Subgroups — C₇×#21278 #13198494

Order 1792.

798 conjugacy classes of subgroups.

The subgroup lattice has 270 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₄×C₇×Q₁₆	C₂	1	1	yes	maximal, characteristic
2	C₇×#2295	C₂	1	2	yes	maximal
3	C₇×#1971	C₂	1	2	yes	maximal
4	C₇×#1943	C₂	1	2	yes	maximal
5	C₇×#1935	C₂	1	4	yes	maximal
6	C₄×C₇×#106	C₂	1	1	yes	maximal, characteristic
7	C₄×C₇×#103	C₂	1	2	yes	maximal
8	C₄×C₇×#97	C₂	1	1	yes	maximal, characteristic
9	C₂²×C₇×Q₁₆	C₂²	1	1	yes	characteristic
10	C₂×C₄×C₇×Q₈	C₂²	1	2	yes	—
11	C₄×C₇×Q₁₆	C₄	1	8	yes	complemented
12	C₂×C₇×#107	C₂²	1	1	yes	characteristic
13	C₂×C₇×#106	C₂²	1	2	yes	—
14	C₂×C₇×#103	C₂²	1	4	yes	—
15	C₂×C₇×#103	C₂²	1	2	yes	—
16	C₂×C₇×#97	C₂²	1	2	yes	—
17	C₂×C₇×#97	C₂²	1	2	yes	—
18	C₂×C₄×C₇×C₈	C₂²	1	1	yes	characteristic
19	C₂×C₄×C₇×C₈	C₂²	1	1	yes	characteristic
20	C₇×#391	C₂²	1	4	yes	—
21	C₇×#389	C₂²	1	4	yes	—
22	C₇×#387	—	2	4	no	—
23	C₄×C₇×C₄⋊C₄	—	2	2	no	—
24	C₄×C₇×C₄⋊C₄	C₂²	1	2	yes	—
25	C₇×#376	—	2	2	no	—
26	C₄³×C₇	C₂²	1	1	yes	characteristic
27	C₇×#340	C₂²	1	4	yes	—
28	C₇×#340	C₂²	1	2	yes	—
29	#21278	C₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
30	C₂²×C₇×Q₈	C₂³	1	2	yes	—
31	C₂×C₇×Q₁₆	—	2	4	no	—
32	C₂×C₇×Q₁₆	C₂×C₄	1	4	yes	—
33	C₂²×C₇×C₈	C₂³	1	1	yes	characteristic
34	C₂²×C₇×C₈	C₂³	1	2	yes	—
35	C₂²×C₇×C₈	C₂³	1	1	yes	characteristic
36	C₄×C₇×Q₈	—	2	4	no	—
37	C₄×C₇×Q₈	C₂×C₄	1	8	yes	—
38	C₂×C₇×C₄⋊C₄	—	2	2	no	—
39	C₂×C₇×C₄⋊C₄	—	2	4	no	—
40	C₂×C₇×C₄⋊C₄	C₂³	1	4	yes	—
41	C₂×C₇×C₄⋊C₄	C₂³	1	2	yes	—
42	C₂×C₄²×C₇	—	2	2	no	—
43	C₂×C₄²×C₇	—	2	2	no	—
44	C₂×C₄²×C₇	D₄	1	2	yes	—
45	C₂×C₄²×C₇	D₄	1	2	yes	—
46	C₂×C₄²×C₇	C₂³	1	2	yes	—
47	C₂×C₄²×C₇	C₂³	1	1	yes	characteristic
48	C₇×#107	C₂×C₄	1	4	yes	—
49	C₇×#106	—	2	4	no	—
50	C₇×#103	—	2	8	no	—
51	C₇×#103	C₂×C₄	1	8	yes	—
52	C₇×#97	—	2	4	no	—
53	C₇×#97	—	2	4	no	—
54	C₄×C₇×C₈	—	2	2	no	—
55	C₄×C₇×C₈	C₂×C₄	1	4	yes	—
56	C₇×#95	—	2	4	no	—
57	C₇×#95	—	2	4	no	—
58	C₇×#95	—	2	4	no	—
59	C₇×#95	—	2	2	no	—
60	C₂×C₄×Q₁₆	C₂×C₇	1	1	yes	characteristic
61	#2295	C₂×C₇	1	2	yes	—
62	#1971	C₂×C₇	1	2	yes	—
63	#1943	C₂×C₇	1	2	yes	—
64	#1935	C₂×C₇	1	4	yes	—
65	C₄×#106	C₂×C₇	1	1	yes	characteristic
66	C₄×#103	C₂×C₇	1	2	yes	—
67	C₄×#97	C₂×C₇	1	1	yes	characteristic
68	C₂×C₇×Q₈	—	2	2	no	—
69	C₂×C₇×Q₈	—	2	4	no	—
70	C₂×C₇×Q₈	—	2	4	no	—
71	C₂×C₇×Q₈	C₂²×C₄	1	4	yes	—
72	C₂²×C₄×C₇	—	2	2	no	—
73	C₂²×C₄×C₇	—	2	2	no	—
74	C₂²×C₄×C₇	—	2	4	no	—
75	C₂²×C₄×C₇	C₂×D₄	1	4	yes	—
76	C₂²×C₄×C₇	C₂×D₄	1	1	yes	characteristic
77	C₂²×C₄×C₇	C₂×D₄	1	1	yes	characteristic
78	C₂²×C₄×C₇	C₂⁴	1	1	yes	characteristic
79	C₇×Q₁₆	—	2	8	no	—
80	C₂×C₇×C₈	—	2	2	no	—
81	C₂×C₇×C₈	—	2	2	no	—
82	C₂×C₇×C₈	—	2	2	no	—
83	C₂×C₇×C₈	C₂²×C₄	1	2	yes	—
84	C₂×C₇×C₈	C₂²×C₄	1	2	yes	—
85	C₂×C₇×C₈	#41	1	4	yes	—
86	C₂×C₇×C₈	#41	1	4	yes	—
87	C₇×C₄⋊C₄	—	2	8	no	—
88	C₇×C₄⋊C₄	—	2	2	no	—
89	C₇×C₄⋊C₄	—	2	4	no	—
90	C₇×C₄⋊C₄	—	2	4	no	—
91	C₇×C₄⋊C₄	—	4	8	no	—
92	C₇×C₄⋊C₄	C₂²×C₄	1	4	yes	—
93	C₄²×C₇	—	2	8	no	—
94	C₄²×C₇	—	2	4	no	—
95	C₄²×C₇	—	2	2	no	—
96	C₄²×C₇	—	2	2	no	—
97	C₄²×C₇	—	4	2	no	—
98	C₄²×C₇	—	4	4	no	—
99	C₄²×C₇	C₂²×C₄	1	2	yes	—
100	C₄²×C₇	#41	1	2	yes	—
101	C₂²×Q₁₆	C₂²×C₇	1	1	yes	characteristic
102	C₂×C₄×Q₈	C₂²×C₇	1	2	yes	—
103	C₄×Q₁₆	C₄×C₇	1	8	yes	complemented
104	C₂×#107	C₂²×C₇	1	1	yes	characteristic
105	C₂×#106	C₂²×C₇	1	2	yes	—
106	C₂×#103	C₂²×C₇	1	4	yes	—
107	C₂×#103	C₂²×C₇	1	2	yes	—
108	C₂×#97	C₂²×C₇	1	2	yes	—
109	C₂×#97	C₂²×C₇	1	2	yes	—
110	C₂×C₄×C₈	C₂²×C₇	1	1	yes	characteristic
111	C₂×C₄×C₈	C₂²×C₇	1	1	yes	characteristic
112	#391	C₂²×C₇	1	4	yes	—
113	#389	C₂²×C₇	1	4	yes	—
114	#387	—	2	4	no	—
115	C₄×C₄⋊C₄	—	2	2	no	—
116	C₄×C₄⋊C₄	C₂²×C₇	1	2	yes	—
117	#376	—	2	2	no	—
118	C₄³	C₂²×C₇	1	1	yes	characteristic
119	#340	C₂²×C₇	1	2	yes	—
120	#340	C₂²×C₇	1	4	yes	—
121	C₂³×C₇	C₂²×D₄	1	1	yes	characteristic
122	C₇×Q₈	—	2	4	no	—
123	C₇×Q₈	—	2	4	no	—
124	C₇×Q₈	—	4	4	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	—	2	4	no	—
131	C₂×C₄×C₇	—	4	4	no	—
132	C₂×C₄×C₇	—	4	4	no	—
133	C₂×C₄×C₇	—	4	4	no	—
134	C₂×C₄×C₇	—	4	4	no	—
135	C₂×C₄×C₇	—	4	4	no	—
136	C₂×C₄×C₇	C₄×D₄	1	2	yes	—
137	C₂×C₄×C₇	C₄×D₄	1	2	yes	—
138	C₂×C₄×C₇	#137	1	4	yes	—
139	C₂×C₄×C₇	C₂³×C₄	1	1	yes	characteristic
140	C₂×C₄×C₇	C₂×#41	1	2	yes	—
141	C₂×C₄×C₇	C₂×#41	1	2	yes	—
142	C₂×C₄×C₇	C₂×#41	1	1	yes	characteristic
143	C₇×C₈	—	2	2	no	—
144	C₇×C₈	—	2	4	no	—
145	C₇×C₈	#117	1	4	yes	—
146	C₂²×Q₈	C₂³×C₇	1	2	yes	—
147	C₂×Q₁₆	—	2	4	no	—
148	C₂×Q₁₆	C₂×C₄×C₇	1	4	yes	—
149	C₂²×C₈	C₂³×C₇	1	1	yes	characteristic
150	C₂²×C₈	C₂³×C₇	1	2	yes	—
151	C₂²×C₈	C₂³×C₇	1	1	yes	characteristic
152	C₄×Q₈	—	2	4	no	—
153	C₄×Q₈	C₂×C₄×C₇	1	8	yes	—
154	C₂×C₄⋊C₄	—	2	2	no	—
155	C₂×C₄⋊C₄	—	2	4	no	—
156	C₂×C₄⋊C₄	C₂³×C₇	1	4	yes	—
157	C₂×C₄⋊C₄	C₂³×C₇	1	2	yes	—
158	C₂×C₄²	—	2	2	no	—
159	C₂×C₄²	—	2	2	no	—
160	C₂×C₄²	C₇×D₄	1	2	yes	—
161	C₂×C₄²	C₇×D₄	1	2	yes	—
162	C₂×C₄²	C₂³×C₇	1	2	yes	—
163	C₂×C₄²	C₂³×C₇	1	1	yes	characteristic
164	#107	C₂×C₄×C₇	1	4	yes	—
165	#106	—	2	4	no	—
166	#103	—	2	8	no	—
167	#103	C₂×C₄×C₇	1	8	yes	—
168	#97	—	2	4	no	—
169	#97	—	2	4	no	—
170	C₄×C₈	—	2	2	no	—
171	C₄×C₈	C₂×C₄×C₇	1	4	yes	—
172	#95	—	2	4	no	—
173	#95	—	2	4	no	—
174	#95	—	2	4	no	—
175	#95	—	2	2	no	—
176	C₂²×C₇	C₂×C₄×D₄	1	1	yes	characteristic
177	C₂²×C₇	#532	1	2	yes	—
178	C₂²×C₇	C₂×#137	1	2	yes	—
179	C₂²×C₇	#575	1	2	yes	—
180	C₄×C₇	—	2	2	no	—
181	C₄×C₇	—	4	4	no	—
182	C₄×C₇	—	4	4	no	—
183	C₄×C₇	—	4	4	no	—
184	C₄×C₇	—	8	4	no	—
185	C₄×C₇	#441	1	4	yes	—
186	C₄×C₇	C₂×#117	1	1	yes	characteristic
187	C₄×C₇	C₄×#41	1	1	yes	characteristic
188	C₄×C₇	#529	1	2	yes	—
189	C₂×Q₈	—	2	2	no	—
190	C₂×Q₈	—	2	4	no	—
191	C₂×Q₈	—	2	4	no	—
192	C₂×Q₈	C₂²×C₄×C₇	1	4	yes	—
193	C₂²×C₄	—	2	2	no	—
194	C₂²×C₄	—	2	2	no	—
195	C₂²×C₄	—	2	4	no	—
196	C₂²×C₄	C₂×C₇×D₄	1	4	yes	—
197	C₂²×C₄	C₂×C₇×D₄	1	1	yes	characteristic
198	C₂²×C₄	C₂×C₇×D₄	1	1	yes	characteristic
199	C₂²×C₄	C₂⁴×C₇	1	1	yes	characteristic
200	Q₁₆	—	2	8	no	—
201	C₂×C₈	—	2	2	no	—
202	C₂×C₈	—	2	2	no	—
203	C₂×C₈	—	2	2	no	—
204	C₂×C₈	C₂²×C₄×C₇	1	2	yes	—
205	C₂×C₈	C₂²×C₄×C₇	1	2	yes	—
206	C₂×C₈	C₇×#41	1	4	yes	—
207	C₂×C₈	C₇×#41	1	4	yes	—
208	C₄⋊C₄	—	2	8	no	—
209	C₄⋊C₄	—	2	2	no	—
210	C₄⋊C₄	—	2	4	no	—
211	C₄⋊C₄	—	2	4	no	—
212	C₄⋊C₄	—	4	8	no	—
213	C₄⋊C₄	C₂²×C₄×C₇	1	4	yes	—
214	C₄²	—	2	8	no	—
215	C₄²	—	2	4	no	—
216	C₄²	—	2	2	no	—
217	C₄²	—	2	2	no	—
218	C₄²	—	4	2	no	—
219	C₄²	—	4	4	no	—
220	C₄²	C₂²×C₄×C₇	1	2	yes	—
221	C₄²	C₇×#41	1	2	yes	—
222	C₂×C₇	#2293	1	1	yes	characteristic
223	C₂×C₇	C₂×#441	1	1	yes	characteristic
224	C₂×C₇	#2943	1	1	yes	characteristic
225	C₂×C₇	#3172	1	4	yes	—
226	C₂³	C₂²×C₇×D₄	1	1	yes	characteristic
227	Q₈	—	2	4	no	—
228	Q₈	—	2	4	no	—
229	Q₈	—	4	4	no	—
230	C₂×C₄	—	2	2	no	—
231	C₂×C₄	—	2	4	no	—
232	C₂×C₄	—	2	4	no	—
233	C₂×C₄	—	2	4	no	—
234	C₂×C₄	—	2	4	no	—
235	C₂×C₄	—	2	4	no	—
236	C₂×C₄	—	4	4	no	—
237	C₂×C₄	—	4	4	no	—
238	C₂×C₄	—	4	4	no	—
239	C₂×C₄	—	4	4	no	—
240	C₂×C₄	—	4	4	no	—
241	C₂×C₄	C₄×C₇×D₄	1	2	yes	—
242	C₂×C₄	C₄×C₇×D₄	1	2	yes	—
243	C₂×C₄	C₇×#137	1	4	yes	—
244	C₂×C₄	C₂³×C₄×C₇	1	1	yes	characteristic
245	C₂×C₄	C₂×C₇×#41	1	2	yes	—
246	C₂×C₄	C₂×C₇×#41	1	2	yes	—
247	C₂×C₄	C₂×C₇×#41	1	1	yes	characteristic
248	C₈	—	2	2	no	—
249	C₈	—	2	4	no	—
250	C₈	C₇×#117	1	4	yes	—
251	C₇	#21278	1	1	yes	characteristic, direct product factor, Sylow
252	C₂²	C₂×C₄×C₇×D₄	1	1	yes	characteristic
253	C₂²	C₇×#532	1	2	yes	—
254	C₂²	C₂×C₇×#137	1	2	yes	—
255	C₂²	C₇×#575	1	2	yes	—
256	C₄	—	2	2	no	—
257	C₄	—	4	4	no	—
258	C₄	—	4	4	no	—
259	C₄	—	4	4	no	—
260	C₄	—	8	4	no	—
261	C₄	C₇×#441	1	4	yes	—
262	C₄	C₂×C₇×#117	1	1	yes	characteristic
263	C₄	C₄×C₇×#41	1	1	yes	characteristic
264	C₄	C₇×#529	1	2	yes	—
265	C₂	C₇×#2293	1	1	yes	characteristic
266	C₂	C₂×C₇×#441	1	1	yes	characteristic
267	C₂	C₇×#2943	1	1	yes	characteristic
268	C₂	C₇×#3172	1	4	yes	—

	subgroup	G/H
center	C₂²×C₄×C₇	C₂×D₄
commutator	C₂×C₄	C₂³×C₄×C₇
Frattini, cosocle	C₂²×C₄	C₂⁴×C₇
socle, prime radical	C₂³×C₇	C₂²×D₄
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₄×C₇ ◁ C₄³×C₇ ◁ G

Complements

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

N	H ≅ G/N	type
11. C₄×C₇×Q₁₆	C₄	semidirect
29. #21278	C₇	direct (N × H)
103. C₄×Q₁₆	C₄×C₇	semidirect
251. C₇	#21278	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×C₄×C₇×Q₁₆	2
C₇×#2295	2
C₇×#1971	2
C₇×#1943	2
C₇×#1935	2
C₄×C₇×#106	2
C₄×C₇×#103	2
C₄×C₇×#97	2
#21278	7

Subgroups — C7×#21278 #13198494

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₇×#21278 #13198494