Subgroups — #21462505

Order 7808.

384 conjugacy classes of subgroups.

The subgroup lattice has 292 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#16840787	C₂	1	2	yes	maximal, complemented
2	#16840786	C₂	1	2	yes	maximal, complemented
3	#16840785	C₂	1	1	yes	maximal, characteristic, complemented
4	#16840784	C₂	1	1	yes	maximal, characteristic, complemented
5	#16840783	C₂	1	1	yes	maximal, characteristic, complemented
6	#16840782	C₂	1	1	yes	maximal, characteristic, complemented
7	#16840768	C₂	1	2	yes	maximal, complemented
8	#16840735	C₂	1	2	yes	maximal
9	C₄×#11758217	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂×#14519832	C₂	1	1	yes	maximal, characteristic, complemented
11	C₄×#11758224	C₂	1	1	yes	maximal, characteristic
12	C₂×#11758217	C₂²	1	2	yes	—
13	C₂²×#92420	C₂²	1	2	yes	—
14	#14519844	C₂²	1	2	yes	—
15	#14519844	C₂²	1	2	yes	—
16	C₄⋊#92420	C₂²	1	2	yes	—
17	C₄⋊#92420	C₂²	1	2	yes	—
18	#14519834	C₂²	1	2	yes	—
19	#14519834	C₂²	1	2	yes	—
20	C₂²⋊#92420	C₂²	1	2	yes	—
21	C₂²⋊#92420	C₂²	1	2	yes	—
22	#14519832	C₂²	1	1	yes	characteristic
23	#14519832	C₂²	1	1	yes	characteristic
24	#14519832	C₂²	1	2	yes	—
25	#14519832	C₂²	1	1	yes	characteristic
26	#14519832	C₂²	1	1	yes	characteristic
27	C₄×#92420	C₂²	1	2	yes	—
28	C₂×#11758224	C₂²	1	1	yes	characteristic
29	C₄×Dic₁₂₂	C₄	1	1	yes	characteristic
30	#14519742	C₄	1	1	yes	characteristic
31	D₄×Dic₆₁	C₄	1	1	yes	characteristic
32	C₄×C₂²⋊D₆₁	C₄	1	1	yes	characteristic
33	C₄×C₂²⋊D₆₁	C₄	1	1	yes	characteristic
34	#14519737	C₄	1	1	yes	characteristic
35	#14519737	C₄	1	1	yes	characteristic
36	#14519735	C₂²	1	1	yes	characteristic
37	#14519733	C₂²	1	1	yes	characteristic
38	#14519733	C₂²	1	1	yes	characteristic
39	C₂×C₄×Dic₆₁	C₂²	1	1	yes	characteristic
40	C₂×C₄×Dic₆₁	C₂²	1	1	yes	characteristic
41	C₄²×D₆₁	C₂²	1	1	yes	characteristic
42	C₄×D₄×C₆₁	C₄	1	1	yes	characteristic
43	C₂×D₄×C₆₁	C₂×C₄	1	1	yes	characteristic
44	C₂²×C₄×C₆₁	C₂×C₄	1	1	yes	characteristic
45	C₂²×C₄×C₆₁	C₂×C₄	1	1	yes	characteristic
46	C₂×C₂²⋊D₆₁	C₂×C₄	1	1	yes	characteristic
47	C₂×C₂²⋊D₆₁	C₂×C₄	1	1	yes	characteristic
48	C₂²×Dic₆₁	C₂×C₄	1	1	yes	characteristic
49	C₂²×Dic₆₁	C₂×C₄	1	1	yes	characteristic
50	C₂²×Dic₆₁	C₂³	1	1	yes	characteristic
51	C₂²×Dic₆₁	C₂³	1	1	yes	characteristic
52	#11758224	—	2	4	no	—
53	C₂×C₄×D₆₁	C₂×C₄	1	1	yes	characteristic
54	C₂×C₄×D₆₁	C₂×C₄	1	1	yes	characteristic
55	C₂×C₄×D₆₁	C₂³	1	1	yes	characteristic
56	C₂×Dic₁₂₂	C₂×C₄	1	1	yes	characteristic
57	#11758217	—	2	4	no	—
58	C₂×#92420	—	2	4	no	—
59	C₂×#92420	C₂³	1	2	yes	—
60	C₂×#92420	C₂³	1	2	yes	—
61	C₂×#92420	C₂³	1	2	yes	—
62	C₂×#92420	C₂³	1	2	yes	—
63	C₄⋊C₄×C₆₁	C₂×C₄	1	1	yes	characteristic
64	C₂²⋊C₄×C₆₁	C₂×C₄	1	1	yes	characteristic
65	C₂²⋊C₄×C₆₁	C₂×C₄	1	1	yes	characteristic
66	C₄²×C₆₁	C₂×C₄	1	1	yes	characteristic
67	C₂²⋊Dic₆₁	C₂×C₄	1	1	yes	characteristic
68	C₂²⋊Dic₆₁	C₂×C₄	1	1	yes	characteristic
69	#11758198	C₂×C₄	1	1	yes	characteristic
70	#11758198	C₂×C₄	1	1	yes	characteristic
71	C₄⋊Dic₆₁	C₂×C₄	1	1	yes	characteristic
72	#11758196	C₂×C₄	1	1	yes	characteristic
73	#11758196	C₂×C₄	1	1	yes	characteristic
74	#11758196	C₂×C₄	1	1	yes	characteristic
75	#11758196	C₂×C₄	1	1	yes	characteristic
76	C₄×Dic₆₁	C₂×C₄	1	1	yes	characteristic
77	C₄×Dic₆₁	C₂×C₄	1	1	yes	characteristic
78	C₄×Dic₆₁	C₂×C₄	1	1	yes	characteristic
79	C₄×Dic₆₁	C₂×C₄	1	1	yes	characteristic
80	C₄×Dic₆₁	C₂×C₄	1	1	yes	characteristic
81	C₄×Dic₆₁	C₂³	1	1	yes	characteristic
82	C₄×Dic₆₁	C₂³	1	1	yes	characteristic
83	C₄×Dic₆₁	C₂³	1	1	yes	characteristic
84	C₄×Dic₆₁	C₂³	1	1	yes	characteristic
85	C₂³×C₆₁	C₂²×C₄	1	1	yes	characteristic
86	C₂³×C₆₁	C₂²×C₄	1	1	yes	characteristic
87	C₂²×D₆₁	C₂²×C₄	1	1	yes	characteristic
88	D₄×C₆₁	—	2	2	no	—
89	C₂×C₄×C₆₁	—	4	1	no	—
90	C₂×C₄×C₆₁	—	4	1	no	—
91	C₂×C₄×C₆₁	C₂²×C₄	1	1	yes	characteristic
92	C₂×C₄×C₆₁	C₂²×C₄	1	1	yes	characteristic
93	C₂×C₄×C₆₁	C₂²×C₄	1	1	yes	characteristic
94	C₂×C₄×C₆₁	C₂²×C₄	1	1	yes	characteristic
95	C₂×C₄×C₆₁	C₂²×C₄	1	1	yes	characteristic
96	C₂²⋊D₆₁	—	2	2	no	—
97	C₂²⋊D₆₁	—	2	2	no	—
98	C₂×Dic₆₁	—	2	2	no	—
99	C₂×Dic₆₁	—	4	1	no	—
100	C₂×Dic₆₁	—	4	1	no	—
101	C₂×Dic₆₁	C₂²×C₄	1	1	yes	characteristic
102	C₂×Dic₆₁	C₂²×C₄	1	1	yes	characteristic
103	C₂×Dic₆₁	C₂²×C₄	1	1	yes	characteristic
104	C₂×Dic₆₁	C₂²×C₄	1	1	yes	characteristic
105	C₂×Dic₆₁	C₂²×C₄	1	1	yes	characteristic
106	C₂×Dic₆₁	C₂²×C₄	1	1	yes	characteristic
107	C₂×Dic₆₁	#41	1	4	yes	—
108	C₂×Dic₆₁	C₂⁴	1	1	yes	characteristic
109	C₄×D₆₁	—	2	2	no	—
110	C₄×D₆₁	—	4	1	no	—
111	C₄×D₆₁	—	4	1	no	—
112	Dic₁₂₂	—	2	2	no	—
113	#92420	—	2	2	no	—
114	#92420	—	2	2	no	—
115	#92420	—	2	2	no	—
116	#92420	—	2	2	no	—
117	C₂²×C₆₁	—	2	1	no	—
118	C₂²×C₆₁	—	2	1	no	—
119	C₂²×C₆₁	—	2	1	no	—
120	C₂²×C₆₁	—	2	1	no	—
121	C₂²×C₆₁	—	2	1	no	—
122	C₂²×C₆₁	#117	1	2	yes	—
123	C₂²×C₆₁	C₂³×C₄	1	1	yes	characteristic
124	C₂×D₆₁	—	2	1	no	—
125	C₂×D₆₁	—	2	1	no	—
126	C₂×D₆₁	—	2	1	no	—
127	C₄×C₆₁	—	2	1	no	—
128	C₄×C₆₁	—	2	1	no	—
129	C₄×C₆₁	—	2	1	no	—
130	C₄×C₆₁	—	2	1	no	—
131	C₄×C₆₁	—	2	1	no	—
132	Dic₆₁	—	2	1	no	—
133	Dic₆₁	—	2	1	no	—
134	Dic₆₁	—	2	1	no	—
135	Dic₆₁	—	2	1	no	—
136	Dic₆₁	—	2	1	no	—
137	Dic₆₁	#117	1	2	yes	—
138	Dic₆₁	C₂×#41	1	2	yes	—
139	#2923	—	61	1	no	maximal, Sylow
140	C₂×C₆₁	—	2	2	no	—
141	C₂×C₆₁	—	4	1	no	—
142	C₂×C₆₁	C₂×#117	1	1	yes	characteristic
143	C₂×C₆₁	#568	1	1	yes	characteristic
144	C₂×C₆₁	#568	1	1	yes	characteristic
145	D₆₁	—	4	1	no	—
146	C₄×#41	—	61	1	no	—
147	#433	—	61	2	no	—
148	#433	—	61	2	no	—
149	#432	—	61	1	no	—
150	#432	—	61	1	no	—
151	#432	—	61	1	no	—
152	#432	—	61	1	no	—
153	#424	—	61	2	no	—
154	#408	—	61	2	no	—
155	C₄×M₁₆	—	61	1	no	—
156	C₂×#97	—	61	1	no	—
157	C₆₁	#2923	1	1	yes	characteristic, complemented, Sylow
158	C₂×#41	—	61	1	no	—
159	C₂×M₁₆	—	61	2	no	—
160	C₂²×C₈	—	61	2	no	—
161	C₄×Q₈	—	61	1	no	—
162	C₄×Q₈	—	61	1	no	—
163	C₄×D₄	—	61	1	no	—
164	C₄×D₄	—	61	1	no	—
165	C₄×D₄	—	61	1	no	—
166	C₄×D₄	—	61	1	no	—
167	C₄×D₄	—	61	1	no	—
168	C₄×D₄	#6923	1	1	yes	characteristic
169	#117	—	61	1	no	—
170	#117	—	61	1	no	—
171	#117	—	61	1	no	—
172	C₂×C₄²	—	61	1	no	—
173	C₂×C₄²	—	61	1	no	—
174	C₂×C₄²	—	61	1	no	—
175	C₄⋊C₈	—	61	2	no	—
176	C₄⋊C₈	—	61	2	no	—
177	C₄⋊C₈	—	61	2	no	—
178	C₄⋊C₈	—	61	2	no	—
179	C₂²⋊C₈	—	61	2	no	—
180	C₂²⋊C₈	—	61	2	no	—
181	C₂²⋊C₈	—	61	2	no	—
182	C₂²⋊C₈	—	61	2	no	—
183	#97	—	61	1	no	—
184	#97	—	61	1	no	—
185	#97	—	61	2	no	—
186	#97	—	61	1	no	—
187	#97	—	61	1	no	—
188	C₄×C₈	—	61	2	no	—
189	#41	—	122	4	no	—
190	C₂×Q₈	—	61	1	no	—
191	C₂×D₄	—	61	1	no	—
192	C₂×D₄	—	61	1	no	—
193	C₂×D₄	C₂×#6923	1	1	yes	characteristic
194	C₂²×C₄	—	61	1	no	—
195	C₂²×C₄	—	61	1	no	—
196	C₂²×C₄	—	61	1	no	—
197	C₂²×C₄	—	61	1	no	—
198	C₂²×C₄	—	61	1	no	—
199	C₂²×C₄	—	61	1	no	—
200	C₂²×C₄	—	61	1	no	—
201	C₂²×C₄	C₂×#6923	1	1	yes	characteristic
202	C₂²×C₄	C₂×#6923	1	1	yes	characteristic
203	M₁₆	—	122	4	no	—
204	C₂×C₈	—	61	2	no	—
205	C₂×C₈	—	61	2	no	—
206	C₂×C₈	—	61	2	no	—
207	C₂×C₈	—	61	2	no	—
208	C₂×C₈	—	122	4	no	—
209	C₄⋊C₄	—	61	1	no	—
210	C₄⋊C₄	—	61	1	no	—
211	C₄⋊C₄	—	61	1	no	—
212	C₄⋊C₄	—	61	1	no	—
213	C₄⋊C₄	—	61	1	no	—
214	C₄⋊C₄	C₂×#6923	1	1	yes	characteristic
215	C₂²⋊C₄	—	61	1	no	—
216	C₂²⋊C₄	—	61	1	no	—
217	C₂²⋊C₄	—	61	1	no	—
218	C₂²⋊C₄	—	61	1	no	—
219	C₂²⋊C₄	C₂×#6923	1	1	yes	characteristic
220	C₂²⋊C₄	C₂×#6923	1	1	yes	characteristic
221	C₄²	—	61	1	no	—
222	C₄²	—	61	1	no	—
223	C₄²	—	61	1	no	—
224	C₄²	—	61	1	no	—
225	C₄²	—	61	1	no	—
226	C₄²	—	61	1	no	—
227	C₄²	—	61	1	no	—
228	C₄²	—	61	1	no	—
229	C₄²	—	61	1	no	—
230	C₄²	C₂×#6923	1	1	yes	characteristic
231	C₂³	—	61	1	no	—
232	C₂³	C₂²×#6923	1	1	yes	characteristic
233	C₂³	C₂²×#6923	1	1	yes	characteristic
234	Q₈	—	122	2	no	—
235	D₄	—	2	2	no	—
236	D₄	—	122	2	no	—
237	D₄	—	122	2	no	—
238	C₂×C₄	—	4	1	no	—
239	C₂×C₄	—	4	1	no	—
240	C₂×C₄	—	61	4	no	—
241	C₂×C₄	—	61	1	no	—
242	C₂×C₄	—	61	1	no	—
243	C₂×C₄	—	61	1	no	—
244	C₂×C₄	—	61	1	no	—
245	C₂×C₄	—	61	1	no	—
246	C₂×C₄	—	61	1	no	—
247	C₂×C₄	—	61	1	no	—
248	C₂×C₄	—	122	2	no	—
249	C₂×C₄	—	122	2	no	—
250	C₂×C₄	—	244	1	no	—
251	C₂×C₄	—	244	1	no	—
252	C₂×C₄	—	244	1	no	—
253	C₂×C₄	—	244	1	no	—
254	C₂×C₄	C₂²×#6923	1	1	yes	characteristic
255	C₂×C₄	C₂²×#6923	1	1	yes	characteristic
256	C₂×C₄	C₂²×#6923	1	1	yes	characteristic
257	C₂×C₄	C₂²×#6923	1	1	yes	characteristic
258	C₂×C₄	C₂²×#6923	1	1	yes	characteristic
259	C₈	—	122	2	no	—
260	C₈	—	122	2	no	—
261	C₈	—	122	2	no	—
262	C₈	—	122	2	no	—
263	C₂²	—	2	1	no	—
264	C₂²	—	2	1	no	—
265	C₂²	—	2	1	no	—
266	C₂²	—	2	1	no	—
267	C₂²	—	2	1	no	—
268	C₂²	—	122	1	no	—
269	C₂²	—	122	1	no	—
270	C₂²	—	122	1	no	—
271	C₂²	#14519856	1	2	yes	—
272	C₂²	C₂³×#6923	1	1	yes	characteristic
273	C₄	—	2	1	no	—
274	C₄	—	2	1	no	—
275	C₄	—	2	1	no	—
276	C₄	—	2	1	no	—
277	C₄	—	2	1	no	—
278	C₄	—	61	2	no	—
279	C₄	—	61	2	no	—
280	C₄	—	122	1	no	—
281	C₄	—	122	1	no	—
282	C₄	—	122	1	no	—
283	C₄	—	122	1	no	—
284	C₄	—	122	1	no	—
285	C₂	—	2	2	no	—
286	C₂	—	4	1	no	—
287	C₂	—	244	1	no	—
288	C₂	C₂×#14519856	1	1	yes	characteristic
289	C₂	#16840840	1	1	yes	characteristic
290	C₂	#16840841	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂²	C₂³×#6923
commutator, socle	C₂²×C₆₁	C₂³×C₄
Fitting	C₄×D₄×C₆₁	C₄
hypercenter	C₄×D₄	#6923
hypocenter	C₆₁	#2923
LCS	G ▷ C₂²×C₆₁ ▷ C₆₁
UCS	C₁ ◁ C₂² ◁ C₄×D₄

Complements

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

N	H ≅ G/N	type
1. #16840787	C₂	semidirect
2. #16840786	C₂	semidirect
3. #16840785	C₂	semidirect
4. #16840784	C₂	semidirect
5. #16840783	C₂	semidirect
6. #16840782	C₂	semidirect
7. #16840768	C₂	semidirect
9. C₄×#11758217	C₂	semidirect
10. C₂×#14519832	C₂	semidirect
157. C₆₁	#2923	semidirect