Subgroups — #31819006

Order 17792.

360 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	#22581651	C₂	1	1	yes	maximal, characteristic, complemented
2	#22581647	C₂	1	1	yes	maximal, characteristic, complemented
3	#22581353	C₂	1	1	yes	maximal, characteristic, complemented
4	#22581279	C₂	1	1	yes	maximal, characteristic, complemented
5	#22581277	C₂	1	1	yes	maximal, characteristic, complemented
6	#22581268	C₂	1	1	yes	maximal, characteristic, complemented
7	#22581266	C₂	1	1	yes	maximal, characteristic, complemented
8	#22581256	C₂	1	1	yes	maximal, characteristic, complemented
9	#22581252	C₂	1	1	yes	maximal, characteristic, complemented
10	#22581207	C₂	1	1	yes	maximal, characteristic, complemented
11	#22581206	C₂	1	1	yes	maximal, characteristic, complemented
12	#22581035	C₂	1	1	yes	maximal, characteristic, complemented
13	#22581029	C₂	1	1	yes	maximal, characteristic, complemented
14	#22580994	C₂	1	1	yes	maximal, characteristic, complemented
15	#428×C₁₃₉	C₂	1	1	yes	maximal, characteristic, complemented
16	C₂×D₁₁₁₂	C₂²	1	1	yes	characteristic
17	C₂×#14555768	C₂²	1	1	yes	characteristic
18	#17984837	—	2	1	no	—
19	#17984836	—	2	1	no	—
20	#17984833	C₂²	1	1	yes	characteristic
21	#17984833	C₂²	1	1	yes	characteristic
22	#17984827	—	2	1	no	—
23	#17984824	—	2	1	no	—
24	#17984817	C₂²	1	1	yes	characteristic
25	#17984817	C₂²	1	1	yes	characteristic
26	#17984816	—	2	1	no	—
27	#17984816	C₂²	1	1	yes	characteristic
28	#17984816	C₂²	1	1	yes	characteristic
29	#17984815	—	2	1	no	—
30	C₄×D₅₅₆	—	2	1	no	—
31	#17984804	C₂²	1	1	yes	characteristic
32	#17984804	C₂²	1	1	yes	characteristic
33	C₄×C₂²⋊D₁₃₉	—	2	1	no	—
34	C₄×C₂²⋊D₁₃₉	C₂²	1	1	yes	characteristic
35	#17984800	C₂²	1	1	yes	characteristic
36	C₄⋊C₄×D₁₃₉	C₂²	1	1	yes	characteristic
37	#17984775	C₂²	1	1	yes	characteristic
38	#17984775	C₂²	1	1	yes	characteristic
39	#17984775	C₂²	1	1	yes	characteristic
40	#17984774	C₂²	1	1	yes	characteristic
41	#17984773	C₂²	1	1	yes	characteristic
42	Dic₁₃₉⋊C₈	C₂²	1	1	yes	characteristic
43	#17984764	C₂²	1	1	yes	characteristic
44	#17984764	C₂²	1	1	yes	characteristic
45	#17984762	C₂²	1	1	yes	characteristic
46	#17984762	C₂²	1	1	yes	characteristic
47	#17984762	C₂²	1	1	yes	characteristic
48	#17984762	C₂²	1	1	yes	characteristic
49	C₂²⋊(C₁₃₉⋊C₈)	C₂²	1	1	yes	characteristic
50	(C₂×D₁₃₉)⋊C₈	C₂²	1	1	yes	characteristic
51	C₈×Dic₁₃₉	C₂²	1	1	yes	characteristic
52	C₂×M₁₆×C₁₃₉	C₂²	1	1	yes	characteristic
53	#117×C₁₃₉	C₂²	1	1	yes	characteristic
54	C₂×C₄⋊C₄×C₁₃₉	C₂²	1	1	yes	characteristic
55	#107×C₁₃₉	C₂²	1	1	yes	characteristic
56	#107×C₁₃₉	C₂²	1	1	yes	characteristic
57	#106×C₁₃₉	C₂²	1	1	yes	characteristic
58	#106×C₁₃₉	C₂²	1	1	yes	characteristic
59	C₂×D₅₅₆	—	2	1	no	—
60	C₂×D₅₅₆	—	2	1	no	—
61	C₂×D₅₅₆	C₂³	1	1	yes	characteristic
62	C₂×D₅₅₆	C₂³	1	1	yes	characteristic
63	C₂×C₂²⋊D₁₃₉	—	2	1	no	—
64	C₂×C₂²⋊D₁₃₉	—	2	1	no	—
65	C₂×C₂²⋊D₁₃₉	D₄	1	1	yes	characteristic
66	C₂×C₄×D₁₃₉	—	2	1	no	—
67	C₂×C₄×D₁₃₉	—	2	1	no	—
68	C₂×C₄×D₁₃₉	—	2	1	no	—
69	C₂×C₄×D₁₃₉	C₂³	1	1	yes	characteristic
70	D₁₁₁₂	—	2	2	no	—
71	#14555768	—	2	2	no	—
72	C₂×C₁₃₉⋊C₈	C₂³	1	1	yes	characteristic
73	C₂×C₁₃₉⋊C₈	C₂³	1	1	yes	characteristic
74	C₄⋊Dic₁₃₉	—	2	1	no	—
75	C₄⋊Dic₁₃₉	—	2	1	no	—
76	C₄⋊Dic₁₃₉	C₂³	1	1	yes	characteristic
77	C₄⋊Dic₁₃₉	C₂³	1	1	yes	characteristic
78	Dic₁₃₉⋊C₄	—	2	1	no	—
79	Dic₁₃₉⋊C₄	—	2	1	no	—
80	Dic₁₃₉⋊C₄	—	2	1	no	—
81	Dic₁₃₉⋊C₄	D₄	1	1	yes	characteristic
82	C₂²⋊Dic₁₃₉	D₄	1	1	yes	characteristic
83	(C₂×D₁₃₉)⋊C₄	—	2	1	no	—
84	(C₂×D₁₃₉)⋊C₄	—	2	1	no	—
85	(C₂×D₁₃₉)⋊C₄	—	2	1	no	—
86	(C₂×D₁₃₉)⋊C₄	—	2	1	no	—
87	(C₂×D₁₃₉)⋊C₄	—	2	1	no	—
88	(C₂×D₁₃₉)⋊C₄	—	2	1	no	—
89	(C₂×D₁₃₉)⋊C₄	—	2	1	no	—
90	(C₂×D₁₃₉)⋊C₄	D₄	1	1	yes	characteristic
91	C₄×Dic₁₃₉	—	2	1	no	—
92	C₄×Dic₁₃₉	C₂³	1	1	yes	characteristic
93	C₂²×C₄×C₁₃₉	—	2	1	no	—
94	C₂²×C₄×C₁₃₉	C₂³	1	1	yes	characteristic
95	M₁₆×C₁₃₉	—	2	2	no	—
96	C₂×C₈×C₁₃₉	C₂³	1	1	yes	characteristic
97	C₂×C₈×C₁₃₉	C₂³	1	1	yes	characteristic
98	C₄⋊C₄×C₁₃₉	—	2	1	no	—
99	C₄⋊C₄×C₁₃₉	C₂³	1	1	yes	characteristic
100	C₄⋊C₄×C₁₃₉	C₂³	1	1	yes	characteristic
101	C₄⋊C₄×C₁₃₉	C₂³	1	1	yes	characteristic
102	C₄⋊C₄×C₁₃₉	C₂³	1	1	yes	characteristic
103	C₂²⋊C₄×C₁₃₉	—	2	1	no	—
104	C₄²×C₁₃₉	—	2	1	no	—
105	C₂²×D₁₃₉	—	2	1	no	—
106	C₂²×D₁₃₉	—	2	1	no	—
107	C₂²×D₁₃₉	C₂×D₄	1	1	yes	characteristic
108	D₅₅₆	—	2	1	no	—
109	D₅₅₆	—	2	1	no	—
110	D₅₅₆	—	2	1	no	—
111	D₅₅₆	—	2	1	no	—
112	D₅₅₆	—	4	2	no	—
113	D₅₅₆	—	8	1	no	—
114	C₂²⋊D₁₃₉	—	4	1	no	—
115	C₂²⋊D₁₃₉	—	4	2	no	—
116	C₂²⋊D₁₃₉	—	8	1	no	—
117	C₂×Dic₁₃₉	—	2	1	no	—
118	C₂×Dic₁₃₉	—	2	1	no	—
119	C₂×Dic₁₃₉	C₂×D₄	1	1	yes	characteristic
120	C₂×Dic₁₃₉	C₂×D₄	1	1	yes	characteristic
121	C₂×Dic₁₃₉	C₂×D₄	1	1	yes	characteristic
122	C₄×D₁₃₉	—	2	1	no	—
123	C₄×D₁₃₉	—	2	1	no	—
124	C₄×D₁₃₉	—	8	1	no	—
125	C₄×D₁₃₉	—	8	1	no	—
126	C₄×D₁₃₉	—	8	1	no	—
127	C₁₃₉⋊C₈	—	2	1	no	—
128	C₁₃₉⋊C₈	—	2	1	no	—
129	C₂³×C₁₃₉	C₂×D₄	1	1	yes	characteristic
130	C₂×C₄×C₁₃₉	—	2	1	no	—
131	C₂×C₄×C₁₃₉	—	2	1	no	—
132	C₂×C₄×C₁₃₉	—	2	1	no	—
133	C₂×C₄×C₁₃₉	—	2	1	no	—
134	C₂×C₄×C₁₃₉	—	2	1	no	—
135	C₂×C₄×C₁₃₉	—	2	1	no	—
136	C₂×C₄×C₁₃₉	—	8	1	no	—
137	C₂×C₄×C₁₃₉	C₂×D₄	1	1	yes	characteristic
138	C₂×C₄×C₁₃₉	C₂⁴	1	1	yes	characteristic
139	C₈×C₁₃₉	—	2	1	no	—
140	C₈×C₁₃₉	#41	1	2	yes	—
141	C₂²×C₁₃₉	—	2	1	no	—
142	C₂²×C₁₃₉	—	2	1	no	—
143	C₂²×C₁₃₉	—	2	1	no	—
144	C₂²×C₁₃₉	C₂²×D₄	1	1	yes	characteristic
145	C₂×D₁₃₉	—	2	1	no	—
146	C₂×D₁₃₉	—	2	1	no	—
147	C₂×D₁₃₉	—	2	1	no	—
148	C₂×D₁₃₉	—	4	1	no	—
149	C₂×D₁₃₉	—	4	1	no	—
150	C₂×D₁₃₉	—	4	1	no	—
151	C₂×D₁₃₉	—	4	1	no	—
152	C₂×D₁₃₉	—	4	1	no	—
153	C₂×D₁₃₉	—	4	1	no	—
154	C₄×C₁₃₉	—	2	1	no	—
155	C₄×C₁₃₉	—	4	1	no	—
156	C₄×C₁₃₉	—	4	1	no	—
157	C₄×C₁₃₉	—	4	1	no	—
158	C₄×C₁₃₉	—	4	1	no	—
159	C₄×C₁₃₉	C₂×#41	1	1	yes	characteristic
160	C₄×C₁₃₉	2₋¹⁺⁴	1	1	yes	characteristic
161	Dic₁₃₉	—	2	1	no	—
162	Dic₁₃₉	—	2	1	no	—
163	Dic₁₃₉	—	2	1	no	—
164	Dic₁₃₉	—	4	1	no	—
165	Dic₁₃₉	—	4	1	no	—
166	C₂×C₁₃₉	—	4	1	no	—
167	C₂×C₁₃₉	#547	1	1	yes	characteristic
168	C₂×C₁₃₉	#575	1	1	yes	characteristic
169	C₂×C₁₃₉	#576	1	1	yes	characteristic
170	D₁₃₉	—	4	1	no	—
171	D₁₃₉	—	8	1	no	—
172	D₁₃₉	—	8	1	no	—
173	C₁₃₉	#3355	1	1	yes	characteristic, complemented, Sylow
174	#3355	—	139	1	no	maximal, Sylow
175	#553	—	139	1	no	—
176	#553	—	139	1	no	—
177	#499	—	139	1	no	—
178	#482	—	139	1	no	—
179	#482	—	139	1	no	—
180	#480	—	139	1	no	—
181	#480	—	139	1	no	—
182	#478	—	139	1	no	—
183	#478	—	139	1	no	—
184	#469	—	139	1	no	—
185	#469	—	139	1	no	—
186	C₄×D₈	—	139	1	no	—
187	#436	—	139	1	no	—
188	#428	—	139	1	no	—
189	#428	D₁₃₉	1	1	yes	characteristic, complemented
190	C₂×D₈	—	139	1	no	—
191	C₂×M₁₆	—	139	1	no	—
192	C₂×M₁₆	C₂×D₁₃₉	1	1	yes	characteristic
193	#126	—	278	1	no	—
194	#126	—	278	1	no	—
195	#125	—	139	1	no	—
196	#125	—	139	1	no	—
197	#123	—	278	1	no	—
198	#123	—	278	1	no	—
199	#121	—	278	1	no	—
200	#121	—	278	1	no	—
201	#121	—	139	1	no	—
202	#121	—	139	1	no	—
203	#121	—	139	1	no	—
204	#121	—	139	1	no	—
205	C₄×D₄	—	278	1	no	—
206	C₄×D₄	—	278	1	no	—
207	C₄×D₄	—	139	1	no	—
208	C₄×D₄	—	139	1	no	—
209	C₄×D₄	—	139	1	no	—
210	#117	—	139	1	no	—
211	#117	C₂×D₁₃₉	1	1	yes	characteristic
212	C₂×C₄⋊C₄	—	139	1	no	—
213	C₂×C₄⋊C₄	C₂×D₁₃₉	1	1	yes	characteristic
214	#107	—	139	1	no	—
215	#107	—	139	1	no	—
216	#107	—	139	1	no	—
217	#107	C₂×D₁₃₉	1	1	yes	characteristic
218	#107	C₂×D₁₃₉	1	1	yes	characteristic
219	#106	—	139	1	no	—
220	#106	—	139	1	no	—
221	#106	C₂×D₁₃₉	1	1	yes	characteristic
222	#106	C₂×D₁₃₉	1	1	yes	characteristic
223	C₄⋊C₈	—	139	1	no	—
224	#102	—	139	1	no	—
225	#102	—	139	1	no	—
226	#102	—	139	1	no	—
227	#102	—	139	1	no	—
228	#102	—	139	1	no	—
229	#102	—	139	1	no	—
230	C₂²⋊C₈	—	139	1	no	—
231	C₂²⋊C₈	—	139	1	no	—
232	C₄×C₈	—	139	1	no	—
233	C₂×D₄	—	278	1	no	—
234	C₂×D₄	—	278	1	no	—
235	C₂×D₄	—	278	1	no	—
236	C₂×D₄	—	278	1	no	—
237	C₂×D₄	—	139	1	no	—
238	C₂×D₄	—	139	1	no	—
239	C₂×D₄	—	139	1	no	—
240	C₂²×C₄	—	2	1	no	—
241	C₂²×C₄	—	278	1	no	—
242	C₂²×C₄	—	278	1	no	—
243	C₂²×C₄	—	278	1	no	—
244	C₂²×C₄	—	139	1	no	—
245	C₂²×C₄	C₂²×D₁₃₉	1	1	yes	characteristic
246	D₈	—	278	2	no	—
247	M₁₆	—	2	2	no	—
248	M₁₆	—	278	2	no	—
249	C₂×C₈	—	139	1	no	—
250	C₂×C₈	—	139	1	no	—
251	C₂×C₈	C₂²×D₁₃₉	1	1	yes	characteristic
252	C₂×C₈	C₂²×D₁₃₉	1	1	yes	characteristic
253	C₄⋊C₄	—	2	1	no	—
254	C₄⋊C₄	—	278	1	no	—
255	C₄⋊C₄	—	278	1	no	—
256	C₄⋊C₄	—	278	1	no	—
257	C₄⋊C₄	—	278	1	no	—
258	C₄⋊C₄	—	278	1	no	—
259	C₄⋊C₄	—	139	1	no	—
260	C₄⋊C₄	—	139	1	no	—
261	C₄⋊C₄	—	139	1	no	—
262	C₄⋊C₄	C₂²×D₁₃₉	1	1	yes	characteristic
263	C₄⋊C₄	C₂²×D₁₃₉	1	1	yes	characteristic
264	C₄⋊C₄	C₂²×D₁₃₉	1	1	yes	characteristic
265	C₄⋊C₄	C₂²×D₁₃₉	1	1	yes	characteristic
266	C₂²⋊C₄	—	2	1	no	—
267	C₂²⋊C₄	—	278	1	no	—
268	C₂²⋊C₄	—	278	1	no	—
269	C₂²⋊C₄	—	278	1	no	—
270	C₂²⋊C₄	—	278	1	no	—
271	C₂²⋊C₄	—	278	1	no	—
272	C₂²⋊C₄	—	278	1	no	—
273	C₂²⋊C₄	—	278	1	no	—
274	C₂²⋊C₄	—	139	1	no	—
275	C₂²⋊C₄	—	139	1	no	—
276	C₄²	—	2	1	no	—
277	C₄²	—	278	1	no	—
278	C₄²	—	139	1	no	—
279	C₂³	—	278	1	no	—
280	C₂³	—	278	1	no	—
281	C₂³	—	139	1	no	—
282	C₂³	D₄×D₁₃₉	1	1	yes	characteristic
283	D₄	—	278	1	no	—
284	D₄	—	278	1	no	—
285	D₄	—	278	1	no	—
286	D₄	—	278	1	no	—
287	D₄	—	556	2	no	—
288	D₄	—	556	1	no	—
289	D₄	—	556	2	no	—
290	D₄	—	1112	1	no	—
291	D₄	—	1112	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	—	8	1	no	—
299	C₂×C₄	—	278	1	no	—
300	C₂×C₄	—	278	1	no	—
301	C₂×C₄	—	278	1	no	—
302	C₂×C₄	—	278	1	no	—
303	C₂×C₄	—	1112	1	no	—
304	C₂×C₄	—	1112	1	no	—
305	C₂×C₄	—	1112	1	no	—
306	C₂×C₄	—	139	1	no	—
307	C₂×C₄	—	139	1	no	—
308	C₂×C₄	—	139	1	no	—
309	C₂×C₄	D₄×D₁₃₉	1	1	yes	characteristic
310	C₂×C₄	C₂³×D₁₃₉	1	1	yes	characteristic
311	C₈	—	2	1	no	—
312	C₈	—	278	1	no	—
313	C₈	—	278	1	no	—
314	C₈	#14555785	1	2	yes	—
315	C₂²	—	2	1	no	—
316	C₂²	—	2	1	no	—
317	C₂²	—	2	1	no	—
318	C₂²	—	278	1	no	—
319	C₂²	—	278	1	no	—
320	C₂²	—	278	1	no	—
321	C₂²	—	556	1	no	—
322	C₂²	—	556	1	no	—
323	C₂²	—	556	1	no	—
324	C₂²	—	556	1	no	—
325	C₂²	—	556	1	no	—
326	C₂²	—	556	1	no	—
327	C₂²	C₂×D₄×D₁₃₉	1	1	yes	characteristic
328	C₄	—	2	1	no	—
329	C₄	—	4	1	no	—
330	C₄	—	4	1	no	—
331	C₄	—	4	1	no	—
332	C₄	—	4	1	no	—
333	C₄	—	278	1	no	—
334	C₄	—	278	1	no	—
335	C₄	—	278	1	no	—
336	C₄	—	556	1	no	—
337	C₄	—	556	1	no	—
338	C₄	C₂×#14555785	1	1	yes	characteristic
339	C₄	#17984890	1	1	yes	characteristic
340	C₂	—	4	1	no	—
341	C₂	—	556	1	no	—
342	C₂	—	1112	1	no	—
343	C₂	—	1112	1	no	—
344	C₂	#22581604	1	1	yes	characteristic
345	C₂	#22581752	1	1	yes	characteristic
346	C₂	#22581757	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×D₄×D₁₃₉
commutator, cosocle	C₂×C₄×C₁₃₉	C₂⁴
Frattini	C₂×C₄	C₂³×D₁₃₉
Fitting	#428×C₁₃₉	C₂
socle	C₂²×C₁₃₉	C₂²×D₄
hypercenter	#428	D₁₃₉
hypocenter	C₁₃₉	#3355
LCS	G ▷ C₂×C₄×C₁₃₉ ▷ C₂×C₁₃₉ ▷ C₁₃₉
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ #428

Complements

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

N	H ≅ G/N	type
1. #22581651	C₂	semidirect
2. #22581647	C₂	semidirect
3. #22581353	C₂	semidirect
4. #22581279	C₂	semidirect
5. #22581277	C₂	semidirect
6. #22581268	C₂	semidirect
7. #22581266	C₂	semidirect
8. #22581256	C₂	semidirect
9. #22581252	C₂	semidirect
10. #22581207	C₂	semidirect
11. #22581206	C₂	semidirect
12. #22581035	C₂	semidirect
13. #22581029	C₂	semidirect
14. #22580994	C₂	semidirect
15. #428×C₁₃₉	C₂	semidirect
173. C₁₃₉	#3355	semidirect
189. #428	D₁₃₉	semidirect

Maximal subgroups

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

subgroup	index
#22581651	2
#22581647	2
#22581353	2
#22581279	2
#22581277	2
#22581268	2
#22581266	2
#22581256	2
#22581252	2
#22581207	2
#22581206	2
#22581035	2
#22581029	2
#22580994	2
#428×C₁₃₉	2
#3355	139