Subgroups — C₂²×#89696 #14217408

Order 1792.

3438 conjugacy classes of subgroups.

The subgroup lattice has 216 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂³×C₄²×C₇	C₂	1	1	yes	maximal, characteristic
2	C₂³×#1136	C₂	1	2	yes	maximal
3	C₂×#89696	C₂	1	24	yes	maximal, direct product factor
4	C₂²×C₄⋊(C₇⋊C₈)	C₂	1	4	yes	maximal, complemented
5	C₂⁴×C₄×C₇	C₄	1	2	yes	—
6	C₂⁴×C₄×C₇	C₂²	1	1	yes	characteristic
7	C₂²×C₄²×C₇	C₄	1	1	yes	characteristic
8	C₂²×C₄²×C₇	C₄	1	12	yes	—
9	C₂²×C₄²×C₇	C₄	1	1	yes	characteristic
10	C₂²×C₄²×C₇	C₂²	1	2	yes	—
11	C₂²×C₄²×C₇	C₂²	1	12	yes	—
12	C₂²×#1136	—	2	16	no	—
13	C₂²×#1136	C₂²	1	24	yes	—
14	C₂³×C₇⋊C₈	C₂²	1	4	yes	—
15	#89696	C₂²	1	64	yes	direct product factor
16	C₂×C₄⋊(C₇⋊C₈)	C₂²	1	48	yes	complemented
17	C₂²×#423	—	7	1	no	maximal, Sylow
18	C₂⁵×C₇	C₂×C₄	1	1	yes	characteristic
19	C₂³×C₄×C₇	—	2	16	no	—
20	C₂³×C₄×C₇	C₂×C₄	1	2	yes	—
21	C₂³×C₄×C₇	C₂×C₄	1	24	yes	—
22	C₂³×C₄×C₇	C₂×C₄	1	6	yes	—
23	C₂³×C₄×C₇	C₂×C₄	1	1	yes	characteristic
24	C₂³×C₄×C₇	C₂×C₄	1	2	yes	—
25	C₂³×C₄×C₇	D₄	1	8	yes	—
26	C₂³×C₄×C₇	Q₈	1	8	yes	—
27	C₂³×C₄×C₇	C₂³	1	6	yes	—
28	C₂³×C₄×C₇	C₂³	1	1	yes	characteristic
29	C₂×C₄²×C₇	C₂×C₄	1	12	yes	—
30	C₂×C₄²×C₇	C₂×C₄	1	48	yes	—
31	C₂×C₄²×C₇	C₂×C₄	1	12	yes	—
32	C₂×C₄²×C₇	C₂×C₄	1	12	yes	—
33	C₂×C₄²×C₇	C₂³	1	12	yes	—
34	C₂×C₄²×C₇	C₂³	1	16	yes	—
35	C₂×#1136	—	2	96	no	—
36	C₂×#1136	C₂³	1	32	yes	—
37	C₂²×C₇⋊C₈	—	2	16	no	—
38	C₂²×C₇⋊C₈	C₂³	1	24	yes	—
39	C₄⋊(C₇⋊C₈)	C₂³	1	64	yes	complemented
40	C₂³×M₁₆	—	7	2	no	—
41	C₂³×C₄²	D₇	1	1	yes	characteristic
42	C₂×#423	—	7	24	no	—
43	C₂²×C₄⋊C₈	—	7	4	no	—
44	C₂⁴×C₇	—	2	8	no	—
45	C₂⁴×C₇	C₄⋊C₄	1	8	yes	—
46	C₂⁴×C₇	C₂²×C₄	1	6	yes	—
47	C₂⁴×C₇	C₂²×C₄	1	1	yes	characteristic
48	C₂²×C₄×C₇	—	2	8	no	—
49	C₂²×C₄×C₇	—	2	96	no	—
50	C₂²×C₄×C₇	C₄⋊C₄	1	48	yes	—
51	C₂²×C₄×C₇	C₄⋊C₄	1	8	yes	—
52	C₂²×C₄×C₇	M₁₆	1	16	yes	complemented
53	C₂²×C₄×C₇	C₂²×C₄	1	12	yes	—
54	C₂²×C₄×C₇	C₂²×C₄	1	32	yes	—
55	C₂²×C₄×C₇	C₂²×C₄	1	12	yes	—
56	C₂²×C₄×C₇	C₂²×C₄	1	6	yes	—
57	C₂²×C₄×C₇	C₂²×C₄	1	3	yes	—
58	C₂²×C₄×C₇	C₂²×C₄	1	12	yes	—
59	C₂²×C₄×C₇	C₂×D₄	1	4	yes	—
60	C₂²×C₄×C₇	C₂×D₄	1	24	yes	—
61	C₂²×C₄×C₇	C₂×Q₈	1	24	yes	—
62	C₂²×C₄×C₇	C₂×Q₈	1	4	yes	—
63	C₂²×C₄×C₇	C₂⁴	1	3	yes	—
64	C₂²×C₄×C₇	C₂⁴	1	4	yes	—
65	C₄²×C₇	C₂²×C₄	1	16	yes	—
66	C₄²×C₇	C₂²×C₄	1	24	yes	—
67	C₄²×C₇	C₂²×C₄	1	16	yes	—
68	C₄²×C₇	C₂⁴	1	8	yes	—
69	#1136	—	2	64	no	—
70	C₂×C₇⋊C₈	—	2	48	no	—
71	C₂×C₇⋊C₈	C₂⁴	1	16	yes	—
72	C₂⁴×C₄	Dic₇	1	2	yes	—
73	C₂⁴×C₄	C₂×D₇	1	1	yes	characteristic
74	C₂²×M₁₆	—	7	24	no	—
75	C₂²×M₁₆	—	14	16	no	—
76	C₂³×C₈	—	7	4	no	—
77	C₂²×C₄²	Dic₇	1	1	yes	characteristic
78	C₂²×C₄²	Dic₇	1	12	yes	—
79	C₂²×C₄²	Dic₇	1	1	yes	characteristic
80	C₂²×C₄²	C₂×D₇	1	12	yes	—
81	C₂²×C₄²	C₂×D₇	1	2	yes	—
82	#423	—	7	64	no	—
83	C₂×C₄⋊C₈	—	7	48	no	—
84	C₂³×C₇	—	2	24	no	—
85	C₂³×C₇	—	2	32	no	—
86	C₂³×C₇	C₂×C₄⋊C₄	1	24	yes	—
87	C₂³×C₇	C₂×C₄⋊C₄	1	4	yes	—
88	C₂³×C₇	C₂×M₁₆	1	8	yes	—
89	C₂³×C₇	C₂³×C₄	1	4	yes	—
90	C₂³×C₇	C₂³×C₄	1	3	yes	—
91	C₂×C₄×C₇	—	2	24	no	—
92	C₂×C₄×C₇	—	2	64	no	—
93	C₂×C₄×C₇	C₂×C₄⋊C₄	1	48	yes	—
94	C₂×C₄×C₇	C₂×C₄⋊C₄	1	12	yes	—
95	C₂×C₄×C₇	C₂×C₄⋊C₄	1	24	yes	—
96	C₂×C₄×C₇	C₂×M₁₆	1	48	yes	complemented
97	C₂×C₄×C₇	C₂³×C₄	1	8	yes	—
98	C₂×C₄×C₇	C₂³×C₄	1	3	yes	—
99	C₂×C₄×C₇	C₂³×C₄	1	4	yes	—
100	C₂×C₄×C₇	C₂³×C₄	1	8	yes	—
101	C₂×C₄×C₇	C₂²×D₄	1	6	yes	—
102	C₂×C₄×C₇	C₂²×D₄	1	8	yes	—
103	C₂×C₄×C₇	C₂²×Q₈	1	8	yes	—
104	C₂×C₄×C₇	C₂²×Q₈	1	6	yes	—
105	C₂×C₄×C₇	C₂⁵	1	1	yes	characteristic
106	C₇⋊C₈	—	2	16	no	—
107	C₂⁵	C₂×Dic₇	1	1	yes	characteristic
108	C₂³×C₄	—	2	16	no	—
109	C₂³×C₄	Dic₁₄	1	8	yes	—
110	C₂³×C₄	D₂₈	1	8	yes	—
111	C₂³×C₄	C₂×Dic₇	1	2	yes	—
112	C₂³×C₄	C₂×Dic₇	1	24	yes	—
113	C₂³×C₄	C₂×Dic₇	1	2	yes	—
114	C₂³×C₄	C₂×Dic₇	1	1	yes	characteristic
115	C₂³×C₄	C₂×Dic₇	1	6	yes	—
116	C₂³×C₄	C₂²×D₇	1	6	yes	—
117	C₂³×C₄	C₂²×D₇	1	1	yes	characteristic
118	C₂×M₁₆	—	7	32	no	—
119	C₂×M₁₆	—	14	96	no	—
120	C₂²×C₈	—	7	24	no	—
121	C₂²×C₈	—	14	16	no	—
122	C₂×C₄²	C₂×Dic₇	1	12	yes	—
123	C₂×C₄²	C₂×Dic₇	1	48	yes	—
124	C₂×C₄²	C₂×Dic₇	1	12	yes	—
125	C₂×C₄²	C₂×Dic₇	1	12	yes	—
126	C₂×C₄²	C₂²×D₇	1	12	yes	—
127	C₂×C₄²	C₂²×D₇	1	16	yes	—
128	C₄⋊C₈	—	7	64	no	—
129	C₂²×C₇	—	2	48	no	—
130	C₂²×C₇	—	2	8	no	—
131	C₂²×C₇	#423	1	16	yes	complemented
132	C₂²×C₇	C₂²×C₄⋊C₄	1	8	yes	—
133	C₂²×C₇	C₂²×C₄⋊C₄	1	6	yes	—
134	C₂²×C₇	C₂²×M₁₆	1	12	yes	—
135	C₂²×C₇	C₂⁴×C₄	1	1	yes	characteristic
136	C₄×C₇	—	2	8	no	—
137	C₄×C₇	C₂²×C₄⋊C₄	1	6	yes	—
138	C₄×C₇	C₂²×C₄⋊C₄	1	8	yes	—
139	C₄×C₇	C₂²×M₁₆	1	16	yes	complemented
140	C₄×C₇	C₂³×D₄	1	1	yes	characteristic
141	C₄×C₇	C₂³×Q₈	1	1	yes	characteristic
142	C₂⁴	—	2	8	no	—
143	C₂⁴	C₄⋊Dic₇	1	8	yes	—
144	C₂⁴	C₂²×Dic₇	1	6	yes	—
145	C₂⁴	C₂²×Dic₇	1	1	yes	characteristic
146	C₂²×C₄	—	2	8	no	—
147	C₂²×C₄	—	2	96	no	—
148	C₂²×C₄	#1136	1	16	yes	complemented
149	C₂²×C₄	C₄⋊Dic₇	1	48	yes	—
150	C₂²×C₄	C₄⋊Dic₇	1	8	yes	—
151	C₂²×C₄	C₂×Dic₁₄	1	24	yes	—
152	C₂²×C₄	C₂×Dic₁₄	1	4	yes	—
153	C₂²×C₄	C₂×D₂₈	1	4	yes	—
154	C₂²×C₄	C₂×D₂₈	1	24	yes	—
155	C₂²×C₄	C₂²×Dic₇	1	12	yes	—
156	C₂²×C₄	C₂²×Dic₇	1	32	yes	—
157	C₂²×C₄	C₂²×Dic₇	1	12	yes	—
158	C₂²×C₄	C₂²×Dic₇	1	12	yes	—
159	C₂²×C₄	C₂²×Dic₇	1	6	yes	—
160	C₂²×C₄	C₂²×Dic₇	1	3	yes	—
161	C₂²×C₄	C₂³×D₇	1	3	yes	—
162	C₂²×C₄	C₂³×D₇	1	4	yes	—
163	M₁₆	—	14	64	no	—
164	C₂×C₈	—	7	16	no	—
165	C₂×C₈	—	14	48	no	—
166	C₄²	C₂²×Dic₇	1	16	yes	—
167	C₄²	C₂²×Dic₇	1	24	yes	—
168	C₄²	C₂²×Dic₇	1	16	yes	—
169	C₄²	C₂³×D₇	1	8	yes	—
170	C₂×C₇	—	2	8	no	—
171	C₂×C₇	C₂×#423	1	12	yes	complemented
172	C₂×C₇	C₂³×C₄⋊C₄	1	1	yes	characteristic
173	C₂×C₇	C₂³×M₁₆	1	2	yes	—
174	C₂³	—	2	32	no	—
175	C₂³	—	2	24	no	—
176	C₂³	C₂×#1136	1	8	yes	—
177	C₂³	C₂×C₄⋊Dic₇	1	24	yes	—
178	C₂³	C₂×C₄⋊Dic₇	1	4	yes	—
179	C₂³	C₂³×Dic₇	1	4	yes	—
180	C₂³	C₂³×Dic₇	1	3	yes	—
181	C₂×C₄	—	2	24	no	—
182	C₂×C₄	—	2	64	no	—
183	C₂×C₄	C₂×#1136	1	48	yes	complemented
184	C₂×C₄	C₂×C₄⋊Dic₇	1	48	yes	—
185	C₂×C₄	C₂×C₄⋊Dic₇	1	24	yes	—
186	C₂×C₄	C₂×C₄⋊Dic₇	1	12	yes	—
187	C₂×C₄	C₂²×Dic₁₄	1	8	yes	—
188	C₂×C₄	C₂²×Dic₁₄	1	6	yes	—
189	C₂×C₄	C₂²×D₂₈	1	6	yes	—
190	C₂×C₄	C₂²×D₂₈	1	8	yes	—
191	C₂×C₄	C₂³×Dic₇	1	8	yes	—
192	C₂×C₄	C₂³×Dic₇	1	8	yes	—
193	C₂×C₄	C₂³×Dic₇	1	4	yes	—
194	C₂×C₄	C₂³×Dic₇	1	3	yes	—
195	C₂×C₄	C₂⁴×D₇	1	1	yes	characteristic
196	C₈	—	14	16	no	—
197	C₇	C₂²×#423	1	1	yes	characteristic, complemented, Sylow
198	C₂²	—	2	8	no	—
199	C₂²	—	2	48	no	—
200	C₂²	#89696	1	16	yes	direct product factor
201	C₂²	C₂²×#1136	1	12	yes	—
202	C₂²	C₂²×C₄⋊Dic₇	1	8	yes	—
203	C₂²	C₂²×C₄⋊Dic₇	1	6	yes	—
204	C₂²	C₂⁴×Dic₇	1	1	yes	characteristic
205	C₄	—	2	8	no	—
206	C₄	C₂²×#1136	1	16	yes	complemented
207	C₄	C₂²×C₄⋊Dic₇	1	6	yes	—
208	C₄	C₂²×C₄⋊Dic₇	1	8	yes	—
209	C₄	C₂³×Dic₁₄	1	1	yes	characteristic
210	C₄	C₂³×D₂₈	1	1	yes	characteristic
211	C₂	—	2	8	no	—
212	C₂	C₂×#89696	1	12	yes	direct product factor
213	C₂	C₂³×#1136	1	2	yes	—
214	C₂	C₂³×C₄⋊Dic₇	1	1	yes	characteristic

	subgroup	G/H
center	C₂³×C₄	C₂²×D₇
commutator	C₂²×C₇	C₂⁴×C₄
Frattini	C₂×C₄	C₂⁴×D₇
Fitting	C₂³×C₄²×C₇	C₂
socle	C₂⁴×C₇	C₂²×C₄
prime radical	C₂⁵×C₇	C₂×C₄
cosocle	C₂×C₄×C₇	C₂⁵
hypercenter	C₂³×C₄²	D₇
hypocenter	C₇	C₂²×#423
LCS	G ▷ C₂²×C₇ ▷ C₇
UCS	C₁ ◁ C₂³×C₄ ◁ C₂³×C₄²

Complements

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

N	H ≅ G/N	type
3. C₂×#89696	C₂	direct (N × H)
4. C₂²×C₄⋊(C₇⋊C₈)	C₂	semidirect
15. #89696	C₂²	direct (N × H)
16. C₂×C₄⋊(C₇⋊C₈)	C₂²	semidirect
39. C₄⋊(C₇⋊C₈)	C₂³	semidirect
52. C₂²×C₄×C₇	M₁₆	semidirect
96. C₂×C₄×C₇	C₂×M₁₆	semidirect
131. C₂²×C₇	#423	semidirect
139. C₄×C₇	C₂²×M₁₆	semidirect
148. C₂²×C₄	#1136	semidirect
171. C₂×C₇	C₂×#423	semidirect
183. C₂×C₄	C₂×#1136	semidirect
197. C₇	C₂²×#423	semidirect
200. C₂²	#89696	direct (N × H)
206. C₄	C₂²×#1136	semidirect
212. C₂	C₂×#89696	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×C₄²×C₇	2
C₂³×#1136	2
C₂×#89696	2
C₂²×C₄⋊(C₇⋊C₈)	2
C₂²×#423	7

Subgroups — C22×#89696 #14217408

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂²×#89696 #14217408