Subgroups — #31814605

Order 17792.

358 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	#22580894	C₂	1	1	yes	maximal, characteristic, complemented
2	#22580892	C₂	1	1	yes	maximal, characteristic, complemented
3	#22580889	C₂	1	1	yes	maximal, characteristic
4	#22580886	C₂	1	1	yes	maximal, characteristic, complemented
5	#22580885	C₂	1	1	yes	maximal, characteristic, complemented
6	#22580884	C₂	1	1	yes	maximal, characteristic, complemented
7	#22580883	C₂	1	1	yes	maximal, characteristic, complemented
8	#22580881	C₂	1	1	yes	maximal, characteristic
9	#22580881	C₂	1	1	yes	maximal, characteristic
10	#22580878	C₂	1	1	yes	maximal, characteristic
11	#22580876	C₂	1	1	yes	maximal, characteristic
12	#22580849	C₂	1	1	yes	maximal, characteristic, complemented
13	#22580846	C₂	1	1	yes	maximal, characteristic
14	#22580839	C₂	1	1	yes	maximal, characteristic, complemented
15	#396×C₁₃₉	C₂	1	1	yes	maximal, characteristic
16	C₂×C₄⋊Dic₁₃₉	C₂²	1	1	yes	characteristic
17	C₂×C₄⋊Dic₁₃₉	C₂²	1	1	yes	characteristic
18	C₂×C₄⋊Dic₁₃₉	C₂²	1	1	yes	characteristic
19	C₂×C₄⋊Dic₁₃₉	C₂²	1	1	yes	characteristic
20	C₂×Dic₁₃₉⋊C₄	C₂²	1	1	yes	characteristic
21	C₂×Dic₁₃₉⋊C₄	C₂²	1	1	yes	characteristic
22	C₂×Dic₁₃₉⋊C₄	C₂²	1	1	yes	characteristic
23	C₂×Dic₁₃₉⋊C₄	C₂²	1	1	yes	characteristic
24	C₂×Dic₁₃₉⋊C₄	C₂²	1	1	yes	characteristic
25	C₂×Dic₁₃₉⋊C₄	C₂²	1	1	yes	characteristic
26	C₂×C₂²⋊Dic₁₃₉	C₂²	1	1	yes	characteristic
27	C₂×C₂²⋊Dic₁₃₉	C₂²	1	1	yes	characteristic
28	C₂×C₂²⋊Dic₁₃₉	C₂²	1	1	yes	characteristic
29	C₂×C₂²⋊Dic₁₃₉	C₂²	1	1	yes	characteristic
30	C₂×C₄×Dic₁₃₉	C₂²	1	1	yes	characteristic
31	C₁₃₉⋊#95	C₂²	1	1	yes	characteristic
32	C₁₃₉⋊#95	C₂²	1	1	yes	characteristic
33	C₁₃₉⋊#95	C₂²	1	1	yes	characteristic
34	C₁₃₉⋊#95	C₂²	1	1	yes	characteristic
35	C₁₃₉⋊#95	C₂²	1	1	yes	characteristic
36	C₁₃₉⋊#95	C₂²	1	1	yes	characteristic
37	C₁₃₉⋊#95	C₂²	1	1	yes	characteristic
38	C₁₃₉⋊#95	C₂²	1	1	yes	characteristic
39	C₁₃₉⋊#95	C₂²	1	1	yes	characteristic
40	C₁₃₉⋊#95	C₂²	1	1	yes	characteristic
41	C₁₃₉⋊#95	C₂²	1	1	yes	characteristic
42	C₁₃₉⋊#95	C₂²	1	1	yes	characteristic
43	C₁₃₉⋊#95	C₂²	1	1	yes	characteristic
44	C₂×C₄⋊C₄×C₁₃₉	C₂²	1	1	yes	characteristic
45	C₂×C₄⋊C₄×C₁₃₉	C₂²	1	1	yes	characteristic
46	C₂×C₄⋊C₄×C₁₃₉	C₂²	1	1	yes	characteristic
47	C₂×C₂²⋊C₄×C₁₃₉	C₂²	1	1	yes	characteristic
48	C₂×C₂²⋊C₄×C₁₃₉	C₂²	1	1	yes	characteristic
49	C₂×C₂²⋊C₄×C₁₃₉	C₂²	1	1	yes	characteristic
50	#95×C₁₃₉	C₂²	1	1	yes	characteristic
51	C₂²×Dic₁₃₉	C₂³	1	1	yes	characteristic
52	C₂²×Dic₁₃₉	C₂³	1	1	yes	characteristic
53	C₂²×Dic₁₃₉	C₂³	1	1	yes	characteristic
54	C₂²×Dic₁₃₉	C₂³	1	1	yes	characteristic
55	C₂²×Dic₁₃₉	C₂³	1	1	yes	characteristic
56	C₂²×Dic₁₃₉	C₂³	1	1	yes	characteristic
57	C₂²×Dic₁₃₉	C₂³	1	1	yes	characteristic
58	C₂²×Dic₁₃₉	C₂³	1	1	yes	characteristic
59	C₄⋊Dic₁₃₉	—	2	2	no	—
60	C₄⋊Dic₁₃₉	—	4	1	no	—
61	C₄⋊Dic₁₃₉	—	4	1	no	—
62	C₄⋊Dic₁₃₉	—	4	1	no	—
63	Dic₁₃₉⋊C₄	—	4	1	no	—
64	Dic₁₃₉⋊C₄	—	4	1	no	—
65	Dic₁₃₉⋊C₄	—	4	1	no	—
66	Dic₁₃₉⋊C₄	—	4	1	no	—
67	Dic₁₃₉⋊C₄	—	4	1	no	—
68	Dic₁₃₉⋊C₄	—	4	1	no	—
69	C₂²⋊Dic₁₃₉	—	4	1	no	—
70	C₂²⋊Dic₁₃₉	—	4	1	no	—
71	C₂²⋊Dic₁₃₉	—	4	1	no	—
72	C₂²⋊Dic₁₃₉	—	4	1	no	—
73	C₄×Dic₁₃₉	—	4	1	no	—
74	C₂⁴×C₁₃₉	C₂³	1	1	yes	characteristic
75	C₂²×C₄×C₁₃₉	C₂³	1	1	yes	characteristic
76	C₂²×C₄×C₁₃₉	C₂³	1	1	yes	characteristic
77	C₂²×C₄×C₁₃₉	C₂³	1	1	yes	characteristic
78	C₂²×C₄×C₁₃₉	C₂³	1	1	yes	characteristic
79	C₂²×C₄×C₁₃₉	C₂³	1	1	yes	characteristic
80	C₂²×C₄×C₁₃₉	C₂³	1	1	yes	characteristic
81	C₄⋊C₄×C₁₃₉	—	2	2	no	—
82	C₄⋊C₄×C₁₃₉	—	4	1	no	—
83	C₄⋊C₄×C₁₃₉	—	4	1	no	—
84	C₂²⋊C₄×C₁₃₉	—	2	2	no	—
85	C₂²⋊C₄×C₁₃₉	—	4	1	no	—
86	C₂²⋊C₄×C₁₃₉	—	4	1	no	—
87	C₂×Dic₁₃₉	—	2	1	no	—
88	C₂×Dic₁₃₉	—	2	1	no	—
89	C₂×Dic₁₃₉	—	2	1	no	—
90	C₂×Dic₁₃₉	—	2	1	no	—
91	C₂×Dic₁₃₉	—	2	1	no	—
92	C₂×Dic₁₃₉	—	2	1	no	—
93	C₂×Dic₁₃₉	—	2	1	no	—
94	C₂×Dic₁₃₉	—	2	1	no	—
95	C₂×Dic₁₃₉	—	2	1	no	—
96	C₂×Dic₁₃₉	—	2	1	no	—
97	C₂×Dic₁₃₉	—	2	1	no	—
98	C₂×Dic₁₃₉	—	2	1	no	—
99	C₂×Dic₁₃₉	—	2	1	no	—
100	C₂×Dic₁₃₉	—	2	1	no	—
101	C₂×Dic₁₃₉	—	2	1	no	—
102	C₂×Dic₁₃₉	—	2	1	no	—
103	C₂×Dic₁₃₉	—	2	1	no	—
104	C₂×Dic₁₃₉	—	2	1	no	—
105	C₂×Dic₁₃₉	—	2	1	no	—
106	C₂×Dic₁₃₉	—	2	1	no	—
107	C₂×Dic₁₃₉	—	2	1	no	—
108	C₂×Dic₁₃₉	—	2	1	no	—
109	C₂×Dic₁₃₉	—	2	1	no	—
110	C₂×Dic₁₃₉	—	2	1	no	—
111	C₂³×C₁₃₉	—	2	1	no	—
112	C₂³×C₁₃₉	—	2	1	no	—
113	C₂³×C₁₃₉	—	2	1	no	—
114	C₂³×C₁₃₉	—	2	1	no	—
115	C₂³×C₁₃₉	—	2	1	no	—
116	C₂³×C₁₃₉	—	2	1	no	—
117	C₂³×C₁₃₉	—	2	1	no	—
118	C₂³×C₁₃₉	C₂⁴	1	1	yes	characteristic
119	C₂×C₄×C₁₃₉	—	2	1	no	—
120	C₂×C₄×C₁₃₉	—	2	1	no	—
121	C₂×C₄×C₁₃₉	—	2	1	no	—
122	C₂×C₄×C₁₃₉	—	2	1	no	—
123	C₂×C₄×C₁₃₉	—	2	1	no	—
124	C₂×C₄×C₁₃₉	—	2	1	no	—
125	C₂×C₄×C₁₃₉	—	2	1	no	—
126	C₂×C₄×C₁₃₉	—	2	1	no	—
127	C₂×C₄×C₁₃₉	—	2	1	no	—
128	C₂×C₄×C₁₃₉	—	2	1	no	—
129	C₂×C₄×C₁₃₉	—	2	1	no	—
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	#41	1	2	yes	—
137	C₂²×C₁₃₉	—	4	1	no	—
138	C₂²×C₁₃₉	—	4	1	no	—
139	C₂²×C₁₃₉	—	4	1	no	—
140	C₂²×C₁₃₉	—	4	1	no	—
141	C₂²×C₁₃₉	—	4	1	no	—
142	C₂²×C₁₃₉	—	4	1	no	—
143	C₂²×C₁₃₉	—	4	1	no	—
144	C₂²×C₁₃₉	C₂×#41	1	1	yes	characteristic
145	C₂²×C₁₃₉	H(5,𝔽₂)	1	1	yes	characteristic
146	C₂²×C₁₃₉	H(5,𝔽₂)	1	1	yes	characteristic
147	C₂²×C₁₃₉	H(5,𝔽₂)	1	1	yes	characteristic
148	C₂²×C₁₃₉	2₋¹⁺⁴	1	1	yes	characteristic
149	C₂²×C₁₃₉	2₋¹⁺⁴	1	1	yes	characteristic
150	C₂²×C₁₃₉	2₋¹⁺⁴	1	1	yes	characteristic
151	C₄×C₁₃₉	—	2	2	no	—
152	C₄×C₁₃₉	—	4	1	no	—
153	C₄×C₁₃₉	—	4	1	no	—
154	C₄×C₁₃₉	—	4	1	no	—
155	C₄×C₁₃₉	—	4	1	no	—
156	C₄×C₁₃₉	—	4	1	no	—
157	Dic₁₃₉	—	4	1	no	—
158	Dic₁₃₉	—	4	1	no	—
159	Dic₁₃₉	—	4	1	no	—
160	Dic₁₃₉	—	4	1	no	—
161	Dic₁₃₉	—	4	1	no	—
162	Dic₁₃₉	—	4	1	no	—
163	Dic₁₃₉	—	4	1	no	—
164	Dic₁₃₉	—	4	1	no	—
165	C₂×C₁₃₉	—	8	1	no	—
166	C₂×C₁₃₉	#539	1	1	yes	characteristic
167	C₂×C₁₃₉	#540	1	1	yes	characteristic
168	C₂×C₁₃₉	#541	1	1	yes	characteristic
169	C₂×C₁₃₉	#562	1	1	yes	characteristic
170	C₂×C₁₃₉	#562	1	1	yes	characteristic
171	C₂×C₁₃₉	#563	1	1	yes	characteristic
172	C₂×C₁₃₉	#563	1	1	yes	characteristic
173	C₁₃₉	#2836	1	1	yes	characteristic, complemented, Sylow
174	#2836	—	139	1	no	maximal, Sylow
175	#400	—	139	1	no	—
176	#400	—	139	1	no	—
177	#399	—	139	1	no	—
178	#398	—	139	1	no	—
179	#398	—	139	1	no	—
180	#398	—	139	1	no	—
181	#398	—	139	1	no	—
182	#397	—	139	1	no	—
183	#397	—	139	1	no	—
184	#397	—	139	1	no	—
185	#396	—	139	1	no	—
186	#396	D₁₃₉	1	1	yes	characteristic
187	#389	—	139	1	no	—
188	#388	—	139	1	no	—
189	#387	—	139	1	no	—
190	C₂×C₄⋊C₄	—	139	1	no	—
191	C₂×C₄⋊C₄	—	139	1	no	—
192	C₂×C₄⋊C₄	—	139	1	no	—
193	C₂×C₄⋊C₄	—	139	1	no	—
194	C₂×C₄⋊C₄	—	139	1	no	—
195	C₂×C₄⋊C₄	—	139	1	no	—
196	C₂×C₄⋊C₄	—	139	1	no	—
197	C₂×C₄⋊C₄	—	139	1	no	—
198	C₂×C₄⋊C₄	—	139	1	no	—
199	C₂×C₄⋊C₄	—	139	1	no	—
200	C₂×C₄⋊C₄	C₂×D₁₃₉	1	1	yes	characteristic
201	C₂×C₄⋊C₄	C₂×D₁₃₉	1	1	yes	characteristic
202	C₂×C₄⋊C₄	C₂×D₁₃₉	1	1	yes	characteristic
203	C₂×C₂²⋊C₄	—	139	1	no	—
204	C₂×C₂²⋊C₄	—	139	1	no	—
205	C₂×C₂²⋊C₄	—	139	1	no	—
206	C₂×C₂²⋊C₄	—	139	1	no	—
207	C₂×C₂²⋊C₄	C₂×D₁₃₉	1	1	yes	characteristic
208	C₂×C₂²⋊C₄	C₂×D₁₃₉	1	1	yes	characteristic
209	C₂×C₂²⋊C₄	C₂×D₁₃₉	1	1	yes	characteristic
210	C₂×C₄²	—	139	1	no	—
211	#95	—	139	1	no	—
212	#95	—	139	1	no	—
213	#95	—	139	1	no	—
214	#95	—	139	1	no	—
215	#95	—	139	1	no	—
216	#95	—	139	1	no	—
217	#95	—	139	1	no	—
218	#95	—	139	1	no	—
219	#95	—	139	1	no	—
220	#95	—	139	1	no	—
221	#95	—	139	1	no	—
222	#95	—	139	1	no	—
223	#95	—	139	1	no	—
224	#95	C₂×D₁₃₉	1	1	yes	characteristic
225	C₂⁴	C₂²×D₁₃₉	1	1	yes	characteristic
226	C₂²×C₄	—	139	1	no	—
227	C₂²×C₄	—	139	1	no	—
228	C₂²×C₄	—	139	1	no	—
229	C₂²×C₄	—	139	1	no	—
230	C₂²×C₄	—	139	1	no	—
231	C₂²×C₄	—	139	1	no	—
232	C₂²×C₄	—	139	1	no	—
233	C₂²×C₄	—	139	1	no	—
234	C₂²×C₄	C₂²×D₁₃₉	1	1	yes	characteristic
235	C₂²×C₄	C₂²×D₁₃₉	1	1	yes	characteristic
236	C₂²×C₄	C₂²×D₁₃₉	1	1	yes	characteristic
237	C₂²×C₄	C₂²×D₁₃₉	1	1	yes	characteristic
238	C₂²×C₄	C₂²×D₁₃₉	1	1	yes	characteristic
239	C₂²×C₄	C₂²×D₁₃₉	1	1	yes	characteristic
240	C₄⋊C₄	—	2	2	no	—
241	C₄⋊C₄	—	4	1	no	—
242	C₄⋊C₄	—	4	1	no	—
243	C₄⋊C₄	—	278	2	no	—
244	C₄⋊C₄	—	556	1	no	—
245	C₄⋊C₄	—	556	1	no	—
246	C₄⋊C₄	—	556	1	no	—
247	C₄⋊C₄	—	556	1	no	—
248	C₄⋊C₄	—	556	1	no	—
249	C₄⋊C₄	—	556	1	no	—
250	C₄⋊C₄	—	556	1	no	—
251	C₄⋊C₄	—	556	1	no	—
252	C₄⋊C₄	—	556	1	no	—
253	C₂²⋊C₄	—	2	2	no	—
254	C₂²⋊C₄	—	4	1	no	—
255	C₂²⋊C₄	—	4	1	no	—
256	C₂²⋊C₄	—	556	1	no	—
257	C₂²⋊C₄	—	556	1	no	—
258	C₂²⋊C₄	—	556	1	no	—
259	C₂²⋊C₄	—	556	1	no	—
260	C₄²	—	556	1	no	—
261	C₂³	—	2	1	no	—
262	C₂³	—	2	1	no	—
263	C₂³	—	2	1	no	—
264	C₂³	—	2	1	no	—
265	C₂³	—	2	1	no	—
266	C₂³	—	2	1	no	—
267	C₂³	—	2	1	no	—
268	C₂³	C₂³×D₁₃₉	1	1	yes	characteristic
269	C₂×C₄	—	2	1	no	—
270	C₂×C₄	—	2	1	no	—
271	C₂×C₄	—	2	1	no	—
272	C₂×C₄	—	2	1	no	—
273	C₂×C₄	—	2	1	no	—
274	C₂×C₄	—	2	1	no	—
275	C₂×C₄	—	2	1	no	—
276	C₂×C₄	—	2	1	no	—
277	C₂×C₄	—	2	1	no	—
278	C₂×C₄	—	2	1	no	—
279	C₂×C₄	—	2	1	no	—
280	C₂×C₄	—	2	1	no	—
281	C₂×C₄	—	2	1	no	—
282	C₂×C₄	—	2	1	no	—
283	C₂×C₄	—	2	1	no	—
284	C₂×C₄	—	2	1	no	—
285	C₂×C₄	—	2	1	no	—
286	C₂×C₄	—	278	1	no	—
287	C₂×C₄	—	278	1	no	—
288	C₂×C₄	—	278	1	no	—
289	C₂×C₄	—	278	1	no	—
290	C₂×C₄	—	278	1	no	—
291	C₂×C₄	—	278	1	no	—
292	C₂×C₄	—	278	1	no	—
293	C₂×C₄	—	278	1	no	—
294	C₂×C₄	—	278	1	no	—
295	C₂×C₄	—	278	1	no	—
296	C₂×C₄	—	278	1	no	—
297	C₂×C₄	—	278	1	no	—
298	C₂×C₄	—	278	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	—	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	—	278	1	no	—
309	C₂×C₄	—	278	1	no	—
310	C₂×C₄	#14555784	1	2	yes	—
311	C₂²	—	4	1	no	—
312	C₂²	—	4	1	no	—
313	C₂²	—	4	1	no	—
314	C₂²	—	4	1	no	—
315	C₂²	—	4	1	no	—
316	C₂²	—	4	1	no	—
317	C₂²	—	4	1	no	—
318	C₂²	C₂×#14555784	1	1	yes	characteristic
319	C₂²	#17984887	1	1	yes	characteristic
320	C₂²	#17984887	1	1	yes	characteristic
321	C₂²	#17984888	1	1	yes	characteristic
322	C₂²	#17984889	1	1	yes	characteristic
323	C₂²	#17984889	1	1	yes	characteristic
324	C₂²	#17984890	1	1	yes	characteristic
325	C₄	—	2	2	no	—
326	C₄	—	4	1	no	—
327	C₄	—	4	1	no	—
328	C₄	—	4	1	no	—
329	C₄	—	4	1	no	—
330	C₄	—	4	1	no	—
331	C₄	—	556	1	no	—
332	C₄	—	556	1	no	—
333	C₄	—	556	1	no	—
334	C₄	—	556	1	no	—
335	C₄	—	556	1	no	—
336	C₄	—	556	1	no	—
337	C₄	—	556	1	no	—
338	C₄	—	556	1	no	—
339	C₂	—	8	1	no	—
340	C₂	#22581548	1	1	yes	characteristic
341	C₂	#22581556	1	1	yes	characteristic
342	C₂	#22581563	1	1	yes	characteristic
343	C₂	#22581698	1	1	yes	characteristic
344	C₂	#22581699	1	1	yes	characteristic
345	C₂	#22581704	1	1	yes	characteristic
346	C₂	#22581705	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂³	C₂³×D₁₃₉
commutator, socle, cosocle	C₂³×C₁₃₉	C₂⁴
Fitting	#396×C₁₃₉	C₂
prime radical	C₂⁴×C₁₃₉	C₂³
hypercenter	#396	D₁₃₉
hypocenter	C₁₃₉	#2836
LCS	G ▷ C₂³×C₁₃₉ ▷ C₁₃₉
UCS	C₁ ◁ C₂³ ◁ #396

Complements

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

N	H ≅ G/N	type
1. #22580894	C₂	semidirect
2. #22580892	C₂	semidirect
4. #22580886	C₂	semidirect
5. #22580885	C₂	semidirect
6. #22580884	C₂	semidirect
7. #22580883	C₂	semidirect
12. #22580849	C₂	semidirect
14. #22580839	C₂	semidirect
173. C₁₃₉	#2836	semidirect

Maximal subgroups

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

subgroup	index
#22580894	2
#22580892	2
#22580889	2
#22580886	2
#22580885	2
#22580884	2
#22580883	2
#22580881	2
#22580878	2
#22580876	2
#22580849	2
#22580846	2
#22580839	2
#396×C₁₃₉	2
#2836	139