Subgroups — #13272009

Order 1792.

614 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	C₂×#89467	C₂	1	1	yes	maximal, characteristic
2	C₇×#1845	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×#89286	C₂	1	4	yes	maximal, complemented
4	C₂×C₄×C₇⋊C₁₆	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂²×D₅₆	—	2	1	no	—
6	C₂²×D₅₆	C₂²	1	2	yes	—
7	C₂×#6437	—	2	1	no	—
8	C₂×C₇×#107	—	2	1	no	—
9	C₂×C₇×#107	C₂²	1	2	yes	—
10	C₂×C₄×C₇×C₈	C₂²	1	1	yes	characteristic
11	C₇×#384	—	2	1	no	—
12	C₂²×C₇⋊C₁₆	C₂²	1	2	yes	—
13	#89467	—	2	2	no	—
14	#89467	C₄	1	4	yes	complemented
15	#89286	—	2	8	no	—
16	C₄×C₇⋊C₁₆	—	2	2	no	—
17	#13202	—	7	1	no	maximal, Sylow
18	C₂²×D₂₈	—	2	2	no	—
19	C₂²×D₂₈	—	4	2	no	—
20	C₂²×C₇×C₈	D₄	1	1	yes	characteristic
21	C₂²×C₇×C₈	C₂³	1	1	yes	characteristic
22	C₂×C₇×C₄⋊C₄	—	2	2	no	—
23	C₂×C₇×C₄⋊C₄	—	4	2	no	—
24	C₂×C₄²×C₇	D₄	1	1	yes	characteristic
25	C₂×D₅₆	—	2	4	no	—
26	C₂×D₅₆	—	2	4	no	—
27	C₂×D₅₆	—	2	4	no	—
28	C₂×D₅₆	—	2	2	no	—
29	C₂×D₅₆	—	4	2	no	—
30	C₂×D₅₆	—	4	2	no	—
31	C₂×D₅₆	C₂×C₄	1	4	yes	—
32	#6437	—	2	4	no	—
33	#6437	—	4	2	no	—
34	C₇×#107	—	2	4	no	—
35	C₇×#107	—	4	2	no	—
36	C₄×C₇×C₈	C₂×C₄	1	2	yes	—
37	C₄×C₇×C₈	D₄	1	2	yes	—
38	C₂×C₇⋊C₁₆	—	2	2	no	—
39	C₂×C₇⋊C₁₆	—	2	2	no	—
40	C₂×C₇⋊C₁₆	D₄	1	4	yes	complemented
41	C₂×#493	—	7	1	no	—
42	C₂×#357	—	7	4	no	—
43	C₂×C₄×C₁₆	—	7	1	no	—
44	#1845	D₇	1	1	yes	characteristic, complemented
45	C₂³×D₇	—	4	2	no	—
46	C₂²×C₄×C₇	—	4	2	no	—
47	C₂²×C₄×C₇	D₈	1	2	yes	—
48	C₂²×C₄×C₇	C₂×D₄	1	1	yes	characteristic
49	C₂×D₂₈	—	2	4	no	—
50	C₂×D₂₈	—	4	2	no	—
51	C₂×D₂₈	—	4	2	no	—
52	C₂×D₂₈	—	4	2	no	—
53	C₂×D₂₈	—	4	2	no	—
54	C₂×D₂₈	—	4	8	no	—
55	C₂×D₂₈	—	4	2	no	—
56	C₂×D₂₈	—	8	4	no	—
57	C₂×D₂₈	—	8	4	no	—
58	C₂×C₇×C₈	C₂²⋊C₄	1	2	yes	—
59	C₂×C₇×C₈	D₈	1	2	yes	—
60	C₂×C₇×C₈	SD₁₆	1	2	yes	—
61	C₂×C₇×C₈	C₂²×C₄	1	1	yes	characteristic
62	C₂×C₇×C₈	C₂×D₄	1	2	yes	—
63	C₂×C₇×C₈	C₂×D₄	1	1	yes	characteristic
64	C₂×C₇×C₈	#41	1	2	yes	—
65	C₇×C₄⋊C₄	—	4	2	no	—
66	C₇×C₄⋊C₄	—	4	2	no	—
67	C₇×C₄⋊C₄	—	8	4	no	—
68	C₄²×C₇	C₂²⋊C₄	1	2	yes	—
69	C₄²×C₇	SD₁₆	1	2	yes	—
70	D₅₆	—	2	8	no	—
71	D₅₆	—	4	8	no	—
72	D₅₆	—	4	4	no	—
73	C₇⋊C₁₆	—	2	4	no	—
74	C₂²×D₈	—	7	2	no	—
75	C₂²×D₈	—	14	1	no	—
76	C₂×#127	—	14	1	no	—
77	C₂²×C₁₆	—	7	2	no	—
78	#493	—	7	4	no	—
79	#493	—	14	2	no	—
80	C₂×#107	—	2	1	no	—
81	C₂×#107	C₂×D₇	1	2	yes	—
82	C₂×C₄×C₈	C₂×D₇	1	1	yes	characteristic
83	#384	—	2	1	no	—
84	#357	—	14	8	no	—
85	C₄×C₁₆	—	14	2	no	—
86	C₂³×C₇	C₂×D₈	1	1	yes	characteristic
87	C₂²×D₇	—	4	4	no	—
88	C₂²×D₇	—	8	2	no	—
89	C₂²×D₇	—	8	2	no	—
90	C₂²×D₇	—	8	4	no	—
91	C₂²×D₇	—	8	4	no	—
92	C₂×C₄×C₇	—	8	2	no	—
93	C₂×C₄×C₇	—	8	4	no	—
94	C₂×C₄×C₇	#102	1	4	yes	—
95	C₂×C₄×C₇	D₁₆	1	4	yes	—
96	C₂×C₄×C₇	SD₃₂	1	4	yes	—
97	C₂×C₄×C₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
98	C₂×C₄×C₇	#124	1	1	yes	characteristic
99	C₂×C₄×C₇	#127	1	1	yes	characteristic
100	C₂×C₄×C₇	C₂×D₈	1	1	yes	characteristic
101	C₂×C₄×C₇	C₂×SD₁₆	1	1	yes	characteristic
102	C₂×C₄×C₇	C₂×SD₁₆	1	1	yes	characteristic
103	D₂₈	—	4	4	no	—
104	D₂₈	—	4	4	no	—
105	D₂₈	—	8	16	no	—
106	D₂₈	—	8	2	no	—
107	D₂₈	—	8	2	no	—
108	C₇×C₈	#102	1	4	yes	—
109	C₇×C₈	C₄×D₄	1	2	yes	—
110	C₇×C₈	#121	1	2	yes	—
111	C₂²×D₄	—	14	2	no	—
112	C₂²×D₄	—	28	2	no	—
113	C₂×D₈	—	7	4	no	—
114	C₂×D₈	—	14	4	no	—
115	C₂×D₈	—	14	2	no	—
116	C₂×D₈	—	14	4	no	—
117	C₂×D₈	—	14	4	no	—
118	C₂×D₈	—	28	2	no	—
119	C₂×D₈	—	28	2	no	—
120	C₂²×C₈	C₂²⋊D₇	1	1	yes	characteristic
121	C₂²×C₈	C₂²×D₇	1	1	yes	characteristic
122	#127	—	14	4	no	—
123	#127	—	28	2	no	—
124	C₂×C₄⋊C₄	—	2	2	no	—
125	C₂×C₄⋊C₄	—	4	2	no	—
126	C₂×C₄²	C₂²⋊D₇	1	1	yes	characteristic
127	C₂×C₁₆	—	7	4	no	—
128	C₂×C₁₆	—	14	2	no	—
129	C₂×C₁₆	—	14	2	no	—
130	#107	—	2	4	no	—
131	#107	—	4	2	no	—
132	C₄×C₈	C₄×D₇	1	2	yes	—
133	C₄×C₈	D₂₈	1	2	yes	—
134	C₂²×C₇	C₂×#102	1	1	yes	characteristic
135	C₂²×C₇	#486	1	1	yes	characteristic
136	C₂²×C₇	#493	1	1	yes	characteristic
137	C₂²×C₇	C₂×D₁₆	1	2	yes	—
138	C₂²×C₇	C₂×SD₃₂	1	2	yes	—
139	C₂×D₇	—	8	8	no	—
140	C₂×D₇	—	8	4	no	—
141	C₂×D₇	—	16	4	no	—
142	C₂×D₇	—	16	2	no	—
143	C₂×D₇	—	16	2	no	—
144	C₄×C₇	—	16	2	no	—
145	C₄×C₇	#357	1	8	yes	complemented
146	C₄×C₇	#390	1	1	yes	characteristic
147	C₄×C₇	C₂×#102	1	1	yes	characteristic
148	C₄×C₇	#486	1	1	yes	characteristic
149	C₄×C₇	#492	1	1	yes	characteristic
150	C₂⁴	—	28	2	no	—
151	C₂×D₄	—	14	4	no	—
152	C₂×D₄	—	28	2	no	—
153	C₂×D₄	—	28	2	no	—
154	C₂×D₄	—	28	2	no	—
155	C₂×D₄	—	28	2	no	—
156	C₂×D₄	—	28	2	no	—
157	C₂×D₄	—	28	8	no	—
158	C₂×D₄	—	56	4	no	—
159	C₂×D₄	—	56	4	no	—
160	C₂²×C₄	—	4	2	no	—
161	C₂²×C₄	#1141	1	2	yes	—
162	C₂²×C₄	C₂×C₂²⋊D₇	1	1	yes	characteristic
163	D₈	—	14	8	no	—
164	D₈	—	28	4	no	—
165	D₈	—	28	8	no	—
166	C₂×C₈	(C₂×D₇)⋊C₄	1	2	yes	—
167	C₂×C₈	#1141	1	2	yes	—
168	C₂×C₈	#1143	1	2	yes	—
169	C₂×C₈	C₂×C₄×D₇	1	1	yes	characteristic
170	C₂×C₈	C₂×D₂₈	1	1	yes	characteristic
171	C₂×C₈	D₄×D₇	1	2	yes	—
172	C₂×C₈	#1161	1	2	yes	—
173	C₄⋊C₄	—	4	2	no	—
174	C₄⋊C₄	—	4	2	no	—
175	C₄⋊C₄	—	8	4	no	—
176	C₄²	(C₂×D₇)⋊C₄	1	2	yes	—
177	C₄²	#1143	1	2	yes	—
178	C₁₆	—	14	4	no	—
179	C₂×C₇	#1957	1	1	yes	characteristic
180	C₂×C₇	C₂×#357	1	2	yes	—
181	C₂×C₇	#2240	1	2	yes	—
182	C₂×C₇	#2246	1	1	yes	characteristic
183	C₂×C₇	#2248	1	1	yes	characteristic
184	D₇	—	16	4	no	—
185	C₂³	—	28	4	no	—
186	C₂³	—	56	2	no	—
187	C₂³	—	56	4	no	—
188	C₂³	—	56	4	no	—
189	C₂³	—	56	2	no	—
190	C₂³	C₂×#1141	1	1	yes	characteristic
191	D₄	—	28	4	no	—
192	D₄	—	28	4	no	—
193	D₄	—	56	2	no	—
194	D₄	—	56	2	no	—
195	D₄	—	56	16	no	—
196	C₂×C₄	—	8	4	no	—
197	C₂×C₄	—	8	2	no	—
198	C₂×C₄	#6383	1	4	yes	—
199	C₂×C₄	#6400	1	4	yes	—
200	C₂×C₄	#6402	1	4	yes	—
201	C₂×C₄	C₂×(C₂×D₇)⋊C₄	1	1	yes	characteristic
202	C₂×C₄	C₂×#1141	1	1	yes	characteristic
203	C₂×C₄	#6503	1	1	yes	characteristic
204	C₂×C₄	C₂×#1143	1	1	yes	characteristic
205	C₂×C₄	C₂×#1143	1	1	yes	characteristic
206	C₂×C₄	#6510	1	1	yes	characteristic
207	C₈	#6383	1	4	yes	—
208	C₈	#6456	1	2	yes	—
209	C₈	#6458	1	2	yes	—
210	C₇	#13202	1	1	yes	characteristic, complemented, Sylow
211	C₂²	—	56	8	no	—
212	C₂²	—	56	4	no	—
213	C₂²	—	112	2	no	—
214	C₂²	—	112	2	no	—
215	C₂²	—	112	4	no	—
216	C₂²	C₂×#6383	1	1	yes	characteristic
217	C₂²	#89845	1	1	yes	characteristic
218	C₂²	#89860	1	1	yes	characteristic
219	C₂²	C₂×#6400	1	2	yes	—
220	C₂²	C₂×#6402	1	2	yes	—
221	C₄	—	16	2	no	—
222	C₄	#89286	1	8	yes	complemented
223	C₄	C₂×#6383	1	1	yes	characteristic
224	C₄	#89760	1	1	yes	characteristic
225	C₄	#89857	1	1	yes	characteristic
226	C₄	#89860	1	1	yes	characteristic
227	C₂	—	112	4	no	—
228	C₂	#11727334	1	1	yes	characteristic
229	C₂	C₂×#89286	1	2	yes	—
230	C₂	#11729044	1	1	yes	characteristic
231	C₂	#11729067	1	1	yes	characteristic
232	C₂	#11729135	1	2	yes	—

	subgroup	G/H
center	C₂³	C₂×#1141
commutator	C₂×C₇×C₈	C₂²×C₄
Frattini	C₂²×C₈	C₂²×D₇
Fitting	C₇×#1845	C₂
socle	C₂³×C₇	C₂×D₈
prime radical	C₂×#89467	C₂
cosocle	C₂²×C₇×C₈	C₂³
hypercenter	#1845	D₇
hypocenter	C₇	#13202
LCS	G ▷ C₂×C₇×C₈ ▷ C₄×C₇ ▷ C₂×C₇ ▷ C₇
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ #1845

Complements

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

N	H ≅ G/N	type
2. C₇×#1845	C₂	semidirect
3. C₂×#89286	C₂	semidirect
4. C₂×C₄×C₇⋊C₁₆	C₂	semidirect
14. #89467	C₄	semidirect
40. C₂×C₇⋊C₁₆	D₄	semidirect
44. #1845	D₇	semidirect
145. C₄×C₇	#357	semidirect
210. C₇	#13202	semidirect
222. C₄	#89286	semidirect

Maximal subgroups

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

subgroup	index
C₂×#89467	2
C₇×#1845	2
C₂×#89286	2
C₂×C₄×C₇⋊C₁₆	2
#13202	7