Subgroups — #90316⋊C₃ #13057363

Order 1344.

384 conjugacy classes of subgroups.

The subgroup lattice has 272 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	ΓL(1,𝔽₈)×C₂²⋊Q₈	C₂	1	1	yes	maximal, characteristic
2	Q₈×ΓL(1,𝔽₈)⋊C₄	C₂	1	1	yes	maximal, characteristic, complemented
3	#6507⋊C₃	C₂	1	1	yes	maximal, characteristic, complemented
4	#6489⋊C₃	C₂	1	1	yes	maximal, characteristic
5	#6487⋊C₃	C₂	1	1	yes	maximal, characteristic
6	#6453⋊C₃	C₂	1	1	yes	maximal, characteristic, complemented
7	#6453⋊C₃	C₂	1	2	yes	maximal, complemented
8	#6452⋊C₃	C₂	1	2	yes	maximal, complemented
9	#6450⋊C₃	C₂	1	1	yes	maximal, characteristic, complemented
10	#6442⋊C₃	C₂	1	2	yes	maximal
11	#6442⋊C₃	C₂	1	2	yes	maximal
12	#90316	C₃	1	1	yes	maximal, characteristic, complemented, Hall
13	C₂×Q₈×ΓL(1,𝔽₈)	C₂²	1	1	yes	characteristic
14	C₂²×C₄×ΓL(1,𝔽₈)	C₂²	1	1	yes	characteristic
15	C₂×ΓL(1,𝔽₈)⋊Q₈	C₂²	1	1	yes	characteristic
16	C₄⋊C₄×ΓL(1,𝔽₈)	C₂²	1	1	yes	characteristic
17	C₄⋊C₄×ΓL(1,𝔽₈)	C₂²	1	2	yes	—
18	C₂²⋊C₄×ΓL(1,𝔽₈)	C₂²	1	2	yes	—
19	(C₂²⋊Dic₇)⋊C₃	C₂²	1	2	yes	—
20	(C₂²⋊Dic₇)⋊C₃	C₂²	1	1	yes	characteristic
21	(C₂²⋊Dic₇)⋊C₃	C₂²	1	1	yes	characteristic
22	C₄⋊(ΓL(1,𝔽₈)⋊C₄)	C₂²	1	2	yes	—
23	C₄⋊(ΓL(1,𝔽₈)⋊C₄)	C₂²	1	2	yes	—
24	C₄⋊(ΓL(1,𝔽₈)⋊C₄)	C₂²	1	1	yes	characteristic
25	C₄⋊(ΓL(1,𝔽₈)⋊C₄)	C₂²	1	1	yes	characteristic
26	C₄⋊(ΓL(1,𝔽₈)⋊C₄)	C₂²	1	1	yes	characteristic
27	(ΓL(1,𝔽₈)⋊C₄)⋊C₄	C₂²	1	2	yes	—
28	(ΓL(1,𝔽₈)⋊C₄)⋊C₄	C₂²	1	2	yes	—
29	(ΓL(1,𝔽₈)⋊C₄)⋊C₄	C₂²	1	2	yes	—
30	(ΓL(1,𝔽₈)⋊C₄)⋊C₄	C₂²	1	2	yes	—
31	(ΓL(1,𝔽₈)⋊C₄)⋊C₄	C₂²	1	2	yes	—
32	C₄×ΓL(1,𝔽₈)⋊C₄	C₂²	1	2	yes	—
33	C₄×ΓL(1,𝔽₈)⋊C₄	C₂²	1	2	yes	—
34	C₄×ΓL(1,𝔽₈)⋊C₄	C₂²	1	1	yes	characteristic
35	C₄×ΓL(1,𝔽₈)⋊C₄	C₂²	1	1	yes	characteristic
36	C₇×C₂²⋊Q₈	C₂×C₃	1	1	yes	characteristic
37	Q₈×Dic₇	C₂×C₃	1	1	yes	characteristic, complemented
38	#6507	C₂×C₃	1	1	yes	characteristic, complemented
39	#6489	C₂×C₃	1	1	yes	characteristic
40	#6487	C₂×C₃	1	1	yes	characteristic
41	#6453	C₂×C₃	1	2	yes	complemented
42	#6453	C₂×C₃	1	1	yes	characteristic, complemented
43	#6452	C₂×C₃	1	2	yes	complemented
44	#6450	C₂×C₃	1	1	yes	characteristic, complemented
45	#6442	C₂×C₃	1	2	yes	—
46	#6442	C₂×C₃	1	2	yes	—
47	C₃×#541	—	7	1	no	maximal, Hall
48	C₂³×ΓL(1,𝔽₈)	C₂³	1	1	yes	characteristic
49	Q₈×ΓL(1,𝔽₈)	—	2	2	no	—
50	C₂×C₄×ΓL(1,𝔽₈)	—	2	2	no	—
51	C₂×C₄×ΓL(1,𝔽₈)	C₂³	1	2	yes	—
52	C₂×C₄×ΓL(1,𝔽₈)	C₂³	1	2	yes	—
53	C₂×C₄×ΓL(1,𝔽₈)	C₂³	1	1	yes	characteristic
54	C₂×C₄×ΓL(1,𝔽₈)	C₂³	1	1	yes	characteristic
55	C₂×ΓL(1,𝔽₈)⋊C₄	C₂³	1	2	yes	—
56	C₂×ΓL(1,𝔽₈)⋊C₄	C₂³	1	1	yes	characteristic
57	C₂×ΓL(1,𝔽₈)⋊C₄	C₂³	1	1	yes	characteristic
58	C₂×ΓL(1,𝔽₈)⋊C₄	C₂³	1	1	yes	characteristic
59	C₂×ΓL(1,𝔽₈)⋊C₄	C₂³	1	2	yes	—
60	C₂×ΓL(1,𝔽₈)⋊C₄	C₂³	1	1	yes	characteristic
61	ΓL(1,𝔽₈)⋊Q₈	—	2	2	no	—
62	C₂×C₇×Q₈	C₂²×C₃	1	1	yes	characteristic
63	C₂²×C₄×C₇	C₂²×C₃	1	1	yes	characteristic
64	C₂×Dic₁₄	C₂²×C₃	1	1	yes	characteristic
65	C₇×C₄⋊C₄	C₂²×C₃	1	1	yes	characteristic
66	C₇×C₄⋊C₄	C₂²×C₃	1	2	yes	—
67	C₇×C₂²⋊C₄	C₂²×C₃	1	2	yes	—
68	C₂²⋊Dic₇	C₂²×C₃	1	2	yes	—
69	C₂²⋊Dic₇	C₂²×C₃	1	1	yes	characteristic
70	C₂²⋊Dic₇	C₂²×C₃	1	1	yes	characteristic
71	C₄⋊Dic₇	C₂²×C₃	1	2	yes	—
72	C₄⋊Dic₇	C₂²×C₃	1	2	yes	—
73	C₄⋊Dic₇	C₂²×C₃	1	1	yes	characteristic
74	C₄⋊Dic₇	C₂²×C₃	1	1	yes	characteristic
75	C₄⋊Dic₇	C₂²×C₃	1	1	yes	characteristic
76	Dic₇⋊C₄	C₂²×C₃	1	2	yes	—
77	Dic₇⋊C₄	C₂²×C₃	1	2	yes	—
78	Dic₇⋊C₄	C₂²×C₃	1	2	yes	—
79	Dic₇⋊C₄	C₂²×C₃	1	2	yes	—
80	Dic₇⋊C₄	C₂²×C₃	1	2	yes	—
81	C₄×Dic₇	C₂²×C₃	1	2	yes	—
82	C₄×Dic₇	C₂²×C₃	1	2	yes	—
83	C₄×Dic₇	C₂²×C₃	1	1	yes	characteristic
84	C₄×Dic₇	C₂²×C₃	1	1	yes	characteristic
85	C₃×C₄⋊Q₈	—	7	1	no	—
86	C₃×#126	—	7	2	no	—
87	C₃×#126	—	7	2	no	—
88	C₃×#125	—	7	2	no	—
89	C₃×#125	—	7	1	no	—
90	C₃×#125	—	7	2	no	—
91	C₃×C₂²⋊Q₈	—	7	1	no	—
92	C₃×C₂²⋊Q₈	—	7	1	no	—
93	C₃×C₄×Q₈	—	7	1	no	—
94	C₃×C₄×Q₈	—	7	1	no	—
95	C₃×#117	—	7	1	no	—
96	C₂²×ΓL(1,𝔽₈)	—	2	1	no	—
97	C₂²×ΓL(1,𝔽₈)	—	2	1	no	—
98	C₂²×ΓL(1,𝔽₈)	—	2	1	no	—
99	C₂²×ΓL(1,𝔽₈)	C₂⁴	1	1	yes	characteristic
100	C₄×ΓL(1,𝔽₈)	—	2	1	no	—
101	C₄×ΓL(1,𝔽₈)	—	2	2	no	—
102	C₄×ΓL(1,𝔽₈)	—	2	2	no	—
103	C₄×ΓL(1,𝔽₈)	#41	1	2	yes	—
104	ΓL(1,𝔽₈)⋊C₄	—	2	2	no	—
105	ΓL(1,𝔽₈)⋊C₄	—	2	1	no	—
106	ΓL(1,𝔽₈)⋊C₄	—	2	2	no	—
107	ΓL(1,𝔽₈)⋊C₄	—	2	1	no	—
108	ΓL(1,𝔽₈)⋊C₄	—	2	1	no	—
109	ΓL(1,𝔽₈)⋊C₄	—	2	1	no	—
110	#541	—	7	1	no	Sylow
111	C₂³×C₇	C₂³×C₃	1	1	yes	characteristic
112	C₇×Q₈	—	2	2	no	—
113	C₂×C₄×C₇	—	2	2	no	—
114	C₂×C₄×C₇	C₂³×C₃	1	2	yes	—
115	C₂×C₄×C₇	C₂³×C₃	1	2	yes	—
116	C₂×C₄×C₇	C₂³×C₃	1	1	yes	characteristic
117	C₂×C₄×C₇	C₂³×C₃	1	1	yes	characteristic
118	C₂×Dic₇	C₂³×C₃	1	1	yes	characteristic
119	C₂×Dic₇	C₂³×C₃	1	1	yes	characteristic
120	C₂×Dic₇	C₂³×C₃	1	1	yes	characteristic
121	C₂×Dic₇	C₂³×C₃	1	2	yes	—
122	C₂×Dic₇	C₂³×C₃	1	2	yes	—
123	C₂×Dic₇	C₂³×C₃	1	1	yes	characteristic
124	Dic₁₄	—	2	2	no	—
125	C₂×C₃×Q₈	—	7	1	no	—
126	C₂×C₃×Q₈	—	7	1	no	—
127	C₂²×C₃×C₄	—	7	1	no	—
128	C₃×C₄⋊C₄	—	7	1	no	—
129	C₃×C₄⋊C₄	—	7	1	no	—
130	C₃×C₄⋊C₄	—	7	1	no	—
131	C₃×C₄⋊C₄	—	7	2	no	—
132	C₃×C₄⋊C₄	—	7	2	no	—
133	C₃×C₄⋊C₄	—	7	2	no	—
134	C₃×C₄⋊C₄	—	7	2	no	—
135	C₃×C₄⋊C₄	—	7	2	no	—
136	C₃×C₄⋊C₄	—	7	2	no	—
137	C₃×C₄⋊C₄	—	7	2	no	—
138	C₃×C₄⋊C₄	—	7	1	no	—
139	C₃×C₄⋊C₄	—	7	2	no	—
140	C₃×C₂²⋊C₄	—	7	1	no	—
141	C₃×C₂²⋊C₄	—	7	1	no	—
142	C₃×C₂²⋊C₄	—	7	2	no	—
143	C₃×C₂²⋊C₄	—	7	2	no	—
144	C₃×C₄²	—	7	1	no	—
145	C₃×C₄²	—	7	1	no	—
146	C₃×C₄²	—	7	2	no	—
147	C₃×C₄²	—	7	2	no	—
148	C₂×ΓL(1,𝔽₈)	—	4	1	no	—
149	C₂×ΓL(1,𝔽₈)	C₂×#41	1	1	yes	characteristic
150	C₂×ΓL(1,𝔽₈)	2₋¹⁺⁴	1	1	yes	characteristic
151	C₂×ΓL(1,𝔽₈)	2₋¹⁺⁴	1	1	yes	characteristic
152	C₄⋊Q₈	—	7	1	no	—
153	#126	—	7	2	no	—
154	#126	—	7	2	no	—
155	#125	—	7	2	no	—
156	#125	—	7	1	no	—
157	#125	—	7	2	no	—
158	C₂²⋊Q₈	—	7	1	no	—
159	C₂²⋊Q₈	AGL(1,𝔽₇)	1	1	yes	characteristic
160	C₄×Q₈	—	7	1	no	—
161	C₄×Q₈	—	7	1	no	—
162	#117	—	7	1	no	—
163	C₂²×C₇	—	2	1	no	—
164	C₂²×C₇	—	2	1	no	—
165	C₂²×C₇	—	2	1	no	—
166	C₂²×C₇	C₂⁴×C₃	1	1	yes	characteristic
167	C₄×C₇	—	2	1	no	—
168	C₄×C₇	—	2	2	no	—
169	C₄×C₇	—	2	2	no	—
170	C₄×C₇	C₃×#41	1	2	yes	—
171	Dic₇	—	2	2	no	—
172	Dic₇	—	2	1	no	—
173	Dic₇	—	2	2	no	—
174	Dic₇	—	2	1	no	—
175	Dic₇	—	2	1	no	—
176	Dic₇	—	2	1	no	—
177	C₂³×C₃	—	7	1	no	—
178	C₃×Q₈	—	14	2	no	—
179	C₃×Q₈	—	14	2	no	—
180	C₂×C₃×C₄	—	7	1	no	—
181	C₂×C₃×C₄	—	7	1	no	—
182	C₂×C₃×C₄	—	7	1	no	—
183	C₂×C₃×C₄	—	7	1	no	—
184	C₂×C₃×C₄	—	7	2	no	—
185	C₂×C₃×C₄	—	7	2	no	—
186	C₂×C₃×C₄	—	7	1	no	—
187	C₂×C₃×C₄	—	7	2	no	—
188	C₂×C₃×C₄	—	7	2	no	—
189	C₂×C₃×C₄	—	7	1	no	—
190	C₂×C₃×C₄	—	14	2	no	—
191	ΓL(1,𝔽₈)	#541	1	1	yes	characteristic, complemented, Hall
192	C₂×Q₈	—	7	1	no	—
193	C₂×Q₈	C₂×AGL(1,𝔽₇)	1	1	yes	characteristic
194	C₂²×C₄	C₂×AGL(1,𝔽₇)	1	1	yes	characteristic
195	C₄⋊C₄	—	7	2	no	—
196	C₄⋊C₄	—	7	2	no	—
197	C₄⋊C₄	—	7	2	no	—
198	C₄⋊C₄	—	7	2	no	—
199	C₄⋊C₄	—	7	2	no	—
200	C₄⋊C₄	—	7	1	no	—
201	C₄⋊C₄	—	7	1	no	—
202	C₄⋊C₄	—	7	1	no	—
203	C₄⋊C₄	—	7	2	no	—
204	C₄⋊C₄	—	7	2	no	—
205	C₄⋊C₄	C₂×AGL(1,𝔽₇)	1	1	yes	characteristic
206	C₄⋊C₄	C₂×AGL(1,𝔽₇)	1	2	yes	—
207	C₂²⋊C₄	—	7	1	no	—
208	C₂²⋊C₄	—	7	1	no	—
209	C₂²⋊C₄	—	7	2	no	—
210	C₂²⋊C₄	C₂×AGL(1,𝔽₇)	1	2	yes	—
211	C₄²	—	7	1	no	—
212	C₄²	—	7	1	no	—
213	C₄²	—	7	2	no	—
214	C₄²	—	7	2	no	—
215	C₂×C₇	—	4	1	no	—
216	C₂×C₇	C₂×C₃×#41	1	1	yes	characteristic
217	C₂×C₇	C₃×2₋¹⁺⁴	1	1	yes	characteristic
218	C₂×C₇	C₃×2₋¹⁺⁴	1	1	yes	characteristic
219	C₂²×C₃	—	7	1	no	—
220	C₂²×C₃	—	14	1	no	—
221	C₂²×C₃	—	14	1	no	—
222	C₂²×C₃	—	14	1	no	—
223	C₃×C₄	—	7	2	no	—
224	C₃×C₄	—	14	1	no	—
225	C₃×C₄	—	14	2	no	—
226	C₃×C₄	—	14	2	no	—
227	C₃×C₄	—	14	2	no	—
228	C₃×C₄	—	14	1	no	—
229	C₃×C₄	—	14	1	no	—
230	C₃×C₄	—	14	2	no	—
231	C₃×C₄	—	14	1	no	—
232	C₃×C₄	—	14	1	no	—
233	C₂³	C₂²×AGL(1,𝔽₇)	1	1	yes	characteristic
234	Q₈	—	2	2	no	—
235	Q₈	—	14	2	no	—
236	C₂×C₄	—	2	2	no	—
237	C₂×C₄	—	7	1	no	—
238	C₂×C₄	—	7	1	no	—
239	C₂×C₄	—	7	1	no	—
240	C₂×C₄	—	7	2	no	—
241	C₂×C₄	—	7	1	no	—
242	C₂×C₄	—	7	2	no	—
243	C₂×C₄	C₂²×AGL(1,𝔽₇)	1	2	yes	—
244	C₂×C₄	C₂²×AGL(1,𝔽₇)	1	2	yes	—
245	C₂×C₄	C₂²×AGL(1,𝔽₇)	1	1	yes	characteristic
246	C₂×C₄	C₂²×AGL(1,𝔽₇)	1	1	yes	characteristic
247	C₇	C₃×#541	1	1	yes	characteristic, complemented, Sylow
248	C₂×C₃	—	7	1	no	—
249	C₂×C₃	—	7	1	no	—
250	C₂×C₃	—	7	1	no	—
251	C₂×C₃	—	28	1	no	—
252	C₂²	—	2	1	no	—
253	C₂²	—	2	1	no	—
254	C₂²	—	2	1	no	—
255	C₂²	C₂³×AGL(1,𝔽₇)	1	1	yes	characteristic
256	C₄	—	2	1	no	—
257	C₄	—	2	2	no	—
258	C₄	—	2	2	no	—
259	C₄	—	14	2	no	—
260	C₄	—	14	1	no	—
261	C₄	—	14	1	no	—
262	C₄	—	14	2	no	—
263	C₄	—	14	1	no	—
264	C₄	—	14	1	no	—
265	C₄	#1159⋊C₃	1	2	yes	—
266	C₃	—	7	1	no	Sylow
267	C₂	—	4	1	no	—
268	C₂	C₂×#1159⋊C₃	1	1	yes	characteristic
269	C₂	#10621516	1	1	yes	characteristic
270	C₂	#10621519	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂²	C₂³×AGL(1,𝔽₇)
commutator, socle, cosocle	C₂²×C₇	C₂⁴×C₃
Fitting	C₇×C₂²⋊Q₈	C₂×C₃
prime radical	C₂³×ΓL(1,𝔽₈)	C₂³
hypercenter	C₂²⋊Q₈	AGL(1,𝔽₇)
hypocenter	C₇	C₃×#541
LCS	G ▷ C₂²×C₇ ▷ C₇
UCS	C₁ ◁ C₂² ◁ C₂²⋊Q₈

Complements

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

N	H ≅ G/N	type
2. Q₈×ΓL(1,𝔽₈)⋊C₄	C₂	semidirect
3. #6507⋊C₃	C₂	semidirect
6. #6453⋊C₃	C₂	semidirect
7. #6453⋊C₃	C₂	semidirect
8. #6452⋊C₃	C₂	semidirect
9. #6450⋊C₃	C₂	semidirect
12. #90316	C₃	semidirect
37. Q₈×Dic₇	C₂×C₃	semidirect
38. #6507	C₂×C₃	semidirect
41. #6453	C₂×C₃	semidirect
42. #6453	C₂×C₃	semidirect
43. #6452	C₂×C₃	semidirect
44. #6450	C₂×C₃	semidirect
191. ΓL(1,𝔽₈)	#541	semidirect
247. C₇	C₃×#541	semidirect

Maximal subgroups

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

subgroup	index
ΓL(1,𝔽₈)×C₂²⋊Q₈	2
Q₈×ΓL(1,𝔽₈)⋊C₄	2
#6507⋊C₃	2
#6489⋊C₃	2
#6487⋊C₃	2
#6453⋊C₃	2
#6452⋊C₃	2
#6450⋊C₃	2
#6442⋊C₃	2
#90316	3
C₃×#541	7

Subgroups — #90316⋊C3 #13057363

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — #90316⋊C₃ #13057363