Subgroups — C₂×#23761283 #34178352

Order 20224.

878 conjugacy classes of subgroups.

The subgroup lattice has 260 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₄⋊(C₇₉⋊C₈)	C₂	1	2	yes	maximal
2	#23761283	C₂	1	4	yes	maximal, direct product factor
3	C₂×#384×C₇₉	C₂	1	1	yes	maximal, characteristic
4	C₂³×C₇₉⋊C₈	—	2	2	no	—
5	C₂×C₄⋊(C₇₉⋊C₈)	—	2	2	no	—
6	C₂×C₄⋊(C₇₉⋊C₈)	—	2	4	no	—
7	C₂×C₄⋊(C₇₉⋊C₈)	C₄	1	8	yes	complemented
8	C₂×C₄⋊(C₇₉⋊C₈)	C₂²	1	4	yes	—
9	C₂²×C₄⋊C₄×C₇₉	—	2	2	no	—
10	C₂²×C₄⋊C₄×C₇₉	C₄	1	2	yes	—
11	C₂²×C₄²×C₇₉	C₂²	1	1	yes	characteristic
12	#384×C₇₉	—	2	2	no	—
13	#384×C₇₉	C₄	1	2	yes	—
14	#384×C₇₉	C₂²	1	2	yes	—
15	C₂²×C₇₉⋊C₈	—	2	8	no	—
16	C₂²×C₇₉⋊C₈	—	2	4	no	—
17	C₂²×C₇₉⋊C₈	—	4	4	no	—
18	C₂²×C₇₉⋊C₈	—	4	4	no	—
19	C₄⋊(C₇₉⋊C₈)	—	2	8	no	—
20	C₄⋊(C₇₉⋊C₈)	—	2	4	no	—
21	C₄⋊(C₇₉⋊C₈)	C₂×C₄	1	8	yes	complemented
22	C₂³×C₄×C₇₉	—	2	2	no	—
23	C₂³×C₄×C₇₉	C₂×C₄	1	1	yes	characteristic
24	C₂³×C₄×C₇₉	D₄	1	2	yes	—
25	C₂×C₄⋊C₄×C₇₉	—	2	4	no	—
26	C₂×C₄⋊C₄×C₇₉	—	2	2	no	—
27	C₂×C₄⋊C₄×C₇₉	—	2	4	no	—
28	C₂×C₄⋊C₄×C₇₉	—	4	4	no	—
29	C₂×C₄⋊C₄×C₇₉	—	4	4	no	—
30	C₂×C₄⋊C₄×C₇₉	—	4	4	no	—
31	C₂×C₄⋊C₄×C₇₉	C₂×C₄	1	4	yes	—
32	C₂×C₄²×C₇₉	C₂×C₄	1	4	yes	—
33	C₂×C₄²×C₇₉	C₂×C₄	1	1	yes	characteristic
34	C₂×C₄²×C₇₉	D₄	1	2	yes	—
35	C₂×C₄²×C₇₉	D₄	1	2	yes	—
36	C₂×C₄²×C₇₉	Q₈	1	2	yes	—
37	C₂×C₄²×C₇₉	C₂³	1	1	yes	characteristic
38	C₂×C₇₉⋊C₈	—	2	8	no	—
39	C₂×C₇₉⋊C₈	—	4	16	no	—
40	C₂×C₇₉⋊C₈	—	4	4	no	—
41	C₂×C₇₉⋊C₈	—	4	4	no	—
42	C₂⁴×C₇₉	C₂²⋊C₄	1	1	yes	characteristic
43	C₂²×C₄×C₇₉	—	2	2	no	—
44	C₂²×C₄×C₇₉	—	2	2	no	—
45	C₂²×C₄×C₇₉	—	2	4	no	—
46	C₂²×C₄×C₇₉	—	4	4	no	—
47	C₂²×C₄×C₇₉	—	4	8	no	—
48	C₂²×C₄×C₇₉	C₄²	1	2	yes	—
49	C₂²×C₄×C₇₉	C₂²⋊C₄	1	1	yes	characteristic
50	C₂²×C₄×C₇₉	C₂²⋊C₄	1	2	yes	—
51	C₂²×C₄×C₇₉	C₂²⋊C₄	1	4	yes	—
52	C₂²×C₄×C₇₉	C₄⋊C₄	1	2	yes	—
53	C₂²×C₄×C₇₉	C₄⋊C₄	1	4	yes	—
54	C₂²×C₄×C₇₉	D₈	1	4	yes	—
55	C₂²×C₄×C₇₉	SD₁₆	1	4	yes	—
56	C₂²×C₄×C₇₉	SD₁₆	1	4	yes	—
57	C₂²×C₄×C₇₉	Q₁₆	1	4	yes	—
58	C₂²×C₄×C₇₉	C₂²×C₄	1	1	yes	characteristic
59	C₂²×C₄×C₇₉	C₂×D₄	1	2	yes	—
60	C₄⋊C₄×C₇₉	—	4	4	no	—
61	C₄⋊C₄×C₇₉	—	4	4	no	—
62	C₄⋊C₄×C₇₉	—	4	8	no	—
63	C₄⋊C₄×C₇₉	—	4	4	no	—
64	C₄⋊C₄×C₇₉	—	4	4	no	—
65	C₄²×C₇₉	C₄²	1	2	yes	—
66	C₄²×C₇₉	C₂²⋊C₄	1	4	yes	—
67	C₄²×C₇₉	C₄⋊C₄	1	4	yes	—
68	C₄²×C₇₉	C₄⋊C₄	1	2	yes	—
69	C₄²×C₇₉	C₂²×C₄	1	2	yes	—
70	C₄²×C₇₉	C₂×D₄	1	1	yes	characteristic
71	C₄²×C₇₉	C₂×Q₈	1	1	yes	characteristic
72	C₂³×C₇₉	#95	1	2	yes	—
73	C₂³×C₇₉	#100	1	2	yes	—
74	C₂³×C₇₉	#101	1	2	yes	—
75	C₂³×C₇₉	#102	1	4	yes	—
76	C₂³×C₇₉	#103	1	4	yes	—
77	C₂³×C₇₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
78	C₂×C₄×C₇₉	—	2	4	no	—
79	C₂×C₄×C₇₉	—	2	4	no	—
80	C₂×C₄×C₇₉	—	2	1	no	—
81	C₂×C₄×C₇₉	—	2	1	no	—
82	C₂×C₄×C₇₉	—	2	1	no	—
83	C₂×C₄×C₇₉	—	2	1	no	—
84	C₂×C₄×C₇₉	—	4	4	no	—
85	C₂×C₄×C₇₉	—	4	8	no	—
86	C₂×C₄×C₇₉	—	8	8	no	—
87	C₂×C₄×C₇₉	#95	1	2	yes	—
88	C₂×C₄×C₇₉	#95	1	4	yes	—
89	C₂×C₄×C₇₉	#102	1	8	yes	—
90	C₂×C₄×C₇₉	#102	1	4	yes	—
91	C₂×C₄×C₇₉	#103	1	8	yes	—
92	C₂×C₄×C₇₉	#103	1	4	yes	—
93	C₂×C₄×C₇₉	#106	1	8	yes	—
94	C₂×C₄×C₇₉	#107	1	8	yes	—
95	C₂×C₄×C₇₉	C₂×C₄²	1	1	yes	characteristic
96	C₂×C₄×C₇₉	C₂×C₂²⋊C₄	1	2	yes	—
97	C₂×C₄×C₇₉	C₂×C₄⋊C₄	1	2	yes	—
98	C₂×C₄×C₇₉	C₂×C₄⋊C₄	1	1	yes	characteristic
99	C₂×C₄×C₇₉	C₂×D₈	1	2	yes	—
100	C₂×C₄×C₇₉	C₂×SD₁₆	1	2	yes	—
101	C₂×C₄×C₇₉	C₂×SD₁₆	1	2	yes	—
102	C₂×C₄×C₇₉	C₂×Q₁₆	1	2	yes	—
103	C₇₉⋊C₈	—	4	8	no	—
104	C₂²×C₇₉	#331	1	4	yes	—
105	C₂²×C₇₉	#332	1	8	yes	—
106	C₂²×C₇₉	#333	1	4	yes	—
107	C₂²×C₇₉	#340	1	8	yes	—
108	C₂²×C₇₉	#343	1	4	yes	—
109	C₂²×C₇₉	C₂×#95	1	1	yes	characteristic
110	C₂²×C₇₉	C₂×#100	1	1	yes	characteristic
111	C₂²×C₇₉	C₂×#101	1	1	yes	characteristic
112	C₂²×C₇₉	C₂×#102	1	2	yes	—
113	C₂²×C₇₉	C₂×#103	1	2	yes	—
114	C₄×C₇₉	—	2	2	no	—
115	C₄×C₇₉	—	2	1	no	—
116	C₄×C₇₉	—	2	1	no	—
117	C₄×C₇₉	—	8	4	no	—
118	C₄×C₇₉	#340	1	8	yes	—
119	C₄×C₇₉	C₂×#102	1	2	yes	—
120	C₄×C₇₉	C₂×#103	1	2	yes	—
121	C₄×C₇₉	C₂×#106	1	2	yes	—
122	C₄×C₇₉	C₂×#107	1	2	yes	—
123	C₂×#1296	—	79	1	no	maximal, Sylow
124	C₂×C₇₉	#1296	1	8	yes	complemented
125	C₂×C₇₉	C₂×#331	1	1	yes	characteristic
126	C₂×C₇₉	C₂×#332	1	2	yes	—
127	C₂×C₇₉	C₂×#333	1	1	yes	characteristic
128	C₂×C₇₉	C₂×#340	1	2	yes	—
129	C₂×C₇₉	C₂×#343	1	1	yes	characteristic
130	C₂²×C₄⋊C₈	—	79	2	no	—
131	C₂×#384	D₇₉	1	1	yes	characteristic
132	#1296	—	79	4	no	—
133	C₇₉	C₂×#1296	1	1	yes	characteristic, complemented, Sylow
134	C₂³×C₈	—	158	2	no	—
135	C₂²×C₄⋊C₄	—	2	2	no	—
136	C₂²×C₄⋊C₄	Dic₇₉	1	2	yes	—
137	C₂²×C₄²	C₂×D₇₉	1	1	yes	characteristic
138	C₂×C₄⋊C₈	—	79	4	no	—
139	C₂×C₄⋊C₈	—	79	8	no	—
140	C₂×C₄⋊C₈	—	158	2	no	—
141	C₂×C₄⋊C₈	—	158	4	no	—
142	#384	—	2	2	no	—
143	#384	Dic₇₉	1	2	yes	—
144	#384	C₂×D₇₉	1	2	yes	—
145	C₂³×C₄	—	2	2	no	—
146	C₂³×C₄	C₂×Dic₇₉	1	1	yes	characteristic
147	C₂³×C₄	C₂²⋊D₇₉	1	2	yes	—
148	C₂²×C₈	—	316	4	no	—
149	C₂²×C₈	—	316	4	no	—
150	C₂²×C₈	—	158	8	no	—
151	C₂²×C₈	—	158	4	no	—
152	C₂×C₄⋊C₄	—	2	4	no	—
153	C₂×C₄⋊C₄	—	2	2	no	—
154	C₂×C₄⋊C₄	—	2	4	no	—
155	C₂×C₄⋊C₄	—	4	4	no	—
156	C₂×C₄⋊C₄	—	4	4	no	—
157	C₂×C₄⋊C₄	—	4	4	no	—
158	C₂×C₄⋊C₄	C₂×Dic₇₉	1	4	yes	—
159	C₂×C₄²	Dic₁₅₈	1	2	yes	—
160	C₂×C₄²	C₄×D₇₉	1	4	yes	—
161	C₂×C₄²	D₃₁₆	1	2	yes	—
162	C₂×C₄²	C₂×Dic₇₉	1	1	yes	characteristic
163	C₂×C₄²	C₂²⋊D₇₉	1	2	yes	—
164	C₂×C₄²	C₂²×D₇₉	1	1	yes	characteristic
165	C₄⋊C₈	—	79	8	no	—
166	C₄⋊C₈	—	158	8	no	—
167	C₄⋊C₈	—	158	4	no	—
168	C₂⁴	C₂²⋊Dic₇₉	1	1	yes	characteristic
169	C₂²×C₄	—	2	2	no	—
170	C₂²×C₄	—	2	2	no	—
171	C₂²×C₄	—	2	4	no	—
172	C₂²×C₄	—	4	4	no	—
173	C₂²×C₄	—	4	8	no	—
174	C₂²×C₄	C₄×Dic₇₉	1	2	yes	—
175	C₂²×C₄	(C₂×D₇₉)⋊C₄	1	4	yes	—
176	C₂²×C₄	C₂²⋊Dic₇₉	1	1	yes	characteristic
177	C₂²×C₄	C₂²⋊Dic₇₉	1	2	yes	—
178	C₂²×C₄	Dic₇₉⋊C₄	1	4	yes	—
179	C₂²×C₄	C₄⋊Dic₇₉	1	2	yes	—
180	C₂²×C₄	#11929609	1	4	yes	—
181	C₂²×C₄	#11929611	1	4	yes	—
182	C₂²×C₄	#11929612	1	4	yes	—
183	C₂²×C₄	#11929614	1	4	yes	—
184	C₂²×C₄	C₂²×Dic₇₉	1	1	yes	characteristic
185	C₂²×C₄	C₂×C₂²⋊D₇₉	1	2	yes	—
186	C₂×C₈	—	316	16	no	—
187	C₂×C₈	—	316	4	no	—
188	C₂×C₈	—	316	4	no	—
189	C₂×C₈	—	158	8	no	—
190	C₄⋊C₄	—	4	4	no	—
191	C₄⋊C₄	—	4	4	no	—
192	C₄⋊C₄	—	4	8	no	—
193	C₄⋊C₄	—	4	4	no	—
194	C₄⋊C₄	—	4	4	no	—
195	C₄²	C₄×Dic₇₉	1	2	yes	—
196	C₄²	(C₂×D₇₉)⋊C₄	1	4	yes	—
197	C₄²	Dic₇₉⋊C₄	1	4	yes	—
198	C₄²	C₄⋊Dic₇₉	1	2	yes	—
199	C₄²	C₂×C₄×D₇₉	1	2	yes	—
200	C₄²	C₂×D₃₁₆	1	1	yes	characteristic
201	C₄²	C₂×Dic₁₅₈	1	1	yes	characteristic
202	C₂³	C₇₉⋊#95	1	2	yes	—
203	C₂³	#14579078	1	2	yes	—
204	C₂³	#14579080	1	2	yes	—
205	C₂³	#14579082	1	4	yes	—
206	C₂³	#14579085	1	4	yes	—
207	C₂³	C₂×C₂²⋊Dic₇₉	1	1	yes	characteristic
208	C₂×C₄	—	2	1	no	—
209	C₂×C₄	—	2	1	no	—
210	C₂×C₄	—	2	1	no	—
211	C₂×C₄	—	2	1	no	—
212	C₂×C₄	—	2	4	no	—
213	C₂×C₄	—	2	4	no	—
214	C₂×C₄	—	4	4	no	—
215	C₂×C₄	—	4	8	no	—
216	C₂×C₄	—	8	8	no	—
217	C₂×C₄	C₇₉⋊#95	1	2	yes	—
218	C₂×C₄	C₇₉⋊#95	1	4	yes	—
219	C₂×C₄	#14579081	1	8	yes	—
220	C₂×C₄	#14579082	1	4	yes	—
221	C₂×C₄	#14579084	1	8	yes	—
222	C₂×C₄	#14579085	1	4	yes	—
223	C₂×C₄	#14579092	1	8	yes	—
224	C₂×C₄	#14579094	1	8	yes	—
225	C₂×C₄	C₂×C₄×Dic₇₉	1	1	yes	characteristic
226	C₂×C₄	C₂×(C₂×D₇₉)⋊C₄	1	2	yes	—
227	C₂×C₄	C₂×Dic₇₉⋊C₄	1	2	yes	—
228	C₂×C₄	C₂×C₄⋊Dic₇₉	1	1	yes	characteristic
229	C₂×C₄	C₂×#11929609	1	2	yes	—
230	C₂×C₄	C₂×#11929611	1	2	yes	—
231	C₂×C₄	C₂×#11929612	1	2	yes	—
232	C₂×C₄	C₂×#11929614	1	2	yes	—
233	C₈	—	316	8	no	—
234	C₂²	#19098307	1	4	yes	—
235	C₂²	#19098309	1	8	yes	—
236	C₂²	#19098312	1	4	yes	—
237	C₂²	#19098325	1	8	yes	—
238	C₂²	#19098332	1	4	yes	—
239	C₂²	C₂×C₇₉⋊#95	1	1	yes	characteristic
240	C₂²	C₂×#14579078	1	1	yes	characteristic
241	C₂²	C₂×#14579080	1	1	yes	characteristic
242	C₂²	C₂×#14579082	1	2	yes	—
243	C₂²	C₂×#14579085	1	2	yes	—
244	C₄	—	2	2	no	—
245	C₄	—	2	1	no	—
246	C₄	—	2	1	no	—
247	C₄	—	8	4	no	—
248	C₄	#19098325	1	8	yes	—
249	C₄	C₂×#14579081	1	2	yes	—
250	C₄	C₂×#14579084	1	2	yes	—
251	C₄	C₂×#14579092	1	2	yes	—
252	C₄	C₂×#14579094	1	2	yes	—
253	C₂	#23761283	1	8	yes	direct product factor
254	C₂	C₂×#19098307	1	1	yes	characteristic
255	C₂	C₂×#19098309	1	2	yes	—
256	C₂	C₂×#19098312	1	1	yes	characteristic
257	C₂	C₂×#19098325	1	2	yes	—
258	C₂	C₂×#19098332	1	1	yes	characteristic

	subgroup	G/H
center	C₂⁴	C₂²⋊Dic₇₉
commutator	C₂×C₄×C₇₉	C₂×C₄²
Frattini	C₂×C₄²	C₂²×D₇₉
Fitting	C₂×#384×C₇₉	C₂
socle, prime radical	C₂⁴×C₇₉	C₂²⋊C₄
cosocle	C₂×C₄²×C₇₉	C₂³
hypercenter	C₂×#384	D₇₉
hypocenter	C₇₉	C₂×#1296
LCS	G ▷ C₂×C₄×C₇₉ ▷ C₂²×C₇₉ ▷ C₇₉
UCS	C₁ ◁ C₂⁴ ◁ C₂²×C₄² ◁ C₂×#384

Complements

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

N	H ≅ G/N	type
2. #23761283	C₂	direct (N × H)
7. C₂×C₄⋊(C₇₉⋊C₈)	C₄	semidirect
21. C₄⋊(C₇₉⋊C₈)	C₂×C₄	semidirect
124. C₂×C₇₉	#1296	semidirect
133. C₇₉	C₂×#1296	semidirect
253. C₂	#23761283	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂²×C₄⋊(C₇₉⋊C₈)	2
#23761283	2
C₂×#384×C₇₉	2
C₂×#1296	79

Subgroups — C2×#23761283 #34178352

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#23761283 #34178352