Subgroups — C₇×#22739 #13199955

Order 1792.

734 conjugacy classes of subgroups.

The subgroup lattice has 274 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₇×C₈×Q₈	C₂	1	1	yes	maximal, characteristic
2	C₇×#2355	C₂	1	2	yes	maximal
3	C₇×#1949	C₂	1	4	yes	maximal
4	C₇×#1947	C₂	1	2	yes	maximal
5	C₇×#1835	C₂	1	2	yes	maximal
6	C₇×#1834	C₂	1	1	yes	maximal, characteristic
7	C₄×C₇×#107	C₂	1	2	yes	maximal
8	C₄×C₇×#106	C₂	1	1	yes	maximal, characteristic
9	C₂×C₄×C₇×Q₈	C₂²	1	1	yes	characteristic
10	C₇×C₈×Q₈	C₄	1	8	yes	complemented
11	C₂×C₇×#107	C₂²	1	4	yes	—
12	C₂×C₇×#106	C₂²	1	2	yes	—
13	C₂×C₇×C₄⋊C₈	C₂²	1	1	yes	characteristic
14	C₂×C₇×C₄⋊C₈	C₂²	1	2	yes	—
15	C₂×C₄×C₇×C₈	C₂²	1	1	yes	characteristic
16	C₂×C₄×C₇×C₈	C₂²	1	2	yes	—
17	C₇×#391	—	2	2	no	—
18	C₇×#387	—	2	4	no	—
19	C₇×#384	C₂²	1	2	yes	—
20	C₇×#382	C₂²	1	4	yes	—
21	C₄×C₇×C₄⋊C₄	—	2	2	no	—
22	C₄×C₇×C₄⋊C₄	C₂²	1	2	yes	—
23	C₄×C₇×C₄⋊C₄	C₂²	1	4	yes	—
24	C₇×#340	C₂²	1	1	yes	characteristic
25	C₇×#340	C₂²	1	1	yes	characteristic
26	C₇×#340	C₂²	1	2	yes	—
27	C₇×#340	C₂²	1	2	yes	—
28	C₇×#340	C₂²	1	4	yes	—
29	#22739	C₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
30	C₂²×C₇×Q₈	D₄	1	1	yes	characteristic
31	C₂²×C₇×C₈	C₂³	1	1	yes	characteristic
32	C₂²×C₇×C₈	C₂³	1	1	yes	characteristic
33	C₂²×C₇×C₈	C₂³	1	2	yes	—
34	C₄×C₇×Q₈	C₂×C₄	1	4	yes	—
35	C₄×C₇×Q₈	Q₈	1	4	yes	—
36	C₂×C₇×C₄⋊C₄	—	2	2	no	—
37	C₂×C₇×C₄⋊C₄	—	2	2	no	—
38	C₂×C₇×C₄⋊C₄	D₄	1	1	yes	characteristic
39	C₂×C₇×C₄⋊C₄	D₄	1	2	yes	—
40	C₂×C₇×C₄⋊C₄	C₂³	1	2	yes	—
41	C₂×C₇×C₄⋊C₄	C₂³	1	4	yes	—
42	C₂×C₇×C₄⋊C₄	C₂³	1	2	yes	—
43	C₂×C₄²×C₇	—	2	2	no	—
44	C₂×C₄²×C₇	—	2	4	no	—
45	C₂×C₄²×C₇	C₂³	1	2	yes	—
46	C₂×C₄²×C₇	C₂³	1	1	yes	characteristic
47	C₇×#107	—	2	8	no	—
48	C₇×#106	—	2	4	no	—
49	C₇×C₄⋊C₈	C₂×C₄	1	4	yes	—
50	C₇×C₄⋊C₈	C₂×C₄	1	8	yes	—
51	C₄×C₇×C₈	C₂×C₄	1	4	yes	—
52	C₄×C₇×C₈	C₂×C₄	1	8	yes	—
53	C₇×#95	—	2	4	no	—
54	C₇×#95	—	2	2	no	—
55	C₇×#95	—	2	4	no	—
56	C₇×#95	—	2	4	no	—
57	C₂×C₈×Q₈	C₂×C₇	1	1	yes	characteristic
58	#2355	C₂×C₇	1	2	yes	—
59	#1949	C₂×C₇	1	4	yes	—
60	#1947	C₂×C₇	1	2	yes	—
61	#1835	C₂×C₇	1	2	yes	—
62	#1834	C₂×C₇	1	1	yes	characteristic
63	C₄×#107	C₂×C₇	1	2	yes	—
64	C₄×#106	C₂×C₇	1	1	yes	characteristic
65	C₂×C₇×Q₈	—	2	4	no	—
66	C₂×C₇×Q₈	C₄⋊C₄	1	4	yes	—
67	C₂²×C₄×C₇	—	2	2	no	—
68	C₂²×C₄×C₇	—	2	2	no	—
69	C₂²×C₄×C₇	—	2	4	no	—
70	C₂²×C₄×C₇	C₂×D₄	1	2	yes	—
71	C₂²×C₄×C₇	C₂×D₄	1	2	yes	—
72	C₂²×C₄×C₇	C₂×D₄	1	1	yes	characteristic
73	C₂²×C₄×C₇	C₂×D₄	1	1	yes	characteristic
74	C₂²×C₄×C₇	C₂⁴	1	1	yes	characteristic
75	C₂×C₇×C₈	—	2	2	no	—
76	C₂×C₇×C₈	—	2	4	no	—
77	C₂×C₇×C₈	—	2	2	no	—
78	C₂×C₇×C₈	C₂²×C₄	1	2	yes	—
79	C₂×C₇×C₈	C₂²×C₄	1	2	yes	—
80	C₂×C₇×C₈	C₂²×C₄	1	4	yes	—
81	C₇×C₄⋊C₄	—	2	2	no	—
82	C₇×C₄⋊C₄	—	2	2	no	—
83	C₇×C₄⋊C₄	—	2	4	no	—
84	C₇×C₄⋊C₄	—	2	2	no	—
85	C₇×C₄⋊C₄	—	2	4	no	—
86	C₇×C₄⋊C₄	—	4	4	no	—
87	C₇×C₄⋊C₄	—	8	2	no	—
88	C₇×C₄⋊C₄	C₄⋊C₄	1	8	yes	—
89	C₇×C₄⋊C₄	C₄⋊C₄	1	4	yes	—
90	C₇×C₄⋊C₄	#41	1	4	yes	—
91	C₄²×C₇	—	4	4	no	—
92	C₄²×C₇	—	8	4	no	—
93	C₄²×C₇	C₂²×C₄	1	4	yes	—
94	C₄²×C₇	C₂²×C₄	1	2	yes	—
95	C₄²×C₇	C₂×Q₈	1	4	yes	—
96	C₄²×C₇	C₂×Q₈	1	2	yes	—
97	C₂×C₄×Q₈	C₂²×C₇	1	1	yes	characteristic
98	C₈×Q₈	C₄×C₇	1	8	yes	complemented
99	C₂×#107	C₂²×C₇	1	4	yes	—
100	C₂×#106	C₂²×C₇	1	2	yes	—
101	C₂×C₄⋊C₈	C₂²×C₇	1	1	yes	characteristic
102	C₂×C₄⋊C₈	C₂²×C₇	1	2	yes	—
103	C₂×C₄×C₈	C₂²×C₇	1	1	yes	characteristic
104	C₂×C₄×C₈	C₂²×C₇	1	2	yes	—
105	#391	—	2	2	no	—
106	#387	—	2	4	no	—
107	#384	C₂²×C₇	1	2	yes	—
108	#382	C₂²×C₇	1	4	yes	—
109	C₄×C₄⋊C₄	—	2	2	no	—
110	C₄×C₄⋊C₄	C₂²×C₇	1	2	yes	—
111	C₄×C₄⋊C₄	C₂²×C₇	1	4	yes	—
112	#340	C₂²×C₇	1	2	yes	—
113	#340	C₂²×C₇	1	2	yes	—
114	#340	C₂²×C₇	1	1	yes	characteristic
115	#340	C₂²×C₇	1	4	yes	—
116	#340	C₂²×C₇	1	1	yes	characteristic
117	C₂³×C₇	C₂²×D₄	1	1	yes	characteristic
118	C₇×Q₈	—	2	8	no	—
119	C₂×C₄×C₇	—	2	4	no	—
120	C₂×C₄×C₇	—	2	2	no	—
121	C₂×C₄×C₇	—	2	1	no	—
122	C₂×C₄×C₇	—	2	2	no	—
123	C₂×C₄×C₇	—	2	1	no	—
124	C₂×C₄×C₇	—	2	4	no	—
125	C₂×C₄×C₇	—	4	2	no	—
126	C₂×C₄×C₇	—	4	2	no	—
127	C₂×C₄×C₇	—	4	4	no	—
128	C₂×C₄×C₇	—	4	2	no	—
129	C₂×C₄×C₇	—	4	2	no	—
130	C₂×C₄×C₇	—	4	4	no	—
131	C₂×C₄×C₇	—	4	4	no	—
132	C₂×C₄×C₇	—	4	2	no	—
133	C₂×C₄×C₇	C₂×C₄⋊C₄	1	4	yes	—
134	C₂×C₄×C₇	C₂×C₄⋊C₄	1	2	yes	—
135	C₂×C₄×C₇	C₂×C₄⋊C₄	1	4	yes	—
136	C₂×C₄×C₇	C₂×C₄⋊C₄	1	2	yes	—
137	C₂×C₄×C₇	#135	1	4	yes	—
138	C₂×C₄×C₇	C₂³×C₄	1	1	yes	characteristic
139	C₂×C₄×C₇	C₂²×Q₈	1	1	yes	characteristic
140	C₂×C₄×C₇	C₂×#41	1	2	yes	—
141	C₂×C₄×C₇	2₋¹⁺⁴	1	2	yes	—
142	C₇×C₈	—	2	2	no	—
143	C₇×C₈	—	2	4	no	—
144	C₇×C₈	—	2	2	no	—
145	C₂²×Q₈	C₇×D₄	1	1	yes	characteristic
146	C₂²×C₈	C₂³×C₇	1	1	yes	characteristic
147	C₂²×C₈	C₂³×C₇	1	1	yes	characteristic
148	C₂²×C₈	C₂³×C₇	1	2	yes	—
149	C₄×Q₈	C₂×C₄×C₇	1	4	yes	—
150	C₄×Q₈	C₇×Q₈	1	4	yes	—
151	C₂×C₄⋊C₄	—	2	2	no	—
152	C₂×C₄⋊C₄	—	2	2	no	—
153	C₂×C₄⋊C₄	C₇×D₄	1	1	yes	characteristic
154	C₂×C₄⋊C₄	C₇×D₄	1	2	yes	—
155	C₂×C₄⋊C₄	C₂³×C₇	1	2	yes	—
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	4	no	—
160	C₂×C₄²	C₂³×C₇	1	2	yes	—
161	C₂×C₄²	C₂³×C₇	1	1	yes	characteristic
162	#107	—	2	8	no	—
163	#106	—	2	4	no	—
164	C₄⋊C₈	C₂×C₄×C₇	1	4	yes	—
165	C₄⋊C₈	C₂×C₄×C₇	1	8	yes	—
166	C₄×C₈	C₂×C₄×C₇	1	4	yes	—
167	C₄×C₈	C₂×C₄×C₇	1	8	yes	—
168	#95	—	2	4	no	—
169	#95	—	2	2	no	—
170	#95	—	2	4	no	—
171	#95	—	2	4	no	—
172	C₂²×C₇	C₂²×C₄⋊C₄	1	1	yes	characteristic
173	C₂²×C₇	#548	1	2	yes	—
174	C₂²×C₇	C₂×#135	1	2	yes	—
175	C₂²×C₇	#576	1	1	yes	characteristic
176	C₂²×C₇	#578	1	1	yes	characteristic
177	C₄×C₇	—	2	4	no	—
178	C₄×C₇	—	2	4	no	—
179	C₄×C₇	—	2	2	no	—
180	C₄×C₇	—	4	4	no	—
181	C₄×C₇	—	8	4	no	—
182	C₄×C₇	—	8	2	no	—
183	C₄×C₇	#427	1	4	yes	—
184	C₄×C₇	C₄×#41	1	1	yes	characteristic
185	C₄×C₇	#519	1	1	yes	characteristic
186	C₄×C₇	#557	1	2	yes	—
187	C₂×Q₈	—	2	4	no	—
188	C₂×Q₈	C₇×C₄⋊C₄	1	4	yes	—
189	C₂²×C₄	—	2	2	no	—
190	C₂²×C₄	—	2	2	no	—
191	C₂²×C₄	—	2	4	no	—
192	C₂²×C₄	C₂×C₇×D₄	1	2	yes	—
193	C₂²×C₄	C₂×C₇×D₄	1	2	yes	—
194	C₂²×C₄	C₂×C₇×D₄	1	1	yes	characteristic
195	C₂²×C₄	C₂×C₇×D₄	1	1	yes	characteristic
196	C₂²×C₄	C₂⁴×C₇	1	1	yes	characteristic
197	C₂×C₈	—	2	2	no	—
198	C₂×C₈	—	2	4	no	—
199	C₂×C₈	—	2	2	no	—
200	C₂×C₈	C₂²×C₄×C₇	1	2	yes	—
201	C₂×C₈	C₂²×C₄×C₇	1	2	yes	—
202	C₂×C₈	C₂²×C₄×C₇	1	4	yes	—
203	C₄⋊C₄	—	2	2	no	—
204	C₄⋊C₄	—	2	2	no	—
205	C₄⋊C₄	—	2	4	no	—
206	C₄⋊C₄	—	2	2	no	—
207	C₄⋊C₄	—	2	4	no	—
208	C₄⋊C₄	—	4	4	no	—
209	C₄⋊C₄	—	8	2	no	—
210	C₄⋊C₄	C₇×C₄⋊C₄	1	8	yes	—
211	C₄⋊C₄	C₇×C₄⋊C₄	1	4	yes	—
212	C₄⋊C₄	C₇×#41	1	4	yes	—
213	C₄²	—	4	4	no	—
214	C₄²	—	8	4	no	—
215	C₄²	C₂²×C₄×C₇	1	4	yes	—
216	C₄²	C₂²×C₄×C₇	1	2	yes	—
217	C₄²	C₂×C₇×Q₈	1	4	yes	—
218	C₄²	C₂×C₇×Q₈	1	2	yes	—
219	C₂×C₇	#2351	1	1	yes	characteristic
220	C₂×C₇	C₂×#427	1	1	yes	characteristic
221	C₂×C₇	#2913	1	1	yes	characteristic
222	C₂×C₇	#3361	1	2	yes	—
223	C₂×C₇	#3364	1	2	yes	—
224	C₂³	C₂²×C₇×D₄	1	1	yes	characteristic
225	Q₈	—	2	8	no	—
226	C₂×C₄	—	2	4	no	—
227	C₂×C₄	—	2	2	no	—
228	C₂×C₄	—	2	1	no	—
229	C₂×C₄	—	2	2	no	—
230	C₂×C₄	—	2	1	no	—
231	C₂×C₄	—	2	4	no	—
232	C₂×C₄	—	4	2	no	—
233	C₂×C₄	—	4	2	no	—
234	C₂×C₄	—	4	4	no	—
235	C₂×C₄	—	4	2	no	—
236	C₂×C₄	—	4	2	no	—
237	C₂×C₄	—	4	4	no	—
238	C₂×C₄	—	4	4	no	—
239	C₂×C₄	—	4	2	no	—
240	C₂×C₄	C₂×C₇×C₄⋊C₄	1	4	yes	—
241	C₂×C₄	C₂×C₇×C₄⋊C₄	1	2	yes	—
242	C₂×C₄	C₂×C₇×C₄⋊C₄	1	4	yes	—
243	C₂×C₄	C₂×C₇×C₄⋊C₄	1	2	yes	—
244	C₂×C₄	C₇×#135	1	4	yes	—
245	C₂×C₄	C₂³×C₄×C₇	1	1	yes	characteristic
246	C₂×C₄	C₂²×C₇×Q₈	1	1	yes	characteristic
247	C₂×C₄	C₂×C₇×#41	1	2	yes	—
248	C₂×C₄	C₇×2₋¹⁺⁴	1	2	yes	—
249	C₈	—	2	2	no	—
250	C₈	—	2	4	no	—
251	C₈	—	2	2	no	—
252	C₇	#22739	1	1	yes	characteristic, direct product factor, Sylow
253	C₂²	C₂²×C₇×C₄⋊C₄	1	1	yes	characteristic
254	C₂²	C₇×#548	1	2	yes	—
255	C₂²	C₂×C₇×#135	1	2	yes	—
256	C₂²	C₇×#576	1	1	yes	characteristic
257	C₂²	C₇×#578	1	1	yes	characteristic
258	C₄	—	2	4	no	—
259	C₄	—	2	4	no	—
260	C₄	—	2	2	no	—
261	C₄	—	4	4	no	—
262	C₄	—	8	4	no	—
263	C₄	—	8	2	no	—
264	C₄	C₇×#427	1	4	yes	—
265	C₄	C₄×C₇×#41	1	1	yes	characteristic
266	C₄	C₇×#519	1	1	yes	characteristic
267	C₄	C₇×#557	1	2	yes	—
268	C₂	C₇×#2351	1	1	yes	characteristic
269	C₂	C₂×C₇×#427	1	1	yes	characteristic
270	C₂	C₇×#2913	1	1	yes	characteristic
271	C₂	C₇×#3361	1	2	yes	—
272	C₂	C₇×#3364	1	2	yes	—

	subgroup	G/H
center, socle, prime radical	C₂³×C₇	C₂²×D₄
commutator	C₂×C₄	C₂³×C₄×C₇
Frattini, cosocle	C₂²×C₄	C₂⁴×C₇
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₇ ◁ C₂×C₄×C₇×Q₈ ◁ G

Complements

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

N	H ≅ G/N	type
10. C₇×C₈×Q₈	C₄	semidirect
29. #22739	C₇	direct (N × H)
98. C₈×Q₈	C₄×C₇	semidirect
252. C₇	#22739	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×C₈×Q₈	2
C₇×#2355	2
C₇×#1949	2
C₇×#1947	2
C₇×#1835	2
C₇×#1834	2
C₄×C₇×#107	2
C₄×C₇×#106	2
#22739	7

Subgroups — C7×#22739 #13199955

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₇×#22739 #13199955