Subgroups — C₄×#17984874 #31815224

Order 17792.

484 conjugacy classes of subgroups.

The subgroup lattice has 356 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
2	C₄×#14555786	C₂	1	1	yes	maximal, characteristic
3	C₄×Q₈×D₁₃₉	C₂	1	1	yes	maximal, characteristic, complemented
4	#22581060	C₂	1	1	yes	maximal, characteristic, complemented
5	#22581057	C₂	1	1	yes	maximal, characteristic, complemented
6	#22581056	C₂	1	1	yes	maximal, characteristic, complemented
7	#22581052	C₂	1	1	yes	maximal, characteristic, complemented
8	C₄×#14555775	C₂	1	1	yes	maximal, characteristic, complemented
9	C₄×#14555774	C₂	1	1	yes	maximal, characteristic, complemented
10	C₄×#14555772	C₂	1	1	yes	maximal, characteristic, complemented
11	#22580994	C₂	1	1	yes	maximal, characteristic, complemented
12	#22580960	C₂	1	1	yes	maximal, characteristic, complemented
13	#22580953	C₂	1	1	yes	maximal, characteristic
14	C₄×#14555768	C₂	1	1	yes	maximal, characteristic, complemented
15	C₄×Q₁₆×C₁₃₉	C₂	1	1	yes	maximal, characteristic, complemented
16	C₂×#14555786	C₂²	1	1	yes	characteristic
17	C₂×Q₈×D₁₃₉	C₂²	1	1	yes	characteristic
18	#17984874	C₄	1	8	yes	direct product factor
19	C₂×#14555775	C₂²	1	1	yes	characteristic
20	C₂×#14555774	C₂²	1	1	yes	characteristic
21	C₂×#14555772	C₂²	1	1	yes	characteristic
22	C₂×#14555768	C₂²	1	1	yes	characteristic
23	C₂×#14555768	C₂²	1	1	yes	characteristic
24	Q₈×Dic₁₃₉	C₂²	1	1	yes	characteristic
25	Q₈×Dic₁₃₉	C₂²	1	1	yes	characteristic
26	C₄×Dic₂₇₈	—	2	1	no	—
27	C₄×Dic₂₇₈	C₂²	1	1	yes	characteristic
28	#17984807	—	2	1	no	—
29	#17984807	C₂²	1	1	yes	characteristic
30	C₄×D₅₅₆	—	2	1	no	—
31	C₄×D₅₅₆	C₂²	1	1	yes	characteristic
32	#17984804	—	2	1	no	—
33	#17984804	C₂²	1	1	yes	characteristic
34	#17984800	—	2	1	no	—
35	#17984800	C₂²	1	1	yes	characteristic
36	C₄⋊C₄×D₁₃₉	—	2	1	no	—
37	C₄⋊C₄×D₁₃₉	C₂²	1	1	yes	characteristic
38	C₄²×D₁₃₉	—	2	1	no	—
39	C₄²×D₁₃₉	—	2	1	no	—
40	C₄²×D₁₃₉	C₂²	1	1	yes	characteristic
41	#17984775	C₂²	1	1	yes	characteristic
42	#17984774	C₂²	1	1	yes	characteristic
43	#17984773	C₂²	1	1	yes	characteristic
44	#17984767	C₂²	1	1	yes	characteristic
45	#17984766	C₂²	1	1	yes	characteristic
46	#17984766	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	#17984752	C₂²	1	1	yes	characteristic
51	#17984752	C₂²	1	1	yes	characteristic
52	C₄×C₁₃₉⋊C₈	C₂²	1	1	yes	characteristic
53	C₂×Q₁₆×C₁₃₉	C₂²	1	1	yes	characteristic
54	C₄×Q₈×C₁₃₉	C₂²	1	1	yes	characteristic
55	C₄×Q₈×C₁₃₉	C₂²	1	1	yes	characteristic
56	#107×C₁₃₉	C₂²	1	1	yes	characteristic
57	#103×C₁₃₉	C₂²	1	1	yes	characteristic
58	#103×C₁₃₉	C₂²	1	1	yes	characteristic
59	C₄×C₈×C₁₃₉	C₂²	1	1	yes	characteristic
60	#14555786	—	2	2	no	—
61	#14555786	C₂×C₄	1	4	yes	—
62	C₂×Dic₂₇₈	—	2	1	no	—
63	C₂×Dic₂₇₈	C₂³	1	1	yes	characteristic
64	Q₈×D₁₃₉	—	2	2	no	—
65	Q₈×D₁₃₉	C₂×C₄	1	4	yes	complemented
66	C₂×D₅₅₆	—	2	1	no	—
67	C₂×D₅₅₆	C₂³	1	1	yes	characteristic
68	C₂×C₄×D₁₃₉	—	2	1	no	—
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	—	2	1	no	—
73	C₂×C₄×D₁₃₉	D₄	1	1	yes	characteristic
74	C₂×C₄×D₁₃₉	D₄	1	1	yes	characteristic
75	C₂×C₄×D₁₃₉	C₂³	1	1	yes	characteristic
76	#14555775	C₂×C₄	1	4	yes	complemented
77	#14555774	C₂×C₄	1	4	yes	complemented
78	#14555772	C₂×C₄	1	4	yes	complemented
79	#14555768	—	2	2	no	—
80	#14555768	C₂×C₄	1	4	yes	complemented
81	C₂×C₁₃₉⋊C₈	C₂³	1	1	yes	characteristic
82	C₂×C₁₃₉⋊C₈	C₂³	1	1	yes	characteristic
83	C₄⋊Dic₁₃₉	—	2	1	no	—
84	C₄⋊Dic₁₃₉	—	2	1	no	—
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	(C₂×D₁₃₉)⋊C₄	—	2	1	no	—
91	(C₂×D₁₃₉)⋊C₄	—	2	1	no	—
92	(C₂×D₁₃₉)⋊C₄	—	2	1	no	—
93	C₄×Dic₁₃₉	—	2	1	no	—
94	C₄×Dic₁₃₉	—	2	1	no	—
95	C₄×Dic₁₃₉	—	2	1	no	—
96	C₄×Dic₁₃₉	—	2	1	no	—
97	C₄×Dic₁₃₉	—	2	1	no	—
98	C₄×Dic₁₃₉	D₄	1	1	yes	characteristic
99	C₄×Dic₁₃₉	D₄	1	1	yes	characteristic
100	C₄×Dic₁₃₉	C₂³	1	1	yes	characteristic
101	C₂×Q₈×C₁₃₉	C₂³	1	1	yes	characteristic
102	C₂×Q₈×C₁₃₉	C₂³	1	1	yes	characteristic
103	Q₁₆×C₁₃₉	C₂×C₄	1	4	yes	complemented
104	C₂×C₈×C₁₃₉	C₂³	1	1	yes	characteristic
105	C₂×C₈×C₁₃₉	C₂³	1	1	yes	characteristic
106	C₄⋊C₄×C₁₃₉	—	2	1	no	—
107	C₄⋊C₄×C₁₃₉	—	2	1	no	—
108	C₄⋊C₄×C₁₃₉	C₂³	1	1	yes	characteristic
109	C₄⋊C₄×C₁₃₉	C₂³	1	1	yes	characteristic
110	C₄²×C₁₃₉	—	2	1	no	—
111	C₄²×C₁₃₉	—	2	1	no	—
112	C₄²×C₁₃₉	C₂³	1	1	yes	characteristic
113	C₂²×D₁₃₉	—	2	1	no	—
114	C₂²×D₁₃₉	C₂×D₄	1	1	yes	characteristic
115	Dic₂₇₈	—	2	1	no	—
116	Dic₂₇₈	—	2	4	no	—
117	Dic₂₇₈	C₂²×C₄	1	2	yes	—
118	D₅₅₆	—	2	4	no	—
119	D₅₅₆	—	2	1	no	—
120	D₅₅₆	C₂²×C₄	1	2	yes	—
121	C₂×Dic₁₃₉	—	2	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	4	no	—
128	C₄×D₁₃₉	—	2	2	no	—
129	C₄×D₁₃₉	—	2	4	no	—
130	C₄×D₁₃₉	—	4	1	no	—
131	C₄×D₁₃₉	—	4	2	no	—
132	C₄×D₁₃₉	—	4	2	no	—
133	C₄×D₁₃₉	—	4	2	no	—
134	C₄×D₁₃₉	C₂²×C₄	1	2	yes	—
135	C₄×D₁₃₉	#41	1	2	yes	—
136	C₁₃₉⋊C₈	—	2	1	no	—
137	C₁₃₉⋊C₈	C₂²×C₄	1	2	yes	—
138	Q₈×C₁₃₉	—	2	1	no	—
139	Q₈×C₁₃₉	—	2	1	no	—
140	Q₈×C₁₃₉	C₂²×C₄	1	2	yes	—
141	Q₈×C₁₃₉	C₂²×C₄	1	2	yes	—
142	C₂×C₄×C₁₃₉	—	2	1	no	—
143	C₂×C₄×C₁₃₉	—	2	1	no	—
144	C₂×C₄×C₁₃₉	—	2	1	no	—
145	C₂×C₄×C₁₃₉	—	2	1	no	—
146	C₂×C₄×C₁₃₉	C₂×D₄	1	1	yes	characteristic
147	C₂×C₄×C₁₃₉	C₂×D₄	1	1	yes	characteristic
148	C₂×C₄×C₁₃₉	C₂⁴	1	1	yes	characteristic
149	C₈×C₁₃₉	—	2	1	no	—
150	C₈×C₁₃₉	C₂²×C₄	1	2	yes	—
151	C₂²×C₁₃₉	C₂²×D₄	1	1	yes	characteristic
152	C₂×D₁₃₉	—	2	1	no	—
153	C₂×D₁₃₉	—	2	1	no	—
154	C₂×D₁₃₉	—	2	2	no	—
155	C₂×D₁₃₉	—	4	1	no	—
156	C₂×D₁₃₉	—	4	1	no	—
157	C₂×D₁₃₉	C₄×D₄	1	2	yes	—
158	C₄×C₁₃₉	—	2	2	no	—
159	C₄×C₁₃₉	—	2	2	no	—
160	C₄×C₁₃₉	—	2	1	no	—
161	C₄×C₁₃₉	—	4	1	no	—
162	C₄×C₁₃₉	—	4	1	no	—
163	C₄×C₁₃₉	#137	1	2	yes	complemented
164	C₄×C₁₃₉	C₂³×C₄	1	1	yes	characteristic
165	C₄×C₁₃₉	C₂×#41	1	1	yes	characteristic
166	Dic₁₃₉	—	2	2	no	—
167	Dic₁₃₉	—	2	1	no	—
168	Dic₁₃₉	—	2	1	no	—
169	Dic₁₃₉	—	4	1	no	—
170	Dic₁₃₉	—	4	1	no	—
171	Dic₁₃₉	C₄×D₄	1	2	yes	—
172	C₂×C₁₃₉	C₂×C₄×D₄	1	1	yes	characteristic
173	C₂×C₁₃₉	C₂×#137	1	1	yes	characteristic
174	C₂×C₁₃₉	#575	1	1	yes	characteristic
175	D₁₃₉	—	2	2	no	—
176	D₁₃₉	—	4	2	no	—
177	C₁₃₉	C₄×#137	1	1	yes	characteristic, complemented, Sylow
178	C₄×#137	—	139	1	no	maximal, Sylow
179	C₂×#137	—	139	1	no	—
180	C₄×#41	—	139	1	no	—
181	C₂×C₄×Q₈	—	139	1	no	—
182	#441	—	139	1	no	—
183	#441	—	139	1	no	—
184	#440	—	139	1	no	—
185	#440	—	139	1	no	—
186	C₄×Q₁₆	—	139	1	no	—
187	C₄×Q₁₆	D₁₃₉	1	1	yes	characteristic, complemented
188	C₄×SD₁₆	—	139	1	no	—
189	C₄×SD₁₆	—	139	1	no	—
190	#428	—	139	1	no	—
191	#419	—	139	1	no	—
192	#417	—	139	1	no	—
193	C₄×M₁₆	—	139	1	no	—
194	C₂×#41	—	139	1	no	—
195	C₂²×Q₈	—	139	1	no	—
196	#137	—	139	8	no	—
197	C₂×Q₁₆	—	139	1	no	—
198	C₂×Q₁₆	C₂×D₁₃₉	1	1	yes	characteristic
199	C₂×SD₁₆	—	139	1	no	—
200	C₂×SD₁₆	—	139	1	no	—
201	C₂×M₁₆	—	139	1	no	—
202	C₂×M₁₆	—	139	1	no	—
203	C₄×Q₈	—	278	1	no	—
204	C₄×Q₈	—	278	1	no	—
205	C₄×Q₈	—	139	1	no	—
206	C₄×Q₈	—	139	1	no	—
207	C₄×Q₈	—	139	1	no	—
208	C₄×Q₈	—	139	1	no	—
209	C₄×Q₈	C₂×D₁₃₉	1	1	yes	characteristic
210	C₄×Q₈	C₂×D₁₃₉	1	1	yes	characteristic
211	C₄×D₄	—	278	1	no	—
212	C₄×D₄	—	278	1	no	—
213	C₄×D₄	—	139	1	no	—
214	C₄×D₄	—	139	1	no	—
215	#117	—	278	1	no	—
216	#117	—	139	1	no	—
217	C₂×C₄⋊C₄	—	278	1	no	—
218	C₂×C₄⋊C₄	—	139	1	no	—
219	C₂×C₄²	—	278	1	no	—
220	C₂×C₄²	—	278	1	no	—
221	C₂×C₄²	—	139	1	no	—
222	#107	—	139	1	no	—
223	#107	C₂×D₁₃₉	1	1	yes	characteristic
224	#106	—	139	1	no	—
225	#106	—	139	1	no	—
226	#103	—	139	1	no	—
227	#103	—	139	1	no	—
228	#103	—	139	1	no	—
229	#103	—	139	1	no	—
230	#103	C₂×D₁₃₉	1	1	yes	characteristic
231	#103	C₂×D₁₃₉	1	1	yes	characteristic
232	#102	—	139	1	no	—
233	#102	—	139	1	no	—
234	#97	—	139	1	no	—
235	#97	—	139	1	no	—
236	C₄×C₈	—	139	1	no	—
237	C₄×C₈	C₂×D₁₃₉	1	1	yes	characteristic
238	#41	—	278	2	no	—
239	#41	—	139	4	no	—
240	C₂×Q₈	—	278	1	no	—
241	C₂×Q₈	—	278	2	no	—
242	C₂×Q₈	—	139	4	no	—
243	C₂×Q₈	—	139	1	no	—
244	C₂×Q₈	C₂²×D₁₃₉	1	1	yes	characteristic
245	C₂×Q₈	C₂²×D₁₃₉	1	1	yes	characteristic
246	C₂×D₄	—	278	1	no	—
247	C₂×D₄	—	139	1	no	—
248	C₂²×C₄	—	278	1	no	—
249	C₂²×C₄	—	278	1	no	—
250	C₂²×C₄	—	278	1	no	—
251	C₂²×C₄	—	278	1	no	—
252	C₂²×C₄	—	278	1	no	—
253	C₂²×C₄	—	139	1	no	—
254	C₂²×C₄	—	139	1	no	—
255	C₂²×C₄	—	139	1	no	—
256	Q₁₆	—	139	4	no	—
257	Q₁₆	C₄×D₁₃₉	1	4	yes	complemented
258	SD₁₆	—	139	4	no	—
259	SD₁₆	—	139	4	no	—
260	M₁₆	—	278	2	no	—
261	M₁₆	—	139	4	no	—
262	C₂×C₈	—	139	1	no	—
263	C₂×C₈	—	139	1	no	—
264	C₂×C₈	C₂²×D₁₃₉	1	1	yes	characteristic
265	C₂×C₈	C₂²×D₁₃₉	1	1	yes	characteristic
266	C₄⋊C₄	—	2	1	no	—
267	C₄⋊C₄	—	2	1	no	—
268	C₄⋊C₄	—	278	1	no	—
269	C₄⋊C₄	—	278	1	no	—
270	C₄⋊C₄	—	278	1	no	—
271	C₄⋊C₄	—	278	1	no	—
272	C₄⋊C₄	—	278	1	no	—
273	C₄⋊C₄	—	139	1	no	—
274	C₄⋊C₄	—	139	1	no	—
275	C₄⋊C₄	C₂²×D₁₃₉	1	1	yes	characteristic
276	C₄⋊C₄	C₂²×D₁₃₉	1	1	yes	characteristic
277	C₂²⋊C₄	—	278	1	no	—
278	C₂²⋊C₄	—	278	1	no	—
279	C₂²⋊C₄	—	278	1	no	—
280	C₄²	—	2	1	no	—
281	C₄²	—	2	1	no	—
282	C₄²	—	278	1	no	—
283	C₄²	—	278	1	no	—
284	C₄²	—	278	1	no	—
285	C₄²	—	278	1	no	—
286	C₄²	—	278	1	no	—
287	C₄²	—	139	1	no	—
288	C₄²	—	139	1	no	—
289	C₄²	—	139	1	no	—
290	C₄²	C₂²×D₁₃₉	1	1	yes	characteristic
291	C₂³	—	278	1	no	—
292	C₂³	—	139	1	no	—
293	Q₈	—	2	1	no	—
294	Q₈	—	2	1	no	—
295	Q₈	—	278	4	no	—
296	Q₈	—	278	1	no	—
297	Q₈	—	139	2	no	—
298	Q₈	C₂×C₄×D₁₃₉	1	2	yes	—
299	Q₈	C₂×C₄×D₁₃₉	1	2	yes	—
300	D₄	—	278	4	no	—
301	D₄	—	278	1	no	—
302	D₄	—	139	2	no	—
303	C₂×C₄	—	2	1	no	—
304	C₂×C₄	—	2	1	no	—
305	C₂×C₄	—	2	1	no	—
306	C₂×C₄	—	2	1	no	—
307	C₂×C₄	—	278	4	no	—
308	C₂×C₄	—	278	4	no	—
309	C₂×C₄	—	278	1	no	—
310	C₂×C₄	—	278	2	no	—
311	C₂×C₄	—	278	1	no	—
312	C₂×C₄	—	278	1	no	—
313	C₂×C₄	—	556	1	no	—
314	C₂×C₄	—	556	2	no	—
315	C₂×C₄	—	556	2	no	—
316	C₂×C₄	—	556	2	no	—
317	C₂×C₄	—	139	1	no	—
318	C₂×C₄	—	139	1	no	—
319	C₂×C₄	—	139	1	no	—
320	C₂×C₄	—	139	2	no	—
321	C₂×C₄	—	139	2	no	—
322	C₂×C₄	D₄×D₁₃₉	1	1	yes	characteristic
323	C₂×C₄	D₄×D₁₃₉	1	1	yes	characteristic
324	C₂×C₄	C₂³×D₁₃₉	1	1	yes	characteristic
325	C₈	—	2	1	no	—
326	C₈	—	278	1	no	—
327	C₈	—	139	2	no	—
328	C₈	C₂×C₄×D₁₃₉	1	2	yes	—
329	C₂²	—	278	1	no	—
330	C₂²	—	278	1	no	—
331	C₂²	—	278	2	no	—
332	C₂²	—	556	1	no	—
333	C₂²	—	556	1	no	—
334	C₂²	—	139	2	no	—
335	C₂²	C₂×D₄×D₁₃₉	1	1	yes	characteristic
336	C₄	—	2	2	no	—
337	C₄	—	2	2	no	—
338	C₄	—	2	1	no	—
339	C₄	—	4	1	no	—
340	C₄	—	4	1	no	—
341	C₄	—	278	2	no	—
342	C₄	—	278	1	no	—
343	C₄	—	278	1	no	—
344	C₄	—	556	1	no	—
345	C₄	—	556	1	no	—
346	C₄	—	139	2	no	—
347	C₄	#17984874	1	2	yes	direct product factor
348	C₄	C₂²×C₄×D₁₃₉	1	1	yes	characteristic
349	C₄	#41×D₁₃₉	1	1	yes	characteristic
350	C₂	—	278	2	no	—
351	C₂	—	556	2	no	—
352	C₂	C₄×D₄×D₁₃₉	1	1	yes	characteristic
353	C₂	C₂×#17984874	1	1	yes	characteristic
354	C₂	#22581750	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	D₄×D₁₃₉
commutator	C₄×C₁₃₉	C₂³×C₄
Frattini	C₂×C₄	C₂³×D₁₃₉
Fitting	C₄×Q₁₆×C₁₃₉	C₂
socle	C₂²×C₁₃₉	C₂²×D₄
prime radical	C₂×#14555786	C₂²
cosocle	C₂×C₄×C₁₃₉	C₂⁴
hypercenter	C₄×Q₁₆	D₁₃₉
hypocenter	C₁₃₉	C₄×#137
LCS	G ▷ C₄×C₁₃₉ ▷ C₂×C₁₃₉ ▷ C₁₃₉
UCS	C₁ ◁ C₂×C₄ ◁ C₄² ◁ C₄×Q₁₆

Complements

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

N	H ≅ G/N	type
3. C₄×Q₈×D₁₃₉	C₂	semidirect
4. #22581060	C₂	semidirect
5. #22581057	C₂	semidirect
6. #22581056	C₂	semidirect
7. #22581052	C₂	semidirect
8. C₄×#14555775	C₂	semidirect
9. C₄×#14555774	C₂	semidirect
10. C₄×#14555772	C₂	semidirect
11. #22580994	C₂	semidirect
12. #22580960	C₂	semidirect
14. C₄×#14555768	C₂	semidirect
15. C₄×Q₁₆×C₁₃₉	C₂	semidirect
18. #17984874	C₄	direct (N × H)
65. Q₈×D₁₃₉	C₂×C₄	semidirect
76. #14555775	C₂×C₄	semidirect
77. #14555774	C₂×C₄	semidirect
78. #14555772	C₂×C₄	semidirect
80. #14555768	C₂×C₄	semidirect
103. Q₁₆×C₁₃₉	C₂×C₄	semidirect
163. C₄×C₁₃₉	#137	semidirect
177. C₁₃₉	C₄×#137	semidirect
187. C₄×Q₁₆	D₁₃₉	semidirect
257. Q₁₆	C₄×D₁₃₉	semidirect
347. C₄	#17984874	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂×#17984874	2
C₄×#14555786	2
C₄×Q₈×D₁₃₉	2
#22581060	2
#22581057	2
#22581056	2
#22581052	2
C₄×#14555775	2
C₄×#14555774	2
C₄×#14555772	2
#22580994	2
#22580960	2
#22580953	2
C₄×#14555768	2
C₄×Q₁₆×C₁₃₉	2
C₄×#137	139

Subgroups — C4×#17984874 #31815224

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₄×#17984874 #31815224