Subgroups — #13243284

Order 1792.

618 conjugacy classes of subgroups.

The subgroup lattice has 276 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₇×#431	C₂	1	1	yes	maximal, characteristic
2	C₂×#89697	C₂	1	1	yes	maximal, characteristic
3	C₂×#89696	C₂	1	1	yes	maximal, characteristic
4	#11725460	C₂	1	4	yes	maximal, complemented
5	C₂²×C₄²×C₇	C₂²	1	1	yes	characteristic
6	C₂²×#1136	—	2	1	no	—
7	C₇×#431	—	2	2	no	—
8	C₇×#431	—	2	2	no	—
9	C₂×C₇×C₄⋊C₈	C₄	1	2	yes	—
10	C₂×C₇×C₂²⋊C₈	C₄	1	2	yes	—
11	C₂×C₄×C₇×C₈	C₂²	1	2	yes	—
12	C₂×C₂²⋊(C₇⋊C₈)	—	2	1	no	—
13	#89697	—	2	2	no	—
14	#89697	C₄	1	4	yes	—
15	#89696	—	2	2	no	—
16	#89696	C₄	1	4	yes	—
17	C₂×C₄⋊(C₇⋊C₈)	—	2	1	no	—
18	C₂×C₄⋊(C₇⋊C₈)	C₂²	1	2	yes	—
19	C₂×C₄⋊(C₇⋊C₈)	C₂²	1	2	yes	—
20	C₂×C₄×C₇⋊C₈	—	2	1	no	—
21	#7880	—	7	1	no	maximal, Sylow
22	C₂³×C₄×C₇	C₂×C₄	1	1	yes	characteristic
23	C₂³×C₄×C₇	D₄	1	1	yes	characteristic
24	C₂³×C₄×C₇	Q₈	1	1	yes	characteristic
25	C₂²×C₇×C₈	C₂×C₄	1	4	yes	—
26	C₂×C₄²×C₇	C₂×C₄	1	2	yes	—
27	C₂×C₄²×C₇	C₂×C₄	1	2	yes	—
28	C₂×C₄²×C₇	C₂×C₄	1	1	yes	characteristic
29	C₂×C₄²×C₇	D₄	1	1	yes	characteristic
30	C₂×C₄²×C₇	D₄	1	2	yes	—
31	C₂×C₄²×C₇	D₄	1	2	yes	—
32	C₂×C₄²×C₇	Q₈	1	1	yes	characteristic
33	C₂×C₄²×C₇	C₂³	1	1	yes	characteristic
34	C₂×#1136	—	2	4	no	—
35	C₂×#1136	—	4	4	no	—
36	C₂²×C₇⋊C₈	—	2	2	no	—
37	C₂²×C₇⋊C₈	—	2	2	no	—
38	C₇×C₄⋊C₈	—	2	4	no	—
39	C₇×C₂²⋊C₈	—	2	4	no	—
40	C₄×C₇×C₈	—	2	2	no	—
41	C₄×C₇×C₈	—	2	2	no	—
42	C₂²⋊(C₇⋊C₈)	—	2	4	no	—
43	C₄⋊(C₇⋊C₈)	—	2	4	no	—
44	C₄⋊(C₇⋊C₈)	—	2	2	no	—
45	C₄⋊(C₇⋊C₈)	—	2	2	no	—
46	C₄⋊(C₇⋊C₈)	C₂×C₄	1	4	yes	—
47	C₄⋊(C₇⋊C₈)	C₂×C₄	1	4	yes	—
48	C₄×C₇⋊C₈	—	2	4	no	—
49	C₂×#431	—	7	1	no	—
50	C₂×#431	D₇	1	1	yes	characteristic
51	C₂×#423	—	7	1	no	—
52	#1277	—	7	4	no	—
53	C₂⁴×C₇	C₄⋊C₄	1	1	yes	characteristic
54	C₂²×C₄×C₇	—	2	4	no	—
55	C₂²×C₄×C₇	—	2	4	no	—
56	C₂²×C₄×C₇	—	2	2	no	—
57	C₂²×C₄×C₇	—	2	2	no	—
58	C₂²×C₄×C₇	C₄²	1	2	yes	—
59	C₂²×C₄×C₇	C₂²⋊C₄	1	2	yes	—
60	C₂²×C₄×C₇	C₂²⋊C₄	1	2	yes	—
61	C₂²×C₄×C₇	C₂²⋊C₄	1	2	yes	—
62	C₂²×C₄×C₇	C₄⋊C₄	1	1	yes	characteristic
63	C₂²×C₄×C₇	C₄⋊C₄	1	1	yes	characteristic
64	C₂²×C₄×C₇	C₄⋊C₄	1	2	yes	—
65	C₂²×C₄×C₇	C₄⋊C₄	1	2	yes	—
66	C₂²×C₄×C₇	C₄⋊C₄	1	1	yes	characteristic
67	C₂²×C₄×C₇	C₂²×C₄	1	1	yes	characteristic
68	C₂²×C₄×C₇	C₂×D₄	1	1	yes	characteristic
69	C₂²×C₄×C₇	C₂×Q₈	1	1	yes	characteristic
70	C₂×C₇×C₈	—	2	4	no	—
71	C₂×C₇×C₈	—	2	8	no	—
72	C₄²×C₇	C₄²	1	2	yes	—
73	C₄²×C₇	C₂²⋊C₄	1	2	yes	—
74	C₄²×C₇	C₂²⋊C₄	1	2	yes	—
75	C₄²×C₇	C₂²⋊C₄	1	2	yes	—
76	C₄²×C₇	C₄⋊C₄	1	2	yes	—
77	C₄²×C₇	C₄⋊C₄	1	2	yes	—
78	C₄²×C₇	C₂²×C₄	1	1	yes	characteristic
79	C₄²×C₇	C₂²×C₄	1	1	yes	characteristic
80	C₄²×C₇	C₂×D₄	1	1	yes	characteristic
81	C₄²×C₇	C₂×D₄	1	1	yes	characteristic
82	#1136	—	4	8	no	—
83	C₂×C₇⋊C₈	—	2	4	no	—
84	C₂×C₇⋊C₈	—	2	4	no	—
85	C₂×C₇⋊C₈	—	4	2	no	—
86	C₂×C₇⋊C₈	—	4	4	no	—
87	C₂×C₇⋊C₈	—	4	2	no	—
88	C₂²×M₁₆	—	14	1	no	—
89	C₂²×C₄²	C₂×D₇	1	1	yes	characteristic
90	#431	—	2	2	no	—
91	#431	—	2	2	no	—
92	#431	—	7	4	no	—
93	#431	—	14	2	no	—
94	#423	—	7	4	no	—
95	#423	—	14	2	no	—
96	C₂×C₄⋊C₈	—	7	2	no	—
97	C₂×C₄⋊C₈	—	7	2	no	—
98	C₂×C₄⋊C₈	—	14	1	no	—
99	C₂×C₄⋊C₈	Dic₇	1	2	yes	—
100	C₂×C₂²⋊C₈	—	14	1	no	—
101	C₂×C₂²⋊C₈	Dic₇	1	2	yes	—
102	C₂×C₄×C₈	—	14	1	no	—
103	C₂×C₄×C₈	C₂×D₇	1	2	yes	—
104	C₂³×C₇	—	2	2	no	—
105	C₂³×C₇	—	2	2	no	—
106	C₂³×C₇	—	2	2	no	—
107	C₂³×C₇	#95	1	2	yes	—
108	C₂³×C₇	C₂×C₄⋊C₄	1	1	yes	characteristic
109	C₂×C₄×C₇	—	2	2	no	—
110	C₂×C₄×C₇	—	2	2	no	—
111	C₂×C₄×C₇	—	2	2	no	—
112	C₂×C₄×C₇	—	2	4	no	—
113	C₂×C₄×C₇	—	2	4	no	—
114	C₂×C₄×C₇	—	2	4	no	—
115	C₂×C₄×C₇	—	2	4	no	—
116	C₂×C₄×C₇	—	2	4	no	—
117	C₂×C₄×C₇	—	2	2	no	—
118	C₂×C₄×C₇	—	2	2	no	—
119	C₂×C₄×C₇	#95	1	2	yes	—
120	C₂×C₄×C₇	#95	1	2	yes	—
121	C₂×C₄×C₇	#95	1	2	yes	—
122	C₂×C₄×C₇	#108	1	4	yes	—
123	C₂×C₄×C₇	C₂×C₄²	1	1	yes	characteristic
124	C₂×C₄×C₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
125	C₂×C₄×C₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
126	C₂×C₄×C₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
127	C₂×C₄×C₇	C₂×C₄⋊C₄	1	1	yes	characteristic
128	C₂×C₄×C₇	C₂×C₄⋊C₄	1	1	yes	characteristic
129	C₂×C₄×C₇	#131	1	4	yes	—
130	C₂×C₄×C₇	AGL(1,ℤ/8)	1	2	yes	—
131	C₂×C₄×C₇	#137	1	2	yes	—
132	C₇×C₈	—	4	4	no	—
133	C₇⋊C₈	—	4	4	no	—
134	C₇⋊C₈	—	4	4	no	—
135	C₂³×C₄	Dic₁₄	1	1	yes	characteristic
136	C₂³×C₄	D₂₈	1	1	yes	characteristic
137	C₂³×C₄	C₂×Dic₇	1	1	yes	characteristic
138	C₂×M₁₆	—	14	4	no	—
139	C₂×M₁₆	—	28	4	no	—
140	C₂²×C₈	—	14	2	no	—
141	C₂²×C₈	—	14	2	no	—
142	C₂²×C₈	C₂×Dic₇	1	4	yes	—
143	C₂×C₄²	Dic₁₄	1	1	yes	characteristic
144	C₂×C₄²	C₄×D₇	1	2	yes	—
145	C₂×C₄²	C₄×D₇	1	2	yes	—
146	C₂×C₄²	D₂₈	1	1	yes	characteristic
147	C₂×C₄²	C₂×Dic₇	1	1	yes	characteristic
148	C₂×C₄²	C₂²⋊D₇	1	2	yes	—
149	C₂×C₄²	C₂²⋊D₇	1	2	yes	—
150	C₂×C₄²	C₂²×D₇	1	1	yes	characteristic
151	C₄⋊C₈	—	2	4	no	—
152	C₄⋊C₈	—	7	4	no	—
153	C₄⋊C₈	—	7	4	no	—
154	C₄⋊C₈	—	14	4	no	—
155	C₄⋊C₈	—	14	2	no	—
156	C₄⋊C₈	—	14	2	no	—
157	C₂²⋊C₈	—	2	4	no	—
158	C₂²⋊C₈	—	14	4	no	—
159	C₄×C₈	—	2	2	no	—
160	C₄×C₈	—	2	2	no	—
161	C₄×C₈	—	14	4	no	—
162	C₂²×C₇	—	2	2	no	—
163	C₂²×C₇	—	2	2	no	—
164	C₂²×C₇	—	2	2	no	—
165	C₂²×C₇	—	4	4	no	—
166	C₂²×C₇	C₂×#95	1	1	yes	characteristic
167	C₂²×C₇	#424	1	2	yes	—
168	C₂²×C₇	#428	1	2	yes	—
169	C₂²×C₇	C₂×#108	1	2	yes	—
170	C₄×C₇	—	2	2	no	—
171	C₄×C₇	—	2	2	no	—
172	C₄×C₇	—	2	1	no	—
173	C₄×C₇	—	2	1	no	—
174	C₄×C₇	#341	1	4	yes	—
175	C₄×C₇	#405	1	2	yes	—
176	C₄×C₇	#408	1	2	yes	—
177	C₄×C₇	#417	1	2	yes	—
178	C₄×C₇	#418	1	1	yes	characteristic
179	C₄×C₇	#419	1	1	yes	characteristic
180	C₂⁴	C₄⋊Dic₇	1	1	yes	characteristic
181	C₂²×C₄	—	2	2	no	—
182	C₂²×C₄	—	2	2	no	—
183	C₂²×C₄	—	2	4	no	—
184	C₂²×C₄	—	2	4	no	—
185	C₂²×C₄	C₄×Dic₇	1	2	yes	—
186	C₂²×C₄	Dic₇⋊C₄	1	2	yes	—
187	C₂²×C₄	Dic₇⋊C₄	1	2	yes	—
188	C₂²×C₄	C₄⋊Dic₇	1	1	yes	characteristic
189	C₂²×C₄	C₄⋊Dic₇	1	1	yes	characteristic
190	C₂²×C₄	C₄⋊Dic₇	1	1	yes	characteristic
191	C₂²×C₄	(C₂×D₇)⋊C₄	1	2	yes	—
192	C₂²×C₄	(C₂×D₇)⋊C₄	1	2	yes	—
193	C₂²×C₄	C₂²⋊Dic₇	1	2	yes	—
194	C₂²×C₄	C₂×Dic₁₄	1	1	yes	characteristic
195	C₂²×C₄	C₂×D₂₈	1	1	yes	characteristic
196	C₂²×C₄	C₂²×Dic₇	1	1	yes	characteristic
197	M₁₆	—	28	8	no	—
198	C₂×C₈	—	2	8	no	—
199	C₂×C₈	—	2	4	no	—
200	C₂×C₈	—	14	4	no	—
201	C₂×C₈	—	14	4	no	—
202	C₂×C₈	—	28	2	no	—
203	C₂×C₈	—	28	4	no	—
204	C₂×C₈	—	28	2	no	—
205	C₄²	C₄×Dic₇	1	2	yes	—
206	C₄²	Dic₇⋊C₄	1	2	yes	—
207	C₄²	Dic₇⋊C₄	1	2	yes	—
208	C₄²	(C₂×D₇)⋊C₄	1	2	yes	—
209	C₄²	(C₂×D₇)⋊C₄	1	2	yes	—
210	C₄²	C₂²⋊Dic₇	1	2	yes	—
211	C₄²	C₂×C₄×D₇	1	1	yes	characteristic
212	C₄²	C₂×C₄×D₇	1	1	yes	characteristic
213	C₄²	C₂×C₂²⋊D₇	1	1	yes	characteristic
214	C₄²	C₂×C₂²⋊D₇	1	1	yes	characteristic
215	C₂×C₇	—	4	2	no	—
216	C₂×C₇	#1574	1	4	yes	—
217	C₂×C₇	#1729	1	1	yes	characteristic
218	C₂×C₇	#1736	1	1	yes	characteristic
219	C₂×C₇	C₂×#341	1	1	yes	characteristic
220	C₂³	—	2	2	no	—
221	C₂³	—	2	2	no	—
222	C₂³	—	2	2	no	—
223	C₂³	C₇⋊#95	1	2	yes	—
224	C₂³	C₂×C₄⋊Dic₇	1	1	yes	characteristic
225	C₂×C₄	—	2	2	no	—
226	C₂×C₄	—	2	2	no	—
227	C₂×C₄	—	2	2	no	—
228	C₂×C₄	—	2	4	no	—
229	C₂×C₄	—	2	4	no	—
230	C₂×C₄	—	2	2	no	—
231	C₂×C₄	—	2	2	no	—
232	C₂×C₄	—	2	4	no	—
233	C₂×C₄	—	2	4	no	—
234	C₂×C₄	—	2	4	no	—
235	C₂×C₄	#6393	1	4	yes	—
236	C₂×C₄	C₇⋊#95	1	2	yes	—
237	C₂×C₄	C₇⋊#95	1	2	yes	—
238	C₂×C₄	C₇⋊#95	1	2	yes	—
239	C₂×C₄	#6471	1	2	yes	—
240	C₂×C₄	#6472	1	2	yes	—
241	C₂×C₄	C₂×C₄×Dic₇	1	1	yes	characteristic
242	C₂×C₄	C₂×Dic₇⋊C₄	1	1	yes	characteristic
243	C₂×C₄	C₂×Dic₇⋊C₄	1	1	yes	characteristic
244	C₂×C₄	C₂×(C₂×D₇)⋊C₄	1	1	yes	characteristic
245	C₂×C₄	C₂×(C₂×D₇)⋊C₄	1	1	yes	characteristic
246	C₂×C₄	#6511	1	4	yes	—
247	C₂×C₄	C₂×C₂²⋊Dic₇	1	1	yes	characteristic
248	C₈	—	4	4	no	—
249	C₈	—	28	4	no	—
250	C₈	—	28	4	no	—
251	C₇	#7880	1	1	yes	characteristic, complemented, Sylow
252	C₂²	—	2	2	no	—
253	C₂²	—	2	2	no	—
254	C₂²	—	2	2	no	—
255	C₂²	—	4	4	no	—
256	C₂²	#89771	1	2	yes	—
257	C₂²	C₂×#6393	1	2	yes	—
258	C₂²	#89893	1	2	yes	—
259	C₂²	C₂×C₇⋊#95	1	1	yes	characteristic
260	C₄	—	2	1	no	—
261	C₄	—	2	1	no	—
262	C₄	—	2	2	no	—
263	C₄	—	2	2	no	—
264	C₄	#89347	1	4	yes	—
265	C₄	#89769	1	2	yes	—
266	C₄	#89892	1	1	yes	characteristic
267	C₄	#89903	1	1	yes	characteristic
268	C₄	#89905	1	2	yes	—
269	C₄	#90006	1	2	yes	—
270	C₂	—	4	2	no	—
271	C₂	#11728507	1	4	yes	—
272	C₂	#11729333	1	1	yes	characteristic
273	C₂	C₂×#89347	1	1	yes	characteristic
274	C₂	#11729739	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄	C₂×D₂₈
commutator	C₂×C₄×C₇	C₂×C₄²
Frattini	C₂×C₄²	C₂²×D₇
Fitting	C₂×C₇×#431	C₂
socle	C₂³×C₇	C₂×C₄⋊C₄
prime radical	C₂⁴×C₇	C₄⋊C₄
cosocle	C₂×C₄²×C₇	C₂³
hypercenter	C₂×#431	D₇
hypocenter	C₇	#7880
LCS	G ▷ C₂×C₄×C₇ ▷ C₂×C₇ ▷ C₇
UCS	C₁ ◁ C₂²×C₄ ◁ C₂²×C₄² ◁ C₂×#431

Complements

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

N	H ≅ G/N	type
4. #11725460	C₂	semidirect
251. C₇	#7880	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₇×#431	2
C₂×#89697	2
C₂×#89696	2
#11725460	2
#7880	7