Subgroups — #31816896

Order 17792.

532 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	C₂×#17984874	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×#17984871	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×#17984863	C₂	1	1	yes	maximal, characteristic
4	C₂×Q₁₆×D₁₃₉	C₂	1	1	yes	maximal, characteristic, complemented
5	#22581528	C₂	1	1	yes	maximal, characteristic, complemented
6	#22581523	C₂	1	1	yes	maximal, characteristic, complemented
7	#22581347	C₂	1	1	yes	maximal, characteristic, complemented
8	#22581346	C₂	1	1	yes	maximal, characteristic, complemented
9	#22581344	C₂	1	1	yes	maximal, characteristic, complemented
10	#22581335	C₂	1	1	yes	maximal, characteristic, complemented
11	#22581334	C₂	1	1	yes	maximal, characteristic, complemented
12	#22581332	C₂	1	1	yes	maximal, characteristic, complemented
13	#22581331	C₂	1	1	yes	maximal, characteristic, complemented
14	#22580910	C₂	1	1	yes	maximal, characteristic, complemented
15	#497×C₁₃₉	C₂	1	1	yes	maximal, characteristic, complemented
16	C₂×#14555786	C₂²	1	1	yes	characteristic
17	C₂×#14555784	C₂²	1	1	yes	characteristic
18	C₂×Q₈×D₁₃₉	C₂²	1	1	yes	characteristic
19	C₂×Q₈×D₁₃₉	C₂²	1	1	yes	characteristic
20	#17984874	—	2	4	no	—
21	#17984871	—	2	4	no	—
22	#17984863	—	2	4	no	—
23	C₂×Dic₅₅₆	C₂²	1	1	yes	characteristic
24	C₂×Dic₅₅₆	C₂²	1	1	yes	characteristic
25	C₂×#14555775	C₂²	1	1	yes	characteristic
26	C₂×#14555775	C₂²	1	1	yes	characteristic
27	C₂×#14555775	C₂²	1	1	yes	characteristic
28	C₂×#14555775	C₂²	1	1	yes	characteristic
29	C₂×#14555775	C₂²	1	1	yes	characteristic
30	Q₁₆×D₁₃₉	—	2	4	no	—
31	C₂×#14555774	C₂²	1	1	yes	characteristic
32	C₂×#14555774	C₂²	1	1	yes	characteristic
33	C₂×#14555773	C₂²	1	1	yes	characteristic
34	C₂×#14555772	C₂²	1	1	yes	characteristic
35	C₂×#14555770	C₂²	1	1	yes	characteristic
36	C₂×#14555768	C₂²	1	1	yes	characteristic
37	C₂×C₈×D₁₃₉	C₂²	1	1	yes	characteristic
38	#17984842	C₂²	1	1	yes	characteristic
39	#17984840	C₂²	1	1	yes	characteristic
40	#17984840	C₂²	1	1	yes	characteristic
41	#17984831	C₂²	1	1	yes	characteristic
42	#17984830	C₂²	1	1	yes	characteristic
43	#17984829	C₂²	1	1	yes	characteristic
44	#17984827	—	2	2	no	—
45	#17984824	—	2	2	no	—
46	#17984820	—	2	2	no	—
47	#17984819	—	2	2	no	—
48	#17984799	C₂²	1	1	yes	characteristic
49	#17984752	C₂²	1	1	yes	characteristic
50	C₄×C₁₃₉⋊C₈	C₂²	1	1	yes	characteristic
51	C₈×Dic₁₃₉	C₂²	1	1	yes	characteristic
52	C₂×Q₁₆×C₁₃₉	C₂²	1	1	yes	characteristic
53	C₂×Q₁₆×C₁₃₉	C₂²	1	1	yes	characteristic
54	C₂×SD₁₆×C₁₃₉	C₂²	1	1	yes	characteristic
55	C₂×SD₁₆×C₁₃₉	C₂²	1	1	yes	characteristic
56	C₄⋊Q₈×C₁₃₉	C₂²	1	1	yes	characteristic
57	#124×C₁₃₉	C₂²	1	1	yes	characteristic
58	#97×C₁₃₉	C₂²	1	1	yes	characteristic
59	#14555786	—	2	2	no	—
60	#14555786	—	2	2	no	—
61	#14555784	—	2	2	no	—
62	#14555784	—	2	2	no	—
63	C₂×Dic₂₇₈	—	2	1	no	—
64	C₂×Dic₂₇₈	—	2	1	no	—
65	C₂×Dic₂₇₈	C₂³	1	1	yes	characteristic
66	C₂×Dic₂₇₈	C₂³	1	1	yes	characteristic
67	C₂×Dic₂₇₈	C₂³	1	1	yes	characteristic
68	Q₈×D₁₃₉	—	2	2	no	—
69	Q₈×D₁₃₉	—	2	2	no	—
70	Q₈×D₁₃₉	—	2	2	no	—
71	Q₈×D₁₃₉	—	2	2	no	—
72	C₂×D₅₅₆	—	2	1	no	—
73	C₂×D₅₅₆	C₂³	1	1	yes	characteristic
74	C₂×C₂²⋊D₁₃₉	—	2	1	no	—
75	C₂²×Dic₁₃₉	—	2	1	no	—
76	C₂×C₄×D₁₃₉	—	2	1	no	—
77	C₂×C₄×D₁₃₉	—	2	1	no	—
78	C₂×C₄×D₁₃₉	—	2	1	no	—
79	C₂×C₄×D₁₃₉	C₂³	1	1	yes	characteristic
80	Dic₅₅₆	—	2	2	no	—
81	Dic₅₅₆	—	2	2	no	—
82	#14555775	—	2	2	no	—
83	#14555775	—	2	2	no	—
84	#14555775	—	2	2	no	—
85	#14555775	—	2	2	no	—
86	#14555775	—	2	2	no	—
87	#14555774	—	2	2	no	—
88	#14555774	—	2	2	no	—
89	#14555773	—	2	2	no	—
90	#14555772	—	2	2	no	—
91	#14555770	—	2	2	no	—
92	#14555768	D₄	1	4	yes	—
93	C₂×C₁₃₉⋊C₈	C₂³	1	1	yes	characteristic
94	C₂×C₁₃₉⋊C₈	C₂³	1	1	yes	characteristic
95	C₈×D₁₃₉	D₄	1	4	yes	—
96	C₄⋊Dic₁₃₉	—	2	2	no	—
97	Dic₁₃₉⋊C₄	—	2	2	no	—
98	Dic₁₃₉⋊C₄	—	2	2	no	—
99	Dic₁₃₉⋊C₄	D₄	1	2	yes	—
100	C₂²⋊Dic₁₃₉	—	2	2	no	—
101	(C₂×D₁₃₉)⋊C₄	—	2	2	no	—
102	(C₂×D₁₃₉)⋊C₄	—	2	2	no	—
103	(C₂×D₁₃₉)⋊C₄	D₄	1	2	yes	—
104	C₄×Dic₁₃₉	C₂³	1	1	yes	characteristic
105	C₂×Q₈×C₁₃₉	C₂³	1	1	yes	characteristic
106	C₂×Q₈×C₁₃₉	C₂³	1	1	yes	characteristic
107	C₂×Q₈×C₁₃₉	C₂³	1	1	yes	characteristic
108	C₂×D₄×C₁₃₉	C₂³	1	1	yes	characteristic
109	Q₁₆×C₁₃₉	—	2	2	no	—
110	Q₁₆×C₁₃₉	—	2	2	no	—
111	SD₁₆×C₁₃₉	—	2	2	no	—
112	SD₁₆×C₁₃₉	—	2	2	no	—
113	C₂×C₈×C₁₃₉	C₂³	1	1	yes	characteristic
114	C₂×C₈×C₁₃₉	C₂³	1	1	yes	characteristic
115	C₄⋊C₄×C₁₃₉	—	2	2	no	—
116	C₂²⋊C₄×C₁₃₉	—	2	2	no	—
117	C₄²×C₁₃₉	C₂³	1	1	yes	characteristic
118	C₂²×D₁₃₉	—	2	1	no	—
119	C₂²×D₁₃₉	C₂×D₄	1	1	yes	characteristic
120	Dic₂₇₈	—	2	1	no	—
121	Dic₂₇₈	—	2	1	no	—
122	Dic₂₇₈	—	2	1	no	—
123	Dic₂₇₈	—	2	1	no	—
124	Dic₂₇₈	—	2	1	no	—
125	Dic₂₇₈	—	2	1	no	—
126	Dic₂₇₈	—	4	2	no	—
127	Dic₂₇₈	—	4	2	no	—
128	D₅₅₆	—	2	1	no	—
129	D₅₅₆	—	2	1	no	—
130	D₅₅₆	—	4	2	no	—
131	C₂²⋊D₁₃₉	—	4	2	no	—
132	C₂×Dic₁₃₉	—	2	1	no	—
133	C₂×Dic₁₃₉	—	2	1	no	—
134	C₂×Dic₁₃₉	—	2	1	no	—
135	C₂×Dic₁₃₉	—	4	2	no	—
136	C₂×Dic₁₃₉	C₂×D₄	1	2	yes	—
137	C₂×Dic₁₃₉	C₂×D₄	1	1	yes	characteristic
138	C₄×D₁₃₉	—	4	2	no	—
139	C₄×D₁₃₉	—	4	2	no	—
140	C₄×D₁₃₉	—	4	2	no	—
141	C₄×D₁₃₉	C₂×D₄	1	2	yes	—
142	C₄×D₁₃₉	C₂×D₄	1	2	yes	—
143	C₁₃₉⋊C₈	C₂×D₄	1	2	yes	—
144	C₁₃₉⋊C₈	C₂×D₄	1	2	yes	—
145	C₂³×C₁₃₉	—	2	1	no	—
146	Q₈×C₁₃₉	—	2	1	no	—
147	Q₈×C₁₃₉	—	2	1	no	—
148	Q₈×C₁₃₉	—	2	1	no	—
149	Q₈×C₁₃₉	—	2	1	no	—
150	Q₈×C₁₃₉	—	2	1	no	—
151	Q₈×C₁₃₉	—	2	1	no	—
152	D₄×C₁₃₉	—	2	1	no	—
153	D₄×C₁₃₉	—	2	1	no	—
154	C₂×C₄×C₁₃₉	—	2	1	no	—
155	C₂×C₄×C₁₃₉	—	2	1	no	—
156	C₂×C₄×C₁₃₉	—	2	1	no	—
157	C₂×C₄×C₁₃₉	C₂×D₄	1	2	yes	—
158	C₂×C₄×C₁₃₉	C₂⁴	1	1	yes	characteristic
159	C₈×C₁₃₉	C₂×D₄	1	2	yes	—
160	C₈×C₁₃₉	C₂×D₄	1	2	yes	—
161	C₂²×C₁₃₉	—	4	1	no	—
162	C₂²×C₁₃₉	—	4	2	no	—
163	C₂²×C₁₃₉	C₂²×D₄	1	1	yes	characteristic
164	C₂×D₁₃₉	—	2	2	no	—
165	C₂×D₁₃₉	—	4	1	no	—
166	C₂×D₁₃₉	—	4	2	no	—
167	C₂×D₁₃₉	#127	1	2	yes	—
168	C₄×C₁₃₉	—	2	2	no	—
169	C₄×C₁₃₉	—	4	1	no	—
170	C₄×C₁₃₉	—	4	1	no	—
171	C₄×C₁₃₉	—	4	1	no	—
172	C₄×C₁₃₉	C₂²×D₄	1	1	yes	characteristic
173	C₄×C₁₃₉	C₂²×D₄	1	1	yes	characteristic
174	Dic₁₃₉	—	2	2	no	—
175	Dic₁₃₉	—	4	1	no	—
176	Dic₁₃₉	—	4	1	no	—
177	Dic₁₃₉	—	4	1	no	—
178	Dic₁₃₉	#127	1	2	yes	—
179	C₂×C₁₃₉	—	8	1	no	—
180	C₂×C₁₃₉	C₂×#127	1	1	yes	characteristic
181	C₂×C₁₃₉	#578	1	2	yes	—
182	D₁₃₉	—	2	2	no	—
183	D₁₃₉	—	8	1	no	—
184	C₁₃₉	#3156	1	1	yes	characteristic, complemented, Sylow
185	#3156	—	139	1	no	maximal, Sylow
186	C₂×#137	—	139	1	no	—
187	C₂×#137	—	139	1	no	—
188	C₂×#135	—	139	1	no	—
189	C₂²×Q₁₆	—	139	1	no	—
190	#536	—	139	1	no	—
191	#536	—	139	1	no	—
192	#497	—	139	1	no	—
193	#497	—	139	1	no	—
194	#497	—	139	1	no	—
195	#497	D₁₃₉	1	1	yes	characteristic, complemented
196	#495	—	139	1	no	—
197	#495	—	139	1	no	—
198	#494	—	139	1	no	—
199	#494	—	139	1	no	—
200	#405	—	139	1	no	—
201	C₂×#41	—	139	1	no	—
202	C₂×#41	—	139	1	no	—
203	C₂²×Q₈	—	139	1	no	—
204	C₂²×Q₈	—	139	1	no	—
205	#137	—	278	4	no	—
206	#137	—	278	4	no	—
207	#135	—	278	4	no	—
208	C₂×Q₁₆	—	278	4	no	—
209	C₂×Q₁₆	—	139	1	no	—
210	C₂×Q₁₆	—	139	1	no	—
211	C₂×Q₁₆	—	139	1	no	—
212	C₂×Q₁₆	—	139	1	no	—
213	C₂×Q₁₆	—	139	1	no	—
214	C₂×Q₁₆	—	139	1	no	—
215	C₂×Q₁₆	—	139	1	no	—
216	C₂×Q₁₆	C₂×D₁₃₉	1	1	yes	characteristic
217	C₂×Q₁₆	C₂×D₁₃₉	1	1	yes	characteristic
218	C₂×SD₁₆	—	139	1	no	—
219	C₂×SD₁₆	—	139	1	no	—
220	C₂×SD₁₆	—	139	1	no	—
221	C₂×SD₁₆	—	139	1	no	—
222	C₂×SD₁₆	C₂×D₁₃₉	1	1	yes	characteristic
223	C₂×SD₁₆	C₂×D₁₃₉	1	1	yes	characteristic
224	C₂×D₈	—	139	1	no	—
225	C₂×M₁₆	—	139	1	no	—
226	C₂²×C₈	—	139	1	no	—
227	C₄⋊Q₈	—	139	1	no	—
228	C₄⋊Q₈	—	139	1	no	—
229	C₄⋊Q₈	—	139	1	no	—
230	C₄⋊Q₈	C₂×D₁₃₉	1	1	yes	characteristic
231	#124	—	139	1	no	—
232	#124	—	139	1	no	—
233	#124	—	139	1	no	—
234	#124	C₂×D₁₃₉	1	1	yes	characteristic
235	#123	—	278	2	no	—
236	#123	—	278	2	no	—
237	C₂²⋊Q₈	—	278	2	no	—
238	C₂²⋊Q₈	—	278	2	no	—
239	#117	—	139	1	no	—
240	#97	—	139	1	no	—
241	#97	C₂×D₁₃₉	1	1	yes	characteristic
242	C₄×C₈	—	139	1	no	—
243	C₄×C₈	—	139	1	no	—
244	#41	—	278	2	no	—
245	#41	—	278	2	no	—
246	#41	—	278	2	no	—
247	#41	—	278	2	no	—
248	C₂×Q₈	—	278	1	no	—
249	C₂×Q₈	—	278	1	no	—
250	C₂×Q₈	—	278	2	no	—
251	C₂×Q₈	—	278	2	no	—
252	C₂×Q₈	—	278	2	no	—
253	C₂×Q₈	—	278	2	no	—
254	C₂×Q₈	—	139	1	no	—
255	C₂×Q₈	—	139	1	no	—
256	C₂×Q₈	—	139	1	no	—
257	C₂×Q₈	C₂²×D₁₃₉	1	1	yes	characteristic
258	C₂×Q₈	C₂²×D₁₃₉	1	1	yes	characteristic
259	C₂×Q₈	C₂²×D₁₃₉	1	1	yes	characteristic
260	C₂×D₄	—	278	1	no	—
261	C₂×D₄	—	278	1	no	—
262	C₂×D₄	—	139	1	no	—
263	C₂×D₄	C₂²×D₁₃₉	1	1	yes	characteristic
264	C₂²×C₄	—	278	1	no	—
265	C₂²×C₄	—	278	1	no	—
266	C₂²×C₄	—	278	1	no	—
267	C₂²×C₄	—	278	1	no	—
268	C₂²×C₄	—	139	1	no	—
269	Q₁₆	—	2	2	no	—
270	Q₁₆	—	2	2	no	—
271	Q₁₆	—	278	2	no	—
272	Q₁₆	—	278	2	no	—
273	Q₁₆	—	278	2	no	—
274	Q₁₆	—	278	2	no	—
275	Q₁₆	—	278	2	no	—
276	Q₁₆	—	278	2	no	—
277	Q₁₆	—	278	2	no	—
278	SD₁₆	—	2	2	no	—
279	SD₁₆	—	2	2	no	—
280	SD₁₆	—	278	2	no	—
281	SD₁₆	—	278	2	no	—
282	SD₁₆	—	278	2	no	—
283	SD₁₆	—	278	2	no	—
284	D₈	—	278	2	no	—
285	M₁₆	—	139	4	no	—
286	C₂×C₈	—	139	1	no	—
287	C₂×C₈	—	139	1	no	—
288	C₂×C₈	—	139	4	no	—
289	C₂×C₈	C₂²×D₁₃₉	1	1	yes	characteristic
290	C₂×C₈	C₂²×D₁₃₉	1	1	yes	characteristic
291	C₄⋊C₄	—	2	2	no	—
292	C₄⋊C₄	—	278	2	no	—
293	C₄⋊C₄	—	278	2	no	—
294	C₄⋊C₄	—	278	2	no	—
295	C₄⋊C₄	—	139	2	no	—
296	C₂²⋊C₄	—	2	2	no	—
297	C₂²⋊C₄	—	278	2	no	—
298	C₂²⋊C₄	—	278	2	no	—
299	C₂²⋊C₄	—	278	2	no	—
300	C₂²⋊C₄	—	139	2	no	—
301	C₄²	—	139	1	no	—
302	C₄²	C₂²×D₁₃₉	1	1	yes	characteristic
303	C₂³	—	2	1	no	—
304	C₂³	—	278	1	no	—
305	C₂³	—	139	1	no	—
306	Q₈	—	2	1	no	—
307	Q₈	—	2	1	no	—
308	Q₈	—	2	1	no	—
309	Q₈	—	2	1	no	—
310	Q₈	—	2	1	no	—
311	Q₈	—	2	1	no	—
312	Q₈	—	278	1	no	—
313	Q₈	—	278	1	no	—
314	Q₈	—	278	1	no	—
315	Q₈	—	278	1	no	—
316	Q₈	—	278	1	no	—
317	Q₈	—	278	1	no	—
318	Q₈	—	556	2	no	—
319	Q₈	—	556	2	no	—
320	D₄	—	2	1	no	—
321	D₄	—	2	1	no	—
322	D₄	—	278	1	no	—
323	D₄	—	278	1	no	—
324	D₄	—	556	2	no	—
325	D₄	—	556	2	no	—
326	C₂×C₄	—	2	1	no	—
327	C₂×C₄	—	2	1	no	—
328	C₂×C₄	—	2	1	no	—
329	C₂×C₄	—	278	1	no	—
330	C₂×C₄	—	278	1	no	—
331	C₂×C₄	—	278	1	no	—
332	C₂×C₄	—	556	2	no	—
333	C₂×C₄	—	556	2	no	—
334	C₂×C₄	—	556	2	no	—
335	C₂×C₄	—	556	2	no	—
336	C₂×C₄	—	139	2	no	—
337	C₂×C₄	—	139	1	no	—
338	C₂×C₄	—	139	2	no	—
339	C₂×C₄	—	139	2	no	—
340	C₂×C₄	D₄×D₁₃₉	1	2	yes	—
341	C₂×C₄	C₂³×D₁₃₉	1	1	yes	characteristic
342	C₈	—	139	2	no	—
343	C₈	—	139	2	no	—
344	C₈	D₄×D₁₃₉	1	2	yes	—
345	C₈	D₄×D₁₃₉	1	2	yes	—
346	C₂²	—	4	2	no	—
347	C₂²	—	4	1	no	—
348	C₂²	—	278	2	no	—
349	C₂²	—	556	2	no	—
350	C₂²	—	556	1	no	—
351	C₂²	—	139	2	no	—
352	C₂²	C₂×D₄×D₁₃₉	1	1	yes	characteristic
353	C₄	—	2	2	no	—
354	C₄	—	4	1	no	—
355	C₄	—	4	1	no	—
356	C₄	—	4	1	no	—
357	C₄	—	278	2	no	—
358	C₄	—	556	1	no	—
359	C₄	—	556	1	no	—
360	C₄	—	556	1	no	—
361	C₄	—	139	2	no	—
362	C₄	C₂×D₄×D₁₃₉	1	1	yes	characteristic
363	C₄	C₂×D₄×D₁₃₉	1	1	yes	characteristic
364	C₂	—	8	1	no	—
365	C₂	—	278	2	no	—
366	C₂	—	1112	1	no	—
367	C₂	#127×D₁₃₉	1	1	yes	characteristic
368	C₂	#22581769	1	2	yes	—

	subgroup	G/H
center	C₂²	C₂×D₄×D₁₃₉
commutator, cosocle	C₂×C₄×C₁₃₉	C₂⁴
Frattini	C₂×C₄	C₂³×D₁₃₉
Fitting	#497×C₁₃₉	C₂
socle	C₂²×C₁₃₉	C₂²×D₄
prime radical	C₂×#17984863	C₂
hypercenter	#497	D₁₃₉
hypocenter	C₁₃₉	#3156
LCS	G ▷ C₂×C₄×C₁₃₉ ▷ C₂×C₁₃₉ ▷ C₁₃₉
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #497

Complements

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

N	H ≅ G/N	type
1. C₂×#17984874	C₂	semidirect
2. C₂×#17984871	C₂	semidirect
4. C₂×Q₁₆×D₁₃₉	C₂	semidirect
5. #22581528	C₂	semidirect
6. #22581523	C₂	semidirect
7. #22581347	C₂	semidirect
8. #22581346	C₂	semidirect
9. #22581344	C₂	semidirect
10. #22581335	C₂	semidirect
11. #22581334	C₂	semidirect
12. #22581332	C₂	semidirect
13. #22581331	C₂	semidirect
14. #22580910	C₂	semidirect
15. #497×C₁₃₉	C₂	semidirect
184. C₁₃₉	#3156	semidirect
195. #497	D₁₃₉	semidirect

Maximal subgroups

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

subgroup	index
C₂×#17984874	2
C₂×#17984871	2
C₂×#17984863	2
C₂×Q₁₆×D₁₃₉	2
#22581528	2
#22581523	2
#22581347	2
#22581346	2
#22581344	2
#22581335	2
#22581334	2
#22581332	2
#22581331	2
#22580910	2
#497×C₁₃₉	2
#3156	139