Subgroups — #90213⋊C₃ #13057259

Order 1344.

620 conjugacy classes of subgroups.

The subgroup lattice has 228 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	#6456⋊C₃	C₂	1	2	yes	maximal, complemented
5	#6450⋊C₃	C₂	1	2	yes	maximal, complemented
6	((C₂²⋊D₇)⋊C₃)⋊C₄	C₂	1	4	yes	maximal, complemented
7	#6435⋊C₃	C₂	1	2	yes	maximal, complemented
8	C₄²×AGL(1,𝔽₇)	C₂	1	2	yes	maximal, complemented
9	#90213	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	2	yes	—
19	C₂²⋊C₄×ΓL(1,𝔽₈)	C₂²	1	2	yes	—
20	C₄²×ΓL(1,𝔽₈)	C₂²	1	2	yes	—
21	(C₂×AGL(1,𝔽₇))⋊C₄	C₂²	1	4	yes	—
22	(ΓL(1,𝔽₈)⋊C₄)⋊C₄	C₂²	1	4	yes	—
23	C₄×ΓL(1,𝔽₈)⋊C₄	C₂²	1	1	yes	characteristic
24	C₄×ΓL(1,𝔽₈)⋊C₄	C₂²	1	4	yes	—
25	C₄×ΓL(1,𝔽₈)⋊C₄	C₂²	1	1	yes	characteristic
26	C₄×ΓL(1,𝔽₈)⋊C₄	C₂²	1	2	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	#6456	C₂×C₃	1	2	yes	complemented
31	#6450	C₂×C₃	1	2	yes	complemented
32	(C₂²⋊D₇)⋊C₄	C₂×C₃	1	4	yes	complemented
33	#6435	C₂×C₃	1	2	yes	complemented
34	C₄²×D₇	C₂×C₃	1	2	yes	complemented
35	C₃×C₄×#41	—	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	4	yes	—
40	C₂×C₄×ΓL(1,𝔽₈)	C₂³	1	1	yes	characteristic
41	C₂×C₄×ΓL(1,𝔽₈)	C₂³	1	1	yes	characteristic
42	(C₂²⋊D₇)⋊C₃	C₂×C₄	1	8	yes	complemented
43	C₂×ΓL(1,𝔽₈)⋊C₄	—	2	4	no	—
44	C₂×ΓL(1,𝔽₈)⋊C₄	C₂³	1	2	yes	—
45	C₂×ΓL(1,𝔽₈)⋊C₄	C₂³	1	2	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)	—	2	8	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	2	yes	—
60	C₇×C₂²⋊C₄	C₂²×C₃	1	2	yes	—
61	C₄²×C₇	C₂²×C₃	1	2	yes	—
62	(C₂×D₇)⋊C₄	C₂²×C₃	1	4	yes	—
63	Dic₇⋊C₄	C₂²×C₃	1	4	yes	—
64	C₄×Dic₇	C₂²×C₃	1	1	yes	characteristic
65	C₄×Dic₇	C₂²×C₃	1	4	yes	—
66	C₄×Dic₇	C₂²×C₃	1	1	yes	characteristic
67	C₄×Dic₇	C₂²×C₃	1	2	yes	—
68	C₂×C₃×#41	—	7	1	no	—
69	C₃×C₄×Q₈	—	7	2	no	—
70	C₃×C₄×D₄	—	7	4	no	—
71	C₃×C₄×D₄	—	7	2	no	—
72	C₃×#117	—	7	2	no	—
73	C₃×#117	—	7	1	no	—
74	C₂×C₃×C₄²	—	7	2	no	—
75	C₂×C₃×C₄²	—	7	1	no	—
76	C₂²×ΓL(1,𝔽₈)	—	2	2	no	—
77	C₂²×ΓL(1,𝔽₈)	C₂²×C₄	1	2	yes	—
78	C₂²×ΓL(1,𝔽₈)	C₂⁴	1	1	yes	characteristic
79	C₂×AGL(1,𝔽₇)	—	2	4	no	—
80	C₂×AGL(1,𝔽₇)	C₂²×C₄	1	4	yes	—
81	C₄×ΓL(1,𝔽₈)	—	2	4	no	—
82	C₄×ΓL(1,𝔽₈)	C₂²×C₄	1	2	yes	—
83	C₄×ΓL(1,𝔽₈)	C₂²×C₄	1	2	yes	—
84	ΓL(1,𝔽₈)⋊C₄	—	2	2	no	—
85	ΓL(1,𝔽₈)⋊C₄	C₂²×C₄	1	4	yes	—
86	ΓL(1,𝔽₈)⋊C₄	#41	1	4	yes	complemented
87	C₄×#41	—	7	1	no	Sylow
88	C₂³×C₇	C₂³×C₃	1	1	yes	characteristic
89	C₂²×D₇	C₂³×C₃	1	2	yes	—
90	C₂×C₄×C₇	C₂×C₃×C₄	1	4	yes	complemented
91	C₂×C₄×C₇	C₂³×C₃	1	4	yes	—
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	8	yes	complemented
95	C₂×Dic₇	—	2	4	no	—
96	C₂×Dic₇	C₂³×C₃	1	2	yes	—
97	C₂×Dic₇	C₂³×C₃	1	2	yes	—
98	C₂×Dic₇	C₂³×C₃	1	2	yes	—
99	D₂₈	C₂×C₃×C₄	1	4	yes	complemented
100	C₄×D₇	—	2	8	no	—
101	C₄×D₇	C₂×C₃×C₄	1	8	yes	complemented
102	Dic₁₄	C₂×C₃×C₄	1	4	yes	complemented
103	C₃×#41	—	7	8	no	—
104	C₂×C₃×Q₈	—	7	1	no	—
105	C₂×C₃×D₄	—	7	1	no	—
106	C₂×C₃×D₄	—	7	2	no	—
107	C₂²×C₃×C₄	—	7	2	no	—
108	C₂²×C₃×C₄	—	7	4	no	—
109	C₂²×C₃×C₄	—	7	2	no	—
110	C₂²×C₃×C₄	—	7	1	no	—
111	C₃×C₄⋊C₄	—	7	4	no	—
112	C₃×C₄⋊C₄	—	7	2	no	—
113	C₃×C₂²⋊C₄	—	7	4	no	—
114	C₃×C₂²⋊C₄	—	7	2	no	—
115	C₃×C₄²	—	7	1	no	—
116	C₃×C₄²	—	7	1	no	—
117	C₃×C₄²	—	7	4	no	—
118	C₃×C₄²	—	7	2	no	—
119	C₃×C₄²	—	7	2	no	—
120	C₂×ΓL(1,𝔽₈)	—	2	2	no	—
121	C₂×ΓL(1,𝔽₈)	C₂³×C₄	1	1	yes	characteristic
122	C₂×ΓL(1,𝔽₈)	C₂×#41	1	2	yes	—
123	AGL(1,𝔽₇)	—	2	4	no	—
124	C₂×#41	—	7	1	no	—
125	C₄×Q₈	—	7	2	no	—
126	C₄×D₄	—	7	2	no	—
127	C₄×D₄	—	7	4	no	—
128	#117	—	7	2	no	—
129	#117	AGL(1,𝔽₇)	1	1	yes	characteristic, complemented
130	C₂×C₄²	—	7	1	no	—
131	C₂×C₄²	—	7	2	no	—
132	C₂²×C₇	—	2	2	no	—
133	C₂²×C₇	C₂²×C₃×C₄	1	2	yes	—
134	C₂²×C₇	C₂⁴×C₃	1	1	yes	characteristic
135	C₂×D₇	—	2	4	no	—
136	C₂×D₇	C₂²×C₃×C₄	1	4	yes	—
137	C₄×C₇	—	2	4	no	—
138	C₄×C₇	C₂²×C₃×C₄	1	2	yes	—
139	C₄×C₇	C₂²×C₃×C₄	1	2	yes	—
140	Dic₇	—	2	2	no	—
141	Dic₇	C₂²×C₃×C₄	1	4	yes	—
142	Dic₇	C₃×#41	1	4	yes	complemented
143	C₂³×C₃	—	7	2	no	—
144	C₂³×C₃	—	7	1	no	—
145	C₃×Q₈	—	7	4	no	—
146	C₃×D₄	—	7	4	no	—
147	C₃×D₄	—	7	8	no	—
148	C₂×C₃×C₄	—	7	8	no	—
149	C₂×C₃×C₄	—	7	2	no	—
150	C₂×C₃×C₄	—	7	2	no	—
151	C₂×C₃×C₄	—	7	2	no	—
152	C₂×C₃×C₄	—	7	4	no	—
153	C₂×C₃×C₄	—	7	1	no	—
154	C₂×C₃×C₄	—	7	4	no	—
155	C₂×C₃×C₄	—	7	1	no	—
156	C₂×C₃×C₄	—	14	8	no	—
157	C₂×C₃×C₄	—	14	4	no	—
158	ΓL(1,𝔽₈)	C₄×#41	1	1	yes	characteristic, complemented, Hall
159	#41	—	7	8	no	—
160	C₂×Q₈	—	7	1	no	—
161	C₂×D₄	—	7	2	no	—
162	C₂×D₄	—	7	1	no	—
163	C₂²×C₄	—	7	2	no	—
164	C₂²×C₄	—	7	2	no	—
165	C₂²×C₄	—	7	4	no	—
166	C₂²×C₄	C₂×AGL(1,𝔽₇)	1	1	yes	characteristic
167	C₄⋊C₄	—	7	4	no	—
168	C₄⋊C₄	C₂×AGL(1,𝔽₇)	1	2	yes	—
169	C₂²⋊C₄	—	7	4	no	—
170	C₂²⋊C₄	C₂×AGL(1,𝔽₇)	1	2	yes	—
171	C₄²	—	7	4	no	—
172	C₄²	—	7	2	no	—
173	C₄²	—	7	1	no	—
174	C₄²	—	7	1	no	—
175	C₄²	C₂×AGL(1,𝔽₇)	1	2	yes	—
176	C₂×C₇	—	2	2	no	—
177	C₂×C₇	C₂³×C₃×C₄	1	1	yes	characteristic
178	C₂×C₇	C₂×C₃×#41	1	2	yes	—
179	D₇	—	2	4	no	—
180	C₂²×C₃	—	7	4	no	—
181	C₂²×C₃	—	7	2	no	—
182	C₂²×C₃	—	7	1	no	—
183	C₂²×C₃	—	14	2	no	—
184	C₂²×C₃	—	14	4	no	—
185	C₃×C₄	—	7	4	no	—
186	C₃×C₄	—	7	4	no	—
187	C₃×C₄	—	7	2	no	—
188	C₃×C₄	—	7	2	no	—
189	C₃×C₄	—	14	4	no	—
190	C₃×C₄	—	14	2	no	—
191	C₂³	—	7	2	no	—
192	C₂³	C₂²×AGL(1,𝔽₇)	1	1	yes	characteristic
193	Q₈	—	7	4	no	—
194	D₄	—	7	8	no	—
195	D₄	—	7	4	no	—
196	C₂×C₄	—	7	2	no	—
197	C₂×C₄	—	7	2	no	—
198	C₂×C₄	—	7	8	no	—
199	C₂×C₄	—	7	2	no	—
200	C₂×C₄	—	14	8	no	—
201	C₂×C₄	—	14	4	no	—
202	C₂×C₄	C₄×AGL(1,𝔽₇)	1	4	yes	complemented
203	C₂×C₄	C₂²×AGL(1,𝔽₇)	1	4	yes	—
204	C₂×C₄	C₂²×AGL(1,𝔽₇)	1	1	yes	characteristic
205	C₂×C₄	C₂²×AGL(1,𝔽₇)	1	1	yes	characteristic
206	C₇	C₃×C₄×#41	1	1	yes	characteristic, complemented, Sylow
207	C₂×C₃	—	7	1	no	—
208	C₂×C₃	—	7	2	no	—
209	C₂×C₃	—	14	2	no	—
210	C₂×C₃	—	14	4	no	—
211	C₂²	—	2	2	no	—
212	C₂²	—	7	4	no	—
213	C₂²	—	14	4	no	—
214	C₂²	C₂×C₄×AGL(1,𝔽₇)	1	2	yes	—
215	C₂²	C₂³×AGL(1,𝔽₇)	1	1	yes	characteristic
216	C₄	—	2	4	no	—
217	C₄	—	7	4	no	—
218	C₄	—	7	4	no	—
219	C₄	—	14	2	no	—
220	C₄	C₂×C₄×AGL(1,𝔽₇)	1	2	yes	—
221	C₄	C₂×C₄×AGL(1,𝔽₇)	1	2	yes	—
222	C₃	—	7	1	no	Sylow
223	C₂	—	2	2	no	—
224	C₂	—	14	4	no	—
225	C₂	C₂²×C₄×AGL(1,𝔽₇)	1	1	yes	characteristic
226	C₂	#41×AGL(1,𝔽₇)	1	2	yes	—

	subgroup	G/H
center	C₂×C₄	C₂²×AGL(1,𝔽₇)
commutator	C₂×C₇	C₂³×C₃×C₄
Frattini	C₂²	C₂³×AGL(1,𝔽₇)
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×C₄×#41
LCS	G ▷ C₂×C₇ ▷ C₇
UCS	C₁ ◁ C₂×C₄ ◁ #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. #6456⋊C₃	C₂	semidirect
5. #6450⋊C₃	C₂	semidirect
6. ((C₂²⋊D₇)⋊C₃)⋊C₄	C₂	semidirect
7. #6435⋊C₃	C₂	semidirect
8. C₄²×AGL(1,𝔽₇)	C₂	semidirect
9. #90213	C₃	semidirect
13. #1157⋊C₃	C₄	semidirect
28. C₇×#117	C₂×C₃	semidirect
29. C₂×C₄×Dic₇	C₂×C₃	semidirect
30. #6456	C₂×C₃	semidirect
31. #6450	C₂×C₃	semidirect
32. (C₂²⋊D₇)⋊C₄	C₂×C₃	semidirect
33. #6435	C₂×C₃	semidirect
34. C₄²×D₇	C₂×C₃	semidirect
38. C₂×C₄×ΓL(1,𝔽₈)	C₂×C₄	semidirect
42. (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
86. ΓL(1,𝔽₈)⋊C₄	#41	semidirect
90. C₂×C₄×C₇	C₂×C₃×C₄	semidirect
94. C₂²⋊D₇	C₂×C₃×C₄	semidirect
99. D₂₈	C₂×C₃×C₄	semidirect
101. C₄×D₇	C₂×C₃×C₄	semidirect
102. Dic₁₄	C₂×C₃×C₄	semidirect
129. #117	AGL(1,𝔽₇)	semidirect
142. Dic₇	C₃×#41	semidirect
158. ΓL(1,𝔽₈)	C₄×#41	semidirect
202. C₂×C₄	C₄×AGL(1,𝔽₇)	semidirect
206. C₇	C₃×C₄×#41	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
#6456⋊C₃	2
#6450⋊C₃	2
((C₂²⋊D₇)⋊C₃)⋊C₄	2
#6435⋊C₃	2
C₄²×AGL(1,𝔽₇)	2
#90213	3
C₃×C₄×#41	7

Subgroups — #90213⋊C3 #13057259

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — #90213⋊C₃ #13057259