Subgroups — #31818994

Order 17792.

386 conjugacy classes of subgroups.

The subgroup lattice has 354 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#22581667	C₂	1	1	yes	maximal, characteristic, complemented
2	#22581603	C₂	1	1	yes	maximal, characteristic, complemented
3	#22581350	C₂	1	1	yes	maximal, characteristic, complemented
4	#22581275	C₂	1	1	yes	maximal, characteristic, complemented
5	#22581274	C₂	1	1	yes	maximal, characteristic, complemented
6	#22581270	C₂	1	1	yes	maximal, characteristic, complemented
7	#22581269	C₂	1	1	yes	maximal, characteristic, complemented
8	#22581241	C₂	1	1	yes	maximal, characteristic, complemented
9	#22581231	C₂	1	1	yes	maximal, characteristic, complemented
10	#22581203	C₂	1	1	yes	maximal, characteristic, complemented
11	D₈×Dic₁₃₉	C₂	1	1	yes	maximal, characteristic, complemented
12	#22581029	C₂	1	1	yes	maximal, characteristic, complemented
13	#22580995	C₂	1	1	yes	maximal, characteristic, complemented
14	#22580992	C₂	1	1	yes	maximal, characteristic, complemented
15	#469×C₁₃₉	C₂	1	1	yes	maximal, characteristic, complemented
16	C₂×#14555784	—	2	1	no	—
17	C₂×#14555768	C₂²	1	1	yes	characteristic
18	#17984841	C₂²	1	1	yes	characteristic
19	#17984841	C₂²	1	1	yes	characteristic
20	#17984829	—	2	1	no	—
21	#17984828	—	2	1	no	—
22	#17984824	—	2	1	no	—
23	#17984823	—	2	1	no	—
24	#17984819	—	2	1	no	—
25	#17984814	—	2	1	no	—
26	#17984813	C₂²	1	1	yes	characteristic
27	#17984813	C₂²	1	1	yes	characteristic
28	D₄×Dic₁₃₉	C₂²	1	1	yes	characteristic
29	D₄×Dic₁₃₉	C₂²	1	1	yes	characteristic
30	C₄×C₂²⋊D₁₃₉	C₂²	1	1	yes	characteristic
31	#17984801	C₂²	1	1	yes	characteristic
32	#17984800	C₂²	1	1	yes	characteristic
33	#17984798	—	2	1	no	—
34	C₂×C₄⋊Dic₁₃₉	C₂²	1	1	yes	characteristic
35	C₄⋊C₄×D₁₃₉	C₂²	1	1	yes	characteristic
36	#17984776	C₂²	1	1	yes	characteristic
37	#17984776	C₂²	1	1	yes	characteristic
38	#17984775	C₂²	1	1	yes	characteristic
39	#17984774	C₂²	1	1	yes	characteristic
40	#17984774	C₂²	1	1	yes	characteristic
41	#17984773	C₂²	1	1	yes	characteristic
42	#17984773	C₂²	1	1	yes	characteristic
43	#17984773	C₂²	1	1	yes	characteristic
44	Dic₁₃₉⋊C₈	C₂²	1	1	yes	characteristic
45	#17984763	C₂²	1	1	yes	characteristic
46	#17984763	C₂²	1	1	yes	characteristic
47	#17984763	C₂²	1	1	yes	characteristic
48	#17984763	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×D₈×C₁₃₉	C₂²	1	1	yes	characteristic
53	C₂×M₁₆×C₁₃₉	C₂²	1	1	yes	characteristic
54	#121×C₁₃₉	C₂²	1	1	yes	characteristic
55	#121×C₁₃₉	C₂²	1	1	yes	characteristic
56	#106×C₁₃₉	C₂²	1	1	yes	characteristic
57	#102×C₁₃₉	C₂²	1	1	yes	characteristic
58	#102×C₁₃₉	C₂²	1	1	yes	characteristic
59	#14555784	—	4	4	no	—
60	C₂×Dic₂₇₈	—	2	1	no	—
61	C₂×Dic₂₇₈	—	2	1	no	—
62	C₂×C₂²⋊D₁₃₉	—	2	1	no	—
63	C₂×C₂²⋊D₁₃₉	—	2	1	no	—
64	C₂×C₂²⋊D₁₃₉	D₄	1	1	yes	characteristic
65	C₂²×Dic₁₃₉	—	2	1	no	—
66	C₂²×Dic₁₃₉	—	2	1	no	—
67	C₂²×Dic₁₃₉	—	2	1	no	—
68	C₂×C₄×D₁₃₉	—	2	1	no	—
69	C₂×C₄×D₁₃₉	C₂³	1	1	yes	characteristic
70	#14555768	—	2	2	no	—
71	C₂×C₁₃₉⋊C₈	C₂³	1	1	yes	characteristic
72	C₂×C₁₃₉⋊C₈	C₂³	1	1	yes	characteristic
73	C₄⋊Dic₁₃₉	—	2	1	no	—
74	C₄⋊Dic₁₃₉	C₂³	1	1	yes	characteristic
75	C₄⋊Dic₁₃₉	C₂³	1	1	yes	characteristic
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	—	2	1	no	—
83	C₂²⋊Dic₁₃₉	—	2	1	no	—
84	C₂²⋊Dic₁₃₉	—	2	1	no	—
85	C₂²⋊Dic₁₃₉	—	2	1	no	—
86	C₂²⋊Dic₁₃₉	—	2	1	no	—
87	C₂²⋊Dic₁₃₉	D₄	1	1	yes	characteristic
88	(C₂×D₁₃₉)⋊C₄	—	2	1	no	—
89	(C₂×D₁₃₉)⋊C₄	D₄	1	1	yes	characteristic
90	C₄×Dic₁₃₉	—	2	1	no	—
91	C₄×Dic₁₃₉	—	2	1	no	—
92	C₄×Dic₁₃₉	C₂³	1	1	yes	characteristic
93	C₂×D₄×C₁₃₉	—	2	1	no	—
94	C₂×D₄×C₁₃₉	—	2	1	no	—
95	C₂×D₄×C₁₃₉	C₂³	1	1	yes	characteristic
96	C₂×D₄×C₁₃₉	C₂³	1	1	yes	characteristic
97	C₂²×C₄×C₁₃₉	C₂³	1	1	yes	characteristic
98	D₈×C₁₃₉	—	2	2	no	—
99	M₁₆×C₁₃₉	—	2	2	no	—
100	C₂×C₈×C₁₃₉	C₂³	1	1	yes	characteristic
101	C₂×C₈×C₁₃₉	C₂³	1	1	yes	characteristic
102	C₄⋊C₄×C₁₃₉	C₂³	1	1	yes	characteristic
103	C₄⋊C₄×C₁₃₉	C₂³	1	1	yes	characteristic
104	C₂²⋊C₄×C₁₃₉	—	2	1	no	—
105	C₂²⋊C₄×C₁₃₉	—	2	1	no	—
106	C₂²×D₁₃₉	C₂×D₄	1	1	yes	characteristic
107	Dic₂₇₈	—	4	2	no	—
108	Dic₂₇₈	—	8	1	no	—
109	C₂²⋊D₁₃₉	—	2	2	no	—
110	C₂²⋊D₁₃₉	—	4	2	no	—
111	C₂²⋊D₁₃₉	—	8	1	no	—
112	C₂×Dic₁₃₉	—	2	1	no	—
113	C₂×Dic₁₃₉	—	2	1	no	—
114	C₂×Dic₁₃₉	—	2	1	no	—
115	C₂×Dic₁₃₉	—	2	1	no	—
116	C₂×Dic₁₃₉	—	4	2	no	—
117	C₂×Dic₁₃₉	—	4	2	no	—
118	C₂×Dic₁₃₉	—	4	2	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	—	4	2	no	—
125	C₁₃₉⋊C₈	—	2	1	no	—
126	C₁₃₉⋊C₈	—	2	1	no	—
127	C₂³×C₁₃₉	—	2	1	no	—
128	C₂³×C₁₃₉	—	2	1	no	—
129	C₂³×C₁₃₉	C₂×D₄	1	1	yes	characteristic
130	D₄×C₁₃₉	—	2	1	no	—
131	D₄×C₁₃₉	—	2	1	no	—
132	D₄×C₁₃₉	—	2	1	no	—
133	D₄×C₁₃₉	—	2	1	no	—
134	D₄×C₁₃₉	—	4	2	no	—
135	D₄×C₁₃₉	—	8	1	no	—
136	C₂×C₄×C₁₃₉	—	2	1	no	—
137	C₂×C₄×C₁₃₉	—	2	1	no	—
138	C₂×C₄×C₁₃₉	—	2	1	no	—
139	C₂×C₄×C₁₃₉	—	2	1	no	—
140	C₂×C₄×C₁₃₉	C₂×D₄	1	1	yes	characteristic
141	C₂×C₄×C₁₃₉	C₂⁴	1	1	yes	characteristic
142	C₈×C₁₃₉	—	2	1	no	—
143	C₈×C₁₃₉	#41	1	2	yes	complemented
144	C₂²×C₁₃₉	—	2	1	no	—
145	C₂²×C₁₃₉	—	2	1	no	—
146	C₂²×C₁₃₉	—	2	1	no	—
147	C₂²×C₁₃₉	—	4	1	no	—
148	C₂²×C₁₃₉	—	4	1	no	—
149	C₂²×C₁₃₉	—	4	1	no	—
150	C₂²×C₁₃₉	—	4	1	no	—
151	C₂²×C₁₃₉	—	4	1	no	—
152	C₂²×C₁₃₉	—	4	1	no	—
153	C₂²×C₁₃₉	C₂²×D₄	1	1	yes	characteristic
154	C₂×D₁₃₉	—	2	1	no	—
155	C₂×D₁₃₉	—	2	1	no	—
156	C₂×D₁₃₉	—	2	1	no	—
157	C₄×C₁₃₉	—	2	1	no	—
158	C₄×C₁₃₉	—	4	1	no	—
159	C₄×C₁₃₉	—	4	1	no	—
160	C₄×C₁₃₉	C₂×#41	1	1	yes	characteristic
161	C₄×C₁₃₉	2₋¹⁺⁴	1	1	yes	characteristic
162	Dic₁₃₉	—	2	1	no	—
163	Dic₁₃₉	—	2	1	no	—
164	Dic₁₃₉	—	4	1	no	—
165	Dic₁₃₉	—	4	1	no	—
166	Dic₁₃₉	—	4	1	no	—
167	Dic₁₃₉	—	4	1	no	—
168	Dic₁₃₉	AGL(1,ℤ/8)	1	2	yes	—
169	C₂×C₁₃₉	—	4	1	no	—
170	C₂×C₁₃₉	—	8	1	no	—
171	C₂×C₁₃₉	—	8	1	no	—
172	C₂×C₁₃₉	#547	1	1	yes	characteristic
173	C₂×C₁₃₉	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
174	C₂×C₁₃₉	#577	1	1	yes	characteristic
175	D₁₃₉	—	4	1	no	—
176	C₁₃₉	#3354	1	1	yes	characteristic, complemented, Sylow
177	#3354	—	139	1	no	maximal, Sylow
178	#555	—	139	1	no	—
179	#547	—	139	1	no	—
180	#498	—	139	1	no	—
181	#482	—	139	1	no	—
182	#482	—	139	1	no	—
183	#481	—	139	1	no	—
184	#481	—	139	1	no	—
185	#476	—	139	1	no	—
186	#474	—	139	1	no	—
187	#469	—	139	1	no	—
188	#469	D₁₃₉	1	1	yes	characteristic, complemented
189	C₄×D₈	—	139	1	no	—
190	#436	—	139	1	no	—
191	#428	—	139	1	no	—
192	#428	—	139	1	no	—
193	C₂×#41	—	278	1	no	—
194	C₂×D₈	C₂×D₁₃₉	1	1	yes	characteristic
195	C₂×M₁₆	—	139	1	no	—
196	C₂×M₁₆	C₂×D₁₃₉	1	1	yes	characteristic
197	C₄⋊Q₈	—	139	1	no	—
198	C₄⋊Q₈	—	139	1	no	—
199	#124	—	278	1	no	—
200	#124	—	278	1	no	—
201	#123	—	278	1	no	—
202	#123	—	278	1	no	—
203	C₂²⋊Q₈	—	278	1	no	—
204	#121	—	278	1	no	—
205	#121	—	139	1	no	—
206	#121	—	139	1	no	—
207	#121	C₂×D₁₃₉	1	1	yes	characteristic
208	#121	C₂×D₁₃₉	1	1	yes	characteristic
209	C₄×D₄	—	139	1	no	—
210	C₄×D₄	—	139	1	no	—
211	C₄×D₄	—	139	1	no	—
212	#117	—	278	1	no	—
213	#117	—	139	1	no	—
214	#117	—	139	1	no	—
215	C₂×C₄⋊C₄	—	139	1	no	—
216	C₂×C₄⋊C₄	—	139	1	no	—
217	#107	—	139	1	no	—
218	#107	—	139	1	no	—
219	#107	—	139	1	no	—
220	#106	—	139	1	no	—
221	#106	—	139	1	no	—
222	#106	—	139	1	no	—
223	#106	—	139	1	no	—
224	#106	—	139	1	no	—
225	#106	C₂×D₁₃₉	1	1	yes	characteristic
226	C₄⋊C₈	—	139	1	no	—
227	#102	—	139	1	no	—
228	#102	—	139	1	no	—
229	#102	—	139	1	no	—
230	#102	—	139	1	no	—
231	#102	C₂×D₁₃₉	1	1	yes	characteristic
232	#102	C₂×D₁₃₉	1	1	yes	characteristic
233	C₂²⋊C₈	—	139	1	no	—
234	C₂²⋊C₈	—	139	1	no	—
235	C₄×C₈	—	139	1	no	—
236	#41	—	556	4	no	—
237	C₂×Q₈	—	278	1	no	—
238	C₂×Q₈	—	278	1	no	—
239	C₂×D₄	—	2	1	no	—
240	C₂×D₄	—	2	1	no	—
241	C₂×D₄	—	278	1	no	—
242	C₂×D₄	—	278	1	no	—
243	C₂×D₄	—	139	1	no	—
244	C₂×D₄	C₂²×D₁₃₉	1	1	yes	characteristic
245	C₂×D₄	C₂²×D₁₃₉	1	1	yes	characteristic
246	C₂²×C₄	—	278	1	no	—
247	C₂²×C₄	—	278	1	no	—
248	C₂²×C₄	—	278	1	no	—
249	C₂²×C₄	—	278	1	no	—
250	C₂²×C₄	—	139	1	no	—
251	C₂²×C₄	C₂²×D₁₃₉	1	1	yes	characteristic
252	D₈	—	2	2	no	—
253	M₁₆	—	2	2	no	—
254	M₁₆	—	278	2	no	—
255	C₂×C₈	—	139	1	no	—
256	C₂×C₈	—	139	1	no	—
257	C₂×C₈	C₂²×D₁₃₉	1	1	yes	characteristic
258	C₂×C₈	C₂²×D₁₃₉	1	1	yes	characteristic
259	C₄⋊C₄	—	278	1	no	—
260	C₄⋊C₄	—	278	1	no	—
261	C₄⋊C₄	—	278	1	no	—
262	C₄⋊C₄	—	278	1	no	—
263	C₄⋊C₄	—	139	1	no	—
264	C₄⋊C₄	—	139	1	no	—
265	C₄⋊C₄	—	139	1	no	—
266	C₄⋊C₄	—	139	1	no	—
267	C₄⋊C₄	—	139	1	no	—
268	C₄⋊C₄	C₂²×D₁₃₉	1	1	yes	characteristic
269	C₄⋊C₄	C₂²×D₁₃₉	1	1	yes	characteristic
270	C₂²⋊C₄	—	2	1	no	—
271	C₂²⋊C₄	—	2	1	no	—
272	C₂²⋊C₄	—	278	1	no	—
273	C₂²⋊C₄	—	278	1	no	—
274	C₂²⋊C₄	—	278	1	no	—
275	C₂²⋊C₄	—	278	1	no	—
276	C₂²⋊C₄	—	278	1	no	—
277	C₂²⋊C₄	—	278	1	no	—
278	C₂²⋊C₄	—	139	1	no	—
279	C₂²⋊C₄	—	139	1	no	—
280	C₄²	—	278	1	no	—
281	C₄²	—	278	1	no	—
282	C₄²	—	139	1	no	—
283	C₂³	—	2	1	no	—
284	C₂³	—	2	1	no	—
285	C₂³	—	139	1	no	—
286	C₂³	D₄×D₁₃₉	1	1	yes	characteristic
287	Q₈	—	556	2	no	—
288	Q₈	—	1112	1	no	—
289	D₄	—	2	1	no	—
290	D₄	—	2	1	no	—
291	D₄	—	2	1	no	—
292	D₄	—	2	1	no	—
293	D₄	—	4	2	no	—
294	D₄	—	8	1	no	—
295	D₄	—	278	2	no	—
296	D₄	—	556	2	no	—
297	D₄	—	1112	1	no	—
298	C₂×C₄	—	2	1	no	—
299	C₂×C₄	—	2	1	no	—
300	C₂×C₄	—	2	1	no	—
301	C₂×C₄	—	2	1	no	—
302	C₂×C₄	—	278	1	no	—
303	C₂×C₄	—	278	1	no	—
304	C₂×C₄	—	278	1	no	—
305	C₂×C₄	—	278	1	no	—
306	C₂×C₄	—	278	1	no	—
307	C₂×C₄	—	278	1	no	—
308	C₂×C₄	—	556	2	no	—
309	C₂×C₄	—	556	2	no	—
310	C₂×C₄	—	556	2	no	—
311	C₂×C₄	—	556	2	no	—
312	C₂×C₄	—	139	1	no	—
313	C₂×C₄	—	139	1	no	—
314	C₂×C₄	—	139	1	no	—
315	C₂×C₄	D₄×D₁₃₉	1	1	yes	characteristic
316	C₂×C₄	C₂³×D₁₃₉	1	1	yes	characteristic
317	C₈	—	2	1	no	—
318	C₈	—	278	1	no	—
319	C₈	—	278	1	no	—
320	C₈	#14555784	1	2	yes	complemented
321	C₂²	—	2	1	no	—
322	C₂²	—	2	1	no	—
323	C₂²	—	2	1	no	—
324	C₂²	—	4	1	no	—
325	C₂²	—	4	1	no	—
326	C₂²	—	4	1	no	—
327	C₂²	—	4	1	no	—
328	C₂²	—	4	1	no	—
329	C₂²	—	4	1	no	—
330	C₂²	—	278	1	no	—
331	C₂²	—	278	1	no	—
332	C₂²	—	278	1	no	—
333	C₂²	C₂×D₄×D₁₃₉	1	1	yes	characteristic
334	C₄	—	2	1	no	—
335	C₄	—	4	1	no	—
336	C₄	—	4	1	no	—
337	C₄	—	278	1	no	—
338	C₄	—	278	1	no	—
339	C₄	—	556	1	no	—
340	C₄	—	556	1	no	—
341	C₄	—	556	1	no	—
342	C₄	—	556	1	no	—
343	C₄	—	139	2	no	—
344	C₄	C₂×#14555784	1	1	yes	characteristic
345	C₄	#17984889	1	1	yes	characteristic
346	C₂	—	4	1	no	—
347	C₂	—	8	1	no	—
348	C₂	—	8	1	no	—
349	C₂	—	556	1	no	—
350	C₂	#22581603	1	1	yes	characteristic
351	C₂	AGL(1,ℤ/8)×D₁₃₉	1	1	yes	characteristic
352	C₂	#22581763	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×D₄×D₁₃₉
commutator, cosocle	C₂×C₄×C₁₃₉	C₂⁴
Frattini	C₂×C₄	C₂³×D₁₃₉
Fitting	#469×C₁₃₉	C₂
socle	C₂²×C₁₃₉	C₂²×D₄
hypercenter	#469	D₁₃₉
hypocenter	C₁₃₉	#3354
LCS	G ▷ C₂×C₄×C₁₃₉ ▷ C₂×C₁₃₉ ▷ C₁₃₉
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ #469

Complements

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

N	H ≅ G/N	type
1. #22581667	C₂	semidirect
2. #22581603	C₂	semidirect
3. #22581350	C₂	semidirect
4. #22581275	C₂	semidirect
5. #22581274	C₂	semidirect
6. #22581270	C₂	semidirect
7. #22581269	C₂	semidirect
8. #22581241	C₂	semidirect
9. #22581231	C₂	semidirect
10. #22581203	C₂	semidirect
11. D₈×Dic₁₃₉	C₂	semidirect
12. #22581029	C₂	semidirect
13. #22580995	C₂	semidirect
14. #22580992	C₂	semidirect
15. #469×C₁₃₉	C₂	semidirect
143. C₈×C₁₃₉	#41	semidirect
176. C₁₃₉	#3354	semidirect
188. #469	D₁₃₉	semidirect
320. C₈	#14555784	semidirect

Maximal subgroups

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

subgroup	index
#22581667	2
#22581603	2
#22581350	2
#22581275	2
#22581274	2
#22581270	2
#22581269	2
#22581241	2
#22581231	2
#22581203	2
D₈×Dic₁₃₉	2
#22581029	2
#22580995	2
#22580992	2
#469×C₁₃₉	2
#3354	139