Subgroups — #34206808

Order 20224.

350 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	C₂×#19098693	C₂	1	1	yes	maximal, characteristic
2	#23765156	C₂	1	1	yes	maximal, characteristic
3	#23761483	C₂	1	4	yes	maximal
4	C₄×M₃₂×C₇₉	C₂	1	1	yes	maximal, characteristic
5	C₂×C₄×Dic₁₅₈	C₄	1	1	yes	characteristic
6	C₂×#14579101	C₂²	1	2	yes	—
7	#19098693	—	2	1	no	—
8	#19098693	—	2	1	no	—
9	#19098693	—	2	1	no	—
10	#19098693	—	2	1	no	—
11	C₂×C₄⋊(C₇₉⋊C₈)	C₄	1	1	yes	characteristic
12	C₂×Dic₇₉⋊C₈	C₄	1	2	yes	—
13	C₂×#14579071	C₂²	1	2	yes	—
14	C₄⋊(C₇₉⋊C₁₆)	—	2	1	no	—
15	C₄⋊(C₇₉⋊C₁₆)	—	2	1	no	—
16	C₂×M₃₂×C₇₉	C₂²	1	2	yes	—
17	C₂×C₄×C₈×C₇₉	C₂²	1	1	yes	characteristic
18	#346×C₇₉	—	2	1	no	—
19	C₄×C₁₆×C₇₉	—	2	1	no	—
20	C₂²×Dic₁₅₈	C₈	1	1	yes	characteristic
21	C₂²×C₇₉⋊C₈	C₂×C₄	1	2	yes	—
22	C₄×Dic₁₅₈	—	2	1	no	—
23	C₄×Dic₁₅₈	—	2	1	no	—
24	C₄×Dic₁₅₈	—	2	1	no	—
25	C₄×Dic₁₅₈	—	2	1	no	—
26	C₂×C₄⋊Dic₇₉	C₈	1	1	yes	characteristic
27	C₂×Dic₇₉⋊C₄	C₈	1	2	yes	—
28	C₂×C₄×Dic₇₉	C₂×C₄	1	2	yes	—
29	#14579101	—	2	4	no	—
30	C₂×C₇₉⋊C₁₆	—	2	2	no	—
31	C₄⋊(C₇₉⋊C₈)	—	2	1	no	—
32	C₄⋊(C₇₉⋊C₈)	—	2	1	no	—
33	Dic₇₉⋊C₈	—	2	2	no	—
34	Dic₇₉⋊C₈	—	2	2	no	—
35	#14579071	—	2	2	no	—
36	#14579071	—	2	2	no	—
37	C₂²×C₈×C₇₉	C₂×C₄	1	1	yes	characteristic
38	C₂²×C₈×C₇₉	C₂³	1	1	yes	characteristic
39	C₂×C₄²×C₇₉	C₂×C₄	1	1	yes	characteristic
40	M₃₂×C₇₉	—	2	2	no	—
41	M₃₂×C₇₉	Q₈	1	4	yes	complemented
42	C₂×C₁₆×C₇₉	—	2	2	no	—
43	C₄×C₈×C₇₉	D₄	1	1	yes	characteristic
44	C₄×C₈×C₇₉	D₄	1	1	yes	characteristic
45	C₄×C₈×C₇₉	D₄	1	1	yes	characteristic
46	C₄×C₈×C₇₉	D₄	1	1	yes	characteristic
47	C₂×Dic₁₅₈	—	2	1	no	—
48	C₂×Dic₁₅₈	—	2	1	no	—
49	C₂×Dic₁₅₈	—	4	2	no	—
50	C₂²×Dic₇₉	C₂×C₈	1	2	yes	—
51	C₂²×Dic₇₉	C₂×C₈	1	2	yes	—
52	C₂×C₇₉⋊C₈	—	2	2	no	—
53	C₂×C₇₉⋊C₈	—	2	4	no	—
54	C₄⋊Dic₇₉	—	2	1	no	—
55	C₄⋊Dic₇₉	—	2	1	no	—
56	Dic₇₉⋊C₄	—	2	2	no	—
57	Dic₇₉⋊C₄	—	2	2	no	—
58	C₄×Dic₇₉	—	2	2	no	—
59	C₄×Dic₇₉	—	2	2	no	—
60	C₇₉⋊C₁₆	—	4	2	no	—
61	C₂²×C₄×C₇₉	C₂×C₈	1	1	yes	characteristic
62	C₂²×C₄×C₇₉	C₂×C₈	1	1	yes	characteristic
63	C₂²×C₄×C₇₉	C₂²×C₄	1	1	yes	characteristic
64	C₂×C₈×C₇₉	—	2	2	no	—
65	C₂×C₈×C₇₉	C₂²⋊C₄	1	1	yes	characteristic
66	C₂×C₈×C₇₉	C₂²⋊C₄	1	1	yes	characteristic
67	C₂×C₈×C₇₉	C₂×D₄	1	1	yes	characteristic
68	C₂×C₈×C₇₉	C₂×D₄	1	1	yes	characteristic
69	C₂×C₈×C₇₉	C₂×Q₈	1	2	yes	—
70	C₂×C₈×C₇₉	#41	1	2	yes	—
71	C₄²×C₇₉	C₂²⋊C₄	1	1	yes	characteristic
72	C₄²×C₇₉	C₂²⋊C₄	1	1	yes	characteristic
73	C₄²×C₇₉	M₁₆	1	1	yes	characteristic
74	C₄²×C₇₉	M₁₆	1	1	yes	characteristic
75	C₁₆×C₇₉	—	2	4	no	—
76	C₂³×C₇₉	C₂²×C₈	1	1	yes	characteristic
77	C₂×C₄×C₇₉	—	2	2	no	—
78	C₂×C₄×C₇₉	—	4	1	no	—
79	C₂×C₄×C₇₉	C₂²⋊C₈	1	1	yes	characteristic
80	C₂×C₄×C₇₉	C₂²⋊C₈	1	1	yes	characteristic
81	C₂×C₄×C₇₉	C₂²⋊C₈	1	1	yes	characteristic
82	C₂×C₄×C₇₉	C₂²⋊C₈	1	1	yes	characteristic
83	C₂×C₄×C₇₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
84	C₂×C₄×C₇₉	C₄×Q₈	1	2	yes	—
85	C₂×C₄×C₇₉	C₂×M₁₆	1	1	yes	characteristic
86	C₂×C₄×C₇₉	#131	1	2	yes	—
87	C₂×Dic₇₉	—	2	2	no	—
88	C₂×Dic₇₉	—	2	2	no	—
89	C₂×Dic₇₉	—	4	2	no	—
90	C₂×Dic₇₉	—	4	2	no	—
91	Dic₁₅₈	—	4	2	no	—
92	Dic₁₅₈	—	4	2	no	—
93	C₈×C₇₉	—	2	1	no	—
94	C₈×C₇₉	—	2	1	no	—
95	C₈×C₇₉	C₂²⋊Q₈	1	2	yes	—
96	C₈×C₇₉	#124	1	1	yes	characteristic
97	C₈×C₇₉	C₄⋊Q₈	1	1	yes	characteristic
98	C₇₉⋊C₈	—	4	2	no	—
99	C₂²×C₇₉	—	2	1	no	—
100	C₂²×C₇₉	—	2	1	no	—
101	C₂²×C₇₉	C₂×C₂²⋊C₈	1	1	yes	characteristic
102	C₂²×C₇₉	C₈×Q₈	1	2	yes	—
103	C₄×C₇₉	—	2	1	no	—
104	C₄×C₇₉	—	2	1	no	—
105	C₄×C₇₉	—	2	1	no	—
106	C₄×C₇₉	#349	1	2	yes	—
107	C₄×C₇₉	#391	1	1	yes	characteristic
108	C₄×C₇₉	#408	1	1	yes	characteristic
109	C₄×C₇₉	#446	1	2	yes	—
110	Dic₇₉	—	4	2	no	—
111	Dic₇₉	—	4	2	no	—
112	#13268	—	79	1	no	maximal, Sylow
113	C₂×C₇₉	—	2	2	no	—
114	C₂×C₇₉	#1984	1	1	yes	characteristic
115	C₂×C₇₉	C₂×#349	1	1	yes	characteristic
116	C₂×C₇₉	#2115	1	1	yes	characteristic
117	C₂×#446	—	79	1	no	—
118	#2150	—	79	1	no	—
119	C₄×M₃₂	D₇₉	1	1	yes	characteristic
120	#1377	—	79	4	no	—
121	C₇₉	#13268	1	1	yes	characteristic, complemented, Sylow
122	C₂×C₄×Q₈	—	79	1	no	—
123	C₂×M₃₂	—	79	2	no	—
124	C₂×M₃₂	C₂×D₇₉	1	2	yes	—
125	#446	—	158	1	no	—
126	#446	—	158	1	no	—
127	#446	—	158	1	no	—
128	#446	—	158	1	no	—
129	C₂×C₄⋊C₈	—	79	2	no	—
130	C₂×C₄⋊C₈	—	79	1	no	—
131	C₂×#97	—	79	2	no	—
132	C₂×C₄×C₈	C₂×D₇₉	1	1	yes	characteristic
133	C₄⋊C₁₆	—	158	1	no	—
134	C₄⋊C₁₆	—	158	1	no	—
135	#346	—	2	1	no	—
136	C₄×C₁₆	—	2	1	no	—
137	C₂²×Q₈	—	79	1	no	—
138	C₂²×C₈	—	79	2	no	—
139	C₂²×C₈	C₄×D₇₉	1	1	yes	characteristic
140	C₂²×C₈	C₂²×D₇₉	1	1	yes	characteristic
141	C₄×Q₈	—	158	1	no	—
142	C₄×Q₈	—	158	1	no	—
143	C₄×Q₈	—	158	1	no	—
144	C₄×Q₈	—	158	1	no	—
145	C₂×C₄⋊C₄	—	79	2	no	—
146	C₂×C₄⋊C₄	—	79	1	no	—
147	C₂×C₄²	—	79	2	no	—
148	C₂×C₄²	C₄×D₇₉	1	1	yes	characteristic
149	M₃₂	—	2	2	no	—
150	M₃₂	—	158	4	no	—
151	M₃₂	Dic₁₅₈	1	4	yes	complemented
152	C₂×C₁₆	—	2	2	no	—
153	C₂×C₁₆	—	158	2	no	—
154	C₄⋊C₈	—	158	2	no	—
155	C₄⋊C₈	—	158	2	no	—
156	C₄⋊C₈	—	158	1	no	—
157	C₄⋊C₈	—	158	1	no	—
158	#97	—	158	2	no	—
159	#97	—	158	2	no	—
160	C₄×C₈	D₃₁₆	1	1	yes	characteristic
161	C₄×C₈	D₃₁₆	1	1	yes	characteristic
162	C₄×C₈	C₂²⋊D₇₉	1	1	yes	characteristic
163	C₄×C₈	C₂²⋊D₇₉	1	1	yes	characteristic
164	C₂×Q₈	—	316	2	no	—
165	C₂×Q₈	—	158	1	no	—
166	C₂×Q₈	—	158	1	no	—
167	C₂²×C₄	—	79	2	no	—
168	C₂²×C₄	—	79	2	no	—
169	C₂²×C₄	C₈×D₇₉	1	1	yes	characteristic
170	C₂²×C₄	C₈×D₇₉	1	1	yes	characteristic
171	C₂²×C₄	C₂×C₄×D₇₉	1	1	yes	characteristic
172	C₂×C₈	—	2	2	no	—
173	C₂×C₈	—	158	2	no	—
174	C₂×C₈	—	158	4	no	—
175	C₂×C₈	(C₂×D₇₉)⋊C₄	1	1	yes	characteristic
176	C₂×C₈	(C₂×D₇₉)⋊C₄	1	1	yes	characteristic
177	C₂×C₈	C₂×C₂²⋊D₇₉	1	1	yes	characteristic
178	C₂×C₈	C₂×D₃₁₆	1	1	yes	characteristic
179	C₂×C₈	C₂×Dic₁₅₈	1	2	yes	—
180	C₂×C₈	#11929624	1	2	yes	—
181	C₄⋊C₄	—	158	2	no	—
182	C₄⋊C₄	—	158	2	no	—
183	C₄⋊C₄	—	158	1	no	—
184	C₄⋊C₄	—	158	1	no	—
185	C₄²	—	158	2	no	—
186	C₄²	—	158	2	no	—
187	C₄²	(C₂×D₇₉)⋊C₄	1	1	yes	characteristic
188	C₄²	(C₂×D₇₉)⋊C₄	1	1	yes	characteristic
189	C₄²	#11929607	1	1	yes	characteristic
190	C₄²	#11929607	1	1	yes	characteristic
191	C₁₆	—	2	4	no	—
192	C₁₆	—	316	2	no	—
193	C₂³	C₂×C₈×D₇₉	1	1	yes	characteristic
194	Q₈	—	316	2	no	—
195	Q₈	—	316	2	no	—
196	C₂×C₄	—	2	2	no	—
197	C₂×C₄	—	4	1	no	—
198	C₂×C₄	—	316	2	no	—
199	C₂×C₄	—	316	2	no	—
200	C₂×C₄	—	158	2	no	—
201	C₂×C₄	—	158	2	no	—
202	C₂×C₄	(C₂×D₇₉)⋊C₈	1	1	yes	characteristic
203	C₂×C₄	(C₂×D₇₉)⋊C₈	1	1	yes	characteristic
204	C₂×C₄	(C₂×D₇₉)⋊C₈	1	1	yes	characteristic
205	C₂×C₄	(C₂×D₇₉)⋊C₈	1	1	yes	characteristic
206	C₂×C₄	C₂×(C₂×D₇₉)⋊C₄	1	1	yes	characteristic
207	C₂×C₄	C₄×Dic₁₅₈	1	2	yes	—
208	C₂×C₄	C₂×#11929607	1	1	yes	characteristic
209	C₂×C₄	#14579167	1	2	yes	—
210	C₈	—	2	1	no	—
211	C₈	—	2	1	no	—
212	C₈	—	316	2	no	—
213	C₈	#14579140	1	2	yes	—
214	C₈	#14579149	1	1	yes	characteristic
215	C₈	#14579161	1	1	yes	characteristic
216	C₂²	—	2	1	no	—
217	C₂²	—	2	1	no	—
218	C₂²	C₂×(C₂×D₇₉)⋊C₈	1	1	yes	characteristic
219	C₂²	C₈×Dic₁₅₈	1	2	yes	—
220	C₄	—	2	1	no	—
221	C₄	—	2	1	no	—
222	C₄	—	2	1	no	—
223	C₄	—	316	2	no	—
224	C₄	—	316	2	no	—
225	C₄	#19098344	1	2	yes	—
226	C₄	#19098469	1	1	yes	characteristic
227	C₄	#19098529	1	1	yes	characteristic
228	C₄	#19098693	1	2	yes	—
229	C₂	—	2	2	no	—
230	C₂	#23764336	1	1	yes	characteristic
231	C₂	C₂×#19098344	1	1	yes	characteristic
232	C₂	#23765020	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₂×(C₂×D₇₉)⋊C₄
commutator, prime radical	C₂³×C₇₉	C₂²×C₈
Frattini	C₂²×C₈	C₂²×D₇₉
Fitting	C₄×M₃₂×C₇₉	C₂
socle	C₂²×C₇₉	C₂×C₂²⋊C₈
cosocle	C₂²×C₈×C₇₉	C₂³
hypercenter	C₄×M₃₂	D₇₉
hypocenter	C₇₉	#13268
LCS	G ▷ C₂³×C₇₉ ▷ C₂×C₇₉ ▷ C₇₉
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄×C₈ ◁ C₄×M₃₂

Complements

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

N	H ≅ G/N	type
41. M₃₂×C₇₉	Q₈	semidirect
121. C₇₉	#13268	semidirect
151. M₃₂	Dic₁₅₈	semidirect

Maximal subgroups

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

subgroup	index
C₂×#19098693	2
#23765156	2
#23761483	2
C₄×M₃₂×C₇₉	2
#13268	79