Subgroups — C₂×#11741230 #14217469

Order 1792.

2766 conjugacy classes of subgroups.

The subgroup lattice has 258 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₇×#131	C₂	1	1	yes	maximal, characteristic
2	C₂×#90518	C₂	1	2	yes	maximal
3	#11741230	C₂	1	16	yes	maximal, direct product factor
4	C₂×#89893	C₂	1	6	yes	maximal, complemented
5	C₂×#89878	C₂	1	3	yes	maximal, complemented
6	C₂²×#6392	C₂	1	3	yes	maximal, complemented
7	C₂²×C₇×#41	C₂²	1	1	yes	characteristic
8	C₂×C₇×#131	C₄	1	16	yes	complemented
9	C₂×C₇×#131	C₂²	1	8	yes	—
10	C₂²×C₇×M₁₆	C₂²	1	3	yes	—
11	C₂³×C₇×C₈	C₂²	1	3	yes	—
12	#90518	—	2	8	no	—
13	#90518	C₂²	1	16	yes	—
14	C₂×Q₈×Dic₇	—	2	2	no	—
15	C₂×D₄×Dic₇	—	2	6	no	—
16	C₂×#6489	C₂²	1	6	yes	—
17	C₂²×C₄⋊Dic₇	C₂²	1	6	yes	—
18	#89893	C₂²	1	48	yes	complemented
19	#89878	C₂²	1	24	yes	complemented
20	C₂×#6392	C₂²	1	6	yes	—
21	C₂×#6392	C₂²	1	3	yes	—
22	C₂×#6392	C₂²	1	1	yes	characteristic
23	C₂×#6392	C₂²	1	24	yes	complemented
24	C₂×#6391	C₂²	1	6	yes	—
25	C₂×#2913	—	7	1	no	maximal, Sylow
26	C₂×C₇×#41	C₂×C₄	1	8	yes	—
27	C₂×C₇×#41	D₄	1	4	yes	—
28	C₂×C₇×#41	Q₈	1	8	yes	—
29	C₂×C₇×#41	C₂³	1	4	yes	—
30	C₂²×C₇×Q₈	D₄	1	1	yes	characteristic
31	C₂²×C₇×D₄	D₄	1	3	yes	—
32	C₂³×C₄×C₇	C₂³	1	3	yes	—
33	C₂³×Dic₇	—	2	6	no	—
34	C₇×#131	C₂×C₄	1	64	yes	complemented
35	C₂×C₇×M₁₆	C₂×C₄	1	24	yes	complemented
36	C₂×C₇×M₁₆	C₂³	1	12	yes	—
37	C₂²×C₇×C₈	C₂×C₄	1	24	yes	complemented
38	C₂²×C₇×C₈	C₂³	1	3	yes	—
39	C₂²×C₇×C₈	C₂³	1	1	yes	characteristic
40	C₂²×C₇×C₈	C₂³	1	12	yes	—
41	C₂×C₂²⋊Dic₇	—	2	6	no	—
42	Q₈×Dic₇	—	2	16	no	—
43	D₄×Dic₇	—	2	48	no	—
44	#6489	—	2	12	no	—
45	#6489	C₂³	1	24	yes	—
46	C₂×C₄⋊Dic₇	—	2	6	no	—
47	C₂×C₄⋊Dic₇	—	2	12	no	—
48	C₂×C₄⋊Dic₇	C₂³	1	2	yes	—
49	C₂×C₄⋊Dic₇	C₂³	1	6	yes	—
50	C₂×C₄⋊Dic₇	C₂³	1	24	yes	—
51	C₂×C₄×Dic₇	—	2	6	no	—
52	#6392	C₂³	1	12	yes	—
53	#6392	C₂³	1	24	yes	—
54	#6392	C₂³	1	4	yes	—
55	#6391	C₂³	1	24	yes	—
56	C₂²×#131	D₇	1	1	yes	characteristic
57	C₂×#520	—	7	2	no	—
58	#2913	—	7	16	no	—
59	C₂×#428	—	7	6	no	—
60	C₂×#427	—	7	3	no	—
61	C₂²×#107	—	7	3	no	—
62	C₂⁴×C₇	C₂×D₄	1	3	yes	—
63	C₇×#41	C₄⋊C₄	1	32	yes	—
64	C₇×#41	C₂²×C₄	1	16	yes	—
65	C₇×#41	C₂×Q₈	1	16	yes	—
66	C₂×C₇×Q₈	C₄⋊C₄	1	8	yes	—
67	C₂×C₇×Q₈	C₂×D₄	1	4	yes	—
68	C₂×C₇×D₄	C₄⋊C₄	1	24	yes	—
69	C₂×C₇×D₄	C₂×D₄	1	12	yes	—
70	C₂²×C₄×C₇	C₂²×C₄	1	12	yes	—
71	C₂²×C₄×C₇	C₂×D₄	1	3	yes	—
72	C₂²×C₄×C₇	C₂×D₄	1	6	yes	—
73	C₂²×C₄×C₇	C₂×Q₈	1	12	yes	—
74	C₂²×C₄×C₇	C₂⁴	1	1	yes	characteristic
75	C₂²×C₄×C₇	C₂⁴	1	6	yes	—
76	C₂²×Dic₇	—	2	2	no	—
77	C₂²×Dic₇	—	2	6	no	—
78	C₂²×Dic₇	—	2	24	no	—
79	C₂²×Dic₇	—	8	12	no	—
80	C₇×M₁₆	C₂²×C₄	1	48	yes	complemented
81	C₂×C₇×C₈	C₂²×C₄	1	4	yes	—
82	C₂×C₇×C₈	C₂²×C₄	1	12	yes	—
83	C₂×C₇×C₈	C₂²×C₄	1	48	yes	complemented
84	C₂×C₇×C₈	C₂⁴	1	2	yes	—
85	C₂×C₇×C₈	C₂⁴	1	6	yes	—
86	C₂²⋊Dic₇	—	2	24	no	—
87	C₄⋊Dic₇	—	2	6	no	—
88	C₄⋊Dic₇	—	2	24	no	—
89	C₄⋊Dic₇	—	2	2	no	—
90	C₄⋊Dic₇	C₂⁴	1	12	yes	—
91	C₄⋊Dic₇	C₂⁴	1	4	yes	—
92	C₄×Dic₇	—	2	24	no	—
93	C₂²×#41	C₂×D₇	1	1	yes	characteristic
94	C₂×#131	Dic₇	1	16	yes	complemented
95	C₂×#131	C₂×D₇	1	8	yes	—
96	C₂²×M₁₆	C₂×D₇	1	3	yes	—
97	C₂³×C₈	C₂×D₇	1	3	yes	—
98	#520	—	7	16	no	—
99	#520	—	14	8	no	—
100	C₂×C₄×Q₈	—	14	2	no	—
101	C₂×C₄×D₄	—	14	6	no	—
102	C₂×#117	—	7	6	no	—
103	C₂²×C₄⋊C₄	—	7	6	no	—
104	#428	—	7	48	no	—
105	#427	—	7	24	no	—
106	C₂×#107	—	7	3	no	—
107	C₂×#107	—	7	6	no	—
108	C₂×#107	—	7	1	no	—
109	C₂×#107	—	7	24	no	—
110	C₂×#106	—	7	6	no	—
111	C₂³×C₇	—	2	6	no	—
112	C₂³×C₇	—	2	6	no	—
113	C₂³×C₇	C₂×C₄⋊C₄	1	12	yes	—
114	C₂³×C₇	C₂²×D₄	1	6	yes	—
115	C₂³×C₇	C₂²×D₄	1	1	yes	characteristic
116	C₇×Q₈	C₂×C₄⋊C₄	1	16	yes	—
117	C₇×D₄	C₂×C₄⋊C₄	1	48	yes	—
118	C₂×C₄×C₇	C₂×C₄⋊C₄	1	12	yes	—
119	C₂×C₄×C₇	C₂×C₄⋊C₄	1	24	yes	—
120	C₂×C₄×C₇	C₂³×C₄	1	2	yes	—
121	C₂×C₄×C₇	C₂³×C₄	1	12	yes	—
122	C₂×C₄×C₇	C₂²×D₄	1	6	yes	—
123	C₂×C₄×C₇	C₂²×D₄	1	1	yes	characteristic
124	C₂×C₄×C₇	C₂²×Q₈	1	2	yes	—
125	C₂×C₄×C₇	C₂²×Q₈	1	12	yes	—
126	C₂×C₄×C₇	C₂⁵	1	1	yes	characteristic
127	C₂×Dic₇	—	2	4	no	—
128	C₂×Dic₇	—	2	12	no	—
129	C₂×Dic₇	—	4	12	no	—
130	C₂×Dic₇	—	4	4	no	—
131	C₂×Dic₇	—	8	24	no	—
132	C₇×C₈	C₂³×C₄	1	4	yes	—
133	C₇×C₈	C₂³×C₄	1	12	yes	—
134	C₂×#41	Dic₁₄	1	8	yes	—
135	C₂×#41	D₂₈	1	4	yes	—
136	C₂×#41	C₂×Dic₇	1	8	yes	—
137	C₂×#41	C₂²×D₇	1	4	yes	—
138	C₂²×Q₈	D₂₈	1	1	yes	characteristic
139	C₂²×D₄	D₂₈	1	3	yes	—
140	C₂³×C₄	—	14	6	no	—
141	C₂³×C₄	C₂²×D₇	1	3	yes	—
142	#131	C₂×Dic₇	1	64	yes	complemented
143	C₂×M₁₆	C₂×Dic₇	1	24	yes	complemented
144	C₂×M₁₆	C₂²×D₇	1	12	yes	—
145	C₂²×C₈	C₂×Dic₇	1	24	yes	complemented
146	C₂²×C₈	C₂²×D₇	1	1	yes	characteristic
147	C₂²×C₈	C₂²×D₇	1	3	yes	—
148	C₂²×C₈	C₂²×D₇	1	12	yes	—
149	C₄×Q₈	—	14	16	no	—
150	C₄×D₄	—	14	48	no	—
151	#117	—	7	24	no	—
152	#117	—	14	12	no	—
153	C₂×C₄⋊C₄	—	7	6	no	—
154	C₂×C₄⋊C₄	—	7	24	no	—
155	C₂×C₄⋊C₄	—	7	2	no	—
156	C₂×C₄⋊C₄	—	14	6	no	—
157	C₂×C₄⋊C₄	—	14	12	no	—
158	C₂×C₂²⋊C₄	—	14	6	no	—
159	C₂×C₄²	—	14	6	no	—
160	#107	—	7	12	no	—
161	#107	—	7	24	no	—
162	#107	—	7	4	no	—
163	#106	—	7	24	no	—
164	C₂²×C₇	—	2	24	no	—
165	C₂²×C₇	—	2	6	no	—
166	C₂²×C₇	—	2	6	no	—
167	C₂²×C₇	C₂²×C₄⋊C₄	1	12	yes	—
168	C₂²×C₇	C₂²×C₄⋊C₄	1	2	yes	—
169	C₂²×C₇	#576	1	2	yes	—
170	C₂²×C₇	#578	1	2	yes	—
171	C₂²×C₇	C₂³×D₄	1	1	yes	characteristic
172	C₄×C₇	C₂²×C₄⋊C₄	1	2	yes	—
173	C₄×C₇	C₂²×C₄⋊C₄	1	12	yes	—
174	C₄×C₇	C₂⁴×C₄	1	1	yes	characteristic
175	C₄×C₇	C₂³×Q₈	1	1	yes	characteristic
176	Dic₇	—	4	12	no	—
177	Dic₇	—	4	4	no	—
178	C₂⁴	C₂×D₂₈	1	3	yes	—
179	#41	C₄⋊Dic₇	1	32	yes	—
180	#41	C₂×Dic₁₄	1	16	yes	—
181	#41	C₂²×Dic₇	1	16	yes	—
182	C₂×Q₈	C₄⋊Dic₇	1	8	yes	—
183	C₂×Q₈	C₂×D₂₈	1	4	yes	—
184	C₂×D₄	C₄⋊Dic₇	1	24	yes	—
185	C₂×D₄	C₂×D₂₈	1	12	yes	—
186	C₂²×C₄	—	14	2	no	—
187	C₂²×C₄	—	14	6	no	—
188	C₂²×C₄	—	14	24	no	—
189	C₂²×C₄	—	56	12	no	—
190	C₂²×C₄	C₂×Dic₁₄	1	12	yes	—
191	C₂²×C₄	C₂×D₂₈	1	6	yes	—
192	C₂²×C₄	C₂×D₂₈	1	3	yes	—
193	C₂²×C₄	C₂²×Dic₇	1	12	yes	—
194	C₂²×C₄	C₂³×D₇	1	6	yes	—
195	C₂²×C₄	C₂³×D₇	1	1	yes	characteristic
196	M₁₆	C₂²×Dic₇	1	48	yes	complemented
197	C₂×C₈	C₂²×Dic₇	1	4	yes	—
198	C₂×C₈	C₂²×Dic₇	1	12	yes	—
199	C₂×C₈	C₂²×Dic₇	1	48	yes	complemented
200	C₂×C₈	C₂³×D₇	1	6	yes	—
201	C₂×C₈	C₂³×D₇	1	2	yes	—
202	C₄⋊C₄	—	7	4	no	—
203	C₄⋊C₄	—	7	12	no	—
204	C₄⋊C₄	—	14	6	no	—
205	C₄⋊C₄	—	14	24	no	—
206	C₄⋊C₄	—	14	2	no	—
207	C₂²⋊C₄	—	14	24	no	—
208	C₄²	—	14	24	no	—
209	C₂×C₇	—	2	12	no	—
210	C₂×C₇	#2913	1	4	yes	complemented
211	C₂×C₇	C₂³×C₄⋊C₄	1	1	yes	characteristic
212	C₂×C₇	C₂×#576	1	1	yes	characteristic
213	C₂×C₇	C₂×#578	1	1	yes	characteristic
214	C₂³	—	2	6	no	—
215	C₂³	—	2	6	no	—
216	C₂³	C₂×C₄⋊Dic₇	1	12	yes	—
217	C₂³	C₂²×D₂₈	1	6	yes	—
218	C₂³	C₂²×D₂₈	1	1	yes	characteristic
219	Q₈	C₂×C₄⋊Dic₇	1	16	yes	—
220	D₄	C₂×C₄⋊Dic₇	1	48	yes	—
221	C₂×C₄	—	14	4	no	—
222	C₂×C₄	—	14	12	no	—
223	C₂×C₄	—	28	12	no	—
224	C₂×C₄	—	28	4	no	—
225	C₂×C₄	—	56	24	no	—
226	C₂×C₄	C₂×C₄⋊Dic₇	1	12	yes	—
227	C₂×C₄	C₂×C₄⋊Dic₇	1	24	yes	—
228	C₂×C₄	C₂²×Dic₁₄	1	2	yes	—
229	C₂×C₄	C₂²×Dic₁₄	1	12	yes	—
230	C₂×C₄	C₂²×D₂₈	1	6	yes	—
231	C₂×C₄	C₂²×D₂₈	1	1	yes	characteristic
232	C₂×C₄	C₂³×Dic₇	1	2	yes	—
233	C₂×C₄	C₂³×Dic₇	1	12	yes	—
234	C₂×C₄	C₂⁴×D₇	1	1	yes	characteristic
235	C₈	C₂³×Dic₇	1	4	yes	—
236	C₈	C₂³×Dic₇	1	12	yes	—
237	C₇	C₂×#2913	1	1	yes	characteristic, complemented, Sylow
238	C₂²	—	2	24	no	—
239	C₂²	—	2	6	no	—
240	C₂²	—	2	6	no	—
241	C₂²	#90443	1	2	yes	—
242	C₂²	#90444	1	2	yes	—
243	C₂²	C₂²×C₄⋊Dic₇	1	12	yes	—
244	C₂²	C₂²×C₄⋊Dic₇	1	2	yes	—
245	C₂²	C₂³×D₂₈	1	1	yes	characteristic
246	C₄	—	28	12	no	—
247	C₄	—	28	4	no	—
248	C₄	C₂²×C₄⋊Dic₇	1	2	yes	—
249	C₄	C₂²×C₄⋊Dic₇	1	12	yes	—
250	C₄	C₂³×Dic₁₄	1	1	yes	characteristic
251	C₄	C₂⁴×Dic₇	1	1	yes	characteristic
252	C₂	—	2	12	no	—
253	C₂	#11741230	1	4	yes	direct product factor
254	C₂	C₂×#90443	1	1	yes	characteristic
255	C₂	C₂×#90444	1	1	yes	characteristic
256	C₂	C₂³×C₄⋊Dic₇	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	C₂²×D₂₈
commutator	C₄×C₇	C₂⁴×C₄
Frattini	C₂×C₄	C₂⁴×D₇
Fitting	C₂²×C₇×#131	C₂
socle	C₂³×C₇	C₂²×D₄
prime radical	C₂²×C₇×#41	C₂²
cosocle	C₂×C₄×C₇	C₂⁵
hypercenter	C₂²×#131	D₇
hypocenter	C₇	C₂×#2913
LCS	G ▷ C₄×C₇ ▷ C₂×C₇ ▷ C₇
UCS	C₁ ◁ C₂³ ◁ C₂²×#41 ◁ C₂²×#131

Complements

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

N	H ≅ G/N	type
3. #11741230	C₂	direct (N × H)
4. C₂×#89893	C₂	semidirect
5. C₂×#89878	C₂	semidirect
6. C₂²×#6392	C₂	semidirect
8. C₂×C₇×#131	C₄	semidirect
18. #89893	C₂²	semidirect
19. #89878	C₂²	semidirect
23. C₂×#6392	C₂²	semidirect
34. C₇×#131	C₂×C₄	semidirect
35. C₂×C₇×M₁₆	C₂×C₄	semidirect
37. C₂²×C₇×C₈	C₂×C₄	semidirect
80. C₇×M₁₆	C₂²×C₄	semidirect
83. C₂×C₇×C₈	C₂²×C₄	semidirect
94. C₂×#131	Dic₇	semidirect
142. #131	C₂×Dic₇	semidirect
143. C₂×M₁₆	C₂×Dic₇	semidirect
145. C₂²×C₈	C₂×Dic₇	semidirect
196. M₁₆	C₂²×Dic₇	semidirect
199. C₂×C₈	C₂²×Dic₇	semidirect
210. C₂×C₇	#2913	semidirect
237. C₇	C₂×#2913	semidirect
253. C₂	#11741230	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×C₇×#131	2
C₂×#90518	2
#11741230	2
C₂×#89893	2
C₂×#89878	2
C₂²×#6392	2
C₂×#2913	7

Subgroups — C2×#11741230 #14217469

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#11741230 #14217469