Subgroups — #34445067

Order 20224.

646 conjugacy classes of subgroups.

The subgroup lattice has 416 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#23776538	C₂	1	1	yes	maximal, characteristic, complemented
2	#23775186	C₂	1	2	yes	maximal, complemented
3	#23765662	C₂	1	1	yes	maximal, characteristic, complemented
4	#23765522	C₂	1	1	yes	maximal, characteristic, complemented
5	#23765520	C₂	1	1	yes	maximal, characteristic, complemented
6	#23765471	C₂	1	2	yes	maximal, complemented
7	#23765368	C₂	1	2	yes	maximal, complemented
8	#23765364	C₂	1	2	yes	maximal, complemented
9	#23765281	C₂	1	1	yes	maximal, characteristic, complemented
10	#23765235	C₂	1	1	yes	maximal, characteristic, complemented
11	C₇₉×#3417	C₂	1	1	yes	maximal, characteristic, complemented
12	#19099371	—	2	2	no	—
13	#19099365	C₂²	1	2	yes	—
14	#19099000	—	2	2	no	—
15	#19098998	C₂²	1	2	yes	—
16	#19098996	—	2	1	no	—
17	C₂×#14579104	C₂²	1	2	yes	complemented
18	C₂×D₁₂₆₄	C₂²	1	1	yes	characteristic
19	D₁₆×D₇₉	—	2	1	no	—
20	#19098979	C₂²	1	1	yes	characteristic
21	#19098948	C₂²	1	2	yes	complemented
22	#19098835	C₂²	1	2	yes	—
23	#19098833	C₂²	1	2	yes	complemented
24	#19098828	—	2	2	no	—
25	#19098739	—	2	2	no	—
26	#19098736	—	2	2	no	—
27	#19098680	C₂²	1	2	yes	complemented
28	#19098678	C₂²	1	1	yes	characteristic
29	#19098393	C₂²	1	1	yes	characteristic, complemented
30	#19098383	C₂²	1	1	yes	characteristic, complemented
31	#19098379	C₂²	1	2	yes	complemented
32	#19098377	C₂²	1	2	yes	complemented
33	#19098370	C₂²	1	2	yes	complemented
34	#19098348	C₂²	1	1	yes	characteristic
35	#19098347	C₂²	1	1	yes	characteristic, complemented
36	C₁₆×Dic₇₉	C₂²	1	1	yes	characteristic, complemented
37	#578×C₇₉	C₂²	1	2	yes	complemented
38	#510×C₇₉	—	2	2	no	—
39	#510×C₇₉	C₂²	1	2	yes	complemented
40	#508×C₇₉	—	2	1	no	—
41	#508×C₇₉	C₂²	1	1	yes	characteristic
42	C₂×Q₃₂×C₇₉	—	2	1	no	—
43	C₂×Q₃₂×C₇₉	C₂²	1	1	yes	characteristic, complemented
44	#504×C₇₉	C₂²	1	1	yes	characteristic, complemented
45	#14579207	—	2	2	no	—
46	#14579192	—	2	2	no	—
47	#14579192	—	4	2	no	—
48	#14579188	—	4	2	no	—
49	#14579186	—	2	2	no	—
50	#14579186	—	2	2	no	—
51	#14579186	—	4	2	no	—
52	#14579184	—	2	2	no	—
53	#14579181	—	4	2	no	—
54	#14579181	D₄	1	1	yes	characteristic
55	C₂×#11929611	—	2	2	no	—
56	C₂×#11929611	—	4	2	no	—
57	C₂×D₆₃₂	C₂³	1	2	yes	—
58	D₈×D₇₉	—	2	2	no	—
59	#14579168	—	2	2	no	—
60	#14579167	C₂³	1	1	yes	characteristic
61	#14579150	—	2	2	no	—
62	#14579104	—	2	2	no	—
63	#14579104	—	4	2	no	—
64	D₁₂₆₄	—	2	1	no	—
65	D₁₂₆₄	—	2	1	no	—
66	#14579102	—	2	2	no	—
67	#14579101	C₂³	1	1	yes	characteristic
68	#14579100	—	2	1	no	—
69	C₂×C₇₉⋊C₁₆	C₂³	1	1	yes	characteristic
70	C₁₆×D₇₉	—	2	1	no	—
71	#14579097	C₂³	1	2	yes	—
72	#14579096	D₄	1	1	yes	characteristic
73	#14579088	—	2	2	no	—
74	#14579088	D₄	1	1	yes	characteristic
75	#14579087	D₄	1	1	yes	characteristic
76	#14579080	—	2	2	no	—
77	#14579077	—	2	2	no	—
78	C₈×Dic₇₉	C₂³	1	1	yes	characteristic
79	2₋¹⁺⁴×C₇₉	—	2	2	no	—
80	#137×C₇₉	—	2	2	no	—
81	#137×C₇₉	—	4	2	no	—
82	#135×C₇₉	—	2	2	no	—
83	#135×C₇₉	C₂³	1	2	yes	—
84	C₂×Q₁₆×C₇₉	—	2	2	no	—
85	C₂×Q₁₆×C₇₉	C₂³	1	2	yes	—
86	#131×C₇₉	C₂³	1	1	yes	characteristic
87	Q₃₂×C₇₉	—	2	2	no	—
88	Q₃₂×C₇₉	—	2	1	no	—
89	Q₃₂×C₇₉	—	2	1	no	—
90	Q₃₂×C₇₉	D₄	1	1	yes	characteristic
91	SD₃₂×C₇₉	—	2	2	no	—
92	SD₃₂×C₇₉	D₄	1	2	yes	—
93	D₁₆×C₇₉	D₄	1	1	yes	characteristic
94	M₃₂×C₇₉	—	2	1	no	—
95	M₃₂×C₇₉	C₂³	1	1	yes	characteristic
96	C₂×C₁₆×C₇₉	—	2	1	no	—
97	C₂×C₁₆×C₇₉	C₂³	1	1	yes	characteristic
98	#11929625	—	4	2	no	—
99	#11929624	—	4	2	no	—
100	#11929624	C₂×D₄	1	1	yes	characteristic
101	C₂×D₃₁₆	—	2	2	no	—
102	C₂×D₃₁₆	—	4	2	no	—
103	D₄×D₇₉	—	4	2	no	—
104	D₄×D₇₉	—	8	2	no	—
105	#11929614	—	4	2	no	—
106	#11929614	D₈	1	1	yes	characteristic
107	#11929612	D₈	1	1	yes	characteristic
108	#11929611	—	4	2	no	—
109	#11929611	—	4	2	no	—
110	#11929611	—	4	2	no	—
111	#11929611	—	4	2	no	—
112	#11929611	—	4	2	no	—
113	#11929611	D₈	1	1	yes	characteristic
114	D₆₃₂	—	2	2	no	—
115	D₆₃₂	—	2	2	no	—
116	#11929609	—	4	2	no	—
117	#11929609	—	4	2	no	—
118	#11929609	D₈	1	1	yes	characteristic
119	#11929608	—	2	2	no	—
120	#11929608	—	4	2	no	—
121	#11929608	C₂×D₄	1	1	yes	characteristic
122	#11929607	—	2	1	no	—
123	C₂×C₇₉⋊C₈	—	4	2	no	—
124	C₂×C₇₉⋊C₈	C₂×D₄	1	1	yes	characteristic
125	C₈×D₇₉	—	2	1	no	—
126	(C₂×D₇₉)⋊C₄	—	8	2	no	—
127	C₄×Dic₇₉	C₂×D₄	1	1	yes	characteristic
128	C₇₉⋊C₁₆	—	2	1	no	—
129	C₇₉⋊C₁₆	—	2	1	no	—
130	#41×C₇₉	—	2	2	no	—
131	#41×C₇₉	—	4	2	no	—
132	#41×C₇₉	—	4	2	no	—
133	#41×C₇₉	—	8	2	no	—
134	#41×C₇₉	C₂×D₄	1	1	yes	characteristic
135	C₂×Q₈×C₇₉	—	2	2	no	—
136	C₂×Q₈×C₇₉	—	4	2	no	—
137	C₂×Q₈×C₇₉	—	4	2	no	—
138	Q₁₆×C₇₉	—	2	2	no	—
139	Q₁₆×C₇₉	—	2	2	no	—
140	Q₁₆×C₇₉	—	4	2	no	—
141	Q₁₆×C₇₉	C₂×D₄	1	2	yes	—
142	SD₁₆×C₇₉	—	2	2	no	—
143	SD₁₆×C₇₉	—	4	2	no	—
144	D₈×C₇₉	C₂×D₄	1	2	yes	—
145	M₁₆×C₇₉	—	2	1	no	—
146	M₁₆×C₇₉	C₂×D₄	1	1	yes	characteristic
147	C₂×C₈×C₇₉	—	2	1	no	—
148	C₂×C₈×C₇₉	C₂⁴	1	1	yes	characteristic
149	C₁₆×C₇₉	—	2	1	no	—
150	C₁₆×C₇₉	C₂×D₄	1	1	yes	characteristic
151	C₁₆×C₇₉	C₂×D₄	1	1	yes	characteristic
152	C₂²×D₇₉	—	4	2	no	—
153	C₂²×D₇₉	—	8	2	no	—
154	Q₈×C₇₉	—	2	2	no	—
155	Q₈×C₇₉	—	4	2	no	—
156	Q₈×C₇₉	—	4	2	no	—
157	Q₈×C₇₉	—	8	2	no	—
158	Q₈×C₇₉	C₂×D₈	1	1	yes	characteristic
159	D₄×C₇₉	—	2	1	no	—
160	D₄×C₇₉	—	2	2	no	—
161	D₄×C₇₉	—	4	2	no	—
162	D₄×C₇₉	—	8	2	no	—
163	D₄×C₇₉	C₂×D₈	1	1	yes	characteristic
164	C₂×C₄×C₇₉	—	2	1	no	—
165	C₂×C₄×C₇₉	—	4	2	no	—
166	C₂×C₄×C₇₉	—	4	2	no	—
167	C₂×C₄×C₇₉	—	8	2	no	—
168	C₂×C₄×C₇₉	—	8	2	no	—
169	C₂×C₄×C₇₉	C₂²×D₄	1	1	yes	characteristic
170	C₂²⋊D₇₉	—	2	1	no	—
171	C₂²⋊D₇₉	—	8	2	no	—
172	C₂×Dic₇₉	—	2	1	no	—
173	D₃₁₆	—	4	2	no	—
174	D₃₁₆	—	4	2	no	—
175	D₃₁₆	—	8	2	no	—
176	D₃₁₆	C₂×D₈	1	1	yes	characteristic
177	C₄×D₇₉	—	2	1	no	—
178	C₄×D₇₉	—	8	2	no	—
179	Dic₁₅₈	C₂×D₈	1	1	yes	characteristic
180	C₈×C₇₉	—	2	1	no	—
181	C₈×C₇₉	C₂²×D₄	1	1	yes	characteristic
182	C₈×C₇₉	H(5,𝔽₂)	1	1	yes	characteristic
183	C₇₉⋊C₈	—	2	1	no	—
184	C₇₉⋊C₈	—	4	2	no	—
185	C₇₉⋊C₈	C₂×D₈	1	1	yes	characteristic
186	C₇₉⋊C₈	C₂×D₈	1	1	yes	characteristic
187	C₂²×C₇₉	—	2	1	no	—
188	C₂²×C₇₉	—	4	2	no	—
189	C₂²×C₇₉	#576	1	1	yes	characteristic
190	C₂×D₇₉	—	2	1	no	—
191	C₂×D₇₉	—	8	2	no	—
192	C₂×D₇₉	—	16	2	no	—
193	C₂×D₇₉	—	32	2	no	—
194	C₄×C₇₉	—	2	1	no	—
195	C₄×C₇₉	—	4	2	no	—
196	C₄×C₇₉	—	8	2	no	—
197	C₄×C₇₉	D₄²	1	1	yes	characteristic
198	C₄×C₇₉	C₂²×D₈	1	1	yes	characteristic
199	Dic₇₉	—	2	1	no	—
200	Dic₇₉	—	4	1	no	—
201	#33859	—	79	1	no	maximal, Sylow
202	C₂×C₇₉	—	2	1	no	—
203	C₂×C₇₉	—	4	1	no	—
204	C₂×C₇₉	—	8	2	no	—
205	C₂×C₇₉	D₄×D₈	1	1	yes	characteristic
206	D₇₉	—	4	1	no	—
207	D₇₉	—	16	2	no	—
208	#3417	D₇₉	1	1	yes	characteristic, complemented
209	#3415	—	79	1	no	—
210	#3290	—	79	2	no	—
211	#2249	—	79	1	no	—
212	#2222	—	79	1	no	—
213	#2221	—	79	1	no	—
214	#2212	—	79	2	no	—
215	#2194	—	79	2	no	—
216	#2193	—	79	2	no	—
217	#2178	—	79	1	no	—
218	#2170	—	79	1	no	—
219	C₇₉	#33859	1	1	yes	characteristic, complemented, Sylow
220	#578	C₂×D₇₉	1	2	yes	complemented
221	#577	—	158	2	no	—
222	#576	—	79	2	no	—
223	#510	—	2	2	no	—
224	#510	C₂×D₇₉	1	2	yes	complemented
225	#509	—	79	2	no	—
226	#509	—	158	2	no	—
227	#508	—	2	1	no	—
228	#508	—	158	1	no	—
229	#508	C₂×D₇₉	1	1	yes	characteristic
230	C₂×Q₃₂	—	2	1	no	—
231	C₂×Q₃₂	C₂×D₇₉	1	1	yes	characteristic, complemented
232	C₂×SD₃₂	—	79	2	no	—
233	C₂×D₁₆	—	79	1	no	—
234	C₂×D₁₆	—	158	1	no	—
235	#504	—	79	1	no	—
236	#504	C₂×D₇₉	1	1	yes	characteristic, complemented
237	#495	—	79	2	no	—
238	#473	—	79	2	no	—
239	#472	—	79	2	no	—
240	#471	—	158	2	no	—
241	#455	—	158	2	no	—
242	#454	—	158	2	no	—
243	#443	—	79	1	no	—
244	#443	—	79	2	no	—
245	#368	—	79	1	no	—
246	#364	—	79	1	no	—
247	#362	—	79	2	no	—
248	#361	—	79	2	no	—
249	#359	—	79	2	no	—
250	#350	—	79	1	no	—
251	#350	—	79	1	no	—
252	C₄×C₁₆	—	79	1	no	—
253	2₋¹⁺⁴	—	2	2	no	—
254	H(5,𝔽₂)	—	158	2	no	—
255	#137	—	2	2	no	—
256	#137	—	4	2	no	—
257	#137	—	316	2	no	—
258	#137	—	158	2	no	—
259	AGL(1,ℤ/8)	—	316	2	no	—
260	AGL(1,ℤ/8)	—	316	2	no	—
261	AGL(1,ℤ/8)	—	158	2	no	—
262	AGL(1,ℤ/8)	—	158	2	no	—
263	#135	—	2	2	no	—
264	#135	—	316	2	no	—
265	#135	—	79	1	no	—
266	#135	—	158	2	no	—
267	#135	C₂²×D₇₉	1	2	yes	—
268	C₂×Q₁₆	—	2	2	no	—
269	C₂×Q₁₆	C₂²×D₇₉	1	2	yes	—
270	C₂×SD₁₆	—	316	2	no	—
271	C₂×SD₁₆	—	158	2	no	—
272	C₂×D₈	—	79	2	no	—
273	C₂×D₈	—	158	2	no	—
274	#131	—	79	1	no	—
275	#131	—	158	2	no	—
276	#131	C₂²×D₇₉	1	1	yes	characteristic
277	#124	—	158	2	no	—
278	Q₃₂	—	2	2	no	—
279	Q₃₂	—	2	1	no	—
280	Q₃₂	—	2	1	no	—
281	Q₃₂	C₂²⋊D₇₉	1	1	yes	characteristic
282	SD₃₂	—	2	2	no	—
283	SD₃₂	—	316	2	no	—
284	SD₃₂	—	158	2	no	—
285	SD₃₂	C₂²⋊D₇₉	1	2	yes	—
286	D₁₆	—	158	2	no	—
287	D₁₆	—	158	1	no	—
288	D₁₆	—	158	1	no	—
289	D₁₆	C₂²⋊D₇₉	1	1	yes	characteristic
290	M₃₂	—	2	1	no	—
291	M₃₂	—	79	1	no	—
292	M₃₂	—	158	1	no	—
293	M₃₂	C₂²×D₇₉	1	1	yes	characteristic
294	C₂×C₁₆	—	2	1	no	—
295	C₂×C₁₆	—	79	1	no	—
296	C₂×C₁₆	—	158	1	no	—
297	C₂×C₁₆	C₂²×D₇₉	1	1	yes	characteristic
298	#108	—	79	1	no	—
299	#108	—	79	2	no	—
300	#104	—	79	1	no	—
301	#104	—	79	1	no	—
302	#104	—	158	2	no	—
303	#101	—	158	2	no	—
304	#100	—	158	2	no	—
305	C₄×C₈	—	79	1	no	—
306	#41	—	2	2	no	—
307	#41	—	4	2	no	—
308	#41	—	4	2	no	—
309	#41	—	8	2	no	—
310	#41	—	316	2	no	—
311	#41	—	316	2	no	—
312	#41	—	79	1	no	—
313	#41	D₄×D₇₉	1	1	yes	characteristic
314	C₂×Q₈	—	2	2	no	—
315	C₂×Q₈	—	4	2	no	—
316	C₂×Q₈	—	4	2	no	—
317	C₂×D₄	—	316	2	no	—
318	C₂×D₄	—	316	2	no	—
319	C₂×D₄	—	632	2	no	—
320	C₂×D₄	—	158	2	no	—
321	Q₁₆	—	2	2	no	—
322	Q₁₆	—	2	2	no	—
323	Q₁₆	—	4	2	no	—
324	Q₁₆	—	316	2	no	—
325	Q₁₆	—	79	1	no	—
326	Q₁₆	C₂×C₂²⋊D₇₉	1	2	yes	—
327	SD₁₆	—	2	2	no	—
328	SD₁₆	—	4	2	no	—
329	SD₁₆	—	316	2	no	—
330	SD₁₆	—	316	2	no	—
331	SD₁₆	—	316	2	no	—
332	SD₁₆	—	316	2	no	—
333	SD₁₆	—	316	2	no	—
334	SD₁₆	—	79	1	no	—
335	SD₁₆	—	79	1	no	—
336	D₈	—	316	2	no	—
337	D₈	—	316	2	no	—
338	D₈	—	79	1	no	—
339	D₈	—	158	2	no	—
340	D₈	—	158	2	no	—
341	D₈	C₂×C₂²⋊D₇₉	1	2	yes	—
342	M₁₆	—	2	1	no	—
343	M₁₆	—	316	2	no	—
344	M₁₆	—	79	1	no	—
345	M₁₆	—	158	2	no	—
346	M₁₆	—	158	1	no	—
347	M₁₆	D₄×D₇₉	1	1	yes	characteristic
348	C₂×C₈	—	2	1	no	—
349	C₂×C₈	—	316	2	no	—
350	C₂×C₈	—	79	1	no	—
351	C₂×C₈	—	158	1	no	—
352	C₂×C₈	C₂³×D₇₉	1	1	yes	characteristic
353	C₂²⋊C₄	—	632	2	no	—
354	C₄²	—	79	1	no	—
355	C₁₆	—	2	1	no	—
356	C₁₆	—	158	1	no	—
357	C₁₆	—	158	1	no	—
358	C₁₆	C₂×C₂²⋊D₇₉	1	1	yes	characteristic
359	C₁₆	C₂×C₂²⋊D₇₉	1	1	yes	characteristic
360	C₂³	—	316	2	no	—
361	C₂³	—	632	2	no	—
362	Q₈	—	2	2	no	—
363	Q₈	—	4	2	no	—
364	Q₈	—	4	2	no	—
365	Q₈	—	8	2	no	—
366	Q₈	—	79	1	no	—
367	Q₈	D₈×D₇₉	1	1	yes	characteristic
368	D₄	—	2	1	no	—
369	D₄	—	2	2	no	—
370	D₄	—	4	2	no	—
371	D₄	—	8	2	no	—
372	D₄	—	316	2	no	—
373	D₄	—	316	2	no	—
374	D₄	—	79	1	no	—
375	D₄	—	632	2	no	—
376	D₄	—	632	2	no	—
377	D₄	—	158	1	no	—
378	D₄	D₈×D₇₉	1	1	yes	characteristic
379	C₂×C₄	—	2	1	no	—
380	C₂×C₄	—	4	2	no	—
381	C₂×C₄	—	4	2	no	—
382	C₂×C₄	—	8	2	no	—
383	C₂×C₄	—	8	2	no	—
384	C₂×C₄	—	632	2	no	—
385	C₂×C₄	—	158	1	no	—
386	C₂×C₄	—	158	1	no	—
387	C₂×C₄	C₂×D₄×D₇₉	1	1	yes	characteristic
388	C₈	—	2	1	no	—
389	C₈	—	316	2	no	—
390	C₈	—	79	1	no	—
391	C₈	—	79	1	no	—
392	C₈	—	158	1	no	—
393	C₈	C₂²×C₂²⋊D₇₉	1	1	yes	characteristic
394	C₈	#14579206	1	1	yes	characteristic
395	C₂²	—	2	1	no	—
396	C₂²	—	4	2	no	—
397	C₂²	—	632	2	no	—
398	C₂²	—	158	1	no	—
399	C₂²	—	2528	2	no	—
400	C₂²	—	1264	2	no	—
401	C₂²	#19099364	1	1	yes	characteristic
402	C₄	—	2	1	no	—
403	C₄	—	4	2	no	—
404	C₄	—	8	2	no	—
405	C₄	—	316	1	no	—
406	C₄	—	158	1	no	—
407	C₄	D₄×C₂²⋊D₇₉	1	1	yes	characteristic
408	C₄	C₂×D₈×D₇₉	1	1	yes	characteristic
409	C₂	—	2	1	no	—
410	C₂	—	4	1	no	—
411	C₂	—	8	2	no	—
412	C₂	—	316	1	no	—
413	C₂	—	1264	2	no	—
414	C₂	D₈×C₂²⋊D₇₉	1	1	yes	characteristic

	subgroup	G/H
center	C₂	D₈×C₂²⋊D₇₉
commutator, cosocle	C₂×C₈×C₇₉	C₂⁴
Frattini	C₂×C₈	C₂³×D₇₉
Fitting	C₇₉×#3417	C₂
socle	C₂×C₇₉	D₄×D₈
hypercenter	#3417	D₇₉
hypocenter	C₇₉	#33859
LCS	G ▷ C₂×C₈×C₇₉ ▷ C₄×C₇₉ ▷ C₂×C₇₉ ▷ C₇₉
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ #131 ◁ #3417

Complements

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

N	H ≅ G/N	type
1. #23776538	C₂	semidirect
2. #23775186	C₂	semidirect
3. #23765662	C₂	semidirect
4. #23765522	C₂	semidirect
5. #23765520	C₂	semidirect
6. #23765471	C₂	semidirect
7. #23765368	C₂	semidirect
8. #23765364	C₂	semidirect
9. #23765281	C₂	semidirect
10. #23765235	C₂	semidirect
11. C₇₉×#3417	C₂	semidirect
17. C₂×#14579104	C₂²	semidirect
21. #19098948	C₂²	semidirect
23. #19098833	C₂²	semidirect
27. #19098680	C₂²	semidirect
29. #19098393	C₂²	semidirect
30. #19098383	C₂²	semidirect
31. #19098379	C₂²	semidirect
32. #19098377	C₂²	semidirect
33. #19098370	C₂²	semidirect
35. #19098347	C₂²	semidirect
36. C₁₆×Dic₇₉	C₂²	semidirect
37. #578×C₇₉	C₂²	semidirect
39. #510×C₇₉	C₂²	semidirect
43. C₂×Q₃₂×C₇₉	C₂²	semidirect
44. #504×C₇₉	C₂²	semidirect
208. #3417	D₇₉	semidirect
219. C₇₉	#33859	semidirect
220. #578	C₂×D₇₉	semidirect
224. #510	C₂×D₇₉	semidirect
231. C₂×Q₃₂	C₂×D₇₉	semidirect
236. #504	C₂×D₇₉	semidirect

Maximal subgroups

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

subgroup	index
#23776538	2
#23775186	2
#23765662	2
#23765522	2
#23765520	2
#23765471	2
#23765368	2
#23765364	2
#23765281	2
#23765235	2
C₇₉×#3417	2
#33859	79