Subgroups — #34204221

Order 20224.

482 conjugacy classes of subgroups.

The subgroup lattice has 264 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×M₁₆×Dic₇₉	C₂	1	1	yes	maximal, characteristic
2	C₂×#19098383	C₂	1	2	yes	maximal, complemented
3	C₂×#19098345	C₂	1	1	yes	maximal, characteristic
4	#23761479	C₂	1	2	yes	maximal, complemented
5	C₂×#349×C₇₉	C₂	1	1	yes	maximal, characteristic
6	C₂²×#11929608	C₄	1	1	yes	characteristic
7	C₂²×C₄×Dic₇₉	C₄	1	1	yes	characteristic
8	C₂×#14579101	C₂²	1	2	yes	—
9	M₁₆×Dic₇₉	—	2	4	no	—
10	C₂×#14579071	C₄	1	1	yes	characteristic
11	C₂×#14579071	C₂²	1	1	yes	characteristic
12	C₂×C₈×Dic₇₉	C₄	1	1	yes	characteristic
13	C₂×C₈×Dic₇₉	C₂²	1	1	yes	characteristic
14	#19098383	—	2	4	no	—
15	#19098345	—	2	2	no	—
16	C₂²×M₁₆×C₇₉	C₂²	1	1	yes	characteristic
17	C₂×M₃₂×C₇₉	C₂²	1	2	yes	—
18	#349×C₇₉	—	2	2	no	—
19	C₂³×Dic₇₉	—	2	1	no	—
20	C₂×#11929608	—	2	2	no	—
21	C₂×#11929608	—	2	2	no	—
22	C₂×#11929608	C₈	1	4	yes	—
23	C₂²×C₇₉⋊C₈	C₂×C₄	1	1	yes	characteristic
24	C₂²×C₇₉⋊C₈	C₂×C₄	1	1	yes	characteristic
25	C₂×C₄×Dic₇₉	—	2	1	no	—
26	C₂×C₄×Dic₇₉	—	2	2	no	—
27	C₂×C₄×Dic₇₉	—	2	2	no	—
28	C₂×C₄×Dic₇₉	C₂×C₄	1	1	yes	characteristic
29	C₂×C₄×Dic₇₉	C₂×C₄	1	1	yes	characteristic
30	#14579101	—	2	2	no	—
31	#14579101	—	2	2	no	—
32	C₂×C₇₉⋊C₁₆	—	2	2	no	—
33	#14579071	—	2	2	no	—
34	#14579071	—	2	2	no	—
35	C₈×Dic₇₉	—	2	2	no	—
36	C₈×Dic₇₉	D₄	1	4	yes	—
37	C₂³×C₄×C₇₉	C₂×C₄	1	1	yes	characteristic
38	C₂×M₁₆×C₇₉	—	2	2	no	—
39	C₂×M₁₆×C₇₉	—	2	2	no	—
40	C₂×M₁₆×C₇₉	D₄	1	2	yes	—
41	C₂×M₁₆×C₇₉	Q₈	1	2	yes	—
42	C₂²×C₈×C₇₉	C₂×C₄	1	1	yes	characteristic
43	C₂²×C₈×C₇₉	C₂³	1	1	yes	characteristic
44	M₃₂×C₇₉	—	2	2	no	—
45	M₃₂×C₇₉	—	2	2	no	—
46	C₂×C₁₆×C₇₉	—	2	2	no	—
47	C₂²×Dic₇₉	—	2	4	no	—
48	C₂²×Dic₇₉	—	2	1	no	—
49	C₂²×Dic₇₉	—	2	1	no	—
50	C₂²×Dic₇₉	—	4	4	no	—
51	#11929608	—	2	8	no	—
52	C₂×C₇₉⋊C₈	—	2	1	no	—
53	C₂×C₇₉⋊C₈	—	2	1	no	—
54	C₂×C₇₉⋊C₈	C₂²⋊C₄	1	2	yes	—
55	C₂×C₇₉⋊C₈	C₂×C₈	1	2	yes	—
56	C₂×C₇₉⋊C₈	C₂×C₈	1	2	yes	—
57	C₂×C₇₉⋊C₈	M₁₆	1	2	yes	—
58	C₄×Dic₇₉	—	2	4	no	—
59	C₄×Dic₇₉	—	2	1	no	—
60	C₄×Dic₇₉	—	2	1	no	—
61	C₄×Dic₇₉	C₂²⋊C₄	1	2	yes	—
62	C₄×Dic₇₉	M₁₆	1	2	yes	—
63	C₇₉⋊C₁₆	—	4	2	no	—
64	C₂⁴×C₇₉	M₁₆	1	1	yes	characteristic
65	C₂²×C₄×C₇₉	—	2	1	no	—
66	C₂²×C₄×C₇₉	—	2	1	no	—
67	C₂²×C₄×C₇₉	—	2	1	no	—
68	C₂²×C₄×C₇₉	—	2	1	no	—
69	C₂²×C₄×C₇₉	C₄⋊C₄	1	2	yes	—
70	C₂²×C₄×C₇₉	C₂×C₈	1	2	yes	—
71	C₂²×C₄×C₇₉	M₁₆	1	1	yes	characteristic
72	C₂²×C₄×C₇₉	C₂²×C₄	1	1	yes	characteristic
73	M₁₆×C₇₉	—	2	4	no	—
74	M₁₆×C₇₉	—	2	4	no	—
75	C₂×C₈×C₇₉	—	2	1	no	—
76	C₂×C₈×C₇₉	—	2	1	no	—
77	C₂×C₈×C₇₉	C₄⋊C₄	1	2	yes	—
78	C₂×C₈×C₇₉	C₂×D₄	1	2	yes	—
79	C₂×C₈×C₇₉	C₂×D₄	1	1	yes	characteristic
80	C₂×C₈×C₇₉	C₂×Q₈	1	1	yes	characteristic
81	C₂×C₈×C₇₉	#41	1	2	yes	—
82	C₁₆×C₇₉	—	4	2	no	—
83	C₂³×C₇₉	—	2	1	no	—
84	C₂³×C₇₉	—	2	1	no	—
85	C₂³×C₇₉	—	4	2	no	—
86	C₂³×C₇₉	C₄⋊C₈	1	2	yes	—
87	C₂³×C₇₉	C₂×M₁₆	1	1	yes	characteristic
88	C₂×C₄×C₇₉	—	2	1	no	—
89	C₂×C₄×C₇₉	—	2	1	no	—
90	C₂×C₄×C₇₉	—	2	2	no	—
91	C₂×C₄×C₇₉	—	2	2	no	—
92	C₂×C₄×C₇₉	—	2	2	no	—
93	C₂×C₄×C₇₉	—	2	2	no	—
94	C₂×C₄×C₇₉	C₄⋊C₈	1	2	yes	—
95	C₂×C₄×C₇₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
96	C₂×C₄×C₇₉	C₂×C₄⋊C₄	1	1	yes	characteristic
97	C₂×C₄×C₇₉	#117	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	4	no	—
102	C₂×Dic₇₉	—	2	2	no	—
103	C₂×Dic₇₉	—	2	2	no	—
104	C₂×Dic₇₉	—	4	8	no	—
105	C₂×Dic₇₉	—	4	2	no	—
106	C₈×C₇₉	—	2	2	no	—
107	C₈×C₇₉	#121	1	2	yes	—
108	C₈×C₇₉	C₂²⋊Q₈	1	2	yes	—
109	C₇₉⋊C₈	—	2	2	no	—
110	C₇₉⋊C₈	C₂²⋊C₈	1	4	yes	—
111	C₂²×C₇₉	—	2	2	no	—
112	C₂²×C₇₉	—	2	2	no	—
113	C₂²×C₇₉	—	4	2	no	—
114	C₂²×C₇₉	—	4	2	no	—
115	C₂²×C₇₉	—	4	2	no	—
116	C₂²×C₇₉	#407	1	1	yes	characteristic
117	C₂²×C₇₉	C₂×C₄⋊C₈	1	1	yes	characteristic
118	C₂²×C₇₉	#432	1	1	yes	characteristic
119	C₄×C₇₉	—	2	2	no	—
120	C₄×C₇₉	#380	1	1	yes	characteristic
121	C₄×C₇₉	C₂×C₂²⋊C₈	1	1	yes	characteristic
122	C₄×C₇₉	#423	1	1	yes	characteristic
123	C₄×C₇₉	#431	1	1	yes	characteristic
124	Dic₇₉	—	2	4	no	—
125	Dic₇₉	—	4	2	no	—
126	#12791	—	79	1	no	maximal, Sylow
127	C₂×C₇₉	—	2	1	no	—
128	C₂×C₇₉	—	2	1	no	—
129	C₂×C₇₉	—	4	2	no	—
130	C₂×C₇₉	#1797	1	1	yes	characteristic
131	C₂×C₇₉	#2165	1	2	yes	—
132	C₂×C₄×M₁₆	—	79	1	no	—
133	C₂×#364	—	79	2	no	—
134	C₂×#349	—	79	1	no	—
135	C₂×#349	D₇₉	1	1	yes	characteristic
136	#1375	—	79	2	no	—
137	C₇₉	#12791	1	1	yes	characteristic, complemented, Sylow
138	C₂²×M₁₆	—	79	1	no	—
139	C₂²×M₁₆	C₂×D₇₉	1	1	yes	characteristic
140	C₂²×C₄²	—	79	1	no	—
141	C₂×M₃₂	—	79	2	no	—
142	C₂×M₃₂	C₂×D₇₉	1	2	yes	—
143	C₄×M₁₆	—	158	4	no	—
144	C₂×#97	—	79	1	no	—
145	C₂×#97	—	79	1	no	—
146	C₂×C₄×C₈	—	79	1	no	—
147	C₂×C₄×C₈	—	79	1	no	—
148	#364	—	158	4	no	—
149	#349	—	2	2	no	—
150	#349	—	158	2	no	—
151	C₂³×C₄	—	158	1	no	—
152	C₂³×C₄	C₄×D₇₉	1	1	yes	characteristic
153	C₂×M₁₆	—	2	2	no	—
154	C₂×M₁₆	—	2	2	no	—
155	C₂×M₁₆	—	79	4	no	—
156	C₂×M₁₆	—	158	2	no	—
157	C₂×M₁₆	—	158	2	no	—
158	C₂×M₁₆	Dic₁₅₈	1	2	yes	—
159	C₂×M₁₆	C₂²⋊D₇₉	1	2	yes	—
160	C₂²×C₈	—	79	1	no	—
161	C₂²×C₈	—	79	1	no	—
162	C₂²×C₈	C₄×D₇₉	1	1	yes	characteristic
163	C₂²×C₈	C₂²×D₇₉	1	1	yes	characteristic
164	C₂×C₄²	—	79	1	no	—
165	C₂×C₄²	—	79	1	no	—
166	C₂×C₄²	—	158	2	no	—
167	C₂×C₄²	—	158	2	no	—
168	C₂×C₄²	—	158	1	no	—
169	M₃₂	—	2	2	no	—
170	M₃₂	—	2	2	no	—
171	M₃₂	—	158	2	no	—
172	M₃₂	—	158	2	no	—
173	C₂×C₁₆	—	2	2	no	—
174	C₂×C₁₆	—	158	2	no	—
175	#97	—	158	2	no	—
176	#97	—	158	2	no	—
177	C₄×C₈	—	79	4	no	—
178	C₄×C₈	—	158	2	no	—
179	C₂⁴	#11929607	1	1	yes	characteristic
180	C₂²×C₄	—	2	1	no	—
181	C₂²×C₄	—	2	1	no	—
182	C₂²×C₄	—	2	1	no	—
183	C₂²×C₄	—	2	1	no	—
184	C₂²×C₄	—	316	4	no	—
185	C₂²×C₄	—	158	4	no	—
186	C₂²×C₄	—	158	1	no	—
187	C₂²×C₄	—	158	1	no	—
188	C₂²×C₄	Dic₇₉⋊C₄	1	2	yes	—
189	C₂²×C₄	C₈×D₇₉	1	2	yes	—
190	C₂²×C₄	#11929607	1	1	yes	characteristic
191	C₂²×C₄	C₂×C₄×D₇₉	1	1	yes	characteristic
192	M₁₆	—	2	4	no	—
193	M₁₆	—	2	4	no	—
194	M₁₆	—	158	8	no	—
195	C₂×C₈	—	2	1	no	—
196	C₂×C₈	—	2	1	no	—
197	C₂×C₈	—	79	2	no	—
198	C₂×C₈	—	79	2	no	—
199	C₂×C₈	—	79	2	no	—
200	C₂×C₈	—	79	2	no	—
201	C₂×C₈	—	158	1	no	—
202	C₂×C₈	—	158	1	no	—
203	C₂×C₈	Dic₇₉⋊C₄	1	2	yes	—
204	C₂×C₈	D₄×D₇₉	1	2	yes	—
205	C₂×C₈	C₂×C₂²⋊D₇₉	1	1	yes	characteristic
206	C₂×C₈	C₂×Dic₁₅₈	1	1	yes	characteristic
207	C₂×C₈	#11929623	1	2	yes	—
208	C₄²	—	79	2	no	—
209	C₄²	—	79	2	no	—
210	C₄²	—	158	4	no	—
211	C₄²	—	158	1	no	—
212	C₄²	—	158	1	no	—
213	C₁₆	—	4	2	no	—
214	C₁₆	—	316	2	no	—
215	C₂³	—	2	1	no	—
216	C₂³	—	2	1	no	—
217	C₂³	—	4	2	no	—
218	C₂³	Dic₇₉⋊C₈	1	2	yes	—
219	C₂³	C₂×#11929607	1	1	yes	characteristic
220	C₂×C₄	—	2	1	no	—
221	C₂×C₄	—	2	1	no	—
222	C₂×C₄	—	2	2	no	—
223	C₂×C₄	—	2	2	no	—
224	C₂×C₄	—	2	2	no	—
225	C₂×C₄	—	2	2	no	—
226	C₂×C₄	—	316	2	no	—
227	C₂×C₄	—	316	8	no	—
228	C₂×C₄	—	158	2	no	—
229	C₂×C₄	—	158	4	no	—
230	C₂×C₄	—	158	2	no	—
231	C₂×C₄	Dic₇₉⋊C₈	1	2	yes	—
232	C₂×C₄	C₂²⋊C₄×D₇₉	1	1	yes	characteristic
233	C₂×C₄	C₂×Dic₇₉⋊C₄	1	1	yes	characteristic
234	C₂×C₄	#14579117	1	1	yes	characteristic
235	C₂×C₄	C₂×C₈×D₇₉	1	1	yes	characteristic
236	C₂×C₄	M₁₆×D₇₉	1	1	yes	characteristic
237	C₂×C₄	M₁₆×D₇₉	1	1	yes	characteristic
238	C₈	—	2	2	no	—
239	C₈	—	79	4	no	—
240	C₈	—	158	2	no	—
241	C₈	#14579133	1	2	yes	—
242	C₈	#14579138	1	2	yes	—
243	C₂²	—	2	2	no	—
244	C₂²	—	2	2	no	—
245	C₂²	—	4	2	no	—
246	C₂²	—	4	2	no	—
247	C₂²	—	4	2	no	—
248	C₂²	#19098525	1	1	yes	characteristic
249	C₂²	C₂×Dic₇₉⋊C₈	1	1	yes	characteristic
250	C₂²	#19098617	1	1	yes	characteristic
251	C₄	—	2	2	no	—
252	C₄	—	316	2	no	—
253	C₄	—	158	4	no	—
254	C₄	#19098421	1	1	yes	characteristic
255	C₄	C₂²⋊C₈×D₇₉	1	1	yes	characteristic
256	C₄	#19098586	1	1	yes	characteristic
257	C₄	#19098613	1	1	yes	characteristic
258	C₂	—	2	1	no	—
259	C₂	—	2	1	no	—
260	C₂	—	4	2	no	—
261	C₂	#23763378	1	1	yes	characteristic
262	C₂	#23765209	1	2	yes	—

	subgroup	G/H
center	C₂×C₄	#14579117
commutator	C₂×C₄×C₇₉	C₂²×C₈
Frattini	C₂²×C₈	C₂²×D₇₉
Fitting	C₂×#349×C₇₉	C₂
socle	C₂²×C₇₉	#432
prime radical	C₂⁴×C₇₉	M₁₆
cosocle	C₂²×C₈×C₇₉	C₂³
hypercenter	C₂×#349	D₇₉
hypocenter	C₇₉	#12791
LCS	G ▷ C₂×C₄×C₇₉ ▷ C₂×C₇₉ ▷ C₇₉
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₈ ◁ C₂×#349

Complements

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

N	H ≅ G/N	type
2. C₂×#19098383	C₂	semidirect
4. #23761479	C₂	semidirect
137. C₇₉	#12791	semidirect

Maximal subgroups

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

subgroup	index
C₂×M₁₆×Dic₇₉	2
C₂×#19098383	2
C₂×#19098345	2
#23761479	2
C₂×#349×C₇₉	2
#12791	79