Subgroups — #31819475

Order 17792.

368 conjugacy classes of subgroups.

The subgroup lattice has 356 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#22581686	C₂	1	1	yes	maximal, characteristic, complemented
2	#22581673	C₂	1	1	yes	maximal, characteristic, complemented
3	#22581347	C₂	1	1	yes	maximal, characteristic, complemented
4	#22581341	C₂	1	1	yes	maximal, characteristic
5	#22581336	C₂	1	1	yes	maximal, characteristic, complemented
6	#22581174	C₂	1	1	yes	maximal, characteristic, complemented
7	#22581174	C₂	1	1	yes	maximal, characteristic, complemented
8	#22581168	C₂	1	1	yes	maximal, characteristic, complemented
9	#22581167	C₂	1	1	yes	maximal, characteristic, complemented
10	#22581158	C₂	1	1	yes	maximal, characteristic, complemented
11	#22581152	C₂	1	1	yes	maximal, characteristic, complemented
12	#22581081	C₂	1	1	yes	maximal, characteristic, complemented
13	#22581059	C₂	1	1	yes	maximal, characteristic, complemented
14	#22581040	C₂	1	1	yes	maximal, characteristic, complemented
15	#442×C₁₃₉	C₂	1	1	yes	maximal, characteristic, complemented
16	C₂×Dic₅₅₆	C₂²	1	1	yes	characteristic
17	C₂×Dic₅₅₆	C₂²	1	1	yes	characteristic
18	C₂×#14555774	C₂²	1	1	yes	characteristic
19	C₂×#14555774	C₂²	1	1	yes	characteristic
20	C₂×#14555773	C₂²	1	1	yes	characteristic
21	C₂×#14555773	C₂²	1	1	yes	characteristic
22	C₂×#14555773	C₂²	1	1	yes	characteristic
23	C₂×#14555770	C₂²	1	1	yes	characteristic
24	#17984842	C₂²	1	1	yes	characteristic
25	#17984838	C₂²	1	1	yes	characteristic
26	#17984835	—	2	1	no	—
27	#17984835	—	2	1	no	—
28	#17984830	C₂²	1	1	yes	characteristic
29	#17984829	C₂²	1	1	yes	characteristic
30	#17984829	C₂²	1	1	yes	characteristic
31	#17984829	C₂²	1	1	yes	characteristic
32	#17984828	—	2	1	no	—
33	#17984825	—	2	1	no	—
34	#17984824	—	2	1	no	—
35	#17984821	—	2	1	no	—
36	C₄×Dic₂₇₈	C₂²	1	1	yes	characteristic
37	#17984807	C₂²	1	1	yes	characteristic
38	#17984807	C₂²	1	1	yes	characteristic
39	#17984807	C₂²	1	1	yes	characteristic
40	#17984804	C₂²	1	1	yes	characteristic
41	C₄×C₂²⋊D₁₃₉	—	2	1	no	—
42	#17984801	—	2	1	no	—
43	C₄⋊(C₁₃₉⋊C₈)	C₂²	1	1	yes	characteristic
44	Dic₁₃₉⋊C₈	C₂²	1	1	yes	characteristic
45	Dic₁₃₉⋊C₈	C₂²	1	1	yes	characteristic
46	#17984765	C₂²	1	1	yes	characteristic
47	#17984765	C₂²	1	1	yes	characteristic
48	#17984765	C₂²	1	1	yes	characteristic
49	#17984762	C₂²	1	1	yes	characteristic
50	#17984752	C₂²	1	1	yes	characteristic
51	C₈×Dic₁₃₉	C₂²	1	1	yes	characteristic
52	C₂×D₈×C₁₃₉	C₂²	1	1	yes	characteristic
53	C₄×D₄×C₁₃₉	C₂²	1	1	yes	characteristic
54	C₄×D₄×C₁₃₉	C₂²	1	1	yes	characteristic
55	#106×C₁₃₉	C₂²	1	1	yes	characteristic
56	#102×C₁₃₉	C₂²	1	1	yes	characteristic
57	#102×C₁₃₉	C₂²	1	1	yes	characteristic
58	#97×C₁₃₉	C₂²	1	1	yes	characteristic
59	C₂×Dic₂₇₈	D₄	1	1	yes	characteristic
60	C₂×Dic₂₇₈	C₂³	1	1	yes	characteristic
61	C₂×Dic₂₇₈	C₂³	1	1	yes	characteristic
62	C₂×Dic₂₇₈	C₂³	1	1	yes	characteristic
63	C₂×D₅₅₆	C₂³	1	1	yes	characteristic
64	C₂×C₂²⋊D₁₃₉	—	2	1	no	—
65	C₂×C₂²⋊D₁₃₉	—	2	1	no	—
66	C₂×C₄×D₁₃₉	—	2	1	no	—
67	Dic₅₅₆	—	2	2	no	—
68	Dic₅₅₆	—	4	1	no	—
69	#14555774	—	2	2	no	—
70	#14555774	—	4	1	no	—
71	#14555773	—	4	1	no	—
72	#14555773	—	4	1	no	—
73	#14555773	—	4	1	no	—
74	#14555770	—	4	1	no	—
75	C₂×C₁₃₉⋊C₈	C₂³	1	1	yes	characteristic
76	C₂×C₁₃₉⋊C₈	C₂³	1	1	yes	characteristic
77	C₄⋊Dic₁₃₉	—	2	1	no	—
78	C₄⋊Dic₁₃₉	—	2	1	no	—
79	C₄⋊Dic₁₃₉	D₄	1	1	yes	characteristic
80	Dic₁₃₉⋊C₄	—	2	1	no	—
81	Dic₁₃₉⋊C₄	—	2	1	no	—
82	Dic₁₃₉⋊C₄	—	2	1	no	—
83	Dic₁₃₉⋊C₄	D₄	1	1	yes	characteristic
84	Dic₁₃₉⋊C₄	D₄	1	1	yes	characteristic
85	C₂²⋊Dic₁₃₉	—	2	1	no	—
86	C₂²⋊Dic₁₃₉	—	2	1	no	—
87	C₂²⋊Dic₁₃₉	—	2	1	no	—
88	C₂²⋊Dic₁₃₉	—	2	1	no	—
89	C₂²⋊Dic₁₃₉	—	2	1	no	—
90	C₂²⋊Dic₁₃₉	—	2	1	no	—
91	(C₂×D₁₃₉)⋊C₄	—	2	1	no	—
92	(C₂×D₁₃₉)⋊C₄	—	2	1	no	—
93	(C₂×D₁₃₉)⋊C₄	—	2	1	no	—
94	C₄×Dic₁₃₉	—	2	1	no	—
95	C₄×Dic₁₃₉	—	2	1	no	—
96	C₄×Dic₁₃₉	—	2	1	no	—
97	C₄×Dic₁₃₉	C₂³	1	1	yes	characteristic
98	C₄×Dic₁₃₉	C₂³	1	1	yes	characteristic
99	C₂×D₄×C₁₃₉	C₂³	1	1	yes	characteristic
100	C₂×D₄×C₁₃₉	C₂³	1	1	yes	characteristic
101	C₂²×C₄×C₁₃₉	—	2	1	no	—
102	C₂²×C₄×C₁₃₉	—	2	1	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	C₄⋊C₄×C₁₃₉	C₂³	1	1	yes	characteristic
108	C₂²⋊C₄×C₁₃₉	—	2	1	no	—
109	C₂²⋊C₄×C₁₃₉	—	2	1	no	—
110	C₄²×C₁₃₉	C₂³	1	1	yes	characteristic
111	C₂²×D₁₃₉	—	2	1	no	—
112	Dic₂₇₈	—	2	1	no	—
113	Dic₂₇₈	—	2	1	no	—
114	Dic₂₇₈	—	2	1	no	—
115	Dic₂₇₈	—	2	1	no	—
116	Dic₂₇₈	—	2	1	no	—
117	Dic₂₇₈	—	2	1	no	—
118	Dic₂₇₈	—	4	1	no	—
119	D₅₅₆	—	2	1	no	—
120	D₅₅₆	—	2	1	no	—
121	C₂²⋊D₁₃₉	—	4	2	no	—
122	C₂²⋊D₁₃₉	—	8	1	no	—
123	C₂×Dic₁₃₉	—	2	1	no	—
124	C₂×Dic₁₃₉	—	2	1	no	—
125	C₂×Dic₁₃₉	—	2	1	no	—
126	C₂×Dic₁₃₉	C₂×D₄	1	1	yes	characteristic
127	C₂×Dic₁₃₉	C₂×D₄	1	1	yes	characteristic
128	C₂×Dic₁₃₉	C₂×D₄	1	1	yes	characteristic
129	C₂×Dic₁₃₉	C₂×D₄	1	1	yes	characteristic
130	C₄×D₁₃₉	—	8	1	no	—
131	C₁₃₉⋊C₈	—	2	1	no	—
132	C₁₃₉⋊C₈	—	2	1	no	—
133	C₂³×C₁₃₉	—	2	1	no	—
134	C₂³×C₁₃₉	—	2	1	no	—
135	D₄×C₁₃₉	—	2	1	no	—
136	D₄×C₁₃₉	—	2	1	no	—
137	D₄×C₁₃₉	—	2	1	no	—
138	D₄×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	—	8	1	no	—
143	C₂×C₄×C₁₃₉	C₂×D₄	1	1	yes	characteristic
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	#41	1	2	yes	—
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	—	4	1	no	—
154	C₂²×C₁₃₉	C₂²×D₄	1	1	yes	characteristic
155	C₂×D₁₃₉	—	4	1	no	—
156	C₂×D₁₃₉	—	4	1	no	—
157	C₂×D₁₃₉	—	4	1	no	—
158	C₄×C₁₃₉	—	2	1	no	—
159	C₄×C₁₃₉	—	2	1	no	—
160	C₄×C₁₃₉	—	4	1	no	—
161	C₄×C₁₃₉	—	4	1	no	—
162	C₄×C₁₃₉	C₂×#41	1	1	yes	characteristic
163	C₄×C₁₃₉	H(5,𝔽₂)	1	1	yes	characteristic
164	Dic₁₃₉	—	2	1	no	—
165	Dic₁₃₉	—	2	1	no	—
166	Dic₁₃₉	—	2	1	no	—
167	Dic₁₃₉	—	2	1	no	—
168	Dic₁₃₉	—	4	1	no	—
169	Dic₁₃₉	—	4	1	no	—
170	Dic₁₃₉	—	4	1	no	—
171	C₂×C₁₃₉	—	8	1	no	—
172	C₂×C₁₃₉	—	8	1	no	—
173	C₂×C₁₃₉	#550	1	1	yes	characteristic
174	C₂×C₁₃₉	#575	1	1	yes	characteristic
175	C₂×C₁₃₉	#577	1	1	yes	characteristic
176	D₁₃₉	—	8	1	no	—
177	C₁₃₉	#3401	1	1	yes	characteristic, complemented, Sylow
178	#3401	—	139	1	no	maximal, Sylow
179	#559	—	139	1	no	—
180	#556	—	139	1	no	—
181	#497	—	139	1	no	—
182	#496	—	139	1	no	—
183	#495	—	139	1	no	—
184	#464	—	139	1	no	—
185	#464	—	139	1	no	—
186	#463	—	139	1	no	—
187	#463	—	139	1	no	—
188	#461	—	139	1	no	—
189	#460	—	139	1	no	—
190	#446	—	139	1	no	—
191	#442	D₁₃₉	1	1	yes	characteristic, complemented
192	#441	—	139	1	no	—
193	C₄×SD₁₆	—	139	1	no	—
194	C₂×Q₁₆	—	139	1	no	—
195	C₂×Q₁₆	—	139	1	no	—
196	C₂×SD₁₆	—	139	1	no	—
197	C₂×SD₁₆	—	139	1	no	—
198	C₂×SD₁₆	—	139	1	no	—
199	C₂×SD₁₆	—	139	1	no	—
200	C₂×SD₁₆	—	139	1	no	—
201	C₂×D₈	—	139	1	no	—
202	C₂×D₈	C₂×D₁₃₉	1	1	yes	characteristic
203	C₄⋊Q₈	—	139	1	no	—
204	#127	—	139	1	no	—
205	#126	—	278	1	no	—
206	#126	—	278	1	no	—
207	#124	—	278	1	no	—
208	#124	—	139	1	no	—
209	#124	—	139	1	no	—
210	#124	—	139	1	no	—
211	#124	—	139	1	no	—
212	#123	—	278	1	no	—
213	#123	—	278	1	no	—
214	C₂²⋊Q₈	—	278	1	no	—
215	C₄×Q₈	—	139	1	no	—
216	C₄×Q₈	—	139	1	no	—
217	C₄×Q₈	—	139	1	no	—
218	C₄×Q₈	—	139	1	no	—
219	C₄×D₄	—	278	1	no	—
220	C₄×D₄	—	139	1	no	—
221	C₄×D₄	C₂×D₁₃₉	1	1	yes	characteristic
222	C₄×D₄	C₂×D₁₃₉	1	1	yes	characteristic
223	#117	—	278	1	no	—
224	#106	C₂×D₁₃₉	1	1	yes	characteristic
225	C₄⋊C₈	—	139	1	no	—
226	C₄⋊C₈	—	139	1	no	—
227	C₄⋊C₈	—	139	1	no	—
228	#103	—	139	1	no	—
229	#103	—	139	1	no	—
230	#103	—	139	1	no	—
231	#102	—	139	1	no	—
232	#102	C₂×D₁₃₉	1	1	yes	characteristic
233	#102	C₂×D₁₃₉	1	1	yes	characteristic
234	#97	—	139	1	no	—
235	#97	C₂×D₁₃₉	1	1	yes	characteristic
236	C₄×C₈	—	139	1	no	—
237	C₂×Q₈	—	139	1	no	—
238	C₂×Q₈	—	139	1	no	—
239	C₂×Q₈	—	139	1	no	—
240	C₂×Q₈	—	139	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	—	2	1	no	—
247	C₂²×C₄	—	2	1	no	—
248	C₂²×C₄	—	278	1	no	—
249	Q₁₆	—	278	2	no	—
250	Q₁₆	—	556	1	no	—
251	SD₁₆	—	278	2	no	—
252	SD₁₆	—	556	1	no	—
253	SD₁₆	—	556	1	no	—
254	SD₁₆	—	556	1	no	—
255	SD₁₆	—	556	1	no	—
256	D₈	—	2	2	no	—
257	D₈	—	556	1	no	—
258	C₂×C₈	—	139	1	no	—
259	C₂×C₈	—	139	1	no	—
260	C₂×C₈	C₂²×D₁₃₉	1	1	yes	characteristic
261	C₂×C₈	C₂²×D₁₃₉	1	1	yes	characteristic
262	C₄⋊C₄	—	278	1	no	—
263	C₄⋊C₄	—	278	1	no	—
264	C₄⋊C₄	—	278	1	no	—
265	C₄⋊C₄	—	278	1	no	—
266	C₄⋊C₄	—	278	1	no	—
267	C₄⋊C₄	—	139	1	no	—
268	C₄⋊C₄	—	139	1	no	—
269	C₄⋊C₄	—	139	1	no	—
270	C₄⋊C₄	C₂²×D₁₃₉	1	1	yes	characteristic
271	C₄⋊C₄	C₂²×D₁₃₉	1	1	yes	characteristic
272	C₂²⋊C₄	—	2	1	no	—
273	C₂²⋊C₄	—	2	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	—	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	—	278	1	no	—
284	C₄²	—	278	1	no	—
285	C₄²	—	278	1	no	—
286	C₄²	—	139	1	no	—
287	C₄²	—	139	1	no	—
288	C₄²	C₂²×D₁₃₉	1	1	yes	characteristic
289	C₂³	—	2	1	no	—
290	C₂³	—	2	1	no	—
291	C₂³	—	278	1	no	—
292	Q₈	—	278	1	no	—
293	Q₈	—	278	1	no	—
294	Q₈	—	278	1	no	—
295	Q₈	—	278	1	no	—
296	Q₈	—	278	1	no	—
297	Q₈	—	278	1	no	—
298	Q₈	—	556	1	no	—
299	D₄	—	2	1	no	—
300	D₄	—	2	1	no	—
301	D₄	—	2	1	no	—
302	D₄	—	2	1	no	—
303	D₄	—	278	1	no	—
304	D₄	—	278	1	no	—
305	D₄	—	556	2	no	—
306	D₄	—	1112	1	no	—
307	C₂×C₄	—	2	1	no	—
308	C₂×C₄	—	2	1	no	—
309	C₂×C₄	—	4	2	no	—
310	C₂×C₄	—	8	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	—	139	1	no	—
316	C₂×C₄	—	139	1	no	—
317	C₂×C₄	—	139	1	no	—
318	C₂×C₄	—	139	1	no	—
319	C₂×C₄	D₄×D₁₃₉	1	1	yes	characteristic
320	C₂×C₄	D₄×D₁₃₉	1	1	yes	characteristic
321	C₂×C₄	C₂³×D₁₃₉	1	1	yes	characteristic
322	C₈	—	2	1	no	—
323	C₈	—	278	1	no	—
324	C₈	—	278	1	no	—
325	C₈	#14555785	1	2	yes	—
326	C₂²	—	4	1	no	—
327	C₂²	—	4	1	no	—
328	C₂²	—	4	1	no	—
329	C₂²	—	4	1	no	—
330	C₂²	—	4	1	no	—
331	C₂²	—	4	1	no	—
332	C₂²	—	556	1	no	—
333	C₂²	—	556	1	no	—
334	C₂²	—	556	1	no	—
335	C₂²	C₂×D₄×D₁₃₉	1	1	yes	characteristic
336	C₄	—	2	1	no	—
337	C₄	—	2	1	no	—
338	C₄	—	4	1	no	—
339	C₄	—	4	1	no	—
340	C₄	—	278	1	no	—
341	C₄	—	278	1	no	—
342	C₄	—	278	1	no	—
343	C₄	—	278	1	no	—
344	C₄	—	556	1	no	—
345	C₄	—	556	1	no	—
346	C₄	—	556	1	no	—
347	C₄	C₂×#14555785	1	1	yes	characteristic
348	C₄	#17984888	1	1	yes	characteristic
349	C₂	—	8	1	no	—
350	C₂	—	8	1	no	—
351	C₂	—	1112	1	no	—
352	C₂	#22581624	1	1	yes	characteristic
353	C₂	#22581749	1	1	yes	characteristic
354	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	#442×C₁₃₉	C₂
socle	C₂²×C₁₃₉	C₂²×D₄
prime radical	#22581341	C₂
hypercenter	#442	D₁₃₉
hypocenter	C₁₃₉	#3401
LCS	G ▷ C₂×C₄×C₁₃₉ ▷ C₂×C₁₃₉ ▷ C₁₃₉
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #442

Complements

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

N	H ≅ G/N	type
1. #22581686	C₂	semidirect
2. #22581673	C₂	semidirect
3. #22581347	C₂	semidirect
5. #22581336	C₂	semidirect
6. #22581174	C₂	semidirect
7. #22581174	C₂	semidirect
8. #22581168	C₂	semidirect
9. #22581167	C₂	semidirect
10. #22581158	C₂	semidirect
11. #22581152	C₂	semidirect
12. #22581081	C₂	semidirect
13. #22581059	C₂	semidirect
14. #22581040	C₂	semidirect
15. #442×C₁₃₉	C₂	semidirect
177. C₁₃₉	#3401	semidirect
191. #442	D₁₃₉	semidirect

Maximal subgroups

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

subgroup	index
#22581686	2
#22581673	2
#22581347	2
#22581341	2
#22581336	2
#22581174	2
#22581168	2
#22581167	2
#22581158	2
#22581152	2
#22581081	2
#22581059	2
#22581040	2
#442×C₁₃₉	2
#3401	139