Subgroups — #34445138

Order 20224.

580 conjugacy classes of subgroups.

The subgroup lattice has 362 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	D₄×Dic₃₁₆	C₂	1	1	yes	maximal, characteristic, complemented
2	#23765629	C₂	1	1	yes	maximal, characteristic, complemented
3	#23765597	C₂	1	2	yes	maximal, complemented
4	#23765539	C₂	1	1	yes	maximal, characteristic, complemented
5	#23765494	C₂	1	2	yes	maximal, complemented
6	#23765445	C₂	1	1	yes	maximal, characteristic, complemented
7	#23765316	C₂	1	2	yes	maximal, complemented
8	C₄×#14579105	C₂	1	1	yes	maximal, characteristic, complemented
9	D₄×C₇₉⋊C₁₆	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂×#19098366	C₂	1	2	yes	maximal, complemented
11	C₇₉×#3334	C₂	1	1	yes	maximal, characteristic
12	C₂²×Dic₃₁₆	C₂²	1	2	yes	—
13	D₄×Dic₁₅₈	—	2	1	no	—
14	C₂×#14579105	C₂²	1	1	yes	characteristic
15	C₂×#14579105	C₂²	1	2	yes	—
16	C₂²×C₇₉⋊C₁₆	C₂²	1	2	yes	—
17	#19098943	C₂²	1	1	yes	characteristic
18	#19098804	C₂²	1	2	yes	—
19	#19098774	—	2	1	no	—
20	#19098721	—	2	2	no	—
21	C₄×Dic₃₁₆	C₂²	1	1	yes	characteristic
22	#19098390	C₂²	1	1	yes	characteristic
23	#19098390	C₂²	1	2	yes	—
24	C₄⋊(C₇₉⋊C₁₆)	C₂²	1	1	yes	characteristic
25	#19098366	C₂²	1	1	yes	characteristic
26	#19098366	C₂²	1	2	yes	—
27	#19098366	C₂²	1	2	yes	—
28	#19098366	C₂²	1	2	yes	—
29	#19098366	C₂²	1	2	yes	—
30	#19098364	C₂²	1	1	yes	characteristic
31	C₂²⋊(C₇₉⋊C₁₆)	C₂²	1	2	yes	—
32	C₄×C₇₉⋊C₁₆	C₂²	1	1	yes	characteristic
33	#547×C₇₉	—	2	1	no	—
34	#500×C₇₉	C₂²	1	1	yes	characteristic
35	#480×C₇₉	—	2	2	no	—
36	#474×C₇₉	—	2	1	no	—
37	#466×C₇₉	C₂²	1	2	yes	—
38	C₄×D₈×C₇₉	C₂²	1	1	yes	characteristic
39	C₈×D₄×C₇₉	C₂²	1	1	yes	characteristic
40	C₂×#107×C₇₉	C₂²	1	2	yes	—
41	C₂²×Dic₁₅₈	—	2	2	no	—
42	C₂×Dic₃₁₆	—	2	4	no	—
43	C₂×Dic₃₁₆	—	2	2	no	—
44	C₂×Dic₃₁₆	—	2	1	no	—
45	C₂×Dic₃₁₆	—	4	2	no	—
46	C₂×Dic₃₁₆	C₂³	1	2	yes	—
47	C₂×Dic₃₁₆	C₂³	1	1	yes	characteristic
48	#14579161	—	2	1	no	—
49	#14579160	—	4	1	no	—
50	#14579140	—	2	2	no	—
51	#14579138	—	4	2	no	—
52	C₄×Dic₁₅₈	—	2	1	no	—
53	D₄×Dic₇₉	—	4	1	no	—
54	#14579105	—	2	2	no	—
55	#14579105	—	4	2	no	—
56	C₂×C₇₉⋊C₁₆	—	4	2	no	—
57	C₂×C₇₉⋊C₁₆	C₂³	1	2	yes	—
58	C₂×C₇₉⋊C₁₆	C₂³	1	1	yes	characteristic
59	C₂×C₇₉⋊C₁₆	C₂³	1	1	yes	characteristic
60	#14579095	C₂³	1	1	yes	characteristic
61	#14579086	—	2	1	no	—
62	#14579086	—	2	2	no	—
63	C₂×#41×C₇₉	—	4	1	no	—
64	C₂×D₈×C₇₉	C₂³	1	1	yes	characteristic
65	C₂²×C₈×C₇₉	C₂³	1	2	yes	—
66	C₄⋊Q₈×C₇₉	—	2	1	no	—
67	#123×C₇₉	—	4	2	no	—
68	C₂²⋊Q₈×C₇₉	—	4	1	no	—
69	#121×C₇₉	—	2	2	no	—
70	C₄×D₄×C₇₉	—	2	1	no	—
71	C₄×D₄×C₇₉	D₄	1	1	yes	characteristic
72	C₂×C₄⋊C₄×C₇₉	—	2	2	no	—
73	#107×C₇₉	—	2	2	no	—
74	#107×C₇₉	—	2	1	no	—
75	#107×C₇₉	C₂³	1	1	yes	characteristic
76	#107×C₇₉	C₂³	1	2	yes	—
77	C₄⋊C₈×C₇₉	D₄	1	1	yes	characteristic
78	#102×C₇₉	—	2	1	no	—
79	#102×C₇₉	—	2	2	no	—
80	C₂²⋊C₈×C₇₉	D₄	1	2	yes	—
81	C₄×C₈×C₇₉	C₂³	1	1	yes	characteristic
82	C₂×Dic₁₅₈	—	2	1	no	—
83	C₂×Dic₁₅₈	—	2	2	no	—
84	C₂×Dic₁₅₈	—	4	4	no	—
85	C₂×Dic₁₅₈	—	4	2	no	—
86	C₂×Dic₁₅₈	—	8	2	no	—
87	C₂²×Dic₇₉	—	4	2	no	—
88	Dic₃₁₆	—	2	2	no	—
89	Dic₃₁₆	—	2	1	no	—
90	Dic₃₁₆	—	2	2	no	—
91	Dic₃₁₆	—	4	4	no	—
92	Dic₃₁₆	—	8	1	no	—
93	Dic₃₁₆	#41	1	2	yes	complemented
94	C₄⋊Dic₇₉	—	2	1	no	—
95	C₄⋊Dic₇₉	—	4	1	no	—
96	C₄⋊Dic₇₉	—	4	1	no	—
97	Dic₇₉⋊C₄	—	4	1	no	—
98	Dic₇₉⋊C₄	—	4	2	no	—
99	C₂²⋊Dic₇₉	—	4	2	no	—
100	C₄×Dic₇₉	—	4	1	no	—
101	C₇₉⋊C₁₆	—	2	1	no	—
102	C₇₉⋊C₁₆	—	2	1	no	—
103	C₇₉⋊C₁₆	—	2	2	no	—
104	#41×C₇₉	—	8	4	no	—
105	C₂×Q₈×C₇₉	—	4	1	no	—
106	C₂×D₄×C₇₉	—	2	1	no	—
107	C₂×D₄×C₇₉	—	4	2	no	—
108	C₂×D₄×C₇₉	D₈	1	1	yes	characteristic
109	C₂²×C₄×C₇₉	—	4	2	no	—
110	C₂²×C₄×C₇₉	—	4	1	no	—
111	C₂²×C₄×C₇₉	C₂×D₄	1	2	yes	—
112	D₈×C₇₉	—	2	1	no	—
113	D₈×C₇₉	—	2	1	no	—
114	C₂×C₈×C₇₉	—	2	2	no	—
115	C₂×C₈×C₇₉	—	2	2	no	—
116	C₂×C₈×C₇₉	C₂×D₄	1	1	yes	characteristic
117	C₂×C₈×C₇₉	C₂×D₄	1	2	yes	—
118	C₂×C₈×C₇₉	C₂⁴	1	1	yes	characteristic
119	C₄⋊C₄×C₇₉	—	2	1	no	—
120	C₄⋊C₄×C₇₉	—	2	2	no	—
121	C₄⋊C₄×C₇₉	—	4	2	no	—
122	C₄⋊C₄×C₇₉	—	4	1	no	—
123	C₄⋊C₄×C₇₉	D₈	1	1	yes	characteristic
124	C₂²⋊C₄×C₇₉	—	4	1	no	—
125	C₂²⋊C₄×C₇₉	—	4	2	no	—
126	C₂²⋊C₄×C₇₉	D₈	1	2	yes	—
127	C₄²×C₇₉	C₂×D₄	1	1	yes	characteristic
128	C₂³×C₇₉	—	4	1	no	—
129	C₂³×C₇₉	C₂×D₈	1	2	yes	—
130	Q₈×C₇₉	—	8	2	no	—
131	D₄×C₇₉	—	4	1	no	—
132	D₄×C₇₉	—	4	1	no	—
133	D₄×C₇₉	—	8	4	no	—
134	D₄×C₇₉	SD₃₂	1	4	yes	complemented
135	C₂×C₄×C₇₉	—	2	2	no	—
136	C₂×C₄×C₇₉	—	2	2	no	—
137	C₂×C₄×C₇₉	—	4	1	no	—
138	C₂×C₄×C₇₉	—	4	2	no	—
139	C₂×C₄×C₇₉	—	8	4	no	—
140	C₂×C₄×C₇₉	—	8	2	no	—
141	C₂×C₄×C₇₉	C₂×D₈	1	1	yes	characteristic
142	C₂×C₄×C₇₉	C₂×D₈	1	2	yes	—
143	C₂×C₄×C₇₉	C₂×D₈	1	1	yes	characteristic
144	C₂×C₄×C₇₉	C₂²×D₄	1	1	yes	characteristic
145	C₂×Dic₇₉	—	4	1	no	—
146	C₂×Dic₇₉	—	4	1	no	—
147	C₂×Dic₇₉	—	4	2	no	—
148	C₂×Dic₇₉	—	8	4	no	—
149	Dic₁₅₈	—	2	2	no	—
150	Dic₁₅₈	—	4	1	no	—
151	Dic₁₅₈	—	4	2	no	—
152	Dic₁₅₈	—	4	2	no	—
153	Dic₁₅₈	—	8	4	no	—
154	C₈×C₇₉	—	2	2	no	—
155	C₈×C₇₉	—	2	1	no	—
156	C₈×C₇₉	C₂×#41	1	1	yes	characteristic
157	C₈×C₇₉	H(5,𝔽₂)	1	1	yes	characteristic
158	C₂²×C₇₉	—	2	2	no	—
159	C₂²×C₇₉	—	2	2	no	—
160	C₂²×C₇₉	—	8	1	no	—
161	C₂²×C₇₉	—	8	1	no	—
162	C₂²×C₇₉	—	8	1	no	—
163	C₂²×C₇₉	C₂×SD₃₂	1	4	yes	—
164	C₂²×C₇₉	C₂²×D₈	1	1	yes	characteristic
165	C₄×C₇₉	—	2	1	no	—
166	C₄×C₇₉	—	2	2	no	—
167	C₄×C₇₉	—	8	1	no	—
168	C₄×C₇₉	—	8	2	no	—
169	C₄×C₇₉	C₂×SD₃₂	1	2	yes	—
170	C₄×C₇₉	#546	1	1	yes	characteristic
171	C₄×C₇₉	#577	1	1	yes	characteristic
172	Dic₇₉	—	4	2	no	—
173	Dic₇₉	—	8	1	no	—
174	Dic₇₉	—	8	2	no	—
175	#33865	—	79	1	no	maximal, Sylow
176	C₂×C₇₉	—	2	4	no	—
177	C₂×C₇₉	—	16	1	no	—
178	C₂×C₇₉	#3294	1	1	yes	characteristic
179	C₂×C₇₉	C₂²×SD₃₂	1	1	yes	characteristic
180	C₂×C₇₉	#3417	1	1	yes	characteristic
181	#3334	D₇₉	1	1	yes	characteristic
182	D₄×Q₁₆	—	79	1	no	—
183	#2241	—	79	1	no	—
184	#2235	—	79	2	no	—
185	#2225	—	79	1	no	—
186	#2217	—	79	2	no	—
187	#2208	—	79	1	no	—
188	#2185	—	79	2	no	—
189	C₄×SD₃₂	—	79	1	no	—
190	D₄×C₁₆	—	79	1	no	—
191	C₂×#358	—	79	2	no	—
192	C₇₉	#33865	1	1	yes	characteristic, complemented, Sylow
193	C₂²×Q₁₆	—	79	2	no	—
194	D₄×Q₈	—	158	1	no	—
195	#547	—	2	1	no	—
196	C₂×SD₃₂	—	79	2	no	—
197	C₂×SD₃₂	—	79	1	no	—
198	C₂²×C₁₆	—	79	2	no	—
199	#500	C₂×D₇₉	1	1	yes	characteristic
200	#494	—	79	1	no	—
201	#480	—	2	2	no	—
202	#474	—	2	1	no	—
203	#467	—	79	2	no	—
204	#466	C₂×D₇₉	1	2	yes	—
205	#462	—	158	1	no	—
206	#451	—	158	2	no	—
207	C₄×Q₁₆	—	79	1	no	—
208	C₄×D₈	C₂×D₇₉	1	1	yes	characteristic
209	C₈×D₄	C₂×D₇₉	1	1	yes	characteristic
210	C₂×#107	C₂×D₇₉	1	2	yes	—
211	#367	—	79	1	no	—
212	#367	—	79	2	no	—
213	C₄⋊C₁₆	—	79	1	no	—
214	#358	—	79	1	no	—
215	#358	—	79	2	no	—
216	#358	—	79	2	no	—
217	#358	—	79	2	no	—
218	#358	—	79	2	no	—
219	#357	—	79	1	no	—
220	C₂²⋊C₁₆	—	79	2	no	—
221	C₄×C₁₆	—	79	1	no	—
222	C₂×#41	—	4	1	no	—
223	C₂²×Q₈	—	158	2	no	—
224	C₂×Q₁₆	—	316	2	no	—
225	C₂×Q₁₆	—	79	1	no	—
226	C₂×Q₁₆	—	79	2	no	—
227	C₂×Q₁₆	—	158	4	no	—
228	C₂×Q₁₆	—	158	2	no	—
229	C₂×Q₁₆	—	158	1	no	—
230	C₂×D₈	C₂²×D₇₉	1	1	yes	characteristic
231	C₂²×C₈	C₂²×D₇₉	1	2	yes	—
232	C₄⋊Q₈	—	2	1	no	—
233	C₄⋊Q₈	—	316	1	no	—
234	C₄⋊Q₈	—	158	1	no	—
235	#123	—	4	2	no	—
236	C₂²⋊Q₈	—	4	1	no	—
237	C₂²⋊Q₈	—	316	2	no	—
238	C₂²⋊Q₈	—	158	2	no	—
239	#121	—	2	2	no	—
240	C₄×Q₈	—	158	1	no	—
241	C₄×D₄	—	2	1	no	—
242	C₄×D₄	—	316	1	no	—
243	C₄×D₄	C₂²⋊D₇₉	1	1	yes	characteristic
244	C₂×C₄⋊C₄	—	2	2	no	—
245	SD₃₂	—	316	2	no	—
246	SD₃₂	—	158	2	no	—
247	C₂×C₁₆	—	316	2	no	—
248	C₂×C₁₆	—	79	1	no	—
249	C₂×C₁₆	—	79	1	no	—
250	C₂×C₁₆	—	79	2	no	—
251	#107	—	2	2	no	—
252	#107	—	2	1	no	—
253	#107	—	79	1	no	—
254	#107	C₂²×D₇₉	1	1	yes	characteristic
255	#107	C₂²×D₇₉	1	2	yes	—
256	C₄⋊C₈	C₂²⋊D₇₉	1	1	yes	characteristic
257	#103	—	158	1	no	—
258	#103	—	158	2	no	—
259	#102	—	2	1	no	—
260	#102	—	2	2	no	—
261	C₂²⋊C₈	C₂²⋊D₇₉	1	2	yes	—
262	C₄×C₈	C₂²×D₇₉	1	1	yes	characteristic
263	#41	—	8	4	no	—
264	C₂×Q₈	—	4	1	no	—
265	C₂×Q₈	—	316	4	no	—
266	C₂×Q₈	—	316	2	no	—
267	C₂×Q₈	—	632	2	no	—
268	C₂×Q₈	—	158	2	no	—
269	C₂×Q₈	—	158	1	no	—
270	C₂×D₄	—	2	1	no	—
271	C₂×D₄	—	4	2	no	—
272	C₂×D₄	#11929609	1	1	yes	characteristic
273	C₂²×C₄	—	4	2	no	—
274	C₂²×C₄	—	4	1	no	—
275	C₂²×C₄	—	316	2	no	—
276	C₂²×C₄	C₂×C₂²⋊D₇₉	1	2	yes	—
277	Q₁₆	—	316	4	no	—
278	Q₁₆	—	79	2	no	—
279	Q₁₆	—	632	1	no	—
280	Q₁₆	—	158	2	no	—
281	Q₁₆	—	158	2	no	—
282	Q₁₆	—	158	1	no	—
283	D₈	—	2	1	no	—
284	D₈	—	2	1	no	—
285	C₂×C₈	—	2	2	no	—
286	C₂×C₈	—	2	2	no	—
287	C₂×C₈	C₂×C₂²⋊D₇₉	1	1	yes	characteristic
288	C₂×C₈	C₂×C₂²⋊D₇₉	1	2	yes	—
289	C₂×C₈	C₂³×D₇₉	1	1	yes	characteristic
290	C₄⋊C₄	—	2	1	no	—
291	C₄⋊C₄	—	2	2	no	—
292	C₄⋊C₄	—	4	2	no	—
293	C₄⋊C₄	—	4	1	no	—
294	C₄⋊C₄	—	316	1	no	—
295	C₄⋊C₄	—	316	2	no	—
296	C₄⋊C₄	—	316	1	no	—
297	C₄⋊C₄	—	316	1	no	—
298	C₄⋊C₄	—	158	1	no	—
299	C₄⋊C₄	#11929609	1	1	yes	characteristic
300	C₂²⋊C₄	—	4	1	no	—
301	C₂²⋊C₄	—	4	2	no	—
302	C₂²⋊C₄	—	316	2	no	—
303	C₂²⋊C₄	#11929609	1	2	yes	—
304	C₄²	—	316	1	no	—
305	C₄²	C₂×C₂²⋊D₇₉	1	1	yes	characteristic
306	C₁₆	—	158	1	no	—
307	C₁₆	—	158	1	no	—
308	C₁₆	—	158	2	no	—
309	C₂³	—	4	1	no	—
310	C₂³	C₂×#11929609	1	2	yes	—
311	Q₈	—	8	2	no	—
312	Q₈	—	316	1	no	—
313	Q₈	—	316	2	no	—
314	Q₈	—	316	2	no	—
315	Q₈	—	632	4	no	—
316	Q₈	—	158	2	no	—
317	D₄	—	4	1	no	—
318	D₄	—	4	1	no	—
319	D₄	—	8	4	no	—
320	D₄	#14579105	1	4	yes	complemented
321	C₂×C₄	—	2	2	no	—
322	C₂×C₄	—	2	2	no	—
323	C₂×C₄	—	4	1	no	—
324	C₂×C₄	—	4	2	no	—
325	C₂×C₄	—	8	4	no	—
326	C₂×C₄	—	8	2	no	—
327	C₂×C₄	—	316	1	no	—
328	C₂×C₄	—	316	1	no	—
329	C₂×C₄	—	316	2	no	—
330	C₂×C₄	—	632	4	no	—
331	C₂×C₄	C₂×#11929609	1	1	yes	characteristic
332	C₂×C₄	C₂×#11929609	1	2	yes	—
333	C₂×C₄	C₂×#11929609	1	1	yes	characteristic
334	C₂×C₄	C₂²×C₂²⋊D₇₉	1	1	yes	characteristic
335	C₈	—	2	2	no	—
336	C₈	—	2	1	no	—
337	C₈	#41×D₇₉	1	1	yes	characteristic
338	C₈	#14579207	1	1	yes	characteristic
339	C₂²	—	2	2	no	—
340	C₂²	—	2	2	no	—
341	C₂²	—	8	1	no	—
342	C₂²	—	8	1	no	—
343	C₂²	—	8	1	no	—
344	C₂²	C₂×#14579105	1	4	yes	—
345	C₂²	C₂²×#11929609	1	1	yes	characteristic
346	C₄	—	2	1	no	—
347	C₄	—	2	2	no	—
348	C₄	—	8	1	no	—
349	C₄	—	8	2	no	—
350	C₄	—	316	2	no	—
351	C₄	—	632	1	no	—
352	C₄	—	632	2	no	—
353	C₄	C₂×#14579105	1	2	yes	—
354	C₄	#19099201	1	1	yes	characteristic
355	C₄	#19099371	1	1	yes	characteristic
356	C₂	—	2	4	no	—
357	C₂	—	16	1	no	—
358	C₂	#23775207	1	1	yes	characteristic
359	C₂	C₂²×#14579105	1	1	yes	characteristic
360	C₂	#23776551	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×#3334	C₂
socle	C₂²×C₇₉	C₂²×D₈
hypercenter	#3334	D₇₉
hypocenter	C₇₉	#33865
LCS	G ▷ C₂×C₈×C₇₉ ▷ C₄×C₇₉ ▷ C₂×C₇₉ ▷ C₇₉
UCS	C₁ ◁ C₂² ◁ C₄×D₄ ◁ C₈×D₄ ◁ #3334

Complements

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

N	H ≅ G/N	type
1. D₄×Dic₃₁₆	C₂	semidirect
2. #23765629	C₂	semidirect
3. #23765597	C₂	semidirect
4. #23765539	C₂	semidirect
5. #23765494	C₂	semidirect
6. #23765445	C₂	semidirect
7. #23765316	C₂	semidirect
8. C₄×#14579105	C₂	semidirect
9. D₄×C₇₉⋊C₁₆	C₂	semidirect
10. C₂×#19098366	C₂	semidirect
93. Dic₃₁₆	#41	semidirect
134. D₄×C₇₉	SD₃₂	semidirect
192. C₇₉	#33865	semidirect
320. D₄	#14579105	semidirect

Maximal subgroups

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

subgroup	index
D₄×Dic₃₁₆	2
#23765629	2
#23765597	2
#23765539	2
#23765494	2
#23765445	2
#23765316	2
C₄×#14579105	2
D₄×C₇₉⋊C₁₆	2
C₂×#19098366	2
C₇₉×#3334	2
#33865	79