Subgroups — C₂×#11726825 #13506426

Order 1792.

918 conjugacy classes of subgroups.

The subgroup lattice has 240 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×#6372	C₂	1	2	yes	maximal
2	C₂×C₄×C₈×D₇	C₂	1	1	yes	maximal, characteristic
3	C₂×C₇×#346	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂×C₁₆×Dic₇	C₂	1	2	yes	maximal, complemented
5	C₂×#89256	C₂	1	1	yes	maximal, characteristic, complemented
6	#11726825	C₂	1	8	yes	maximal, direct product factor
7	C₂²×C₈×D₇	C₄	1	1	yes	characteristic
8	C₂²×C₈×D₇	C₂²	1	1	yes	characteristic
9	C₂×C₄²×D₇	C₄	1	1	yes	characteristic
10	C₂²×C₇×C₁₆	C₂²	1	2	yes	—
11	C₂×C₄×C₇×C₈	C₂²	1	1	yes	characteristic
12	C₂×C₈×Dic₇	C₄	1	1	yes	characteristic
13	C₂×C₈×Dic₇	C₂²	1	1	yes	characteristic
14	C₂²×C₇⋊C₁₆	C₂²	1	2	yes	—
15	C₂×C₄×C₇⋊C₈	C₄	1	1	yes	characteristic
16	C₂×#6372	C₄	1	16	yes	complemented
17	C₂×#6372	C₂²	1	8	yes	—
18	C₄×C₈×D₇	C₄	1	4	yes	—
19	C₄×C₈×D₇	C₂²	1	4	yes	—
20	C₇×#346	C₂²	1	4	yes	complemented
21	C₁₆×Dic₇	C₂²	1	8	yes	complemented
22	#89256	C₂²	1	4	yes	complemented
23	C₂×C₄×M₃₂	—	7	1	no	maximal, Sylow
24	C₂²×C₄×D₇	C₈	1	2	yes	—
25	C₂²×C₄×D₇	C₂×C₄	1	1	yes	characteristic
26	C₂²×C₇×C₈	C₂×C₄	1	1	yes	characteristic
27	C₂²×C₇×C₈	C₂³	1	1	yes	characteristic
28	C₂×C₄²×C₇	C₂×C₄	1	1	yes	characteristic
29	C₂×C₄×Dic₇	C₈	1	2	yes	—
30	C₂×C₄×Dic₇	C₂×C₄	1	1	yes	characteristic
31	C₂²×C₇⋊C₈	C₂×C₄	1	1	yes	characteristic
32	C₂²×C₇⋊C₈	C₂×C₄	1	1	yes	characteristic
33	C₂×C₈×D₇	C₈	1	8	yes	—
34	C₂×C₈×D₇	C₂×C₄	1	2	yes	—
35	C₂×C₈×D₇	C₂×C₄	1	8	yes	—
36	C₂×C₈×D₇	C₂×C₄	1	4	yes	—
37	C₂×C₈×D₇	C₂³	1	2	yes	—
38	C₄²×D₇	C₈	1	4	yes	—
39	C₄²×D₇	C₂×C₄	1	4	yes	—
40	C₂×C₇×C₁₆	C₂×C₄	1	8	yes	complemented
41	C₂×C₇×C₁₆	C₂³	1	4	yes	—
42	C₄×C₇×C₈	C₂×C₄	1	2	yes	—
43	C₄×C₇×C₈	C₂³	1	2	yes	—
44	C₈×Dic₇	C₂×C₄	1	4	yes	—
45	C₈×Dic₇	C₂×C₄	1	2	yes	—
46	C₈×Dic₇	C₂³	1	2	yes	—
47	C₂×C₇⋊C₁₆	C₂×C₄	1	8	yes	complemented
48	C₂×C₇⋊C₁₆	C₂³	1	4	yes	—
49	C₄×C₇⋊C₈	C₂×C₄	1	4	yes	—
50	#6372	C₂×C₄	1	32	yes	complemented
51	C₂²×M₃₂	—	7	2	no	—
52	C₂²×C₄×C₈	—	7	1	no	—
53	C₄×M₃₂	—	7	8	no	—
54	C₂×#346	—	7	1	no	—
55	C₂×#346	D₇	1	1	yes	characteristic, complemented
56	C₂×C₄×C₁₆	—	7	2	no	—
57	C₂³×D₇	C₂×C₈	1	1	yes	characteristic
58	C₂²×C₄×C₇	C₂×C₈	1	2	yes	—
59	C₂²×C₄×C₇	C₂²×C₄	1	1	yes	characteristic
60	C₂²×Dic₇	C₂×C₈	1	1	yes	characteristic
61	C₂²×Dic₇	C₂×C₈	1	2	yes	—
62	C₂×C₄×D₇	—	2	8	no	—
63	C₂×C₄×D₇	C₄²	1	4	yes	—
64	C₂×C₄×D₇	C₂×C₈	1	8	yes	—
65	C₂×C₄×D₇	C₂×C₈	1	2	yes	—
66	C₂×C₄×D₇	C₂×C₈	1	4	yes	—
67	C₂×C₄×D₇	C₂²×C₄	1	2	yes	—
68	C₂×C₇×C₈	C₂×C₈	1	4	yes	—
69	C₂×C₇×C₈	C₂²×C₄	1	1	yes	characteristic
70	C₂×C₇×C₈	C₂²×C₄	1	4	yes	—
71	C₂×C₇×C₈	C₂²×C₄	1	2	yes	—
72	C₂×C₇×C₈	C₂⁴	1	1	yes	characteristic
73	C₄²×C₇	C₂×C₈	1	2	yes	—
74	C₄²×C₇	C₂²×C₄	1	2	yes	—
75	C₄×Dic₇	C₂×C₈	1	2	yes	—
76	C₄×Dic₇	C₂×C₈	1	8	yes	—
77	C₄×Dic₇	C₂²×C₄	1	2	yes	—
78	C₂×C₇⋊C₈	C₄²	1	4	yes	—
79	C₂×C₇⋊C₈	C₂×C₈	1	4	yes	—
80	C₂×C₇⋊C₈	C₂²×C₄	1	2	yes	—
81	C₂×C₇⋊C₈	C₂²×C₄	1	2	yes	—
82	C₈×D₇	C₄²	1	8	yes	—
83	C₈×D₇	C₂×C₈	1	16	yes	—
84	C₈×D₇	C₂²×C₄	1	8	yes	—
85	C₇×C₁₆	C₂²×C₄	1	8	yes	complemented
86	C₇⋊C₁₆	C₂²×C₄	1	8	yes	complemented
87	C₂³×C₈	—	7	1	no	—
88	C₂³×C₈	—	7	1	no	—
89	C₂²×C₄²	—	7	1	no	—
90	C₂×M₃₂	—	7	16	no	—
91	C₂×M₃₂	—	7	8	no	—
92	C₂²×C₁₆	—	7	2	no	—
93	C₂²×C₁₆	C₂×D₇	1	2	yes	—
94	C₂×C₄×C₈	—	7	1	no	—
95	C₂×C₄×C₈	—	7	1	no	—
96	C₂×C₄×C₈	—	7	1	no	—
97	C₂×C₄×C₈	—	7	4	no	—
98	C₂×C₄×C₈	—	7	4	no	—
99	C₂×C₄×C₈	C₂×D₇	1	1	yes	characteristic
100	#346	—	7	4	no	—
101	#346	C₂×D₇	1	4	yes	complemented
102	C₄×C₁₆	—	7	8	no	—
103	C₂³×C₇	C₂²×C₈	1	1	yes	characteristic
104	C₂²×D₇	—	2	4	no	—
105	C₂²×D₇	C₄×C₈	1	4	yes	—
106	C₂²×D₇	C₂²×C₈	1	2	yes	—
107	C₂×C₄×C₇	—	2	4	no	—
108	C₂×C₄×C₇	C₂×C₄²	1	2	yes	—
109	C₂×C₄×C₇	C₂²×C₈	1	4	yes	—
110	C₂×C₄×C₇	C₂²×C₈	1	2	yes	—
111	C₂×C₄×C₇	C₂²×C₈	1	1	yes	characteristic
112	C₂×C₄×C₇	C₂³×C₄	1	1	yes	characteristic
113	C₂×Dic₇	C₄×C₈	1	4	yes	—
114	C₂×Dic₇	M₃₂	1	8	yes	complemented
115	C₂×Dic₇	C₂²×C₈	1	2	yes	—
116	C₂×Dic₇	C₂²×C₈	1	4	yes	—
117	C₄×D₇	—	2	16	no	—
118	C₄×D₇	C₄×C₈	1	8	yes	—
119	C₄×D₇	C₂×C₄²	1	4	yes	—
120	C₄×D₇	C₂²×C₈	1	4	yes	—
121	C₇×C₈	C₂×C₄²	1	2	yes	—
122	C₇×C₈	C₂²×C₈	1	4	yes	—
123	C₇×C₈	C₂³×C₄	1	2	yes	—
124	C₇⋊C₈	C₂×C₄²	1	4	yes	—
125	C₇⋊C₈	C₂²×C₈	1	4	yes	—
126	C₂³×C₄	—	7	1	no	—
127	C₂³×C₄	—	7	2	no	—
128	C₂²×C₈	—	7	1	no	—
129	C₂²×C₈	—	7	1	no	—
130	C₂²×C₈	—	7	8	no	—
131	C₂²×C₈	—	7	2	no	—
132	C₂²×C₈	—	7	4	no	—
133	C₂²×C₈	—	7	8	no	—
134	C₂²×C₈	—	7	2	no	—
135	C₂²×C₈	C₄×D₇	1	1	yes	characteristic
136	C₂²×C₈	C₂²×D₇	1	1	yes	characteristic
137	C₂×C₄²	—	7	2	no	—
138	C₂×C₄²	—	7	1	no	—
139	C₂×C₄²	—	7	4	no	—
140	C₂×C₄²	—	7	4	no	—
141	C₂×C₄²	C₄×D₇	1	1	yes	characteristic
142	M₃₂	—	7	32	no	—
143	C₂×C₁₆	—	7	8	no	—
144	C₂×C₁₆	—	7	4	no	—
145	C₂×C₁₆	C₄×D₇	1	8	yes	complemented
146	C₂×C₁₆	C₂²×D₇	1	4	yes	—
147	C₄×C₈	—	7	2	no	—
148	C₄×C₈	—	7	4	no	—
149	C₄×C₈	—	7	2	no	—
150	C₄×C₈	—	7	4	no	—
151	C₄×C₈	C₄×D₇	1	2	yes	—
152	C₄×C₈	C₂²×D₇	1	2	yes	—
153	C₂²×C₇	C₂×C₄×C₈	1	2	yes	—
154	C₂²×C₇	C₂×M₃₂	1	4	yes	—
155	C₂²×C₇	C₂³×C₈	1	1	yes	characteristic
156	C₂×D₇	—	2	8	no	—
157	C₂×D₇	—	2	4	no	—
158	C₂×D₇	C₂×C₄×C₈	1	4	yes	—
159	C₄×C₇	—	2	4	no	—
160	C₄×C₇	C₂×C₄×C₈	1	2	yes	—
161	C₄×C₇	C₂²×C₄²	1	1	yes	characteristic
162	C₄×C₇	C₂³×C₈	1	1	yes	characteristic
163	Dic₇	C₂×C₄×C₈	1	4	yes	—
164	Dic₇	C₂×M₃₂	1	8	yes	complemented
165	C₂⁴	—	7	1	no	—
166	C₂²×C₄	—	7	2	no	—
167	C₂²×C₄	—	7	1	no	—
168	C₂²×C₄	—	7	4	no	—
169	C₂²×C₄	—	7	2	no	—
170	C₂²×C₄	—	7	4	no	—
171	C₂²×C₄	—	7	2	no	—
172	C₂²×C₄	—	7	8	no	—
173	C₂²×C₄	—	14	8	no	—
174	C₂²×C₄	C₈×D₇	1	2	yes	—
175	C₂²×C₄	C₂×C₄×D₇	1	1	yes	characteristic
176	C₂×C₈	—	7	4	no	—
177	C₂×C₈	—	7	4	no	—
178	C₂×C₈	—	7	2	no	—
179	C₂×C₈	—	7	2	no	—
180	C₂×C₈	—	7	8	no	—
181	C₂×C₈	—	7	16	no	—
182	C₂×C₈	—	7	8	no	—
183	C₂×C₈	C₈×D₇	1	4	yes	—
184	C₂×C₈	C₂×C₄×D₇	1	1	yes	characteristic
185	C₂×C₈	C₂×C₄×D₇	1	4	yes	—
186	C₂×C₈	C₂×C₄×D₇	1	2	yes	—
187	C₂×C₈	C₂³×D₇	1	1	yes	characteristic
188	C₄²	—	7	8	no	—
189	C₄²	—	7	2	no	—
190	C₄²	—	7	2	no	—
191	C₄²	C₈×D₇	1	2	yes	—
192	C₄²	C₂×C₄×D₇	1	2	yes	—
193	C₁₆	—	7	8	no	—
194	C₁₆	C₂×C₄×D₇	1	8	yes	complemented
195	C₂×C₇	C₄×M₃₂	1	4	yes	complemented
196	C₂×C₇	C₂²×C₄×C₈	1	1	yes	characteristic
197	C₂×C₇	C₂²×M₃₂	1	2	yes	—
198	D₇	—	2	4	no	—
199	C₂³	—	7	2	no	—
200	C₂³	—	7	4	no	—
201	C₂³	—	14	4	no	—
202	C₂³	C₂×C₈×D₇	1	1	yes	characteristic
203	C₂×C₄	—	2	4	no	—
204	C₂×C₄	—	7	4	no	—
205	C₂×C₄	—	7	2	no	—
206	C₂×C₄	—	7	8	no	—
207	C₂×C₄	—	7	4	no	—
208	C₂×C₄	—	7	4	no	—
209	C₂×C₄	—	7	4	no	—
210	C₂×C₄	—	7	8	no	—
211	C₂×C₄	—	14	16	no	—
212	C₂×C₄	C₄²×D₇	1	2	yes	—
213	C₂×C₄	C₂×C₈×D₇	1	4	yes	—
214	C₂×C₄	C₂×C₈×D₇	1	2	yes	—
215	C₂×C₄	C₂×C₈×D₇	1	1	yes	characteristic
216	C₂×C₄	C₂²×C₄×D₇	1	1	yes	characteristic
217	C₈	—	7	4	no	—
218	C₈	—	7	4	no	—
219	C₈	C₄²×D₇	1	2	yes	—
220	C₈	C₂×C₈×D₇	1	4	yes	—
221	C₈	C₂²×C₄×D₇	1	2	yes	—
222	C₇	C₂×C₄×M₃₂	1	1	yes	characteristic, complemented, Sylow
223	C₂²	—	7	4	no	—
224	C₂²	—	14	4	no	—
225	C₂²	—	14	8	no	—
226	C₂²	C₄×C₈×D₇	1	2	yes	—
227	C₂²	D₇×M₃₂	1	4	yes	—
228	C₂²	C₂²×C₈×D₇	1	1	yes	characteristic
229	C₄	—	2	4	no	—
230	C₄	—	7	8	no	—
231	C₄	—	7	4	no	—
232	C₄	C₄×C₈×D₇	1	2	yes	—
233	C₄	C₂×C₄²×D₇	1	1	yes	characteristic
234	C₄	C₂²×C₈×D₇	1	1	yes	characteristic
235	C₂	—	14	4	no	—
236	C₂	#11726825	1	4	yes	direct product factor
237	C₂	C₂×C₄×C₈×D₇	1	1	yes	characteristic
238	C₂	C₂×D₇×M₃₂	1	2	yes	—

	subgroup	G/H
center	C₂²×C₈	C₂²×D₇
commutator	C₂×C₇	C₂²×C₄×C₈
Frattini	C₂×C₈	C₂³×D₇
Fitting	C₂×C₇×#346	C₂
socle	C₂³×C₇	C₂²×C₈
prime radical	C₂³×D₇	C₂×C₈
cosocle	C₂×C₇×C₈	C₂⁴
hypercenter	C₂×#346	D₇
hypocenter	C₇	C₂×C₄×M₃₂
LCS	G ▷ C₂×C₇ ▷ C₇
UCS	C₁ ◁ C₂²×C₈ ◁ C₂×#346

Complements

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

N	H ≅ G/N	type
3. C₂×C₇×#346	C₂	semidirect
4. C₂×C₁₆×Dic₇	C₂	semidirect
5. C₂×#89256	C₂	semidirect
6. #11726825	C₂	direct (N × H)
16. C₂×#6372	C₄	semidirect
20. C₇×#346	C₂²	semidirect
21. C₁₆×Dic₇	C₂²	semidirect
22. #89256	C₂²	semidirect
40. C₂×C₇×C₁₆	C₂×C₄	semidirect
47. C₂×C₇⋊C₁₆	C₂×C₄	semidirect
50. #6372	C₂×C₄	semidirect
55. C₂×#346	D₇	semidirect
85. C₇×C₁₆	C₂²×C₄	semidirect
86. C₇⋊C₁₆	C₂²×C₄	semidirect
101. #346	C₂×D₇	semidirect
114. C₂×Dic₇	M₃₂	semidirect
145. C₂×C₁₆	C₄×D₇	semidirect
164. Dic₇	C₂×M₃₂	semidirect
194. C₁₆	C₂×C₄×D₇	semidirect
195. C₂×C₇	C₄×M₃₂	semidirect
222. C₇	C₂×C₄×M₃₂	semidirect
236. C₂	#11726825	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×#6372	2
C₂×C₄×C₈×D₇	2
C₂×C₇×#346	2
C₂×C₁₆×Dic₇	2
C₂×#89256	2
#11726825	2
C₂×C₄×M₃₂	7

Subgroups — C2×#11726825 #13506426

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#11726825 #13506426