Subgroups — #34307227

Order 20224.

734 conjugacy classes of subgroups.

The subgroup lattice has 276 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#23766357	C₂	1	2	yes	maximal
2	#23764170	C₂	1	4	yes	maximal
3	#23764164	C₂	1	2	yes	maximal
4	#23763531	C₂	1	2	yes	maximal
5	#23763528	C₂	1	1	yes	maximal, characteristic
6	C₄×#14579095	C₂	1	1	yes	maximal, characteristic
7	C₄×#14579093	C₂	1	2	yes	maximal
8	C₂×C₈×Q₈×C₇₉	C₂	1	1	yes	maximal, characteristic
9	C₂×#14579095	C₂²	1	2	yes	—
10	C₂×#14579093	C₂²	1	4	yes	—
11	#19098470	—	2	2	no	—
12	#19098449	—	2	4	no	—
13	#19098433	C₂²	1	2	yes	—
14	#19098429	C₂²	1	4	yes	—
15	C₄⋊C₄×Dic₇₉	—	2	2	no	—
16	C₄×C₄⋊Dic₇₉	C₂²	1	4	yes	—
17	C₄×C₄⋊Dic₇₉	C₂²	1	2	yes	—
18	#19098326	C₂²	1	2	yes	—
19	#19098326	C₂²	1	2	yes	—
20	#19098326	C₂²	1	1	yes	characteristic
21	#19098326	C₂²	1	4	yes	—
22	#19098326	C₂²	1	1	yes	characteristic
23	C₂×C₄×Q₈×C₇₉	C₂²	1	1	yes	characteristic
24	C₈×Q₈×C₇₉	C₄	1	8	yes	complemented
25	C₂×C₄⋊C₈×C₇₉	C₂²	1	1	yes	characteristic
26	C₂×C₄⋊C₈×C₇₉	C₂²	1	2	yes	—
27	C₂×C₄×C₈×C₇₉	C₂²	1	1	yes	characteristic
28	C₂×C₄×C₈×C₇₉	C₂²	1	2	yes	—
29	C₂×C₄⋊Dic₇₉	—	2	2	no	—
30	C₂×C₄⋊Dic₇₉	—	2	2	no	—
31	C₂×C₄⋊Dic₇₉	C₂³	1	2	yes	—
32	C₂×C₄⋊Dic₇₉	C₂³	1	4	yes	—
33	C₂×C₄⋊Dic₇₉	C₂³	1	2	yes	—
34	C₂×C₄×Dic₇₉	—	2	2	no	—
35	C₂×C₄×Dic₇₉	—	2	4	no	—
36	#14579095	—	2	4	no	—
37	#14579093	—	2	8	no	—
38	C₇₉⋊#95	—	2	2	no	—
39	C₇₉⋊#95	—	2	4	no	—
40	C₇₉⋊#95	—	2	4	no	—
41	C₇₉⋊#95	—	2	4	no	—
42	C₂²×Q₈×C₇₉	D₄	1	1	yes	characteristic
43	C₂²×C₈×C₇₉	C₂³	1	2	yes	—
44	C₂²×C₈×C₇₉	C₂³	1	1	yes	characteristic
45	C₂²×C₈×C₇₉	C₂³	1	1	yes	characteristic
46	C₄×Q₈×C₇₉	C₂×C₄	1	4	yes	—
47	C₄×Q₈×C₇₉	Q₈	1	4	yes	—
48	C₂×C₄⋊C₄×C₇₉	D₄	1	1	yes	characteristic
49	C₂×C₄⋊C₄×C₇₉	D₄	1	2	yes	—
50	C₂×C₄²×C₇₉	C₂³	1	2	yes	—
51	C₂×C₄²×C₇₉	C₂³	1	1	yes	characteristic
52	C₄⋊C₈×C₇₉	C₂×C₄	1	4	yes	—
53	C₄⋊C₈×C₇₉	C₂×C₄	1	8	yes	—
54	C₄×C₈×C₇₉	C₂×C₄	1	4	yes	—
55	C₄×C₈×C₇₉	C₂×C₄	1	8	yes	—
56	C₂²×Dic₇₉	—	2	2	no	—
57	C₂²×Dic₇₉	—	2	2	no	—
58	C₂²×Dic₇₉	—	2	4	no	—
59	C₄⋊Dic₇₉	—	2	4	no	—
60	C₄⋊Dic₇₉	—	2	2	no	—
61	C₄⋊Dic₇₉	—	2	2	no	—
62	C₄⋊Dic₇₉	—	2	2	no	—
63	C₄⋊Dic₇₉	—	2	4	no	—
64	C₄⋊Dic₇₉	—	4	4	no	—
65	C₄⋊Dic₇₉	—	8	2	no	—
66	C₄⋊Dic₇₉	#41	1	4	yes	—
67	C₄×Dic₇₉	—	4	4	no	—
68	C₄×Dic₇₉	—	8	4	no	—
69	C₂×Q₈×C₇₉	—	2	2	no	—
70	C₂×Q₈×C₇₉	—	2	2	no	—
71	C₂×Q₈×C₇₉	C₄⋊C₄	1	4	yes	—
72	C₂²×C₄×C₇₉	C₂×D₄	1	2	yes	—
73	C₂²×C₄×C₇₉	C₂×D₄	1	1	yes	characteristic
74	C₂²×C₄×C₇₉	C₂×D₄	1	2	yes	—
75	C₂²×C₄×C₇₉	C₂×D₄	1	1	yes	characteristic
76	C₂²×C₄×C₇₉	C₂⁴	1	1	yes	characteristic
77	C₂×C₈×C₇₉	—	2	4	no	—
78	C₂×C₈×C₇₉	—	2	2	no	—
79	C₂×C₈×C₇₉	—	2	2	no	—
80	C₂×C₈×C₇₉	C₂²×C₄	1	2	yes	—
81	C₂×C₈×C₇₉	C₂²×C₄	1	4	yes	—
82	C₂×C₈×C₇₉	C₂²×C₄	1	2	yes	—
83	C₄⋊C₄×C₇₉	C₄⋊C₄	1	8	yes	—
84	C₄⋊C₄×C₇₉	C₄⋊C₄	1	4	yes	—
85	C₄²×C₇₉	C₂²×C₄	1	4	yes	—
86	C₄²×C₇₉	C₂²×C₄	1	2	yes	—
87	C₄²×C₇₉	C₂×Q₈	1	4	yes	—
88	C₄²×C₇₉	C₂×Q₈	1	2	yes	—
89	C₂³×C₇₉	C₂²×D₄	1	1	yes	characteristic
90	Q₈×C₇₉	—	2	8	no	—
91	C₂×C₄×C₇₉	—	2	2	no	—
92	C₂×C₄×C₇₉	—	2	2	no	—
93	C₂×C₄×C₇₉	—	2	4	no	—
94	C₂×C₄×C₇₉	—	2	2	no	—
95	C₂×C₄×C₇₉	C₂×C₄⋊C₄	1	4	yes	—
96	C₂×C₄×C₇₉	C₂×C₄⋊C₄	1	2	yes	—
97	C₂×C₄×C₇₉	C₂×C₄⋊C₄	1	4	yes	—
98	C₂×C₄×C₇₉	C₂×C₄⋊C₄	1	2	yes	—
99	C₂×C₄×C₇₉	#135	1	2	yes	—
100	C₂×C₄×C₇₉	#135	1	2	yes	—
101	C₂×C₄×C₇₉	C₂³×C₄	1	1	yes	characteristic
102	C₂×C₄×C₇₉	C₂²×Q₈	1	1	yes	characteristic
103	C₂×C₄×C₇₉	C₂×#41	1	2	yes	—
104	C₂×C₄×C₇₉	2₋¹⁺⁴	1	2	yes	—
105	C₂×Dic₇₉	—	2	4	no	—
106	C₂×Dic₇₉	—	4	2	no	—
107	C₂×Dic₇₉	—	4	2	no	—
108	C₂×Dic₇₉	—	4	2	no	—
109	C₂×Dic₇₉	—	4	4	no	—
110	C₂×Dic₇₉	—	4	2	no	—
111	C₂×Dic₇₉	—	4	4	no	—
112	C₂×Dic₇₉	—	4	4	no	—
113	C₂×Dic₇₉	—	4	2	no	—
114	C₈×C₇₉	—	2	4	no	—
115	C₈×C₇₉	—	2	2	no	—
116	C₈×C₇₉	—	2	2	no	—
117	C₂²×C₇₉	C₂²×C₄⋊C₄	1	1	yes	characteristic
118	C₂²×C₇₉	#548	1	2	yes	—
119	C₂²×C₇₉	C₂×#135	1	1	yes	characteristic
120	C₂²×C₇₉	C₂×#135	1	1	yes	characteristic
121	C₂²×C₇₉	#577	1	2	yes	—
122	C₄×C₇₉	—	2	4	no	—
123	C₄×C₇₉	—	2	4	no	—
124	C₄×C₇₉	—	2	2	no	—
125	C₄×C₇₉	#427	1	4	yes	—
126	C₄×C₇₉	C₄×#41	1	1	yes	characteristic
127	C₄×C₇₉	#519	1	1	yes	characteristic
128	C₄×C₇₉	#557	1	2	yes	—
129	Dic₇₉	—	4	4	no	—
130	Dic₇₉	—	8	2	no	—
131	Dic₇₉	—	8	4	no	—
132	#22741	—	79	1	no	maximal, Sylow
133	C₂×C₇₉	#2351	1	1	yes	characteristic
134	C₂×C₇₉	C₂×#427	1	1	yes	characteristic
135	C₂×C₇₉	#2912	1	1	yes	characteristic
136	C₂×C₇₉	#3360	1	2	yes	—
137	C₂×C₇₉	#3365	1	2	yes	—
138	C₂×C₈×Q₈	D₇₉	1	1	yes	characteristic
139	#2355	—	79	2	no	—
140	#1949	—	79	4	no	—
141	#1948	—	79	2	no	—
142	#1835	—	79	2	no	—
143	#1834	—	79	1	no	—
144	C₄×#107	—	79	1	no	—
145	C₄×#106	—	79	2	no	—
146	C₇₉	#22741	1	1	yes	characteristic, complemented, Sylow
147	C₂×C₄×Q₈	C₂×D₇₉	1	1	yes	characteristic
148	C₈×Q₈	Dic₇₉	1	8	yes	complemented
149	C₂×#107	—	79	2	no	—
150	C₂×#106	—	79	4	no	—
151	C₂×C₄⋊C₈	C₂×D₇₉	1	1	yes	characteristic
152	C₂×C₄⋊C₈	C₂×D₇₉	1	2	yes	—
153	C₂×C₄×C₈	C₂×D₇₉	1	1	yes	characteristic
154	C₂×C₄×C₈	C₂×D₇₉	1	2	yes	—
155	#391	—	158	2	no	—
156	#387	—	158	4	no	—
157	#384	—	79	2	no	—
158	#382	—	79	4	no	—
159	C₄×C₄⋊C₄	—	79	2	no	—
160	C₄×C₄⋊C₄	—	79	4	no	—
161	C₄×C₄⋊C₄	—	158	2	no	—
162	#340	—	79	2	no	—
163	#340	—	79	2	no	—
164	#340	—	79	1	no	—
165	#340	—	79	4	no	—
166	#340	—	79	1	no	—
167	C₂²×Q₈	D₃₁₆	1	1	yes	characteristic
168	C₂²×C₈	C₂²×D₇₉	1	1	yes	characteristic
169	C₂²×C₈	C₂²×D₇₉	1	2	yes	—
170	C₂²×C₈	C₂²×D₇₉	1	1	yes	characteristic
171	C₄×Q₈	Dic₁₅₈	1	4	yes	—
172	C₄×Q₈	C₂×Dic₇₉	1	4	yes	—
173	C₂×C₄⋊C₄	—	79	2	no	—
174	C₂×C₄⋊C₄	—	79	2	no	—
175	C₂×C₄⋊C₄	—	79	4	no	—
176	C₂×C₄⋊C₄	—	158	2	no	—
177	C₂×C₄⋊C₄	—	158	2	no	—
178	C₂×C₄⋊C₄	D₃₁₆	1	1	yes	characteristic
179	C₂×C₄⋊C₄	D₃₁₆	1	2	yes	—
180	C₂×C₄²	—	158	2	no	—
181	C₂×C₄²	—	158	4	no	—
182	C₂×C₄²	C₂²×D₇₉	1	2	yes	—
183	C₂×C₄²	C₂²×D₇₉	1	1	yes	characteristic
184	#107	—	158	4	no	—
185	#106	—	158	8	no	—
186	C₄⋊C₈	C₂×Dic₇₉	1	4	yes	—
187	C₄⋊C₈	C₂×Dic₇₉	1	8	yes	—
188	C₄×C₈	C₂×Dic₇₉	1	4	yes	—
189	C₄×C₈	C₂×Dic₇₉	1	8	yes	—
190	#95	—	158	2	no	—
191	#95	—	158	4	no	—
192	#95	—	158	4	no	—
193	#95	—	158	4	no	—
194	C₂×Q₈	—	2	2	no	—
195	C₂×Q₈	—	2	2	no	—
196	C₂×Q₈	C₄⋊Dic₇₉	1	4	yes	—
197	C₂²×C₄	—	158	2	no	—
198	C₂²×C₄	—	158	4	no	—
199	C₂²×C₄	—	158	2	no	—
200	C₂²×C₄	C₂×D₃₁₆	1	2	yes	—
201	C₂²×C₄	C₂×D₃₁₆	1	2	yes	—
202	C₂²×C₄	C₂×D₃₁₆	1	1	yes	characteristic
203	C₂²×C₄	C₂×D₃₁₆	1	1	yes	characteristic
204	C₂²×C₄	C₂³×D₇₉	1	1	yes	characteristic
205	C₂×C₈	—	2	4	no	—
206	C₂×C₈	—	2	2	no	—
207	C₂×C₈	—	2	2	no	—
208	C₂×C₈	C₂²×Dic₇₉	1	2	yes	—
209	C₂×C₈	C₂²×Dic₇₉	1	4	yes	—
210	C₂×C₈	C₂²×Dic₇₉	1	2	yes	—
211	C₄⋊C₄	—	316	4	no	—
212	C₄⋊C₄	—	79	4	no	—
213	C₄⋊C₄	—	632	2	no	—
214	C₄⋊C₄	—	158	4	no	—
215	C₄⋊C₄	—	158	2	no	—
216	C₄⋊C₄	—	158	2	no	—
217	C₄⋊C₄	—	158	4	no	—
218	C₄⋊C₄	—	158	2	no	—
219	C₄⋊C₄	C₄⋊Dic₇₉	1	8	yes	—
220	C₄⋊C₄	C₄⋊Dic₇₉	1	4	yes	—
221	C₄²	—	316	4	no	—
222	C₄²	—	632	4	no	—
223	C₄²	C₂²×Dic₇₉	1	4	yes	—
224	C₄²	C₂²×Dic₇₉	1	2	yes	—
225	C₄²	C₂×Dic₁₅₈	1	4	yes	—
226	C₄²	C₂×Dic₁₅₈	1	2	yes	—
227	C₂³	C₂²×D₃₁₆	1	1	yes	characteristic
228	Q₈	—	2	8	no	—
229	C₂×C₄	—	2	2	no	—
230	C₂×C₄	—	2	2	no	—
231	C₂×C₄	—	2	4	no	—
232	C₂×C₄	—	2	2	no	—
233	C₂×C₄	—	316	2	no	—
234	C₂×C₄	—	316	2	no	—
235	C₂×C₄	—	316	2	no	—
236	C₂×C₄	—	316	4	no	—
237	C₂×C₄	—	316	2	no	—
238	C₂×C₄	—	316	4	no	—
239	C₂×C₄	—	316	4	no	—
240	C₂×C₄	—	316	2	no	—
241	C₂×C₄	—	158	4	no	—
242	C₂×C₄	C₂×C₄⋊Dic₇₉	1	4	yes	—
243	C₂×C₄	C₂×C₄⋊Dic₇₉	1	2	yes	—
244	C₂×C₄	C₂×C₄⋊Dic₇₉	1	4	yes	—
245	C₂×C₄	C₂×C₄⋊Dic₇₉	1	2	yes	—
246	C₂×C₄	#14579183	1	2	yes	—
247	C₂×C₄	#14579183	1	2	yes	—
248	C₂×C₄	C₂³×Dic₇₉	1	1	yes	characteristic
249	C₂×C₄	C₂²×Dic₁₅₈	1	1	yes	characteristic
250	C₂×C₄	#41×D₇₉	1	2	yes	—
251	C₂×C₄	#14579209	1	2	yes	—
252	C₈	—	2	4	no	—
253	C₈	—	2	2	no	—
254	C₈	—	2	2	no	—
255	C₂²	C₂²×C₄⋊Dic₇₉	1	1	yes	characteristic
256	C₂²	#19099225	1	2	yes	—
257	C₂²	C₂×#14579183	1	1	yes	characteristic
258	C₂²	C₂×#14579183	1	1	yes	characteristic
259	C₂²	#19099375	1	2	yes	—
260	C₄	—	2	4	no	—
261	C₄	—	2	4	no	—
262	C₄	—	2	2	no	—
263	C₄	—	316	4	no	—
264	C₄	—	632	2	no	—
265	C₄	—	632	4	no	—
266	C₄	#19098601	1	4	yes	—
267	C₄	#41×Dic₇₉	1	1	yes	characteristic
268	C₄	#19099041	1	1	yes	characteristic
269	C₄	#19099289	1	2	yes	—
270	C₂	#23766321	1	1	yes	characteristic
271	C₂	C₂×#19098601	1	1	yes	characteristic
272	C₂	#23771995	1	1	yes	characteristic
273	C₂	#23776051	1	2	yes	—
274	C₂	#23776098	1	2	yes	—

	subgroup	G/H
center	C₂³	C₂²×D₃₁₆
commutator	C₂×C₄×C₇₉	C₂³×C₄
Frattini	C₂²×C₄	C₂³×D₇₉
Fitting	C₂×C₈×Q₈×C₇₉	C₂
socle, prime radical	C₂³×C₇₉	C₂²×D₄
cosocle	C₂²×C₄×C₇₉	C₂⁴
hypercenter	C₂×C₈×Q₈	D₇₉
hypocenter	C₇₉	#22741
LCS	G ▷ C₂×C₄×C₇₉ ▷ C₂×C₇₉ ▷ C₇₉
UCS	C₁ ◁ C₂³ ◁ C₂×C₄×Q₈ ◁ C₂×C₈×Q₈

Complements

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

N	H ≅ G/N	type
24. C₈×Q₈×C₇₉	C₄	semidirect
146. C₇₉	#22741	semidirect
148. C₈×Q₈	Dic₇₉	semidirect

Maximal subgroups

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

subgroup	index
#23766357	2
#23764170	2
#23764164	2
#23763531	2
#23763528	2
C₄×#14579095	2
C₄×#14579093	2
C₂×C₈×Q₈×C₇₉	2
#22741	79