Subgroups — C₂×C₇×#1296 #13185253

Order 1792.

878 conjugacy classes of subgroups.

The subgroup lattice has 260 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₇×C₄⋊C₈	C₂	1	2	yes	maximal
2	C₂×C₇×#384	C₂	1	1	yes	maximal, characteristic
3	C₇×#1296	C₂	1	4	yes	maximal, direct product factor
4	C₂³×C₇×C₈	—	2	2	no	—
5	C₂²×C₇×C₄⋊C₄	—	2	2	no	—
6	C₂²×C₇×C₄⋊C₄	C₄	1	2	yes	—
7	C₂²×C₄²×C₇	C₂²	1	1	yes	characteristic
8	C₂×C₇×C₄⋊C₈	—	2	2	no	—
9	C₂×C₇×C₄⋊C₈	—	2	4	no	—
10	C₂×C₇×C₄⋊C₈	C₄	1	8	yes	complemented
11	C₂×C₇×C₄⋊C₈	C₂²	1	4	yes	—
12	C₇×#384	—	2	2	no	—
13	C₇×#384	C₄	1	2	yes	—
14	C₇×#384	C₂²	1	2	yes	—
15	C₂×#1296	C₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
16	C₂³×C₄×C₇	—	2	2	no	—
17	C₂³×C₄×C₇	C₂×C₄	1	1	yes	characteristic
18	C₂³×C₄×C₇	D₄	1	2	yes	—
19	C₂²×C₇×C₈	—	2	8	no	—
20	C₂²×C₇×C₈	—	2	4	no	—
21	C₂²×C₇×C₈	—	4	4	no	—
22	C₂²×C₇×C₈	—	4	4	no	—
23	C₂×C₇×C₄⋊C₄	—	2	4	no	—
24	C₂×C₇×C₄⋊C₄	—	2	2	no	—
25	C₂×C₇×C₄⋊C₄	—	2	4	no	—
26	C₂×C₇×C₄⋊C₄	—	4	4	no	—
27	C₂×C₇×C₄⋊C₄	—	4	4	no	—
28	C₂×C₇×C₄⋊C₄	—	4	4	no	—
29	C₂×C₇×C₄⋊C₄	C₂×C₄	1	4	yes	—
30	C₂×C₄²×C₇	C₂×C₄	1	1	yes	characteristic
31	C₂×C₄²×C₇	C₂×C₄	1	4	yes	—
32	C₂×C₄²×C₇	D₄	1	2	yes	—
33	C₂×C₄²×C₇	D₄	1	2	yes	—
34	C₂×C₄²×C₇	Q₈	1	2	yes	—
35	C₂×C₄²×C₇	C₂³	1	1	yes	characteristic
36	C₇×C₄⋊C₈	—	2	8	no	—
37	C₇×C₄⋊C₈	—	2	4	no	—
38	C₇×C₄⋊C₈	C₂×C₄	1	8	yes	complemented
39	C₂²×C₄⋊C₈	C₂×C₇	1	2	yes	—
40	C₂×#384	C₂×C₇	1	1	yes	characteristic
41	#1296	C₂×C₇	1	4	yes	direct product factor
42	C₂⁴×C₇	C₂²⋊C₄	1	1	yes	characteristic
43	C₂²×C₄×C₇	—	2	2	no	—
44	C₂²×C₄×C₇	—	2	2	no	—
45	C₂²×C₄×C₇	—	2	4	no	—
46	C₂²×C₄×C₇	—	4	4	no	—
47	C₂²×C₄×C₇	—	4	8	no	—
48	C₂²×C₄×C₇	C₄²	1	2	yes	—
49	C₂²×C₄×C₇	C₂²⋊C₄	1	1	yes	characteristic
50	C₂²×C₄×C₇	C₂²⋊C₄	1	2	yes	—
51	C₂²×C₄×C₇	C₂²⋊C₄	1	4	yes	—
52	C₂²×C₄×C₇	C₄⋊C₄	1	2	yes	—
53	C₂²×C₄×C₇	C₄⋊C₄	1	4	yes	—
54	C₂²×C₄×C₇	D₈	1	4	yes	—
55	C₂²×C₄×C₇	SD₁₆	1	4	yes	—
56	C₂²×C₄×C₇	SD₁₆	1	4	yes	—
57	C₂²×C₄×C₇	Q₁₆	1	4	yes	—
58	C₂²×C₄×C₇	C₂²×C₄	1	1	yes	characteristic
59	C₂²×C₄×C₇	C₂×D₄	1	2	yes	—
60	C₂×C₇×C₈	—	2	8	no	—
61	C₂×C₇×C₈	—	4	16	no	—
62	C₂×C₇×C₈	—	4	4	no	—
63	C₂×C₇×C₈	—	4	4	no	—
64	C₇×C₄⋊C₄	—	4	4	no	—
65	C₇×C₄⋊C₄	—	4	4	no	—
66	C₇×C₄⋊C₄	—	4	8	no	—
67	C₇×C₄⋊C₄	—	4	4	no	—
68	C₇×C₄⋊C₄	—	4	4	no	—
69	C₄²×C₇	C₄²	1	2	yes	—
70	C₄²×C₇	C₂²⋊C₄	1	4	yes	—
71	C₄²×C₇	C₄⋊C₄	1	4	yes	—
72	C₄²×C₇	C₄⋊C₄	1	2	yes	—
73	C₄²×C₇	C₂²×C₄	1	2	yes	—
74	C₄²×C₇	C₂×D₄	1	1	yes	characteristic
75	C₄²×C₇	C₂×Q₈	1	1	yes	characteristic
76	C₂³×C₈	—	2	2	no	—
77	C₂²×C₄⋊C₄	—	2	2	no	—
78	C₂²×C₄⋊C₄	C₄×C₇	1	2	yes	—
79	C₂²×C₄²	C₂²×C₇	1	1	yes	characteristic
80	C₂×C₄⋊C₈	—	2	2	no	—
81	C₂×C₄⋊C₈	—	2	4	no	—
82	C₂×C₄⋊C₈	C₄×C₇	1	8	yes	complemented
83	C₂×C₄⋊C₈	C₂²×C₇	1	4	yes	—
84	#384	—	2	2	no	—
85	#384	C₄×C₇	1	2	yes	—
86	#384	C₂²×C₇	1	2	yes	—
87	C₂³×C₇	#95	1	2	yes	—
88	C₂³×C₇	#100	1	2	yes	—
89	C₂³×C₇	#101	1	2	yes	—
90	C₂³×C₇	#102	1	4	yes	—
91	C₂³×C₇	#103	1	4	yes	—
92	C₂³×C₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
93	C₂×C₄×C₇	—	2	4	no	—
94	C₂×C₄×C₇	—	2	4	no	—
95	C₂×C₄×C₇	—	2	1	no	—
96	C₂×C₄×C₇	—	2	1	no	—
97	C₂×C₄×C₇	—	2	1	no	—
98	C₂×C₄×C₇	—	2	1	no	—
99	C₂×C₄×C₇	—	4	4	no	—
100	C₂×C₄×C₇	—	4	8	no	—
101	C₂×C₄×C₇	—	8	8	no	—
102	C₂×C₄×C₇	#95	1	2	yes	—
103	C₂×C₄×C₇	#95	1	4	yes	—
104	C₂×C₄×C₇	#102	1	4	yes	—
105	C₂×C₄×C₇	#102	1	8	yes	—
106	C₂×C₄×C₇	#103	1	4	yes	—
107	C₂×C₄×C₇	#103	1	8	yes	—
108	C₂×C₄×C₇	#106	1	8	yes	—
109	C₂×C₄×C₇	#107	1	8	yes	—
110	C₂×C₄×C₇	C₂×C₄²	1	1	yes	characteristic
111	C₂×C₄×C₇	C₂×C₂²⋊C₄	1	2	yes	—
112	C₂×C₄×C₇	C₂×C₄⋊C₄	1	1	yes	characteristic
113	C₂×C₄×C₇	C₂×C₄⋊C₄	1	2	yes	—
114	C₂×C₄×C₇	C₂×D₈	1	2	yes	—
115	C₂×C₄×C₇	C₂×SD₁₆	1	2	yes	—
116	C₂×C₄×C₇	C₂×SD₁₆	1	2	yes	—
117	C₂×C₄×C₇	C₂×Q₁₆	1	2	yes	—
118	C₇×C₈	—	4	8	no	—
119	C₂³×C₄	—	2	2	no	—
120	C₂³×C₄	C₂×C₄×C₇	1	1	yes	characteristic
121	C₂³×C₄	C₇×D₄	1	2	yes	—
122	C₂²×C₈	—	2	8	no	—
123	C₂²×C₈	—	2	4	no	—
124	C₂²×C₈	—	4	4	no	—
125	C₂²×C₈	—	4	4	no	—
126	C₂×C₄⋊C₄	—	2	4	no	—
127	C₂×C₄⋊C₄	—	2	2	no	—
128	C₂×C₄⋊C₄	—	2	4	no	—
129	C₂×C₄⋊C₄	—	4	4	no	—
130	C₂×C₄⋊C₄	—	4	4	no	—
131	C₂×C₄⋊C₄	—	4	4	no	—
132	C₂×C₄⋊C₄	C₂×C₄×C₇	1	4	yes	—
133	C₂×C₄²	C₂×C₄×C₇	1	1	yes	characteristic
134	C₂×C₄²	C₂×C₄×C₇	1	4	yes	—
135	C₂×C₄²	C₇×D₄	1	2	yes	—
136	C₂×C₄²	C₇×D₄	1	2	yes	—
137	C₂×C₄²	C₇×Q₈	1	2	yes	—
138	C₂×C₄²	C₂³×C₇	1	1	yes	characteristic
139	C₄⋊C₈	—	2	8	no	—
140	C₄⋊C₈	—	2	4	no	—
141	C₄⋊C₈	C₂×C₄×C₇	1	8	yes	complemented
142	C₂²×C₇	#331	1	4	yes	—
143	C₂²×C₇	#332	1	8	yes	—
144	C₂²×C₇	#333	1	4	yes	—
145	C₂²×C₇	#340	1	8	yes	—
146	C₂²×C₇	#343	1	4	yes	—
147	C₂²×C₇	C₂×#95	1	1	yes	characteristic
148	C₂²×C₇	C₂×#100	1	1	yes	characteristic
149	C₂²×C₇	C₂×#101	1	1	yes	characteristic
150	C₂²×C₇	C₂×#102	1	2	yes	—
151	C₂²×C₇	C₂×#103	1	2	yes	—
152	C₄×C₇	—	2	2	no	—
153	C₄×C₇	—	2	1	no	—
154	C₄×C₇	—	2	1	no	—
155	C₄×C₇	—	8	4	no	—
156	C₄×C₇	#340	1	8	yes	—
157	C₄×C₇	C₂×#102	1	2	yes	—
158	C₄×C₇	C₂×#103	1	2	yes	—
159	C₄×C₇	C₂×#106	1	2	yes	—
160	C₄×C₇	C₂×#107	1	2	yes	—
161	C₂⁴	C₇×C₂²⋊C₄	1	1	yes	characteristic
162	C₂²×C₄	—	2	2	no	—
163	C₂²×C₄	—	2	2	no	—
164	C₂²×C₄	—	2	4	no	—
165	C₂²×C₄	—	4	4	no	—
166	C₂²×C₄	—	4	8	no	—
167	C₂²×C₄	C₄²×C₇	1	2	yes	—
168	C₂²×C₄	C₇×C₂²⋊C₄	1	1	yes	characteristic
169	C₂²×C₄	C₇×C₂²⋊C₄	1	2	yes	—
170	C₂²×C₄	C₇×C₂²⋊C₄	1	4	yes	—
171	C₂²×C₄	C₇×C₄⋊C₄	1	2	yes	—
172	C₂²×C₄	C₇×C₄⋊C₄	1	4	yes	—
173	C₂²×C₄	C₇×D₈	1	4	yes	—
174	C₂²×C₄	C₇×SD₁₆	1	4	yes	—
175	C₂²×C₄	C₇×SD₁₆	1	4	yes	—
176	C₂²×C₄	C₇×Q₁₆	1	4	yes	—
177	C₂²×C₄	C₂²×C₄×C₇	1	1	yes	characteristic
178	C₂²×C₄	C₂×C₇×D₄	1	2	yes	—
179	C₂×C₈	—	2	8	no	—
180	C₂×C₈	—	4	16	no	—
181	C₂×C₈	—	4	4	no	—
182	C₂×C₈	—	4	4	no	—
183	C₄⋊C₄	—	4	4	no	—
184	C₄⋊C₄	—	4	4	no	—
185	C₄⋊C₄	—	4	8	no	—
186	C₄⋊C₄	—	4	4	no	—
187	C₄⋊C₄	—	4	4	no	—
188	C₄²	C₄²×C₇	1	2	yes	—
189	C₄²	C₇×C₂²⋊C₄	1	4	yes	—
190	C₄²	C₇×C₄⋊C₄	1	4	yes	—
191	C₄²	C₇×C₄⋊C₄	1	2	yes	—
192	C₄²	C₂²×C₄×C₇	1	2	yes	—
193	C₄²	C₂×C₇×D₄	1	1	yes	characteristic
194	C₄²	C₂×C₇×Q₈	1	1	yes	characteristic
195	C₂×C₇	#1296	1	8	yes	direct product factor
196	C₂×C₇	C₂×#331	1	1	yes	characteristic
197	C₂×C₇	C₂×#332	1	2	yes	—
198	C₂×C₇	C₂×#333	1	1	yes	characteristic
199	C₂×C₇	C₂×#340	1	2	yes	—
200	C₂×C₇	C₂×#343	1	1	yes	characteristic
201	C₂³	C₇×#95	1	2	yes	—
202	C₂³	C₇×#100	1	2	yes	—
203	C₂³	C₇×#101	1	2	yes	—
204	C₂³	C₇×#102	1	4	yes	—
205	C₂³	C₇×#103	1	4	yes	—
206	C₂³	C₂×C₇×C₂²⋊C₄	1	1	yes	characteristic
207	C₂×C₄	—	2	4	no	—
208	C₂×C₄	—	2	4	no	—
209	C₂×C₄	—	2	1	no	—
210	C₂×C₄	—	2	1	no	—
211	C₂×C₄	—	2	1	no	—
212	C₂×C₄	—	2	1	no	—
213	C₂×C₄	—	4	4	no	—
214	C₂×C₄	—	4	8	no	—
215	C₂×C₄	—	8	8	no	—
216	C₂×C₄	C₇×#95	1	2	yes	—
217	C₂×C₄	C₇×#95	1	4	yes	—
218	C₂×C₄	C₇×#102	1	4	yes	—
219	C₂×C₄	C₇×#102	1	8	yes	—
220	C₂×C₄	C₇×#103	1	4	yes	—
221	C₂×C₄	C₇×#103	1	8	yes	—
222	C₂×C₄	C₇×#106	1	8	yes	—
223	C₂×C₄	C₇×#107	1	8	yes	—
224	C₂×C₄	C₂×C₄²×C₇	1	1	yes	characteristic
225	C₂×C₄	C₂×C₇×C₂²⋊C₄	1	2	yes	—
226	C₂×C₄	C₂×C₇×C₄⋊C₄	1	1	yes	characteristic
227	C₂×C₄	C₂×C₇×C₄⋊C₄	1	2	yes	—
228	C₂×C₄	C₂×C₇×D₈	1	2	yes	—
229	C₂×C₄	C₂×C₇×SD₁₆	1	2	yes	—
230	C₂×C₄	C₂×C₇×SD₁₆	1	2	yes	—
231	C₂×C₄	C₂×C₇×Q₁₆	1	2	yes	—
232	C₈	—	4	8	no	—
233	C₇	C₂×#1296	1	1	yes	characteristic, direct product factor, Sylow
234	C₂²	C₇×#331	1	4	yes	—
235	C₂²	C₇×#332	1	8	yes	—
236	C₂²	C₇×#333	1	4	yes	—
237	C₂²	C₇×#340	1	8	yes	—
238	C₂²	C₇×#343	1	4	yes	—
239	C₂²	C₂×C₇×#95	1	1	yes	characteristic
240	C₂²	C₂×C₇×#100	1	1	yes	characteristic
241	C₂²	C₂×C₇×#101	1	1	yes	characteristic
242	C₂²	C₂×C₇×#102	1	2	yes	—
243	C₂²	C₂×C₇×#103	1	2	yes	—
244	C₄	—	2	2	no	—
245	C₄	—	2	1	no	—
246	C₄	—	2	1	no	—
247	C₄	—	8	4	no	—
248	C₄	C₇×#340	1	8	yes	—
249	C₄	C₂×C₇×#102	1	2	yes	—
250	C₄	C₂×C₇×#103	1	2	yes	—
251	C₄	C₂×C₇×#106	1	2	yes	—
252	C₄	C₂×C₇×#107	1	2	yes	—
253	C₂	C₇×#1296	1	8	yes	direct product factor
254	C₂	C₂×C₇×#331	1	1	yes	characteristic
255	C₂	C₂×C₇×#332	1	2	yes	—
256	C₂	C₂×C₇×#333	1	1	yes	characteristic
257	C₂	C₂×C₇×#340	1	2	yes	—
258	C₂	C₂×C₇×#343	1	1	yes	characteristic

	subgroup	G/H
center, socle, prime radical	C₂⁴×C₇	C₂²⋊C₄
commutator	C₂×C₄	C₂×C₄²×C₇
Frattini, cosocle	C₂×C₄²	C₂³×C₇
LCS	G ▷ C₂×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂⁴×C₇ ◁ C₂²×C₄²×C₇ ◁ G

Complements

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

N	H ≅ G/N	type
3. C₇×#1296	C₂	direct (N × H)
10. C₂×C₇×C₄⋊C₈	C₄	semidirect
15. C₂×#1296	C₇	direct (N × H)
38. C₇×C₄⋊C₈	C₂×C₄	semidirect
41. #1296	C₂×C₇	direct (N × H)
82. C₂×C₄⋊C₈	C₄×C₇	semidirect
141. C₄⋊C₈	C₂×C₄×C₇	semidirect
195. C₂×C₇	#1296	direct (N × H)
233. C₇	C₂×#1296	direct (N × H)
253. C₂	C₇×#1296	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂²×C₇×C₄⋊C₈	2
C₂×C₇×#384	2
C₇×#1296	2
C₂×#1296	7

Subgroups — C2×C7×#1296 #13185253

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₇×#1296 #13185253