Subgroups — #34203108

Order 20224.

678 conjugacy classes of subgroups.

The subgroup lattice has 262 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×#14579101	C₂	1	1	yes	maximal, characteristic
2	C₂²×C₈×Dic₇₉	C₂	1	1	yes	maximal, characteristic
3	#23761477	C₂	1	4	yes	maximal, complemented
4	C₂²×M₃₂×C₇₉	C₂	1	1	yes	maximal, characteristic
5	C₂³×C₇₉⋊C₈	C₄	1	1	yes	characteristic
6	C₂²×C₄×Dic₇₉	C₄	1	1	yes	characteristic
7	C₂×#14579101	—	2	2	no	—
8	C₂×#14579101	—	2	2	no	—
9	C₂×#14579101	C₄	1	4	yes	complemented
10	C₂²×C₇₉⋊C₁₆	C₂²	1	2	yes	—
11	C₂×C₈×Dic₇₉	—	2	2	no	—
12	C₂×C₈×Dic₇₉	—	2	2	no	—
13	C₂×C₈×Dic₇₉	C₄	1	2	yes	—
14	C₂×C₈×Dic₇₉	C₂²	1	2	yes	—
15	C₂³×C₈×C₇₉	C₂²	1	1	yes	characteristic
16	C₂×M₃₂×C₇₉	—	2	2	no	—
17	C₂×M₃₂×C₇₉	—	2	2	no	—
18	C₂×M₃₂×C₇₉	C₄	1	4	yes	complemented
19	C₂²×C₁₆×C₇₉	C₂²	1	2	yes	—
20	C₂³×Dic₇₉	C₈	1	2	yes	—
21	C₂²×C₇₉⋊C₈	—	2	2	no	—
22	C₂²×C₇₉⋊C₈	—	2	2	no	—
23	C₂²×C₇₉⋊C₈	C₈	1	4	yes	—
24	C₂²×C₇₉⋊C₈	C₂×C₄	1	2	yes	—
25	C₂×C₄×Dic₇₉	—	2	2	no	—
26	C₂×C₄×Dic₇₉	—	2	2	no	—
27	C₂×C₄×Dic₇₉	C₈	1	2	yes	—
28	C₂×C₄×Dic₇₉	C₂×C₄	1	2	yes	—
29	#14579101	—	2	8	no	—
30	C₂×C₇₉⋊C₁₆	—	2	2	no	—
31	C₂×C₇₉⋊C₁₆	—	2	2	no	—
32	C₂×C₇₉⋊C₁₆	C₂×C₄	1	4	yes	complemented
33	C₈×Dic₇₉	—	2	4	no	—
34	C₈×Dic₇₉	—	2	2	no	—
35	C₈×Dic₇₉	—	2	2	no	—
36	C₂³×C₄×C₇₉	C₂×C₄	1	1	yes	characteristic
37	C₂²×C₈×C₇₉	C₂×C₄	1	1	yes	characteristic
38	C₂²×C₈×C₇₉	C₂×C₄	1	2	yes	—
39	C₂²×C₈×C₇₉	C₂×C₄	1	2	yes	—
40	C₂²×C₈×C₇₉	D₄	1	2	yes	—
41	C₂²×C₈×C₇₉	D₄	1	2	yes	—
42	C₂²×C₈×C₇₉	D₄	1	2	yes	—
43	C₂²×C₈×C₇₉	Q₈	1	2	yes	—
44	C₂²×C₈×C₇₉	C₂³	1	1	yes	characteristic
45	M₃₂×C₇₉	—	2	8	no	—
46	C₂×C₁₆×C₇₉	—	2	2	no	—
47	C₂×C₁₆×C₇₉	—	2	2	no	—
48	C₂×C₁₆×C₇₉	C₂×C₄	1	4	yes	complemented
49	C₂²×Dic₇₉	—	2	8	no	—
50	C₂²×Dic₇₉	—	2	4	no	—
51	C₂²×Dic₇₉	C₂×C₈	1	4	yes	—
52	C₂×C₇₉⋊C₈	—	2	8	no	—
53	C₂×C₇₉⋊C₈	—	2	2	no	—
54	C₂×C₇₉⋊C₈	—	2	2	no	—
55	C₂×C₇₉⋊C₈	C₂×C₈	1	4	yes	—
56	C₄×Dic₇₉	—	2	4	no	—
57	C₄×Dic₇₉	—	2	2	no	—
58	C₄×Dic₇₉	—	2	2	no	—
59	C₇₉⋊C₁₆	—	2	4	no	—
60	C₂⁴×C₇₉	C₂×C₈	1	1	yes	characteristic
61	C₂²×C₄×C₇₉	C₄²	1	2	yes	—
62	C₂²×C₄×C₇₉	C₂²⋊C₄	1	2	yes	—
63	C₂²×C₄×C₇₉	C₄⋊C₄	1	2	yes	—
64	C₂²×C₄×C₇₉	C₂×C₈	1	1	yes	characteristic
65	C₂²×C₄×C₇₉	C₂×C₈	1	2	yes	—
66	C₂²×C₄×C₇₉	M₁₆	1	2	yes	—
67	C₂²×C₄×C₇₉	M₁₆	1	2	yes	—
68	C₂²×C₄×C₇₉	C₂²×C₄	1	1	yes	characteristic
69	C₂×C₈×C₇₉	C₄²	1	2	yes	—
70	C₂×C₈×C₇₉	C₂²⋊C₄	1	4	yes	—
71	C₂×C₈×C₇₉	C₂²⋊C₄	1	4	yes	—
72	C₂×C₈×C₇₉	C₂²⋊C₄	1	2	yes	—
73	C₂×C₈×C₇₉	C₄⋊C₄	1	4	yes	—
74	C₂×C₈×C₇₉	C₄⋊C₄	1	4	yes	—
75	C₂×C₈×C₇₉	C₄⋊C₄	1	2	yes	—
76	C₂×C₈×C₇₉	C₂²×C₄	1	1	yes	characteristic
77	C₂×C₈×C₇₉	C₂²×C₄	1	1	yes	characteristic
78	C₂×C₈×C₇₉	C₂×D₄	1	1	yes	characteristic
79	C₂×C₈×C₇₉	C₂×D₄	1	1	yes	characteristic
80	C₂×C₈×C₇₉	C₂×D₄	1	1	yes	characteristic
81	C₂×C₈×C₇₉	C₂×Q₈	1	1	yes	characteristic
82	C₁₆×C₇₉	—	2	4	no	—
83	C₂³×C₇₉	—	2	4	no	—
84	C₂³×C₇₉	C₄×C₈	1	2	yes	—
85	C₂³×C₇₉	C₂²⋊C₈	1	2	yes	—
86	C₂³×C₇₉	C₄⋊C₈	1	2	yes	—
87	C₂³×C₇₉	C₂²×C₈	1	1	yes	characteristic
88	C₂×C₄×C₇₉	#95	1	4	yes	—
89	C₂×C₄×C₇₉	C₄×C₈	1	2	yes	—
90	C₂×C₄×C₇₉	#97	1	4	yes	—
91	C₂×C₄×C₇₉	C₂²⋊C₈	1	2	yes	—
92	C₂×C₄×C₇₉	C₂²⋊C₈	1	4	yes	—
93	C₂×C₄×C₇₉	C₄⋊C₈	1	2	yes	—
94	C₂×C₄×C₇₉	C₄⋊C₈	1	4	yes	—
95	C₂×C₄×C₇₉	C₂×C₄²	1	1	yes	characteristic
96	C₂×C₄×C₇₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
97	C₂×C₄×C₇₉	C₂×C₄⋊C₄	1	1	yes	characteristic
98	C₂×C₄×C₇₉	C₂²×C₈	1	1	yes	characteristic
99	C₂×C₄×C₇₉	C₂×M₁₆	1	1	yes	characteristic
100	C₂×C₄×C₇₉	C₂×M₁₆	1	1	yes	characteristic
101	C₂×Dic₇₉	—	2	16	no	—
102	C₂×Dic₇₉	—	2	4	no	—
103	C₂×Dic₇₉	—	2	4	no	—
104	C₂×Dic₇₉	M₃₂	1	8	yes	complemented
105	C₈×C₇₉	#95	1	4	yes	—
106	C₈×C₇₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
107	C₈×C₇₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
108	C₈×C₇₉	C₂×C₄⋊C₄	1	1	yes	characteristic
109	C₈×C₇₉	C₂×C₄⋊C₄	1	1	yes	characteristic
110	C₇₉⋊C₈	—	2	4	no	—
111	C₂²×C₇₉	—	2	4	no	—
112	C₂²×C₇₉	—	2	4	no	—
113	C₂²×C₇₉	—	2	4	no	—
114	C₂²×C₇₉	#336	1	4	yes	—
115	C₂²×C₇₉	C₂×C₄×C₈	1	1	yes	characteristic
116	C₂²×C₇₉	C₂×C₂²⋊C₈	1	1	yes	characteristic
117	C₂²×C₇₉	C₂×C₄⋊C₈	1	1	yes	characteristic
118	C₂²×C₇₉	C₂×M₃₂	1	4	yes	—
119	C₄×C₇₉	#336	1	4	yes	—
120	C₄×C₇₉	C₂×#95	1	1	yes	characteristic
121	C₄×C₇₉	C₂×#97	1	1	yes	characteristic
122	C₄×C₇₉	C₂×C₂²⋊C₈	1	1	yes	characteristic
123	C₄×C₇₉	C₂×C₄⋊C₈	1	1	yes	characteristic
124	Dic₇₉	—	2	8	no	—
125	#12544	—	79	1	no	maximal, Sylow
126	C₂×C₇₉	—	2	4	no	—
127	C₂×C₇₉	C₂×#336	1	1	yes	characteristic
128	C₂×C₇₉	C₄×M₃₂	1	2	yes	—
129	C₂×C₇₉	#2112	1	2	yes	—
130	C₂×C₇₉	#2150	1	2	yes	—
131	C₂²×M₃₂	—	79	1	no	—
132	C₂²×M₃₂	D₇₉	1	1	yes	characteristic
133	C₂²×C₄×C₈	—	79	1	no	—
134	#1374	—	79	4	no	—
135	C₇₉	#12544	1	1	yes	characteristic, complemented, Sylow
136	C₂³×C₈	—	79	1	no	—
137	C₂³×C₈	C₂×D₇₉	1	1	yes	characteristic
138	C₂²×C₄²	—	79	1	no	—
139	C₂×M₃₂	—	2	2	no	—
140	C₂×M₃₂	—	2	2	no	—
141	C₂×M₃₂	—	79	4	no	—
142	C₂×M₃₂	—	158	2	no	—
143	C₂×M₃₂	—	158	2	no	—
144	C₂×M₃₂	Dic₇₉	1	4	yes	complemented
145	C₂²×C₁₆	—	79	2	no	—
146	C₂²×C₁₆	C₂×D₇₉	1	2	yes	—
147	C₂×C₄×C₈	—	79	2	no	—
148	C₂×C₄×C₈	—	79	2	no	—
149	C₂×C₄×C₈	—	158	2	no	—
150	C₂×C₄×C₈	—	158	2	no	—
151	C₂³×C₄	—	79	2	no	—
152	C₂³×C₄	C₄×D₇₉	1	1	yes	characteristic
153	C₂²×C₈	—	79	4	no	—
154	C₂²×C₈	—	79	2	no	—
155	C₂²×C₈	—	158	2	no	—
156	C₂²×C₈	—	158	2	no	—
157	C₂²×C₈	Dic₁₅₈	1	2	yes	—
158	C₂²×C₈	C₄×D₇₉	1	1	yes	characteristic
159	C₂²×C₈	C₄×D₇₉	1	2	yes	—
160	C₂²×C₈	D₃₁₆	1	2	yes	—
161	C₂²×C₈	C₂×Dic₇₉	1	2	yes	—
162	C₂²×C₈	C₂²⋊D₇₉	1	2	yes	—
163	C₂²×C₈	C₂²⋊D₇₉	1	2	yes	—
164	C₂²×C₈	C₂²×D₇₉	1	1	yes	characteristic
165	C₂×C₄²	—	79	2	no	—
166	C₂×C₄²	—	79	2	no	—
167	C₂×C₄²	—	158	2	no	—
168	C₂×C₄²	—	158	2	no	—
169	M₃₂	—	2	8	no	—
170	M₃₂	—	158	8	no	—
171	C₂×C₁₆	—	2	2	no	—
172	C₂×C₁₆	—	2	2	no	—
173	C₂×C₁₆	—	79	4	no	—
174	C₂×C₁₆	—	158	2	no	—
175	C₂×C₁₆	—	158	2	no	—
176	C₂×C₁₆	C₂×Dic₇₉	1	4	yes	complemented
177	C₄×C₈	—	158	4	no	—
178	C₄×C₈	—	158	2	no	—
179	C₄×C₈	—	158	2	no	—
180	C₂⁴	C₈×D₇₉	1	1	yes	characteristic
181	C₂²×C₄	—	79	4	no	—
182	C₂²×C₄	—	158	8	no	—
183	C₂²×C₄	—	158	4	no	—
184	C₂²×C₄	C₄×Dic₇₉	1	2	yes	—
185	C₂²×C₄	(C₂×D₇₉)⋊C₄	1	2	yes	—
186	C₂²×C₄	Dic₇₉⋊C₄	1	2	yes	—
187	C₂²×C₄	C₈×D₇₉	1	1	yes	characteristic
188	C₂²×C₄	C₈×D₇₉	1	2	yes	—
189	C₂²×C₄	#11929607	1	2	yes	—
190	C₂²×C₄	#11929607	1	2	yes	—
191	C₂²×C₄	C₂×C₄×D₇₉	1	1	yes	characteristic
192	C₂×C₈	—	79	4	no	—
193	C₂×C₈	—	158	8	no	—
194	C₂×C₈	—	158	2	no	—
195	C₂×C₈	—	158	2	no	—
196	C₂×C₈	C₄×Dic₇₉	1	2	yes	—
197	C₂×C₈	(C₂×D₇₉)⋊C₄	1	2	yes	—
198	C₂×C₈	(C₂×D₇₉)⋊C₄	1	4	yes	—
199	C₂×C₈	C₂²⋊Dic₇₉	1	4	yes	—
200	C₂×C₈	Dic₇₉⋊C₄	1	2	yes	—
201	C₂×C₈	Dic₇₉⋊C₄	1	4	yes	—
202	C₂×C₈	C₄⋊Dic₇₉	1	4	yes	—
203	C₂×C₈	C₂×C₄×D₇₉	1	1	yes	characteristic
204	C₂×C₈	C₂²×Dic₇₉	1	1	yes	characteristic
205	C₂×C₈	C₂×C₂²⋊D₇₉	1	1	yes	characteristic
206	C₂×C₈	C₂×C₂²⋊D₇₉	1	1	yes	characteristic
207	C₂×C₈	C₂×D₃₁₆	1	1	yes	characteristic
208	C₂×C₈	C₂×Dic₁₅₈	1	1	yes	characteristic
209	C₄²	—	158	4	no	—
210	C₄²	—	158	2	no	—
211	C₄²	—	158	2	no	—
212	C₁₆	—	2	4	no	—
213	C₁₆	—	158	4	no	—
214	C₂³	—	2	4	no	—
215	C₂³	C₈×Dic₇₉	1	2	yes	—
216	C₂³	(C₂×D₇₉)⋊C₈	1	2	yes	—
217	C₂³	Dic₇₉⋊C₈	1	2	yes	—
218	C₂³	C₂×C₈×D₇₉	1	1	yes	characteristic
219	C₂×C₄	—	79	8	no	—
220	C₂×C₄	—	158	16	no	—
221	C₂×C₄	—	158	4	no	—
222	C₂×C₄	—	158	4	no	—
223	C₂×C₄	C₇₉⋊#95	1	4	yes	—
224	C₂×C₄	C₈×Dic₇₉	1	2	yes	—
225	C₂×C₄	#14579071	1	4	yes	—
226	C₂×C₄	(C₂×D₇₉)⋊C₈	1	4	yes	—
227	C₂×C₄	(C₂×D₇₉)⋊C₈	1	2	yes	—
228	C₂×C₄	Dic₇₉⋊C₈	1	2	yes	—
229	C₂×C₄	Dic₇₉⋊C₈	1	4	yes	—
230	C₂×C₄	C₂×C₄×Dic₇₉	1	1	yes	characteristic
231	C₂×C₄	C₂×(C₂×D₇₉)⋊C₄	1	1	yes	characteristic
232	C₂×C₄	C₂×Dic₇₉⋊C₄	1	1	yes	characteristic
233	C₂×C₄	C₂×C₈×D₇₉	1	1	yes	characteristic
234	C₂×C₄	C₂×#11929607	1	1	yes	characteristic
235	C₂×C₄	C₂×#11929607	1	1	yes	characteristic
236	C₈	—	158	4	no	—
237	C₈	C₇₉⋊#95	1	4	yes	—
238	C₈	C₂×(C₂×D₇₉)⋊C₄	1	1	yes	characteristic
239	C₈	C₂×C₂²⋊Dic₇₉	1	1	yes	characteristic
240	C₈	C₂×Dic₇₉⋊C₄	1	1	yes	characteristic
241	C₈	C₂×C₄⋊Dic₇₉	1	1	yes	characteristic
242	C₂²	—	2	4	no	—
243	C₂²	—	2	4	no	—
244	C₂²	—	2	4	no	—
245	C₂²	#19098317	1	4	yes	—
246	C₂²	C₂×C₈×Dic₇₉	1	1	yes	characteristic
247	C₂²	C₂×(C₂×D₇₉)⋊C₈	1	1	yes	characteristic
248	C₂²	C₂×Dic₇₉⋊C₈	1	1	yes	characteristic
249	C₂²	M₃₂×D₇₉	1	4	yes	—
250	C₄	—	158	8	no	—
251	C₄	#19098317	1	4	yes	—
252	C₄	C₂×C₇₉⋊#95	1	1	yes	characteristic
253	C₄	C₂×#14579071	1	1	yes	characteristic
254	C₄	C₂×(C₂×D₇₉)⋊C₈	1	1	yes	characteristic
255	C₄	C₂×Dic₇₉⋊C₈	1	1	yes	characteristic
256	C₂	—	2	4	no	—
257	C₂	C₂×#19098317	1	1	yes	characteristic
258	C₂	M₃₂×Dic₇₉	1	2	yes	—
259	C₂	#23765010	1	2	yes	—
260	C₂	#23765155	1	2	yes	—

	subgroup	G/H
center, Frattini	C₂²×C₈	C₂²×D₇₉
commutator	C₂²×C₇₉	C₂×C₄×C₈
Fitting	C₂²×M₃₂×C₇₉	C₂
socle	C₂³×C₇₉	C₂²×C₈
prime radical	C₂⁴×C₇₉	C₂×C₈
cosocle	C₂²×C₈×C₇₉	C₂³
hypercenter	C₂²×M₃₂	D₇₉
hypocenter	C₇₉	#12544
LCS	G ▷ C₂²×C₇₉ ▷ C₇₉
UCS	C₁ ◁ C₂²×C₈ ◁ C₂²×M₃₂

Complements

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

N	H ≅ G/N	type
3. #23761477	C₂	semidirect
9. C₂×#14579101	C₄	semidirect
18. C₂×M₃₂×C₇₉	C₄	semidirect
32. C₂×C₇₉⋊C₁₆	C₂×C₄	semidirect
48. C₂×C₁₆×C₇₉	C₂×C₄	semidirect
104. C₂×Dic₇₉	M₃₂	semidirect
135. C₇₉	#12544	semidirect
144. C₂×M₃₂	Dic₇₉	semidirect
176. C₂×C₁₆	C₂×Dic₇₉	semidirect

Maximal subgroups

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

subgroup	index
C₂²×#14579101	2
C₂²×C₈×Dic₇₉	2
#23761477	2
C₂²×M₃₂×C₇₉	2
#12544	79