Subgroups — #30677822

Order 16768.

420 conjugacy classes of subgroups.

The subgroup lattice has 368 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	Q₈×C₂²⋊D₁₃₁	C₂	1	1	yes	maximal, characteristic, complemented
2	#22553963	C₂	1	1	yes	maximal, characteristic
3	#22553661	C₂	1	1	yes	maximal, characteristic, complemented
4	#22553642	C₂	1	1	yes	maximal, characteristic, complemented
5	#22553641	C₂	1	1	yes	maximal, characteristic, complemented
6	#22553595	C₂	1	1	yes	maximal, characteristic, complemented
7	#22553544	C₂	1	1	yes	maximal, characteristic, complemented
8	#22553541	C₂	1	1	yes	maximal, characteristic, complemented
9	#22553516	C₂	1	1	yes	maximal, characteristic, complemented
10	#22553452	C₂	1	1	yes	maximal, characteristic, complemented
11	#22553451	C₂	1	1	yes	maximal, characteristic, complemented
12	#22553397	C₂	1	1	yes	maximal, characteristic, complemented
13	C₈×C₂²⋊D₁₃₁	C₂	1	1	yes	maximal, characteristic, complemented
14	#103×D₁₃₁	C₂	1	1	yes	maximal, characteristic, complemented
15	C₂×#103×C₁₃₁	C₂	1	1	yes	maximal, characteristic, complemented
16	C₂×#14523513	—	2	1	no	—
17	C₂×Q₈×D₁₃₁	C₂²	1	1	yes	characteristic
18	C₂×#14523502	C₂²	1	1	yes	characteristic
19	C₂×#14523500	C₂²	1	1	yes	characteristic
20	C₂×#14523500	C₂²	1	1	yes	characteristic
21	C₂×C₈×D₁₃₁	C₂²	1	1	yes	characteristic
22	#16862245	C₂²	1	1	yes	characteristic
23	#16862244	—	2	1	no	—
24	#16862244	C₂²	1	1	yes	characteristic
25	#16862231	—	2	1	no	—
26	#16862229	—	2	1	no	—
27	#16862225	—	2	1	no	—
28	#16862225	C₂²	1	1	yes	characteristic
29	#16862224	—	2	1	no	—
30	#16862222	—	2	1	no	—
31	#16862222	C₂²	1	1	yes	characteristic
32	#16862221	C₂²	1	1	yes	characteristic
33	#16862220	C₂²	1	1	yes	characteristic
34	Q₈×Dic₁₃₁	C₂²	1	1	yes	characteristic
35	#16862208	C₂²	1	1	yes	characteristic
36	C₄×C₂²⋊D₁₃₁	—	2	1	no	—
37	C₄×C₂²⋊D₁₃₁	C₂²	1	1	yes	characteristic
38	C₄⋊C₄×D₁₃₁	C₂²	1	1	yes	characteristic
39	#16862178	C₂²	1	1	yes	characteristic
40	#16862177	C₂²	1	1	yes	characteristic
41	#16862177	C₂²	1	1	yes	characteristic
42	Dic₁₃₁⋊C₈	C₂²	1	1	yes	characteristic
43	#16862171	C₂²	1	1	yes	characteristic
44	#16862170	C₂²	1	1	yes	characteristic
45	#16862170	C₂²	1	1	yes	characteristic
46	#16862169	C₂²	1	1	yes	characteristic
47	#16862169	C₂²	1	1	yes	characteristic
48	#16862168	C₂²	1	1	yes	characteristic
49	C₂²⋊(C₁₃₁⋊C₈)	C₂²	1	1	yes	characteristic
50	(C₂×D₁₃₁)⋊C₈	C₂²	1	1	yes	characteristic
51	C₈×Dic₁₃₁	C₂²	1	1	yes	characteristic
52	C₂²×Q₈×C₁₃₁	C₂²	1	1	yes	characteristic
53	C₂²×C₈×C₁₃₁	C₂²	1	1	yes	characteristic
54	C₂×C₄⋊C₄×C₁₃₁	C₂²	1	1	yes	characteristic
55	#103×C₁₃₁	C₂²	1	1	yes	characteristic
56	#103×C₁₃₁	C₂²	1	1	yes	characteristic
57	#103×C₁₃₁	C₂²	1	1	yes	characteristic
58	#103×C₁₃₁	C₂²	1	1	yes	characteristic
59	#14523513	—	4	4	no	—
60	C₂×Dic₂₆₂	—	2	1	no	—
61	C₂×Dic₂₆₂	—	2	1	no	—
62	C₂×Dic₂₆₂	C₂³	1	1	yes	characteristic
63	Q₈×D₁₃₁	—	2	2	no	—
64	Q₈×D₁₃₁	—	4	1	no	—
65	C₂×D₅₂₄	—	2	1	no	—
66	C₂×D₅₂₄	C₂³	1	1	yes	characteristic
67	C₂×C₂²⋊D₁₃₁	—	2	1	no	—
68	C₂×C₂²⋊D₁₃₁	D₄	1	1	yes	characteristic
69	C₂×C₄×D₁₃₁	—	2	1	no	—
70	C₂×C₄×D₁₃₁	—	2	1	no	—
71	C₂×C₄×D₁₃₁	—	2	1	no	—
72	C₂×C₄×D₁₃₁	C₂³	1	1	yes	characteristic
73	#14523502	—	4	1	no	—
74	#14523500	—	2	2	no	—
75	#14523500	—	4	1	no	—
76	C₂×C₁₃₁⋊C₈	C₂³	1	1	yes	characteristic
77	C₂×C₁₃₁⋊C₈	C₂³	1	1	yes	characteristic
78	C₈×D₁₃₁	—	4	1	no	—
79	C₄⋊Dic₁₃₁	—	2	1	no	—
80	C₄⋊Dic₁₃₁	C₂³	1	1	yes	characteristic
81	C₄⋊Dic₁₃₁	C₂³	1	1	yes	characteristic
82	Dic₁₃₁⋊C₄	—	2	1	no	—
83	Dic₁₃₁⋊C₄	—	2	1	no	—
84	Dic₁₃₁⋊C₄	—	2	1	no	—
85	Dic₁₃₁⋊C₄	—	2	1	no	—
86	Dic₁₃₁⋊C₄	—	2	1	no	—
87	Dic₁₃₁⋊C₄	—	2	1	no	—
88	Dic₁₃₁⋊C₄	D₄	1	1	yes	characteristic
89	C₂²⋊Dic₁₃₁	—	2	1	no	—
90	C₂²⋊Dic₁₃₁	D₄	1	1	yes	characteristic
91	(C₂×D₁₃₁)⋊C₄	—	2	1	no	—
92	(C₂×D₁₃₁)⋊C₄	—	2	1	no	—
93	(C₂×D₁₃₁)⋊C₄	—	2	1	no	—
94	(C₂×D₁₃₁)⋊C₄	—	2	1	no	—
95	(C₂×D₁₃₁)⋊C₄	D₄	1	1	yes	characteristic
96	C₄×Dic₁₃₁	—	2	1	no	—
97	C₄×Dic₁₃₁	C₂³	1	1	yes	characteristic
98	C₂×Q₈×C₁₃₁	—	2	1	no	—
99	C₂×Q₈×C₁₃₁	—	2	2	no	—
100	C₂×Q₈×C₁₃₁	—	4	1	no	—
101	C₂×Q₈×C₁₃₁	C₂³	1	1	yes	characteristic
102	C₂×Q₈×C₁₃₁	C₂³	1	1	yes	characteristic
103	C₂²×C₄×C₁₃₁	—	2	1	no	—
104	C₂²×C₄×C₁₃₁	—	2	1	no	—
105	C₂²×C₄×C₁₃₁	C₂³	1	1	yes	characteristic
106	C₂×C₈×C₁₃₁	—	4	1	no	—
107	C₂×C₈×C₁₃₁	C₂³	1	1	yes	characteristic
108	C₂×C₈×C₁₃₁	C₂³	1	1	yes	characteristic
109	C₄⋊C₄×C₁₃₁	—	2	1	no	—
110	C₄⋊C₄×C₁₃₁	C₂³	1	1	yes	characteristic
111	C₄⋊C₄×C₁₃₁	C₂³	1	1	yes	characteristic
112	C₂²×D₁₃₁	—	2	1	no	—
113	C₂²×D₁₃₁	C₂×D₄	1	1	yes	characteristic
114	Dic₂₆₂	—	2	1	no	—
115	Dic₂₆₂	—	2	1	no	—
116	Dic₂₆₂	—	4	2	no	—
117	Dic₂₆₂	—	4	2	no	—
118	D₅₂₄	—	2	1	no	—
119	D₅₂₄	—	2	1	no	—
120	D₅₂₄	—	4	2	no	—
121	C₂²⋊D₁₃₁	—	4	2	no	—
122	C₂²⋊D₁₃₁	SD₁₆	1	4	yes	complemented
123	C₂×Dic₁₃₁	—	2	1	no	—
124	C₂×Dic₁₃₁	—	2	1	no	—
125	C₂×Dic₁₃₁	—	2	1	no	—
126	C₂×Dic₁₃₁	C₂×D₄	1	1	yes	characteristic
127	C₂×Dic₁₃₁	C₂×D₄	1	1	yes	characteristic
128	C₂×Dic₁₃₁	C₂×D₄	1	1	yes	characteristic
129	C₄×D₁₃₁	—	2	1	no	—
130	C₄×D₁₃₁	—	2	1	no	—
131	C₄×D₁₃₁	—	4	2	no	—
132	C₄×D₁₃₁	—	4	2	no	—
133	C₄×D₁₃₁	—	4	2	no	—
134	C₁₃₁⋊C₈	—	2	1	no	—
135	C₁₃₁⋊C₈	—	2	1	no	—
136	C₂³×C₁₃₁	C₂×D₄	1	1	yes	characteristic
137	Q₈×C₁₃₁	—	2	1	no	—
138	Q₈×C₁₃₁	—	2	1	no	—
139	Q₈×C₁₃₁	—	2	1	no	—
140	Q₈×C₁₃₁	—	4	2	no	—
141	Q₈×C₁₃₁	#41	1	2	yes	complemented
142	C₂×C₄×C₁₃₁	—	2	1	no	—
143	C₂×C₄×C₁₃₁	—	2	1	no	—
144	C₂×C₄×C₁₃₁	—	2	1	no	—
145	C₂×C₄×C₁₃₁	—	2	1	no	—
146	C₂×C₄×C₁₃₁	—	2	1	no	—
147	C₂×C₄×C₁₃₁	—	2	1	no	—
148	C₂×C₄×C₁₃₁	—	4	2	no	—
149	C₂×C₄×C₁₃₁	—	4	2	no	—
150	C₂×C₄×C₁₃₁	C₂×D₄	1	1	yes	characteristic
151	C₂×C₄×C₁₃₁	C₂⁴	1	1	yes	characteristic
152	C₈×C₁₃₁	—	2	1	no	—
153	C₈×C₁₃₁	—	2	1	no	—
154	C₂²×C₁₃₁	—	2	1	no	—
155	C₂²×C₁₃₁	—	2	1	no	—
156	C₂²×C₁₃₁	C₂×SD₁₆	1	2	yes	—
157	C₂²×C₁₃₁	C₂²×D₄	1	1	yes	characteristic
158	C₂×D₁₃₁	—	2	1	no	—
159	C₂×D₁₃₁	—	2	1	no	—
160	C₂×D₁₃₁	—	4	1	no	—
161	C₂×D₁₃₁	—	4	1	no	—
162	C₂×D₁₃₁	—	4	1	no	—
163	C₂×D₁₃₁	C₂×SD₁₆	1	2	yes	—
164	C₄×C₁₃₁	—	2	1	no	—
165	C₄×C₁₃₁	—	2	2	no	—
166	C₄×C₁₃₁	—	4	1	no	—
167	C₄×C₁₃₁	—	4	1	no	—
168	C₄×C₁₃₁	—	4	1	no	—
169	C₄×C₁₃₁	C₂×#41	1	1	yes	characteristic
170	C₄×C₁₃₁	H(5,𝔽₂)	1	1	yes	characteristic
171	Dic₁₃₁	—	2	1	no	—
172	Dic₁₃₁	—	2	1	no	—
173	Dic₁₃₁	—	4	1	no	—
174	Dic₁₃₁	—	4	1	no	—
175	Dic₁₃₁	—	4	1	no	—
176	Dic₁₃₁	C₂×SD₁₆	1	2	yes	—
177	C₂×C₁₃₁	—	2	2	no	—
178	C₂×C₁₃₁	#546	1	1	yes	characteristic
179	C₂×C₁₃₁	C₂²×SD₁₆	1	1	yes	characteristic
180	C₂×C₁₃₁	#578	1	1	yes	characteristic
181	D₁₃₁	—	2	2	no	—
182	D₁₃₁	—	8	1	no	—
183	C₁₃₁	#3298	1	1	yes	characteristic, complemented, Sylow
184	#3298	—	131	1	no	maximal, Sylow
185	D₄×Q₈	—	131	1	no	—
186	#547	—	131	1	no	—
187	#487	—	131	1	no	—
188	#483	—	131	1	no	—
189	#483	—	131	1	no	—
190	#475	—	131	1	no	—
191	#465	—	131	1	no	—
192	#465	—	131	1	no	—
193	#461	—	131	1	no	—
194	#448	—	131	1	no	—
195	#448	—	131	1	no	—
196	C₄×SD₁₆	—	131	1	no	—
197	C₈×D₄	—	131	1	no	—
198	C₂×#103	—	131	1	no	—
199	C₂×#103	D₁₃₁	1	1	yes	characteristic, complemented
200	C₂×#41	—	262	1	no	—
201	C₂²×Q₈	—	131	1	no	—
202	C₂²×Q₈	C₂×D₁₃₁	1	1	yes	characteristic
203	C₂×SD₁₆	—	131	1	no	—
204	C₂×SD₁₆	—	131	1	no	—
205	C₂×SD₁₆	—	131	1	no	—
206	C₂²×C₈	—	131	1	no	—
207	C₂²×C₈	C₂×D₁₃₁	1	1	yes	characteristic
208	C₄⋊Q₈	—	262	1	no	—
209	C₄⋊Q₈	—	131	1	no	—
210	C₄⋊Q₈	—	131	1	no	—
211	#123	—	262	1	no	—
212	#123	—	262	1	no	—
213	C₂²⋊Q₈	—	262	1	no	—
214	C₂²⋊Q₈	—	262	1	no	—
215	C₂²⋊Q₈	—	262	1	no	—
216	C₂²⋊Q₈	—	131	1	no	—
217	C₂²⋊Q₈	—	131	1	no	—
218	#121	—	131	1	no	—
219	#121	—	131	1	no	—
220	C₄×Q₈	—	131	1	no	—
221	C₄×D₄	—	262	1	no	—
222	C₄×D₄	—	131	1	no	—
223	C₄×D₄	—	131	1	no	—
224	C₂×C₄⋊C₄	—	131	1	no	—
225	C₂×C₄⋊C₄	C₂×D₁₃₁	1	1	yes	characteristic
226	#106	—	131	1	no	—
227	#106	—	131	1	no	—
228	#106	—	131	1	no	—
229	C₄⋊C₈	—	131	1	no	—
230	#103	—	131	1	no	—
231	#103	—	131	1	no	—
232	#103	—	131	1	no	—
233	#103	—	131	1	no	—
234	#103	—	131	1	no	—
235	#103	C₂×D₁₃₁	1	1	yes	characteristic
236	#103	C₂×D₁₃₁	1	1	yes	characteristic
237	#103	C₂×D₁₃₁	1	1	yes	characteristic
238	#103	C₂×D₁₃₁	1	1	yes	characteristic
239	#102	—	131	1	no	—
240	C₂²⋊C₈	—	131	1	no	—
241	C₂²⋊C₈	—	131	1	no	—
242	C₄×C₈	—	131	1	no	—
243	#41	—	524	4	no	—
244	C₂×Q₈	—	2	1	no	—
245	C₂×Q₈	—	2	2	no	—
246	C₂×Q₈	—	4	1	no	—
247	C₂×Q₈	—	262	1	no	—
248	C₂×Q₈	—	262	1	no	—
249	C₂×Q₈	—	262	2	no	—
250	C₂×Q₈	—	524	1	no	—
251	C₂×Q₈	—	131	1	no	—
252	C₂×Q₈	C₂²×D₁₃₁	1	1	yes	characteristic
253	C₂×Q₈	C₂²×D₁₃₁	1	1	yes	characteristic
254	C₂×D₄	—	262	1	no	—
255	C₂×D₄	—	262	1	no	—
256	C₂×D₄	—	131	1	no	—
257	C₂×D₄	—	131	1	no	—
258	C₂²×C₄	—	2	1	no	—
259	C₂²×C₄	—	2	1	no	—
260	C₂²×C₄	—	262	1	no	—
261	C₂²×C₄	—	262	1	no	—
262	C₂²×C₄	—	262	1	no	—
263	C₂²×C₄	—	131	1	no	—
264	C₂²×C₄	C₂²×D₁₃₁	1	1	yes	characteristic
265	SD₁₆	—	262	2	no	—
266	SD₁₆	—	524	1	no	—
267	SD₁₆	—	524	1	no	—
268	C₂×C₈	—	4	1	no	—
269	C₂×C₈	—	524	1	no	—
270	C₂×C₈	—	131	1	no	—
271	C₂×C₈	—	131	1	no	—
272	C₂×C₈	C₂²×D₁₃₁	1	1	yes	characteristic
273	C₂×C₈	C₂²×D₁₃₁	1	1	yes	characteristic
274	C₄⋊C₄	—	2	1	no	—
275	C₄⋊C₄	—	262	1	no	—
276	C₄⋊C₄	—	262	1	no	—
277	C₄⋊C₄	—	262	1	no	—
278	C₄⋊C₄	—	262	1	no	—
279	C₄⋊C₄	—	262	1	no	—
280	C₄⋊C₄	—	262	1	no	—
281	C₄⋊C₄	—	262	1	no	—
282	C₄⋊C₄	—	131	1	no	—
283	C₄⋊C₄	—	131	1	no	—
284	C₄⋊C₄	—	131	1	no	—
285	C₄⋊C₄	C₂²×D₁₃₁	1	1	yes	characteristic
286	C₄⋊C₄	C₂²×D₁₃₁	1	1	yes	characteristic
287	C₂²⋊C₄	—	262	1	no	—
288	C₂²⋊C₄	—	262	1	no	—
289	C₂²⋊C₄	—	262	1	no	—
290	C₂²⋊C₄	—	262	1	no	—
291	C₂²⋊C₄	—	262	1	no	—
292	C₂²⋊C₄	—	131	1	no	—
293	C₂²⋊C₄	—	131	1	no	—
294	C₄²	—	262	1	no	—
295	C₄²	—	131	1	no	—
296	C₂³	—	262	1	no	—
297	C₂³	—	131	1	no	—
298	C₂³	D₄×D₁₃₁	1	1	yes	characteristic
299	Q₈	—	2	1	no	—
300	Q₈	—	2	1	no	—
301	Q₈	—	2	1	no	—
302	Q₈	—	4	2	no	—
303	Q₈	—	262	1	no	—
304	Q₈	—	262	1	no	—
305	Q₈	—	524	2	no	—
306	Q₈	—	524	2	no	—
307	Q₈	#14523513	1	2	yes	complemented
308	D₄	—	262	1	no	—
309	D₄	—	262	1	no	—
310	D₄	—	524	2	no	—
311	D₄	—	524	2	no	—
312	D₄	—	131	4	no	—
313	C₂×C₄	—	2	1	no	—
314	C₂×C₄	—	2	1	no	—
315	C₂×C₄	—	2	1	no	—
316	C₂×C₄	—	2	1	no	—
317	C₂×C₄	—	2	1	no	—
318	C₂×C₄	—	2	1	no	—
319	C₂×C₄	—	4	2	no	—
320	C₂×C₄	—	4	2	no	—
321	C₂×C₄	—	262	1	no	—
322	C₂×C₄	—	262	1	no	—
323	C₂×C₄	—	262	1	no	—
324	C₂×C₄	—	262	1	no	—
325	C₂×C₄	—	262	1	no	—
326	C₂×C₄	—	524	2	no	—
327	C₂×C₄	—	524	2	no	—
328	C₂×C₄	—	524	2	no	—
329	C₂×C₄	—	131	1	no	—
330	C₂×C₄	—	131	1	no	—
331	C₂×C₄	—	131	1	no	—
332	C₂×C₄	D₄×D₁₃₁	1	1	yes	characteristic
333	C₂×C₄	C₂³×D₁₃₁	1	1	yes	characteristic
334	C₈	—	2	1	no	—
335	C₈	—	2	1	no	—
336	C₈	—	262	1	no	—
337	C₈	—	262	1	no	—
338	C₂²	—	2	1	no	—
339	C₂²	—	2	1	no	—
340	C₂²	—	262	1	no	—
341	C₂²	—	262	1	no	—
342	C₂²	—	524	1	no	—
343	C₂²	—	524	1	no	—
344	C₂²	—	524	1	no	—
345	C₂²	—	131	2	no	—
346	C₂²	SD₁₆×D₁₃₁	1	2	yes	—
347	C₂²	C₂×D₄×D₁₃₁	1	1	yes	characteristic
348	C₄	—	2	1	no	—
349	C₄	—	2	2	no	—
350	C₄	—	4	1	no	—
351	C₄	—	4	1	no	—
352	C₄	—	4	1	no	—
353	C₄	—	262	1	no	—
354	C₄	—	262	1	no	—
355	C₄	—	524	1	no	—
356	C₄	—	524	1	no	—
357	C₄	—	524	1	no	—
358	C₄	—	131	2	no	—
359	C₄	C₂×#14523513	1	1	yes	characteristic
360	C₄	#16862291	1	1	yes	characteristic
361	C₂	—	2	2	no	—
362	C₂	—	262	2	no	—
363	C₂	—	1048	1	no	—
364	C₂	#22553952	1	1	yes	characteristic
365	C₂	C₂×SD₁₆×D₁₃₁	1	1	yes	characteristic
366	C₂	#22554125	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×D₄×D₁₃₁
commutator, cosocle	C₂×C₄×C₁₃₁	C₂⁴
Frattini	C₂×C₄	C₂³×D₁₃₁
Fitting	C₂×#103×C₁₃₁	C₂
socle	C₂²×C₁₃₁	C₂²×D₄
prime radical	#22553963	C₂
hypercenter	C₂×#103	D₁₃₁
hypocenter	C₁₃₁	#3298
LCS	G ▷ C₂×C₄×C₁₃₁ ▷ C₂×C₁₃₁ ▷ C₁₃₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×#103

Complements

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

N	H ≅ G/N	type
1. Q₈×C₂²⋊D₁₃₁	C₂	semidirect
3. #22553661	C₂	semidirect
4. #22553642	C₂	semidirect
5. #22553641	C₂	semidirect
6. #22553595	C₂	semidirect
7. #22553544	C₂	semidirect
8. #22553541	C₂	semidirect
9. #22553516	C₂	semidirect
10. #22553452	C₂	semidirect
11. #22553451	C₂	semidirect
12. #22553397	C₂	semidirect
13. C₈×C₂²⋊D₁₃₁	C₂	semidirect
14. #103×D₁₃₁	C₂	semidirect
15. C₂×#103×C₁₃₁	C₂	semidirect
122. C₂²⋊D₁₃₁	SD₁₆	semidirect
141. Q₈×C₁₃₁	#41	semidirect
183. C₁₃₁	#3298	semidirect
199. C₂×#103	D₁₃₁	semidirect
307. Q₈	#14523513	semidirect

Maximal subgroups

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

subgroup	index
Q₈×C₂²⋊D₁₃₁	2
#22553963	2
#22553661	2
#22553642	2
#22553641	2
#22553595	2
#22553544	2
#22553541	2
#22553516	2
#22553452	2
#22553451	2
#22553397	2
C₈×C₂²⋊D₁₃₁	2
#103×D₁₃₁	2
C₂×#103×C₁₃₁	2
#3298	131