Subgroups — #34198076

Order 20224.

362 conjugacy classes of subgroups.

The subgroup lattice has 234 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#23765156	C₂	1	1	yes	maximal, characteristic
2	C₄×#14579101	C₂	1	1	yes	maximal, characteristic
3	Q₈⋊(C₇₉⋊C₁₆)	C₂	1	4	yes	maximal, complemented
4	C₂×C₈×Q₈×C₇₉	C₂	1	1	yes	maximal, characteristic
5	C₂×#14579101	—	2	1	no	—
6	C₂×#14579101	—	2	1	no	—
7	C₄⋊(C₇₉⋊C₁₆)	—	2	1	no	—
8	C₄⋊(C₇₉⋊C₁₆)	C₂²	1	2	yes	—
9	#19098339	—	2	1	no	—
10	C₄×C₇₉⋊C₁₆	C₂²	1	2	yes	—
11	C₂×C₄×Q₈×C₇₉	C₄	1	1	yes	characteristic
12	C₈×Q₈×C₇₉	—	2	1	no	—
13	C₈×Q₈×C₇₉	—	2	1	no	—
14	C₈×Q₈×C₇₉	C₄	1	2	yes	—
15	C₈×Q₈×C₇₉	C₂²	1	2	yes	—
16	C₂×C₄⋊C₈×C₇₉	—	2	1	no	—
17	C₂×C₄⋊C₈×C₇₉	C₄	1	1	yes	characteristic
18	C₂×C₄×C₈×C₇₉	—	2	1	no	—
19	C₂×C₄×C₈×C₇₉	C₂²	1	1	yes	characteristic
20	#14579101	—	2	4	no	—
21	#14579101	—	4	2	no	—
22	C₂×C₇₉⋊C₁₆	—	2	2	no	—
23	C₂×C₇₉⋊C₁₆	—	2	2	no	—
24	C₂²×Q₈×C₇₉	C₈	1	1	yes	characteristic
25	C₂²×C₈×C₇₉	—	2	1	no	—
26	C₂²×C₈×C₇₉	D₄	1	1	yes	characteristic
27	C₂²×C₈×C₇₉	D₄	1	1	yes	characteristic
28	C₄×Q₈×C₇₉	—	2	1	no	—
29	C₄×Q₈×C₇₉	—	2	1	no	—
30	C₄×Q₈×C₇₉	C₈	1	2	yes	—
31	C₄×Q₈×C₇₉	C₂×C₄	1	2	yes	—
32	C₂×C₄⋊C₄×C₇₉	—	2	1	no	—
33	C₂×C₄⋊C₄×C₇₉	C₈	1	1	yes	characteristic
34	C₂×C₄²×C₇₉	—	2	1	no	—
35	C₂×C₄²×C₇₉	C₂×C₄	1	1	yes	characteristic
36	C₄⋊C₈×C₇₉	—	2	2	no	—
37	C₄⋊C₈×C₇₉	—	2	2	no	—
38	C₄⋊C₈×C₇₉	—	2	1	no	—
39	C₄⋊C₈×C₇₉	C₂×C₄	1	2	yes	—
40	C₄×C₈×C₇₉	—	2	2	no	—
41	C₄×C₈×C₇₉	—	2	2	no	—
42	C₄×C₈×C₇₉	C₂×C₄	1	1	yes	characteristic
43	C₄×C₈×C₇₉	D₄	1	1	yes	characteristic
44	C₄×C₈×C₇₉	D₄	1	1	yes	characteristic
45	C₄×C₈×C₇₉	C₂³	1	1	yes	characteristic
46	C₇₉⋊C₁₆	—	2	4	no	—
47	C₇₉⋊C₁₆	—	4	2	no	—
48	C₂×Q₈×C₇₉	—	2	4	no	—
49	C₂×Q₈×C₇₉	—	2	1	no	—
50	C₂×Q₈×C₇₉	C₂×C₈	1	2	yes	—
51	C₂²×C₄×C₇₉	—	2	1	no	—
52	C₂²×C₄×C₇₉	—	2	1	no	—
53	C₂²×C₄×C₇₉	C₂²⋊C₄	1	1	yes	characteristic
54	C₂²×C₄×C₇₉	C₂×C₈	1	1	yes	characteristic
55	C₂²×C₄×C₇₉	M₁₆	1	1	yes	characteristic
56	C₂×C₈×C₇₉	—	2	2	no	—
57	C₂×C₈×C₇₉	—	2	2	no	—
58	C₂×C₈×C₇₉	—	4	2	no	—
59	C₂×C₈×C₇₉	C₂²⋊C₄	1	1	yes	characteristic
60	C₂×C₈×C₇₉	C₂²⋊C₄	1	1	yes	characteristic
61	C₂×C₈×C₇₉	SD₁₆	1	2	yes	—
62	C₂×C₈×C₇₉	Q₁₆	1	2	yes	—
63	C₂×C₈×C₇₉	C₂×D₄	1	1	yes	characteristic
64	C₂×C₈×C₇₉	C₂×D₄	1	1	yes	characteristic
65	C₄⋊C₄×C₇₉	—	2	2	no	—
66	C₄⋊C₄×C₇₉	—	2	1	no	—
67	C₄⋊C₄×C₇₉	—	2	2	no	—
68	C₄⋊C₄×C₇₉	C₂×C₈	1	2	yes	—
69	C₄²×C₇₉	—	2	2	no	—
70	C₄²×C₇₉	—	2	2	no	—
71	C₄²×C₇₉	C₂²⋊C₄	1	1	yes	characteristic
72	C₄²×C₇₉	C₂×C₈	1	1	yes	characteristic
73	C₄²×C₇₉	M₁₆	1	1	yes	characteristic
74	C₄²×C₇₉	C₂²×C₄	1	1	yes	characteristic
75	C₂³×C₇₉	C₂²⋊C₈	1	1	yes	characteristic
76	Q₈×C₇₉	—	2	4	no	—
77	Q₈×C₇₉	—	2	2	no	—
78	Q₈×C₇₉	M₃₂	1	4	yes	complemented
79	C₂×C₄×C₇₉	—	2	4	no	—
80	C₂×C₄×C₇₉	—	2	2	no	—
81	C₂×C₄×C₇₉	—	2	2	no	—
82	C₂×C₄×C₇₉	—	2	2	no	—
83	C₂×C₄×C₇₉	—	4	1	no	—
84	C₂×C₄×C₇₉	—	4	2	no	—
85	C₂×C₄×C₇₉	C₂²⋊C₈	1	1	yes	characteristic
86	C₂×C₄×C₇₉	C₂²⋊C₈	1	1	yes	characteristic
87	C₂×C₄×C₇₉	C₂²⋊C₈	1	1	yes	characteristic
88	C₂×C₄×C₇₉	#103	1	2	yes	—
89	C₂×C₄×C₇₉	#104	1	2	yes	—
90	C₂×C₄×C₇₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
91	C₂×C₄×C₇₉	C₂²×C₈	1	1	yes	characteristic
92	C₂×C₄×C₇₉	C₂×M₁₆	1	1	yes	characteristic
93	C₈×C₇₉	—	2	1	no	—
94	C₈×C₇₉	—	2	1	no	—
95	C₈×C₇₉	—	4	2	no	—
96	C₈×C₇₉	#103	1	2	yes	—
97	C₈×C₇₉	C₂×SD₁₆	1	1	yes	characteristic
98	C₈×C₇₉	C₂×Q₁₆	1	1	yes	characteristic
99	C₂²×C₇₉	—	2	1	no	—
100	C₂²×C₇₉	—	2	1	no	—
101	C₂²×C₇₉	Q₈⋊C₈	1	2	yes	—
102	C₂²×C₇₉	C₂×C₂²⋊C₈	1	1	yes	characteristic
103	C₄×C₇₉	—	2	4	no	—
104	C₄×C₇₉	—	2	1	no	—
105	C₄×C₇₉	—	2	1	no	—
106	C₄×C₇₉	—	2	1	no	—
107	C₄×C₇₉	—	4	2	no	—
108	C₄×C₇₉	Q₈⋊C₈	1	2	yes	—
109	C₄×C₇₉	C₂×#103	1	1	yes	characteristic
110	C₄×C₇₉	C₂×#104	1	1	yes	characteristic
111	C₄×C₇₉	C₂×M₃₂	1	2	yes	—
112	#11607	—	79	1	no	maximal, Sylow
113	C₂×C₇₉	—	2	2	no	—
114	C₂×C₇₉	C₂×Q₈⋊C₈	1	1	yes	characteristic
115	C₂×C₇₉	#2112	1	1	yes	characteristic
116	C₂×C₇₉	#2117	1	1	yes	characteristic
117	C₂×C₈×Q₈	D₇₉	1	1	yes	characteristic
118	#2150	—	79	1	no	—
119	C₄×M₃₂	—	79	1	no	—
120	Q₈⋊C₁₆	—	79	4	no	—
121	C₇₉	#11607	1	1	yes	characteristic, complemented, Sylow
122	C₂×C₄×Q₈	Dic₇₉	1	1	yes	characteristic
123	C₂×M₃₂	—	158	1	no	—
124	C₂×M₃₂	—	158	1	no	—
125	C₈×Q₈	—	2	1	no	—
126	C₈×Q₈	—	2	1	no	—
127	C₈×Q₈	Dic₇₉	1	2	yes	—
128	C₈×Q₈	C₂×D₇₉	1	2	yes	—
129	C₂×C₄⋊C₈	—	2	1	no	—
130	C₂×C₄⋊C₈	Dic₇₉	1	1	yes	characteristic
131	C₂×C₄×C₈	—	2	1	no	—
132	C₂×C₄×C₈	C₂×D₇₉	1	1	yes	characteristic
133	C₄⋊C₁₆	—	79	2	no	—
134	C₄⋊C₁₆	—	158	1	no	—
135	#346	—	158	1	no	—
136	C₄×C₁₆	—	79	2	no	—
137	C₂²×Q₈	C₇₉⋊C₈	1	1	yes	characteristic
138	C₂²×C₈	—	2	1	no	—
139	C₂²×C₈	C₂²⋊D₇₉	1	1	yes	characteristic
140	C₂²×C₈	C₂²⋊D₇₉	1	1	yes	characteristic
141	C₄×Q₈	—	2	1	no	—
142	C₄×Q₈	—	2	1	no	—
143	C₄×Q₈	C₇₉⋊C₈	1	2	yes	—
144	C₄×Q₈	C₂×Dic₇₉	1	2	yes	—
145	C₂×C₄⋊C₄	—	2	1	no	—
146	C₂×C₄⋊C₄	C₇₉⋊C₈	1	1	yes	characteristic
147	C₂×C₄²	—	2	1	no	—
148	C₂×C₄²	C₂×Dic₇₉	1	1	yes	characteristic
149	M₃₂	—	316	2	no	—
150	M₃₂	—	158	4	no	—
151	C₂×C₁₆	—	158	2	no	—
152	C₂×C₁₆	—	158	2	no	—
153	C₄⋊C₈	—	2	2	no	—
154	C₄⋊C₈	—	2	2	no	—
155	C₄⋊C₈	—	2	1	no	—
156	C₄⋊C₈	C₂×Dic₇₉	1	2	yes	—
157	C₄×C₈	—	2	2	no	—
158	C₄×C₈	—	2	2	no	—
159	C₄×C₈	C₂×Dic₇₉	1	1	yes	characteristic
160	C₄×C₈	C₂²⋊D₇₉	1	1	yes	characteristic
161	C₄×C₈	C₂²⋊D₇₉	1	1	yes	characteristic
162	C₄×C₈	C₂²×D₇₉	1	1	yes	characteristic
163	C₂×Q₈	—	2	4	no	—
164	C₂×Q₈	—	2	1	no	—
165	C₂×Q₈	C₂×C₇₉⋊C₈	1	2	yes	—
166	C₂²×C₄	—	2	1	no	—
167	C₂²×C₄	—	2	1	no	—
168	C₂²×C₄	C₂²⋊Dic₇₉	1	1	yes	characteristic
169	C₂²×C₄	C₂×C₇₉⋊C₈	1	1	yes	characteristic
170	C₂²×C₄	#11929608	1	1	yes	characteristic
171	C₂×C₈	—	2	2	no	—
172	C₂×C₈	—	2	2	no	—
173	C₂×C₈	—	4	2	no	—
174	C₂×C₈	C₂²⋊Dic₇₉	1	1	yes	characteristic
175	C₂×C₈	C₂²⋊Dic₇₉	1	1	yes	characteristic
176	C₂×C₈	#11929611	1	2	yes	—
177	C₂×C₈	#11929614	1	2	yes	—
178	C₂×C₈	C₂×C₂²⋊D₇₉	1	1	yes	characteristic
179	C₂×C₈	C₂×C₂²⋊D₇₉	1	1	yes	characteristic
180	C₄⋊C₄	—	2	2	no	—
181	C₄⋊C₄	—	2	1	no	—
182	C₄⋊C₄	—	2	2	no	—
183	C₄⋊C₄	C₂×C₇₉⋊C₈	1	2	yes	—
184	C₄²	—	2	2	no	—
185	C₄²	—	2	2	no	—
186	C₄²	C₂²⋊Dic₇₉	1	1	yes	characteristic
187	C₄²	C₂×C₇₉⋊C₈	1	1	yes	characteristic
188	C₄²	#11929608	1	1	yes	characteristic
189	C₄²	C₂²×Dic₇₉	1	1	yes	characteristic
190	C₁₆	—	316	2	no	—
191	C₁₆	—	158	4	no	—
192	C₂³	C₂²⋊(C₇₉⋊C₈)	1	1	yes	characteristic
193	Q₈	—	2	4	no	—
194	Q₈	—	2	2	no	—
195	Q₈	#14579101	1	4	yes	complemented
196	C₂×C₄	—	2	4	no	—
197	C₂×C₄	—	2	2	no	—
198	C₂×C₄	—	2	2	no	—
199	C₂×C₄	—	2	2	no	—
200	C₂×C₄	—	4	1	no	—
201	C₂×C₄	—	4	2	no	—
202	C₂×C₄	C₂²⋊(C₇₉⋊C₈)	1	1	yes	characteristic
203	C₂×C₄	C₂²⋊(C₇₉⋊C₈)	1	1	yes	characteristic
204	C₂×C₄	C₂²⋊(C₇₉⋊C₈)	1	1	yes	characteristic
205	C₂×C₄	#14579085	1	2	yes	—
206	C₂×C₄	#14579088	1	2	yes	—
207	C₂×C₄	C₂×C₂²⋊Dic₇₉	1	1	yes	characteristic
208	C₂×C₄	C₂²×C₇₉⋊C₈	1	1	yes	characteristic
209	C₂×C₄	C₂×#11929608	1	1	yes	characteristic
210	C₈	—	2	1	no	—
211	C₈	—	2	1	no	—
212	C₈	—	4	2	no	—
213	C₈	#14579085	1	2	yes	—
214	C₈	C₂×#11929611	1	1	yes	characteristic
215	C₈	C₂×#11929614	1	1	yes	characteristic
216	C₂²	—	2	1	no	—
217	C₂²	—	2	1	no	—
218	C₂²	Q₈⋊(C₇₉⋊C₈)	1	2	yes	—
219	C₂²	C₂×C₂²⋊(C₇₉⋊C₈)	1	1	yes	characteristic
220	C₄	—	2	4	no	—
221	C₄	—	2	1	no	—
222	C₄	—	2	1	no	—
223	C₄	—	2	1	no	—
224	C₄	—	4	2	no	—
225	C₄	Q₈⋊(C₇₉⋊C₈)	1	2	yes	—
226	C₄	C₂×#14579085	1	1	yes	characteristic
227	C₄	C₂×#14579088	1	1	yes	characteristic
228	C₄	C₂×#14579101	1	2	yes	—
229	C₂	—	2	2	no	—
230	C₂	C₂×Q₈⋊(C₇₉⋊C₈)	1	1	yes	characteristic
231	C₂	#23765011	1	1	yes	characteristic
232	C₂	#23765028	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₈	C₂×C₂²⋊D₇₉
commutator	C₂×C₄×C₇₉	C₂²×C₈
Frattini	C₄×C₈	C₂²×D₇₉
Fitting	C₂×C₈×Q₈×C₇₉	C₂
socle	C₂²×C₇₉	C₂×C₂²⋊C₈
prime radical	C₂³×C₇₉	C₂²⋊C₈
cosocle	C₄×C₈×C₇₉	C₂³
hypercenter	C₂×C₈×Q₈	D₇₉
hypocenter	C₇₉	#11607
LCS	G ▷ C₂×C₄×C₇₉ ▷ C₂×C₇₉ ▷ C₇₉
UCS	C₁ ◁ C₂×C₈ ◁ C₂×C₄×C₈ ◁ C₂×C₈×Q₈

Complements

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

N	H ≅ G/N	type
3. Q₈⋊(C₇₉⋊C₁₆)	C₂	semidirect
78. Q₈×C₇₉	M₃₂	semidirect
121. C₇₉	#11607	semidirect
195. Q₈	#14579101	semidirect

Maximal subgroups

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

subgroup	index
#23765156	2
C₄×#14579101	2
Q₈⋊(C₇₉⋊C₁₆)	2
C₂×C₈×Q₈×C₇₉	2
#11607	79