Subgroups — #31819493

Order 17792.

380 conjugacy classes of subgroups.

The subgroup lattice has 368 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#22581688	C₂	1	1	yes	maximal, characteristic, complemented
2	#22581686	C₂	1	1	yes	maximal, characteristic, complemented
3	#22581342	C₂	1	1	yes	maximal, characteristic, complemented
4	#22581339	C₂	1	1	yes	maximal, characteristic, complemented
5	#22581334	C₂	1	1	yes	maximal, characteristic, complemented
6	#22581178	C₂	1	1	yes	maximal, characteristic, complemented
7	#22581175	C₂	1	1	yes	maximal, characteristic, complemented
8	#22581171	C₂	1	1	yes	maximal, characteristic, complemented
9	#22581167	C₂	1	1	yes	maximal, characteristic, complemented
10	#22581148	C₂	1	1	yes	maximal, characteristic, complemented
11	#22581147	C₂	1	1	yes	maximal, characteristic, complemented
12	#22581081	C₂	1	1	yes	maximal, characteristic, complemented
13	#22581052	C₂	1	1	yes	maximal, characteristic, complemented
14	#22581035	C₂	1	1	yes	maximal, characteristic, complemented
15	#440×C₁₃₉	C₂	1	1	yes	maximal, characteristic, complemented
16	C₂×#14555775	C₂²	1	1	yes	characteristic
17	C₂×#14555774	C₂²	1	1	yes	characteristic
18	C₂×#14555774	C₂²	1	1	yes	characteristic
19	C₂×#14555772	C₂²	1	1	yes	characteristic
20	C₂×D₁₁₁₂	C₂²	1	1	yes	characteristic
21	C₂×D₁₁₁₂	C₂²	1	1	yes	characteristic
22	C₂×#14555770	C₂²	1	1	yes	characteristic
23	C₂×#14555770	C₂²	1	1	yes	characteristic
24	#17984839	C₂²	1	1	yes	characteristic
25	#17984838	C₂²	1	1	yes	characteristic
26	#17984831	C₂²	1	1	yes	characteristic
27	#17984831	C₂²	1	1	yes	characteristic
28	#17984830	—	2	1	no	—
29	#17984830	C₂²	1	1	yes	characteristic
30	#17984829	C₂²	1	1	yes	characteristic
31	#17984828	—	2	1	no	—
32	#17984827	—	2	1	no	—
33	#17984827	—	2	1	no	—
34	#17984825	—	2	1	no	—
35	#17984824	—	2	1	no	—
36	C₄×Dic₂₇₈	C₂²	1	1	yes	characteristic
37	#17984807	C₂²	1	1	yes	characteristic
38	C₄×D₅₅₆	—	2	1	no	—
39	#17984804	C₂²	1	1	yes	characteristic
40	#17984804	C₂²	1	1	yes	characteristic
41	#17984804	C₂²	1	1	yes	characteristic
42	C₄×C₂²⋊D₁₃₉	—	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	#17984762	C₂²	1	1	yes	characteristic
48	#17984762	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×SD₁₆×C₁₃₉	C₂²	1	1	yes	characteristic
53	C₄×Q₈×C₁₃₉	C₂²	1	1	yes	characteristic
54	C₄×D₄×C₁₃₉	C₂²	1	1	yes	characteristic
55	#107×C₁₃₉	C₂²	1	1	yes	characteristic
56	#103×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×D₅₅₆	—	2	1	no	—
62	C₂×D₅₅₆	—	2	1	no	—
63	C₂×D₅₅₆	C₂³	1	1	yes	characteristic
64	C₂×D₅₅₆	C₂³	1	1	yes	characteristic
65	C₂×D₅₅₆	C₂³	1	1	yes	characteristic
66	C₂×C₂²⋊D₁₃₉	—	2	1	no	—
67	C₂×C₂²⋊D₁₃₉	—	2	1	no	—
68	C₂×C₄×D₁₃₉	—	2	1	no	—
69	C₂×C₄×D₁₃₉	—	2	1	no	—
70	C₂×C₄×D₁₃₉	—	2	1	no	—
71	#14555775	—	4	1	no	—
72	#14555774	—	2	2	no	—
73	#14555774	—	4	1	no	—
74	#14555772	—	4	1	no	—
75	D₁₁₁₂	—	2	2	no	—
76	D₁₁₁₂	—	4	1	no	—
77	#14555770	—	4	1	no	—
78	#14555770	—	4	1	no	—
79	C₂×C₁₃₉⋊C₈	C₂³	1	1	yes	characteristic
80	C₂×C₁₃₉⋊C₈	C₂³	1	1	yes	characteristic
81	C₄⋊Dic₁₃₉	D₄	1	1	yes	characteristic
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	(C₂×D₁₃₉)⋊C₄	—	2	1	no	—
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	(C₂×D₁₃₉)⋊C₄	—	2	1	no	—
94	(C₂×D₁₃₉)⋊C₄	—	2	1	no	—
95	(C₂×D₁₃₉)⋊C₄	—	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×Q₈×C₁₃₉	C₂³	1	1	yes	characteristic
100	C₂×D₄×C₁₃₉	C₂³	1	1	yes	characteristic
101	C₂²×C₄×C₁₃₉	—	2	1	no	—
102	SD₁₆×C₁₃₉	—	2	2	no	—
103	C₂×C₈×C₁₃₉	C₂³	1	1	yes	characteristic
104	C₂×C₈×C₁₃₉	C₂³	1	1	yes	characteristic
105	C₄⋊C₄×C₁₃₉	—	2	1	no	—
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	C₄²×C₁₃₉	—	2	1	no	—
110	C₄²×C₁₃₉	C₂³	1	1	yes	characteristic
111	C₂²×D₁₃₉	—	2	1	no	—
112	C₂²×D₁₃₉	—	2	1	no	—
113	C₂²×D₁₃₉	—	2	1	no	—
114	Dic₂₇₈	—	2	1	no	—
115	Dic₂₇₈	—	2	1	no	—
116	Dic₂₇₈	—	4	1	no	—
117	D₅₅₆	—	2	1	no	—
118	D₅₅₆	—	2	1	no	—
119	D₅₅₆	—	2	1	no	—
120	D₅₅₆	—	2	1	no	—
121	D₅₅₆	—	2	1	no	—
122	D₅₅₆	—	2	1	no	—
123	D₅₅₆	—	4	2	no	—
124	D₅₅₆	—	8	1	no	—
125	C₂²⋊D₁₃₉	—	4	2	no	—
126	C₂²⋊D₁₃₉	—	8	1	no	—
127	C₂×Dic₁₃₉	—	2	1	no	—
128	C₂×Dic₁₃₉	C₂×D₄	1	1	yes	characteristic
129	C₂×Dic₁₃₉	C₂×D₄	1	1	yes	characteristic
130	C₂×Dic₁₃₉	C₂×D₄	1	1	yes	characteristic
131	C₂×Dic₁₃₉	C₂×D₄	1	1	yes	characteristic
132	C₄×D₁₃₉	—	8	1	no	—
133	C₄×D₁₃₉	—	8	1	no	—
134	C₄×D₁₃₉	—	8	1	no	—
135	C₁₃₉⋊C₈	—	2	1	no	—
136	C₁₃₉⋊C₈	—	2	1	no	—
137	C₂³×C₁₃₉	—	2	1	no	—
138	Q₈×C₁₃₉	—	2	1	no	—
139	Q₈×C₁₃₉	—	2	1	no	—
140	D₄×C₁₃₉	—	2	1	no	—
141	D₄×C₁₃₉	—	2	1	no	—
142	C₂×C₄×C₁₃₉	—	2	1	no	—
143	C₂×C₄×C₁₃₉	—	2	1	no	—
144	C₂×C₄×C₁₃₉	—	2	1	no	—
145	C₂×C₄×C₁₃₉	—	8	1	no	—
146	C₂×C₄×C₁₃₉	C₂×D₄	1	1	yes	characteristic
147	C₂×C₄×C₁₃₉	C₂×D₄	1	1	yes	characteristic
148	C₂×C₄×C₁₃₉	C₂⁴	1	1	yes	characteristic
149	C₈×C₁₃₉	—	2	1	no	—
150	C₈×C₁₃₉	#41	1	2	yes	—
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	C₂×D₁₃₉	—	4	1	no	—
159	C₂×D₁₃₉	—	4	1	no	—
160	C₂×D₁₃₉	—	4	1	no	—
161	C₂×D₁₃₉	—	4	1	no	—
162	C₂×D₁₃₉	—	4	1	no	—
163	C₂×D₁₃₉	—	4	1	no	—
164	C₄×C₁₃₉	—	2	1	no	—
165	C₄×C₁₃₉	—	2	1	no	—
166	C₄×C₁₃₉	—	4	1	no	—
167	C₄×C₁₃₉	—	4	1	no	—
168	C₄×C₁₃₉	—	4	1	no	—
169	C₄×C₁₃₉	C₂×#41	1	1	yes	characteristic
170	C₄×C₁₃₉	H(5,𝔽₂)	1	1	yes	characteristic
171	Dic₁₃₉	—	2	1	no	—
172	Dic₁₃₉	—	2	1	no	—
173	Dic₁₃₉	—	2	1	no	—
174	Dic₁₃₉	—	2	1	no	—
175	Dic₁₃₉	—	4	1	no	—
176	C₂×C₁₃₉	—	8	1	no	—
177	C₂×C₁₃₉	#550	1	1	yes	characteristic
178	C₂×C₁₃₉	#575	1	1	yes	characteristic
179	C₂×C₁₃₉	#576	1	1	yes	characteristic
180	D₁₃₉	—	8	1	no	—
181	D₁₃₉	—	8	1	no	—
182	D₁₃₉	—	8	1	no	—
183	C₁₃₉	#3403	1	1	yes	characteristic, complemented, Sylow
184	#3403	—	139	1	no	maximal, Sylow
185	#559	—	139	1	no	—
186	#559	—	139	1	no	—
187	#496	—	139	1	no	—
188	#496	—	139	1	no	—
189	#495	—	139	1	no	—
190	#464	—	139	1	no	—
191	#464	—	139	1	no	—
192	#463	—	139	1	no	—
193	#463	—	139	1	no	—
194	#459	—	139	1	no	—
195	#459	—	139	1	no	—
196	#446	—	139	1	no	—
197	#440	—	139	1	no	—
198	#440	D₁₃₉	1	1	yes	characteristic, complemented
199	C₄×D₈	—	139	1	no	—
200	C₂×Q₁₆	—	139	1	no	—
201	C₂×SD₁₆	—	139	1	no	—
202	C₂×SD₁₆	—	139	1	no	—
203	C₂×SD₁₆	—	139	1	no	—
204	C₂×SD₁₆	C₂×D₁₃₉	1	1	yes	characteristic
205	C₂×D₈	—	139	1	no	—
206	C₂×D₈	—	139	1	no	—
207	C₂×D₈	—	139	1	no	—
208	C₂×D₈	—	139	1	no	—
209	#127	—	139	1	no	—
210	#127	—	139	1	no	—
211	#124	—	278	1	no	—
212	#124	—	278	1	no	—
213	#124	—	139	1	no	—
214	#124	—	139	1	no	—
215	#124	—	139	1	no	—
216	#124	—	139	1	no	—
217	#123	—	278	1	no	—
218	#123	—	278	1	no	—
219	#123	—	278	1	no	—
220	#123	—	278	1	no	—
221	C₄×Q₈	—	139	1	no	—
222	C₄×Q₈	—	139	1	no	—
223	C₄×Q₈	C₂×D₁₃₉	1	1	yes	characteristic
224	C₄×D₄	—	278	1	no	—
225	C₄×D₄	—	278	1	no	—
226	C₄×D₄	—	139	1	no	—
227	C₄×D₄	—	139	1	no	—
228	C₄×D₄	—	139	1	no	—
229	C₄×D₄	C₂×D₁₃₉	1	1	yes	characteristic
230	#107	C₂×D₁₃₉	1	1	yes	characteristic
231	C₄⋊C₈	—	139	1	no	—
232	C₄⋊C₈	—	139	1	no	—
233	C₄⋊C₈	—	139	1	no	—
234	#103	—	139	1	no	—
235	#103	C₂×D₁₃₉	1	1	yes	characteristic
236	#102	—	139	1	no	—
237	#102	—	139	1	no	—
238	#102	—	139	1	no	—
239	#102	C₂×D₁₃₉	1	1	yes	characteristic
240	#97	—	139	1	no	—
241	#97	C₂×D₁₃₉	1	1	yes	characteristic
242	C₄×C₈	—	139	1	no	—
243	C₂×Q₈	—	139	1	no	—
244	C₂×Q₈	—	139	1	no	—
245	C₂×Q₈	C₂²×D₁₃₉	1	1	yes	characteristic
246	C₂×D₄	—	278	1	no	—
247	C₂×D₄	—	278	1	no	—
248	C₂×D₄	—	278	1	no	—
249	C₂×D₄	—	278	1	no	—
250	C₂×D₄	—	139	1	no	—
251	C₂×D₄	—	139	1	no	—
252	C₂×D₄	—	139	1	no	—
253	C₂×D₄	C₂²×D₁₃₉	1	1	yes	characteristic
254	C₂²×C₄	—	2	1	no	—
255	C₂²×C₄	—	278	1	no	—
256	C₂²×C₄	—	278	1	no	—
257	C₂²×C₄	—	278	1	no	—
258	Q₁₆	—	556	1	no	—
259	SD₁₆	—	2	2	no	—
260	SD₁₆	—	278	2	no	—
261	SD₁₆	—	556	1	no	—
262	SD₁₆	—	556	1	no	—
263	D₈	—	278	2	no	—
264	D₈	—	556	1	no	—
265	D₈	—	556	1	no	—
266	D₈	—	556	1	no	—
267	C₂×C₈	—	139	1	no	—
268	C₂×C₈	—	139	1	no	—
269	C₂×C₈	C₂²×D₁₃₉	1	1	yes	characteristic
270	C₂×C₈	C₂²×D₁₃₉	1	1	yes	characteristic
271	C₄⋊C₄	—	2	1	no	—
272	C₄⋊C₄	—	278	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	C₄⋊C₄	C₂²×D₁₃₉	1	1	yes	characteristic
278	C₂²⋊C₄	—	2	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	—	278	1	no	—
284	C₂²⋊C₄	—	278	1	no	—
285	C₂²⋊C₄	—	278	1	no	—
286	C₂²⋊C₄	—	278	1	no	—
287	C₂²⋊C₄	—	278	1	no	—
288	C₂²⋊C₄	—	278	1	no	—
289	C₂²⋊C₄	—	278	1	no	—
290	C₄²	—	2	1	no	—
291	C₄²	—	278	1	no	—
292	C₄²	—	139	1	no	—
293	C₄²	—	139	1	no	—
294	C₄²	C₂²×D₁₃₉	1	1	yes	characteristic
295	C₂³	—	2	1	no	—
296	C₂³	—	278	1	no	—
297	C₂³	—	278	1	no	—
298	C₂³	—	278	1	no	—
299	Q₈	—	2	1	no	—
300	Q₈	—	2	1	no	—
301	Q₈	—	278	1	no	—
302	Q₈	—	278	1	no	—
303	Q₈	—	556	1	no	—
304	D₄	—	2	1	no	—
305	D₄	—	2	1	no	—
306	D₄	—	278	1	no	—
307	D₄	—	278	1	no	—
308	D₄	—	278	1	no	—
309	D₄	—	278	1	no	—
310	D₄	—	278	1	no	—
311	D₄	—	278	1	no	—
312	D₄	—	556	2	no	—
313	D₄	—	556	2	no	—
314	D₄	—	1112	1	no	—
315	D₄	—	1112	1	no	—
316	C₂×C₄	—	2	1	no	—
317	C₂×C₄	—	2	1	no	—
318	C₂×C₄	—	2	1	no	—
319	C₂×C₄	—	8	1	no	—
320	C₂×C₄	—	278	1	no	—
321	C₂×C₄	—	1112	1	no	—
322	C₂×C₄	—	1112	1	no	—
323	C₂×C₄	—	1112	1	no	—
324	C₂×C₄	—	139	1	no	—
325	C₂×C₄	—	139	1	no	—
326	C₂×C₄	—	139	1	no	—
327	C₂×C₄	—	139	1	no	—
328	C₂×C₄	D₄×D₁₃₉	1	1	yes	characteristic
329	C₂×C₄	D₄×D₁₃₉	1	1	yes	characteristic
330	C₂×C₄	C₂³×D₁₃₉	1	1	yes	characteristic
331	C₈	—	2	1	no	—
332	C₈	—	278	1	no	—
333	C₈	—	278	1	no	—
334	C₈	#14555785	1	2	yes	—
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	—	556	1	no	—
342	C₂²	—	556	1	no	—
343	C₂²	—	556	1	no	—
344	C₂²	—	556	1	no	—
345	C₂²	—	556	1	no	—
346	C₂²	—	556	1	no	—
347	C₂²	C₂×D₄×D₁₃₉	1	1	yes	characteristic
348	C₄	—	2	1	no	—
349	C₄	—	2	1	no	—
350	C₄	—	4	1	no	—
351	C₄	—	4	1	no	—
352	C₄	—	4	1	no	—
353	C₄	—	278	1	no	—
354	C₄	—	278	1	no	—
355	C₄	—	278	1	no	—
356	C₄	—	278	1	no	—
357	C₄	—	556	1	no	—
358	C₄	C₂×#14555785	1	1	yes	characteristic
359	C₄	#17984888	1	1	yes	characteristic
360	C₂	—	8	1	no	—
361	C₂	—	1112	1	no	—
362	C₂	—	1112	1	no	—
363	C₂	—	1112	1	no	—
364	C₂	#22581624	1	1	yes	characteristic
365	C₂	#22581752	1	1	yes	characteristic
366	C₂	#22581757	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×D₄×D₁₃₉
commutator, cosocle	C₂×C₄×C₁₃₉	C₂⁴
Frattini	C₂×C₄	C₂³×D₁₃₉
Fitting	#440×C₁₃₉	C₂
socle	C₂²×C₁₃₉	C₂²×D₄
hypercenter	#440	D₁₃₉
hypocenter	C₁₃₉	#3403
LCS	G ▷ C₂×C₄×C₁₃₉ ▷ C₂×C₁₃₉ ▷ C₁₃₉
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #440

Complements

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

N	H ≅ G/N	type
1. #22581688	C₂	semidirect
2. #22581686	C₂	semidirect
3. #22581342	C₂	semidirect
4. #22581339	C₂	semidirect
5. #22581334	C₂	semidirect
6. #22581178	C₂	semidirect
7. #22581175	C₂	semidirect
8. #22581171	C₂	semidirect
9. #22581167	C₂	semidirect
10. #22581148	C₂	semidirect
11. #22581147	C₂	semidirect
12. #22581081	C₂	semidirect
13. #22581052	C₂	semidirect
14. #22581035	C₂	semidirect
15. #440×C₁₃₉	C₂	semidirect
183. C₁₃₉	#3403	semidirect
198. #440	D₁₃₉	semidirect

Maximal subgroups

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

subgroup	index
#22581688	2
#22581686	2
#22581342	2
#22581339	2
#22581334	2
#22581178	2
#22581175	2
#22581171	2
#22581167	2
#22581148	2
#22581147	2
#22581081	2
#22581052	2
#22581035	2
#440×C₁₃₉	2
#3403	139