Subgroups — #13059427

Order 1344.

516 conjugacy classes of subgroups.

The subgroup lattice has 240 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#66728	C₂	1	1	yes	maximal, characteristic
2	C₂×C₇×#208	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×C₃×#1136	C₂	1	1	yes	maximal, characteristic, complemented
4	#10620854	C₂	1	4	yes	maximal, complemented
5	#10620853	C₂	1	4	yes	maximal, complemented
6	#10620852	C₂	1	4	yes	maximal, complemented
7	#90436	—	3	1	no	maximal, Hall
8	C₂²×C₃×C₄×C₇	C₂²	1	1	yes	characteristic
9	C₂×#4268	C₄	1	2	yes	—
10	#66728	C₄	1	2	yes	—
11	#66728	C₄	1	2	yes	—
12	#66728	C₂²	1	2	yes	—
13	#66728	C₂²	1	2	yes	—
14	C₂×D₈₄	C₄	1	1	yes	characteristic
15	C₂×C₄×D₂₁	C₂²	1	2	yes	—
16	C₂×Dic₄₂	C₄	1	1	yes	characteristic
17	C₇×#208	C₂²	1	2	yes	—
18	C₇×#208	C₂²	1	2	yes	complemented
19	C₂×C₇×C₃⋊C₈	C₂²	1	2	yes	complemented
20	C₃×#1136	C₂²	1	2	yes	—
21	C₃×#1136	C₂²	1	2	yes	complemented
22	C₂×C₃×C₇⋊C₈	C₂²	1	2	yes	complemented
23	#66559	C₂²	1	8	yes	complemented
24	D₂₁⋊C₈	C₂²	1	8	yes	complemented
25	C₂×#1157	—	3	1	no	—
26	C₂×C₇×M₁₆	—	3	1	no	—
27	C₂×#1136	D₃	1	1	yes	characteristic, complemented
28	#6470	—	3	4	no	—
29	D₇×M₁₆	—	3	4	no	—
30	#6464	—	3	4	no	—
31	#5745	—	7	1	no	maximal, Hall
32	C₂³×C₃×C₇	C₂×C₄	1	1	yes	characteristic
33	C₂²×D₂₁	C₂×C₄	1	2	yes	—
34	C₂×C₃×C₄×C₇	C₂×C₄	1	1	yes	characteristic
35	C₂×C₃×C₄×C₇	C₂×C₄	1	1	yes	characteristic
36	C₂×C₃×C₄×C₇	C₂×C₄	1	1	yes	characteristic
37	C₂×C₃×C₄×C₇	C₂³	1	1	yes	characteristic
38	C₂×C₃×C₄×C₇	C₂³	1	1	yes	characteristic
39	C₂×C₃×C₄×C₇	C₂³	1	1	yes	characteristic
40	#4268	C₂×C₄	1	4	yes	—
41	#4268	C₂×C₄	1	4	yes	—
42	C₂×Dic₂₁	C₂×C₄	1	2	yes	—
43	D₈₄	C₂×C₄	1	2	yes	—
44	D₈₄	C₂×C₄	1	2	yes	—
45	C₄×D₂₁	C₂×C₄	1	4	yes	—
46	C₄×D₂₁	C₂³	1	4	yes	—
47	Dic₄₂	C₂×C₄	1	2	yes	—
48	Dic₄₂	C₂×C₄	1	2	yes	—
49	C₃×C₇⋊C₈	C₂³	1	4	yes	—
50	C₇×C₃⋊C₈	C₂³	1	4	yes	—
51	C₂²×C₄×C₇	C₂×D₃	1	1	yes	characteristic
52	C₂×C₂²⋊D₇	—	3	2	no	—
53	#1157	—	3	2	no	—
54	#1157	—	3	2	no	—
55	#1157	—	3	2	no	—
56	#1157	—	3	2	no	—
57	C₂×D₂₈	—	3	1	no	—
58	C₂×C₄×D₇	—	3	2	no	—
59	C₂×Dic₁₄	—	3	1	no	—
60	C₇×M₁₆	—	3	2	no	—
61	C₇×M₁₆	—	3	2	no	—
62	C₂×C₇×C₈	—	3	2	no	—
63	#1136	C₂×D₃	1	2	yes	—
64	#1136	C₂×D₃	1	2	yes	complemented
65	C₂×C₇⋊C₈	C₂×D₃	1	2	yes	complemented
66	#1131	—	3	8	no	—
67	C₈×D₇	—	3	8	no	—
68	C₂×#235	—	7	1	no	—
69	C₂×C₃×M₁₆	—	7	1	no	—
70	C₂×#208	D₇	1	1	yes	characteristic, complemented
71	#907	—	7	4	no	—
72	D₃×M₁₆	—	7	4	no	—
73	#901	—	7	4	no	—
74	C₂²×C₃×C₇	—	4	1	no	—
75	C₂²×C₃×C₇	C₂²×C₄	1	1	yes	characteristic
76	C₂²×C₃×C₇	C₂²×C₄	1	1	yes	characteristic
77	C₂²×C₃×C₇	C₂²×C₄	1	1	yes	characteristic
78	C₂×D₂₁	—	4	2	no	—
79	C₂×D₂₁	C₂²×C₄	1	4	yes	—
80	C₃×C₄×C₇	C₂²×C₄	1	1	yes	characteristic
81	C₃×C₄×C₇	C₂²×C₄	1	1	yes	characteristic
82	C₃×C₄×C₇	C₂²×C₄	1	1	yes	characteristic
83	C₃×C₄×C₇	C₂⁴	1	1	yes	characteristic
84	Dic₂₁	C₂²×C₄	1	4	yes	—
85	#568	—	21	1	no	Sylow
86	C₂³×C₇	C₄×D₃	1	1	yes	characteristic
87	C₂²×D₇	—	3	2	no	—
88	C₂×C₄×C₇	C₄×D₃	1	1	yes	characteristic
89	C₂×C₄×C₇	C₄×D₃	1	1	yes	characteristic
90	C₂×C₄×C₇	C₄×D₃	1	1	yes	characteristic
91	C₂×C₄×C₇	C₂²×D₃	1	1	yes	characteristic
92	C₂×C₄×C₇	C₂²×D₃	1	1	yes	characteristic
93	C₂×C₄×C₇	C₂²×D₃	1	1	yes	characteristic
94	C₂²⋊D₇	—	3	4	no	—
95	C₂²⋊D₇	—	3	4	no	—
96	C₂×Dic₇	—	3	2	no	—
97	D₂₈	—	3	2	no	—
98	D₂₈	—	3	2	no	—
99	C₄×D₇	—	3	4	no	—
100	C₄×D₇	—	3	4	no	—
101	Dic₁₄	—	3	2	no	—
102	Dic₁₄	—	3	2	no	—
103	C₇×C₈	—	3	4	no	—
104	C₇⋊C₈	C₂²×D₃	1	4	yes	—
105	C₂²×C₃×C₄	C₂×D₇	1	1	yes	characteristic
106	C₂×#67	—	7	2	no	—
107	#235	—	7	2	no	—
108	#235	—	7	2	no	—
109	#235	—	7	2	no	—
110	#235	—	7	2	no	—
111	C₂×D₁₂	—	7	1	no	—
112	C₂×C₄×D₃	—	7	2	no	—
113	C₂×Dic₆	—	7	1	no	—
114	C₃×M₁₆	—	7	2	no	—
115	C₃×M₁₆	—	7	2	no	—
116	C₂×C₃×C₈	—	7	2	no	—
117	#208	C₂×D₇	1	2	yes	—
118	#208	C₂×D₇	1	2	yes	complemented
119	C₂×C₃⋊C₈	C₂×D₇	1	2	yes	complemented
120	ΓL(1,𝔽₂₅)	—	7	8	no	—
121	D₃×C₈	—	7	8	no	—
122	C₂×C₃×C₇	—	2	1	no	—
123	C₂×C₃×C₇	—	2	1	no	—
124	C₂×C₃×C₇	—	2	1	no	—
125	C₂×C₃×C₇	C₂³×C₄	1	1	yes	characteristic
126	D₂₁	—	2	4	no	—
127	C₂×#41	—	21	1	no	—
128	#131	—	21	4	no	—
129	#131	—	21	4	no	—
130	C₂×M₁₆	—	3	1	no	—
131	C₂×M₁₆	—	7	1	no	—
132	C₂×M₁₆	—	21	4	no	—
133	C₂²×C₇	—	4	1	no	—
134	C₂²×C₇	C₂×C₄×D₃	1	1	yes	characteristic
135	C₂²×C₇	C₂×C₄×D₃	1	1	yes	characteristic
136	C₂²×C₇	C₂×C₄×D₃	1	1	yes	characteristic
137	C₂×D₇	—	3	4	no	—
138	C₂×D₇	—	12	2	no	—
139	C₄×C₇	C₂×C₄×D₃	1	1	yes	characteristic
140	C₄×C₇	C₂×C₄×D₃	1	1	yes	characteristic
141	C₄×C₇	C₂×C₄×D₃	1	1	yes	characteristic
142	C₄×C₇	C₂³×D₃	1	1	yes	characteristic
143	Dic₇	—	3	4	no	—
144	C₂³×C₃	C₄×D₇	1	1	yes	characteristic
145	C₂²×D₃	—	7	2	no	—
146	C₂×C₃×C₄	C₄×D₇	1	1	yes	characteristic
147	C₂×C₃×C₄	C₄×D₇	1	1	yes	characteristic
148	C₂×C₃×C₄	C₄×D₇	1	1	yes	characteristic
149	C₂×C₃×C₄	C₂²×D₇	1	1	yes	characteristic
150	C₂×C₃×C₄	C₂²×D₇	1	1	yes	characteristic
151	C₂×C₃×C₄	C₂²×D₇	1	1	yes	characteristic
152	#67	—	7	4	no	—
153	#67	—	7	4	no	—
154	C₂×Dic₃	—	7	2	no	—
155	D₁₂	—	7	2	no	—
156	D₁₂	—	7	2	no	—
157	C₄×D₃	—	7	4	no	—
158	C₄×D₃	—	7	4	no	—
159	Dic₆	—	7	2	no	—
160	Dic₆	—	7	2	no	—
161	C₃×C₈	—	7	4	no	—
162	C₃⋊C₈	C₂²×D₇	1	4	yes	—
163	C₃×C₇	#568	1	1	yes	characteristic, complemented, Hall
164	#41	—	21	2	no	—
165	#41	—	21	2	no	—
166	#41	—	21	2	no	—
167	#41	—	21	2	no	—
168	C₂×Q₈	—	21	1	no	—
169	C₂×D₄	—	21	2	no	—
170	C₂×D₄	—	21	1	no	—
171	C₂²×C₄	—	21	2	no	—
172	C₂²×C₄	D₃×D₇	1	1	yes	characteristic
173	M₁₆	—	3	2	no	—
174	M₁₆	—	3	2	no	—
175	M₁₆	—	7	2	no	—
176	M₁₆	—	7	2	no	—
177	M₁₆	—	21	8	no	—
178	C₂×C₈	—	3	2	no	—
179	C₂×C₈	—	7	2	no	—
180	C₂×C₈	—	21	8	no	—
181	C₂×C₇	—	2	1	no	—
182	C₂×C₇	—	2	1	no	—
183	C₂×C₇	—	2	1	no	—
184	C₂×C₇	C₂²×C₄×D₃	1	1	yes	characteristic
185	D₇	—	6	4	no	—
186	C₂²×C₃	—	4	1	no	—
187	C₂²×C₃	C₂×C₄×D₇	1	1	yes	characteristic
188	C₂²×C₃	C₂×C₄×D₇	1	1	yes	characteristic
189	C₂²×C₃	C₂×C₄×D₇	1	1	yes	characteristic
190	C₂×D₃	—	7	4	no	—
191	C₂×D₃	—	28	2	no	—
192	C₃×C₄	C₂×C₄×D₇	1	1	yes	characteristic
193	C₃×C₄	C₂×C₄×D₇	1	1	yes	characteristic
194	C₃×C₄	C₂×C₄×D₇	1	1	yes	characteristic
195	C₃×C₄	C₂³×D₇	1	1	yes	characteristic
196	Dic₃	—	7	4	no	—
197	C₂³	—	21	2	no	—
198	C₂³	D₂₁⋊C₄	1	1	yes	characteristic
199	Q₈	—	21	2	no	—
200	Q₈	—	21	2	no	—
201	D₄	—	21	4	no	—
202	D₄	—	21	4	no	—
203	D₄	—	21	2	no	—
204	D₄	—	21	2	no	—
205	C₂×C₄	—	21	2	no	—
206	C₂×C₄	—	21	4	no	—
207	C₂×C₄	—	21	4	no	—
208	C₂×C₄	D₂₁⋊C₄	1	1	yes	characteristic
209	C₂×C₄	D₂₁⋊C₄	1	1	yes	characteristic
210	C₂×C₄	D₂₁⋊C₄	1	1	yes	characteristic
211	C₂×C₄	C₂×D₃×D₇	1	1	yes	characteristic
212	C₂×C₄	C₂×D₃×D₇	1	1	yes	characteristic
213	C₂×C₄	C₂×D₃×D₇	1	1	yes	characteristic
214	C₈	—	3	4	no	—
215	C₈	—	7	4	no	—
216	C₇	#5745	1	1	yes	characteristic, complemented, Sylow
217	C₂×C₃	—	2	1	no	—
218	C₂×C₃	—	2	1	no	—
219	C₂×C₃	—	2	1	no	—
220	C₂×C₃	C₂²×C₄×D₇	1	1	yes	characteristic
221	D₃	—	14	4	no	—
222	C₂²	—	4	1	no	—
223	C₂²	—	21	4	no	—
224	C₂²	—	84	2	no	—
225	C₂²	C₂×D₂₁⋊C₄	1	1	yes	characteristic
226	C₂²	C₂×D₂₁⋊C₄	1	1	yes	characteristic
227	C₂²	C₂×D₂₁⋊C₄	1	1	yes	characteristic
228	C₄	—	21	4	no	—
229	C₄	C₂×D₂₁⋊C₄	1	1	yes	characteristic
230	C₄	C₂×D₂₁⋊C₄	1	1	yes	characteristic
231	C₄	C₂×D₂₁⋊C₄	1	1	yes	characteristic
232	C₄	C₂²×D₃×D₇	1	1	yes	characteristic
233	C₃	#90436	1	1	yes	characteristic, complemented, Sylow
234	C₂	—	2	1	no	—
235	C₂	—	2	1	no	—
236	C₂	—	2	1	no	—
237	C₂	—	42	4	no	—
238	C₂	C₂²×D₂₁⋊C₄	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₄	C₂²×D₃×D₇
commutator, socle	C₂×C₃×C₇	C₂³×C₄
Fitting	C₂²×C₃×C₄×C₇	C₂²
prime radical	C₂×#66728	C₂
cosocle	C₃×C₄×C₇	C₂⁴
hypercenter	C₂²×C₄	D₃×D₇
hypocenter	C₃×C₇	#568
LCS	G ▷ C₂×C₃×C₇ ▷ C₃×C₇
UCS	C₁ ◁ C₄ ◁ C₂²×C₄

Complements

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

N	H ≅ G/N	type
2. C₂×C₇×#208	C₂	semidirect
3. C₂×C₃×#1136	C₂	semidirect
4. #10620854	C₂	semidirect
5. #10620853	C₂	semidirect
6. #10620852	C₂	semidirect
18. C₇×#208	C₂²	semidirect
19. C₂×C₇×C₃⋊C₈	C₂²	semidirect
21. C₃×#1136	C₂²	semidirect
22. C₂×C₃×C₇⋊C₈	C₂²	semidirect
23. #66559	C₂²	semidirect
24. D₂₁⋊C₈	C₂²	semidirect
27. C₂×#1136	D₃	semidirect
64. #1136	C₂×D₃	semidirect
65. C₂×C₇⋊C₈	C₂×D₃	semidirect
70. C₂×#208	D₇	semidirect
118. #208	C₂×D₇	semidirect
119. C₂×C₃⋊C₈	C₂×D₇	semidirect
163. C₃×C₇	#568	semidirect
216. C₇	#5745	semidirect
233. C₃	#90436	semidirect

Maximal subgroups

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

subgroup	index
C₂×#66728	2
C₂×C₇×#208	2
C₂×C₃×#1136	2
#10620854	2
#10620853	2
#10620852	2
#90436	3
#5745	7