Subgroups — #90415⋊C₃ #13057459

Order 1344.

540 conjugacy classes of subgroups.

The subgroup lattice has 256 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#66659	C₂	1	1	yes	maximal, characteristic
2	C₂×Q₈×AGL(1,𝔽₇)	C₂	1	1	yes	maximal, characteristic, complemented
3	ΓL(1,𝔽₈)×C₄⋊Q₈	C₂	1	1	yes	maximal, characteristic, complemented
4	#6510⋊C₃	C₂	1	1	yes	maximal, characteristic, complemented
5	#6507⋊C₃	C₂	1	1	yes	maximal, characteristic, complemented
6	#6459⋊C₃	C₂	1	4	yes	maximal, complemented
7	#6457⋊C₃	C₂	1	4	yes	maximal, complemented
8	#6438⋊C₃	C₂	1	1	yes	maximal, characteristic, complemented
9	#6435⋊C₃	C₂	1	1	yes	maximal, characteristic, complemented
10	#90415	C₃	1	1	yes	maximal, characteristic, complemented, Hall
11	C₂×Q₈×ΓL(1,𝔽₈)	C₂²	1	1	yes	characteristic
12	C₂×Q₈×ΓL(1,𝔽₈)	C₂²	1	1	yes	characteristic
13	#66659	—	2	4	no	—
14	Q₈×AGL(1,𝔽₇)	—	2	4	no	—
15	C₂×C₄⋊AGL(1,𝔽₇)	C₂²	1	1	yes	characteristic
16	C₂×C₄⋊AGL(1,𝔽₇)	C₂²	1	2	yes	—
17	C₂×C₄×AGL(1,𝔽₇)	C₂²	1	2	yes	—
18	C₂×C₄×AGL(1,𝔽₇)	C₂²	1	2	yes	—
19	C₂×C₄×AGL(1,𝔽₇)	C₂²	1	1	yes	characteristic
20	C₂×ΓL(1,𝔽₈)⋊Q₈	C₂²	1	1	yes	characteristic
21	C₂×ΓL(1,𝔽₈)⋊Q₈	C₂²	1	2	yes	—
22	C₄⋊C₄×ΓL(1,𝔽₈)	C₂²	1	4	yes	—
23	C₄²×ΓL(1,𝔽₈)	C₂²	1	1	yes	characteristic
24	(C₂×AGL(1,𝔽₇))⋊C₄	C₂²	1	2	yes	—
25	(C₂×AGL(1,𝔽₇))⋊C₄	C₂²	1	4	yes	—
26	(C₂×AGL(1,𝔽₇))⋊C₄	C₂²	1	4	yes	—
27	(ΓL(1,𝔽₈)⋊C₄)⋊C₄	C₂²	1	2	yes	—
28	(ΓL(1,𝔽₈)⋊C₄)⋊C₄	C₂²	1	4	yes	—
29	C₄×ΓL(1,𝔽₈)⋊C₄	C₂²	1	1	yes	characteristic
30	C₂×#1161	C₂×C₃	1	1	yes	characteristic
31	C₂×Q₈×D₇	C₂×C₃	1	1	yes	characteristic, complemented
32	C₇×C₄⋊Q₈	C₂×C₃	1	1	yes	characteristic, complemented
33	#6510	C₂×C₃	1	1	yes	characteristic, complemented
34	#6507	C₂×C₃	1	1	yes	characteristic, complemented
35	#6459	C₂×C₃	1	4	yes	complemented
36	#6457	C₂×C₃	1	4	yes	complemented
37	#6438	C₂×C₃	1	1	yes	characteristic, complemented
38	#6435	C₂×C₃	1	1	yes	characteristic, complemented
39	C₃×#536	—	7	1	no	maximal, Hall
40	C₂²×AGL(1,𝔽₇)	C₂³	1	2	yes	—
41	C₂²×AGL(1,𝔽₇)	C₂³	1	1	yes	characteristic
42	Q₈×ΓL(1,𝔽₈)	—	2	2	no	—
43	Q₈×ΓL(1,𝔽₈)	—	2	2	no	—
44	C₂×C₄×ΓL(1,𝔽₈)	C₂³	1	2	yes	—
45	C₂×C₄×ΓL(1,𝔽₈)	C₂³	1	2	yes	—
46	C₂×C₄×ΓL(1,𝔽₈)	C₂³	1	2	yes	—
47	C₂×C₄×ΓL(1,𝔽₈)	C₂³	1	1	yes	characteristic
48	C₂×ΓL(1,𝔽₈)⋊C₄	C₂³	1	2	yes	—
49	C₂×ΓL(1,𝔽₈)⋊C₄	C₂³	1	2	yes	—
50	C₂×ΓL(1,𝔽₈)⋊C₄	C₂³	1	1	yes	characteristic
51	C₄⋊AGL(1,𝔽₇)	—	2	4	no	—
52	C₄⋊AGL(1,𝔽₇)	—	2	2	no	—
53	C₄×AGL(1,𝔽₇)	—	2	4	no	—
54	C₄×AGL(1,𝔽₇)	—	2	4	no	—
55	C₄×AGL(1,𝔽₇)	D₄	1	4	yes	—
56	ΓL(1,𝔽₈)⋊Q₈	—	2	2	no	—
57	ΓL(1,𝔽₈)⋊Q₈	—	2	4	no	—
58	C₂×C₇×Q₈	C₂²×C₃	1	1	yes	characteristic
59	C₂×C₇×Q₈	C₂²×C₃	1	1	yes	characteristic
60	#1161	—	2	4	no	—
61	Q₈×D₇	—	2	4	no	—
62	C₂×D₂₈	C₂²×C₃	1	1	yes	characteristic
63	C₂×D₂₈	C₂²×C₃	1	2	yes	—
64	C₂×C₄×D₇	C₂²×C₃	1	2	yes	—
65	C₂×C₄×D₇	C₂²×C₃	1	2	yes	—
66	C₂×C₄×D₇	C₂²×C₃	1	1	yes	characteristic
67	C₂×Dic₁₄	C₂²×C₃	1	1	yes	characteristic
68	C₂×Dic₁₄	C₂²×C₃	1	2	yes	—
69	C₇×C₄⋊C₄	C₂²×C₃	1	4	yes	—
70	C₄²×C₇	C₂²×C₃	1	1	yes	characteristic
71	(C₂×D₇)⋊C₄	C₂²×C₃	1	2	yes	—
72	(C₂×D₇)⋊C₄	C₂²×C₃	1	4	yes	—
73	(C₂×D₇)⋊C₄	C₂²×C₃	1	4	yes	—
74	Dic₇⋊C₄	C₂²×C₃	1	2	yes	—
75	Dic₇⋊C₄	C₂²×C₃	1	4	yes	—
76	C₄×Dic₇	C₂²×C₃	1	1	yes	characteristic
77	C₂×C₃×#41	—	7	1	no	—
78	C₂²×C₃×Q₈	—	7	1	no	—
79	C₃×C₄⋊Q₈	—	7	1	no	—
80	C₃×C₄⋊Q₈	—	7	1	no	—
81	C₃×#124	—	7	1	no	—
82	C₃×#124	—	7	1	no	—
83	C₃×#123	—	7	4	no	—
84	C₃×C₂²⋊Q₈	—	7	4	no	—
85	C₃×#117	—	7	1	no	—
86	C₂²×ΓL(1,𝔽₈)	C₂⁴	1	1	yes	characteristic
87	C₂×AGL(1,𝔽₇)	—	2	2	no	—
88	C₂×AGL(1,𝔽₇)	—	2	2	no	—
89	C₂×AGL(1,𝔽₇)	—	2	4	no	—
90	C₂×AGL(1,𝔽₇)	C₂×D₄	1	2	yes	—
91	C₄×ΓL(1,𝔽₈)	—	2	2	no	—
92	C₄×ΓL(1,𝔽₈)	—	2	2	no	—
93	C₄×ΓL(1,𝔽₈)	—	2	2	no	—
94	C₄×ΓL(1,𝔽₈)	C₂×D₄	1	2	yes	—
95	ΓL(1,𝔽₈)⋊C₄	—	2	2	no	—
96	ΓL(1,𝔽₈)⋊C₄	—	2	2	no	—
97	ΓL(1,𝔽₈)⋊C₄	C₂×D₄	1	2	yes	—
98	#536	—	7	1	no	Sylow
99	C₂²×D₇	C₂³×C₃	1	2	yes	—
100	C₂²×D₇	C₂³×C₃	1	1	yes	characteristic
101	C₇×Q₈	—	2	2	no	—
102	C₇×Q₈	—	2	2	no	—
103	C₂×C₄×C₇	C₂³×C₃	1	2	yes	—
104	C₂×C₄×C₇	C₂³×C₃	1	2	yes	—
105	C₂×C₄×C₇	C₂³×C₃	1	2	yes	—
106	C₂×C₄×C₇	C₂³×C₃	1	1	yes	characteristic
107	C₂×Dic₇	C₂³×C₃	1	2	yes	—
108	C₂×Dic₇	C₂³×C₃	1	2	yes	—
109	C₂×Dic₇	C₂³×C₃	1	1	yes	characteristic
110	D₂₈	—	2	4	no	—
111	D₂₈	—	2	2	no	—
112	C₄×D₇	—	2	4	no	—
113	C₄×D₇	—	2	4	no	—
114	C₄×D₇	C₃×D₄	1	4	yes	—
115	Dic₁₄	—	2	2	no	—
116	Dic₁₄	—	2	4	no	—
117	C₃×#41	—	14	4	no	—
118	C₂×C₃×Q₈	—	7	2	no	—
119	C₂×C₃×Q₈	—	7	1	no	—
120	C₂×C₃×Q₈	—	7	1	no	—
121	C₂×C₃×Q₈	—	7	1	no	—
122	C₂×C₃×Q₈	—	14	4	no	—
123	C₂×C₃×D₄	—	7	2	no	—
124	C₂×C₃×D₄	—	7	1	no	—
125	C₂²×C₃×C₄	—	7	1	no	—
126	C₂²×C₃×C₄	—	7	2	no	—
127	C₂²×C₃×C₄	—	7	2	no	—
128	C₃×C₄⋊C₄	—	7	2	no	—
129	C₃×C₄⋊C₄	—	7	4	no	—
130	C₃×C₄⋊C₄	—	7	4	no	—
131	C₃×C₂²⋊C₄	—	7	2	no	—
132	C₃×C₂²⋊C₄	—	7	4	no	—
133	C₃×C₂²⋊C₄	—	7	4	no	—
134	C₃×C₄²	—	7	1	no	—
135	C₃×C₄²	—	7	1	no	—
136	C₂×ΓL(1,𝔽₈)	C₂²×D₄	1	1	yes	characteristic
137	C₂×ΓL(1,𝔽₈)	2₋¹⁺⁴	1	2	yes	—
138	AGL(1,𝔽₇)	—	2	2	no	—
139	AGL(1,𝔽₇)	—	4	2	no	—
140	C₂×#41	—	7	1	no	—
141	C₂²×Q₈	—	7	1	no	—
142	C₄⋊Q₈	—	7	1	no	—
143	C₄⋊Q₈	AGL(1,𝔽₇)	1	1	yes	characteristic, complemented
144	#124	—	7	1	no	—
145	#124	—	7	1	no	—
146	#123	—	7	4	no	—
147	C₂²⋊Q₈	—	7	4	no	—
148	#117	—	7	1	no	—
149	C₂²×C₇	C₂⁴×C₃	1	1	yes	characteristic
150	C₂×D₇	—	2	2	no	—
151	C₂×D₇	—	2	2	no	—
152	C₂×D₇	—	2	4	no	—
153	C₂×D₇	C₂×C₃×D₄	1	2	yes	—
154	C₄×C₇	—	2	2	no	—
155	C₄×C₇	—	2	2	no	—
156	C₄×C₇	—	2	2	no	—
157	C₄×C₇	C₂×C₃×D₄	1	2	yes	—
158	Dic₇	—	2	2	no	—
159	Dic₇	—	2	2	no	—
160	Dic₇	C₂×C₃×D₄	1	2	yes	—
161	C₂³×C₃	—	7	1	no	—
162	C₂³×C₃	—	7	2	no	—
163	C₃×Q₈	—	14	2	no	—
164	C₃×Q₈	—	14	2	no	—
165	C₃×Q₈	—	14	4	no	—
166	C₃×Q₈	—	14	2	no	—
167	C₃×D₄	—	14	2	no	—
168	C₃×D₄	—	14	4	no	—
169	C₂×C₃×C₄	—	7	4	no	—
170	C₂×C₃×C₄	—	7	1	no	—
171	C₂×C₃×C₄	—	7	2	no	—
172	C₂×C₃×C₄	—	7	2	no	—
173	C₂×C₃×C₄	—	7	1	no	—
174	C₂×C₃×C₄	—	7	2	no	—
175	C₂×C₃×C₄	—	7	2	no	—
176	C₂×C₃×C₄	—	7	2	no	—
177	C₂×C₃×C₄	—	14	4	no	—
178	C₂×C₃×C₄	—	14	4	no	—
179	ΓL(1,𝔽₈)	#536	1	1	yes	characteristic, complemented, Hall
180	#41	—	14	4	no	—
181	C₂×Q₈	—	7	2	no	—
182	C₂×Q₈	—	7	1	no	—
183	C₂×Q₈	—	14	4	no	—
184	C₂×Q₈	C₂×AGL(1,𝔽₇)	1	1	yes	characteristic
185	C₂×Q₈	C₂×AGL(1,𝔽₇)	1	1	yes	characteristic
186	C₂×D₄	—	7	2	no	—
187	C₂×D₄	—	7	1	no	—
188	C₂²×C₄	—	7	1	no	—
189	C₂²×C₄	—	7	2	no	—
190	C₂²×C₄	—	7	2	no	—
191	C₄⋊C₄	—	7	2	no	—
192	C₄⋊C₄	—	7	4	no	—
193	C₄⋊C₄	C₂×AGL(1,𝔽₇)	1	4	yes	—
194	C₂²⋊C₄	—	7	2	no	—
195	C₂²⋊C₄	—	7	4	no	—
196	C₂²⋊C₄	—	7	4	no	—
197	C₄²	—	7	1	no	—
198	C₄²	C₂×AGL(1,𝔽₇)	1	1	yes	characteristic
199	C₂×C₇	C₂²×C₃×D₄	1	1	yes	characteristic
200	C₂×C₇	C₃×2₋¹⁺⁴	1	2	yes	—
201	D₇	—	2	2	no	—
202	D₇	—	4	2	no	—
203	C₂²×C₃	—	7	2	no	—
204	C₂²×C₃	—	7	1	no	—
205	C₂²×C₃	—	14	2	no	—
206	C₂²×C₃	—	14	2	no	—
207	C₂²×C₃	—	14	4	no	—
208	C₃×C₄	—	7	2	no	—
209	C₃×C₄	—	7	2	no	—
210	C₃×C₄	—	14	2	no	—
211	C₃×C₄	—	14	2	no	—
212	C₃×C₄	—	14	2	no	—
213	C₃×C₄	—	14	2	no	—
214	C₃×C₄	—	14	2	no	—
215	C₂³	—	7	1	no	—
216	C₂³	—	7	2	no	—
217	Q₈	—	2	2	no	—
218	Q₈	—	2	2	no	—
219	Q₈	—	14	4	no	—
220	Q₈	—	14	2	no	—
221	D₄	—	14	4	no	—
222	D₄	—	14	2	no	—
223	C₂×C₄	—	7	1	no	—
224	C₂×C₄	—	7	4	no	—
225	C₂×C₄	—	7	2	no	—
226	C₂×C₄	—	7	2	no	—
227	C₂×C₄	—	14	4	no	—
228	C₂×C₄	—	14	4	no	—
229	C₂×C₄	C₂²×AGL(1,𝔽₇)	1	2	yes	—
230	C₂×C₄	C₂²×AGL(1,𝔽₇)	1	2	yes	—
231	C₂×C₄	C₂²×AGL(1,𝔽₇)	1	2	yes	—
232	C₂×C₄	C₂²×AGL(1,𝔽₇)	1	1	yes	characteristic
233	C₇	C₃×#536	1	1	yes	characteristic, complemented, Sylow
234	C₂×C₃	—	7	1	no	—
235	C₂×C₃	—	7	2	no	—
236	C₂×C₃	—	14	2	no	—
237	C₂×C₃	—	28	2	no	—
238	C₂²	—	7	2	no	—
239	C₂²	—	14	2	no	—
240	C₂²	—	14	2	no	—
241	C₂²	—	14	4	no	—
242	C₂²	C₂³×AGL(1,𝔽₇)	1	1	yes	characteristic
243	C₄	—	2	2	no	—
244	C₄	—	2	2	no	—
245	C₄	—	2	2	no	—
246	C₄	—	7	2	no	—
247	C₄	—	14	2	no	—
248	C₄	—	14	2	no	—
249	C₄	D₄×AGL(1,𝔽₇)	1	2	yes	—
250	C₃	—	7	1	no	Sylow
251	C₂	—	14	2	no	—
252	C₂	—	28	2	no	—
253	C₂	C₂×D₄×AGL(1,𝔽₇)	1	1	yes	characteristic
254	C₂	#10621516	1	2	yes	—

	subgroup	G/H
center, Frattini	C₂²	C₂³×AGL(1,𝔽₇)
commutator, socle, cosocle	C₂²×C₇	C₂⁴×C₃
Fitting	C₇×C₄⋊Q₈	C₂×C₃
prime radical	C₂×#66659	C₂
hypercenter	C₄⋊Q₈	AGL(1,𝔽₇)
hypocenter	C₇	C₃×#536
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
2. C₂×Q₈×AGL(1,𝔽₇)	C₂	semidirect
3. ΓL(1,𝔽₈)×C₄⋊Q₈	C₂	semidirect
4. #6510⋊C₃	C₂	semidirect
5. #6507⋊C₃	C₂	semidirect
6. #6459⋊C₃	C₂	semidirect
7. #6457⋊C₃	C₂	semidirect
8. #6438⋊C₃	C₂	semidirect
9. #6435⋊C₃	C₂	semidirect
10. #90415	C₃	semidirect
31. C₂×Q₈×D₇	C₂×C₃	semidirect
32. C₇×C₄⋊Q₈	C₂×C₃	semidirect
33. #6510	C₂×C₃	semidirect
34. #6507	C₂×C₃	semidirect
35. #6459	C₂×C₃	semidirect
36. #6457	C₂×C₃	semidirect
37. #6438	C₂×C₃	semidirect
38. #6435	C₂×C₃	semidirect
143. C₄⋊Q₈	AGL(1,𝔽₇)	semidirect
179. ΓL(1,𝔽₈)	#536	semidirect
233. C₇	C₃×#536	semidirect

Maximal subgroups

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

subgroup	index
C₂×#66659	2
C₂×Q₈×AGL(1,𝔽₇)	2
ΓL(1,𝔽₈)×C₄⋊Q₈	2
#6510⋊C₃	2
#6507⋊C₃	2
#6459⋊C₃	2
#6457⋊C₃	2
#6438⋊C₃	2
#6435⋊C₃	2
#90415	3
C₃×#536	7

Subgroups — #90415⋊C3 #13057459

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — #90415⋊C₃ #13057459