Subgroups — #34175225

Order 20224.

326 conjugacy classes of subgroups.

The subgroup lattice has 272 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
2	#23763096	C₂	1	1	yes	maximal, characteristic
3	C₂×C₂²⋊C₁₆×C₇₉	C₂	1	1	yes	maximal, characteristic
4	C₂²×#11929608	C₄	1	1	yes	characteristic
5	C₂×#14579120	C₄	1	1	yes	characteristic
6	C₂²×C₇₉⋊C₁₆	—	2	1	no	—
7	C₂²⋊(C₇₉⋊C₁₆)	—	2	1	no	—
8	C₂²⋊(C₇₉⋊C₁₆)	C₄	1	2	yes	complemented
9	#19098317	—	2	1	no	—
10	#19098317	—	2	1	no	—
11	C₂³×C₈×C₇₉	C₂²	1	1	yes	characteristic
12	C₂²×C₁₆×C₇₉	—	2	1	no	—
13	C₂²⋊C₁₆×C₇₉	—	2	1	no	—
14	C₂²⋊C₁₆×C₇₉	C₄	1	2	yes	complemented
15	C₂×#11929608	—	2	1	no	—
16	C₂×#11929608	—	4	1	no	—
17	C₂×#11929608	—	4	1	no	—
18	C₂×#11929608	C₈	1	2	yes	—
19	C₂²×C₇₉⋊C₈	—	2	1	no	—
20	#14579120	—	4	1	no	—
21	#14579120	—	4	1	no	—
22	C₂×C₄⋊Dic₇₉	C₈	1	2	yes	—
23	C₂×C₂²⋊Dic₇₉	—	2	1	no	—
24	C₂×C₄×Dic₇₉	—	2	1	no	—
25	C₂×C₇₉⋊C₁₆	—	2	2	no	—
26	C₂×C₇₉⋊C₁₆	—	4	1	no	—
27	C₂×C₇₉⋊C₁₆	—	4	1	no	—
28	C₂³×C₄×C₇₉	C₂×C₄	1	1	yes	characteristic
29	C₂²×C₈×C₇₉	—	4	1	no	—
30	C₂²×C₈×C₇₉	—	4	1	no	—
31	C₂²×C₈×C₇₉	C₂×C₄	1	1	yes	characteristic
32	C₂²×C₈×C₇₉	C₂×C₄	1	1	yes	characteristic
33	C₂²×C₈×C₇₉	D₄	1	1	yes	characteristic
34	C₂²×C₈×C₇₉	D₄	1	1	yes	characteristic
35	C₂²×C₈×C₇₉	D₄	1	1	yes	characteristic
36	C₂²×C₈×C₇₉	Q₈	1	1	yes	characteristic
37	C₂×C₁₆×C₇₉	—	2	2	no	—
38	C₂×C₁₆×C₇₉	—	4	1	no	—
39	C₂×C₁₆×C₇₉	—	4	1	no	—
40	C₂²×Dic₇₉	—	2	2	no	—
41	#11929608	—	4	2	no	—
42	#11929608	—	4	2	no	—
43	C₂×C₇₉⋊C₈	—	2	2	no	—
44	C₂×C₇₉⋊C₈	—	4	1	no	—
45	C₂×C₇₉⋊C₈	—	4	1	no	—
46	C₄⋊Dic₇₉	—	4	2	no	—
47	C₂²⋊Dic₇₉	—	4	2	no	—
48	C₄×Dic₇₉	—	4	1	no	—
49	C₄×Dic₇₉	—	4	1	no	—
50	C₇₉⋊C₁₆	—	4	2	no	—
51	C₂⁴×C₇₉	M₁₆	1	1	yes	characteristic
52	C₂²×C₄×C₇₉	—	2	1	no	—
53	C₂²×C₄×C₇₉	—	2	1	no	—
54	C₂²×C₄×C₇₉	—	2	1	no	—
55	C₂²×C₄×C₇₉	—	2	1	no	—
56	C₂²×C₄×C₇₉	C₄²	1	1	yes	characteristic
57	C₂²×C₄×C₇₉	C₂²⋊C₄	1	1	yes	characteristic
58	C₂²×C₄×C₇₉	C₄⋊C₄	1	1	yes	characteristic
59	C₂²×C₄×C₇₉	C₂×C₈	1	1	yes	characteristic
60	C₂²×C₄×C₇₉	C₂×C₈	1	1	yes	characteristic
61	C₂²×C₄×C₇₉	M₁₆	1	1	yes	characteristic
62	C₂×C₈×C₇₉	—	2	1	no	—
63	C₂×C₈×C₇₉	—	2	1	no	—
64	C₂×C₈×C₇₉	—	2	1	no	—
65	C₂×C₈×C₇₉	—	2	1	no	—
66	C₂×C₈×C₇₉	—	4	1	no	—
67	C₂×C₈×C₇₉	—	4	1	no	—
68	C₂×C₈×C₇₉	—	4	1	no	—
69	C₂×C₈×C₇₉	—	4	1	no	—
70	C₂×C₈×C₇₉	C₂²⋊C₄	1	1	yes	characteristic
71	C₂×C₈×C₇₉	C₂²⋊C₄	1	1	yes	characteristic
72	C₂×C₈×C₇₉	C₄⋊C₄	1	1	yes	characteristic
73	C₂×C₈×C₇₉	C₄⋊C₄	1	1	yes	characteristic
74	C₁₆×C₇₉	—	4	2	no	—
75	C₂³×C₇₉	—	2	1	no	—
76	C₂³×C₇₉	—	2	1	no	—
77	C₂³×C₇₉	—	2	2	no	—
78	C₂³×C₇₉	—	2	2	no	—
79	C₂³×C₇₉	#97	1	1	yes	characteristic
80	C₂³×C₇₉	C₂²⋊C₈	1	1	yes	characteristic
81	C₂³×C₇₉	C₄⋊C₈	1	1	yes	characteristic
82	C₂×C₄×C₇₉	—	2	1	no	—
83	C₂×C₄×C₇₉	—	2	1	no	—
84	C₂×C₄×C₇₉	—	2	1	no	—
85	C₂×C₄×C₇₉	—	2	1	no	—
86	C₂×C₄×C₇₉	—	2	1	no	—
87	C₂×C₄×C₇₉	—	2	1	no	—
88	C₂×C₄×C₇₉	—	2	1	no	—
89	C₂×C₄×C₇₉	—	2	1	no	—
90	C₂×C₄×C₇₉	—	2	1	no	—
91	C₂×C₄×C₇₉	—	2	1	no	—
92	C₂×C₄×C₇₉	#95	1	1	yes	characteristic
93	C₂×C₄×C₇₉	C₄×C₈	1	1	yes	characteristic
94	C₂×C₄×C₇₉	C₂²⋊C₈	1	1	yes	characteristic
95	C₂×C₄×C₇₉	#99	1	1	yes	characteristic
96	C₂×C₄×C₇₉	#99	1	1	yes	characteristic
97	C₂×C₄×C₇₉	#100	1	1	yes	characteristic
98	C₂×C₄×C₇₉	#101	1	1	yes	characteristic
99	C₂×C₄×C₇₉	C₄⋊C₈	1	1	yes	characteristic
100	C₂×Dic₇₉	—	4	2	no	—
101	C₂×Dic₇₉	—	4	2	no	—
102	C₂×Dic₇₉	—	4	2	no	—
103	C₈×C₇₉	—	2	1	no	—
104	C₈×C₇₉	—	2	1	no	—
105	C₈×C₇₉	—	2	1	no	—
106	C₈×C₇₉	—	2	1	no	—
107	C₇₉⋊C₈	—	4	2	no	—
108	C₂²×C₇₉	—	2	1	no	—
109	C₂²×C₇₉	—	2	1	no	—
110	C₂²×C₇₉	—	2	1	no	—
111	C₂²×C₇₉	—	2	1	no	—
112	C₂²×C₇₉	—	2	1	no	—
113	C₂²×C₇₉	—	2	1	no	—
114	C₂²×C₇₉	—	4	2	no	—
115	C₂²×C₇₉	—	4	2	no	—
116	C₂²×C₇₉	#323	1	1	yes	characteristic
117	C₂²×C₇₉	#324	1	1	yes	characteristic
118	C₂²×C₇₉	#336	1	1	yes	characteristic
119	C₂²×C₇₉	#350	1	2	yes	—
120	C₂²×C₇₉	#350	1	2	yes	—
121	C₄×C₇₉	—	2	1	no	—
122	C₄×C₇₉	—	2	1	no	—
123	C₄×C₇₉	#323	1	1	yes	characteristic
124	C₄×C₇₉	#324	1	1	yes	characteristic
125	C₄×C₇₉	#342	1	1	yes	characteristic
126	C₄×C₇₉	#343	1	1	yes	characteristic
127	Dic₇₉	—	8	2	no	—
128	#7186	—	79	1	no	maximal, Sylow
129	C₂×C₇₉	—	4	1	no	—
130	C₂×C₇₉	—	4	1	no	—
131	C₂×C₇₉	#1280	1	1	yes	characteristic
132	C₂×C₇₉	#1315	1	2	yes	—
133	C₂×C₇₉	#1323	1	2	yes	—
134	C₂×C₇₉	#1379	1	1	yes	characteristic
135	C₂×C₇₉	#1379	1	1	yes	characteristic
136	C₂×C₂²⋊C₁₆	—	79	1	no	—
137	C₂×C₂²⋊C₁₆	D₇₉	1	1	yes	characteristic
138	#1729	—	79	1	no	—
139	C₇₉	#7186	1	1	yes	characteristic, complemented, Sylow
140	C₂²×M₁₆	—	79	1	no	—
141	C₂³×C₈	C₂×D₇₉	1	1	yes	characteristic
142	C₂×#117	—	79	1	no	—
143	C₂²×C₁₆	—	2	1	no	—
144	C₂²×C₁₆	—	158	1	no	—
145	C₂²⋊C₁₆	—	2	1	no	—
146	C₂²⋊C₁₆	—	79	2	no	—
147	C₂²⋊C₁₆	—	158	1	no	—
148	C₂²⋊C₁₆	Dic₇₉	1	2	yes	complemented
149	#336	—	158	1	no	—
150	#336	—	158	1	no	—
151	C₂³×C₄	C₄×D₇₉	1	1	yes	characteristic
152	C₂×M₁₆	—	316	1	no	—
153	C₂×M₁₆	—	316	1	no	—
154	C₂×M₁₆	—	79	2	no	—
155	C₂×M₁₆	—	158	1	no	—
156	C₂²×C₈	—	4	1	no	—
157	C₂²×C₈	—	4	1	no	—
158	C₂²×C₈	—	158	1	no	—
159	C₂²×C₈	Dic₁₅₈	1	1	yes	characteristic
160	C₂²×C₈	C₄×D₇₉	1	1	yes	characteristic
161	C₂²×C₈	D₃₁₆	1	1	yes	characteristic
162	C₂²×C₈	C₂×Dic₇₉	1	1	yes	characteristic
163	C₂²×C₈	C₂²⋊D₇₉	1	1	yes	characteristic
164	C₂²×C₈	C₂²⋊D₇₉	1	1	yes	characteristic
165	#117	—	316	1	no	—
166	#117	—	316	1	no	—
167	C₂×C₄⋊C₄	—	79	2	no	—
168	C₂×C₂²⋊C₄	—	158	1	no	—
169	C₂×C₄²	—	158	1	no	—
170	C₂×C₁₆	—	2	2	no	—
171	C₂×C₁₆	—	4	1	no	—
172	C₂×C₁₆	—	4	1	no	—
173	C₂×C₁₆	—	316	1	no	—
174	C₂×C₁₆	—	316	1	no	—
175	C₂×C₁₆	—	158	2	no	—
176	C₂⁴	#11929607	1	1	yes	characteristic
177	C₂²×C₄	—	2	1	no	—
178	C₂²×C₄	—	2	1	no	—
179	C₂²×C₄	—	2	1	no	—
180	C₂²×C₄	—	2	1	no	—
181	C₂²×C₄	—	158	2	no	—
182	C₂²×C₄	C₄×Dic₇₉	1	1	yes	characteristic
183	C₂²×C₄	(C₂×D₇₉)⋊C₄	1	1	yes	characteristic
184	C₂²×C₄	Dic₇₉⋊C₄	1	1	yes	characteristic
185	C₂²×C₄	C₈×D₇₉	1	1	yes	characteristic
186	C₂²×C₄	C₈×D₇₉	1	1	yes	characteristic
187	C₂²×C₄	#11929607	1	1	yes	characteristic
188	M₁₆	—	316	2	no	—
189	M₁₆	—	316	2	no	—
190	C₂×C₈	—	2	1	no	—
191	C₂×C₈	—	2	1	no	—
192	C₂×C₈	—	2	1	no	—
193	C₂×C₈	—	2	1	no	—
194	C₂×C₈	—	4	1	no	—
195	C₂×C₈	—	4	1	no	—
196	C₂×C₈	—	4	1	no	—
197	C₂×C₈	—	4	1	no	—
198	C₂×C₈	—	316	1	no	—
199	C₂×C₈	—	316	1	no	—
200	C₂×C₈	—	158	2	no	—
201	C₂×C₈	(C₂×D₇₉)⋊C₄	1	1	yes	characteristic
202	C₂×C₈	C₂²⋊Dic₇₉	1	1	yes	characteristic
203	C₂×C₈	Dic₇₉⋊C₄	1	1	yes	characteristic
204	C₂×C₈	C₄⋊Dic₇₉	1	1	yes	characteristic
205	C₄⋊C₄	—	316	2	no	—
206	C₂²⋊C₄	—	316	2	no	—
207	C₄²	—	316	1	no	—
208	C₄²	—	316	1	no	—
209	C₁₆	—	4	2	no	—
210	C₁₆	—	316	2	no	—
211	C₂³	—	2	1	no	—
212	C₂³	—	2	1	no	—
213	C₂³	—	2	2	no	—
214	C₂³	—	2	2	no	—
215	C₂³	#14579071	1	1	yes	characteristic
216	C₂³	(C₂×D₇₉)⋊C₈	1	1	yes	characteristic
217	C₂³	Dic₇₉⋊C₈	1	1	yes	characteristic
218	C₂×C₄	—	2	1	no	—
219	C₂×C₄	—	2	1	no	—
220	C₂×C₄	—	2	1	no	—
221	C₂×C₄	—	2	1	no	—
222	C₂×C₄	—	2	1	no	—
223	C₂×C₄	—	2	1	no	—
224	C₂×C₄	—	2	1	no	—
225	C₂×C₄	—	2	1	no	—
226	C₂×C₄	—	2	1	no	—
227	C₂×C₄	—	2	1	no	—
228	C₂×C₄	—	316	2	no	—
229	C₂×C₄	—	316	2	no	—
230	C₂×C₄	—	316	2	no	—
231	C₂×C₄	C₇₉⋊#95	1	1	yes	characteristic
232	C₂×C₄	C₈×Dic₇₉	1	1	yes	characteristic
233	C₂×C₄	(C₂×D₇₉)⋊C₈	1	1	yes	characteristic
234	C₂×C₄	#14579075	1	1	yes	characteristic
235	C₂×C₄	#14579075	1	1	yes	characteristic
236	C₂×C₄	#14579077	1	1	yes	characteristic
237	C₂×C₄	#14579079	1	1	yes	characteristic
238	C₂×C₄	Dic₇₉⋊C₈	1	1	yes	characteristic
239	C₈	—	2	1	no	—
240	C₈	—	2	1	no	—
241	C₈	—	2	1	no	—
242	C₈	—	2	1	no	—
243	C₈	—	316	2	no	—
244	C₂²	—	2	1	no	—
245	C₂²	—	2	1	no	—
246	C₂²	—	2	1	no	—
247	C₂²	—	2	1	no	—
248	C₂²	—	2	1	no	—
249	C₂²	—	2	1	no	—
250	C₂²	—	4	2	no	—
251	C₂²	—	4	2	no	—
252	C₂²	#19098286	1	1	yes	characteristic
253	C₂²	#19098288	1	1	yes	characteristic
254	C₂²	#19098317	1	1	yes	characteristic
255	C₂²	#19098346	1	2	yes	—
256	C₂²	#19098348	1	2	yes	—
257	C₄	—	2	1	no	—
258	C₄	—	2	1	no	—
259	C₄	—	632	2	no	—
260	C₄	#19098286	1	1	yes	characteristic
261	C₄	#19098288	1	1	yes	characteristic
262	C₄	#19098329	1	1	yes	characteristic
263	C₄	#19098331	1	1	yes	characteristic
264	C₂	—	4	1	no	—
265	C₂	—	4	1	no	—
266	C₂	#23761242	1	1	yes	characteristic
267	C₂	#23761325	1	2	yes	—
268	C₂	#23761341	1	2	yes	—
269	C₂	#23761488	1	1	yes	characteristic
270	C₂	#23761490	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄	(C₂×D₇₉)⋊C₄
commutator	C₂×C₄×C₇₉	C₄×C₈
Frattini	C₂³×C₈	C₂×D₇₉
Fitting	C₂×C₂²⋊C₁₆×C₇₉	C₂
socle	C₂³×C₇₉	C₂²⋊C₈
prime radical	C₂⁴×C₇₉	M₁₆
cosocle	C₂³×C₈×C₇₉	C₂²
hypercenter	C₂×C₂²⋊C₁₆	D₇₉
hypocenter	C₇₉	#7186
LCS	G ▷ C₂×C₄×C₇₉ ▷ C₂²×C₇₉ ▷ C₇₉
UCS	C₁ ◁ C₂²×C₄ ◁ C₂³×C₈ ◁ C₂×C₂²⋊C₁₆

Complements

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

N	H ≅ G/N	type
8. C₂²⋊(C₇₉⋊C₁₆)	C₄	semidirect
14. C₂²⋊C₁₆×C₇₉	C₄	semidirect
139. C₇₉	#7186	semidirect
148. C₂²⋊C₁₆	Dic₇₉	semidirect

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
#23763096	2
C₂×C₂²⋊C₁₆×C₇₉	2
#7186	79