Subgroups — #34445079

Order 20224.

636 conjugacy classes of subgroups.

The subgroup lattice has 422 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#23776552	C₂	1	2	yes	maximal, complemented
2	#23775205	C₂	1	1	yes	maximal, characteristic, complemented
3	#23775204	C₂	1	1	yes	maximal, characteristic, complemented
4	#23765652	C₂	1	1	yes	maximal, characteristic, complemented
5	#23765524	C₂	1	2	yes	maximal, complemented
6	#23765482	C₂	1	1	yes	maximal, characteristic, complemented
7	#23765381	C₂	1	2	yes	maximal, complemented
8	#23765375	C₂	1	2	yes	maximal, complemented
9	#23765279	C₂	1	1	yes	maximal, characteristic, complemented
10	#23765236	C₂	1	1	yes	maximal, characteristic, complemented
11	C₇₉×#2214	C₂	1	1	yes	maximal, characteristic, complemented
12	#19099380	—	2	1	no	—
13	#19099377	C₂²	1	2	yes	—
14	#19099376	—	2	1	no	—
15	#19099376	C₂²	1	2	yes	—
16	#19099006	C₂²	1	2	yes	—
17	#19099005	—	2	2	no	—
18	#19099002	—	2	2	no	—
19	#19098994	—	2	2	no	—
20	#19098993	C₂²	1	1	yes	characteristic
21	C₂×#14579107	C₂²	1	2	yes	—
22	C₂×#14579107	C₂²	1	2	yes	—
23	Q₃₂×D₇₉	—	2	2	no	—
24	#19098981	C₂²	1	2	yes	—
25	#19098943	C₂²	1	1	yes	characteristic
26	#19098839	C₂²	1	2	yes	—
27	#19098838	—	2	1	no	—
28	#19098835	—	2	1	no	—
29	#19098835	C₂²	1	2	yes	—
30	#19098747	—	2	2	no	—
31	#19098745	—	2	2	no	—
32	#19098681	C₂²	1	1	yes	characteristic
33	#19098681	C₂²	1	1	yes	characteristic
34	#19098392	C₂²	1	1	yes	characteristic
35	#19098380	C₂²	1	2	yes	—
36	#19098370	C₂²	1	1	yes	characteristic
37	#19098369	C₂²	1	1	yes	characteristic
38	#19098348	C₂²	1	2	yes	—
39	C₄×C₇₉⋊C₁₆	C₂²	1	1	yes	characteristic
40	#510×C₇₉	C₂²	1	2	yes	—
41	#494×C₇₉	C₂²	1	1	yes	characteristic
42	#443×C₇₉	C₂²	1	1	yes	characteristic
43	#364×C₇₉	C₂²	1	1	yes	characteristic
44	#362×C₇₉	C₂²	1	2	yes	—
45	#14579209	—	2	2	no	—
46	#14579208	—	2	2	no	—
47	#14579194	—	2	2	no	—
48	#14579194	—	4	2	no	—
49	#14579193	—	4	2	no	—
50	#14579193	—	4	2	no	—
51	#14579192	—	2	2	no	—
52	#14579192	—	2	2	no	—
53	#14579191	—	2	2	no	—
54	#14579190	—	4	2	no	—
55	#14579184	—	2	2	no	—
56	#14579184	—	4	1	no	—
57	#14579183	—	4	1	no	—
58	#14579183	D₄	1	1	yes	characteristic
59	#14579183	C₂³	1	1	yes	characteristic
60	#14579180	—	2	2	no	—
61	C₂×Dic₃₁₆	—	2	1	no	—
62	C₂×Dic₃₁₆	—	4	1	no	—
63	C₂×Dic₃₁₆	C₂³	1	2	yes	—
64	Q₁₆×D₇₉	—	2	2	no	—
65	Q₁₆×D₇₉	—	2	2	no	—
66	Q₁₆×D₇₉	—	4	1	no	—
67	#14579167	—	2	1	no	—
68	#14579167	C₂³	1	2	yes	—
69	#14579161	—	2	1	no	—
70	Dic₆₃₂	—	2	2	no	—
71	#14579107	—	2	2	no	—
72	#14579107	—	2	2	no	—
73	#14579107	—	4	2	no	—
74	#14579107	D₄	1	1	yes	characteristic
75	#14579106	—	2	2	no	—
76	#14579105	D₄	1	1	yes	characteristic
77	#14579104	D₄	1	1	yes	characteristic
78	#14579102	D₄	1	1	yes	characteristic
79	#14579100	—	2	2	no	—
80	C₂×C₇₉⋊C₁₆	C₂³	1	1	yes	characteristic
81	C₂×C₇₉⋊C₁₆	C₂³	1	1	yes	characteristic
82	C₁₆×D₇₉	—	2	2	no	—
83	#14579097	D₄	1	1	yes	characteristic
84	#14579097	C₂³	1	1	yes	characteristic
85	#14579089	—	2	1	no	—
86	#14579089	D₄	1	2	yes	—
87	#14579080	—	2	2	no	—
88	#14579079	—	2	2	no	—
89	#135×C₇₉	C₂³	1	1	yes	characteristic
90	C₂×Q₁₆×C₇₉	—	2	1	no	—
91	C₂×Q₁₆×C₇₉	C₂³	1	2	yes	—
92	#131×C₇₉	—	2	1	no	—
93	C₄⋊Q₈×C₇₉	—	2	1	no	—
94	Q₃₂×C₇₉	—	2	2	no	—
95	SD₃₂×C₇₉	—	2	2	no	—
96	M₃₂×C₇₉	C₂³	1	2	yes	—
97	#108×C₇₉	C₂³	1	1	yes	characteristic
98	#104×C₇₉	—	2	1	no	—
99	C₄×C₈×C₇₉	C₂³	1	1	yes	characteristic
100	#11929625	—	4	2	no	—
101	#11929625	—	4	2	no	—
102	#11929624	—	2	1	no	—
103	#11929624	—	4	2	no	—
104	#11929624	—	8	2	no	—
105	#11929624	C₂×D₄	1	2	yes	—
106	#11929623	—	4	2	no	—
107	#11929623	—	8	2	no	—
108	C₂×Dic₁₅₈	—	2	2	no	—
109	C₂×Dic₁₅₈	—	4	2	no	—
110	Q₈×D₇₉	—	4	2	no	—
111	Q₈×D₇₉	—	4	2	no	—
112	Q₈×D₇₉	—	4	2	no	—
113	Dic₃₁₆	—	2	2	no	—
114	Dic₃₁₆	—	2	2	no	—
115	Dic₃₁₆	—	4	2	no	—
116	Dic₃₁₆	—	4	1	no	—
117	Dic₃₁₆	—	4	1	no	—
118	Dic₃₁₆	D₈	1	1	yes	characteristic
119	Dic₃₁₆	C₂×D₄	1	1	yes	characteristic
120	#11929614	—	4	2	no	—
121	#11929614	—	4	2	no	—
122	#11929614	—	4	2	no	—
123	#11929613	—	2	1	no	—
124	#11929613	—	4	2	no	—
125	#11929613	D₈	1	2	yes	—
126	#11929612	—	4	2	no	—
127	#11929611	—	4	2	no	—
128	#11929611	—	4	2	no	—
129	D₆₃₂	D₈	1	1	yes	characteristic
130	D₆₃₂	C₂×D₄	1	1	yes	characteristic
131	#11929608	—	2	1	no	—
132	#11929608	C₂×D₄	1	2	yes	—
133	#11929607	—	2	2	no	—
134	#11929607	—	4	1	no	—
135	C₈×D₇₉	—	2	2	no	—
136	C₈×D₇₉	—	4	1	no	—
137	C₄⋊Dic₇₉	—	8	1	no	—
138	C₇₉⋊C₁₆	—	2	1	no	—
139	C₇₉⋊C₁₆	C₂×D₄	1	1	yes	characteristic
140	C₇₉⋊C₁₆	C₂×D₄	1	1	yes	characteristic
141	#41×C₇₉	—	2	1	no	—
142	C₂×Q₈×C₇₉	—	2	2	no	—
143	Q₁₆×C₇₉	—	2	2	no	—
144	Q₁₆×C₇₉	—	2	2	no	—
145	Q₁₆×C₇₉	—	4	1	no	—
146	Q₁₆×C₇₉	—	4	1	no	—
147	Q₁₆×C₇₉	C₂×D₄	1	1	yes	characteristic
148	SD₁₆×C₇₉	—	2	1	no	—
149	D₈×C₇₉	C₂×D₄	1	1	yes	characteristic
150	M₁₆×C₇₉	—	2	1	no	—
151	M₁₆×C₇₉	—	4	1	no	—
152	C₂×C₈×C₇₉	—	4	1	no	—
153	C₂×C₈×C₇₉	C₂×D₄	1	1	yes	characteristic
154	C₂×C₈×C₇₉	C₂⁴	1	1	yes	characteristic
155	C₄⋊C₄×C₇₉	—	8	1	no	—
156	C₄²×C₇₉	C₂×D₄	1	1	yes	characteristic
157	C₁₆×C₇₉	—	2	2	no	—
158	Q₈×C₇₉	—	2	1	no	—
159	Q₈×C₇₉	—	4	2	no	—
160	Q₈×C₇₉	—	4	2	no	—
161	D₄×C₇₉	—	2	1	no	—
162	D₄×C₇₉	—	4	1	no	—
163	C₂×C₄×C₇₉	—	2	1	no	—
164	C₂×C₄×C₇₉	—	4	1	no	—
165	C₂×C₄×C₇₉	—	4	2	no	—
166	C₂×C₄×C₇₉	C₂²×D₄	1	1	yes	characteristic
167	C₂²⋊D₇₉	—	2	2	no	—
168	C₂²⋊D₇₉	—	4	1	no	—
169	C₂²⋊D₇₉	—	8	2	no	—
170	C₂×Dic₇₉	—	4	2	no	—
171	C₂×Dic₇₉	—	8	2	no	—
172	D₃₁₆	—	2	1	no	—
173	D₃₁₆	—	8	2	no	—
174	D₃₁₆	C₂×D₈	1	2	yes	—
175	C₄×D₇₉	—	2	2	no	—
176	C₄×D₇₉	—	4	1	no	—
177	C₄×D₇₉	—	8	2	no	—
178	C₄×D₇₉	—	8	2	no	—
179	C₄×D₇₉	—	8	2	no	—
180	Dic₁₅₈	—	2	1	no	—
181	Dic₁₅₈	—	4	2	no	—
182	Dic₁₅₈	—	4	2	no	—
183	Dic₁₅₈	—	8	2	no	—
184	Dic₁₅₈	—	8	2	no	—
185	Dic₁₅₈	C₂×D₈	1	2	yes	—
186	C₈×C₇₉	—	4	1	no	—
187	C₈×C₇₉	C₂×D₈	1	1	yes	characteristic
188	C₈×C₇₉	C₂×D₈	1	1	yes	characteristic
189	C₈×C₇₉	C₂²×D₄	1	1	yes	characteristic
190	C₈×C₇₉	H(5,𝔽₂)	1	1	yes	characteristic
191	C₇₉⋊C₈	—	2	2	no	—
192	C₇₉⋊C₈	—	4	1	no	—
193	C₂²×C₇₉	—	4	1	no	—
194	C₂²×C₇₉	#576	1	1	yes	characteristic
195	C₂×D₇₉	—	2	2	no	—
196	C₂×D₇₉	—	4	1	no	—
197	C₄×C₇₉	—	2	2	no	—
198	C₄×C₇₉	—	4	1	no	—
199	C₄×C₇₉	—	8	2	no	—
200	C₄×C₇₉	D₄²	1	1	yes	characteristic
201	C₄×C₇₉	C₂²×D₈	1	1	yes	characteristic
202	Dic₇₉	—	2	2	no	—
203	Dic₇₉	—	4	1	no	—
204	Dic₇₉	—	8	2	no	—
205	#33860	—	79	1	no	maximal, Sylow
206	C₂×C₇₉	—	2	1	no	—
207	C₂×C₇₉	—	8	1	no	—
208	C₂×C₇₉	D₄×D₈	1	1	yes	characteristic
209	D₇₉	—	4	2	no	—
210	D₇₉	—	8	1	no	—
211	#3417	—	79	2	no	—
212	#3293	—	79	1	no	—
213	#3293	—	79	1	no	—
214	#2247	—	79	1	no	—
215	#2222	—	79	2	no	—
216	#2214	—	79	1	no	—
217	#2214	D₇₉	1	1	yes	characteristic, complemented
218	#2195	—	79	2	no	—
219	#2195	—	79	2	no	—
220	#2178	—	79	1	no	—
221	#2170	—	79	1	no	—
222	C₇₉	#33860	1	1	yes	characteristic, complemented, Sylow
223	#578	—	79	2	no	—
224	#578	—	79	2	no	—
225	#578	—	158	1	no	—
226	#578	—	158	1	no	—
227	#510	—	79	2	no	—
228	#510	—	158	2	no	—
229	#510	—	158	2	no	—
230	#510	C₂×D₇₉	1	2	yes	—
231	#508	—	79	1	no	—
232	#508	—	158	2	no	—
233	C₂×Q₃₂	—	79	2	no	—
234	C₂×Q₃₂	—	79	2	no	—
235	C₂×Q₃₂	—	158	2	no	—
236	#504	—	79	2	no	—
237	#494	—	79	1	no	—
238	#494	C₂×D₇₉	1	1	yes	characteristic
239	#473	—	79	2	no	—
240	#473	—	79	2	no	—
241	#473	—	158	1	no	—
242	#473	—	158	1	no	—
243	#456	—	158	2	no	—
244	#456	—	158	2	no	—
245	#443	—	79	1	no	—
246	#443	—	79	1	no	—
247	#443	C₂×D₇₉	1	1	yes	characteristic
248	#368	—	79	1	no	—
249	#364	C₂×D₇₉	1	1	yes	characteristic
250	#362	—	79	2	no	—
251	#362	C₂×D₇₉	1	2	yes	—
252	#359	—	79	1	no	—
253	#359	—	79	1	no	—
254	#350	—	79	2	no	—
255	C₄×C₁₆	—	79	1	no	—
256	2₋¹⁺⁴	—	158	2	no	—
257	2₋¹⁺⁴	—	158	2	no	—
258	#137	—	316	2	no	—
259	#137	—	316	2	no	—
260	#137	—	316	2	no	—
261	#137	—	316	2	no	—
262	#137	—	158	2	no	—
263	#137	—	158	2	no	—
264	#137	—	158	2	no	—
265	#137	—	158	2	no	—
266	#135	—	316	1	no	—
267	#135	—	316	1	no	—
268	#135	—	79	1	no	—
269	#135	—	79	1	no	—
270	#135	—	158	2	no	—
271	#135	—	158	2	no	—
272	#135	C₂²×D₇₉	1	1	yes	characteristic
273	C₂×Q₁₆	—	2	1	no	—
274	C₂×Q₁₆	—	316	1	no	—
275	C₂×Q₁₆	—	316	1	no	—
276	C₂×Q₁₆	—	79	2	no	—
277	C₂×Q₁₆	—	158	2	no	—
278	C₂×Q₁₆	—	158	2	no	—
279	C₂×Q₁₆	—	158	1	no	—
280	C₂×Q₁₆	C₂²×D₇₉	1	2	yes	—
281	#131	—	2	1	no	—
282	#131	—	79	2	no	—
283	#131	—	158	1	no	—
284	C₄⋊Q₈	—	2	1	no	—
285	C₄⋊Q₈	—	158	1	no	—
286	Q₃₂	—	2	2	no	—
287	Q₃₂	—	316	2	no	—
288	Q₃₂	—	79	1	no	—
289	Q₃₂	—	158	2	no	—
290	Q₃₂	—	158	2	no	—
291	Q₃₂	—	158	2	no	—
292	SD₃₂	—	2	2	no	—
293	SD₃₂	—	79	1	no	—
294	SD₃₂	—	79	1	no	—
295	SD₃₂	—	158	2	no	—
296	D₁₆	—	79	1	no	—
297	M₃₂	—	158	2	no	—
298	M₃₂	C₂²×D₇₉	1	2	yes	—
299	C₂×C₁₆	—	79	1	no	—
300	C₂×C₁₆	—	79	1	no	—
301	C₂×C₁₆	—	158	2	no	—
302	#108	—	79	1	no	—
303	#108	—	79	1	no	—
304	#108	C₂²×D₇₉	1	1	yes	characteristic
305	#104	—	2	1	no	—
306	#104	—	79	2	no	—
307	#104	—	158	1	no	—
308	#101	—	158	2	no	—
309	#101	—	158	2	no	—
310	C₄×C₈	C₂²×D₇₉	1	1	yes	characteristic
311	#41	—	2	1	no	—
312	#41	—	316	2	no	—
313	#41	—	316	2	no	—
314	#41	—	316	2	no	—
315	#41	—	316	2	no	—
316	#41	—	79	2	no	—
317	#41	—	632	2	no	—
318	#41	—	632	2	no	—
319	#41	—	158	1	no	—
320	C₂×Q₈	—	2	2	no	—
321	C₂×Q₈	—	316	2	no	—
322	C₂×Q₈	—	316	2	no	—
323	C₂×Q₈	—	316	2	no	—
324	C₂×Q₈	—	316	2	no	—
325	C₂×Q₈	—	158	2	no	—
326	Q₁₆	—	2	2	no	—
327	Q₁₆	—	2	2	no	—
328	Q₁₆	—	4	1	no	—
329	Q₁₆	—	4	1	no	—
330	Q₁₆	—	316	2	no	—
331	Q₁₆	—	316	2	no	—
332	Q₁₆	—	316	2	no	—
333	Q₁₆	—	316	2	no	—
334	Q₁₆	—	316	1	no	—
335	Q₁₆	—	316	1	no	—
336	Q₁₆	—	79	1	no	—
337	Q₁₆	—	79	1	no	—
338	Q₁₆	—	158	2	no	—
339	Q₁₆	—	158	2	no	—
340	Q₁₆	D₄×D₇₉	1	1	yes	characteristic
341	SD₁₆	—	2	1	no	—
342	SD₁₆	—	316	2	no	—
343	SD₁₆	—	316	2	no	—
344	SD₁₆	—	316	2	no	—
345	SD₁₆	—	316	2	no	—
346	SD₁₆	—	79	2	no	—
347	SD₁₆	—	158	1	no	—
348	D₈	—	79	1	no	—
349	D₈	—	79	1	no	—
350	D₈	D₄×D₇₉	1	1	yes	characteristic
351	M₁₆	—	2	1	no	—
352	M₁₆	—	4	1	no	—
353	M₁₆	—	316	1	no	—
354	M₁₆	—	79	2	no	—
355	M₁₆	—	158	1	no	—
356	M₁₆	—	158	2	no	—
357	C₂×C₈	—	4	1	no	—
358	C₂×C₈	—	316	1	no	—
359	C₂×C₈	—	158	2	no	—
360	C₂×C₈	D₄×D₇₉	1	1	yes	characteristic
361	C₂×C₈	C₂³×D₇₉	1	1	yes	characteristic
362	C₄⋊C₄	—	8	1	no	—
363	C₄⋊C₄	—	632	1	no	—
364	C₄²	D₄×D₇₉	1	1	yes	characteristic
365	C₁₆	—	2	2	no	—
366	C₁₆	—	79	1	no	—
367	C₁₆	—	79	1	no	—
368	C₁₆	—	158	1	no	—
369	Q₈	—	2	1	no	—
370	Q₈	—	4	2	no	—
371	Q₈	—	4	2	no	—
372	Q₈	—	316	2	no	—
373	Q₈	—	316	2	no	—
374	Q₈	—	79	2	no	—
375	Q₈	—	632	2	no	—
376	Q₈	—	632	2	no	—
377	Q₈	—	158	1	no	—
378	D₄	—	2	1	no	—
379	D₄	—	4	1	no	—
380	D₄	—	316	1	no	—
381	D₄	—	79	2	no	—
382	D₄	—	632	2	no	—
383	D₄	—	632	2	no	—
384	D₄	—	158	2	no	—
385	D₄	—	158	1	no	—
386	C₂×C₄	—	2	1	no	—
387	C₂×C₄	—	4	1	no	—
388	C₂×C₄	—	4	2	no	—
389	C₂×C₄	—	316	1	no	—
390	C₂×C₄	—	316	2	no	—
391	C₂×C₄	—	632	2	no	—
392	C₂×C₄	—	632	2	no	—
393	C₂×C₄	—	632	2	no	—
394	C₂×C₄	—	632	2	no	—
395	C₂×C₄	—	158	2	no	—
396	C₂×C₄	C₂×D₄×D₇₉	1	1	yes	characteristic
397	C₈	—	4	1	no	—
398	C₈	—	316	1	no	—
399	C₈	—	158	2	no	—
400	C₈	D₈×D₇₉	1	1	yes	characteristic
401	C₈	D₈×D₇₉	1	1	yes	characteristic
402	C₈	C₂×D₄×D₇₉	1	1	yes	characteristic
403	C₈	#14579207	1	1	yes	characteristic
404	C₂²	—	4	1	no	—
405	C₂²	—	316	1	no	—
406	C₂²	—	158	2	no	—
407	C₂²	#19099367	1	1	yes	characteristic
408	C₄	—	2	2	no	—
409	C₄	—	4	1	no	—
410	C₄	—	8	2	no	—
411	C₄	—	316	1	no	—
412	C₄	—	632	2	no	—
413	C₄	—	158	2	no	—
414	C₄	#19099197	1	1	yes	characteristic
415	C₄	C₂×D₈×D₇₉	1	1	yes	characteristic
416	C₂	—	2	1	no	—
417	C₂	—	8	1	no	—
418	C₂	—	316	2	no	—
419	C₂	—	632	1	no	—
420	C₂	#23775072	1	1	yes	characteristic

	subgroup	G/H
center	C₂	#23775072
commutator, cosocle	C₂×C₈×C₇₉	C₂⁴
Frattini	C₂×C₈	C₂³×D₇₉
Fitting	C₇₉×#2214	C₂
socle	C₂×C₇₉	D₄×D₈
hypercenter	#2214	D₇₉
hypocenter	C₇₉	#33860
LCS	G ▷ C₂×C₈×C₇₉ ▷ C₄×C₇₉ ▷ C₂×C₇₉ ▷ C₇₉
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ C₄×C₈ ◁ #2214

Complements

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

N	H ≅ G/N	type
1. #23776552	C₂	semidirect
2. #23775205	C₂	semidirect
3. #23775204	C₂	semidirect
4. #23765652	C₂	semidirect
5. #23765524	C₂	semidirect
6. #23765482	C₂	semidirect
7. #23765381	C₂	semidirect
8. #23765375	C₂	semidirect
9. #23765279	C₂	semidirect
10. #23765236	C₂	semidirect
11. C₇₉×#2214	C₂	semidirect
217. #2214	D₇₉	semidirect
222. C₇₉	#33860	semidirect

Maximal subgroups

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

subgroup	index
#23776552	2
#23775205	2
#23775204	2
#23765652	2
#23765524	2
#23765482	2
#23765381	2
#23765375	2
#23765279	2
#23765236	2
C₇₉×#2214	2
#33860	79