Subgroups — Q₈×#14579096 #34307159

Order 20224.

658 conjugacy classes of subgroups.

The subgroup lattice has 250 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	1	yes	maximal, characteristic
2	C₂×Q₈×C₇₉⋊C₈	C₂	1	1	yes	maximal, characteristic
3	#23764175	C₂	1	3	yes	maximal
4	#23764173	C₂	1	3	yes	maximal
5	#23763571	C₂	1	3	yes	maximal
6	C₄×#14579096	C₂	1	3	yes	maximal
7	Q₈×M₁₆×C₇₉	C₂	1	1	yes	maximal, characteristic
8	#19098694	—	2	3	no	—
9	Q₈×C₇₉⋊C₈	—	2	1	no	—
10	Q₈×C₇₉⋊C₈	C₄	1	6	yes	—
11	Q₈×C₇₉⋊C₈	C₄	1	2	yes	—
12	C₂×#14579096	C₂²	1	3	yes	—
13	C₂×#14579096	C₂²	1	6	yes	—
14	C₂×#14579096	C₂²	1	1	yes	characteristic
15	#19098587	C₂²	1	3	yes	—
16	C₂×C₄⋊(C₇₉⋊C₈)	C₂²	1	3	yes	—
17	C₄×#11929608	C₂²	1	3	yes	—
18	C₂×C₄×C₇₉⋊C₈	C₂²	1	3	yes	—
19	#19098327	C₂²	1	3	yes	—
20	#19098327	C₂²	1	3	yes	—
21	C₂×C₄×Q₈×C₇₉	C₂²	1	1	yes	characteristic
22	#446×C₇₉	—	2	3	no	—
23	C₈×Q₈×C₇₉	—	2	1	no	—
24	#423×C₇₉	C₂²	1	3	yes	—
25	C₄×M₁₆×C₇₉	C₂²	1	3	yes	—
26	C₂×#11929608	C₂³	1	3	yes	—
27	C₂×#11929608	C₂³	1	1	yes	characteristic
28	C₂²×C₇₉⋊C₈	C₂³	1	3	yes	—
29	C₂²×C₇₉⋊C₈	C₂³	1	1	yes	characteristic
30	#14579096	—	2	6	no	—
31	#14579096	—	4	6	no	—
32	#14579096	Q₈	1	4	yes	direct product factor
33	C₄⋊(C₇₉⋊C₈)	—	2	3	no	—
34	C₄⋊(C₇₉⋊C₈)	—	2	3	no	—
35	C₄⋊(C₇₉⋊C₈)	C₂×C₄	1	6	yes	—
36	C₄⋊(C₇₉⋊C₈)	C₂×C₄	1	6	yes	—
37	#14579072	—	2	3	no	—
38	C₄×C₇₉⋊C₈	—	2	3	no	—
39	C₄×C₇₉⋊C₈	C₂×C₄	1	6	yes	—
40	C₄×C₇₉⋊C₈	C₂×C₄	1	6	yes	—
41	C₂²×Q₈×C₇₉	Q₈	1	1	yes	characteristic
42	C₂×M₁₆×C₇₉	C₂³	1	3	yes	—
43	C₂×M₁₆×C₇₉	C₂³	1	1	yes	characteristic
44	C₄×Q₈×C₇₉	C₂×C₄	1	3	yes	—
45	C₄×Q₈×C₇₉	C₂×C₄	1	1	yes	characteristic
46	C₄×Q₈×C₇₉	D₄	1	1	yes	characteristic
47	C₄×Q₈×C₇₉	D₄	1	3	yes	—
48	C₂×C₄⋊C₄×C₇₉	Q₈	1	3	yes	—
49	C₂×C₄²×C₇₉	C₂³	1	3	yes	—
50	C₄⋊C₈×C₇₉	—	2	3	no	—
51	C₄⋊C₈×C₇₉	—	2	3	no	—
52	#97×C₇₉	—	2	3	no	—
53	C₄×C₈×C₇₉	—	2	3	no	—
54	#11929608	—	2	3	no	—
55	#11929608	—	2	3	no	—
56	#11929608	—	2	1	no	—
57	#11929608	C₂×Q₈	1	2	yes	—
58	C₂×C₇₉⋊C₈	—	2	1	no	—
59	C₂×C₇₉⋊C₈	—	2	3	no	—
60	C₂×C₇₉⋊C₈	—	2	3	no	—
61	C₂×C₇₉⋊C₈	—	2	3	no	—
62	C₂×C₇₉⋊C₈	C₂²×C₄	1	6	yes	—
63	C₂×C₇₉⋊C₈	C₂²×C₄	1	2	yes	—
64	C₂×C₇₉⋊C₈	C₂×Q₈	1	2	yes	—
65	C₂×C₇₉⋊C₈	#41	1	2	yes	—
66	C₂×Q₈×C₇₉	—	2	4	no	—
67	C₂×Q₈×C₇₉	C₄⋊C₄	1	3	yes	—
68	C₂×Q₈×C₇₉	C₄⋊C₄	1	1	yes	characteristic
69	C₂²×C₄×C₇₉	C₂×Q₈	1	3	yes	—
70	C₂²×C₄×C₇₉	C₂×Q₈	1	3	yes	—
71	C₂²×C₄×C₇₉	C₂⁴	1	1	yes	characteristic
72	M₁₆×C₇₉	—	2	3	no	—
73	M₁₆×C₇₉	—	2	3	no	—
74	M₁₆×C₇₉	—	2	1	no	—
75	M₁₆×C₇₉	C₂×Q₈	1	2	yes	—
76	C₂×C₈×C₇₉	—	2	1	no	—
77	C₂×C₈×C₇₉	—	2	3	no	—
78	C₄⋊C₄×C₇₉	C₄⋊C₄	1	6	yes	—
79	C₄⋊C₄×C₇₉	C₄⋊C₄	1	3	yes	—
80	C₄⋊C₄×C₇₉	C₄⋊C₄	1	3	yes	—
81	C₄²×C₇₉	C₂²×C₄	1	3	yes	—
82	C₄²×C₇₉	C₂²×C₄	1	3	yes	—
83	C₄²×C₇₉	C₂×D₄	1	3	yes	—
84	C₄²×C₇₉	C₂×D₄	1	3	yes	—
85	C₂³×C₇₉	C₂²×Q₈	1	1	yes	characteristic
86	Q₈×C₇₉	—	2	6	no	—
87	Q₈×C₇₉	#108	1	4	yes	complemented
88	C₂×C₄×C₇₉	—	2	6	no	—
89	C₂×C₄×C₇₉	—	4	3	no	—
90	C₂×C₄×C₇₉	C₂×C₄⋊C₄	1	3	yes	—
91	C₂×C₄×C₇₉	C₂×C₄⋊C₄	1	3	yes	—
92	C₂×C₄×C₇₉	C₂×C₄⋊C₄	1	3	yes	—
93	C₂×C₄×C₇₉	C₂×C₄⋊C₄	1	3	yes	—
94	C₂×C₄×C₇₉	C₂³×C₄	1	1	yes	characteristic
95	C₂×C₄×C₇₉	C₂²×D₄	1	1	yes	characteristic
96	C₂×C₄×C₇₉	C₂²×Q₈	1	1	yes	characteristic
97	C₂×C₄×C₇₉	C₂×#41	1	1	yes	characteristic
98	C₂×C₄×C₇₉	H(5,𝔽₂)	1	1	yes	characteristic
99	C₂×C₄×C₇₉	2₋¹⁺⁴	1	1	yes	characteristic
100	C₈×C₇₉	—	2	2	no	—
101	C₈×C₇₉	—	4	3	no	—
102	C₇₉⋊C₈	—	2	6	no	—
103	C₇₉⋊C₈	—	2	2	no	—
104	C₇₉⋊C₈	—	4	3	no	—
105	C₇₉⋊C₈	C₄×Q₈	1	4	yes	—
106	C₂²×C₇₉	—	2	1	no	—
107	C₂²×C₇₉	—	2	1	no	—
108	C₂²×C₇₉	C₂²×C₄⋊C₄	1	1	yes	characteristic
109	C₂²×C₇₉	#557	1	1	yes	characteristic
110	C₂²×C₇₉	Q₈²	1	1	yes	characteristic
111	C₄×C₇₉	—	2	3	no	—
112	C₄×C₇₉	—	2	3	no	—
113	C₄×C₇₉	—	2	3	no	—
114	C₄×C₇₉	C₂×#108	1	6	yes	—
115	C₄×C₇₉	C₂×C₄×Q₈	1	1	yes	characteristic
116	C₄×C₇₉	#520	1	1	yes	characteristic
117	C₄×C₇₉	D₄×Q₈	1	1	yes	characteristic
118	C₄×C₇₉	#550	1	1	yes	characteristic
119	Q₈×#108	—	79	1	no	maximal, Sylow
120	C₂×C₇₉	—	2	1	no	—
121	C₂×C₇₉	—	2	1	no	—
122	C₂×C₇₉	Q₈×C₄⋊C₄	1	1	yes	characteristic
123	C₂×C₇₉	C₂²×#108	1	1	yes	characteristic
124	C₂×C₇₉	#2916	1	1	yes	characteristic
125	Q₈×M₁₆	—	79	1	no	—
126	Q₈×M₁₆	D₇₉	1	1	yes	characteristic
127	C₂×C₈×Q₈	—	79	1	no	—
128	#1950	—	79	3	no	—
129	#1950	—	79	3	no	—
130	#1848	—	79	3	no	—
131	C₄×#108	—	79	3	no	—
132	C₇₉	Q₈×#108	1	1	yes	characteristic, complemented, Sylow
133	C₂×C₄×Q₈	C₂×D₇₉	1	1	yes	characteristic
134	#446	—	2	3	no	—
135	#446	—	158	3	no	—
136	C₈×Q₈	—	2	1	no	—
137	C₈×Q₈	—	79	6	no	—
138	C₈×Q₈	—	79	2	no	—
139	C₈×Q₈	—	158	1	no	—
140	C₂×#108	—	79	6	no	—
141	C₂×#108	—	79	1	no	—
142	C₂×#108	—	79	3	no	—
143	#423	—	79	3	no	—
144	#423	C₂×D₇₉	1	3	yes	—
145	C₂×C₄⋊C₈	—	79	3	no	—
146	C₄×M₁₆	—	79	3	no	—
147	C₄×M₁₆	C₂×D₇₉	1	3	yes	—
148	C₂×C₄×C₈	—	79	3	no	—
149	#341	—	79	3	no	—
150	#341	—	79	3	no	—
151	C₂²×Q₈	Dic₁₅₈	1	1	yes	characteristic
152	C₂×M₁₆	—	79	1	no	—
153	C₂×M₁₆	—	79	3	no	—
154	C₂×M₁₆	C₂²×D₇₉	1	3	yes	—
155	C₂×M₁₆	C₂²×D₇₉	1	1	yes	characteristic
156	C₂²×C₈	—	79	1	no	—
157	C₂²×C₈	—	79	3	no	—
158	C₄×Q₈	C₄×D₇₉	1	3	yes	—
159	C₄×Q₈	C₄×D₇₉	1	1	yes	characteristic
160	C₄×Q₈	C₂²⋊D₇₉	1	1	yes	characteristic
161	C₄×Q₈	C₂²⋊D₇₉	1	3	yes	—
162	C₂×C₄⋊C₄	Dic₁₅₈	1	3	yes	—
163	C₂×C₄²	C₂²×D₇₉	1	3	yes	—
164	#108	—	316	6	no	—
165	#108	—	79	4	no	—
166	#108	—	158	6	no	—
167	C₄⋊C₈	—	2	3	no	—
168	C₄⋊C₈	—	2	3	no	—
169	C₄⋊C₈	—	79	6	no	—
170	C₄⋊C₈	—	79	6	no	—
171	C₄⋊C₈	—	158	3	no	—
172	C₄⋊C₈	—	158	3	no	—
173	#97	—	2	3	no	—
174	#97	—	158	3	no	—
175	C₄×C₈	—	2	3	no	—
176	C₄×C₈	—	79	6	no	—
177	C₄×C₈	—	79	6	no	—
178	C₄×C₈	—	158	3	no	—
179	C₂×Q₈	—	2	4	no	—
180	C₂×Q₈	Dic₇₉⋊C₄	1	3	yes	—
181	C₂×Q₈	Dic₇₉⋊C₄	1	1	yes	characteristic
182	C₂²×C₄	C₂×Dic₁₅₈	1	3	yes	—
183	C₂²×C₄	C₂×Dic₁₅₈	1	3	yes	—
184	C₂²×C₄	C₂³×D₇₉	1	1	yes	characteristic
185	M₁₆	—	2	1	no	—
186	M₁₆	—	2	3	no	—
187	M₁₆	—	2	3	no	—
188	M₁₆	—	79	2	no	—
189	M₁₆	—	158	3	no	—
190	M₁₆	—	158	3	no	—
191	M₁₆	—	158	1	no	—
192	M₁₆	Q₈×D₇₉	1	2	yes	—
193	C₂×C₈	—	2	1	no	—
194	C₂×C₈	—	2	3	no	—
195	C₂×C₈	—	79	2	no	—
196	C₂×C₈	—	79	2	no	—
197	C₂×C₈	—	79	2	no	—
198	C₂×C₈	—	79	6	no	—
199	C₂×C₈	—	158	3	no	—
200	C₂×C₈	—	158	3	no	—
201	C₂×C₈	—	158	3	no	—
202	C₂×C₈	—	158	1	no	—
203	C₄⋊C₄	Dic₇₉⋊C₄	1	6	yes	—
204	C₄⋊C₄	Dic₇₉⋊C₄	1	3	yes	—
205	C₄⋊C₄	Dic₇₉⋊C₄	1	3	yes	—
206	C₄²	C₂×C₄×D₇₉	1	3	yes	—
207	C₄²	C₂×C₄×D₇₉	1	3	yes	—
208	C₄²	C₂×C₂²⋊D₇₉	1	3	yes	—
209	C₄²	C₂×C₂²⋊D₇₉	1	3	yes	—
210	C₂³	C₂²×Dic₁₅₈	1	1	yes	characteristic
211	Q₈	—	2	6	no	—
212	Q₈	#14579096	1	4	yes	direct product factor
213	C₂×C₄	—	2	6	no	—
214	C₂×C₄	—	4	3	no	—
215	C₂×C₄	C₂×Dic₇₉⋊C₄	1	3	yes	—
216	C₂×C₄	C₂×Dic₇₉⋊C₄	1	3	yes	—
217	C₂×C₄	C₂×Dic₇₉⋊C₄	1	3	yes	—
218	C₂×C₄	C₂×Dic₇₉⋊C₄	1	3	yes	—
219	C₂×C₄	C₂²×C₄×D₇₉	1	1	yes	characteristic
220	C₂×C₄	C₂²×C₂²⋊D₇₉	1	1	yes	characteristic
221	C₂×C₄	C₂×Q₈×D₇₉	1	1	yes	characteristic
222	C₂×C₄	#41×D₇₉	1	1	yes	characteristic
223	C₂×C₄	#14579207	1	1	yes	characteristic
224	C₂×C₄	#14579209	1	1	yes	characteristic
225	C₈	—	2	2	no	—
226	C₈	—	4	3	no	—
227	C₈	—	316	3	no	—
228	C₈	—	79	4	no	—
229	C₈	—	158	2	no	—
230	C₈	—	158	6	no	—
231	C₂²	—	2	1	no	—
232	C₂²	—	2	1	no	—
233	C₂²	C₂²×Dic₇₉⋊C₄	1	1	yes	characteristic
234	C₂²	#19099289	1	1	yes	characteristic
235	C₂²	Q₈×Dic₁₅₈	1	1	yes	characteristic
236	C₄	—	2	3	no	—
237	C₄	—	2	3	no	—
238	C₄	—	2	3	no	—
239	C₄	C₂×#14579096	1	6	yes	—
240	C₄	C₄×Q₈×D₇₉	1	1	yes	characteristic
241	C₄	#19099045	1	1	yes	characteristic
242	C₄	Q₈×C₂²⋊D₇₉	1	1	yes	characteristic
243	C₄	#19099232	1	1	yes	characteristic
244	C₂	—	2	1	no	—
245	C₂	—	2	1	no	—
246	C₂	Q₈×Dic₇₉⋊C₄	1	1	yes	characteristic
247	C₂	C₂²×#14579096	1	1	yes	characteristic
248	C₂	#23772009	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₂²×C₂²⋊D₇₉
commutator	C₂×C₄×C₇₉	C₂³×C₄
Frattini	C₂²×C₄	C₂³×D₇₉
Fitting	Q₈×M₁₆×C₇₉	C₂
socle	C₂²×C₇₉	C₂²×C₄⋊C₄
prime radical	C₂³×C₇₉	C₂²×Q₈
cosocle	C₂²×C₄×C₇₉	C₂⁴
hypercenter	Q₈×M₁₆	D₇₉
hypocenter	C₇₉	Q₈×#108
LCS	G ▷ C₂×C₄×C₇₉ ▷ C₂×C₇₉ ▷ C₇₉
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄×Q₈ ◁ Q₈×M₁₆

Complements

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

N	H ≅ G/N	type
32. #14579096	Q₈	direct (N × H)
87. Q₈×C₇₉	#108	semidirect
132. C₇₉	Q₈×#108	semidirect
212. Q₈	#14579096	direct (N × H)

Maximal subgroups

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

subgroup	index
Q₈×#11929608	2
C₂×Q₈×C₇₉⋊C₈	2
#23764175	2
#23764173	2
#23763571	2
C₄×#14579096	2
Q₈×M₁₆×C₇₉	2
Q₈×#108	79

Subgroups — Q8×#14579096 #34307159

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — Q₈×#14579096 #34307159