Subgroups — #13265088

Order 1792.

334 conjugacy classes of subgroups.

The subgroup lattice has 208 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#89454	C₂	1	1	yes	maximal, characteristic
2	C₇×#1564	C₂	1	1	yes	maximal, characteristic
3	#11727497	C₂	1	1	yes	maximal, characteristic
4	#11725281	C₂	1	4	yes	maximal, complemented
5	C₂×#6432	—	2	1	no	—
6	C₇×#431	—	2	1	no	—
7	C₂×C₄×C₇×C₈	C₂²	1	1	yes	characteristic
8	C₂×#6392	—	2	1	no	—
9	C₂×#6392	C₄	1	2	yes	—
10	C₂×#6386	—	2	1	no	—
11	#89454	—	2	1	no	—
12	#89454	—	2	1	no	—
13	#89454	C₄	1	2	yes	—
14	#89454	C₂²	1	2	yes	—
15	C₇×#335	C₂²	1	2	yes	—
16	C₇×#334	—	2	1	no	—
17	C₄⋊(C₇⋊C₁₆)	—	2	1	no	—
18	C₄⋊(C₇⋊C₁₆)	C₂²	1	2	yes	—
19	#11940	—	7	1	no	maximal, Sylow
20	C₂²×Dic₁₄	—	4	1	no	—
21	C₂²×C₇×C₈	C₂×C₄	1	1	yes	characteristic
22	C₂²×C₇×C₈	D₄	1	1	yes	characteristic
23	C₂×C₄²×C₇	D₄	1	1	yes	characteristic
24	C₂×C₄⋊Dic₇	—	2	2	no	—
25	C₂×C₄⋊Dic₇	—	4	1	no	—
26	#6432	—	2	2	no	—
27	#6432	—	2	2	no	—
28	#6432	—	4	1	no	—
29	#6432	—	4	1	no	—
30	C₇×C₄⋊C₈	—	2	2	no	—
31	C₇×C₂²⋊C₈	—	4	1	no	—
32	C₄×C₇×C₈	—	4	1	no	—
33	C₄×C₇×C₈	C₂×C₄	1	1	yes	characteristic
34	C₄×C₇×C₈	D₄	1	1	yes	characteristic
35	C₄×C₇×C₈	D₄	1	1	yes	characteristic
36	C₄×C₇×C₈	C₂³	1	1	yes	characteristic
37	#6392	—	2	2	no	—
38	#6392	—	2	2	no	—
39	#6392	—	2	2	no	—
40	#6392	C₂×C₄	1	4	yes	—
41	#6386	—	4	2	no	—
42	C₂×C₇⋊C₁₆	—	2	2	no	—
43	C₂×#500	—	7	1	no	—
44	#2150	—	7	1	no	—
45	#1564	D₇	1	1	yes	characteristic
46	#1364	—	7	4	no	—
47	C₂²×C₄×C₇	C₂²⋊C₄	1	1	yes	characteristic
48	C₂²×C₄×C₇	D₈	1	1	yes	characteristic
49	C₂²×C₄×C₇	SD₁₆	1	1	yes	characteristic
50	C₂²×Dic₇	—	4	2	no	—
51	C₂×Dic₁₄	—	4	2	no	—
52	C₂×Dic₁₄	—	4	2	no	—
53	C₂×Dic₁₄	—	8	4	no	—
54	C₂×C₇×C₈	—	2	2	no	—
55	C₂×C₇×C₈	—	4	2	no	—
56	C₂×C₇×C₈	C₂²⋊C₄	1	1	yes	characteristic
57	C₂×C₇×C₈	C₂²⋊C₄	1	1	yes	characteristic
58	C₂×C₇×C₈	SD₁₆	1	2	yes	—
59	C₂×C₇×C₈	Q₁₆	1	2	yes	—
60	C₂×C₇×C₈	C₂²×C₄	1	1	yes	characteristic
61	C₂×C₇×C₈	C₂×D₄	1	1	yes	characteristic
62	C₄²×C₇	C₂²⋊C₄	1	1	yes	characteristic
63	C₄²×C₇	D₈	1	1	yes	characteristic
64	C₄²×C₇	SD₁₆	1	1	yes	characteristic
65	C₄²×C₇	C₂×D₄	1	1	yes	characteristic
66	C₄⋊Dic₇	—	2	4	no	—
67	C₄⋊Dic₇	—	4	2	no	—
68	C₄⋊Dic₇	—	4	2	no	—
69	C₄⋊Dic₇	—	4	2	no	—
70	C₇⋊C₁₆	—	4	2	no	—
71	C₂×C₄⋊Q₈	—	14	1	no	—
72	C₂×M₃₂	—	14	1	no	—
73	#500	—	7	2	no	—
74	#500	—	7	2	no	—
75	#500	—	14	1	no	—
76	#500	—	14	1	no	—
77	#431	—	2	1	no	—
78	C₂×#107	—	7	2	no	—
79	C₂×#107	—	14	1	no	—
80	C₂×C₄×C₈	C₂×D₇	1	1	yes	characteristic
81	C₄⋊C₁₆	—	7	2	no	—
82	C₄⋊C₁₆	—	14	1	no	—
83	#335	C₂×D₇	1	2	yes	—
84	#334	—	2	1	no	—
85	C₂³×C₇	#102	1	1	yes	characteristic
86	C₂×C₄×C₇	—	2	2	no	—
87	C₂×C₄×C₇	—	2	2	no	—
88	C₂×C₄×C₇	#101	1	2	yes	—
89	C₂×C₄×C₇	#102	1	1	yes	characteristic
90	C₂×C₄×C₇	#102	1	1	yes	characteristic
91	C₂×C₄×C₇	#102	1	1	yes	characteristic
92	C₂×C₄×C₇	#103	1	2	yes	—
93	C₂×C₄×C₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
94	C₂×C₄×C₇	C₂×D₈	1	1	yes	characteristic
95	C₂×C₄×C₇	C₂×SD₁₆	1	1	yes	characteristic
96	C₂×Dic₇	—	4	4	no	—
97	C₂×Dic₇	—	8	4	no	—
98	Dic₁₄	—	8	4	no	—
99	Dic₁₄	—	8	4	no	—
100	C₇×C₈	—	2	1	no	—
101	C₇×C₈	—	2	1	no	—
102	C₇×C₈	—	8	2	no	—
103	C₇×C₈	#103	1	2	yes	—
104	C₇×C₈	C₂×SD₁₆	1	1	yes	characteristic
105	C₇×C₈	C₂×Q₁₆	1	1	yes	characteristic
106	C₂²×Q₈	—	28	1	no	—
107	C₂²×C₈	C₄×D₇	1	1	yes	characteristic
108	C₂²×C₈	D₂₈	1	1	yes	characteristic
109	C₄⋊Q₈	—	14	2	no	—
110	C₄⋊Q₈	—	14	2	no	—
111	C₄⋊Q₈	—	28	1	no	—
112	C₄⋊Q₈	—	28	1	no	—
113	C₂×C₄⋊C₄	—	14	2	no	—
114	C₂×C₄⋊C₄	—	28	1	no	—
115	C₂×C₄²	C₂²⋊D₇	1	1	yes	characteristic
116	M₃₂	—	28	2	no	—
117	C₂×C₁₆	—	14	2	no	—
118	#107	—	7	4	no	—
119	#107	—	14	2	no	—
120	#107	—	14	2	no	—
121	#107	—	14	2	no	—
122	C₄⋊C₈	—	2	2	no	—
123	C₂²⋊C₈	—	4	1	no	—
124	C₄×C₈	—	4	1	no	—
125	C₄×C₈	C₄×D₇	1	1	yes	characteristic
126	C₄×C₈	D₂₈	1	1	yes	characteristic
127	C₄×C₈	C₂²⋊D₇	1	1	yes	characteristic
128	C₄×C₈	C₂²×D₇	1	1	yes	characteristic
129	C₂²×C₇	—	2	1	no	—
130	C₂²×C₇	—	2	1	no	—
131	C₂²×C₇	#332	1	2	yes	—
132	C₂²×C₇	C₂×#102	1	1	yes	characteristic
133	C₄×C₇	—	2	1	no	—
134	C₄×C₇	—	2	1	no	—
135	C₄×C₇	#332	1	2	yes	—
136	C₄×C₇	#362	1	2	yes	—
137	C₄×C₇	C₂×#101	1	1	yes	characteristic
138	C₄×C₇	C₂×#103	1	1	yes	characteristic
139	C₄×C₇	#509	1	2	yes	—
140	Dic₇	—	8	4	no	—
141	C₂×Q₈	—	28	2	no	—
142	C₂×Q₈	—	28	2	no	—
143	C₂×Q₈	—	56	4	no	—
144	C₂²×C₄	—	28	2	no	—
145	C₂²×C₄	(C₂×D₇)⋊C₄	1	1	yes	characteristic
146	C₂²×C₄	#1141	1	1	yes	characteristic
147	C₂²×C₄	#1143	1	1	yes	characteristic
148	C₂×C₈	—	2	2	no	—
149	C₂×C₈	—	4	2	no	—
150	C₂×C₈	#1132	1	2	yes	—
151	C₂×C₈	Dic₂₈	1	2	yes	—
152	C₂×C₈	(C₂×D₇)⋊C₄	1	1	yes	characteristic
153	C₂×C₈	(C₂×D₇)⋊C₄	1	1	yes	characteristic
154	C₂×C₈	C₂×C₄×D₇	1	1	yes	characteristic
155	C₂×C₈	C₂×D₂₈	1	1	yes	characteristic
156	C₄⋊C₄	—	14	4	no	—
157	C₄⋊C₄	—	28	2	no	—
158	C₄⋊C₄	—	28	2	no	—
159	C₄⋊C₄	—	28	2	no	—
160	C₄²	(C₂×D₇)⋊C₄	1	1	yes	characteristic
161	C₄²	#1141	1	1	yes	characteristic
162	C₄²	#1143	1	1	yes	characteristic
163	C₄²	C₂×C₂²⋊D₇	1	1	yes	characteristic
164	C₁₆	—	28	2	no	—
165	C₂×C₇	—	2	2	no	—
166	C₂×C₇	C₂×#332	1	1	yes	characteristic
167	C₂×C₇	#2140	1	1	yes	characteristic
168	C₂×C₇	C₂×#362	1	1	yes	characteristic
169	C₂³	#6383	1	1	yes	characteristic
170	Q₈	—	56	4	no	—
171	Q₈	—	56	4	no	—
172	C₂×C₄	—	2	2	no	—
173	C₂×C₄	—	2	2	no	—
174	C₂×C₄	—	28	4	no	—
175	C₂×C₄	—	56	4	no	—
176	C₂×C₄	#6383	1	1	yes	characteristic
177	C₂×C₄	#6383	1	1	yes	characteristic
178	C₂×C₄	#6383	1	1	yes	characteristic
179	C₂×C₄	#6390	1	2	yes	—
180	C₂×C₄	#6398	1	2	yes	—
181	C₂×C₄	C₂×(C₂×D₇)⋊C₄	1	1	yes	characteristic
182	C₂×C₄	C₂×#1141	1	1	yes	characteristic
183	C₂×C₄	C₂×#1143	1	1	yes	characteristic
184	C₈	—	2	1	no	—
185	C₈	—	2	1	no	—
186	C₈	—	8	2	no	—
187	C₈	#6390	1	2	yes	—
188	C₈	C₂×#1132	1	1	yes	characteristic
189	C₈	C₂×Dic₂₈	1	1	yes	characteristic
190	C₇	#11940	1	1	yes	characteristic, complemented, Sylow
191	C₂²	—	2	1	no	—
192	C₂²	—	2	1	no	—
193	C₂²	#89277	1	2	yes	—
194	C₂²	C₂×#6383	1	1	yes	characteristic
195	C₄	—	2	1	no	—
196	C₄	—	2	1	no	—
197	C₄	—	56	4	no	—
198	C₄	#89277	1	2	yes	—
199	C₄	#89291	1	2	yes	—
200	C₄	C₂×#6390	1	1	yes	characteristic
201	C₄	C₂×#6398	1	1	yes	characteristic
202	C₄	#89965	1	2	yes	—
203	C₂	—	2	2	no	—
204	C₂	C₂×#89277	1	1	yes	characteristic
205	C₂	#11728406	1	1	yes	characteristic
206	C₂	C₂×#89291	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×#6383
commutator	C₂×C₇×C₈	C₂²×C₄
Frattini	C₄×C₈	C₂²×D₇
Fitting	C₇×#1564	C₂
socle	C₂²×C₇	C₂×#102
prime radical	C₂³×C₇	#102
cosocle	C₄×C₇×C₈	C₂³
hypercenter	#1564	D₇
hypocenter	C₇	#11940
LCS	G ▷ C₂×C₇×C₈ ▷ C₂×C₄×C₇ ▷ C₂×C₇ ▷ C₇
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ #1564

Complements

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

N	H ≅ G/N	type
4. #11725281	C₂	semidirect
190. C₇	#11940	semidirect

Maximal subgroups

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

subgroup	index
C₂×#89454	2
C₇×#1564	2
#11727497	2
#11725281	2
#11940	7