Subgroups — C₂³×C₇×#102 #13237293

Order 1792.

7646 conjugacy classes of subgroups.

The subgroup lattice has 206 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	C₂⁴×C₇×C₈	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂³×C₇×C₄⋊C₄	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂²×C₇×#102	C₂	1	28	yes	maximal, direct product factor
5	C₂⁶×C₇	—	2	1	no	—
6	C₂³×C₇×D₄	—	2	7	no	—
7	C₂³×C₇×D₄	—	2	8	no	—
8	C₂³×C₇×D₄	C₄	1	16	yes	complemented
9	C₂³×C₇×D₄	C₂²	1	14	yes	—
10	C₂⁴×C₄×C₇	—	2	1	no	—
11	C₂⁴×C₄×C₇	C₂²	1	1	yes	characteristic
12	C₂³×C₇×C₈	—	2	8	no	—
13	C₂³×C₇×C₈	C₂²	1	14	yes	complemented
14	C₂²×C₇×C₄⋊C₄	—	2	7	no	—
15	C₂²×C₇×C₄⋊C₄	C₂²	1	14	yes	complemented
16	C₂×C₇×#102	C₂²	1	112	yes	direct product factor
17	C₂³×#102	C₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
18	C₂⁵×C₇	—	2	16	no	—
19	C₂⁵×C₇	—	2	14	no	—
20	C₂⁵×C₇	—	4	8	no	—
21	C₂⁵×C₇	—	4	8	no	—
22	C₂⁵×C₇	D₄	1	1	yes	characteristic
23	C₂²×C₇×D₄	—	2	112	no	—
24	C₂²×C₇×D₄	—	2	42	no	—
25	C₂²×C₇×D₄	—	2	56	no	—
26	C₂²×C₇×D₄	C₂×C₄	1	112	yes	complemented
27	C₂²×C₇×D₄	C₂³	1	28	yes	—
28	C₂³×C₄×C₇	—	2	14	no	—
29	C₂³×C₄×C₇	—	4	8	no	—
30	C₂³×C₄×C₇	C₂×C₄	1	8	yes	—
31	C₂³×C₄×C₇	D₄	1	7	yes	—
32	C₂³×C₄×C₇	D₄	1	8	yes	—
33	C₂³×C₄×C₇	C₂³	1	7	yes	—
34	C₂²×C₇×C₈	—	2	56	no	—
35	C₂²×C₇×C₈	C₂³	1	28	yes	complemented
36	C₂×C₇×C₄⋊C₄	—	2	42	no	—
37	C₂×C₇×C₄⋊C₄	C₂³	1	28	yes	complemented
38	C₇×#102	C₂³	1	64	yes	direct product factor
39	C₂⁴×D₄	C₂×C₇	1	1	yes	characteristic
40	C₂⁴×C₈	C₂×C₇	1	1	yes	characteristic, complemented
41	C₂³×C₄⋊C₄	C₂×C₇	1	1	yes	characteristic, complemented
42	C₂²×#102	C₂×C₇	1	28	yes	direct product factor
43	C₂⁴×C₇	—	2	112	no	—
44	C₂⁴×C₇	—	2	28	no	—
45	C₂⁴×C₇	—	4	128	no	—
46	C₂⁴×C₇	—	4	56	no	—
47	C₂⁴×C₇	—	4	56	no	—
48	C₂⁴×C₇	C₂²⋊C₄	1	8	yes	—
49	C₂⁴×C₇	D₈	1	8	yes	—
50	C₂⁴×C₇	SD₁₆	1	8	yes	—
51	C₂⁴×C₇	C₂×D₄	1	7	yes	—
52	C₂×C₇×D₄	—	2	336	no	—
53	C₂×C₇×D₄	—	2	28	no	—
54	C₂×C₇×D₄	—	2	56	no	—
55	C₂×C₇×D₄	C₂²×C₄	1	112	yes	complemented
56	C₂×C₇×D₄	C₂⁴	1	8	yes	—
57	C₂²×C₄×C₇	—	2	28	no	—
58	C₂²×C₄×C₇	—	4	56	no	—
59	C₂²×C₄×C₇	C₂²⋊C₄	1	56	yes	—
60	C₂²×C₄×C₇	C₂²×C₄	1	28	yes	—
61	C₂²×C₄×C₇	C₂×D₄	1	21	yes	—
62	C₂²×C₄×C₇	C₂×D₄	1	28	yes	—
63	C₂²×C₄×C₇	C₂⁴	1	7	yes	—
64	C₂×C₇×C₈	—	2	56	no	—
65	C₂×C₇×C₈	C₂⁴	1	8	yes	complemented
66	C₇×C₄⋊C₄	—	2	28	no	—
67	C₇×C₄⋊C₄	C₂⁴	1	8	yes	complemented
68	C₂⁶	—	2	1	no	—
69	C₂³×D₄	—	2	7	no	—
70	C₂³×D₄	—	2	8	no	—
71	C₂³×D₄	C₄×C₇	1	16	yes	complemented
72	C₂³×D₄	C₂²×C₇	1	14	yes	—
73	C₂⁴×C₄	—	2	1	no	—
74	C₂⁴×C₄	C₂²×C₇	1	1	yes	characteristic
75	C₂³×C₈	—	2	8	no	—
76	C₂³×C₈	C₂²×C₇	1	14	yes	complemented
77	C₂²×C₄⋊C₄	—	2	7	no	—
78	C₂²×C₄⋊C₄	C₂²×C₇	1	14	yes	complemented
79	C₂×#102	C₂²×C₇	1	112	yes	direct product factor
80	C₂³×C₇	—	2	112	no	—
81	C₂³×C₇	—	2	8	no	—
82	C₂³×C₇	—	4	56	no	—
83	C₂³×C₇	—	4	56	no	—
84	C₂³×C₇	—	4	448	no	—
85	C₂³×C₇	#102	1	64	yes	direct product factor
86	C₂³×C₇	C₂×C₂²⋊C₄	1	28	yes	—
87	C₂³×C₇	C₂×D₈	1	28	yes	—
88	C₂³×C₇	C₂×SD₁₆	1	28	yes	—
89	C₂³×C₇	C₂²×D₄	1	7	yes	—
90	C₇×D₄	—	2	112	no	—
91	C₇×D₄	—	2	8	no	—
92	C₇×D₄	C₂³×C₄	1	16	yes	complemented
93	C₂×C₄×C₇	—	2	8	no	—
94	C₂×C₄×C₇	—	4	56	no	—
95	C₂×C₄×C₇	C₂×C₂²⋊C₄	1	84	yes	—
96	C₂×C₄×C₇	C₂³×C₄	1	14	yes	—
97	C₂×C₄×C₇	C₂²×D₄	1	7	yes	—
98	C₂×C₄×C₇	C₂²×D₄	1	14	yes	—
99	C₂×C₄×C₇	C₂⁵	1	1	yes	characteristic
100	C₇×C₈	—	2	8	no	—
101	C₂⁵	—	2	16	no	—
102	C₂⁵	—	2	14	no	—
103	C₂⁵	—	4	8	no	—
104	C₂⁵	—	4	8	no	—
105	C₂⁵	C₇×D₄	1	1	yes	characteristic
106	C₂²×D₄	—	2	112	no	—
107	C₂²×D₄	—	2	42	no	—
108	C₂²×D₄	—	2	56	no	—
109	C₂²×D₄	C₂×C₄×C₇	1	112	yes	complemented
110	C₂²×D₄	C₂³×C₇	1	28	yes	—
111	C₂³×C₄	—	2	14	no	—
112	C₂³×C₄	—	4	8	no	—
113	C₂³×C₄	C₂×C₄×C₇	1	8	yes	—
114	C₂³×C₄	C₇×D₄	1	7	yes	—
115	C₂³×C₄	C₇×D₄	1	8	yes	—
116	C₂³×C₄	C₂³×C₇	1	7	yes	—
117	C₂²×C₈	—	2	56	no	—
118	C₂²×C₈	C₂³×C₇	1	28	yes	complemented
119	C₂×C₄⋊C₄	—	2	42	no	—
120	C₂×C₄⋊C₄	C₂³×C₇	1	28	yes	complemented
121	#102	C₂³×C₇	1	64	yes	direct product factor
122	C₂²×C₇	—	2	16	no	—
123	C₂²×C₇	—	4	224	no	—
124	C₂²×C₇	—	4	8	no	—
125	C₂²×C₇	—	4	8	no	—
126	C₂²×C₇	C₂×#102	1	112	yes	direct product factor
127	C₂²×C₇	C₂²×C₂²⋊C₄	1	14	yes	—
128	C₂²×C₇	C₂²×D₈	1	14	yes	—
129	C₂²×C₇	C₂²×SD₁₆	1	14	yes	—
130	C₂²×C₇	C₂³×D₄	1	1	yes	characteristic
131	C₄×C₇	—	4	8	no	—
132	C₄×C₇	C₂²×C₂²⋊C₄	1	14	yes	—
133	C₄×C₇	C₂⁴×C₄	1	1	yes	characteristic
134	C₄×C₇	C₂³×D₄	1	1	yes	characteristic
135	C₂⁴	—	2	112	no	—
136	C₂⁴	—	2	28	no	—
137	C₂⁴	—	4	128	no	—
138	C₂⁴	—	4	56	no	—
139	C₂⁴	—	4	56	no	—
140	C₂⁴	C₇×C₂²⋊C₄	1	8	yes	—
141	C₂⁴	C₇×D₈	1	8	yes	—
142	C₂⁴	C₇×SD₁₆	1	8	yes	—
143	C₂⁴	C₂×C₇×D₄	1	7	yes	—
144	C₂×D₄	—	2	336	no	—
145	C₂×D₄	—	2	56	no	—
146	C₂×D₄	—	2	28	no	—
147	C₂×D₄	C₂²×C₄×C₇	1	112	yes	complemented
148	C₂×D₄	C₂⁴×C₇	1	8	yes	—
149	C₂²×C₄	—	2	28	no	—
150	C₂²×C₄	—	4	56	no	—
151	C₂²×C₄	C₇×C₂²⋊C₄	1	56	yes	—
152	C₂²×C₄	C₂²×C₄×C₇	1	28	yes	—
153	C₂²×C₄	C₂×C₇×D₄	1	21	yes	—
154	C₂²×C₄	C₂×C₇×D₄	1	28	yes	—
155	C₂²×C₄	C₂⁴×C₇	1	7	yes	—
156	C₂×C₈	—	2	56	no	—
157	C₂×C₈	C₂⁴×C₇	1	8	yes	complemented
158	C₄⋊C₄	—	2	28	no	—
159	C₄⋊C₄	C₂⁴×C₇	1	8	yes	complemented
160	C₂×C₇	—	4	16	no	—
161	C₂×C₇	C₂²×#102	1	28	yes	direct product factor
162	C₂×C₇	C₂³×C₂²⋊C₄	1	1	yes	characteristic
163	C₂×C₇	C₂³×D₈	1	1	yes	characteristic
164	C₂×C₇	C₂³×SD₁₆	1	1	yes	characteristic
165	C₂³	—	2	8	no	—
166	C₂³	—	2	112	no	—
167	C₂³	—	4	448	no	—
168	C₂³	—	4	56	no	—
169	C₂³	—	4	56	no	—
170	C₂³	C₇×#102	1	64	yes	direct product factor
171	C₂³	C₂×C₇×C₂²⋊C₄	1	28	yes	—
172	C₂³	C₂×C₇×D₈	1	28	yes	—
173	C₂³	C₂×C₇×SD₁₆	1	28	yes	—
174	C₂³	C₂²×C₇×D₄	1	7	yes	—
175	D₄	—	2	8	no	—
176	D₄	—	2	112	no	—
177	D₄	C₂³×C₄×C₇	1	16	yes	complemented
178	C₂×C₄	—	2	8	no	—
179	C₂×C₄	—	4	56	no	—
180	C₂×C₄	C₂×C₇×C₂²⋊C₄	1	84	yes	—
181	C₂×C₄	C₂³×C₄×C₇	1	14	yes	—
182	C₂×C₄	C₂²×C₇×D₄	1	14	yes	—
183	C₂×C₄	C₂²×C₇×D₄	1	7	yes	—
184	C₂×C₄	C₂⁵×C₇	1	1	yes	characteristic
185	C₈	—	2	8	no	—
186	C₇	C₂³×#102	1	1	yes	characteristic, direct product factor, Sylow
187	C₂²	—	2	16	no	—
188	C₂²	—	4	8	no	—
189	C₂²	—	4	8	no	—
190	C₂²	—	4	224	no	—
191	C₂²	C₂×C₇×#102	1	112	yes	direct product factor
192	C₂²	C₂²×C₇×C₂²⋊C₄	1	14	yes	—
193	C₂²	C₂²×C₇×D₈	1	14	yes	—
194	C₂²	C₂²×C₇×SD₁₆	1	14	yes	—
195	C₂²	C₂³×C₇×D₄	1	1	yes	characteristic
196	C₄	—	4	8	no	—
197	C₄	C₂²×C₇×C₂²⋊C₄	1	14	yes	—
198	C₄	C₂⁴×C₄×C₇	1	1	yes	characteristic
199	C₄	C₂³×C₇×D₄	1	1	yes	characteristic
200	C₂	—	4	16	no	—
201	C₂	C₂²×C₇×#102	1	28	yes	direct product factor
202	C₂	C₂³×C₇×C₂²⋊C₄	1	1	yes	characteristic
203	C₂	C₂³×C₇×D₈	1	1	yes	characteristic
204	C₂	C₂³×C₇×SD₁₆	1	1	yes	characteristic

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

Complements

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

N	H ≅ G/N	type
2. C₂⁴×C₇×C₈	C₂	semidirect
3. C₂³×C₇×C₄⋊C₄	C₂	semidirect
4. C₂²×C₇×#102	C₂	direct (N × H)
8. C₂³×C₇×D₄	C₄	semidirect
13. C₂³×C₇×C₈	C₂²	semidirect
15. C₂²×C₇×C₄⋊C₄	C₂²	semidirect
16. C₂×C₇×#102	C₂²	direct (N × H)
17. C₂³×#102	C₇	direct (N × H)
26. C₂²×C₇×D₄	C₂×C₄	semidirect
35. C₂²×C₇×C₈	C₂³	semidirect
37. C₂×C₇×C₄⋊C₄	C₂³	semidirect
38. C₇×#102	C₂³	direct (N × H)
40. C₂⁴×C₈	C₂×C₇	semidirect
41. C₂³×C₄⋊C₄	C₂×C₇	semidirect
42. C₂²×#102	C₂×C₇	direct (N × H)
55. C₂×C₇×D₄	C₂²×C₄	semidirect
65. C₂×C₇×C₈	C₂⁴	semidirect
67. C₇×C₄⋊C₄	C₂⁴	semidirect
71. C₂³×D₄	C₄×C₇	semidirect
76. C₂³×C₈	C₂²×C₇	semidirect
78. C₂²×C₄⋊C₄	C₂²×C₇	semidirect
79. C₂×#102	C₂²×C₇	direct (N × H)
85. C₂³×C₇	#102	direct (N × H)
92. C₇×D₄	C₂³×C₄	semidirect
109. C₂²×D₄	C₂×C₄×C₇	semidirect
118. C₂²×C₈	C₂³×C₇	semidirect
120. C₂×C₄⋊C₄	C₂³×C₇	semidirect
121. #102	C₂³×C₇	direct (N × H)
126. C₂²×C₇	C₂×#102	direct (N × H)
147. C₂×D₄	C₂²×C₄×C₇	semidirect
157. C₂×C₈	C₂⁴×C₇	semidirect
159. C₄⋊C₄	C₂⁴×C₇	semidirect
161. C₂×C₇	C₂²×#102	direct (N × H)
170. C₂³	C₇×#102	direct (N × H)
177. D₄	C₂³×C₄×C₇	semidirect
186. C₇	C₂³×#102	direct (N × H)
191. C₂²	C₂×C₇×#102	direct (N × H)
201. C₂	C₂²×C₇×#102	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
C₂⁴×C₇×C₈	2
C₂³×C₇×C₄⋊C₄	2
C₂²×C₇×#102	2
C₂³×#102	7

Subgroups — C23×C7×#102 #13237293

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂³×C₇×#102 #13237293