Subgroups — #72245331

Order 7056.

560 conjugacy classes of subgroups.

The subgroup lattice has 390 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#72245313	C₂	1	1	yes	maximal, characteristic, complemented
2	#72245228	C₃	1	1	yes	maximal, characteristic, complemented
3	#72245179	C₃	1	1	yes	maximal, characteristic, complemented
4	#72245179	C₃	1	1	yes	maximal, characteristic, complemented
5	C₂×ΓL(1,𝔽₈)⋊(ΓL(1,𝔽₈)⋊C₄)	C₂	1	2	yes	maximal, complemented
6	#72228759	C₃	1	1	yes	maximal, characteristic, complemented
7	#72222016	C₂	1	2	yes	maximal, complemented
8	D₄×ΓL(1,𝔽₈)²	C₂	1	1	yes	maximal, characteristic, complemented
9	C₄×ΓL(1,𝔽₈)⋊AGL(1,𝔽₇)	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂²×ΓL(1,𝔽₈)²	C₂²	1	2	yes	—
11	C₂×ΓL(1,𝔽₈)⋊AGL(1,𝔽₇)	C₂²	1	1	yes	characteristic
12	C₄×ΓL(1,𝔽₈)²	C₂²	1	1	yes	characteristic
13	ΓL(1,𝔽₈)⋊(ΓL(1,𝔽₈)⋊C₄)	C₂²	1	1	yes	characteristic
14	ΓL(1,𝔽₈)⋊(ΓL(1,𝔽₈)⋊C₄)	C₂²	1	2	yes	—
15	D₄×#3862	C₂×C₃	1	1	yes	characteristic, complemented
16	D₄×#3861	C₂×C₃	1	1	yes	characteristic, complemented
17	C₇×D₄×ΓL(1,𝔽₈)	C₂×C₃	1	1	yes	characteristic, complemented
18	C₇×D₄×ΓL(1,𝔽₈)	C₂×C₃	1	1	yes	characteristic, complemented
19	#11924222	C₂×C₃	1	2	yes	complemented
20	C₂×#10596875	C₂×C₃	1	2	yes	complemented
21	C₄×#64425	C₂×C₃	1	1	yes	characteristic, complemented
22	#11924218	C₂×C₃	1	1	yes	characteristic, complemented
23	#11924217	C₂×C₃	1	2	yes	complemented
24	C₂×#10596874	C₂×C₃	1	2	yes	complemented
25	C₄×#64424	C₂×C₃	1	1	yes	characteristic, complemented
26	#11924213	C₂×C₃	1	1	yes	characteristic, complemented
27	#11924202	C₂×C₃	1	2	yes	complemented
28	#11924202	C₂×C₃	1	2	yes	complemented
29	C₂×ΓL(1,𝔽₈)⋊Dic₇	C₂×C₃	1	2	yes	complemented
30	C₂×ΓL(1,𝔽₈)⋊Dic₇	C₂×C₃	1	2	yes	complemented
31	C₄×ΓL(1,𝔽₈)⋊D₇	C₂×C₃	1	1	yes	characteristic, complemented
32	C₄×ΓL(1,𝔽₈)⋊D₇	C₂×C₃	1	1	yes	characteristic, complemented
33	#11924198	C₂×C₃	1	1	yes	characteristic, complemented
34	#11924198	C₂×C₃	1	1	yes	characteristic, complemented
35	C₃×#1159⋊C₃	—	7	1	no	maximal
36	C₃×#1159⋊C₃	—	7	1	no	maximal
37	C₂×ΓL(1,𝔽₈)²	—	2	2	no	—
38	C₂×ΓL(1,𝔽₈)²	C₂³	1	1	yes	characteristic
39	ΓL(1,𝔽₈)⋊AGL(1,𝔽₇)	—	2	1	no	—
40	#11716131	C₃²	1	1	yes	characteristic, complemented, Hall
41	C₂²×#3862	C₂²×C₃	1	2	yes	—
42	C₂²×#3861	C₂²×C₃	1	2	yes	—
43	C₂²×C₇×ΓL(1,𝔽₈)	C₂²×C₃	1	2	yes	—
44	C₂²×C₇×ΓL(1,𝔽₈)	C₂²×C₃	1	2	yes	—
45	C₂×#64425	C₂²×C₃	1	1	yes	characteristic
46	C₂×#64424	C₂²×C₃	1	1	yes	characteristic
47	C₂×ΓL(1,𝔽₈)⋊D₇	C₂²×C₃	1	1	yes	characteristic
48	C₂×ΓL(1,𝔽₈)⋊D₇	C₂²×C₃	1	1	yes	characteristic
49	C₄×#3862	C₂²×C₃	1	1	yes	characteristic
50	C₄×#3861	C₂²×C₃	1	1	yes	characteristic
51	C₄×C₇×ΓL(1,𝔽₈)	C₂²×C₃	1	1	yes	characteristic
52	C₄×C₇×ΓL(1,𝔽₈)	C₂²×C₃	1	1	yes	characteristic
53	#10596875	C₂²×C₃	1	2	yes	—
54	#10596875	C₂²×C₃	1	1	yes	characteristic
55	#10596874	C₂²×C₃	1	2	yes	—
56	#10596874	C₂²×C₃	1	1	yes	characteristic
57	ΓL(1,𝔽₈)⋊Dic₇	C₂²×C₃	1	2	yes	—
58	ΓL(1,𝔽₈)⋊Dic₇	C₂²×C₃	1	2	yes	—
59	ΓL(1,𝔽₈)⋊Dic₇	C₂²×C₃	1	1	yes	characteristic
60	ΓL(1,𝔽₈)⋊Dic₇	C₂²×C₃	1	1	yes	characteristic
61	C₃×D₄×ΓL(1,𝔽₈)	—	14	1	no	—
62	C₃×D₄×ΓL(1,𝔽₈)	—	14	1	no	—
63	C₃×(C₂²⋊D₇)⋊C₃	—	7	2	no	—
64	C₃×(C₂²⋊D₇)⋊C₃	—	7	2	no	—
65	C₂×C₃×ΓL(1,𝔽₈)⋊C₄	—	7	2	no	—
66	C₂×C₃×ΓL(1,𝔽₈)⋊C₄	—	7	2	no	—
67	C₃×C₄×AGL(1,𝔽₇)	—	7	1	no	—
68	C₃×C₄×AGL(1,𝔽₇)	—	7	1	no	—
69	C₃×ΓL(1,𝔽₈)⋊Q₈	—	7	1	no	—
70	C₃×ΓL(1,𝔽₈)⋊Q₈	—	7	1	no	—
71	ΓL(1,𝔽₈)²	#41	1	1	yes	characteristic, complemented, Hall
72	C₇²×D₄	C₂×C₃²	1	1	yes	characteristic, complemented
73	C₂²⋊(C₇⋊D₇)	C₂×C₃²	1	2	yes	complemented
74	C₂×C₇⋊Dic₇	C₂×C₃²	1	2	yes	complemented
75	C₄×C₇⋊D₇	C₂×C₃²	1	1	yes	characteristic, complemented
76	#87648	C₂×C₃²	1	1	yes	characteristic, complemented
77	C₃×#1159	—	7	1	no	—
78	C₃×#1159	—	7	1	no	—
79	#1159⋊C₃	—	7	1	no	—
80	#1159⋊C₃	—	7	1	no	—
81	#1159⋊C₃	—	7	1	no	—
82	#1159⋊C₃	—	7	1	no	—
83	#1159⋊C₃	—	7	1	no	—
84	#1159⋊C₃	—	7	1	no	—
85	#1159⋊C₃	—	21	2	no	—
86	C₂×#3862	—	2	2	no	—
87	C₂×#3862	C₂³×C₃	1	1	yes	characteristic
88	C₂×#3861	—	2	2	no	—
89	C₂×#3861	C₂³×C₃	1	1	yes	characteristic
90	C₂×C₇×ΓL(1,𝔽₈)	—	2	2	no	—
91	C₂×C₇×ΓL(1,𝔽₈)	—	2	2	no	—
92	C₂×C₇×ΓL(1,𝔽₈)	C₂³×C₃	1	1	yes	characteristic
93	C₂×C₇×ΓL(1,𝔽₈)	C₂³×C₃	1	1	yes	characteristic
94	#64425	—	2	1	no	—
95	#64424	—	2	1	no	—
96	ΓL(1,𝔽₈)⋊D₇	—	2	1	no	—
97	ΓL(1,𝔽₈)⋊D₇	—	2	1	no	—
98	C₂²×C₃×ΓL(1,𝔽₈)	—	14	2	no	—
99	C₂²×C₃×ΓL(1,𝔽₈)	—	14	2	no	—
100	C₂×C₃×AGL(1,𝔽₇)	—	7	1	no	—
101	C₂×C₃×AGL(1,𝔽₇)	—	7	1	no	—
102	C₃×C₄×ΓL(1,𝔽₈)	—	14	1	no	—
103	C₃×C₄×ΓL(1,𝔽₈)	—	14	1	no	—
104	C₃×ΓL(1,𝔽₈)⋊C₄	—	7	2	no	—
105	C₃×ΓL(1,𝔽₈)⋊C₄	—	7	2	no	—
106	C₃×ΓL(1,𝔽₈)⋊C₄	—	7	1	no	—
107	C₃×ΓL(1,𝔽₈)⋊C₄	—	7	1	no	—
108	C₂²×C₇²	C₂²×C₃²	1	2	yes	—
109	C₂×C₇⋊D₇	C₂²×C₃²	1	1	yes	characteristic
110	C₄×C₇²	C₂²×C₃²	1	1	yes	characteristic
111	C₇⋊Dic₇	C₂²×C₃²	1	2	yes	—
112	C₇⋊Dic₇	C₂²×C₃²	1	1	yes	characteristic
113	C₃×C₇×D₄	—	14	1	no	—
114	C₃×C₇×D₄	—	14	1	no	—
115	C₃×C₂²⋊D₇	—	7	2	no	—
116	C₃×C₂²⋊D₇	—	7	2	no	—
117	C₂×C₃×Dic₇	—	7	2	no	—
118	C₂×C₃×Dic₇	—	7	2	no	—
119	C₃×C₄×D₇	—	7	1	no	—
120	C₃×C₄×D₇	—	7	1	no	—
121	C₃×Dic₁₄	—	7	1	no	—
122	C₃×Dic₁₄	—	7	1	no	—
123	D₄×ΓL(1,𝔽₈)	—	7	1	no	—
124	D₄×ΓL(1,𝔽₈)	—	14	1	no	—
125	D₄×ΓL(1,𝔽₈)	—	14	1	no	—
126	D₄×ΓL(1,𝔽₈)	—	14	1	no	—
127	D₄×ΓL(1,𝔽₈)	—	42	2	no	—
128	D₄×ΓL(1,𝔽₈)	AGL(1,𝔽₇)	1	1	yes	characteristic, complemented
129	D₄×ΓL(1,𝔽₈)	AGL(1,𝔽₇)	1	1	yes	characteristic, complemented
130	(C₂²⋊D₇)⋊C₃	—	7	2	no	—
131	(C₂²⋊D₇)⋊C₃	—	7	2	no	—
132	(C₂²⋊D₇)⋊C₃	—	7	2	no	—
133	(C₂²⋊D₇)⋊C₃	—	7	2	no	—
134	(C₂²⋊D₇)⋊C₃	—	7	2	no	—
135	(C₂²⋊D₇)⋊C₃	—	7	2	no	—
136	(C₂²⋊D₇)⋊C₃	—	21	4	no	—
137	C₂×ΓL(1,𝔽₈)⋊C₄	—	7	2	no	—
138	C₂×ΓL(1,𝔽₈)⋊C₄	—	7	2	no	—
139	C₂×ΓL(1,𝔽₈)⋊C₄	—	7	2	no	—
140	C₂×ΓL(1,𝔽₈)⋊C₄	—	7	2	no	—
141	C₂×ΓL(1,𝔽₈)⋊C₄	—	7	2	no	—
142	C₂×ΓL(1,𝔽₈)⋊C₄	—	7	2	no	—
143	C₂×ΓL(1,𝔽₈)⋊C₄	—	21	4	no	—
144	C₄×AGL(1,𝔽₇)	—	7	1	no	—
145	C₄×AGL(1,𝔽₇)	—	7	1	no	—
146	C₄×AGL(1,𝔽₇)	—	7	1	no	—
147	C₄×AGL(1,𝔽₇)	—	7	1	no	—
148	C₄×AGL(1,𝔽₇)	—	7	1	no	—
149	C₄×AGL(1,𝔽₇)	—	7	1	no	—
150	C₄×AGL(1,𝔽₇)	—	21	2	no	—
151	ΓL(1,𝔽₈)⋊Q₈	—	7	1	no	—
152	ΓL(1,𝔽₈)⋊Q₈	—	7	1	no	—
153	ΓL(1,𝔽₈)⋊Q₈	—	7	1	no	—
154	ΓL(1,𝔽₈)⋊Q₈	—	7	1	no	—
155	ΓL(1,𝔽₈)⋊Q₈	—	7	1	no	—
156	ΓL(1,𝔽₈)⋊Q₈	—	7	1	no	—
157	ΓL(1,𝔽₈)⋊Q₈	—	21	2	no	—
158	#3862	C₃×#41	1	1	yes	characteristic, complemented
159	#3861	C₃×#41	1	1	yes	characteristic, complemented
160	C₇×ΓL(1,𝔽₈)	C₃×#41	1	1	yes	characteristic, complemented
161	C₇×ΓL(1,𝔽₈)	C₃×#41	1	1	yes	characteristic, complemented
162	C₃²×#41	—	49	1	no	Hall
163	C₂×C₃×ΓL(1,𝔽₈)	—	14	1	no	—
164	C₂×C₃×ΓL(1,𝔽₈)	—	14	1	no	—
165	C₂×C₃×ΓL(1,𝔽₈)	—	28	2	no	—
166	C₂×C₃×ΓL(1,𝔽₈)	—	28	2	no	—
167	C₃×AGL(1,𝔽₇)	—	14	1	no	—
168	C₃×AGL(1,𝔽₇)	—	14	1	no	—
169	#1159	—	7	1	no	—
170	#1159	—	7	1	no	—
171	#1159	—	21	2	no	—
172	C₂×C₇²	—	2	2	no	—
173	C₂×C₇²	C₂³×C₃²	1	1	yes	characteristic
174	C₇⋊D₇	—	2	1	no	—
175	C₂²×C₃×C₇	—	14	2	no	—
176	C₂²×C₃×C₇	—	14	2	no	—
177	C₂×C₃×D₇	—	7	1	no	—
178	C₂×C₃×D₇	—	7	1	no	—
179	C₂²×ΓL(1,𝔽₈)	—	7	2	no	—
180	C₂²×ΓL(1,𝔽₈)	—	14	2	no	—
181	C₂²×ΓL(1,𝔽₈)	—	14	2	no	—
182	C₂²×ΓL(1,𝔽₈)	—	14	2	no	—
183	C₂²×ΓL(1,𝔽₈)	—	42	4	no	—
184	C₂²×ΓL(1,𝔽₈)	C₂×AGL(1,𝔽₇)	1	2	yes	—
185	C₂²×ΓL(1,𝔽₈)	C₂×AGL(1,𝔽₇)	1	2	yes	—
186	C₂×AGL(1,𝔽₇)	—	7	1	no	—
187	C₂×AGL(1,𝔽₇)	—	7	1	no	—
188	C₂×AGL(1,𝔽₇)	—	7	1	no	—
189	C₂×AGL(1,𝔽₇)	—	7	1	no	—
190	C₂×AGL(1,𝔽₇)	—	7	1	no	—
191	C₂×AGL(1,𝔽₇)	—	7	1	no	—
192	C₂×AGL(1,𝔽₇)	—	21	2	no	—
193	C₃×C₄×C₇	—	14	1	no	—
194	C₃×C₄×C₇	—	14	1	no	—
195	C₃×Dic₇	—	7	2	no	—
196	C₃×Dic₇	—	7	2	no	—
197	C₃×Dic₇	—	7	1	no	—
198	C₃×Dic₇	—	7	1	no	—
199	C₄×ΓL(1,𝔽₈)	—	7	1	no	—
200	C₄×ΓL(1,𝔽₈)	—	14	1	no	—
201	C₄×ΓL(1,𝔽₈)	—	14	1	no	—
202	C₄×ΓL(1,𝔽₈)	—	14	1	no	—
203	C₄×ΓL(1,𝔽₈)	—	42	2	no	—
204	C₄×ΓL(1,𝔽₈)	C₂×AGL(1,𝔽₇)	1	1	yes	characteristic
205	C₄×ΓL(1,𝔽₈)	C₂×AGL(1,𝔽₇)	1	1	yes	characteristic
206	ΓL(1,𝔽₈)⋊C₄	—	7	2	no	—
207	ΓL(1,𝔽₈)⋊C₄	—	7	1	no	—
208	ΓL(1,𝔽₈)⋊C₄	—	7	2	no	—
209	ΓL(1,𝔽₈)⋊C₄	—	7	2	no	—
210	ΓL(1,𝔽₈)⋊C₄	—	7	1	no	—
211	ΓL(1,𝔽₈)⋊C₄	—	7	1	no	—
212	ΓL(1,𝔽₈)⋊C₄	—	7	2	no	—
213	ΓL(1,𝔽₈)⋊C₄	—	7	1	no	—
214	ΓL(1,𝔽₈)⋊C₄	—	7	2	no	—
215	ΓL(1,𝔽₈)⋊C₄	—	7	2	no	—
216	ΓL(1,𝔽₈)⋊C₄	—	7	1	no	—
217	ΓL(1,𝔽₈)⋊C₄	—	7	1	no	—
218	ΓL(1,𝔽₈)⋊C₄	—	21	4	no	—
219	ΓL(1,𝔽₈)⋊C₄	—	21	2	no	—
220	C₃²×Q₈	—	49	1	no	—
221	C₃²×D₄	—	49	2	no	—
222	C₃²×D₄	—	98	1	no	—
223	C₂×C₃²×C₄	—	49	2	no	—
224	C₂×C₃²×C₄	—	49	1	no	—
225	C₃×ΓL(1,𝔽₈)	—	14	1	no	—
226	C₃×ΓL(1,𝔽₈)	—	14	1	no	—
227	C₇×D₄	—	3	2	no	—
228	C₇×D₄	C₃×AGL(1,𝔽₇)	1	1	yes	characteristic, complemented
229	C₇×D₄	C₃×AGL(1,𝔽₇)	1	1	yes	characteristic, complemented
230	C₂²⋊D₇	—	7	2	no	—
231	C₂²⋊D₇	—	7	2	no	—
232	C₂²⋊D₇	—	21	4	no	—
233	C₂×Dic₇	—	7	2	no	—
234	C₂×Dic₇	—	7	2	no	—
235	C₂×Dic₇	—	21	4	no	—
236	C₄×D₇	—	7	1	no	—
237	C₄×D₇	—	7	1	no	—
238	C₄×D₇	—	21	2	no	—
239	Dic₁₄	—	7	1	no	—
240	Dic₁₄	—	7	1	no	—
241	Dic₁₄	—	21	2	no	—
242	C₇²	C₃²×#41	1	1	yes	characteristic, complemented, Sylow
243	C₃×#41	—	49	1	no	—
244	C₃×#41	—	49	1	no	—
245	C₃×#41	—	49	1	no	—
246	C₃×#41	—	49	1	no	—
247	C₂×C₃×C₇	—	14	1	no	—
248	C₂×C₃×C₇	—	14	1	no	—
249	C₂×C₃×C₇	—	28	2	no	—
250	C₂×C₃×C₇	—	28	2	no	—
251	C₃×D₇	—	14	1	no	—
252	C₃×D₇	—	14	1	no	—
253	C₂×ΓL(1,𝔽₈)	—	2	2	no	—
254	C₂×ΓL(1,𝔽₈)	—	2	2	no	—
255	C₂×ΓL(1,𝔽₈)	—	7	1	no	—
256	C₂×ΓL(1,𝔽₈)	—	14	1	no	—
257	C₂×ΓL(1,𝔽₈)	—	14	1	no	—
258	C₂×ΓL(1,𝔽₈)	—	14	1	no	—
259	C₂×ΓL(1,𝔽₈)	—	14	2	no	—
260	C₂×ΓL(1,𝔽₈)	—	28	2	no	—
261	C₂×ΓL(1,𝔽₈)	—	28	2	no	—
262	C₂×ΓL(1,𝔽₈)	—	28	2	no	—
263	C₂×ΓL(1,𝔽₈)	—	42	2	no	—
264	C₂×ΓL(1,𝔽₈)	—	84	4	no	—
265	C₂×ΓL(1,𝔽₈)	C₂²×AGL(1,𝔽₇)	1	1	yes	characteristic
266	C₂×ΓL(1,𝔽₈)	C₂²×AGL(1,𝔽₇)	1	1	yes	characteristic
267	AGL(1,𝔽₇)	—	14	1	no	—
268	AGL(1,𝔽₇)	—	14	1	no	—
269	AGL(1,𝔽₇)	—	14	1	no	—
270	AGL(1,𝔽₇)	—	14	1	no	—
271	AGL(1,𝔽₇)	—	14	1	no	—
272	AGL(1,𝔽₇)	—	14	1	no	—
273	AGL(1,𝔽₇)	—	42	2	no	—
274	C₂²×C₃²	—	49	1	no	—
275	C₂²×C₃²	—	98	2	no	—
276	C₃²×C₄	—	49	1	no	—
277	C₃²×C₄	—	49	2	no	—
278	C₃²×C₄	—	98	1	no	—
279	C₂²×C₇	—	3	4	no	—
280	C₂²×C₇	C₂×C₃×AGL(1,𝔽₇)	1	2	yes	—
281	C₂²×C₇	C₂×C₃×AGL(1,𝔽₇)	1	2	yes	—
282	C₂×D₇	—	7	1	no	—
283	C₂×D₇	—	7	1	no	—
284	C₂×D₇	—	21	2	no	—
285	C₄×C₇	—	3	2	no	—
286	C₄×C₇	C₂×C₃×AGL(1,𝔽₇)	1	1	yes	characteristic
287	C₄×C₇	C₂×C₃×AGL(1,𝔽₇)	1	1	yes	characteristic
288	Dic₇	—	7	2	no	—
289	Dic₇	—	7	2	no	—
290	Dic₇	—	7	1	no	—
291	Dic₇	—	7	1	no	—
292	Dic₇	—	21	4	no	—
293	Dic₇	—	21	2	no	—
294	C₃×Q₈	—	49	1	no	—
295	C₃×Q₈	—	49	1	no	—
296	C₃×Q₈	—	49	1	no	—
297	C₃×Q₈	—	49	1	no	—
298	C₃×D₄	—	14	1	no	—
299	C₃×D₄	—	14	1	no	—
300	C₃×D₄	—	49	2	no	—
301	C₃×D₄	—	49	2	no	—
302	C₃×D₄	—	49	2	no	—
303	C₃×D₄	—	49	2	no	—
304	C₃×D₄	—	98	1	no	—
305	C₃×D₄	—	98	1	no	—
306	C₂×C₃×C₄	—	49	1	no	—
307	C₂×C₃×C₄	—	49	1	no	—
308	C₂×C₃×C₄	—	49	2	no	—
309	C₂×C₃×C₄	—	49	2	no	—
310	C₂×C₃×C₄	—	49	2	no	—
311	C₂×C₃×C₄	—	49	2	no	—
312	C₂×C₃×C₄	—	49	1	no	—
313	C₂×C₃×C₄	—	49	1	no	—
314	C₃×C₇	—	14	1	no	—
315	C₃×C₇	—	14	1	no	—
316	ΓL(1,𝔽₈)	—	7	1	no	—
317	ΓL(1,𝔽₈)	—	14	1	no	—
318	ΓL(1,𝔽₈)	—	14	1	no	—
319	ΓL(1,𝔽₈)	—	14	1	no	—
320	ΓL(1,𝔽₈)	—	42	2	no	—
321	ΓL(1,𝔽₈)	#1159⋊C₃	1	1	yes	characteristic, complemented
322	ΓL(1,𝔽₈)	#1159⋊C₃	1	1	yes	characteristic, complemented
323	C₂×C₃²	—	98	1	no	—
324	C₂×C₃²	—	98	1	no	—
325	C₂×C₃²	—	196	2	no	—
326	#41	—	49	1	no	Sylow
327	C₂×C₇	—	2	2	no	—
328	C₂×C₇	—	2	2	no	—
329	C₂×C₇	—	3	2	no	—
330	C₂×C₇	—	6	4	no	—
331	C₂×C₇	C₂²×C₃×AGL(1,𝔽₇)	1	1	yes	characteristic
332	C₂×C₇	C₂²×C₃×AGL(1,𝔽₇)	1	1	yes	characteristic
333	D₇	—	14	1	no	—
334	D₇	—	14	1	no	—
335	D₇	—	42	2	no	—
336	C₂²×C₃	—	14	2	no	—
337	C₂²×C₃	—	14	2	no	—
338	C₂²×C₃	—	49	1	no	—
339	C₂²×C₃	—	49	1	no	—
340	C₂²×C₃	—	49	1	no	—
341	C₂²×C₃	—	49	1	no	—
342	C₂²×C₃	—	98	2	no	—
343	C₂²×C₃	—	98	2	no	—
344	C₃×C₄	—	14	1	no	—
345	C₃×C₄	—	14	1	no	—
346	C₃×C₄	—	49	1	no	—
347	C₃×C₄	—	49	1	no	—
348	C₃×C₄	—	49	2	no	—
349	C₃×C₄	—	49	2	no	—
350	C₃×C₄	—	49	1	no	—
351	C₃×C₄	—	49	1	no	—
352	C₃×C₄	—	49	2	no	—
353	C₃×C₄	—	49	2	no	—
354	C₃×C₄	—	98	1	no	—
355	C₃×C₄	—	98	1	no	—
356	C₃²	—	98	1	no	Sylow
357	Q₈	—	49	1	no	—
358	D₄	—	49	2	no	—
359	D₄	ΓL(1,𝔽₈)⋊AGL(1,𝔽₇)	1	1	yes	characteristic, complemented
360	C₂×C₄	—	49	2	no	—
361	C₂×C₄	—	49	1	no	—
362	C₇	—	3	2	no	—
363	C₇	C₃×#1159⋊C₃	1	1	yes	characteristic, complemented
364	C₇	C₃×#1159⋊C₃	1	1	yes	characteristic, complemented
365	C₂×C₃	—	14	1	no	—
366	C₂×C₃	—	14	1	no	—
367	C₂×C₃	—	28	2	no	—
368	C₂×C₃	—	28	2	no	—
369	C₂×C₃	—	98	1	no	—
370	C₂×C₃	—	98	1	no	—
371	C₂×C₃	—	98	1	no	—
372	C₂×C₃	—	98	1	no	—
373	C₂×C₃	—	98	1	no	—
374	C₂×C₃	—	98	1	no	—
375	C₂×C₃	—	196	2	no	—
376	C₂×C₃	—	196	2	no	—
377	C₂²	—	49	1	no	—
378	C₂²	C₂×ΓL(1,𝔽₈)⋊AGL(1,𝔽₇)	1	2	yes	—
379	C₄	—	49	1	no	—
380	C₄	—	49	2	no	—
381	C₄	C₂×ΓL(1,𝔽₈)⋊AGL(1,𝔽₇)	1	1	yes	characteristic
382	C₃	—	14	1	no	—
383	C₃	—	14	1	no	—
384	C₃	—	98	1	no	—
385	C₃	—	98	1	no	—
386	C₂	—	2	2	no	—
387	C₂	—	98	1	no	—
388	C₂	C₂²×ΓL(1,𝔽₈)⋊AGL(1,𝔽₇)	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂	C₂²×ΓL(1,𝔽₈)⋊AGL(1,𝔽₇)
commutator, socle, cosocle	C₂×C₇²	C₂³×C₃²
Fitting	C₇²×D₄	C₂×C₃²
hypercenter	D₄	ΓL(1,𝔽₈)⋊AGL(1,𝔽₇)
hypocenter	C₇²	C₃²×#41
LCS	G ▷ C₂×C₇² ▷ C₇²
UCS	C₁ ◁ C₂ ◁ D₄

Complements

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

N	H ≅ G/N	type
1. #72245313	C₂	semidirect
2. #72245228	C₃	semidirect
3. #72245179	C₃	semidirect
4. #72245179	C₃	semidirect
5. C₂×ΓL(1,𝔽₈)⋊(ΓL(1,𝔽₈)⋊C₄)	C₂	semidirect
6. #72228759	C₃	semidirect
7. #72222016	C₂	semidirect
8. D₄×ΓL(1,𝔽₈)²	C₂	semidirect
9. C₄×ΓL(1,𝔽₈)⋊AGL(1,𝔽₇)	C₂	semidirect
15. D₄×#3862	C₂×C₃	semidirect
16. D₄×#3861	C₂×C₃	semidirect
17. C₇×D₄×ΓL(1,𝔽₈)	C₂×C₃	semidirect
18. C₇×D₄×ΓL(1,𝔽₈)	C₂×C₃	semidirect
19. #11924222	C₂×C₃	semidirect
20. C₂×#10596875	C₂×C₃	semidirect
21. C₄×#64425	C₂×C₃	semidirect
22. #11924218	C₂×C₃	semidirect
23. #11924217	C₂×C₃	semidirect
24. C₂×#10596874	C₂×C₃	semidirect
25. C₄×#64424	C₂×C₃	semidirect
26. #11924213	C₂×C₃	semidirect
27. #11924202	C₂×C₃	semidirect
28. #11924202	C₂×C₃	semidirect
29. C₂×ΓL(1,𝔽₈)⋊Dic₇	C₂×C₃	semidirect
30. C₂×ΓL(1,𝔽₈)⋊Dic₇	C₂×C₃	semidirect
31. C₄×ΓL(1,𝔽₈)⋊D₇	C₂×C₃	semidirect
32. C₄×ΓL(1,𝔽₈)⋊D₇	C₂×C₃	semidirect
33. #11924198	C₂×C₃	semidirect
34. #11924198	C₂×C₃	semidirect
40. #11716131	C₃²	semidirect
71. ΓL(1,𝔽₈)²	#41	semidirect
72. C₇²×D₄	C₂×C₃²	semidirect
73. C₂²⋊(C₇⋊D₇)	C₂×C₃²	semidirect
74. C₂×C₇⋊Dic₇	C₂×C₃²	semidirect
75. C₄×C₇⋊D₇	C₂×C₃²	semidirect
76. #87648	C₂×C₃²	semidirect
128. D₄×ΓL(1,𝔽₈)	AGL(1,𝔽₇)	semidirect
129. D₄×ΓL(1,𝔽₈)	AGL(1,𝔽₇)	semidirect
158. #3862	C₃×#41	semidirect
159. #3861	C₃×#41	semidirect
160. C₇×ΓL(1,𝔽₈)	C₃×#41	semidirect
161. C₇×ΓL(1,𝔽₈)	C₃×#41	semidirect
228. C₇×D₄	C₃×AGL(1,𝔽₇)	semidirect
229. C₇×D₄	C₃×AGL(1,𝔽₇)	semidirect
242. C₇²	C₃²×#41	semidirect
321. ΓL(1,𝔽₈)	#1159⋊C₃	semidirect
322. ΓL(1,𝔽₈)	#1159⋊C₃	semidirect
359. D₄	ΓL(1,𝔽₈)⋊AGL(1,𝔽₇)	semidirect
363. C₇	C₃×#1159⋊C₃	semidirect
364. C₇	C₃×#1159⋊C₃	semidirect

Maximal subgroups

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

subgroup	index
#72245313	2
#72245228	3
#72245179	3
C₂×ΓL(1,𝔽₈)⋊(ΓL(1,𝔽₈)⋊C₄)	2
#72228759	3
#72222016	2
D₄×ΓL(1,𝔽₈)²	2
C₄×ΓL(1,𝔽₈)⋊AGL(1,𝔽₇)	2
C₃×#1159⋊C₃	7