Subgroups — #90214⋊C₃ #13057260

Order 1344.

588 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	ΓL(1,𝔽₈)×#117	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×C₄⋊(ΓL(1,𝔽₈)⋊C₄)	C₂	1	1	yes	maximal, characteristic, complemented
4	#6455⋊C₃	C₂	1	2	yes	maximal, complemented
5	C₄⋊C₄×AGL(1,𝔽₇)	C₂	1	2	yes	maximal, complemented
6	((C₂²⋊D₇)⋊C₃)⋊C₄	C₂	1	4	yes	maximal, complemented
7	C₄×C₄⋊AGL(1,𝔽₇)	C₂	1	2	yes	maximal, complemented
8	C₄×ΓL(1,𝔽₈)⋊Q₈	C₂	1	2	yes	maximal, complemented
9	#90214	C₃	1	1	yes	maximal, characteristic, complemented, Hall
10	C₂²×C₄×ΓL(1,𝔽₈)	C₂²	1	1	yes	characteristic
11	C₂×(C₂²⋊D₇)⋊C₃	C₂²	1	2	yes	—
12	C₂²×ΓL(1,𝔽₈)⋊C₄	C₂²	1	2	yes	—
13	#1157⋊C₃	C₄	1	8	yes	complemented
14	C₂×C₄⋊AGL(1,𝔽₇)	C₂²	1	1	yes	characteristic
15	C₂×C₄×AGL(1,𝔽₇)	C₂²	1	4	yes	—
16	C₂×C₄×AGL(1,𝔽₇)	C₂²	1	2	yes	—
17	C₂×ΓL(1,𝔽₈)⋊Q₈	C₂²	1	1	yes	characteristic
18	C₄⋊C₄×ΓL(1,𝔽₈)	C₂²	1	1	yes	characteristic
19	C₄⋊C₄×ΓL(1,𝔽₈)	C₂²	1	1	yes	characteristic
20	C₂²⋊C₄×ΓL(1,𝔽₈)	C₂²	1	2	yes	—
21	C₄²×ΓL(1,𝔽₈)	C₂²	1	2	yes	—
22	(C₂×AGL(1,𝔽₇))⋊C₄	C₂²	1	4	yes	—
23	C₄⋊(ΓL(1,𝔽₈)⋊C₄)	C₂²	1	2	yes	—
24	C₄⋊(ΓL(1,𝔽₈)⋊C₄)	C₂²	1	2	yes	—
25	(ΓL(1,𝔽₈)⋊C₄)⋊C₄	C₂²	1	4	yes	—
26	C₄×ΓL(1,𝔽₈)⋊C₄	C₂²	1	4	yes	—
27	C₂×#1157	C₂×C₃	1	1	yes	characteristic
28	C₇×#117	C₂×C₃	1	1	yes	characteristic, complemented
29	C₂×C₄⋊Dic₇	C₂×C₃	1	1	yes	characteristic, complemented
30	#6455	C₂×C₃	1	2	yes	complemented
31	D₇×C₄⋊C₄	C₂×C₃	1	2	yes	complemented
32	(C₂²⋊D₇)⋊C₄	C₂×C₃	1	4	yes	complemented
33	C₄×D₂₈	C₂×C₃	1	2	yes	complemented
34	C₄×Dic₁₄	C₂×C₃	1	2	yes	complemented
35	C₃×#520	—	7	1	no	maximal, Hall
36	C₂³×ΓL(1,𝔽₈)	C₂³	1	1	yes	characteristic
37	C₂²×AGL(1,𝔽₇)	C₂³	1	2	yes	—
38	C₂×C₄×ΓL(1,𝔽₈)	C₂×C₄	1	4	yes	complemented
39	C₂×C₄×ΓL(1,𝔽₈)	C₂³	1	2	yes	—
40	C₂×C₄×ΓL(1,𝔽₈)	C₂³	1	2	yes	—
41	C₂×C₄×ΓL(1,𝔽₈)	C₂³	1	1	yes	characteristic
42	C₂×C₄×ΓL(1,𝔽₈)	C₂³	1	1	yes	characteristic
43	(C₂²⋊D₇)⋊C₃	C₂×C₄	1	8	yes	complemented
44	C₂×ΓL(1,𝔽₈)⋊C₄	—	4	2	no	—
45	C₂×ΓL(1,𝔽₈)⋊C₄	C₂³	1	4	yes	—
46	C₂×ΓL(1,𝔽₈)⋊C₄	C₂³	1	2	yes	—
47	C₄⋊AGL(1,𝔽₇)	C₂×C₄	1	4	yes	complemented
48	C₄×AGL(1,𝔽₇)	—	4	4	no	—
49	C₄×AGL(1,𝔽₇)	C₂×C₄	1	8	yes	complemented
50	ΓL(1,𝔽₈)⋊Q₈	C₂×C₄	1	4	yes	complemented
51	C₂²×C₄×C₇	C₂²×C₃	1	1	yes	characteristic
52	C₂×C₂²⋊D₇	C₂²×C₃	1	2	yes	—
53	C₂²×Dic₇	C₂²×C₃	1	2	yes	—
54	#1157	C₃×C₄	1	8	yes	complemented
55	C₂×D₂₈	C₂²×C₃	1	1	yes	characteristic
56	C₂×C₄×D₇	C₂²×C₃	1	4	yes	—
57	C₂×C₄×D₇	C₂²×C₃	1	2	yes	—
58	C₂×Dic₁₄	C₂²×C₃	1	1	yes	characteristic
59	C₇×C₄⋊C₄	C₂²×C₃	1	1	yes	characteristic
60	C₇×C₄⋊C₄	C₂²×C₃	1	1	yes	characteristic
61	C₇×C₂²⋊C₄	C₂²×C₃	1	2	yes	—
62	C₄²×C₇	C₂²×C₃	1	2	yes	—
63	(C₂×D₇)⋊C₄	C₂²×C₃	1	4	yes	—
64	C₄⋊Dic₇	C₂²×C₃	1	2	yes	—
65	C₄⋊Dic₇	C₂²×C₃	1	2	yes	—
66	Dic₇⋊C₄	C₂²×C₃	1	4	yes	—
67	C₄×Dic₇	C₂²×C₃	1	4	yes	—
68	C₂×C₃×#41	—	7	1	no	—
69	C₃×C₄×Q₈	—	7	2	no	—
70	C₃×C₄×D₄	—	7	2	no	—
71	C₃×C₄×D₄	—	7	4	no	—
72	C₃×#117	—	7	2	no	—
73	C₃×#117	—	7	1	no	—
74	C₂×C₃×C₄⋊C₄	—	7	2	no	—
75	C₂×C₃×C₄⋊C₄	—	7	1	no	—
76	C₂²×ΓL(1,𝔽₈)	—	2	1	no	—
77	C₂²×ΓL(1,𝔽₈)	—	2	1	no	—
78	C₂²×ΓL(1,𝔽₈)	C₂²×C₄	1	2	yes	—
79	C₂²×ΓL(1,𝔽₈)	C₂⁴	1	1	yes	characteristic
80	C₂×AGL(1,𝔽₇)	—	2	2	no	—
81	C₂×AGL(1,𝔽₇)	—	2	2	no	—
82	C₂×AGL(1,𝔽₇)	C₂²×C₄	1	4	yes	—
83	C₄×ΓL(1,𝔽₈)	—	2	2	no	—
84	C₄×ΓL(1,𝔽₈)	—	2	2	no	—
85	C₄×ΓL(1,𝔽₈)	C₂²×C₄	1	2	yes	—
86	C₄×ΓL(1,𝔽₈)	C₂²×C₄	1	2	yes	—
87	ΓL(1,𝔽₈)⋊C₄	—	2	4	no	—
88	ΓL(1,𝔽₈)⋊C₄	C₂²×C₄	1	4	yes	—
89	#520	—	7	1	no	Sylow
90	C₂³×C₇	C₂³×C₃	1	1	yes	characteristic
91	C₂²×D₇	C₂³×C₃	1	2	yes	—
92	C₂×C₄×C₇	C₂×C₃×C₄	1	4	yes	complemented
93	C₂×C₄×C₇	C₂³×C₃	1	2	yes	—
94	C₂×C₄×C₇	C₂³×C₃	1	2	yes	—
95	C₂×C₄×C₇	C₂³×C₃	1	1	yes	characteristic
96	C₂×C₄×C₇	C₂³×C₃	1	1	yes	characteristic
97	C₂²⋊D₇	C₂×C₃×C₄	1	8	yes	complemented
98	C₂×Dic₇	—	4	2	no	—
99	C₂×Dic₇	C₂³×C₃	1	4	yes	—
100	C₂×Dic₇	C₂³×C₃	1	2	yes	—
101	D₂₈	C₂×C₃×C₄	1	4	yes	complemented
102	C₄×D₇	—	4	4	no	—
103	C₄×D₇	C₂×C₃×C₄	1	8	yes	complemented
104	Dic₁₄	C₂×C₃×C₄	1	4	yes	complemented
105	C₃×#41	—	7	8	no	—
106	C₂×C₃×Q₈	—	7	1	no	—
107	C₂×C₃×D₄	—	7	1	no	—
108	C₂×C₃×D₄	—	7	2	no	—
109	C₂²×C₃×C₄	—	7	2	no	—
110	C₂²×C₃×C₄	—	7	4	no	—
111	C₂²×C₃×C₄	—	7	2	no	—
112	C₂²×C₃×C₄	—	7	1	no	—
113	C₃×C₄⋊C₄	—	7	2	no	—
114	C₃×C₄⋊C₄	—	7	4	no	—
115	C₃×C₄⋊C₄	—	7	2	no	—
116	C₃×C₄⋊C₄	—	7	1	no	—
117	C₃×C₄⋊C₄	—	7	1	no	—
118	C₃×C₂²⋊C₄	—	7	4	no	—
119	C₃×C₂²⋊C₄	—	7	2	no	—
120	C₃×C₄²	—	7	4	no	—
121	C₃×C₄²	—	7	2	no	—
122	C₂×ΓL(1,𝔽₈)	—	2	2	no	—
123	C₂×ΓL(1,𝔽₈)	C₂³×C₄	1	1	yes	characteristic
124	C₂×ΓL(1,𝔽₈)	H(5,𝔽₂)	1	1	yes	characteristic
125	C₂×ΓL(1,𝔽₈)	2₋¹⁺⁴	1	1	yes	characteristic
126	AGL(1,𝔽₇)	—	2	4	no	—
127	C₂×#41	—	7	1	no	—
128	C₄×Q₈	—	7	2	no	—
129	C₄×D₄	—	7	4	no	—
130	C₄×D₄	—	7	2	no	—
131	#117	—	7	2	no	—
132	#117	AGL(1,𝔽₇)	1	1	yes	characteristic, complemented
133	C₂×C₄⋊C₄	—	7	1	no	—
134	C₂×C₄⋊C₄	—	7	2	no	—
135	C₂²×C₇	—	2	1	no	—
136	C₂²×C₇	—	2	1	no	—
137	C₂²×C₇	C₂²×C₃×C₄	1	2	yes	—
138	C₂²×C₇	C₂⁴×C₃	1	1	yes	characteristic
139	C₂×D₇	—	2	2	no	—
140	C₂×D₇	—	2	2	no	—
141	C₂×D₇	C₂²×C₃×C₄	1	4	yes	—
142	C₄×C₇	—	2	2	no	—
143	C₄×C₇	—	2	2	no	—
144	C₄×C₇	C₂²×C₃×C₄	1	2	yes	—
145	C₄×C₇	C₂²×C₃×C₄	1	2	yes	—
146	Dic₇	—	2	4	no	—
147	Dic₇	C₂²×C₃×C₄	1	4	yes	—
148	C₂³×C₃	—	7	2	no	—
149	C₂³×C₃	—	7	1	no	—
150	C₃×Q₈	—	7	4	no	—
151	C₃×D₄	—	7	4	no	—
152	C₃×D₄	—	7	8	no	—
153	C₂×C₃×C₄	—	7	8	no	—
154	C₂×C₃×C₄	—	7	2	no	—
155	C₂×C₃×C₄	—	7	4	no	—
156	C₂×C₃×C₄	—	7	4	no	—
157	C₂×C₃×C₄	—	7	1	no	—
158	C₂×C₃×C₄	—	7	1	no	—
159	C₂×C₃×C₄	—	7	2	no	—
160	C₂×C₃×C₄	—	7	2	no	—
161	C₂×C₃×C₄	—	28	4	no	—
162	C₂×C₃×C₄	—	28	2	no	—
163	ΓL(1,𝔽₈)	#520	1	1	yes	characteristic, complemented, Hall
164	#41	—	7	8	no	—
165	C₂×Q₈	—	7	1	no	—
166	C₂×D₄	—	7	2	no	—
167	C₂×D₄	—	7	1	no	—
168	C₂²×C₄	—	7	2	no	—
169	C₂²×C₄	—	7	2	no	—
170	C₂²×C₄	—	7	4	no	—
171	C₂²×C₄	C₂×AGL(1,𝔽₇)	1	1	yes	characteristic
172	C₄⋊C₄	—	7	4	no	—
173	C₄⋊C₄	—	7	2	no	—
174	C₄⋊C₄	—	7	2	no	—
175	C₄⋊C₄	C₂×AGL(1,𝔽₇)	1	1	yes	characteristic
176	C₄⋊C₄	C₂×AGL(1,𝔽₇)	1	1	yes	characteristic
177	C₂²⋊C₄	—	7	4	no	—
178	C₂²⋊C₄	C₂×AGL(1,𝔽₇)	1	2	yes	—
179	C₄²	—	7	4	no	—
180	C₄²	C₂×AGL(1,𝔽₇)	1	2	yes	—
181	C₂×C₇	—	2	2	no	—
182	C₂×C₇	C₂³×C₃×C₄	1	1	yes	characteristic
183	C₂×C₇	C₃×H(5,𝔽₂)	1	1	yes	characteristic
184	C₂×C₇	C₃×2₋¹⁺⁴	1	1	yes	characteristic
185	D₇	—	2	4	no	—
186	C₂²×C₃	—	7	4	no	—
187	C₂²×C₃	—	7	2	no	—
188	C₂²×C₃	—	7	1	no	—
189	C₂²×C₃	—	14	1	no	—
190	C₂²×C₃	—	14	1	no	—
191	C₂²×C₃	—	14	2	no	—
192	C₂²×C₃	—	14	2	no	—
193	C₃×C₄	—	7	4	no	—
194	C₃×C₄	—	7	2	no	—
195	C₃×C₄	—	7	2	no	—
196	C₃×C₄	—	14	2	no	—
197	C₃×C₄	—	14	2	no	—
198	C₃×C₄	—	14	4	no	—
199	C₂³	—	7	2	no	—
200	C₂³	C₂²×AGL(1,𝔽₇)	1	1	yes	characteristic
201	Q₈	—	7	4	no	—
202	D₄	—	7	8	no	—
203	D₄	—	7	4	no	—
204	C₂×C₄	—	7	2	no	—
205	C₂×C₄	—	7	8	no	—
206	C₂×C₄	—	7	4	no	—
207	C₂×C₄	—	28	4	no	—
208	C₂×C₄	—	28	2	no	—
209	C₂×C₄	C₄×AGL(1,𝔽₇)	1	4	yes	complemented
210	C₂×C₄	C₂²×AGL(1,𝔽₇)	1	2	yes	—
211	C₂×C₄	C₂²×AGL(1,𝔽₇)	1	2	yes	—
212	C₂×C₄	C₂²×AGL(1,𝔽₇)	1	1	yes	characteristic
213	C₂×C₄	C₂²×AGL(1,𝔽₇)	1	1	yes	characteristic
214	C₇	C₃×#520	1	1	yes	characteristic, complemented, Sylow
215	C₂×C₃	—	7	1	no	—
216	C₂×C₃	—	7	1	no	—
217	C₂×C₃	—	7	1	no	—
218	C₂×C₃	—	14	2	no	—
219	C₂×C₃	—	14	4	no	—
220	C₂²	—	2	1	no	—
221	C₂²	—	2	1	no	—
222	C₂²	—	7	4	no	—
223	C₂²	—	14	2	no	—
224	C₂²	—	14	2	no	—
225	C₂²	C₂×C₄×AGL(1,𝔽₇)	1	2	yes	—
226	C₂²	C₂³×AGL(1,𝔽₇)	1	1	yes	characteristic
227	C₄	—	2	2	no	—
228	C₄	—	2	2	no	—
229	C₄	—	7	4	no	—
230	C₄	—	14	4	no	—
231	C₄	C₂×C₄×AGL(1,𝔽₇)	1	2	yes	—
232	C₄	C₂×C₄×AGL(1,𝔽₇)	1	2	yes	—
233	C₃	—	7	1	no	Sylow
234	C₂	—	2	2	no	—
235	C₂	—	14	4	no	—
236	C₂	C₂²×C₄×AGL(1,𝔽₇)	1	1	yes	characteristic
237	C₂	#10621518	1	1	yes	characteristic
238	C₂	#10621519	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂²	C₂³×AGL(1,𝔽₇)
commutator	C₂×C₇	C₂³×C₃×C₄
Fitting	C₇×#117	C₂×C₃
socle, cosocle	C₂²×C₇	C₂⁴×C₃
prime radical	C₂×#1157⋊C₃	C₂
hypercenter	#117	AGL(1,𝔽₇)
hypocenter	C₇	C₃×#520
LCS	G ▷ C₂×C₇ ▷ C₇
UCS	C₁ ◁ C₂² ◁ #117

Complements

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

N	H ≅ G/N	type
2. ΓL(1,𝔽₈)×#117	C₂	semidirect
3. C₂×C₄⋊(ΓL(1,𝔽₈)⋊C₄)	C₂	semidirect
4. #6455⋊C₃	C₂	semidirect
5. C₄⋊C₄×AGL(1,𝔽₇)	C₂	semidirect
6. ((C₂²⋊D₇)⋊C₃)⋊C₄	C₂	semidirect
7. C₄×C₄⋊AGL(1,𝔽₇)	C₂	semidirect
8. C₄×ΓL(1,𝔽₈)⋊Q₈	C₂	semidirect
9. #90214	C₃	semidirect
13. #1157⋊C₃	C₄	semidirect
28. C₇×#117	C₂×C₃	semidirect
29. C₂×C₄⋊Dic₇	C₂×C₃	semidirect
30. #6455	C₂×C₃	semidirect
31. D₇×C₄⋊C₄	C₂×C₃	semidirect
32. (C₂²⋊D₇)⋊C₄	C₂×C₃	semidirect
33. C₄×D₂₈	C₂×C₃	semidirect
34. C₄×Dic₁₄	C₂×C₃	semidirect
38. C₂×C₄×ΓL(1,𝔽₈)	C₂×C₄	semidirect
43. (C₂²⋊D₇)⋊C₃	C₂×C₄	semidirect
47. C₄⋊AGL(1,𝔽₇)	C₂×C₄	semidirect
49. C₄×AGL(1,𝔽₇)	C₂×C₄	semidirect
50. ΓL(1,𝔽₈)⋊Q₈	C₂×C₄	semidirect
54. #1157	C₃×C₄	semidirect
92. C₂×C₄×C₇	C₂×C₃×C₄	semidirect
97. C₂²⋊D₇	C₂×C₃×C₄	semidirect
101. D₂₈	C₂×C₃×C₄	semidirect
103. C₄×D₇	C₂×C₃×C₄	semidirect
104. Dic₁₄	C₂×C₃×C₄	semidirect
132. #117	AGL(1,𝔽₇)	semidirect
163. ΓL(1,𝔽₈)	#520	semidirect
209. C₂×C₄	C₄×AGL(1,𝔽₇)	semidirect
214. C₇	C₃×#520	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
ΓL(1,𝔽₈)×#117	2
C₂×C₄⋊(ΓL(1,𝔽₈)⋊C₄)	2
#6455⋊C₃	2
C₄⋊C₄×AGL(1,𝔽₇)	2
((C₂²⋊D₇)⋊C₃)⋊C₄	2
C₄×C₄⋊AGL(1,𝔽₇)	2
C₄×ΓL(1,𝔽₈)⋊Q₈	2
#90214	3
C₃×#520	7

Subgroups — #90214⋊C3 #13057260

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — #90214⋊C₃ #13057260