Subgroups — C₂×#11729028 #13341436

Order 1792.

1046 conjugacy classes of subgroups.

The subgroup lattice has 272 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₇×#493	C₂	1	1	yes	maximal, characteristic
2	C₂×#89848	C₂	1	2	yes	maximal, complemented
3	C₂×#89842	C₂	1	2	yes	maximal, complemented
4	C₂×#89454	C₂	1	1	yes	maximal, characteristic, complemented
5	#11729028	C₂	1	8	yes	maximal, direct product factor
6	C₂×C₈×C₇⋊C₈	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂²×C₇×D₈	—	2	2	no	—
8	C₂×C₇×#127	C₂²	1	2	yes	—
9	C₂²×#1142	—	2	4	no	—
10	C₂×#6432	C₂²	1	2	yes	—
11	C₇×#493	—	2	2	no	—
12	C₇×#493	C₂²	1	4	yes	—
13	C₂×C₄×C₇×C₈	C₂²	1	1	yes	characteristic
14	C₂×#6406	—	2	4	no	—
15	C₂×#6392	—	2	2	no	—
16	#89848	—	2	4	no	—
17	#89848	C₂²	1	8	yes	complemented
18	#89842	—	2	4	no	—
19	#89842	C₂²	1	8	yes	complemented
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	complemented
24	C₈×C₇⋊C₈	C₂²	1	4	yes	complemented
25	C₂×#1706	—	7	1	no	maximal, Sylow
26	C₂²×C₇×D₄	—	2	2	no	—
27	C₂²×C₇×D₄	—	2	2	no	—
28	C₂²×C₇×D₄	—	2	2	no	—
29	C₂²×Dic₁₄	—	2	2	no	—
30	C₂×C₇×D₈	—	2	8	no	—
31	C₂×C₇×D₈	—	4	8	no	—
32	C₂²×C₇×C₈	D₄	1	2	yes	—
33	C₇×#127	—	2	2	no	—
34	C₇×#127	—	2	2	no	—
35	C₇×#127	—	2	2	no	—
36	C₇×#127	C₂³	1	4	yes	—
37	C₂×C₄²×C₇	C₂³	1	1	yes	characteristic
38	C₂×#1142	—	2	16	no	—
39	C₂×#1142	—	4	16	no	—
40	C₂×C₄⋊Dic₇	—	2	2	no	—
41	C₂×C₄⋊Dic₇	—	2	2	no	—
42	C₂²×C₇⋊C₈	D₄	1	2	yes	—
43	C₂²×C₇⋊C₈	D₄	1	2	yes	—
44	#6432	—	2	2	no	—
45	#6432	—	2	2	no	—
46	#6432	—	2	2	no	—
47	#6432	C₂³	1	4	yes	—
48	C₄×C₇×C₈	D₄	1	2	yes	—
49	C₄×C₇×C₈	C₂³	1	2	yes	—
50	#6406	—	2	16	no	—
51	#6392	—	2	8	no	—
52	C₄×C₇⋊C₈	D₄	1	2	yes	—
53	C₄×C₇⋊C₈	D₄	1	2	yes	—
54	C₄×C₇⋊C₈	C₂³	1	2	yes	—
55	C₄×C₇⋊C₈	C₂³	1	2	yes	—
56	C₂×#500	—	7	1	no	—
57	C₂×#493	D₇	1	1	yes	characteristic
58	C₂×#492	—	7	2	no	—
59	C₂×#486	—	7	2	no	—
60	#1706	—	7	8	no	—
61	C₂×C₈²	—	7	1	no	—
62	C₂⁴×C₇	—	4	2	no	—
63	C₂×C₇×D₄	—	2	4	no	—
64	C₂×C₇×D₄	—	2	4	no	—
65	C₂×C₇×D₄	—	2	4	no	—
66	C₂×C₇×D₄	—	4	2	no	—
67	C₂×C₇×D₄	—	4	2	no	—
68	C₂×C₇×D₄	—	4	2	no	—
69	C₂×C₇×D₄	—	4	8	no	—
70	C₂×C₇×D₄	—	4	8	no	—
71	C₂×C₇×D₄	—	4	8	no	—
72	C₂²×C₄×C₇	C₂×D₄	1	1	yes	characteristic
73	C₂²×C₄×C₇	C₂×D₄	1	1	yes	characteristic
74	C₂²×C₄×C₇	C₂×D₄	1	1	yes	characteristic
75	C₂²×Dic₇	—	4	2	no	—
76	C₂×Dic₁₄	—	2	4	no	—
77	C₂×Dic₁₄	—	4	8	no	—
78	C₂×Dic₁₄	—	4	2	no	—
79	C₇×D₈	—	4	16	no	—
80	C₂×C₇×C₈	SD₁₆	1	8	yes	complemented
81	C₂×C₇×C₈	C₂×D₄	1	4	yes	—
82	C₄²×C₇	C₂×D₄	1	1	yes	characteristic
83	C₄²×C₇	C₂×D₄	1	1	yes	characteristic
84	C₄²×C₇	C₂×D₄	1	1	yes	characteristic
85	C₄²×C₇	C₂⁴	1	1	yes	characteristic
86	#1142	—	4	32	no	—
87	C₄⋊Dic₇	—	2	4	no	—
88	C₄⋊Dic₇	—	2	4	no	—
89	C₄⋊Dic₇	—	4	2	no	—
90	C₄⋊Dic₇	—	4	2	no	—
91	C₂×C₇⋊C₈	—	2	4	no	—
92	C₂×C₇⋊C₈	D₈	1	8	yes	complemented
93	C₂×C₇⋊C₈	C₂×D₄	1	4	yes	—
94	C₂×C₇⋊C₈	C₂×D₄	1	4	yes	—
95	C₂²×SD₁₆	—	14	4	no	—
96	C₂²×D₈	—	2	2	no	—
97	C₂×C₄⋊Q₈	—	7	2	no	—
98	C₂×#127	C₂×D₇	1	2	yes	—
99	#500	—	7	4	no	—
100	#500	—	14	2	no	—
101	#493	—	2	2	no	—
102	#493	C₂×D₇	1	4	yes	—
103	#492	—	7	8	no	—
104	#492	—	14	4	no	—
105	#486	—	7	8	no	—
106	#486	—	14	4	no	—
107	C₂×#107	—	14	2	no	—
108	C₂×#102	—	14	4	no	—
109	C₂×C₄×C₈	—	7	1	no	—
110	C₂×C₄×C₈	—	7	1	no	—
111	C₂×C₄×C₈	C₂×D₇	1	1	yes	characteristic
112	C₈²	—	7	4	no	—
113	C₂³×C₇	—	4	4	no	—
114	C₂³×C₇	—	8	4	no	—
115	C₂³×C₇	—	8	4	no	—
116	C₂³×C₇	—	8	4	no	—
117	C₂³×C₇	#127	1	1	yes	characteristic
118	C₇×D₄	—	4	8	no	—
119	C₇×D₄	—	4	8	no	—
120	C₇×D₄	—	4	8	no	—
121	C₇×D₄	—	8	4	no	—
122	C₇×D₄	—	8	4	no	—
123	C₇×D₄	—	8	4	no	—
124	C₂×C₄×C₇	#127	1	1	yes	characteristic
125	C₂×C₄×C₇	#127	1	1	yes	characteristic
126	C₂×C₄×C₇	#127	1	1	yes	characteristic
127	C₂×C₄×C₇	C₂×D₈	1	4	yes	—
128	C₂×C₄×C₇	C₂×SD₁₆	1	4	yes	—
129	C₂×C₄×C₇	#135	1	4	yes	—
130	C₂×C₄×C₇	C₂²×D₄	1	1	yes	characteristic
131	C₂×C₄×C₇	C₂²×D₄	1	1	yes	characteristic
132	C₂×C₄×C₇	C₂²×D₄	1	1	yes	characteristic
133	C₂×Dic₇	—	4	4	no	—
134	C₂×Dic₇	—	8	4	no	—
135	Dic₁₄	—	4	8	no	—
136	Dic₁₄	—	8	4	no	—
137	C₇×C₈	C₂×SD₁₆	1	8	yes	complemented
138	C₇⋊C₈	—	2	4	no	—
139	C₇⋊C₈	C₂×D₈	1	8	yes	complemented
140	C₂²×Q₈	—	14	2	no	—
141	C₂²×D₄	—	2	2	no	—
142	C₂²×D₄	—	2	2	no	—
143	C₂²×D₄	—	2	2	no	—
144	C₂×SD₁₆	—	14	16	no	—
145	C₂×SD₁₆	—	28	16	no	—
146	C₂×D₈	—	2	8	no	—
147	C₂×D₈	—	4	8	no	—
148	C₂²×C₈	—	7	2	no	—
149	C₂²×C₈	—	7	2	no	—
150	C₂²×C₈	C₂²⋊D₇	1	2	yes	—
151	C₄⋊Q₈	—	7	4	no	—
152	C₄⋊Q₈	—	14	2	no	—
153	C₄⋊Q₈	—	14	2	no	—
154	C₄⋊Q₈	—	14	2	no	—
155	#127	—	2	2	no	—
156	#127	—	2	2	no	—
157	#127	—	2	2	no	—
158	#127	C₂²×D₇	1	4	yes	—
159	C₂×C₄⋊C₄	—	14	2	no	—
160	C₂×C₄⋊C₄	—	14	2	no	—
161	C₂×C₄²	C₂²×D₇	1	1	yes	characteristic
162	#107	—	14	8	no	—
163	#102	—	14	16	no	—
164	C₄×C₈	—	7	2	no	—
165	C₄×C₈	—	7	2	no	—
166	C₄×C₈	—	7	2	no	—
167	C₄×C₈	—	7	2	no	—
168	C₄×C₈	C₂²⋊D₇	1	2	yes	—
169	C₄×C₈	C₂²×D₇	1	2	yes	—
170	C₂²×C₇	—	8	4	no	—
171	C₂²×C₇	—	8	4	no	—
172	C₂²×C₇	—	8	4	no	—
173	C₂²×C₇	—	16	8	no	—
174	C₂²×C₇	#492	1	2	yes	—
175	C₂²×C₇	#493	1	2	yes	—
176	C₂²×C₇	#495	1	2	yes	—
177	C₂²×C₇	C₂×#127	1	1	yes	characteristic
178	C₄×C₇	#492	1	2	yes	—
179	C₄×C₇	#493	1	2	yes	—
180	C₄×C₇	#495	1	2	yes	—
181	C₄×C₇	C₂²×D₈	1	2	yes	—
182	C₄×C₇	C₂²×SD₁₆	1	2	yes	—
183	C₄×C₇	C₂×#135	1	2	yes	—
184	Dic₇	—	8	4	no	—
185	C₂⁴	—	4	2	no	—
186	C₂×Q₈	—	14	4	no	—
187	C₂×Q₈	—	28	2	no	—
188	C₂×Q₈	—	28	8	no	—
189	C₂×D₄	—	2	4	no	—
190	C₂×D₄	—	2	4	no	—
191	C₂×D₄	—	2	4	no	—
192	C₂×D₄	—	4	8	no	—
193	C₂×D₄	—	4	8	no	—
194	C₂×D₄	—	4	8	no	—
195	C₂×D₄	—	4	2	no	—
196	C₂×D₄	—	4	2	no	—
197	C₂×D₄	—	4	2	no	—
198	C₂²×C₄	—	28	2	no	—
199	C₂²×C₄	D₄×D₇	1	1	yes	characteristic
200	C₂²×C₄	D₄×D₇	1	1	yes	characteristic
201	C₂²×C₄	C₂×C₂²⋊D₇	1	1	yes	characteristic
202	SD₁₆	—	28	32	no	—
203	D₈	—	4	16	no	—
204	C₂×C₈	—	7	8	no	—
205	C₂×C₈	—	7	4	no	—
206	C₂×C₈	—	7	4	no	—
207	C₂×C₈	—	14	4	no	—
208	C₂×C₈	#1142	1	8	yes	complemented
209	C₂×C₈	C₂×C₂²⋊D₇	1	4	yes	—
210	C₄⋊C₄	—	14	4	no	—
211	C₄⋊C₄	—	14	4	no	—
212	C₄⋊C₄	—	28	2	no	—
213	C₄⋊C₄	—	28	2	no	—
214	C₄²	D₄×D₇	1	1	yes	characteristic
215	C₄²	D₄×D₇	1	1	yes	characteristic
216	C₄²	C₂×C₂²⋊D₇	1	1	yes	characteristic
217	C₄²	C₂³×D₇	1	1	yes	characteristic
218	C₂×C₇	—	16	4	no	—
219	C₂×C₇	#1706	1	4	yes	complemented
220	C₂×C₇	C₂×#492	1	1	yes	characteristic
221	C₂×C₇	C₂×#493	1	1	yes	characteristic
222	C₂×C₇	C₂×#495	1	1	yes	characteristic
223	C₂³	—	4	4	no	—
224	C₂³	—	8	4	no	—
225	C₂³	—	8	4	no	—
226	C₂³	—	8	4	no	—
227	C₂³	#6503	1	1	yes	characteristic
228	Q₈	—	28	8	no	—
229	Q₈	—	56	4	no	—
230	D₄	—	4	8	no	—
231	D₄	—	4	8	no	—
232	D₄	—	4	8	no	—
233	D₄	—	8	4	no	—
234	D₄	—	8	4	no	—
235	D₄	—	8	4	no	—
236	C₂×C₄	—	28	4	no	—
237	C₂×C₄	—	56	4	no	—
238	C₂×C₄	D₇×D₈	1	4	yes	—
239	C₂×C₄	#6475	1	4	yes	—
240	C₂×C₄	C₂×#1142	1	4	yes	—
241	C₂×C₄	#6503	1	1	yes	characteristic
242	C₂×C₄	#6503	1	1	yes	characteristic
243	C₂×C₄	#6503	1	1	yes	characteristic
244	C₂×C₄	C₂×D₄×D₇	1	1	yes	characteristic
245	C₂×C₄	C₂×D₄×D₇	1	1	yes	characteristic
246	C₂×C₄	C₂²×C₂²⋊D₇	1	1	yes	characteristic
247	C₈	—	7	8	no	—
248	C₈	—	14	4	no	—
249	C₈	C₂×#1142	1	8	yes	complemented
250	C₇	C₂×#1706	1	1	yes	characteristic, complemented, Sylow
251	C₂²	—	8	4	no	—
252	C₂²	—	8	4	no	—
253	C₂²	—	8	4	no	—
254	C₂²	—	16	8	no	—
255	C₂²	#89848	1	2	yes	—
256	C₂²	#89923	1	2	yes	—
257	C₂²	#89927	1	2	yes	—
258	C₂²	C₂×#6503	1	1	yes	characteristic
259	C₄	—	56	4	no	—
260	C₄	#89848	1	2	yes	—
261	C₄	#89923	1	2	yes	—
262	C₄	#89927	1	2	yes	—
263	C₄	C₂×D₇×D₈	1	2	yes	—
264	C₄	C₂×#6475	1	2	yes	—
265	C₄	C₂²×#1142	1	2	yes	—
266	C₂	—	16	4	no	—
267	C₂	#11729028	1	4	yes	direct product factor
268	C₂	C₂×#89848	1	1	yes	characteristic
269	C₂	C₂×#89923	1	1	yes	characteristic
270	C₂	C₂×#89927	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	#6503
commutator, cosocle	C₄²×C₇	C₂⁴
Frattini	C₄²	C₂³×D₇
Fitting, prime radical	C₂×C₇×#493	C₂
socle	C₂³×C₇	#127
hypercenter	C₂×#493	D₇
hypocenter	C₇	C₂×#1706
LCS	G ▷ C₄²×C₇ ▷ C₂²×C₇ ▷ C₇
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×#493

Complements

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

N	H ≅ G/N	type
2. C₂×#89848	C₂	semidirect
3. C₂×#89842	C₂	semidirect
4. C₂×#89454	C₂	semidirect
5. #11729028	C₂	direct (N × H)
6. C₂×C₈×C₇⋊C₈	C₂	semidirect
17. #89848	C₂²	semidirect
19. #89842	C₂²	semidirect
23. #89454	C₂²	semidirect
24. C₈×C₇⋊C₈	C₂²	semidirect
80. C₂×C₇×C₈	SD₁₆	semidirect
92. C₂×C₇⋊C₈	D₈	semidirect
137. C₇×C₈	C₂×SD₁₆	semidirect
139. C₇⋊C₈	C₂×D₈	semidirect
208. C₂×C₈	#1142	semidirect
219. C₂×C₇	#1706	semidirect
249. C₈	C₂×#1142	semidirect
250. C₇	C₂×#1706	semidirect
267. C₂	#11729028	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×#493	2
C₂×#89848	2
C₂×#89842	2
C₂×#89454	2
#11729028	2
C₂×C₈×C₇⋊C₈	2
C₂×#1706	7

Subgroups — C2×#11729028 #13341436

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#11729028 #13341436