Subgroups — C₂²×#89796 #14218154

Order 1792.

2222 conjugacy classes of subgroups.

The subgroup lattice has 224 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₄×C₇×Q₈	C₂	1	1	yes	maximal, characteristic
2	C₂×#89796	C₂	1	24	yes	maximal, direct product factor
3	C₂²×C₄⋊(C₇⋊C₈)	C₂	1	2	yes	maximal
4	C₂²×C₄⋊(C₇⋊C₈)	C₂	1	1	yes	maximal, characteristic
5	C₂²×#6377	C₂	1	2	yes	maximal
6	C₂²×C₄×C₇⋊C₈	C₂	1	1	yes	maximal, characteristic
7	C₂³×C₇×Q₈	C₄	1	1	yes	characteristic
8	C₂×C₄×C₇×Q₈	C₄	1	12	yes	—
9	C₂×C₄×C₇×Q₈	C₂²	1	12	yes	—
10	C₂²×C₇×C₄⋊C₄	C₄	1	2	yes	—
11	C₂²×C₇×C₄⋊C₄	C₄	1	1	yes	characteristic
12	C₂²×C₄²×C₇	C₂²	1	1	yes	characteristic
13	C₂²×C₄²×C₇	C₂²	1	2	yes	—
14	C₂³×C₇⋊C₈	C₂²	1	1	yes	characteristic
15	C₂³×C₇⋊C₈	C₂²	1	2	yes	—
16	C₂³×C₇⋊C₈	C₂²	1	1	yes	characteristic
17	#89796	C₂²	1	64	yes	direct product factor
18	C₂×C₄⋊(C₇⋊C₈)	C₂²	1	24	yes	—
19	C₂×C₄⋊(C₇⋊C₈)	C₂²	1	12	yes	—
20	C₂×#6377	C₂²	1	24	yes	—
21	C₂×C₄×C₇⋊C₈	C₂²	1	12	yes	—
22	C₂²×#446	—	7	1	no	maximal, Sylow
23	C₂²×C₇×Q₈	—	2	8	no	—
24	C₂²×C₇×Q₈	C₂×C₄	1	12	yes	—
25	C₂³×C₄×C₇	C₂×C₄	1	2	yes	—
26	C₂³×C₄×C₇	C₂×C₄	1	1	yes	characteristic
27	C₂³×C₄×C₇	C₂×C₄	1	2	yes	—
28	C₂³×C₄×C₇	C₂×C₄	1	1	yes	characteristic
29	C₂³×C₄×C₇	C₂³	1	1	yes	characteristic
30	C₄×C₇×Q₈	C₂×C₄	1	48	yes	—
31	C₄×C₇×Q₈	C₂³	1	16	yes	—
32	C₂×C₇×C₄⋊C₄	C₂×C₄	1	24	yes	—
33	C₂×C₇×C₄⋊C₄	C₂×C₄	1	12	yes	—
34	C₂×C₄²×C₇	C₂×C₄	1	12	yes	—
35	C₂×C₄²×C₇	C₂×C₄	1	6	yes	—
36	C₂×C₄²×C₇	C₂³	1	6	yes	—
37	C₂×C₄²×C₇	C₂³	1	12	yes	—
38	C₂²×C₇⋊C₈	—	2	4	no	—
39	C₂²×C₇⋊C₈	—	2	8	no	—
40	C₂²×C₇⋊C₈	Q₈	1	8	yes	complemented
41	C₂²×C₇⋊C₈	C₂³	1	6	yes	—
42	C₂²×C₇⋊C₈	C₂³	1	12	yes	—
43	C₂²×C₇⋊C₈	C₂³	1	6	yes	—
44	C₄⋊(C₇⋊C₈)	C₂³	1	32	yes	—
45	C₄⋊(C₇⋊C₈)	C₂³	1	16	yes	—
46	#6377	C₂³	1	32	yes	—
47	C₄×C₇⋊C₈	C₂³	1	16	yes	—
48	C₂²×C₄×Q₈	D₇	1	1	yes	characteristic
49	C₂×#446	—	7	24	no	—
50	C₂²×C₄⋊C₈	—	7	2	no	—
51	C₂²×C₄⋊C₈	—	7	1	no	—
52	C₂²×#97	—	7	2	no	—
53	C₂²×C₄×C₈	—	7	1	no	—
54	C₂⁴×C₇	C₂²×C₄	1	1	yes	characteristic
55	C₂×C₇×Q₈	—	2	48	no	—
56	C₂×C₇×Q₈	C₂²×C₄	1	16	yes	—
57	C₂²×C₄×C₇	—	2	8	no	—
58	C₂²×C₄×C₇	—	2	4	no	—
59	C₂²×C₄×C₇	—	2	8	no	—
60	C₂²×C₄×C₇	M₁₆	1	8	yes	—
61	C₂²×C₄×C₇	C₂²×C₄	1	6	yes	—
62	C₂²×C₄×C₇	C₂²×C₄	1	12	yes	—
63	C₂²×C₄×C₇	C₂²×C₄	1	12	yes	—
64	C₂²×C₄×C₇	C₂²×C₄	1	3	yes	—
65	C₂²×C₄×C₇	C₂²×C₄	1	6	yes	—
66	C₂²×C₄×C₇	C₂×Q₈	1	4	yes	—
67	C₂²×C₄×C₇	#41	1	4	yes	—
68	C₂²×C₄×C₇	C₂⁴	1	3	yes	—
69	C₇×C₄⋊C₄	C₂²×C₄	1	32	yes	—
70	C₇×C₄⋊C₄	C₂²×C₄	1	16	yes	—
71	C₄²×C₇	C₂²×C₄	1	24	yes	—
72	C₄²×C₇	C₂²×C₄	1	12	yes	—
73	C₄²×C₇	C₂⁴	1	4	yes	—
74	C₄²×C₇	C₂⁴	1	8	yes	—
75	C₂×C₇⋊C₈	—	2	12	no	—
76	C₂×C₇⋊C₈	—	2	24	no	—
77	C₂×C₇⋊C₈	C₂×Q₈	1	24	yes	complemented
78	C₂×C₇⋊C₈	C₂⁴	1	4	yes	—
79	C₂×C₇⋊C₈	C₂⁴	1	8	yes	—
80	C₂×C₇⋊C₈	C₂⁴	1	4	yes	—
81	C₂³×Q₈	Dic₇	1	1	yes	characteristic
82	C₂³×C₈	—	7	1	no	—
83	C₂³×C₈	—	7	1	no	—
84	C₂³×C₈	—	7	2	no	—
85	C₂×C₄×Q₈	Dic₇	1	12	yes	—
86	C₂×C₄×Q₈	C₂×D₇	1	12	yes	—
87	C₂²×C₄⋊C₄	Dic₇	1	2	yes	—
88	C₂²×C₄⋊C₄	Dic₇	1	1	yes	characteristic
89	C₂²×C₄²	C₂×D₇	1	1	yes	characteristic
90	C₂²×C₄²	C₂×D₇	1	2	yes	—
91	#446	—	7	64	no	—
92	C₂×C₄⋊C₈	—	7	12	no	—
93	C₂×C₄⋊C₈	—	7	24	no	—
94	C₂×#97	—	7	24	no	—
95	C₂×C₄×C₈	—	7	12	no	—
96	C₂³×C₇	C₄×Q₈	1	4	yes	—
97	C₂³×C₇	C₂×M₁₆	1	4	yes	—
98	C₂³×C₇	#131	1	4	yes	—
99	C₂³×C₇	C₂³×C₄	1	3	yes	—
100	C₇×Q₈	—	2	32	no	—
101	C₂×C₄×C₇	—	2	24	no	—
102	C₂×C₄×C₇	—	2	12	no	—
103	C₂×C₄×C₇	—	2	24	no	—
104	C₂×C₄×C₇	C₄×Q₈	1	12	yes	—
105	C₂×C₄×C₇	C₂×M₁₆	1	24	yes	—
106	C₂×C₄×C₇	C₂³×C₄	1	4	yes	—
107	C₂×C₄×C₇	C₂³×C₄	1	8	yes	—
108	C₂×C₄×C₇	C₂³×C₄	1	8	yes	—
109	C₂×C₄×C₇	C₂³×C₄	1	3	yes	—
110	C₂×C₄×C₇	C₂³×C₄	1	4	yes	—
111	C₂×C₄×C₇	C₂²×Q₈	1	6	yes	—
112	C₂×C₄×C₇	C₂×#41	1	6	yes	—
113	C₂×C₄×C₇	C₂⁵	1	1	yes	characteristic
114	C₇⋊C₈	—	2	4	no	—
115	C₇⋊C₈	—	2	8	no	—
116	C₇⋊C₈	C₂²×Q₈	1	8	yes	complemented
117	C₂²×Q₈	—	2	8	no	—
118	C₂²×Q₈	C₂×Dic₇	1	12	yes	—
119	C₂³×C₄	C₂×Dic₇	1	1	yes	characteristic
120	C₂³×C₄	C₂×Dic₇	1	2	yes	—
121	C₂³×C₄	C₂×Dic₇	1	2	yes	—
122	C₂³×C₄	C₂×Dic₇	1	1	yes	characteristic
123	C₂³×C₄	C₂²×D₇	1	1	yes	characteristic
124	C₂²×C₈	—	7	8	no	—
125	C₂²×C₈	—	7	6	no	—
126	C₂²×C₈	—	7	6	no	—
127	C₂²×C₈	—	7	12	no	—
128	C₂²×C₈	—	14	4	no	—
129	C₂²×C₈	—	14	8	no	—
130	C₄×Q₈	C₂×Dic₇	1	48	yes	—
131	C₄×Q₈	C₂²×D₇	1	16	yes	—
132	C₂×C₄⋊C₄	C₂×Dic₇	1	24	yes	—
133	C₂×C₄⋊C₄	C₂×Dic₇	1	12	yes	—
134	C₂×C₄²	C₂×Dic₇	1	12	yes	—
135	C₂×C₄²	C₂×Dic₇	1	6	yes	—
136	C₂×C₄²	C₂²×D₇	1	6	yes	—
137	C₂×C₄²	C₂²×D₇	1	12	yes	—
138	C₄⋊C₈	—	7	16	no	—
139	C₄⋊C₈	—	7	32	no	—
140	#97	—	7	32	no	—
141	C₄×C₈	—	7	16	no	—
142	C₂²×C₇	#446	1	16	yes	complemented
143	C₂²×C₇	C₂×C₄×Q₈	1	6	yes	—
144	C₂²×C₇	C₂²×M₁₆	1	6	yes	—
145	C₂²×C₇	C₂×#131	1	6	yes	—
146	C₂²×C₇	C₂⁴×C₄	1	1	yes	characteristic
147	C₄×C₇	—	2	8	no	—
148	C₄×C₇	—	2	4	no	—
149	C₄×C₇	—	2	8	no	—
150	C₄×C₇	C₂×C₄×Q₈	1	6	yes	—
151	C₄×C₇	C₂²×M₁₆	1	8	yes	—
152	C₄×C₇	C₂³×Q₈	1	1	yes	characteristic
153	C₄×C₇	C₂²×#41	1	1	yes	characteristic
154	C₂⁴	C₂²×Dic₇	1	1	yes	characteristic
155	C₂×Q₈	—	2	48	no	—
156	C₂×Q₈	C₂²×Dic₇	1	16	yes	—
157	C₂²×C₄	—	2	8	no	—
158	C₂²×C₄	—	2	4	no	—
159	C₂²×C₄	—	2	8	no	—
160	C₂²×C₄	#1136	1	8	yes	—
161	C₂²×C₄	Q₈×D₇	1	4	yes	—
162	C₂²×C₄	#1161	1	4	yes	—
163	C₂²×C₄	C₂²×Dic₇	1	6	yes	—
164	C₂²×C₄	C₂²×Dic₇	1	12	yes	—
165	C₂²×C₄	C₂²×Dic₇	1	12	yes	—
166	C₂²×C₄	C₂²×Dic₇	1	6	yes	—
167	C₂²×C₄	C₂²×Dic₇	1	3	yes	—
168	C₂²×C₄	C₂³×D₇	1	3	yes	—
169	C₂×C₈	—	7	24	no	—
170	C₂×C₈	—	7	4	no	—
171	C₂×C₈	—	7	4	no	—
172	C₂×C₈	—	7	8	no	—
173	C₂×C₈	—	14	12	no	—
174	C₂×C₈	—	14	24	no	—
175	C₄⋊C₄	C₂²×Dic₇	1	32	yes	—
176	C₄⋊C₄	C₂²×Dic₇	1	16	yes	—
177	C₄²	C₂²×Dic₇	1	24	yes	—
178	C₄²	C₂²×Dic₇	1	12	yes	—
179	C₄²	C₂³×D₇	1	4	yes	—
180	C₄²	C₂³×D₇	1	8	yes	—
181	C₂×C₇	C₂×#446	1	12	yes	complemented
182	C₂×C₇	C₂²×C₄×Q₈	1	1	yes	characteristic
183	C₂×C₇	C₂³×M₁₆	1	1	yes	characteristic
184	C₂×C₇	C₂²×#131	1	1	yes	characteristic
185	C₂³	C₂×#1136	1	4	yes	—
186	C₂³	Q₈×Dic₇	1	4	yes	—
187	C₂³	#6511	1	4	yes	—
188	C₂³	C₂³×Dic₇	1	3	yes	—
189	Q₈	—	2	32	no	—
190	C₂×C₄	—	2	24	no	—
191	C₂×C₄	—	2	12	no	—
192	C₂×C₄	—	2	24	no	—
193	C₂×C₄	C₂×#1136	1	24	yes	—
194	C₂×C₄	Q₈×Dic₇	1	12	yes	—
195	C₂×C₄	C₂×Q₈×D₇	1	6	yes	—
196	C₂×C₄	C₂×#1161	1	6	yes	—
197	C₂×C₄	C₂³×Dic₇	1	4	yes	—
198	C₂×C₄	C₂³×Dic₇	1	8	yes	—
199	C₂×C₄	C₂³×Dic₇	1	8	yes	—
200	C₂×C₄	C₂³×Dic₇	1	4	yes	—
201	C₂×C₄	C₂³×Dic₇	1	3	yes	—
202	C₂×C₄	C₂⁴×D₇	1	1	yes	characteristic
203	C₈	—	7	8	no	—
204	C₈	—	14	4	no	—
205	C₈	—	14	8	no	—
206	C₇	C₂²×#446	1	1	yes	characteristic, complemented, Sylow
207	C₂²	#89796	1	16	yes	direct product factor
208	C₂²	C₂²×#1136	1	6	yes	—
209	C₂²	C₂×Q₈×Dic₇	1	6	yes	—
210	C₂²	C₂×#6511	1	6	yes	—
211	C₂²	C₂⁴×Dic₇	1	1	yes	characteristic
212	C₄	—	2	8	no	—
213	C₄	—	2	4	no	—
214	C₄	—	2	8	no	—
215	C₄	C₂²×#1136	1	8	yes	—
216	C₄	C₂×Q₈×Dic₇	1	6	yes	—
217	C₄	C₂²×Q₈×D₇	1	1	yes	characteristic
218	C₄	C₂²×#1161	1	1	yes	characteristic
219	C₂	C₂×#89796	1	12	yes	direct product factor
220	C₂	C₂³×#1136	1	1	yes	characteristic
221	C₂	C₂²×Q₈×Dic₇	1	1	yes	characteristic
222	C₂	C₂²×#6511	1	1	yes	characteristic

	subgroup	G/H
center	C₂³×C₄	C₂²×D₇
commutator	C₂²×C₇	C₂⁴×C₄
Frattini	C₂×C₄	C₂⁴×D₇
Fitting	C₂²×C₄×C₇×Q₈	C₂
socle, prime radical	C₂⁴×C₇	C₂²×C₄
cosocle	C₂×C₄×C₇	C₂⁵
hypercenter	C₂²×C₄×Q₈	D₇
hypocenter	C₇	C₂²×#446
LCS	G ▷ C₂²×C₇ ▷ C₇
UCS	C₁ ◁ C₂³×C₄ ◁ C₂²×C₄×Q₈

Complements

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

N	H ≅ G/N	type
2. C₂×#89796	C₂	direct (N × H)
17. #89796	C₂²	direct (N × H)
40. C₂²×C₇⋊C₈	Q₈	semidirect
77. C₂×C₇⋊C₈	C₂×Q₈	semidirect
116. C₇⋊C₈	C₂²×Q₈	semidirect
142. C₂²×C₇	#446	semidirect
181. C₂×C₇	C₂×#446	semidirect
206. C₇	C₂²×#446	semidirect
207. C₂²	#89796	direct (N × H)
219. C₂	C₂×#89796	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂²×C₄×C₇×Q₈	2
C₂×#89796	2
C₂²×C₄⋊(C₇⋊C₈)	2
C₂²×#6377	2
C₂²×C₄×C₇⋊C₈	2
C₂²×#446	7

Subgroups — C22×#89796 #14218154

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂²×#89796 #14218154