Subgroups — #34174823

Order 20224.

442 conjugacy classes of subgroups.

The subgroup lattice has 268 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄×C₂²⋊(C₇₉⋊C₈)	C₂	1	1	yes	maximal, characteristic
2	#23763094	C₂	1	1	yes	maximal, characteristic
3	C₄×C₂²⋊C₈×C₇₉	C₂	1	1	yes	maximal, characteristic
4	C₂²×#11929608	C₄	1	2	yes	—
5	C₂×C₂²⋊(C₇₉⋊C₈)	—	2	1	no	—
6	C₂×C₄×C₇₉⋊C₈	—	2	1	no	—
7	#19098318	—	2	1	no	—
8	#19098318	—	2	1	no	—
9	#19098318	C₄	1	2	yes	—
10	C₂²×C₄²×C₇₉	C₂²	1	1	yes	characteristic
11	C₂×C₂²⋊C₈×C₇₉	—	2	1	no	—
12	C₂×C₄×C₈×C₇₉	—	2	1	no	—
13	#336×C₇₉	C₄	1	2	yes	—
14	C₂×#11929608	—	2	2	no	—
15	C₂×#11929608	—	4	2	no	—
16	C₂×#11929608	—	4	2	no	—
17	C₂×#11929608	C₈	1	4	yes	complemented
18	C₂²×C₇₉⋊C₈	—	2	2	no	—
19	C₂²×C₇₉⋊C₈	—	2	2	no	—
20	C₂²⋊(C₇₉⋊C₈)	—	2	4	no	—
21	C₄×C₇₉⋊C₈	—	4	2	no	—
22	C₂³×C₄×C₇₉	C₂×C₄	1	1	yes	characteristic
23	C₂³×C₄×C₇₉	D₄	1	1	yes	characteristic
24	C₂³×C₄×C₇₉	Q₈	1	1	yes	characteristic
25	C₂²×C₈×C₇₉	—	2	2	no	—
26	C₂×C₄²×C₇₉	—	4	2	no	—
27	C₂×C₄²×C₇₉	C₂×C₄	1	1	yes	characteristic
28	C₂×C₄²×C₇₉	C₂×C₄	1	1	yes	characteristic
29	C₂×C₄²×C₇₉	D₄	1	1	yes	characteristic
30	C₂×C₄²×C₇₉	D₄	1	1	yes	characteristic
31	C₂²⋊C₈×C₇₉	—	2	4	no	—
32	C₄×C₈×C₇₉	—	4	2	no	—
33	#11929608	—	4	4	no	—
34	#11929608	—	4	4	no	—
35	C₂×C₇₉⋊C₈	—	2	4	no	—
36	C₂×C₇₉⋊C₈	—	2	4	no	—
37	C₂×C₇₉⋊C₈	—	4	2	no	—
38	C₂×C₇₉⋊C₈	—	4	2	no	—
39	C₂×C₇₉⋊C₈	—	4	2	no	—
40	C₂×C₇₉⋊C₈	—	4	2	no	—
41	C₂⁴×C₇₉	C₄⋊C₄	1	1	yes	characteristic
42	C₂²×C₄×C₇₉	—	2	2	no	—
43	C₂²×C₄×C₇₉	—	2	1	no	—
44	C₂²×C₄×C₇₉	—	2	1	no	—
45	C₂²×C₄×C₇₉	—	2	2	no	—
46	C₂²×C₄×C₇₉	—	2	2	no	—
47	C₂²×C₄×C₇₉	—	2	2	no	—
48	C₂²×C₄×C₇₉	—	2	2	no	—
49	C₂²×C₄×C₇₉	—	4	1	no	—
50	C₂²×C₄×C₇₉	—	4	1	no	—
51	C₂²×C₄×C₇₉	C₄²	1	1	yes	characteristic
52	C₂²×C₄×C₇₉	C₂²⋊C₄	1	1	yes	characteristic
53	C₂²×C₄×C₇₉	C₂²⋊C₄	1	1	yes	characteristic
54	C₂²×C₄×C₇₉	C₂²⋊C₄	1	1	yes	characteristic
55	C₂²×C₄×C₇₉	C₄⋊C₄	1	1	yes	characteristic
56	C₂²×C₄×C₇₉	C₄⋊C₄	1	1	yes	characteristic
57	C₂²×C₄×C₇₉	C₂×C₈	1	2	yes	—
58	C₂²×C₄×C₇₉	M₁₆	1	2	yes	—
59	C₂×C₈×C₇₉	—	2	4	no	—
60	C₂×C₈×C₇₉	—	4	2	no	—
61	C₂×C₈×C₇₉	—	4	2	no	—
62	C₄²×C₇₉	—	2	2	no	—
63	C₄²×C₇₉	—	2	2	no	—
64	C₄²×C₇₉	—	4	1	no	—
65	C₄²×C₇₉	—	4	1	no	—
66	C₂³×C₇₉	—	2	1	no	—
67	C₂³×C₇₉	—	2	1	no	—
68	C₂³×C₇₉	—	2	1	no	—
69	C₂³×C₇₉	—	2	1	no	—
70	C₂³×C₇₉	—	2	1	no	—
71	C₂³×C₇₉	—	2	1	no	—
72	C₂³×C₇₉	#95	1	1	yes	characteristic
73	C₂³×C₇₉	C₄⋊C₈	1	2	yes	—
74	C₂×C₄×C₇₉	—	2	1	no	—
75	C₂×C₄×C₇₉	—	2	1	no	—
76	C₂×C₄×C₇₉	—	2	1	no	—
77	C₂×C₄×C₇₉	—	2	1	no	—
78	C₂×C₄×C₇₉	—	2	1	no	—
79	C₂×C₄×C₇₉	—	2	1	no	—
80	C₂×C₄×C₇₉	—	2	1	no	—
81	C₂×C₄×C₇₉	—	2	1	no	—
82	C₂×C₄×C₇₉	—	2	1	no	—
83	C₂×C₄×C₇₉	—	2	1	no	—
84	C₂×C₄×C₇₉	—	2	2	no	—
85	C₂×C₄×C₇₉	—	2	2	no	—
86	C₂×C₄×C₇₉	—	2	2	no	—
87	C₂×C₄×C₇₉	—	2	2	no	—
88	C₂×C₄×C₇₉	—	4	4	no	—
89	C₂×C₄×C₇₉	—	4	4	no	—
90	C₂×C₄×C₇₉	C₄×C₈	1	1	yes	characteristic
91	C₂×C₄×C₇₉	#97	1	1	yes	characteristic
92	C₂×C₄×C₇₉	C₂²⋊C₈	1	2	yes	—
93	C₂×C₄×C₇₉	#99	1	2	yes	—
94	C₂×C₄×C₇₉	#100	1	1	yes	characteristic
95	C₂×C₄×C₇₉	#101	1	1	yes	characteristic
96	C₂×C₄×C₇₉	#104	1	2	yes	—
97	C₂×C₄×C₇₉	#104	1	2	yes	—
98	C₂×C₄×C₇₉	#108	1	2	yes	—
99	C₂×C₄×C₇₉	#108	1	2	yes	—
100	C₈×C₇₉	—	4	4	no	—
101	C₇₉⋊C₈	—	4	4	no	—
102	C₇₉⋊C₈	—	4	4	no	—
103	C₂²×C₇₉	—	2	2	no	—
104	C₂²×C₇₉	—	2	2	no	—
105	C₂²×C₇₉	—	2	2	no	—
106	C₂²×C₇₉	—	4	1	no	—
107	C₂²×C₇₉	—	4	1	no	—
108	C₂²×C₇₉	—	4	1	no	—
109	C₂²×C₇₉	—	4	1	no	—
110	C₂²×C₇₉	#336	1	1	yes	characteristic
111	C₂²×C₇₉	#339	1	1	yes	characteristic
112	C₂²×C₇₉	#339	1	1	yes	characteristic
113	C₂²×C₇₉	#341	1	1	yes	characteristic
114	C₂²×C₇₉	#341	1	1	yes	characteristic
115	C₂²×C₇₉	#342	1	1	yes	characteristic
116	C₂²×C₇₉	#343	1	1	yes	characteristic
117	C₄×C₇₉	—	2	1	no	—
118	C₄×C₇₉	—	2	1	no	—
119	C₄×C₇₉	—	2	2	no	—
120	C₄×C₇₉	—	2	2	no	—
121	C₄×C₇₉	—	2	2	no	—
122	C₄×C₇₉	—	2	2	no	—
123	C₄×C₇₉	#323	1	2	yes	—
124	C₄×C₇₉	#324	1	2	yes	—
125	#7026	—	79	1	no	maximal, Sylow
126	C₂×C₇₉	—	4	2	no	—
127	C₂×C₇₉	#1278	1	1	yes	characteristic
128	C₂×C₇₉	#1278	1	1	yes	characteristic
129	C₂×C₇₉	#1280	1	1	yes	characteristic
130	C₂×C₇₉	#1297	1	1	yes	characteristic
131	C₂×C₇₉	#1298	1	1	yes	characteristic
132	C₂×C₇₉	#1299	1	1	yes	characteristic
133	C₂×C₇₉	#1299	1	1	yes	characteristic
134	C₄×C₂²⋊C₈	—	79	1	no	—
135	C₄×C₂²⋊C₈	D₇₉	1	1	yes	characteristic
136	#1728	—	79	1	no	—
137	C₇₉	#7026	1	1	yes	characteristic, complemented, Sylow
138	C₂²×M₁₆	—	79	2	no	—
139	C₂²×C₄²	C₂×D₇₉	1	1	yes	characteristic
140	C₂×C₂²⋊C₈	—	2	1	no	—
141	C₂×C₂²⋊C₈	—	158	1	no	—
142	C₂×C₄×C₈	—	2	1	no	—
143	C₂×C₄×C₈	—	158	1	no	—
144	#336	—	79	2	no	—
145	#336	—	158	1	no	—
146	#336	—	158	1	no	—
147	#336	Dic₇₉	1	2	yes	—
148	C₂³×C₄	Dic₁₅₈	1	1	yes	characteristic
149	C₂³×C₄	C₄×D₇₉	1	1	yes	characteristic
150	C₂³×C₄	C₂²⋊D₇₉	1	1	yes	characteristic
151	C₂×M₁₆	—	316	2	no	—
152	C₂×M₁₆	—	316	2	no	—
153	C₂×M₁₆	—	79	4	no	—
154	C₂×M₁₆	—	158	2	no	—
155	C₂²×C₈	—	2	2	no	—
156	C₂²×C₈	—	158	2	no	—
157	C₂²×C₈	—	158	2	no	—
158	C₂×C₄²	—	4	2	no	—
159	C₂×C₄²	C₄×D₇₉	1	1	yes	characteristic
160	C₂×C₄²	D₃₁₆	1	1	yes	characteristic
161	C₂×C₄²	C₂×Dic₇₉	1	1	yes	characteristic
162	C₂×C₄²	C₂²⋊D₇₉	1	1	yes	characteristic
163	C₂²⋊C₈	—	2	4	no	—
164	C₂²⋊C₈	—	158	4	no	—
165	C₄×C₈	—	4	2	no	—
166	C₄×C₈	—	316	2	no	—
167	C₂⁴	Dic₇₉⋊C₄	1	1	yes	characteristic
168	C₂²×C₄	—	2	2	no	—
169	C₂²×C₄	—	2	1	no	—
170	C₂²×C₄	—	2	1	no	—
171	C₂²×C₄	—	2	2	no	—
172	C₂²×C₄	—	2	2	no	—
173	C₂²×C₄	—	2	2	no	—
174	C₂²×C₄	—	2	2	no	—
175	C₂²×C₄	—	4	1	no	—
176	C₂²×C₄	—	4	1	no	—
177	C₂²×C₄	C₄×Dic₇₉	1	1	yes	characteristic
178	C₂²×C₄	(C₂×D₇₉)⋊C₄	1	1	yes	characteristic
179	C₂²×C₄	(C₂×D₇₉)⋊C₄	1	1	yes	characteristic
180	C₂²×C₄	C₂²⋊Dic₇₉	1	1	yes	characteristic
181	C₂²×C₄	Dic₇₉⋊C₄	1	1	yes	characteristic
182	C₂²×C₄	C₄⋊Dic₇₉	1	1	yes	characteristic
183	C₂²×C₄	C₈×D₇₉	1	2	yes	—
184	C₂²×C₄	#11929607	1	2	yes	—
185	M₁₆	—	316	4	no	—
186	M₁₆	—	316	4	no	—
187	C₂×C₈	—	2	4	no	—
188	C₂×C₈	—	4	2	no	—
189	C₂×C₈	—	4	2	no	—
190	C₂×C₈	—	316	2	no	—
191	C₂×C₈	—	316	2	no	—
192	C₂×C₈	—	316	2	no	—
193	C₂×C₈	—	316	2	no	—
194	C₂×C₈	—	158	4	no	—
195	C₂×C₈	—	158	4	no	—
196	C₄²	—	2	2	no	—
197	C₄²	—	2	2	no	—
198	C₄²	—	4	1	no	—
199	C₄²	—	4	1	no	—
200	C₂³	—	2	1	no	—
201	C₂³	—	2	1	no	—
202	C₂³	—	2	1	no	—
203	C₂³	—	2	1	no	—
204	C₂³	—	2	1	no	—
205	C₂³	—	2	1	no	—
206	C₂³	C₇₉⋊#95	1	1	yes	characteristic
207	C₂³	Dic₇₉⋊C₈	1	2	yes	—
208	C₂×C₄	—	2	1	no	—
209	C₂×C₄	—	2	1	no	—
210	C₂×C₄	—	2	1	no	—
211	C₂×C₄	—	2	1	no	—
212	C₂×C₄	—	2	1	no	—
213	C₂×C₄	—	2	1	no	—
214	C₂×C₄	—	2	1	no	—
215	C₂×C₄	—	2	1	no	—
216	C₂×C₄	—	2	1	no	—
217	C₂×C₄	—	2	1	no	—
218	C₂×C₄	—	2	2	no	—
219	C₂×C₄	—	2	2	no	—
220	C₂×C₄	—	2	2	no	—
221	C₂×C₄	—	2	2	no	—
222	C₂×C₄	—	4	4	no	—
223	C₂×C₄	—	4	4	no	—
224	C₂×C₄	C₈×Dic₇₉	1	1	yes	characteristic
225	C₂×C₄	#14579071	1	1	yes	characteristic
226	C₂×C₄	(C₂×D₇₉)⋊C₈	1	2	yes	—
227	C₂×C₄	#14579075	1	2	yes	—
228	C₂×C₄	#14579077	1	1	yes	characteristic
229	C₂×C₄	#14579079	1	1	yes	characteristic
230	C₂×C₄	#14579088	1	2	yes	—
231	C₂×C₄	#14579089	1	2	yes	—
232	C₂×C₄	#14579096	1	2	yes	—
233	C₂×C₄	#14579097	1	2	yes	—
234	C₈	—	4	4	no	—
235	C₈	—	316	4	no	—
236	C₈	—	316	4	no	—
237	C₂²	—	2	2	no	—
238	C₂²	—	2	2	no	—
239	C₂²	—	2	2	no	—
240	C₂²	—	4	1	no	—
241	C₂²	—	4	1	no	—
242	C₂²	—	4	1	no	—
243	C₂²	—	4	1	no	—
244	C₂²	#19098317	1	1	yes	characteristic
245	C₂²	#19098322	1	1	yes	characteristic
246	C₂²	#19098323	1	1	yes	characteristic
247	C₂²	#19098327	1	1	yes	characteristic
248	C₂²	#19098328	1	1	yes	characteristic
249	C₂²	#19098329	1	1	yes	characteristic
250	C₂²	#19098331	1	1	yes	characteristic
251	C₄	—	2	1	no	—
252	C₄	—	2	1	no	—
253	C₄	—	2	2	no	—
254	C₄	—	2	2	no	—
255	C₄	—	2	2	no	—
256	C₄	—	2	2	no	—
257	C₄	#19098286	1	2	yes	—
258	C₄	#19098288	1	2	yes	—
259	C₂	—	4	2	no	—
260	C₂	#23761236	1	1	yes	characteristic
261	C₂	#23761237	1	1	yes	characteristic
262	C₂	#23761242	1	1	yes	characteristic
263	C₂	#23761286	1	1	yes	characteristic
264	C₂	#23761288	1	1	yes	characteristic
265	C₂	#23761290	1	1	yes	characteristic
266	C₂	#23761291	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄	(C₂×D₇₉)⋊C₄
commutator	C₂×C₄×C₇₉	C₄×C₈
Frattini	C₂²×C₄²	C₂×D₇₉
Fitting	C₄×C₂²⋊C₈×C₇₉	C₂
socle	C₂³×C₇₉	#95
prime radical	C₂⁴×C₇₉	C₄⋊C₄
cosocle	C₂²×C₄²×C₇₉	C₂²
hypercenter	C₄×C₂²⋊C₈	D₇₉
hypocenter	C₇₉	#7026
LCS	G ▷ C₂×C₄×C₇₉ ▷ C₂²×C₇₉ ▷ C₇₉
UCS	C₁ ◁ C₂²×C₄ ◁ C₂²×C₄² ◁ C₄×C₂²⋊C₈

Complements

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

N	H ≅ G/N	type
17. C₂×#11929608	C₈	semidirect
137. C₇₉	#7026	semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
C₄×C₂²⋊(C₇₉⋊C₈)	2
#23763094	2
C₄×C₂²⋊C₈×C₇₉	2
#7026	79