Subgroups — #21454939

Order 7808.

398 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	C₄×#11758224	C₂	1	1	yes	maximal, characteristic
2	#16839891	C₂	1	1	yes	maximal, characteristic, complemented
3	#16839890	C₂	1	1	yes	maximal, characteristic, complemented
4	#16839836	C₂	1	1	yes	maximal, characteristic, complemented
5	#16839834	C₂	1	1	yes	maximal, characteristic, complemented
6	#16839831	C₂	1	1	yes	maximal, characteristic, complemented
7	C₈×C₂²⋊D₆₁	C₂	1	1	yes	maximal, characteristic, complemented
8	#16839825	C₂	1	1	yes	maximal, characteristic, complemented
9	#16839817	C₂	1	1	yes	maximal, characteristic, complemented
10	#16839816	C₂	1	1	yes	maximal, characteristic, complemented
11	#16839812	C₂	1	1	yes	maximal, characteristic, complemented
12	#16839785	C₂	1	1	yes	maximal, characteristic, complemented
13	C₂×#14519706	C₂	1	1	yes	maximal, characteristic, complemented
14	C₄⋊C₈×D₆₁	C₂	1	1	yes	maximal, characteristic, complemented
15	C₆₁×#435	C₂	1	1	yes	maximal, characteristic, complemented
16	C₂×#11758224	C₄	1	1	yes	characteristic
17	C₂×#11758194	C₂²	1	1	yes	characteristic
18	C₂²×#92418	C₂²	1	1	yes	characteristic
19	C₂×C₈×D₆₁	C₂²	1	1	yes	characteristic
20	C₂×C₈×D₆₁	C₂²	1	1	yes	characteristic
21	C₄×Dic₁₂₂	C₂²	1	1	yes	characteristic
22	#14519742	C₄	1	1	yes	characteristic
23	D₄×Dic₆₁	C₄	1	1	yes	characteristic
24	C₄×C₂²⋊D₆₁	C₂²	1	1	yes	characteristic
25	C₄×C₂²⋊D₆₁	C₂²	1	1	yes	characteristic
26	#14519737	C₄	1	1	yes	characteristic
27	#14519737	C₄	1	1	yes	characteristic
28	#14519735	C₄	1	1	yes	characteristic
29	#14519733	C₄	1	1	yes	characteristic
30	#14519733	C₄	1	1	yes	characteristic
31	C₂×C₄×Dic₆₁	C₂²	1	1	yes	characteristic
32	C₂×C₄×Dic₆₁	C₂²	1	1	yes	characteristic
33	C₄²×D₆₁	C₂²	1	1	yes	characteristic
34	C₄⋊#92418	C₂²	1	1	yes	characteristic
35	#14519706	C₂²	1	1	yes	characteristic
36	#14519706	C₂²	1	1	yes	characteristic
37	#14519706	C₂²	1	1	yes	characteristic
38	#14519706	C₂²	1	1	yes	characteristic
39	#14519706	C₂²	1	1	yes	characteristic
40	#14519706	C₂²	1	1	yes	characteristic
41	#14519706	C₂²	1	1	yes	characteristic
42	#14519706	C₂²	1	1	yes	characteristic
43	C₂²⋊#92418	C₂²	1	1	yes	characteristic
44	C₂²⋊#92418	C₂²	1	1	yes	characteristic
45	#14519689	C₂²	1	1	yes	characteristic
46	#14519689	C₂²	1	1	yes	characteristic
47	#14519688	C₂²	1	1	yes	characteristic
48	#14519687	C₂²	1	1	yes	characteristic
49	#14519687	C₂²	1	1	yes	characteristic
50	C₈×Dic₆₁	C₂²	1	1	yes	characteristic
51	C₈×Dic₆₁	C₂²	1	1	yes	characteristic
52	C₂×M₁₆×C₆₁	C₂²	1	1	yes	characteristic
53	C₂²×C₈×C₆₁	C₂²	1	1	yes	characteristic
54	C₄×D₄×C₆₁	C₂²	1	1	yes	characteristic
55	C₄⋊C₈×C₆₁	C₂²	1	1	yes	characteristic
56	C₂²⋊C₈×C₆₁	C₂²	1	1	yes	characteristic
57	C₂²⋊C₈×C₆₁	C₂²	1	1	yes	characteristic
58	#97×C₆₁	C₂²	1	1	yes	characteristic
59	C₂×D₄×C₆₁	C₂×C₄	1	1	yes	characteristic
60	C₂²×C₄×C₆₁	C₂³	1	1	yes	characteristic
61	C₂²×C₄×C₆₁	C₂³	1	1	yes	characteristic
62	C₂×C₂²⋊D₆₁	C₂×C₄	1	1	yes	characteristic
63	C₂×C₂²⋊D₆₁	C₂×C₄	1	1	yes	characteristic
64	C₂²×Dic₆₁	C₂×C₄	1	1	yes	characteristic
65	C₂²×Dic₆₁	C₂×C₄	1	1	yes	characteristic
66	C₂²×Dic₆₁	C₂×C₄	1	1	yes	characteristic
67	C₂²×Dic₆₁	C₂×C₄	1	1	yes	characteristic
68	#11758224	—	2	4	no	—
69	C₂×C₄×D₆₁	C₂×C₄	1	1	yes	characteristic
70	C₂×C₄×D₆₁	C₂×C₄	1	1	yes	characteristic
71	C₂×C₄×D₆₁	C₂³	1	1	yes	characteristic
72	C₂×Dic₁₂₂	C₂×C₄	1	1	yes	characteristic
73	M₁₆×C₆₁	—	4	1	no	—
74	C₂×C₈×C₆₁	—	4	1	no	—
75	C₂×C₈×C₆₁	C₂³	1	1	yes	characteristic
76	C₂×C₈×C₆₁	C₂³	1	1	yes	characteristic
77	C₂×C₈×C₆₁	C₂³	1	1	yes	characteristic
78	C₂×C₈×C₆₁	C₂³	1	1	yes	characteristic
79	C₄⋊C₄×C₆₁	C₂×C₄	1	1	yes	characteristic
80	C₂²⋊C₄×C₆₁	C₂×C₄	1	1	yes	characteristic
81	C₂²⋊C₄×C₆₁	C₂×C₄	1	1	yes	characteristic
82	C₄²×C₆₁	C₂³	1	1	yes	characteristic
83	C₂²⋊Dic₆₁	C₂×C₄	1	1	yes	characteristic
84	C₂²⋊Dic₆₁	C₂×C₄	1	1	yes	characteristic
85	#11758198	C₂×C₄	1	1	yes	characteristic
86	#11758198	C₂×C₄	1	1	yes	characteristic
87	C₄⋊Dic₆₁	C₂×C₄	1	1	yes	characteristic
88	#11758196	C₂×C₄	1	1	yes	characteristic
89	#11758196	C₂×C₄	1	1	yes	characteristic
90	#11758196	C₂×C₄	1	1	yes	characteristic
91	#11758196	C₂×C₄	1	1	yes	characteristic
92	C₄×Dic₆₁	C₂×C₄	1	1	yes	characteristic
93	C₄×Dic₆₁	C₂×C₄	1	1	yes	characteristic
94	C₄×Dic₆₁	C₂×C₄	1	1	yes	characteristic
95	C₄×Dic₆₁	C₂×C₄	1	1	yes	characteristic
96	C₄×Dic₆₁	C₂×C₄	1	1	yes	characteristic
97	C₄×Dic₆₁	C₂×C₄	1	1	yes	characteristic
98	C₄×Dic₆₁	C₂³	1	1	yes	characteristic
99	C₄×Dic₆₁	C₂³	1	1	yes	characteristic
100	C₄×Dic₆₁	C₂³	1	1	yes	characteristic
101	#11758194	—	4	1	no	—
102	C₂×#92418	—	4	1	no	—
103	C₂×#92418	C₂³	1	1	yes	characteristic
104	C₂×#92418	C₂³	1	1	yes	characteristic
105	C₂×#92418	C₂³	1	1	yes	characteristic
106	C₂×#92418	C₂³	1	1	yes	characteristic
107	C₈×D₆₁	—	4	1	no	—
108	C₈×D₆₁	—	4	1	no	—
109	C₂³×C₆₁	C₂²×C₄	1	1	yes	characteristic
110	C₂³×C₆₁	C₂²×C₄	1	1	yes	characteristic
111	C₂²×D₆₁	C₂²×C₄	1	1	yes	characteristic
112	D₄×C₆₁	—	2	2	no	—
113	C₂×C₄×C₆₁	—	2	1	no	—
114	C₂×C₄×C₆₁	—	2	1	no	—
115	C₂×C₄×C₆₁	—	2	1	no	—
116	C₂×C₄×C₆₁	—	2	1	no	—
117	C₂×C₄×C₆₁	C₂²×C₄	1	1	yes	characteristic
118	C₂×C₄×C₆₁	C₂²×C₄	1	1	yes	characteristic
119	C₂×C₄×C₆₁	C₂²×C₄	1	1	yes	characteristic
120	C₂×C₄×C₆₁	C₂²×C₄	1	1	yes	characteristic
121	C₂×C₄×C₆₁	C₂⁴	1	1	yes	characteristic
122	C₂²⋊D₆₁	—	2	2	no	—
123	C₂²⋊D₆₁	M₁₆	1	4	yes	complemented
124	C₂×Dic₆₁	—	2	2	no	—
125	C₂×Dic₆₁	—	2	2	no	—
126	C₂×Dic₆₁	—	2	2	no	—
127	C₂×Dic₆₁	—	2	2	no	—
128	C₂×Dic₆₁	C₂²×C₄	1	1	yes	characteristic
129	C₂×Dic₆₁	C₂²×C₄	1	1	yes	characteristic
130	C₂×Dic₆₁	C₂²×C₄	1	1	yes	characteristic
131	C₂×Dic₆₁	C₂²×C₄	1	1	yes	characteristic
132	C₂×Dic₆₁	C₂²×C₄	1	1	yes	characteristic
133	C₂×Dic₆₁	C₂²×C₄	1	1	yes	characteristic
134	C₂×Dic₆₁	C₂²×C₄	1	1	yes	characteristic
135	C₄×D₆₁	—	2	2	no	—
136	C₄×D₆₁	—	2	1	no	—
137	C₄×D₆₁	—	2	1	no	—
138	C₄×D₆₁	—	2	2	no	—
139	Dic₁₂₂	—	2	2	no	—
140	C₈×C₆₁	—	2	1	no	—
141	C₈×C₆₁	—	2	1	no	—
142	C₈×C₆₁	—	2	1	no	—
143	C₈×C₆₁	—	2	1	no	—
144	#92418	—	2	1	no	—
145	#92418	—	2	1	no	—
146	#92418	—	2	1	no	—
147	#92418	—	2	1	no	—
148	C₂²×C₆₁	—	2	1	no	—
149	C₂²×C₆₁	—	2	1	no	—
150	C₂²×C₆₁	—	2	1	no	—
151	C₂²×C₆₁	—	2	1	no	—
152	C₂²×C₆₁	—	2	1	no	—
153	C₂²×C₆₁	C₂×M₁₆	1	2	yes	—
154	C₂²×C₆₁	C₂³×C₄	1	1	yes	characteristic
155	C₂×D₆₁	—	2	1	no	—
156	C₂×D₆₁	—	2	1	no	—
157	C₂×D₆₁	C₂×M₁₆	1	2	yes	—
158	C₄×C₆₁	—	2	1	no	—
159	C₄×C₆₁	—	2	1	no	—
160	C₄×C₆₁	—	2	1	no	—
161	C₄×C₆₁	—	2	1	no	—
162	C₄×C₆₁	H(5,𝔽₂)	1	1	yes	characteristic
163	C₄×C₆₁	2₋¹⁺⁴	1	1	yes	characteristic
164	Dic₆₁	—	2	1	no	—
165	Dic₆₁	—	2	1	no	—
166	Dic₆₁	—	2	1	no	—
167	Dic₆₁	—	2	1	no	—
168	Dic₆₁	—	2	1	no	—
169	Dic₆₁	C₂×M₁₆	1	2	yes	—
170	Dic₆₁	#131	1	2	yes	—
171	#2990	—	61	1	no	maximal, Sylow
172	C₂×C₆₁	—	2	2	no	—
173	C₂×C₆₁	—	4	1	no	—
174	C₂×C₆₁	#520	1	1	yes	characteristic
175	C₂×C₆₁	C₂²×M₁₆	1	1	yes	characteristic
176	C₂×C₆₁	C₂×#131	1	1	yes	characteristic
177	D₆₁	—	2	2	no	—
178	C₄×#41	—	61	1	no	—
179	#446	—	61	1	no	—
180	#446	—	61	1	no	—
181	#435	—	61	1	no	—
182	#435	—	61	1	no	—
183	#435	—	61	1	no	—
184	#435	D₆₁	1	1	yes	characteristic, complemented
185	C₈×D₄	—	61	1	no	—
186	C₈×D₄	—	61	1	no	—
187	#432	—	61	1	no	—
188	#432	—	61	1	no	—
189	#431	—	61	1	no	—
190	#423	—	61	1	no	—
191	C₂×C₄⋊C₈	—	61	1	no	—
192	C₂×C₄⋊C₈	—	61	1	no	—
193	C₆₁	#2990	1	1	yes	characteristic, complemented, Sylow
194	C₂×#41	—	61	1	no	—
195	C₂×M₁₆	—	61	1	no	—
196	C₂×M₁₆	C₂×D₆₁	1	1	yes	characteristic
197	C₂²×C₈	—	61	1	no	—
198	C₂²×C₈	—	61	1	no	—
199	C₂²×C₈	—	61	1	no	—
200	C₂²×C₈	C₂×D₆₁	1	1	yes	characteristic
201	C₄×Q₈	—	61	1	no	—
202	C₄×Q₈	—	61	1	no	—
203	C₄×D₄	—	61	1	no	—
204	C₄×D₄	—	61	1	no	—
205	C₄×D₄	—	61	1	no	—
206	C₄×D₄	—	61	1	no	—
207	C₄×D₄	—	61	1	no	—
208	C₄×D₄	C₂×D₆₁	1	1	yes	characteristic
209	#117	—	61	1	no	—
210	#117	—	61	1	no	—
211	#117	—	61	1	no	—
212	C₂×C₄²	—	61	1	no	—
213	C₂×C₄²	—	61	1	no	—
214	C₂×C₄²	—	61	1	no	—
215	C₄⋊C₈	—	61	1	no	—
216	C₄⋊C₈	—	61	1	no	—
217	C₄⋊C₈	—	61	1	no	—
218	C₄⋊C₈	—	61	1	no	—
219	C₄⋊C₈	—	61	1	no	—
220	C₄⋊C₈	—	61	1	no	—
221	C₄⋊C₈	—	61	1	no	—
222	C₄⋊C₈	—	61	1	no	—
223	C₄⋊C₈	—	61	1	no	—
224	C₄⋊C₈	C₂×D₆₁	1	1	yes	characteristic
225	C₂²⋊C₈	—	61	1	no	—
226	C₂²⋊C₈	—	61	1	no	—
227	C₂²⋊C₈	—	61	1	no	—
228	C₂²⋊C₈	—	61	1	no	—
229	C₂²⋊C₈	C₂×D₆₁	1	1	yes	characteristic
230	C₂²⋊C₈	C₂×D₆₁	1	1	yes	characteristic
231	#97	—	61	1	no	—
232	#97	—	61	1	no	—
233	#97	—	61	1	no	—
234	#97	C₂×D₆₁	1	1	yes	characteristic
235	C₄×C₈	—	61	1	no	—
236	C₄×C₈	—	61	1	no	—
237	#41	—	122	4	no	—
238	C₂×Q₈	—	61	1	no	—
239	C₂×D₄	—	61	1	no	—
240	C₂×D₄	—	61	1	no	—
241	C₂×D₄	C₄×D₆₁	1	1	yes	characteristic
242	C₂²×C₄	—	61	1	no	—
243	C₂²×C₄	—	61	1	no	—
244	C₂²×C₄	—	61	1	no	—
245	C₂²×C₄	—	61	1	no	—
246	C₂²×C₄	—	61	1	no	—
247	C₂²×C₄	—	61	1	no	—
248	C₂²×C₄	—	61	1	no	—
249	C₂²×C₄	C₂²×D₆₁	1	1	yes	characteristic
250	C₂²×C₄	C₂²×D₆₁	1	1	yes	characteristic
251	M₁₆	—	4	1	no	—
252	M₁₆	—	244	1	no	—
253	C₂×C₈	—	4	1	no	—
254	C₂×C₈	—	61	1	no	—
255	C₂×C₈	—	61	1	no	—
256	C₂×C₈	—	61	1	no	—
257	C₂×C₈	—	61	1	no	—
258	C₂×C₈	—	244	1	no	—
259	C₂×C₈	—	244	1	no	—
260	C₂×C₈	—	244	1	no	—
261	C₂×C₈	C₂²×D₆₁	1	1	yes	characteristic
262	C₂×C₈	C₂²×D₆₁	1	1	yes	characteristic
263	C₂×C₈	C₂²×D₆₁	1	1	yes	characteristic
264	C₂×C₈	C₂²×D₆₁	1	1	yes	characteristic
265	C₄⋊C₄	—	61	1	no	—
266	C₄⋊C₄	—	61	1	no	—
267	C₄⋊C₄	—	61	1	no	—
268	C₄⋊C₄	—	61	1	no	—
269	C₄⋊C₄	—	61	1	no	—
270	C₄⋊C₄	C₄×D₆₁	1	1	yes	characteristic
271	C₂²⋊C₄	—	61	1	no	—
272	C₂²⋊C₄	—	61	1	no	—
273	C₂²⋊C₄	—	61	1	no	—
274	C₂²⋊C₄	—	61	1	no	—
275	C₂²⋊C₄	C₄×D₆₁	1	1	yes	characteristic
276	C₂²⋊C₄	C₄×D₆₁	1	1	yes	characteristic
277	C₄²	—	61	1	no	—
278	C₄²	—	61	1	no	—
279	C₄²	—	61	1	no	—
280	C₄²	—	61	1	no	—
281	C₄²	—	61	1	no	—
282	C₄²	—	61	1	no	—
283	C₄²	—	61	1	no	—
284	C₄²	—	61	1	no	—
285	C₄²	—	61	1	no	—
286	C₄²	C₂²×D₆₁	1	1	yes	characteristic
287	C₂³	—	61	1	no	—
288	C₂³	C₂×C₄×D₆₁	1	1	yes	characteristic
289	C₂³	C₂×C₄×D₆₁	1	1	yes	characteristic
290	Q₈	—	122	2	no	—
291	D₄	—	2	2	no	—
292	D₄	—	61	4	no	—
293	D₄	—	122	2	no	—
294	C₂×C₄	—	2	1	no	—
295	C₂×C₄	—	2	1	no	—
296	C₂×C₄	—	2	1	no	—
297	C₂×C₄	—	2	1	no	—
298	C₂×C₄	—	61	1	no	—
299	C₂×C₄	—	61	1	no	—
300	C₂×C₄	—	61	1	no	—
301	C₂×C₄	—	61	1	no	—
302	C₂×C₄	—	61	1	no	—
303	C₂×C₄	—	61	1	no	—
304	C₂×C₄	—	61	1	no	—
305	C₂×C₄	—	122	2	no	—
306	C₂×C₄	—	122	2	no	—
307	C₂×C₄	—	122	2	no	—
308	C₂×C₄	—	122	2	no	—
309	C₂×C₄	—	122	2	no	—
310	C₂×C₄	—	122	2	no	—
311	C₂×C₄	—	122	1	no	—
312	C₂×C₄	—	122	1	no	—
313	C₂×C₄	C₂×C₄×D₆₁	1	1	yes	characteristic
314	C₂×C₄	C₂×C₄×D₆₁	1	1	yes	characteristic
315	C₂×C₄	C₂×C₄×D₆₁	1	1	yes	characteristic
316	C₂×C₄	C₂×C₄×D₆₁	1	1	yes	characteristic
317	C₂×C₄	C₂³×D₆₁	1	1	yes	characteristic
318	C₈	—	2	1	no	—
319	C₈	—	2	1	no	—
320	C₈	—	2	1	no	—
321	C₈	—	2	1	no	—
322	C₈	—	122	1	no	—
323	C₈	—	122	1	no	—
324	C₈	—	122	1	no	—
325	C₈	—	122	1	no	—
326	C₂²	—	2	1	no	—
327	C₂²	—	2	1	no	—
328	C₂²	—	2	1	no	—
329	C₂²	—	2	1	no	—
330	C₂²	—	2	1	no	—
331	C₂²	—	61	2	no	—
332	C₂²	—	122	1	no	—
333	C₂²	—	122	1	no	—
334	C₂²	M₁₆×D₆₁	1	2	yes	—
335	C₂²	C₂²×C₄×D₆₁	1	1	yes	characteristic
336	C₄	—	2	1	no	—
337	C₄	—	2	1	no	—
338	C₄	—	2	1	no	—
339	C₄	—	2	1	no	—
340	C₄	—	61	2	no	—
341	C₄	—	61	2	no	—
342	C₄	—	122	1	no	—
343	C₄	—	122	1	no	—
344	C₄	—	122	1	no	—
345	C₄	—	122	1	no	—
346	C₄	—	122	1	no	—
347	C₄	#14519822	1	1	yes	characteristic
348	C₄	#14519824	1	1	yes	characteristic
349	C₂	—	2	2	no	—
350	C₂	—	4	1	no	—
351	C₂	—	122	2	no	—
352	C₂	#16840241	1	1	yes	characteristic
353	C₂	C₂×M₁₆×D₆₁	1	1	yes	characteristic
354	C₂	#131×D₆₁	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂×C₄	C₂³×D₆₁
commutator, socle	C₂²×C₆₁	C₂³×C₄
Fitting	C₆₁×#435	C₂
hypercenter	#435	D₆₁
hypocenter	C₆₁	#2990
LCS	G ▷ C₂²×C₆₁ ▷ C₆₁
UCS	C₁ ◁ C₂×C₄ ◁ #435

Complements

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

N	H ≅ G/N	type
2. #16839891	C₂	semidirect
3. #16839890	C₂	semidirect
4. #16839836	C₂	semidirect
5. #16839834	C₂	semidirect
6. #16839831	C₂	semidirect
7. C₈×C₂²⋊D₆₁	C₂	semidirect
8. #16839825	C₂	semidirect
9. #16839817	C₂	semidirect
10. #16839816	C₂	semidirect
11. #16839812	C₂	semidirect
12. #16839785	C₂	semidirect
13. C₂×#14519706	C₂	semidirect
14. C₄⋊C₈×D₆₁	C₂	semidirect
15. C₆₁×#435	C₂	semidirect
123. C₂²⋊D₆₁	M₁₆	semidirect
184. #435	D₆₁	semidirect
193. C₆₁	#2990	semidirect