Subgroups — #21455607

Order 7808.

388 conjugacy classes of subgroups.

The subgroup lattice has 270 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#16840319	C₂	1	1	yes	maximal, characteristic, complemented
2	#16840069	C₂	1	2	yes	maximal, complemented
3	#16840062	C₂	1	2	yes	maximal, complemented
4	#16840057	C₂	1	1	yes	maximal, characteristic, complemented
5	#16840052	C₂	1	1	yes	maximal, characteristic, complemented
6	#16840047	C₂	1	1	yes	maximal, characteristic, complemented
7	#16840042	C₂	1	1	yes	maximal, characteristic, complemented
8	#16839798	C₂	1	2	yes	maximal, complemented
9	C₂×C₄⋊#92418	C₂	1	1	yes	maximal, characteristic, complemented
10	#16839758	C₂	1	2	yes	maximal
11	C₄×#41×C₆₁	C₂	1	1	yes	maximal, characteristic
12	C₂²×#92418	C₂²	1	2	yes	—
13	#14519777	C₂²	1	1	yes	characteristic
14	#14519777	C₂²	1	1	yes	characteristic
15	#14519769	C₂²	1	2	yes	—
16	#14519756	—	2	1	no	—
17	#14519756	—	2	1	no	—
18	C₄×Dic₁₂₂	—	2	1	no	—
19	C₄×Dic₁₂₂	—	2	1	no	—
20	#14519736	C₂²	1	2	yes	—
21	#14519710	C₂²	1	2	yes	—
22	#14519710	C₂²	1	2	yes	—
23	#14519708	C₂²	1	2	yes	—
24	#14519708	C₂²	1	2	yes	—
25	C₄⋊#92418	C₂²	1	1	yes	characteristic
26	C₄⋊#92418	C₂²	1	1	yes	characteristic
27	C₄⋊#92418	C₂²	1	2	yes	—
28	#14519701	C₂²	1	2	yes	—
29	#14519701	C₂²	1	2	yes	—
30	#14519698	C₂²	1	2	yes	—
31	#14519698	C₂²	1	2	yes	—
32	C₂×#41×C₆₁	C₂²	1	1	yes	characteristic
33	C₄×Q₈×C₆₁	C₂²	1	1	yes	characteristic
34	C₄×Q₈×C₆₁	C₂²	1	1	yes	characteristic
35	C₄×D₄×C₆₁	—	2	1	no	—
36	C₄×D₄×C₆₁	—	2	1	no	—
37	C₄×D₄×C₆₁	C₂²	1	1	yes	characteristic
38	C₄×D₄×C₆₁	C₂²	1	1	yes	characteristic
39	#117×C₆₁	—	2	1	no	—
40	#117×C₆₁	C₂²	1	1	yes	characteristic
41	C₂×C₄²×C₆₁	—	2	1	no	—
42	C₂×C₄²×C₆₁	C₂²	1	1	yes	characteristic
43	#41×C₆₁	—	2	2	no	—
44	#41×C₆₁	Q₈	1	4	yes	complemented
45	C₂×Q₈×C₆₁	C₂³	1	1	yes	characteristic
46	C₂×D₄×C₆₁	—	2	1	no	—
47	C₂×D₄×C₆₁	C₂³	1	1	yes	characteristic
48	C₂²×C₄×C₆₁	—	2	1	no	—
49	C₂²×C₄×C₆₁	—	2	1	no	—
50	C₂²×C₄×C₆₁	—	2	1	no	—
51	C₂²×C₄×C₆₁	D₄	1	1	yes	characteristic
52	C₂²×C₄×C₆₁	D₄	1	1	yes	characteristic
53	C₂²×C₄×C₆₁	C₂³	1	1	yes	characteristic
54	C₂×Dic₁₂₂	—	2	1	no	—
55	C₂×Dic₁₂₂	—	2	1	no	—
56	C₄⋊C₄×C₆₁	—	2	1	no	—
57	C₄⋊C₄×C₆₁	—	2	1	no	—
58	C₄⋊C₄×C₆₁	C₂³	1	1	yes	characteristic
59	C₄⋊C₄×C₆₁	C₂³	1	1	yes	characteristic
60	C₂²⋊C₄×C₆₁	—	2	1	no	—
61	C₂²⋊C₄×C₆₁	—	2	1	no	—
62	C₂²⋊C₄×C₆₁	—	2	1	no	—
63	C₄²×C₆₁	—	2	1	no	—
64	C₄²×C₆₁	—	2	1	no	—
65	C₄²×C₆₁	—	2	1	no	—
66	C₄²×C₆₁	D₄	1	1	yes	characteristic
67	C₄²×C₆₁	D₄	1	1	yes	characteristic
68	C₄²×C₆₁	C₂³	1	1	yes	characteristic
69	C₄²×C₆₁	C₂³	1	1	yes	characteristic
70	C₂²⋊Dic₆₁	—	2	2	no	—
71	C₄⋊Dic₆₁	—	2	1	no	—
72	C₄⋊Dic₆₁	—	2	1	no	—
73	C₄⋊Dic₆₁	C₂³	1	2	yes	—
74	C₄⋊Dic₆₁	C₂³	1	2	yes	—
75	#11758196	—	2	2	no	—
76	#11758196	—	2	2	no	—
77	C₄×Dic₆₁	—	2	2	no	—
78	C₂×#92418	—	2	4	no	—
79	C₂×#92418	C₂³	1	2	yes	—
80	C₂×#92418	C₂³	1	2	yes	—
81	C₂³×C₆₁	—	2	1	no	—
82	C₂³×C₆₁	C₂×D₄	1	1	yes	characteristic
83	Q₈×C₆₁	—	2	1	no	—
84	Q₈×C₆₁	C₂×Q₈	1	2	yes	—
85	D₄×C₆₁	—	2	4	no	—
86	D₄×C₆₁	—	2	1	no	—
87	D₄×C₆₁	C₂×Q₈	1	2	yes	—
88	C₂×C₄×C₆₁	—	2	1	no	—
89	C₂×C₄×C₆₁	—	2	4	no	—
90	C₂×C₄×C₆₁	—	2	2	no	—
91	C₂×C₄×C₆₁	—	2	1	no	—
92	C₂×C₄×C₆₁	—	2	1	no	—
93	C₂×C₄×C₆₁	—	4	1	no	—
94	C₂×C₄×C₆₁	—	4	2	no	—
95	C₂×C₄×C₆₁	—	4	2	no	—
96	C₂×C₄×C₆₁	C₂×D₄	1	1	yes	characteristic
97	C₂×C₄×C₆₁	C₂×D₄	1	1	yes	characteristic
98	C₂×C₄×C₆₁	C₂×D₄	1	1	yes	characteristic
99	C₂×C₄×C₆₁	C₂×D₄	1	1	yes	characteristic
100	C₂×C₄×C₆₁	C₂×D₄	1	1	yes	characteristic
101	C₂×C₄×C₆₁	C₂×Q₈	1	2	yes	—
102	C₂×C₄×C₆₁	#41	1	2	yes	—
103	C₂×C₄×C₆₁	C₂⁴	1	1	yes	characteristic
104	C₂×Dic₆₁	—	2	2	no	—
105	C₂×Dic₆₁	—	2	2	no	—
106	Dic₁₂₂	—	4	2	no	—
107	Dic₁₂₂	—	4	2	no	—
108	#92418	—	2	2	no	—
109	#92418	—	2	2	no	—
110	C₂²×C₆₁	—	2	1	no	—
111	C₂²×C₆₁	—	2	1	no	—
112	C₂²×C₆₁	—	2	2	no	—
113	C₂²×C₆₁	—	4	1	no	—
114	C₂²×C₆₁	—	4	1	no	—
115	C₂²×C₆₁	C₂²⋊Q₈	1	2	yes	—
116	C₂²×C₆₁	C₂²×D₄	1	1	yes	characteristic
117	C₄×C₆₁	—	2	2	no	—
118	C₄×C₆₁	—	2	1	no	—
119	C₄×C₆₁	—	2	1	no	—
120	C₄×C₆₁	—	2	1	no	—
121	C₄×C₆₁	—	4	1	no	—
122	C₄×C₆₁	—	4	1	no	—
123	C₄×C₆₁	C₂²⋊Q₈	1	2	yes	—
124	C₄×C₆₁	#135	1	2	yes	—
125	C₄×C₆₁	C₂²×Q₈	1	1	yes	characteristic
126	C₄×C₆₁	C₂×#41	1	1	yes	characteristic
127	Dic₆₁	—	4	2	no	—
128	Dic₆₁	—	4	2	no	—
129	#3076	—	61	1	no	maximal, Sylow
130	C₂×C₆₁	—	2	2	no	—
131	C₂×C₆₁	—	4	2	no	—
132	C₂×C₆₁	C₂×C₂²⋊Q₈	1	1	yes	characteristic
133	C₂×C₆₁	C₂×#135	1	1	yes	characteristic
134	C₂×C₆₁	#575	1	1	yes	characteristic
135	#533	—	61	1	no	—
136	C₄×#41	D₆₁	1	1	yes	characteristic
137	#479	—	61	2	no	—
138	#478	—	61	2	no	—
139	#477	—	61	1	no	—
140	#476	—	61	1	no	—
141	#475	—	61	1	no	—
142	#474	—	61	1	no	—
143	#427	—	61	2	no	—
144	C₂×C₄⋊C₈	—	61	1	no	—
145	#416	—	61	2	no	—
146	C₆₁	#3076	1	1	yes	characteristic, complemented, Sylow
147	C₂×#41	C₂×D₆₁	1	1	yes	characteristic
148	C₂²×C₈	—	61	2	no	—
149	C₄⋊Q₈	—	61	1	no	—
150	C₄⋊Q₈	—	61	1	no	—
151	#125	—	61	2	no	—
152	C₂²⋊Q₈	—	122	1	no	—
153	C₂²⋊Q₈	—	122	1	no	—
154	C₄×Q₈	—	122	1	no	—
155	C₄×Q₈	—	122	1	no	—
156	C₄×Q₈	C₂×D₆₁	1	1	yes	characteristic
157	C₄×Q₈	C₂×D₆₁	1	1	yes	characteristic
158	C₄×D₄	—	2	1	no	—
159	C₄×D₄	—	2	1	no	—
160	C₄×D₄	C₂×D₆₁	1	1	yes	characteristic
161	C₄×D₄	C₂×D₆₁	1	1	yes	characteristic
162	#117	—	2	1	no	—
163	#117	—	61	2	no	—
164	#117	C₂×D₆₁	1	1	yes	characteristic
165	C₂×C₄²	—	2	1	no	—
166	C₂×C₄²	C₂×D₆₁	1	1	yes	characteristic
167	#107	—	61	2	no	—
168	#107	—	61	2	no	—
169	#106	—	61	2	no	—
170	#106	—	61	2	no	—
171	C₄⋊C₈	—	61	1	no	—
172	C₄⋊C₈	—	61	1	no	—
173	C₄⋊C₈	—	61	2	no	—
174	#103	—	61	2	no	—
175	#103	—	61	2	no	—
176	#102	—	61	2	no	—
177	#102	—	61	2	no	—
178	#41	—	2	2	no	—
179	#41	Dic₁₂₂	1	4	yes	complemented
180	C₂×Q₈	—	122	1	no	—
181	C₂×Q₈	—	122	1	no	—
182	C₂×Q₈	C₂²×D₆₁	1	1	yes	characteristic
183	C₂×D₄	—	2	1	no	—
184	C₂×D₄	C₂²×D₆₁	1	1	yes	characteristic
185	C₂²×C₄	—	2	1	no	—
186	C₂²×C₄	—	2	1	no	—
187	C₂²×C₄	—	2	1	no	—
188	C₂²×C₄	C₂²⋊D₆₁	1	1	yes	characteristic
189	C₂²×C₄	C₂²⋊D₆₁	1	1	yes	characteristic
190	C₂²×C₄	C₂²×D₆₁	1	1	yes	characteristic
191	C₂×C₈	—	61	2	no	—
192	C₂×C₈	—	61	2	no	—
193	C₂×C₈	—	122	4	no	—
194	C₄⋊C₄	—	2	1	no	—
195	C₄⋊C₄	—	2	1	no	—
196	C₄⋊C₄	—	61	2	no	—
197	C₄⋊C₄	—	61	2	no	—
198	C₄⋊C₄	—	122	2	no	—
199	C₄⋊C₄	—	122	2	no	—
200	C₄⋊C₄	—	122	1	no	—
201	C₄⋊C₄	—	122	1	no	—
202	C₄⋊C₄	C₂²×D₆₁	1	1	yes	characteristic
203	C₄⋊C₄	C₂²×D₆₁	1	1	yes	characteristic
204	C₂²⋊C₄	—	2	1	no	—
205	C₂²⋊C₄	—	2	1	no	—
206	C₂²⋊C₄	—	2	1	no	—
207	C₂²⋊C₄	—	122	2	no	—
208	C₄²	—	2	1	no	—
209	C₄²	—	2	1	no	—
210	C₄²	—	2	1	no	—
211	C₄²	—	122	2	no	—
212	C₄²	C₂²⋊D₆₁	1	1	yes	characteristic
213	C₄²	C₂²⋊D₆₁	1	1	yes	characteristic
214	C₄²	C₂²×D₆₁	1	1	yes	characteristic
215	C₄²	C₂²×D₆₁	1	1	yes	characteristic
216	C₂³	—	2	1	no	—
217	C₂³	C₂×C₂²⋊D₆₁	1	1	yes	characteristic
218	Q₈	—	2	1	no	—
219	Q₈	—	244	2	no	—
220	Q₈	—	244	2	no	—
221	Q₈	C₂×Dic₁₂₂	1	2	yes	—
222	D₄	—	2	4	no	—
223	D₄	—	2	1	no	—
224	D₄	C₂×Dic₁₂₂	1	2	yes	—
225	C₂×C₄	—	2	1	no	—
226	C₂×C₄	—	2	4	no	—
227	C₂×C₄	—	2	2	no	—
228	C₂×C₄	—	2	1	no	—
229	C₂×C₄	—	2	1	no	—
230	C₂×C₄	—	4	1	no	—
231	C₂×C₄	—	4	2	no	—
232	C₂×C₄	—	4	2	no	—
233	C₂×C₄	—	122	2	no	—
234	C₂×C₄	—	122	2	no	—
235	C₂×C₄	C₂×Dic₁₂₂	1	2	yes	—
236	C₂×C₄	#11758222	1	2	yes	—
237	C₂×C₄	C₂×C₂²⋊D₆₁	1	1	yes	characteristic
238	C₂×C₄	C₂×C₂²⋊D₆₁	1	1	yes	characteristic
239	C₂×C₄	C₂×C₂²⋊D₆₁	1	1	yes	characteristic
240	C₂×C₄	C₂×C₂²⋊D₆₁	1	1	yes	characteristic
241	C₂×C₄	C₂×C₂²⋊D₆₁	1	1	yes	characteristic
242	C₂×C₄	C₂³×D₆₁	1	1	yes	characteristic
243	C₈	—	122	2	no	—
244	C₈	—	122	2	no	—
245	C₂²	—	2	1	no	—
246	C₂²	—	2	1	no	—
247	C₂²	—	2	2	no	—
248	C₂²	—	4	1	no	—
249	C₂²	—	4	1	no	—
250	C₂²	#14519756	1	2	yes	—
251	C₂²	C₂²×C₂²⋊D₆₁	1	1	yes	characteristic
252	C₄	—	2	2	no	—
253	C₄	—	2	1	no	—
254	C₄	—	2	1	no	—
255	C₄	—	2	1	no	—
256	C₄	—	4	1	no	—
257	C₄	—	4	1	no	—
258	C₄	—	244	2	no	—
259	C₄	—	244	2	no	—
260	C₄	#14519756	1	2	yes	—
261	C₄	#14519797	1	2	yes	—
262	C₄	C₂²×Dic₁₂₂	1	1	yes	characteristic
263	C₄	C₂×#11758222	1	1	yes	characteristic
264	C₂	—	2	2	no	—
265	C₂	—	4	2	no	—
266	C₂	C₂×#14519756	1	1	yes	characteristic
267	C₂	C₂×#14519797	1	1	yes	characteristic
268	C₂	#16840560	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₂×C₂²⋊D₆₁
commutator	C₂×C₄×C₆₁	C₂⁴
Frattini	C₂×C₄	C₂³×D₆₁
Fitting	C₄×#41×C₆₁	C₂
socle	C₂²×C₆₁	C₂²×D₄
hypercenter	C₄×#41	D₆₁
hypocenter	C₆₁	#3076
LCS	G ▷ C₂×C₄×C₆₁ ▷ C₂×C₆₁ ▷ C₆₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ C₄×#41

Complements

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

N	H ≅ G/N	type
1. #16840319	C₂	semidirect
2. #16840069	C₂	semidirect
3. #16840062	C₂	semidirect
4. #16840057	C₂	semidirect
5. #16840052	C₂	semidirect
6. #16840047	C₂	semidirect
7. #16840042	C₂	semidirect
8. #16839798	C₂	semidirect
9. C₂×C₄⋊#92418	C₂	semidirect
44. #41×C₆₁	Q₈	semidirect
146. C₆₁	#3076	semidirect
179. #41	Dic₁₂₂	semidirect