Subgroups — C₇×#12784 #13190000

Order 1792.

478 conjugacy classes of subgroups.

The subgroup lattice has 246 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₇×#405	C₂	1	1	yes	maximal, characteristic
2	C₂×C₇×C₄⋊C₁₆	C₂	1	2	yes	maximal, complemented
3	C₂×C₇×C₂²⋊C₁₆	C₂	1	2	yes	maximal
4	C₇×#1374	C₂	1	2	yes	maximal, complemented
5	C₂²×C₇×M₁₆	C₄	1	1	yes	characteristic
6	C₂³×C₇×C₈	C₂²	1	1	yes	characteristic
7	C₂×C₇×#117	C₄	1	1	yes	characteristic
8	C₂²×C₇×C₁₆	C₂²	1	4	yes	—
9	C₇×#405	—	2	4	no	—
10	C₂×C₇×#97	C₄	1	1	yes	characteristic
11	C₂×C₇×#97	C₄	1	1	yes	characteristic
12	C₂×C₄×C₇×C₈	C₂²	1	1	yes	characteristic
13	C₂×C₄×C₇×C₈	C₂²	1	1	yes	characteristic
14	C₇×C₄⋊C₁₆	—	2	4	no	—
15	C₇×C₂²⋊C₁₆	—	2	4	no	—
16	#12784	C₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
17	C₂³×C₄×C₇	C₂×C₄	1	1	yes	characteristic
18	C₂×C₇×M₁₆	—	2	2	no	—
19	C₂×C₇×M₁₆	—	2	2	no	—
20	C₂×C₇×M₁₆	—	2	2	no	—
21	C₂²×C₇×C₈	—	2	2	no	—
22	C₂²×C₇×C₈	—	2	2	no	—
23	C₂²×C₇×C₈	C₂×C₄	1	1	yes	characteristic
24	C₂²×C₇×C₈	C₂×C₄	1	1	yes	characteristic
25	C₂²×C₇×C₈	C₂×C₄	1	1	yes	characteristic
26	C₂²×C₇×C₈	D₄	1	2	yes	—
27	C₂²×C₇×C₈	Q₈	1	2	yes	—
28	C₂²×C₇×C₈	C₂³	1	1	yes	characteristic
29	C₇×#117	—	2	2	no	—
30	C₇×#117	—	2	2	no	—
31	C₂×C₇×C₄⋊C₄	C₈	1	2	yes	—
32	C₂×C₇×C₂²⋊C₄	C₈	1	2	yes	—
33	C₂×C₄²×C₇	C₂×C₄	1	1	yes	characteristic
34	C₂×C₄²×C₇	C₂×C₄	1	1	yes	characteristic
35	C₂×C₇×C₁₆	—	2	4	no	—
36	C₂×C₇×C₁₆	—	2	4	no	—
37	C₂×C₇×C₁₆	—	2	4	no	—
38	C₇×#97	—	2	2	no	—
39	C₇×#97	—	2	2	no	—
40	C₄×C₇×C₈	—	2	2	no	—
41	C₄×C₇×C₈	D₄	1	4	yes	—
42	C₂×#405	C₂×C₇	1	1	yes	characteristic
43	C₂×C₄⋊C₁₆	C₂×C₇	1	2	yes	complemented
44	C₂×C₂²⋊C₁₆	C₂×C₇	1	2	yes	—
45	#1374	C₂×C₇	1	2	yes	complemented
46	C₂⁴×C₇	C₂×C₈	1	1	yes	characteristic
47	C₂²×C₄×C₇	—	2	1	no	—
48	C₂²×C₄×C₇	—	2	1	no	—
49	C₂²×C₄×C₇	—	2	1	no	—
50	C₂²×C₄×C₇	—	2	1	no	—
51	C₂²×C₄×C₇	C₄⋊C₄	1	2	yes	—
52	C₂²×C₄×C₇	C₂×C₈	1	2	yes	—
53	C₂²×C₄×C₇	C₂×C₈	1	2	yes	—
54	C₂²×C₄×C₇	C₂×C₈	1	1	yes	characteristic
55	C₂²×C₄×C₇	M₁₆	1	2	yes	—
56	C₂²×C₄×C₇	C₂²×C₄	1	1	yes	characteristic
57	C₇×M₁₆	—	2	8	no	—
58	C₂×C₇×C₈	—	2	1	no	—
59	C₂×C₇×C₈	—	2	1	no	—
60	C₂×C₇×C₈	—	2	1	no	—
61	C₂×C₇×C₈	—	2	1	no	—
62	C₂×C₇×C₈	—	2	1	no	—
63	C₂×C₇×C₈	—	2	1	no	—
64	C₂×C₇×C₈	—	2	4	no	—
65	C₂×C₇×C₈	—	2	4	no	—
66	C₂×C₇×C₈	C₂²⋊C₄	1	2	yes	—
67	C₂×C₇×C₈	C₄⋊C₄	1	2	yes	—
68	C₂×C₇×C₈	M₁₆	1	2	yes	—
69	C₂×C₇×C₈	C₂×D₄	1	1	yes	characteristic
70	C₂×C₇×C₈	C₂×D₄	1	2	yes	—
71	C₂×C₇×C₈	C₂×Q₈	1	1	yes	characteristic
72	C₂×C₇×C₈	#41	1	2	yes	—
73	C₇×C₄⋊C₄	—	2	4	no	—
74	C₇×C₂²⋊C₄	—	2	4	no	—
75	C₄²×C₇	—	2	1	no	—
76	C₄²×C₇	—	2	1	no	—
77	C₄²×C₇	C₂²⋊C₄	1	2	yes	—
78	C₄²×C₇	M₁₆	1	2	yes	—
79	C₇×C₁₆	—	4	4	no	—
80	C₂²×M₁₆	C₄×C₇	1	1	yes	characteristic
81	C₂³×C₈	C₂²×C₇	1	1	yes	characteristic
82	C₂×#117	C₄×C₇	1	1	yes	characteristic
83	C₂²×C₁₆	C₂²×C₇	1	4	yes	—
84	#405	—	2	4	no	—
85	C₂×#97	C₄×C₇	1	1	yes	characteristic
86	C₂×#97	C₄×C₇	1	1	yes	characteristic
87	C₂×C₄×C₈	C₂²×C₇	1	1	yes	characteristic
88	C₂×C₄×C₈	C₂²×C₇	1	1	yes	characteristic
89	C₄⋊C₁₆	—	2	4	no	—
90	C₂²⋊C₁₆	—	2	4	no	—
91	C₂³×C₇	—	2	1	no	—
92	C₂³×C₇	—	2	1	no	—
93	C₂³×C₇	—	2	4	no	—
94	C₂³×C₇	C₄⋊C₈	1	2	yes	—
95	C₂³×C₇	C₂²×C₈	1	1	yes	characteristic
96	C₂×C₄×C₇	—	2	1	no	—
97	C₂×C₄×C₇	—	2	1	no	—
98	C₂×C₄×C₇	—	2	4	no	—
99	C₂×C₄×C₇	—	2	2	no	—
100	C₂×C₄×C₇	—	2	2	no	—
101	C₂×C₄×C₇	—	2	2	no	—
102	C₂×C₄×C₇	—	2	2	no	—
103	C₂×C₄×C₇	—	2	2	no	—
104	C₂×C₄×C₇	—	2	4	no	—
105	C₂×C₄×C₇	C₂²⋊C₈	1	4	yes	—
106	C₂×C₄×C₇	C₄⋊C₈	1	2	yes	—
107	C₂×C₄×C₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
108	C₂×C₄×C₇	C₂×C₄⋊C₄	1	1	yes	characteristic
109	C₂×C₄×C₇	#117	1	1	yes	characteristic
110	C₂×C₄×C₇	C₂×M₁₆	1	1	yes	characteristic
111	C₂×C₄×C₇	C₂×M₁₆	1	1	yes	characteristic
112	C₂×C₄×C₇	C₂×M₁₆	1	1	yes	characteristic
113	C₇×C₈	—	2	2	no	—
114	C₇×C₈	—	2	2	no	—
115	C₇×C₈	—	2	2	no	—
116	C₇×C₈	#121	1	2	yes	—
117	C₇×C₈	C₂²⋊Q₈	1	2	yes	—
118	C₂³×C₄	C₂×C₄×C₇	1	1	yes	characteristic
119	C₂×M₁₆	—	2	2	no	—
120	C₂×M₁₆	—	2	2	no	—
121	C₂×M₁₆	—	2	2	no	—
122	C₂²×C₈	—	2	2	no	—
123	C₂²×C₈	—	2	2	no	—
124	C₂²×C₈	C₂×C₄×C₇	1	1	yes	characteristic
125	C₂²×C₈	C₂×C₄×C₇	1	1	yes	characteristic
126	C₂²×C₈	C₂×C₄×C₇	1	1	yes	characteristic
127	C₂²×C₈	C₇×D₄	1	2	yes	—
128	C₂²×C₈	C₇×Q₈	1	2	yes	—
129	C₂²×C₈	C₂³×C₇	1	1	yes	characteristic
130	#117	—	2	2	no	—
131	#117	—	2	2	no	—
132	C₂×C₄⋊C₄	C₇×C₈	1	2	yes	—
133	C₂×C₂²⋊C₄	C₇×C₈	1	2	yes	—
134	C₂×C₄²	C₂×C₄×C₇	1	1	yes	characteristic
135	C₂×C₄²	C₂×C₄×C₇	1	1	yes	characteristic
136	C₂×C₁₆	—	2	4	no	—
137	C₂×C₁₆	—	2	4	no	—
138	C₂×C₁₆	—	2	4	no	—
139	#97	—	2	2	no	—
140	#97	—	2	2	no	—
141	C₄×C₈	—	2	2	no	—
142	C₄×C₈	C₇×D₄	1	4	yes	—
143	C₂²×C₇	—	2	2	no	—
144	C₂²×C₇	—	2	4	no	—
145	C₂²×C₇	—	4	2	no	—
146	C₂²×C₇	—	4	2	no	—
147	C₂²×C₇	C₂×C₂²⋊C₈	1	1	yes	characteristic
148	C₂²×C₇	C₂×C₄⋊C₈	1	1	yes	characteristic
149	C₂²×C₇	#431	1	1	yes	characteristic
150	C₂²×C₇	#504	1	4	yes	—
151	C₄×C₇	—	2	2	no	—
152	C₄×C₇	—	2	4	no	—
153	C₄×C₇	—	4	2	no	—
154	C₄×C₇	#380	1	1	yes	characteristic
155	C₄×C₇	#407	1	1	yes	characteristic
156	C₄×C₇	#423	1	1	yes	characteristic
157	C₄×C₇	#432	1	1	yes	characteristic
158	C₂⁴	C₂×C₇×C₈	1	1	yes	characteristic
159	C₂²×C₄	—	2	1	no	—
160	C₂²×C₄	—	2	1	no	—
161	C₂²×C₄	—	2	1	no	—
162	C₂²×C₄	—	2	1	no	—
163	C₂²×C₄	C₇×C₄⋊C₄	1	2	yes	—
164	C₂²×C₄	C₂×C₇×C₈	1	2	yes	—
165	C₂²×C₄	C₂×C₇×C₈	1	2	yes	—
166	C₂²×C₄	C₂×C₇×C₈	1	1	yes	characteristic
167	C₂²×C₄	C₇×M₁₆	1	2	yes	—
168	C₂²×C₄	C₂²×C₄×C₇	1	1	yes	characteristic
169	M₁₆	—	2	8	no	—
170	C₂×C₈	—	2	1	no	—
171	C₂×C₈	—	2	1	no	—
172	C₂×C₈	—	2	1	no	—
173	C₂×C₈	—	2	1	no	—
174	C₂×C₈	—	2	1	no	—
175	C₂×C₈	—	2	1	no	—
176	C₂×C₈	—	2	4	no	—
177	C₂×C₈	—	2	4	no	—
178	C₂×C₈	C₇×C₂²⋊C₄	1	2	yes	—
179	C₂×C₈	C₇×C₄⋊C₄	1	2	yes	—
180	C₂×C₈	C₇×M₁₆	1	2	yes	—
181	C₂×C₈	C₂×C₇×D₄	1	1	yes	characteristic
182	C₂×C₈	C₂×C₇×D₄	1	2	yes	—
183	C₂×C₈	C₂×C₇×Q₈	1	1	yes	characteristic
184	C₂×C₈	C₇×#41	1	2	yes	—
185	C₄⋊C₄	—	2	4	no	—
186	C₂²⋊C₄	—	2	4	no	—
187	C₄²	—	2	1	no	—
188	C₄²	—	2	1	no	—
189	C₄²	C₇×C₂²⋊C₄	1	2	yes	—
190	C₄²	C₇×M₁₆	1	2	yes	—
191	C₁₆	—	4	4	no	—
192	C₂×C₇	—	4	2	no	—
193	C₂×C₇	#1797	1	1	yes	characteristic
194	C₂×C₇	#2113	1	2	yes	—
195	C₂×C₇	#2151	1	2	yes	—
196	C₂×C₇	#2164	1	2	yes	—
197	C₂³	—	2	4	no	—
198	C₂³	—	2	1	no	—
199	C₂³	—	2	1	no	—
200	C₂³	C₇×C₄⋊C₈	1	2	yes	—
201	C₂³	C₂²×C₇×C₈	1	1	yes	characteristic
202	C₂×C₄	—	2	4	no	—
203	C₂×C₄	—	2	1	no	—
204	C₂×C₄	—	2	1	no	—
205	C₂×C₄	—	2	2	no	—
206	C₂×C₄	—	2	2	no	—
207	C₂×C₄	—	2	2	no	—
208	C₂×C₄	—	2	2	no	—
209	C₂×C₄	—	2	4	no	—
210	C₂×C₄	—	2	2	no	—
211	C₂×C₄	C₇×C₂²⋊C₈	1	4	yes	—
212	C₂×C₄	C₇×C₄⋊C₈	1	2	yes	—
213	C₂×C₄	C₂×C₇×C₂²⋊C₄	1	1	yes	characteristic
214	C₂×C₄	C₂×C₇×C₄⋊C₄	1	1	yes	characteristic
215	C₂×C₄	C₇×#117	1	1	yes	characteristic
216	C₂×C₄	C₂×C₇×M₁₆	1	1	yes	characteristic
217	C₂×C₄	C₂×C₇×M₁₆	1	1	yes	characteristic
218	C₂×C₄	C₂×C₇×M₁₆	1	1	yes	characteristic
219	C₈	—	2	2	no	—
220	C₈	—	2	2	no	—
221	C₈	—	2	2	no	—
222	C₈	C₇×#121	1	2	yes	—
223	C₈	C₇×C₂²⋊Q₈	1	2	yes	—
224	C₇	#12784	1	1	yes	characteristic, direct product factor, Sylow
225	C₂²	—	2	2	no	—
226	C₂²	—	2	4	no	—
227	C₂²	—	4	2	no	—
228	C₂²	—	4	2	no	—
229	C₂²	C₂×C₇×C₂²⋊C₈	1	1	yes	characteristic
230	C₂²	C₂×C₇×C₄⋊C₈	1	1	yes	characteristic
231	C₂²	C₇×#431	1	1	yes	characteristic
232	C₂²	C₇×#504	1	4	yes	—
233	C₄	—	2	2	no	—
234	C₄	—	2	4	no	—
235	C₄	—	4	2	no	—
236	C₄	C₇×#380	1	1	yes	characteristic
237	C₄	C₇×#407	1	1	yes	characteristic
238	C₄	C₇×#423	1	1	yes	characteristic
239	C₄	C₇×#432	1	1	yes	characteristic
240	C₂	—	4	2	no	—
241	C₂	C₇×#1797	1	1	yes	characteristic
242	C₂	C₇×#2113	1	2	yes	—
243	C₂	C₇×#2151	1	2	yes	—
244	C₂	C₇×#2164	1	2	yes	—

	subgroup	G/H
center	C₂²×C₇×C₈	C₂³
commutator	C₂³	C₂²×C₇×C₈
Frattini, cosocle	C₂²×C₈	C₂³×C₇
socle	C₂³×C₇	C₂²×C₈
prime radical	C₂⁴×C₇	C₂×C₈
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂²×C₇×C₈ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₂×C₇×C₄⋊C₁₆	C₂	semidirect
4. C₇×#1374	C₂	semidirect
16. #12784	C₇	direct (N × H)
43. C₂×C₄⋊C₁₆	C₂×C₇	semidirect
45. #1374	C₂×C₇	semidirect
224. C₇	#12784	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂×C₇×#405	2
C₂×C₇×C₄⋊C₁₆	2
C₂×C₇×C₂²⋊C₁₆	2
C₇×#1374	2
#12784	7

Subgroups — C7×#12784 #13190000

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₇×#12784 #13190000