Subgroups — C₂×#24905434 #35314166

Order 21248.

398 conjugacy classes of subgroups.

The subgroup lattice has 244 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₄×Dic₃₃₂	C₂	1	1	yes	maximal, characteristic
2	C₂×#19127452	C₂	1	1	yes	maximal, characteristic
3	#24905434	C₂	1	4	yes	maximal, direct product factor
4	C₂×#346×C₈₃	C₂	1	1	yes	maximal, characteristic
5	C₂²×Dic₃₃₂	C₄	1	1	yes	characteristic
6	C₂×C₄×Dic₁₆₆	—	2	1	no	—
7	C₄×Dic₃₃₂	—	2	1	no	—
8	C₄×Dic₃₃₂	—	2	1	no	—
9	C₄×Dic₃₃₂	C₄	1	2	yes	—
10	C₄×Dic₃₃₂	C₂²	1	2	yes	—
11	C₂×#14581726	C₄	1	1	yes	characteristic
12	C₂×C₄⋊(C₈₃⋊C₈)	—	2	1	no	—
13	C₂×#14581717	—	2	1	no	—
14	#19127452	—	2	1	no	—
15	#19127452	C₂²	1	2	yes	—
16	C₂²×C₁₆×C₈₃	—	2	1	no	—
17	C₂×C₄×C₈×C₈₃	C₂²	1	1	yes	characteristic
18	#346×C₈₃	—	2	1	no	—
19	#346×C₈₃	C₂²	1	2	yes	—
20	C₂²×Dic₁₆₆	—	2	1	no	—
21	C₂×Dic₃₃₂	—	2	1	no	—
22	C₂×Dic₃₃₂	—	2	2	no	—
23	C₂×Dic₃₃₂	C₈	1	4	yes	complemented
24	C₂×Dic₃₃₂	C₂×C₄	1	2	yes	—
25	C₂²×C₈₃⋊C₈	—	4	1	no	—
26	C₄×Dic₁₆₆	—	2	2	no	—
27	C₄×Dic₁₆₆	—	2	2	no	—
28	C₄×Dic₁₆₆	—	4	1	no	—
29	C₄×Dic₁₆₆	—	4	1	no	—
30	C₂×C₄⋊Dic₈₃	—	2	1	no	—
31	C₂×Dic₈₃⋊C₄	—	4	1	no	—
32	C₂×C₄×Dic₈₃	—	4	1	no	—
33	#14581726	—	2	1	no	—
34	#14581726	C₂×C₄	1	2	yes	—
35	C₄⋊(C₈₃⋊C₈)	—	2	2	no	—
36	C₄⋊(C₈₃⋊C₈)	—	4	1	no	—
37	#14581717	—	2	2	no	—
38	#14581717	—	2	2	no	—
39	C₂²×C₈×C₈₃	C₂×C₄	1	1	yes	characteristic
40	C₂²×C₈×C₈₃	D₄	1	1	yes	characteristic
41	C₂×C₄²×C₈₃	D₄	1	1	yes	characteristic
42	C₂×C₁₆×C₈₃	—	2	4	no	—
43	C₂×C₁₆×C₈₃	—	2	2	no	—
44	C₄×C₈×C₈₃	C₂×C₄	1	1	yes	characteristic
45	C₄×C₈×C₈₃	D₄	1	1	yes	characteristic
46	C₄×C₈×C₈₃	D₄	1	1	yes	characteristic
47	C₄×C₈×C₈₃	C₂³	1	1	yes	characteristic
48	C₂×Dic₁₆₆	—	2	4	no	—
49	C₂×Dic₁₆₆	—	2	2	no	—
50	C₂×Dic₁₆₆	—	4	1	no	—
51	C₂×Dic₁₆₆	—	4	2	no	—
52	C₂²×Dic₈₃	—	4	1	no	—
53	C₂²×Dic₈₃	—	4	1	no	—
54	Dic₃₃₂	—	2	4	no	—
55	Dic₃₃₂	—	2	2	no	—
56	Dic₃₃₂	C₂×C₈	1	4	yes	complemented
57	C₂×C₈₃⋊C₈	—	4	2	no	—
58	C₂×C₈₃⋊C₈	—	8	2	no	—
59	C₄⋊Dic₈₃	—	2	2	no	—
60	C₄⋊Dic₈₃	—	4	1	no	—
61	Dic₈₃⋊C₄	—	4	2	no	—
62	Dic₈₃⋊C₄	—	4	2	no	—
63	C₄×Dic₈₃	—	4	2	no	—
64	C₄×Dic₈₃	—	4	2	no	—
65	C₂²×C₄×C₈₃	C₂²⋊C₄	1	1	yes	characteristic
66	C₂²×C₄×C₈₃	D₈	1	1	yes	characteristic
67	C₂²×C₄×C₈₃	SD₁₆	1	1	yes	characteristic
68	C₂×C₈×C₈₃	—	2	2	no	—
69	C₂×C₈×C₈₃	C₂²⋊C₄	1	1	yes	characteristic
70	C₂×C₈×C₈₃	C₂²⋊C₄	1	1	yes	characteristic
71	C₂×C₈×C₈₃	C₂×C₈	1	2	yes	—
72	C₂×C₈×C₈₃	M₁₆	1	2	yes	—
73	C₂×C₈×C₈₃	C₂²×C₄	1	1	yes	characteristic
74	C₂×C₈×C₈₃	C₂×D₄	1	1	yes	characteristic
75	C₄²×C₈₃	C₂²⋊C₄	1	1	yes	characteristic
76	C₄²×C₈₃	D₈	1	1	yes	characteristic
77	C₄²×C₈₃	SD₁₆	1	1	yes	characteristic
78	C₄²×C₈₃	C₂×D₄	1	1	yes	characteristic
79	C₁₆×C₈₃	—	2	4	no	—
80	C₂³×C₈₃	#102	1	1	yes	characteristic
81	C₂×C₄×C₈₃	—	2	2	no	—
82	C₂×C₄×C₈₃	—	2	2	no	—
83	C₂×C₄×C₈₃	C₂²⋊C₈	1	2	yes	—
84	C₂×C₄×C₈₃	#102	1	1	yes	characteristic
85	C₂×C₄×C₈₃	#102	1	1	yes	characteristic
86	C₂×C₄×C₈₃	#102	1	1	yes	characteristic
87	C₂×C₄×C₈₃	#104	1	2	yes	—
88	C₂×C₄×C₈₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
89	C₂×C₄×C₈₃	C₂×D₈	1	1	yes	characteristic
90	C₂×C₄×C₈₃	C₂×SD₁₆	1	1	yes	characteristic
91	C₂×Dic₈₃	—	4	4	no	—
92	C₂×Dic₈₃	—	4	2	no	—
93	C₂×Dic₈₃	—	4	2	no	—
94	C₂×Dic₈₃	—	8	2	no	—
95	Dic₁₆₆	—	2	4	no	—
96	Dic₁₆₆	—	4	4	no	—
97	Dic₁₆₆	—	4	2	no	—
98	C₈×C₈₃	—	2	1	no	—
99	C₈×C₈₃	—	2	1	no	—
100	C₈×C₈₃	C₂²⋊C₈	1	2	yes	—
101	C₈×C₈₃	C₂²×C₈	1	1	yes	characteristic
102	C₈×C₈₃	C₂×M₁₆	1	1	yes	characteristic
103	C₈₃⋊C₈	—	8	2	no	—
104	C₂²×C₈₃	D₄⋊C₈	1	2	yes	—
105	C₂²×C₈₃	#360	1	2	yes	—
106	C₂²×C₈₃	#362	1	2	yes	—
107	C₂²×C₈₃	C₂×#102	1	1	yes	characteristic
108	C₄×C₈₃	—	2	1	no	—
109	C₄×C₈₃	—	2	1	no	—
110	C₄×C₈₃	—	2	1	no	—
111	C₄×C₈₃	—	2	1	no	—
112	C₄×C₈₃	D₄⋊C₈	1	2	yes	—
113	C₄×C₈₃	C₂×C₂²⋊C₈	1	1	yes	characteristic
114	C₄×C₈₃	C₂×#104	1	1	yes	characteristic
115	Dic₈₃	—	4	4	no	—
116	Dic₈₃	—	8	2	no	—
117	C₂×#1334	—	83	1	no	maximal, Sylow
118	C₂×C₈₃	#1334	1	4	yes	complemented
119	C₂×C₈₃	C₂×D₄⋊C₈	1	1	yes	characteristic
120	C₂×C₈₃	C₂×#360	1	1	yes	characteristic
121	C₂×C₈₃	C₂×#362	1	1	yes	characteristic
122	C₂×C₄×Q₁₆	—	83	1	no	—
123	C₂×#346	D₈₃	1	1	yes	characteristic
124	C₂×#334	—	83	1	no	—
125	#1334	—	83	4	no	—
126	C₈₃	C₂×#1334	1	1	yes	characteristic, complemented, Sylow
127	C₂²×Q₁₆	—	83	1	no	—
128	C₂×C₄×Q₈	—	166	1	no	—
129	C₂²×C₁₆	—	2	1	no	—
130	C₄×Q₁₆	—	83	2	no	—
131	C₄×Q₁₆	—	83	2	no	—
132	C₄×Q₁₆	—	166	1	no	—
133	C₄×Q₁₆	—	166	1	no	—
134	C₂×#107	—	83	1	no	—
135	C₂×C₄⋊C₈	—	166	1	no	—
136	C₂×#103	—	166	1	no	—
137	C₂×C₄×C₈	C₂×D₈₃	1	1	yes	characteristic
138	#346	—	2	1	no	—
139	#346	C₂×D₈₃	1	2	yes	—
140	#334	—	83	2	no	—
141	#334	—	166	1	no	—
142	C₂²×Q₈	—	166	1	no	—
143	C₂×Q₁₆	—	83	4	no	—
144	C₂×Q₁₆	—	83	2	no	—
145	C₂×Q₁₆	—	166	1	no	—
146	C₂×Q₁₆	—	166	2	no	—
147	C₂²×C₈	—	332	1	no	—
148	C₂²×C₈	C₄×D₈₃	1	1	yes	characteristic
149	C₂²×C₈	C₂²⋊D₈₃	1	1	yes	characteristic
150	C₄×Q₈	—	332	1	no	—
151	C₄×Q₈	—	332	1	no	—
152	C₄×Q₈	—	166	2	no	—
153	C₄×Q₈	—	166	2	no	—
154	C₂×C₄⋊C₄	—	332	1	no	—
155	C₂×C₄⋊C₄	—	166	1	no	—
156	C₂×C₄²	—	332	1	no	—
157	C₂×C₄²	D₃₃₂	1	1	yes	characteristic
158	C₂×C₁₆	—	2	4	no	—
159	C₂×C₁₆	—	2	2	no	—
160	#107	—	83	2	no	—
161	#107	—	166	1	no	—
162	C₄⋊C₈	—	332	1	no	—
163	C₄⋊C₈	—	166	2	no	—
164	#103	—	166	2	no	—
165	#103	—	166	2	no	—
166	C₄×C₈	C₄×D₈₃	1	1	yes	characteristic
167	C₄×C₈	D₃₃₂	1	1	yes	characteristic
168	C₄×C₈	C₂²⋊D₈₃	1	1	yes	characteristic
169	C₄×C₈	C₂²×D₈₃	1	1	yes	characteristic
170	C₂×Q₈	—	332	1	no	—
171	C₂×Q₈	—	332	2	no	—
172	C₂×Q₈	—	166	4	no	—
173	C₂×Q₈	—	166	2	no	—
174	C₂²×C₄	—	332	1	no	—
175	C₂²×C₄	—	332	1	no	—
176	C₂²×C₄	(C₂×D₈₃)⋊C₄	1	1	yes	characteristic
177	C₂²×C₄	D₆₆₄	1	1	yes	characteristic
178	C₂²×C₄	#13050579	1	1	yes	characteristic
179	Q₁₆	—	83	4	no	—
180	Q₁₆	—	166	2	no	—
181	Q₁₆	—	166	4	no	—
182	C₂×C₈	—	2	2	no	—
183	C₂×C₈	—	332	2	no	—
184	C₂×C₈	—	664	2	no	—
185	C₂×C₈	(C₂×D₈₃)⋊C₄	1	1	yes	characteristic
186	C₂×C₈	(C₂×D₈₃)⋊C₄	1	1	yes	characteristic
187	C₂×C₈	C₈×D₈₃	1	2	yes	—
188	C₂×C₈	#13050573	1	2	yes	—
189	C₂×C₈	C₂×C₄×D₈₃	1	1	yes	characteristic
190	C₂×C₈	C₂×C₂²⋊D₈₃	1	1	yes	characteristic
191	C₄⋊C₄	—	332	2	no	—
192	C₄⋊C₄	—	332	2	no	—
193	C₄⋊C₄	—	332	1	no	—
194	C₄⋊C₄	—	166	2	no	—
195	C₄²	—	332	2	no	—
196	C₄²	—	332	2	no	—
197	C₄²	(C₂×D₈₃)⋊C₄	1	1	yes	characteristic
198	C₄²	D₆₆₄	1	1	yes	characteristic
199	C₄²	#13050579	1	1	yes	characteristic
200	C₄²	C₂×D₃₃₂	1	1	yes	characteristic
201	C₁₆	—	2	4	no	—
202	C₂³	#14581714	1	1	yes	characteristic
203	Q₈	—	332	4	no	—
204	Q₈	—	332	2	no	—
205	Q₈	—	166	4	no	—
206	C₂×C₄	—	2	2	no	—
207	C₂×C₄	—	2	2	no	—
208	C₂×C₄	—	332	4	no	—
209	C₂×C₄	—	332	2	no	—
210	C₂×C₄	—	332	2	no	—
211	C₂×C₄	—	664	2	no	—
212	C₂×C₄	(C₂×D₈₃)⋊C₈	1	2	yes	—
213	C₂×C₄	#14581714	1	1	yes	characteristic
214	C₂×C₄	#14581714	1	1	yes	characteristic
215	C₂×C₄	#14581714	1	1	yes	characteristic
216	C₂×C₄	#14581720	1	2	yes	—
217	C₂×C₄	C₂×(C₂×D₈₃)⋊C₄	1	1	yes	characteristic
218	C₂×C₄	C₂×D₆₆₄	1	1	yes	characteristic
219	C₂×C₄	C₂×#13050579	1	1	yes	characteristic
220	C₈	—	2	1	no	—
221	C₈	—	2	1	no	—
222	C₈	—	664	2	no	—
223	C₈	(C₂×D₈₃)⋊C₈	1	2	yes	—
224	C₈	C₂×C₈×D₈₃	1	1	yes	characteristic
225	C₈	C₂×#13050573	1	1	yes	characteristic
226	C₂²	#19127430	1	2	yes	—
227	C₂²	#19127512	1	2	yes	—
228	C₂²	#19127518	1	2	yes	—
229	C₂²	C₂×#14581714	1	1	yes	characteristic
230	C₄	—	2	1	no	—
231	C₄	—	2	1	no	—
232	C₄	—	2	1	no	—
233	C₄	—	2	1	no	—
234	C₄	—	332	4	no	—
235	C₄	—	664	2	no	—
236	C₄	#19127430	1	2	yes	—
237	C₄	C₂×(C₂×D₈₃)⋊C₈	1	1	yes	characteristic
238	C₄	C₂×#14581720	1	1	yes	characteristic
239	C₂	#24905434	1	4	yes	direct product factor
240	C₂	C₂×#19127430	1	1	yes	characteristic
241	C₂	C₂×#19127512	1	1	yes	characteristic
242	C₂	C₂×#19127518	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	#14581714
commutator	C₈×C₈₃	C₂²×C₈
Frattini	C₄×C₈	C₂²×D₈₃
Fitting	C₂×#346×C₈₃	C₂
socle, prime radical	C₂³×C₈₃	#102
cosocle	C₄×C₈×C₈₃	C₂³
hypercenter	C₂×#346	D₈₃
hypocenter	C₈₃	C₂×#1334
LCS	G ▷ C₈×C₈₃ ▷ C₄×C₈₃ ▷ C₂×C₈₃ ▷ C₈₃
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ C₂×#346

Complements

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

N	H ≅ G/N	type
3. #24905434	C₂	direct (N × H)
23. C₂×Dic₃₃₂	C₈	semidirect
56. Dic₃₃₂	C₂×C₈	semidirect
118. C₂×C₈₃	#1334	semidirect
126. C₈₃	C₂×#1334	semidirect
239. C₂	#24905434	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂×C₄×Dic₃₃₂	2
C₂×#19127452	2
#24905434	2
C₂×#346×C₈₃	2
C₂×#1334	83

Subgroups — C2×#24905434 #35314166

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#24905434 #35314166