Subgroups — #31816596

Order 17792.

396 conjugacy classes of subgroups.

The subgroup lattice has 362 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#22581556	C₂	1	1	yes	maximal, characteristic, complemented
2	#22581436	C₂	1	1	yes	maximal, characteristic, complemented
3	#22581269	C₂	1	1	yes	maximal, characteristic, complemented
4	#22581267	C₂	1	1	yes	maximal, characteristic, complemented
5	#22581262	C₂	1	1	yes	maximal, characteristic, complemented
6	#22581251	C₂	1	1	yes	maximal, characteristic, complemented
7	#22581151	C₂	1	1	yes	maximal, characteristic, complemented
8	#22581146	C₂	1	1	yes	maximal, characteristic, complemented
9	#22581098	C₂	1	1	yes	maximal, characteristic, complemented
10	#22581062	C₂	1	1	yes	maximal, characteristic, complemented
11	#22581054	C₂	1	1	yes	maximal, characteristic, complemented
12	#22581038	C₂	1	1	yes	maximal, characteristic, complemented
13	D₈×Dic₁₃₉	C₂	1	1	yes	maximal, characteristic, complemented
14	#22581011	C₂	1	1	yes	maximal, characteristic, complemented
15	#449×C₁₃₉	C₂	1	1	yes	maximal, characteristic, complemented
16	C₂×#14555772	C₂²	1	1	yes	characteristic
17	C₂×#14555770	C₂²	1	1	yes	characteristic
18	#17984838	C₂²	1	1	yes	characteristic
19	#17984833	C₂²	1	1	yes	characteristic
20	#17984826	—	2	1	no	—
21	#17984824	—	2	1	no	—
22	#17984816	C₂²	1	1	yes	characteristic
23	#17984814	—	2	1	no	—
24	#17984813	—	2	1	no	—
25	Q₈×Dic₁₃₉	C₂²	1	1	yes	characteristic
26	Q₈×Dic₁₃₉	C₂²	1	1	yes	characteristic
27	D₄×Dic₁₃₉	—	2	1	no	—
28	D₄×Dic₁₃₉	—	2	1	no	—
29	D₄×Dic₁₃₉	C₂²	1	1	yes	characteristic
30	D₄×Dic₁₃₉	C₂²	1	1	yes	characteristic
31	D₄×Dic₁₃₉	C₂²	1	1	yes	characteristic
32	#17984804	C₂²	1	1	yes	characteristic
33	#17984801	—	2	1	no	—
34	#17984801	C₂²	1	1	yes	characteristic
35	C₂×C₄⋊Dic₁₃₉	—	2	1	no	—
36	C₂×C₄⋊Dic₁₃₉	C₂²	1	1	yes	characteristic
37	#17984776	C₂²	1	1	yes	characteristic
38	#17984776	C₂²	1	1	yes	characteristic
39	#17984773	C₂²	1	1	yes	characteristic
40	#17984773	C₂²	1	1	yes	characteristic
41	Dic₁₃₉⋊C₈	C₂²	1	1	yes	characteristic
42	Dic₁₃₉⋊C₈	C₂²	1	1	yes	characteristic
43	#17984766	C₂²	1	1	yes	characteristic
44	#17984764	C₂²	1	1	yes	characteristic
45	#17984764	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	#17984752	C₂²	1	1	yes	characteristic
51	C₈×Dic₁₃₉	C₂²	1	1	yes	characteristic
52	C₂×#41×C₁₃₉	C₂²	1	1	yes	characteristic
53	C₂×SD₁₆×C₁₃₉	C₂²	1	1	yes	characteristic
54	C₂×D₈×C₁₃₉	C₂²	1	1	yes	characteristic
55	#121×C₁₃₉	C₂²	1	1	yes	characteristic
56	#103×C₁₃₉	C₂²	1	1	yes	characteristic
57	#102×C₁₃₉	C₂²	1	1	yes	characteristic
58	C₂²⋊C₈×C₁₃₉	C₂²	1	1	yes	characteristic
59	C₂×D₅₅₆	C₂³	1	1	yes	characteristic
60	C₂×C₂²⋊D₁₃₉	—	2	1	no	—
61	C₂×C₂²⋊D₁₃₉	—	2	1	no	—
62	C₂×C₂²⋊D₁₃₉	—	2	1	no	—
63	C₂²×Dic₁₃₉	—	2	1	no	—
64	C₂²×Dic₁₃₉	—	2	1	no	—
65	C₂²×Dic₁₃₉	—	2	1	no	—
66	C₂²×Dic₁₃₉	—	2	1	no	—
67	C₂×C₄×D₁₃₉	—	2	1	no	—
68	#14555772	—	2	2	no	—
69	#14555770	—	2	2	no	—
70	C₂×C₁₃₉⋊C₈	C₂³	1	1	yes	characteristic
71	C₂×C₁₃₉⋊C₈	C₂³	1	1	yes	characteristic
72	C₄⋊Dic₁₃₉	—	2	1	no	—
73	C₄⋊Dic₁₃₉	—	2	1	no	—
74	C₄⋊Dic₁₃₉	—	2	1	no	—
75	C₄⋊Dic₁₃₉	—	2	1	no	—
76	C₄⋊Dic₁₃₉	D₄	1	1	yes	characteristic
77	C₄⋊Dic₁₃₉	D₄	1	1	yes	characteristic
78	C₄⋊Dic₁₃₉	C₂³	1	1	yes	characteristic
79	C₄⋊Dic₁₃₉	C₂³	1	1	yes	characteristic
80	C₄⋊Dic₁₃₉	C₂³	1	1	yes	characteristic
81	Dic₁₃₉⋊C₄	—	2	1	no	—
82	C₂²⋊Dic₁₃₉	—	2	1	no	—
83	C₂²⋊Dic₁₃₉	—	2	1	no	—
84	C₂²⋊Dic₁₃₉	—	2	1	no	—
85	C₂²⋊Dic₁₃₉	D₄	1	1	yes	characteristic
86	C₂²⋊Dic₁₃₉	D₄	1	1	yes	characteristic
87	(C₂×D₁₃₉)⋊C₄	—	2	1	no	—
88	(C₂×D₁₃₉)⋊C₄	—	2	1	no	—
89	C₄×Dic₁₃₉	—	2	1	no	—
90	C₄×Dic₁₃₉	—	2	1	no	—
91	C₄×Dic₁₃₉	C₂³	1	1	yes	characteristic
92	C₄×Dic₁₃₉	C₂³	1	1	yes	characteristic
93	#41×C₁₃₉	—	2	2	no	—
94	#41×C₁₃₉	—	2	2	no	—
95	C₂×Q₈×C₁₃₉	C₂³	1	1	yes	characteristic
96	C₂×D₄×C₁₃₉	—	2	1	no	—
97	C₂×D₄×C₁₃₉	—	2	1	no	—
98	C₂×D₄×C₁₃₉	C₂³	1	1	yes	characteristic
99	C₂×D₄×C₁₃₉	C₂³	1	1	yes	characteristic
100	C₂²×C₄×C₁₃₉	—	2	1	no	—
101	C₂²×C₄×C₁₃₉	C₂³	1	1	yes	characteristic
102	SD₁₆×C₁₃₉	—	2	2	no	—
103	D₈×C₁₃₉	—	2	2	no	—
104	C₂×C₈×C₁₃₉	C₂³	1	1	yes	characteristic
105	C₂×C₈×C₁₃₉	C₂³	1	1	yes	characteristic
106	C₄⋊C₄×C₁₃₉	C₂³	1	1	yes	characteristic
107	C₂²⋊C₄×C₁₃₉	—	2	1	no	—
108	C₂²×D₁₃₉	—	2	1	no	—
109	D₅₅₆	—	2	1	no	—
110	D₅₅₆	—	2	1	no	—
111	C₂²⋊D₁₃₉	—	8	1	no	—
112	C₂²⋊D₁₃₉	—	8	1	no	—
113	C₂²⋊D₁₃₉	—	8	1	no	—
114	C₂×Dic₁₃₉	—	2	1	no	—
115	C₂×Dic₁₃₉	—	2	1	no	—
116	C₂×Dic₁₃₉	—	2	1	no	—
117	C₂×Dic₁₃₉	—	8	1	no	—
118	C₂×Dic₁₃₉	—	8	1	no	—
119	C₂×Dic₁₃₉	—	8	1	no	—
120	C₂×Dic₁₃₉	—	8	1	no	—
121	C₂×Dic₁₃₉	C₂×D₄	1	1	yes	characteristic
122	C₂×Dic₁₃₉	C₂×D₄	1	1	yes	characteristic
123	C₂×Dic₁₃₉	C₂×D₄	1	1	yes	characteristic
124	C₂×Dic₁₃₉	C₂×D₄	1	1	yes	characteristic
125	C₄×D₁₃₉	—	8	1	no	—
126	C₁₃₉⋊C₈	—	2	1	no	—
127	C₁₃₉⋊C₈	—	2	1	no	—
128	C₂³×C₁₃₉	—	2	1	no	—
129	C₂³×C₁₃₉	—	2	1	no	—
130	C₂³×C₁₃₉	C₂×D₄	1	1	yes	characteristic
131	Q₈×C₁₃₉	—	2	1	no	—
132	Q₈×C₁₃₉	#41	1	2	yes	—
133	D₄×C₁₃₉	—	2	4	no	—
134	D₄×C₁₃₉	—	2	1	no	—
135	D₄×C₁₃₉	—	2	1	no	—
136	D₄×C₁₃₉	—	2	1	no	—
137	D₄×C₁₃₉	—	8	1	no	—
138	D₄×C₁₃₉	#41	1	2	yes	—
139	C₂×C₄×C₁₃₉	—	2	4	no	—
140	C₂×C₄×C₁₃₉	—	2	1	no	—
141	C₂×C₄×C₁₃₉	—	2	1	no	—
142	C₂×C₄×C₁₃₉	—	2	1	no	—
143	C₂×C₄×C₁₃₉	—	2	1	no	—
144	C₂×C₄×C₁₃₉	C₂×D₄	1	1	yes	characteristic
145	C₂×C₄×C₁₃₉	C₂⁴	1	1	yes	characteristic
146	C₈×C₁₃₉	—	2	1	no	—
147	C₈×C₁₃₉	—	2	1	no	—
148	C₂²×C₁₃₉	—	2	1	no	—
149	C₂²×C₁₃₉	—	2	2	no	—
150	C₂²×C₁₃₉	—	2	1	no	—
151	C₂²×C₁₃₉	—	2	1	no	—
152	C₂²×C₁₃₉	—	4	1	no	—
153	C₂²×C₁₃₉	—	4	1	no	—
154	C₂²×C₁₃₉	—	4	1	no	—
155	C₂²×C₁₃₉	—	4	1	no	—
156	C₂²×C₁₃₉	—	4	1	no	—
157	C₂²×C₁₃₉	C₂²×D₄	1	1	yes	characteristic
158	C₂×D₁₃₉	—	4	1	no	—
159	C₂×D₁₃₉	—	4	1	no	—
160	C₂×D₁₃₉	—	4	1	no	—
161	C₄×C₁₃₉	—	2	1	no	—
162	C₄×C₁₃₉	—	2	2	no	—
163	C₄×C₁₃₉	—	4	1	no	—
164	C₄×C₁₃₉	C₂×#41	1	1	yes	characteristic
165	C₄×C₁₃₉	C₂×#41	1	1	yes	characteristic
166	Dic₁₃₉	—	2	1	no	—
167	Dic₁₃₉	—	2	1	no	—
168	Dic₁₃₉	—	2	1	no	—
169	Dic₁₃₉	—	2	1	no	—
170	Dic₁₃₉	—	4	1	no	—
171	Dic₁₃₉	—	4	1	no	—
172	Dic₁₃₉	—	4	1	no	—
173	C₂×C₁₃₉	—	4	1	no	—
174	C₂×C₁₃₉	—	4	2	no	—
175	C₂×C₁₃₉	—	8	1	no	—
176	C₂×C₁₃₉	#525	1	1	yes	characteristic
177	C₂×C₁₃₉	#576	1	1	yes	characteristic
178	C₂×C₁₃₉	#577	1	1	yes	characteristic
179	D₁₃₉	—	8	1	no	—
180	C₁₃₉	#3119	1	1	yes	characteristic, complemented, Sylow
181	#3119	—	139	1	no	maximal, Sylow
182	#540	—	139	1	no	—
183	#520	—	139	1	no	—
184	#481	—	139	1	no	—
185	#480	—	139	1	no	—
186	#479	—	139	1	no	—
187	#478	—	139	1	no	—
188	#460	—	139	1	no	—
189	#459	—	139	1	no	—
190	#449	—	139	1	no	—
191	#449	D₁₃₉	1	1	yes	characteristic, complemented
192	#442	—	139	1	no	—
193	#440	—	139	1	no	—
194	C₄×SD₁₆	—	139	1	no	—
195	C₄×D₈	—	139	1	no	—
196	#433	—	139	1	no	—
197	C₂×#41	C₂×D₁₃₉	1	1	yes	characteristic
198	C₂×SD₁₆	—	139	1	no	—
199	C₂×SD₁₆	C₂×D₁₃₉	1	1	yes	characteristic
200	C₂×D₈	—	139	1	no	—
201	C₂×D₈	C₂×D₁₃₉	1	1	yes	characteristic
202	#127	—	139	1	no	—
203	#125	—	139	1	no	—
204	#123	—	278	1	no	—
205	#123	—	278	1	no	—
206	#121	—	278	1	no	—
207	#121	—	278	1	no	—
208	#121	—	139	1	no	—
209	#121	C₂×D₁₃₉	1	1	yes	characteristic
210	C₄×Q₈	—	139	1	no	—
211	C₄×Q₈	—	139	1	no	—
212	C₄×D₄	—	278	1	no	—
213	C₄×D₄	—	278	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	#117	—	278	1	no	—
219	#117	—	139	1	no	—
220	C₂×C₄⋊C₄	—	278	1	no	—
221	C₂×C₄⋊C₄	—	139	1	no	—
222	#107	—	139	1	no	—
223	#107	—	139	1	no	—
224	#106	—	139	1	no	—
225	#106	—	139	1	no	—
226	C₄⋊C₈	—	139	1	no	—
227	C₄⋊C₈	—	139	1	no	—
228	#103	—	139	1	no	—
229	#103	C₂×D₁₃₉	1	1	yes	characteristic
230	#102	—	139	1	no	—
231	#102	—	139	1	no	—
232	#102	—	139	1	no	—
233	#102	—	139	1	no	—
234	#102	—	139	1	no	—
235	#102	C₂×D₁₃₉	1	1	yes	characteristic
236	C₂²⋊C₈	—	139	1	no	—
237	C₂²⋊C₈	C₂×D₁₃₉	1	1	yes	characteristic
238	#97	—	139	1	no	—
239	C₄×C₈	—	139	1	no	—
240	#41	—	2	2	no	—
241	#41	—	2	2	no	—
242	C₂×Q₈	C₂²×D₁₃₉	1	1	yes	characteristic
243	C₂×D₄	—	2	1	no	—
244	C₂×D₄	—	2	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	C₂²×D₁₃₉	1	1	yes	characteristic
250	C₂×D₄	C₂²×D₁₃₉	1	1	yes	characteristic
251	C₂²×C₄	—	2	1	no	—
252	C₂²×C₄	—	278	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	C₂²×D₁₃₉	1	1	yes	characteristic
258	SD₁₆	—	2	2	no	—
259	SD₁₆	—	278	2	no	—
260	D₈	—	2	2	no	—
261	D₈	—	278	2	no	—
262	C₂×C₈	—	139	1	no	—
263	C₂×C₈	—	139	1	no	—
264	C₂×C₈	C₂²×D₁₃₉	1	1	yes	characteristic
265	C₂×C₈	C₂²×D₁₃₉	1	1	yes	characteristic
266	C₄⋊C₄	—	278	1	no	—
267	C₄⋊C₄	—	278	1	no	—
268	C₄⋊C₄	—	278	1	no	—
269	C₄⋊C₄	—	278	1	no	—
270	C₄⋊C₄	—	278	1	no	—
271	C₄⋊C₄	—	139	1	no	—
272	C₄⋊C₄	—	139	1	no	—
273	C₄⋊C₄	—	139	1	no	—
274	C₄⋊C₄	—	139	1	no	—
275	C₄⋊C₄	—	139	1	no	—
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	C₂²⋊C₄	—	139	1	no	—
284	C₂²⋊C₄	—	139	1	no	—
285	C₄²	—	278	1	no	—
286	C₄²	—	278	1	no	—
287	C₄²	—	139	1	no	—
288	C₄²	—	139	1	no	—
289	C₂³	—	2	1	no	—
290	C₂³	—	2	1	no	—
291	C₂³	—	278	1	no	—
292	C₂³	D₄×D₁₃₉	1	1	yes	characteristic
293	Q₈	—	2	1	no	—
294	Q₈	#14555784	1	2	yes	—
295	D₄	—	2	4	no	—
296	D₄	—	2	1	no	—
297	D₄	—	2	1	no	—
298	D₄	—	2	1	no	—
299	D₄	—	8	1	no	—
300	D₄	—	278	1	no	—
301	D₄	—	278	1	no	—
302	D₄	—	1112	1	no	—
303	D₄	—	1112	1	no	—
304	D₄	—	1112	1	no	—
305	D₄	#14555784	1	2	yes	—
306	C₂×C₄	—	2	4	no	—
307	C₂×C₄	—	2	1	no	—
308	C₂×C₄	—	2	1	no	—
309	C₂×C₄	—	2	1	no	—
310	C₂×C₄	—	2	1	no	—
311	C₂×C₄	—	278	1	no	—
312	C₂×C₄	—	278	1	no	—
313	C₂×C₄	—	278	1	no	—
314	C₂×C₄	—	1112	1	no	—
315	C₂×C₄	—	1112	1	no	—
316	C₂×C₄	—	1112	1	no	—
317	C₂×C₄	—	1112	1	no	—
318	C₂×C₄	—	1112	1	no	—
319	C₂×C₄	—	139	1	no	—
320	C₂×C₄	—	139	1	no	—
321	C₂×C₄	—	139	1	no	—
322	C₂×C₄	—	139	1	no	—
323	C₂×C₄	D₄×D₁₃₉	1	1	yes	characteristic
324	C₂×C₄	C₂³×D₁₃₉	1	1	yes	characteristic
325	C₈	—	2	1	no	—
326	C₈	—	2	1	no	—
327	C₈	—	278	1	no	—
328	C₈	—	278	1	no	—
329	C₂²	—	2	1	no	—
330	C₂²	—	2	1	no	—
331	C₂²	—	2	2	no	—
332	C₂²	—	2	1	no	—
333	C₂²	—	4	1	no	—
334	C₂²	—	4	1	no	—
335	C₂²	—	4	1	no	—
336	C₂²	—	4	1	no	—
337	C₂²	—	4	1	no	—
338	C₂²	—	556	1	no	—
339	C₂²	—	556	1	no	—
340	C₂²	—	556	1	no	—
341	C₂²	C₂×D₄×D₁₃₉	1	1	yes	characteristic
342	C₄	—	2	1	no	—
343	C₄	—	2	2	no	—
344	C₄	—	4	1	no	—
345	C₄	—	278	1	no	—
346	C₄	—	278	1	no	—
347	C₄	—	278	1	no	—
348	C₄	—	278	1	no	—
349	C₄	—	556	1	no	—
350	C₄	—	556	1	no	—
351	C₄	—	556	1	no	—
352	C₄	C₂×#14555784	1	1	yes	characteristic
353	C₄	C₂×#14555784	1	1	yes	characteristic
354	C₂	—	4	1	no	—
355	C₂	—	4	2	no	—
356	C₂	—	8	1	no	—
357	C₂	—	1112	1	no	—
358	C₂	#22581459	1	1	yes	characteristic
359	C₂	#22581755	1	1	yes	characteristic
360	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	#449×C₁₃₉	C₂
socle	C₂²×C₁₃₉	C₂²×D₄
hypercenter	#449	D₁₃₉
hypocenter	C₁₃₉	#3119
LCS	G ▷ C₂×C₄×C₁₃₉ ▷ C₂×C₁₃₉ ▷ C₁₃₉
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ #449

Complements

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

N	H ≅ G/N	type
1. #22581556	C₂	semidirect
2. #22581436	C₂	semidirect
3. #22581269	C₂	semidirect
4. #22581267	C₂	semidirect
5. #22581262	C₂	semidirect
6. #22581251	C₂	semidirect
7. #22581151	C₂	semidirect
8. #22581146	C₂	semidirect
9. #22581098	C₂	semidirect
10. #22581062	C₂	semidirect
11. #22581054	C₂	semidirect
12. #22581038	C₂	semidirect
13. D₈×Dic₁₃₉	C₂	semidirect
14. #22581011	C₂	semidirect
15. #449×C₁₃₉	C₂	semidirect
180. C₁₃₉	#3119	semidirect
191. #449	D₁₃₉	semidirect

Maximal subgroups

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

subgroup	index
#22581556	2
#22581436	2
#22581269	2
#22581267	2
#22581262	2
#22581251	2
#22581151	2
#22581146	2
#22581098	2
#22581062	2
#22581054	2
#22581038	2
D₈×Dic₁₃₉	2
#22581011	2
#449×C₁₃₉	2
#3119	139