Subgroups — C₇₉×#33757 #34145446

Order 20224.

598 conjugacy classes of subgroups.

The subgroup lattice has 338 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₇₉×#3050	C₂	1	2	yes	maximal, complemented
2	C₇₉×#2237	C₂	1	2	yes	maximal, complemented
3	C₇₉×#2234	C₂	1	2	yes	maximal, complemented
4	C₇₉×#2217	C₂	1	2	yes	maximal, complemented
5	C₇₉×#2216	C₂	1	1	yes	maximal, characteristic, complemented
6	C₇₉×#2215	C₂	1	1	yes	maximal, characteristic, complemented
7	C₇₉×#2189	C₂	1	2	yes	maximal, complemented
8	C₇₉×#2186	C₂	1	2	yes	maximal, complemented
9	C₂×C₂²⋊C₁₆×C₇₉	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂×#135×C₇₉	C₂²	1	2	yes	—
11	C₂³×C₈×C₇₉	C₂²	1	1	yes	characteristic
12	#525×C₇₉	—	2	2	no	—
13	C₂×Q₃₂×C₇₉	C₂²	1	1	yes	characteristic, complemented
14	C₂×SD₃₂×C₇₉	C₂²	1	2	yes	complemented
15	C₂×D₁₆×C₇₉	C₂²	1	1	yes	characteristic, complemented
16	C₂²×C₁₆×C₇₉	C₂²	1	1	yes	characteristic
17	C₂²×C₁₆×C₇₉	C₂²	1	1	yes	characteristic
18	#467×C₇₉	C₂²	1	2	yes	—
19	#467×C₇₉	C₂²	1	2	yes	—
20	#466×C₇₉	C₂²	1	2	yes	—
21	#466×C₇₉	C₂²	1	2	yes	—
22	#465×C₇₉	—	2	2	no	—
23	#465×C₇₉	—	2	2	no	—
24	#427×C₇₉	C₂²	1	2	yes	—
25	#416×C₇₉	—	2	2	no	—
26	#367×C₇₉	C₂²	1	2	yes	complemented
27	#366×C₇₉	C₂²	1	1	yes	characteristic, complemented
28	#366×C₇₉	C₂²	1	1	yes	characteristic, complemented
29	#358×C₇₉	C₂²	1	2	yes	complemented
30	#358×C₇₉	C₂²	1	2	yes	complemented
31	#357×C₇₉	C₂²	1	2	yes	complemented
32	#357×C₇₉	C₂²	1	2	yes	complemented
33	C₂²⋊C₁₆×C₇₉	C₂²	1	2	yes	—
34	C₂²⋊C₁₆×C₇₉	C₂²	1	2	yes	—
35	C₂×#41×C₇₉	—	2	2	no	—
36	C₂³×C₄×C₇₉	D₄	1	1	yes	characteristic
37	#135×C₇₉	—	4	2	no	—
38	#135×C₇₉	—	4	2	no	—
39	C₂×Q₁₆×C₇₉	C₂³	1	2	yes	—
40	C₂×SD₁₆×C₇₉	—	2	2	no	—
41	C₂×D₈×C₇₉	C₂³	1	2	yes	—
42	C₂²×C₈×C₇₉	—	4	1	no	—
43	C₂²×C₈×C₇₉	—	4	1	no	—
44	C₂²×C₈×C₇₉	D₄	1	1	yes	characteristic
45	C₂²×C₈×C₇₉	D₄	1	2	yes	—
46	C₂²×C₈×C₇₉	C₂³	1	2	yes	—
47	C₂²×C₈×C₇₉	C₂³	1	1	yes	characteristic
48	#123×C₇₉	—	4	2	no	—
49	C₂²⋊Q₈×C₇₉	—	2	2	no	—
50	C₂²⋊Q₈×C₇₉	—	2	2	no	—
51	#121×C₇₉	—	2	2	no	—
52	#121×C₇₉	—	2	2	no	—
53	#120×C₇₉	—	4	2	no	—
54	C₄×D₄×C₇₉	—	4	2	no	—
55	C₄×D₄×C₇₉	—	4	2	no	—
56	#117×C₇₉	—	2	2	no	—
57	Q₃₂×C₇₉	—	4	1	no	—
58	SD₃₂×C₇₉	—	4	2	no	—
59	D₁₆×C₇₉	—	4	1	no	—
60	C₂×C₁₆×C₇₉	—	4	1	no	—
61	C₂×C₁₆×C₇₉	—	4	1	no	—
62	C₂×C₁₆×C₇₉	C₂³	1	1	yes	characteristic
63	C₂×C₁₆×C₇₉	C₂³	1	2	yes	—
64	C₂×C₁₆×C₇₉	C₂³	1	1	yes	characteristic
65	#107×C₇₉	C₂³	1	2	yes	—
66	#107×C₇₉	C₂³	1	2	yes	—
67	#106×C₇₉	—	2	2	no	—
68	#103×C₇₉	—	2	2	no	—
69	#103×C₇₉	—	2	2	no	—
70	#102×C₇₉	—	2	2	no	—
71	#102×C₇₉	—	2	2	no	—
72	C₂⁴×C₇₉	D₈	1	1	yes	characteristic
73	#41×C₇₉	—	4	4	no	—
74	#41×C₇₉	—	4	4	no	—
75	C₂×Q₈×C₇₉	—	2	2	no	—
76	C₂×D₄×C₇₉	—	2	2	no	—
77	C₂×D₄×C₇₉	—	4	2	no	—
78	C₂×D₄×C₇₉	—	4	2	no	—
79	C₂×D₄×C₇₉	—	4	2	no	—
80	C₂²×C₄×C₇₉	—	4	1	no	—
81	C₂²×C₄×C₇₉	—	4	1	no	—
82	C₂²×C₄×C₇₉	—	4	2	no	—
83	C₂²×C₄×C₇₉	D₈	1	1	yes	characteristic
84	C₂²×C₄×C₇₉	D₈	1	2	yes	—
85	C₂²×C₄×C₇₉	C₂×D₄	1	2	yes	—
86	C₂²×C₄×C₇₉	C₂×D₄	1	1	yes	characteristic
87	Q₁₆×C₇₉	—	2	2	no	—
88	Q₁₆×C₇₉	—	2	2	no	—
89	SD₁₆×C₇₉	—	4	4	no	—
90	D₈×C₇₉	—	2	2	no	—
91	D₈×C₇₉	—	2	2	no	—
92	C₂×C₈×C₇₉	—	2	1	no	—
93	C₂×C₈×C₇₉	—	2	1	no	—
94	C₂×C₈×C₇₉	—	2	2	no	—
95	C₂×C₈×C₇₉	—	2	2	no	—
96	C₂×C₈×C₇₉	—	4	1	no	—
97	C₂×C₈×C₇₉	—	4	2	no	—
98	C₂×C₈×C₇₉	C₂×D₄	1	1	yes	characteristic
99	C₂×C₈×C₇₉	C₂×D₄	1	2	yes	—
100	C₂×C₈×C₇₉	C₂⁴	1	1	yes	characteristic
101	C₄⋊C₄×C₇₉	—	2	2	no	—
102	C₄⋊C₄×C₇₉	—	2	2	no	—
103	C₄⋊C₄×C₇₉	—	4	2	no	—
104	C₄⋊C₄×C₇₉	—	4	2	no	—
105	C₂²⋊C₄×C₇₉	—	4	2	no	—
106	C₂²⋊C₄×C₇₉	—	4	2	no	—
107	C₂²⋊C₄×C₇₉	—	4	2	no	—
108	C₂²⋊C₄×C₇₉	—	4	2	no	—
109	C₂²⋊C₄×C₇₉	—	4	2	no	—
110	C₄²×C₇₉	—	4	2	no	—
111	C₁₆×C₇₉	—	2	1	no	—
112	C₁₆×C₇₉	—	2	2	no	—
113	C₁₆×C₇₉	—	2	1	no	—
114	C₂³×C₇₉	—	2	2	no	—
115	C₂³×C₇₉	—	4	1	no	—
116	C₂³×C₇₉	—	4	1	no	—
117	C₂³×C₇₉	—	4	2	no	—
118	C₂³×C₇₉	C₂×D₈	1	2	yes	—
119	C₂³×C₇₉	C₂×D₈	1	1	yes	characteristic
120	Q₈×C₇₉	—	4	2	no	—
121	Q₈×C₇₉	—	4	2	no	—
122	D₄×C₇₉	—	4	2	no	—
123	D₄×C₇₉	—	4	2	no	—
124	D₄×C₇₉	—	8	4	no	—
125	D₄×C₇₉	—	8	4	no	—
126	D₄×C₇₉	—	8	4	no	—
127	C₂×C₄×C₇₉	—	2	1	no	—
128	C₂×C₄×C₇₉	—	2	2	no	—
129	C₂×C₄×C₇₉	—	2	1	no	—
130	C₂×C₄×C₇₉	—	2	2	no	—
131	C₂×C₄×C₇₉	—	4	1	no	—
132	C₂×C₄×C₇₉	—	4	2	no	—
133	C₂×C₄×C₇₉	—	4	2	no	—
134	C₂×C₄×C₇₉	—	4	2	no	—
135	C₂×C₄×C₇₉	—	4	2	no	—
136	C₂×C₄×C₇₉	—	8	4	no	—
137	C₂×C₄×C₇₉	C₂×D₈	1	2	yes	—
138	C₂×C₄×C₇₉	C₂×D₈	1	1	yes	characteristic
139	C₂×C₄×C₇₉	C₂²×D₄	1	1	yes	characteristic
140	C₈×C₇₉	—	2	1	no	—
141	C₈×C₇₉	—	2	2	no	—
142	C₈×C₇₉	H(5,𝔽₂)	1	1	yes	characteristic
143	C₈×C₇₉	H(5,𝔽₂)	1	1	yes	characteristic
144	C₂²×C₇₉	—	2	1	no	—
145	C₂²×C₇₉	—	2	1	no	—
146	C₂²×C₇₉	—	2	2	no	—
147	C₂²×C₇₉	—	2	2	no	—
148	C₂²×C₇₉	—	2	2	no	—
149	C₂²×C₇₉	—	4	4	no	—
150	C₂²×C₇₉	—	8	2	no	—
151	C₂²×C₇₉	—	8	2	no	—
152	C₂²×C₇₉	—	8	2	no	—
153	C₂²×C₇₉	#508	1	2	yes	—
154	C₂²×C₇₉	C₂²×D₈	1	1	yes	characteristic
155	C₄×C₇₉	—	2	2	no	—
156	C₄×C₇₉	—	2	1	no	—
157	C₄×C₇₉	—	8	2	no	—
158	C₄×C₇₉	—	8	2	no	—
159	C₄×C₇₉	—	8	2	no	—
160	C₄×C₇₉	#534	1	1	yes	characteristic
161	C₄×C₇₉	#575	1	1	yes	characteristic
162	#33757	C₇₉	1	1	yes	maximal, characteristic, direct product factor, Sylow
163	C₂×C₇₉	—	2	2	no	—
164	C₂×C₇₉	—	4	2	no	—
165	C₂×C₇₉	—	16	2	no	—
166	C₂×C₇₉	#3186	1	1	yes	characteristic
167	C₂×C₇₉	C₂×#508	1	1	yes	characteristic
168	C₂×C₇₉	#3414	1	1	yes	characteristic
169	#3050	C₂×C₇₉	1	2	yes	complemented
170	#2237	C₂×C₇₉	1	2	yes	complemented
171	#2234	C₂×C₇₉	1	2	yes	complemented
172	#2217	C₂×C₇₉	1	2	yes	complemented
173	#2216	C₂×C₇₉	1	1	yes	characteristic, complemented
174	#2215	C₂×C₇₉	1	1	yes	characteristic, complemented
175	#2189	C₂×C₇₉	1	2	yes	complemented
176	#2186	C₂×C₇₉	1	2	yes	complemented
177	C₂×C₂²⋊C₁₆	C₂×C₇₉	1	1	yes	characteristic, complemented
178	C₇₉	#33757	1	1	yes	characteristic, direct product factor, Sylow
179	C₂×#135	C₂²×C₇₉	1	2	yes	—
180	C₂³×C₈	C₂²×C₇₉	1	1	yes	characteristic
181	#525	—	2	2	no	—
182	C₂×Q₃₂	C₂²×C₇₉	1	1	yes	characteristic, complemented
183	C₂×SD₃₂	C₂²×C₇₉	1	2	yes	complemented
184	C₂×D₁₆	C₂²×C₇₉	1	1	yes	characteristic, complemented
185	C₂²×C₁₆	C₂²×C₇₉	1	1	yes	characteristic
186	C₂²×C₁₆	C₂²×C₇₉	1	1	yes	characteristic
187	#467	C₂²×C₇₉	1	2	yes	—
188	#467	C₂²×C₇₉	1	2	yes	—
189	#466	C₂²×C₇₉	1	2	yes	—
190	#466	C₂²×C₇₉	1	2	yes	—
191	#465	—	2	2	no	—
192	#465	—	2	2	no	—
193	#427	C₂²×C₇₉	1	2	yes	—
194	#416	—	2	2	no	—
195	#367	C₂²×C₇₉	1	2	yes	complemented
196	#366	C₂²×C₇₉	1	1	yes	characteristic, complemented
197	#366	C₂²×C₇₉	1	1	yes	characteristic, complemented
198	#358	C₂²×C₇₉	1	2	yes	complemented
199	#358	C₂²×C₇₉	1	2	yes	complemented
200	#357	C₂²×C₇₉	1	2	yes	complemented
201	#357	C₂²×C₇₉	1	2	yes	complemented
202	C₂²⋊C₁₆	C₂²×C₇₉	1	2	yes	—
203	C₂²⋊C₁₆	C₂²×C₇₉	1	2	yes	—
204	C₂×#41	—	2	2	no	—
205	C₂³×C₄	D₄×C₇₉	1	1	yes	characteristic
206	#135	—	4	2	no	—
207	#135	—	4	2	no	—
208	C₂×Q₁₆	C₂³×C₇₉	1	2	yes	—
209	C₂×SD₁₆	—	2	2	no	—
210	C₂×D₈	C₂³×C₇₉	1	2	yes	—
211	C₂²×C₈	—	4	1	no	—
212	C₂²×C₈	—	4	1	no	—
213	C₂²×C₈	D₄×C₇₉	1	2	yes	—
214	C₂²×C₈	D₄×C₇₉	1	1	yes	characteristic
215	C₂²×C₈	C₂³×C₇₉	1	2	yes	—
216	C₂²×C₈	C₂³×C₇₉	1	1	yes	characteristic
217	#123	—	4	2	no	—
218	C₂²⋊Q₈	—	2	2	no	—
219	C₂²⋊Q₈	—	2	2	no	—
220	#121	—	2	2	no	—
221	#121	—	2	2	no	—
222	#120	—	4	2	no	—
223	C₄×D₄	—	4	2	no	—
224	C₄×D₄	—	4	2	no	—
225	#117	—	2	2	no	—
226	Q₃₂	—	4	1	no	—
227	SD₃₂	—	4	2	no	—
228	D₁₆	—	4	1	no	—
229	C₂×C₁₆	—	4	1	no	—
230	C₂×C₁₆	—	4	1	no	—
231	C₂×C₁₆	C₂³×C₇₉	1	1	yes	characteristic
232	C₂×C₁₆	C₂³×C₇₉	1	2	yes	—
233	C₂×C₁₆	C₂³×C₇₉	1	1	yes	characteristic
234	#107	C₂³×C₇₉	1	2	yes	—
235	#107	C₂³×C₇₉	1	2	yes	—
236	#106	—	2	2	no	—
237	#103	—	2	2	no	—
238	#103	—	2	2	no	—
239	#102	—	2	2	no	—
240	#102	—	2	2	no	—
241	C₂⁴	D₈×C₇₉	1	1	yes	characteristic
242	#41	—	4	4	no	—
243	#41	—	4	4	no	—
244	C₂×Q₈	—	2	2	no	—
245	C₂×D₄	—	2	2	no	—
246	C₂×D₄	—	4	2	no	—
247	C₂×D₄	—	4	2	no	—
248	C₂×D₄	—	4	2	no	—
249	C₂²×C₄	—	4	1	no	—
250	C₂²×C₄	—	4	1	no	—
251	C₂²×C₄	—	4	2	no	—
252	C₂²×C₄	D₈×C₇₉	1	1	yes	characteristic
253	C₂²×C₄	D₈×C₇₉	1	2	yes	—
254	C₂²×C₄	C₂×D₄×C₇₉	1	2	yes	—
255	C₂²×C₄	C₂×D₄×C₇₉	1	1	yes	characteristic
256	Q₁₆	—	2	2	no	—
257	Q₁₆	—	2	2	no	—
258	SD₁₆	—	4	4	no	—
259	D₈	—	2	2	no	—
260	D₈	—	2	2	no	—
261	C₂×C₈	—	2	1	no	—
262	C₂×C₈	—	2	1	no	—
263	C₂×C₈	—	2	2	no	—
264	C₂×C₈	—	2	2	no	—
265	C₂×C₈	—	4	1	no	—
266	C₂×C₈	—	4	2	no	—
267	C₂×C₈	C₂×D₄×C₇₉	1	1	yes	characteristic
268	C₂×C₈	C₂×D₄×C₇₉	1	2	yes	—
269	C₂×C₈	C₂⁴×C₇₉	1	1	yes	characteristic
270	C₄⋊C₄	—	2	2	no	—
271	C₄⋊C₄	—	2	2	no	—
272	C₄⋊C₄	—	4	2	no	—
273	C₄⋊C₄	—	4	2	no	—
274	C₂²⋊C₄	—	4	2	no	—
275	C₂²⋊C₄	—	4	2	no	—
276	C₂²⋊C₄	—	4	2	no	—
277	C₂²⋊C₄	—	4	2	no	—
278	C₂²⋊C₄	—	4	2	no	—
279	C₄²	—	4	2	no	—
280	C₁₆	—	2	1	no	—
281	C₁₆	—	2	2	no	—
282	C₁₆	—	2	1	no	—
283	C₂³	—	2	2	no	—
284	C₂³	—	4	1	no	—
285	C₂³	—	4	1	no	—
286	C₂³	—	4	2	no	—
287	C₂³	C₂×D₈×C₇₉	1	2	yes	—
288	C₂³	C₂×D₈×C₇₉	1	1	yes	characteristic
289	Q₈	—	4	2	no	—
290	Q₈	—	4	2	no	—
291	D₄	—	4	2	no	—
292	D₄	—	4	2	no	—
293	D₄	—	8	4	no	—
294	D₄	—	8	4	no	—
295	D₄	—	8	4	no	—
296	C₂×C₄	—	2	1	no	—
297	C₂×C₄	—	2	2	no	—
298	C₂×C₄	—	2	1	no	—
299	C₂×C₄	—	2	2	no	—
300	C₂×C₄	—	4	1	no	—
301	C₂×C₄	—	4	2	no	—
302	C₂×C₄	—	4	2	no	—
303	C₂×C₄	—	4	2	no	—
304	C₂×C₄	—	4	2	no	—
305	C₂×C₄	—	8	4	no	—
306	C₂×C₄	C₂×D₈×C₇₉	1	2	yes	—
307	C₂×C₄	C₂×D₈×C₇₉	1	1	yes	characteristic
308	C₂×C₄	C₂²×D₄×C₇₉	1	1	yes	characteristic
309	C₈	—	2	2	no	—
310	C₈	—	2	1	no	—
311	C₈	H(5,𝔽₂)×C₇₉	1	1	yes	characteristic
312	C₈	H(5,𝔽₂)×C₇₉	1	1	yes	characteristic
313	C₂²	—	2	1	no	—
314	C₂²	—	2	1	no	—
315	C₂²	—	2	2	no	—
316	C₂²	—	2	2	no	—
317	C₂²	—	2	2	no	—
318	C₂²	—	4	4	no	—
319	C₂²	—	8	2	no	—
320	C₂²	—	8	2	no	—
321	C₂²	—	8	2	no	—
322	C₂²	#508×C₇₉	1	2	yes	—
323	C₂²	C₂²×D₈×C₇₉	1	1	yes	characteristic
324	C₄	—	2	2	no	—
325	C₄	—	2	1	no	—
326	C₄	—	8	2	no	—
327	C₄	—	8	2	no	—
328	C₄	—	8	2	no	—
329	C₄	#534×C₇₉	1	1	yes	characteristic
330	C₄	#575×C₇₉	1	1	yes	characteristic
331	C₂	—	2	2	no	—
332	C₂	—	4	2	no	—
333	C₂	—	16	2	no	—
334	C₂	C₇₉×#3186	1	1	yes	characteristic
335	C₂	C₂×#508×C₇₉	1	1	yes	characteristic
336	C₂	C₇₉×#3414	1	1	yes	characteristic

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

Complements

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

N	H ≅ G/N	type
1. C₇₉×#3050	C₂	semidirect
2. C₇₉×#2237	C₂	semidirect
3. C₇₉×#2234	C₂	semidirect
4. C₇₉×#2217	C₂	semidirect
5. C₇₉×#2216	C₂	semidirect
6. C₇₉×#2215	C₂	semidirect
7. C₇₉×#2189	C₂	semidirect
8. C₇₉×#2186	C₂	semidirect
9. C₂×C₂²⋊C₁₆×C₇₉	C₂	semidirect
13. C₂×Q₃₂×C₇₉	C₂²	semidirect
14. C₂×SD₃₂×C₇₉	C₂²	semidirect
15. C₂×D₁₆×C₇₉	C₂²	semidirect
26. #367×C₇₉	C₂²	semidirect
27. #366×C₇₉	C₂²	semidirect
28. #366×C₇₉	C₂²	semidirect
29. #358×C₇₉	C₂²	semidirect
30. #358×C₇₉	C₂²	semidirect
31. #357×C₇₉	C₂²	semidirect
32. #357×C₇₉	C₂²	semidirect
162. #33757	C₇₉	direct (N × H)
169. #3050	C₂×C₇₉	semidirect
170. #2237	C₂×C₇₉	semidirect
171. #2234	C₂×C₇₉	semidirect
172. #2217	C₂×C₇₉	semidirect
173. #2216	C₂×C₇₉	semidirect
174. #2215	C₂×C₇₉	semidirect
175. #2189	C₂×C₇₉	semidirect
176. #2186	C₂×C₇₉	semidirect
177. C₂×C₂²⋊C₁₆	C₂×C₇₉	semidirect
178. C₇₉	#33757	direct (N × H)
182. C₂×Q₃₂	C₂²×C₇₉	semidirect
183. C₂×SD₃₂	C₂²×C₇₉	semidirect
184. C₂×D₁₆	C₂²×C₇₉	semidirect
195. #367	C₂²×C₇₉	semidirect
196. #366	C₂²×C₇₉	semidirect
197. #366	C₂²×C₇₉	semidirect
198. #358	C₂²×C₇₉	semidirect
199. #358	C₂²×C₇₉	semidirect
200. #357	C₂²×C₇₉	semidirect
201. #357	C₂²×C₇₉	semidirect

Maximal subgroups

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

subgroup	index
C₇₉×#3050	2
C₇₉×#2237	2
C₇₉×#2234	2
C₇₉×#2217	2
C₇₉×#2216	2
C₇₉×#2215	2
C₇₉×#2189	2
C₇₉×#2186	2
C₂×C₂²⋊C₁₆×C₇₉	2
#33757	79

Subgroups — C79×#33757 #34145446

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₇₉×#33757 #34145446