Subgroups — #34443601

Order 20224.

602 conjugacy classes of subgroups.

The subgroup lattice has 388 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#23773939	C₂	1	1	yes	maximal, characteristic, complemented
2	#23765641	C₂	1	2	yes	maximal, complemented
3	#23765636	C₂	1	2	yes	maximal, complemented
4	#23765631	C₂	1	2	yes	maximal, complemented
5	#23765623	C₂	1	2	yes	maximal, complemented
6	#23765116	C₂	1	1	yes	maximal, characteristic, complemented
7	#23765112	C₂	1	1	yes	maximal, characteristic, complemented
8	#23765107	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂×#19098364	C₂	1	1	yes	maximal, characteristic, complemented
10	#23764997	C₂	1	1	yes	maximal, characteristic, complemented
11	C₇₉×#3155	C₂	1	1	yes	maximal, characteristic
12	#19099191	—	2	1	no	—
13	C₂×#14579101	C₂²	1	1	yes	characteristic
14	C₂²×C₇₉⋊C₁₆	C₂²	1	1	yes	characteristic
15	#19098972	—	2	2	no	—
16	#19098968	C₂²	1	2	yes	complemented
17	#19098965	C₂²	1	2	yes	complemented
18	#19098605	—	2	1	no	—
19	#19098601	C₂²	1	1	yes	characteristic
20	C₂×#14579095	C₂²	1	1	yes	characteristic
21	#19098367	C₂²	1	1	yes	characteristic, complemented
22	#19098367	C₂²	1	1	yes	characteristic, complemented
23	#19098367	C₂²	1	2	yes	complemented
24	#19098364	C₂²	1	2	yes	—
25	#19098364	C₂²	1	2	yes	—
26	#19098364	C₂²	1	2	yes	—
27	#19098364	C₂²	1	2	yes	—
28	#19098364	C₂²	1	1	yes	characteristic, complemented
29	#19098364	C₂²	1	1	yes	characteristic, complemented
30	#19098364	C₂²	1	2	yes	complemented
31	#19098339	C₂²	1	2	yes	complemented
32	C₄×C₇₉⋊C₁₆	C₂²	1	2	yes	complemented
33	C₂×AGL(1,ℤ/8)×C₇₉	—	2	1	no	—
34	C₂×#135×C₇₉	C₂²	1	1	yes	characteristic
35	C₂²×D₈×C₇₉	C₂²	1	1	yes	characteristic
36	#535×C₇₉	—	2	1	no	—
37	#496×C₇₉	—	2	2	no	—
38	#495×C₇₉	C₂²	1	2	yes	—
39	#493×C₇₉	C₂²	1	2	yes	—
40	#405×C₇₉	C₂²	1	1	yes	characteristic
41	#14579161	—	2	2	no	—
42	#14579160	—	4	1	no	—
43	#14579153	—	2	2	no	—
44	#14579152	—	4	1	no	—
45	#14579138	—	4	2	no	—
46	#14579120	—	2	1	no	—
47	C₂×C₄⋊Dic₇₉	—	2	1	no	—
48	#14579101	—	2	2	no	—
49	C₂×C₇₉⋊C₁₆	—	2	2	no	—
50	C₂×C₇₉⋊C₁₆	C₂³	1	1	yes	characteristic
51	C₂×C₇₉⋊C₁₆	C₂³	1	1	yes	characteristic
52	C₂×C₇₉⋊C₁₆	C₂³	1	2	yes	—
53	#14579095	—	2	2	no	—
54	#14579095	—	2	1	no	—
55	#14579095	C₂³	1	1	yes	characteristic
56	#14579095	C₂³	1	2	yes	—
57	#14579095	C₂³	1	1	yes	characteristic
58	#14579093	—	2	2	no	—
59	#14579093	—	2	1	no	—
60	C₂×#41×C₇₉	—	2	1	no	—
61	C₂²×D₄×C₇₉	—	2	1	no	—
62	AGL(1,ℤ/8)×C₇₉	—	4	4	no	—
63	#135×C₇₉	—	2	2	no	—
64	#135×C₇₉	—	2	2	no	—
65	C₂×Q₁₆×C₇₉	C₂³	1	1	yes	characteristic
66	C₂×SD₁₆×C₇₉	—	2	2	no	—
67	C₂×SD₁₆×C₇₉	—	2	1	no	—
68	C₂×D₈×C₇₉	—	2	2	no	—
69	C₂×D₈×C₇₉	—	2	1	no	—
70	C₂×D₈×C₇₉	—	2	2	no	—
71	C₂×D₈×C₇₉	—	2	2	no	—
72	C₂×D₈×C₇₉	C₂³	1	1	yes	characteristic
73	C₂×D₈×C₇₉	C₂³	1	2	yes	—
74	C₂×M₁₆×C₇₉	D₄	1	1	yes	characteristic
75	C₂²×C₈×C₇₉	C₂³	1	1	yes	characteristic
76	#127×C₇₉	—	2	2	no	—
77	#124×C₇₉	—	2	2	no	—
78	#121×C₇₉	—	4	1	no	—
79	#121×C₇₉	—	4	1	no	—
80	#120×C₇₉	—	4	2	no	—
81	#117×C₇₉	D₄	1	1	yes	characteristic
82	#97×C₇₉	D₄	1	2	yes	—
83	C₄×C₈×C₇₉	C₂³	1	2	yes	—
84	C₂×Dic₁₅₈	—	4	2	no	—
85	C₂²×Dic₇₉	—	4	1	no	—
86	C₄⋊Dic₇₉	—	2	1	no	—
87	C₄⋊Dic₇₉	—	2	2	no	—
88	C₄⋊Dic₇₉	—	2	1	no	—
89	C₄⋊Dic₇₉	—	4	1	no	—
90	C₄⋊Dic₇₉	—	4	2	no	—
91	Dic₇₉⋊C₄	—	4	2	no	—
92	Dic₇₉⋊C₄	—	4	2	no	—
93	C₂²⋊Dic₇₉	—	4	1	no	—
94	C₄×Dic₇₉	—	4	1	no	—
95	C₇₉⋊C₁₆	—	2	1	no	—
96	C₇₉⋊C₁₆	—	2	1	no	—
97	C₇₉⋊C₁₆	—	2	2	no	—
98	C₂⁴×C₇₉	—	4	1	no	—
99	#41×C₇₉	—	4	2	no	—
100	#41×C₇₉	—	4	2	no	—
101	C₂×Q₈×C₇₉	—	2	1	no	—
102	C₂×D₄×C₇₉	—	2	2	no	—
103	C₂×D₄×C₇₉	—	2	1	no	—
104	C₂×D₄×C₇₉	—	4	1	no	—
105	C₂×D₄×C₇₉	—	4	2	no	—
106	C₂×D₄×C₇₉	—	4	1	no	—
107	C₂×D₄×C₇₉	—	4	2	no	—
108	C₂×D₄×C₇₉	—	4	2	no	—
109	C₂×D₄×C₇₉	—	4	2	no	—
110	C₂²×C₄×C₇₉	—	4	1	no	—
111	C₂²×C₄×C₇₉	C₂×D₄	1	1	yes	characteristic
112	Q₁₆×C₇₉	—	2	1	no	—
113	Q₁₆×C₇₉	—	2	1	no	—
114	SD₁₆×C₇₉	—	4	2	no	—
115	SD₁₆×C₇₉	—	4	4	no	—
116	D₈×C₇₉	—	2	2	no	—
117	D₈×C₇₉	—	2	2	no	—
118	D₈×C₇₉	—	2	1	no	—
119	D₈×C₇₉	—	2	1	no	—
120	D₈×C₇₉	—	4	2	no	—
121	D₈×C₇₉	—	4	4	no	—
122	M₁₆×C₇₉	SD₁₆	1	4	yes	—
123	C₂×C₈×C₇₉	C₂×D₄	1	1	yes	characteristic
124	C₂×C₈×C₇₉	C₂×D₄	1	2	yes	—
125	C₂×C₈×C₇₉	#41	1	2	yes	—
126	C₂×C₈×C₇₉	#41	1	2	yes	—
127	C₂×C₈×C₇₉	C₂⁴	1	1	yes	characteristic
128	C₄⋊C₄×C₇₉	D₈	1	1	yes	characteristic
129	C₄⋊C₄×C₇₉	D₈	1	1	yes	characteristic
130	C₂²⋊C₄×C₇₉	—	4	2	no	—
131	C₂²⋊C₄×C₇₉	—	4	2	no	—
132	C₂²⋊C₄×C₇₉	D₈	1	2	yes	—
133	C₄²×C₇₉	C₂×D₄	1	2	yes	—
134	C₂³×C₇₉	—	4	2	no	—
135	C₂³×C₇₉	—	4	1	no	—
136	C₂³×C₇₉	—	8	2	no	—
137	C₂³×C₇₉	—	16	1	no	—
138	C₂³×C₇₉	—	16	1	no	—
139	C₂³×C₇₉	C₂×D₈	1	1	yes	characteristic
140	Q₈×C₇₉	—	4	1	no	—
141	Q₈×C₇₉	—	4	1	no	—
142	D₄×C₇₉	—	4	2	no	—
143	D₄×C₇₉	—	4	1	no	—
144	D₄×C₇₉	—	4	1	no	—
145	D₄×C₇₉	—	4	2	no	—
146	D₄×C₇₉	—	8	2	no	—
147	D₄×C₇₉	—	8	2	no	—
148	D₄×C₇₉	—	16	2	no	—
149	D₄×C₇₉	—	16	2	no	—
150	C₂×C₄×C₇₉	—	4	1	no	—
151	C₂×C₄×C₇₉	—	8	2	no	—
152	C₂×C₄×C₇₉	#124	1	2	yes	—
153	C₂×C₄×C₇₉	C₂×D₈	1	2	yes	—
154	C₂×C₄×C₇₉	C₂×D₈	1	2	yes	—
155	C₂×C₄×C₇₉	C₂×D₈	1	1	yes	characteristic
156	C₂×C₄×C₇₉	C₂×SD₁₆	1	2	yes	—
157	C₂×C₄×C₇₉	C₂²×D₄	1	1	yes	characteristic
158	C₂×Dic₇₉	—	4	1	no	—
159	C₂×Dic₇₉	—	4	1	no	—
160	C₂×Dic₇₉	—	4	2	no	—
161	C₂×Dic₇₉	—	16	1	no	—
162	Dic₁₅₈	—	16	2	no	—
163	C₈×C₇₉	#124	1	2	yes	—
164	C₈×C₇₉	C₂×SD₁₆	1	4	yes	—
165	C₈×C₇₉	C₂×#41	1	1	yes	characteristic
166	C₈×C₇₉	C₂×#41	1	1	yes	characteristic
167	C₂²×C₇₉	—	2	1	no	—
168	C₂²×C₇₉	—	2	1	no	—
169	C₂²×C₇₉	—	8	1	no	—
170	C₂²×C₇₉	—	8	2	no	—
171	C₂²×C₇₉	—	8	1	no	—
172	C₂²×C₇₉	—	8	2	no	—
173	C₂²×C₇₉	—	8	1	no	—
174	C₂²×C₇₉	—	8	2	no	—
175	C₂²×C₇₉	—	16	4	no	—
176	C₂²×C₇₉	#486	1	2	yes	—
177	C₂²×C₇₉	C₂²×D₈	1	1	yes	characteristic
178	C₄×C₇₉	—	2	2	no	—
179	C₄×C₇₉	—	2	2	no	—
180	C₄×C₇₉	—	8	1	no	—
181	C₄×C₇₉	#486	1	2	yes	—
182	C₄×C₇₉	C₂×#124	1	1	yes	characteristic
183	C₄×C₇₉	C₂²×SD₁₆	1	1	yes	characteristic
184	Dic₇₉	—	8	1	no	—
185	Dic₇₉	—	8	2	no	—
186	Dic₇₉	—	8	1	no	—
187	#33702	—	79	1	no	maximal, Sylow
188	C₂×C₇₉	—	2	2	no	—
189	C₂×C₇₉	—	16	1	no	—
190	C₂×C₇₉	—	16	2	no	—
191	C₂×C₇₉	C₂×#486	1	1	yes	characteristic
192	C₂×C₇₉	#3415	1	1	yes	characteristic
193	C₂×C₇₉	#3416	1	1	yes	characteristic
194	#3164	—	79	1	no	—
195	#3155	D₇₉	1	1	yes	characteristic
196	#2244	—	79	2	no	—
197	#2243	—	79	2	no	—
198	#2242	—	79	2	no	—
199	#2240	—	79	2	no	—
200	#2140	—	79	1	no	—
201	#2139	—	79	1	no	—
202	#2138	—	79	1	no	—
203	C₂×#357	—	79	1	no	—
204	#2108	—	79	1	no	—
205	C₇₉	#33702	1	1	yes	characteristic, complemented, Sylow
206	C₂×AGL(1,ℤ/8)	—	2	1	no	—
207	C₂×#135	C₂×D₇₉	1	1	yes	characteristic
208	C₂²×D₈	C₂×D₇₉	1	1	yes	characteristic
209	#544	—	158	1	no	—
210	#535	—	2	1	no	—
211	C₂×M₃₂	—	79	1	no	—
212	C₂²×C₁₆	—	79	1	no	—
213	#501	—	158	2	no	—
214	#500	—	79	2	no	—
215	#499	—	79	2	no	—
216	#496	—	2	2	no	—
217	#495	C₂×D₇₉	1	2	yes	—
218	#493	C₂×D₇₉	1	2	yes	—
219	#428	—	158	1	no	—
220	#427	—	79	1	no	—
221	C₂×#107	—	79	1	no	—
222	#405	C₂×D₇₉	1	1	yes	characteristic
223	#358	—	79	1	no	—
224	#358	—	79	1	no	—
225	#358	—	79	2	no	—
226	#357	—	79	2	no	—
227	#357	—	79	2	no	—
228	#357	—	79	1	no	—
229	#357	—	79	1	no	—
230	#357	—	79	2	no	—
231	#357	—	79	2	no	—
232	#357	—	79	2	no	—
233	#346	—	79	2	no	—
234	C₄×C₁₆	—	79	2	no	—
235	C₂×#41	—	2	1	no	—
236	C₂²×D₄	—	2	1	no	—
237	AGL(1,ℤ/8)	—	4	4	no	—
238	#135	—	2	2	no	—
239	#135	—	2	2	no	—
240	C₂×Q₁₆	C₂²×D₇₉	1	1	yes	characteristic
241	C₂×SD₁₆	—	2	1	no	—
242	C₂×SD₁₆	—	2	2	no	—
243	C₂×D₈	—	2	1	no	—
244	C₂×D₈	—	2	2	no	—
245	C₂×D₈	—	2	2	no	—
246	C₂×D₈	—	2	2	no	—
247	C₂×D₈	C₂²×D₇₉	1	2	yes	—
248	C₂×D₈	C₂²×D₇₉	1	1	yes	characteristic
249	C₂×M₁₆	C₂²⋊D₇₉	1	1	yes	characteristic
250	C₂²×C₈	C₂²×D₇₉	1	1	yes	characteristic
251	C₄⋊Q₈	—	316	1	no	—
252	C₄⋊Q₈	—	158	2	no	—
253	#127	—	2	2	no	—
254	#125	—	316	1	no	—
255	#125	—	158	2	no	—
256	#124	—	2	2	no	—
257	C₂²⋊Q₈	—	316	2	no	—
258	#121	—	4	1	no	—
259	#121	—	4	1	no	—
260	#120	—	4	2	no	—
261	#117	—	158	1	no	—
262	#117	C₂²⋊D₇₉	1	1	yes	characteristic
263	C₂×C₄⋊C₄	—	158	1	no	—
264	M₃₂	—	158	2	no	—
265	C₂×C₁₆	—	79	1	no	—
266	C₂×C₁₆	—	79	2	no	—
267	C₂×C₁₆	—	79	1	no	—
268	C₂×C₁₆	—	158	2	no	—
269	#107	—	79	2	no	—
270	#107	—	79	1	no	—
271	#107	—	79	1	no	—
272	#107	—	158	1	no	—
273	#107	—	158	2	no	—
274	#106	—	158	1	no	—
275	#106	—	158	2	no	—
276	#97	C₂²⋊D₇₉	1	2	yes	—
277	C₄×C₈	C₂²×D₇₉	1	2	yes	—
278	C₂⁴	—	4	1	no	—
279	#41	—	4	2	no	—
280	#41	—	4	2	no	—
281	C₂×Q₈	—	2	1	no	—
282	C₂×Q₈	—	316	2	no	—
283	C₂×D₄	—	2	2	no	—
284	C₂×D₄	—	2	1	no	—
285	C₂×D₄	—	4	1	no	—
286	C₂×D₄	—	4	2	no	—
287	C₂×D₄	—	4	1	no	—
288	C₂×D₄	—	4	2	no	—
289	C₂×D₄	—	4	2	no	—
290	C₂×D₄	—	4	2	no	—
291	C₂²×C₄	—	4	1	no	—
292	C₂²×C₄	—	316	1	no	—
293	C₂²×C₄	C₂×C₂²⋊D₇₉	1	1	yes	characteristic
294	Q₁₆	—	2	1	no	—
295	Q₁₆	—	2	1	no	—
296	SD₁₆	—	4	2	no	—
297	SD₁₆	—	4	4	no	—
298	D₈	—	2	2	no	—
299	D₈	—	2	2	no	—
300	D₈	—	2	1	no	—
301	D₈	—	2	1	no	—
302	D₈	—	4	2	no	—
303	D₈	—	4	4	no	—
304	M₁₆	#11929612	1	4	yes	—
305	C₂×C₈	C₂×C₂²⋊D₇₉	1	1	yes	characteristic
306	C₂×C₈	C₂×C₂²⋊D₇₉	1	2	yes	—
307	C₂×C₈	#11929623	1	2	yes	—
308	C₂×C₈	#11929623	1	2	yes	—
309	C₂×C₈	C₂³×D₇₉	1	1	yes	characteristic
310	C₄⋊C₄	—	316	2	no	—
311	C₄⋊C₄	—	316	2	no	—
312	C₄⋊C₄	—	316	1	no	—
313	C₄⋊C₄	—	316	2	no	—
314	C₄⋊C₄	—	158	1	no	—
315	C₄⋊C₄	—	158	2	no	—
316	C₄⋊C₄	—	158	1	no	—
317	C₄⋊C₄	#11929609	1	1	yes	characteristic
318	C₄⋊C₄	#11929609	1	1	yes	characteristic
319	C₂²⋊C₄	—	4	2	no	—
320	C₂²⋊C₄	—	4	2	no	—
321	C₂²⋊C₄	—	316	1	no	—
322	C₂²⋊C₄	#11929609	1	2	yes	—
323	C₄²	—	316	1	no	—
324	C₄²	C₂×C₂²⋊D₇₉	1	2	yes	—
325	C₁₆	—	158	1	no	—
326	C₁₆	—	158	1	no	—
327	C₁₆	—	158	2	no	—
328	C₂³	—	4	2	no	—
329	C₂³	—	4	1	no	—
330	C₂³	—	8	2	no	—
331	C₂³	—	16	1	no	—
332	C₂³	—	16	1	no	—
333	C₂³	C₂×#11929609	1	1	yes	characteristic
334	Q₈	—	4	1	no	—
335	Q₈	—	4	1	no	—
336	Q₈	—	1264	2	no	—
337	D₄	—	4	2	no	—
338	D₄	—	4	1	no	—
339	D₄	—	4	1	no	—
340	D₄	—	4	2	no	—
341	D₄	—	8	2	no	—
342	D₄	—	8	2	no	—
343	D₄	—	16	2	no	—
344	D₄	—	16	2	no	—
345	C₂×C₄	—	4	1	no	—
346	C₂×C₄	—	8	2	no	—
347	C₂×C₄	—	316	1	no	—
348	C₂×C₄	—	316	1	no	—
349	C₂×C₄	—	316	2	no	—
350	C₂×C₄	—	1264	1	no	—
351	C₂×C₄	#14579148	1	2	yes	—
352	C₂×C₄	C₂×#11929609	1	2	yes	—
353	C₂×C₄	C₂×#11929609	1	2	yes	—
354	C₂×C₄	C₂×#11929609	1	1	yes	characteristic
355	C₂×C₄	C₂×#11929612	1	2	yes	—
356	C₂×C₄	C₂²×C₂²⋊D₇₉	1	1	yes	characteristic
357	C₈	#14579148	1	2	yes	—
358	C₈	C₂×#11929612	1	4	yes	—
359	C₈	C₂×#11929623	1	1	yes	characteristic
360	C₈	C₂×#11929623	1	1	yes	characteristic
361	C₂²	—	2	1	no	—
362	C₂²	—	2	1	no	—
363	C₂²	—	8	1	no	—
364	C₂²	—	8	2	no	—
365	C₂²	—	8	1	no	—
366	C₂²	—	8	2	no	—
367	C₂²	—	8	1	no	—
368	C₂²	—	8	2	no	—
369	C₂²	—	16	4	no	—
370	C₂²	#19098909	1	2	yes	—
371	C₂²	C₂²×#11929609	1	1	yes	characteristic
372	C₄	—	2	2	no	—
373	C₄	—	2	2	no	—
374	C₄	—	8	1	no	—
375	C₄	—	632	2	no	—
376	C₄	—	632	1	no	—
377	C₄	—	632	1	no	—
378	C₄	#19098909	1	2	yes	—
379	C₄	C₂×#14579148	1	1	yes	characteristic
380	C₄	C₂²×#11929612	1	1	yes	characteristic
381	C₂	—	2	2	no	—
382	C₂	—	16	2	no	—
383	C₂	—	16	1	no	—
384	C₂	C₂×#19098909	1	1	yes	characteristic
385	C₂	#23776539	1	1	yes	characteristic
386	C₂	#23776547	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂²×#11929609
commutator, cosocle	C₂×C₈×C₇₉	C₂⁴
Frattini	C₂×C₈	C₂³×D₇₉
Fitting, prime radical	C₇₉×#3155	C₂
socle	C₂²×C₇₉	C₂²×D₈
hypercenter	#3155	D₇₉
hypocenter	C₇₉	#33702
LCS	G ▷ C₂×C₈×C₇₉ ▷ C₄×C₇₉ ▷ C₂×C₇₉ ▷ C₇₉
UCS	C₁ ◁ C₂² ◁ #117 ◁ #405 ◁ #3155

Complements

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

N	H ≅ G/N	type
1. #23773939	C₂	semidirect
2. #23765641	C₂	semidirect
3. #23765636	C₂	semidirect
4. #23765631	C₂	semidirect
5. #23765623	C₂	semidirect
6. #23765116	C₂	semidirect
7. #23765112	C₂	semidirect
8. #23765107	C₂	semidirect
9. C₂×#19098364	C₂	semidirect
10. #23764997	C₂	semidirect
16. #19098968	C₂²	semidirect
17. #19098965	C₂²	semidirect
21. #19098367	C₂²	semidirect
22. #19098367	C₂²	semidirect
23. #19098367	C₂²	semidirect
28. #19098364	C₂²	semidirect
29. #19098364	C₂²	semidirect
30. #19098364	C₂²	semidirect
31. #19098339	C₂²	semidirect
32. C₄×C₇₉⋊C₁₆	C₂²	semidirect
205. C₇₉	#33702	semidirect

Maximal subgroups

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

subgroup	index
#23773939	2
#23765641	2
#23765636	2
#23765631	2
#23765623	2
#23765116	2
#23765112	2
#23765107	2
C₂×#19098364	2
#23764997	2
C₇₉×#3155	2
#33702	79