Subgroups — #34307280

Order 20224.

622 conjugacy classes of subgroups.

The subgroup lattice has 270 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	Q₈×#11929608	C₂	1	2	yes	maximal
2	#23764176	C₂	1	2	yes	maximal
3	#23764176	C₂	1	4	yes	maximal
4	#23763541	C₂	1	2	yes	maximal
5	#23763538	C₂	1	1	yes	maximal, characteristic
6	#23763306	C₂	1	2	yes	maximal
7	C₄×#14579097	C₂	1	1	yes	maximal, characteristic
8	C₂×#446×C₇₉	C₂	1	1	yes	maximal, characteristic
9	#19098694	—	2	2	no	—
10	#19098694	—	2	4	no	—
11	Q₈×C₇₉⋊C₈	—	2	2	no	—
12	C₂×#14579097	C₂²	1	2	yes	—
13	C₂×#14579097	C₂²	1	4	yes	—
14	#19098587	C₂²	1	2	yes	—
15	#19098587	C₂²	1	4	yes	—
16	C₄×#11929608	C₂²	1	2	yes	—
17	C₄×#11929608	C₂²	1	4	yes	—
18	#19098328	C₂²	1	2	yes	—
19	#19098328	C₂²	1	2	yes	—
20	#19098328	C₂²	1	1	yes	characteristic
21	#19098328	C₂²	1	4	yes	—
22	#19098328	C₂²	1	1	yes	characteristic
23	C₂×C₄×Q₈×C₇₉	C₂²	1	1	yes	characteristic
24	#446×C₇₉	C₄	1	4	yes	—
25	#446×C₇₉	C₄	1	4	yes	—
26	C₂×C₄⋊C₈×C₇₉	C₂²	1	1	yes	characteristic
27	C₂×C₄⋊C₈×C₇₉	C₂²	1	2	yes	—
28	C₂×#97×C₇₉	C₂²	1	2	yes	—
29	C₂×C₄×C₈×C₇₉	C₂²	1	1	yes	characteristic
30	C₂×#11929608	C₂³	1	2	yes	—
31	C₂×#11929608	C₂³	1	4	yes	—
32	C₂×#11929608	C₂³	1	2	yes	—
33	#14579097	—	2	4	no	—
34	#14579097	—	2	8	no	—
35	C₄⋊(C₇₉⋊C₈)	—	2	2	no	—
36	C₄⋊(C₇₉⋊C₈)	—	2	4	no	—
37	C₄⋊(C₇₉⋊C₈)	—	2	4	no	—
38	C₄⋊(C₇₉⋊C₈)	—	2	2	no	—
39	#14579072	—	2	4	no	—
40	#14579072	—	2	2	no	—
41	C₄×C₇₉⋊C₈	—	2	2	no	—
42	C₄×C₇₉⋊C₈	—	2	4	no	—
43	C₂²×Q₈×C₇₉	Q₈	1	1	yes	characteristic
44	C₂²×C₈×C₇₉	C₂³	1	1	yes	characteristic
45	C₂²×C₈×C₇₉	C₂³	1	2	yes	—
46	C₂²×C₈×C₇₉	C₂³	1	1	yes	characteristic
47	C₄×Q₈×C₇₉	C₂×C₄	1	2	yes	—
48	C₄×Q₈×C₇₉	C₂×C₄	1	2	yes	—
49	C₄×Q₈×C₇₉	D₄	1	2	yes	—
50	C₄×Q₈×C₇₉	D₄	1	2	yes	—
51	C₂×C₄⋊C₄×C₇₉	Q₈	1	1	yes	characteristic
52	C₂×C₄⋊C₄×C₇₉	Q₈	1	2	yes	—
53	C₂×C₄²×C₇₉	C₂³	1	1	yes	characteristic
54	C₂×C₄²×C₇₉	C₂³	1	2	yes	—
55	C₄⋊C₈×C₇₉	C₂×C₄	1	2	yes	—
56	C₄⋊C₈×C₇₉	C₂×C₄	1	2	yes	—
57	C₄⋊C₈×C₇₉	C₂×C₄	1	8	yes	—
58	#97×C₇₉	C₂×C₄	1	8	yes	—
59	C₄×C₈×C₇₉	C₂×C₄	1	2	yes	—
60	C₄×C₈×C₇₉	C₂×C₄	1	2	yes	—
61	#11929608	—	2	2	no	—
62	#11929608	—	2	4	no	—
63	#11929608	—	2	2	no	—
64	#11929608	—	2	2	no	—
65	#11929608	—	2	4	no	—
66	#11929608	#41	1	4	yes	—
67	C₂×C₇₉⋊C₈	—	2	2	no	—
68	C₂×C₇₉⋊C₈	—	2	2	no	—
69	C₂×C₇₉⋊C₈	—	2	4	no	—
70	C₂×Q₈×C₇₉	—	4	2	no	—
71	C₂×Q₈×C₇₉	C₄⋊C₄	1	2	yes	—
72	C₂×Q₈×C₇₉	C₄⋊C₄	1	2	yes	—
73	C₂²×C₄×C₇₉	C₂×Q₈	1	1	yes	characteristic
74	C₂²×C₄×C₇₉	C₂×Q₈	1	1	yes	characteristic
75	C₂²×C₄×C₇₉	C₂×Q₈	1	2	yes	—
76	C₂²×C₄×C₇₉	C₂×Q₈	1	2	yes	—
77	C₂²×C₄×C₇₉	C₂⁴	1	1	yes	characteristic
78	C₂×C₈×C₇₉	—	2	2	no	—
79	C₂×C₈×C₇₉	—	2	4	no	—
80	C₂×C₈×C₇₉	—	2	2	no	—
81	C₂×C₈×C₇₉	C₂²×C₄	1	2	yes	—
82	C₂×C₈×C₇₉	C₂²×C₄	1	2	yes	—
83	C₂×C₈×C₇₉	C₂²×C₄	1	4	yes	—
84	C₄⋊C₄×C₇₉	C₄⋊C₄	1	4	yes	—
85	C₄⋊C₄×C₇₉	C₄⋊C₄	1	4	yes	—
86	C₄⋊C₄×C₇₉	C₄⋊C₄	1	2	yes	—
87	C₄⋊C₄×C₇₉	C₄⋊C₄	1	2	yes	—
88	C₄²×C₇₉	C₂²×C₄	1	1	yes	characteristic
89	C₄²×C₇₉	C₂²×C₄	1	1	yes	characteristic
90	C₄²×C₇₉	C₂²×C₄	1	4	yes	—
91	C₄²×C₇₉	C₂×D₄	1	1	yes	characteristic
92	C₄²×C₇₉	C₂×D₄	1	4	yes	—
93	C₄²×C₇₉	C₂×D₄	1	1	yes	characteristic
94	C₂³×C₇₉	C₂²×Q₈	1	1	yes	characteristic
95	Q₈×C₇₉	—	2	4	no	—
96	Q₈×C₇₉	—	4	2	no	—
97	C₂×C₄×C₇₉	—	2	2	no	—
98	C₂×C₄×C₇₉	—	4	1	no	—
99	C₂×C₄×C₇₉	—	4	2	no	—
100	C₂×C₄×C₇₉	—	4	2	no	—
101	C₂×C₄×C₇₉	C₂×C₄⋊C₄	1	1	yes	characteristic
102	C₂×C₄×C₇₉	C₂×C₄⋊C₄	1	1	yes	characteristic
103	C₂×C₄×C₇₉	C₂×C₄⋊C₄	1	4	yes	—
104	C₂×C₄×C₇₉	C₂×C₄⋊C₄	1	1	yes	characteristic
105	C₂×C₄×C₇₉	C₂×C₄⋊C₄	1	4	yes	—
106	C₂×C₄×C₇₉	C₂×C₄⋊C₄	1	1	yes	characteristic
107	C₂×C₄×C₇₉	C₂³×C₄	1	1	yes	characteristic
108	C₂×C₄×C₇₉	C₂²×D₄	1	1	yes	characteristic
109	C₂×C₄×C₇₉	C₂×#41	1	2	yes	—
110	C₂×C₄×C₇₉	2₋¹⁺⁴	1	2	yes	—
111	C₈×C₇₉	—	2	2	no	—
112	C₈×C₇₉	—	2	4	no	—
113	C₈×C₇₉	—	2	2	no	—
114	C₇₉⋊C₈	—	2	4	no	—
115	C₇₉⋊C₈	—	4	2	no	—
116	C₇₉⋊C₈	—	4	4	no	—
117	C₂²×C₇₉	—	2	1	no	—
118	C₂²×C₇₉	—	2	1	no	—
119	C₂²×C₇₉	C₂²×C₄⋊C₄	1	1	yes	characteristic
120	C₂²×C₇₉	#557	1	2	yes	—
121	C₄×C₇₉	—	2	1	no	—
122	C₄×C₇₉	—	2	4	no	—
123	C₄×C₇₉	—	2	1	no	—
124	C₄×C₇₉	—	2	1	no	—
125	C₄×C₇₉	—	2	4	no	—
126	C₄×C₇₉	#430	1	2	yes	—
127	C₄×C₇₉	C₄×#41	1	1	yes	characteristic
128	C₄×C₇₉	#519	1	1	yes	characteristic
129	C₄×C₇₉	#548	1	2	yes	—
130	#22747	—	79	1	no	maximal, Sylow
131	C₂×C₇₉	—	2	2	no	—
132	C₂×C₇₉	#2351	1	1	yes	characteristic
133	C₂×C₇₉	C₂×#430	1	1	yes	characteristic
134	C₂×C₇₉	#2916	1	1	yes	characteristic
135	Q₈×M₁₆	—	79	2	no	—
136	C₂×#446	D₇₉	1	1	yes	characteristic
137	#1950	—	79	4	no	—
138	#1950	—	79	2	no	—
139	#1838	—	79	2	no	—
140	#1837	—	79	1	no	—
141	#1778	—	79	2	no	—
142	C₄×#108	—	79	1	no	—
143	C₇₉	#22747	1	1	yes	characteristic, complemented, Sylow
144	C₂×C₄×Q₈	C₂×D₇₉	1	1	yes	characteristic
145	#446	—	158	2	no	—
146	#446	—	158	4	no	—
147	#446	Dic₇₉	1	4	yes	—
148	#446	Dic₇₉	1	4	yes	—
149	C₈×Q₈	—	158	2	no	—
150	C₂×#108	—	79	2	no	—
151	C₂×#108	—	79	4	no	—
152	#423	—	79	2	no	—
153	#423	—	79	4	no	—
154	C₂×C₄⋊C₈	C₂×D₇₉	1	1	yes	characteristic
155	C₂×C₄⋊C₈	C₂×D₇₉	1	2	yes	—
156	C₄×M₁₆	—	79	2	no	—
157	C₄×M₁₆	—	79	4	no	—
158	C₂×#97	C₂×D₇₉	1	2	yes	—
159	C₂×C₄×C₈	C₂×D₇₉	1	1	yes	characteristic
160	#341	—	79	2	no	—
161	#341	—	79	2	no	—
162	#341	—	79	1	no	—
163	#341	—	79	4	no	—
164	#341	—	79	1	no	—
165	C₂²×Q₈	Dic₁₅₈	1	1	yes	characteristic
166	C₂×M₁₆	—	79	2	no	—
167	C₂×M₁₆	—	79	4	no	—
168	C₂×M₁₆	—	79	2	no	—
169	C₂²×C₈	C₂²×D₇₉	1	1	yes	characteristic
170	C₂²×C₈	C₂²×D₇₉	1	1	yes	characteristic
171	C₂²×C₈	C₂²×D₇₉	1	2	yes	—
172	C₄×Q₈	D₃₁₆	1	2	yes	—
173	C₄×Q₈	D₃₁₆	1	2	yes	—
174	C₄×Q₈	C₂×Dic₇₉	1	2	yes	—
175	C₄×Q₈	C₂×Dic₇₉	1	2	yes	—
176	C₂×C₄⋊C₄	Dic₁₅₈	1	1	yes	characteristic
177	C₂×C₄⋊C₄	Dic₁₅₈	1	2	yes	—
178	C₂×C₄²	C₂²×D₇₉	1	1	yes	characteristic
179	C₂×C₄²	C₂²×D₇₉	1	2	yes	—
180	#108	—	158	4	no	—
181	#108	—	158	8	no	—
182	C₄⋊C₈	—	158	2	no	—
183	C₄⋊C₈	—	158	4	no	—
184	C₄⋊C₈	—	158	4	no	—
185	C₄⋊C₈	—	158	2	no	—
186	C₄⋊C₈	C₂×Dic₇₉	1	2	yes	—
187	C₄⋊C₈	C₂×Dic₇₉	1	2	yes	—
188	C₄⋊C₈	C₂×Dic₇₉	1	8	yes	—
189	#97	—	158	4	no	—
190	#97	—	158	2	no	—
191	#97	C₂×Dic₇₉	1	8	yes	—
192	C₄×C₈	—	158	2	no	—
193	C₄×C₈	—	158	4	no	—
194	C₄×C₈	C₂×Dic₇₉	1	2	yes	—
195	C₄×C₈	C₂×Dic₇₉	1	2	yes	—
196	C₂×Q₈	—	4	2	no	—
197	C₂×Q₈	C₄⋊Dic₇₉	1	2	yes	—
198	C₂×Q₈	C₄⋊Dic₇₉	1	2	yes	—
199	C₂²×C₄	C₂×Dic₁₅₈	1	1	yes	characteristic
200	C₂²×C₄	C₂×Dic₁₅₈	1	1	yes	characteristic
201	C₂²×C₄	C₂×Dic₁₅₈	1	2	yes	—
202	C₂²×C₄	C₂×Dic₁₅₈	1	2	yes	—
203	C₂²×C₄	C₂³×D₇₉	1	1	yes	characteristic
204	M₁₆	—	79	4	no	—
205	M₁₆	—	158	2	no	—
206	M₁₆	—	158	4	no	—
207	M₁₆	—	158	4	no	—
208	M₁₆	—	158	2	no	—
209	M₁₆	—	158	2	no	—
210	C₂×C₈	—	2	2	no	—
211	C₂×C₈	—	2	4	no	—
212	C₂×C₈	—	2	2	no	—
213	C₂×C₈	—	158	2	no	—
214	C₂×C₈	—	158	4	no	—
215	C₂×C₈	—	158	2	no	—
216	C₂×C₈	C₂²×Dic₇₉	1	2	yes	—
217	C₂×C₈	C₂²×Dic₇₉	1	2	yes	—
218	C₂×C₈	C₂²×Dic₇₉	1	4	yes	—
219	C₄⋊C₄	C₄⋊Dic₇₉	1	4	yes	—
220	C₄⋊C₄	C₄⋊Dic₇₉	1	4	yes	—
221	C₄⋊C₄	C₄⋊Dic₇₉	1	2	yes	—
222	C₄⋊C₄	C₄⋊Dic₇₉	1	2	yes	—
223	C₄²	C₂²×Dic₇₉	1	1	yes	characteristic
224	C₄²	C₂²×Dic₇₉	1	1	yes	characteristic
225	C₄²	C₂²×Dic₇₉	1	4	yes	—
226	C₄²	C₂×D₃₁₆	1	1	yes	characteristic
227	C₄²	C₂×D₃₁₆	1	4	yes	—
228	C₄²	C₂×D₃₁₆	1	1	yes	characteristic
229	C₂³	C₂²×Dic₁₅₈	1	1	yes	characteristic
230	Q₈	—	2	4	no	—
231	Q₈	—	4	2	no	—
232	C₂×C₄	—	2	2	no	—
233	C₂×C₄	—	4	1	no	—
234	C₂×C₄	—	4	2	no	—
235	C₂×C₄	—	4	2	no	—
236	C₂×C₄	C₂×C₄⋊Dic₇₉	1	1	yes	characteristic
237	C₂×C₄	C₂×C₄⋊Dic₇₉	1	1	yes	characteristic
238	C₂×C₄	C₂×C₄⋊Dic₇₉	1	4	yes	—
239	C₂×C₄	C₂×C₄⋊Dic₇₉	1	1	yes	characteristic
240	C₂×C₄	C₂×C₄⋊Dic₇₉	1	4	yes	—
241	C₂×C₄	C₂×C₄⋊Dic₇₉	1	1	yes	characteristic
242	C₂×C₄	C₂³×Dic₇₉	1	1	yes	characteristic
243	C₂×C₄	C₂²×D₃₁₆	1	1	yes	characteristic
244	C₂×C₄	#41×D₇₉	1	2	yes	—
245	C₂×C₄	#14579209	1	2	yes	—
246	C₈	—	2	2	no	—
247	C₈	—	2	4	no	—
248	C₈	—	2	2	no	—
249	C₈	—	316	2	no	—
250	C₈	—	316	4	no	—
251	C₈	—	158	4	no	—
252	C₂²	—	2	1	no	—
253	C₂²	—	2	1	no	—
254	C₂²	C₂²×C₄⋊Dic₇₉	1	1	yes	characteristic
255	C₂²	#19099289	1	2	yes	—
256	C₄	—	2	1	no	—
257	C₄	—	2	4	no	—
258	C₄	—	2	1	no	—
259	C₄	—	2	1	no	—
260	C₄	—	2	4	no	—
261	C₄	#19098612	1	2	yes	—
262	C₄	#41×Dic₇₉	1	1	yes	characteristic
263	C₄	#19099041	1	1	yes	characteristic
264	C₄	#19099225	1	2	yes	—
265	C₂	—	2	2	no	—
266	C₂	#23766321	1	1	yes	characteristic
267	C₂	C₂×#19098612	1	1	yes	characteristic
268	C₂	#23772011	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₂²×D₃₁₆
commutator	C₂×C₄×C₇₉	C₂³×C₄
Frattini	C₂²×C₄	C₂³×D₇₉
Fitting	C₂×#446×C₇₉	C₂
socle	C₂²×C₇₉	C₂²×C₄⋊C₄
prime radical	C₂³×C₇₉	C₂²×Q₈
cosocle	C₂²×C₄×C₇₉	C₂⁴
hypercenter	C₂×#446	D₇₉
hypocenter	C₇₉	#22747
LCS	G ▷ C₂×C₄×C₇₉ ▷ C₂×C₇₉ ▷ C₇₉
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄×Q₈ ◁ C₂×#446

Complements

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

N	H ≅ G/N	type
143. C₇₉	#22747	semidirect

Maximal subgroups

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

subgroup	index
Q₈×#11929608	2
#23764176	2
#23763541	2
#23763538	2
#23763306	2
C₄×#14579097	2
C₂×#446×C₇₉	2
#22747	79