Subgroups — #13271930

Order 1792.

306 conjugacy classes of subgroups.

The subgroup lattice has 234 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₇×#1848	C₂	1	1	yes	maximal, characteristic
2	C₂×#89289	C₂	1	2	yes	maximal
3	C₄×#6391	C₂	1	1	yes	maximal, characteristic
4	#11727497	C₂	1	1	yes	maximal, characteristic
5	#11725536	C₂	1	2	yes	maximal
6	C₂×C₇×#108	—	2	1	no	—
7	C₂×C₇×#108	C₂²	1	2	yes	—
8	C₇×#423	—	2	1	no	—
9	C₂×C₄×C₇×C₈	C₂²	1	1	yes	characteristic
10	C₂×#6391	C₂²	1	1	yes	characteristic
11	C₂×#6391	C₂²	1	1	yes	characteristic
12	C₂×#6386	C₂²	1	2	yes	—
13	C₄×C₄⋊Dic₇	—	2	1	no	—
14	#89346	—	2	1	no	—
15	#89289	—	2	4	no	—
16	C₄⋊(C₇⋊C₁₆)	C₄	1	2	yes	complemented
17	C₄⋊(C₇⋊C₁₆)	C₄	1	2	yes	complemented
18	#13185	—	7	1	no	maximal, Sylow
19	C₂×C₇×M₁₆	—	2	2	no	—
20	C₂²×C₇×C₈	D₄	1	1	yes	characteristic
21	C₂²×C₇×C₈	C₂³	1	1	yes	characteristic
22	C₂×C₄²×C₇	D₄	1	1	yes	characteristic
23	C₂×C₄⋊Dic₇	—	2	1	no	—
24	C₂×C₄⋊Dic₇	—	2	1	no	—
25	C₂×C₄×Dic₇	—	4	1	no	—
26	C₇×#108	—	2	2	no	—
27	C₇×#108	—	2	2	no	—
28	C₇×#108	—	4	2	no	—
29	C₇×C₄⋊C₈	—	4	1	no	—
30	C₇×C₄⋊C₈	—	4	1	no	—
31	C₄×C₇×C₈	C₂×C₄	1	1	yes	characteristic
32	C₄×C₇×C₈	C₂×C₄	1	1	yes	characteristic
33	C₄×C₇×C₈	Q₈	1	1	yes	characteristic
34	C₄×C₇×C₈	Q₈	1	1	yes	characteristic
35	C₇⋊#95	—	4	1	no	—
36	#6391	—	2	1	no	—
37	#6391	—	2	1	no	—
38	#6391	D₄	1	2	yes	—
39	#6391	Q₈	1	2	yes	—
40	#6386	—	2	2	no	—
41	#6386	—	2	2	no	—
42	C₂×C₇⋊C₁₆	C₂×C₄	1	4	yes	—
43	C₂×#365	—	7	2	no	—
44	#2150	—	7	1	no	—
45	#1848	D₇	1	1	yes	characteristic
46	C₄×#106	—	7	1	no	—
47	#1383	—	7	2	no	—
48	C₂²×C₄×C₇	SD₁₆	1	1	yes	characteristic
49	C₂²×C₄×C₇	SD₁₆	1	1	yes	characteristic
50	C₂²×C₄×C₇	C₂×D₄	1	1	yes	characteristic
51	C₂²×Dic₇	—	4	1	no	—
52	C₂²×Dic₇	—	4	1	no	—
53	C₇×M₁₆	—	4	2	no	—
54	C₇×M₁₆	—	4	2	no	—
55	C₂×C₇×C₈	—	2	1	no	—
56	C₂×C₇×C₈	—	2	1	no	—
57	C₂×C₇×C₈	—	4	2	no	—
58	C₂×C₇×C₈	C₄⋊C₄	1	1	yes	characteristic
59	C₂×C₇×C₈	C₄⋊C₄	1	1	yes	characteristic
60	C₂×C₇×C₈	C₂²×C₄	1	1	yes	characteristic
61	C₂×C₇×C₈	C₂×D₄	1	1	yes	characteristic
62	C₂×C₇×C₈	C₂×Q₈	1	1	yes	characteristic
63	C₂×C₇×C₈	C₂×Q₈	1	1	yes	characteristic
64	C₂×C₇×C₈	#41	1	1	yes	characteristic
65	C₂×C₇×C₈	#41	1	1	yes	characteristic
66	C₄²×C₇	C₄⋊C₄	1	1	yes	characteristic
67	C₄²×C₇	C₄⋊C₄	1	1	yes	characteristic
68	C₄²×C₇	SD₁₆	1	1	yes	characteristic
69	C₄²×C₇	SD₁₆	1	1	yes	characteristic
70	C₄⋊Dic₇	—	2	2	no	—
71	C₄⋊Dic₇	—	2	2	no	—
72	C₄⋊Dic₇	—	4	1	no	—
73	C₄⋊Dic₇	—	4	1	no	—
74	C₄×Dic₇	—	8	2	no	—
75	C₇⋊C₁₆	—	2	4	no	—
76	C₂×M₃₂	—	7	2	no	—
77	C₂×#108	—	2	1	no	—
78	C₂×#108	C₂×D₇	1	2	yes	—
79	C₂×#106	—	7	1	no	—
80	C₂×#106	—	7	1	no	—
81	#423	—	2	1	no	—
82	C₂×C₄×C₈	C₂×D₇	1	1	yes	characteristic
83	C₄×C₄⋊C₄	—	14	1	no	—
84	#365	—	14	4	no	—
85	C₄⋊C₁₆	—	7	2	no	—
86	C₄⋊C₁₆	—	7	2	no	—
87	#340	—	14	1	no	—
88	C₂³×C₇	C₂×SD₁₆	1	1	yes	characteristic
89	C₂×C₄×C₇	—	2	2	no	—
90	C₂×C₄×C₇	—	4	1	no	—
91	C₂×C₄×C₇	#106	1	1	yes	characteristic
92	C₂×C₄×C₇	#106	1	1	yes	characteristic
93	C₂×C₄×C₇	#106	1	1	yes	characteristic
94	C₂×C₄×C₇	#106	1	1	yes	characteristic
95	C₂×C₄×C₇	C₂×C₄⋊C₄	1	1	yes	characteristic
96	C₂×C₄×C₇	#121	1	1	yes	characteristic
97	C₂×C₄×C₇	C₂²⋊Q₈	1	1	yes	characteristic
98	C₂×C₄×C₇	C₂×SD₁₆	1	1	yes	characteristic
99	C₂×C₄×C₇	AGL(1,ℤ/8)	1	1	yes	characteristic
100	C₂×C₄×C₇	#137	1	1	yes	characteristic
101	C₂×Dic₇	—	4	2	no	—
102	C₂×Dic₇	—	4	2	no	—
103	C₂×Dic₇	—	8	1	no	—
104	C₂×Dic₇	—	8	1	no	—
105	C₂×Dic₇	—	8	1	no	—
106	C₂×Dic₇	—	8	1	no	—
107	C₇×C₈	—	2	1	no	—
108	C₇×C₈	—	2	1	no	—
109	C₇×C₈	—	8	2	no	—
110	C₇×C₈	C₄×D₄	1	1	yes	characteristic
111	C₇×C₈	C₄×Q₈	1	1	yes	characteristic
112	C₇×C₈	#125	1	1	yes	characteristic
113	C₇×C₈	C₄⋊Q₈	1	1	yes	characteristic
114	C₂×M₁₆	—	2	2	no	—
115	C₂²×C₈	C₂²⋊D₇	1	1	yes	characteristic
116	C₂²×C₈	C₂²×D₇	1	1	yes	characteristic
117	C₂×C₄⋊C₄	—	14	1	no	—
118	C₂×C₄⋊C₄	—	14	1	no	—
119	C₂×C₄²	—	28	1	no	—
120	C₂×C₄²	C₂²⋊D₇	1	1	yes	characteristic
121	M₃₂	—	14	2	no	—
122	M₃₂	—	14	2	no	—
123	C₂×C₁₆	—	7	4	no	—
124	#108	—	2	2	no	—
125	#108	—	2	2	no	—
126	#108	—	4	2	no	—
127	#106	—	7	2	no	—
128	#106	—	7	2	no	—
129	#106	—	14	1	no	—
130	#106	—	14	1	no	—
131	C₄⋊C₈	—	4	1	no	—
132	C₄⋊C₈	—	4	1	no	—
133	C₄×C₈	Dic₁₄	1	1	yes	characteristic
134	C₄×C₈	Dic₁₄	1	1	yes	characteristic
135	C₄×C₈	C₄×D₇	1	1	yes	characteristic
136	C₄×C₈	C₄×D₇	1	1	yes	characteristic
137	#95	—	28	1	no	—
138	C₂²×C₇	—	2	1	no	—
139	C₂²×C₇	—	2	1	no	—
140	C₂²×C₇	C₂×#106	1	1	yes	characteristic
141	C₂²×C₇	#461	1	1	yes	characteristic
142	C₂²×C₇	#476	1	1	yes	characteristic
143	C₄×C₇	—	2	1	no	—
144	C₄×C₇	—	2	1	no	—
145	C₄×C₇	—	2	1	no	—
146	C₄×C₇	#365	1	2	yes	complemented
147	C₄×C₇	#389	1	1	yes	characteristic
148	C₄×C₇	#428	1	1	yes	characteristic
149	C₄×C₇	#460	1	1	yes	characteristic
150	C₄×C₇	#475	1	1	yes	characteristic
151	Dic₇	—	8	2	no	—
152	Dic₇	—	8	2	no	—
153	C₂²×C₄	—	28	1	no	—
154	C₂²×C₄	—	28	1	no	—
155	C₂²×C₄	#1143	1	1	yes	characteristic
156	C₂²×C₄	#1143	1	1	yes	characteristic
157	C₂²×C₄	C₂×C₂²⋊D₇	1	1	yes	characteristic
158	M₁₆	—	4	2	no	—
159	M₁₆	—	4	2	no	—
160	C₂×C₈	—	2	1	no	—
161	C₂×C₈	—	2	1	no	—
162	C₂×C₈	—	4	2	no	—
163	C₂×C₈	Dic₇⋊C₄	1	1	yes	characteristic
164	C₂×C₈	Dic₇⋊C₄	1	1	yes	characteristic
165	C₂×C₈	C₂×Dic₁₄	1	1	yes	characteristic
166	C₂×C₈	C₂×C₄×D₇	1	1	yes	characteristic
167	C₂×C₈	D₄×D₇	1	1	yes	characteristic
168	C₂×C₈	#1159	1	1	yes	characteristic
169	C₂×C₈	Q₈×D₇	1	1	yes	characteristic
170	C₂×C₈	#1161	1	1	yes	characteristic
171	C₄⋊C₄	—	14	2	no	—
172	C₄⋊C₄	—	14	2	no	—
173	C₄⋊C₄	—	28	1	no	—
174	C₄⋊C₄	—	28	1	no	—
175	C₄²	—	56	2	no	—
176	C₄²	Dic₇⋊C₄	1	1	yes	characteristic
177	C₄²	Dic₇⋊C₄	1	1	yes	characteristic
178	C₄²	#1142	1	1	yes	characteristic
179	C₄²	#1142	1	1	yes	characteristic
180	C₁₆	—	14	4	no	—
181	C₂×C₇	—	2	1	no	—
182	C₂×C₇	—	2	1	no	—
183	C₂×C₇	#1948	1	1	yes	characteristic
184	C₂×C₇	C₂×#365	1	1	yes	characteristic
185	C₂×C₇	#2155	1	1	yes	characteristic
186	C₂³	C₂×#1143	1	1	yes	characteristic
187	C₂×C₄	—	2	2	no	—
188	C₂×C₄	—	4	1	no	—
189	C₂×C₄	—	28	2	no	—
190	C₂×C₄	—	28	2	no	—
191	C₂×C₄	—	56	1	no	—
192	C₂×C₄	—	56	1	no	—
193	C₂×C₄	—	56	1	no	—
194	C₂×C₄	—	56	1	no	—
195	C₂×C₄	#6382	1	1	yes	characteristic
196	C₂×C₄	#6382	1	1	yes	characteristic
197	C₂×C₄	#6382	1	1	yes	characteristic
198	C₂×C₄	#6382	1	1	yes	characteristic
199	C₂×C₄	C₂×Dic₇⋊C₄	1	1	yes	characteristic
200	C₂×C₄	#6495	1	1	yes	characteristic
201	C₂×C₄	C₂×#1142	1	1	yes	characteristic
202	C₂×C₄	#6501	1	1	yes	characteristic
203	C₂×C₄	#6505	1	1	yes	characteristic
204	C₂×C₄	#6509	1	1	yes	characteristic
205	C₈	—	2	1	no	—
206	C₈	—	2	1	no	—
207	C₈	—	8	2	no	—
208	C₈	#6450	1	1	yes	characteristic
209	C₈	#6451	1	1	yes	characteristic
210	C₈	#6453	1	1	yes	characteristic
211	C₈	#6456	1	1	yes	characteristic
212	C₇	#13185	1	1	yes	characteristic, complemented, Sylow
213	C₂²	—	2	1	no	—
214	C₂²	—	2	1	no	—
215	C₂²	C₂×#6382	1	1	yes	characteristic
216	C₂²	#89855	1	1	yes	characteristic
217	C₂²	#89856	1	1	yes	characteristic
218	C₄	—	2	1	no	—
219	C₄	—	2	1	no	—
220	C₄	—	2	1	no	—
221	C₄	—	56	2	no	—
222	C₄	—	56	2	no	—
223	C₄	#89289	1	2	yes	complemented
224	C₄	#89736	1	1	yes	characteristic
225	C₄	#89745	1	1	yes	characteristic
226	C₄	#89847	1	1	yes	characteristic
227	C₄	#89866	1	1	yes	characteristic
228	C₂	—	2	1	no	—
229	C₂	—	2	1	no	—
230	C₂	#11727322	1	1	yes	characteristic
231	C₂	C₂×#89289	1	1	yes	characteristic
232	C₂	#11728439	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×#6382
commutator	C₂×C₇×C₈	C₂²×C₄
Frattini	C₂²×C₈	C₂²×D₇
Fitting	C₇×#1848	C₂
socle	C₂²×C₇	C₂×#106
prime radical	C₂³×C₇	C₂×SD₁₆
cosocle	C₂²×C₇×C₈	C₂³
hypercenter	#1848	D₇
hypocenter	C₇	#13185
LCS	G ▷ C₂×C₇×C₈ ▷ C₄×C₇ ▷ C₂×C₇ ▷ C₇
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ #1848

Complements

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

N	H ≅ G/N	type
16. C₄⋊(C₇⋊C₁₆)	C₄	semidirect
17. C₄⋊(C₇⋊C₁₆)	C₄	semidirect
146. C₄×C₇	#365	semidirect
212. C₇	#13185	semidirect
223. C₄	#89289	semidirect

Maximal subgroups

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

subgroup	index
C₇×#1848	2
C₂×#89289	2
C₄×#6391	2
#11727497	2
#11725536	2
#13185	7