Subgroups — C₂×#11727546 #13354252

Order 1792.

1046 conjugacy classes of subgroups.

The subgroup lattice has 230 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₄×C₇×C₈	C₂	1	1	yes	maximal, characteristic
2	C₂²×#6393	C₂	1	4	yes	maximal
3	C₂×#89696	C₂	1	2	yes	maximal
4	#11727546	C₂	1	8	yes	maximal, direct product factor
5	C₂³×C₇×C₈	C₂²	1	2	yes	—
6	C₂²×C₄²×C₇	C₂²	1	1	yes	characteristic
7	C₂²×#1136	C₂²	1	4	yes	—
8	C₂×C₄×C₇×C₈	C₄	1	2	yes	—
9	C₂×C₄×C₇×C₈	C₄	1	4	yes	—
10	C₂×C₄×C₇×C₈	C₄	1	2	yes	—
11	C₂×C₄×C₇×C₈	C₂²	1	4	yes	—
12	C₂×#6393	—	2	16	no	—
13	C₂×#6393	C₂²	1	16	yes	—
14	#89696	—	2	4	no	—
15	#89696	C₂²	1	8	yes	—
16	C₂×C₄⋊(C₇⋊C₈)	—	2	4	no	—
17	C₂×#1848	—	7	1	no	maximal, Sylow
18	C₂³×C₄×C₇	Q₈	1	2	yes	—
19	C₂³×C₄×C₇	C₂³	1	1	yes	characteristic
20	C₂²×C₇×C₈	C₂×C₄	1	4	yes	—
21	C₂²×C₇×C₈	C₂×C₄	1	4	yes	—
22	C₂²×C₇×C₈	D₄	1	8	yes	—
23	C₂²×C₇×C₈	Q₈	1	8	yes	—
24	C₂²×C₇×C₈	C₂³	1	4	yes	—
25	C₂×C₄²×C₇	C₂×C₄	1	1	yes	characteristic
26	C₂×C₄²×C₇	C₂×C₄	1	2	yes	—
27	C₂×C₄²×C₇	C₂×C₄	1	1	yes	characteristic
28	C₂×C₄²×C₇	D₄	1	2	yes	—
29	C₂×C₄²×C₇	D₄	1	2	yes	—
30	C₂×C₄²×C₇	Q₈	1	2	yes	—
31	C₂×C₄²×C₇	C₂³	1	2	yes	—
32	C₂×#1136	—	2	4	no	—
33	C₂×#1136	—	2	16	no	—
34	C₂×#1136	C₂³	1	8	yes	—
35	C₂²×C₇⋊C₈	—	2	4	no	—
36	C₄×C₇×C₈	C₂×C₄	1	8	yes	—
37	C₄×C₇×C₈	C₂×C₄	1	8	yes	—
38	#6393	—	2	32	no	—
39	C₄⋊(C₇⋊C₈)	—	2	16	no	—
40	C₂²×#108	—	7	4	no	—
41	C₂×#423	—	7	2	no	—
42	C₂²×C₄×C₈	D₇	1	1	yes	characteristic
43	#1848	—	7	8	no	—
44	C₂⁴×C₇	C₂×Q₈	1	1	yes	characteristic
45	C₂²×C₄×C₇	—	2	8	no	—
46	C₂²×C₄×C₇	C₄⋊C₄	1	2	yes	—
47	C₂²×C₄×C₇	C₄⋊C₄	1	4	yes	—
48	C₂²×C₄×C₇	C₄⋊C₄	1	2	yes	—
49	C₂²×C₄×C₇	C₂²×C₄	1	1	yes	characteristic
50	C₂²×C₄×C₇	C₂²×C₄	1	1	yes	characteristic
51	C₂²×C₄×C₇	C₂×D₄	1	4	yes	—
52	C₂²×C₄×C₇	C₂×D₄	1	1	yes	characteristic
53	C₂²×C₄×C₇	C₂×D₄	1	1	yes	characteristic
54	C₂²×C₄×C₇	C₂×Q₈	1	4	yes	—
55	C₂²×C₄×C₇	C₂×Q₈	1	1	yes	characteristic
56	C₂²×C₄×C₇	C₂×Q₈	1	4	yes	—
57	C₂²×C₄×C₇	C₂⁴	1	1	yes	characteristic
58	C₂×C₇×C₈	C₄⋊C₄	1	16	yes	—
59	C₂×C₇×C₈	C₄⋊C₄	1	16	yes	—
60	C₂×C₇×C₈	C₂²×C₄	1	8	yes	—
61	C₂×C₇×C₈	C₂×D₄	1	8	yes	—
62	C₂×C₇×C₈	C₂×Q₈	1	8	yes	—
63	C₄²×C₇	C₄⋊C₄	1	2	yes	—
64	C₄²×C₇	C₄⋊C₄	1	4	yes	—
65	C₄²×C₇	C₄⋊C₄	1	2	yes	—
66	C₄²×C₇	C₂²×C₄	1	2	yes	—
67	C₄²×C₇	C₂²×C₄	1	2	yes	—
68	C₄²×C₇	C₂×D₄	1	4	yes	—
69	#1136	—	2	16	no	—
70	#1136	—	4	8	no	—
71	C₂×C₇⋊C₈	—	2	8	no	—
72	C₂×C₇⋊C₈	—	4	8	no	—
73	C₂²×M₁₆	—	7	4	no	—
74	C₂³×C₈	C₂×D₇	1	2	yes	—
75	C₂²×C₄²	C₂×D₇	1	1	yes	characteristic
76	C₂×#108	—	7	16	no	—
77	C₂×#108	—	14	16	no	—
78	#423	—	7	8	no	—
79	#423	—	14	4	no	—
80	C₂×C₄⋊C₈	—	14	4	no	—
81	C₂×C₄×C₈	Dic₇	1	2	yes	—
82	C₂×C₄×C₈	Dic₇	1	4	yes	—
83	C₂×C₄×C₈	Dic₇	1	2	yes	—
84	C₂×C₄×C₈	C₂×D₇	1	4	yes	—
85	C₂³×C₇	—	2	4	no	—
86	C₂³×C₇	C₂×C₄⋊C₄	1	1	yes	characteristic
87	C₂³×C₇	C₂×C₄⋊C₄	1	1	yes	characteristic
88	C₂³×C₇	C₄⋊Q₈	1	4	yes	—
89	C₂³×C₇	C₂²×Q₈	1	1	yes	characteristic
90	C₂×C₄×C₇	—	2	4	no	—
91	C₂×C₄×C₇	—	2	8	no	—
92	C₂×C₄×C₇	—	2	8	no	—
93	C₂×C₄×C₇	#108	1	8	yes	complemented
94	C₂×C₄×C₇	C₂×C₄⋊C₄	1	4	yes	—
95	C₂×C₄×C₇	C₂×C₄⋊C₄	1	4	yes	—
96	C₂×C₄×C₇	C₂×C₄⋊C₄	1	4	yes	—
97	C₂×C₄×C₇	C₂×C₄⋊C₄	1	1	yes	characteristic
98	C₂×C₄×C₇	C₂×C₄⋊C₄	1	1	yes	characteristic
99	C₂×C₄×C₇	C₂×C₄⋊C₄	1	4	yes	—
100	C₂×C₄×C₇	#127	1	2	yes	—
101	C₂×C₄×C₇	#127	1	2	yes	—
102	C₂×C₄×C₇	C₄⋊Q₈	1	2	yes	—
103	C₂×C₄×C₇	C₄⋊Q₈	1	2	yes	—
104	C₂×C₄×C₇	C₄⋊Q₈	1	4	yes	—
105	C₂×C₄×C₇	C₂³×C₄	1	1	yes	characteristic
106	C₂×C₄×C₇	C₂²×D₄	1	2	yes	—
107	C₂×C₄×C₇	C₂²×D₄	1	1	yes	characteristic
108	C₂×C₄×C₇	C₂²×Q₈	1	2	yes	—
109	C₇×C₈	C₂×C₄⋊C₄	1	16	yes	—
110	C₇⋊C₈	—	4	8	no	—
111	C₂³×C₄	Dic₁₄	1	2	yes	—
112	C₂³×C₄	C₂²×D₇	1	1	yes	characteristic
113	C₂×M₁₆	—	7	8	no	—
114	C₂×M₁₆	—	14	4	no	—
115	C₂×M₁₆	—	14	16	no	—
116	C₂²×C₈	—	14	4	no	—
117	C₂²×C₈	Dic₁₄	1	8	yes	—
118	C₂²×C₈	D₂₈	1	8	yes	—
119	C₂²×C₈	C₂×Dic₇	1	4	yes	—
120	C₂²×C₈	C₂×Dic₇	1	4	yes	—
121	C₂²×C₈	C₂²×D₇	1	4	yes	—
122	C₂×C₄²	Dic₁₄	1	2	yes	—
123	C₂×C₄²	D₂₈	1	2	yes	—
124	C₂×C₄²	D₂₈	1	2	yes	—
125	C₂×C₄²	C₂×Dic₇	1	1	yes	characteristic
126	C₂×C₄²	C₂×Dic₇	1	2	yes	—
127	C₂×C₄²	C₂×Dic₇	1	1	yes	characteristic
128	C₂×C₄²	C₂²×D₇	1	2	yes	—
129	#108	—	14	32	no	—
130	C₄⋊C₈	—	14	16	no	—
131	C₄×C₈	C₂×Dic₇	1	8	yes	—
132	C₄×C₈	C₂×Dic₇	1	8	yes	—
133	C₂²×C₇	—	2	8	no	—
134	C₂²×C₇	—	2	2	no	—
135	C₂²×C₇	—	2	2	no	—
136	C₂²×C₇	#384	1	2	yes	—
137	C₂²×C₇	#384	1	2	yes	—
138	C₂²×C₇	C₂×#108	1	4	yes	—
139	C₂²×C₇	C₂²×C₄⋊C₄	1	1	yes	characteristic
140	C₂²×C₇	C₂×C₄⋊Q₈	1	2	yes	—
141	C₄×C₇	—	2	4	no	—
142	C₄×C₇	#384	1	2	yes	—
143	C₄×C₇	#384	1	2	yes	—
144	C₄×C₇	C₂×#108	1	8	yes	complemented
145	C₄×C₇	C₂²×C₄⋊C₄	1	2	yes	—
146	C₄×C₇	C₂×#127	1	1	yes	characteristic
147	C₄×C₇	C₂×C₄⋊Q₈	1	1	yes	characteristic
148	C₂⁴	C₂×Dic₁₄	1	1	yes	characteristic
149	C₂²×C₄	—	2	8	no	—
150	C₂²×C₄	C₄⋊Dic₇	1	2	yes	—
151	C₂²×C₄	C₄⋊Dic₇	1	4	yes	—
152	C₂²×C₄	C₄⋊Dic₇	1	2	yes	—
153	C₂²×C₄	C₂×Dic₁₄	1	4	yes	—
154	C₂²×C₄	C₂×Dic₁₄	1	1	yes	characteristic
155	C₂²×C₄	C₂×Dic₁₄	1	4	yes	—
156	C₂²×C₄	C₂×D₂₈	1	1	yes	characteristic
157	C₂²×C₄	C₂×D₂₈	1	4	yes	—
158	C₂²×C₄	C₂×D₂₈	1	1	yes	characteristic
159	C₂²×C₄	C₂²×Dic₇	1	1	yes	characteristic
160	C₂²×C₄	C₂²×Dic₇	1	1	yes	characteristic
161	C₂²×C₄	C₂³×D₇	1	1	yes	characteristic
162	M₁₆	—	14	16	no	—
163	M₁₆	—	28	8	no	—
164	C₂×C₈	—	14	8	no	—
165	C₂×C₈	—	28	8	no	—
166	C₂×C₈	C₄⋊Dic₇	1	16	yes	—
167	C₂×C₈	C₄⋊Dic₇	1	16	yes	—
168	C₂×C₈	C₂×Dic₁₄	1	8	yes	—
169	C₂×C₈	C₂×D₂₈	1	8	yes	—
170	C₂×C₈	C₂²×Dic₇	1	8	yes	—
171	C₄²	C₄⋊Dic₇	1	2	yes	—
172	C₄²	C₄⋊Dic₇	1	4	yes	—
173	C₄²	C₄⋊Dic₇	1	2	yes	—
174	C₄²	C₂×D₂₈	1	4	yes	—
175	C₄²	C₂²×Dic₇	1	2	yes	—
176	C₄²	C₂²×Dic₇	1	2	yes	—
177	C₂×C₇	—	2	2	no	—
178	C₂×C₇	—	2	2	no	—
179	C₂×C₇	#1848	1	4	yes	complemented
180	C₂×C₇	C₂×#384	1	1	yes	characteristic
181	C₂×C₇	C₂²×#108	1	2	yes	—
182	C₂³	—	2	4	no	—
183	C₂³	#6432	1	4	yes	—
184	C₂³	C₂×C₄⋊Dic₇	1	1	yes	characteristic
185	C₂³	C₂×C₄⋊Dic₇	1	1	yes	characteristic
186	C₂³	C₂²×Dic₁₄	1	1	yes	characteristic
187	C₂×C₄	—	2	4	no	—
188	C₂×C₄	—	2	8	no	—
189	C₂×C₄	—	2	8	no	—
190	C₂×C₄	#6393	1	8	yes	complemented
191	C₂×C₄	#6432	1	2	yes	—
192	C₂×C₄	#6432	1	2	yes	—
193	C₂×C₄	#6432	1	4	yes	—
194	C₂×C₄	#6437	1	2	yes	—
195	C₂×C₄	#6437	1	2	yes	—
196	C₂×C₄	C₂×C₄⋊Dic₇	1	4	yes	—
197	C₂×C₄	C₂×C₄⋊Dic₇	1	4	yes	—
198	C₂×C₄	C₂×C₄⋊Dic₇	1	1	yes	characteristic
199	C₂×C₄	C₂×C₄⋊Dic₇	1	1	yes	characteristic
200	C₂×C₄	C₂×C₄⋊Dic₇	1	4	yes	—
201	C₂×C₄	C₂×C₄⋊Dic₇	1	4	yes	—
202	C₂×C₄	C₂²×Dic₁₄	1	2	yes	—
203	C₂×C₄	C₂²×D₂₈	1	1	yes	characteristic
204	C₂×C₄	C₂²×D₂₈	1	2	yes	—
205	C₂×C₄	C₂³×Dic₇	1	1	yes	characteristic
206	C₈	—	28	8	no	—
207	C₈	C₂×C₄⋊Dic₇	1	16	yes	—
208	C₇	C₂×#1848	1	1	yes	characteristic, complemented, Sylow
209	C₂²	—	2	8	no	—
210	C₂²	—	2	2	no	—
211	C₂²	—	2	2	no	—
212	C₂²	#89707	1	2	yes	—
213	C₂²	#89707	1	2	yes	—
214	C₂²	C₂×#6393	1	4	yes	—
215	C₂²	C₂×#6432	1	2	yes	—
216	C₂²	C₂²×C₄⋊Dic₇	1	1	yes	characteristic
217	C₄	—	2	4	no	—
218	C₄	#89707	1	2	yes	—
219	C₄	#89707	1	2	yes	—
220	C₄	C₂×#6393	1	8	yes	complemented
221	C₄	C₂×#6432	1	1	yes	characteristic
222	C₄	C₂×#6437	1	1	yes	characteristic
223	C₄	C₂²×C₄⋊Dic₇	1	2	yes	—
224	C₂	—	2	2	no	—
225	C₂	—	2	2	no	—
226	C₂	#11727546	1	4	yes	direct product factor
227	C₂	C₂×#89707	1	1	yes	characteristic
228	C₂	C₂²×#6393	1	2	yes	—

	subgroup	G/H
center	C₂²×C₄	C₂×D₂₈
commutator	C₂×C₄×C₇	C₂³×C₄
Frattini	C₂²×C₄	C₂³×D₇
Fitting	C₂²×C₄×C₇×C₈	C₂
socle	C₂³×C₇	C₂×C₄⋊C₄
prime radical	C₂⁴×C₇	C₂×Q₈
cosocle	C₂²×C₄×C₇	C₂⁴
hypercenter	C₂²×C₄×C₈	D₇
hypocenter	C₇	C₂×#1848
LCS	G ▷ C₂×C₄×C₇ ▷ C₂×C₇ ▷ C₇
UCS	C₁ ◁ C₂²×C₄ ◁ C₂²×C₄² ◁ C₂²×C₄×C₈

Complements

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

N	H ≅ G/N	type
4. #11727546	C₂	direct (N × H)
93. C₂×C₄×C₇	#108	semidirect
144. C₄×C₇	C₂×#108	semidirect
179. C₂×C₇	#1848	semidirect
190. C₂×C₄	#6393	semidirect
208. C₇	C₂×#1848	semidirect
220. C₄	C₂×#6393	semidirect
226. C₂	#11727546	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×C₈	2
C₂²×#6393	2
C₂×#89696	2
#11727546	2
C₂×#1848	7

Subgroups — C2×#11727546 #13354252

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#11727546 #13354252