Subgroups — #90436⋊C₃ #13057479

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×#1157⋊C₃	C₂	1	1	yes	maximal, characteristic
2	C₂×M₁₆×ΓL(1,𝔽₈)	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×#1136⋊C₃	C₂	1	1	yes	maximal, characteristic, complemented
4	#6470⋊C₃	C₂	1	4	yes	maximal, complemented
5	M₁₆×AGL(1,𝔽₇)	C₂	1	4	yes	maximal, complemented
6	#6464⋊C₃	C₂	1	4	yes	maximal, complemented
7	#90436	C₃	1	1	yes	maximal, characteristic, complemented, Hall
8	C₂²×C₄×ΓL(1,𝔽₈)	C₂²	1	1	yes	characteristic
9	C₂×(C₂²⋊D₇)⋊C₃	C₄	1	2	yes	—
10	#1157⋊C₃	C₄	1	2	yes	—
11	#1157⋊C₃	C₄	1	2	yes	—
12	#1157⋊C₃	C₂²	1	2	yes	—
13	#1157⋊C₃	C₂²	1	2	yes	—
14	C₂×C₄⋊AGL(1,𝔽₇)	C₄	1	1	yes	characteristic
15	C₂×C₄×AGL(1,𝔽₇)	C₂²	1	2	yes	—
16	C₂×ΓL(1,𝔽₈)⋊Q₈	C₄	1	1	yes	characteristic
17	M₁₆×ΓL(1,𝔽₈)	C₂²	1	2	yes	—
18	M₁₆×ΓL(1,𝔽₈)	C₂²	1	2	yes	complemented
19	C₂×C₈×ΓL(1,𝔽₈)	C₂²	1	2	yes	complemented
20	#1136⋊C₃	C₂²	1	2	yes	—
21	#1136⋊C₃	C₂²	1	2	yes	complemented
22	C₂×ΓL(1,𝔽₈)⋊C₈	C₂²	1	2	yes	complemented
23	#1131⋊C₃	C₂²	1	8	yes	complemented
24	C₈×AGL(1,𝔽₇)	C₂²	1	8	yes	complemented
25	C₂×#1157	C₂×C₃	1	1	yes	characteristic
26	C₂×C₇×M₁₆	C₂×C₃	1	1	yes	characteristic, complemented
27	C₂×#1136	C₂×C₃	1	1	yes	characteristic, complemented
28	#6470	C₂×C₃	1	4	yes	complemented
29	D₇×M₁₆	C₂×C₃	1	4	yes	complemented
30	#6464	C₂×C₃	1	4	yes	complemented
31	C₃×#568	—	7	1	no	maximal, Hall
32	C₂³×ΓL(1,𝔽₈)	C₂×C₄	1	1	yes	characteristic
33	C₂²×AGL(1,𝔽₇)	C₂×C₄	1	2	yes	—
34	C₂×C₄×ΓL(1,𝔽₈)	C₂×C₄	1	1	yes	characteristic
35	C₂×C₄×ΓL(1,𝔽₈)	C₂×C₄	1	1	yes	characteristic
36	C₂×C₄×ΓL(1,𝔽₈)	C₂×C₄	1	1	yes	characteristic
37	C₂×C₄×ΓL(1,𝔽₈)	C₂³	1	1	yes	characteristic
38	C₂×C₄×ΓL(1,𝔽₈)	C₂³	1	1	yes	characteristic
39	C₂×C₄×ΓL(1,𝔽₈)	C₂³	1	1	yes	characteristic
40	(C₂²⋊D₇)⋊C₃	C₂×C₄	1	4	yes	—
41	(C₂²⋊D₇)⋊C₃	C₂×C₄	1	4	yes	—
42	C₂×ΓL(1,𝔽₈)⋊C₄	C₂×C₄	1	2	yes	—
43	C₄⋊AGL(1,𝔽₇)	C₂×C₄	1	2	yes	—
44	C₄⋊AGL(1,𝔽₇)	C₂×C₄	1	2	yes	—
45	C₄×AGL(1,𝔽₇)	C₂×C₄	1	4	yes	—
46	C₄×AGL(1,𝔽₇)	C₂³	1	4	yes	—
47	ΓL(1,𝔽₈)⋊Q₈	C₂×C₄	1	2	yes	—
48	ΓL(1,𝔽₈)⋊Q₈	C₂×C₄	1	2	yes	—
49	C₈×ΓL(1,𝔽₈)	C₂³	1	4	yes	—
50	ΓL(1,𝔽₈)⋊C₈	C₂³	1	4	yes	—
51	C₂²×C₄×C₇	C₂²×C₃	1	1	yes	characteristic
52	C₂×C₂²⋊D₇	C₃×C₄	1	2	yes	—
53	#1157	C₃×C₄	1	2	yes	—
54	#1157	C₃×C₄	1	2	yes	—
55	#1157	C₂²×C₃	1	2	yes	—
56	#1157	C₂²×C₃	1	2	yes	—
57	C₂×D₂₈	C₃×C₄	1	1	yes	characteristic
58	C₂×C₄×D₇	C₂²×C₃	1	2	yes	—
59	C₂×Dic₁₄	C₃×C₄	1	1	yes	characteristic
60	C₇×M₁₆	C₂²×C₃	1	2	yes	—
61	C₇×M₁₆	C₂²×C₃	1	2	yes	complemented
62	C₂×C₇×C₈	C₂²×C₃	1	2	yes	complemented
63	#1136	C₂²×C₃	1	2	yes	—
64	#1136	C₂²×C₃	1	2	yes	complemented
65	C₂×C₇⋊C₈	C₂²×C₃	1	2	yes	complemented
66	#1131	C₂²×C₃	1	8	yes	complemented
67	C₈×D₇	C₂²×C₃	1	8	yes	complemented
68	C₂×C₃×#41	—	7	1	no	—
69	C₃×#131	—	7	4	no	—
70	C₃×#131	—	7	4	no	—
71	C₂×C₃×M₁₆	—	7	4	no	—
72	C₂×C₃×M₁₆	—	7	1	no	—
73	C₂×C₃×M₁₆	—	7	1	no	—
74	C₂²×ΓL(1,𝔽₈)	—	4	1	no	—
75	C₂²×ΓL(1,𝔽₈)	C₂²×C₄	1	1	yes	characteristic
76	C₂²×ΓL(1,𝔽₈)	C₂²×C₄	1	1	yes	characteristic
77	C₂²×ΓL(1,𝔽₈)	C₂²×C₄	1	1	yes	characteristic
78	C₂×AGL(1,𝔽₇)	—	4	2	no	—
79	C₂×AGL(1,𝔽₇)	C₂²×C₄	1	4	yes	—
80	C₄×ΓL(1,𝔽₈)	C₂²×C₄	1	1	yes	characteristic
81	C₄×ΓL(1,𝔽₈)	C₂²×C₄	1	1	yes	characteristic
82	C₄×ΓL(1,𝔽₈)	C₂²×C₄	1	1	yes	characteristic
83	C₄×ΓL(1,𝔽₈)	C₂⁴	1	1	yes	characteristic
84	ΓL(1,𝔽₈)⋊C₄	C₂²×C₄	1	4	yes	—
85	#568	—	7	1	no	Sylow
86	C₂³×C₇	C₂×C₃×C₄	1	1	yes	characteristic
87	C₂²×D₇	C₂×C₃×C₄	1	2	yes	—
88	C₂×C₄×C₇	C₂×C₃×C₄	1	1	yes	characteristic
89	C₂×C₄×C₇	C₂×C₃×C₄	1	1	yes	characteristic
90	C₂×C₄×C₇	C₂×C₃×C₄	1	1	yes	characteristic
91	C₂×C₄×C₇	C₂³×C₃	1	1	yes	characteristic
92	C₂×C₄×C₇	C₂³×C₃	1	1	yes	characteristic
93	C₂×C₄×C₇	C₂³×C₃	1	1	yes	characteristic
94	C₂²⋊D₇	C₂×C₃×C₄	1	4	yes	—
95	C₂²⋊D₇	C₂×C₃×C₄	1	4	yes	—
96	C₂×Dic₇	C₂×C₃×C₄	1	2	yes	—
97	D₂₈	C₂×C₃×C₄	1	2	yes	—
98	D₂₈	C₂×C₃×C₄	1	2	yes	—
99	C₄×D₇	C₂×C₃×C₄	1	4	yes	—
100	C₄×D₇	C₂³×C₃	1	4	yes	—
101	Dic₁₄	C₂×C₃×C₄	1	2	yes	—
102	Dic₁₄	C₂×C₃×C₄	1	2	yes	—
103	C₇×C₈	C₂³×C₃	1	4	yes	—
104	C₇⋊C₈	C₂³×C₃	1	4	yes	—
105	C₃×#41	—	7	2	no	—
106	C₃×#41	—	7	2	no	—
107	C₃×#41	—	7	2	no	—
108	C₃×#41	—	7	2	no	—
109	C₂×C₃×Q₈	—	7	1	no	—
110	C₂×C₃×D₄	—	7	1	no	—
111	C₂×C₃×D₄	—	7	2	no	—
112	C₂²×C₃×C₄	—	7	2	no	—
113	C₂²×C₃×C₄	—	7	1	no	—
114	C₃×M₁₆	—	7	2	no	—
115	C₃×M₁₆	—	7	8	no	—
116	C₃×M₁₆	—	7	2	no	—
117	C₃×M₁₆	—	7	2	no	—
118	C₃×M₁₆	—	7	2	no	—
119	C₂×C₃×C₈	—	7	8	no	—
120	C₂×C₃×C₈	—	7	2	no	—
121	C₂×C₃×C₈	—	7	2	no	—
122	C₂×ΓL(1,𝔽₈)	—	2	1	no	—
123	C₂×ΓL(1,𝔽₈)	—	2	1	no	—
124	C₂×ΓL(1,𝔽₈)	—	2	1	no	—
125	C₂×ΓL(1,𝔽₈)	C₂³×C₄	1	1	yes	characteristic
126	AGL(1,𝔽₇)	—	2	4	no	—
127	C₂×#41	—	7	1	no	—
128	#131	—	7	4	no	—
129	#131	—	7	4	no	—
130	C₂×M₁₆	—	7	1	no	—
131	C₂×M₁₆	—	7	4	no	—
132	C₂×M₁₆	AGL(1,𝔽₇)	1	1	yes	characteristic, complemented
133	C₂²×C₇	—	4	1	no	—
134	C₂²×C₇	C₂²×C₃×C₄	1	1	yes	characteristic
135	C₂²×C₇	C₂²×C₃×C₄	1	1	yes	characteristic
136	C₂²×C₇	C₂²×C₃×C₄	1	1	yes	characteristic
137	C₂×D₇	—	4	2	no	—
138	C₂×D₇	C₂²×C₃×C₄	1	4	yes	—
139	C₄×C₇	C₂²×C₃×C₄	1	1	yes	characteristic
140	C₄×C₇	C₂²×C₃×C₄	1	1	yes	characteristic
141	C₄×C₇	C₂²×C₃×C₄	1	1	yes	characteristic
142	C₄×C₇	C₂⁴×C₃	1	1	yes	characteristic
143	Dic₇	C₂²×C₃×C₄	1	4	yes	—
144	C₂³×C₃	—	7	2	no	—
145	C₂³×C₃	—	7	1	no	—
146	C₃×Q₈	—	7	2	no	—
147	C₃×Q₈	—	7	2	no	—
148	C₃×D₄	—	7	2	no	—
149	C₃×D₄	—	7	2	no	—
150	C₃×D₄	—	7	4	no	—
151	C₃×D₄	—	7	4	no	—
152	C₂×C₃×C₄	—	7	4	no	—
153	C₂×C₃×C₄	—	7	4	no	—
154	C₂×C₃×C₄	—	7	2	no	—
155	C₂×C₃×C₄	—	7	1	no	—
156	C₂×C₃×C₄	—	7	1	no	—
157	C₂×C₃×C₄	—	7	1	no	—
158	C₂×C₃×C₄	—	7	1	no	—
159	C₂×C₃×C₄	—	7	1	no	—
160	C₂×C₃×C₄	—	7	1	no	—
161	C₃×C₈	—	7	4	no	—
162	C₃×C₈	—	7	4	no	—
163	ΓL(1,𝔽₈)	#568	1	1	yes	characteristic, complemented, Hall
164	#41	—	7	2	no	—
165	#41	—	7	2	no	—
166	#41	—	7	2	no	—
167	#41	—	7	2	no	—
168	C₂×Q₈	—	7	1	no	—
169	C₂×D₄	—	7	2	no	—
170	C₂×D₄	—	7	1	no	—
171	C₂²×C₄	—	7	2	no	—
172	C₂²×C₄	C₂×AGL(1,𝔽₇)	1	1	yes	characteristic
173	M₁₆	—	7	2	no	—
174	M₁₆	—	7	2	no	—
175	M₁₆	—	7	8	no	—
176	M₁₆	C₂×AGL(1,𝔽₇)	1	2	yes	—
177	M₁₆	C₂×AGL(1,𝔽₇)	1	2	yes	complemented
178	C₂×C₈	—	7	2	no	—
179	C₂×C₈	—	7	8	no	—
180	C₂×C₈	C₂×AGL(1,𝔽₇)	1	2	yes	complemented
181	C₂×C₇	—	2	1	no	—
182	C₂×C₇	—	2	1	no	—
183	C₂×C₇	—	2	1	no	—
184	C₂×C₇	C₂³×C₃×C₄	1	1	yes	characteristic
185	D₇	—	2	4	no	—
186	C₂²×C₃	—	7	4	no	—
187	C₂²×C₃	—	7	1	no	—
188	C₂²×C₃	—	7	1	no	—
189	C₂²×C₃	—	7	1	no	—
190	C₂²×C₃	—	28	1	no	—
191	C₂²×C₃	—	28	2	no	—
192	C₃×C₄	—	7	4	no	—
193	C₃×C₄	—	7	1	no	—
194	C₃×C₄	—	7	1	no	—
195	C₃×C₄	—	7	1	no	—
196	C₃×C₄	—	7	1	no	—
197	C₂³	—	7	2	no	—
198	C₂³	C₄×AGL(1,𝔽₇)	1	1	yes	characteristic
199	Q₈	—	7	2	no	—
200	Q₈	—	7	2	no	—
201	D₄	—	7	4	no	—
202	D₄	—	7	4	no	—
203	D₄	—	7	2	no	—
204	D₄	—	7	2	no	—
205	C₂×C₄	—	7	2	no	—
206	C₂×C₄	—	7	4	no	—
207	C₂×C₄	—	7	4	no	—
208	C₂×C₄	C₄×AGL(1,𝔽₇)	1	1	yes	characteristic
209	C₂×C₄	C₄×AGL(1,𝔽₇)	1	1	yes	characteristic
210	C₂×C₄	C₄×AGL(1,𝔽₇)	1	1	yes	characteristic
211	C₂×C₄	C₂²×AGL(1,𝔽₇)	1	1	yes	characteristic
212	C₂×C₄	C₂²×AGL(1,𝔽₇)	1	1	yes	characteristic
213	C₂×C₄	C₂²×AGL(1,𝔽₇)	1	1	yes	characteristic
214	C₈	—	7	4	no	—
215	C₈	C₂²×AGL(1,𝔽₇)	1	4	yes	—
216	C₇	C₃×#568	1	1	yes	characteristic, complemented, Sylow
217	C₂×C₃	—	7	1	no	—
218	C₂×C₃	—	14	1	no	—
219	C₂×C₃	—	14	1	no	—
220	C₂×C₃	—	14	1	no	—
221	C₂×C₃	—	14	4	no	—
222	C₂²	—	4	1	no	—
223	C₂²	—	7	4	no	—
224	C₂²	—	28	2	no	—
225	C₂²	C₂×C₄×AGL(1,𝔽₇)	1	1	yes	characteristic
226	C₂²	C₂×C₄×AGL(1,𝔽₇)	1	1	yes	characteristic
227	C₂²	C₂×C₄×AGL(1,𝔽₇)	1	1	yes	characteristic
228	C₄	—	7	4	no	—
229	C₄	C₂×C₄×AGL(1,𝔽₇)	1	1	yes	characteristic
230	C₄	C₂×C₄×AGL(1,𝔽₇)	1	1	yes	characteristic
231	C₄	C₂×C₄×AGL(1,𝔽₇)	1	1	yes	characteristic
232	C₄	C₂³×AGL(1,𝔽₇)	1	1	yes	characteristic
233	C₃	—	7	1	no	Sylow
234	C₂	—	2	1	no	—
235	C₂	—	2	1	no	—
236	C₂	—	2	1	no	—
237	C₂	—	14	4	no	—
238	C₂	C₂²×C₄×AGL(1,𝔽₇)	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₄	C₂³×AGL(1,𝔽₇)
commutator, socle	C₂×C₇	C₂³×C₃×C₄
Fitting	C₂×C₇×M₁₆	C₂×C₃
prime radical	C₂×#1157⋊C₃	C₂
cosocle	C₄×C₇	C₂⁴×C₃
hypercenter	C₂×M₁₆	AGL(1,𝔽₇)
hypocenter	C₇	C₃×#568
LCS	G ▷ C₂×C₇ ▷ C₇
UCS	C₁ ◁ C₄ ◁ C₂×M₁₆

Complements

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

N	H ≅ G/N	type
2. C₂×M₁₆×ΓL(1,𝔽₈)	C₂	semidirect
3. C₂×#1136⋊C₃	C₂	semidirect
4. #6470⋊C₃	C₂	semidirect
5. M₁₆×AGL(1,𝔽₇)	C₂	semidirect
6. #6464⋊C₃	C₂	semidirect
7. #90436	C₃	semidirect
18. M₁₆×ΓL(1,𝔽₈)	C₂²	semidirect
19. C₂×C₈×ΓL(1,𝔽₈)	C₂²	semidirect
21. #1136⋊C₃	C₂²	semidirect
22. C₂×ΓL(1,𝔽₈)⋊C₈	C₂²	semidirect
23. #1131⋊C₃	C₂²	semidirect
24. C₈×AGL(1,𝔽₇)	C₂²	semidirect
26. C₂×C₇×M₁₆	C₂×C₃	semidirect
27. C₂×#1136	C₂×C₃	semidirect
28. #6470	C₂×C₃	semidirect
29. D₇×M₁₆	C₂×C₃	semidirect
30. #6464	C₂×C₃	semidirect
61. C₇×M₁₆	C₂²×C₃	semidirect
62. C₂×C₇×C₈	C₂²×C₃	semidirect
64. #1136	C₂²×C₃	semidirect
65. C₂×C₇⋊C₈	C₂²×C₃	semidirect
66. #1131	C₂²×C₃	semidirect
67. C₈×D₇	C₂²×C₃	semidirect
132. C₂×M₁₆	AGL(1,𝔽₇)	semidirect
163. ΓL(1,𝔽₈)	#568	semidirect
177. M₁₆	C₂×AGL(1,𝔽₇)	semidirect
180. C₂×C₈	C₂×AGL(1,𝔽₇)	semidirect
216. C₇	C₃×#568	semidirect

Maximal subgroups

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

subgroup	index
C₂×#1157⋊C₃	2
C₂×M₁₆×ΓL(1,𝔽₈)	2
C₂×#1136⋊C₃	2
#6470⋊C₃	2
M₁₆×AGL(1,𝔽₇)	2
#6464⋊C₃	2
#90436	3
C₃×#568	7

Subgroups — #90436⋊C3 #13057479

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — #90436⋊C₃ #13057479