Subgroups — #31815620

Order 17792.

764 conjugacy classes of subgroups.

The subgroup lattice has 352 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×#14555785	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×#17984862	C₂	1	2	yes	maximal, complemented
3	#22581113	C₂	1	2	yes	maximal, complemented
4	#22581108	C₂	1	1	yes	maximal, characteristic, complemented
5	#22581103	C₂	1	1	yes	maximal, characteristic, complemented
6	#22581096	C₂	1	2	yes	maximal, complemented
7	#22581091	C₂	1	1	yes	maximal, characteristic, complemented
8	#22581086	C₂	1	1	yes	maximal, characteristic, complemented
9	#22580947	C₂	1	2	yes	maximal, complemented
10	C₂×C₂²⋊(C₁₃₉⋊C₈)	C₂	1	1	yes	maximal, characteristic, complemented
11	#525×C₁₃₉	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×#14555785	—	2	8	no	—
13	C₂×#14555785	—	2	1	no	—
14	C₂×#14555785	—	2	1	no	—
15	C₂×#14555785	C₂²	1	2	yes	—
16	C₂×#14555785	C₂²	1	2	yes	—
17	C₂²×Dic₂₇₈	C₂²	1	1	yes	characteristic
18	C₂²×D₅₅₆	C₂²	1	1	yes	characteristic
19	C₂²×C₂²⋊D₁₃₉	—	2	1	no	—
20	C₂²×C₄×D₁₃₉	—	2	1	no	—
21	#17984862	—	2	8	no	—
22	C₂×#14555775	C₂²	1	2	yes	complemented
23	C₂×#14555773	C₂²	1	2	yes	complemented
24	C₂×#14555772	C₂²	1	2	yes	complemented
25	C₂×#14555770	C₂²	1	2	yes	complemented
26	C₂²×C₁₃₉⋊C₈	C₂²	1	2	yes	complemented
27	#17984765	C₂²	1	2	yes	complemented
28	#17984765	C₂²	1	2	yes	complemented
29	#17984762	C₂²	1	2	yes	complemented
30	#17984762	C₂²	1	2	yes	complemented
31	C₂²⋊(C₁₃₉⋊C₈)	C₂²	1	1	yes	characteristic
32	C₂²⋊(C₁₃₉⋊C₈)	C₂²	1	1	yes	characteristic
33	C₂²⋊(C₁₃₉⋊C₈)	C₂²	1	2	yes	complemented
34	C₂×#41×C₁₃₉	C₂²	1	1	yes	characteristic, complemented
35	C₂³×C₄×C₁₃₉	C₂²	1	1	yes	characteristic
36	#123×C₁₃₉	—	2	1	no	—
37	C₂²⋊Q₈×C₁₃₉	C₂²	1	1	yes	characteristic
38	C₂²⋊Q₈×C₁₃₉	C₂²	1	1	yes	characteristic, complemented
39	#121×C₁₃₉	C₂²	1	1	yes	characteristic
40	#121×C₁₃₉	C₂²	1	1	yes	characteristic, complemented
41	#120×C₁₃₉	—	2	1	no	—
42	C₄×D₄×C₁₃₉	—	2	1	no	—
43	C₄×D₄×C₁₃₉	—	2	1	no	—
44	#117×C₁₃₉	C₂²	1	1	yes	characteristic, complemented
45	C₂³×D₁₃₉	—	2	1	no	—
46	#14555785	—	2	8	no	—
47	#14555785	—	2	8	no	—
48	#14555785	—	2	4	no	—
49	#14555785	—	2	4	no	—
50	#14555785	—	2	4	no	—
51	#14555785	D₄	1	8	yes	complemented
52	C₂×Dic₂₇₈	—	2	4	no	—
53	C₂×Dic₂₇₈	—	2	1	no	—
54	C₂×Dic₂₇₈	C₂³	1	2	yes	—
55	C₂×D₅₅₆	—	2	4	no	—
56	C₂×D₅₅₆	—	2	1	no	—
57	C₂×D₅₅₆	C₂³	1	2	yes	—
58	C₂×C₂²⋊D₁₃₉	—	2	8	no	—
59	C₂×C₂²⋊D₁₃₉	—	2	2	no	—
60	C₂×C₂²⋊D₁₃₉	—	2	2	no	—
61	C₂²×Dic₁₃₉	—	2	1	no	—
62	C₂×C₄×D₁₃₉	—	2	8	no	—
63	C₂×C₄×D₁₃₉	—	2	2	no	—
64	C₂×C₄×D₁₃₉	—	2	2	no	—
65	#14555775	—	2	4	no	—
66	#14555773	—	2	4	no	—
67	#14555772	—	2	4	no	—
68	#14555770	—	2	4	no	—
69	C₂×C₁₃₉⋊C₈	—	2	4	no	—
70	C₂×C₁₃₉⋊C₈	C₂³	1	2	yes	—
71	C₂×C₁₃₉⋊C₈	C₂³	1	2	yes	—
72	C₂⁴×C₁₃₉	D₄	1	1	yes	characteristic
73	#41×C₁₃₉	—	2	4	no	—
74	C₂×Q₈×C₁₃₉	C₂³	1	1	yes	characteristic
75	C₂×D₄×C₁₃₉	—	2	1	no	—
76	C₂×D₄×C₁₃₉	—	2	1	no	—
77	C₂×D₄×C₁₃₉	—	2	1	no	—
78	C₂×D₄×C₁₃₉	C₂³	1	1	yes	characteristic
79	C₂²×C₄×C₁₃₉	—	2	4	no	—
80	C₂²×C₄×C₁₃₉	—	2	1	no	—
81	C₂²×C₄×C₁₃₉	D₄	1	1	yes	characteristic
82	C₂²×C₄×C₁₃₉	D₄	1	1	yes	characteristic
83	C₂²×C₄×C₁₃₉	D₄	1	1	yes	characteristic
84	C₂²×C₄×C₁₃₉	C₂³	1	1	yes	characteristic
85	C₂²×C₄×C₁₃₉	C₂³	1	1	yes	characteristic
86	C₂²×C₄×C₁₃₉	C₂³	1	1	yes	characteristic
87	C₄⋊C₄×C₁₃₉	—	2	1	no	—
88	C₄⋊C₄×C₁₃₉	—	2	1	no	—
89	C₄⋊C₄×C₁₃₉	C₂³	1	1	yes	characteristic
90	C₄⋊C₄×C₁₃₉	C₂³	1	1	yes	characteristic
91	C₂²⋊C₄×C₁₃₉	—	2	1	no	—
92	C₂²⋊C₄×C₁₃₉	—	2	1	no	—
93	C₂²⋊C₄×C₁₃₉	—	2	1	no	—
94	C₂²⋊C₄×C₁₃₉	—	2	1	no	—
95	C₂²⋊C₄×C₁₃₉	—	2	1	no	—
96	C₄²×C₁₃₉	—	2	1	no	—
97	C₂²×D₁₃₉	—	2	4	no	—
98	C₂²×D₁₃₉	—	2	2	no	—
99	C₂²×D₁₃₉	—	4	2	no	—
100	C₂²×D₁₃₉	—	4	2	no	—
101	Dic₂₇₈	—	2	4	no	—
102	Dic₂₇₈	—	2	2	no	—
103	Dic₂₇₈	C₂×D₄	1	4	yes	—
104	D₅₅₆	—	2	4	no	—
105	D₅₅₆	—	2	2	no	—
106	D₅₅₆	C₂×D₄	1	4	yes	—
107	C₂²⋊D₁₃₉	—	2	8	no	—
108	C₂²⋊D₁₃₉	—	2	8	no	—
109	C₂×Dic₁₃₉	—	2	4	no	—
110	C₂×Dic₁₃₉	—	2	2	no	—
111	C₄×D₁₃₉	—	2	8	no	—
112	C₄×D₁₃₉	—	2	8	no	—
113	C₁₃₉⋊C₈	—	2	2	no	—
114	C₁₃₉⋊C₈	—	2	2	no	—
115	C₂³×C₁₃₉	—	2	2	no	—
116	C₂³×C₁₃₉	—	2	2	no	—
117	C₂³×C₁₃₉	—	2	1	no	—
118	C₂³×C₁₃₉	—	4	1	no	—
119	C₂³×C₁₃₉	C₂×D₄	1	1	yes	characteristic
120	C₂³×C₁₃₉	C₂×D₄	1	1	yes	characteristic
121	C₂³×C₁₃₉	C₂×D₄	1	1	yes	characteristic
122	Q₈×C₁₃₉	—	2	2	no	—
123	D₄×C₁₃₉	—	2	2	no	—
124	D₄×C₁₃₉	—	4	2	no	—
125	D₄×C₁₃₉	—	4	2	no	—
126	D₄×C₁₃₉	—	4	2	no	—
127	C₂×C₄×C₁₃₉	—	2	2	no	—
128	C₂×C₄×C₁₃₉	—	2	2	no	—
129	C₂×C₄×C₁₃₉	—	2	2	no	—
130	C₂×C₄×C₁₃₉	—	2	2	no	—
131	C₂×C₄×C₁₃₉	—	2	2	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	—	4	2	no	—
136	C₂×C₄×C₁₃₉	C₂×D₄	1	4	yes	—
137	C₂×C₄×C₁₃₉	C₂×D₄	1	1	yes	characteristic
138	C₂×C₄×C₁₃₉	C₂×D₄	1	1	yes	characteristic
139	C₂×C₄×C₁₃₉	C₂×D₄	1	1	yes	characteristic
140	C₂×C₄×C₁₃₉	C₂⁴	1	1	yes	characteristic
141	C₂²×C₁₃₉	—	2	1	no	—
142	C₂²×C₁₃₉	—	2	1	no	—
143	C₂²×C₁₃₉	—	2	1	no	—
144	C₂²×C₁₃₉	—	2	1	no	—
145	C₂²×C₁₃₉	—	2	1	no	—
146	C₂²×C₁₃₉	—	2	1	no	—
147	C₂²×C₁₃₉	—	2	1	no	—
148	C₂²×C₁₃₉	—	4	4	no	—
149	C₂²×C₁₃₉	—	4	1	no	—
150	C₂²×C₁₃₉	—	4	1	no	—
151	C₂²×C₁₃₉	—	4	1	no	—
152	C₂²×C₁₃₉	#120	1	2	yes	—
153	C₂²×C₁₃₉	#135	1	2	yes	complemented
154	C₂²×C₁₃₉	C₂²×D₄	1	1	yes	characteristic
155	C₂×D₁₃₉	—	2	4	no	—
156	C₂×D₁₃₉	—	4	8	no	—
157	C₂×D₁₃₉	—	4	2	no	—
158	C₂×D₁₃₉	—	4	2	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	#120	1	2	yes	—
165	C₄×C₁₃₉	C₂²×D₄	1	2	yes	—
166	Dic₁₃₉	—	2	4	no	—
167	C₂×C₁₃₉	—	2	2	no	—
168	C₂×C₁₃₉	—	2	2	no	—
169	C₂×C₁₃₉	—	4	1	no	—
170	C₂×C₁₃₉	—	8	1	no	—
171	C₂×C₁₃₉	C₂×#120	1	1	yes	characteristic
172	C₂×C₁₃₉	C₂×#135	1	1	yes	characteristic
173	C₂×C₁₃₉	#575	1	1	yes	characteristic
174	D₁₃₉	—	4	4	no	—
175	C₁₃₉	#3002	1	1	yes	characteristic, complemented, Sylow
176	#3002	—	139	1	no	maximal, Sylow
177	C₂²×#41	—	139	1	no	—
178	C₂×#135	—	139	2	no	—
179	#525	D₁₃₉	1	1	yes	characteristic, complemented
180	#452	—	139	2	no	—
181	#451	—	139	1	no	—
182	#450	—	139	1	no	—
183	#449	—	139	2	no	—
184	#448	—	139	1	no	—
185	#447	—	139	1	no	—
186	#416	—	139	2	no	—
187	C₂×C₂²⋊C₈	—	139	1	no	—
188	C₂×#41	—	278	8	no	—
189	C₂×#41	—	278	1	no	—
190	C₂×#41	—	278	1	no	—
191	C₂×#41	—	139	2	no	—
192	C₂×#41	—	139	2	no	—
193	C₂×#41	C₂×D₁₃₉	1	1	yes	characteristic, complemented
194	C₂²×Q₈	—	139	1	no	—
195	C₂²×D₄	—	278	1	no	—
196	C₂²×D₄	—	139	1	no	—
197	C₂³×C₄	—	278	1	no	—
198	C₂³×C₄	C₂×D₁₃₉	1	1	yes	characteristic
199	#135	—	278	8	no	—
200	C₂×Q₁₆	—	139	2	no	—
201	C₂×SD₁₆	—	139	2	no	—
202	C₂×SD₁₆	—	139	2	no	—
203	C₂×D₈	—	139	2	no	—
204	C₂²×C₈	—	139	2	no	—
205	#123	—	2	1	no	—
206	C₂²⋊Q₈	C₂×D₁₃₉	1	1	yes	characteristic
207	C₂²⋊Q₈	C₂×D₁₃₉	1	1	yes	characteristic, complemented
208	#121	C₂×D₁₃₉	1	1	yes	characteristic
209	#121	C₂×D₁₃₉	1	1	yes	characteristic, complemented
210	#120	—	2	1	no	—
211	C₄×D₄	—	2	1	no	—
212	C₄×D₄	—	2	1	no	—
213	#117	C₂×D₁₃₉	1	1	yes	characteristic, complemented
214	#103	—	139	2	no	—
215	#103	—	139	2	no	—
216	#102	—	139	2	no	—
217	#102	—	139	2	no	—
218	C₂²⋊C₈	—	139	1	no	—
219	C₂²⋊C₈	—	139	1	no	—
220	C₂²⋊C₈	—	139	2	no	—
221	C₂⁴	—	278	1	no	—
222	C₂⁴	C₂²⋊D₁₃₉	1	1	yes	characteristic
223	#41	—	2	4	no	—
224	#41	—	278	8	no	—
225	#41	—	278	8	no	—
226	#41	—	278	4	no	—
227	#41	—	278	4	no	—
228	#41	—	278	4	no	—
229	#41	—	139	8	no	—
230	C₂×Q₈	—	278	4	no	—
231	C₂×Q₈	—	278	1	no	—
232	C₂×Q₈	—	139	2	no	—
233	C₂×Q₈	C₂²×D₁₃₉	1	1	yes	characteristic
234	C₂×D₄	—	2	1	no	—
235	C₂×D₄	—	2	1	no	—
236	C₂×D₄	—	2	1	no	—
237	C₂×D₄	—	278	4	no	—
238	C₂×D₄	—	278	8	no	—
239	C₂×D₄	—	278	2	no	—
240	C₂×D₄	—	278	2	no	—
241	C₂×D₄	—	278	1	no	—
242	C₂×D₄	—	139	2	no	—
243	C₂×D₄	C₂²×D₁₃₉	1	1	yes	characteristic
244	C₂²×C₄	—	2	4	no	—
245	C₂²×C₄	—	2	1	no	—
246	C₂²×C₄	—	278	8	no	—
247	C₂²×C₄	—	278	1	no	—
248	C₂²×C₄	—	278	2	no	—
249	C₂²×C₄	—	278	2	no	—
250	C₂²×C₄	C₂²⋊D₁₃₉	1	1	yes	characteristic
251	C₂²×C₄	C₂²⋊D₁₃₉	1	1	yes	characteristic
252	C₂²×C₄	C₂²⋊D₁₃₉	1	1	yes	characteristic
253	C₂²×C₄	C₂²×D₁₃₉	1	1	yes	characteristic
254	C₂²×C₄	C₂²×D₁₃₉	1	1	yes	characteristic
255	C₂²×C₄	C₂²×D₁₃₉	1	1	yes	characteristic
256	Q₁₆	—	278	4	no	—
257	SD₁₆	—	278	4	no	—
258	SD₁₆	—	278	4	no	—
259	D₈	—	278	4	no	—
260	C₂×C₈	—	278	4	no	—
261	C₂×C₈	—	139	2	no	—
262	C₂×C₈	—	139	2	no	—
263	C₄⋊C₄	—	2	1	no	—
264	C₄⋊C₄	—	2	1	no	—
265	C₄⋊C₄	C₂²×D₁₃₉	1	1	yes	characteristic
266	C₄⋊C₄	C₂²×D₁₃₉	1	1	yes	characteristic
267	C₂²⋊C₄	—	2	1	no	—
268	C₂²⋊C₄	—	2	1	no	—
269	C₂²⋊C₄	—	2	1	no	—
270	C₂²⋊C₄	—	2	1	no	—
271	C₂²⋊C₄	—	2	1	no	—
272	C₄²	—	2	1	no	—
273	C₂³	—	2	2	no	—
274	C₂³	—	2	2	no	—
275	C₂³	—	2	1	no	—
276	C₂³	—	4	1	no	—
277	C₂³	—	278	4	no	—
278	C₂³	—	278	2	no	—
279	C₂³	—	556	2	no	—
280	C₂³	—	556	2	no	—
281	C₂³	C₂×C₂²⋊D₁₃₉	1	1	yes	characteristic
282	C₂³	C₂×C₂²⋊D₁₃₉	1	1	yes	characteristic
283	C₂³	C₂×C₂²⋊D₁₃₉	1	1	yes	characteristic
284	Q₈	—	2	2	no	—
285	Q₈	—	278	4	no	—
286	Q₈	—	278	2	no	—
287	Q₈	—	139	4	no	—
288	D₄	—	2	2	no	—
289	D₄	—	4	2	no	—
290	D₄	—	4	2	no	—
291	D₄	—	4	2	no	—
292	D₄	—	278	4	no	—
293	D₄	—	278	8	no	—
294	D₄	—	278	8	no	—
295	D₄	—	278	2	no	—
296	D₄	—	139	4	no	—
297	C₂×C₄	—	2	2	no	—
298	C₂×C₄	—	2	2	no	—
299	C₂×C₄	—	2	2	no	—
300	C₂×C₄	—	2	2	no	—
301	C₂×C₄	—	2	2	no	—
302	C₂×C₄	—	2	1	no	—
303	C₂×C₄	—	2	1	no	—
304	C₂×C₄	—	2	1	no	—
305	C₂×C₄	—	4	2	no	—
306	C₂×C₄	—	278	8	no	—
307	C₂×C₄	—	278	4	no	—
308	C₂×C₄	—	278	2	no	—
309	C₂×C₄	—	278	8	no	—
310	C₂×C₄	D₄×D₁₃₉	1	4	yes	—
311	C₂×C₄	C₂×C₂²⋊D₁₃₉	1	1	yes	characteristic
312	C₂×C₄	C₂×C₂²⋊D₁₃₉	1	1	yes	characteristic
313	C₂×C₄	C₂×C₂²⋊D₁₃₉	1	1	yes	characteristic
314	C₂×C₄	C₂³×D₁₃₉	1	1	yes	characteristic
315	C₈	—	278	2	no	—
316	C₈	—	278	2	no	—
317	C₂²	—	2	1	no	—
318	C₂²	—	2	1	no	—
319	C₂²	—	2	1	no	—
320	C₂²	—	2	1	no	—
321	C₂²	—	2	1	no	—
322	C₂²	—	2	1	no	—
323	C₂²	—	2	1	no	—
324	C₂²	—	4	4	no	—
325	C₂²	—	4	1	no	—
326	C₂²	—	4	1	no	—
327	C₂²	—	4	1	no	—
328	C₂²	—	278	4	no	—
329	C₂²	—	556	8	no	—
330	C₂²	—	556	2	no	—
331	C₂²	—	556	2	no	—
332	C₂²	#17984810	1	2	yes	—
333	C₂²	#17984862	1	2	yes	complemented
334	C₂²	C₂²×C₂²⋊D₁₃₉	1	1	yes	characteristic
335	C₄	—	2	1	no	—
336	C₄	—	2	1	no	—
337	C₄	—	4	1	no	—
338	C₄	—	4	1	no	—
339	C₄	—	4	1	no	—
340	C₄	—	278	4	no	—
341	C₄	#17984810	1	2	yes	—
342	C₄	C₂×D₄×D₁₃₉	1	2	yes	—
343	C₂	—	2	2	no	—
344	C₂	—	2	2	no	—
345	C₂	—	4	1	no	—
346	C₂	—	8	1	no	—
347	C₂	—	556	4	no	—
348	C₂	C₂×#17984810	1	1	yes	characteristic
349	C₂	C₂×#17984862	1	1	yes	characteristic
350	C₂	#22581751	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₂×C₂²⋊D₁₃₉
commutator, cosocle	C₂×C₄×C₁₃₉	C₂⁴
Frattini	C₂×C₄	C₂³×D₁₃₉
Fitting	#525×C₁₃₉	C₂
socle	C₂²×C₁₃₉	C₂²×D₄
hypercenter	#525	D₁₃₉
hypocenter	C₁₃₉	#3002
LCS	G ▷ C₂×C₄×C₁₃₉ ▷ C₂×C₁₃₉ ▷ C₁₃₉
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₄ ◁ #525

Complements

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

N	H ≅ G/N	type
1. C₂²×#14555785	C₂	semidirect
2. C₂×#17984862	C₂	semidirect
3. #22581113	C₂	semidirect
4. #22581108	C₂	semidirect
5. #22581103	C₂	semidirect
6. #22581096	C₂	semidirect
7. #22581091	C₂	semidirect
8. #22581086	C₂	semidirect
9. #22580947	C₂	semidirect
10. C₂×C₂²⋊(C₁₃₉⋊C₈)	C₂	semidirect
11. #525×C₁₃₉	C₂	semidirect
22. C₂×#14555775	C₂²	semidirect
23. C₂×#14555773	C₂²	semidirect
24. C₂×#14555772	C₂²	semidirect
25. C₂×#14555770	C₂²	semidirect
26. C₂²×C₁₃₉⋊C₈	C₂²	semidirect
27. #17984765	C₂²	semidirect
28. #17984765	C₂²	semidirect
29. #17984762	C₂²	semidirect
30. #17984762	C₂²	semidirect
33. C₂²⋊(C₁₃₉⋊C₈)	C₂²	semidirect
34. C₂×#41×C₁₃₉	C₂²	semidirect
38. C₂²⋊Q₈×C₁₃₉	C₂²	semidirect
40. #121×C₁₃₉	C₂²	semidirect
44. #117×C₁₃₉	C₂²	semidirect
51. #14555785	D₄	semidirect
153. C₂²×C₁₃₉	#135	semidirect
175. C₁₃₉	#3002	semidirect
179. #525	D₁₃₉	semidirect
193. C₂×#41	C₂×D₁₃₉	semidirect
207. C₂²⋊Q₈	C₂×D₁₃₉	semidirect
209. #121	C₂×D₁₃₉	semidirect
213. #117	C₂×D₁₃₉	semidirect
333. C₂²	#17984862	semidirect

Maximal subgroups

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

subgroup	index
C₂²×#14555785	2
C₂×#17984862	2
#22581113	2
#22581108	2
#22581103	2
#22581096	2
#22581091	2
#22581086	2
#22580947	2
C₂×C₂²⋊(C₁₃₉⋊C₈)	2
#525×C₁₃₉	2
#3002	139