Subgroups — #31816121

Order 17792.

452 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	#22581751	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×#17984862	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×#17984849	C₂	1	1	yes	maximal, characteristic, complemented
4	#22581226	C₂	1	2	yes	maximal, complemented
5	#22581221	C₂	1	2	yes	maximal, complemented
6	#22581218	C₂	1	2	yes	maximal, complemented
7	#22581214	C₂	1	2	yes	maximal, complemented
8	C₂×#17984777	C₂	1	1	yes	maximal, characteristic, complemented
9	#22580937	C₂	1	1	yes	maximal, characteristic, complemented
10	#22580935	C₂	1	1	yes	maximal, characteristic, complemented
11	#575×C₁₃₉	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×#14555785	C₂²	1	1	yes	characteristic
13	#17984873	—	2	1	no	—
14	#17984872	C₂²	1	2	yes	—
15	#17984868	—	2	1	no	—
16	#17984867	C₂²	1	2	yes	—
17	#17984862	—	2	1	no	—
18	#17984862	—	2	1	no	—
19	#17984862	—	2	1	no	—
20	#17984862	—	2	1	no	—
21	#17984862	—	2	2	no	—
22	C₂×#14555775	C₂²	1	1	yes	characteristic, complemented
23	C₂×#14555773	C₂²	1	1	yes	characteristic, complemented
24	C₂×#14555772	C₂²	1	1	yes	characteristic, complemented
25	C₂×#14555770	C₂²	1	1	yes	characteristic, complemented
26	#17984849	—	2	1	no	—
27	#17984849	—	2	1	no	—
28	#17984849	C₂²	1	2	yes	—
29	#17984849	C₂²	1	2	yes	—
30	C₂×#14555769	—	2	1	no	—
31	C₂×#14555769	C₂²	1	1	yes	characteristic
32	C₂×#14555769	C₂²	1	1	yes	characteristic
33	C₂²×C₁₃₉⋊C₈	—	2	1	no	—
34	C₂²×C₁₃₉⋊C₈	C₂²	1	1	yes	characteristic
35	#17984777	C₂²	1	2	yes	complemented
36	#17984777	C₂²	1	2	yes	complemented
37	#17984761	C₂²	1	2	yes	complemented
38	#17984760	C₂²	1	2	yes	complemented
39	#17984759	C₂²	1	2	yes	complemented
40	#17984758	C₂²	1	2	yes	complemented
41	C₂×#41×C₁₃₉	C₂²	1	1	yes	characteristic
42	C₂×#41×C₁₃₉	C₂²	1	1	yes	characteristic
43	#137×C₁₃₉	—	2	1	no	—
44	#137×C₁₃₉	C₂²	1	2	yes	—
45	AGL(1,ℤ/8)×C₁₃₉	—	2	1	no	—
46	AGL(1,ℤ/8)×C₁₃₉	C₂²	1	2	yes	—
47	#135×C₁₃₉	—	2	2	no	—
48	C₂×M₁₆×C₁₃₉	C₂²	1	1	yes	characteristic
49	#14555785	—	2	1	no	—
50	#14555785	—	2	1	no	—
51	#14555785	—	2	1	no	—
52	#14555785	—	2	1	no	—
53	C₂×Dic₂₇₈	C₂³	1	1	yes	characteristic
54	C₂×D₅₅₆	C₂³	1	1	yes	characteristic
55	C₂×C₂²⋊D₁₃₉	—	2	1	no	—
56	C₂×C₄×D₁₃₉	—	2	1	no	—
57	#14555775	—	2	1	no	—
58	#14555775	—	2	1	no	—
59	#14555775	—	2	2	no	—
60	#14555773	—	2	1	no	—
61	#14555773	—	2	1	no	—
62	#14555773	—	2	2	no	—
63	#14555772	—	2	1	no	—
64	#14555772	—	2	1	no	—
65	#14555772	—	2	2	no	—
66	#14555770	—	2	1	no	—
67	#14555770	—	2	1	no	—
68	#14555770	—	2	2	no	—
69	#14555769	—	2	2	no	—
70	#14555769	—	2	2	no	—
71	#14555769	—	2	1	no	—
72	#14555769	—	2	1	no	—
73	#14555769	C₂³	1	2	yes	—
74	#14555769	C₂³	1	2	yes	—
75	C₂×C₁₃₉⋊C₈	—	2	2	no	—
76	C₂×C₁₃₉⋊C₈	—	2	2	no	—
77	C₂×C₁₃₉⋊C₈	—	2	1	no	—
78	C₂×C₁₃₉⋊C₈	—	2	1	no	—
79	C₂×C₁₃₉⋊C₈	D₄	1	1	yes	characteristic
80	C₂×C₁₃₉⋊C₈	D₄	1	1	yes	characteristic
81	C₂×C₁₃₉⋊C₈	D₄	1	1	yes	characteristic
82	C₂×C₁₃₉⋊C₈	D₄	1	1	yes	characteristic
83	C₂×C₁₃₉⋊C₈	C₂³	1	1	yes	characteristic
84	C₂×C₁₃₉⋊C₈	C₂³	1	1	yes	characteristic
85	#41×C₁₃₉	—	2	1	no	—
86	#41×C₁₃₉	—	2	1	no	—
87	#41×C₁₃₉	—	2	1	no	—
88	#41×C₁₃₉	—	2	1	no	—
89	#41×C₁₃₉	—	2	1	no	—
90	#41×C₁₃₉	—	2	1	no	—
91	#41×C₁₃₉	D₄	1	2	yes	—
92	#41×C₁₃₉	C₂³	1	2	yes	—
93	C₂×Q₈×C₁₃₉	D₄	1	1	yes	characteristic
94	C₂×Q₈×C₁₃₉	C₂³	1	1	yes	characteristic
95	C₂×D₄×C₁₃₉	—	2	1	no	—
96	C₂×D₄×C₁₃₉	—	2	1	no	—
97	C₂×D₄×C₁₃₉	D₄	1	1	yes	characteristic
98	C₂×D₄×C₁₃₉	C₂³	1	1	yes	characteristic
99	C₂²×C₄×C₁₃₉	—	2	1	no	—
100	C₂²×C₄×C₁₃₉	—	2	1	no	—
101	C₂²×C₄×C₁₃₉	C₂³	1	1	yes	characteristic
102	Q₁₆×C₁₃₉	—	2	2	no	—
103	SD₁₆×C₁₃₉	—	2	2	no	—
104	SD₁₆×C₁₃₉	—	2	2	no	—
105	D₈×C₁₃₉	—	2	2	no	—
106	M₁₆×C₁₃₉	—	2	1	no	—
107	M₁₆×C₁₃₉	C₂³	1	2	yes	—
108	C₂×C₈×C₁₃₉	—	2	1	no	—
109	C₂²×D₁₃₉	—	2	1	no	—
110	Dic₂₇₈	—	2	1	no	—
111	Dic₂₇₈	—	2	1	no	—
112	D₅₅₆	—	2	1	no	—
113	D₅₅₆	—	2	1	no	—
114	C₂²⋊D₁₃₉	—	4	1	no	—
115	C₂²⋊D₁₃₉	—	4	1	no	—
116	C₂×Dic₁₃₉	—	2	1	no	—
117	C₄×D₁₃₉	—	4	1	no	—
118	C₄×D₁₃₉	—	4	1	no	—
119	C₁₃₉⋊C₈	—	2	2	no	—
120	C₁₃₉⋊C₈	—	2	2	no	—
121	C₁₃₉⋊C₈	C₂×D₄	1	1	yes	characteristic
122	C₁₃₉⋊C₈	C₂×D₄	1	1	yes	characteristic
123	C₁₃₉⋊C₈	C₂×D₄	1	1	yes	characteristic
124	C₁₃₉⋊C₈	C₂×D₄	1	1	yes	characteristic
125	C₂³×C₁₃₉	—	2	1	no	—
126	C₂³×C₁₃₉	—	2	1	no	—
127	C₂³×C₁₃₉	#41	1	1	yes	characteristic
128	Q₈×C₁₃₉	—	2	1	no	—
129	Q₈×C₁₃₉	—	2	1	no	—
130	Q₈×C₁₃₉	—	2	1	no	—
131	Q₈×C₁₃₉	C₂×D₄	1	2	yes	—
132	D₄×C₁₃₉	—	2	2	no	—
133	D₄×C₁₃₉	—	2	2	no	—
134	D₄×C₁₃₉	—	2	1	no	—
135	D₄×C₁₃₉	—	2	1	no	—
136	D₄×C₁₃₉	—	2	1	no	—
137	D₄×C₁₃₉	—	4	1	no	—
138	D₄×C₁₃₉	—	4	1	no	—
139	D₄×C₁₃₉	C₂×D₄	1	2	yes	—
140	C₂×C₄×C₁₃₉	—	2	1	no	—
141	C₂×C₄×C₁₃₉	—	2	2	no	—
142	C₂×C₄×C₁₃₉	—	2	2	no	—
143	C₂×C₄×C₁₃₉	—	2	1	no	—
144	C₂×C₄×C₁₃₉	—	4	1	no	—
145	C₂×C₄×C₁₃₉	—	4	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	C₂×D₄	1	1	yes	characteristic
149	C₂×C₄×C₁₃₉	C₂×D₄	1	1	yes	characteristic
150	C₂×C₄×C₁₃₉	#41	1	1	yes	characteristic
151	C₂×C₄×C₁₃₉	C₂⁴	1	1	yes	characteristic
152	C₈×C₁₃₉	—	2	2	no	—
153	C₂²×C₁₃₉	—	2	2	no	—
154	C₂²×C₁₃₉	—	4	1	no	—
155	C₂²×C₁₃₉	—	4	2	no	—
156	C₂²×C₁₃₉	—	4	1	no	—
157	C₂²×C₁₃₉	—	8	1	no	—
158	C₂²×C₁₃₉	#121	1	1	yes	characteristic
159	C₂²×C₁₃₉	#121	1	1	yes	characteristic
160	C₂²×C₁₃₉	C₂×#41	1	1	yes	characteristic
161	C₂×D₁₃₉	—	4	1	no	—
162	C₂×D₁₃₉	—	8	1	no	—
163	C₄×C₁₃₉	—	2	2	no	—
164	C₄×C₁₃₉	—	4	1	no	—
165	C₄×C₁₃₉	#121	1	1	yes	characteristic
166	C₄×C₁₃₉	#121	1	1	yes	characteristic
167	C₄×C₁₃₉	C₂²×D₄	1	1	yes	characteristic
168	C₄×C₁₃₉	C₂²×D₄	1	1	yes	characteristic
169	Dic₁₃₉	—	4	1	no	—
170	C₂×C₁₃₉	—	2	1	no	—
171	C₂×C₁₃₉	—	2	1	no	—
172	C₂×C₁₃₉	—	2	1	no	—
173	C₂×C₁₃₉	—	4	2	no	—
174	C₂×C₁₃₉	—	8	1	no	—
175	C₂×C₁₃₉	C₂×#121	1	1	yes	characteristic
176	D₁₃₉	—	8	1	no	—
177	C₁₃₉	#3067	1	1	yes	characteristic, complemented, Sylow
178	#3067	—	139	1	no	maximal, Sylow
179	#575	—	139	1	no	—
180	#575	D₁₃₉	1	1	yes	characteristic, complemented
181	C₂×#135	—	139	1	no	—
182	C₂×#131	—	139	1	no	—
183	#473	—	139	2	no	—
184	#472	—	139	2	no	—
185	#471	—	139	2	no	—
186	#471	—	139	2	no	—
187	C₂×#108	—	139	1	no	—
188	#413	—	139	1	no	—
189	#413	—	139	1	no	—
190	C₂×#41	—	139	1	no	—
191	C₂×#41	C₂×D₁₃₉	1	1	yes	characteristic
192	C₂×#41	C₂×D₁₃₉	1	1	yes	characteristic
193	#137	—	2	1	no	—
194	#137	—	278	1	no	—
195	#137	—	139	2	no	—
196	#137	C₂×D₁₃₉	1	2	yes	—
197	AGL(1,ℤ/8)	—	2	1	no	—
198	AGL(1,ℤ/8)	—	278	1	no	—
199	AGL(1,ℤ/8)	—	139	2	no	—
200	AGL(1,ℤ/8)	C₂×D₁₃₉	1	2	yes	—
201	#135	—	2	2	no	—
202	#135	—	278	2	no	—
203	#135	—	278	1	no	—
204	#135	—	278	1	no	—
205	#135	—	278	1	no	—
206	#135	—	278	1	no	—
207	C₂×Q₁₆	—	139	1	no	—
208	C₂×SD₁₆	—	139	1	no	—
209	C₂×SD₁₆	—	139	1	no	—
210	C₂×D₈	—	139	1	no	—
211	#131	—	278	1	no	—
212	#131	—	278	1	no	—
213	#131	—	139	2	no	—
214	#131	—	139	2	no	—
215	C₂×M₁₆	—	278	1	no	—
216	C₂×M₁₆	—	139	1	no	—
217	C₂×M₁₆	—	139	1	no	—
218	C₂×M₁₆	C₂×D₁₃₉	1	1	yes	characteristic
219	C₂²×C₈	—	278	1	no	—
220	C₂²×C₈	—	139	1	no	—
221	#108	—	139	2	no	—
222	#108	—	139	2	no	—
223	#101	—	139	2	no	—
224	#101	—	139	2	no	—
225	#100	—	139	2	no	—
226	#100	—	139	2	no	—
227	#41	—	2	1	no	—
228	#41	—	2	1	no	—
229	#41	—	2	1	no	—
230	#41	—	2	1	no	—
231	#41	—	2	1	no	—
232	#41	—	2	1	no	—
233	#41	—	278	1	no	—
234	#41	—	278	1	no	—
235	#41	—	278	1	no	—
236	#41	—	278	1	no	—
237	#41	C₂²⋊D₁₃₉	1	2	yes	—
238	#41	C₂²×D₁₃₉	1	2	yes	—
239	C₂×Q₈	—	139	1	no	—
240	C₂×Q₈	C₂²⋊D₁₃₉	1	1	yes	characteristic
241	C₂×Q₈	C₂²×D₁₃₉	1	1	yes	characteristic
242	C₂×D₄	—	2	1	no	—
243	C₂×D₄	—	2	1	no	—
244	C₂×D₄	—	278	1	no	—
245	C₂×D₄	—	139	1	no	—
246	C₂×D₄	C₂²⋊D₁₃₉	1	1	yes	characteristic
247	C₂×D₄	C₂²×D₁₃₉	1	1	yes	characteristic
248	C₂²×C₄	—	2	1	no	—
249	C₂²×C₄	—	2	1	no	—
250	C₂²×C₄	—	278	1	no	—
251	C₂²×C₄	C₂²×D₁₃₉	1	1	yes	characteristic
252	Q₁₆	—	2	2	no	—
253	Q₁₆	—	278	2	no	—
254	Q₁₆	—	278	1	no	—
255	Q₁₆	—	278	1	no	—
256	SD₁₆	—	2	2	no	—
257	SD₁₆	—	2	2	no	—
258	SD₁₆	—	278	2	no	—
259	SD₁₆	—	278	2	no	—
260	SD₁₆	—	278	1	no	—
261	SD₁₆	—	278	1	no	—
262	SD₁₆	—	278	1	no	—
263	SD₁₆	—	278	1	no	—
264	D₈	—	2	2	no	—
265	D₈	—	278	2	no	—
266	D₈	—	278	1	no	—
267	D₈	—	278	1	no	—
268	M₁₆	—	2	1	no	—
269	M₁₆	—	278	1	no	—
270	M₁₆	—	278	1	no	—
271	M₁₆	—	278	2	no	—
272	M₁₆	—	278	2	no	—
273	M₁₆	—	139	2	no	—
274	M₁₆	—	139	2	no	—
275	M₁₆	C₂²×D₁₃₉	1	2	yes	—
276	C₂×C₈	—	2	1	no	—
277	C₂×C₈	—	278	1	no	—
278	C₂×C₈	—	278	1	no	—
279	C₂×C₈	—	278	2	no	—
280	C₂×C₈	—	278	2	no	—
281	C₂×C₈	—	139	1	no	—
282	C₂×C₈	—	139	1	no	—
283	C₂×C₈	—	139	1	no	—
284	C₂×C₈	—	139	1	no	—
285	C₂×C₈	—	139	1	no	—
286	C₂×C₈	—	139	1	no	—
287	C₂³	—	2	1	no	—
288	C₂³	—	2	1	no	—
289	C₂³	—	278	1	no	—
290	C₂³	#14555784	1	1	yes	characteristic
291	Q₈	—	2	1	no	—
292	Q₈	—	2	1	no	—
293	Q₈	—	2	1	no	—
294	Q₈	—	278	1	no	—
295	Q₈	—	278	1	no	—
296	Q₈	C₂×C₂²⋊D₁₃₉	1	2	yes	—
297	D₄	—	2	2	no	—
298	D₄	—	2	2	no	—
299	D₄	—	2	1	no	—
300	D₄	—	2	1	no	—
301	D₄	—	2	1	no	—
302	D₄	—	4	1	no	—
303	D₄	—	4	1	no	—
304	D₄	—	278	1	no	—
305	D₄	—	278	1	no	—
306	D₄	—	556	1	no	—
307	D₄	—	556	1	no	—
308	D₄	C₂×C₂²⋊D₁₃₉	1	2	yes	—
309	C₂×C₄	—	2	1	no	—
310	C₂×C₄	—	2	2	no	—
311	C₂×C₄	—	2	2	no	—
312	C₂×C₄	—	2	1	no	—
313	C₂×C₄	—	4	1	no	—
314	C₂×C₄	—	4	1	no	—
315	C₂×C₄	—	278	1	no	—
316	C₂×C₄	—	556	1	no	—
317	C₂×C₄	—	556	1	no	—
318	C₂×C₄	D₄×D₁₃₉	1	1	yes	characteristic
319	C₂×C₄	D₄×D₁₃₉	1	1	yes	characteristic
320	C₂×C₄	C₂×C₂²⋊D₁₃₉	1	1	yes	characteristic
321	C₂×C₄	C₂×C₂²⋊D₁₃₉	1	1	yes	characteristic
322	C₂×C₄	#14555784	1	1	yes	characteristic
323	C₂×C₄	C₂³×D₁₃₉	1	1	yes	characteristic
324	C₈	—	2	2	no	—
325	C₈	—	278	2	no	—
326	C₈	—	278	2	no	—
327	C₈	—	139	1	no	—
328	C₈	—	139	1	no	—
329	C₈	—	139	1	no	—
330	C₈	—	139	1	no	—
331	C₂²	—	2	2	no	—
332	C₂²	—	4	1	no	—
333	C₂²	—	4	2	no	—
334	C₂²	—	4	1	no	—
335	C₂²	—	8	1	no	—
336	C₂²	—	556	1	no	—
337	C₂²	—	1112	1	no	—
338	C₂²	#17984814	1	1	yes	characteristic
339	C₂²	#17984814	1	1	yes	characteristic
340	C₂²	C₂×#14555784	1	1	yes	characteristic
341	C₄	—	2	2	no	—
342	C₄	—	4	1	no	—
343	C₄	—	556	1	no	—
344	C₄	#17984814	1	1	yes	characteristic
345	C₄	#17984814	1	1	yes	characteristic
346	C₄	C₂×D₄×D₁₃₉	1	1	yes	characteristic
347	C₄	C₂²×C₂²⋊D₁₃₉	1	1	yes	characteristic
348	C₂	—	2	1	no	—
349	C₂	—	2	1	no	—
350	C₂	—	2	1	no	—
351	C₂	—	4	2	no	—
352	C₂	—	8	1	no	—
353	C₂	—	1112	1	no	—
354	C₂	C₂×#17984814	1	1	yes	characteristic

	subgroup	G/H
center	C₄	#17984814
commutator, cosocle	C₂×C₄×C₁₃₉	C₂⁴
Frattini	C₂×C₄	C₂³×D₁₃₉
Fitting	#575×C₁₃₉	C₂
socle	C₂×C₁₃₉	C₂×#121
hypercenter	#575	D₁₃₉
hypocenter	C₁₃₉	#3067
LCS	G ▷ C₂×C₄×C₁₃₉ ▷ C₂×C₁₃₉ ▷ C₁₃₉
UCS	C₁ ◁ C₄ ◁ C₂²×C₄ ◁ #575

Complements

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

N	H ≅ G/N	type
1. #22581751	C₂	semidirect
2. C₂×#17984862	C₂	semidirect
3. C₂×#17984849	C₂	semidirect
4. #22581226	C₂	semidirect
5. #22581221	C₂	semidirect
6. #22581218	C₂	semidirect
7. #22581214	C₂	semidirect
8. C₂×#17984777	C₂	semidirect
9. #22580937	C₂	semidirect
10. #22580935	C₂	semidirect
11. #575×C₁₃₉	C₂	semidirect
22. C₂×#14555775	C₂²	semidirect
23. C₂×#14555773	C₂²	semidirect
24. C₂×#14555772	C₂²	semidirect
25. C₂×#14555770	C₂²	semidirect
35. #17984777	C₂²	semidirect
36. #17984777	C₂²	semidirect
37. #17984761	C₂²	semidirect
38. #17984760	C₂²	semidirect
39. #17984759	C₂²	semidirect
40. #17984758	C₂²	semidirect
177. C₁₃₉	#3067	semidirect
180. #575	D₁₃₉	semidirect

Maximal subgroups

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

subgroup	index
#22581751	2
C₂×#17984862	2
C₂×#17984849	2
#22581226	2
#22581221	2
#22581218	2
#22581214	2
C₂×#17984777	2
#22580937	2
#22580935	2
#575×C₁₃₉	2
#3067	139