Subgroups — C₄×#19098670 #34286771

Order 20224.

1086 conjugacy classes of subgroups.

The subgroup lattice has 280 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#19098670	C₂	1	2	yes	maximal
2	C₄×Q₈×Dic₇₉	C₂	1	2	yes	maximal
3	#23763299	C₂	1	2	yes	maximal
4	C₄×#14579093	C₂	1	1	yes	maximal, characteristic
5	#23763232	C₂	1	4	yes	maximal
6	C₄×#14579085	C₂	1	2	yes	maximal
7	C₄×#14579071	C₂	1	1	yes	maximal, characteristic
8	C₂×C₄×Q₁₆×C₇₉	C₂	1	1	yes	maximal, characteristic
9	C₂×Q₈×Dic₇₉	C₂²	1	4	yes	—
10	#19098670	C₄	1	16	yes	direct product factor
11	C₂×#14579093	C₂²	1	2	yes	—
12	C₂×#14579085	C₂²	1	4	yes	—
13	C₂×#14579071	C₂²	1	2	yes	—
14	C₂×#14579071	C₂²	1	2	yes	—
15	C₂×C₄×C₇₉⋊C₈	C₂²	1	1	yes	characteristic
16	C₄⋊C₄×Dic₇₉	—	2	4	no	—
17	C₄⋊C₄×Dic₇₉	C₂²	1	4	yes	—
18	C₄×C₄⋊Dic₇₉	—	2	2	no	—
19	C₄×C₄⋊Dic₇₉	C₂²	1	2	yes	—
20	C₄²×Dic₇₉	—	2	2	no	—
21	C₄²×Dic₇₉	C₂²	1	1	yes	characteristic
22	#19098326	C₂²	1	2	yes	—
23	#19098325	C₂²	1	4	yes	—
24	C₂²×Q₁₆×C₇₉	C₂²	1	1	yes	characteristic
25	C₂×C₄×Q₈×C₇₉	C₂²	1	2	yes	—
26	C₄×Q₁₆×C₇₉	C₄	1	8	yes	complemented
27	C₂×#107×C₇₉	C₂²	1	1	yes	characteristic
28	C₂×#103×C₇₉	C₂²	1	2	yes	—
29	C₂×C₄×C₈×C₇₉	C₂²	1	1	yes	characteristic
30	C₂²×C₇₉⋊C₈	C₂³	1	2	yes	—
31	Q₈×Dic₇₉	—	2	8	no	—
32	Q₈×Dic₇₉	C₂×C₄	1	16	yes	—
33	C₂×C₄⋊Dic₇₉	—	2	4	no	—
34	C₂×C₄⋊Dic₇₉	C₂³	1	4	yes	—
35	C₂×C₄×Dic₇₉	—	2	4	no	—
36	C₂×C₄×Dic₇₉	—	2	2	no	—
37	C₂×C₄×Dic₇₉	—	2	4	no	—
38	C₂×C₄×Dic₇₉	D₄	1	2	yes	—
39	C₂×C₄×Dic₇₉	D₄	1	2	yes	—
40	C₂×C₄×Dic₇₉	C₂³	1	2	yes	—
41	#14579093	C₂×C₄	1	8	yes	—
42	#14579085	C₂×C₄	1	16	yes	—
43	#14579071	—	2	4	no	—
44	#14579071	C₂×C₄	1	8	yes	—
45	C₄×C₇₉⋊C₈	—	2	2	no	—
46	C₇₉⋊#95	—	2	2	no	—
47	C₇₉⋊#95	—	2	4	no	—
48	C₂²×Q₈×C₇₉	C₂³	1	2	yes	—
49	C₂×Q₁₆×C₇₉	C₂×C₄	1	8	yes	—
50	C₂×Q₁₆×C₇₉	C₂×C₄	1	4	yes	—
51	C₂²×C₈×C₇₉	C₂³	1	1	yes	characteristic
52	C₂²×C₈×C₇₉	C₂³	1	1	yes	characteristic
53	C₄×Q₈×C₇₉	—	2	4	no	—
54	C₄×Q₈×C₇₉	C₂×C₄	1	8	yes	—
55	C₂×C₄⋊C₄×C₇₉	—	2	2	no	—
56	C₂×C₄⋊C₄×C₇₉	C₂³	1	2	yes	—
57	C₂×C₄²×C₇₉	—	2	2	no	—
58	C₂×C₄²×C₇₉	C₂³	1	1	yes	characteristic
59	#107×C₇₉	C₂×C₄	1	4	yes	—
60	#103×C₇₉	C₂×C₄	1	8	yes	—
61	C₄×C₈×C₇₉	C₂×C₄	1	4	yes	—
62	C₂²×Dic₇₉	—	2	4	no	—
63	C₂²×Dic₇₉	C₂×D₄	1	4	yes	—
64	C₂×C₇₉⋊C₈	—	2	2	no	—
65	C₂×C₇₉⋊C₈	—	2	2	no	—
66	C₂×C₇₉⋊C₈	C₂²×C₄	1	4	yes	—
67	C₄⋊Dic₇₉	—	2	16	no	—
68	C₄⋊Dic₇₉	—	2	4	no	—
69	C₄⋊Dic₇₉	C₂²×C₄	1	8	yes	—
70	C₄×Dic₇₉	—	2	16	no	—
71	C₄×Dic₇₉	—	2	4	no	—
72	C₄×Dic₇₉	—	4	2	no	—
73	C₄×Dic₇₉	—	4	8	no	—
74	C₄×Dic₇₉	—	4	4	no	—
75	C₄×Dic₇₉	C₂²×C₄	1	4	yes	—
76	C₄×Dic₇₉	#41	1	4	yes	—
77	C₂×Q₈×C₇₉	—	2	2	no	—
78	C₂×Q₈×C₇₉	—	2	4	no	—
79	C₂×Q₈×C₇₉	C₂²×C₄	1	8	yes	—
80	C₂×Q₈×C₇₉	C₂²×C₄	1	4	yes	—
81	C₂²×C₄×C₇₉	—	2	2	no	—
82	C₂²×C₄×C₇₉	—	2	2	no	—
83	C₂²×C₄×C₇₉	C₂×D₄	1	1	yes	characteristic
84	C₂²×C₄×C₇₉	C₂×D₄	1	1	yes	characteristic
85	C₂²×C₄×C₇₉	C₂⁴	1	1	yes	characteristic
86	Q₁₆×C₇₉	C₄²	1	16	yes	complemented
87	C₂×C₈×C₇₉	—	2	2	no	—
88	C₂×C₈×C₇₉	C₂²×C₄	1	4	yes	—
89	C₂×C₈×C₇₉	C₂²×C₄	1	2	yes	—
90	C₂×C₈×C₇₉	C₂²×C₄	1	2	yes	—
91	C₄⋊C₄×C₇₉	—	2	8	no	—
92	C₄⋊C₄×C₇₉	—	2	2	no	—
93	C₄⋊C₄×C₇₉	C₂²×C₄	1	4	yes	—
94	C₄²×C₇₉	—	2	8	no	—
95	C₄²×C₇₉	C₂²×C₄	1	2	yes	—
96	C₄²×C₇₉	#41	1	2	yes	—
97	C₂³×C₇₉	C₂²×D₄	1	1	yes	characteristic
98	Q₈×C₇₉	—	2	8	no	—
99	Q₈×C₇₉	—	2	4	no	—
100	Q₈×C₇₉	C₂×C₄²	1	8	yes	—
101	C₂×C₄×C₇₉	—	2	8	no	—
102	C₂×C₄×C₇₉	—	2	4	no	—
103	C₂×C₄×C₇₉	—	2	2	no	—
104	C₂×C₄×C₇₉	—	2	4	no	—
105	C₂×C₄×C₇₉	—	4	4	no	—
106	C₂×C₄×C₇₉	C₄×D₄	1	2	yes	—
107	C₂×C₄×C₇₉	C₄×D₄	1	2	yes	—
108	C₂×C₄×C₇₉	#137	1	4	yes	—
109	C₂×C₄×C₇₉	C₂³×C₄	1	2	yes	—
110	C₂×C₄×C₇₉	C₂³×C₄	1	1	yes	characteristic
111	C₂×C₄×C₇₉	C₂×#41	1	2	yes	—
112	C₂×C₄×C₇₉	C₂×#41	1	1	yes	characteristic
113	C₂×Dic₇₉	—	2	4	no	—
114	C₂×Dic₇₉	—	2	4	no	—
115	C₂×Dic₇₉	—	2	8	no	—
116	C₂×Dic₇₉	—	4	4	no	—
117	C₂×Dic₇₉	—	4	4	no	—
118	C₂×Dic₇₉	C₄×D₄	1	8	yes	—
119	C₈×C₇₉	—	2	2	no	—
120	C₈×C₇₉	C₂×C₄²	1	4	yes	—
121	C₇₉⋊C₈	—	2	4	no	—
122	C₂²×C₇₉	C₂×C₄×D₄	1	2	yes	—
123	C₂²×C₇₉	C₂×C₄×D₄	1	1	yes	characteristic
124	C₂²×C₇₉	C₂×#137	1	2	yes	—
125	C₂²×C₇₉	#575	1	2	yes	—
126	C₄×C₇₉	—	2	2	no	—
127	C₄×C₇₉	—	2	8	no	—
128	C₄×C₇₉	—	4	4	no	—
129	C₄×C₇₉	#441	1	4	yes	complemented
130	C₄×C₇₉	C₂²×C₄²	1	1	yes	characteristic
131	C₄×C₇₉	C₄×#41	1	2	yes	—
132	C₄×C₇₉	C₄×#41	1	1	yes	characteristic
133	Dic₇₉	—	2	8	no	—
134	Dic₇₉	—	4	8	no	—
135	C₄×#441	—	79	1	no	maximal, Sylow
136	C₂×C₇₉	C₄²×D₄	1	1	yes	characteristic
137	C₂×C₇₉	C₂×#441	1	1	yes	characteristic
138	C₂×C₇₉	#2943	1	1	yes	characteristic
139	C₂×C₇₉	C₄×#137	1	4	yes	—
140	C₂×#441	—	79	2	no	—
141	C₂×C₄×Q₁₆	D₇₉	1	1	yes	characteristic
142	C₄²×Q₈	—	79	2	no	—
143	#1776	—	79	2	no	—
144	C₄×#106	—	79	1	no	—
145	#1763	—	79	4	no	—
146	C₄×#103	—	79	2	no	—
147	C₄×#97	—	79	1	no	—
148	C₇₉	C₄×#441	1	1	yes	characteristic, complemented, Sylow
149	C₂²×Q₁₆	C₂×D₇₉	1	1	yes	characteristic
150	C₂×C₄×Q₈	—	79	4	no	—
151	C₂×C₄×Q₈	C₂×D₇₉	1	2	yes	—
152	#441	—	79	16	no	—
153	C₄×Q₁₆	Dic₇₉	1	8	yes	complemented
154	C₂×#107	C₂×D₇₉	1	1	yes	characteristic
155	C₂×#106	—	79	2	no	—
156	C₂×#103	—	79	4	no	—
157	C₂×#103	C₂×D₇₉	1	2	yes	—
158	C₂×#97	—	79	2	no	—
159	C₂×#97	—	79	2	no	—
160	C₂×C₄×C₈	—	79	1	no	—
161	C₂×C₄×C₈	C₂×D₇₉	1	1	yes	characteristic
162	C₄×C₄⋊C₄	—	79	4	no	—
163	C₄×C₄⋊C₄	—	79	2	no	—
164	C₄×C₄⋊C₄	—	158	2	no	—
165	C₄×C₄⋊C₄	—	158	4	no	—
166	C₄³	—	79	1	no	—
167	C₄³	—	158	2	no	—
168	#340	—	79	2	no	—
169	#340	—	79	4	no	—
170	C₂²×Q₈	C₂²×D₇₉	1	2	yes	—
171	C₂×Q₁₆	C₄×D₇₉	1	8	yes	—
172	C₂×Q₁₆	C₂×Dic₇₉	1	4	yes	—
173	C₂²×C₈	—	79	2	no	—
174	C₂²×C₈	C₂²×D₇₉	1	1	yes	characteristic
175	C₂²×C₈	C₂²×D₇₉	1	1	yes	characteristic
176	C₄×Q₈	—	2	4	no	—
177	C₄×Q₈	—	79	16	no	—
178	C₄×Q₈	—	158	8	no	—
179	C₄×Q₈	C₂×Dic₇₉	1	8	yes	—
180	C₂×C₄⋊C₄	—	2	2	no	—
181	C₂×C₄⋊C₄	—	79	4	no	—
182	C₂×C₄⋊C₄	—	158	4	no	—
183	C₂×C₄⋊C₄	C₂²×D₇₉	1	2	yes	—
184	C₂×C₄²	—	2	2	no	—
185	C₂×C₄²	—	79	2	no	—
186	C₂×C₄²	—	79	2	no	—
187	C₂×C₄²	—	79	2	no	—
188	C₂×C₄²	—	158	4	no	—
189	C₂×C₄²	—	158	2	no	—
190	C₂×C₄²	—	158	4	no	—
191	C₂×C₄²	C₂²×D₇₉	1	1	yes	characteristic
192	#107	C₂×Dic₇₉	1	4	yes	—
193	#106	—	79	8	no	—
194	#103	—	79	16	no	—
195	#103	C₂×Dic₇₉	1	8	yes	—
196	#97	—	79	8	no	—
197	#97	—	158	4	no	—
198	C₄×C₈	—	158	2	no	—
199	C₄×C₈	C₂×Dic₇₉	1	4	yes	—
200	#95	—	158	4	no	—
201	#95	—	158	2	no	—
202	C₂×Q₈	—	2	4	no	—
203	C₂×Q₈	—	2	2	no	—
204	C₂×Q₈	C₂×C₄×D₇₉	1	8	yes	—
205	C₂×Q₈	C₂²×Dic₇₉	1	4	yes	—
206	C₂²×C₄	—	2	2	no	—
207	C₂²×C₄	—	2	2	no	—
208	C₂²×C₄	—	79	4	no	—
209	C₂²×C₄	—	158	4	no	—
210	C₂²×C₄	D₄×D₇₉	1	1	yes	characteristic
211	C₂²×C₄	D₄×D₇₉	1	1	yes	characteristic
212	C₂²×C₄	C₂³×D₇₉	1	1	yes	characteristic
213	Q₁₆	C₄×Dic₇₉	1	16	yes	complemented
214	C₂×C₈	—	2	2	no	—
215	C₂×C₈	—	79	4	no	—
216	C₂×C₈	—	158	2	no	—
217	C₂×C₈	—	158	2	no	—
218	C₂×C₈	C₂×C₄×D₇₉	1	4	yes	—
219	C₂×C₈	C₂²×Dic₇₉	1	2	yes	—
220	C₂×C₈	C₂²×Dic₇₉	1	2	yes	—
221	C₄⋊C₄	—	2	2	no	—
222	C₄⋊C₄	—	2	8	no	—
223	C₄⋊C₄	—	79	8	no	—
224	C₄⋊C₄	—	158	16	no	—
225	C₄⋊C₄	—	158	4	no	—
226	C₄⋊C₄	C₂²×Dic₇₉	1	4	yes	—
227	C₄²	—	2	8	no	—
228	C₄²	—	316	2	no	—
229	C₄²	—	316	8	no	—
230	C₄²	—	316	4	no	—
231	C₄²	—	79	4	no	—
232	C₄²	—	79	4	no	—
233	C₄²	—	158	16	no	—
234	C₄²	—	158	4	no	—
235	C₄²	C₂²×Dic₇₉	1	2	yes	—
236	C₄²	#11929623	1	2	yes	—
237	C₂³	C₂×D₄×D₇₉	1	1	yes	characteristic
238	Q₈	—	2	8	no	—
239	Q₈	—	2	4	no	—
240	Q₈	C₂×C₄×Dic₇₉	1	8	yes	—
241	C₂×C₄	—	2	8	no	—
242	C₂×C₄	—	2	4	no	—
243	C₂×C₄	—	2	2	no	—
244	C₂×C₄	—	2	4	no	—
245	C₂×C₄	—	4	4	no	—
246	C₂×C₄	—	316	4	no	—
247	C₂×C₄	—	316	4	no	—
248	C₂×C₄	—	79	8	no	—
249	C₂×C₄	—	158	8	no	—
250	C₂×C₄	—	158	4	no	—
251	C₂×C₄	—	158	4	no	—
252	C₂×C₄	D₄×Dic₇₉	1	2	yes	—
253	C₂×C₄	D₄×Dic₇₉	1	2	yes	—
254	C₂×C₄	#14579193	1	4	yes	—
255	C₂×C₄	C₂²×C₄×D₇₉	1	2	yes	—
256	C₂×C₄	C₂³×Dic₇₉	1	1	yes	characteristic
257	C₂×C₄	#41×D₇₉	1	2	yes	—
258	C₂×C₄	C₂×#11929623	1	1	yes	characteristic
259	C₈	—	2	2	no	—
260	C₈	—	158	4	no	—
261	C₈	C₂×C₄×Dic₇₉	1	4	yes	—
262	C₂²	C₄×D₄×D₇₉	1	2	yes	—
263	C₂²	C₂×D₄×Dic₇₉	1	1	yes	characteristic
264	C₂²	C₂×#14579193	1	2	yes	—
265	C₂²	#19099360	1	2	yes	—
266	C₄	—	2	2	no	—
267	C₄	—	2	8	no	—
268	C₄	—	4	4	no	—
269	C₄	—	316	8	no	—
270	C₄	—	158	8	no	—
271	C₄	#19098670	1	4	yes	direct product factor
272	C₄	C₂²×C₄×Dic₇₉	1	1	yes	characteristic
273	C₄	C₄×#11929623	1	2	yes	—
274	C₄	#41×Dic₇₉	1	1	yes	characteristic
275	C₂	C₄×D₄×Dic₇₉	1	1	yes	characteristic
276	C₂	C₂×#19098670	1	1	yes	characteristic
277	C₂	#23772201	1	1	yes	characteristic
278	C₂	C₄×#14579193	1	4	yes	—

	subgroup	G/H
center	C₂²×C₄	D₄×D₇₉
commutator	C₄×C₇₉	C₂²×C₄²
Frattini	C₂²×C₄	C₂³×D₇₉
Fitting	C₂×C₄×Q₁₆×C₇₉	C₂
socle, prime radical	C₂³×C₇₉	C₂²×D₄
cosocle	C₂²×C₄×C₇₉	C₂⁴
hypercenter	C₂×C₄×Q₁₆	D₇₉
hypocenter	C₇₉	C₄×#441
LCS	G ▷ C₄×C₇₉ ▷ C₂×C₇₉ ▷ C₇₉
UCS	C₁ ◁ C₂²×C₄ ◁ C₂×C₄² ◁ C₂×C₄×Q₁₆

Complements

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

N	H ≅ G/N	type
10. #19098670	C₄	direct (N × H)
26. C₄×Q₁₆×C₇₉	C₄	semidirect
86. Q₁₆×C₇₉	C₄²	semidirect
129. C₄×C₇₉	#441	semidirect
148. C₇₉	C₄×#441	semidirect
153. C₄×Q₁₆	Dic₇₉	semidirect
213. Q₁₆	C₄×Dic₇₉	semidirect
271. C₄	#19098670	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂×#19098670	2
C₄×Q₈×Dic₇₉	2
#23763299	2
C₄×#14579093	2
#23763232	2
C₄×#14579085	2
C₄×#14579071	2
C₂×C₄×Q₁₆×C₇₉	2
C₄×#441	79

Subgroups — C4×#19098670 #34286771

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₄×#19098670 #34286771