Subgroups — C₇×#12820 #13190036

Order 1792.

414 conjugacy classes of subgroups.

The subgroup lattice has 268 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₇×#443	C₂	1	1	yes	maximal, characteristic
2	C₂×C₇×#364	C₂	1	1	yes	maximal, characteristic, complemented
3	C₇×#2139	C₂	1	2	yes	maximal
4	C₇×#1846	C₂	1	1	yes	maximal, characteristic, complemented
5	C₇×#1387	C₂	1	2	yes	maximal, complemented
6	C₂×C₇×#135	C₂²	1	1	yes	characteristic
7	C₂×C₇×#131	—	2	1	no	—
8	C₂×C₇×M₃₂	C₂²	1	2	yes	—
9	C₇×#443	—	2	2	no	—
10	C₇×#443	C₄	1	4	yes	complemented
11	C₂×C₇×#108	C₂²	1	1	yes	characteristic
12	C₂×C₇×#107	C₂²	1	2	yes	—
13	C₂×C₇×#106	—	2	1	no	—
14	C₂×C₇×#104	—	2	1	no	—
15	C₂×C₄×C₇×C₈	C₂²	1	1	yes	characteristic
16	C₇×#382	—	2	1	no	—
17	C₇×#364	—	2	2	no	—
18	C₇×#358	—	2	2	no	—
19	C₇×#357	—	2	2	no	—
20	#12820	C₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
21	C₂×C₇×#41	—	2	1	no	—
22	C₇×#135	—	2	1	no	—
23	C₇×#135	—	2	1	no	—
24	C₇×#135	C₂×C₄	1	2	yes	—
25	C₇×#135	Q₈	1	2	yes	—
26	C₂×C₇×Q₁₆	D₄	1	1	yes	characteristic
27	C₂×C₇×SD₁₆	—	2	1	no	—
28	C₂×C₇×D₈	D₄	1	1	yes	characteristic
29	C₇×#131	—	2	4	no	—
30	C₇×#131	—	4	2	no	—
31	C₂×C₇×M₁₆	—	2	1	no	—
32	C₂×C₇×M₁₆	—	4	1	no	—
33	C₂²×C₇×C₈	—	4	1	no	—
34	C₂²×C₇×C₈	D₄	1	1	yes	characteristic
35	C₂²×C₇×C₈	C₂³	1	1	yes	characteristic
36	C₂×C₇×C₄⋊C₄	—	2	2	no	—
37	C₂×C₄²×C₇	D₄	1	1	yes	characteristic
38	C₇×M₃₂	—	2	2	no	—
39	C₇×M₃₂	—	2	2	no	—
40	C₂×C₇×C₁₆	—	2	2	no	—
41	C₇×#108	—	2	1	no	—
42	C₇×#108	C₂×C₄	1	2	yes	—
43	C₇×#107	—	2	2	no	—
44	C₇×#107	—	2	2	no	—
45	C₇×#106	—	4	1	no	—
46	C₇×#106	—	4	1	no	—
47	C₇×#104	—	2	4	no	—
48	C₄×C₇×C₈	C₂×C₄	1	2	yes	—
49	C₄×C₇×C₈	D₄	1	2	yes	—
50	C₇×#95	—	4	2	no	—
51	C₂×#443	C₂×C₇	1	1	yes	characteristic
52	C₂×#364	C₂×C₇	1	1	yes	characteristic, complemented
53	#2139	C₂×C₇	1	2	yes	—
54	#1846	C₂×C₇	1	1	yes	characteristic, complemented
55	#1387	C₂×C₇	1	2	yes	complemented
56	C₇×#41	—	2	2	no	—
57	C₇×#41	—	2	2	no	—
58	C₇×#41	—	4	1	no	—
59	C₇×#41	—	4	1	no	—
60	C₂×C₇×Q₈	—	2	1	no	—
61	C₂×C₇×D₄	—	2	1	no	—
62	C₂×C₇×D₄	—	4	1	no	—
63	C₂²×C₄×C₇	—	2	1	no	—
64	C₂²×C₄×C₇	—	4	1	no	—
65	C₂²×C₄×C₇	—	4	2	no	—
66	C₂²×C₄×C₇	C₂×D₄	1	1	yes	characteristic
67	C₇×Q₁₆	—	2	1	no	—
68	C₇×Q₁₆	C₄⋊C₄	1	2	yes	—
69	C₇×SD₁₆	—	2	2	no	—
70	C₇×SD₁₆	—	2	2	no	—
71	C₇×D₈	—	2	1	no	—
72	C₇×D₈	C₄⋊C₄	1	2	yes	—
73	C₇×M₁₆	—	2	2	no	—
74	C₇×M₁₆	—	4	1	no	—
75	C₇×M₁₆	—	4	4	no	—
76	C₂×C₇×C₈	—	4	1	no	—
77	C₂×C₇×C₈	—	4	4	no	—
78	C₂×C₇×C₈	C₂²⋊C₄	1	2	yes	—
79	C₂×C₇×C₈	D₈	1	1	yes	characteristic
80	C₂×C₇×C₈	D₈	1	1	yes	characteristic
81	C₂×C₇×C₈	SD₁₆	1	2	yes	—
82	C₂×C₇×C₈	C₂²×C₄	1	1	yes	characteristic
83	C₂×C₇×C₈	C₂×D₄	1	1	yes	characteristic
84	C₂×C₇×C₈	C₂×D₄	1	1	yes	characteristic
85	C₂×C₇×C₈	C₂×Q₈	1	1	yes	characteristic
86	C₂×C₇×C₈	#41	1	1	yes	characteristic
87	C₂×C₇×C₈	#41	1	1	yes	characteristic
88	C₇×C₄⋊C₄	—	4	2	no	—
89	C₇×C₄⋊C₄	—	4	2	no	—
90	C₄²×C₇	—	2	1	no	—
91	C₄²×C₇	C₂²⋊C₄	1	2	yes	—
92	C₇×C₁₆	—	4	2	no	—
93	C₂×#135	C₂²×C₇	1	1	yes	characteristic
94	C₂×#131	—	2	1	no	—
95	C₂×M₃₂	C₂²×C₇	1	2	yes	—
96	#443	—	2	2	no	—
97	#443	C₄×C₇	1	4	yes	complemented
98	C₂×#108	C₂²×C₇	1	1	yes	characteristic
99	C₂×#107	C₂²×C₇	1	2	yes	—
100	C₂×#106	—	2	1	no	—
101	C₂×#104	—	2	1	no	—
102	C₂×C₄×C₈	C₂²×C₇	1	1	yes	characteristic
103	#382	—	2	1	no	—
104	#364	—	2	2	no	—
105	#358	—	2	2	no	—
106	#357	—	2	2	no	—
107	C₂³×C₇	—	4	1	no	—
108	C₂³×C₇	AGL(1,ℤ/8)	1	1	yes	characteristic
109	C₇×Q₈	—	2	2	no	—
110	C₇×Q₈	—	4	1	no	—
111	C₇×D₄	—	2	2	no	—
112	C₇×D₄	—	4	2	no	—
113	C₇×D₄	—	4	2	no	—
114	C₇×D₄	—	4	1	no	—
115	C₂×C₄×C₇	—	2	2	no	—
116	C₂×C₄×C₇	—	4	2	no	—
117	C₂×C₄×C₇	—	4	1	no	—
118	C₂×C₄×C₇	—	4	2	no	—
119	C₂×C₄×C₇	—	4	2	no	—
120	C₂×C₄×C₇	—	8	4	no	—
121	C₂×C₄×C₇	—	8	2	no	—
122	C₂×C₄×C₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
123	C₂×C₄×C₇	#121	1	1	yes	characteristic
124	C₂×C₄×C₇	C₂²⋊Q₈	1	1	yes	characteristic
125	C₂×C₄×C₇	C₂×D₈	1	1	yes	characteristic
126	C₂×C₄×C₇	C₂×SD₁₆	1	1	yes	characteristic
127	C₂×C₄×C₇	#137	1	1	yes	characteristic
128	C₇×C₈	—	4	2	no	—
129	C₇×C₈	#102	1	2	yes	—
130	C₇×C₈	#102	1	2	yes	—
131	C₇×C₈	C₂×C₄⋊C₄	1	1	yes	characteristic
132	C₇×C₈	#117	1	1	yes	characteristic
133	C₇×C₈	#121	1	1	yes	characteristic
134	C₇×C₈	C₂²⋊Q₈	1	1	yes	characteristic
135	C₂×#41	—	2	1	no	—
136	#135	—	2	1	no	—
137	#135	—	2	1	no	—
138	#135	C₂×C₄×C₇	1	2	yes	—
139	#135	C₇×Q₈	1	2	yes	—
140	C₂×Q₁₆	C₇×D₄	1	1	yes	characteristic
141	C₂×SD₁₆	—	2	1	no	—
142	C₂×D₈	C₇×D₄	1	1	yes	characteristic
143	#131	—	2	4	no	—
144	#131	—	4	2	no	—
145	C₂×M₁₆	—	2	1	no	—
146	C₂×M₁₆	—	4	1	no	—
147	C₂²×C₈	—	4	1	no	—
148	C₂²×C₈	C₇×D₄	1	1	yes	characteristic
149	C₂²×C₈	C₂³×C₇	1	1	yes	characteristic
150	C₂×C₄⋊C₄	—	2	2	no	—
151	C₂×C₄²	C₇×D₄	1	1	yes	characteristic
152	M₃₂	—	2	2	no	—
153	M₃₂	—	2	2	no	—
154	C₂×C₁₆	—	2	2	no	—
155	#108	—	2	1	no	—
156	#108	C₂×C₄×C₇	1	2	yes	—
157	#107	—	2	2	no	—
158	#107	—	2	2	no	—
159	#106	—	4	1	no	—
160	#106	—	4	1	no	—
161	#104	—	2	4	no	—
162	C₄×C₈	C₂×C₄×C₇	1	2	yes	—
163	C₄×C₈	C₇×D₄	1	2	yes	—
164	#95	—	4	2	no	—
165	C₂²×C₇	—	2	2	no	—
166	C₂²×C₇	—	4	2	no	—
167	C₂²×C₇	—	8	2	no	—
168	C₂²×C₇	#417	1	1	yes	characteristic
169	C₂²×C₇	#459	1	1	yes	characteristic
170	C₂²×C₇	#476	1	1	yes	characteristic
171	C₄×C₇	—	4	2	no	—
172	C₄×C₇	—	4	2	no	—
173	C₄×C₇	—	16	2	no	—
174	C₄×C₇	#380	1	1	yes	characteristic
175	C₄×C₇	C₂×#102	1	1	yes	characteristic
176	C₄×C₇	#461	1	1	yes	characteristic
177	C₄×C₇	#474	1	1	yes	characteristic
178	#41	—	2	2	no	—
179	#41	—	2	2	no	—
180	#41	—	4	1	no	—
181	#41	—	4	1	no	—
182	C₂×Q₈	—	2	1	no	—
183	C₂×D₄	—	2	1	no	—
184	C₂×D₄	—	4	1	no	—
185	C₂²×C₄	—	2	1	no	—
186	C₂²×C₄	—	4	1	no	—
187	C₂²×C₄	—	4	2	no	—
188	C₂²×C₄	C₂×C₇×D₄	1	1	yes	characteristic
189	Q₁₆	—	2	1	no	—
190	Q₁₆	C₇×C₄⋊C₄	1	2	yes	—
191	SD₁₆	—	2	2	no	—
192	SD₁₆	—	2	2	no	—
193	D₈	—	2	1	no	—
194	D₈	C₇×C₄⋊C₄	1	2	yes	—
195	M₁₆	—	2	2	no	—
196	M₁₆	—	4	1	no	—
197	M₁₆	—	4	4	no	—
198	C₂×C₈	—	4	1	no	—
199	C₂×C₈	—	4	4	no	—
200	C₂×C₈	C₇×C₂²⋊C₄	1	2	yes	—
201	C₂×C₈	C₇×D₈	1	1	yes	characteristic
202	C₂×C₈	C₇×D₈	1	1	yes	characteristic
203	C₂×C₈	C₇×SD₁₆	1	2	yes	—
204	C₂×C₈	C₂²×C₄×C₇	1	1	yes	characteristic
205	C₂×C₈	C₂×C₇×D₄	1	1	yes	characteristic
206	C₂×C₈	C₂×C₇×D₄	1	1	yes	characteristic
207	C₂×C₈	C₂×C₇×Q₈	1	1	yes	characteristic
208	C₂×C₈	C₇×#41	1	1	yes	characteristic
209	C₂×C₈	C₇×#41	1	1	yes	characteristic
210	C₄⋊C₄	—	4	2	no	—
211	C₄⋊C₄	—	4	2	no	—
212	C₄²	—	2	1	no	—
213	C₄²	C₇×C₂²⋊C₄	1	2	yes	—
214	C₁₆	—	4	2	no	—
215	C₂×C₇	—	2	1	no	—
216	C₂×C₇	—	2	1	no	—
217	C₂×C₇	—	8	2	no	—
218	C₂×C₇	#1802	1	1	yes	characteristic
219	C₂×C₇	#2212	1	2	yes	—
220	C₂³	—	4	1	no	—
221	C₂³	C₇×AGL(1,ℤ/8)	1	1	yes	characteristic
222	Q₈	—	2	2	no	—
223	Q₈	—	4	1	no	—
224	D₄	—	2	2	no	—
225	D₄	—	4	2	no	—
226	D₄	—	4	2	no	—
227	D₄	—	4	1	no	—
228	C₂×C₄	—	2	2	no	—
229	C₂×C₄	—	4	2	no	—
230	C₂×C₄	—	4	1	no	—
231	C₂×C₄	—	4	2	no	—
232	C₂×C₄	—	4	2	no	—
233	C₂×C₄	—	8	4	no	—
234	C₂×C₄	—	8	2	no	—
235	C₂×C₄	C₂×C₇×C₂²⋊C₄	1	1	yes	characteristic
236	C₂×C₄	C₇×#121	1	1	yes	characteristic
237	C₂×C₄	C₇×C₂²⋊Q₈	1	1	yes	characteristic
238	C₂×C₄	C₂×C₇×D₈	1	1	yes	characteristic
239	C₂×C₄	C₂×C₇×SD₁₆	1	1	yes	characteristic
240	C₂×C₄	C₇×#137	1	1	yes	characteristic
241	C₈	—	4	2	no	—
242	C₈	C₇×#102	1	2	yes	—
243	C₈	C₇×#102	1	2	yes	—
244	C₈	C₂×C₇×C₄⋊C₄	1	1	yes	characteristic
245	C₈	C₇×#117	1	1	yes	characteristic
246	C₈	C₇×#121	1	1	yes	characteristic
247	C₈	C₇×C₂²⋊Q₈	1	1	yes	characteristic
248	C₇	#12820	1	1	yes	characteristic, direct product factor, Sylow
249	C₂²	—	2	2	no	—
250	C₂²	—	4	2	no	—
251	C₂²	—	8	2	no	—
252	C₂²	C₇×#417	1	1	yes	characteristic
253	C₂²	C₇×#459	1	1	yes	characteristic
254	C₂²	C₇×#476	1	1	yes	characteristic
255	C₄	—	4	2	no	—
256	C₄	—	4	2	no	—
257	C₄	—	16	2	no	—
258	C₄	C₇×#380	1	1	yes	characteristic
259	C₄	C₂×C₇×#102	1	1	yes	characteristic
260	C₄	C₇×#461	1	1	yes	characteristic
261	C₄	C₇×#474	1	1	yes	characteristic
262	C₂	—	2	1	no	—
263	C₂	—	2	1	no	—
264	C₂	—	8	2	no	—
265	C₂	C₇×#1802	1	1	yes	characteristic
266	C₂	C₇×#2212	1	2	yes	—

	subgroup	G/H
center, socle	C₂²×C₇	#459
commutator	C₂×C₈	C₂²×C₄×C₇
Frattini, cosocle	C₂²×C₈	C₂³×C₇
prime radical	C₂×C₇×#135	C₂²
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₇ ◁ C₂²×C₄×C₇ ◁ C₂×C₄×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×#1846	C₂	semidirect
5. C₇×#1387	C₂	semidirect
10. C₇×#443	C₄	semidirect
20. #12820	C₇	direct (N × H)
52. C₂×#364	C₂×C₇	semidirect
54. #1846	C₂×C₇	semidirect
55. #1387	C₂×C₇	semidirect
97. #443	C₄×C₇	semidirect
248. C₇	#12820	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×#443	2
C₂×C₇×#364	2
C₇×#2139	2
C₇×#1846	2
C₇×#1387	2
#12820	7

Subgroups — C7×#12820 #13190036

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₇×#12820 #13190036