Subgroups — #34444909

Order 20224.

674 conjugacy classes of subgroups.

The subgroup lattice has 390 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#19099004	C₂	1	2	yes	maximal, complemented
2	#23775160	C₂	1	1	yes	maximal, characteristic
3	#23765657	C₂	1	1	yes	maximal, characteristic, complemented
4	#23765515	C₂	1	2	yes	maximal, complemented
5	#23765447	C₂	1	1	yes	maximal, characteristic, complemented
6	#23765440	C₂	1	1	yes	maximal, characteristic, complemented
7	#23765338	C₂	1	2	yes	maximal, complemented
8	#23765317	C₂	1	2	yes	maximal, complemented
9	C₄×#14579107	C₂	1	1	yes	maximal, characteristic, complemented
10	#23765232	C₂	1	1	yes	maximal, characteristic, complemented
11	D₄×Q₁₆×C₇₉	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×#14579183	C₂²	1	2	yes	—
13	#19099234	—	2	1	no	—
14	#19099004	—	2	8	no	—
15	C₂×#14579107	C₂²	1	2	yes	—
16	C₂×#14579107	C₂²	1	1	yes	characteristic
17	C₂×#14579104	C₂²	1	2	yes	—
18	C₂×#14579104	C₂²	1	2	yes	—
19	C₂×#14579104	C₂²	1	2	yes	—
20	C₂×#14579101	C₂²	1	2	yes	—
21	#19098940	C₂²	1	1	yes	characteristic
22	#19098799	C₂²	1	2	yes	—
23	#19098764	—	2	1	no	—
24	#19098710	—	2	2	no	—
25	C₄×Dic₃₁₆	C₂²	1	1	yes	characteristic
26	#19098388	C₂²	1	1	yes	characteristic
27	C₄⋊(C₇₉⋊C₁₆)	C₂²	1	1	yes	characteristic
28	#19098367	C₂²	1	1	yes	characteristic
29	#19098367	C₂²	1	2	yes	—
30	#19098366	C₂²	1	1	yes	characteristic
31	#19098363	C₂²	1	2	yes	—
32	C₂²⋊(C₇₉⋊C₁₆)	C₂²	1	2	yes	—
33	C₄×C₇₉⋊C₁₆	C₂²	1	1	yes	characteristic
34	C₂²×Q₁₆×C₇₉	C₂²	1	2	yes	—
35	D₄×Q₈×C₇₉	—	2	1	no	—
36	#494×C₇₉	C₂²	1	1	yes	characteristic
37	#467×C₇₉	C₂²	1	2	yes	—
38	#462×C₇₉	—	2	1	no	—
39	#451×C₇₉	—	2	2	no	—
40	C₄×Q₁₆×C₇₉	C₂²	1	1	yes	characteristic
41	C₈×D₄×C₇₉	C₂²	1	1	yes	characteristic
42	C₂×#11929624	—	2	2	no	—
43	#14579183	—	2	4	no	—
44	#14579183	—	4	2	no	—
45	C₂×Dic₃₁₆	C₂³	1	1	yes	characteristic
46	C₂×#11929613	—	2	2	no	—
47	C₂×D₆₃₂	—	2	1	no	—
48	C₂×D₆₃₂	C₂³	1	2	yes	—
49	#14579158	—	2	1	no	—
50	#14579157	—	4	1	no	—
51	#14579135	—	2	2	no	—
52	#14579134	—	4	2	no	—
53	C₄×Dic₁₅₈	—	2	1	no	—
54	#14579123	—	4	1	no	—
55	#14579107	—	2	4	no	—
56	#14579107	D₄	1	4	yes	complemented
57	#14579104	—	2	4	no	—
58	#14579104	—	2	4	no	—
59	#14579104	—	4	2	no	—
60	#14579101	—	2	4	no	—
61	C₂×C₇₉⋊C₁₆	C₂³	1	2	yes	—
62	C₂×C₇₉⋊C₁₆	C₂³	1	1	yes	characteristic
63	C₂×C₇₉⋊C₁₆	C₂³	1	1	yes	characteristic
64	#14579095	C₂³	1	1	yes	characteristic
65	#14579086	—	2	1	no	—
66	#14579083	—	2	2	no	—
67	C₂²×Q₈×C₇₉	—	2	2	no	—
68	C₂×Q₁₆×C₇₉	—	2	4	no	—
69	C₂×Q₁₆×C₇₉	—	2	2	no	—
70	C₂×Q₁₆×C₇₉	—	2	1	no	—
71	C₂×Q₁₆×C₇₉	—	4	2	no	—
72	C₂×Q₁₆×C₇₉	C₂³	1	2	yes	—
73	C₂×Q₁₆×C₇₉	C₂³	1	1	yes	characteristic
74	C₂²×C₈×C₇₉	C₂³	1	2	yes	—
75	C₄⋊Q₈×C₇₉	—	2	1	no	—
76	C₄⋊Q₈×C₇₉	—	4	1	no	—
77	C₂²⋊Q₈×C₇₉	—	2	2	no	—
78	C₂²⋊Q₈×C₇₉	—	4	2	no	—
79	C₄×Q₈×C₇₉	—	2	1	no	—
80	C₄×D₄×C₇₉	—	4	1	no	—
81	C₄×D₄×C₇₉	D₄	1	1	yes	characteristic
82	#107×C₇₉	C₂³	1	1	yes	characteristic
83	C₄⋊C₈×C₇₉	D₄	1	1	yes	characteristic
84	#103×C₇₉	—	2	1	no	—
85	#103×C₇₉	—	2	2	no	—
86	C₂²⋊C₈×C₇₉	D₄	1	2	yes	—
87	C₄×C₈×C₇₉	C₂³	1	1	yes	characteristic
88	#11929624	—	4	4	no	—
89	#11929624	—	8	2	no	—
90	C₂×Dic₁₅₈	—	2	1	no	—
91	C₂×D₃₁₆	—	2	2	no	—
92	C₂×D₃₁₆	—	4	1	no	—
93	C₂×D₃₁₆	—	4	1	no	—
94	C₂×C₂²⋊D₇₉	—	4	2	no	—
95	C₂×C₂²⋊D₇₉	—	4	2	no	—
96	C₂×C₄×D₇₉	—	4	2	no	—
97	Dic₃₁₆	—	2	1	no	—
98	Dic₃₁₆	C₂×D₄	1	2	yes	—
99	#11929613	—	4	4	no	—
100	D₆₃₂	—	2	2	no	—
101	D₆₃₂	—	2	2	no	—
102	D₆₃₂	—	8	1	no	—
103	C₄⋊Dic₇₉	—	2	1	no	—
104	Dic₇₉⋊C₄	—	4	1	no	—
105	(C₂×D₇₉)⋊C₄	—	4	2	no	—
106	C₄×Dic₇₉	—	4	1	no	—
107	C₇₉⋊C₁₆	—	2	2	no	—
108	C₇₉⋊C₁₆	—	2	1	no	—
109	C₇₉⋊C₁₆	C₂×D₄	1	2	yes	—
110	C₂×Q₈×C₇₉	—	2	1	no	—
111	C₂×Q₈×C₇₉	—	2	2	no	—
112	C₂×Q₈×C₇₉	—	4	4	no	—
113	C₂×Q₈×C₇₉	—	4	2	no	—
114	C₂×Q₈×C₇₉	—	8	2	no	—
115	C₂×D₄×C₇₉	D₈	1	1	yes	characteristic
116	C₂²×C₄×C₇₉	—	4	2	no	—
117	C₂²×C₄×C₇₉	C₂×D₄	1	2	yes	—
118	Q₁₆×C₇₉	—	2	1	no	—
119	Q₁₆×C₇₉	—	2	2	no	—
120	Q₁₆×C₇₉	—	2	2	no	—
121	Q₁₆×C₇₉	—	4	4	no	—
122	Q₁₆×C₇₉	—	8	1	no	—
123	Q₁₆×C₇₉	C₂×D₄	1	2	yes	—
124	C₂×C₈×C₇₉	—	2	2	no	—
125	C₂×C₈×C₇₉	—	2	2	no	—
126	C₂×C₈×C₇₉	C₂×D₄	1	1	yes	characteristic
127	C₂×C₈×C₇₉	C₂×D₄	1	2	yes	—
128	C₂×C₈×C₇₉	C₂⁴	1	1	yes	characteristic
129	C₄⋊C₄×C₇₉	—	2	1	no	—
130	C₄⋊C₄×C₇₉	—	4	1	no	—
131	C₄⋊C₄×C₇₉	—	4	1	no	—
132	C₄⋊C₄×C₇₉	—	4	2	no	—
133	C₄⋊C₄×C₇₉	—	4	1	no	—
134	C₄⋊C₄×C₇₉	D₈	1	1	yes	characteristic
135	C₂²⋊C₄×C₇₉	—	4	2	no	—
136	C₂²⋊C₄×C₇₉	D₈	1	2	yes	—
137	C₄²×C₇₉	—	4	1	no	—
138	C₄²×C₇₉	C₂×D₄	1	1	yes	characteristic
139	C₂³×C₇₉	C₂×D₈	1	2	yes	—
140	C₂²×D₇₉	—	4	2	no	—
141	Q₈×C₇₉	—	2	2	no	—
142	Q₈×C₇₉	—	4	1	no	—
143	Q₈×C₇₉	—	4	2	no	—
144	Q₈×C₇₉	—	4	2	no	—
145	Q₈×C₇₉	—	8	4	no	—
146	D₄×C₇₉	—	2	2	no	—
147	C₂×C₄×C₇₉	—	2	2	no	—
148	C₂×C₄×C₇₉	—	2	2	no	—
149	C₂×C₄×C₇₉	—	4	1	no	—
150	C₂×C₄×C₇₉	—	4	1	no	—
151	C₂×C₄×C₇₉	—	4	2	no	—
152	C₂×C₄×C₇₉	—	8	4	no	—
153	C₂×C₄×C₇₉	C₂×D₈	1	2	yes	—
154	C₂×C₄×C₇₉	C₂×D₈	1	1	yes	characteristic
155	C₂×C₄×C₇₉	C₂×D₈	1	1	yes	characteristic
156	C₂×C₄×C₇₉	C₂²×D₄	1	1	yes	characteristic
157	C₂²⋊D₇₉	—	8	4	no	—
158	C₂²⋊D₇₉	—	16	2	no	—
159	C₂×Dic₇₉	—	4	1	no	—
160	C₂×Dic₇₉	—	4	1	no	—
161	D₃₁₆	—	4	2	no	—
162	D₃₁₆	—	4	2	no	—
163	D₃₁₆	—	8	2	no	—
164	D₃₁₆	—	16	1	no	—
165	C₄×D₇₉	—	8	4	no	—
166	Dic₁₅₈	—	2	2	no	—
167	Dic₁₅₈	—	4	1	no	—
168	C₈×C₇₉	—	2	2	no	—
169	C₈×C₇₉	—	2	1	no	—
170	C₈×C₇₉	C₂²×D₄	1	1	yes	characteristic
171	C₈×C₇₉	H(5,𝔽₂)	1	1	yes	characteristic
172	C₂²×C₇₉	—	2	2	no	—
173	C₂²×C₇₉	—	2	2	no	—
174	C₂²×C₇₉	—	2	2	no	—
175	C₂²×C₇₉	C₂²×D₈	1	1	yes	characteristic
176	C₂×D₇₉	—	8	2	no	—
177	C₂×D₇₉	—	8	2	no	—
178	C₂×D₇₉	—	8	2	no	—
179	C₄×C₇₉	—	2	1	no	—
180	C₄×C₇₉	—	2	2	no	—
181	C₄×C₇₉	—	4	2	no	—
182	C₄×C₇₉	—	8	1	no	—
183	C₄×C₇₉	—	8	2	no	—
184	C₄×C₇₉	#510	1	2	yes	complemented
185	C₄×C₇₉	D₄²	1	1	yes	characteristic
186	C₄×C₇₉	#576	1	1	yes	characteristic
187	Dic₇₉	—	4	2	no	—
188	Dic₇₉	—	8	1	no	—
189	#33844	—	79	1	no	maximal, Sylow
190	C₂×C₇₉	—	4	2	no	—
191	C₂×C₇₉	D₄×D₈	1	1	yes	characteristic
192	C₂×C₇₉	C₂×#510	1	1	yes	characteristic
193	C₂×C₇₉	#3416	1	1	yes	characteristic
194	D₇₉	—	16	2	no	—
195	C₂×#510	—	79	2	no	—
196	#3287	—	79	1	no	—
197	D₄×Q₁₆	D₇₉	1	1	yes	characteristic, complemented
198	#2248	—	79	1	no	—
199	#2220	—	79	2	no	—
200	#2209	—	79	1	no	—
201	#2207	—	79	1	no	—
202	#2189	—	79	2	no	—
203	#2185	—	79	2	no	—
204	C₄×Q₃₂	—	79	1	no	—
205	#2169	—	79	1	no	—
206	C₇₉	#33844	1	1	yes	characteristic, complemented, Sylow
207	C₂×#135	—	79	2	no	—
208	C₂²×Q₁₆	C₂×D₇₉	1	2	yes	—
209	#550	—	158	1	no	—
210	D₄×Q₈	—	2	1	no	—
211	#510	—	158	8	no	—
212	C₂×Q₃₂	—	79	1	no	—
213	C₂×Q₃₂	—	79	2	no	—
214	C₂×SD₃₂	—	79	2	no	—
215	C₂×SD₃₂	—	79	2	no	—
216	C₂×SD₃₂	—	79	2	no	—
217	C₂×M₃₂	—	79	2	no	—
218	#494	C₂×D₇₉	1	1	yes	characteristic
219	#493	—	79	1	no	—
220	#467	C₂×D₇₉	1	2	yes	—
221	#466	—	79	2	no	—
222	#462	—	2	1	no	—
223	#460	—	158	1	no	—
224	#451	—	2	2	no	—
225	#449	—	158	2	no	—
226	C₄×Q₁₆	—	79	1	no	—
227	C₄×Q₁₆	C₂×D₇₉	1	1	yes	characteristic
228	C₈×D₄	C₂×D₇₉	1	1	yes	characteristic
229	#366	—	79	1	no	—
230	C₄⋊C₁₆	—	79	1	no	—
231	#358	—	79	1	no	—
232	#358	—	79	2	no	—
233	#358	—	79	1	no	—
234	#357	—	79	2	no	—
235	C₂²⋊C₁₆	—	79	2	no	—
236	C₄×C₁₆	—	79	1	no	—
237	C₂×#41	—	158	2	no	—
238	C₂²×Q₈	—	2	2	no	—
239	#135	—	316	2	no	—
240	#135	—	158	4	no	—
241	C₂×Q₁₆	—	2	4	no	—
242	C₂×Q₁₆	—	2	2	no	—
243	C₂×Q₁₆	—	2	1	no	—
244	C₂×Q₁₆	—	4	2	no	—
245	C₂×Q₁₆	—	79	1	no	—
246	C₂×Q₁₆	C₂²×D₇₉	1	2	yes	—
247	C₂×Q₁₆	C₂²×D₇₉	1	1	yes	characteristic
248	C₂×SD₁₆	—	158	2	no	—
249	C₂×D₈	—	79	2	no	—
250	C₂×D₈	—	158	1	no	—
251	C₂²×C₈	C₂²×D₇₉	1	2	yes	—
252	C₄⋊Q₈	—	2	1	no	—
253	C₄⋊Q₈	—	4	1	no	—
254	#127	—	316	1	no	—
255	#127	—	158	1	no	—
256	C₂²⋊Q₈	—	2	2	no	—
257	C₂²⋊Q₈	—	4	2	no	—
258	#121	—	316	2	no	—
259	#121	—	158	2	no	—
260	C₄×Q₈	—	2	1	no	—
261	C₄×Q₈	—	158	1	no	—
262	C₄×D₄	—	4	1	no	—
263	C₄×D₄	—	316	1	no	—
264	C₄×D₄	C₂²⋊D₇₉	1	1	yes	characteristic
265	Q₃₂	—	79	4	no	—
266	Q₃₂	—	158	4	no	—
267	SD₃₂	—	316	2	no	—
268	SD₃₂	—	158	4	no	—
269	SD₃₂	—	158	4	no	—
270	M₃₂	—	158	4	no	—
271	C₂×C₁₆	—	79	1	no	—
272	C₂×C₁₆	—	79	1	no	—
273	C₂×C₁₆	—	79	2	no	—
274	#107	—	79	1	no	—
275	#107	C₂²×D₇₉	1	1	yes	characteristic
276	C₄⋊C₈	C₂²⋊D₇₉	1	1	yes	characteristic
277	#103	—	2	1	no	—
278	#103	—	2	2	no	—
279	#103	—	158	1	no	—
280	#102	—	158	2	no	—
281	C₂²⋊C₈	C₂²⋊D₇₉	1	2	yes	—
282	C₄×C₈	C₂²×D₇₉	1	1	yes	characteristic
283	#41	—	316	4	no	—
284	#41	—	632	2	no	—
285	C₂×Q₈	—	2	1	no	—
286	C₂×Q₈	—	2	2	no	—
287	C₂×Q₈	—	4	4	no	—
288	C₂×Q₈	—	4	2	no	—
289	C₂×Q₈	—	8	2	no	—
290	C₂×Q₈	—	158	1	no	—
291	C₂×D₄	—	316	2	no	—
292	C₂×D₄	—	316	2	no	—
293	C₂×D₄	—	316	1	no	—
294	C₂×D₄	—	316	1	no	—
295	C₂×D₄	—	158	2	no	—
296	C₂×D₄	#11929609	1	1	yes	characteristic
297	C₂²×C₄	—	4	2	no	—
298	C₂²×C₄	—	316	2	no	—
299	C₂²×C₄	C₂×C₂²⋊D₇₉	1	2	yes	—
300	Q₁₆	—	2	1	no	—
301	Q₁₆	—	2	2	no	—
302	Q₁₆	—	2	2	no	—
303	Q₁₆	—	4	4	no	—
304	Q₁₆	—	8	1	no	—
305	Q₁₆	—	79	2	no	—
306	Q₁₆	—	158	1	no	—
307	Q₁₆	D₄×D₇₉	1	2	yes	—
308	SD₁₆	—	316	4	no	—
309	D₈	—	632	1	no	—
310	D₈	—	158	2	no	—
311	D₈	—	158	2	no	—
312	C₂×C₈	—	2	2	no	—
313	C₂×C₈	—	2	2	no	—
314	C₂×C₈	C₂×C₂²⋊D₇₉	1	1	yes	characteristic
315	C₂×C₈	C₂×C₂²⋊D₇₉	1	2	yes	—
316	C₂×C₈	C₂³×D₇₉	1	1	yes	characteristic
317	C₄⋊C₄	—	2	1	no	—
318	C₄⋊C₄	—	4	1	no	—
319	C₄⋊C₄	—	4	1	no	—
320	C₄⋊C₄	—	4	2	no	—
321	C₄⋊C₄	—	4	1	no	—
322	C₄⋊C₄	—	316	1	no	—
323	C₄⋊C₄	—	158	1	no	—
324	C₄⋊C₄	#11929609	1	1	yes	characteristic
325	C₂²⋊C₄	—	4	2	no	—
326	C₂²⋊C₄	—	316	2	no	—
327	C₂²⋊C₄	#11929609	1	2	yes	—
328	C₄²	—	4	1	no	—
329	C₄²	—	316	1	no	—
330	C₄²	C₂×C₂²⋊D₇₉	1	1	yes	characteristic
331	C₁₆	—	79	2	no	—
332	C₁₆	—	158	1	no	—
333	C₁₆	—	158	2	no	—
334	C₂³	—	316	2	no	—
335	C₂³	C₂×#11929609	1	2	yes	—
336	Q₈	—	2	2	no	—
337	Q₈	—	4	1	no	—
338	Q₈	—	4	2	no	—
339	Q₈	—	4	2	no	—
340	Q₈	—	8	4	no	—
341	Q₈	—	316	1	no	—
342	Q₈	—	158	2	no	—
343	D₄	—	2	2	no	—
344	D₄	—	316	2	no	—
345	D₄	—	316	2	no	—
346	D₄	—	632	4	no	—
347	D₄	—	632	2	no	—
348	D₄	—	1264	1	no	—
349	D₄	—	1264	2	no	—
350	C₂×C₄	—	2	2	no	—
351	C₂×C₄	—	2	2	no	—
352	C₂×C₄	—	4	1	no	—
353	C₂×C₄	—	4	1	no	—
354	C₂×C₄	—	4	2	no	—
355	C₂×C₄	—	8	4	no	—
356	C₂×C₄	—	316	1	no	—
357	C₂×C₄	—	316	1	no	—
358	C₂×C₄	—	632	4	no	—
359	C₂×C₄	C₂×#11929609	1	1	yes	characteristic
360	C₂×C₄	C₂×#11929609	1	2	yes	—
361	C₂×C₄	C₂×#11929609	1	1	yes	characteristic
362	C₂×C₄	C₂²×C₂²⋊D₇₉	1	1	yes	characteristic
363	C₈	—	2	2	no	—
364	C₈	—	2	1	no	—
365	C₈	C₂×D₄×D₇₉	1	1	yes	characteristic
366	C₈	#14579206	1	1	yes	characteristic
367	C₂²	—	2	2	no	—
368	C₂²	—	2	2	no	—
369	C₂²	—	2	2	no	—
370	C₂²	—	632	2	no	—
371	C₂²	—	632	2	no	—
372	C₂²	—	632	2	no	—
373	C₂²	C₂²×#11929609	1	1	yes	characteristic
374	C₄	—	2	1	no	—
375	C₄	—	2	2	no	—
376	C₄	—	4	2	no	—
377	C₄	—	8	1	no	—
378	C₄	—	8	2	no	—
379	C₄	—	316	2	no	—
380	C₄	—	632	1	no	—
381	C₄	#19099004	1	2	yes	complemented
382	C₄	D₄×C₂²⋊D₇₉	1	1	yes	characteristic
383	C₄	#19099366	1	1	yes	characteristic
384	C₂	—	4	2	no	—
385	C₂	—	1264	2	no	—
386	C₂	D₄×#11929609	1	1	yes	characteristic
387	C₂	C₂×#19099004	1	1	yes	characteristic
388	C₂	#23776544	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂²×#11929609
commutator, cosocle	C₂×C₈×C₇₉	C₂⁴
Frattini	C₂×C₈	C₂³×D₇₉
Fitting	D₄×Q₁₆×C₇₉	C₂
socle	C₂²×C₇₉	C₂²×D₈
prime radical	#23775160	C₂
hypercenter	D₄×Q₁₆	D₇₉
hypocenter	C₇₉	#33844
LCS	G ▷ C₂×C₈×C₇₉ ▷ C₄×C₇₉ ▷ C₂×C₇₉ ▷ C₇₉
UCS	C₁ ◁ C₂² ◁ C₄×D₄ ◁ C₈×D₄ ◁ D₄×Q₁₆

Complements

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

N	H ≅ G/N	type
1. C₂×#19099004	C₂	semidirect
3. #23765657	C₂	semidirect
4. #23765515	C₂	semidirect
5. #23765447	C₂	semidirect
6. #23765440	C₂	semidirect
7. #23765338	C₂	semidirect
8. #23765317	C₂	semidirect
9. C₄×#14579107	C₂	semidirect
10. #23765232	C₂	semidirect
11. D₄×Q₁₆×C₇₉	C₂	semidirect
56. #14579107	D₄	semidirect
184. C₄×C₇₉	#510	semidirect
197. D₄×Q₁₆	D₇₉	semidirect
206. C₇₉	#33844	semidirect
381. C₄	#19099004	semidirect

Maximal subgroups

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

subgroup	index
C₂×#19099004	2
#23775160	2
#23765657	2
#23765515	2
#23765447	2
#23765440	2
#23765338	2
#23765317	2
C₄×#14579107	2
#23765232	2
D₄×Q₁₆×C₇₉	2
#33844	79