Subgroups — C₂²×C₇×#441 #13237369

Order 1792.

2790 conjugacy classes of subgroups.

The subgroup lattice has 214 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂³×C₇×Q₁₆	C₂	1	1	yes	maximal, characteristic
2	C₂²×C₄×C₇×Q₈	C₂	1	2	yes	maximal
3	C₂×C₇×#441	C₂	1	24	yes	maximal, direct product factor
4	C₂²×C₇×#106	C₂	1	1	yes	maximal, characteristic
5	C₂²×C₇×#103	C₂	1	2	yes	maximal
6	C₂²×C₇×#97	C₂	1	1	yes	maximal, characteristic
7	C₂³×C₇×Q₈	C₂²	1	2	yes	—
8	C₂²×C₇×Q₁₆	C₄	1	16	yes	complemented
9	C₂²×C₇×Q₁₆	C₂²	1	12	yes	—
10	C₂³×C₇×C₈	C₂²	1	1	yes	characteristic
11	C₂³×C₇×C₈	C₂²	1	1	yes	characteristic
12	C₂×C₄×C₇×Q₈	—	2	12	no	—
13	C₂×C₄×C₇×Q₈	C₂²	1	24	yes	—
14	C₂²×C₇×C₄⋊C₄	—	2	2	no	—
15	C₂²×C₇×C₄⋊C₄	C₂²	1	2	yes	—
16	C₂²×C₄²×C₇	—	2	2	no	—
17	C₂²×C₄²×C₇	C₂²	1	1	yes	characteristic
18	C₇×#441	C₂²	1	64	yes	direct product factor
19	C₂×C₇×#106	C₂²	1	12	yes	—
20	C₂×C₇×#103	C₂²	1	24	yes	—
21	C₂×C₇×#97	C₂²	1	12	yes	—
22	C₂²×#441	C₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
23	C₂²×C₇×Q₈	—	2	6	no	—
24	C₂²×C₇×Q₈	—	2	8	no	—
25	C₂²×C₇×Q₈	C₂×C₄	1	16	yes	—
26	C₂²×C₇×Q₈	C₂³	1	12	yes	—
27	C₂³×C₄×C₇	—	2	2	no	—
28	C₂³×C₄×C₇	—	2	2	no	—
29	C₂³×C₄×C₇	D₄	1	2	yes	—
30	C₂³×C₄×C₇	C₂³	1	1	yes	characteristic
31	C₂×C₇×Q₁₆	C₂×C₄	1	96	yes	complemented
32	C₂×C₇×Q₁₆	C₂³	1	16	yes	—
33	C₂²×C₇×C₈	—	2	4	no	—
34	C₂²×C₇×C₈	C₂×C₄	1	8	yes	—
35	C₂²×C₇×C₈	C₂³	1	6	yes	—
36	C₂²×C₇×C₈	C₂³	1	6	yes	—
37	C₄×C₇×Q₈	—	2	48	no	—
38	C₄×C₇×Q₈	C₂³	1	32	yes	—
39	C₂×C₇×C₄⋊C₄	—	2	24	no	—
40	C₂×C₇×C₄⋊C₄	—	2	6	no	—
41	C₂×C₇×C₄⋊C₄	C₂³	1	12	yes	—
42	C₂×C₄²×C₇	—	2	24	no	—
43	C₂×C₄²×C₇	D₄	1	6	yes	—
44	C₂×C₄²×C₇	C₂³	1	6	yes	—
45	C₇×#106	C₂³	1	16	yes	—
46	C₇×#103	C₂³	1	32	yes	—
47	C₇×#97	C₂³	1	16	yes	—
48	C₂³×Q₁₆	C₂×C₇	1	1	yes	characteristic
49	C₂²×C₄×Q₈	C₂×C₇	1	2	yes	—
50	C₂×#441	C₂×C₇	1	24	yes	direct product factor
51	C₂²×#106	C₂×C₇	1	1	yes	characteristic
52	C₂²×#103	C₂×C₇	1	2	yes	—
53	C₂²×#97	C₂×C₇	1	1	yes	characteristic
54	C₂⁴×C₇	C₂×D₄	1	1	yes	characteristic
55	C₂×C₇×Q₈	—	2	48	no	—
56	C₂×C₇×Q₈	—	2	12	no	—
57	C₂×C₇×Q₈	—	2	24	no	—
58	C₂×C₇×Q₈	C₂²×C₄	1	48	yes	—
59	C₂×C₇×Q₈	C₂⁴	1	8	yes	—
60	C₂²×C₄×C₇	—	2	16	no	—
61	C₂²×C₄×C₇	—	2	8	no	—
62	C₂²×C₄×C₇	—	2	12	no	—
63	C₂²×C₄×C₇	—	2	12	no	—
64	C₂²×C₄×C₇	—	4	8	no	—
65	C₂²×C₄×C₇	C₂²×C₄	1	4	yes	—
66	C₂²×C₄×C₇	C₂×D₄	1	12	yes	—
67	C₂²×C₄×C₇	C₂×D₄	1	3	yes	—
68	C₂²×C₄×C₇	#41	1	4	yes	—
69	C₂²×C₄×C₇	C₂⁴	1	3	yes	—
70	C₇×Q₁₆	C₂²×C₄	1	64	yes	complemented
71	C₂×C₇×C₈	—	2	12	no	—
72	C₂×C₇×C₈	C₂²×C₄	1	24	yes	—
73	C₂×C₇×C₈	C₂⁴	1	4	yes	—
74	C₂×C₇×C₈	C₂⁴	1	4	yes	—
75	C₇×C₄⋊C₄	—	2	32	no	—
76	C₇×C₄⋊C₄	—	2	12	no	—
77	C₇×C₄⋊C₄	C₂⁴	1	8	yes	—
78	C₄²×C₇	—	2	32	no	—
79	C₄²×C₇	C₂×D₄	1	12	yes	—
80	C₄²×C₇	C₂⁴	1	4	yes	—
81	C₂³×Q₈	C₂²×C₇	1	2	yes	—
82	C₂²×Q₁₆	C₄×C₇	1	16	yes	complemented
83	C₂²×Q₁₆	C₂²×C₇	1	12	yes	—
84	C₂³×C₈	C₂²×C₇	1	1	yes	characteristic
85	C₂³×C₈	C₂²×C₇	1	1	yes	characteristic
86	C₂×C₄×Q₈	—	2	12	no	—
87	C₂×C₄×Q₈	C₂²×C₇	1	24	yes	—
88	C₂²×C₄⋊C₄	—	2	2	no	—
89	C₂²×C₄⋊C₄	C₂²×C₇	1	2	yes	—
90	C₂²×C₄²	—	2	2	no	—
91	C₂²×C₄²	C₂²×C₇	1	1	yes	characteristic
92	#441	C₂²×C₇	1	64	yes	direct product factor
93	C₂×#106	C₂²×C₇	1	12	yes	—
94	C₂×#103	C₂²×C₇	1	24	yes	—
95	C₂×#97	C₂²×C₇	1	12	yes	—
96	C₂³×C₇	C₄×D₄	1	4	yes	—
97	C₂³×C₇	#137	1	8	yes	—
98	C₂³×C₇	C₂²×D₄	1	3	yes	—
99	C₇×Q₈	—	2	48	no	—
100	C₇×Q₈	—	2	8	no	—
101	C₇×Q₈	C₂³×C₄	1	16	yes	—
102	C₂×C₄×C₇	—	2	48	no	—
103	C₂×C₄×C₇	—	2	24	no	—
104	C₂×C₄×C₇	—	2	8	no	—
105	C₂×C₄×C₇	—	2	8	no	—
106	C₂×C₄×C₇	—	4	24	no	—
107	C₂×C₄×C₇	C₄×D₄	1	12	yes	—
108	C₂×C₄×C₇	C₂³×C₄	1	6	yes	—
109	C₂×C₄×C₇	C₂²×D₄	1	8	yes	—
110	C₂×C₄×C₇	C₂²×D₄	1	3	yes	—
111	C₂×C₄×C₇	C₂×#41	1	6	yes	—
112	C₂×C₄×C₇	C₂⁵	1	1	yes	characteristic
113	C₇×C₈	—	2	4	no	—
114	C₇×C₈	C₂³×C₄	1	8	yes	—
115	C₂²×Q₈	—	2	6	no	—
116	C₂²×Q₈	—	2	8	no	—
117	C₂²×Q₈	C₂×C₄×C₇	1	16	yes	—
118	C₂²×Q₈	C₂³×C₇	1	12	yes	—
119	C₂³×C₄	—	2	2	no	—
120	C₂³×C₄	—	2	2	no	—
121	C₂³×C₄	C₇×D₄	1	2	yes	—
122	C₂³×C₄	C₂³×C₇	1	1	yes	characteristic
123	C₂×Q₁₆	C₂×C₄×C₇	1	96	yes	complemented
124	C₂×Q₁₆	C₂³×C₇	1	16	yes	—
125	C₂²×C₈	—	2	4	no	—
126	C₂²×C₈	C₂×C₄×C₇	1	8	yes	—
127	C₂²×C₈	C₂³×C₇	1	6	yes	—
128	C₂²×C₈	C₂³×C₇	1	6	yes	—
129	C₄×Q₈	—	2	48	no	—
130	C₄×Q₈	C₂³×C₇	1	32	yes	—
131	C₂×C₄⋊C₄	—	2	24	no	—
132	C₂×C₄⋊C₄	—	2	6	no	—
133	C₂×C₄⋊C₄	C₂³×C₇	1	12	yes	—
134	C₂×C₄²	—	2	24	no	—
135	C₂×C₄²	C₇×D₄	1	6	yes	—
136	C₂×C₄²	C₂³×C₇	1	6	yes	—
137	#106	C₂³×C₇	1	16	yes	—
138	#103	C₂³×C₇	1	32	yes	—
139	#97	C₂³×C₇	1	16	yes	—
140	C₂²×C₇	#441	1	16	yes	direct product factor
141	C₂²×C₇	C₂×C₄×D₄	1	6	yes	—
142	C₂²×C₇	C₂×#137	1	12	yes	—
143	C₂²×C₇	C₂³×D₄	1	1	yes	characteristic
144	C₄×C₇	—	2	16	no	—
145	C₄×C₇	—	2	8	no	—
146	C₄×C₇	—	4	8	no	—
147	C₄×C₇	C₂×C₄×D₄	1	6	yes	—
148	C₄×C₇	C₂⁴×C₄	1	1	yes	characteristic
149	C₄×C₇	C₂²×#41	1	1	yes	characteristic
150	C₂⁴	C₂×C₇×D₄	1	1	yes	characteristic
151	C₂×Q₈	—	2	48	no	—
152	C₂×Q₈	—	2	12	no	—
153	C₂×Q₈	—	2	24	no	—
154	C₂×Q₈	C₂²×C₄×C₇	1	48	yes	—
155	C₂×Q₈	C₂⁴×C₇	1	8	yes	—
156	C₂²×C₄	—	2	16	no	—
157	C₂²×C₄	—	2	8	no	—
158	C₂²×C₄	—	2	12	no	—
159	C₂²×C₄	—	2	12	no	—
160	C₂²×C₄	—	4	8	no	—
161	C₂²×C₄	C₂²×C₄×C₇	1	4	yes	—
162	C₂²×C₄	C₂×C₇×D₄	1	12	yes	—
163	C₂²×C₄	C₂×C₇×D₄	1	3	yes	—
164	C₂²×C₄	C₇×#41	1	4	yes	—
165	C₂²×C₄	C₂⁴×C₇	1	3	yes	—
166	Q₁₆	C₂²×C₄×C₇	1	64	yes	complemented
167	C₂×C₈	—	2	12	no	—
168	C₂×C₈	C₂²×C₄×C₇	1	24	yes	—
169	C₂×C₈	C₂⁴×C₇	1	4	yes	—
170	C₂×C₈	C₂⁴×C₇	1	4	yes	—
171	C₄⋊C₄	—	2	32	no	—
172	C₄⋊C₄	—	2	12	no	—
173	C₄⋊C₄	C₂⁴×C₇	1	8	yes	—
174	C₄²	—	2	32	no	—
175	C₄²	C₂×C₇×D₄	1	12	yes	—
176	C₄²	C₂⁴×C₇	1	4	yes	—
177	C₂×C₇	C₂×#441	1	12	yes	direct product factor
178	C₂×C₇	C₂²×C₄×D₄	1	1	yes	characteristic
179	C₂×C₇	C₂²×#137	1	2	yes	—
180	C₂³	C₄×C₇×D₄	1	4	yes	—
181	C₂³	C₇×#137	1	8	yes	—
182	C₂³	C₂²×C₇×D₄	1	3	yes	—
183	Q₈	—	2	8	no	—
184	Q₈	—	2	48	no	—
185	Q₈	C₂³×C₄×C₇	1	16	yes	—
186	C₂×C₄	—	2	8	no	—
187	C₂×C₄	—	2	48	no	—
188	C₂×C₄	—	2	24	no	—
189	C₂×C₄	—	2	8	no	—
190	C₂×C₄	—	4	24	no	—
191	C₂×C₄	C₄×C₇×D₄	1	12	yes	—
192	C₂×C₄	C₂³×C₄×C₇	1	6	yes	—
193	C₂×C₄	C₂²×C₇×D₄	1	8	yes	—
194	C₂×C₄	C₂²×C₇×D₄	1	3	yes	—
195	C₂×C₄	C₂×C₇×#41	1	6	yes	—
196	C₂×C₄	C₂⁵×C₇	1	1	yes	characteristic
197	C₈	—	2	4	no	—
198	C₈	C₂³×C₄×C₇	1	8	yes	—
199	C₇	C₂²×#441	1	1	yes	characteristic, direct product factor, Sylow
200	C₂²	C₇×#441	1	16	yes	direct product factor
201	C₂²	C₂×C₄×C₇×D₄	1	6	yes	—
202	C₂²	C₂×C₇×#137	1	12	yes	—
203	C₂²	C₂³×C₇×D₄	1	1	yes	characteristic
204	C₄	—	2	8	no	—
205	C₄	—	2	16	no	—
206	C₄	—	4	8	no	—
207	C₄	C₂×C₄×C₇×D₄	1	6	yes	—
208	C₄	C₂⁴×C₄×C₇	1	1	yes	characteristic
209	C₄	C₂²×C₇×#41	1	1	yes	characteristic
210	C₂	C₂×C₇×#441	1	12	yes	direct product factor
211	C₂	C₂²×C₄×C₇×D₄	1	1	yes	characteristic
212	C₂	C₂²×C₇×#137	1	2	yes	—

	subgroup	G/H
center, socle, prime radical	C₂⁴×C₇	C₂×D₄
commutator	C₄	C₂⁴×C₄×C₇
Frattini, cosocle	C₂×C₄	C₂⁵×C₇
LCS	G ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂⁴×C₇ ◁ C₂²×C₄²×C₇ ◁ G

Complements

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

N	H ≅ G/N	type
3. C₂×C₇×#441	C₂	direct (N × H)
8. C₂²×C₇×Q₁₆	C₄	semidirect
18. C₇×#441	C₂²	direct (N × H)
22. C₂²×#441	C₇	direct (N × H)
31. C₂×C₇×Q₁₆	C₂×C₄	semidirect
50. C₂×#441	C₂×C₇	direct (N × H)
70. C₇×Q₁₆	C₂²×C₄	semidirect
82. C₂²×Q₁₆	C₄×C₇	semidirect
92. #441	C₂²×C₇	direct (N × H)
123. C₂×Q₁₆	C₂×C₄×C₇	semidirect
140. C₂²×C₇	#441	direct (N × H)
166. Q₁₆	C₂²×C₄×C₇	semidirect
177. C₂×C₇	C₂×#441	direct (N × H)
199. C₇	C₂²×#441	direct (N × H)
200. C₂²	C₇×#441	direct (N × H)
210. C₂	C₂×C₇×#441	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂³×C₇×Q₁₆	2
C₂²×C₄×C₇×Q₈	2
C₂×C₇×#441	2
C₂²×C₇×#106	2
C₂²×C₇×#103	2
C₂²×C₇×#97	2
C₂²×#441	7

Subgroups — C22×C7×#441 #13237369

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂²×C₇×#441 #13237369