Subgroups — C₇×#21342 #13198558

Order 1792.

646 conjugacy classes of subgroups.

The subgroup lattice has 264 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₇×#2295	C₂	1	1	yes	maximal, characteristic
2	C₇×#1985	C₂	1	2	yes	maximal
3	C₇×#1940	C₂	1	2	yes	maximal
4	C₇×#1917	C₂	1	4	yes	maximal
5	C₇×#1768	C₂	1	2	yes	maximal
6	C₇×#1767	C₂	1	1	yes	maximal, characteristic
7	C₇×#1746	C₂	1	2	yes	maximal
8	C₄×C₇×#97	C₂	1	1	yes	maximal, characteristic
9	C₂×C₄×C₇×Q₈	C₂²	1	1	yes	characteristic
10	C₂×C₇×C₄⋊C₈	C₂²	1	2	yes	—
11	C₂×C₇×C₄⋊C₈	C₂²	1	4	yes	—
12	C₂×C₇×#97	C₂²	1	2	yes	—
13	C₂×C₇×#97	C₂²	1	2	yes	—
14	C₂×C₇×#97	C₂²	1	2	yes	—
15	C₂×C₄×C₇×C₈	C₂²	1	1	yes	characteristic
16	C₂×C₄×C₇×C₈	C₂²	1	1	yes	characteristic
17	C₇×#391	C₄	1	2	yes	—
18	C₇×#389	C₄	1	2	yes	—
19	C₇×#387	C₄	1	4	yes	—
20	C₄×C₇×C₄⋊C₄	C₂²	1	2	yes	—
21	C₄×C₇×C₄⋊C₄	C₂²	1	1	yes	characteristic
22	C₇×#376	C₂²	1	2	yes	—
23	C₄³×C₇	C₂²	1	1	yes	characteristic
24	C₇×#336	C₂²	1	4	yes	—
25	C₇×#336	C₂²	1	4	yes	—
26	C₇×#336	C₂²	1	4	yes	—
27	C₇×#336	C₂²	1	2	yes	—
28	#21342	C₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
29	C₂²×C₇×Q₈	C₂×C₄	1	1	yes	characteristic
30	C₂²×C₇×C₈	C₂³	1	4	yes	—
31	C₂²×C₇×C₈	C₂³	1	2	yes	—
32	C₂²×C₇×C₈	C₂³	1	2	yes	—
33	C₄×C₇×Q₈	—	2	4	no	—
34	C₂×C₇×C₄⋊C₄	C₂×C₄	1	2	yes	—
35	C₂×C₇×C₄⋊C₄	C₂×C₄	1	1	yes	characteristic
36	C₂×C₇×C₄⋊C₄	C₂×C₄	1	4	yes	—
37	C₂×C₇×C₄⋊C₄	C₂×C₄	1	2	yes	—
38	C₂×C₄²×C₇	C₂×C₄	1	2	yes	—
39	C₂×C₄²×C₇	C₂×C₄	1	2	yes	—
40	C₂×C₄²×C₇	C₂³	1	2	yes	—
41	C₂×C₄²×C₇	C₂³	1	1	yes	characteristic
42	C₂×C₄²×C₇	C₂³	1	2	yes	—
43	C₂×C₄²×C₇	C₂³	1	1	yes	characteristic
44	C₂×C₄²×C₇	C₂³	1	1	yes	characteristic
45	C₇×C₄⋊C₈	—	2	8	no	—
46	C₇×C₄⋊C₈	—	2	4	no	—
47	C₇×#97	—	2	4	no	—
48	C₇×#97	—	2	4	no	—
49	C₇×#97	—	2	4	no	—
50	C₄×C₇×C₈	—	2	2	no	—
51	C₄×C₇×C₈	Q₈	1	4	yes	complemented
52	C₇×#95	C₂×C₄	1	4	yes	—
53	C₇×#95	C₂×C₄	1	4	yes	—
54	C₇×#95	C₂×C₄	1	2	yes	—
55	C₇×#95	C₂×C₄	1	4	yes	—
56	#2295	C₂×C₇	1	1	yes	characteristic
57	#1985	C₂×C₇	1	2	yes	—
58	#1940	C₂×C₇	1	2	yes	—
59	#1917	C₂×C₇	1	4	yes	—
60	#1768	C₂×C₇	1	2	yes	—
61	#1767	C₂×C₇	1	1	yes	characteristic
62	#1746	C₂×C₇	1	2	yes	—
63	C₄×#97	C₂×C₇	1	1	yes	characteristic
64	C₂×C₇×Q₈	—	2	2	no	—
65	C₂×C₇×Q₈	—	4	2	no	—
66	C₂²×C₄×C₇	C₂²×C₄	1	4	yes	—
67	C₂²×C₄×C₇	C₂²×C₄	1	2	yes	—
68	C₂²×C₄×C₇	C₂²×C₄	1	2	yes	—
69	C₂²×C₄×C₇	C₂²×C₄	1	2	yes	—
70	C₂²×C₄×C₇	C₂²×C₄	1	1	yes	characteristic
71	C₂²×C₄×C₇	C₂²×C₄	1	1	yes	characteristic
72	C₂²×C₄×C₇	C₂²×C₄	1	2	yes	—
73	C₂²×C₄×C₇	C₂⁴	1	1	yes	characteristic
74	C₂×C₇×C₈	—	2	2	no	—
75	C₂×C₇×C₈	—	2	2	no	—
76	C₂×C₇×C₈	—	2	2	no	—
77	C₂×C₇×C₈	—	2	2	no	—
78	C₂×C₇×C₈	—	2	4	no	—
79	C₂×C₇×C₈	—	2	4	no	—
80	C₂×C₇×C₈	—	2	4	no	—
81	C₂×C₇×C₈	C₂×Q₈	1	4	yes	—
82	C₂×C₇×C₈	#41	1	4	yes	—
83	C₇×C₄⋊C₄	—	2	2	no	—
84	C₇×C₄⋊C₄	—	2	4	no	—
85	C₇×C₄⋊C₄	—	4	2	no	—
86	C₇×C₄⋊C₄	—	4	4	no	—
87	C₄²×C₇	—	2	2	no	—
88	C₄²×C₇	—	2	4	no	—
89	C₄²×C₇	—	2	4	no	—
90	C₄²×C₇	—	4	2	no	—
91	C₄²×C₇	—	4	2	no	—
92	C₄²×C₇	C₂×Q₈	1	2	yes	—
93	C₄²×C₇	#41	1	4	yes	—
94	C₄²×C₇	#41	1	2	yes	—
95	C₂×C₄×Q₈	C₂²×C₇	1	1	yes	characteristic
96	C₂×C₄⋊C₈	C₂²×C₇	1	4	yes	—
97	C₂×C₄⋊C₈	C₂²×C₇	1	2	yes	—
98	C₂×#97	C₂²×C₇	1	2	yes	—
99	C₂×#97	C₂²×C₇	1	2	yes	—
100	C₂×#97	C₂²×C₇	1	2	yes	—
101	C₂×C₄×C₈	C₂²×C₇	1	1	yes	characteristic
102	C₂×C₄×C₈	C₂²×C₇	1	1	yes	characteristic
103	#391	C₄×C₇	1	2	yes	—
104	#389	C₄×C₇	1	2	yes	—
105	#387	C₄×C₇	1	4	yes	—
106	C₄×C₄⋊C₄	C₂²×C₇	1	2	yes	—
107	C₄×C₄⋊C₄	C₂²×C₇	1	1	yes	characteristic
108	#376	C₂²×C₇	1	2	yes	—
109	C₄³	C₂²×C₇	1	1	yes	characteristic
110	#336	C₂²×C₇	1	2	yes	—
111	#336	C₂²×C₇	1	4	yes	—
112	#336	C₂²×C₇	1	4	yes	—
113	#336	C₂²×C₇	1	4	yes	—
114	C₂³×C₇	C₂³×C₄	1	1	yes	characteristic
115	C₇×Q₈	—	4	4	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	—	2	2	no	—
120	C₂×C₄×C₇	—	2	2	no	—
121	C₂×C₄×C₇	—	2	2	no	—
122	C₂×C₄×C₇	—	2	2	no	—
123	C₂×C₄×C₇	—	2	4	no	—
124	C₂×C₄×C₇	—	2	2	no	—
125	C₂×C₄×C₇	—	2	2	no	—
126	C₂×C₄×C₇	—	2	4	no	—
127	C₂×C₄×C₇	—	2	4	no	—
128	C₂×C₄×C₇	—	2	4	no	—
129	C₂×C₄×C₇	—	2	4	no	—
130	C₂×C₄×C₇	#117	1	2	yes	—
131	C₂×C₄×C₇	#117	1	2	yes	—
132	C₂×C₄×C₇	C₄×Q₈	1	4	yes	—
133	C₂×C₄×C₇	C₂²×Q₈	1	1	yes	characteristic
134	C₂×C₄×C₇	C₂×#41	1	1	yes	characteristic
135	C₂×C₄×C₇	C₂×#41	1	2	yes	—
136	C₂×C₄×C₇	C₂×#41	1	2	yes	—
137	C₇×C₈	—	2	4	no	—
138	C₇×C₈	—	2	4	no	—
139	C₇×C₈	—	4	4	no	—
140	C₂²×Q₈	C₂×C₄×C₇	1	1	yes	characteristic
141	C₂²×C₈	C₂³×C₇	1	4	yes	—
142	C₂²×C₈	C₂³×C₇	1	2	yes	—
143	C₂²×C₈	C₂³×C₇	1	2	yes	—
144	C₄×Q₈	—	2	4	no	—
145	C₂×C₄⋊C₄	C₂×C₄×C₇	1	2	yes	—
146	C₂×C₄⋊C₄	C₂×C₄×C₇	1	1	yes	characteristic
147	C₂×C₄⋊C₄	C₂×C₄×C₇	1	4	yes	—
148	C₂×C₄⋊C₄	C₂×C₄×C₇	1	2	yes	—
149	C₂×C₄²	C₂×C₄×C₇	1	2	yes	—
150	C₂×C₄²	C₂×C₄×C₇	1	2	yes	—
151	C₂×C₄²	C₂³×C₇	1	2	yes	—
152	C₂×C₄²	C₂³×C₇	1	1	yes	characteristic
153	C₂×C₄²	C₂³×C₇	1	2	yes	—
154	C₂×C₄²	C₂³×C₇	1	1	yes	characteristic
155	C₂×C₄²	C₂³×C₇	1	1	yes	characteristic
156	C₄⋊C₈	—	2	8	no	—
157	C₄⋊C₈	—	2	4	no	—
158	#97	—	2	4	no	—
159	#97	—	2	4	no	—
160	#97	—	2	4	no	—
161	C₄×C₈	—	2	2	no	—
162	C₄×C₈	C₇×Q₈	1	4	yes	complemented
163	#95	C₂×C₄×C₇	1	4	yes	—
164	#95	C₂×C₄×C₇	1	4	yes	—
165	#95	C₂×C₄×C₇	1	2	yes	—
166	#95	C₂×C₄×C₇	1	4	yes	—
167	C₂²×C₇	C₂×#117	1	1	yes	characteristic
168	C₂²×C₇	C₂×C₄×Q₈	1	1	yes	characteristic
169	C₂²×C₇	C₄×#41	1	1	yes	characteristic
170	C₂²×C₇	#568	1	2	yes	—
171	C₂²×C₇	#568	1	2	yes	—
172	C₄×C₇	—	2	2	no	—
173	C₄×C₇	—	2	2	no	—
174	C₄×C₇	—	2	4	no	—
175	C₄×C₇	—	4	2	no	—
176	C₄×C₇	—	4	2	no	—
177	C₄×C₇	—	4	2	no	—
178	C₄×C₇	—	4	4	no	—
179	C₄×C₇	#529	1	2	yes	—
180	C₄×C₇	#533	1	2	yes	—
181	C₂×Q₈	—	2	2	no	—
182	C₂×Q₈	—	4	2	no	—
183	C₂²×C₄	C₂²×C₄×C₇	1	4	yes	—
184	C₂²×C₄	C₂²×C₄×C₇	1	2	yes	—
185	C₂²×C₄	C₂²×C₄×C₇	1	2	yes	—
186	C₂²×C₄	C₂²×C₄×C₇	1	2	yes	—
187	C₂²×C₄	C₂²×C₄×C₇	1	1	yes	characteristic
188	C₂²×C₄	C₂²×C₄×C₇	1	1	yes	characteristic
189	C₂²×C₄	C₂²×C₄×C₇	1	2	yes	—
190	C₂²×C₄	C₂⁴×C₇	1	1	yes	characteristic
191	C₂×C₈	—	2	2	no	—
192	C₂×C₈	—	2	2	no	—
193	C₂×C₈	—	2	2	no	—
194	C₂×C₈	—	2	2	no	—
195	C₂×C₈	—	2	4	no	—
196	C₂×C₈	—	2	4	no	—
197	C₂×C₈	—	2	4	no	—
198	C₂×C₈	C₂×C₇×Q₈	1	4	yes	—
199	C₂×C₈	C₇×#41	1	4	yes	—
200	C₄⋊C₄	—	2	2	no	—
201	C₄⋊C₄	—	2	4	no	—
202	C₄⋊C₄	—	4	2	no	—
203	C₄⋊C₄	—	4	4	no	—
204	C₄²	—	2	2	no	—
205	C₄²	—	2	4	no	—
206	C₄²	—	2	4	no	—
207	C₄²	—	4	2	no	—
208	C₄²	—	4	2	no	—
209	C₄²	C₂×C₇×Q₈	1	2	yes	—
210	C₄²	C₇×#41	1	4	yes	—
211	C₄²	C₇×#41	1	2	yes	—
212	C₂×C₇	#2295	1	1	yes	characteristic
213	C₂×C₇	#2921	1	1	yes	characteristic
214	C₂×C₇	#2923	1	1	yes	characteristic
215	C₂×C₇	#2961	1	2	yes	—
216	C₂×C₇	#2967	1	2	yes	—
217	C₂³	C₂³×C₄×C₇	1	1	yes	characteristic
218	Q₈	—	4	4	no	—
219	C₂×C₄	—	2	2	no	—
220	C₂×C₄	—	2	2	no	—
221	C₂×C₄	—	2	2	no	—
222	C₂×C₄	—	2	2	no	—
223	C₂×C₄	—	2	2	no	—
224	C₂×C₄	—	2	2	no	—
225	C₂×C₄	—	2	2	no	—
226	C₂×C₄	—	2	4	no	—
227	C₂×C₄	—	2	2	no	—
228	C₂×C₄	—	2	2	no	—
229	C₂×C₄	—	2	4	no	—
230	C₂×C₄	—	2	4	no	—
231	C₂×C₄	—	2	4	no	—
232	C₂×C₄	—	2	4	no	—
233	C₂×C₄	C₇×#117	1	2	yes	—
234	C₂×C₄	C₇×#117	1	2	yes	—
235	C₂×C₄	C₄×C₇×Q₈	1	4	yes	—
236	C₂×C₄	C₂²×C₇×Q₈	1	1	yes	characteristic
237	C₂×C₄	C₂×C₇×#41	1	1	yes	characteristic
238	C₂×C₄	C₂×C₇×#41	1	2	yes	—
239	C₂×C₄	C₂×C₇×#41	1	2	yes	—
240	C₈	—	2	4	no	—
241	C₈	—	2	4	no	—
242	C₈	—	4	4	no	—
243	C₇	#21342	1	1	yes	characteristic, direct product factor, Sylow
244	C₂²	C₂×C₇×#117	1	1	yes	characteristic
245	C₂²	C₂×C₄×C₇×Q₈	1	1	yes	characteristic
246	C₂²	C₄×C₇×#41	1	1	yes	characteristic
247	C₂²	C₇×#568	1	2	yes	—
248	C₂²	C₇×#568	1	2	yes	—
249	C₄	—	2	2	no	—
250	C₄	—	2	2	no	—
251	C₄	—	2	4	no	—
252	C₄	—	4	2	no	—
253	C₄	—	4	2	no	—
254	C₄	—	4	2	no	—
255	C₄	—	4	4	no	—
256	C₄	C₇×#529	1	2	yes	—
257	C₄	C₇×#533	1	2	yes	—
258	C₂	C₇×#2295	1	1	yes	characteristic
259	C₂	C₇×#2921	1	1	yes	characteristic
260	C₂	C₇×#2923	1	1	yes	characteristic
261	C₂	C₇×#2961	1	2	yes	—
262	C₂	C₇×#2967	1	2	yes	—

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

Complements

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

N	H ≅ G/N	type
28. #21342	C₇	direct (N × H)
51. C₄×C₇×C₈	Q₈	semidirect
162. C₄×C₈	C₇×Q₈	semidirect
243. C₇	#21342	direct (N × H)

Maximal subgroups

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

subgroup	index
C₇×#2295	2
C₇×#1985	2
C₇×#1940	2
C₇×#1917	2
C₇×#1768	2
C₇×#1767	2
C₇×#1746	2
C₄×C₇×#97	2
#21342	7

Subgroups — C7×#21342 #13198558

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₇×#21342 #13198558