Subgroups — Q₈×#14579082 #34304858

Order 20224.

1314 conjugacy classes of subgroups.

The subgroup lattice has 344 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×Q₈×C₇₉⋊C₈	C₂	1	1	yes	maximal, characteristic, complemented
2	Q₈×C₄⋊Dic₇₉	C₂	1	1	yes	maximal, characteristic, complemented
3	#23764362	C₂	1	3	yes	maximal, complemented
4	#23764353	C₂	1	3	yes	maximal, complemented
5	#23763400	C₂	1	3	yes	maximal
6	C₄×#14579082	C₂	1	3	yes	maximal
7	C₂×D₄×Q₈×C₇₉	C₂	1	1	yes	maximal, characteristic
8	C₂×Q₈×Dic₇₉	—	2	1	no	—
9	Q₈×C₇₉⋊C₈	—	2	4	no	—
10	C₂×C₄⋊(C₇₉⋊C₈)	C₂²	1	3	yes	—
11	C₂×#14579082	C₂²	1	3	yes	—
12	C₂×#14579082	C₂²	1	1	yes	characteristic
13	C₂×C₄×C₇₉⋊C₈	C₂²	1	3	yes	—
14	#19098462	—	2	3	no	—
15	#19098433	C₂²	1	3	yes	—
16	C₄×C₄⋊Dic₇₉	C₂²	1	3	yes	—
17	#19098325	C₂²	1	3	yes	—
18	#19098325	C₂²	1	3	yes	—
19	#19098325	C₂²	1	6	yes	—
20	C₂³×Q₈×C₇₉	—	2	1	no	—
21	D₄×Q₈×C₇₉	—	2	4	no	—
22	D₄×Q₈×C₇₉	C₄	1	8	yes	complemented
23	C₂×C₄⋊Q₈×C₇₉	C₂²	1	3	yes	—
24	C₂×C₂²⋊Q₈×C₇₉	—	2	3	no	—
25	C₂×C₄×Q₈×C₇₉	C₂²	1	1	yes	characteristic
26	C₂×C₄×D₄×C₇₉	C₂²	1	3	yes	—
27	C₂²×C₇₉⋊C₈	C₂³	1	3	yes	—
28	C₂²×C₇₉⋊C₈	C₂³	1	1	yes	characteristic
29	Q₈×Dic₇₉	—	4	4	no	—
30	C₂×C₄⋊Dic₇₉	—	2	3	no	—
31	C₂×C₄⋊Dic₇₉	—	2	3	no	—
32	C₂×C₄⋊Dic₇₉	C₂³	1	3	yes	—
33	C₂×C₄⋊Dic₇₉	C₂³	1	1	yes	characteristic
34	C₂×C₄×Dic₇₉	—	2	3	no	—
35	C₄⋊(C₇₉⋊C₈)	—	2	6	no	—
36	#14579082	—	2	6	no	—
37	#14579082	Q₈	1	4	yes	direct product factor
38	C₄×C₇₉⋊C₈	—	2	6	no	—
39	C₇₉⋊#95	—	2	3	no	—
40	C₂²×Q₈×C₇₉	—	2	8	no	—
41	C₂²×Q₈×C₇₉	—	2	3	no	—
42	C₂²×Q₈×C₇₉	—	4	4	no	—
43	C₂²×Q₈×C₇₉	—	4	4	no	—
44	C₂²×Q₈×C₇₉	D₄	1	1	yes	characteristic
45	C₂²×D₄×C₇₉	C₂³	1	1	yes	characteristic
46	C₂³×C₄×C₇₉	—	2	3	no	—
47	C₄⋊Q₈×C₇₉	—	2	6	no	—
48	C₄⋊Q₈×C₇₉	C₂×C₄	1	12	yes	—
49	C₂²⋊Q₈×C₇₉	—	2	24	no	—
50	C₄×Q₈×C₇₉	C₂×C₄	1	4	yes	—
51	C₄×Q₈×C₇₉	D₄	1	4	yes	—
52	C₄×D₄×C₇₉	—	2	6	no	—
53	C₄×D₄×C₇₉	C₂×C₄	1	12	yes	—
54	C₂×C₄⋊C₄×C₇₉	—	2	3	no	—
55	C₂×C₄⋊C₄×C₇₉	D₄	1	3	yes	—
56	C₂×C₄⋊C₄×C₇₉	C₂³	1	3	yes	—
57	C₂×C₂²⋊C₄×C₇₉	—	2	3	no	—
58	C₂×C₄²×C₇₉	C₂³	1	3	yes	—
59	C₂²×Dic₇₉	—	2	1	no	—
60	C₂²×Dic₇₉	—	2	3	no	—
61	C₂×C₇₉⋊C₈	—	2	1	no	—
62	C₂×C₇₉⋊C₈	—	2	3	no	—
63	C₂×C₇₉⋊C₈	—	2	3	no	—
64	C₂×C₇₉⋊C₈	—	2	1	no	—
65	C₂×C₇₉⋊C₈	—	2	3	no	—
66	C₂×C₇₉⋊C₈	C₂×Q₈	1	2	yes	complemented
67	C₄⋊Dic₇₉	—	2	3	no	—
68	C₄⋊Dic₇₉	—	2	1	no	—
69	C₄⋊Dic₇₉	—	2	3	no	—
70	C₄⋊Dic₇₉	—	4	6	no	—
71	C₄⋊Dic₇₉	—	8	3	no	—
72	C₄⋊Dic₇₉	C₂×Q₈	1	2	yes	complemented
73	C₄×Dic₇₉	—	4	6	no	—
74	C₂⁴×C₇₉	—	2	1	no	—
75	C₂×Q₈×C₇₉	—	2	12	no	—
76	C₂×Q₈×C₇₉	—	4	32	no	—
77	C₂×Q₈×C₇₉	—	4	6	no	—
78	C₂×Q₈×C₇₉	—	4	6	no	—
79	C₂×Q₈×C₇₉	C₂²⋊C₄	1	4	yes	—
80	C₂×Q₈×C₇₉	D₈	1	4	yes	—
81	C₂×Q₈×C₇₉	SD₁₆	1	4	yes	—
82	C₂×D₄×C₇₉	—	2	1	no	—
83	C₂×D₄×C₇₉	—	2	1	no	—
84	C₂×D₄×C₇₉	—	2	1	no	—
85	C₂×D₄×C₇₉	C₂²×C₄	1	2	yes	—
86	C₂×D₄×C₇₉	C₂×Q₈	1	2	yes	—
87	C₂×D₄×C₇₉	#41	1	2	yes	—
88	C₂²×C₄×C₇₉	—	2	3	no	—
89	C₂²×C₄×C₇₉	—	2	12	no	—
90	C₂²×C₄×C₇₉	—	4	6	no	—
91	C₂²×C₄×C₇₉	—	4	6	no	—
92	C₂²×C₄×C₇₉	C₂×D₄	1	3	yes	—
93	C₂²×C₄×C₇₉	C₂×D₄	1	3	yes	—
94	C₂²×C₄×C₇₉	C₂⁴	1	1	yes	characteristic
95	C₄⋊C₄×C₇₉	—	2	12	no	—
96	C₄⋊C₄×C₇₉	—	2	3	no	—
97	C₄⋊C₄×C₇₉	C₂²⋊C₄	1	12	yes	—
98	C₄⋊C₄×C₇₉	C₂²×C₄	1	6	yes	—
99	C₂²⋊C₄×C₇₉	—	2	12	no	—
100	C₄²×C₇₉	C₂²×C₄	1	6	yes	—
101	C₄²×C₇₉	C₂×D₄	1	6	yes	—
102	C₂³×C₇₉	—	2	2	no	—
103	C₂³×C₇₉	—	2	2	no	—
104	C₂³×C₇₉	—	2	2	no	—
105	C₂³×C₇₉	—	4	1	no	—
106	C₂³×C₇₉	—	4	1	no	—
107	C₂³×C₇₉	—	4	1	no	—
108	C₂³×C₇₉	—	4	1	no	—
109	C₂³×C₇₉	C₂²×D₄	1	1	yes	characteristic
110	Q₈×C₇₉	—	4	24	no	—
111	Q₈×C₇₉	#102	1	16	yes	complemented
112	D₄×C₇₉	—	2	2	no	—
113	D₄×C₇₉	—	2	2	no	—
114	D₄×C₇₉	—	2	2	no	—
115	D₄×C₇₉	C₄×Q₈	1	4	yes	complemented
116	C₂×C₄×C₇₉	—	2	3	no	—
117	C₂×C₄×C₇₉	—	2	3	no	—
118	C₂×C₄×C₇₉	—	2	6	no	—
119	C₂×C₄×C₇₉	—	4	3	no	—
120	C₂×C₄×C₇₉	—	4	3	no	—
121	C₂×C₄×C₇₉	—	4	24	no	—
122	C₂×C₄×C₇₉	C₂×C₂²⋊C₄	1	6	yes	—
123	C₂×C₄×C₇₉	C₂×C₂²⋊C₄	1	6	yes	—
124	C₂×C₄×C₇₉	C₂×D₈	1	6	yes	—
125	C₂×C₄×C₇₉	C₂×SD₁₆	1	6	yes	—
126	C₂×C₄×C₇₉	C₂³×C₄	1	1	yes	characteristic
127	C₂×C₄×C₇₉	C₂²×D₄	1	1	yes	characteristic
128	C₂×C₄×C₇₉	C₂²×Q₈	1	1	yes	characteristic
129	C₂×C₄×C₇₉	C₂×#41	1	1	yes	characteristic
130	C₂×C₄×C₇₉	2₋¹⁺⁴	1	1	yes	characteristic
131	C₂×C₄×C₇₉	2₋¹⁺⁴	1	1	yes	characteristic
132	C₂×Dic₇₉	—	2	2	no	—
133	C₂×Dic₇₉	—	4	1	no	—
134	C₂×Dic₇₉	—	4	1	no	—
135	C₂×Dic₇₉	—	4	3	no	—
136	C₂×Dic₇₉	—	4	3	no	—
137	C₂×Dic₇₉	—	4	3	no	—
138	C₇₉⋊C₈	—	2	2	no	—
139	C₇₉⋊C₈	—	4	3	no	—
140	C₂²×C₇₉	—	2	4	no	—
141	C₂²×C₇₉	—	4	2	no	—
142	C₂²×C₇₉	—	4	2	no	—
143	C₂²×C₇₉	—	4	2	no	—
144	C₂²×C₇₉	—	4	2	no	—
145	C₂²×C₇₉	—	4	2	no	—
146	C₂²×C₇₉	—	4	2	no	—
147	C₂²×C₇₉	C₂²×C₂²⋊C₄	1	1	yes	characteristic
148	C₂²×C₇₉	#548	1	1	yes	characteristic
149	C₂²×C₇₉	D₄×Q₈	1	1	yes	characteristic
150	C₂²×C₇₉	C₂²×D₈	1	1	yes	characteristic
151	C₂²×C₇₉	C₂²×SD₁₆	1	1	yes	characteristic
152	C₂²×C₇₉	#577	1	1	yes	characteristic
153	C₂²×C₇₉	#578	1	1	yes	characteristic
154	C₄×C₇₉	—	2	6	no	—
155	C₄×C₇₉	—	4	6	no	—
156	C₄×C₇₉	C₂×#102	1	12	yes	—
157	C₄×C₇₉	C₂×C₄×Q₈	1	1	yes	characteristic
158	C₄×C₇₉	#519	1	1	yes	characteristic
159	C₄×C₇₉	#548	1	1	yes	characteristic
160	C₄×C₇₉	D₄×Q₈	1	1	yes	characteristic
161	Dic₇₉	—	4	2	no	—
162	Dic₇₉	—	8	3	no	—
163	Q₈×#102	—	79	1	no	maximal, Sylow
164	C₂×C₇₉	—	4	4	no	—
165	C₂×C₇₉	Q₈×C₂²⋊C₄	1	1	yes	characteristic
166	C₂×C₇₉	C₂²×#102	1	1	yes	characteristic
167	C₂×C₇₉	#2898	1	1	yes	characteristic
168	C₂×C₇₉	#3358	1	1	yes	characteristic
169	C₂×C₇₉	#3359	1	1	yes	characteristic
170	C₂×C₇₉	Q₈×D₈	1	1	yes	characteristic
171	C₂×C₇₉	Q₈×SD₁₆	1	1	yes	characteristic
172	C₂×D₄×Q₈	D₇₉	1	1	yes	characteristic
173	C₂×C₈×Q₈	—	79	1	no	—
174	Q₈×C₄⋊C₄	—	79	1	no	—
175	#1990	—	79	3	no	—
176	#1988	—	79	3	no	—
177	#1802	—	79	3	no	—
178	C₄×#102	—	79	3	no	—
179	C₇₉	Q₈×#102	1	1	yes	characteristic, complemented, Sylow
180	C₂³×Q₈	—	2	1	no	—
181	D₄×Q₈	—	2	4	no	—
182	D₄×Q₈	Dic₇₉	1	8	yes	complemented
183	C₂×C₄⋊Q₈	C₂×D₇₉	1	3	yes	—
184	C₂×C₂²⋊Q₈	—	2	3	no	—
185	C₂×C₄×Q₈	—	158	1	no	—
186	C₂×C₄×Q₈	C₂×D₇₉	1	1	yes	characteristic
187	C₂×C₄×D₄	C₂×D₇₉	1	3	yes	—
188	C₈×Q₈	—	158	4	no	—
189	C₂×C₄⋊C₈	—	79	3	no	—
190	C₂×#102	—	79	1	no	—
191	C₂×#102	—	79	3	no	—
192	C₂×C₄×C₈	—	79	3	no	—
193	#389	—	158	3	no	—
194	#384	—	79	3	no	—
195	C₄×C₄⋊C₄	—	79	3	no	—
196	#340	—	79	6	no	—
197	#340	—	79	3	no	—
198	#340	—	79	3	no	—
199	C₂²×Q₈	—	2	8	no	—
200	C₂²×Q₈	—	2	3	no	—
201	C₂²×Q₈	—	4	4	no	—
202	C₂²×Q₈	—	4	4	no	—
203	C₂²×Q₈	C₂²⋊D₇₉	1	1	yes	characteristic
204	C₂²×D₄	C₂²×D₇₉	1	1	yes	characteristic
205	C₂³×C₄	—	2	3	no	—
206	C₂²×C₈	—	79	1	no	—
207	C₂²×C₈	—	79	3	no	—
208	C₄⋊Q₈	—	2	6	no	—
209	C₄⋊Q₈	C₂×Dic₇₉	1	12	yes	—
210	C₂²⋊Q₈	—	2	24	no	—
211	C₄×Q₈	—	316	4	no	—
212	C₄×Q₈	C₂×Dic₇₉	1	4	yes	—
213	C₄×Q₈	C₂²⋊D₇₉	1	4	yes	—
214	C₄×D₄	—	2	6	no	—
215	C₄×D₄	C₂×Dic₇₉	1	12	yes	—
216	C₂×C₄⋊C₄	—	2	3	no	—
217	C₂×C₄⋊C₄	—	79	1	no	—
218	C₂×C₄⋊C₄	—	79	3	no	—
219	C₂×C₄⋊C₄	—	158	3	no	—
220	C₂×C₄⋊C₄	—	158	3	no	—
221	C₂×C₄⋊C₄	C₂²⋊D₇₉	1	3	yes	—
222	C₂×C₄⋊C₄	C₂²×D₇₉	1	3	yes	—
223	C₂×C₂²⋊C₄	—	2	3	no	—
224	C₂×C₄²	—	158	3	no	—
225	C₂×C₄²	C₂²×D₇₉	1	3	yes	—
226	C₄⋊C₈	—	158	6	no	—
227	#102	—	79	4	no	—
228	#102	—	158	6	no	—
229	C₄×C₈	—	158	6	no	—
230	#95	—	158	3	no	—
231	C₂⁴	—	2	1	no	—
232	C₂×Q₈	—	2	12	no	—
233	C₂×Q₈	—	4	32	no	—
234	C₂×Q₈	—	4	6	no	—
235	C₂×Q₈	—	4	6	no	—
236	C₂×Q₈	C₂²⋊Dic₇₉	1	4	yes	—
237	C₂×Q₈	#11929609	1	4	yes	—
238	C₂×Q₈	#11929612	1	4	yes	—
239	C₂×D₄	—	2	1	no	—
240	C₂×D₄	—	2	1	no	—
241	C₂×D₄	—	2	1	no	—
242	C₂×D₄	C₂²×Dic₇₉	1	2	yes	—
243	C₂×D₄	Q₈×D₇₉	1	2	yes	—
244	C₂×D₄	#11929625	1	2	yes	—
245	C₂²×C₄	—	2	3	no	—
246	C₂²×C₄	—	2	12	no	—
247	C₂²×C₄	—	4	6	no	—
248	C₂²×C₄	—	4	6	no	—
249	C₂²×C₄	—	158	3	no	—
250	C₂²×C₄	—	158	1	no	—
251	C₂²×C₄	C₂×C₂²⋊D₇₉	1	3	yes	—
252	C₂²×C₄	C₂×C₂²⋊D₇₉	1	3	yes	—
253	C₂²×C₄	C₂³×D₇₉	1	1	yes	characteristic
254	C₂×C₈	—	79	2	no	—
255	C₂×C₈	—	158	3	no	—
256	C₂×C₈	—	158	3	no	—
257	C₂×C₈	—	158	3	no	—
258	C₂×C₈	—	158	1	no	—
259	C₂×C₈	—	158	1	no	—
260	C₄⋊C₄	—	2	12	no	—
261	C₄⋊C₄	—	2	3	no	—
262	C₄⋊C₄	—	316	6	no	—
263	C₄⋊C₄	—	79	2	no	—
264	C₄⋊C₄	—	632	3	no	—
265	C₄⋊C₄	—	158	3	no	—
266	C₄⋊C₄	—	158	3	no	—
267	C₄⋊C₄	—	158	1	no	—
268	C₄⋊C₄	C₂²⋊Dic₇₉	1	12	yes	—
269	C₄⋊C₄	C₂²×Dic₇₉	1	6	yes	—
270	C₂²⋊C₄	—	2	12	no	—
271	C₄²	—	316	6	no	—
272	C₄²	C₂²×Dic₇₉	1	6	yes	—
273	C₄²	C₂×C₂²⋊D₇₉	1	6	yes	—
274	C₂³	—	2	2	no	—
275	C₂³	—	2	2	no	—
276	C₂³	—	2	2	no	—
277	C₂³	—	4	1	no	—
278	C₂³	—	4	1	no	—
279	C₂³	—	4	1	no	—
280	C₂³	—	4	1	no	—
281	C₂³	C₂²×C₂²⋊D₇₉	1	1	yes	characteristic
282	Q₈	—	4	24	no	—
283	Q₈	#14579082	1	16	yes	direct product factor
284	D₄	—	2	2	no	—
285	D₄	—	2	2	no	—
286	D₄	—	2	2	no	—
287	D₄	Q₈×Dic₇₉	1	4	yes	complemented
288	C₂×C₄	—	2	3	no	—
289	C₂×C₄	—	2	3	no	—
290	C₂×C₄	—	2	6	no	—
291	C₂×C₄	—	4	3	no	—
292	C₂×C₄	—	4	3	no	—
293	C₂×C₄	—	4	24	no	—
294	C₂×C₄	—	316	1	no	—
295	C₂×C₄	—	316	1	no	—
296	C₂×C₄	—	316	3	no	—
297	C₂×C₄	—	316	3	no	—
298	C₂×C₄	—	316	3	no	—
299	C₂×C₄	—	158	2	no	—
300	C₂×C₄	C₂×C₂²⋊Dic₇₉	1	6	yes	—
301	C₂×C₄	C₂×C₂²⋊Dic₇₉	1	6	yes	—
302	C₂×C₄	C₂×#11929609	1	6	yes	—
303	C₂×C₄	C₂×#11929612	1	6	yes	—
304	C₂×C₄	C₂³×Dic₇₉	1	1	yes	characteristic
305	C₂×C₄	C₂²×C₂²⋊D₇₉	1	1	yes	characteristic
306	C₂×C₄	C₂×Q₈×D₇₉	1	1	yes	characteristic
307	C₂×C₄	C₂×#11929625	1	1	yes	characteristic
308	C₂×C₄	#14579209	1	1	yes	characteristic
309	C₂×C₄	#14579209	1	1	yes	characteristic
310	C₈	—	316	3	no	—
311	C₈	—	158	2	no	—
312	C₂²	—	2	4	no	—
313	C₂²	—	4	2	no	—
314	C₂²	—	4	2	no	—
315	C₂²	—	4	2	no	—
316	C₂²	—	4	2	no	—
317	C₂²	—	4	2	no	—
318	C₂²	—	4	2	no	—
319	C₂²	C₂²×C₂²⋊Dic₇₉	1	1	yes	characteristic
320	C₂²	#19099224	1	1	yes	characteristic
321	C₂²	Q₈×C₂²⋊D₇₉	1	1	yes	characteristic
322	C₂²	C₂²×#11929609	1	1	yes	characteristic
323	C₂²	C₂²×#11929612	1	1	yes	characteristic
324	C₂²	#19099371	1	1	yes	characteristic
325	C₂²	#19099378	1	1	yes	characteristic
326	C₄	—	2	6	no	—
327	C₄	—	4	6	no	—
328	C₄	—	316	2	no	—
329	C₄	—	632	3	no	—
330	C₄	C₂×#14579082	1	12	yes	—
331	C₄	C₂×Q₈×Dic₇₉	1	1	yes	characteristic
332	C₄	#19099041	1	1	yes	characteristic
333	C₄	#19099224	1	1	yes	characteristic
334	C₄	Q₈×C₂²⋊D₇₉	1	1	yes	characteristic
335	C₂	—	4	4	no	—
336	C₂	Q₈×C₂²⋊Dic₇₉	1	1	yes	characteristic
337	C₂	C₂²×#14579082	1	1	yes	characteristic
338	C₂	#23771933	1	1	yes	characteristic
339	C₂	#23776035	1	1	yes	characteristic
340	C₂	#23776042	1	1	yes	characteristic
341	C₂	Q₈×#11929609	1	1	yes	characteristic
342	C₂	Q₈×#11929612	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	C₂²×C₂²⋊D₇₉
commutator	C₂×C₄×C₇₉	C₂³×C₄
Frattini	C₂²×C₄	C₂³×D₇₉
Fitting	C₂×D₄×Q₈×C₇₉	C₂
socle	C₂³×C₇₉	C₂²×D₄
prime radical	C₂²×D₄×C₇₉	C₂³
cosocle	C₂²×C₄×C₇₉	C₂⁴
hypercenter	C₂×D₄×Q₈	D₇₉
hypocenter	C₇₉	Q₈×#102
LCS	G ▷ C₂×C₄×C₇₉ ▷ C₂×C₇₉ ▷ C₇₉
UCS	C₁ ◁ C₂³ ◁ C₂×C₄×Q₈ ◁ C₂×D₄×Q₈

Complements

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

N	H ≅ G/N	type
1. C₂×Q₈×C₇₉⋊C₈	C₂	semidirect
2. Q₈×C₄⋊Dic₇₉	C₂	semidirect
3. #23764362	C₂	semidirect
4. #23764353	C₂	semidirect
22. D₄×Q₈×C₇₉	C₄	semidirect
37. #14579082	Q₈	direct (N × H)
66. C₂×C₇₉⋊C₈	C₂×Q₈	semidirect
72. C₄⋊Dic₇₉	C₂×Q₈	semidirect
111. Q₈×C₇₉	#102	semidirect
115. D₄×C₇₉	C₄×Q₈	semidirect
179. C₇₉	Q₈×#102	semidirect
182. D₄×Q₈	Dic₇₉	semidirect
283. Q₈	#14579082	direct (N × H)
287. D₄	Q₈×Dic₇₉	semidirect

Maximal subgroups

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

subgroup	index
C₂×Q₈×C₇₉⋊C₈	2
Q₈×C₄⋊Dic₇₉	2
#23764362	2
#23764353	2
#23763400	2
C₄×#14579082	2
C₂×D₄×Q₈×C₇₉	2
Q₈×#102	79

Subgroups — Q8×#14579082 #34304858

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — Q₈×#14579082 #34304858