Subgroups — C₇×#12821 #13190037

Order 1792.

430 conjugacy classes of subgroups.

The subgroup lattice has 278 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×#1845	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	—	2	1	no	—
13	C₂×C₇×#107	C₂²	1	2	yes	—
14	C₂×C₇×#104	—	2	1	no	—
15	C₂×C₄×C₇×C₈	C₂²	1	1	yes	characteristic
16	C₇×#384	—	2	1	no	—
17	C₇×#364	—	2	2	no	—
18	C₇×#358	—	2	2	no	—
19	C₇×#357	—	2	2	no	—
20	#12821	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×C₇×C₄⋊C₄	—	4	2	no	—
38	C₂×C₄²×C₇	D₄	1	1	yes	characteristic
39	C₇×M₃₂	—	2	2	no	—
40	C₇×M₃₂	—	2	2	no	—
41	C₂×C₇×C₁₆	—	2	2	no	—
42	C₇×#108	—	2	1	no	—
43	C₇×#108	C₂×C₄	1	2	yes	—
44	C₇×#107	—	2	2	no	—
45	C₇×#107	—	2	2	no	—
46	C₇×#107	—	4	1	no	—
47	C₇×#107	—	4	1	no	—
48	C₇×#104	—	2	4	no	—
49	C₄×C₇×C₈	C₂×C₄	1	2	yes	—
50	C₄×C₇×C₈	D₄	1	2	yes	—
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	#1845	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	complemented
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	complemented
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	C₇×C₄⋊C₄	—	8	4	no	—
91	C₄²×C₇	—	2	1	no	—
92	C₄²×C₇	C₂²⋊C₄	1	2	yes	—
93	C₇×C₁₆	—	4	2	no	—
94	C₂×#135	C₂²×C₇	1	1	yes	characteristic
95	C₂×#131	—	2	1	no	—
96	C₂×M₃₂	C₂²×C₇	1	2	yes	—
97	#443	—	2	2	no	—
98	#443	C₄×C₇	1	4	yes	complemented
99	C₂×#108	C₂²×C₇	1	1	yes	characteristic
100	C₂×#107	—	2	1	no	—
101	C₂×#107	C₂²×C₇	1	2	yes	—
102	C₂×#104	—	2	1	no	—
103	C₂×C₄×C₈	C₂²×C₇	1	1	yes	characteristic
104	#384	—	2	1	no	—
105	#364	—	2	2	no	—
106	#358	—	2	2	no	—
107	#357	—	2	2	no	—
108	C₂³×C₇	—	4	1	no	—
109	C₂³×C₇	AGL(1,ℤ/8)	1	1	yes	characteristic
110	C₇×Q₈	—	2	2	no	—
111	C₇×Q₈	—	4	1	no	—
112	C₇×D₄	—	2	2	no	—
113	C₇×D₄	—	4	2	no	—
114	C₇×D₄	—	4	2	no	—
115	C₇×D₄	—	4	1	no	—
116	C₂×C₄×C₇	—	2	2	no	—
117	C₂×C₄×C₇	—	2	2	no	—
118	C₂×C₄×C₇	—	2	2	no	—
119	C₂×C₄×C₇	—	4	1	no	—
120	C₂×C₄×C₇	—	4	2	no	—
121	C₂×C₄×C₇	—	4	2	no	—
122	C₂×C₄×C₇	—	8	4	no	—
123	C₂×C₄×C₇	—	8	2	no	—
124	C₂×C₄×C₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
125	C₂×C₄×C₇	#121	1	1	yes	characteristic
126	C₂×C₄×C₇	C₂²⋊Q₈	1	1	yes	characteristic
127	C₂×C₄×C₇	C₂×D₈	1	1	yes	characteristic
128	C₂×C₄×C₇	C₂×SD₁₆	1	1	yes	characteristic
129	C₂×C₄×C₇	#137	1	1	yes	characteristic
130	C₇×C₈	—	4	2	no	—
131	C₇×C₈	#102	1	2	yes	—
132	C₇×C₈	#102	1	2	yes	—
133	C₇×C₈	C₂×C₄⋊C₄	1	1	yes	characteristic
134	C₇×C₈	#117	1	1	yes	characteristic
135	C₇×C₈	#121	1	1	yes	characteristic
136	C₇×C₈	C₂²⋊Q₈	1	1	yes	characteristic
137	C₂×#41	—	2	1	no	—
138	#135	—	2	1	no	—
139	#135	—	2	1	no	—
140	#135	C₂×C₄×C₇	1	2	yes	—
141	#135	C₇×Q₈	1	2	yes	—
142	C₂×Q₁₆	C₇×D₄	1	1	yes	characteristic
143	C₂×SD₁₆	—	2	1	no	—
144	C₂×D₈	C₇×D₄	1	1	yes	characteristic
145	#131	—	2	4	no	—
146	#131	—	4	2	no	—
147	C₂×M₁₆	—	2	1	no	—
148	C₂×M₁₆	—	4	1	no	—
149	C₂²×C₈	—	4	1	no	—
150	C₂²×C₈	C₇×D₄	1	1	yes	characteristic
151	C₂²×C₈	C₂³×C₇	1	1	yes	characteristic
152	C₂×C₄⋊C₄	—	2	2	no	—
153	C₂×C₄⋊C₄	—	4	2	no	—
154	C₂×C₄²	C₇×D₄	1	1	yes	characteristic
155	M₃₂	—	2	2	no	—
156	M₃₂	—	2	2	no	—
157	C₂×C₁₆	—	2	2	no	—
158	#108	—	2	1	no	—
159	#108	C₂×C₄×C₇	1	2	yes	—
160	#107	—	2	2	no	—
161	#107	—	2	2	no	—
162	#107	—	4	1	no	—
163	#107	—	4	1	no	—
164	#104	—	2	4	no	—
165	C₄×C₈	C₂×C₄×C₇	1	2	yes	—
166	C₄×C₈	C₇×D₄	1	2	yes	—
167	C₂²×C₇	—	2	1	no	—
168	C₂²×C₇	—	2	1	no	—
169	C₂²×C₇	—	4	2	no	—
170	C₂²×C₇	—	8	1	no	—
171	C₂²×C₇	—	8	1	no	—
172	C₂²×C₇	#417	1	1	yes	characteristic
173	C₂²×C₇	#459	1	1	yes	characteristic
174	C₂²×C₇	#476	1	1	yes	characteristic
175	C₄×C₇	—	2	4	no	—
176	C₄×C₇	—	4	2	no	—
177	C₄×C₇	—	16	2	no	—
178	C₄×C₇	#380	1	1	yes	characteristic
179	C₄×C₇	C₂×#102	1	1	yes	characteristic
180	C₄×C₇	#461	1	1	yes	characteristic
181	C₄×C₇	#474	1	1	yes	characteristic
182	#41	—	2	2	no	—
183	#41	—	2	2	no	—
184	#41	—	4	1	no	—
185	#41	—	4	1	no	—
186	C₂×Q₈	—	2	1	no	—
187	C₂×D₄	—	2	1	no	—
188	C₂×D₄	—	4	1	no	—
189	C₂²×C₄	—	2	1	no	—
190	C₂²×C₄	—	4	1	no	—
191	C₂²×C₄	—	4	2	no	—
192	C₂²×C₄	C₂×C₇×D₄	1	1	yes	characteristic
193	Q₁₆	—	2	1	no	—
194	Q₁₆	C₇×C₄⋊C₄	1	2	yes	complemented
195	SD₁₆	—	2	2	no	—
196	SD₁₆	—	2	2	no	—
197	D₈	—	2	1	no	—
198	D₈	C₇×C₄⋊C₄	1	2	yes	complemented
199	M₁₆	—	2	2	no	—
200	M₁₆	—	4	1	no	—
201	M₁₆	—	4	4	no	—
202	C₂×C₈	—	4	1	no	—
203	C₂×C₈	—	4	4	no	—
204	C₂×C₈	C₇×C₂²⋊C₄	1	2	yes	—
205	C₂×C₈	C₇×D₈	1	1	yes	characteristic
206	C₂×C₈	C₇×D₈	1	1	yes	characteristic
207	C₂×C₈	C₇×SD₁₆	1	2	yes	—
208	C₂×C₈	C₂²×C₄×C₇	1	1	yes	characteristic
209	C₂×C₈	C₂×C₇×D₄	1	1	yes	characteristic
210	C₂×C₈	C₂×C₇×D₄	1	1	yes	characteristic
211	C₂×C₈	C₂×C₇×Q₈	1	1	yes	characteristic
212	C₂×C₈	C₇×#41	1	1	yes	characteristic
213	C₂×C₈	C₇×#41	1	1	yes	characteristic
214	C₄⋊C₄	—	4	2	no	—
215	C₄⋊C₄	—	4	2	no	—
216	C₄⋊C₄	—	8	4	no	—
217	C₄²	—	2	1	no	—
218	C₄²	C₇×C₂²⋊C₄	1	2	yes	—
219	C₁₆	—	4	2	no	—
220	C₂×C₇	—	2	1	no	—
221	C₂×C₇	—	2	1	no	—
222	C₂×C₇	—	8	2	no	—
223	C₂×C₇	#1802	1	1	yes	characteristic
224	C₂×C₇	#2213	1	1	yes	characteristic
225	C₂×C₇	#2214	1	1	yes	characteristic
226	C₂³	—	4	1	no	—
227	C₂³	C₇×AGL(1,ℤ/8)	1	1	yes	characteristic
228	Q₈	—	2	2	no	—
229	Q₈	—	4	1	no	—
230	D₄	—	2	2	no	—
231	D₄	—	4	2	no	—
232	D₄	—	4	2	no	—
233	D₄	—	4	1	no	—
234	C₂×C₄	—	2	2	no	—
235	C₂×C₄	—	2	2	no	—
236	C₂×C₄	—	2	2	no	—
237	C₂×C₄	—	4	1	no	—
238	C₂×C₄	—	4	2	no	—
239	C₂×C₄	—	4	2	no	—
240	C₂×C₄	—	8	4	no	—
241	C₂×C₄	—	8	2	no	—
242	C₂×C₄	C₂×C₇×C₂²⋊C₄	1	1	yes	characteristic
243	C₂×C₄	C₇×#121	1	1	yes	characteristic
244	C₂×C₄	C₇×C₂²⋊Q₈	1	1	yes	characteristic
245	C₂×C₄	C₂×C₇×D₈	1	1	yes	characteristic
246	C₂×C₄	C₂×C₇×SD₁₆	1	1	yes	characteristic
247	C₂×C₄	C₇×#137	1	1	yes	characteristic
248	C₈	—	4	2	no	—
249	C₈	C₇×#102	1	2	yes	—
250	C₈	C₇×#102	1	2	yes	—
251	C₈	C₂×C₇×C₄⋊C₄	1	1	yes	characteristic
252	C₈	C₇×#117	1	1	yes	characteristic
253	C₈	C₇×#121	1	1	yes	characteristic
254	C₈	C₇×C₂²⋊Q₈	1	1	yes	characteristic
255	C₇	#12821	1	1	yes	characteristic, direct product factor, Sylow
256	C₂²	—	2	1	no	—
257	C₂²	—	2	1	no	—
258	C₂²	—	4	2	no	—
259	C₂²	—	8	1	no	—
260	C₂²	—	8	1	no	—
261	C₂²	C₇×#417	1	1	yes	characteristic
262	C₂²	C₇×#459	1	1	yes	characteristic
263	C₂²	C₇×#476	1	1	yes	characteristic
264	C₄	—	2	4	no	—
265	C₄	—	4	2	no	—
266	C₄	—	16	2	no	—
267	C₄	C₇×#380	1	1	yes	characteristic
268	C₄	C₂×C₇×#102	1	1	yes	characteristic
269	C₄	C₇×#461	1	1	yes	characteristic
270	C₄	C₇×#474	1	1	yes	characteristic
271	C₂	—	2	1	no	—
272	C₂	—	2	1	no	—
273	C₂	—	8	2	no	—
274	C₂	C₇×#1802	1	1	yes	characteristic
275	C₂	C₇×#2213	1	1	yes	characteristic
276	C₂	C₇×#2214	1	1	yes	characteristic

	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×#1845	C₂	semidirect
5. C₇×#1387	C₂	semidirect
10. C₇×#443	C₄	semidirect
20. #12821	C₇	direct (N × H)
52. C₂×#364	C₂×C₇	semidirect
54. #1845	C₂×C₇	semidirect
55. #1387	C₂×C₇	semidirect
68. C₇×Q₁₆	C₄⋊C₄	semidirect
72. C₇×D₈	C₄⋊C₄	semidirect
98. #443	C₄×C₇	semidirect
194. Q₁₆	C₇×C₄⋊C₄	semidirect
198. D₈	C₇×C₄⋊C₄	semidirect
255. C₇	#12821	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×#1845	2
C₇×#1387	2
#12821	7

Subgroups — C7×#12821 #13190037

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₇×#12821 #13190037