Subgroups — #31818995

Order 17792.

386 conjugacy classes of subgroups.

The subgroup lattice has 354 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#22581666	C₂	1	1	yes	maximal, characteristic, complemented
2	#22581604	C₂	1	1	yes	maximal, characteristic, complemented
3	#22581350	C₂	1	1	yes	maximal, characteristic, complemented
4	#22581280	C₂	1	1	yes	maximal, characteristic, complemented
5	#22581277	C₂	1	1	yes	maximal, characteristic, complemented
6	#22581273	C₂	1	1	yes	maximal, characteristic, complemented
7	#22581271	C₂	1	1	yes	maximal, characteristic, complemented
8	#22581242	C₂	1	1	yes	maximal, characteristic, complemented
9	#22581232	C₂	1	1	yes	maximal, characteristic, complemented
10	#22581207	C₂	1	1	yes	maximal, characteristic, complemented
11	#22581206	C₂	1	1	yes	maximal, characteristic, complemented
12	#22581035	C₂	1	1	yes	maximal, characteristic, complemented
13	#22581029	C₂	1	1	yes	maximal, characteristic, complemented
14	#22580994	C₂	1	1	yes	maximal, characteristic, complemented
15	#428×C₁₃₉	C₂	1	1	yes	maximal, characteristic, complemented
16	C₂×#14555785	—	2	1	no	—
17	C₂×D₁₁₁₂	C₂²	1	1	yes	characteristic
18	C₂×#14555768	C₂²	1	1	yes	characteristic
19	#17984841	C₂²	1	1	yes	characteristic
20	#17984841	C₂²	1	1	yes	characteristic
21	#17984830	—	2	1	no	—
22	#17984828	—	2	1	no	—
23	#17984827	—	2	1	no	—
24	#17984825	—	2	1	no	—
25	#17984820	—	2	1	no	—
26	#17984817	C₂²	1	1	yes	characteristic
27	#17984817	C₂²	1	1	yes	characteristic
28	#17984816	C₂²	1	1	yes	characteristic
29	#17984816	C₂²	1	1	yes	characteristic
30	#17984815	—	2	1	no	—
31	#17984804	C₂²	1	1	yes	characteristic
32	#17984804	C₂²	1	1	yes	characteristic
33	C₄×C₂²⋊D₁₃₉	C₂²	1	1	yes	characteristic
34	#17984800	C₂²	1	1	yes	characteristic
35	#17984799	—	2	1	no	—
36	C₄⋊C₄×D₁₃₉	C₂²	1	1	yes	characteristic
37	#17984775	C₂²	1	1	yes	characteristic
38	#17984774	C₂²	1	1	yes	characteristic
39	#17984773	C₂²	1	1	yes	characteristic
40	#17984773	C₂²	1	1	yes	characteristic
41	#17984773	C₂²	1	1	yes	characteristic
42	Dic₁₃₉⋊C₈	C₂²	1	1	yes	characteristic
43	#17984764	C₂²	1	1	yes	characteristic
44	#17984764	C₂²	1	1	yes	characteristic
45	#17984762	C₂²	1	1	yes	characteristic
46	#17984762	C₂²	1	1	yes	characteristic
47	#17984762	C₂²	1	1	yes	characteristic
48	#17984762	C₂²	1	1	yes	characteristic
49	C₂²⋊(C₁₃₉⋊C₈)	C₂²	1	1	yes	characteristic
50	(C₂×D₁₃₉)⋊C₈	C₂²	1	1	yes	characteristic
51	C₈×Dic₁₃₉	C₂²	1	1	yes	characteristic
52	C₂×M₁₆×C₁₃₉	C₂²	1	1	yes	characteristic
53	#117×C₁₃₉	C₂²	1	1	yes	characteristic
54	C₂×C₄⋊C₄×C₁₃₉	C₂²	1	1	yes	characteristic
55	#107×C₁₃₉	C₂²	1	1	yes	characteristic
56	#107×C₁₃₉	C₂²	1	1	yes	characteristic
57	#106×C₁₃₉	C₂²	1	1	yes	characteristic
58	#106×C₁₃₉	C₂²	1	1	yes	characteristic
59	#14555785	—	4	4	no	—
60	C₂×Dic₂₇₈	—	2	1	no	—
61	C₂×Dic₂₇₈	—	2	1	no	—
62	C₂×D₅₅₆	—	2	1	no	—
63	C₂×D₅₅₆	—	2	1	no	—
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×C₂²⋊D₁₃₉	D₄	1	1	yes	characteristic
69	C₂×C₄×D₁₃₉	—	2	1	no	—
70	C₂×C₄×D₁₃₉	—	2	1	no	—
71	C₂×C₄×D₁₃₉	—	2	1	no	—
72	C₂×C₄×D₁₃₉	C₂³	1	1	yes	characteristic
73	D₁₁₁₂	—	2	2	no	—
74	#14555768	—	2	2	no	—
75	C₂×C₁₃₉⋊C₈	C₂³	1	1	yes	characteristic
76	C₂×C₁₃₉⋊C₈	C₂³	1	1	yes	characteristic
77	C₄⋊Dic₁₃₉	C₂³	1	1	yes	characteristic
78	C₄⋊Dic₁₃₉	C₂³	1	1	yes	characteristic
79	Dic₁₃₉⋊C₄	—	2	1	no	—
80	Dic₁₃₉⋊C₄	—	2	1	no	—
81	Dic₁₃₉⋊C₄	—	2	1	no	—
82	Dic₁₃₉⋊C₄	D₄	1	1	yes	characteristic
83	C₂²⋊Dic₁₃₉	D₄	1	1	yes	characteristic
84	(C₂×D₁₃₉)⋊C₄	—	2	1	no	—
85	(C₂×D₁₃₉)⋊C₄	—	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	D₄	1	1	yes	characteristic
92	C₄×Dic₁₃₉	—	2	1	no	—
93	C₄×Dic₁₃₉	C₂³	1	1	yes	characteristic
94	C₂²×C₄×C₁₃₉	—	2	1	no	—
95	C₂²×C₄×C₁₃₉	C₂³	1	1	yes	characteristic
96	M₁₆×C₁₃₉	—	2	2	no	—
97	C₂×C₈×C₁₃₉	C₂³	1	1	yes	characteristic
98	C₂×C₈×C₁₃₉	C₂³	1	1	yes	characteristic
99	C₄⋊C₄×C₁₃₉	—	2	1	no	—
100	C₄⋊C₄×C₁₃₉	C₂³	1	1	yes	characteristic
101	C₄⋊C₄×C₁₃₉	C₂³	1	1	yes	characteristic
102	C₄⋊C₄×C₁₃₉	C₂³	1	1	yes	characteristic
103	C₄⋊C₄×C₁₃₉	C₂³	1	1	yes	characteristic
104	C₂²⋊C₄×C₁₃₉	—	2	1	no	—
105	C₄²×C₁₃₉	—	2	1	no	—
106	C₂²×D₁₃₉	—	2	1	no	—
107	C₂²×D₁₃₉	—	2	1	no	—
108	C₂²×D₁₃₉	C₂×D₄	1	1	yes	characteristic
109	Dic₂₇₈	—	4	2	no	—
110	Dic₂₇₈	—	8	1	no	—
111	D₅₅₆	—	2	1	no	—
112	D₅₅₆	—	2	1	no	—
113	D₅₅₆	—	2	1	no	—
114	D₅₅₆	—	2	1	no	—
115	D₅₅₆	—	4	2	no	—
116	D₅₅₆	—	8	1	no	—
117	C₂²⋊D₁₃₉	—	2	2	no	—
118	C₂²⋊D₁₃₉	—	4	2	no	—
119	C₂²⋊D₁₃₉	—	8	1	no	—
120	C₂×Dic₁₃₉	—	2	1	no	—
121	C₂×Dic₁₃₉	—	2	1	no	—
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	—	2	1	no	—
126	C₄×D₁₃₉	—	2	1	no	—
127	C₄×D₁₃₉	—	4	2	no	—
128	C₄×D₁₃₉	—	4	2	no	—
129	C₄×D₁₃₉	—	4	2	no	—
130	C₁₃₉⋊C₈	—	2	1	no	—
131	C₁₃₉⋊C₈	—	2	1	no	—
132	C₂³×C₁₃₉	C₂×D₄	1	1	yes	characteristic
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	—	2	1	no	—
137	C₂×C₄×C₁₃₉	—	2	1	no	—
138	C₂×C₄×C₁₃₉	—	2	1	no	—
139	C₂×C₄×C₁₃₉	—	4	2	no	—
140	C₂×C₄×C₁₃₉	C₂×D₄	1	1	yes	characteristic
141	C₂×C₄×C₁₃₉	C₂⁴	1	1	yes	characteristic
142	C₈×C₁₃₉	—	2	1	no	—
143	C₈×C₁₃₉	#41	1	2	yes	complemented
144	C₂²×C₁₃₉	—	2	1	no	—
145	C₂²×C₁₃₉	—	2	1	no	—
146	C₂²×C₁₃₉	—	2	1	no	—
147	C₂²×C₁₃₉	C₂²×D₄	1	1	yes	characteristic
148	C₂×D₁₃₉	—	2	1	no	—
149	C₂×D₁₃₉	—	2	1	no	—
150	C₂×D₁₃₉	—	2	1	no	—
151	C₂×D₁₃₉	—	4	1	no	—
152	C₂×D₁₃₉	—	4	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	C₄×C₁₃₉	—	2	1	no	—
158	C₄×C₁₃₉	—	4	1	no	—
159	C₄×C₁₃₉	—	4	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	2₋¹⁺⁴	1	1	yes	characteristic
164	Dic₁₃₉	—	2	1	no	—
165	Dic₁₃₉	—	2	1	no	—
166	Dic₁₃₉	—	4	1	no	—
167	Dic₁₃₉	—	4	1	no	—
168	Dic₁₃₉	AGL(1,ℤ/8)	1	2	yes	—
169	C₂×C₁₃₉	—	4	1	no	—
170	C₂×C₁₃₉	#547	1	1	yes	characteristic
171	C₂×C₁₃₉	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
172	C₂×C₁₃₉	#577	1	1	yes	characteristic
173	D₁₃₉	—	4	1	no	—
174	D₁₃₉	—	8	1	no	—
175	D₁₃₉	—	8	1	no	—
176	C₁₃₉	#3354	1	1	yes	characteristic, complemented, Sylow
177	#3354	—	139	1	no	maximal, Sylow
178	#555	—	139	1	no	—
179	#547	—	139	1	no	—
180	#498	—	139	1	no	—
181	#482	—	139	1	no	—
182	#482	—	139	1	no	—
183	#481	—	139	1	no	—
184	#481	—	139	1	no	—
185	#476	—	139	1	no	—
186	#474	—	139	1	no	—
187	#469	—	139	1	no	—
188	#469	—	139	1	no	—
189	C₄×D₈	—	139	1	no	—
190	#436	—	139	1	no	—
191	#428	—	139	1	no	—
192	#428	D₁₃₉	1	1	yes	characteristic, complemented
193	C₂×#41	—	278	1	no	—
194	C₂×D₈	—	139	1	no	—
195	C₂×M₁₆	—	139	1	no	—
196	C₂×M₁₆	C₂×D₁₃₉	1	1	yes	characteristic
197	C₄⋊Q₈	—	139	1	no	—
198	C₄⋊Q₈	—	139	1	no	—
199	#124	—	278	1	no	—
200	#124	—	278	1	no	—
201	#123	—	278	1	no	—
202	#123	—	278	1	no	—
203	C₂²⋊Q₈	—	278	1	no	—
204	#121	—	278	1	no	—
205	#121	—	139	1	no	—
206	#121	—	139	1	no	—
207	#121	—	139	1	no	—
208	#121	—	139	1	no	—
209	C₄×D₄	—	139	1	no	—
210	C₄×D₄	—	139	1	no	—
211	C₄×D₄	—	139	1	no	—
212	#117	—	278	1	no	—
213	#117	—	139	1	no	—
214	#117	C₂×D₁₃₉	1	1	yes	characteristic
215	C₂×C₄⋊C₄	—	139	1	no	—
216	C₂×C₄⋊C₄	C₂×D₁₃₉	1	1	yes	characteristic
217	#107	—	139	1	no	—
218	#107	C₂×D₁₃₉	1	1	yes	characteristic
219	#107	C₂×D₁₃₉	1	1	yes	characteristic
220	#106	—	139	1	no	—
221	#106	—	139	1	no	—
222	#106	—	139	1	no	—
223	#106	—	139	1	no	—
224	#106	C₂×D₁₃₉	1	1	yes	characteristic
225	#106	C₂×D₁₃₉	1	1	yes	characteristic
226	C₄⋊C₈	—	139	1	no	—
227	#102	—	139	1	no	—
228	#102	—	139	1	no	—
229	#102	—	139	1	no	—
230	#102	—	139	1	no	—
231	#102	—	139	1	no	—
232	#102	—	139	1	no	—
233	C₂²⋊C₈	—	139	1	no	—
234	C₂²⋊C₈	—	139	1	no	—
235	C₄×C₈	—	139	1	no	—
236	#41	—	556	4	no	—
237	C₂×Q₈	—	278	1	no	—
238	C₂×Q₈	—	278	1	no	—
239	C₂×D₄	—	278	1	no	—
240	C₂×D₄	—	278	1	no	—
241	C₂×D₄	—	278	1	no	—
242	C₂×D₄	—	278	1	no	—
243	C₂×D₄	—	139	1	no	—
244	C₂×D₄	—	139	1	no	—
245	C₂×D₄	—	139	1	no	—
246	C₂²×C₄	—	2	1	no	—
247	C₂²×C₄	—	278	1	no	—
248	C₂²×C₄	—	278	1	no	—
249	C₂²×C₄	—	278	1	no	—
250	C₂²×C₄	—	139	1	no	—
251	C₂²×C₄	C₂²×D₁₃₉	1	1	yes	characteristic
252	D₈	—	278	2	no	—
253	M₁₆	—	2	2	no	—
254	M₁₆	—	278	2	no	—
255	C₂×C₈	—	139	1	no	—
256	C₂×C₈	—	139	1	no	—
257	C₂×C₈	C₂²×D₁₃₉	1	1	yes	characteristic
258	C₂×C₈	C₂²×D₁₃₉	1	1	yes	characteristic
259	C₄⋊C₄	—	2	1	no	—
260	C₄⋊C₄	—	278	1	no	—
261	C₄⋊C₄	—	278	1	no	—
262	C₄⋊C₄	—	278	1	no	—
263	C₄⋊C₄	—	139	1	no	—
264	C₄⋊C₄	—	139	1	no	—
265	C₄⋊C₄	—	139	1	no	—
266	C₄⋊C₄	C₂²×D₁₃₉	1	1	yes	characteristic
267	C₄⋊C₄	C₂²×D₁₃₉	1	1	yes	characteristic
268	C₄⋊C₄	C₂²×D₁₃₉	1	1	yes	characteristic
269	C₄⋊C₄	C₂²×D₁₃₉	1	1	yes	characteristic
270	C₂²⋊C₄	—	2	1	no	—
271	C₂²⋊C₄	—	278	1	no	—
272	C₂²⋊C₄	—	278	1	no	—
273	C₂²⋊C₄	—	278	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	—	139	1	no	—
279	C₂²⋊C₄	—	139	1	no	—
280	C₄²	—	2	1	no	—
281	C₄²	—	278	1	no	—
282	C₄²	—	139	1	no	—
283	C₂³	—	278	1	no	—
284	C₂³	—	278	1	no	—
285	C₂³	—	139	1	no	—
286	C₂³	D₄×D₁₃₉	1	1	yes	characteristic
287	Q₈	—	556	2	no	—
288	Q₈	—	1112	1	no	—
289	D₄	—	278	2	no	—
290	D₄	—	278	1	no	—
291	D₄	—	278	1	no	—
292	D₄	—	278	1	no	—
293	D₄	—	278	1	no	—
294	D₄	—	556	2	no	—
295	D₄	—	556	2	no	—
296	D₄	—	1112	1	no	—
297	D₄	—	1112	1	no	—
298	C₂×C₄	—	2	1	no	—
299	C₂×C₄	—	2	1	no	—
300	C₂×C₄	—	2	1	no	—
301	C₂×C₄	—	2	1	no	—
302	C₂×C₄	—	2	1	no	—
303	C₂×C₄	—	2	1	no	—
304	C₂×C₄	—	4	2	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	—	556	2	no	—
310	C₂×C₄	—	556	2	no	—
311	C₂×C₄	—	556	2	no	—
312	C₂×C₄	—	139	1	no	—
313	C₂×C₄	—	139	1	no	—
314	C₂×C₄	—	139	1	no	—
315	C₂×C₄	D₄×D₁₃₉	1	1	yes	characteristic
316	C₂×C₄	C₂³×D₁₃₉	1	1	yes	characteristic
317	C₈	—	2	1	no	—
318	C₈	—	278	1	no	—
319	C₈	—	278	1	no	—
320	C₈	#14555785	1	2	yes	complemented
321	C₂²	—	2	1	no	—
322	C₂²	—	2	1	no	—
323	C₂²	—	2	1	no	—
324	C₂²	—	278	1	no	—
325	C₂²	—	278	1	no	—
326	C₂²	—	278	1	no	—
327	C₂²	—	556	1	no	—
328	C₂²	—	556	1	no	—
329	C₂²	—	556	1	no	—
330	C₂²	—	556	1	no	—
331	C₂²	—	556	1	no	—
332	C₂²	—	556	1	no	—
333	C₂²	C₂×D₄×D₁₃₉	1	1	yes	characteristic
334	C₄	—	2	1	no	—
335	C₄	—	4	1	no	—
336	C₄	—	4	1	no	—
337	C₄	—	4	1	no	—
338	C₄	—	4	1	no	—
339	C₄	—	278	1	no	—
340	C₄	—	278	1	no	—
341	C₄	—	556	1	no	—
342	C₄	—	556	1	no	—
343	C₄	—	139	2	no	—
344	C₄	C₂×#14555785	1	1	yes	characteristic
345	C₄	#17984890	1	1	yes	characteristic
346	C₂	—	4	1	no	—
347	C₂	—	556	1	no	—
348	C₂	—	1112	1	no	—
349	C₂	—	1112	1	no	—
350	C₂	#22581604	1	1	yes	characteristic
351	C₂	AGL(1,ℤ/8)×D₁₃₉	1	1	yes	characteristic
352	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	#428×C₁₃₉	C₂
socle	C₂²×C₁₃₉	C₂²×D₄
hypercenter	#428	D₁₃₉
hypocenter	C₁₃₉	#3354
LCS	G ▷ C₂×C₄×C₁₃₉ ▷ C₂×C₁₃₉ ▷ C₁₃₉
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ #428

Complements

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

N	H ≅ G/N	type
1. #22581666	C₂	semidirect
2. #22581604	C₂	semidirect
3. #22581350	C₂	semidirect
4. #22581280	C₂	semidirect
5. #22581277	C₂	semidirect
6. #22581273	C₂	semidirect
7. #22581271	C₂	semidirect
8. #22581242	C₂	semidirect
9. #22581232	C₂	semidirect
10. #22581207	C₂	semidirect
11. #22581206	C₂	semidirect
12. #22581035	C₂	semidirect
13. #22581029	C₂	semidirect
14. #22580994	C₂	semidirect
15. #428×C₁₃₉	C₂	semidirect
143. C₈×C₁₃₉	#41	semidirect
176. C₁₃₉	#3354	semidirect
192. #428	D₁₃₉	semidirect
320. C₈	#14555785	semidirect

Maximal subgroups

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

subgroup	index
#22581666	2
#22581604	2
#22581350	2
#22581280	2
#22581277	2
#22581273	2
#22581271	2
#22581242	2
#22581232	2
#22581207	2
#22581206	2
#22581035	2
#22581029	2
#22580994	2
#428×C₁₃₉	2
#3354	139