Subgroups — C₂×#11729050 #13341442

Order 1792.

814 conjugacy classes of subgroups.

The subgroup lattice has 244 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₇×#494	C₂	1	1	yes	maximal, characteristic
2	C₂×#89851	C₂	1	2	yes	maximal
3	C₂×#89865	C₂	1	2	yes	maximal
4	C₂×#89454	C₂	1	1	yes	maximal, characteristic
5	#11729050	C₂	1	8	yes	maximal, direct product factor
6	C₂×C₈×C₇⋊C₈	C₂	1	1	yes	maximal, characteristic
7	C₂²×C₇×Q₁₆	—	2	2	no	—
8	C₂×C₇×C₄⋊Q₈	C₂²	1	2	yes	—
9	C₂×#6432	C₂²	1	2	yes	—
10	C₇×#494	—	2	2	no	—
11	C₇×#494	C₂²	1	4	yes	—
12	C₂×C₄×C₇×C₈	C₂²	1	1	yes	characteristic
13	C₂×#6392	—	2	2	no	—
14	#89865	—	2	4	no	—
15	#89865	C₂²	1	8	yes	—
16	#89851	—	2	4	no	—
17	#89851	C₂²	1	8	yes	—
18	C₂×#6384	—	2	4	no	—
19	C₂×#6382	—	2	4	no	—
20	C₂×C₄×C₇⋊C₈	C₂²	1	1	yes	characteristic
21	C₂×C₄×C₇⋊C₈	C₂²	1	1	yes	characteristic
22	#89454	—	2	2	no	—
23	#89454	C₂²	1	4	yes	—
24	C₈×C₇⋊C₈	C₂²	1	4	yes	—
25	C₂×#1707	—	7	1	no	maximal, Sylow
26	C₂²×C₇×Q₈	—	2	2	no	—
27	C₂²×Dic₁₄	—	2	2	no	—
28	C₂×C₇×Q₁₆	—	2	8	no	—
29	C₂×C₇×Q₁₆	—	4	8	no	—
30	C₂²×C₇×C₈	D₄	1	2	yes	—
31	C₇×C₄⋊Q₈	—	2	2	no	—
32	C₇×C₄⋊Q₈	—	2	2	no	—
33	C₇×C₄⋊Q₈	—	2	2	no	—
34	C₇×C₄⋊Q₈	C₂³	1	4	yes	—
35	C₂×C₇×C₄⋊C₄	—	2	2	no	—
36	C₂×C₇×C₄⋊C₄	—	2	2	no	—
37	C₂×C₄²×C₇	C₂³	1	1	yes	characteristic
38	C₂×C₄⋊Dic₇	—	2	2	no	—
39	C₂×C₄⋊Dic₇	—	2	2	no	—
40	C₂²×C₇⋊C₈	D₄	1	2	yes	—
41	C₂²×C₇⋊C₈	D₄	1	2	yes	—
42	#6432	—	2	2	no	—
43	#6432	—	2	2	no	—
44	#6432	—	2	2	no	—
45	#6432	C₂³	1	4	yes	—
46	C₄×C₇×C₈	D₄	1	2	yes	—
47	C₄×C₇×C₈	C₂³	1	2	yes	—
48	#6392	—	2	8	no	—
49	#6384	—	2	16	no	—
50	#6382	—	2	16	no	—
51	C₄×C₇⋊C₈	D₄	1	2	yes	—
52	C₄×C₇⋊C₈	D₄	1	2	yes	—
53	C₄×C₇⋊C₈	C₂³	1	2	yes	—
54	C₄×C₇⋊C₈	C₂³	1	2	yes	—
55	C₂×#500	—	7	1	no	—
56	C₂×#498	—	7	2	no	—
57	C₂×#494	D₇	1	1	yes	characteristic
58	C₂×#487	—	7	2	no	—
59	#1707	—	7	8	no	—
60	C₂×C₈²	—	7	1	no	—
61	C₂×C₇×Q₈	—	2	4	no	—
62	C₂×C₇×Q₈	—	4	2	no	—
63	C₂×C₇×Q₈	—	4	8	no	—
64	C₂²×C₄×C₇	—	4	2	no	—
65	C₂²×C₄×C₇	C₂×D₄	1	1	yes	characteristic
66	C₂²×C₄×C₇	C₂×D₄	1	1	yes	characteristic
67	C₂²×C₄×C₇	C₂×D₄	1	1	yes	characteristic
68	C₂²×Dic₇	—	4	2	no	—
69	C₂×Dic₁₄	—	2	4	no	—
70	C₂×Dic₁₄	—	4	8	no	—
71	C₂×Dic₁₄	—	4	2	no	—
72	C₇×Q₁₆	—	4	16	no	—
73	C₂×C₇×C₈	SD₁₆	1	8	yes	—
74	C₂×C₇×C₈	C₂×D₄	1	4	yes	—
75	C₇×C₄⋊C₄	—	2	4	no	—
76	C₇×C₄⋊C₄	—	2	4	no	—
77	C₇×C₄⋊C₄	—	4	2	no	—
78	C₇×C₄⋊C₄	—	4	2	no	—
79	C₄²×C₇	C₂×D₄	1	1	yes	characteristic
80	C₄²×C₇	C₂×D₄	1	1	yes	characteristic
81	C₄²×C₇	C₂×D₄	1	1	yes	characteristic
82	C₄²×C₇	C₂⁴	1	1	yes	characteristic
83	C₄⋊Dic₇	—	2	4	no	—
84	C₄⋊Dic₇	—	2	4	no	—
85	C₄⋊Dic₇	—	4	2	no	—
86	C₄⋊Dic₇	—	4	2	no	—
87	C₂×C₇⋊C₈	—	2	4	no	—
88	C₂×C₇⋊C₈	Q₁₆	1	8	yes	complemented
89	C₂×C₇⋊C₈	C₂×D₄	1	4	yes	—
90	C₂×C₇⋊C₈	C₂×D₄	1	4	yes	—
91	C₂²×Q₁₆	—	2	2	no	—
92	C₂×C₄⋊Q₈	—	7	2	no	—
93	C₂×C₄⋊Q₈	C₂×D₇	1	2	yes	—
94	#500	—	7	4	no	—
95	#500	—	14	2	no	—
96	#498	—	7	8	no	—
97	#498	—	14	4	no	—
98	#494	—	2	2	no	—
99	#494	C₂×D₇	1	4	yes	—
100	#487	—	7	8	no	—
101	#487	—	14	4	no	—
102	C₂×#107	—	14	2	no	—
103	C₂×#106	—	14	4	no	—
104	C₂×#103	—	14	4	no	—
105	C₂×C₄×C₈	—	7	1	no	—
106	C₂×C₄×C₈	—	7	1	no	—
107	C₂×C₄×C₈	C₂×D₇	1	1	yes	characteristic
108	C₈²	—	7	4	no	—
109	C₂³×C₇	#127	1	1	yes	characteristic
110	C₇×Q₈	—	4	8	no	—
111	C₇×Q₈	—	8	4	no	—
112	C₂×C₄×C₇	—	4	4	no	—
113	C₂×C₄×C₇	—	8	4	no	—
114	C₂×C₄×C₇	#127	1	1	yes	characteristic
115	C₂×C₄×C₇	#127	1	1	yes	characteristic
116	C₂×C₄×C₇	#127	1	1	yes	characteristic
117	C₂×C₄×C₇	C₂×SD₁₆	1	4	yes	—
118	C₂×C₄×C₇	C₂×Q₁₆	1	4	yes	—
119	C₂×C₄×C₇	#135	1	4	yes	—
120	C₂×C₄×C₇	C₂²×D₄	1	1	yes	characteristic
121	C₂×C₄×C₇	C₂²×D₄	1	1	yes	characteristic
122	C₂×C₄×C₇	C₂²×D₄	1	1	yes	characteristic
123	C₂×Dic₇	—	4	4	no	—
124	C₂×Dic₇	—	8	4	no	—
125	Dic₁₄	—	4	8	no	—
126	Dic₁₄	—	8	4	no	—
127	C₇×C₈	C₂×SD₁₆	1	8	yes	—
128	C₇⋊C₈	—	2	4	no	—
129	C₇⋊C₈	C₂×Q₁₆	1	8	yes	complemented
130	C₂²×Q₈	—	2	2	no	—
131	C₂²×Q₈	—	14	2	no	—
132	C₂×Q₁₆	—	2	8	no	—
133	C₂×Q₁₆	—	4	8	no	—
134	C₂²×C₈	—	7	2	no	—
135	C₂²×C₈	—	7	2	no	—
136	C₂²×C₈	C₂²⋊D₇	1	2	yes	—
137	C₄⋊Q₈	—	2	2	no	—
138	C₄⋊Q₈	—	2	2	no	—
139	C₄⋊Q₈	—	2	2	no	—
140	C₄⋊Q₈	—	7	4	no	—
141	C₄⋊Q₈	—	14	2	no	—
142	C₄⋊Q₈	—	14	2	no	—
143	C₄⋊Q₈	—	14	2	no	—
144	C₄⋊Q₈	C₂²×D₇	1	4	yes	—
145	C₂×C₄⋊C₄	—	2	2	no	—
146	C₂×C₄⋊C₄	—	2	2	no	—
147	C₂×C₄⋊C₄	—	14	2	no	—
148	C₂×C₄⋊C₄	—	14	2	no	—
149	C₂×C₄²	C₂²×D₇	1	1	yes	characteristic
150	#107	—	14	8	no	—
151	#106	—	14	16	no	—
152	#103	—	14	16	no	—
153	C₄×C₈	—	7	2	no	—
154	C₄×C₈	—	7	2	no	—
155	C₄×C₈	—	7	2	no	—
156	C₄×C₈	—	7	2	no	—
157	C₄×C₈	C₂²⋊D₇	1	2	yes	—
158	C₄×C₈	C₂²×D₇	1	2	yes	—
159	C₂²×C₇	#492	1	2	yes	—
160	C₂²×C₇	#494	1	2	yes	—
161	C₂²×C₇	#495	1	2	yes	—
162	C₂²×C₇	C₂×#127	1	1	yes	characteristic
163	C₄×C₇	—	8	4	no	—
164	C₄×C₇	#492	1	2	yes	—
165	C₄×C₇	#494	1	2	yes	—
166	C₄×C₇	#495	1	2	yes	—
167	C₄×C₇	C₂²×SD₁₆	1	2	yes	—
168	C₄×C₇	C₂²×Q₁₆	1	2	yes	—
169	C₄×C₇	C₂×#135	1	2	yes	—
170	Dic₇	—	8	4	no	—
171	C₂×Q₈	—	2	4	no	—
172	C₂×Q₈	—	4	2	no	—
173	C₂×Q₈	—	4	8	no	—
174	C₂×Q₈	—	14	4	no	—
175	C₂×Q₈	—	28	2	no	—
176	C₂×Q₈	—	28	8	no	—
177	C₂²×C₄	—	4	2	no	—
178	C₂²×C₄	—	28	2	no	—
179	C₂²×C₄	D₄×D₇	1	1	yes	characteristic
180	C₂²×C₄	D₄×D₇	1	1	yes	characteristic
181	C₂²×C₄	C₂×C₂²⋊D₇	1	1	yes	characteristic
182	Q₁₆	—	4	16	no	—
183	C₂×C₈	—	7	8	no	—
184	C₂×C₈	—	7	4	no	—
185	C₂×C₈	—	7	4	no	—
186	C₂×C₈	—	14	4	no	—
187	C₂×C₈	#1142	1	8	yes	—
188	C₂×C₈	C₂×C₂²⋊D₇	1	4	yes	—
189	C₄⋊C₄	—	2	4	no	—
190	C₄⋊C₄	—	2	4	no	—
191	C₄⋊C₄	—	4	2	no	—
192	C₄⋊C₄	—	4	2	no	—
193	C₄⋊C₄	—	14	4	no	—
194	C₄⋊C₄	—	14	4	no	—
195	C₄⋊C₄	—	28	2	no	—
196	C₄⋊C₄	—	28	2	no	—
197	C₄²	D₄×D₇	1	1	yes	characteristic
198	C₄²	D₄×D₇	1	1	yes	characteristic
199	C₄²	C₂×C₂²⋊D₇	1	1	yes	characteristic
200	C₄²	C₂³×D₇	1	1	yes	characteristic
201	C₂×C₇	#1707	1	4	yes	complemented
202	C₂×C₇	C₂×#492	1	1	yes	characteristic
203	C₂×C₇	C₂×#494	1	1	yes	characteristic
204	C₂×C₇	C₂×#495	1	1	yes	characteristic
205	C₂³	#6503	1	1	yes	characteristic
206	Q₈	—	4	8	no	—
207	Q₈	—	8	4	no	—
208	Q₈	—	28	8	no	—
209	Q₈	—	56	4	no	—
210	C₂×C₄	—	4	4	no	—
211	C₂×C₄	—	8	4	no	—
212	C₂×C₄	—	28	4	no	—
213	C₂×C₄	—	56	4	no	—
214	C₂×C₄	D₇×Q₁₆	1	4	yes	—
215	C₂×C₄	#6482	1	4	yes	—
216	C₂×C₄	C₂×#1142	1	4	yes	—
217	C₂×C₄	#6503	1	1	yes	characteristic
218	C₂×C₄	#6503	1	1	yes	characteristic
219	C₂×C₄	#6503	1	1	yes	characteristic
220	C₂×C₄	C₂×D₄×D₇	1	1	yes	characteristic
221	C₂×C₄	C₂×D₄×D₇	1	1	yes	characteristic
222	C₂×C₄	C₂²×C₂²⋊D₇	1	1	yes	characteristic
223	C₈	—	7	8	no	—
224	C₈	—	14	4	no	—
225	C₈	C₂×#1142	1	8	yes	—
226	C₇	C₂×#1707	1	1	yes	characteristic, complemented, Sylow
227	C₂²	#89848	1	2	yes	—
228	C₂²	#89953	1	2	yes	—
229	C₂²	#89963	1	2	yes	—
230	C₂²	C₂×#6503	1	1	yes	characteristic
231	C₄	—	8	4	no	—
232	C₄	—	56	4	no	—
233	C₄	#89848	1	2	yes	—
234	C₄	#89953	1	2	yes	—
235	C₄	#89963	1	2	yes	—
236	C₄	C₂×D₇×Q₁₆	1	2	yes	—
237	C₄	C₂×#6482	1	2	yes	—
238	C₄	C₂²×#1142	1	2	yes	—
239	C₂	#11729050	1	4	yes	direct product factor
240	C₂	C₂×#89848	1	1	yes	characteristic
241	C₂	C₂×#89953	1	1	yes	characteristic
242	C₂	C₂×#89963	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	#6503
commutator, cosocle	C₄²×C₇	C₂⁴
Frattini	C₄²	C₂³×D₇
Fitting	C₂×C₇×#494	C₂
socle, prime radical	C₂³×C₇	#127
hypercenter	C₂×#494	D₇
hypocenter	C₇	C₂×#1707
LCS	G ▷ C₄²×C₇ ▷ C₂²×C₇ ▷ C₇
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×#494

Complements

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

N	H ≅ G/N	type
5. #11729050	C₂	direct (N × H)
88. C₂×C₇⋊C₈	Q₁₆	semidirect
129. C₇⋊C₈	C₂×Q₁₆	semidirect
201. C₂×C₇	#1707	semidirect
226. C₇	C₂×#1707	semidirect
239. C₂	#11729050	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂×C₇×#494	2
C₂×#89851	2
C₂×#89865	2
C₂×#89454	2
#11729050	2
C₂×C₈×C₇⋊C₈	2
C₂×#1707	7

Subgroups — C2×#11729050 #13341442

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#11729050 #13341442