Subgroups — AGL(1,𝔽₇)×#6910 #72249136

Order 10206.

2176 conjugacy classes of subgroups.

The subgroup lattice has 576 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#72249135	C₃	1	9	yes	maximal, complemented
2	#72249134	C₃	1	9	yes	maximal, complemented
3	AGL(1,𝔽₇)×#731	C₃	1	9	yes	maximal, complemented
4	#72249132	C₃	1	2	yes	maximal, complemented
5	#72249121	C₃	1	2	yes	maximal, complemented
6	#72249115	C₃	1	4	yes	maximal, complemented
7	C₃×H(3,𝔽₃)×AGL(1,𝔽₇)	C₃	1	1	yes	maximal, characteristic
8	D₇×#6910	C₃	1	1	yes	maximal, characteristic, complemented
9	C₃×3₋¹⁺²×AGL(1,𝔽₇)	C₃	1	2	yes	maximal, complemented
10	C₃×3₋¹⁺²×AGL(1,𝔽₇)	C₃	1	1	yes	maximal, characteristic, complemented
11	ΓL(1,𝔽₈)×#6910	C₂	1	1	yes	maximal, characteristic, complemented, Hall
12	C₃×ΓL(1,𝔽₈)×3₋¹⁺²	C₂×C₃	1	2	yes	complemented
13	C₃×ΓL(1,𝔽₈)×3₋¹⁺²	C₂×C₃	1	1	yes	characteristic, complemented
14	C₃×ΓL(1,𝔽₈)×H(3,𝔽₃)	C₂×C₃	1	1	yes	characteristic
15	#13133636	C₂×C₃	1	4	yes	complemented
16	#13133635	C₂×C₃	1	2	yes	complemented
17	#13133634	C₂×C₃	1	2	yes	complemented
18	#13133633	C₂×C₃	1	9	yes	complemented
19	#13133632	C₂×C₃	1	9	yes	complemented
20	ΓL(1,𝔽₈)×#731	C₂×C₃	1	9	yes	complemented
21	C₇×#6910	C₂×C₃	1	1	yes	characteristic, complemented
22	C₂×C₃×#6910	—	7	1	no	maximal, Hall
23	C₃³×AGL(1,𝔽₇)	—	3	1	no	—
24	C₃³×AGL(1,𝔽₇)	C₃²	1	1	yes	characteristic
25	C₃×D₇×3₋¹⁺²	C₃²	1	2	yes	complemented
26	C₃×D₇×3₋¹⁺²	C₃²	1	1	yes	characteristic, complemented
27	C₃×D₇×H(3,𝔽₃)	C₃²	1	1	yes	characteristic
28	C₃×D₇⋊H(3,𝔽₃)	—	3	2	no	—
29	C₃×D₇⋊H(3,𝔽₃)	C₃²	1	2	yes	—
30	C₃×#4430⋊C₂	C₃²	1	4	yes	complemented
31	C₃×#4430⋊C₂	C₃²	1	2	yes	complemented
32	3₋¹⁺²×AGL(1,𝔽₇)	—	3	2	no	—
33	3₋¹⁺²×AGL(1,𝔽₇)	—	3	2	no	—
34	3₋¹⁺²×AGL(1,𝔽₇)	—	3	2	no	—
35	3₋¹⁺²×AGL(1,𝔽₇)	—	3	2	no	—
36	3₋¹⁺²×AGL(1,𝔽₇)	—	3	2	no	—
37	3₋¹⁺²×AGL(1,𝔽₇)	—	3	2	no	—
38	3₋¹⁺²×AGL(1,𝔽₇)	—	3	1	no	—
39	3₋¹⁺²×AGL(1,𝔽₇)	—	3	2	no	—
40	3₋¹⁺²×AGL(1,𝔽₇)	—	3	2	no	—
41	3₋¹⁺²×AGL(1,𝔽₇)	—	3	2	no	—
42	3₋¹⁺²×AGL(1,𝔽₇)	—	3	2	no	—
43	3₋¹⁺²×AGL(1,𝔽₇)	C₃²	1	6	yes	complemented
44	3₋¹⁺²×AGL(1,𝔽₇)	C₃²	1	6	yes	complemented
45	3₋¹⁺²×AGL(1,𝔽₇)	C₃²	1	6	yes	complemented
46	H(3,𝔽₃)×AGL(1,𝔽₇)	—	3	2	no	—
47	H(3,𝔽₃)×AGL(1,𝔽₇)	—	3	2	no	—
48	H(3,𝔽₃)×AGL(1,𝔽₇)	—	3	2	no	—
49	H(3,𝔽₃)×AGL(1,𝔽₇)	C₃²	1	3	yes	—
50	H(3,𝔽₃)×AGL(1,𝔽₇)	C₃²	1	3	yes	—
51	H(3,𝔽₃)×AGL(1,𝔽₇)	C₃²	1	3	yes	—
52	C₃×#67356	—	3	2	no	—
53	C₃×#67356	C₃²	1	3	yes	complemented
54	C₃×#67355	—	3	2	no	—
55	C₃×#67355	C₃²	1	3	yes	complemented
56	C₃×C₉×AGL(1,𝔽₇)	—	3	2	no	—
57	C₃×C₉×AGL(1,𝔽₇)	C₃²	1	3	yes	complemented
58	D₇×#731	C₃²	1	9	yes	complemented
59	#10588045⋊C₂	C₃²	1	36	yes	complemented
60	#11765590	C₃²	1	18	yes	complemented
61	#11765586	C₃²	1	18	yes	complemented
62	C₃×#6910	—	7	1	no	Sylow
63	C₃³×ΓL(1,𝔽₈)	—	3	1	no	—
64	C₃³×ΓL(1,𝔽₈)	C₂×C₃²	1	1	yes	characteristic
65	C₃×C₇×3₋¹⁺²	C₂×C₃²	1	2	yes	complemented
66	C₃×C₇×3₋¹⁺²	C₂×C₃²	1	1	yes	characteristic, complemented
67	C₃×C₇×H(3,𝔽₃)	C₂×C₃²	1	1	yes	characteristic
68	C₃×C₇⋊H(3,𝔽₃)	—	3	2	no	—
69	C₃×C₇⋊H(3,𝔽₃)	C₂×C₃²	1	2	yes	—
70	C₃×#4430	C₂×C₃²	1	4	yes	complemented
71	C₃×#4430	C₂×C₃²	1	2	yes	complemented
72	ΓL(1,𝔽₈)×3₋¹⁺²	—	3	2	no	—
73	ΓL(1,𝔽₈)×3₋¹⁺²	—	3	2	no	—
74	ΓL(1,𝔽₈)×3₋¹⁺²	—	3	2	no	—
75	ΓL(1,𝔽₈)×3₋¹⁺²	—	3	2	no	—
76	ΓL(1,𝔽₈)×3₋¹⁺²	—	3	2	no	—
77	ΓL(1,𝔽₈)×3₋¹⁺²	—	3	2	no	—
78	ΓL(1,𝔽₈)×3₋¹⁺²	—	3	2	no	—
79	ΓL(1,𝔽₈)×3₋¹⁺²	—	3	1	no	—
80	ΓL(1,𝔽₈)×3₋¹⁺²	—	3	2	no	—
81	ΓL(1,𝔽₈)×3₋¹⁺²	—	3	2	no	—
82	ΓL(1,𝔽₈)×3₋¹⁺²	—	3	2	no	—
83	ΓL(1,𝔽₈)×3₋¹⁺²	C₂×C₃²	1	6	yes	complemented
84	ΓL(1,𝔽₈)×3₋¹⁺²	C₂×C₃²	1	6	yes	complemented
85	ΓL(1,𝔽₈)×3₋¹⁺²	C₂×C₃²	1	6	yes	complemented
86	ΓL(1,𝔽₈)×H(3,𝔽₃)	—	3	2	no	—
87	ΓL(1,𝔽₈)×H(3,𝔽₃)	—	3	2	no	—
88	ΓL(1,𝔽₈)×H(3,𝔽₃)	—	3	2	no	—
89	ΓL(1,𝔽₈)×H(3,𝔽₃)	C₂×C₃²	1	3	yes	—
90	ΓL(1,𝔽₈)×H(3,𝔽₃)	C₂×C₃²	1	3	yes	—
91	ΓL(1,𝔽₈)×H(3,𝔽₃)	C₂×C₃²	1	3	yes	—
92	C₃×#4428	—	3	2	no	—
93	C₃×#4428	C₂×C₃²	1	3	yes	complemented
94	C₃×#4427	—	3	2	no	—
95	C₃×#4427	C₂×C₃²	1	3	yes	complemented
96	C₃×C₉×ΓL(1,𝔽₈)	—	3	2	no	—
97	C₃×C₉×ΓL(1,𝔽₈)	C₂×C₃²	1	3	yes	complemented
98	C₇×#731	C₂×C₃²	1	9	yes	complemented
99	#10588045	C₂×C₃²	1	36	yes	complemented
100	#10588041	C₂×C₃²	1	18	yes	complemented
101	#10588037	C₂×C₃²	1	18	yes	complemented
102	C₂×C₃²×3₋¹⁺²	—	7	1	no	—
103	C₂×C₃²×3₋¹⁺²	—	7	2	no	—
104	C₂×C₃²×H(3,𝔽₃)	—	7	1	no	—
105	C₂×#6910	—	7	4	no	—
106	C₂×#6910	—	7	2	no	—
107	C₂×#6910	—	7	2	no	—
108	C₂×#6910	—	7	9	no	—
109	C₂×#6910	—	7	9	no	—
110	C₂×#6910	—	7	1	no	—
111	C₂×C₃×#731	—	7	9	no	—
112	C₃³×D₇	—	3	1	no	—
113	C₃³×D₇	C₃³	1	1	yes	characteristic
114	C₃²×AGL(1,𝔽₇)	—	3	2	no	—
115	C₃²×AGL(1,𝔽₇)	—	3	3	no	—
116	C₃²×AGL(1,𝔽₇)	—	3	3	no	—
117	C₃²×AGL(1,𝔽₇)	—	3	3	no	—
118	C₃²×AGL(1,𝔽₇)	—	9	3	no	—
119	C₃²×AGL(1,𝔽₇)	—	9	9	no	—
120	C₃²×AGL(1,𝔽₇)	H(3,𝔽₃)	1	1	yes	characteristic
121	C₃²×AGL(1,𝔽₇)	H(3,𝔽₃)	1	2	yes	—
122	C₃²×AGL(1,𝔽₇)	H(3,𝔽₃)	1	2	yes	—
123	C₃²×AGL(1,𝔽₇)	H(3,𝔽₃)	1	2	yes	—
124	C₃²×AGL(1,𝔽₇)	H(3,𝔽₃)	1	2	yes	—
125	C₃²×AGL(1,𝔽₇)	C₃³	1	1	yes	characteristic
126	C₃²×AGL(1,𝔽₇)	C₃³	1	1	yes	characteristic
127	C₃²×AGL(1,𝔽₇)	C₃³	1	1	yes	characteristic
128	D₇×3₋¹⁺²	—	3	2	no	—
129	D₇×3₋¹⁺²	—	3	2	no	—
130	D₇×3₋¹⁺²	—	3	2	no	—
131	D₇×3₋¹⁺²	—	3	1	no	—
132	D₇×3₋¹⁺²	C₃³	1	6	yes	complemented
133	D₇×H(3,𝔽₃)	—	3	2	no	—
134	D₇×H(3,𝔽₃)	C₃³	1	3	yes	—
135	C₃×C₉×D₇	—	3	2	no	—
136	C₃×C₉×D₇	C₃³	1	3	yes	complemented
137	D₇⋊H(3,𝔽₃)	—	3	4	no	—
138	D₇⋊H(3,𝔽₃)	—	3	6	no	—
139	D₇⋊H(3,𝔽₃)	—	3	6	no	—
140	D₇⋊H(3,𝔽₃)	—	3	6	no	—
141	D₇⋊H(3,𝔽₃)	C₃³	1	6	yes	—
142	#4430⋊C₂	—	3	4	no	—
143	#4430⋊C₂	—	3	4	no	—
144	#4430⋊C₂	—	3	4	no	—
145	#4430⋊C₂	—	3	2	no	—
146	#4430⋊C₂	C₃³	1	12	yes	complemented
147	C₃×D₇⋊C₉	—	3	4	no	—
148	C₃×D₇⋊C₉	C₃³	1	6	yes	complemented
149	#67356	—	3	6	no	—
150	#67356	—	3	6	no	—
151	#67356	—	3	6	no	—
152	#67356	—	3	6	no	—
153	#67356	—	3	3	no	—
154	#67355	—	3	6	no	—
155	#67355	—	3	6	no	—
156	#67355	—	3	6	no	—
157	#67355	—	3	6	no	—
158	#67355	—	3	3	no	—
159	C₉×AGL(1,𝔽₇)	—	3	6	no	—
160	C₉×AGL(1,𝔽₇)	—	3	6	no	—
161	C₉×AGL(1,𝔽₇)	—	3	6	no	—
162	C₉×AGL(1,𝔽₇)	—	3	6	no	—
163	C₉×AGL(1,𝔽₇)	—	3	3	no	—
164	C₃²×3₋¹⁺²	—	7	1	no	—
165	C₃²×3₋¹⁺²	—	7	2	no	—
166	C₃²×H(3,𝔽₃)	—	7	1	no	—
167	#6910	—	7	2	no	—
168	#6910	—	7	4	no	—
169	#6910	—	7	2	no	—
170	#6910	—	7	9	no	—
171	#6910	—	7	9	no	—
172	#6910	AGL(1,𝔽₇)	1	1	yes	characteristic, direct product factor
173	C₃×#731	—	7	9	no	—
174	C₃³×C₇	—	3	1	no	—
175	C₃³×C₇	C₂×C₃³	1	1	yes	characteristic
176	C₃²×ΓL(1,𝔽₈)	—	3	2	no	—
177	C₃²×ΓL(1,𝔽₈)	—	3	3	no	—
178	C₃²×ΓL(1,𝔽₈)	—	3	3	no	—
179	C₃²×ΓL(1,𝔽₈)	—	3	3	no	—
180	C₃²×ΓL(1,𝔽₈)	—	9	3	no	—
181	C₃²×ΓL(1,𝔽₈)	—	9	9	no	—
182	C₃²×ΓL(1,𝔽₈)	C₂×H(3,𝔽₃)	1	2	yes	—
183	C₃²×ΓL(1,𝔽₈)	C₂×H(3,𝔽₃)	1	1	yes	characteristic
184	C₃²×ΓL(1,𝔽₈)	C₂×H(3,𝔽₃)	1	2	yes	—
185	C₃²×ΓL(1,𝔽₈)	C₂×H(3,𝔽₃)	1	2	yes	—
186	C₃²×ΓL(1,𝔽₈)	C₂×H(3,𝔽₃)	1	2	yes	—
187	C₃²×ΓL(1,𝔽₈)	C₂×C₃³	1	1	yes	characteristic
188	C₃²×ΓL(1,𝔽₈)	C₂×C₃³	1	1	yes	characteristic
189	C₃²×ΓL(1,𝔽₈)	C₂×C₃³	1	1	yes	characteristic
190	C₇×3₋¹⁺²	—	3	2	no	—
191	C₇×3₋¹⁺²	—	3	2	no	—
192	C₇×3₋¹⁺²	—	3	2	no	—
193	C₇×3₋¹⁺²	—	3	1	no	—
194	C₇×3₋¹⁺²	C₂×C₃³	1	6	yes	complemented
195	C₇×H(3,𝔽₃)	—	3	2	no	—
196	C₇×H(3,𝔽₃)	C₂×C₃³	1	3	yes	—
197	C₃×C₇×C₉	—	3	2	no	—
198	C₃×C₇×C₉	C₂×C₃³	1	3	yes	complemented
199	C₇⋊H(3,𝔽₃)	—	3	4	no	—
200	C₇⋊H(3,𝔽₃)	—	3	6	no	—
201	C₇⋊H(3,𝔽₃)	—	3	6	no	—
202	C₇⋊H(3,𝔽₃)	—	3	6	no	—
203	C₇⋊H(3,𝔽₃)	C₂×C₃³	1	6	yes	—
204	#4430	—	3	4	no	—
205	#4430	—	3	4	no	—
206	#4430	—	3	4	no	—
207	#4430	—	3	2	no	—
208	#4430	C₂×C₃³	1	12	yes	complemented
209	C₃×C₇⋊C₉	—	3	4	no	—
210	C₃×C₇⋊C₉	C₂×C₃³	1	6	yes	complemented
211	#4428	—	3	6	no	—
212	#4428	—	3	6	no	—
213	#4428	—	3	6	no	—
214	#4428	—	3	6	no	—
215	#4428	—	3	3	no	—
216	#4427	—	3	6	no	—
217	#4427	—	3	6	no	—
218	#4427	—	3	6	no	—
219	#4427	—	3	6	no	—
220	#4427	—	3	3	no	—
221	C₉×ΓL(1,𝔽₈)	—	3	6	no	—
222	C₉×ΓL(1,𝔽₈)	—	3	6	no	—
223	C₉×ΓL(1,𝔽₈)	—	3	6	no	—
224	C₉×ΓL(1,𝔽₈)	—	3	6	no	—
225	C₉×ΓL(1,𝔽₈)	—	3	3	no	—
226	C₂×C₃⁴	—	7	1	no	—
227	C₂×C₃⁴	—	21	1	no	—
228	C₂×C₃×3₋¹⁺²	—	7	3	no	—
229	C₂×C₃×3₋¹⁺²	—	7	3	no	—
230	C₂×C₃×3₋¹⁺²	—	7	2	no	—
231	C₂×C₃×3₋¹⁺²	—	7	1	no	—
232	C₂×C₃×3₋¹⁺²	—	7	6	no	—
233	C₂×C₃×3₋¹⁺²	—	7	6	no	—
234	C₂×C₃×3₋¹⁺²	—	7	6	no	—
235	C₂×C₃×3₋¹⁺²	—	7	4	no	—
236	C₂×C₃×3₋¹⁺²	—	7	2	no	—
237	C₂×C₃×3₋¹⁺²	—	21	2	no	—
238	C₂×C₃×3₋¹⁺²	—	21	2	no	—
239	C₂×C₃×3₋¹⁺²	—	21	2	no	—
240	C₂×C₃×3₋¹⁺²	—	21	1	no	—
241	C₂×C₃×3₋¹⁺²	—	21	2	no	—
242	C₂×C₃×3₋¹⁺²	—	21	2	no	—
243	C₂×C₃×3₋¹⁺²	—	21	2	no	—
244	C₂×C₃×3₋¹⁺²	—	21	2	no	—
245	C₂×C₃×3₋¹⁺²	—	21	2	no	—
246	C₂×C₃×3₋¹⁺²	—	21	2	no	—
247	C₂×C₃×3₋¹⁺²	—	21	2	no	—
248	C₂×C₃×3₋¹⁺²	—	21	2	no	—
249	C₂×C₃×3₋¹⁺²	—	21	2	no	—
250	C₂×C₃×H(3,𝔽₃)	—	7	3	no	—
251	C₂×C₃×H(3,𝔽₃)	—	7	3	no	—
252	C₂×C₃×H(3,𝔽₃)	—	7	3	no	—
253	C₂×C₃×H(3,𝔽₃)	—	7	2	no	—
254	C₂×C₃×H(3,𝔽₃)	—	7	1	no	—
255	C₂×C₃×H(3,𝔽₃)	—	21	2	no	—
256	C₂×C₃×H(3,𝔽₃)	—	21	2	no	—
257	C₂×C₃×H(3,𝔽₃)	—	21	2	no	—
258	C₂×C₃×H(3,𝔽₃)	—	21	2	no	—
259	C₂×C₃²×C₉	—	7	3	no	—
260	C₂×C₃²×C₉	—	21	2	no	—
261	C₂×#731	—	7	18	no	—
262	C₂×#731	—	7	18	no	—
263	C₂×#731	—	7	36	no	—
264	C₂×#731	—	7	9	no	—
265	C₃²×D₇	—	3	3	no	—
266	C₃²×D₇	—	9	1	no	—
267	C₃²×D₇	—	9	3	no	—
268	C₃²×D₇	C₃×H(3,𝔽₃)	1	2	yes	—
269	C₃²×D₇	C₃×H(3,𝔽₃)	1	1	yes	characteristic
270	C₃²×D₇	C₃⁴	1	1	yes	characteristic
271	C₃×AGL(1,𝔽₇)	—	3	3	no	—
272	C₃×AGL(1,𝔽₇)	—	3	6	no	—
273	C₃×AGL(1,𝔽₇)	—	3	3	no	—
274	C₃×AGL(1,𝔽₇)	—	3	6	no	—
275	C₃×AGL(1,𝔽₇)	—	9	2	no	—
276	C₃×AGL(1,𝔽₇)	—	9	2	no	—
277	C₃×AGL(1,𝔽₇)	—	9	2	no	—
278	C₃×AGL(1,𝔽₇)	—	9	1	no	—
279	C₃×AGL(1,𝔽₇)	—	9	1	no	—
280	C₃×AGL(1,𝔽₇)	—	9	9	no	—
281	C₃×AGL(1,𝔽₇)	—	9	9	no	—
282	C₃×AGL(1,𝔽₇)	—	9	6	no	—
283	C₃×AGL(1,𝔽₇)	—	9	9	no	—
284	C₃×AGL(1,𝔽₇)	C₃×H(3,𝔽₃)	1	2	yes	—
285	C₃×AGL(1,𝔽₇)	C₃×H(3,𝔽₃)	1	2	yes	—
286	C₃×AGL(1,𝔽₇)	C₃×H(3,𝔽₃)	1	2	yes	—
287	C₃×AGL(1,𝔽₇)	C₃×H(3,𝔽₃)	1	1	yes	characteristic
288	C₃×AGL(1,𝔽₇)	C₃×H(3,𝔽₃)	1	1	yes	characteristic
289	C₃×AGL(1,𝔽₇)	C₃×H(3,𝔽₃)	1	1	yes	characteristic
290	C₉×D₇	—	3	6	no	—
291	C₉×D₇	—	3	3	no	—
292	D₇⋊C₉	—	3	12	no	—
293	D₇⋊C₉	—	3	6	no	—
294	C₃⁴	—	7	1	no	—
295	C₃⁴	—	21	1	no	—
296	C₃×3₋¹⁺²	—	7	2	no	—
297	C₃×3₋¹⁺²	—	7	3	no	—
298	C₃×3₋¹⁺²	—	7	3	no	—
299	C₃×3₋¹⁺²	—	7	4	no	—
300	C₃×3₋¹⁺²	—	7	6	no	—
301	C₃×3₋¹⁺²	—	7	6	no	—
302	C₃×3₋¹⁺²	—	7	6	no	—
303	C₃×3₋¹⁺²	—	21	2	no	—
304	C₃×3₋¹⁺²	—	21	2	no	—
305	C₃×3₋¹⁺²	—	21	2	no	—
306	C₃×3₋¹⁺²	—	21	1	no	—
307	C₃×3₋¹⁺²	—	21	2	no	—
308	C₃×3₋¹⁺²	—	21	2	no	—
309	C₃×3₋¹⁺²	—	21	2	no	—
310	C₃×3₋¹⁺²	—	21	2	no	—
311	C₃×3₋¹⁺²	—	21	2	no	—
312	C₃×3₋¹⁺²	—	21	2	no	—
313	C₃×3₋¹⁺²	—	21	2	no	—
314	C₃×3₋¹⁺²	—	21	2	no	—
315	C₃×3₋¹⁺²	—	21	2	no	—
316	C₃×3₋¹⁺²	C₃×AGL(1,𝔽₇)	1	2	yes	complemented
317	C₃×3₋¹⁺²	C₃×AGL(1,𝔽₇)	1	1	yes	characteristic, complemented
318	C₃×H(3,𝔽₃)	—	7	2	no	—
319	C₃×H(3,𝔽₃)	—	7	3	no	—
320	C₃×H(3,𝔽₃)	—	7	3	no	—
321	C₃×H(3,𝔽₃)	—	7	3	no	—
322	C₃×H(3,𝔽₃)	—	21	2	no	—
323	C₃×H(3,𝔽₃)	—	21	2	no	—
324	C₃×H(3,𝔽₃)	—	21	2	no	—
325	C₃×H(3,𝔽₃)	—	21	2	no	—
326	C₃×H(3,𝔽₃)	C₃×AGL(1,𝔽₇)	1	1	yes	characteristic
327	C₃²×C₉	—	7	3	no	—
328	C₃²×C₉	—	21	2	no	—
329	#731	—	7	36	no	—
330	#731	—	7	18	no	—
331	#731	—	7	18	no	—
332	#731	C₃×AGL(1,𝔽₇)	1	9	yes	complemented
333	C₃²×C₇	—	3	3	no	—
334	C₃²×C₇	—	9	1	no	—
335	C₃²×C₇	—	9	3	no	—
336	C₃²×C₇	C₂×C₃×H(3,𝔽₃)	1	2	yes	—
337	C₃²×C₇	C₂×C₃×H(3,𝔽₃)	1	1	yes	characteristic
338	C₃²×C₇	C₂×C₃⁴	1	1	yes	characteristic
339	C₃×ΓL(1,𝔽₈)	—	3	3	no	—
340	C₃×ΓL(1,𝔽₈)	—	3	6	no	—
341	C₃×ΓL(1,𝔽₈)	—	3	3	no	—
342	C₃×ΓL(1,𝔽₈)	—	3	6	no	—
343	C₃×ΓL(1,𝔽₈)	—	9	2	no	—
344	C₃×ΓL(1,𝔽₈)	—	9	2	no	—
345	C₃×ΓL(1,𝔽₈)	—	9	2	no	—
346	C₃×ΓL(1,𝔽₈)	—	9	1	no	—
347	C₃×ΓL(1,𝔽₈)	—	9	1	no	—
348	C₃×ΓL(1,𝔽₈)	—	9	9	no	—
349	C₃×ΓL(1,𝔽₈)	—	9	9	no	—
350	C₃×ΓL(1,𝔽₈)	—	9	6	no	—
351	C₃×ΓL(1,𝔽₈)	—	9	9	no	—
352	C₃×ΓL(1,𝔽₈)	C₂×C₃×H(3,𝔽₃)	1	2	yes	—
353	C₃×ΓL(1,𝔽₈)	C₂×C₃×H(3,𝔽₃)	1	2	yes	—
354	C₃×ΓL(1,𝔽₈)	C₂×C₃×H(3,𝔽₃)	1	2	yes	—
355	C₃×ΓL(1,𝔽₈)	C₂×C₃×H(3,𝔽₃)	1	1	yes	characteristic
356	C₃×ΓL(1,𝔽₈)	C₂×C₃×H(3,𝔽₃)	1	1	yes	characteristic
357	C₃×ΓL(1,𝔽₈)	C₂×C₃×H(3,𝔽₃)	1	1	yes	characteristic
358	C₇×C₉	—	3	6	no	—
359	C₇×C₉	—	3	3	no	—
360	C₇⋊C₉	—	3	12	no	—
361	C₇⋊C₉	—	3	6	no	—
362	C₂×C₃³	—	7	2	no	—
363	C₂×C₃³	—	7	1	no	—
364	C₂×C₃³	—	7	1	no	—
365	C₂×C₃³	—	7	2	no	—
366	C₂×C₃³	—	7	2	no	—
367	C₂×C₃³	—	7	2	no	—
368	C₂×C₃³	—	7	1	no	—
369	C₂×C₃³	—	7	1	no	—
370	C₂×C₃³	—	7	1	no	—
371	C₂×C₃³	—	21	1	no	—
372	C₂×C₃³	—	21	3	no	—
373	C₂×C₃³	—	21	3	no	—
374	C₂×C₃³	—	21	3	no	—
375	C₂×C₃³	—	21	2	no	—
376	C₂×C₃³	—	63	3	no	—
377	C₂×C₃³	—	63	9	no	—
378	C₂×3₋¹⁺²	—	7	12	no	—
379	C₂×3₋¹⁺²	—	7	6	no	—
380	C₂×3₋¹⁺²	—	21	2	no	—
381	C₂×3₋¹⁺²	—	21	1	no	—
382	C₂×3₋¹⁺²	—	21	2	no	—
383	C₂×3₋¹⁺²	—	21	2	no	—
384	C₂×3₋¹⁺²	—	21	6	no	—
385	C₂×3₋¹⁺²	—	21	3	no	—
386	C₂×3₋¹⁺²	—	21	6	no	—
387	C₂×3₋¹⁺²	—	21	6	no	—
388	C₂×3₋¹⁺²	—	21	6	no	—
389	C₂×3₋¹⁺²	—	21	6	no	—
390	C₂×3₋¹⁺²	—	21	3	no	—
391	C₂×3₋¹⁺²	—	21	6	no	—
392	C₂×3₋¹⁺²	—	21	6	no	—
393	C₂×3₋¹⁺²	—	21	6	no	—
394	C₂×3₋¹⁺²	—	21	4	no	—
395	C₂×3₋¹⁺²	—	21	2	no	—
396	C₂×3₋¹⁺²	—	21	4	no	—
397	C₂×3₋¹⁺²	—	21	4	no	—
398	C₂×H(3,𝔽₃)	—	7	6	no	—
399	C₂×H(3,𝔽₃)	—	7	3	no	—
400	C₂×H(3,𝔽₃)	—	21	2	no	—
401	C₂×H(3,𝔽₃)	—	21	6	no	—
402	C₂×H(3,𝔽₃)	—	21	6	no	—
403	C₂×H(3,𝔽₃)	—	21	6	no	—
404	C₂×H(3,𝔽₃)	—	21	4	no	—
405	C₂×C₃×C₉	—	7	6	no	—
406	C₂×C₃×C₉	—	7	3	no	—
407	C₂×C₃×C₉	—	21	2	no	—
408	C₂×C₃×C₉	—	21	3	no	—
409	C₂×C₃×C₉	—	21	6	no	—
410	C₂×C₃×C₉	—	21	6	no	—
411	C₂×C₃×C₉	—	21	6	no	—
412	C₂×C₃×C₉	—	21	6	no	—
413	C₂×C₃×C₉	—	21	4	no	—
414	C₃×D₇	—	3	1	no	—
415	C₃×D₇	—	3	2	no	—
416	C₃×D₇	—	3	1	no	—
417	C₃×D₇	—	9	3	no	—
418	C₃×D₇	C₃²×H(3,𝔽₃)	1	1	yes	characteristic
419	AGL(1,𝔽₇)	—	3	1	no	—
420	AGL(1,𝔽₇)	—	3	1	no	—
421	AGL(1,𝔽₇)	—	3	2	no	—
422	AGL(1,𝔽₇)	—	3	2	no	—
423	AGL(1,𝔽₇)	—	3	2	no	—
424	AGL(1,𝔽₇)	—	9	6	no	—
425	AGL(1,𝔽₇)	#6910	1	3	yes	direct product factor
426	C₃³	—	3	1	no	—
427	C₃³	—	7	1	no	—
428	C₃³	—	7	2	no	—
429	C₃³	—	7	1	no	—
430	C₃³	—	7	1	no	—
431	C₃³	—	7	1	no	—
432	C₃³	—	7	2	no	—
433	C₃³	—	7	2	no	—
434	C₃³	—	7	2	no	—
435	C₃³	—	21	3	no	—
436	C₃³	—	21	3	no	—
437	C₃³	—	21	3	no	—
438	C₃³	—	21	2	no	—
439	C₃³	—	63	3	no	—
440	C₃³	—	63	9	no	—
441	C₃³	C₃²×AGL(1,𝔽₇)	1	1	yes	characteristic
442	3₋¹⁺²	—	3	2	no	—
443	3₋¹⁺²	—	3	2	no	—
444	3₋¹⁺²	—	3	2	no	—
445	3₋¹⁺²	—	3	1	no	—
446	3₋¹⁺²	—	7	12	no	—
447	3₋¹⁺²	—	21	6	no	—
448	3₋¹⁺²	—	21	6	no	—
449	3₋¹⁺²	—	21	6	no	—
450	3₋¹⁺²	—	21	6	no	—
451	3₋¹⁺²	—	21	3	no	—
452	3₋¹⁺²	—	21	6	no	—
453	3₋¹⁺²	—	21	6	no	—
454	3₋¹⁺²	—	21	6	no	—
455	3₋¹⁺²	—	21	6	no	—
456	3₋¹⁺²	—	21	3	no	—
457	3₋¹⁺²	—	21	4	no	—
458	3₋¹⁺²	—	21	2	no	—
459	3₋¹⁺²	—	21	4	no	—
460	3₋¹⁺²	—	21	4	no	—
461	3₋¹⁺²	C₃²×AGL(1,𝔽₇)	1	6	yes	complemented
462	H(3,𝔽₃)	—	3	2	no	—
463	H(3,𝔽₃)	—	7	6	no	—
464	H(3,𝔽₃)	—	21	6	no	—
465	H(3,𝔽₃)	—	21	6	no	—
466	H(3,𝔽₃)	—	21	6	no	—
467	H(3,𝔽₃)	—	21	4	no	—
468	H(3,𝔽₃)	C₃²×AGL(1,𝔽₇)	1	3	yes	—
469	C₃×C₉	—	3	2	no	—
470	C₃×C₉	—	7	6	no	—
471	C₃×C₉	—	21	3	no	—
472	C₃×C₉	—	21	6	no	—
473	C₃×C₉	—	21	6	no	—
474	C₃×C₉	—	21	6	no	—
475	C₃×C₉	—	21	6	no	—
476	C₃×C₉	—	21	4	no	—
477	C₃×C₉	C₃²×AGL(1,𝔽₇)	1	3	yes	complemented
478	C₃×C₇	—	3	1	no	—
479	C₃×C₇	—	3	2	no	—
480	C₃×C₇	—	3	1	no	—
481	C₃×C₇	—	9	3	no	—
482	C₃×C₇	C₂×C₃²×H(3,𝔽₃)	1	1	yes	characteristic
483	ΓL(1,𝔽₈)	—	3	1	no	—
484	ΓL(1,𝔽₈)	—	3	1	no	—
485	ΓL(1,𝔽₈)	—	3	2	no	—
486	ΓL(1,𝔽₈)	—	3	2	no	—
487	ΓL(1,𝔽₈)	—	3	2	no	—
488	ΓL(1,𝔽₈)	—	9	6	no	—
489	ΓL(1,𝔽₈)	C₂×#6910	1	3	yes	complemented
490	C₂×C₃²	—	7	1	no	—
491	C₂×C₃²	—	7	2	no	—
492	C₂×C₃²	—	7	2	no	—
493	C₂×C₃²	—	7	2	no	—
494	C₂×C₃²	—	7	1	no	—
495	C₂×C₃²	—	7	1	no	—
496	C₂×C₃²	—	7	1	no	—
497	C₂×C₃²	—	7	2	no	—
498	C₂×C₃²	—	7	1	no	—
499	C₂×C₃²	—	21	3	no	—
500	C₂×C₃²	—	21	6	no	—
501	C₂×C₃²	—	21	3	no	—
502	C₂×C₃²	—	21	3	no	—
503	C₂×C₃²	—	21	6	no	—
504	C₂×C₃²	—	63	9	no	—
505	C₂×C₃²	—	63	1	no	—
506	C₂×C₃²	—	63	2	no	—
507	C₂×C₃²	—	63	2	no	—
508	C₂×C₃²	—	63	2	no	—
509	C₂×C₃²	—	63	1	no	—
510	C₂×C₃²	—	63	1	no	—
511	C₂×C₃²	—	63	3	no	—
512	C₂×C₃²	—	63	6	no	—
513	C₂×C₃²	—	63	9	no	—
514	C₂×C₃²	—	63	9	no	—
515	C₂×C₉	—	21	3	no	—
516	C₂×C₉	—	21	6	no	—
517	C₂×C₉	—	21	6	no	—
518	C₂×C₉	—	21	12	no	—
519	D₇	C₃×#6910	1	1	yes	characteristic, complemented, Hall
520	C₃²	—	3	3	no	—
521	C₃²	—	7	1	no	—
522	C₃²	—	7	2	no	—
523	C₃²	—	7	2	no	—
524	C₃²	—	7	2	no	—
525	C₃²	—	7	1	no	—
526	C₃²	—	7	1	no	—
527	C₃²	—	9	1	no	—
528	C₃²	—	9	3	no	—
529	C₃²	—	21	3	no	—
530	C₃²	—	21	6	no	—
531	C₃²	—	21	3	no	—
532	C₃²	—	21	6	no	—
533	C₃²	—	63	2	no	—
534	C₃²	—	63	1	no	—
535	C₃²	—	63	1	no	—
536	C₃²	—	63	2	no	—
537	C₃²	—	63	2	no	—
538	C₃²	—	63	6	no	—
539	C₃²	—	63	9	no	—
540	C₃²	—	63	9	no	—
541	C₃²	—	63	9	no	—
542	C₃²	H(3,𝔽₃)×AGL(1,𝔽₇)	1	2	yes	—
543	C₃²	H(3,𝔽₃)×AGL(1,𝔽₇)	1	1	yes	characteristic
544	C₃²	C₃³×AGL(1,𝔽₇)	1	1	yes	characteristic
545	C₉	—	3	6	no	—
546	C₉	—	3	3	no	—
547	C₉	—	21	6	no	—
548	C₉	—	21	12	no	—
549	C₇	C₂×C₃×#6910	1	1	yes	characteristic, complemented, Sylow
550	C₂×C₃	—	7	3	no	—
551	C₂×C₃	—	7	1	no	—
552	C₂×C₃	—	21	1	no	—
553	C₂×C₃	—	21	1	no	—
554	C₂×C₃	—	21	2	no	—
555	C₂×C₃	—	21	1	no	—
556	C₂×C₃	—	21	1	no	—
557	C₂×C₃	—	21	2	no	—
558	C₂×C₃	—	21	2	no	—
559	C₂×C₃	—	21	2	no	—
560	C₂×C₃	—	63	3	no	—
561	C₂×C₃	—	63	6	no	—
562	C₃	—	3	1	no	—
563	C₃	—	3	1	no	—
564	C₃	—	3	2	no	—
565	C₃	—	7	3	no	—
566	C₃	—	9	3	no	—
567	C₃	—	21	1	no	—
568	C₃	—	21	1	no	—
569	C₃	—	21	2	no	—
570	C₃	—	21	2	no	—
571	C₃	—	21	2	no	—
572	C₃	—	63	6	no	—
573	C₃	C₃×H(3,𝔽₃)×AGL(1,𝔽₇)	1	1	yes	characteristic
574	C₂	—	7	1	no	Sylow

	subgroup	G/H
center	C₃	C₃×H(3,𝔽₃)×AGL(1,𝔽₇)
commutator, cosocle	C₃²×C₇	C₂×C₃⁴
Frattini	C₃²	C₃³×AGL(1,𝔽₇)
Fitting	C₇×#6910	C₂×C₃
socle	C₃×C₇	C₂×C₃²×H(3,𝔽₃)
prime radical	C₃×H(3,𝔽₃)×AGL(1,𝔽₇)	C₃
hypercenter	#6910	AGL(1,𝔽₇)
hypocenter	C₇	C₂×C₃×#6910
LCS	G ▷ C₃²×C₇ ▷ C₃×C₇ ▷ C₇
UCS	C₁ ◁ C₃ ◁ C₃³ ◁ #6910

Complements

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

N	H ≅ G/N	type
1. #72249135	C₃	semidirect
2. #72249134	C₃	semidirect
3. AGL(1,𝔽₇)×#731	C₃	semidirect
4. #72249132	C₃	semidirect
5. #72249121	C₃	semidirect
6. #72249115	C₃	semidirect
8. D₇×#6910	C₃	semidirect
9. C₃×3₋¹⁺²×AGL(1,𝔽₇)	C₃	semidirect
10. C₃×3₋¹⁺²×AGL(1,𝔽₇)	C₃	semidirect
11. ΓL(1,𝔽₈)×#6910	C₂	semidirect
12. C₃×ΓL(1,𝔽₈)×3₋¹⁺²	C₂×C₃	semidirect
13. C₃×ΓL(1,𝔽₈)×3₋¹⁺²	C₂×C₃	semidirect
15. #13133636	C₂×C₃	semidirect
16. #13133635	C₂×C₃	semidirect
17. #13133634	C₂×C₃	semidirect
18. #13133633	C₂×C₃	semidirect
19. #13133632	C₂×C₃	semidirect
20. ΓL(1,𝔽₈)×#731	C₂×C₃	semidirect
21. C₇×#6910	C₂×C₃	semidirect
25. C₃×D₇×3₋¹⁺²	C₃²	semidirect
26. C₃×D₇×3₋¹⁺²	C₃²	semidirect
30. C₃×#4430⋊C₂	C₃²	semidirect
31. C₃×#4430⋊C₂	C₃²	semidirect
43. 3₋¹⁺²×AGL(1,𝔽₇)	C₃²	semidirect
44. 3₋¹⁺²×AGL(1,𝔽₇)	C₃²	semidirect
45. 3₋¹⁺²×AGL(1,𝔽₇)	C₃²	semidirect
53. C₃×#67356	C₃²	semidirect
55. C₃×#67355	C₃²	semidirect
57. C₃×C₉×AGL(1,𝔽₇)	C₃²	semidirect
58. D₇×#731	C₃²	semidirect
59. #10588045⋊C₂	C₃²	semidirect
60. #11765590	C₃²	semidirect
61. #11765586	C₃²	semidirect
65. C₃×C₇×3₋¹⁺²	C₂×C₃²	semidirect
66. C₃×C₇×3₋¹⁺²	C₂×C₃²	semidirect
70. C₃×#4430	C₂×C₃²	semidirect
71. C₃×#4430	C₂×C₃²	semidirect
83. ΓL(1,𝔽₈)×3₋¹⁺²	C₂×C₃²	semidirect
84. ΓL(1,𝔽₈)×3₋¹⁺²	C₂×C₃²	semidirect
85. ΓL(1,𝔽₈)×3₋¹⁺²	C₂×C₃²	semidirect
93. C₃×#4428	C₂×C₃²	semidirect
95. C₃×#4427	C₂×C₃²	semidirect
97. C₃×C₉×ΓL(1,𝔽₈)	C₂×C₃²	semidirect
98. C₇×#731	C₂×C₃²	semidirect
99. #10588045	C₂×C₃²	semidirect
100. #10588041	C₂×C₃²	semidirect
101. #10588037	C₂×C₃²	semidirect
132. D₇×3₋¹⁺²	C₃³	semidirect
136. C₃×C₉×D₇	C₃³	semidirect
146. #4430⋊C₂	C₃³	semidirect
148. C₃×D₇⋊C₉	C₃³	semidirect
172. #6910	AGL(1,𝔽₇)	direct (N × H)
194. C₇×3₋¹⁺²	C₂×C₃³	semidirect
198. C₃×C₇×C₉	C₂×C₃³	semidirect
208. #4430	C₂×C₃³	semidirect
210. C₃×C₇⋊C₉	C₂×C₃³	semidirect
316. C₃×3₋¹⁺²	C₃×AGL(1,𝔽₇)	semidirect
317. C₃×3₋¹⁺²	C₃×AGL(1,𝔽₇)	semidirect
332. #731	C₃×AGL(1,𝔽₇)	semidirect
425. AGL(1,𝔽₇)	#6910	direct (N × H)
461. 3₋¹⁺²	C₃²×AGL(1,𝔽₇)	semidirect
477. C₃×C₉	C₃²×AGL(1,𝔽₇)	semidirect
489. ΓL(1,𝔽₈)	C₂×#6910	semidirect
519. D₇	C₃×#6910	semidirect
549. C₇	C₂×C₃×#6910	semidirect

Maximal subgroups

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

subgroup	index
#72249135	3
#72249134	3
AGL(1,𝔽₇)×#731	3
#72249132	3
#72249121	3
#72249115	3
C₃×H(3,𝔽₃)×AGL(1,𝔽₇)	3
D₇×#6910	3
C₃×3₋¹⁺²×AGL(1,𝔽₇)	3
ΓL(1,𝔽₈)×#6910	2
C₂×C₃×#6910	7

Subgroups — AGL(1,𝔽7)×#6910 #72249136

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — AGL(1,𝔽₇)×#6910 #72249136