Subgroups — #13252425

Order 1792.

382 conjugacy classes of subgroups.

The subgroup lattice has 230 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₇×#1844	C₂	1	1	yes	maximal, characteristic
2	C₈×C₄⋊Dic₇	C₂	1	1	yes	maximal, characteristic
3	C₂×#89252	C₂	1	1	yes	maximal, characteristic
4	#11727321	C₂	1	2	yes	maximal
5	#11725440	C₂	1	2	yes	maximal
6	C₂×C₇×#106	—	2	1	no	—
7	C₂×C₇×#106	—	2	1	no	—
8	C₂×C₄×C₇×C₈	C₂²	1	1	yes	characteristic
9	C₇×#384	C₂²	1	2	yes	—
10	C₂×C₈×Dic₇	—	2	1	no	—
11	C₂×#6381	—	2	2	no	—
12	C₄×C₄⋊Dic₇	C₂²	1	1	yes	characteristic
13	C₂×C₄⋊(C₇⋊C₈)	C₂²	1	2	yes	—
14	C₂×C₄⋊(C₇⋊C₈)	C₂²	1	1	yes	characteristic
15	#89345	—	2	1	no	—
16	#89324	—	2	2	no	—
17	#89252	C₄	1	4	yes	complemented
18	#9611	—	7	1	no	maximal, Sylow
19	C₂²×C₇×C₈	D₄	1	1	yes	characteristic
20	C₂²×C₇×C₈	D₄	1	1	yes	characteristic
21	C₂×C₇×C₄⋊C₄	—	2	2	no	—
22	C₂×C₇×C₄⋊C₄	—	2	2	no	—
23	C₂×C₇×C₄⋊C₄	—	2	2	no	—
24	C₂×C₄²×C₇	C₂³	1	1	yes	characteristic
25	C₂×C₄⋊Dic₇	D₄	1	2	yes	—
26	C₂×C₄×Dic₇	—	2	1	no	—
27	C₂²×C₇⋊C₈	—	2	1	no	—
28	C₂²×C₇⋊C₈	—	2	2	no	—
29	C₇×#106	—	4	2	no	—
30	C₇×#106	—	4	2	no	—
31	C₄×C₇×C₈	C₂×C₄	1	2	yes	—
32	C₄×C₇×C₈	Q₈	1	2	yes	—
33	C₇⋊#95	—	2	1	no	—
34	C₈×Dic₇	—	4	2	no	—
35	#6381	—	4	4	no	—
36	C₄⋊(C₇⋊C₈)	—	2	2	no	—
37	C₄⋊(C₇⋊C₈)	—	2	1	no	—
38	C₄⋊(C₇⋊C₈)	C₂×C₄	1	4	yes	—
39	C₄⋊(C₇⋊C₈)	Q₈	1	2	yes	—
40	#1947	—	7	2	no	—
41	#1844	D₇	1	1	yes	characteristic
42	C₈×C₄⋊C₄	—	7	1	no	—
43	C₂×#334	—	7	1	no	—
44	#1296	—	7	2	no	—
45	C₂²×C₄×C₇	—	4	2	no	—
46	C₂²×C₄×C₇	C₂×D₄	1	1	yes	characteristic
47	C₂²×C₄×C₇	C₂×D₄	1	1	yes	characteristic
48	C₂²×C₄×C₇	#41	1	1	yes	characteristic
49	C₂²×Dic₇	—	2	2	no	—
50	C₂×C₇×C₈	—	2	1	no	—
51	C₂×C₇×C₈	—	2	1	no	—
52	C₂×C₇×C₈	C₄⋊C₄	1	2	yes	—
53	C₂×C₇×C₈	C₄⋊C₄	1	2	yes	—
54	C₂×C₇×C₈	D₈	1	2	yes	—
55	C₂×C₇×C₈	Q₁₆	1	2	yes	—
56	C₇×C₄⋊C₄	—	4	2	no	—
57	C₇×C₄⋊C₄	—	4	2	no	—
58	C₇×C₄⋊C₄	—	4	2	no	—
59	C₇×C₄⋊C₄	—	4	2	no	—
60	C₇×C₄⋊C₄	—	8	2	no	—
61	C₄²×C₇	C₂²×C₄	1	1	yes	characteristic
62	C₄²×C₇	C₂×Q₈	1	1	yes	characteristic
63	C₄²×C₇	C₂×Q₈	1	1	yes	characteristic
64	C₄²×C₇	#41	1	1	yes	characteristic
65	C₄⋊Dic₇	—	2	2	no	—
66	C₄⋊Dic₇	SD₁₆	1	4	yes	—
67	C₄×Dic₇	—	4	1	no	—
68	C₄×Dic₇	—	4	1	no	—
69	C₂×C₇⋊C₈	—	2	4	no	—
70	C₂×C₇⋊C₈	—	2	2	no	—
71	C₂×C₇⋊C₈	—	4	1	no	—
72	C₂×C₇⋊C₈	—	4	2	no	—
73	C₂×C₇⋊C₈	—	4	2	no	—
74	C₂×C₇⋊C₈	—	4	1	no	—
75	C₂×#107	—	14	2	no	—
76	C₂×#106	—	2	1	no	—
77	C₂×#106	—	2	1	no	—
78	C₂×C₄⋊C₈	—	7	1	no	—
79	C₂×C₄⋊C₈	—	7	2	no	—
80	C₂×C₄×C₈	—	14	1	no	—
81	C₂×C₄×C₈	C₂×D₇	1	1	yes	characteristic
82	#384	C₂×D₇	1	2	yes	—
83	C₄×C₄⋊C₄	—	7	1	no	—
84	#340	—	14	2	no	—
85	#336	—	14	1	no	—
86	#334	—	7	4	no	—
87	C₂³×C₇	#121	1	1	yes	characteristic
88	C₂×C₄×C₇	—	2	2	no	—
89	C₂×C₄×C₇	—	8	2	no	—
90	C₂×C₄×C₇	—	8	2	no	—
91	C₂×C₄×C₇	—	8	2	no	—
92	C₂×C₄×C₇	C₂×C₄⋊C₄	1	1	yes	characteristic
93	C₂×C₄×C₇	C₄×D₄	1	1	yes	characteristic
94	C₂×C₄×C₇	C₄×Q₈	1	1	yes	characteristic
95	C₂×C₄×C₇	C₂²⋊Q₈	1	1	yes	characteristic
96	C₂×C₄×C₇	#125	1	1	yes	characteristic
97	C₂×C₄×C₇	C₄⋊Q₈	1	1	yes	characteristic
98	C₂×C₄×C₇	C₂×D₈	1	1	yes	characteristic
99	C₂×C₄×C₇	C₂×SD₁₆	1	2	yes	—
100	C₂×C₄×C₇	C₂×Q₁₆	1	1	yes	characteristic
101	C₂×C₄×C₇	#135	1	2	yes	—
102	C₂×C₄×C₇	AGL(1,ℤ/8)	1	1	yes	characteristic
103	C₂×C₄×C₇	#137	1	1	yes	characteristic
104	C₂×Dic₇	—	2	4	no	—
105	C₂×Dic₇	—	4	2	no	—
106	C₂×Dic₇	—	4	2	no	—
107	C₇×C₈	—	2	2	no	—
108	C₇×C₈	#107	1	4	yes	complemented
109	C₇⋊C₈	—	4	4	no	—
110	C₇⋊C₈	—	4	2	no	—
111	C₂²×C₈	—	14	2	no	—
112	C₂²×C₈	—	14	1	no	—
113	C₂²×C₈	C₂²⋊D₇	1	1	yes	characteristic
114	C₂²×C₈	C₂²⋊D₇	1	1	yes	characteristic
115	C₂×C₄⋊C₄	—	2	2	no	—
116	C₂×C₄⋊C₄	—	2	2	no	—
117	C₂×C₄⋊C₄	—	2	2	no	—
118	C₂×C₄⋊C₄	—	7	2	no	—
119	C₂×C₄²	—	14	1	no	—
120	C₂×C₄²	C₂²×D₇	1	1	yes	characteristic
121	#107	—	28	4	no	—
122	#106	—	4	2	no	—
123	#106	—	4	2	no	—
124	C₄⋊C₈	—	7	4	no	—
125	C₄⋊C₈	—	7	2	no	—
126	C₄⋊C₈	—	14	2	no	—
127	C₄⋊C₈	—	14	1	no	—
128	C₄×C₈	—	28	2	no	—
129	C₄×C₈	Dic₁₄	1	2	yes	—
130	C₄×C₈	C₄×D₇	1	2	yes	—
131	#95	—	14	1	no	—
132	C₂²×C₇	#389	1	1	yes	characteristic
133	C₂²×C₇	#459	1	1	yes	characteristic
134	C₂²×C₇	#462	1	1	yes	characteristic
135	C₂²×C₇	#465	1	2	yes	—
136	C₂²×C₇	#471	1	2	yes	—
137	C₄×C₇	—	2	1	no	—
138	C₄×C₇	—	2	1	no	—
139	C₄×C₇	—	16	2	no	—
140	C₄×C₇	C₂×#107	1	1	yes	characteristic
141	C₄×C₇	#428	1	1	yes	characteristic
142	C₄×C₇	C₄×SD₁₆	1	2	yes	—
143	C₄×C₇	#474	1	1	yes	characteristic
144	C₄×C₇	#477	1	1	yes	characteristic
145	C₄×C₇	#498	1	1	yes	characteristic
146	C₄×C₇	#499	1	1	yes	characteristic
147	Dic₇	—	4	4	no	—
148	C₂²×C₄	—	4	2	no	—
149	C₂²×C₄	—	14	2	no	—
150	C₂²×C₄	D₄×D₇	1	1	yes	characteristic
151	C₂²×C₄	#1159	1	1	yes	characteristic
152	C₂²×C₄	C₂×C₂²⋊D₇	1	1	yes	characteristic
153	C₂×C₈	—	2	1	no	—
154	C₂×C₈	—	2	1	no	—
155	C₂×C₈	—	14	4	no	—
156	C₂×C₈	—	14	2	no	—
157	C₂×C₈	—	28	2	no	—
158	C₂×C₈	—	28	2	no	—
159	C₂×C₈	—	28	1	no	—
160	C₂×C₈	—	28	1	no	—
161	C₂×C₈	Dic₇⋊C₄	1	2	yes	—
162	C₂×C₈	Dic₇⋊C₄	1	2	yes	—
163	C₂×C₈	#1141	1	2	yes	—
164	C₂×C₈	#1144	1	2	yes	—
165	C₄⋊C₄	—	4	2	no	—
166	C₄⋊C₄	—	4	2	no	—
167	C₄⋊C₄	—	4	2	no	—
168	C₄⋊C₄	—	4	2	no	—
169	C₄⋊C₄	—	7	4	no	—
170	C₄⋊C₄	—	8	2	no	—
171	C₄⋊C₄	—	14	2	no	—
172	C₄²	—	28	1	no	—
173	C₄²	—	28	1	no	—
174	C₄²	C₂×Dic₁₄	1	1	yes	characteristic
175	C₄²	C₂×C₄×D₇	1	1	yes	characteristic
176	C₄²	Q₈×D₇	1	1	yes	characteristic
177	C₄²	#1161	1	1	yes	characteristic
178	C₂×C₇	#1665	1	2	yes	—
179	C₂×C₇	#1672	1	2	yes	—
180	C₂×C₇	#1941	1	1	yes	characteristic
181	C₂×C₇	#1947	1	1	yes	characteristic
182	C₂×C₇	#1951	1	1	yes	characteristic
183	C₂³	#6501	1	1	yes	characteristic
184	C₂×C₄	—	2	2	no	—
185	C₂×C₄	—	8	2	no	—
186	C₂×C₄	—	8	2	no	—
187	C₂×C₄	—	8	2	no	—
188	C₂×C₄	—	14	4	no	—
189	C₂×C₄	—	28	2	no	—
190	C₂×C₄	—	28	2	no	—
191	C₂×C₄	#6450	1	1	yes	characteristic
192	C₂×C₄	#6451	1	1	yes	characteristic
193	C₂×C₄	#6453	1	1	yes	characteristic
194	C₂×C₄	#6456	1	1	yes	characteristic
195	C₂×C₄	D₇×SD₁₆	1	2	yes	—
196	C₂×C₄	#6479	1	2	yes	—
197	C₂×C₄	C₂×Dic₇⋊C₄	1	1	yes	characteristic
198	C₂×C₄	C₂×#1141	1	1	yes	characteristic
199	C₂×C₄	#6495	1	1	yes	characteristic
200	C₂×C₄	#6505	1	1	yes	characteristic
201	C₂×C₄	C₂×#1144	1	1	yes	characteristic
202	C₂×C₄	#6509	1	1	yes	characteristic
203	C₈	—	2	2	no	—
204	C₈	—	28	2	no	—
205	C₈	—	28	4	no	—
206	C₈	#6381	1	4	yes	complemented
207	C₇	#9611	1	1	yes	characteristic, complemented, Sylow
208	C₂²	#89745	1	1	yes	characteristic
209	C₂²	#89844	1	1	yes	characteristic
210	C₂²	#89862	1	1	yes	characteristic
211	C₂²	#89943	1	2	yes	—
212	C₂²	#89949	1	2	yes	—
213	C₄	—	2	1	no	—
214	C₄	—	2	1	no	—
215	C₄	—	16	2	no	—
216	C₄	—	28	4	no	—
217	C₄	#89624	1	2	yes	—
218	C₄	#89627	1	1	yes	characteristic
219	C₄	#89630	1	1	yes	characteristic
220	C₄	C₂×#6381	1	1	yes	characteristic
221	C₄	#89736	1	1	yes	characteristic
222	C₄	#89859	1	1	yes	characteristic
223	C₄	#89863	1	1	yes	characteristic
224	C₂	#11727321	1	1	yes	characteristic
225	C₂	#11728313	1	1	yes	characteristic
226	C₂	#11728326	1	1	yes	characteristic
227	C₂	#11729021	1	2	yes	—
228	C₂	#11729102	1	2	yes	—

	subgroup	G/H
center	C₂³	#6501
commutator	C₄²×C₇	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₇
Fitting	C₇×#1844	C₂
socle, prime radical	C₂³×C₇	#121
cosocle	C₂×C₄²×C₇	C₂³
hypercenter	#1844	D₇
hypocenter	C₇	#9611
LCS	G ▷ C₄²×C₇ ▷ C₂²×C₇ ▷ C₇
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ #1844

Complements

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

N	H ≅ G/N	type
17. #89252	C₄	semidirect
108. C₇×C₈	#107	semidirect
206. C₈	#6381	semidirect
207. C₇	#9611	semidirect

Maximal subgroups

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

subgroup	index
C₇×#1844	2
C₈×C₄⋊Dic₇	2
C₂×#89252	2
#11727321	2
#11725440	2
#9611	7