Subgroups — C₂×C₇₉×#2138 #34145065

Order 20224.

1190 conjugacy classes of subgroups.

The subgroup lattice has 328 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×#135×C₇₉	C₂	1	1	yes	maximal, characteristic
2	C₂³×C₁₆×C₇₉	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×#427×C₇₉	C₂	1	1	yes	maximal, characteristic, complemented
4	C₇₉×#2138	C₂	1	8	yes	maximal, direct product factor
5	C₂×#358×C₇₉	C₂	1	2	yes	maximal, complemented
6	C₂×#357×C₇₉	C₂	1	2	yes	maximal, complemented
7	C₂²×#41×C₇₉	—	2	1	no	—
8	C₂×#135×C₇₉	—	2	2	no	—
9	C₂×#135×C₇₉	—	2	4	no	—
10	C₂×#135×C₇₉	C₄	1	8	yes	complemented
11	C₂×#135×C₇₉	C₂²	1	4	yes	—
12	C₂²×Q₁₆×C₇₉	C₂²	1	1	yes	characteristic
13	C₂²×SD₁₆×C₇₉	—	2	1	no	—
14	C₂²×D₈×C₇₉	C₂²	1	1	yes	characteristic
15	C₂³×C₈×C₇₉	C₂²	1	1	yes	characteristic
16	C₂×#117×C₇₉	—	2	1	no	—
17	C₂²×C₁₆×C₇₉	—	2	4	no	—
18	C₂²×C₁₆×C₇₉	C₂²	1	2	yes	—
19	C₂²×C₁₆×C₇₉	C₂²	1	4	yes	complemented
20	#427×C₇₉	—	2	2	no	—
21	#427×C₇₉	C₂²	1	4	yes	complemented
22	C₂×#107×C₇₉	C₂²	1	2	yes	—
23	C₂×#106×C₇₉	—	2	1	no	—
24	#358×C₇₉	C₂²	1	8	yes	complemented
25	#357×C₇₉	C₂²	1	8	yes	complemented
26	C₂×#41×C₇₉	—	2	8	no	—
27	C₂×#41×C₇₉	—	2	4	no	—
28	C₂×#41×C₇₉	—	4	2	no	—
29	C₂×#41×C₇₉	—	4	4	no	—
30	C₂²×Q₈×C₇₉	—	2	1	no	—
31	C₂²×D₄×C₇₉	—	2	1	no	—
32	C₂²×D₄×C₇₉	—	4	1	no	—
33	C₂³×C₄×C₇₉	—	4	1	no	—
34	C₂³×C₄×C₇₉	D₄	1	1	yes	characteristic
35	#135×C₇₉	—	2	16	no	—
36	#135×C₇₉	—	2	8	no	—
37	#135×C₇₉	C₂×C₄	1	16	yes	complemented
38	C₂×Q₁₆×C₇₉	—	2	2	no	—
39	C₂×Q₁₆×C₇₉	—	2	1	no	—
40	C₂×Q₁₆×C₇₉	C₂×C₄	1	4	yes	—
41	C₂×Q₁₆×C₇₉	C₂³	1	2	yes	—
42	C₂×SD₁₆×C₇₉	—	2	8	no	—
43	C₂×SD₁₆×C₇₉	—	2	4	no	—
44	C₂×D₈×C₇₉	—	2	2	no	—
45	C₂×D₈×C₇₉	—	2	1	no	—
46	C₂×D₈×C₇₉	C₂×C₄	1	4	yes	—
47	C₂×D₈×C₇₉	C₂³	1	2	yes	—
48	C₂²×C₈×C₇₉	C₂×C₄	1	4	yes	—
49	C₂²×C₈×C₇₉	D₄	1	1	yes	characteristic
50	C₂²×C₈×C₇₉	D₄	1	2	yes	—
51	C₂²×C₈×C₇₉	D₄	1	4	yes	—
52	C₂²×C₈×C₇₉	C₂³	1	1	yes	characteristic
53	C₂²×C₈×C₇₉	C₂³	1	2	yes	—
54	#117×C₇₉	—	2	4	no	—
55	#117×C₇₉	—	4	2	no	—
56	C₂×C₄⋊C₄×C₇₉	—	2	2	no	—
57	C₂×C₂²⋊C₄×C₇₉	—	4	1	no	—
58	C₂×C₄²×C₇₉	—	4	1	no	—
59	C₂×C₁₆×C₇₉	—	2	4	no	—
60	C₂×C₁₆×C₇₉	—	2	8	no	—
61	C₂×C₁₆×C₇₉	C₂³	1	4	yes	—
62	#107×C₇₉	—	2	2	no	—
63	#107×C₇₉	C₂³	1	4	yes	—
64	#106×C₇₉	—	2	4	no	—
65	C₂⁴×C₇₉	—	4	1	no	—
66	C₂⁴×C₇₉	D₈	1	1	yes	characteristic
67	#41×C₇₉	—	2	16	no	—
68	#41×C₇₉	—	4	16	no	—
69	#41×C₇₉	—	4	8	no	—
70	C₂×Q₈×C₇₉	—	2	4	no	—
71	C₂×Q₈×C₇₉	—	2	2	no	—
72	C₂×Q₈×C₇₉	—	4	1	no	—
73	C₂×Q₈×C₇₉	—	4	2	no	—
74	C₂×D₄×C₇₉	—	2	4	no	—
75	C₂×D₄×C₇₉	—	2	2	no	—
76	C₂×D₄×C₇₉	—	4	8	no	—
77	C₂×D₄×C₇₉	—	4	4	no	—
78	C₂×D₄×C₇₉	—	4	1	no	—
79	C₂×D₄×C₇₉	—	4	2	no	—
80	C₂²×C₄×C₇₉	—	4	1	no	—
81	C₂²×C₄×C₇₉	—	4	8	no	—
82	C₂²×C₄×C₇₉	—	4	4	no	—
83	C₂²×C₄×C₇₉	—	4	2	no	—
84	C₂²×C₄×C₇₉	C₂²⋊C₄	1	4	yes	—
85	C₂²×C₄×C₇₉	D₈	1	2	yes	—
86	C₂²×C₄×C₇₉	D₈	1	1	yes	characteristic
87	C₂²×C₄×C₇₉	SD₁₆	1	4	yes	—
88	C₂²×C₄×C₇₉	C₂×D₄	1	1	yes	characteristic
89	C₂²×C₄×C₇₉	C₂×D₄	1	2	yes	—
90	Q₁₆×C₇₉	—	2	4	no	—
91	Q₁₆×C₇₉	—	2	2	no	—
92	Q₁₆×C₇₉	C₂²×C₄	1	4	yes	—
93	SD₁₆×C₇₉	—	2	16	no	—
94	D₈×C₇₉	—	2	4	no	—
95	D₈×C₇₉	—	2	2	no	—
96	D₈×C₇₉	C₂²×C₄	1	4	yes	—
97	C₂×C₈×C₇₉	C₂²⋊C₄	1	4	yes	—
98	C₂×C₈×C₇₉	C₂²⋊C₄	1	8	yes	—
99	C₂×C₈×C₇₉	C₂²×C₄	1	2	yes	—
100	C₂×C₈×C₇₉	C₂²×C₄	1	4	yes	—
101	C₂×C₈×C₇₉	C₂×D₄	1	2	yes	—
102	C₂×C₈×C₇₉	C₂×D₄	1	4	yes	—
103	C₂×C₈×C₇₉	C₂×D₄	1	2	yes	—
104	C₂×C₈×C₇₉	C₂×D₄	1	1	yes	characteristic
105	C₂×C₈×C₇₉	C₂⁴	1	1	yes	characteristic
106	C₄⋊C₄×C₇₉	—	2	4	no	—
107	C₄⋊C₄×C₇₉	—	4	2	no	—
108	C₂²⋊C₄×C₇₉	—	4	4	no	—
109	C₄²×C₇₉	—	4	4	no	—
110	C₁₆×C₇₉	—	2	4	no	—
111	C₂³×C₇₉	—	2	4	no	—
112	C₂³×C₇₉	—	4	2	no	—
113	C₂³×C₇₉	—	4	4	no	—
114	C₂³×C₇₉	—	8	4	no	—
115	C₂³×C₇₉	#102	1	4	yes	—
116	C₂³×C₇₉	C₂×D₈	1	2	yes	—
117	C₂³×C₇₉	C₂×D₈	1	1	yes	characteristic
118	Q₈×C₇₉	—	2	4	no	—
119	Q₈×C₇₉	—	4	2	no	—
120	Q₈×C₇₉	—	4	4	no	—
121	D₄×C₇₉	—	2	4	no	—
122	D₄×C₇₉	—	4	16	no	—
123	D₄×C₇₉	—	4	2	no	—
124	D₄×C₇₉	—	4	4	no	—
125	C₂×C₄×C₇₉	—	4	2	no	—
126	C₂×C₄×C₇₉	—	4	4	no	—
127	C₂×C₄×C₇₉	—	4	16	no	—
128	C₂×C₄×C₇₉	—	4	4	no	—
129	C₂×C₄×C₇₉	—	8	4	no	—
130	C₂×C₄×C₇₉	#102	1	4	yes	—
131	C₂×C₄×C₇₉	#102	1	8	yes	—
132	C₂×C₄×C₇₉	C₂×C₂²⋊C₄	1	2	yes	—
133	C₂×C₄×C₇₉	C₂×C₂²⋊C₄	1	4	yes	—
134	C₂×C₄×C₇₉	C₂×D₈	1	2	yes	—
135	C₂×C₄×C₇₉	C₂×D₈	1	1	yes	characteristic
136	C₂×C₄×C₇₉	C₂×SD₁₆	1	2	yes	—
137	C₂×C₄×C₇₉	C₂×SD₁₆	1	4	yes	—
138	C₂×C₄×C₇₉	C₂²×D₄	1	1	yes	characteristic
139	C₈×C₇₉	C₂×C₂²⋊C₄	1	2	yes	—
140	C₈×C₇₉	C₂×C₂²⋊C₄	1	4	yes	—
141	C₈×C₇₉	C₂³×C₄	1	1	yes	characteristic
142	C₈×C₇₉	C₂²×D₄	1	1	yes	characteristic
143	C₂²×C₇₉	—	2	4	no	—
144	C₂²×C₇₉	—	2	8	no	—
145	C₂²×C₇₉	—	4	4	no	—
146	C₂²×C₇₉	—	8	4	no	—
147	C₂²×C₇₉	—	8	8	no	—
148	C₂²×C₇₉	C₂×#102	1	4	yes	—
149	C₂²×C₇₉	C₂×#102	1	2	yes	—
150	C₂²×C₇₉	#508	1	4	yes	—
151	C₂²×C₇₉	C₂²×D₈	1	1	yes	characteristic
152	C₄×C₇₉	—	4	4	no	—
153	C₄×C₇₉	—	8	4	no	—
154	C₄×C₇₉	C₂×#102	1	4	yes	—
155	C₄×C₇₉	C₂×#102	1	2	yes	—
156	C₄×C₇₉	C₂²×C₂²⋊C₄	1	1	yes	characteristic
157	C₄×C₇₉	C₂²×SD₁₆	1	1	yes	characteristic
158	C₂×#2138	C₇₉	1	1	yes	maximal, characteristic, direct product factor, Sylow
159	C₂×C₇₉	—	2	4	no	—
160	C₂×C₇₉	—	8	4	no	—
161	C₂×C₇₉	#2138	1	4	yes	direct product factor
162	C₂×C₇₉	C₂²×#102	1	1	yes	characteristic
163	C₂×C₇₉	C₂×#508	1	2	yes	—
164	C₂²×#135	C₂×C₇₉	1	1	yes	characteristic
165	C₂³×C₁₆	C₂×C₇₉	1	1	yes	characteristic, complemented
166	C₂×#427	C₂×C₇₉	1	1	yes	characteristic, complemented
167	#2138	C₂×C₇₉	1	8	yes	direct product factor
168	C₂×#358	C₂×C₇₉	1	2	yes	complemented
169	C₂×#357	C₂×C₇₉	1	2	yes	complemented
170	C₇₉	C₂×#2138	1	1	yes	characteristic, direct product factor, Sylow
171	C₂²×#41	—	2	1	no	—
172	C₂×#135	—	2	2	no	—
173	C₂×#135	—	2	4	no	—
174	C₂×#135	C₄×C₇₉	1	8	yes	complemented
175	C₂×#135	C₂²×C₇₉	1	4	yes	—
176	C₂²×Q₁₆	C₂²×C₇₉	1	1	yes	characteristic
177	C₂²×SD₁₆	—	2	1	no	—
178	C₂²×D₈	C₂²×C₇₉	1	1	yes	characteristic
179	C₂³×C₈	C₂²×C₇₉	1	1	yes	characteristic
180	C₂×#117	—	2	1	no	—
181	C₂²×C₁₆	—	2	4	no	—
182	C₂²×C₁₆	C₂²×C₇₉	1	2	yes	—
183	C₂²×C₁₆	C₂²×C₇₉	1	4	yes	complemented
184	#427	—	2	2	no	—
185	#427	C₂²×C₇₉	1	4	yes	complemented
186	C₂×#107	C₂²×C₇₉	1	2	yes	—
187	C₂×#106	—	2	1	no	—
188	#358	C₂²×C₇₉	1	8	yes	complemented
189	#357	C₂²×C₇₉	1	8	yes	complemented
190	C₂×#41	—	2	8	no	—
191	C₂×#41	—	2	4	no	—
192	C₂×#41	—	4	2	no	—
193	C₂×#41	—	4	4	no	—
194	C₂²×Q₈	—	2	1	no	—
195	C₂²×D₄	—	2	1	no	—
196	C₂²×D₄	—	4	1	no	—
197	C₂³×C₄	—	4	1	no	—
198	C₂³×C₄	D₄×C₇₉	1	1	yes	characteristic
199	#135	—	2	16	no	—
200	#135	—	2	8	no	—
201	#135	C₂×C₄×C₇₉	1	16	yes	complemented
202	C₂×Q₁₆	—	2	1	no	—
203	C₂×Q₁₆	—	2	2	no	—
204	C₂×Q₁₆	C₂×C₄×C₇₉	1	4	yes	—
205	C₂×Q₁₆	C₂³×C₇₉	1	2	yes	—
206	C₂×SD₁₆	—	2	8	no	—
207	C₂×SD₁₆	—	2	4	no	—
208	C₂×D₈	—	2	1	no	—
209	C₂×D₈	—	2	2	no	—
210	C₂×D₈	C₂×C₄×C₇₉	1	4	yes	—
211	C₂×D₈	C₂³×C₇₉	1	2	yes	—
212	C₂²×C₈	C₂×C₄×C₇₉	1	4	yes	—
213	C₂²×C₈	D₄×C₇₉	1	1	yes	characteristic
214	C₂²×C₈	D₄×C₇₉	1	2	yes	—
215	C₂²×C₈	D₄×C₇₉	1	4	yes	—
216	C₂²×C₈	C₂³×C₇₉	1	1	yes	characteristic
217	C₂²×C₈	C₂³×C₇₉	1	2	yes	—
218	#117	—	2	4	no	—
219	#117	—	4	2	no	—
220	C₂×C₄⋊C₄	—	2	2	no	—
221	C₂×C₂²⋊C₄	—	4	1	no	—
222	C₂×C₄²	—	4	1	no	—
223	C₂×C₁₆	—	2	4	no	—
224	C₂×C₁₆	—	2	8	no	—
225	C₂×C₁₆	C₂³×C₇₉	1	4	yes	—
226	#107	—	2	2	no	—
227	#107	C₂³×C₇₉	1	4	yes	—
228	#106	—	2	4	no	—
229	C₂⁴	—	4	1	no	—
230	C₂⁴	D₈×C₇₉	1	1	yes	characteristic
231	#41	—	2	16	no	—
232	#41	—	4	16	no	—
233	#41	—	4	8	no	—
234	C₂×Q₈	—	2	4	no	—
235	C₂×Q₈	—	2	2	no	—
236	C₂×Q₈	—	4	1	no	—
237	C₂×Q₈	—	4	2	no	—
238	C₂×D₄	—	2	4	no	—
239	C₂×D₄	—	2	2	no	—
240	C₂×D₄	—	4	8	no	—
241	C₂×D₄	—	4	4	no	—
242	C₂×D₄	—	4	1	no	—
243	C₂×D₄	—	4	2	no	—
244	C₂²×C₄	—	4	1	no	—
245	C₂²×C₄	—	4	8	no	—
246	C₂²×C₄	—	4	4	no	—
247	C₂²×C₄	—	4	2	no	—
248	C₂²×C₄	C₂²⋊C₄×C₇₉	1	4	yes	—
249	C₂²×C₄	D₈×C₇₉	1	1	yes	characteristic
250	C₂²×C₄	D₈×C₇₉	1	2	yes	—
251	C₂²×C₄	SD₁₆×C₇₉	1	4	yes	—
252	C₂²×C₄	C₂×D₄×C₇₉	1	1	yes	characteristic
253	C₂²×C₄	C₂×D₄×C₇₉	1	2	yes	—
254	Q₁₆	—	2	4	no	—
255	Q₁₆	—	2	2	no	—
256	Q₁₆	C₂²×C₄×C₇₉	1	4	yes	—
257	SD₁₆	—	2	16	no	—
258	D₈	—	2	4	no	—
259	D₈	—	2	2	no	—
260	D₈	C₂²×C₄×C₇₉	1	4	yes	—
261	C₂×C₈	C₂²⋊C₄×C₇₉	1	4	yes	—
262	C₂×C₈	C₂²⋊C₄×C₇₉	1	8	yes	—
263	C₂×C₈	C₂²×C₄×C₇₉	1	2	yes	—
264	C₂×C₈	C₂²×C₄×C₇₉	1	4	yes	—
265	C₂×C₈	C₂×D₄×C₇₉	1	2	yes	—
266	C₂×C₈	C₂×D₄×C₇₉	1	1	yes	characteristic
267	C₂×C₈	C₂×D₄×C₇₉	1	2	yes	—
268	C₂×C₈	C₂×D₄×C₇₉	1	4	yes	—
269	C₂×C₈	C₂⁴×C₇₉	1	1	yes	characteristic
270	C₄⋊C₄	—	2	4	no	—
271	C₄⋊C₄	—	4	2	no	—
272	C₂²⋊C₄	—	4	4	no	—
273	C₄²	—	4	4	no	—
274	C₁₆	—	2	4	no	—
275	C₂³	—	2	4	no	—
276	C₂³	—	4	2	no	—
277	C₂³	—	4	4	no	—
278	C₂³	—	8	4	no	—
279	C₂³	#102×C₇₉	1	4	yes	—
280	C₂³	C₂×D₈×C₇₉	1	2	yes	—
281	C₂³	C₂×D₈×C₇₉	1	1	yes	characteristic
282	Q₈	—	2	4	no	—
283	Q₈	—	4	2	no	—
284	Q₈	—	4	4	no	—
285	D₄	—	2	4	no	—
286	D₄	—	4	16	no	—
287	D₄	—	4	2	no	—
288	D₄	—	4	4	no	—
289	C₂×C₄	—	4	2	no	—
290	C₂×C₄	—	4	4	no	—
291	C₂×C₄	—	4	16	no	—
292	C₂×C₄	—	4	4	no	—
293	C₂×C₄	—	8	4	no	—
294	C₂×C₄	#102×C₇₉	1	4	yes	—
295	C₂×C₄	#102×C₇₉	1	8	yes	—
296	C₂×C₄	C₂×C₂²⋊C₄×C₇₉	1	2	yes	—
297	C₂×C₄	C₂×C₂²⋊C₄×C₇₉	1	4	yes	—
298	C₂×C₄	C₂×D₈×C₇₉	1	2	yes	—
299	C₂×C₄	C₂×D₈×C₇₉	1	1	yes	characteristic
300	C₂×C₄	C₂×SD₁₆×C₇₉	1	2	yes	—
301	C₂×C₄	C₂×SD₁₆×C₇₉	1	4	yes	—
302	C₂×C₄	C₂²×D₄×C₇₉	1	1	yes	characteristic
303	C₈	C₂×C₂²⋊C₄×C₇₉	1	2	yes	—
304	C₈	C₂×C₂²⋊C₄×C₇₉	1	4	yes	—
305	C₈	C₂³×C₄×C₇₉	1	1	yes	characteristic
306	C₈	C₂²×D₄×C₇₉	1	1	yes	characteristic
307	C₂²	—	2	4	no	—
308	C₂²	—	2	8	no	—
309	C₂²	—	4	4	no	—
310	C₂²	—	8	4	no	—
311	C₂²	—	8	8	no	—
312	C₂²	C₂×#102×C₇₉	1	4	yes	—
313	C₂²	C₂×#102×C₇₉	1	2	yes	—
314	C₂²	#508×C₇₉	1	4	yes	—
315	C₂²	C₂²×D₈×C₇₉	1	1	yes	characteristic
316	C₄	—	4	4	no	—
317	C₄	—	8	4	no	—
318	C₄	C₂×#102×C₇₉	1	4	yes	—
319	C₄	C₂×#102×C₇₉	1	2	yes	—
320	C₄	C₂²×C₂²⋊C₄×C₇₉	1	1	yes	characteristic
321	C₄	C₂²×SD₁₆×C₇₉	1	1	yes	characteristic
322	C₂	—	2	4	no	—
323	C₂	—	8	4	no	—
324	C₂	C₇₉×#2138	1	4	yes	direct product factor
325	C₂	C₂²×#102×C₇₉	1	1	yes	characteristic
326	C₂	C₂×#508×C₇₉	1	2	yes	—

	subgroup	G/H
center	C₂²×C₄×C₇₉	D₈
commutator	C₈	C₂³×C₄×C₇₉
Frattini, cosocle	C₂×C₈	C₂⁴×C₇₉
socle	C₂³×C₇₉	C₂×D₈
prime radical	C₂²×#135×C₇₉	C₂
LCS	G ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₄×C₇₉ ◁ C₂³×C₄×C₇₉ ◁ C₂³×C₈×C₇₉ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₂³×C₁₆×C₇₉	C₂	semidirect
3. C₂×#427×C₇₉	C₂	semidirect
4. C₇₉×#2138	C₂	direct (N × H)
5. C₂×#358×C₇₉	C₂	semidirect
6. C₂×#357×C₇₉	C₂	semidirect
10. C₂×#135×C₇₉	C₄	semidirect
19. C₂²×C₁₆×C₇₉	C₂²	semidirect
21. #427×C₇₉	C₂²	semidirect
24. #358×C₇₉	C₂²	semidirect
25. #357×C₇₉	C₂²	semidirect
37. #135×C₇₉	C₂×C₄	semidirect
158. C₂×#2138	C₇₉	direct (N × H)
161. C₂×C₇₉	#2138	direct (N × H)
165. C₂³×C₁₆	C₂×C₇₉	semidirect
166. C₂×#427	C₂×C₇₉	semidirect
167. #2138	C₂×C₇₉	direct (N × H)
168. C₂×#358	C₂×C₇₉	semidirect
169. C₂×#357	C₂×C₇₉	semidirect
170. C₇₉	C₂×#2138	direct (N × H)
174. C₂×#135	C₄×C₇₉	semidirect
183. C₂²×C₁₆	C₂²×C₇₉	semidirect
185. #427	C₂²×C₇₉	semidirect
188. #358	C₂²×C₇₉	semidirect
189. #357	C₂²×C₇₉	semidirect
201. #135	C₂×C₄×C₇₉	semidirect
324. C₂	C₇₉×#2138	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×#135×C₇₉	2
C₂³×C₁₆×C₇₉	2
C₂×#427×C₇₉	2
C₇₉×#2138	2
C₂×#358×C₇₉	2
C₂×#357×C₇₉	2
C₂×#2138	79

Subgroups — C2×C79×#2138 #34145065

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₇₉×#2138 #34145065