Subgroups — #31818305

Order 17792.

398 conjugacy classes of subgroups.

The subgroup lattice has 370 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#22581656	C₂	1	1	yes	maximal, characteristic, complemented
2	#22581611	C₂	1	1	yes	maximal, characteristic, complemented
3	#22581292	C₂	1	1	yes	maximal, characteristic, complemented
4	#22581270	C₂	1	1	yes	maximal, characteristic, complemented
5	#22581257	C₂	1	1	yes	maximal, characteristic, complemented
6	#22581188	C₂	1	1	yes	maximal, characteristic, complemented
7	#22581185	C₂	1	1	yes	maximal, characteristic, complemented
8	#22581165	C₂	1	1	yes	maximal, characteristic, complemented
9	#22581099	C₂	1	1	yes	maximal, characteristic, complemented
10	#22581093	C₂	1	1	yes	maximal, characteristic, complemented
11	C₄×#14555774	C₂	1	1	yes	maximal, characteristic, complemented
12	C₈×D₅₅₆	C₂	1	1	yes	maximal, characteristic, complemented
13	#22580946	C₂	1	1	yes	maximal, characteristic, complemented
14	#103×D₁₃₉	C₂	1	1	yes	maximal, characteristic, complemented
15	#488×C₁₃₉	C₂	1	1	yes	maximal, characteristic, complemented
16	C₂×#14555784	C₂²	1	1	yes	characteristic
17	C₂×Q₈×D₁₃₉	C₂²	1	1	yes	characteristic
18	C₂×#14555774	C₂²	1	1	yes	characteristic
19	C₂×#14555772	C₂²	1	1	yes	characteristic
20	C₂×#14555770	C₂²	1	1	yes	characteristic
21	C₂×C₈×D₁₃₉	C₂²	1	1	yes	characteristic
22	C₂×C₈×D₁₃₉	C₂²	1	1	yes	characteristic
23	#17984837	—	2	1	no	—
24	#17984828	—	2	1	no	—
25	#17984827	—	2	1	no	—
26	#17984819	—	2	1	no	—
27	#17984818	C₂²	1	1	yes	characteristic
28	#17984817	—	2	1	no	—
29	#17984817	C₂²	1	1	yes	characteristic
30	#17984817	C₂²	1	1	yes	characteristic
31	#17984815	—	2	1	no	—
32	#17984813	C₂²	1	1	yes	characteristic
33	C₄×Dic₂₇₈	C₂²	1	1	yes	characteristic
34	C₄×D₅₅₆	C₂²	1	1	yes	characteristic
35	C₄×D₅₅₆	C₂²	1	1	yes	characteristic
36	#17984804	—	2	1	no	—
37	(C₂²⋊D₁₃₉)⋊C₄	—	2	1	no	—
38	#17984800	C₂²	1	1	yes	characteristic
39	C₄⋊C₄×D₁₃₉	C₂²	1	1	yes	characteristic
40	#17984775	C₂²	1	1	yes	characteristic
41	#17984774	C₂²	1	1	yes	characteristic
42	#17984773	C₂²	1	1	yes	characteristic
43	C₄⋊(C₁₃₉⋊C₈)	C₂²	1	1	yes	characteristic
44	#17984767	C₂²	1	1	yes	characteristic
45	#17984766	C₂²	1	1	yes	characteristic
46	#17984765	C₂²	1	1	yes	characteristic
47	#17984765	C₂²	1	1	yes	characteristic
48	#17984764	C₂²	1	1	yes	characteristic
49	#17984763	C₂²	1	1	yes	characteristic
50	(C₂×D₁₃₉)⋊C₈	C₂²	1	1	yes	characteristic
51	(C₂×D₁₃₉)⋊C₈	C₂²	1	1	yes	characteristic
52	#125×C₁₃₉	C₂²	1	1	yes	characteristic
53	#124×C₁₃₉	C₂²	1	1	yes	characteristic
54	#103×C₁₃₉	C₂²	1	1	yes	characteristic
55	#103×C₁₃₉	C₂²	1	1	yes	characteristic
56	#102×C₁₃₉	C₂²	1	1	yes	characteristic
57	#102×C₁₃₉	C₂²	1	1	yes	characteristic
58	C₄×C₈×C₁₃₉	C₂²	1	1	yes	characteristic
59	#14555784	—	2	2	no	—
60	#14555784	—	4	1	no	—
61	C₂×Dic₂₇₈	—	2	1	no	—
62	C₂×Dic₂₇₈	C₂³	1	1	yes	characteristic
63	Q₈×D₁₃₉	—	2	2	no	—
64	Q₈×D₁₃₉	—	4	1	no	—
65	C₂×D₅₅₆	—	2	1	no	—
66	C₂×D₅₅₆	—	2	1	no	—
67	C₂×D₅₅₆	D₄	1	1	yes	characteristic
68	C₂×D₅₅₆	C₂³	1	1	yes	characteristic
69	C₂×C₂²⋊D₁₃₉	—	2	1	no	—
70	C₂×C₂²⋊D₁₃₉	—	2	1	no	—
71	C₂²×Dic₁₃₉	—	2	1	no	—
72	C₂×C₄×D₁₃₉	—	2	1	no	—
73	C₂×C₄×D₁₃₉	—	2	1	no	—
74	C₂×C₄×D₁₃₉	—	2	1	no	—
75	C₂×C₄×D₁₃₉	C₂³	1	1	yes	characteristic
76	C₂×C₄×D₁₃₉	C₂³	1	1	yes	characteristic
77	#14555774	—	2	2	no	—
78	#14555772	—	4	1	no	—
79	#14555770	—	4	1	no	—
80	C₂×C₁₃₉⋊C₈	C₂³	1	1	yes	characteristic
81	C₂×C₁₃₉⋊C₈	C₂³	1	1	yes	characteristic
82	C₈×D₁₃₉	—	4	1	no	—
83	C₈×D₁₃₉	—	4	1	no	—
84	C₄⋊Dic₁₃₉	D₄	1	1	yes	characteristic
85	C₄⋊Dic₁₃₉	C₂³	1	1	yes	characteristic
86	C₄⋊Dic₁₃₉	C₂³	1	1	yes	characteristic
87	Dic₁₃₉⋊C₄	—	2	1	no	—
88	Dic₁₃₉⋊C₄	—	2	1	no	—
89	Dic₁₃₉⋊C₄	—	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	(C₂×D₁₃₉)⋊C₄	—	2	1	no	—
95	(C₂×D₁₃₉)⋊C₄	—	2	1	no	—
96	(C₂×D₁₃₉)⋊C₄	D₄	1	1	yes	characteristic
97	(C₂×D₁₃₉)⋊C₄	D₄	1	1	yes	characteristic
98	C₄×Dic₁₃₉	—	2	1	no	—
99	C₄×Dic₁₃₉	—	2	1	no	—
100	C₂×Q₈×C₁₃₉	C₂³	1	1	yes	characteristic
101	C₂×D₄×C₁₃₉	C₂³	1	1	yes	characteristic
102	C₂×C₈×C₁₃₉	C₂³	1	1	yes	characteristic
103	C₂×C₈×C₁₃₉	C₂³	1	1	yes	characteristic
104	C₄⋊C₄×C₁₃₉	—	2	1	no	—
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	C₂²⋊C₄×C₁₃₉	—	2	1	no	—
110	C₄²×C₁₃₉	C₂³	1	1	yes	characteristic
111	C₂²×D₁₃₉	—	2	1	no	—
112	C₂²×D₁₃₉	C₂×D₄	1	1	yes	characteristic
113	C₂²×D₁₃₉	C₂×D₄	1	1	yes	characteristic
114	Dic₂₇₈	—	2	1	no	—
115	Dic₂₇₈	—	4	2	no	—
116	Dic₂₇₈	#41	1	2	yes	—
117	D₅₅₆	—	2	2	no	—
118	D₅₅₆	—	2	1	no	—
119	D₅₅₆	—	2	1	no	—
120	D₅₅₆	—	4	2	no	—
121	D₅₅₆	—	8	1	no	—
122	C₂²⋊D₁₃₉	—	4	2	no	—
123	C₂²⋊D₁₃₉	—	4	2	no	—
124	C₂×Dic₁₃₉	—	2	1	no	—
125	C₂×Dic₁₃₉	—	2	1	no	—
126	C₂×Dic₁₃₉	—	2	1	no	—
127	C₂×Dic₁₃₉	—	4	2	no	—
128	C₂×Dic₁₃₉	C₂×D₄	1	1	yes	characteristic
129	C₂×Dic₁₃₉	C₂×D₄	1	1	yes	characteristic
130	C₄×D₁₃₉	—	2	1	no	—
131	C₄×D₁₃₉	—	2	1	no	—
132	C₄×D₁₃₉	—	2	1	no	—
133	C₄×D₁₃₉	—	2	1	no	—
134	C₄×D₁₃₉	—	4	2	no	—
135	C₄×D₁₃₉	—	8	1	no	—
136	C₄×D₁₃₉	—	8	1	no	—
137	C₁₃₉⋊C₈	—	2	1	no	—
138	C₁₃₉⋊C₈	—	2	1	no	—
139	C₂³×C₁₃₉	—	2	1	no	—
140	Q₈×C₁₃₉	—	2	1	no	—
141	Q₈×C₁₃₉	—	2	1	no	—
142	D₄×C₁₃₉	—	2	1	no	—
143	D₄×C₁₃₉	—	2	1	no	—
144	C₂×C₄×C₁₃₉	—	2	1	no	—
145	C₂×C₄×C₁₃₉	—	2	1	no	—
146	C₂×C₄×C₁₃₉	—	2	1	no	—
147	C₂×C₄×C₁₃₉	C₂×D₄	1	1	yes	characteristic
148	C₂×C₄×C₁₃₉	C₂×D₄	1	1	yes	characteristic
149	C₂×C₄×C₁₃₉	C₂⁴	1	1	yes	characteristic
150	C₈×C₁₃₉	—	2	1	no	—
151	C₈×C₁₃₉	—	2	1	no	—
152	C₂²×C₁₃₉	—	4	1	no	—
153	C₂²×C₁₃₉	—	4	1	no	—
154	C₂²×C₁₃₉	—	4	1	no	—
155	C₂²×C₁₃₉	C₂²×D₄	1	1	yes	characteristic
156	C₂×D₁₃₉	—	2	1	no	—
157	C₂×D₁₃₉	—	2	1	no	—
158	C₂×D₁₃₉	—	2	1	no	—
159	C₂×D₁₃₉	—	2	1	no	—
160	C₂×D₁₃₉	—	2	1	no	—
161	C₂×D₁₃₉	—	4	1	no	—
162	C₂×D₁₃₉	—	4	1	no	—
163	C₂×D₁₃₉	—	4	1	no	—
164	C₂×D₁₃₉	#135	1	2	yes	—
165	C₄×C₁₃₉	—	2	1	no	—
166	C₄×C₁₃₉	—	2	1	no	—
167	C₄×C₁₃₉	—	4	1	no	—
168	C₄×C₁₃₉	—	4	1	no	—
169	C₄×C₁₃₉	—	4	1	no	—
170	C₄×C₁₃₉	C₂×#41	1	1	yes	characteristic
171	C₄×C₁₃₉	H(5,𝔽₂)	1	1	yes	characteristic
172	Dic₁₃₉	—	2	1	no	—
173	Dic₁₃₉	—	2	1	no	—
174	Dic₁₃₉	—	2	2	no	—
175	Dic₁₃₉	—	4	1	no	—
176	Dic₁₃₉	—	4	1	no	—
177	C₂×C₁₃₉	—	8	1	no	—
178	C₂×C₁₃₉	#546	1	1	yes	characteristic
179	C₂×C₁₃₉	C₂×#135	1	1	yes	characteristic
180	C₂×C₁₃₉	#577	1	1	yes	characteristic
181	D₁₃₉	—	2	2	no	—
182	D₁₃₉	—	4	1	no	—
183	D₁₃₉	—	8	1	no	—
184	C₁₃₉	#3301	1	1	yes	characteristic, complemented, Sylow
185	#3301	—	139	1	no	maximal, Sylow
186	#553	—	139	1	no	—
187	#548	—	139	1	no	—
188	#488	D₁₃₉	1	1	yes	characteristic, complemented
189	#485	—	139	1	no	—
190	#481	—	139	1	no	—
191	#479	—	139	1	no	—
192	#466	—	139	1	no	—
193	#465	—	139	1	no	—
194	#463	—	139	1	no	—
195	#449	—	139	1	no	—
196	#448	—	139	1	no	—
197	C₄×SD₁₆	—	139	1	no	—
198	C₈×D₄	—	139	1	no	—
199	#416	—	139	1	no	—
200	C₂×#103	—	139	1	no	—
201	C₂×#41	—	139	1	no	—
202	C₂²×Q₈	—	139	1	no	—
203	C₂×SD₁₆	—	139	1	no	—
204	C₂×SD₁₆	—	139	1	no	—
205	C₂×D₈	—	139	1	no	—
206	C₂²×C₈	—	139	1	no	—
207	C₂²×C₈	—	139	1	no	—
208	#126	—	278	1	no	—
209	#125	C₂×D₁₃₉	1	1	yes	characteristic
210	#124	—	278	1	no	—
211	#124	C₂×D₁₃₉	1	1	yes	characteristic
212	#123	—	278	1	no	—
213	C₂²⋊Q₈	—	278	1	no	—
214	C₂²⋊Q₈	—	139	1	no	—
215	#121	—	278	1	no	—
216	#121	—	278	1	no	—
217	#121	—	139	1	no	—
218	#121	—	139	1	no	—
219	#121	—	139	1	no	—
220	C₄×Q₈	—	139	1	no	—
221	C₄×D₄	—	278	1	no	—
222	C₄×D₄	—	278	1	no	—
223	C₄×D₄	—	139	1	no	—
224	C₄×D₄	—	139	1	no	—
225	#117	—	139	1	no	—
226	C₂×C₄⋊C₄	—	139	1	no	—
227	#107	—	139	1	no	—
228	#106	—	139	1	no	—
229	#106	—	139	1	no	—
230	C₄⋊C₈	—	139	1	no	—
231	#103	—	139	1	no	—
232	#103	—	139	1	no	—
233	#103	—	139	1	no	—
234	#103	—	139	1	no	—
235	#103	C₂×D₁₃₉	1	1	yes	characteristic
236	#103	C₂×D₁₃₉	1	1	yes	characteristic
237	#102	—	139	1	no	—
238	#102	—	139	1	no	—
239	#102	C₂×D₁₃₉	1	1	yes	characteristic
240	#102	C₂×D₁₃₉	1	1	yes	characteristic
241	C₂²⋊C₈	—	139	1	no	—
242	C₂²⋊C₈	—	139	1	no	—
243	C₄×C₈	C₂×D₁₃₉	1	1	yes	characteristic
244	#41	—	278	2	no	—
245	#41	—	556	1	no	—
246	C₂×Q₈	—	278	1	no	—
247	C₂×Q₈	—	278	2	no	—
248	C₂×Q₈	—	556	1	no	—
249	C₂×Q₈	—	139	1	no	—
250	C₂×Q₈	C₂²×D₁₃₉	1	1	yes	characteristic
251	C₂×D₄	—	278	1	no	—
252	C₂×D₄	—	278	1	no	—
253	C₂×D₄	—	278	1	no	—
254	C₂×D₄	—	278	1	no	—
255	C₂×D₄	—	139	1	no	—
256	C₂×D₄	—	139	1	no	—
257	C₂×D₄	C₂²×D₁₃₉	1	1	yes	characteristic
258	C₂²×C₄	—	278	1	no	—
259	C₂²×C₄	—	278	1	no	—
260	C₂²×C₄	—	278	1	no	—
261	C₂²×C₄	—	278	1	no	—
262	C₂²×C₄	—	139	1	no	—
263	C₂²×C₄	—	139	1	no	—
264	SD₁₆	—	278	2	no	—
265	SD₁₆	—	556	1	no	—
266	D₈	—	556	1	no	—
267	C₂×C₈	—	556	1	no	—
268	C₂×C₈	—	556	1	no	—
269	C₂×C₈	—	139	1	no	—
270	C₂×C₈	—	139	1	no	—
271	C₂×C₈	C₂²×D₁₃₉	1	1	yes	characteristic
272	C₂×C₈	C₂²×D₁₃₉	1	1	yes	characteristic
273	C₄⋊C₄	—	2	1	no	—
274	C₄⋊C₄	—	2	1	no	—
275	C₄⋊C₄	—	278	1	no	—
276	C₄⋊C₄	—	278	1	no	—
277	C₄⋊C₄	—	278	1	no	—
278	C₄⋊C₄	—	139	1	no	—
279	C₄⋊C₄	—	139	1	no	—
280	C₄⋊C₄	—	139	1	no	—
281	C₄⋊C₄	C₂²×D₁₃₉	1	1	yes	characteristic
282	C₄⋊C₄	C₂²×D₁₃₉	1	1	yes	characteristic
283	C₂²⋊C₄	—	2	1	no	—
284	C₂²⋊C₄	—	2	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	C₂²⋊C₄	—	278	1	no	—
291	C₂²⋊C₄	—	139	1	no	—
292	C₂²⋊C₄	—	139	1	no	—
293	C₄²	—	278	1	no	—
294	C₄²	—	278	1	no	—
295	C₄²	C₂²×D₁₃₉	1	1	yes	characteristic
296	C₂³	—	2	1	no	—
297	C₂³	—	278	1	no	—
298	C₂³	—	139	1	no	—
299	C₂³	—	139	1	no	—
300	Q₈	—	2	1	no	—
301	Q₈	—	2	1	no	—
302	Q₈	—	278	1	no	—
303	Q₈	—	556	2	no	—
304	Q₈	—	139	2	no	—
305	D₄	—	2	1	no	—
306	D₄	—	2	1	no	—
307	D₄	—	278	2	no	—
308	D₄	—	278	1	no	—
309	D₄	—	278	1	no	—
310	D₄	—	556	2	no	—
311	D₄	—	556	2	no	—
312	D₄	—	556	2	no	—
313	D₄	—	1112	1	no	—
314	C₂×C₄	—	2	1	no	—
315	C₂×C₄	—	2	1	no	—
316	C₂×C₄	—	2	1	no	—
317	C₂×C₄	—	278	1	no	—
318	C₂×C₄	—	278	1	no	—
319	C₂×C₄	—	278	1	no	—
320	C₂×C₄	—	278	1	no	—
321	C₂×C₄	—	278	1	no	—
322	C₂×C₄	—	278	1	no	—
323	C₂×C₄	—	278	1	no	—
324	C₂×C₄	—	556	2	no	—
325	C₂×C₄	—	556	2	no	—
326	C₂×C₄	—	1112	1	no	—
327	C₂×C₄	—	1112	1	no	—
328	C₂×C₄	—	139	1	no	—
329	C₂×C₄	—	139	1	no	—
330	C₂×C₄	D₄×D₁₃₉	1	1	yes	characteristic
331	C₂×C₄	D₄×D₁₃₉	1	1	yes	characteristic
332	C₂×C₄	C₂³×D₁₃₉	1	1	yes	characteristic
333	C₈	—	2	1	no	—
334	C₈	—	2	1	no	—
335	C₈	—	278	1	no	—
336	C₈	—	278	1	no	—
337	C₂²	—	4	1	no	—
338	C₂²	—	4	1	no	—
339	C₂²	—	4	1	no	—
340	C₂²	—	278	1	no	—
341	C₂²	—	278	1	no	—
342	C₂²	—	278	1	no	—
343	C₂²	—	278	1	no	—
344	C₂²	—	278	1	no	—
345	C₂²	—	556	1	no	—
346	C₂²	—	556	1	no	—
347	C₂²	—	556	1	no	—
348	C₂²	—	139	2	no	—
349	C₂²	C₂×D₄×D₁₃₉	1	1	yes	characteristic
350	C₄	—	2	1	no	—
351	C₄	—	2	1	no	—
352	C₄	—	4	1	no	—
353	C₄	—	4	1	no	—
354	C₄	—	4	1	no	—
355	C₄	—	278	1	no	—
356	C₄	—	278	1	no	—
357	C₄	—	278	2	no	—
358	C₄	—	556	1	no	—
359	C₄	—	556	1	no	—
360	C₄	#41×D₁₃₉	1	1	yes	characteristic
361	C₄	#17984888	1	1	yes	characteristic
362	C₂	—	8	1	no	—
363	C₂	—	278	2	no	—
364	C₂	—	556	1	no	—
365	C₂	—	1112	1	no	—
366	C₂	#22581596	1	1	yes	characteristic
367	C₂	#135×D₁₃₉	1	1	yes	characteristic
368	C₂	#22581760	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×D₄×D₁₃₉
commutator, cosocle	C₂×C₄×C₁₃₉	C₂⁴
Frattini	C₂×C₄	C₂³×D₁₃₉
Fitting	#488×C₁₃₉	C₂
socle	C₂²×C₁₃₉	C₂²×D₄
hypercenter	#488	D₁₃₉
hypocenter	C₁₃₉	#3301
LCS	G ▷ C₂×C₄×C₁₃₉ ▷ C₂×C₁₃₉ ▷ C₁₃₉
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #488

Complements

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

N	H ≅ G/N	type
1. #22581656	C₂	semidirect
2. #22581611	C₂	semidirect
3. #22581292	C₂	semidirect
4. #22581270	C₂	semidirect
5. #22581257	C₂	semidirect
6. #22581188	C₂	semidirect
7. #22581185	C₂	semidirect
8. #22581165	C₂	semidirect
9. #22581099	C₂	semidirect
10. #22581093	C₂	semidirect
11. C₄×#14555774	C₂	semidirect
12. C₈×D₅₅₆	C₂	semidirect
13. #22580946	C₂	semidirect
14. #103×D₁₃₉	C₂	semidirect
15. #488×C₁₃₉	C₂	semidirect
184. C₁₃₉	#3301	semidirect
188. #488	D₁₃₉	semidirect

Maximal subgroups

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

subgroup	index
#22581656	2
#22581611	2
#22581292	2
#22581270	2
#22581257	2
#22581188	2
#22581185	2
#22581165	2
#22581099	2
#22581093	2
C₄×#14555774	2
C₈×D₅₅₆	2
#22580946	2
#103×D₁₃₉	2
#488×C₁₃₉	2
#3301	139