Subgroups — #30675787

Order 16768.

398 conjugacy classes of subgroups.

The subgroup lattice has 346 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#22553939	C₂	1	1	yes	maximal, characteristic, complemented
2	#22553784	C₂	1	1	yes	maximal, characteristic
3	#22553614	C₂	1	1	yes	maximal, characteristic, complemented
4	#22553613	C₂	1	1	yes	maximal, characteristic, complemented
5	#22553612	C₂	1	1	yes	maximal, characteristic, complemented
6	#22553611	C₂	1	1	yes	maximal, characteristic, complemented
7	#22553601	C₂	1	1	yes	maximal, characteristic, complemented
8	#22553590	C₂	1	1	yes	maximal, characteristic, complemented
9	#22553589	C₂	1	1	yes	maximal, characteristic, complemented
10	#22553351	C₂	1	1	yes	maximal, characteristic, complemented
11	#22553347	C₂	1	1	yes	maximal, characteristic, complemented
12	#22553337	C₂	1	1	yes	maximal, characteristic, complemented
13	#22553315	C₂	1	1	yes	maximal, characteristic
14	#102×D₁₃₁	C₂	1	1	yes	maximal, characteristic
15	#476×C₁₃₁	C₂	1	1	yes	maximal, characteristic, complemented
16	C₂×D₄×D₁₃₁	C₂²	1	1	yes	characteristic
17	C₂×#14523496	C₂²	1	1	yes	characteristic
18	C₂×C₈×D₁₃₁	C₂²	1	1	yes	characteristic
19	#16862245	C₂²	1	1	yes	characteristic, complemented
20	#16862244	—	2	1	no	—
21	#16862238	C₂²	1	1	yes	characteristic, complemented
22	#16862237	C₂²	1	1	yes	characteristic, complemented
23	#16862236	—	2	1	no	—
24	#16862224	—	2	1	no	—
25	#16862222	—	2	1	no	—
26	C₄×D₅₂₄	C₂²	1	1	yes	characteristic
27	D₄×Dic₁₃₁	C₂²	1	1	yes	characteristic
28	#16862208	C₂²	1	1	yes	characteristic
29	C₄×C₂²⋊D₁₃₁	—	2	1	no	—
30	(C₂²⋊D₁₃₁)⋊C₄	—	2	1	no	—
31	#16862204	C₂²	1	1	yes	characteristic
32	#16862204	C₂²	1	1	yes	characteristic
33	#16862203	C₂²	1	1	yes	characteristic
34	C₄⋊C₄×D₁₃₁	C₂²	1	1	yes	characteristic
35	C₂×(C₂×D₁₃₁)⋊C₄	—	2	1	no	—
36	C₂²⋊C₄×D₁₃₁	—	2	1	no	—
37	#16862180	C₂²	1	1	yes	characteristic, complemented
38	#16862179	C₂²	1	1	yes	characteristic, complemented
39	#16862179	C₂²	1	1	yes	characteristic, complemented
40	#16862179	C₂²	1	1	yes	characteristic, complemented
41	#16862178	C₂²	1	1	yes	characteristic, complemented
42	#16862177	C₂²	1	1	yes	characteristic, complemented
43	C₄⋊(C₁₃₁⋊C₈)	C₂²	1	1	yes	characteristic, complemented
44	Dic₁₃₁⋊C₈	C₂²	1	1	yes	characteristic, complemented
45	Dic₁₃₁⋊C₈	C₂²	1	1	yes	characteristic, complemented
46	#16862168	C₂²	1	1	yes	characteristic
47	#16862168	C₂²	1	1	yes	characteristic
48	#16862167	C₂²	1	1	yes	characteristic
49	#16862167	C₂²	1	1	yes	characteristic
50	#16862166	C₂²	1	1	yes	characteristic
51	#16862166	C₂²	1	1	yes	characteristic
52	C₄⋊Q₈×C₁₃₁	C₂²	1	1	yes	characteristic, complemented
53	C₄×D₄×C₁₃₁	C₂²	1	1	yes	characteristic
54	#106×C₁₃₁	C₂²	1	1	yes	characteristic, complemented
55	#106×C₁₃₁	C₂²	1	1	yes	characteristic, complemented
56	C₄⋊C₈×C₁₃₁	C₂²	1	1	yes	characteristic, complemented
57	#102×C₁₃₁	C₂²	1	1	yes	characteristic
58	#102×C₁₃₁	C₂²	1	1	yes	characteristic
59	C₂³×D₁₃₁	—	2	1	no	—
60	C₂×Dic₂₆₂	—	2	1	no	—
61	C₂×D₅₂₄	C₂³	1	1	yes	characteristic
62	C₂×C₂²⋊D₁₃₁	—	2	1	no	—
63	D₄×D₁₃₁	—	2	2	no	—
64	D₄×D₁₃₁	Q₈	1	4	yes	complemented
65	C₂²×Dic₁₃₁	—	2	1	no	—
66	C₂×C₄×D₁₃₁	—	2	1	no	—
67	C₂×C₄×D₁₃₁	—	2	1	no	—
68	C₂×C₄×D₁₃₁	—	2	1	no	—
69	C₂×C₄×D₁₃₁	C₂³	1	1	yes	characteristic
70	#14523496	—	2	2	no	—
71	C₂×C₁₃₁⋊C₈	C₂³	1	1	yes	characteristic
72	C₂×C₁₃₁⋊C₈	C₂³	1	1	yes	characteristic
73	C₈×D₁₃₁	—	2	2	no	—
74	C₄⋊Dic₁₃₁	—	2	1	no	—
75	C₄⋊Dic₁₃₁	C₂³	1	1	yes	characteristic
76	C₄⋊Dic₁₃₁	C₂³	1	1	yes	characteristic
77	C₄⋊Dic₁₃₁	C₂³	1	1	yes	characteristic
78	Dic₁₃₁⋊C₄	—	2	1	no	—
79	Dic₁₃₁⋊C₄	—	2	1	no	—
80	Dic₁₃₁⋊C₄	—	2	1	no	—
81	Dic₁₃₁⋊C₄	—	2	1	no	—
82	Dic₁₃₁⋊C₄	—	2	1	no	—
83	Dic₁₃₁⋊C₄	D₄	1	1	yes	characteristic
84	Dic₁₃₁⋊C₄	D₄	1	1	yes	characteristic
85	C₂²⋊Dic₁₃₁	—	2	1	no	—
86	(C₂×D₁₃₁)⋊C₄	—	2	1	no	—
87	(C₂×D₁₃₁)⋊C₄	—	2	1	no	—
88	(C₂×D₁₃₁)⋊C₄	—	2	1	no	—
89	(C₂×D₁₃₁)⋊C₄	—	2	1	no	—
90	(C₂×D₁₃₁)⋊C₄	D₄	1	1	yes	characteristic
91	(C₂×D₁₃₁)⋊C₄	D₄	1	1	yes	characteristic
92	C₄×Dic₁₃₁	—	2	1	no	—
93	C₄×Dic₁₃₁	—	2	1	no	—
94	C₄×Dic₁₃₁	C₂³	1	1	yes	characteristic
95	C₂×Q₈×C₁₃₁	—	2	1	no	—
96	C₂×D₄×C₁₃₁	C₂³	1	1	yes	characteristic
97	C₂²×C₄×C₁₃₁	—	2	1	no	—
98	C₂×C₈×C₁₃₁	C₂³	1	1	yes	characteristic
99	C₂×C₈×C₁₃₁	C₂³	1	1	yes	characteristic
100	C₄⋊C₄×C₁₃₁	—	2	1	no	—
101	C₄⋊C₄×C₁₃₁	C₂³	1	1	yes	characteristic
102	C₄⋊C₄×C₁₃₁	C₂³	1	1	yes	characteristic
103	C₄⋊C₄×C₁₃₁	C₂³	1	1	yes	characteristic
104	C₂²⋊C₄×C₁₃₁	—	2	1	no	—
105	C₄²×C₁₃₁	C₂³	1	1	yes	characteristic
106	C₂²×D₁₃₁	—	2	1	no	—
107	C₂²×D₁₃₁	—	2	4	no	—
108	C₂²×D₁₃₁	—	8	1	no	—
109	C₂²×D₁₃₁	—	8	1	no	—
110	C₂²×D₁₃₁	C₂×D₄	1	1	yes	characteristic
111	Dic₂₆₂	—	8	1	no	—
112	D₅₂₄	—	2	1	no	—
113	D₅₂₄	C₂×Q₈	1	2	yes	—
114	C₂²⋊D₁₃₁	—	2	4	no	—
115	C₂×Dic₁₃₁	—	2	1	no	—
116	C₂×Dic₁₃₁	—	2	1	no	—
117	C₂×Dic₁₃₁	—	2	1	no	—
118	C₂×Dic₁₃₁	—	8	1	no	—
119	C₂×Dic₁₃₁	C₂×D₄	1	1	yes	characteristic
120	C₂×Dic₁₃₁	C₂×D₄	1	1	yes	characteristic
121	C₂×Dic₁₃₁	C₂×D₄	1	1	yes	characteristic
122	C₄×D₁₃₁	—	8	1	no	—
123	C₄×D₁₃₁	—	8	1	no	—
124	C₄×D₁₃₁	—	8	1	no	—
125	C₄×D₁₃₁	C₂×Q₈	1	2	yes	—
126	C₄×D₁₃₁	#41	1	2	yes	—
127	C₁₃₁⋊C₈	—	2	1	no	—
128	C₁₃₁⋊C₈	—	2	1	no	—
129	C₂³×C₁₃₁	—	2	1	no	—
130	Q₈×C₁₃₁	—	8	1	no	—
131	D₄×C₁₃₁	—	2	1	no	—
132	D₄×C₁₃₁	C₂×Q₈	1	2	yes	—
133	C₂×C₄×C₁₃₁	—	2	1	no	—
134	C₂×C₄×C₁₃₁	—	2	1	no	—
135	C₂×C₄×C₁₃₁	—	2	1	no	—
136	C₂×C₄×C₁₃₁	—	8	1	no	—
137	C₂×C₄×C₁₃₁	C₂×D₄	1	1	yes	characteristic
138	C₂×C₄×C₁₃₁	C₂×D₄	1	1	yes	characteristic
139	C₂×C₄×C₁₃₁	C₂⁴	1	1	yes	characteristic
140	C₈×C₁₃₁	—	2	1	no	—
141	C₈×C₁₃₁	—	2	1	no	—
142	C₂²×C₁₃₁	—	2	2	no	—
143	C₂²×C₁₃₁	—	4	1	no	—
144	C₂²×C₁₃₁	—	4	1	no	—
145	C₂²×C₁₃₁	C₂²×D₄	1	1	yes	characteristic
146	C₂×D₁₃₁	—	2	1	no	—
147	C₂×D₁₃₁	—	2	1	no	—
148	C₂×D₁₃₁	—	2	2	no	—
149	C₂×D₁₃₁	—	4	1	no	—
150	C₂×D₁₃₁	—	4	1	no	—
151	C₂×D₁₃₁	—	8	4	no	—
152	C₂×D₁₃₁	C₂²⋊Q₈	1	2	yes	—
153	C₄×C₁₃₁	—	2	1	no	—
154	C₄×C₁₃₁	—	2	1	no	—
155	C₄×C₁₃₁	—	4	1	no	—
156	C₄×C₁₃₁	—	4	1	no	—
157	C₄×C₁₃₁	—	4	1	no	—
158	C₄×C₁₃₁	C₂²×Q₈	1	1	yes	characteristic
159	C₄×C₁₃₁	C₂×#41	1	1	yes	characteristic
160	Dic₁₃₁	—	2	1	no	—
161	Dic₁₃₁	—	2	1	no	—
162	Dic₁₃₁	—	4	1	no	—
163	Dic₁₃₁	—	4	1	no	—
164	Dic₁₃₁	—	4	1	no	—
165	Dic₁₃₁	C₂²⋊Q₈	1	2	yes	—
166	C₂×C₁₃₁	—	4	2	no	—
167	C₂×C₁₃₁	C₂×C₂²⋊Q₈	1	1	yes	characteristic
168	C₂×C₁₃₁	#576	1	1	yes	characteristic
169	C₂×C₁₃₁	#577	1	1	yes	characteristic
170	D₁₃₁	—	2	2	no	—
171	D₁₃₁	—	4	2	no	—
172	C₁₃₁	#3081	1	1	yes	characteristic, complemented, Sylow
173	#3081	—	131	1	no	maximal, Sylow
174	#544	—	131	1	no	—
175	#518	—	131	1	no	—
176	#478	—	131	1	no	—
177	#478	—	131	1	no	—
178	#478	—	131	1	no	—
179	#478	—	131	1	no	—
180	#476	—	131	1	no	—
181	#476	D₁₃₁	1	1	yes	characteristic, complemented
182	#474	—	131	1	no	—
183	#474	—	131	1	no	—
184	#428	—	131	1	no	—
185	#427	—	131	1	no	—
186	#424	—	131	1	no	—
187	#418	—	131	1	no	—
188	C₂×#102	—	131	1	no	—
189	C₂²×D₄	—	131	1	no	—
190	C₂×M₁₆	—	131	1	no	—
191	C₂²×C₈	—	131	1	no	—
192	C₄⋊Q₈	—	262	1	no	—
193	C₄⋊Q₈	—	131	1	no	—
194	C₄⋊Q₈	C₂×D₁₃₁	1	1	yes	characteristic, complemented
195	#125	—	262	1	no	—
196	#125	—	131	1	no	—
197	#125	—	131	1	no	—
198	C₂²⋊Q₈	—	262	1	no	—
199	C₂²⋊Q₈	—	262	1	no	—
200	C₄×D₄	—	262	1	no	—
201	C₄×D₄	—	262	1	no	—
202	C₄×D₄	—	131	1	no	—
203	C₄×D₄	—	131	1	no	—
204	C₄×D₄	—	131	1	no	—
205	C₄×D₄	C₂×D₁₃₁	1	1	yes	characteristic
206	#117	—	131	1	no	—
207	#117	—	131	1	no	—
208	#117	—	131	1	no	—
209	C₂×C₄⋊C₄	—	131	1	no	—
210	C₂×C₂²⋊C₄	—	262	1	no	—
211	C₂×C₂²⋊C₄	—	262	1	no	—
212	#107	—	131	1	no	—
213	#107	—	131	1	no	—
214	#107	—	131	1	no	—
215	#107	—	131	1	no	—
216	#106	—	131	1	no	—
217	#106	—	131	1	no	—
218	#106	C₂×D₁₃₁	1	1	yes	characteristic, complemented
219	#106	C₂×D₁₃₁	1	1	yes	characteristic, complemented
220	C₄⋊C₈	—	131	1	no	—
221	C₄⋊C₈	—	131	1	no	—
222	C₄⋊C₈	—	131	1	no	—
223	C₄⋊C₈	C₂×D₁₃₁	1	1	yes	characteristic, complemented
224	#102	—	131	1	no	—
225	#102	—	131	1	no	—
226	#102	—	131	1	no	—
227	#102	—	131	1	no	—
228	#102	—	131	1	no	—
229	#102	—	131	1	no	—
230	#102	C₂×D₁₃₁	1	1	yes	characteristic
231	#102	C₂×D₁₃₁	1	1	yes	characteristic
232	C₂⁴	—	262	1	no	—
233	C₂×Q₈	—	2	1	no	—
234	C₂×Q₈	—	262	1	no	—
235	C₂×D₄	—	262	2	no	—
236	C₂×D₄	—	262	1	no	—
237	C₂×D₄	—	131	4	no	—
238	C₂×D₄	—	131	1	no	—
239	C₂×D₄	C₂²×D₁₃₁	1	1	yes	characteristic
240	C₂²×C₄	—	2	1	no	—
241	C₂²×C₄	—	262	1	no	—
242	C₂²×C₄	—	262	1	no	—
243	C₂²×C₄	—	262	1	no	—
244	C₂²×C₄	—	262	1	no	—
245	C₂²×C₄	—	131	1	no	—
246	M₁₆	—	262	2	no	—
247	C₂×C₈	—	262	2	no	—
248	C₂×C₈	—	131	1	no	—
249	C₂×C₈	—	131	1	no	—
250	C₂×C₈	C₂²×D₁₃₁	1	1	yes	characteristic
251	C₂×C₈	C₂²×D₁₃₁	1	1	yes	characteristic
252	C₄⋊C₄	—	2	1	no	—
253	C₄⋊C₄	—	262	1	no	—
254	C₄⋊C₄	—	262	1	no	—
255	C₄⋊C₄	—	262	1	no	—
256	C₄⋊C₄	—	262	1	no	—
257	C₄⋊C₄	—	262	1	no	—
258	C₄⋊C₄	—	262	1	no	—
259	C₄⋊C₄	—	131	1	no	—
260	C₄⋊C₄	—	131	1	no	—
261	C₄⋊C₄	—	131	1	no	—
262	C₄⋊C₄	—	131	1	no	—
263	C₄⋊C₄	—	131	1	no	—
264	C₄⋊C₄	C₂²×D₁₃₁	1	1	yes	characteristic
265	C₄⋊C₄	C₂²×D₁₃₁	1	1	yes	characteristic
266	C₄⋊C₄	C₂²×D₁₃₁	1	1	yes	characteristic
267	C₂²⋊C₄	—	2	1	no	—
268	C₂²⋊C₄	—	262	1	no	—
269	C₂²⋊C₄	—	262	1	no	—
270	C₂²⋊C₄	—	262	1	no	—
271	C₂²⋊C₄	—	262	1	no	—
272	C₂²⋊C₄	—	262	1	no	—
273	C₂²⋊C₄	—	131	1	no	—
274	C₂²⋊C₄	—	131	1	no	—
275	C₄²	—	262	1	no	—
276	C₄²	—	262	1	no	—
277	C₄²	—	131	1	no	—
278	C₄²	C₂²×D₁₃₁	1	1	yes	characteristic
279	C₂³	—	2	1	no	—
280	C₂³	—	262	4	no	—
281	C₂³	—	262	1	no	—
282	C₂³	—	1048	1	no	—
283	C₂³	—	1048	1	no	—
284	C₂³	—	131	1	no	—
285	Q₈	—	8	1	no	—
286	Q₈	—	1048	1	no	—
287	D₄	—	2	1	no	—
288	D₄	—	262	4	no	—
289	D₄	—	262	1	no	—
290	D₄	—	131	2	no	—
291	D₄	Q₈×D₁₃₁	1	2	yes	—
292	C₂×C₄	—	2	1	no	—
293	C₂×C₄	—	2	1	no	—
294	C₂×C₄	—	2	1	no	—
295	C₂×C₄	—	262	1	no	—
296	C₂×C₄	—	262	1	no	—
297	C₂×C₄	—	262	1	no	—
298	C₂×C₄	—	8	1	no	—
299	C₂×C₄	—	1048	1	no	—
300	C₂×C₄	—	1048	1	no	—
301	C₂×C₄	—	1048	1	no	—
302	C₂×C₄	—	1048	1	no	—
303	C₂×C₄	—	131	1	no	—
304	C₂×C₄	—	131	1	no	—
305	C₂×C₄	—	131	1	no	—
306	C₂×C₄	—	131	2	no	—
307	C₂×C₄	—	131	2	no	—
308	C₂×C₄	D₄×D₁₃₁	1	1	yes	characteristic
309	C₂×C₄	D₄×D₁₃₁	1	1	yes	characteristic
310	C₂×C₄	C₂³×D₁₃₁	1	1	yes	characteristic
311	C₈	—	2	1	no	—
312	C₈	—	2	1	no	—
313	C₈	—	262	1	no	—
314	C₈	—	262	1	no	—
315	C₂²	—	2	2	no	—
316	C₂²	—	4	1	no	—
317	C₂²	—	4	1	no	—
318	C₂²	—	262	1	no	—
319	C₂²	—	262	1	no	—
320	C₂²	—	262	2	no	—
321	C₂²	—	524	1	no	—
322	C₂²	—	524	1	no	—
323	C₂²	—	1048	4	no	—
324	C₂²	—	131	2	no	—
325	C₂²	C₂×D₄×D₁₃₁	1	1	yes	characteristic
326	C₄	—	2	1	no	—
327	C₄	—	2	1	no	—
328	C₄	—	4	1	no	—
329	C₄	—	4	1	no	—
330	C₄	—	4	1	no	—
331	C₄	—	262	1	no	—
332	C₄	—	262	1	no	—
333	C₄	—	524	1	no	—
334	C₄	—	524	1	no	—
335	C₄	—	524	1	no	—
336	C₄	—	131	2	no	—
337	C₄	C₂×Q₈×D₁₃₁	1	1	yes	characteristic
338	C₄	#41×D₁₃₁	1	1	yes	characteristic
339	C₂	—	4	2	no	—
340	C₂	—	262	2	no	—
341	C₂	—	524	2	no	—
342	C₂	C₂²⋊Q₈×D₁₃₁	1	1	yes	characteristic
343	C₂	#22554116	1	1	yes	characteristic
344	C₂	#22554118	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×D₄×D₁₃₁
commutator, cosocle	C₂×C₄×C₁₃₁	C₂⁴
Frattini	C₂×C₄	C₂³×D₁₃₁
Fitting	#476×C₁₃₁	C₂
socle	C₂²×C₁₃₁	C₂²×D₄
prime radical	C₂×D₄×D₁₃₁	C₂²
hypercenter	#476	D₁₃₁
hypocenter	C₁₃₁	#3081
LCS	G ▷ C₂×C₄×C₁₃₁ ▷ C₂×C₁₃₁ ▷ C₁₃₁
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #476

Complements

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

N	H ≅ G/N	type
1. #22553939	C₂	semidirect
3. #22553614	C₂	semidirect
4. #22553613	C₂	semidirect
5. #22553612	C₂	semidirect
6. #22553611	C₂	semidirect
7. #22553601	C₂	semidirect
8. #22553590	C₂	semidirect
9. #22553589	C₂	semidirect
10. #22553351	C₂	semidirect
11. #22553347	C₂	semidirect
12. #22553337	C₂	semidirect
15. #476×C₁₃₁	C₂	semidirect
19. #16862245	C₂²	semidirect
21. #16862238	C₂²	semidirect
22. #16862237	C₂²	semidirect
37. #16862180	C₂²	semidirect
38. #16862179	C₂²	semidirect
39. #16862179	C₂²	semidirect
40. #16862179	C₂²	semidirect
41. #16862178	C₂²	semidirect
42. #16862177	C₂²	semidirect
43. C₄⋊(C₁₃₁⋊C₈)	C₂²	semidirect
44. Dic₁₃₁⋊C₈	C₂²	semidirect
45. Dic₁₃₁⋊C₈	C₂²	semidirect
52. C₄⋊Q₈×C₁₃₁	C₂²	semidirect
54. #106×C₁₃₁	C₂²	semidirect
55. #106×C₁₃₁	C₂²	semidirect
56. C₄⋊C₈×C₁₃₁	C₂²	semidirect
64. D₄×D₁₃₁	Q₈	semidirect
172. C₁₃₁	#3081	semidirect
181. #476	D₁₃₁	semidirect
194. C₄⋊Q₈	C₂×D₁₃₁	semidirect
218. #106	C₂×D₁₃₁	semidirect
219. #106	C₂×D₁₃₁	semidirect
223. C₄⋊C₈	C₂×D₁₃₁	semidirect

Maximal subgroups

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

subgroup	index
#22553939	2
#22553784	2
#22553614	2
#22553613	2
#22553612	2
#22553611	2
#22553601	2
#22553590	2
#22553589	2
#22553351	2
#22553347	2
#22553337	2
#22553315	2
#102×D₁₃₁	2
#476×C₁₃₁	2
#3081	131