Subgroups — #34319181

Order 20224.

778 conjugacy classes of subgroups.

The subgroup lattice has 348 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄×#14579161	C₂	1	1	yes	maximal, characteristic
2	#23764877	C₂	1	2	yes	maximal
3	#23764876	C₂	1	2	yes	maximal
4	#23764733	C₂	1	2	yes	maximal
5	#23764730	C₂	1	2	yes	maximal
6	C₄×#14579092	C₂	1	4	yes	maximal
7	#23763146	C₂	1	1	yes	maximal, characteristic
8	C₄×C₄⋊Q₈×C₇₉	C₂	1	1	yes	maximal, characteristic
9	C₂×#14579161	C₂²	1	1	yes	characteristic
10	C₂×C₄×Dic₁₅₈	—	2	1	no	—
11	C₂×#14579092	C₂²	1	4	yes	—
12	C₂×#14579092	C₂²	1	4	yes	—
13	C₂×C₄×C₇₉⋊C₈	C₂²	1	2	yes	—
14	#19098469	—	2	1	no	—
15	#19098461	—	2	1	no	—
16	#19098461	C₂²	1	2	yes	—
17	#19098433	C₂²	1	1	yes	characteristic
18	C₄×C₄⋊Dic₇₉	—	2	1	no	—
19	C₄×C₄⋊Dic₇₉	C₂²	1	2	yes	—
20	#19098325	C₂²	1	4	yes	—
21	#19098325	C₂²	1	4	yes	—
22	#19098318	C₂²	1	2	yes	—
23	#19098318	C₂²	1	2	yes	—
24	C₂×C₄⋊Q₈×C₇₉	C₂²	1	1	yes	characteristic
25	C₂×C₄×Q₈×C₇₉	—	2	1	no	—
26	#391×C₇₉	—	2	1	no	—
27	#389×C₇₉	—	2	1	no	—
28	#389×C₇₉	C₂²	1	2	yes	—
29	#384×C₇₉	C₂²	1	1	yes	characteristic
30	C₄×C₄⋊C₄×C₇₉	—	2	1	no	—
31	C₄×C₄⋊C₄×C₇₉	C₂²	1	2	yes	—
32	C₄³×C₇₉	C₂²	1	1	yes	characteristic
33	C₂²×Dic₁₅₈	—	2	1	no	—
34	C₂²×C₇₉⋊C₈	C₂³	1	4	yes	—
35	#14579161	—	2	4	no	—
36	C₄×Dic₁₅₈	—	4	4	no	—
37	C₂×C₄⋊Dic₇₉	—	2	1	no	—
38	C₂×C₄⋊Dic₇₉	—	2	1	no	—
39	C₂×C₄⋊Dic₇₉	—	2	1	no	—
40	C₂×C₄⋊Dic₇₉	C₂³	1	2	yes	—
41	C₂×C₄⋊Dic₇₉	C₂³	1	2	yes	—
42	C₂×Dic₇₉⋊C₄	—	2	2	no	—
43	C₂×C₄×Dic₇₉	—	2	2	no	—
44	#14579092	—	2	8	no	—
45	#14579092	—	2	8	no	—
46	C₄×C₇₉⋊C₈	Q₈	1	8	yes	complemented
47	C₇₉⋊#95	—	2	2	no	—
48	C₇₉⋊#95	—	2	2	no	—
49	C₂²×Q₈×C₇₉	—	2	1	no	—
50	C₄⋊Q₈×C₇₉	—	2	4	no	—
51	C₄×Q₈×C₇₉	—	4	4	no	—
52	C₂×C₄⋊C₄×C₇₉	—	2	1	no	—
53	C₂×C₄⋊C₄×C₇₉	—	2	1	no	—
54	C₂×C₄⋊C₄×C₇₉	—	2	2	no	—
55	C₂×C₄⋊C₄×C₇₉	—	2	1	no	—
56	C₂×C₄⋊C₄×C₇₉	C₂³	1	2	yes	—
57	C₂×C₄⋊C₄×C₇₉	C₂³	1	2	yes	—
58	C₂×C₄²×C₇₉	—	2	2	no	—
59	C₂×C₄²×C₇₉	D₄	1	1	yes	characteristic
60	C₂×C₄²×C₇₉	D₄	1	1	yes	characteristic
61	C₂×C₄²×C₇₉	D₄	1	1	yes	characteristic
62	C₂×C₄²×C₇₉	D₄	1	1	yes	characteristic
63	C₂×C₄²×C₇₉	C₂³	1	1	yes	characteristic
64	C₂×C₄²×C₇₉	C₂³	1	1	yes	characteristic
65	C₂×C₄²×C₇₉	C₂³	1	1	yes	characteristic
66	#95×C₇₉	—	2	2	no	—
67	#95×C₇₉	—	2	2	no	—
68	C₂×Dic₁₅₈	—	2	4	no	—
69	C₂×Dic₁₅₈	—	4	2	no	—
70	C₂×Dic₁₅₈	—	4	2	no	—
71	C₂²×Dic₇₉	—	2	2	no	—
72	C₂²×Dic₇₉	—	2	2	no	—
73	C₂×C₇₉⋊C₈	—	2	4	no	—
74	C₂×C₇₉⋊C₈	—	2	4	no	—
75	C₂×C₇₉⋊C₈	C₂×Q₈	1	8	yes	—
76	C₄⋊Dic₇₉	—	2	4	no	—
77	C₄⋊Dic₇₉	—	2	2	no	—
78	C₄⋊Dic₇₉	—	2	4	no	—
79	C₄⋊Dic₇₉	—	4	2	no	—
80	C₄⋊Dic₇₉	—	4	2	no	—
81	C₄⋊Dic₇₉	#41	1	4	yes	—
82	Dic₇₉⋊C₄	—	4	4	no	—
83	C₄×Dic₇₉	—	4	4	no	—
84	C₂×Q₈×C₇₉	—	2	4	no	—
85	C₂×Q₈×C₇₉	—	4	2	no	—
86	C₂×Q₈×C₇₉	—	4	2	no	—
87	C₂²×C₄×C₇₉	—	2	2	no	—
88	C₂²×C₄×C₇₉	—	2	2	no	—
89	C₂²×C₄×C₇₉	C₂×D₄	1	1	yes	characteristic
90	C₂²×C₄×C₇₉	C₂×D₄	1	1	yes	characteristic
91	C₂²×C₄×C₇₉	C₂×D₄	1	1	yes	characteristic
92	C₂²×C₄×C₇₉	C₂×D₄	1	1	yes	characteristic
93	C₂²×C₄×C₇₉	C₂×D₄	1	1	yes	characteristic
94	C₂²×C₄×C₇₉	C₂×D₄	1	1	yes	characteristic
95	C₂²×C₄×C₇₉	C₂⁴	1	1	yes	characteristic
96	C₄⋊C₄×C₇₉	—	2	4	no	—
97	C₄⋊C₄×C₇₉	—	2	2	no	—
98	C₄⋊C₄×C₇₉	—	2	4	no	—
99	C₄⋊C₄×C₇₉	—	4	2	no	—
100	C₄⋊C₄×C₇₉	—	4	4	no	—
101	C₄⋊C₄×C₇₉	—	4	2	no	—
102	C₄⋊C₄×C₇₉	#41	1	4	yes	—
103	C₄²×C₇₉	—	2	2	no	—
104	C₄²×C₇₉	—	2	2	no	—
105	C₄²×C₇₉	—	4	1	no	—
106	C₄²×C₇₉	—	4	1	no	—
107	C₄²×C₇₉	—	4	4	no	—
108	C₄²×C₇₉	SD₁₆	1	4	yes	—
109	C₄²×C₇₉	SD₁₆	1	4	yes	—
110	C₄²×C₇₉	C₂×Q₈	1	4	yes	—
111	C₂³×C₇₉	C₂²×D₄	1	1	yes	characteristic
112	Q₈×C₇₉	—	4	8	no	—
113	C₂×C₄×C₇₉	—	2	1	no	—
114	C₂×C₄×C₇₉	—	2	1	no	—
115	C₂×C₄×C₇₉	—	2	1	no	—
116	C₂×C₄×C₇₉	—	2	1	no	—
117	C₂×C₄×C₇₉	—	2	1	no	—
118	C₂×C₄×C₇₉	—	2	1	no	—
119	C₂×C₄×C₇₉	—	2	1	no	—
120	C₂×C₄×C₇₉	—	2	1	no	—
121	C₂×C₄×C₇₉	—	2	1	no	—
122	C₂×C₄×C₇₉	—	2	1	no	—
123	C₂×C₄×C₇₉	—	2	4	no	—
124	C₂×C₄×C₇₉	—	4	2	no	—
125	C₂×C₄×C₇₉	—	4	2	no	—
126	C₂×C₄×C₇₉	—	4	2	no	—
127	C₂×C₄×C₇₉	—	4	2	no	—
128	C₂×C₄×C₇₉	—	4	2	no	—
129	C₂×C₄×C₇₉	C₄⋊Q₈	1	2	yes	—
130	C₂×C₄×C₇₉	C₄⋊Q₈	1	2	yes	—
131	C₂×C₄×C₇₉	C₂×SD₁₆	1	2	yes	—
132	C₂×C₄×C₇₉	C₂×SD₁₆	1	2	yes	—
133	C₂×C₄×C₇₉	C₂×SD₁₆	1	2	yes	—
134	C₂×C₄×C₇₉	C₂×SD₁₆	1	2	yes	—
135	C₂×C₄×C₇₉	C₂×SD₁₆	1	2	yes	—
136	C₂×C₄×C₇₉	C₂×SD₁₆	1	2	yes	—
137	C₂×C₄×C₇₉	C₂²×Q₈	1	2	yes	—
138	C₂×C₄×C₇₉	C₂×#41	1	2	yes	—
139	C₂×C₄×C₇₉	C₂×#41	1	2	yes	—
140	C₂×Dic₇₉	—	2	4	no	—
141	C₂×Dic₇₉	—	4	2	no	—
142	C₂×Dic₇₉	—	4	2	no	—
143	C₂×Dic₇₉	—	4	2	no	—
144	C₂×Dic₇₉	—	4	2	no	—
145	C₂×Dic₇₉	—	4	2	no	—
146	Dic₁₅₈	—	4	8	no	—
147	C₇₉⋊C₈	—	2	8	no	—
148	C₂²×C₇₉	#525	1	1	yes	characteristic
149	C₂²×C₇₉	#525	1	1	yes	characteristic
150	C₂²×C₇₉	C₂×C₄⋊Q₈	1	1	yes	characteristic
151	C₂²×C₇₉	C₂²×SD₁₆	1	1	yes	characteristic
152	C₂²×C₇₉	C₂²×SD₁₆	1	1	yes	characteristic
153	C₂²×C₇₉	#575	1	1	yes	characteristic
154	C₂²×C₇₉	#575	1	1	yes	characteristic
155	C₄×C₇₉	—	2	2	no	—
156	C₄×C₇₉	—	2	2	no	—
157	C₄×C₇₉	—	2	2	no	—
158	C₄×C₇₉	—	2	2	no	—
159	C₄×C₇₉	—	2	2	no	—
160	C₄×C₇₉	—	4	4	no	—
161	C₄×C₇₉	—	8	2	no	—
162	C₄×C₇₉	#498	1	4	yes	—
163	C₄×C₇₉	#533	1	4	yes	—
164	Dic₇₉	—	4	4	no	—
165	Dic₇₉	—	8	2	no	—
166	#23837	—	79	1	no	maximal, Sylow
167	C₂×C₇₉	#2399	1	1	yes	characteristic
168	C₂×C₇₉	#3101	1	1	yes	characteristic
169	C₂×C₇₉	#3101	1	1	yes	characteristic
170	C₂×C₇₉	#3103	1	1	yes	characteristic
171	C₂×C₇₉	#3103	1	1	yes	characteristic
172	C₂×C₇₉	C₂×#498	1	1	yes	characteristic
173	C₂×C₇₉	#3162	1	1	yes	characteristic
174	C₄×C₄⋊Q₈	—	79	1	no	—
175	C₄×C₄⋊Q₈	D₇₉	1	1	yes	characteristic
176	#2084	—	79	2	no	—
177	#2084	—	79	2	no	—
178	#2058	—	79	2	no	—
179	#2058	—	79	2	no	—
180	C₄×#106	—	79	4	no	—
181	#1744	—	79	1	no	—
182	C₇₉	#23837	1	1	yes	characteristic, complemented, Sylow
183	C₂×C₄⋊Q₈	—	79	1	no	—
184	C₂×C₄⋊Q₈	C₂×D₇₉	1	1	yes	characteristic
185	C₂×C₄×Q₈	—	2	1	no	—
186	C₂×C₄×Q₈	—	158	1	no	—
187	C₂×#106	—	79	4	no	—
188	C₂×#106	—	79	4	no	—
189	C₂×C₄×C₈	—	79	2	no	—
190	#391	—	2	1	no	—
191	#391	—	158	1	no	—
192	#389	—	2	1	no	—
193	#389	—	79	2	no	—
194	#389	—	158	1	no	—
195	#389	C₂×D₇₉	1	2	yes	—
196	#384	—	79	1	no	—
197	#384	C₂×D₇₉	1	1	yes	characteristic
198	C₄×C₄⋊C₄	—	2	1	no	—
199	C₄×C₄⋊C₄	—	79	2	no	—
200	C₄×C₄⋊C₄	—	158	1	no	—
201	C₄×C₄⋊C₄	C₂×D₇₉	1	2	yes	—
202	C₄³	C₂×D₇₉	1	1	yes	characteristic
203	#340	—	79	4	no	—
204	#340	—	79	4	no	—
205	#336	—	79	2	no	—
206	#336	—	79	2	no	—
207	C₂²×Q₈	—	2	1	no	—
208	C₂²×Q₈	—	158	1	no	—
209	C₂²×C₈	—	79	4	no	—
210	C₄⋊Q₈	—	2	4	no	—
211	C₄⋊Q₈	—	158	4	no	—
212	C₄×Q₈	—	4	4	no	—
213	C₄×Q₈	—	316	4	no	—
214	C₂×C₄⋊C₄	—	2	1	no	—
215	C₂×C₄⋊C₄	—	2	1	no	—
216	C₂×C₄⋊C₄	—	2	2	no	—
217	C₂×C₄⋊C₄	—	2	1	no	—
218	C₂×C₄⋊C₄	—	79	2	no	—
219	C₂×C₄⋊C₄	—	79	2	no	—
220	C₂×C₄⋊C₄	—	158	1	no	—
221	C₂×C₄⋊C₄	—	158	2	no	—
222	C₂×C₄⋊C₄	—	158	1	no	—
223	C₂×C₄⋊C₄	—	158	1	no	—
224	C₂×C₄⋊C₄	C₂²×D₇₉	1	2	yes	—
225	C₂×C₄⋊C₄	C₂²×D₇₉	1	2	yes	—
226	C₂×C₄²	—	2	2	no	—
227	C₂×C₄²	—	158	2	no	—
228	C₂×C₄²	C₂²⋊D₇₉	1	1	yes	characteristic
229	C₂×C₄²	C₂²⋊D₇₉	1	1	yes	characteristic
230	C₂×C₄²	C₂²⋊D₇₉	1	1	yes	characteristic
231	C₂×C₄²	C₂²⋊D₇₉	1	1	yes	characteristic
232	C₂×C₄²	C₂²×D₇₉	1	1	yes	characteristic
233	C₂×C₄²	C₂²×D₇₉	1	1	yes	characteristic
234	C₂×C₄²	C₂²×D₇₉	1	1	yes	characteristic
235	#106	—	158	8	no	—
236	#106	—	158	8	no	—
237	C₄×C₈	—	79	8	no	—
238	#95	—	2	2	no	—
239	#95	—	2	2	no	—
240	#95	—	158	2	no	—
241	#95	—	158	2	no	—
242	C₂×Q₈	—	2	4	no	—
243	C₂×Q₈	—	4	2	no	—
244	C₂×Q₈	—	4	2	no	—
245	C₂×Q₈	—	316	2	no	—
246	C₂×Q₈	—	316	2	no	—
247	C₂×Q₈	—	158	4	no	—
248	C₂²×C₄	—	2	2	no	—
249	C₂²×C₄	—	2	2	no	—
250	C₂²×C₄	—	158	2	no	—
251	C₂²×C₄	—	158	2	no	—
252	C₂²×C₄	C₂×C₂²⋊D₇₉	1	1	yes	characteristic
253	C₂²×C₄	C₂×C₂²⋊D₇₉	1	1	yes	characteristic
254	C₂²×C₄	C₂×C₂²⋊D₇₉	1	1	yes	characteristic
255	C₂²×C₄	C₂×C₂²⋊D₇₉	1	1	yes	characteristic
256	C₂²×C₄	C₂×C₂²⋊D₇₉	1	1	yes	characteristic
257	C₂²×C₄	C₂×C₂²⋊D₇₉	1	1	yes	characteristic
258	C₂²×C₄	C₂³×D₇₉	1	1	yes	characteristic
259	C₂×C₈	—	79	8	no	—
260	C₂×C₈	—	158	4	no	—
261	C₂×C₈	—	158	4	no	—
262	C₄⋊C₄	—	2	4	no	—
263	C₄⋊C₄	—	2	2	no	—
264	C₄⋊C₄	—	2	4	no	—
265	C₄⋊C₄	—	4	2	no	—
266	C₄⋊C₄	—	4	4	no	—
267	C₄⋊C₄	—	4	2	no	—
268	C₄⋊C₄	—	316	2	no	—
269	C₄⋊C₄	—	316	4	no	—
270	C₄⋊C₄	—	316	2	no	—
271	C₄⋊C₄	—	79	4	no	—
272	C₄⋊C₄	—	158	4	no	—
273	C₄⋊C₄	—	158	4	no	—
274	C₄⋊C₄	—	158	2	no	—
275	C₄⋊C₄	#11929624	1	4	yes	—
276	C₄²	—	2	2	no	—
277	C₄²	—	2	2	no	—
278	C₄²	—	4	1	no	—
279	C₄²	—	4	1	no	—
280	C₄²	—	4	4	no	—
281	C₄²	—	316	4	no	—
282	C₄²	#11929611	1	4	yes	—
283	C₄²	#11929612	1	4	yes	—
284	C₄²	Q₈×D₇₉	1	4	yes	—
285	C₂³	C₂²×C₂²⋊D₇₉	1	1	yes	characteristic
286	Q₈	—	4	8	no	—
287	Q₈	—	316	8	no	—
288	C₂×C₄	—	2	1	no	—
289	C₂×C₄	—	2	1	no	—
290	C₂×C₄	—	2	1	no	—
291	C₂×C₄	—	2	1	no	—
292	C₂×C₄	—	2	1	no	—
293	C₂×C₄	—	2	1	no	—
294	C₂×C₄	—	2	1	no	—
295	C₂×C₄	—	2	1	no	—
296	C₂×C₄	—	2	1	no	—
297	C₂×C₄	—	2	1	no	—
298	C₂×C₄	—	2	4	no	—
299	C₂×C₄	—	4	2	no	—
300	C₂×C₄	—	4	2	no	—
301	C₂×C₄	—	4	2	no	—
302	C₂×C₄	—	4	2	no	—
303	C₂×C₄	—	4	2	no	—
304	C₂×C₄	—	316	2	no	—
305	C₂×C₄	—	316	2	no	—
306	C₂×C₄	—	316	2	no	—
307	C₂×C₄	—	316	2	no	—
308	C₂×C₄	—	316	2	no	—
309	C₂×C₄	—	158	4	no	—
310	C₂×C₄	#14579159	1	2	yes	—
311	C₂×C₄	#14579159	1	2	yes	—
312	C₂×C₄	C₂×#11929611	1	2	yes	—
313	C₂×C₄	C₂×#11929611	1	2	yes	—
314	C₂×C₄	C₂×#11929611	1	2	yes	—
315	C₂×C₄	C₂×#11929612	1	2	yes	—
316	C₂×C₄	C₂×#11929612	1	2	yes	—
317	C₂×C₄	C₂×#11929612	1	2	yes	—
318	C₂×C₄	C₂×Q₈×D₇₉	1	2	yes	—
319	C₂×C₄	#41×D₇₉	1	2	yes	—
320	C₂×C₄	C₂×#11929624	1	2	yes	—
321	C₈	—	158	8	no	—
322	C₂²	#19099070	1	1	yes	characteristic
323	C₂²	#19099074	1	1	yes	characteristic
324	C₂²	C₂×#14579159	1	1	yes	characteristic
325	C₂²	C₂²×#11929611	1	1	yes	characteristic
326	C₂²	C₂²×#11929612	1	1	yes	characteristic
327	C₂²	#19099361	1	1	yes	characteristic
328	C₂²	#19099361	1	1	yes	characteristic
329	C₄	—	2	2	no	—
330	C₄	—	2	2	no	—
331	C₄	—	2	2	no	—
332	C₄	—	2	2	no	—
333	C₄	—	2	2	no	—
334	C₄	—	4	4	no	—
335	C₄	—	8	2	no	—
336	C₄	—	316	4	no	—
337	C₄	—	632	2	no	—
338	C₄	#19098959	1	4	yes	—
339	C₄	#19099117	1	4	yes	—
340	C₂	#23766691	1	1	yes	characteristic
341	C₂	#23773404	1	1	yes	characteristic
342	C₂	#23773408	1	1	yes	characteristic
343	C₂	#23773427	1	1	yes	characteristic
344	C₂	#23773431	1	1	yes	characteristic
345	C₂	C₂×#19098959	1	1	yes	characteristic
346	C₂	#23773922	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄	C₂×C₂²⋊D₇₉
commutator, cosocle	C₂²×C₄×C₇₉	C₂⁴
Frattini	C₂²×C₄	C₂³×D₇₉
Fitting	C₄×C₄⋊Q₈×C₇₉	C₂
socle, prime radical	C₂³×C₇₉	C₂²×D₄
hypercenter	C₄×C₄⋊Q₈	D₇₉
hypocenter	C₇₉	#23837
LCS	G ▷ C₂²×C₄×C₇₉ ▷ C₂×C₇₉ ▷ C₇₉
UCS	C₁ ◁ C₂²×C₄ ◁ C₄³ ◁ C₄×C₄⋊Q₈

Complements

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

N	H ≅ G/N	type
46. C₄×C₇₉⋊C₈	Q₈	semidirect
182. C₇₉	#23837	semidirect

Maximal subgroups

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

subgroup	index
C₄×#14579161	2
#23764877	2
#23764876	2
#23764733	2
#23764730	2
C₄×#14579092	2
#23763146	2
C₄×C₄⋊Q₈×C₇₉	2
#23837	79