Subgroups — #90270⋊C₃ #13057313

Order 1344.

456 conjugacy classes of subgroups.

The subgroup lattice has 216 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄×Q₈×ΓL(1,𝔽₈)	C₂	1	1	yes	maximal, characteristic, complemented
2	#6509⋊C₃	C₂	1	2	yes	maximal, complemented
3	#6459⋊C₃	C₂	1	4	yes	maximal, complemented
4	C₄⋊C₄×AGL(1,𝔽₇)	C₂	1	2	yes	maximal, complemented
5	C₄×C₄⋊AGL(1,𝔽₇)	C₂	1	1	yes	maximal, characteristic
6	C₄×C₄⋊AGL(1,𝔽₇)	C₂	1	2	yes	maximal
7	#6433⋊C₃	C₂	1	2	yes	maximal, complemented
8	#6432⋊C₃	C₂	1	1	yes	maximal, characteristic, complemented
9	#90270	C₃	1	1	yes	maximal, characteristic, complemented, Hall
10	C₂×Q₈×ΓL(1,𝔽₈)	C₂²	1	1	yes	characteristic
11	C₂×C₄⋊AGL(1,𝔽₇)	C₂²	1	1	yes	characteristic
12	C₂×C₄×AGL(1,𝔽₇)	C₂²	1	2	yes	—
13	C₂×C₄×AGL(1,𝔽₇)	C₂²	1	4	yes	—
14	C₂×ΓL(1,𝔽₈)⋊Q₈	C₂²	1	2	yes	—
15	C₄⋊C₄×ΓL(1,𝔽₈)	C₂²	1	1	yes	characteristic
16	C₄⋊C₄×ΓL(1,𝔽₈)	C₂²	1	2	yes	—
17	C₄²×ΓL(1,𝔽₈)	C₂²	1	1	yes	characteristic
18	C₄²×ΓL(1,𝔽₈)	C₂²	1	2	yes	—
19	(C₂×AGL(1,𝔽₇))⋊C₄	C₂²	1	2	yes	—
20	(C₂×AGL(1,𝔽₇))⋊C₄	C₂²	1	4	yes	—
21	C₄⋊(ΓL(1,𝔽₈)⋊C₄)	C₂²	1	2	yes	—
22	C₄⋊(ΓL(1,𝔽₈)⋊C₄)	C₂²	1	2	yes	—
23	C₄⋊(ΓL(1,𝔽₈)⋊C₄)	C₂²	1	1	yes	characteristic
24	(ΓL(1,𝔽₈)⋊C₄)⋊C₄	C₂²	1	4	yes	—
25	(ΓL(1,𝔽₈)⋊C₄)⋊C₄	C₂²	1	4	yes	—
26	C₄×C₇×Q₈	C₂×C₃	1	1	yes	characteristic, complemented
27	#6509	C₂×C₃	1	2	yes	complemented
28	#6459	C₂×C₃	1	4	yes	complemented
29	D₇×C₄⋊C₄	C₂×C₃	1	2	yes	complemented
30	C₄×D₂₈	C₂×C₃	1	2	yes	—
31	C₄×D₂₈	C₂×C₃	1	1	yes	characteristic
32	#6433	C₂×C₃	1	2	yes	complemented
33	#6432	C₂×C₃	1	1	yes	characteristic, complemented
34	C₃×#554	—	7	1	no	maximal, Hall
35	C₂²×AGL(1,𝔽₇)	C₂³	1	2	yes	—
36	Q₈×ΓL(1,𝔽₈)	—	2	2	no	—
37	C₂×C₄×ΓL(1,𝔽₈)	C₂³	1	1	yes	characteristic
38	C₂×C₄×ΓL(1,𝔽₈)	C₂³	1	2	yes	—
39	C₂×C₄×ΓL(1,𝔽₈)	C₂³	1	2	yes	—
40	C₂×C₄×ΓL(1,𝔽₈)	C₂³	1	1	yes	characteristic
41	C₂×C₄×ΓL(1,𝔽₈)	C₂³	1	1	yes	characteristic
42	C₂×ΓL(1,𝔽₈)⋊C₄	C₂³	1	4	yes	—
43	C₂×ΓL(1,𝔽₈)⋊C₄	C₂³	1	2	yes	—
44	C₄⋊AGL(1,𝔽₇)	Q₈	1	4	yes	complemented
45	C₄×AGL(1,𝔽₇)	—	2	4	no	—
46	C₄×AGL(1,𝔽₇)	—	4	4	no	—
47	ΓL(1,𝔽₈)⋊Q₈	—	4	2	no	—
48	C₂×C₇×Q₈	C₂²×C₃	1	1	yes	characteristic
49	C₂×D₂₈	C₂²×C₃	1	1	yes	characteristic
50	C₂×C₄×D₇	C₂²×C₃	1	2	yes	—
51	C₂×C₄×D₇	C₂²×C₃	1	4	yes	—
52	C₂×Dic₁₄	C₂²×C₃	1	2	yes	—
53	C₇×C₄⋊C₄	C₂²×C₃	1	1	yes	characteristic
54	C₇×C₄⋊C₄	C₂²×C₃	1	2	yes	—
55	C₄²×C₇	C₂²×C₃	1	1	yes	characteristic
56	C₄²×C₇	C₂²×C₃	1	2	yes	—
57	(C₂×D₇)⋊C₄	C₂²×C₃	1	2	yes	—
58	(C₂×D₇)⋊C₄	C₂²×C₃	1	4	yes	—
59	C₄⋊Dic₇	C₂²×C₃	1	2	yes	—
60	C₄⋊Dic₇	C₂²×C₃	1	2	yes	—
61	C₄⋊Dic₇	C₂²×C₃	1	1	yes	characteristic
62	Dic₇⋊C₄	C₂²×C₃	1	4	yes	—
63	Dic₇⋊C₄	C₂²×C₃	1	4	yes	—
64	C₃×C₄⋊Q₈	—	7	1	no	—
65	C₃×#125	—	7	2	no	—
66	C₃×C₂²⋊Q₈	—	7	4	no	—
67	C₃×C₂²⋊Q₈	—	7	2	no	—
68	C₃×C₄×Q₈	—	7	1	no	—
69	C₃×C₄×D₄	—	7	2	no	—
70	C₃×C₄×D₄	—	7	1	no	—
71	C₂×C₃×C₄⋊C₄	—	7	2	no	—
72	C₂²×ΓL(1,𝔽₈)	C₂⁴	1	1	yes	characteristic
73	C₂×AGL(1,𝔽₇)	—	2	2	no	—
74	C₂×AGL(1,𝔽₇)	—	2	2	no	—
75	C₂×AGL(1,𝔽₇)	C₂×Q₈	1	4	yes	—
76	C₄×ΓL(1,𝔽₈)	—	2	2	no	—
77	C₄×ΓL(1,𝔽₈)	—	2	1	no	—
78	C₄×ΓL(1,𝔽₈)	—	2	2	no	—
79	C₄×ΓL(1,𝔽₈)	C₂×Q₈	1	2	yes	—
80	C₄×ΓL(1,𝔽₈)	#41	1	2	yes	—
81	ΓL(1,𝔽₈)⋊C₄	—	2	2	no	—
82	ΓL(1,𝔽₈)⋊C₄	—	2	4	no	—
83	#554	—	7	1	no	Sylow
84	C₂²×D₇	C₂³×C₃	1	2	yes	—
85	C₇×Q₈	—	2	2	no	—
86	C₂×C₄×C₇	C₂³×C₃	1	2	yes	—
87	C₂×C₄×C₇	C₂³×C₃	1	2	yes	—
88	C₂×C₄×C₇	C₂³×C₃	1	1	yes	characteristic
89	C₂×C₄×C₇	C₂³×C₃	1	1	yes	characteristic
90	C₂×C₄×C₇	C₂³×C₃	1	1	yes	characteristic
91	C₂×Dic₇	C₂³×C₃	1	2	yes	—
92	C₂×Dic₇	C₂³×C₃	1	4	yes	—
93	D₂₈	C₃×Q₈	1	4	yes	complemented
94	C₄×D₇	—	2	4	no	—
95	C₄×D₇	—	4	4	no	—
96	Dic₁₄	—	4	2	no	—
97	C₂×C₃×Q₈	—	7	2	no	—
98	C₂×C₃×Q₈	—	7	1	no	—
99	C₂×C₃×D₄	—	7	1	no	—
100	C₂²×C₃×C₄	—	7	2	no	—
101	C₂²×C₃×C₄	—	7	4	no	—
102	C₃×C₄⋊C₄	—	7	4	no	—
103	C₃×C₄⋊C₄	—	7	2	no	—
104	C₃×C₄⋊C₄	—	7	1	no	—
105	C₃×C₄⋊C₄	—	7	2	no	—
106	C₃×C₄⋊C₄	—	7	4	no	—
107	C₃×C₄⋊C₄	—	7	1	no	—
108	C₃×C₄⋊C₄	—	7	2	no	—
109	C₃×C₂²⋊C₄	—	7	2	no	—
110	C₃×C₂²⋊C₄	—	7	4	no	—
111	C₃×C₄²	—	7	2	no	—
112	C₃×C₄²	—	7	1	no	—
113	C₂×ΓL(1,𝔽₈)	C₂²×Q₈	1	1	yes	characteristic
114	C₂×ΓL(1,𝔽₈)	C₂×#41	1	1	yes	characteristic
115	C₂×ΓL(1,𝔽₈)	H(5,𝔽₂)	1	1	yes	characteristic
116	AGL(1,𝔽₇)	—	2	4	no	—
117	C₄⋊Q₈	—	7	1	no	—
118	#125	—	7	2	no	—
119	C₂²⋊Q₈	—	7	2	no	—
120	C₂²⋊Q₈	—	7	4	no	—
121	C₄×Q₈	AGL(1,𝔽₇)	1	1	yes	characteristic, complemented
122	C₄×D₄	—	7	2	no	—
123	C₄×D₄	—	7	1	no	—
124	C₂×C₄⋊C₄	—	7	2	no	—
125	C₂²×C₇	C₂⁴×C₃	1	1	yes	characteristic
126	C₂×D₇	—	2	2	no	—
127	C₂×D₇	—	2	2	no	—
128	C₂×D₇	C₂×C₃×Q₈	1	4	yes	—
129	C₄×C₇	—	2	2	no	—
130	C₄×C₇	—	2	1	no	—
131	C₄×C₇	—	2	2	no	—
132	C₄×C₇	C₂×C₃×Q₈	1	2	yes	—
133	C₄×C₇	C₃×#41	1	2	yes	—
134	Dic₇	—	2	2	no	—
135	Dic₇	—	2	4	no	—
136	C₂³×C₃	—	7	2	no	—
137	C₃×Q₈	—	14	2	no	—
138	C₃×Q₈	—	28	2	no	—
139	C₃×D₄	—	7	4	no	—
140	C₂×C₃×C₄	—	7	2	no	—
141	C₂×C₃×C₄	—	7	4	no	—
142	C₂×C₃×C₄	—	7	1	no	—
143	C₂×C₃×C₄	—	7	1	no	—
144	C₂×C₃×C₄	—	7	2	no	—
145	C₂×C₃×C₄	—	7	1	no	—
146	C₂×C₃×C₄	—	7	2	no	—
147	C₂×C₃×C₄	—	14	4	no	—
148	C₂×C₃×C₄	—	28	4	no	—
149	ΓL(1,𝔽₈)	#554	1	1	yes	characteristic, complemented, Hall
150	C₂×Q₈	—	7	2	no	—
151	C₂×Q₈	C₂×AGL(1,𝔽₇)	1	1	yes	characteristic
152	C₂×D₄	—	7	1	no	—
153	C₂²×C₄	—	7	2	no	—
154	C₂²×C₄	—	7	4	no	—
155	C₄⋊C₄	—	7	4	no	—
156	C₄⋊C₄	—	7	4	no	—
157	C₄⋊C₄	—	7	2	no	—
158	C₄⋊C₄	—	7	1	no	—
159	C₄⋊C₄	—	7	2	no	—
160	C₄⋊C₄	C₂×AGL(1,𝔽₇)	1	1	yes	characteristic
161	C₄⋊C₄	C₂×AGL(1,𝔽₇)	1	2	yes	—
162	C₂²⋊C₄	—	7	2	no	—
163	C₂²⋊C₄	—	7	4	no	—
164	C₄²	C₂×AGL(1,𝔽₇)	1	2	yes	—
165	C₄²	C₂×AGL(1,𝔽₇)	1	1	yes	characteristic
166	C₂×C₇	C₂²×C₃×Q₈	1	1	yes	characteristic
167	C₂×C₇	C₂×C₃×#41	1	1	yes	characteristic
168	C₂×C₇	C₃×H(5,𝔽₂)	1	1	yes	characteristic
169	D₇	—	2	4	no	—
170	C₂²×C₃	—	7	4	no	—
171	C₂²×C₃	—	7	1	no	—
172	C₂²×C₃	—	14	2	no	—
173	C₂²×C₃	—	14	2	no	—
174	C₃×C₄	—	7	2	no	—
175	C₃×C₄	—	7	2	no	—
176	C₃×C₄	—	14	2	no	—
177	C₃×C₄	—	14	2	no	—
178	C₃×C₄	—	14	1	no	—
179	C₃×C₄	—	14	2	no	—
180	C₃×C₄	—	14	4	no	—
181	C₂³	—	7	2	no	—
182	Q₈	—	2	2	no	—
183	Q₈	—	28	2	no	—
184	D₄	—	7	4	no	—
185	C₂×C₄	—	7	2	no	—
186	C₂×C₄	—	7	4	no	—
187	C₂×C₄	—	14	4	no	—
188	C₂×C₄	—	28	4	no	—
189	C₂×C₄	C₂²×AGL(1,𝔽₇)	1	2	yes	—
190	C₂×C₄	C₂²×AGL(1,𝔽₇)	1	1	yes	characteristic
191	C₂×C₄	C₂²×AGL(1,𝔽₇)	1	2	yes	—
192	C₂×C₄	C₂²×AGL(1,𝔽₇)	1	1	yes	characteristic
193	C₂×C₄	C₂²×AGL(1,𝔽₇)	1	1	yes	characteristic
194	C₇	C₃×#554	1	1	yes	characteristic, complemented, Sylow
195	C₂×C₃	—	7	1	no	—
196	C₂×C₃	—	7	1	no	—
197	C₂×C₃	—	7	1	no	—
198	C₂×C₃	—	14	4	no	—
199	C₂²	—	7	4	no	—
200	C₂²	—	14	2	no	—
201	C₂²	—	14	2	no	—
202	C₂²	C₂³×AGL(1,𝔽₇)	1	1	yes	characteristic
203	C₄	—	2	2	no	—
204	C₄	—	2	2	no	—
205	C₄	—	2	1	no	—
206	C₄	—	14	2	no	—
207	C₄	—	14	4	no	—
208	C₄	#1157⋊C₃	1	2	yes	—
209	C₄	Q₈×AGL(1,𝔽₇)	1	2	yes	—
210	C₃	—	7	1	no	Sylow
211	C₂	—	14	4	no	—
212	C₂	C₂×#1157⋊C₃	1	1	yes	characteristic
213	C₂	C₂×Q₈×AGL(1,𝔽₇)	1	1	yes	characteristic
214	C₂	#10621518	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂²	C₂³×AGL(1,𝔽₇)
commutator, socle, cosocle	C₂²×C₇	C₂⁴×C₃
Fitting	C₄×C₇×Q₈	C₂×C₃
prime radical	C₂×C₄⋊AGL(1,𝔽₇)	C₂²
hypercenter	C₄×Q₈	AGL(1,𝔽₇)
hypocenter	C₇	C₃×#554
LCS	G ▷ C₂²×C₇ ▷ C₇
UCS	C₁ ◁ C₂² ◁ C₄×Q₈

Complements

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

N	H ≅ G/N	type
1. C₄×Q₈×ΓL(1,𝔽₈)	C₂	semidirect
2. #6509⋊C₃	C₂	semidirect
3. #6459⋊C₃	C₂	semidirect
4. C₄⋊C₄×AGL(1,𝔽₇)	C₂	semidirect
7. #6433⋊C₃	C₂	semidirect
8. #6432⋊C₃	C₂	semidirect
9. #90270	C₃	semidirect
26. C₄×C₇×Q₈	C₂×C₃	semidirect
27. #6509	C₂×C₃	semidirect
28. #6459	C₂×C₃	semidirect
29. D₇×C₄⋊C₄	C₂×C₃	semidirect
32. #6433	C₂×C₃	semidirect
33. #6432	C₂×C₃	semidirect
44. C₄⋊AGL(1,𝔽₇)	Q₈	semidirect
93. D₂₈	C₃×Q₈	semidirect
121. C₄×Q₈	AGL(1,𝔽₇)	semidirect
149. ΓL(1,𝔽₈)	#554	semidirect
194. C₇	C₃×#554	semidirect

Maximal subgroups

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

subgroup	index
C₄×Q₈×ΓL(1,𝔽₈)	2
#6509⋊C₃	2
#6459⋊C₃	2
C₄⋊C₄×AGL(1,𝔽₇)	2
C₄×C₄⋊AGL(1,𝔽₇)	2
#6433⋊C₃	2
#6432⋊C₃	2
#90270	3
C₃×#554	7

Subgroups — #90270⋊C3 #13057313

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — #90270⋊C₃ #13057313