Subgroups — #90308⋊C₃ #13057351

Order 1344.

500 conjugacy classes of subgroups.

The subgroup lattice has 300 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	ΓL(1,𝔽₈)×#121	C₂	1	1	yes	maximal, characteristic, complemented
2	#6502⋊C₃	C₂	1	2	yes	maximal, complemented
3	#6500⋊C₃	C₂	1	2	yes	maximal, complemented
4	D₄×ΓL(1,𝔽₈)⋊C₄	C₂	1	2	yes	maximal, complemented
5	C₂×(C₂×AGL(1,𝔽₇))⋊C₄	C₂	1	1	yes	maximal, characteristic, complemented
6	#6487⋊C₃	C₂	1	1	yes	maximal, characteristic, complemented
7	#6461⋊C₃	C₂	1	2	yes	maximal, complemented
8	#6449⋊C₃	C₂	1	2	yes	maximal, complemented
9	#6448⋊C₃	C₂	1	2	yes	maximal, complemented
10	#90308	C₃	1	1	yes	maximal, characteristic, complemented, Hall
11	C₂³×AGL(1,𝔽₇)	C₂²	1	1	yes	characteristic
12	C₂×D₄×ΓL(1,𝔽₈)	C₂²	1	2	yes	—
13	C₂×D₄×ΓL(1,𝔽₈)	C₂²	1	1	yes	characteristic, complemented
14	C₂²×C₄×ΓL(1,𝔽₈)	C₂²	1	1	yes	characteristic
15	C₂×(C₂²⋊D₇)⋊C₃	C₂²	1	2	yes	—
16	C₂×(C₂²⋊D₇)⋊C₃	C₂²	1	2	yes	—
17	C₂²×ΓL(1,𝔽₈)⋊C₄	C₂²	1	1	yes	characteristic
18	C₂²×ΓL(1,𝔽₈)⋊C₄	C₂²	1	2	yes	complemented
19	C₂×ΓL(1,𝔽₈)⋊Q₈	C₂²	1	1	yes	characteristic, complemented
20	C₄⋊C₄×ΓL(1,𝔽₈)	C₂²	1	1	yes	characteristic, complemented
21	C₂²⋊C₄×ΓL(1,𝔽₈)	C₂²	1	2	yes	complemented
22	(C₂²⋊Dic₇)⋊C₃	C₂²	1	2	yes	—
23	(C₂²⋊Dic₇)⋊C₃	C₂²	1	2	yes	complemented
24	(C₂×AGL(1,𝔽₇))⋊C₄	C₂²	1	2	yes	—
25	(C₂×AGL(1,𝔽₇))⋊C₄	C₂²	1	2	yes	—
26	(C₂×AGL(1,𝔽₇))⋊C₄	C₂²	1	2	yes	—
27	(C₂×AGL(1,𝔽₇))⋊C₄	C₂²	1	2	yes	—
28	C₄⋊(ΓL(1,𝔽₈)⋊C₄)	C₂²	1	1	yes	characteristic, complemented
29	C₄⋊(ΓL(1,𝔽₈)⋊C₄)	C₂²	1	2	yes	complemented
30	(ΓL(1,𝔽₈)⋊C₄)⋊C₄	C₂²	1	2	yes	complemented
31	C₄×ΓL(1,𝔽₈)⋊C₄	C₂²	1	2	yes	complemented
32	C₇×#121	C₂×C₃	1	1	yes	characteristic, complemented
33	#6502	C₂×C₃	1	2	yes	complemented
34	#6500	C₂×C₃	1	2	yes	complemented
35	D₄×Dic₇	C₂×C₃	1	2	yes	complemented
36	C₂×(C₂×D₇)⋊C₄	C₂×C₃	1	1	yes	characteristic, complemented
37	#6487	C₂×C₃	1	1	yes	characteristic, complemented
38	#6461	C₂×C₃	1	2	yes	complemented
39	#6449	C₂×C₃	1	2	yes	complemented
40	#6448	C₂×C₃	1	2	yes	complemented
41	C₃×#538	—	7	1	no	maximal, Hall
42	C₂³×ΓL(1,𝔽₈)	C₂³	1	2	yes	—
43	C₂³×ΓL(1,𝔽₈)	C₂³	1	1	yes	characteristic
44	C₂²×AGL(1,𝔽₇)	—	2	2	no	—
45	C₂²×AGL(1,𝔽₇)	—	4	1	no	—
46	C₂²×AGL(1,𝔽₇)	—	4	1	no	—
47	C₂²×AGL(1,𝔽₇)	C₂³	1	2	yes	—
48	D₄×ΓL(1,𝔽₈)	—	2	4	no	—
49	D₄×ΓL(1,𝔽₈)	—	4	1	no	—
50	C₂×C₄×ΓL(1,𝔽₈)	—	4	1	no	—
51	C₂×C₄×ΓL(1,𝔽₈)	C₂³	1	1	yes	characteristic
52	C₂×C₄×ΓL(1,𝔽₈)	C₂³	1	2	yes	—
53	C₂×C₄×ΓL(1,𝔽₈)	C₂³	1	1	yes	characteristic
54	(C₂²⋊D₇)⋊C₃	—	4	2	no	—
55	(C₂²⋊D₇)⋊C₃	—	4	2	no	—
56	C₂×ΓL(1,𝔽₈)⋊C₄	—	4	1	no	—
57	C₂×ΓL(1,𝔽₈)⋊C₄	—	4	2	no	—
58	C₂×ΓL(1,𝔽₈)⋊C₄	C₂³	1	2	yes	—
59	C₂×ΓL(1,𝔽₈)⋊C₄	C₂³	1	2	yes	—
60	C₂×ΓL(1,𝔽₈)⋊C₄	C₂³	1	2	yes	—
61	ΓL(1,𝔽₈)⋊Q₈	—	4	1	no	—
62	C₂³×D₇	C₂²×C₃	1	1	yes	characteristic
63	C₂×C₇×D₄	C₂²×C₃	1	2	yes	—
64	C₂×C₇×D₄	C₂²×C₃	1	1	yes	characteristic, complemented
65	C₂²×C₄×C₇	C₂²×C₃	1	1	yes	characteristic
66	C₂×C₂²⋊D₇	C₂²×C₃	1	2	yes	—
67	C₂×C₂²⋊D₇	C₂²×C₃	1	2	yes	—
68	C₂²×Dic₇	C₂²×C₃	1	1	yes	characteristic
69	C₂²×Dic₇	C₂²×C₃	1	2	yes	complemented
70	C₂×Dic₁₄	C₂²×C₃	1	1	yes	characteristic, complemented
71	C₇×C₄⋊C₄	C₂²×C₃	1	1	yes	characteristic, complemented
72	C₇×C₂²⋊C₄	C₂²×C₃	1	2	yes	complemented
73	C₂²⋊Dic₇	C₂²×C₃	1	2	yes	—
74	C₂²⋊Dic₇	C₂²×C₃	1	2	yes	complemented
75	(C₂×D₇)⋊C₄	C₂²×C₃	1	2	yes	—
76	(C₂×D₇)⋊C₄	C₂²×C₃	1	2	yes	—
77	(C₂×D₇)⋊C₄	C₂²×C₃	1	2	yes	—
78	(C₂×D₇)⋊C₄	C₂²×C₃	1	2	yes	—
79	C₄⋊Dic₇	C₂²×C₃	1	1	yes	characteristic, complemented
80	C₄⋊Dic₇	C₂²×C₃	1	2	yes	complemented
81	Dic₇⋊C₄	C₂²×C₃	1	2	yes	complemented
82	C₄×Dic₇	C₂²×C₃	1	2	yes	complemented
83	C₃×#126	—	7	2	no	—
84	C₃×#124	—	7	2	no	—
85	C₃×#123	—	7	2	no	—
86	C₃×C₂²⋊Q₈	—	7	1	no	—
87	C₃×#121	—	7	2	no	—
88	C₃×#121	—	7	1	no	—
89	C₃×#120	—	7	2	no	—
90	C₃×C₄×D₄	—	7	2	no	—
91	C₂×C₃×C₂²⋊C₄	—	7	1	no	—
92	C₂²×ΓL(1,𝔽₈)	—	2	1	no	—
93	C₂²×ΓL(1,𝔽₈)	—	2	1	no	—
94	C₂²×ΓL(1,𝔽₈)	—	2	2	no	—
95	C₂²×ΓL(1,𝔽₈)	—	2	2	no	—
96	C₂²×ΓL(1,𝔽₈)	—	2	2	no	—
97	C₂²×ΓL(1,𝔽₈)	#41	1	2	yes	—
98	C₂²×ΓL(1,𝔽₈)	C₂⁴	1	1	yes	characteristic
99	C₂×AGL(1,𝔽₇)	—	2	2	no	—
100	C₂×AGL(1,𝔽₇)	—	2	2	no	—
101	C₂×AGL(1,𝔽₇)	—	2	2	no	—
102	C₂×AGL(1,𝔽₇)	—	4	4	no	—
103	C₄×ΓL(1,𝔽₈)	—	2	2	no	—
104	C₄×ΓL(1,𝔽₈)	—	2	1	no	—
105	C₄×ΓL(1,𝔽₈)	—	2	1	no	—
106	ΓL(1,𝔽₈)⋊C₄	—	2	2	no	—
107	ΓL(1,𝔽₈)⋊C₄	—	2	2	no	—
108	ΓL(1,𝔽₈)⋊C₄	—	2	2	no	—
109	#538	—	7	1	no	Sylow
110	C₂³×C₇	C₂³×C₃	1	2	yes	—
111	C₂³×C₇	C₂³×C₃	1	1	yes	characteristic
112	C₂²×D₇	—	2	2	no	—
113	C₂²×D₇	—	4	1	no	—
114	C₂²×D₇	—	4	1	no	—
115	C₂²×D₇	C₂³×C₃	1	2	yes	—
116	C₇×D₄	—	2	4	no	—
117	C₇×D₄	—	4	1	no	—
118	C₂×C₄×C₇	—	4	1	no	—
119	C₂×C₄×C₇	C₂³×C₃	1	2	yes	—
120	C₂×C₄×C₇	C₂³×C₃	1	1	yes	characteristic
121	C₂×C₄×C₇	C₂³×C₃	1	1	yes	characteristic
122	C₂²⋊D₇	—	4	2	no	—
123	C₂²⋊D₇	—	4	2	no	—
124	C₂×Dic₇	—	4	1	no	—
125	C₂×Dic₇	—	4	2	no	—
126	C₂×Dic₇	C₂³×C₃	1	2	yes	—
127	C₂×Dic₇	C₂³×C₃	1	2	yes	—
128	C₂×Dic₇	C₂³×C₃	1	2	yes	—
129	Dic₁₄	—	4	1	no	—
130	C₂⁴×C₃	—	7	1	no	—
131	C₂×C₃×Q₈	—	7	1	no	—
132	C₂×C₃×D₄	—	7	2	no	—
133	C₂×C₃×D₄	—	7	2	no	—
134	C₂×C₃×D₄	—	7	2	no	—
135	C₂×C₃×D₄	—	7	1	no	—
136	C₂²×C₃×C₄	—	7	1	no	—
137	C₂²×C₃×C₄	—	7	2	no	—
138	C₂²×C₃×C₄	—	7	1	no	—
139	C₃×C₄⋊C₄	—	7	2	no	—
140	C₃×C₄⋊C₄	—	7	2	no	—
141	C₃×C₄⋊C₄	—	7	1	no	—
142	C₃×C₄⋊C₄	—	7	1	no	—
143	C₃×C₂²⋊C₄	—	7	2	no	—
144	C₃×C₂²⋊C₄	—	7	2	no	—
145	C₃×C₂²⋊C₄	—	7	2	no	—
146	C₃×C₂²⋊C₄	—	7	2	no	—
147	C₃×C₂²⋊C₄	—	7	2	no	—
148	C₃×C₂²⋊C₄	—	7	2	no	—
149	C₃×C₂²⋊C₄	—	7	2	no	—
150	C₃×C₄²	—	7	2	no	—
151	C₂×ΓL(1,𝔽₈)	—	2	2	no	—
152	C₂×ΓL(1,𝔽₈)	—	4	2	no	—
153	C₂×ΓL(1,𝔽₈)	C₂×#41	1	1	yes	characteristic
154	C₂×ΓL(1,𝔽₈)	H(5,𝔽₂)	1	1	yes	characteristic
155	C₂×ΓL(1,𝔽₈)	H(5,𝔽₂)	1	1	yes	characteristic
156	AGL(1,𝔽₇)	—	4	2	no	—
157	#126	—	7	2	no	—
158	#124	—	7	2	no	—
159	#123	—	7	2	no	—
160	C₂²⋊Q₈	—	7	1	no	—
161	#121	—	7	2	no	—
162	#121	AGL(1,𝔽₇)	1	1	yes	characteristic, complemented
163	#120	—	7	2	no	—
164	C₄×D₄	—	7	2	no	—
165	C₂×C₂²⋊C₄	—	7	1	no	—
166	C₂²×C₇	—	2	1	no	—
167	C₂²×C₇	—	2	1	no	—
168	C₂²×C₇	—	2	2	no	—
169	C₂²×C₇	—	2	2	no	—
170	C₂²×C₇	—	2	2	no	—
171	C₂²×C₇	C₃×#41	1	2	yes	—
172	C₂²×C₇	C₂⁴×C₃	1	1	yes	characteristic
173	C₂×D₇	—	2	2	no	—
174	C₂×D₇	—	2	2	no	—
175	C₂×D₇	—	2	2	no	—
176	C₂×D₇	—	4	4	no	—
177	C₄×C₇	—	2	2	no	—
178	C₄×C₇	—	2	1	no	—
179	C₄×C₇	—	2	1	no	—
180	Dic₇	—	2	2	no	—
181	Dic₇	—	2	2	no	—
182	Dic₇	—	2	2	no	—
183	C₂³×C₃	—	7	2	no	—
184	C₂³×C₃	—	7	1	no	—
185	C₂³×C₃	—	7	2	no	—
186	C₂³×C₃	—	14	2	no	—
187	C₂³×C₃	—	28	1	no	—
188	C₂³×C₃	—	28	1	no	—
189	C₃×Q₈	—	28	1	no	—
190	C₃×D₄	—	14	4	no	—
191	C₃×D₄	—	28	1	no	—
192	C₃×D₄	—	28	2	no	—
193	C₃×D₄	—	28	2	no	—
194	C₂×C₃×C₄	—	7	2	no	—
195	C₂×C₃×C₄	—	7	2	no	—
196	C₂×C₃×C₄	—	7	2	no	—
197	C₂×C₃×C₄	—	7	2	no	—
198	C₂×C₃×C₄	—	7	1	no	—
199	C₂×C₃×C₄	—	7	1	no	—
200	C₂×C₃×C₄	—	28	1	no	—
201	C₂×C₃×C₄	—	28	1	no	—
202	C₂×C₃×C₄	—	28	2	no	—
203	ΓL(1,𝔽₈)	#538	1	1	yes	characteristic, complemented, Hall
204	C₂⁴	—	7	1	no	—
205	C₂×Q₈	—	7	1	no	—
206	C₂×D₄	—	7	2	no	—
207	C₂×D₄	—	7	2	no	—
208	C₂×D₄	C₂×AGL(1,𝔽₇)	1	2	yes	—
209	C₂×D₄	C₂×AGL(1,𝔽₇)	1	1	yes	characteristic, complemented
210	C₂²×C₄	—	7	1	no	—
211	C₂²×C₄	—	7	2	no	—
212	C₂²×C₄	C₂×AGL(1,𝔽₇)	1	1	yes	characteristic
213	C₄⋊C₄	—	7	2	no	—
214	C₄⋊C₄	—	7	2	no	—
215	C₄⋊C₄	—	7	1	no	—
216	C₄⋊C₄	C₂×AGL(1,𝔽₇)	1	1	yes	characteristic, complemented
217	C₂²⋊C₄	—	7	2	no	—
218	C₂²⋊C₄	—	7	2	no	—
219	C₂²⋊C₄	—	7	2	no	—
220	C₂²⋊C₄	—	7	2	no	—
221	C₂²⋊C₄	—	7	2	no	—
222	C₂²⋊C₄	—	7	2	no	—
223	C₂²⋊C₄	C₂×AGL(1,𝔽₇)	1	2	yes	complemented
224	C₄²	—	7	2	no	—
225	C₂×C₇	—	2	2	no	—
226	C₂×C₇	—	4	2	no	—
227	C₂×C₇	C₂×C₃×#41	1	1	yes	characteristic
228	C₂×C₇	C₃×H(5,𝔽₂)	1	1	yes	characteristic
229	C₂×C₇	C₃×H(5,𝔽₂)	1	1	yes	characteristic
230	D₇	—	4	2	no	—
231	C₂²×C₃	—	7	2	no	—
232	C₂²×C₃	—	7	1	no	—
233	C₂²×C₃	—	14	1	no	—
234	C₂²×C₃	—	14	1	no	—
235	C₂²×C₃	—	14	2	no	—
236	C₂²×C₃	—	14	2	no	—
237	C₂²×C₃	—	14	2	no	—
238	C₂²×C₃	—	14	2	no	—
239	C₂²×C₃	—	14	2	no	—
240	C₂²×C₃	—	14	2	no	—
241	C₂²×C₃	—	28	4	no	—
242	C₃×C₄	—	14	1	no	—
243	C₃×C₄	—	14	1	no	—
244	C₃×C₄	—	14	2	no	—
245	C₃×C₄	—	14	2	no	—
246	C₃×C₄	—	14	2	no	—
247	C₃×C₄	—	14	2	no	—
248	C₂³	—	7	2	no	—
249	C₂³	—	14	2	no	—
250	C₂³	—	28	1	no	—
251	C₂³	—	28	1	no	—
252	C₂³	C₂²×AGL(1,𝔽₇)	1	2	yes	—
253	C₂³	C₂²×AGL(1,𝔽₇)	1	1	yes	characteristic
254	Q₈	—	28	1	no	—
255	D₄	—	2	4	no	—
256	D₄	—	4	1	no	—
257	D₄	—	28	2	no	—
258	D₄	—	28	2	no	—
259	C₂×C₄	—	4	1	no	—
260	C₂×C₄	—	7	2	no	—
261	C₂×C₄	—	7	2	no	—
262	C₂×C₄	—	7	2	no	—
263	C₂×C₄	—	28	1	no	—
264	C₂×C₄	—	28	2	no	—
265	C₂×C₄	C₂²×AGL(1,𝔽₇)	1	2	yes	—
266	C₂×C₄	C₂²×AGL(1,𝔽₇)	1	1	yes	characteristic
267	C₂×C₄	C₂²×AGL(1,𝔽₇)	1	1	yes	characteristic
268	C₇	C₃×#538	1	1	yes	characteristic, complemented, Sylow
269	C₂×C₃	—	7	1	no	—
270	C₂×C₃	—	7	1	no	—
271	C₂×C₃	—	7	1	no	—
272	C₂×C₃	—	14	2	no	—
273	C₂×C₃	—	28	2	no	—
274	C₂×C₃	—	28	2	no	—
275	C₂²	—	2	1	no	—
276	C₂²	—	2	1	no	—
277	C₂²	—	2	2	no	—
278	C₂²	—	2	2	no	—
279	C₂²	—	2	2	no	—
280	C₂²	—	14	2	no	—
281	C₂²	—	14	2	no	—
282	C₂²	—	14	2	no	—
283	C₂²	—	28	4	no	—
284	C₂²	#1159⋊C₃	1	2	yes	—
285	C₂²	C₂³×AGL(1,𝔽₇)	1	1	yes	characteristic
286	C₄	—	2	2	no	—
287	C₄	—	2	1	no	—
288	C₄	—	2	1	no	—
289	C₄	—	14	2	no	—
290	C₄	—	14	2	no	—
291	C₄	—	14	2	no	—
292	C₃	—	7	1	no	Sylow
293	C₂	—	2	2	no	—
294	C₂	—	4	2	no	—
295	C₂	—	28	2	no	—
296	C₂	C₂×#1159⋊C₃	1	1	yes	characteristic
297	C₂	#6548⋊C₃	1	1	yes	characteristic
298	C₂	#10621518	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂²	C₂³×AGL(1,𝔽₇)
commutator, socle, cosocle	C₂²×C₇	C₂⁴×C₃
Fitting	C₇×#121	C₂×C₃
hypercenter	#121	AGL(1,𝔽₇)
hypocenter	C₇	C₃×#538
LCS	G ▷ C₂²×C₇ ▷ C₇
UCS	C₁ ◁ C₂² ◁ #121

Complements

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

N	H ≅ G/N	type
1. ΓL(1,𝔽₈)×#121	C₂	semidirect
2. #6502⋊C₃	C₂	semidirect
3. #6500⋊C₃	C₂	semidirect
4. D₄×ΓL(1,𝔽₈)⋊C₄	C₂	semidirect
5. C₂×(C₂×AGL(1,𝔽₇))⋊C₄	C₂	semidirect
6. #6487⋊C₃	C₂	semidirect
7. #6461⋊C₃	C₂	semidirect
8. #6449⋊C₃	C₂	semidirect
9. #6448⋊C₃	C₂	semidirect
10. #90308	C₃	semidirect
13. C₂×D₄×ΓL(1,𝔽₈)	C₂²	semidirect
18. C₂²×ΓL(1,𝔽₈)⋊C₄	C₂²	semidirect
19. C₂×ΓL(1,𝔽₈)⋊Q₈	C₂²	semidirect
20. C₄⋊C₄×ΓL(1,𝔽₈)	C₂²	semidirect
21. C₂²⋊C₄×ΓL(1,𝔽₈)	C₂²	semidirect
23. (C₂²⋊Dic₇)⋊C₃	C₂²	semidirect
28. C₄⋊(ΓL(1,𝔽₈)⋊C₄)	C₂²	semidirect
29. C₄⋊(ΓL(1,𝔽₈)⋊C₄)	C₂²	semidirect
30. (ΓL(1,𝔽₈)⋊C₄)⋊C₄	C₂²	semidirect
31. C₄×ΓL(1,𝔽₈)⋊C₄	C₂²	semidirect
32. C₇×#121	C₂×C₃	semidirect
33. #6502	C₂×C₃	semidirect
34. #6500	C₂×C₃	semidirect
35. D₄×Dic₇	C₂×C₃	semidirect
36. C₂×(C₂×D₇)⋊C₄	C₂×C₃	semidirect
37. #6487	C₂×C₃	semidirect
38. #6461	C₂×C₃	semidirect
39. #6449	C₂×C₃	semidirect
40. #6448	C₂×C₃	semidirect
64. C₂×C₇×D₄	C₂²×C₃	semidirect
69. C₂²×Dic₇	C₂²×C₃	semidirect
70. C₂×Dic₁₄	C₂²×C₃	semidirect
71. C₇×C₄⋊C₄	C₂²×C₃	semidirect
72. C₇×C₂²⋊C₄	C₂²×C₃	semidirect
74. C₂²⋊Dic₇	C₂²×C₃	semidirect
79. C₄⋊Dic₇	C₂²×C₃	semidirect
80. C₄⋊Dic₇	C₂²×C₃	semidirect
81. Dic₇⋊C₄	C₂²×C₃	semidirect
82. C₄×Dic₇	C₂²×C₃	semidirect
162. #121	AGL(1,𝔽₇)	semidirect
203. ΓL(1,𝔽₈)	#538	semidirect
209. C₂×D₄	C₂×AGL(1,𝔽₇)	semidirect
216. C₄⋊C₄	C₂×AGL(1,𝔽₇)	semidirect
223. C₂²⋊C₄	C₂×AGL(1,𝔽₇)	semidirect
268. C₇	C₃×#538	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)×#121	2
#6502⋊C₃	2
#6500⋊C₃	2
D₄×ΓL(1,𝔽₈)⋊C₄	2
C₂×(C₂×AGL(1,𝔽₇))⋊C₄	2
#6487⋊C₃	2
#6461⋊C₃	2
#6449⋊C₃	2
#6448⋊C₃	2
#90308	3
C₃×#538	7

Subgroups — #90308⋊C3 #13057351

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — #90308⋊C₃ #13057351