Subgroups — C₇₉×#13196 #34124885

Order 20224.

430 conjugacy classes of subgroups.

The subgroup lattice has 256 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#443×C₇₉	C₂	1	1	yes	maximal, characteristic
2	C₂×#364×C₇₉	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×#350×C₇₉	C₂	1	2	yes	maximal, complemented
4	C₇₉×#2113	C₂	1	2	yes	maximal, complemented
5	C₂×#346×C₇₉	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×#135×C₇₉	C₄	1	1	yes	characteristic
7	C₂×#131×C₇₉	C₂²	1	2	yes	—
8	C₂×M₃₂×C₇₉	C₂²	1	2	yes	—
9	C₂²×C₁₆×C₇₉	C₂²	1	1	yes	characteristic
10	C₂²×C₁₆×C₇₉	C₂²	1	1	yes	characteristic
11	#443×C₇₉	—	2	2	no	—
12	#443×C₇₉	—	2	2	no	—
13	C₂×#108×C₇₉	C₄	1	1	yes	characteristic
14	C₂×#104×C₇₉	C₄	1	2	yes	—
15	C₂×C₄×C₈×C₇₉	C₂²	1	1	yes	characteristic
16	#364×C₇₉	—	2	2	no	—
17	#350×C₇₉	—	2	4	no	—
18	C₂²⋊C₁₆×C₇₉	—	2	4	no	—
19	#346×C₇₉	—	2	2	no	—
20	C₂×#41×C₇₉	C₂×C₄	1	2	yes	—
21	#135×C₇₉	—	2	2	no	—
22	#135×C₇₉	—	2	2	no	—
23	C₂×Q₁₆×C₇₉	C₈	1	1	yes	characteristic
24	C₂×SD₁₆×C₇₉	C₈	1	2	yes	—
25	C₂×D₈×C₇₉	C₈	1	1	yes	characteristic
26	#131×C₇₉	—	2	2	no	—
27	#131×C₇₉	—	2	2	no	—
28	#131×C₇₉	—	2	2	no	—
29	#131×C₇₉	—	2	2	no	—
30	C₂×M₁₆×C₇₉	—	2	2	no	—
31	C₂×M₁₆×C₇₉	C₂×C₄	1	2	yes	—
32	C₂²×C₈×C₇₉	—	2	2	no	—
33	C₂²×C₈×C₇₉	C₂×C₄	1	1	yes	characteristic
34	C₂²×C₈×C₇₉	C₂³	1	1	yes	characteristic
35	C₂×C₄²×C₇₉	C₂×C₄	1	1	yes	characteristic
36	M₃₂×C₇₉	—	2	2	no	—
37	M₃₂×C₇₉	—	2	2	no	—
38	C₂×C₁₆×C₇₉	—	2	1	no	—
39	C₂×C₁₆×C₇₉	—	2	1	no	—
40	C₂×C₁₆×C₇₉	—	2	1	no	—
41	C₂×C₁₆×C₇₉	—	2	1	no	—
42	C₂×C₁₆×C₇₉	—	2	2	no	—
43	C₂×C₁₆×C₇₉	D₄	1	2	yes	—
44	C₂×C₁₆×C₇₉	D₄	1	2	yes	—
45	#108×C₇₉	—	2	2	no	—
46	#104×C₇₉	—	2	4	no	—
47	C₄×C₈×C₇₉	D₄	1	2	yes	—
48	C₄×C₈×C₇₉	D₄	1	2	yes	—
49	#41×C₇₉	—	2	2	no	—
50	#41×C₇₉	—	2	2	no	—
51	#41×C₇₉	—	2	2	no	—
52	#41×C₇₉	—	2	2	no	—
53	C₂×Q₈×C₇₉	C₂×C₈	1	2	yes	—
54	C₂×D₄×C₇₉	—	2	2	no	—
55	C₂×D₄×C₇₉	C₂×C₈	1	2	yes	—
56	C₂²×C₄×C₇₉	—	2	1	no	—
57	C₂²×C₄×C₇₉	—	2	2	no	—
58	C₂²×C₄×C₇₉	C₂²×C₄	1	1	yes	characteristic
59	Q₁₆×C₇₉	—	2	2	no	—
60	SD₁₆×C₇₉	—	2	4	no	—
61	D₈×C₇₉	—	2	2	no	—
62	M₁₆×C₇₉	—	2	2	no	—
63	M₁₆×C₇₉	—	2	2	no	—
64	M₁₆×C₇₉	—	4	2	no	—
65	M₁₆×C₇₉	—	4	2	no	—
66	C₂×C₈×C₇₉	—	2	2	no	—
67	C₂×C₈×C₇₉	—	4	2	no	—
68	C₂×C₈×C₇₉	—	4	2	no	—
69	C₂×C₈×C₇₉	C₂²⋊C₄	1	2	yes	—
70	C₂×C₈×C₇₉	C₂×C₈	1	1	yes	characteristic
71	C₂×C₈×C₇₉	C₂×C₈	1	1	yes	characteristic
72	C₂×C₈×C₇₉	M₁₆	1	2	yes	—
73	C₂×C₈×C₇₉	C₂×D₄	1	1	yes	characteristic
74	C₂×C₈×C₇₉	C₂×D₄	1	1	yes	characteristic
75	C₂×C₈×C₇₉	C₂×D₄	1	1	yes	characteristic
76	C₂×C₈×C₇₉	C₂×D₄	1	1	yes	characteristic
77	C₂×C₈×C₇₉	#41	1	1	yes	characteristic
78	C₂×C₈×C₇₉	#41	1	1	yes	characteristic
79	C₄²×C₇₉	—	2	1	no	—
80	C₄²×C₇₉	C₂²⋊C₄	1	2	yes	—
81	C₁₆×C₇₉	—	2	2	no	—
82	C₁₆×C₇₉	—	2	2	no	—
83	C₁₆×C₇₉	—	4	2	no	—
84	C₂³×C₇₉	—	2	2	no	—
85	C₂³×C₇₉	#131	1	1	yes	characteristic
86	Q₈×C₇₉	—	2	2	no	—
87	Q₈×C₇₉	—	2	2	no	—
88	D₄×C₇₉	—	2	2	no	—
89	D₄×C₇₉	—	2	2	no	—
90	D₄×C₇₉	—	4	2	no	—
91	D₄×C₇₉	—	4	2	no	—
92	C₂×C₄×C₇₉	—	2	2	no	—
93	C₂×C₄×C₇₉	—	2	2	no	—
94	C₂×C₄×C₇₉	—	4	2	no	—
95	C₂×C₄×C₇₉	—	4	2	no	—
96	C₂×C₄×C₇₉	—	4	2	no	—
97	C₂×C₄×C₇₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
98	C₂×C₄×C₇₉	C₄×D₄	1	1	yes	characteristic
99	C₂×C₄×C₇₉	C₄×D₄	1	1	yes	characteristic
100	C₂×C₄×C₇₉	C₂²×C₈	1	1	yes	characteristic
101	C₂×C₄×C₇₉	C₂×M₁₆	1	1	yes	characteristic
102	C₂×C₄×C₇₉	#131	1	1	yes	characteristic
103	C₈×C₇₉	—	4	2	no	—
104	C₈×C₇₉	C₂²⋊C₈	1	2	yes	—
105	C₈×C₇₉	C₂²⋊C₈	1	2	yes	—
106	C₈×C₇₉	#121	1	1	yes	characteristic
107	C₈×C₇₉	#121	1	1	yes	characteristic
108	C₈×C₇₉	#124	1	1	yes	characteristic
109	C₈×C₇₉	#127	1	1	yes	characteristic
110	C₂²×C₇₉	—	2	2	no	—
111	C₂²×C₇₉	—	4	4	no	—
112	C₂²×C₇₉	—	4	2	no	—
113	C₂²×C₇₉	#408	1	1	yes	characteristic
114	C₂²×C₇₉	C₈×D₄	1	1	yes	characteristic
115	C₂²×C₇₉	#436	1	1	yes	characteristic
116	C₄×C₇₉	—	4	2	no	—
117	C₄×C₇₉	—	4	2	no	—
118	C₄×C₇₉	#390	1	1	yes	characteristic
119	C₄×C₇₉	C₂×C₂²⋊C₈	1	1	yes	characteristic
120	C₄×C₇₉	C₈×D₄	1	1	yes	characteristic
121	C₄×C₇₉	#436	1	1	yes	characteristic
122	#13196	C₇₉	1	1	yes	maximal, characteristic, direct product factor, Sylow
123	C₂×C₇₉	—	2	1	no	—
124	C₂×C₇₉	—	2	1	no	—
125	C₂×C₇₉	—	8	2	no	—
126	C₂×C₇₉	#1954	1	1	yes	characteristic
127	C₂×C₇₉	#2171	1	2	yes	—
128	C₂×#443	C₂×C₇₉	1	1	yes	characteristic
129	C₂×#364	C₂×C₇₉	1	1	yes	characteristic, complemented
130	C₂×#350	C₂×C₇₉	1	2	yes	complemented
131	#2113	C₂×C₇₉	1	2	yes	complemented
132	C₂×#346	C₂×C₇₉	1	1	yes	characteristic, complemented
133	C₇₉	#13196	1	1	yes	characteristic, direct product factor, Sylow
134	C₂×#135	C₄×C₇₉	1	1	yes	characteristic
135	C₂×#131	C₂²×C₇₉	1	2	yes	—
136	C₂×M₃₂	C₂²×C₇₉	1	2	yes	—
137	C₂²×C₁₆	C₂²×C₇₉	1	1	yes	characteristic
138	C₂²×C₁₆	C₂²×C₇₉	1	1	yes	characteristic
139	#443	—	2	2	no	—
140	#443	—	2	2	no	—
141	C₂×#108	C₄×C₇₉	1	1	yes	characteristic
142	C₂×#104	C₄×C₇₉	1	2	yes	—
143	C₂×C₄×C₈	C₂²×C₇₉	1	1	yes	characteristic
144	#364	—	2	2	no	—
145	#350	—	2	4	no	—
146	C₂²⋊C₁₆	—	2	4	no	—
147	#346	—	2	2	no	—
148	C₂×#41	C₂×C₄×C₇₉	1	2	yes	—
149	#135	—	2	2	no	—
150	#135	—	2	2	no	—
151	C₂×Q₁₆	C₈×C₇₉	1	1	yes	characteristic
152	C₂×SD₁₆	C₈×C₇₉	1	2	yes	—
153	C₂×D₈	C₈×C₇₉	1	1	yes	characteristic
154	#131	—	2	2	no	—
155	#131	—	2	2	no	—
156	#131	—	2	2	no	—
157	#131	—	2	2	no	—
158	C₂×M₁₆	—	2	2	no	—
159	C₂×M₁₆	C₂×C₄×C₇₉	1	2	yes	—
160	C₂²×C₈	—	2	2	no	—
161	C₂²×C₈	C₂×C₄×C₇₉	1	1	yes	characteristic
162	C₂²×C₈	C₂³×C₇₉	1	1	yes	characteristic
163	C₂×C₄²	C₂×C₄×C₇₉	1	1	yes	characteristic
164	M₃₂	—	2	2	no	—
165	M₃₂	—	2	2	no	—
166	C₂×C₁₆	—	2	1	no	—
167	C₂×C₁₆	—	2	1	no	—
168	C₂×C₁₆	—	2	1	no	—
169	C₂×C₁₆	—	2	1	no	—
170	C₂×C₁₆	—	2	2	no	—
171	C₂×C₁₆	D₄×C₇₉	1	2	yes	—
172	C₂×C₁₆	D₄×C₇₉	1	2	yes	—
173	#108	—	2	2	no	—
174	#104	—	2	4	no	—
175	C₄×C₈	D₄×C₇₉	1	2	yes	—
176	C₄×C₈	D₄×C₇₉	1	2	yes	—
177	#41	—	2	2	no	—
178	#41	—	2	2	no	—
179	#41	—	2	2	no	—
180	#41	—	2	2	no	—
181	C₂×Q₈	C₂×C₈×C₇₉	1	2	yes	—
182	C₂×D₄	—	2	2	no	—
183	C₂×D₄	C₂×C₈×C₇₉	1	2	yes	—
184	C₂²×C₄	—	2	1	no	—
185	C₂²×C₄	—	2	2	no	—
186	C₂²×C₄	C₂²×C₄×C₇₉	1	1	yes	characteristic
187	Q₁₆	—	2	2	no	—
188	SD₁₆	—	2	4	no	—
189	D₈	—	2	2	no	—
190	M₁₆	—	2	2	no	—
191	M₁₆	—	2	2	no	—
192	M₁₆	—	4	2	no	—
193	M₁₆	—	4	2	no	—
194	C₂×C₈	—	2	2	no	—
195	C₂×C₈	—	4	2	no	—
196	C₂×C₈	—	4	2	no	—
197	C₂×C₈	C₂²⋊C₄×C₇₉	1	2	yes	—
198	C₂×C₈	C₂×C₈×C₇₉	1	1	yes	characteristic
199	C₂×C₈	C₂×C₈×C₇₉	1	1	yes	characteristic
200	C₂×C₈	M₁₆×C₇₉	1	2	yes	—
201	C₂×C₈	C₂×D₄×C₇₉	1	1	yes	characteristic
202	C₂×C₈	C₂×D₄×C₇₉	1	1	yes	characteristic
203	C₂×C₈	C₂×D₄×C₇₉	1	1	yes	characteristic
204	C₂×C₈	C₂×D₄×C₇₉	1	1	yes	characteristic
205	C₂×C₈	#41×C₇₉	1	1	yes	characteristic
206	C₂×C₈	#41×C₇₉	1	1	yes	characteristic
207	C₄²	—	2	1	no	—
208	C₄²	C₂²⋊C₄×C₇₉	1	2	yes	—
209	C₁₆	—	2	2	no	—
210	C₁₆	—	2	2	no	—
211	C₁₆	—	4	2	no	—
212	C₂³	—	2	2	no	—
213	C₂³	#131×C₇₉	1	1	yes	characteristic
214	Q₈	—	2	2	no	—
215	Q₈	—	2	2	no	—
216	D₄	—	2	2	no	—
217	D₄	—	2	2	no	—
218	D₄	—	4	2	no	—
219	D₄	—	4	2	no	—
220	C₂×C₄	—	2	2	no	—
221	C₂×C₄	—	2	2	no	—
222	C₂×C₄	—	4	2	no	—
223	C₂×C₄	—	4	2	no	—
224	C₂×C₄	—	4	2	no	—
225	C₂×C₄	C₂×C₂²⋊C₄×C₇₉	1	1	yes	characteristic
226	C₂×C₄	C₄×D₄×C₇₉	1	1	yes	characteristic
227	C₂×C₄	C₄×D₄×C₇₉	1	1	yes	characteristic
228	C₂×C₄	C₂²×C₈×C₇₉	1	1	yes	characteristic
229	C₂×C₄	C₂×M₁₆×C₇₉	1	1	yes	characteristic
230	C₂×C₄	#131×C₇₉	1	1	yes	characteristic
231	C₈	—	4	2	no	—
232	C₈	C₂²⋊C₈×C₇₉	1	2	yes	—
233	C₈	C₂²⋊C₈×C₇₉	1	2	yes	—
234	C₈	#121×C₇₉	1	1	yes	characteristic
235	C₈	#121×C₇₉	1	1	yes	characteristic
236	C₈	#124×C₇₉	1	1	yes	characteristic
237	C₈	#127×C₇₉	1	1	yes	characteristic
238	C₂²	—	2	2	no	—
239	C₂²	—	4	4	no	—
240	C₂²	—	4	2	no	—
241	C₂²	#408×C₇₉	1	1	yes	characteristic
242	C₂²	C₈×D₄×C₇₉	1	1	yes	characteristic
243	C₂²	#436×C₇₉	1	1	yes	characteristic
244	C₄	—	4	2	no	—
245	C₄	—	4	2	no	—
246	C₄	#390×C₇₉	1	1	yes	characteristic
247	C₄	C₂×C₂²⋊C₈×C₇₉	1	1	yes	characteristic
248	C₄	C₈×D₄×C₇₉	1	1	yes	characteristic
249	C₄	#436×C₇₉	1	1	yes	characteristic
250	C₂	—	2	1	no	—
251	C₂	—	2	1	no	—
252	C₂	—	8	2	no	—
253	C₂	C₇₉×#1954	1	1	yes	characteristic
254	C₂	C₇₉×#2171	1	2	yes	—

	subgroup	G/H
center	C₂×C₈×C₇₉	C₂×D₄
commutator	C₂×C₄	C₂²×C₈×C₇₉
Frattini, cosocle	C₂²×C₈	C₂³×C₇₉
socle	C₂²×C₇₉	C₈×D₄
prime radical	C₂×#135×C₇₉	C₄
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₈×C₇₉ ◁ C₂²×C₁₆×C₇₉ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₂×#364×C₇₉	C₂	semidirect
3. C₂×#350×C₇₉	C₂	semidirect
4. C₇₉×#2113	C₂	semidirect
5. C₂×#346×C₇₉	C₂	semidirect
122. #13196	C₇₉	direct (N × H)
129. C₂×#364	C₂×C₇₉	semidirect
130. C₂×#350	C₂×C₇₉	semidirect
131. #2113	C₂×C₇₉	semidirect
132. C₂×#346	C₂×C₇₉	semidirect
133. C₇₉	#13196	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×#443×C₇₉	2
C₂×#364×C₇₉	2
C₂×#350×C₇₉	2
C₇₉×#2113	2
C₂×#346×C₇₉	2
#13196	79

Subgroups — C79×#13196 #34124885

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₇₉×#13196 #34124885