Subgroups — #21462616

Order 7808.

356 conjugacy classes of subgroups.

The subgroup lattice has 218 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#16840814	C₂	1	2	yes	maximal, complemented
2	#16840814	C₂	1	2	yes	maximal, complemented
3	#16840813	C₂	1	2	yes	maximal, complemented
4	#16840811	C₂	1	2	yes	maximal, complemented
5	#16840768	C₂	1	2	yes	maximal
6	C₂×#14519844	C₂	1	1	yes	maximal, characteristic
7	#16840726	C₂	1	2	yes	maximal
8	C₄×#11758217	C₂	1	1	yes	maximal, characteristic
9	#16840314	C₂	1	1	yes	maximal, characteristic
10	C₂×#11758217	C₂²	1	1	yes	characteristic
11	C₂×#11758217	C₂²	1	1	yes	characteristic
12	C₂²×#92420	C₂²	1	2	yes	—
13	#14519844	C₂²	1	1	yes	characteristic
14	#14519844	C₂²	1	2	yes	—
15	#14519844	C₂²	1	1	yes	characteristic
16	#14519844	C₂²	1	2	yes	—
17	#14519844	C₂²	1	2	yes	—
18	C₄⋊#92420	C₂²	1	2	yes	—
19	C₄⋊#92420	C₂²	1	2	yes	—
20	#14519832	C₂²	1	2	yes	—
21	#14519832	C₂²	1	2	yes	—
22	#14519832	C₂²	1	2	yes	—
23	C₄×#92420	C₂²	1	2	yes	—
24	C₄×#92420	C₂²	1	2	yes	—
25	C₄×#92420	C₂²	1	2	yes	—
26	#14519776	C₄	1	1	yes	characteristic
27	#14519775	C₄	1	1	yes	characteristic
28	#14519767	C₄	1	2	yes	—
29	#14519756	C₄	1	1	yes	characteristic
30	#14519754	C₄	1	2	yes	—
31	Q₈×Dic₆₁	C₂²	1	1	yes	characteristic
32	#14519742	C₂²	1	2	yes	—
33	#14519742	C₂²	1	1	yes	characteristic
34	#14519733	C₂²	1	2	yes	—
35	C₂×C₄×Dic₆₁	C₂²	1	1	yes	characteristic
36	C₂²⋊Q₈×C₆₁	C₄	1	1	yes	characteristic
37	C₂×Q₈×C₆₁	C₂×C₄	1	1	yes	characteristic
38	C₂²×C₄×C₆₁	C₂×C₄	1	1	yes	characteristic
39	C₂²×Dic₆₁	C₂×C₄	1	1	yes	characteristic
40	C₂²×Dic₆₁	C₂³	1	1	yes	characteristic
41	C₂×Dic₁₂₂	C₂×C₄	1	2	yes	—
42	C₂×Dic₁₂₂	C₂×C₄	1	1	yes	characteristic
43	#11758217	—	2	2	no	—
44	#11758217	Q₈	1	4	yes	complemented
45	C₂×#92420	—	2	4	no	—
46	C₂×#92420	C₂³	1	2	yes	—
47	C₂×#92420	C₂³	1	2	yes	—
48	C₂×#92420	C₂³	1	2	yes	—
49	C₂×#92420	C₂³	1	2	yes	—
50	C₄⋊C₄×C₆₁	C₂×C₄	1	2	yes	—
51	C₄⋊C₄×C₆₁	C₂×C₄	1	1	yes	characteristic
52	C₂²⋊C₄×C₆₁	C₂×C₄	1	2	yes	—
53	C₂²⋊Dic₆₁	C₂×C₄	1	2	yes	—
54	C₄⋊Dic₆₁	C₂×C₄	1	2	yes	—
55	C₄⋊Dic₆₁	C₂×C₄	1	1	yes	characteristic
56	#11758196	C₂×C₄	1	2	yes	—
57	#11758196	C₂×C₄	1	2	yes	—
58	#11758196	C₂×C₄	1	2	yes	—
59	#11758196	C₂×C₄	1	2	yes	—
60	#11758196	C₂×C₄	1	2	yes	—
61	C₄×Dic₆₁	C₂×C₄	1	1	yes	characteristic
62	C₄×Dic₆₁	C₂×C₄	1	1	yes	characteristic
63	C₄×Dic₆₁	C₂³	1	1	yes	characteristic
64	C₄×Dic₆₁	C₂³	1	2	yes	—
65	C₄×Dic₆₁	C₂³	1	2	yes	—
66	C₄×Dic₆₁	C₂³	1	1	yes	characteristic
67	C₂³×C₆₁	C₂²×C₄	1	1	yes	characteristic
68	Q₈×C₆₁	—	4	1	no	—
69	C₂×C₄×C₆₁	—	4	1	no	—
70	C₂×C₄×C₆₁	C₂²×C₄	1	1	yes	characteristic
71	C₂×C₄×C₆₁	C₂²×C₄	1	1	yes	characteristic
72	C₂×C₄×C₆₁	C₂²×C₄	1	2	yes	—
73	C₂×C₄×C₆₁	C₂²×C₄	1	2	yes	—
74	C₂×Dic₆₁	—	2	2	no	—
75	C₂×Dic₆₁	C₂²×C₄	1	1	yes	characteristic
76	C₂×Dic₆₁	C₂²×C₄	1	2	yes	—
77	C₂×Dic₆₁	C₂²×C₄	1	2	yes	—
78	C₂×Dic₆₁	C₂²×C₄	1	1	yes	characteristic
79	C₂×Dic₆₁	C₂²×C₄	1	1	yes	characteristic
80	C₂×Dic₆₁	C₂×Q₈	1	2	yes	—
81	C₂×Dic₆₁	#41	1	2	yes	—
82	C₂×Dic₆₁	C₂⁴	1	1	yes	characteristic
83	Dic₁₂₂	—	2	4	no	—
84	Dic₁₂₂	—	4	1	no	—
85	#92420	—	2	2	no	—
86	#92420	—	2	2	no	—
87	#92420	—	2	2	no	—
88	#92420	C₂×Q₈	1	4	yes	—
89	C₂²×C₆₁	—	2	1	no	—
90	C₂²×C₆₁	—	2	1	no	—
91	C₂²×C₆₁	C₄×Q₈	1	2	yes	—
92	C₂²×C₆₁	C₂³×C₄	1	1	yes	characteristic
93	C₄×C₆₁	—	2	2	no	—
94	C₄×C₆₁	—	2	1	no	—
95	C₄×C₆₁	—	2	1	no	—
96	C₄×C₆₁	—	2	2	no	—
97	Dic₆₁	—	2	2	no	—
98	Dic₆₁	—	2	1	no	—
99	Dic₆₁	—	2	2	no	—
100	Dic₆₁	C₄×Q₈	1	2	yes	—
101	Dic₆₁	#131	1	2	yes	—
102	Dic₆₁	C₂²×Q₈	1	1	yes	characteristic
103	Dic₆₁	C₂×#41	1	1	yes	characteristic
104	#2962	—	61	1	no	maximal, Sylow
105	C₂×C₆₁	—	2	2	no	—
106	C₂×C₆₁	C₂×C₄×Q₈	1	1	yes	characteristic
107	C₂×C₆₁	C₂×#131	1	1	yes	characteristic
108	C₂×C₆₁	#568	1	1	yes	characteristic
109	#533	—	61	1	no	—
110	#446	—	61	2	no	—
111	#446	—	61	2	no	—
112	#446	—	61	2	no	—
113	C₈×Q₈	—	61	2	no	—
114	#424	—	61	2	no	—
115	C₂×C₄⋊C₈	—	61	1	no	—
116	#405	—	61	2	no	—
117	C₄×M₁₆	—	61	1	no	—
118	C₆₁	#2962	1	1	yes	characteristic, complemented, Sylow
119	C₂×M₁₆	—	61	1	no	—
120	C₂×M₁₆	—	61	1	no	—
121	C₂²×C₈	—	61	2	no	—
122	C₄⋊Q₈	—	61	1	no	—
123	C₄⋊Q₈	—	61	1	no	—
124	#125	—	61	2	no	—
125	C₂²⋊Q₈	—	61	1	no	—
126	C₂²⋊Q₈	—	61	2	no	—
127	C₂²⋊Q₈	#6923	1	1	yes	characteristic
128	C₄×Q₈	—	61	1	no	—
129	C₄×Q₈	—	61	2	no	—
130	C₄×Q₈	—	61	1	no	—
131	#117	—	61	2	no	—
132	C₂×C₄²	—	61	1	no	—
133	C₄⋊C₈	—	61	2	no	—
134	C₄⋊C₈	—	61	2	no	—
135	C₄⋊C₈	—	61	1	no	—
136	C₄⋊C₈	—	61	1	no	—
137	C₄⋊C₈	—	61	2	no	—
138	C₄⋊C₈	—	61	2	no	—
139	C₄⋊C₈	—	61	2	no	—
140	#97	—	61	2	no	—
141	#97	—	61	2	no	—
142	#97	—	61	2	no	—
143	C₄×C₈	—	61	2	no	—
144	C₄×C₈	—	61	2	no	—
145	C₄×C₈	—	61	2	no	—
146	C₂×Q₈	—	61	2	no	—
147	C₂×Q₈	—	61	1	no	—
148	C₂×Q₈	C₂×#6923	1	1	yes	characteristic
149	C₂²×C₄	—	61	1	no	—
150	C₂²×C₄	—	61	1	no	—
151	C₂²×C₄	C₂×#6923	1	1	yes	characteristic
152	M₁₆	—	61	4	no	—
153	M₁₆	—	122	2	no	—
154	C₂×C₈	—	61	2	no	—
155	C₂×C₈	—	61	2	no	—
156	C₂×C₈	—	61	2	no	—
157	C₂×C₈	—	61	2	no	—
158	C₂×C₈	—	122	4	no	—
159	C₄⋊C₄	—	61	2	no	—
160	C₄⋊C₄	—	61	2	no	—
161	C₄⋊C₄	—	61	2	no	—
162	C₄⋊C₄	—	61	2	no	—
163	C₄⋊C₄	—	61	2	no	—
164	C₄⋊C₄	—	61	2	no	—
165	C₄⋊C₄	—	61	1	no	—
166	C₄⋊C₄	C₂×#6923	1	2	yes	—
167	C₄⋊C₄	C₂×#6923	1	1	yes	characteristic
168	C₂²⋊C₄	—	61	2	no	—
169	C₂²⋊C₄	C₂×#6923	1	2	yes	—
170	C₄²	—	61	1	no	—
171	C₄²	—	61	2	no	—
172	C₄²	—	61	2	no	—
173	C₄²	—	61	1	no	—
174	C₄²	—	61	1	no	—
175	C₄²	—	61	1	no	—
176	C₂³	C₂²×#6923	1	1	yes	characteristic
177	Q₈	—	4	1	no	—
178	Q₈	—	122	4	no	—
179	Q₈	—	244	1	no	—
180	C₂×C₄	—	4	1	no	—
181	C₂×C₄	—	61	2	no	—
182	C₂×C₄	—	61	2	no	—
183	C₂×C₄	—	61	1	no	—
184	C₂×C₄	—	61	1	no	—
185	C₂×C₄	—	61	1	no	—
186	C₂×C₄	—	61	1	no	—
187	C₂×C₄	—	61	2	no	—
188	C₂×C₄	—	61	2	no	—
189	C₂×C₄	—	122	2	no	—
190	C₂×C₄	C₂²×#6923	1	1	yes	characteristic
191	C₂×C₄	C₂²×#6923	1	1	yes	characteristic
192	C₂×C₄	C₂²×#6923	1	2	yes	—
193	C₂×C₄	C₂²×#6923	1	2	yes	—
194	C₈	—	61	4	no	—
195	C₈	—	122	2	no	—
196	C₈	—	122	2	no	—
197	C₈	—	122	2	no	—
198	C₂²	—	2	1	no	—
199	C₂²	—	2	1	no	—
200	C₂²	Q₈×#6923	1	2	yes	—
201	C₂²	C₂³×#6923	1	1	yes	characteristic
202	C₄	—	2	2	no	—
203	C₄	—	2	1	no	—
204	C₄	—	2	1	no	—
205	C₄	—	2	2	no	—
206	C₄	—	61	2	no	—
207	C₄	—	61	1	no	—
208	C₄	—	61	1	no	—
209	C₄	—	61	2	no	—
210	C₄	—	122	1	no	—
211	C₄	—	122	2	no	—
212	C₄	—	122	2	no	—
213	C₂	—	2	2	no	—
214	C₂	C₂×Q₈×#6923	1	1	yes	characteristic
215	C₂	D₆₁⋊#131	1	1	yes	characteristic
216	C₂	#16840840	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂²	C₂³×#6923
commutator, socle	C₂²×C₆₁	C₂³×C₄
Fitting	C₂²⋊Q₈×C₆₁	C₄
hypercenter	C₂²⋊Q₈	#6923
hypocenter	C₆₁	#2962
LCS	G ▷ C₂²×C₆₁ ▷ C₆₁
UCS	C₁ ◁ C₂² ◁ C₂²⋊Q₈

Complements

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

N	H ≅ G/N	type
1. #16840814	C₂	semidirect
2. #16840814	C₂	semidirect
3. #16840813	C₂	semidirect
4. #16840811	C₂	semidirect
44. #11758217	Q₈	semidirect
118. C₆₁	#2962	semidirect