	C₂×D₄×C₇₉	—	4	2	no	—
60	C₂×D₄×C₇₉	—	4	2	no	—
61	C₂×D₄×C₇₉	—	4	2	no	—
62	C₂×D₄×C₇₉	—	4	1	no	—
63	C₂×D₄×C₇₉	—	4	2	no	—
64	C₂×D₄×C₇₉	—	8	4	no	—
65	C₂²×C₄×C₇₉	—	2	2	no	—
66	C₂²×C₄×C₇₉	—	2	2	no	—
67	C₂²×C₄×C₇₉	—	4	1	no	—
68	C₂²×C₄×C₇₉	C₂²⋊C₄	1	1	yes	characteristic
69	C₂²×C₄×C₇₉	C₂²⋊C₄	1	1	yes	characteristic
70	C₂²×C₄×C₇₉	D₈	1	1	yes	characteristic
71	C₂²×C₄×C₇₉	D₈	1	1	yes	characteristic
72	C₂²×C₄×C₇₉	SD₁₆	1	1	yes	characteristic
73	C₂²×C₄×C₇₉	C₂×D₄	1	1	yes	characteristic
74	D₈×C₇₉	—	4	2	no	—
75	D₈×C₇₉	—	4	2	no	—
76	C₂×C₈×C₇₉	—	2	2	no	—
77	C₂×C₈×C₇₉	—	2	2	no	—
78	C₂×C₈×C₇₉	—	4	1	no	—
79	C₂×C₈×C₇₉	—	4	1	no	—
80	C₂×C₈×C₇₉	—	4	1	no	—
81	C₂×C₈×C₇₉	—	4	1	no	—
82	C₂×C₈×C₇₉	C₂²⋊C₄	1	1	yes	characteristic
83	C₂×C₈×C₇₉	C₂²⋊C₄	1	1	yes	characteristic
84	C₂×C₈×C₇₉	C₂²×C₄	1	1	yes	characteristic
85	C₂×C₈×C₇₉	C₂×D₄	1	1	yes	characteristic
86	C₄⋊C₄×C₇₉	—	2	2	no	—
87	C₄⋊C₄×C₇₉	—	4	1	no	—
88	C₂²⋊C₄×C₇₉	—	4	2	no	—
89	C₂²⋊C₄×C₇₉	—	4	2	no	—
90	C₂³×C₇₉	—	2	2	no	—
91	C₂³×C₇₉	—	2	2	no	—
92	C₂³×C₇₉	—	2	2	no	—
93	C₂³×C₇₉	—	4	2	no	—
94	C₂³×C₇₉	—	8	2	no	—
95	C₂³×C₇₉	—	8	2	no	—
96	C₂³×C₇₉	—	8	2	no	—
97	C₂³×C₇₉	#102	1	1	yes	characteristic
98	C₂³×C₇₉	#102	1	1	yes	characteristic
99	C₂³×C₇₉	C₂×SD₁₆	1	1	yes	characteristic
100	D₄×C₇₉	—	4	2	no	—
101	D₄×C₇₉	—	4	2	no	—
102	D₄×C₇₉	—	8	4	no	—
103	D₄×C₇₉	—	8	4	no	—
104	D₄×C₇₉	—	8	2	no	—
105	C₂×C₄×C₇₉	—	2	2	no	—
106	C₂×C₄×C₇₉	—	2	2	no	—
107	C₂×C₄×C₇₉	—	2	2	no	—
108	C₂×C₄×C₇₉	—	2	1	no	—
109	C₂×C₄×C₇₉	—	2	1	no	—
110	C₂×C₄×C₇₉	—	2	1	no	—
111	C₂×C₄×C₇₉	—	2	1	no	—
112	C₂×C₄×C₇₉	—	4	2	no	—
113	C₂×C₄×C₇₉	—	8	2	no	—
114	C₂×C₄×C₇₉	#99	1	2	yes	—
115	C₂×C₄×C₇₉	#102	1	1	yes	characteristic
116	C₂×C₄×C₇₉	#102	1	1	yes	characteristic
117	C₂×C₄×C₇₉	#104	1	2	yes	—
118	C₂×C₄×C₇₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
119	C₂×C₄×C₇₉	C₂×D₈	1	1	yes	characteristic
120	C₈×C₇₉	—	2	1	no	—
121	C₈×C₇₉	—	2	1	no	—
122	C₈×C₇₉	—	2	1	no	—
123	C₈×C₇₉	—	2	1	no	—
124	C₇₉⋊C₈	—	4	4	no	—
125	C₂²×C₇₉	—	2	1	no	—
126	C₂²×C₇₉	—	2	1	no	—
127	C₂²×C₇₉	—	2	1	no	—
128	C₂²×C₇₉	—	2	1	no	—
129	C₂²×C₇₉	—	2	1	no	—
130	C₂²×C₇₉	—	2	1	no	—
131	C₂²×C₇₉	—	4	4	no	—
132	C₂²×C₇₉	—	8	2	no	—
133	C₂²×C₇₉	—	8	2	no	—
134	C₂²×C₇₉	—	8	2	no	—
135	C₂²×C₇₉	—	16	4	no	—
136	C₂²×C₇₉	#327	1	2	yes	—
137	C₂²×C₇₉	#359	1	2	yes	—
138	C₂²×C₇₉	#361	1	2	yes	—
139	C₂²×C₇₉	C₂×#102	1	1	yes	characteristic
140	C₄×C₇₉	—	2	1	no	—
141	C₄×C₇₉	—	2	1	no	—
142	C₄×C₇₉	—	8	2	no	—
143	C₄×C₇₉	#327	1	2	yes	—
144	C₄×C₇₉	C₂×#99	1	1	yes	characteristic
145	C₄×C₇₉	C₂×#104	1	1	yes	characteristic
146	C₂×#1349	—	79	1	no	maximal, Sylow
147	C₂×C₇₉	—	4	1	no	—
148	C₂×C₇₉	—	4	1	no	—
149	C₂×C₇₉	—	16	2	no	—
150	C₂×C₇₉	#1349	1	4	yes	complemented
151	C₂×C₇₉	C₂×#327	1	1	yes	characteristic
152	C₂×C₇₉	C₂×#359	1	1	yes	characteristic
153	C₂×C₇₉	C₂×#361	1	1	yes	characteristic
154	C₂×#466	D₇₉	1	1	yes	characteristic
155	C₂×C₂²⋊C₁₆	—	79	1	no	—
156	C₂×#341	—	79	1	no	—
157	#1349	—	79	4	no	—
158	C₇₉	C₂×#1349	1	1	yes	characteristic, complemented, Sylow
159	C₂²×D₈	Dic₇₉	1	1	yes	characteristic
160	C₂²×M₁₆	—	158	1	no	—
161	C₂³×C₈	C₂×D₇₉	1	1	yes	characteristic
162	C₂×#121	—	2	1	no	—
163	C₂²×C₁₆	—	158	1	no	—
164	#466	—	2	1	no	—
165	#466	—	2	1	no	—
166	#466	Dic₇₉	1	2	yes	—
167	#466	C₂×D₇₉	1	2	yes	—
168	C₂×#107	Dic₇₉	1	1	yes	characteristic
169	C₂×#102	—	2	1	no	—
170	C₂²⋊C₁₆	—	79	2	no	—
171	C₂²⋊C₁₆	—	158	1	no	—
172	#341	—	79	2	no	—
173	#341	—	158	1	no	—
174	C₂²×D₄	—	2	1	no	—
175	C₂²×D₄	—	4	1	no	—
176	C₂³×C₄	C₂²⋊D₇₉	1	1	yes	characteristic
177	C₂×D₈	—	2	1	no	—
178	C₂×D₈	—	4	2	no	—
179	C₂×D₈	C₂×Dic₇₉	1	2	yes	—
180	C₂×M₁₆	—	316	2	no	—
181	C₂×M₁₆	—	316	1	no	—
182	C₂×M₁₆	—	158	4	no	—
183	C₂×M₁₆	—	158	2	no	—
184	C₂²×C₈	—	4	2	no	—
185	C₂²×C₈	—	316	1	no	—
186	C₂²×C₈	C₂×Dic₇₉	1	1	yes	characteristic
187	C₂²×C₈	C₂×Dic₇₉	1	1	yes	characteristic
188	C₂²×C₈	C₂²⋊D₇₉	1	1	yes	characteristic
189	C₂²×C₈	C₂²⋊D₇₉	1	1	yes	characteristic
190	C₂²×C₈	C₂²⋊D₇₉	1	1	yes	characteristic
191	C₂²×C₈	C₂²×D₇₉	1	1	yes	characteristic
192	#121	—	2	2	no	—
193	#121	—	2	2	no	—
194	#121	—	4	1	no	—
195	#121	—	4	1	no	—
196	C₂×C₄⋊C₄	—	2	1	no	—
197	C₂×C₂²⋊C₄	—	4	1	no	—
198	C₂×C₁₆	—	316	2	no	—
199	C₂×C₁₆	—	158	2	no	—
200	#107	—	2	1	no	—
201	#107	C₂×Dic₇₉	1	2	yes	—
202	#102	—	2	2	no	—
203	#102	—	2	2	no	—
204	C₂⁴	—	4	1	no	—
205	C₂⁴	#11929612	1	1	yes	characteristic
206	C₂×D₄	—	2	2	no	—
207	C₂×D₄	—	4	2	no	—
208	C₂×D₄	—	4	2	no	—
209	C₂×D₄	—	4	2	no	—
210	C₂×D₄	—	4	1	no	—
211	C₂×D₄	—	4	2	no	—
212	C₂×D₄	—	8	4	no	—
213	C₂²×C₄	—	2	2	no	—
214	C₂²×C₄	—	2	2	no	—
215	C₂²×C₄	—	4	1	no	—
216	C₂²×C₄	C₂²⋊Dic₇₉	1	1	yes	characteristic
217	C₂²×C₄	C₂²⋊Dic₇₉	1	1	yes	characteristic
218	C₂²×C₄	#11929609	1	1	yes	characteristic
219	C₂²×C₄	#11929609	1	1	yes	characteristic
220	C₂²×C₄	#11929612	1	1	yes	characteristic
221	C₂²×C₄	C₂×C₂²⋊D₇₉	1	1	yes	characteristic
222	D₈	—	4	2	no	—
223	D₈	—	4	2	no	—
224	M₁₆	—	316	2	no	—
225	M₁₆	—	316	4	no	—
226	M₁₆	—	158	4	no	—
227	C₂×C₈	—	2	2	no	—
228	C₂×C₈	—	2	2	no	—
229	C₂×C₈	—	4	1	no	—
230	C₂×C₈	—	4	1	no	—
231	C₂×C₈	—	4	1	no	—
232	C₂×C₈	—	4	1	no	—
233	C₂×C₈	—	316	4	no	—
234	C₂×C₈	—	316	2	no	—
235	C₂×C₈	C₂²⋊Dic₇₉	1	1	yes	characteristic
236	C₂×C₈	C₂²⋊Dic₇₉	1	1	yes	characteristic
237	C₂×C₈	C₂²×Dic₇₉	1	1	yes	characteristic
238	C₂×C₈	C₂×C₂²⋊D₇₉	1	1	yes	characteristic
239	C₄⋊C₄	—	2	2	no	—
240	C₄⋊C₄	—	4	1	no	—
241	C₂²⋊C₄	—	4	2	no	—
242	C₂²⋊C₄	—	4	2	no	—
243	C₁₆	—	316	2	no	—
244	C₂³	—	2	2	no	—
245	C₂³	—	2	2	no	—
246	C₂³	—	2	2	no	—
247	C₂³	—	4	2	no	—
248	C₂³	—	8	2	no	—
249	C₂³	—	8	2	no	—
250	C₂³	—	8	2	no	—
251	C₂³	#14579082	1	1	yes	characteristic
252	C₂³	#14579082	1	1	yes	characteristic
253	C₂³	C₂×#11929612	1	1	yes	characteristic
254	D₄	—	4	2	no	—
255	D₄	—	4	2	no	—
256	D₄	—	8	4	no	—
257	D₄	—	8	4	no	—
258	D₄	—	8	2	no	—
259	C₂×C₄	—	2	2	no	—
260	C₂×C₄	—	2	1	no	—
261	C₂×C₄	—	2	1	no	—
262	C₂×C₄	—	2	2	no	—
263	C₂×C₄	—	2	2	no	—
264	C₂×C₄	—	2	1	no	—
265	C₂×C₄	—	2	1	no	—
266	C₂×C₄	—	4	2	no	—
267	C₂×C₄	—	8	2	no	—
268	C₂×C₄	#14579076	1	2	yes	—
269	C₂×C₄	#14579082	1	1	yes	characteristic
270	C₂×C₄	#14579082	1	1	yes	characteristic
271	C₂×C₄	#14579088	1	2	yes	—
272	C₂×C₄	C₂×C₂²⋊Dic₇₉	1	1	yes	characteristic
273	C₂×C₄	C₂×#11929609	1	1	yes	characteristic
274	C₈	—	2	1	no	—
275	C₈	—	2	1	no	—
276	C₈	—	2	1	no	—
277	C₈	—	2	1	no	—
278	C₈	—	316	4	no	—
279	C₂²	—	2	1	no	—
280	C₂²	—	2	1	no	—
281	C₂²	—	2	1	no	—
282	C₂²	—	2	1	no	—
283	C₂²	—	2	1	no	—
284	C₂²	—	2	1	no	—
285	C₂²	—	4	4	no	—
286	C₂²	—	8	2	no	—
287	C₂²	—	8	2	no	—
288	C₂²	—	8	2	no	—
289	C₂²	—	16	4	no	—
290	C₂²	#19098297	1	2	yes	—
291	C₂²	#19098370	1	2	yes	—
292	C₂²	#19098376	1	2	yes	—
293	C₂²	C₂×#14579082	1	1	yes	characteristic
294	C₄	—	2	1	no	—
295	C₄	—	2	1	no	—
296	C₄	—	8	2	no	—
297	C₄	#19098297	1	2	yes	—
298	C₄	C₂×#14579076	1	1	yes	characteristic
299	C₄	C₂×#14579088	1	1	yes	characteristic
300	C₂	—	4	1	no	—
301	C₂	—	4	1	no	—
302	C₂	—	16	2	no	—
303	C₂	#23761415	1	4	yes	direct product factor
304	C₂	C₂×#19098297	1	1	yes	characteristic
305	C₂	C₂×#19098370	1	1	yes	characteristic
306	C₂	C₂×#19098376	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	#14579082
commutator	C₂×C₈×C₇₉	C₂²×C₄
Frattini	C₂²×C₈	C₂²×D₇₉
Fitting, prime radical	C₂×#466×C₇₉	C₂
socle	C₂³×C₇₉	#102
cosocle	C₂²×C₈×C₇₉	C₂³
hypercenter	C₂×#466	D₇₉
hypocenter	C₇₉	C₂×#1349
LCS	G ▷ C₂×C₈×C₇₉ ▷ C₂×C₄×C₇₉ ▷ C₂×C₇₉ ▷ C₇₉
UCS	C₁ ◁ C₂³ ◁ C₂³×C₄ ◁ C₂³×C₈ ◁ C₂×#466

Complements

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

N	H ≅ G/N	type
1. C₂×C₂²⋊(C₇₉⋊C₁₆)	C₂	semidirect
2. C₂×#19098328	C₂	semidirect
3. #23761415	C₂	direct (N × H)
8. C₂²⋊(C₇₉⋊C₁₆)	C₂²	semidirect
10. #19098328	C₂²	semidirect
150. C₂×C₇₉	#1349	semidirect
158. C₇₉	C₂×#1349	semidirect
303. C₂	#23761415	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×C₂²⋊(C₇₉⋊C₁₆)	2
C₂×#19098328	2
#23761415	2
C₂×#466×C₇₉	2
C₂×#1349	79

Subgroups — C2×#23761415 #34198514

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#23761415 #34198514