Subgroups — #31819114

Order 17792.

396 conjugacy classes of subgroups.

The subgroup lattice has 360 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#22581659	C₂	1	1	yes	maximal, characteristic, complemented
2	#22581624	C₂	1	1	yes	maximal, characteristic, complemented
3	#22581319	C₂	1	1	yes	maximal, characteristic, complemented
4	#22581316	C₂	1	1	yes	maximal, characteristic, complemented
5	#22581298	C₂	1	1	yes	maximal, characteristic, complemented
6	#22581256	C₂	1	1	yes	maximal, characteristic, complemented
7	#22581242	C₂	1	1	yes	maximal, characteristic, complemented
8	#22581149	C₂	1	1	yes	maximal, characteristic, complemented
9	#22581147	C₂	1	1	yes	maximal, characteristic, complemented
10	#22581147	C₂	1	1	yes	maximal, characteristic, complemented
11	#22581082	C₂	1	1	yes	maximal, characteristic, complemented
12	#22581064	C₂	1	1	yes	maximal, characteristic, complemented
13	#22581063	C₂	1	1	yes	maximal, characteristic, complemented
14	#22581035	C₂	1	1	yes	maximal, characteristic, complemented
15	#478×C₁₃₉	C₂	1	1	yes	maximal, characteristic, complemented
16	C₂×#14555785	—	2	1	no	—
17	C₂×D₁₁₁₂	C₂²	1	1	yes	characteristic
18	C₂×D₁₁₁₂	C₂²	1	1	yes	characteristic
19	C₂×#14555770	C₂²	1	1	yes	characteristic
20	#17984841	C₂²	1	1	yes	characteristic
21	#17984839	C₂²	1	1	yes	characteristic
22	#17984838	C₂²	1	1	yes	characteristic
23	#17984838	C₂²	1	1	yes	characteristic
24	#17984833	C₂²	1	1	yes	characteristic
25	#17984822	—	2	1	no	—
26	#17984820	—	2	1	no	—
27	#17984820	—	2	1	no	—
28	#17984816	—	2	1	no	—
29	#17984815	—	2	1	no	—
30	#17984815	—	2	1	no	—
31	C₄×Dic₂₇₈	C₂²	1	1	yes	characteristic
32	C₄×D₅₅₆	C₂²	1	1	yes	characteristic
33	#17984804	C₂²	1	1	yes	characteristic
34	#17984804	C₂²	1	1	yes	characteristic
35	#17984804	C₂²	1	1	yes	characteristic
36	#17984804	C₂²	1	1	yes	characteristic
37	C₄⋊C₄×D₁₃₉	—	2	1	no	—
38	#17984773	C₂²	1	1	yes	characteristic
39	Dic₁₃₉⋊C₈	C₂²	1	1	yes	characteristic
40	Dic₁₃₉⋊C₈	C₂²	1	1	yes	characteristic
41	#17984764	C₂²	1	1	yes	characteristic
42	#17984764	C₂²	1	1	yes	characteristic
43	#17984763	C₂²	1	1	yes	characteristic
44	#17984762	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	#17984753	C₂²	1	1	yes	characteristic
50	#17984752	C₂²	1	1	yes	characteristic
51	C₈×Dic₁₃₉	C₂²	1	1	yes	characteristic
52	#125×C₁₃₉	C₂²	1	1	yes	characteristic
53	C₄×D₄×C₁₃₉	C₂²	1	1	yes	characteristic
54	#107×C₁₃₉	C₂²	1	1	yes	characteristic
55	#106×C₁₃₉	C₂²	1	1	yes	characteristic
56	C₄⋊C₈×C₁₃₉	C₂²	1	1	yes	characteristic
57	#102×C₁₃₉	C₂²	1	1	yes	characteristic
58	#102×C₁₃₉	C₂²	1	1	yes	characteristic
59	#14555785	—	4	4	no	—
60	C₂×Dic₂₇₈	—	2	1	no	—
61	C₂×Dic₂₇₈	D₄	1	1	yes	characteristic
62	C₂×D₅₅₆	—	2	1	no	—
63	C₂×D₅₅₆	—	2	1	no	—
64	C₂×D₅₅₆	C₂³	1	1	yes	characteristic
65	C₂×D₅₅₆	C₂³	1	1	yes	characteristic
66	C₂×D₅₅₆	C₂³	1	1	yes	characteristic
67	C₂×C₂²⋊D₁₃₉	—	2	1	no	—
68	C₂×C₂²⋊D₁₃₉	—	2	1	no	—
69	C₂×C₂²⋊D₁₃₉	—	2	1	no	—
70	C₂×C₂²⋊D₁₃₉	—	2	1	no	—
71	C₂×C₄×D₁₃₉	—	2	1	no	—
72	C₂×C₄×D₁₃₉	—	2	1	no	—
73	C₂×C₄×D₁₃₉	—	2	1	no	—
74	C₂×C₄×D₁₃₉	—	2	1	no	—
75	C₂×C₄×D₁₃₉	—	2	1	no	—
76	D₁₁₁₂	—	2	2	no	—
77	D₁₁₁₂	—	2	2	no	—
78	#14555770	—	2	2	no	—
79	C₂×C₁₃₉⋊C₈	C₂³	1	1	yes	characteristic
80	C₂×C₁₃₉⋊C₈	C₂³	1	1	yes	characteristic
81	C₄⋊Dic₁₃₉	—	2	1	no	—
82	C₄⋊Dic₁₃₉	D₄	1	1	yes	characteristic
83	C₄⋊Dic₁₃₉	C₂³	1	1	yes	characteristic
84	Dic₁₃₉⋊C₄	—	2	1	no	—
85	Dic₁₃₉⋊C₄	—	2	1	no	—
86	Dic₁₃₉⋊C₄	D₄	1	1	yes	characteristic
87	Dic₁₃₉⋊C₄	D₄	1	1	yes	characteristic
88	(C₂×D₁₃₉)⋊C₄	—	2	1	no	—
89	(C₂×D₁₃₉)⋊C₄	—	2	1	no	—
90	(C₂×D₁₃₉)⋊C₄	—	2	1	no	—
91	(C₂×D₁₃₉)⋊C₄	—	2	1	no	—
92	(C₂×D₁₃₉)⋊C₄	—	2	1	no	—
93	C₄×Dic₁₃₉	C₂³	1	1	yes	characteristic
94	C₄×Dic₁₃₉	C₂³	1	1	yes	characteristic
95	C₂×D₄×C₁₃₉	C₂³	1	1	yes	characteristic
96	C₂²×C₄×C₁₃₉	—	2	1	no	—
97	C₂×C₈×C₁₃₉	C₂³	1	1	yes	characteristic
98	C₂×C₈×C₁₃₉	C₂³	1	1	yes	characteristic
99	C₄⋊C₄×C₁₃₉	—	2	1	no	—
100	C₄⋊C₄×C₁₃₉	—	2	1	no	—
101	C₄⋊C₄×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	C₄²×C₁₃₉	C₂³	1	1	yes	characteristic
106	C₂²×D₁₃₉	—	2	1	no	—
107	C₂²×D₁₃₉	—	2	1	no	—
108	C₂²×D₁₃₉	—	2	1	no	—
109	Dic₂₇₈	—	2	2	no	—
110	Dic₂₇₈	—	8	1	no	—
111	D₅₅₆	—	2	1	no	—
112	D₅₅₆	—	2	1	no	—
113	D₅₅₆	—	2	1	no	—
114	D₅₅₆	—	2	1	no	—
115	D₅₅₆	—	2	1	no	—
116	D₅₅₆	—	4	2	no	—
117	D₅₅₆	—	8	1	no	—
118	D₅₅₆	#41	1	2	yes	complemented
119	C₂²⋊D₁₃₉	—	4	2	no	—
120	C₂²⋊D₁₃₉	—	4	2	no	—
121	C₂²⋊D₁₃₉	—	8	1	no	—
122	C₂²⋊D₁₃₉	—	8	1	no	—
123	C₂×Dic₁₃₉	—	2	1	no	—
124	C₂×Dic₁₃₉	C₂×D₄	1	1	yes	characteristic
125	C₂×Dic₁₃₉	C₂×D₄	1	1	yes	characteristic
126	C₂×Dic₁₃₉	C₂×D₄	1	1	yes	characteristic
127	C₂×Dic₁₃₉	C₂×D₄	1	1	yes	characteristic
128	C₄×D₁₃₉	—	4	2	no	—
129	C₄×D₁₃₉	—	4	2	no	—
130	C₄×D₁₃₉	—	4	2	no	—
131	C₄×D₁₃₉	—	8	1	no	—
132	C₄×D₁₃₉	—	8	1	no	—
133	C₁₃₉⋊C₈	—	2	1	no	—
134	C₁₃₉⋊C₈	—	2	1	no	—
135	C₂³×C₁₃₉	—	2	1	no	—
136	D₄×C₁₃₉	—	2	1	no	—
137	D₄×C₁₃₉	—	2	1	no	—
138	C₂×C₄×C₁₃₉	—	2	1	no	—
139	C₂×C₄×C₁₃₉	—	2	1	no	—
140	C₂×C₄×C₁₃₉	—	2	1	no	—
141	C₂×C₄×C₁₃₉	—	4	2	no	—
142	C₂×C₄×C₁₃₉	C₂×D₄	1	1	yes	characteristic
143	C₂×C₄×C₁₃₉	C₂×D₄	1	1	yes	characteristic
144	C₂×C₄×C₁₃₉	C₂⁴	1	1	yes	characteristic
145	C₈×C₁₃₉	—	2	1	no	—
146	C₈×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	C₂²×D₄	1	1	yes	characteristic
151	C₂×D₁₃₉	—	2	2	no	—
152	C₂×D₁₃₉	—	4	1	no	—
153	C₂×D₁₃₉	—	4	1	no	—
154	C₂×D₁₃₉	—	4	1	no	—
155	C₂×D₁₃₉	—	4	1	no	—
156	C₂×D₁₃₉	—	4	1	no	—
157	C₂×D₁₃₉	—	4	1	no	—
158	C₂×D₁₃₉	—	4	1	no	—
159	C₂×D₁₃₉	—	4	1	no	—
160	C₄×C₁₃₉	—	2	1	no	—
161	C₄×C₁₃₉	—	2	1	no	—
162	C₄×C₁₃₉	—	4	1	no	—
163	C₄×C₁₃₉	—	4	1	no	—
164	C₄×C₁₃₉	—	4	1	no	—
165	C₄×C₁₃₉	C₂×#41	1	1	yes	characteristic
166	C₄×C₁₃₉	2₋¹⁺⁴	1	1	yes	characteristic
167	Dic₁₃₉	—	2	1	no	—
168	Dic₁₃₉	—	2	1	no	—
169	Dic₁₃₉	—	2	1	no	—
170	Dic₁₃₉	—	4	1	no	—
171	Dic₁₃₉	AGL(1,ℤ/8)	1	2	yes	—
172	C₂×C₁₃₉	—	8	1	no	—
173	C₂×C₁₃₉	#548	1	1	yes	characteristic
174	C₂×C₁₃₉	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
175	C₂×C₁₃₉	#576	1	1	yes	characteristic
176	D₁₃₉	—	4	2	no	—
177	D₁₃₉	—	8	1	no	—
178	D₁₃₉	—	8	1	no	—
179	C₁₃₉	#3366	1	1	yes	characteristic, complemented, Sylow
180	#3366	—	139	1	no	maximal, Sylow
181	#554	—	139	1	no	—
182	#550	—	139	1	no	—
183	#490	—	139	1	no	—
184	#490	—	139	1	no	—
185	#486	—	139	1	no	—
186	#478	—	139	1	no	—
187	#478	D₁₃₉	1	1	yes	characteristic, complemented
188	#476	—	139	1	no	—
189	#459	—	139	1	no	—
190	#459	—	139	1	no	—
191	#459	—	139	1	no	—
192	#446	—	139	1	no	—
193	#442	—	139	1	no	—
194	#442	—	139	1	no	—
195	C₄×D₈	—	139	1	no	—
196	C₂×#41	—	278	1	no	—
197	C₂×D₈	—	139	1	no	—
198	C₂×D₈	—	139	1	no	—
199	C₂×D₈	—	139	1	no	—
200	C₄⋊Q₈	—	139	1	no	—
201	#127	—	139	1	no	—
202	#127	—	139	1	no	—
203	#127	—	139	1	no	—
204	#125	—	139	1	no	—
205	#125	C₂×D₁₃₉	1	1	yes	characteristic
206	C₂²⋊Q₈	—	278	1	no	—
207	C₂²⋊Q₈	—	278	1	no	—
208	C₂²⋊Q₈	—	278	1	no	—
209	#121	—	278	1	no	—
210	#121	—	278	1	no	—
211	#121	—	278	1	no	—
212	C₄×Q₈	—	139	1	no	—
213	C₄×D₄	—	139	1	no	—
214	C₄×D₄	—	139	1	no	—
215	C₄×D₄	—	139	1	no	—
216	C₄×D₄	—	139	1	no	—
217	C₄×D₄	—	139	1	no	—
218	C₄×D₄	C₂×D₁₃₉	1	1	yes	characteristic
219	C₂×C₄⋊C₄	—	278	1	no	—
220	#107	C₂×D₁₃₉	1	1	yes	characteristic
221	#106	—	139	1	no	—
222	#106	C₂×D₁₃₉	1	1	yes	characteristic
223	C₄⋊C₈	—	139	1	no	—
224	C₄⋊C₈	—	139	1	no	—
225	C₄⋊C₈	C₂×D₁₃₉	1	1	yes	characteristic
226	#102	—	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	—	139	1	no	—
232	#102	—	139	1	no	—
233	#102	—	139	1	no	—
234	#102	C₂×D₁₃₉	1	1	yes	characteristic
235	#102	C₂×D₁₃₉	1	1	yes	characteristic
236	#97	—	139	1	no	—
237	#97	—	139	1	no	—
238	C₄×C₈	—	139	1	no	—
239	#41	—	556	4	no	—
240	C₂×Q₈	—	278	1	no	—
241	C₂×Q₈	—	139	1	no	—
242	C₂×D₄	—	278	1	no	—
243	C₂×D₄	—	278	1	no	—
244	C₂×D₄	—	278	1	no	—
245	C₂×D₄	—	278	1	no	—
246	C₂×D₄	—	278	1	no	—
247	C₂×D₄	—	278	1	no	—
248	C₂×D₄	—	139	1	no	—
249	C₂×D₄	—	139	1	no	—
250	C₂×D₄	—	139	1	no	—
251	C₂×D₄	C₂²×D₁₃₉	1	1	yes	characteristic
252	C₂²×C₄	—	2	1	no	—
253	C₂²×C₄	—	278	1	no	—
254	C₂²×C₄	—	278	1	no	—
255	C₂²×C₄	—	278	1	no	—
256	C₂²×C₄	—	278	1	no	—
257	C₂²×C₄	—	278	1	no	—
258	D₈	—	278	2	no	—
259	D₈	—	278	2	no	—
260	D₈	—	278	2	no	—
261	C₂×C₈	—	139	1	no	—
262	C₂×C₈	—	139	1	no	—
263	C₂×C₈	C₂²×D₁₃₉	1	1	yes	characteristic
264	C₂×C₈	C₂²×D₁₃₉	1	1	yes	characteristic
265	C₄⋊C₄	—	2	1	no	—
266	C₄⋊C₄	—	2	1	no	—
267	C₄⋊C₄	—	278	1	no	—
268	C₄⋊C₄	—	278	1	no	—
269	C₄⋊C₄	—	278	1	no	—
270	C₄⋊C₄	—	139	1	no	—
271	C₄⋊C₄	—	139	1	no	—
272	C₄⋊C₄	—	139	1	no	—
273	C₄⋊C₄	—	139	1	no	—
274	C₄⋊C₄	C₂²×D₁₃₉	1	1	yes	characteristic
275	C₄⋊C₄	C₂²×D₁₃₉	1	1	yes	characteristic
276	C₄⋊C₄	C₂²×D₁₃₉	1	1	yes	characteristic
277	C₂²⋊C₄	—	2	1	no	—
278	C₂²⋊C₄	—	278	1	no	—
279	C₂²⋊C₄	—	278	1	no	—
280	C₂²⋊C₄	—	278	1	no	—
281	C₂²⋊C₄	—	278	1	no	—
282	C₂²⋊C₄	—	278	1	no	—
283	C₄²	—	139	1	no	—
284	C₄²	—	139	1	no	—
285	C₄²	C₂²×D₁₃₉	1	1	yes	characteristic
286	C₂³	—	2	1	no	—
287	C₂³	—	278	1	no	—
288	C₂³	—	278	1	no	—
289	C₂³	—	278	1	no	—
290	Q₈	—	278	2	no	—
291	Q₈	—	1112	1	no	—
292	D₄	—	2	1	no	—
293	D₄	—	2	1	no	—
294	D₄	—	278	1	no	—
295	D₄	—	278	1	no	—
296	D₄	—	278	1	no	—
297	D₄	—	278	1	no	—
298	D₄	—	278	1	no	—
299	D₄	—	556	2	no	—
300	D₄	—	556	2	no	—
301	D₄	—	556	2	no	—
302	D₄	—	1112	1	no	—
303	D₄	—	1112	1	no	—
304	D₄	—	1112	1	no	—
305	D₄	—	139	2	no	—
306	C₂×C₄	—	2	1	no	—
307	C₂×C₄	—	2	1	no	—
308	C₂×C₄	—	2	1	no	—
309	C₂×C₄	—	4	2	no	—
310	C₂×C₄	—	278	1	no	—
311	C₂×C₄	—	556	2	no	—
312	C₂×C₄	—	556	2	no	—
313	C₂×C₄	—	556	2	no	—
314	C₂×C₄	—	1112	1	no	—
315	C₂×C₄	—	1112	1	no	—
316	C₂×C₄	—	139	1	no	—
317	C₂×C₄	—	139	1	no	—
318	C₂×C₄	—	139	1	no	—
319	C₂×C₄	—	139	1	no	—
320	C₂×C₄	D₄×D₁₃₉	1	1	yes	characteristic
321	C₂×C₄	D₄×D₁₃₉	1	1	yes	characteristic
322	C₂×C₄	C₂³×D₁₃₉	1	1	yes	characteristic
323	C₈	—	2	1	no	—
324	C₈	—	2	1	no	—
325	C₈	—	278	1	no	—
326	C₈	—	278	1	no	—
327	C₂²	—	4	1	no	—
328	C₂²	—	4	1	no	—
329	C₂²	—	4	1	no	—
330	C₂²	—	278	2	no	—
331	C₂²	—	556	1	no	—
332	C₂²	—	556	1	no	—
333	C₂²	—	556	1	no	—
334	C₂²	—	556	1	no	—
335	C₂²	—	556	1	no	—
336	C₂²	—	556	1	no	—
337	C₂²	—	556	1	no	—
338	C₂²	—	556	1	no	—
339	C₂²	C₂×D₄×D₁₃₉	1	1	yes	characteristic
340	C₄	—	2	1	no	—
341	C₄	—	2	1	no	—
342	C₄	—	4	1	no	—
343	C₄	—	4	1	no	—
344	C₄	—	4	1	no	—
345	C₄	—	278	1	no	—
346	C₄	—	278	1	no	—
347	C₄	—	278	1	no	—
348	C₄	—	556	1	no	—
349	C₄	—	139	2	no	—
350	C₄	#41×D₁₃₉	1	1	yes	characteristic
351	C₄	#17984890	1	1	yes	characteristic
352	C₂	—	8	1	no	—
353	C₂	—	556	2	no	—
354	C₂	—	1112	1	no	—
355	C₂	—	1112	1	no	—
356	C₂	#22581613	1	1	yes	characteristic
357	C₂	AGL(1,ℤ/8)×D₁₃₉	1	1	yes	characteristic
358	C₂	#22581755	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×D₄×D₁₃₉
commutator, cosocle	C₂×C₄×C₁₃₉	C₂⁴
Frattini	C₂×C₄	C₂³×D₁₃₉
Fitting	#478×C₁₃₉	C₂
socle	C₂²×C₁₃₉	C₂²×D₄
hypercenter	#478	D₁₃₉
hypocenter	C₁₃₉	#3366
LCS	G ▷ C₂×C₄×C₁₃₉ ▷ C₂×C₁₃₉ ▷ C₁₃₉
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #478

Complements

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

N	H ≅ G/N	type
1. #22581659	C₂	semidirect
2. #22581624	C₂	semidirect
3. #22581319	C₂	semidirect
4. #22581316	C₂	semidirect
5. #22581298	C₂	semidirect
6. #22581256	C₂	semidirect
7. #22581242	C₂	semidirect
8. #22581149	C₂	semidirect
9. #22581147	C₂	semidirect
10. #22581147	C₂	semidirect
11. #22581082	C₂	semidirect
12. #22581064	C₂	semidirect
13. #22581063	C₂	semidirect
14. #22581035	C₂	semidirect
15. #478×C₁₃₉	C₂	semidirect
118. D₅₅₆	#41	semidirect
179. C₁₃₉	#3366	semidirect
187. #478	D₁₃₉	semidirect

Maximal subgroups

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

subgroup	index
#22581659	2
#22581624	2
#22581319	2
#22581316	2
#22581298	2
#22581256	2
#22581242	2
#22581149	2
#22581147	2
#22581082	2
#22581064	2
#22581063	2
#22581035	2
#478×C₁₃₉	2
#3366	139