Subgroups — #34442545

Order 20224.

770 conjugacy classes of subgroups.

The subgroup lattice has 386 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#19098999	C₂	1	2	yes	maximal, complemented
2	#23773828	C₂	1	1	yes	maximal, characteristic, complemented
3	#23765455	C₂	1	4	yes	maximal, complemented
4	#23765445	C₂	1	2	yes	maximal, complemented
5	#23765435	C₂	1	2	yes	maximal, complemented
6	#23765156	C₂	1	1	yes	maximal, characteristic, complemented
7	#23765110	C₂	1	2	yes	maximal, complemented
8	C₂×C₄×D₈×C₇₉	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂×#14579183	—	2	1	no	—
10	C₂×#14579183	C₂²	1	2	yes	—
11	#19099113	—	2	1	no	—
12	#19098999	—	2	8	no	—
13	C₂×#14579105	C₂²	1	4	yes	—
14	C₂×#14579102	C₂²	1	4	yes	—
15	C₂×#14579101	C₂²	1	2	yes	—
16	#19098947	—	2	1	no	—
17	#19098947	C₂²	1	2	yes	complemented
18	#19098943	C₂²	1	1	yes	characteristic, complemented
19	#19098940	C₂²	1	1	yes	characteristic, complemented
20	#19098936	—	2	1	no	—
21	C₄⋊(C₇₉⋊C₁₆)	C₂²	1	2	yes	complemented
22	C₄⋊(C₇₉⋊C₁₆)	C₂²	1	2	yes	complemented
23	#19098366	C₂²	1	4	yes	complemented
24	#19098363	C₂²	1	4	yes	complemented
25	C₂²×D₈×C₇₉	C₂²	1	1	yes	characteristic
26	C₂×C₄×D₄×C₇₉	—	2	1	no	—
27	C₄×D₈×C₇₉	—	2	1	no	—
28	C₄×D₈×C₇₉	—	2	1	no	—
29	C₄×D₈×C₇₉	C₂²	1	2	yes	—
30	C₄×D₈×C₇₉	C₂²	1	2	yes	—
31	C₂×#107×C₇₉	C₂²	1	1	yes	characteristic
32	C₂×#102×C₇₉	—	2	1	no	—
33	C₂×C₄×C₈×C₇₉	C₂²	1	1	yes	characteristic
34	C₂×#11929624	—	2	2	no	—
35	#14579183	—	2	4	no	—
36	#14579183	—	2	4	no	—
37	#14579183	—	4	4	no	—
38	C₂×Dic₃₁₆	—	2	1	no	—
39	C₂×Dic₃₁₆	C₂³	1	2	yes	—
40	C₂×#11929613	—	2	2	no	—
41	C₂×#11929613	—	2	2	no	—
42	C₂×D₆₃₂	—	2	1	no	—
43	C₂×D₆₃₂	C₂³	1	2	yes	—
44	#14579161	—	2	1	no	—
45	#14579158	—	2	1	no	—
46	#14579149	—	2	2	no	—
47	#14579135	—	4	1	no	—
48	#14579135	—	4	1	no	—
49	C₄×D₃₁₆	—	4	1	no	—
50	C₄×D₃₁₆	—	4	1	no	—
51	#14579105	—	2	8	no	—
52	#14579102	—	2	8	no	—
53	#14579101	—	2	4	no	—
54	C₂×C₇₉⋊C₁₆	C₂³	1	4	yes	—
55	C₂²×D₄×C₇₉	—	2	1	no	—
56	C₂³×C₄×C₇₉	—	4	1	no	—
57	C₂×D₈×C₇₉	—	2	1	no	—
58	C₂×D₈×C₇₉	—	2	2	no	—
59	C₂×D₈×C₇₉	D₄	1	4	yes	complemented
60	C₂×D₈×C₇₉	C₂³	1	2	yes	—
61	C₂²×C₈×C₇₉	D₄	1	1	yes	characteristic
62	C₂²×C₈×C₇₉	C₂³	1	1	yes	characteristic
63	C₄×D₄×C₇₉	—	2	2	no	—
64	C₄×D₄×C₇₉	—	2	2	no	—
65	C₄×D₄×C₇₉	—	4	1	no	—
66	C₄×D₄×C₇₉	—	4	1	no	—
67	C₂×C₄⋊C₄×C₇₉	—	2	1	no	—
68	C₂×C₂²⋊C₄×C₇₉	—	4	1	no	—
69	C₂×C₄²×C₇₉	D₄	1	1	yes	characteristic
70	#107×C₇₉	—	2	1	no	—
71	#107×C₇₉	C₂³	1	2	yes	—
72	#102×C₇₉	—	2	2	no	—
73	#102×C₇₉	—	2	2	no	—
74	C₄×C₈×C₇₉	D₄	1	1	yes	characteristic
75	C₄×C₈×C₇₉	D₄	1	1	yes	characteristic
76	C₄×C₈×C₇₉	C₂³	1	1	yes	characteristic
77	C₄×C₈×C₇₉	C₂³	1	1	yes	characteristic
78	#11929624	—	4	4	no	—
79	#11929624	—	4	4	no	—
80	C₂×Dic₁₅₈	—	2	2	no	—
81	C₂×D₃₁₆	—	2	2	no	—
82	C₂×D₃₁₆	—	4	1	no	—
83	C₂×D₃₁₆	—	4	1	no	—
84	C₂×C₂²⋊D₇₉	—	4	2	no	—
85	C₂×C₄×D₇₉	—	4	2	no	—
86	Dic₃₁₆	—	2	2	no	—
87	Dic₃₁₆	—	2	2	no	—
88	Dic₃₁₆	—	4	2	no	—
89	#11929613	—	4	4	no	—
90	#11929613	—	4	4	no	—
91	D₆₃₂	—	2	2	no	—
92	D₆₃₂	—	2	2	no	—
93	D₆₃₂	—	4	2	no	—
94	C₄⋊Dic₇₉	—	4	1	no	—
95	C₄⋊Dic₇₉	—	4	1	no	—
96	(C₂×D₇₉)⋊C₄	—	4	2	no	—
97	(C₂×D₇₉)⋊C₄	—	4	2	no	—
98	C₇₉⋊C₁₆	—	2	4	no	—
99	C₂⁴×C₇₉	—	4	1	no	—
100	C₂×D₄×C₇₉	—	2	4	no	—
101	C₂×D₄×C₇₉	—	2	2	no	—
102	C₂×D₄×C₇₉	—	4	1	no	—
103	C₂×D₄×C₇₉	—	4	2	no	—
104	C₂²×C₄×C₇₉	—	4	1	no	—
105	C₂²×C₄×C₇₉	—	4	2	no	—
106	C₂²×C₄×C₇₉	—	4	2	no	—
107	C₂²×C₄×C₇₉	—	8	4	no	—
108	C₂²×C₄×C₇₉	D₈	1	1	yes	characteristic
109	C₂²×C₄×C₇₉	D₈	1	1	yes	characteristic
110	C₂²×C₄×C₇₉	C₂×D₄	1	1	yes	characteristic
111	D₈×C₇₉	—	2	4	no	—
112	D₈×C₇₉	—	2	2	no	—
113	D₈×C₇₉	C₂×D₄	1	4	yes	—
114	C₂×C₈×C₇₉	—	2	2	no	—
115	C₂×C₈×C₇₉	C₂×D₄	1	2	yes	—
116	C₂×C₈×C₇₉	C₂×D₄	1	1	yes	characteristic
117	C₂×C₈×C₇₉	C₂×D₄	1	1	yes	characteristic
118	C₂×C₈×C₇₉	C₂×D₄	1	1	yes	characteristic
119	C₂×C₈×C₇₉	#41	1	2	yes	—
120	C₂×C₈×C₇₉	C₂⁴	1	1	yes	characteristic
121	C₄⋊C₄×C₇₉	—	2	2	no	—
122	C₄⋊C₄×C₇₉	—	4	1	no	—
123	C₂²⋊C₄×C₇₉	—	4	2	no	—
124	C₂²⋊C₄×C₇₉	—	4	2	no	—
125	C₄²×C₇₉	D₈	1	1	yes	characteristic
126	C₄²×C₇₉	D₈	1	1	yes	characteristic
127	C₄²×C₇₉	C₂×D₄	1	1	yes	characteristic
128	C₄²×C₇₉	C₂×D₄	1	1	yes	characteristic
129	C₂³×C₇₉	—	4	2	no	—
130	C₂³×C₇₉	—	4	4	no	—
131	C₂³×C₇₉	—	8	2	no	—
132	C₂³×C₇₉	—	8	2	no	—
133	C₂³×C₇₉	C₂×D₈	1	1	yes	characteristic
134	C₂²×D₇₉	—	4	2	no	—
135	D₄×C₇₉	—	2	4	no	—
136	D₄×C₇₉	—	4	2	no	—
137	D₄×C₇₉	—	4	4	no	—
138	C₂×C₄×C₇₉	—	2	2	no	—
139	C₂×C₄×C₇₉	—	4	1	no	—
140	C₂×C₄×C₇₉	—	4	2	no	—
141	C₂×C₄×C₇₉	—	8	4	no	—
142	C₂×C₄×C₇₉	—	8	2	no	—
143	C₂×C₄×C₇₉	—	8	4	no	—
144	C₂×C₄×C₇₉	#121	1	2	yes	—
145	C₂×C₄×C₇₉	C₂×D₈	1	1	yes	characteristic
146	C₂×C₄×C₇₉	C₂×D₈	1	1	yes	characteristic
147	C₂×C₄×C₇₉	C₂×D₈	1	1	yes	characteristic
148	C₂×C₄×C₇₉	C₂×D₈	1	1	yes	characteristic
149	C₂×C₄×C₇₉	C₂×D₈	1	1	yes	characteristic
150	C₂×C₄×C₇₉	AGL(1,ℤ/8)	1	2	yes	—
151	C₂×C₄×C₇₉	C₂²×D₄	1	1	yes	characteristic
152	C₂²⋊D₇₉	—	8	4	no	—
153	C₂×Dic₇₉	—	4	2	no	—
154	D₃₁₆	—	4	2	no	—
155	D₃₁₆	—	4	2	no	—
156	D₃₁₆	—	8	2	no	—
157	D₃₁₆	—	8	2	no	—
158	C₄×D₇₉	—	8	4	no	—
159	Dic₁₅₈	—	4	2	no	—
160	Dic₁₅₈	—	4	2	no	—
161	C₈×C₇₉	—	2	1	no	—
162	C₈×C₇₉	—	2	1	no	—
163	C₈×C₇₉	#121	1	2	yes	—
164	C₈×C₇₉	C₂²×D₄	1	1	yes	characteristic
165	C₈×C₇₉	C₂×#41	1	1	yes	characteristic
166	C₂²×C₇₉	—	2	1	no	—
167	C₂²×C₇₉	—	2	1	no	—
168	C₂²×C₇₉	—	4	4	no	—
169	C₂²×C₇₉	—	8	8	no	—
170	C₂²×C₇₉	—	8	2	no	—
171	C₂²×C₇₉	—	8	2	no	—
172	C₂²×C₇₉	#459	1	2	yes	—
173	C₂²×C₇₉	C₂²×D₈	1	1	yes	characteristic
174	C₂×D₇₉	—	8	2	no	—
175	C₂×D₇₉	—	8	2	no	—
176	C₂×D₇₉	—	8	2	no	—
177	C₄×C₇₉	—	2	1	no	—
178	C₄×C₇₉	—	2	1	no	—
179	C₄×C₇₉	—	2	1	no	—
180	C₄×C₇₉	—	8	2	no	—
181	C₄×C₇₉	#459	1	2	yes	—
182	C₄×C₇₉	#509	1	2	yes	complemented
183	C₄×C₇₉	C₂×#121	1	1	yes	characteristic
184	C₄×C₇₉	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
185	Dic₇₉	—	8	2	no	—
186	#33580	—	79	1	no	maximal, Sylow
187	C₂×C₇₉	—	2	2	no	—
188	C₂×C₇₉	—	8	4	no	—
189	C₂×C₇₉	C₂×#459	1	1	yes	characteristic
190	C₂×C₇₉	C₂×#509	1	1	yes	characteristic
191	C₂×C₇₉	#3414	1	1	yes	characteristic
192	D₇₉	—	16	2	no	—
193	C₂×#509	—	79	2	no	—
194	#3147	—	79	1	no	—
195	C₂×C₄×D₈	D₇₉	1	1	yes	characteristic, complemented
196	#2210	—	79	4	no	—
197	#2208	—	79	2	no	—
198	#2206	—	79	2	no	—
199	#2150	—	79	1	no	—
200	#2139	—	79	2	no	—
201	C₇₉	#33580	1	1	yes	characteristic, complemented, Sylow
202	C₂×#135	—	79	2	no	—
203	C₂×#135	—	158	1	no	—
204	C₂²×D₈	C₂×D₇₉	1	1	yes	characteristic
205	#532	—	158	1	no	—
206	C₂×C₄×D₄	—	2	1	no	—
207	#509	—	158	8	no	—
208	C₂×SD₃₂	—	79	4	no	—
209	C₂×D₁₆	—	79	4	no	—
210	C₂×M₃₂	—	79	2	no	—
211	#495	—	79	2	no	—
212	#495	—	158	1	no	—
213	#494	—	79	1	no	—
214	#493	—	79	1	no	—
215	#492	—	158	1	no	—
216	C₄×D₈	—	2	1	no	—
217	C₄×D₈	—	2	1	no	—
218	C₄×D₈	C₂×D₇₉	1	2	yes	—
219	C₄×D₈	C₂×D₇₉	1	2	yes	—
220	C₂×#107	C₂×D₇₉	1	1	yes	characteristic
221	C₂×#102	—	2	1	no	—
222	C₂×C₄×C₈	C₂×D₇₉	1	1	yes	characteristic
223	C₄⋊C₁₆	—	79	2	no	—
224	C₄⋊C₁₆	—	79	2	no	—
225	#358	—	79	4	no	—
226	#357	—	79	4	no	—
227	C₂×#41	—	158	2	no	—
228	C₂²×D₄	—	2	1	no	—
229	C₂³×C₄	—	4	1	no	—
230	#135	—	316	4	no	—
231	#135	—	158	4	no	—
232	#135	—	158	4	no	—
233	C₂×Q₁₆	—	79	2	no	—
234	C₂×Q₁₆	—	158	1	no	—
235	C₂×SD₁₆	—	158	2	no	—
236	C₂×SD₁₆	—	158	2	no	—
237	C₂×D₈	—	2	1	no	—
238	C₂×D₈	—	2	2	no	—
239	C₂×D₈	—	79	2	no	—
240	C₂×D₈	—	158	1	no	—
241	C₂×D₈	D₃₁₆	1	4	yes	complemented
242	C₂×D₈	C₂²×D₇₉	1	2	yes	—
243	C₂²×C₈	C₂²⋊D₇₉	1	1	yes	characteristic
244	C₂²×C₈	C₂²×D₇₉	1	1	yes	characteristic
245	C₄⋊Q₈	—	158	1	no	—
246	#127	—	158	1	no	—
247	#124	—	158	2	no	—
248	#121	—	316	1	no	—
249	#121	—	316	1	no	—
250	C₄×D₄	—	2	2	no	—
251	C₄×D₄	—	2	2	no	—
252	C₄×D₄	—	4	1	no	—
253	C₄×D₄	—	4	1	no	—
254	C₄×D₄	—	316	1	no	—
255	C₄×D₄	—	316	1	no	—
256	C₂×C₄⋊C₄	—	2	1	no	—
257	C₂×C₂²⋊C₄	—	4	1	no	—
258	C₂×C₄²	C₂²⋊D₇₉	1	1	yes	characteristic
259	SD₃₂	—	158	8	no	—
260	D₁₆	—	158	8	no	—
261	M₃₂	—	158	4	no	—
262	C₂×C₁₆	—	79	4	no	—
263	#107	—	2	1	no	—
264	#107	C₂²×D₇₉	1	2	yes	—
265	#102	—	2	2	no	—
266	#102	—	2	2	no	—
267	C₄×C₈	C₂²⋊D₇₉	1	1	yes	characteristic
268	C₄×C₈	C₂²⋊D₇₉	1	1	yes	characteristic
269	C₄×C₈	C₂²×D₇₉	1	1	yes	characteristic
270	C₄×C₈	C₂²×D₇₉	1	1	yes	characteristic
271	C₂⁴	—	4	1	no	—
272	#41	—	316	4	no	—
273	#41	—	316	4	no	—
274	C₂×Q₈	—	158	2	no	—
275	C₂×D₄	—	2	4	no	—
276	C₂×D₄	—	2	2	no	—
277	C₂×D₄	—	4	1	no	—
278	C₂×D₄	—	4	2	no	—
279	C₂×D₄	—	316	2	no	—
280	C₂×D₄	—	316	1	no	—
281	C₂×D₄	—	316	1	no	—
282	C₂×D₄	—	158	2	no	—
283	C₂²×C₄	—	4	1	no	—
284	C₂²×C₄	—	4	2	no	—
285	C₂²×C₄	—	4	2	no	—
286	C₂²×C₄	—	8	4	no	—
287	C₂²×C₄	—	316	2	no	—
288	C₂²×C₄	#11929609	1	1	yes	characteristic
289	C₂²×C₄	#11929609	1	1	yes	characteristic
290	C₂²×C₄	C₂×C₂²⋊D₇₉	1	1	yes	characteristic
291	Q₁₆	—	316	2	no	—
292	Q₁₆	—	158	2	no	—
293	Q₁₆	—	158	2	no	—
294	SD₁₆	—	316	4	no	—
295	SD₁₆	—	316	4	no	—
296	D₈	—	2	2	no	—
297	D₈	—	2	4	no	—
298	D₈	—	316	2	no	—
299	D₈	—	158	2	no	—
300	D₈	—	158	2	no	—
301	D₈	C₂×D₃₁₆	1	4	yes	—
302	C₂×C₈	—	2	2	no	—
303	C₂×C₈	C₂×C₂²⋊D₇₉	1	1	yes	characteristic
304	C₂×C₈	C₂×C₂²⋊D₇₉	1	1	yes	characteristic
305	C₂×C₈	C₂×C₂²⋊D₇₉	1	1	yes	characteristic
306	C₂×C₈	C₂×D₃₁₆	1	2	yes	—
307	C₂×C₈	#11929624	1	2	yes	—
308	C₂×C₈	C₂³×D₇₉	1	1	yes	characteristic
309	C₄⋊C₄	—	2	2	no	—
310	C₄⋊C₄	—	4	1	no	—
311	C₄⋊C₄	—	316	1	no	—
312	C₄⋊C₄	—	316	1	no	—
313	C₂²⋊C₄	—	4	2	no	—
314	C₂²⋊C₄	—	4	2	no	—
315	C₂²⋊C₄	—	316	2	no	—
316	C₂²⋊C₄	—	316	2	no	—
317	C₄²	#11929609	1	1	yes	characteristic
318	C₄²	#11929609	1	1	yes	characteristic
319	C₄²	C₂×C₂²⋊D₇₉	1	1	yes	characteristic
320	C₄²	C₂×C₂²⋊D₇₉	1	1	yes	characteristic
321	C₁₆	—	158	4	no	—
322	C₂³	—	4	4	no	—
323	C₂³	—	4	2	no	—
324	C₂³	—	8	2	no	—
325	C₂³	—	8	2	no	—
326	C₂³	—	316	2	no	—
327	C₂³	C₂×#11929609	1	1	yes	characteristic
328	Q₈	—	316	2	no	—
329	Q₈	—	316	2	no	—
330	D₄	—	2	4	no	—
331	D₄	—	4	4	no	—
332	D₄	—	4	2	no	—
333	D₄	—	316	2	no	—
334	D₄	—	316	2	no	—
335	D₄	—	632	2	no	—
336	D₄	—	632	4	no	—
337	D₄	—	632	2	no	—
338	C₂×C₄	—	2	2	no	—
339	C₂×C₄	—	4	1	no	—
340	C₂×C₄	—	4	2	no	—
341	C₂×C₄	—	8	4	no	—
342	C₂×C₄	—	8	2	no	—
343	C₂×C₄	—	8	4	no	—
344	C₂×C₄	—	316	2	no	—
345	C₂×C₄	—	632	4	no	—
346	C₂×C₄	#14579135	1	2	yes	—
347	C₂×C₄	C₂×#11929609	1	1	yes	characteristic
348	C₂×C₄	C₂×#11929609	1	1	yes	characteristic
349	C₂×C₄	C₂×#11929609	1	1	yes	characteristic
350	C₂×C₄	C₂×#11929609	1	1	yes	characteristic
351	C₂×C₄	C₂×#11929609	1	1	yes	characteristic
352	C₂×C₄	#14579186	1	2	yes	—
353	C₂×C₄	C₂²×C₂²⋊D₇₉	1	1	yes	characteristic
354	C₈	—	2	1	no	—
355	C₈	—	2	1	no	—
356	C₈	#14579135	1	2	yes	—
357	C₈	C₂²×D₃₁₆	1	1	yes	characteristic
358	C₈	C₂×#11929624	1	1	yes	characteristic
359	C₂²	—	2	1	no	—
360	C₂²	—	2	1	no	—
361	C₂²	—	4	4	no	—
362	C₂²	—	8	8	no	—
363	C₂²	—	8	2	no	—
364	C₂²	—	8	2	no	—
365	C₂²	—	632	2	no	—
366	C₂²	—	632	2	no	—
367	C₂²	—	632	2	no	—
368	C₂²	#19098758	1	2	yes	—
369	C₂²	C₂²×#11929609	1	1	yes	characteristic
370	C₄	—	2	1	no	—
371	C₄	—	2	1	no	—
372	C₄	—	2	1	no	—
373	C₄	—	8	2	no	—
374	C₄	—	632	2	no	—
375	C₄	#19098758	1	2	yes	—
376	C₄	#19098999	1	2	yes	complemented
377	C₄	C₂×#14579135	1	1	yes	characteristic
378	C₄	C₂×#14579186	1	1	yes	characteristic
379	C₂	—	2	2	no	—
380	C₂	—	8	4	no	—
381	C₂	—	1264	2	no	—
382	C₂	C₂×#19098758	1	1	yes	characteristic
383	C₂	C₂×#19098999	1	1	yes	characteristic
384	C₂	#23776534	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂²×#11929609
commutator, cosocle	C₂×C₈×C₇₉	C₂⁴
Frattini	C₂×C₈	C₂³×D₇₉
Fitting	C₂×C₄×D₈×C₇₉	C₂
socle	C₂²×C₇₉	C₂²×D₈
hypercenter	C₂×C₄×D₈	D₇₉
hypocenter	C₇₉	#33580
LCS	G ▷ C₂×C₈×C₇₉ ▷ C₄×C₇₉ ▷ C₂×C₇₉ ▷ C₇₉
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ C₂×C₄×D₈

Complements

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

N	H ≅ G/N	type
1. C₂×#19098999	C₂	semidirect
2. #23773828	C₂	semidirect
3. #23765455	C₂	semidirect
4. #23765445	C₂	semidirect
5. #23765435	C₂	semidirect
6. #23765156	C₂	semidirect
7. #23765110	C₂	semidirect
8. C₂×C₄×D₈×C₇₉	C₂	semidirect
17. #19098947	C₂²	semidirect
18. #19098943	C₂²	semidirect
19. #19098940	C₂²	semidirect
21. C₄⋊(C₇₉⋊C₁₆)	C₂²	semidirect
22. C₄⋊(C₇₉⋊C₁₆)	C₂²	semidirect
23. #19098366	C₂²	semidirect
24. #19098363	C₂²	semidirect
59. C₂×D₈×C₇₉	D₄	semidirect
182. C₄×C₇₉	#509	semidirect
195. C₂×C₄×D₈	D₇₉	semidirect
201. C₇₉	#33580	semidirect
241. C₂×D₈	D₃₁₆	semidirect
376. C₄	#19098999	semidirect

Maximal subgroups

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

subgroup	index
C₂×#19098999	2
#23773828	2
#23765455	2
#23765445	2
#23765435	2
#23765156	2
#23765110	2
C₂×C₄×D₈×C₇₉	2
#33580	79