Subgroups — #31817822

Order 17792.

358 conjugacy classes of subgroups.

The subgroup lattice has 358 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#22581577	C₂	1	1	yes	maximal, characteristic, complemented
2	#22581554	C₂	1	1	yes	maximal, characteristic, complemented
3	#22581259	C₂	1	1	yes	maximal, characteristic, complemented
4	#22581256	C₂	1	1	yes	maximal, characteristic, complemented
5	#22581240	C₂	1	1	yes	maximal, characteristic, complemented
6	#22581205	C₂	1	1	yes	maximal, characteristic, complemented
7	#22581200	C₂	1	1	yes	maximal, characteristic, complemented
8	#22581190	C₂	1	1	yes	maximal, characteristic, complemented
9	#22581181	C₂	1	1	yes	maximal, characteristic, complemented
10	#22581169	C₂	1	1	yes	maximal, characteristic, complemented
11	#22581168	C₂	1	1	yes	maximal, characteristic, complemented
12	#22581152	C₂	1	1	yes	maximal, characteristic, complemented
13	#22581147	C₂	1	1	yes	maximal, characteristic, complemented
14	#22580978	C₂	1	1	yes	maximal, characteristic, complemented
15	#474×C₁₃₉	C₂	1	1	yes	maximal, characteristic, complemented
16	C₂×#14555774	C₂²	1	1	yes	characteristic
17	C₂×#14555773	C₂²	1	1	yes	characteristic
18	C₂×D₁₁₁₂	C₂²	1	1	yes	characteristic
19	C₂×#14555770	C₂²	1	1	yes	characteristic
20	C₂×#14555768	C₂²	1	1	yes	characteristic
21	C₂×C₈×D₁₃₉	C₂²	1	1	yes	characteristic
22	#17984838	C₂²	1	1	yes	characteristic
23	#17984837	—	2	1	no	—
24	#17984833	C₂²	1	1	yes	characteristic
25	#17984831	—	2	1	no	—
26	#17984830	C₂²	1	1	yes	characteristic
27	#17984827	—	2	1	no	—
28	#17984825	—	2	1	no	—
29	#17984824	—	2	1	no	—
30	#17984822	C₂²	1	1	yes	characteristic
31	#17984818	—	2	1	no	—
32	#17984817	C₂²	1	1	yes	characteristic
33	#17984817	C₂²	1	1	yes	characteristic
34	#17984816	—	2	1	no	—
35	#17984815	—	2	1	no	—
36	#17984813	C₂²	1	1	yes	characteristic
37	#17984807	C₂²	1	1	yes	characteristic
38	C₄×D₅₅₆	C₂²	1	1	yes	characteristic
39	#17984804	C₂²	1	1	yes	characteristic
40	#17984799	C₂²	1	1	yes	characteristic
41	#17984773	C₂²	1	1	yes	characteristic
42	#17984773	C₂²	1	1	yes	characteristic
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	#17984767	C₂²	1	1	yes	characteristic
47	#17984765	C₂²	1	1	yes	characteristic
48	#17984764	C₂²	1	1	yes	characteristic
49	#17984762	C₂²	1	1	yes	characteristic
50	#17984762	C₂²	1	1	yes	characteristic
51	#17984762	C₂²	1	1	yes	characteristic
52	C₄⋊Q₈×C₁₃₉	C₂²	1	1	yes	characteristic
53	C₄×D₄×C₁₃₉	C₂²	1	1	yes	characteristic
54	#107×C₁₃₉	C₂²	1	1	yes	characteristic
55	#107×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	C₂×Dic₂₇₈	C₂³	1	1	yes	characteristic
60	C₂×D₅₅₆	—	2	1	no	—
61	C₂×D₅₅₆	—	2	1	no	—
62	C₂×D₅₅₆	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×C₂²⋊D₁₃₉	—	2	1	no	—
67	C₂×C₄×D₁₃₉	—	2	1	no	—
68	C₂×C₄×D₁₃₉	—	2	1	no	—
69	C₂×C₄×D₁₃₉	C₂³	1	1	yes	characteristic
70	#14555774	—	4	1	no	—
71	#14555773	—	4	1	no	—
72	D₁₁₁₂	—	4	1	no	—
73	#14555770	—	4	1	no	—
74	#14555768	—	4	1	no	—
75	C₂×C₁₃₉⋊C₈	C₂³	1	1	yes	characteristic
76	C₂×C₁₃₉⋊C₈	C₂³	1	1	yes	characteristic
77	C₈×D₁₃₉	—	4	1	no	—
78	C₄⋊Dic₁₃₉	—	2	1	no	—
79	C₄⋊Dic₁₃₉	—	2	1	no	—
80	C₄⋊Dic₁₃₉	C₂³	1	1	yes	characteristic
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	(C₂×D₁₃₉)⋊C₄	—	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	D₄	1	1	yes	characteristic
94	(C₂×D₁₃₉)⋊C₄	D₄	1	1	yes	characteristic
95	C₄×Dic₁₃₉	—	2	1	no	—
96	C₄×Dic₁₃₉	C₂³	1	1	yes	characteristic
97	C₂×Q₈×C₁₃₉	—	2	1	no	—
98	C₂×D₄×C₁₃₉	C₂³	1	1	yes	characteristic
99	C₂²×C₄×C₁₃₉	—	2	1	no	—
100	C₂×C₈×C₁₃₉	C₂³	1	1	yes	characteristic
101	C₂×C₈×C₁₃₉	C₂³	1	1	yes	characteristic
102	C₄⋊C₄×C₁₃₉	—	2	1	no	—
103	C₄⋊C₄×C₁₃₉	C₂³	1	1	yes	characteristic
104	C₄⋊C₄×C₁₃₉	C₂³	1	1	yes	characteristic
105	C₄⋊C₄×C₁₃₉	C₂³	1	1	yes	characteristic
106	C₂²⋊C₄×C₁₃₉	—	2	1	no	—
107	C₄²×C₁₃₉	C₂³	1	1	yes	characteristic
108	C₂²×D₁₃₉	—	2	1	no	—
109	C₂²×D₁₃₉	—	2	1	no	—
110	C₂²×D₁₃₉	C₂×D₄	1	1	yes	characteristic
111	Dic₂₇₈	—	2	1	no	—
112	Dic₂₇₈	—	2	1	no	—
113	D₅₅₆	—	2	1	no	—
114	D₅₅₆	—	2	1	no	—
115	D₅₅₆	—	2	1	no	—
116	D₅₅₆	—	2	1	no	—
117	D₅₅₆	—	8	1	no	—
118	D₅₅₆	—	8	1	no	—
119	C₂²⋊D₁₃₉	—	8	1	no	—
120	C₂²⋊D₁₃₉	—	8	1	no	—
121	C₂²⋊D₁₃₉	—	8	1	no	—
122	C₂×Dic₁₃₉	—	2	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	C₄×D₁₃₉	—	2	1	no	—
128	C₄×D₁₃₉	—	2	1	no	—
129	C₄×D₁₃₉	—	8	1	no	—
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	Q₈×C₁₃₉	—	8	1	no	—
135	D₄×C₁₃₉	—	2	1	no	—
136	D₄×C₁₃₉	—	2	1	no	—
137	C₂×C₄×C₁₃₉	—	2	1	no	—
138	C₂×C₄×C₁₃₉	—	2	1	no	—
139	C₂×C₄×C₁₃₉	—	2	1	no	—
140	C₂×C₄×C₁₃₉	—	8	1	no	—
141	C₂×C₄×C₁₃₉	C₂×D₄	1	1	yes	characteristic
142	C₂×C₄×C₁₃₉	C₂×D₄	1	1	yes	characteristic
143	C₂×C₄×C₁₃₉	C₂⁴	1	1	yes	characteristic
144	C₈×C₁₃₉	—	2	1	no	—
145	C₈×C₁₃₉	—	2	1	no	—
146	C₂²×C₁₃₉	—	4	1	no	—
147	C₂²×C₁₃₉	—	4	1	no	—
148	C₂²×C₁₃₉	—	4	1	no	—
149	C₂²×C₁₃₉	C₂²×D₄	1	1	yes	characteristic
150	C₂×D₁₃₉	—	2	1	no	—
151	C₂×D₁₃₉	—	2	1	no	—
152	C₂×D₁₃₉	—	2	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	C₄×C₁₃₉	—	2	1	no	—
160	C₄×C₁₃₉	—	2	1	no	—
161	C₄×C₁₃₉	—	4	1	no	—
162	C₄×C₁₃₉	—	4	1	no	—
163	C₄×C₁₃₉	—	4	1	no	—
164	C₄×C₁₃₉	H(5,𝔽₂)	1	1	yes	characteristic
165	C₄×C₁₃₉	2₋¹⁺⁴	1	1	yes	characteristic
166	Dic₁₃₉	—	2	1	no	—
167	Dic₁₃₉	—	2	1	no	—
168	Dic₁₃₉	—	2	1	no	—
169	Dic₁₃₉	—	4	1	no	—
170	Dic₁₃₉	—	4	1	no	—
171	C₂×C₁₃₉	—	8	1	no	—
172	C₂×C₁₃₉	#537	1	1	yes	characteristic
173	C₂×C₁₃₉	#576	1	1	yes	characteristic
174	C₂×C₁₃₉	#577	1	1	yes	characteristic
175	D₁₃₉	—	4	1	no	—
176	D₁₃₉	—	8	1	no	—
177	D₁₃₉	—	8	1	no	—
178	C₁₃₉	#3257	1	1	yes	characteristic, complemented, Sylow
179	#3257	—	139	1	no	maximal, Sylow
180	#542	—	139	1	no	—
181	#540	—	139	1	no	—
182	#479	—	139	1	no	—
183	#478	—	139	1	no	—
184	#476	—	139	1	no	—
185	#474	D₁₃₉	1	1	yes	characteristic, complemented
186	#469	—	139	1	no	—
187	#468	—	139	1	no	—
188	#466	—	139	1	no	—
189	#465	—	139	1	no	—
190	#463	—	139	1	no	—
191	#463	—	139	1	no	—
192	#460	—	139	1	no	—
193	#459	—	139	1	no	—
194	#424	—	139	1	no	—
195	C₂×SD₁₆	—	139	1	no	—
196	C₂×SD₁₆	—	139	1	no	—
197	C₂×D₈	—	139	1	no	—
198	C₂×D₈	—	139	1	no	—
199	C₂×M₁₆	—	139	1	no	—
200	C₂²×C₈	—	139	1	no	—
201	C₄⋊Q₈	C₂×D₁₃₉	1	1	yes	characteristic
202	#127	—	139	1	no	—
203	#126	—	278	1	no	—
204	#125	—	139	1	no	—
205	#124	—	278	1	no	—
206	#124	—	139	1	no	—
207	#123	—	278	1	no	—
208	#123	—	278	1	no	—
209	#123	—	278	1	no	—
210	C₂²⋊Q₈	—	278	1	no	—
211	C₂²⋊Q₈	—	139	1	no	—
212	#121	—	278	1	no	—
213	#121	—	278	1	no	—
214	#121	—	139	1	no	—
215	#121	—	139	1	no	—
216	#121	—	139	1	no	—
217	C₄×Q₈	—	139	1	no	—
218	C₄×D₄	—	139	1	no	—
219	C₄×D₄	—	139	1	no	—
220	C₄×D₄	C₂×D₁₃₉	1	1	yes	characteristic
221	#117	—	139	1	no	—
222	#107	C₂×D₁₃₉	1	1	yes	characteristic
223	#107	C₂×D₁₃₉	1	1	yes	characteristic
224	#106	—	139	1	no	—
225	#106	—	139	1	no	—
226	C₄⋊C₈	—	139	1	no	—
227	C₄⋊C₈	—	139	1	no	—
228	C₄⋊C₈	—	139	1	no	—
229	C₄⋊C₈	C₂×D₁₃₉	1	1	yes	characteristic
230	#103	—	139	1	no	—
231	#103	—	139	1	no	—
232	#102	—	139	1	no	—
233	#102	—	139	1	no	—
234	#102	—	139	1	no	—
235	#102	—	139	1	no	—
236	#102	C₂×D₁₃₉	1	1	yes	characteristic
237	#102	C₂×D₁₃₉	1	1	yes	characteristic
238	C₂×Q₈	—	2	1	no	—
239	C₂×Q₈	—	139	1	no	—
240	C₂×D₄	—	278	1	no	—
241	C₂×D₄	—	278	1	no	—
242	C₂×D₄	—	278	1	no	—
243	C₂×D₄	—	278	1	no	—
244	C₂×D₄	—	278	1	no	—
245	C₂×D₄	—	139	1	no	—
246	C₂×D₄	—	139	1	no	—
247	C₂×D₄	C₂²×D₁₃₉	1	1	yes	characteristic
248	C₂²×C₄	—	2	1	no	—
249	C₂²×C₄	—	278	1	no	—
250	C₂²×C₄	—	278	1	no	—
251	C₂²×C₄	—	139	1	no	—
252	SD₁₆	—	556	1	no	—
253	SD₁₆	—	556	1	no	—
254	D₈	—	556	1	no	—
255	D₈	—	556	1	no	—
256	M₁₆	—	556	1	no	—
257	C₂×C₈	—	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	—	2	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	C₄⋊C₄	C₂²×D₁₃₉	1	1	yes	characteristic
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	—	139	1	no	—
283	C₂²⋊C₄	—	139	1	no	—
284	C₄²	—	278	1	no	—
285	C₄²	—	139	1	no	—
286	C₄²	C₂²×D₁₃₉	1	1	yes	characteristic
287	C₂³	—	2	1	no	—
288	C₂³	—	278	1	no	—
289	C₂³	—	278	1	no	—
290	C₂³	—	139	1	no	—
291	Q₈	—	8	1	no	—
292	Q₈	—	278	1	no	—
293	Q₈	—	278	1	no	—
294	D₄	—	2	1	no	—
295	D₄	—	2	1	no	—
296	D₄	—	278	1	no	—
297	D₄	—	278	1	no	—
298	D₄	—	278	1	no	—
299	D₄	—	278	1	no	—
300	D₄	—	1112	1	no	—
301	D₄	—	1112	1	no	—
302	D₄	—	1112	1	no	—
303	D₄	—	1112	1	no	—
304	D₄	—	1112	1	no	—
305	C₂×C₄	—	2	1	no	—
306	C₂×C₄	—	2	1	no	—
307	C₂×C₄	—	2	1	no	—
308	C₂×C₄	—	8	1	no	—
309	C₂×C₄	—	278	1	no	—
310	C₂×C₄	—	278	1	no	—
311	C₂×C₄	—	278	1	no	—
312	C₂×C₄	—	278	1	no	—
313	C₂×C₄	—	1112	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	D₄×D₁₃₉	1	1	yes	characteristic
319	C₂×C₄	D₄×D₁₃₉	1	1	yes	characteristic
320	C₂×C₄	C₂³×D₁₃₉	1	1	yes	characteristic
321	C₈	—	2	1	no	—
322	C₈	—	2	1	no	—
323	C₈	—	278	1	no	—
324	C₈	—	278	1	no	—
325	C₂²	—	4	1	no	—
326	C₂²	—	4	1	no	—
327	C₂²	—	4	1	no	—
328	C₂²	—	278	1	no	—
329	C₂²	—	278	1	no	—
330	C₂²	—	278	1	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	C₂×D₄×D₁₃₉	1	1	yes	characteristic
338	C₄	—	2	1	no	—
339	C₄	—	2	1	no	—
340	C₄	—	4	1	no	—
341	C₄	—	4	1	no	—
342	C₄	—	4	1	no	—
343	C₄	—	278	1	no	—
344	C₄	—	278	1	no	—
345	C₄	—	278	1	no	—
346	C₄	—	556	1	no	—
347	C₄	—	556	1	no	—
348	C₄	#17984888	1	1	yes	characteristic
349	C₄	#17984890	1	1	yes	characteristic
350	C₂	—	8	1	no	—
351	C₂	—	556	1	no	—
352	C₂	—	1112	1	no	—
353	C₂	—	1112	1	no	—
354	C₂	#22581533	1	1	yes	characteristic
355	C₂	#22581754	1	1	yes	characteristic
356	C₂	#22581762	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×D₄×D₁₃₉
commutator, cosocle	C₂×C₄×C₁₃₉	C₂⁴
Frattini	C₂×C₄	C₂³×D₁₃₉
Fitting	#474×C₁₃₉	C₂
socle	C₂²×C₁₃₉	C₂²×D₄
hypercenter	#474	D₁₃₉
hypocenter	C₁₃₉	#3257
LCS	G ▷ C₂×C₄×C₁₃₉ ▷ C₂×C₁₃₉ ▷ C₁₃₉
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #474

Complements

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

N	H ≅ G/N	type
1. #22581577	C₂	semidirect
2. #22581554	C₂	semidirect
3. #22581259	C₂	semidirect
4. #22581256	C₂	semidirect
5. #22581240	C₂	semidirect
6. #22581205	C₂	semidirect
7. #22581200	C₂	semidirect
8. #22581190	C₂	semidirect
9. #22581181	C₂	semidirect
10. #22581169	C₂	semidirect
11. #22581168	C₂	semidirect
12. #22581152	C₂	semidirect
13. #22581147	C₂	semidirect
14. #22580978	C₂	semidirect
15. #474×C₁₃₉	C₂	semidirect
178. C₁₃₉	#3257	semidirect
185. #474	D₁₃₉	semidirect

Maximal subgroups

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

subgroup	index
#22581577	2
#22581554	2
#22581259	2
#22581256	2
#22581240	2
#22581205	2
#22581200	2
#22581190	2
#22581181	2
#22581169	2
#22581168	2
#22581152	2
#22581147	2
#22580978	2
#474×C₁₃₉	2
#3257	139