Subgroups — #35155716

Order 20224.

2578 conjugacy classes of subgroups.

The subgroup lattice has 378 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#23778187	C₂	1	2	yes	maximal, complemented
2	#23778111	C₂	1	1	yes	maximal, characteristic, complemented
3	#23773133	C₂	1	12	yes	maximal, complemented
4	#23773116	C₂	1	6	yes	maximal, complemented
5	C₂×#19098835	C₂	1	6	yes	maximal, complemented
6	#23772011	C₂	1	1	yes	maximal, characteristic, complemented
7	#23771883	C₂	1	2	yes	maximal, complemented
8	C₂×#578×C₇₉	C₂	1	1	yes	maximal, characteristic, complemented
9	#19099399	C₂²	1	1	yes	characteristic
10	#19099378	—	2	8	no	—
11	#19099371	—	2	8	no	—
12	#19099361	C₂²	1	12	yes	—
13	C₂×#14579192	C₂²	1	12	yes	—
14	#19099330	C₂²	1	1	yes	characteristic
15	#19099330	C₂²	1	2	yes	—
16	#19099329	C₂²	1	6	yes	—
17	#19099328	—	2	4	no	—
18	C₂×#14579167	C₂²	1	6	yes	—
19	#131×D₇₉	—	2	4	no	—
20	#19098835	C₂²	1	12	yes	complemented
21	#19098835	C₂²	1	24	yes	complemented
22	#19098830	C₂²	1	4	yes	complemented
23	#19098823	C₂²	1	24	yes	complemented
24	#19098612	C₂²	1	6	yes	complemented
25	C₂×#14579097	C₂²	1	6	yes	complemented
26	#19098547	C₂²	1	12	yes	complemented
27	C₂×#14579080	C₂²	1	12	yes	complemented
28	C₂×2₋¹⁺⁴×C₇₉	C₂²	1	2	yes	—
29	#578×C₇₉	—	2	4	no	—
30	#578×C₇₉	—	2	4	no	—
31	C₂×#137×C₇₉	C₂²	1	6	yes	complemented
32	C₂×#135×C₇₉	C₂²	1	3	yes	complemented
33	C₂²×Q₁₆×C₇₉	C₂²	1	3	yes	complemented
34	C₂×#131×C₇₉	C₂²	1	1	yes	characteristic
35	#14579209	—	2	1	no	—
36	#14579208	D₄	1	4	yes	—
37	#14579207	D₄	1	4	yes	—
38	#14579206	—	2	3	no	—
39	C₂×#11929624	C₂³	1	6	yes	—
40	#41×D₇₉	—	2	4	no	—
41	#14579192	—	2	24	no	—
42	#14579192	C₂³	1	24	yes	—
43	#14579191	—	2	12	no	—
44	#14579187	C₂³	1	8	yes	—
45	#14579186	—	2	12	no	—
46	#14579181	—	2	24	no	—
47	C₂×#11929614	—	2	12	no	—
48	C₂×#11929611	—	2	12	no	—
49	#14579169	—	2	12	no	—
50	#14579168	—	2	4	no	—
51	#14579168	—	2	8	no	—
52	#14579167	—	2	6	no	—
53	#14579167	—	2	6	no	—
54	#14579167	C₂³	1	4	yes	—
55	#14579167	C₂³	1	12	yes	—
56	C₂×#11929608	C₂³	1	6	yes	—
57	C₂×#11929608	C₂³	1	12	yes	—
58	C₂×#11929607	—	2	6	no	—
59	M₁₆×D₇₉	—	2	12	no	—
60	C₂×C₈×D₇₉	—	2	6	no	—
61	#14579097	C₂³	1	12	yes	—
62	#14579097	C₂³	1	4	yes	—
63	#14579080	C₂³	1	24	yes	—
64	#14579078	C₂³	1	8	yes	—
65	2₋¹⁺⁴×C₇₉	—	2	8	no	—
66	2₋¹⁺⁴×C₇₉	—	2	8	no	—
67	C₂×#41×C₇₉	—	2	6	no	—
68	C₂×#41×C₇₉	C₂³	1	6	yes	—
69	C₂×#41×C₇₉	C₂³	1	1	yes	characteristic
70	C₂²×Q₈×C₇₉	—	2	2	no	—
71	C₂²×Q₈×C₇₉	C₂³	1	6	yes	—
72	#137×C₇₉	—	2	24	no	—
73	#135×C₇₉	—	2	6	no	—
74	#135×C₇₉	—	2	6	no	—
75	C₂×Q₁₆×C₇₉	—	2	6	no	—
76	C₂×Q₁₆×C₇₉	—	2	6	no	—
77	C₂×Q₁₆×C₇₉	C₂³	1	3	yes	—
78	C₂×Q₁₆×C₇₉	C₂³	1	6	yes	—
79	C₂×SD₁₆×C₇₉	C₂³	1	6	yes	—
80	C₂×D₈×C₇₉	C₂³	1	1	yes	characteristic
81	#131×C₇₉	D₄	1	4	yes	—
82	#131×C₇₉	D₄	1	4	yes	—
83	C₂×M₁₆×C₇₉	C₂³	1	3	yes	—
84	C₂²×C₈×C₇₉	C₂³	1	3	yes	—
85	#11929625	—	2	4	no	—
86	#11929624	—	2	6	no	—
87	#11929624	—	2	6	no	—
88	#11929624	C₂×D₄	1	12	yes	—
89	#11929624	C₂⁴	1	4	yes	—
90	#11929623	—	2	12	no	—
91	C₂×Dic₁₅₈	C₂×D₄	1	6	yes	—
92	Q₈×D₇₉	—	2	4	no	—
93	C₂×D₃₁₆	C₂×D₄	1	6	yes	—
94	C₂×C₂²⋊D₇₉	—	2	6	no	—
95	D₄×D₇₉	—	2	12	no	—
96	C₂×C₄×D₇₉	—	2	6	no	—
97	#11929614	—	2	24	no	—
98	#11929612	—	2	8	no	—
99	#11929611	—	2	24	no	—
100	#11929609	—	2	8	no	—
101	#11929608	—	2	6	no	—
102	#11929608	—	2	12	no	—
103	#11929608	C₂⁴	1	4	yes	—
104	#11929608	C₂⁴	1	8	yes	—
105	#11929607	—	2	12	no	—
106	#11929607	—	2	4	no	—
107	C₂×C₇₉⋊C₈	—	2	6	no	—
108	C₂×C₇₉⋊C₈	—	2	12	no	—
109	C₈×D₇₉	—	2	4	no	—
110	C₈×D₇₉	—	2	12	no	—
111	#41×C₇₉	—	2	12	no	—
112	#41×C₇₉	—	2	12	no	—
113	#41×C₇₉	—	4	24	no	—
114	#41×C₇₉	C₂×D₄	1	4	yes	—
115	#41×C₇₉	C₂×D₄	1	4	yes	—
116	C₂×Q₈×C₇₉	—	2	6	no	—
117	C₂×Q₈×C₇₉	—	2	12	no	—
118	C₂×Q₈×C₇₉	—	2	12	no	—
119	C₂×Q₈×C₇₉	—	4	8	no	—
120	C₂×Q₈×C₇₉	#41	1	1	yes	characteristic
121	C₂×Q₈×C₇₉	C₂⁴	1	6	yes	—
122	C₂×D₄×C₇₉	—	2	6	no	—
123	C₂×D₄×C₇₉	#41	1	3	yes	—
124	C₂×D₄×C₇₉	C₂⁴	1	2	yes	—
125	C₂²×C₄×C₇₉	—	2	6	no	—
126	C₂²×C₄×C₇₉	—	2	6	no	—
127	C₂²×C₄×C₇₉	C₂⁴	1	3	yes	—
128	Q₁₆×C₇₉	—	2	3	no	—
129	Q₁₆×C₇₉	—	2	3	no	—
130	Q₁₆×C₇₉	—	2	12	no	—
131	SD₁₆×C₇₉	—	2	12	no	—
132	D₈×C₇₉	—	2	1	no	—
133	D₈×C₇₉	—	2	1	no	—
134	M₁₆×C₇₉	C₂×D₄	1	12	yes	—
135	C₂×C₈×C₇₉	C₂×D₄	1	6	yes	—
136	C₂×C₈×C₇₉	C₂×D₄	1	6	yes	—
137	C₂×C₈×C₇₉	C₂⁴	1	1	yes	characteristic
138	C₂×C₈×C₇₉	C₂⁴	1	3	yes	—
139	C₂³×C₇₉	—	2	2	no	—
140	C₂³×C₇₉	C₂×#41	1	3	yes	—
141	C₂²×D₇₉	—	2	6	no	—
142	Q₈×C₇₉	—	2	6	no	—
143	Q₈×C₇₉	—	2	6	no	—
144	Q₈×C₇₉	—	4	12	no	—
145	Q₈×C₇₉	#121	1	4	yes	—
146	D₄×C₇₉	—	2	2	no	—
147	D₄×C₇₉	—	2	2	no	—
148	D₄×C₇₉	—	4	12	no	—
149	D₄×C₇₉	#121	1	12	yes	—
150	C₂×C₄×C₇₉	—	2	6	no	—
151	C₂×C₄×C₇₉	—	4	12	no	—
152	C₂×C₄×C₇₉	—	4	12	no	—
153	C₂×C₄×C₇₉	C₂²×D₄	1	6	yes	—
154	C₂×C₄×C₇₉	C₂²×D₄	1	6	yes	—
155	C₂×C₄×C₇₉	C₂×#41	1	3	yes	—
156	C₂×C₄×C₇₉	C₂⁵	1	1	yes	characteristic
157	C₂²⋊D₇₉	—	2	12	no	—
158	C₂²⋊D₇₉	—	2	4	no	—
159	C₂×Dic₇₉	—	2	6	no	—
160	D₃₁₆	—	2	6	no	—
161	D₃₁₆	C₂²×D₄	1	4	yes	—
162	C₄×D₇₉	—	2	12	no	—
163	C₄×D₇₉	—	2	4	no	—
164	Dic₁₅₈	—	2	6	no	—
165	Dic₁₅₈	C₂²×D₄	1	4	yes	—
166	C₈×C₇₉	C₂²×D₄	1	1	yes	characteristic
167	C₈×C₇₉	C₂²×D₄	1	6	yes	—
168	C₈×C₇₉	C₂²×D₄	1	1	yes	characteristic
169	C₇₉⋊C₈	—	2	4	no	—
170	C₇₉⋊C₈	—	2	8	no	—
171	C₂²×C₇₉	—	4	3	no	—
172	C₂²×C₇₉	—	4	2	no	—
173	C₂²×C₇₉	—	8	2	no	—
174	C₂²×C₇₉	C₂×#121	1	6	yes	—
175	C₂²×C₇₉	C₂²×#41	1	1	yes	characteristic
176	C₂×D₇₉	—	2	4	no	—
177	C₂×D₇₉	—	8	6	no	—
178	C₄×C₇₉	—	4	6	no	—
179	C₄×C₇₉	C₂×#121	1	6	yes	—
180	C₄×C₇₉	C₂³×D₄	1	1	yes	characteristic
181	C₄×C₇₉	C₂³×D₄	1	1	yes	characteristic
182	Dic₇₉	—	2	4	no	—
183	#60472	—	79	1	no	maximal, Sylow
184	C₂×C₇₉	—	2	6	no	—
185	C₂×C₇₉	—	2	1	no	—
186	C₂×C₇₉	—	8	2	no	—
187	C₂×C₇₉	C₂²×#121	1	1	yes	characteristic
188	D₇₉	—	4	4	no	—
189	#3586	—	79	2	no	—
190	C₂×#578	D₇₉	1	1	yes	characteristic, complemented
191	#3573	—	79	1	no	—
192	#3069	—	79	12	no	—
193	#3067	—	79	6	no	—
194	C₂×#473	—	79	6	no	—
195	#2916	—	79	1	no	—
196	#2889	—	79	2	no	—
197	C₇₉	#60472	1	1	yes	characteristic, complemented, Sylow
198	#585	—	79	1	no	—
199	C₂×2₋¹⁺⁴	C₂×D₇₉	1	2	yes	—
200	#578	—	2	4	no	—
201	#578	—	2	4	no	—
202	#578	—	158	8	no	—
203	#577	—	158	8	no	—
204	#575	—	79	12	no	—
205	C₂×#137	—	79	12	no	—
206	C₂×#137	C₂×D₇₉	1	6	yes	complemented
207	C₂×#135	C₂×D₇₉	1	3	yes	complemented
208	C₂²×Q₁₆	C₂×D₇₉	1	3	yes	complemented
209	#568	—	79	2	no	—
210	#568	—	79	1	no	—
211	#568	—	79	6	no	—
212	#568	—	158	4	no	—
213	C₂×#131	—	79	6	no	—
214	C₂×#131	—	158	4	no	—
215	C₂×#131	C₂×D₇₉	1	1	yes	characteristic
216	#473	—	79	12	no	—
217	#473	—	79	24	no	—
218	#472	—	79	4	no	—
219	#471	—	79	24	no	—
220	#430	—	79	6	no	—
221	C₂×#108	—	79	6	no	—
222	#413	—	79	12	no	—
223	C₂×#101	—	79	12	no	—
224	2₋¹⁺⁴	—	2	8	no	—
225	2₋¹⁺⁴	—	2	8	no	—
226	2₋¹⁺⁴	—	79	4	no	—
227	2₋¹⁺⁴	—	158	1	no	—
228	H(5,𝔽₂)	—	79	4	no	—
229	H(5,𝔽₂)	—	158	3	no	—
230	C₂×#41	—	2	6	no	—
231	C₂×#41	—	79	6	no	—
232	C₂×#41	—	158	4	no	—
233	C₂×#41	C₂²×D₇₉	1	6	yes	—
234	C₂×#41	C₂²×D₇₉	1	1	yes	characteristic
235	C₂²×Q₈	—	2	2	no	—
236	C₂²×Q₈	C₂²×D₇₉	1	6	yes	—
237	#137	—	2	24	no	—
238	#137	—	79	24	no	—
239	#137	—	158	24	no	—
240	#137	—	158	12	no	—
241	AGL(1,ℤ/8)	—	79	8	no	—
242	AGL(1,ℤ/8)	—	158	12	no	—
243	#135	—	2	6	no	—
244	#135	—	2	6	no	—
245	#135	—	158	24	no	—
246	C₂×Q₁₆	—	2	6	no	—
247	C₂×Q₁₆	—	2	6	no	—
248	C₂×Q₁₆	—	158	12	no	—
249	C₂×Q₁₆	C₂²×D₇₉	1	3	yes	—
250	C₂×Q₁₆	C₂²×D₇₉	1	6	yes	—
251	C₂×SD₁₆	—	158	12	no	—
252	C₂×SD₁₆	C₂²×D₇₉	1	6	yes	—
253	C₂×D₈	C₂²×D₇₉	1	1	yes	characteristic
254	#131	—	79	4	no	—
255	#131	—	79	12	no	—
256	#131	—	158	8	no	—
257	#131	—	158	4	no	—
258	#131	—	158	12	no	—
259	#131	—	158	6	no	—
260	#131	—	158	6	no	—
261	#131	C₂²⋊D₇₉	1	4	yes	—
262	#131	C₂²⋊D₇₉	1	4	yes	—
263	C₂×M₁₆	—	79	6	no	—
264	C₂×M₁₆	—	79	12	no	—
265	C₂×M₁₆	—	158	12	no	—
266	C₂×M₁₆	—	158	6	no	—
267	C₂×M₁₆	C₂²×D₇₉	1	3	yes	—
268	C₂²×C₈	—	158	6	no	—
269	C₂²×C₈	C₂²×D₇₉	1	3	yes	—
270	#108	—	79	4	no	—
271	#108	—	79	12	no	—
272	#101	—	79	24	no	—
273	#100	—	79	8	no	—
274	#41	—	2	12	no	—
275	#41	—	2	12	no	—
276	#41	—	4	24	no	—
277	#41	—	79	4	no	—
278	#41	—	79	12	no	—
279	#41	—	158	4	no	—
280	#41	—	158	12	no	—
281	#41	—	158	6	no	—
282	#41	—	158	6	no	—
283	#41	D₄×D₇₉	1	4	yes	—
284	#41	C₂×C₂²⋊D₇₉	1	4	yes	—
285	C₂×Q₈	—	2	6	no	—
286	C₂×Q₈	—	2	12	no	—
287	C₂×Q₈	—	2	12	no	—
288	C₂×Q₈	—	4	8	no	—
289	C₂×Q₈	—	79	6	no	—
290	C₂×Q₈	—	158	4	no	—
291	C₂×Q₈	#11929623	1	1	yes	characteristic
292	C₂×Q₈	C₂³×D₇₉	1	6	yes	—
293	C₂×D₄	—	2	6	no	—
294	C₂×D₄	—	79	6	no	—
295	C₂×D₄	—	158	6	no	—
296	C₂×D₄	—	158	12	no	—
297	C₂×D₄	#11929623	1	3	yes	—
298	C₂×D₄	C₂³×D₇₉	1	2	yes	—
299	C₂²×C₄	—	2	6	no	—
300	C₂²×C₄	—	2	6	no	—
301	C₂²×C₄	—	158	6	no	—
302	C₂²×C₄	C₂³×D₇₉	1	3	yes	—
303	Q₁₆	—	2	3	no	—
304	Q₁₆	—	2	3	no	—
305	Q₁₆	—	2	12	no	—
306	Q₁₆	—	158	24	no	—
307	SD₁₆	—	2	12	no	—
308	SD₁₆	—	158	8	no	—
309	SD₁₆	—	158	24	no	—
310	D₈	—	2	1	no	—
311	D₈	—	2	1	no	—
312	D₈	—	158	8	no	—
313	M₁₆	—	79	4	no	—
314	M₁₆	—	79	8	no	—
315	M₁₆	—	158	12	no	—
316	M₁₆	—	158	6	no	—
317	M₁₆	—	158	12	no	—
318	M₁₆	—	158	4	no	—
319	M₁₆	C₂×C₂²⋊D₇₉	1	12	yes	—
320	C₂×C₈	—	158	12	no	—
321	C₂×C₈	—	158	6	no	—
322	C₂×C₈	—	158	4	no	—
323	C₂×C₈	—	158	12	no	—
324	C₂×C₈	C₂×C₂²⋊D₇₉	1	6	yes	—
325	C₂×C₈	C₂×C₂²⋊D₇₉	1	6	yes	—
326	C₂×C₈	C₂³×D₇₉	1	1	yes	characteristic
327	C₂×C₈	C₂³×D₇₉	1	3	yes	—
328	C₂³	—	2	2	no	—
329	C₂³	—	158	6	no	—
330	C₂³	C₂×#11929623	1	3	yes	—
331	Q₈	—	2	6	no	—
332	Q₈	—	2	6	no	—
333	Q₈	—	4	12	no	—
334	Q₈	—	79	4	no	—
335	Q₈	—	158	6	no	—
336	Q₈	#14579132	1	4	yes	—
337	D₄	—	2	2	no	—
338	D₄	—	2	2	no	—
339	D₄	—	4	12	no	—
340	D₄	—	79	4	no	—
341	D₄	—	158	12	no	—
342	D₄	—	158	4	no	—
343	D₄	—	158	6	no	—
344	D₄	#14579132	1	12	yes	—
345	C₂×C₄	—	2	6	no	—
346	C₂×C₄	—	4	12	no	—
347	C₂×C₄	—	4	12	no	—
348	C₂×C₄	—	158	6	no	—
349	C₂×C₄	—	158	12	no	—
350	C₂×C₄	—	158	4	no	—
351	C₂×C₄	C₂×D₄×D₇₉	1	6	yes	—
352	C₂×C₄	C₂²×C₂²⋊D₇₉	1	6	yes	—
353	C₂×C₄	C₂×#11929623	1	3	yes	—
354	C₂×C₄	C₂⁴×D₇₉	1	1	yes	characteristic
355	C₈	—	158	8	no	—
356	C₈	—	158	4	no	—
357	C₈	C₂²×C₂²⋊D₇₉	1	1	yes	characteristic
358	C₈	C₂²×C₂²⋊D₇₉	1	6	yes	—
359	C₈	C₂²×C₂²⋊D₇₉	1	1	yes	characteristic
360	C₂²	—	4	3	no	—
361	C₂²	—	4	2	no	—
362	C₂²	—	8	2	no	—
363	C₂²	—	632	6	no	—
364	C₂²	—	158	4	no	—
365	C₂²	C₂×#14579132	1	6	yes	—
366	C₂²	C₂²×#11929623	1	1	yes	characteristic
367	C₄	—	4	6	no	—
368	C₄	—	158	4	no	—
369	C₄	C₂×#14579132	1	6	yes	—
370	C₄	C₂²×D₄×D₇₉	1	1	yes	characteristic
371	C₄	C₂³×C₂²⋊D₇₉	1	1	yes	characteristic
372	C₂	—	2	1	no	—
373	C₂	—	2	6	no	—
374	C₂	—	8	2	no	—
375	C₂	—	316	4	no	—
376	C₂	C₂²×#14579132	1	1	yes	characteristic

	subgroup	G/H
center	C₂	C₂²×#14579132
commutator, cosocle	C₂×C₄×C₇₉	C₂⁵
Frattini	C₂×C₄	C₂⁴×D₇₉
Fitting	C₂×#578×C₇₉	C₂
socle	C₂×C₇₉	C₂²×#121
hypercenter	C₂×#578	D₇₉
hypocenter	C₇₉	#60472
LCS	G ▷ C₂×C₄×C₇₉ ▷ C₂×C₇₉ ▷ C₇₉
UCS	C₁ ◁ C₂ ◁ C₂×#41 ◁ C₂×#578

Complements

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

N	H ≅ G/N	type
1. #23778187	C₂	semidirect
2. #23778111	C₂	semidirect
3. #23773133	C₂	semidirect
4. #23773116	C₂	semidirect
5. C₂×#19098835	C₂	semidirect
6. #23772011	C₂	semidirect
7. #23771883	C₂	semidirect
8. C₂×#578×C₇₉	C₂	semidirect
20. #19098835	C₂²	semidirect
21. #19098835	C₂²	semidirect
22. #19098830	C₂²	semidirect
23. #19098823	C₂²	semidirect
24. #19098612	C₂²	semidirect
25. C₂×#14579097	C₂²	semidirect
26. #19098547	C₂²	semidirect
27. C₂×#14579080	C₂²	semidirect
31. C₂×#137×C₇₉	C₂²	semidirect
32. C₂×#135×C₇₉	C₂²	semidirect
33. C₂²×Q₁₆×C₇₉	C₂²	semidirect
190. C₂×#578	D₇₉	semidirect
197. C₇₉	#60472	semidirect
206. C₂×#137	C₂×D₇₉	semidirect
207. C₂×#135	C₂×D₇₉	semidirect
208. C₂²×Q₁₆	C₂×D₇₉	semidirect

Maximal subgroups

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

subgroup	index
#23778187	2
#23778111	2
#23773133	2
#23773116	2
C₂×#19098835	2
#23772011	2
#23771883	2
C₂×#578×C₇₉	2
#60472	79