Subgroups — C₂³×C₇×#99 #13237282

Order 1792.

7902 conjugacy classes of subgroups.

The subgroup lattice has 262 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂⁴×C₇×D₄	C₂	1	1	yes	maximal, characteristic
2	C₂³×C₇×C₂²⋊C₄	C₂	1	2	yes	maximal, complemented
3	C₂²×C₇×#99	C₂	1	28	yes	maximal, direct product factor
4	C₂⁶×C₇	C₄	1	1	yes	characteristic, complemented
5	C₂⁶×C₇	C₂²	1	1	yes	characteristic
6	C₂³×C₇×D₄	—	2	14	no	—
7	C₂³×C₇×D₄	—	2	2	no	—
8	C₂³×C₇×D₄	C₄	1	14	yes	complemented
9	C₂³×C₇×D₄	C₂²	1	14	yes	—
10	C₂⁴×C₄×C₇	—	2	2	no	—
11	C₂⁴×C₄×C₇	C₄	1	1	yes	characteristic, complemented
12	C₂²×C₇×C₂²⋊C₄	—	2	14	no	—
13	C₂²×C₇×C₂²⋊C₄	C₂²	1	28	yes	complemented
14	C₂×C₇×#99	C₂²	1	112	yes	direct product factor
15	C₂³×#99	C₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
16	C₂⁵×C₇	—	2	7	no	—
17	C₂⁵×C₇	—	2	1	no	—
18	C₂⁵×C₇	—	4	8	no	—
19	C₂⁵×C₇	—	4	8	no	—
20	C₂⁵×C₇	C₂×C₄	1	1	yes	characteristic
21	C₂⁵×C₇	C₂×C₄	1	7	yes	—
22	C₂⁵×C₇	C₂×C₄	1	14	yes	complemented
23	C₂⁵×C₇	D₄	1	2	yes	—
24	C₂⁵×C₇	D₄	1	14	yes	—
25	C₂⁵×C₇	C₂³	1	7	yes	—
26	C₂²×C₇×D₄	—	2	28	no	—
27	C₂²×C₇×D₄	—	2	84	no	—
28	C₂²×C₇×D₄	—	2	28	no	—
29	C₂²×C₇×D₄	—	2	84	no	—
30	C₂²×C₇×D₄	C₂×C₄	1	84	yes	complemented
31	C₂²×C₇×D₄	C₂³	1	28	yes	—
32	C₂³×C₄×C₇	—	2	7	no	—
33	C₂³×C₄×C₇	—	2	1	no	—
34	C₂³×C₄×C₇	—	2	28	no	—
35	C₂³×C₄×C₇	—	4	16	no	—
36	C₂³×C₄×C₇	C₂×C₄	1	14	yes	complemented
37	C₂×C₇×C₂²⋊C₄	—	2	84	no	—
38	C₂×C₇×C₂²⋊C₄	C₂³	1	56	yes	complemented
39	C₇×#99	C₂³	1	64	yes	direct product factor
40	C₂⁴×D₄	C₂×C₇	1	1	yes	characteristic
41	C₂³×C₂²⋊C₄	C₂×C₇	1	2	yes	complemented
42	C₂²×#99	C₂×C₇	1	28	yes	direct product factor
43	C₂⁴×C₇	—	2	16	no	—
44	C₂⁴×C₇	—	2	8	no	—
45	C₂⁴×C₇	—	2	42	no	—
46	C₂⁴×C₇	—	2	14	no	—
47	C₂⁴×C₇	—	4	28	no	—
48	C₂⁴×C₇	—	4	56	no	—
49	C₂⁴×C₇	—	4	28	no	—
50	C₂⁴×C₇	—	4	56	no	—
51	C₂⁴×C₇	—	4	56	no	—
52	C₂⁴×C₇	—	4	8	no	—
53	C₂⁴×C₇	C₂²⋊C₄	1	7	yes	—
54	C₂⁴×C₇	C₂²⋊C₄	1	14	yes	—
55	C₂⁴×C₇	C₂²⋊C₄	1	42	yes	—
56	C₂⁴×C₇	C₂²⋊C₄	1	1	yes	characteristic
57	C₂⁴×C₇	C₂²×C₄	1	7	yes	—
58	C₂⁴×C₇	C₂²×C₄	1	21	yes	—
59	C₂⁴×C₇	C₂²×C₄	1	28	yes	complemented
60	C₂⁴×C₇	C₂×D₄	1	14	yes	—
61	C₂⁴×C₇	C₂×D₄	1	42	yes	—
62	C₂⁴×C₇	C₂⁴	1	7	yes	—
63	C₂×C₇×D₄	—	2	168	no	—
64	C₂×C₇×D₄	—	2	168	no	—
65	C₂×C₇×D₄	—	2	56	no	—
66	C₂×C₇×D₄	—	2	56	no	—
67	C₂×C₇×D₄	C₂²×C₄	1	56	yes	complemented
68	C₂×C₇×D₄	C₂⁴	1	8	yes	—
69	C₂²×C₄×C₇	—	2	42	no	—
70	C₂²×C₄×C₇	—	2	14	no	—
71	C₂²×C₄×C₇	—	2	56	no	—
72	C₂²×C₄×C₇	—	4	112	no	—
73	C₂²×C₄×C₇	C₂²×C₄	1	28	yes	complemented
74	C₇×C₂²⋊C₄	—	2	56	no	—
75	C₇×C₂²⋊C₄	C₂⁴	1	16	yes	complemented
76	C₂⁶	C₄×C₇	1	1	yes	characteristic, complemented
77	C₂⁶	C₂²×C₇	1	1	yes	characteristic
78	C₂³×D₄	—	2	2	no	—
79	C₂³×D₄	—	2	14	no	—
80	C₂³×D₄	C₄×C₇	1	14	yes	complemented
81	C₂³×D₄	C₂²×C₇	1	14	yes	—
82	C₂⁴×C₄	—	2	2	no	—
83	C₂⁴×C₄	C₄×C₇	1	1	yes	characteristic, complemented
84	C₂²×C₂²⋊C₄	—	2	14	no	—
85	C₂²×C₂²⋊C₄	C₂²×C₇	1	28	yes	complemented
86	C₂×#99	C₂²×C₇	1	112	yes	direct product factor
87	C₂³×C₇	—	2	56	no	—
88	C₂³×C₇	—	2	28	no	—
89	C₂³×C₇	—	2	28	no	—
90	C₂³×C₇	—	2	56	no	—
91	C₂³×C₇	—	2	28	no	—
92	C₂³×C₇	—	2	28	no	—
93	C₂³×C₇	—	4	84	no	—
94	C₂³×C₇	—	4	42	no	—
95	C₂³×C₇	—	4	84	no	—
96	C₂³×C₇	—	4	14	no	—
97	C₂³×C₇	—	4	56	no	—
98	C₂³×C₇	—	4	168	no	—
99	C₂³×C₇	—	4	56	no	—
100	C₂³×C₇	#99	1	8	yes	direct product factor
101	C₂³×C₇	C₂×C₂²⋊C₄	1	21	yes	—
102	C₂³×C₇	C₂×C₂²⋊C₄	1	42	yes	—
103	C₂³×C₇	C₂×C₂²⋊C₄	1	42	yes	—
104	C₂³×C₇	C₂×C₂²⋊C₄	1	7	yes	—
105	C₂³×C₇	C₂³×C₄	1	7	yes	—
106	C₂³×C₇	C₂³×C₄	1	7	yes	—
107	C₂³×C₇	C₂³×C₄	1	8	yes	complemented
108	C₂³×C₇	C₂²×D₄	1	14	yes	—
109	C₂³×C₇	C₂²×D₄	1	14	yes	—
110	C₂³×C₇	C₂⁵	1	1	yes	characteristic
111	C₇×D₄	—	2	112	no	—
112	C₇×D₄	—	2	16	no	—
113	C₂×C₄×C₇	—	2	28	no	—
114	C₂×C₄×C₇	—	2	28	no	—
115	C₂×C₄×C₇	—	2	16	no	—
116	C₂×C₄×C₇	—	4	112	no	—
117	C₂×C₄×C₇	C₂³×C₄	1	8	yes	complemented
118	C₂⁵	—	2	7	no	—
119	C₂⁵	—	2	1	no	—
120	C₂⁵	—	4	8	no	—
121	C₂⁵	—	4	8	no	—
122	C₂⁵	C₂×C₄×C₇	1	1	yes	characteristic
123	C₂⁵	C₂×C₄×C₇	1	7	yes	—
124	C₂⁵	C₂×C₄×C₇	1	14	yes	complemented
125	C₂⁵	C₇×D₄	1	2	yes	—
126	C₂⁵	C₇×D₄	1	14	yes	—
127	C₂⁵	C₂³×C₇	1	7	yes	—
128	C₂²×D₄	—	2	28	no	—
129	C₂²×D₄	—	2	84	no	—
130	C₂²×D₄	—	2	28	no	—
131	C₂²×D₄	—	2	84	no	—
132	C₂²×D₄	C₂×C₄×C₇	1	84	yes	complemented
133	C₂²×D₄	C₂³×C₇	1	28	yes	—
134	C₂³×C₄	—	2	7	no	—
135	C₂³×C₄	—	2	1	no	—
136	C₂³×C₄	—	2	28	no	—
137	C₂³×C₄	—	4	16	no	—
138	C₂³×C₄	C₂×C₄×C₇	1	14	yes	complemented
139	C₂×C₂²⋊C₄	—	2	84	no	—
140	C₂×C₂²⋊C₄	C₂³×C₇	1	56	yes	complemented
141	#99	C₂³×C₇	1	64	yes	direct product factor
142	C₂²×C₇	—	2	28	no	—
143	C₂²×C₇	—	2	42	no	—
144	C₂²×C₇	—	2	14	no	—
145	C₂²×C₇	—	2	84	no	—
146	C₂²×C₇	—	2	8	no	—
147	C₂²×C₇	—	4	42	no	—
148	C₂²×C₇	—	4	7	no	—
149	C₂²×C₇	—	4	14	no	—
150	C₂²×C₇	—	4	1	no	—
151	C₂²×C₇	—	4	8	no	—
152	C₂²×C₇	—	4	56	no	—
153	C₂²×C₇	—	4	56	no	—
154	C₂²×C₇	C₂×#99	1	28	yes	direct product factor
155	C₂²×C₇	C₂²×C₂²⋊C₄	1	7	yes	—
156	C₂²×C₇	C₂²×C₂²⋊C₄	1	14	yes	—
157	C₂²×C₇	C₂²×C₂²⋊C₄	1	7	yes	—
158	C₂²×C₇	C₂⁴×C₄	1	1	yes	characteristic
159	C₂²×C₇	C₂³×D₄	1	2	yes	—
160	C₄×C₇	—	2	8	no	—
161	C₄×C₇	—	4	16	no	—
162	C₂⁴	—	2	16	no	—
163	C₂⁴	—	2	8	no	—
164	C₂⁴	—	2	42	no	—
165	C₂⁴	—	2	14	no	—
166	C₂⁴	—	4	28	no	—
167	C₂⁴	—	4	56	no	—
168	C₂⁴	—	4	28	no	—
169	C₂⁴	—	4	56	no	—
170	C₂⁴	—	4	56	no	—
171	C₂⁴	—	4	8	no	—
172	C₂⁴	C₇×C₂²⋊C₄	1	7	yes	—
173	C₂⁴	C₇×C₂²⋊C₄	1	14	yes	—
174	C₂⁴	C₇×C₂²⋊C₄	1	42	yes	—
175	C₂⁴	C₇×C₂²⋊C₄	1	1	yes	characteristic
176	C₂⁴	C₂²×C₄×C₇	1	7	yes	—
177	C₂⁴	C₂²×C₄×C₇	1	21	yes	—
178	C₂⁴	C₂²×C₄×C₇	1	28	yes	complemented
179	C₂⁴	C₂×C₇×D₄	1	14	yes	—
180	C₂⁴	C₂×C₇×D₄	1	42	yes	—
181	C₂⁴	C₂⁴×C₇	1	7	yes	—
182	C₂×D₄	—	2	168	no	—
183	C₂×D₄	—	2	168	no	—
184	C₂×D₄	—	2	56	no	—
185	C₂×D₄	—	2	56	no	—
186	C₂×D₄	C₂²×C₄×C₇	1	56	yes	complemented
187	C₂×D₄	C₂⁴×C₇	1	8	yes	—
188	C₂²×C₄	—	2	42	no	—
189	C₂²×C₄	—	2	14	no	—
190	C₂²×C₄	—	2	56	no	—
191	C₂²×C₄	—	4	112	no	—
192	C₂²×C₄	C₂²×C₄×C₇	1	28	yes	complemented
193	C₂²⋊C₄	—	2	56	no	—
194	C₂²⋊C₄	C₂⁴×C₇	1	16	yes	complemented
195	C₂×C₇	—	2	2	no	—
196	C₂×C₇	—	2	7	no	—
197	C₂×C₇	—	2	1	no	—
198	C₂×C₇	—	2	14	no	—
199	C₂×C₇	—	4	8	no	—
200	C₂×C₇	C₂²×#99	1	14	yes	direct product factor
201	C₂×C₇	C₂³×C₂²⋊C₄	1	1	yes	characteristic
202	C₂³	—	2	56	no	—
203	C₂³	—	2	28	no	—
204	C₂³	—	2	28	no	—
205	C₂³	—	2	56	no	—
206	C₂³	—	2	28	no	—
207	C₂³	—	2	28	no	—
208	C₂³	—	4	84	no	—
209	C₂³	—	4	42	no	—
210	C₂³	—	4	84	no	—
211	C₂³	—	4	14	no	—
212	C₂³	—	4	56	no	—
213	C₂³	—	4	168	no	—
214	C₂³	—	4	56	no	—
215	C₂³	C₇×#99	1	8	yes	direct product factor
216	C₂³	C₂×C₇×C₂²⋊C₄	1	21	yes	—
217	C₂³	C₂×C₇×C₂²⋊C₄	1	42	yes	—
218	C₂³	C₂×C₇×C₂²⋊C₄	1	42	yes	—
219	C₂³	C₂×C₇×C₂²⋊C₄	1	7	yes	—
220	C₂³	C₂³×C₄×C₇	1	7	yes	—
221	C₂³	C₂³×C₄×C₇	1	7	yes	—
222	C₂³	C₂³×C₄×C₇	1	8	yes	complemented
223	C₂³	C₂²×C₇×D₄	1	14	yes	—
224	C₂³	C₂²×C₇×D₄	1	14	yes	—
225	C₂³	C₂⁵×C₇	1	1	yes	characteristic
226	D₄	—	2	16	no	—
227	D₄	—	2	112	no	—
228	C₂×C₄	—	2	28	no	—
229	C₂×C₄	—	2	28	no	—
230	C₂×C₄	—	2	16	no	—
231	C₂×C₄	—	4	112	no	—
232	C₂×C₄	C₂³×C₄×C₇	1	8	yes	complemented
233	C₇	C₂³×#99	1	1	yes	characteristic, direct product factor, Sylow
234	C₂²	—	2	28	no	—
235	C₂²	—	2	42	no	—
236	C₂²	—	2	14	no	—
237	C₂²	—	2	84	no	—
238	C₂²	—	2	8	no	—
239	C₂²	—	4	42	no	—
240	C₂²	—	4	7	no	—
241	C₂²	—	4	14	no	—
242	C₂²	—	4	1	no	—
243	C₂²	—	4	8	no	—
244	C₂²	—	4	56	no	—
245	C₂²	—	4	56	no	—
246	C₂²	C₂×C₇×#99	1	28	yes	direct product factor
247	C₂²	C₂²×C₇×C₂²⋊C₄	1	7	yes	—
248	C₂²	C₂²×C₇×C₂²⋊C₄	1	14	yes	—
249	C₂²	C₂²×C₇×C₂²⋊C₄	1	7	yes	—
250	C₂²	C₂⁴×C₄×C₇	1	1	yes	characteristic
251	C₂²	C₂³×C₇×D₄	1	2	yes	—
252	C₄	—	2	8	no	—
253	C₄	—	4	16	no	—
254	C₂	—	2	2	no	—
255	C₂	—	2	7	no	—
256	C₂	—	2	1	no	—
257	C₂	—	2	14	no	—
258	C₂	—	4	8	no	—
259	C₂	C₂²×C₇×#99	1	14	yes	direct product factor
260	C₂	C₂³×C₇×C₂²⋊C₄	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂⁴×C₇	C₂²⋊C₄
commutator	C₂²	C₂⁴×C₄×C₇
Frattini, cosocle	C₂³	C₂⁵×C₇
prime radical	C₂⁴×C₇×D₄	C₂
LCS	G ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂⁴×C₇ ◁ C₂⁶×C₇ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₂³×C₇×C₂²⋊C₄	C₂	semidirect
3. C₂²×C₇×#99	C₂	direct (N × H)
4. C₂⁶×C₇	C₄	semidirect
8. C₂³×C₇×D₄	C₄	semidirect
11. C₂⁴×C₄×C₇	C₄	semidirect
13. C₂²×C₇×C₂²⋊C₄	C₂²	semidirect
14. C₂×C₇×#99	C₂²	direct (N × H)
15. C₂³×#99	C₇	direct (N × H)
22. C₂⁵×C₇	C₂×C₄	semidirect
30. C₂²×C₇×D₄	C₂×C₄	semidirect
36. C₂³×C₄×C₇	C₂×C₄	semidirect
38. C₂×C₇×C₂²⋊C₄	C₂³	semidirect
39. C₇×#99	C₂³	direct (N × H)
41. C₂³×C₂²⋊C₄	C₂×C₇	semidirect
42. C₂²×#99	C₂×C₇	direct (N × H)
59. C₂⁴×C₇	C₂²×C₄	semidirect
67. C₂×C₇×D₄	C₂²×C₄	semidirect
73. C₂²×C₄×C₇	C₂²×C₄	semidirect
75. C₇×C₂²⋊C₄	C₂⁴	semidirect
76. C₂⁶	C₄×C₇	semidirect
80. C₂³×D₄	C₄×C₇	semidirect
83. C₂⁴×C₄	C₄×C₇	semidirect
85. C₂²×C₂²⋊C₄	C₂²×C₇	semidirect
86. C₂×#99	C₂²×C₇	direct (N × H)
100. C₂³×C₇	#99	direct (N × H)
107. C₂³×C₇	C₂³×C₄	semidirect
117. C₂×C₄×C₇	C₂³×C₄	semidirect
124. C₂⁵	C₂×C₄×C₇	semidirect
132. C₂²×D₄	C₂×C₄×C₇	semidirect
138. C₂³×C₄	C₂×C₄×C₇	semidirect
140. C₂×C₂²⋊C₄	C₂³×C₇	semidirect
141. #99	C₂³×C₇	direct (N × H)
154. C₂²×C₇	C₂×#99	direct (N × H)
178. C₂⁴	C₂²×C₄×C₇	semidirect
186. C₂×D₄	C₂²×C₄×C₇	semidirect
192. C₂²×C₄	C₂²×C₄×C₇	semidirect
194. C₂²⋊C₄	C₂⁴×C₇	semidirect
200. C₂×C₇	C₂²×#99	direct (N × H)
215. C₂³	C₇×#99	direct (N × H)
222. C₂³	C₂³×C₄×C₇	semidirect
232. C₂×C₄	C₂³×C₄×C₇	semidirect
233. C₇	C₂³×#99	direct (N × H)
246. C₂²	C₂×C₇×#99	direct (N × H)
259. C₂	C₂²×C₇×#99	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂⁴×C₇×D₄	2
C₂³×C₇×C₂²⋊C₄	2
C₂²×C₇×#99	2
C₂³×#99	7

Subgroups — C23×C7×#99 #13237282

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂³×C₇×#99 #13237282