Subgroups — C₇×#12788 #13190004

Order 1792.

538 conjugacy classes of subgroups.

The subgroup lattice has 252 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₄×C₇×C₈	C₂	1	1	yes	maximal, characteristic
2	C₂×C₇×#364	C₂	1	2	yes	maximal, complemented
3	C₇×#2112	C₂	1	2	yes	maximal
4	C₇×#1376	C₂	1	2	yes	maximal, complemented
5	C₂³×C₇×C₈	C₄	1	1	yes	characteristic
6	C₂³×C₇×C₈	C₂²	1	1	yes	characteristic
7	C₂²×C₄²×C₇	C₄	1	1	yes	characteristic
8	C₂×C₇×M₃₂	C₂²	1	4	yes	—
9	C₂×C₄×C₇×C₈	—	2	4	no	—
10	C₂×C₄×C₇×C₈	C₄	1	1	yes	characteristic
11	C₂×C₄×C₇×C₈	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×#364	—	2	4	no	—
15	C₇×C₂²⋊C₁₆	—	2	4	no	—
16	#12788	C₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
17	C₂³×C₄×C₇	—	2	1	no	—
18	C₂³×C₄×C₇	C₂×C₄	1	1	yes	characteristic
19	C₂²×C₇×C₈	—	2	2	no	—
20	C₂²×C₇×C₈	—	2	2	no	—
21	C₂²×C₇×C₈	—	2	2	no	—
22	C₂²×C₇×C₈	—	2	2	no	—
23	C₂²×C₇×C₈	C₈	1	4	yes	—
24	C₂²×C₇×C₈	C₂×C₄	1	1	yes	characteristic
25	C₂²×C₇×C₈	C₂×C₄	1	1	yes	characteristic
26	C₂²×C₇×C₈	C₂×C₄	1	1	yes	characteristic
27	C₂²×C₇×C₈	D₄	1	2	yes	—
28	C₂²×C₇×C₈	Q₈	1	2	yes	—
29	C₂²×C₇×C₈	C₂³	1	1	yes	characteristic
30	C₂×C₄²×C₇	—	2	1	no	—
31	C₂×C₄²×C₇	—	2	2	no	—
32	C₂×C₄²×C₇	—	2	2	no	—
33	C₂×C₄²×C₇	C₂×C₄	1	1	yes	characteristic
34	C₂×C₄²×C₇	C₂×C₄	1	1	yes	characteristic
35	C₇×M₃₂	—	2	4	no	—
36	C₇×M₃₂	—	2	4	no	—
37	C₂×C₇×C₁₆	—	2	4	no	—
38	C₄×C₇×C₈	—	2	2	no	—
39	C₄×C₇×C₈	—	2	2	no	—
40	C₄×C₇×C₈	—	2	2	no	—
41	C₄×C₇×C₈	D₄	1	4	yes	—
42	C₂²×C₄×C₈	C₂×C₇	1	1	yes	characteristic
43	C₂×#364	C₂×C₇	1	2	yes	complemented
44	#2112	C₂×C₇	1	2	yes	—
45	#1376	C₂×C₇	1	2	yes	complemented
46	C₂⁴×C₇	M₁₆	1	1	yes	characteristic
47	C₂²×C₄×C₇	—	2	4	no	—
48	C₂²×C₄×C₇	—	2	8	no	—
49	C₂²×C₄×C₇	—	2	1	no	—
50	C₂²×C₄×C₇	—	2	1	no	—
51	C₂²×C₄×C₇	—	2	1	no	—
52	C₂²×C₄×C₇	—	2	1	no	—
53	C₂²×C₄×C₇	—	2	1	no	—
54	C₂²×C₄×C₇	—	2	1	no	—
55	C₂²×C₄×C₇	C₄⋊C₄	1	2	yes	—
56	C₂²×C₄×C₇	C₂×C₈	1	2	yes	—
57	C₂²×C₄×C₇	M₁₆	1	1	yes	characteristic
58	C₂²×C₄×C₇	C₂²×C₄	1	1	yes	characteristic
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	8	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	C₂×C₈	1	2	yes	—
69	C₂×C₇×C₈	C₂×C₈	1	2	yes	—
70	C₂×C₇×C₈	M₁₆	1	2	yes	—
71	C₂×C₇×C₈	C₂×D₄	1	2	yes	—
72	C₂×C₇×C₈	C₂×D₄	1	1	yes	characteristic
73	C₂×C₇×C₈	C₂×Q₈	1	1	yes	characteristic
74	C₂×C₇×C₈	#41	1	2	yes	—
75	C₄²×C₇	—	2	4	no	—
76	C₄²×C₇	—	2	1	no	—
77	C₄²×C₇	—	2	1	no	—
78	C₄²×C₇	C₂²⋊C₄	1	2	yes	—
79	C₄²×C₇	M₁₆	1	2	yes	—
80	C₇×C₁₆	—	4	4	no	—
81	C₂³×C₈	C₄×C₇	1	1	yes	characteristic
82	C₂³×C₈	C₂²×C₇	1	1	yes	characteristic
83	C₂²×C₄²	C₄×C₇	1	1	yes	characteristic
84	C₂×M₃₂	C₂²×C₇	1	4	yes	—
85	C₂×C₄×C₈	—	2	4	no	—
86	C₂×C₄×C₈	C₄×C₇	1	1	yes	characteristic
87	C₂×C₄×C₈	C₄×C₇	1	1	yes	characteristic
88	C₂×C₄×C₈	C₂²×C₇	1	1	yes	characteristic
89	C₂×C₄×C₈	C₂²×C₇	1	1	yes	characteristic
90	#364	—	2	4	no	—
91	C₂²⋊C₁₆	—	2	4	no	—
92	C₂³×C₇	—	2	4	no	—
93	C₂³×C₇	—	2	1	no	—
94	C₂³×C₇	—	2	1	no	—
95	C₂³×C₇	C₄⋊C₈	1	2	yes	—
96	C₂³×C₇	C₂×M₁₆	1	1	yes	characteristic
97	C₂×C₄×C₇	—	2	16	no	—
98	C₂×C₄×C₇	—	2	4	no	—
99	C₂×C₄×C₇	—	2	4	no	—
100	C₂×C₄×C₇	—	2	1	no	—
101	C₂×C₄×C₇	—	2	2	no	—
102	C₂×C₄×C₇	—	2	2	no	—
103	C₂×C₄×C₇	—	2	1	no	—
104	C₂×C₄×C₇	—	2	2	no	—
105	C₂×C₄×C₇	—	2	2	no	—
106	C₂×C₄×C₇	—	2	2	no	—
107	C₂×C₄×C₇	—	2	2	no	—
108	C₂×C₄×C₇	C₄⋊C₈	1	2	yes	—
109	C₂×C₄×C₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
110	C₂×C₄×C₇	C₂×C₄⋊C₄	1	1	yes	characteristic
111	C₂×C₄×C₇	#117	1	1	yes	characteristic
112	C₂×C₄×C₇	C₂²×C₈	1	1	yes	characteristic
113	C₂×C₄×C₇	C₂×M₁₆	1	1	yes	characteristic
114	C₂×C₄×C₇	C₂×M₁₆	1	1	yes	characteristic
115	C₇×C₈	—	2	2	no	—
116	C₇×C₈	—	2	2	no	—
117	C₇×C₈	C₂²⋊C₈	1	4	yes	—
118	C₇×C₈	#121	1	2	yes	—
119	C₇×C₈	C₂²⋊Q₈	1	2	yes	—
120	C₂³×C₄	—	2	1	no	—
121	C₂³×C₄	C₂×C₄×C₇	1	1	yes	characteristic
122	C₂²×C₈	—	2	2	no	—
123	C₂²×C₈	—	2	2	no	—
124	C₂²×C₈	—	2	2	no	—
125	C₂²×C₈	—	2	2	no	—
126	C₂²×C₈	C₇×C₈	1	4	yes	—
127	C₂²×C₈	C₂×C₄×C₇	1	1	yes	characteristic
128	C₂²×C₈	C₂×C₄×C₇	1	1	yes	characteristic
129	C₂²×C₈	C₂×C₄×C₇	1	1	yes	characteristic
130	C₂²×C₈	C₇×D₄	1	2	yes	—
131	C₂²×C₈	C₇×Q₈	1	2	yes	—
132	C₂²×C₈	C₂³×C₇	1	1	yes	characteristic
133	C₂×C₄²	—	2	1	no	—
134	C₂×C₄²	—	2	2	no	—
135	C₂×C₄²	—	2	2	no	—
136	C₂×C₄²	C₂×C₄×C₇	1	1	yes	characteristic
137	C₂×C₄²	C₂×C₄×C₇	1	1	yes	characteristic
138	M₃₂	—	2	4	no	—
139	M₃₂	—	2	4	no	—
140	C₂×C₁₆	—	2	4	no	—
141	C₄×C₈	—	2	2	no	—
142	C₄×C₈	—	2	2	no	—
143	C₄×C₈	—	2	2	no	—
144	C₄×C₈	C₇×D₄	1	4	yes	—
145	C₂²×C₇	—	2	4	no	—
146	C₂²×C₇	—	2	4	no	—
147	C₂²×C₇	—	2	2	no	—
148	C₂²×C₇	—	2	2	no	—
149	C₂²×C₇	—	2	2	no	—
150	C₂²×C₇	#364	1	4	yes	complemented
151	C₂²×C₇	#407	1	1	yes	characteristic
152	C₂²×C₇	C₂×C₄⋊C₈	1	1	yes	characteristic
153	C₂²×C₇	#432	1	1	yes	characteristic
154	C₄×C₇	—	2	4	no	—
155	C₄×C₇	—	2	4	no	—
156	C₄×C₇	—	2	2	no	—
157	C₄×C₇	#380	1	1	yes	characteristic
158	C₄×C₇	C₂×C₂²⋊C₈	1	1	yes	characteristic
159	C₄×C₇	#423	1	1	yes	characteristic
160	C₄×C₇	#431	1	1	yes	characteristic
161	C₂⁴	C₇×M₁₆	1	1	yes	characteristic
162	C₂²×C₄	—	2	4	no	—
163	C₂²×C₄	—	2	8	no	—
164	C₂²×C₄	—	2	1	no	—
165	C₂²×C₄	—	2	1	no	—
166	C₂²×C₄	—	2	1	no	—
167	C₂²×C₄	—	2	1	no	—
168	C₂²×C₄	—	2	1	no	—
169	C₂²×C₄	—	2	1	no	—
170	C₂²×C₄	C₇×C₄⋊C₄	1	2	yes	—
171	C₂²×C₄	C₂×C₇×C₈	1	2	yes	—
172	C₂²×C₄	C₇×M₁₆	1	1	yes	characteristic
173	C₂²×C₄	C₂²×C₄×C₇	1	1	yes	characteristic
174	C₂×C₈	—	2	1	no	—
175	C₂×C₈	—	2	1	no	—
176	C₂×C₈	—	2	1	no	—
177	C₂×C₈	—	2	1	no	—
178	C₂×C₈	—	2	8	no	—
179	C₂×C₈	—	2	4	no	—
180	C₂×C₈	—	2	4	no	—
181	C₂×C₈	C₇×C₂²⋊C₄	1	2	yes	—
182	C₂×C₈	C₇×C₄⋊C₄	1	2	yes	—
183	C₂×C₈	C₂×C₇×C₈	1	2	yes	—
184	C₂×C₈	C₂×C₇×C₈	1	2	yes	—
185	C₂×C₈	C₇×M₁₆	1	2	yes	—
186	C₂×C₈	C₂×C₇×D₄	1	1	yes	characteristic
187	C₂×C₈	C₂×C₇×D₄	1	2	yes	—
188	C₂×C₈	C₂×C₇×Q₈	1	1	yes	characteristic
189	C₂×C₈	C₇×#41	1	2	yes	—
190	C₄²	—	2	4	no	—
191	C₄²	—	2	1	no	—
192	C₄²	—	2	1	no	—
193	C₄²	C₇×C₂²⋊C₄	1	2	yes	—
194	C₄²	C₇×M₁₆	1	2	yes	—
195	C₁₆	—	4	4	no	—
196	C₂×C₇	—	2	1	no	—
197	C₂×C₇	—	2	4	no	—
198	C₂×C₇	—	2	1	no	—
199	C₂×C₇	#1797	1	1	yes	characteristic
200	C₂×C₇	C₂×#364	1	2	yes	—
201	C₂³	—	2	4	no	—
202	C₂³	—	2	1	no	—
203	C₂³	—	2	1	no	—
204	C₂³	C₇×C₄⋊C₈	1	2	yes	—
205	C₂³	C₂×C₇×M₁₆	1	1	yes	characteristic
206	C₂×C₄	—	2	16	no	—
207	C₂×C₄	—	2	4	no	—
208	C₂×C₄	—	2	4	no	—
209	C₂×C₄	—	2	1	no	—
210	C₂×C₄	—	2	2	no	—
211	C₂×C₄	—	2	2	no	—
212	C₂×C₄	—	2	1	no	—
213	C₂×C₄	—	2	2	no	—
214	C₂×C₄	—	2	2	no	—
215	C₂×C₄	—	2	2	no	—
216	C₂×C₄	—	2	2	no	—
217	C₂×C₄	C₇×C₄⋊C₈	1	2	yes	—
218	C₂×C₄	C₂×C₇×C₂²⋊C₄	1	1	yes	characteristic
219	C₂×C₄	C₂×C₇×C₄⋊C₄	1	1	yes	characteristic
220	C₂×C₄	C₇×#117	1	1	yes	characteristic
221	C₂×C₄	C₂²×C₇×C₈	1	1	yes	characteristic
222	C₂×C₄	C₂×C₇×M₁₆	1	1	yes	characteristic
223	C₂×C₄	C₂×C₇×M₁₆	1	1	yes	characteristic
224	C₈	—	2	2	no	—
225	C₈	—	2	2	no	—
226	C₈	C₇×C₂²⋊C₈	1	4	yes	—
227	C₈	C₇×#121	1	2	yes	—
228	C₈	C₇×C₂²⋊Q₈	1	2	yes	—
229	C₇	#12788	1	1	yes	characteristic, direct product factor, Sylow
230	C₂²	—	2	4	no	—
231	C₂²	—	2	4	no	—
232	C₂²	—	2	2	no	—
233	C₂²	—	2	2	no	—
234	C₂²	—	2	2	no	—
235	C₂²	C₇×#364	1	4	yes	complemented
236	C₂²	C₇×#407	1	1	yes	characteristic
237	C₂²	C₂×C₇×C₄⋊C₈	1	1	yes	characteristic
238	C₂²	C₇×#432	1	1	yes	characteristic
239	C₄	—	2	4	no	—
240	C₄	—	2	4	no	—
241	C₄	—	2	2	no	—
242	C₄	C₇×#380	1	1	yes	characteristic
243	C₄	C₂×C₇×C₂²⋊C₈	1	1	yes	characteristic
244	C₄	C₇×#423	1	1	yes	characteristic
245	C₄	C₇×#431	1	1	yes	characteristic
246	C₂	—	2	1	no	—
247	C₂	—	2	4	no	—
248	C₂	—	2	1	no	—
249	C₂	C₇×#1797	1	1	yes	characteristic
250	C₂	C₂×C₇×#364	1	2	yes	—

	subgroup	G/H
center	C₂×C₇×C₈	C₂×D₄
commutator	C₂×C₄	C₂²×C₇×C₈
Frattini, cosocle	C₂²×C₈	C₂³×C₇
socle	C₂²×C₇	C₂×C₄⋊C₈
prime radical	C₂⁴×C₇	M₁₆
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₇×C₈ ◁ C₂³×C₇×C₈ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₂×C₇×#364	C₂	semidirect
4. C₇×#1376	C₂	semidirect
16. #12788	C₇	direct (N × H)
43. C₂×#364	C₂×C₇	semidirect
45. #1376	C₂×C₇	semidirect
150. C₂²×C₇	#364	semidirect
229. C₇	#12788	direct (N × H)
235. C₂²	C₇×#364	semidirect

Maximal subgroups

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

subgroup	index
C₂²×C₄×C₇×C₈	2
C₂×C₇×#364	2
C₇×#2112	2
C₇×#1376	2
#12788	7

Subgroups — C7×#12788 #13190004

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₇×#12788 #13190004