Subgroups — C₂×#11728656 #13323189

Order 1792.

718 conjugacy classes of subgroups.

The subgroup lattice has 246 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₄×C₇×Q₁₆	C₂	1	1	yes	maximal, characteristic
2	C₂×Q₈×C₇⋊C₈	C₂	1	2	yes	maximal
3	C₂×Q₈⋊(C₇⋊C₈)	C₂	1	2	yes	maximal
4	#11728656	C₂	1	8	yes	maximal, direct product factor
5	C₂×#89252	C₂	1	1	yes	maximal, characteristic
6	C₂×#89250	C₂	1	1	yes	maximal, characteristic
7	C₂²×C₇×Q₁₆	C₄	1	1	yes	characteristic
8	C₂×C₄×C₇×Q₈	C₂²	1	2	yes	—
9	C₄×C₇×Q₁₆	C₄	1	4	yes	—
10	C₄×C₇×Q₁₆	C₂²	1	4	yes	—
11	C₂×C₇×#107	C₄	1	1	yes	characteristic
12	C₂×C₇×#103	C₄	1	2	yes	—
13	C₂×C₄×C₇×C₈	C₂²	1	1	yes	characteristic
14	Q₈×C₇⋊C₈	—	2	4	no	—
15	Q₈×C₇⋊C₈	C₂²	1	8	yes	—
16	C₂×C₄⋊(C₇⋊C₈)	—	2	2	no	—
17	C₂×C₄⋊(C₇⋊C₈)	C₂²	1	2	yes	—
18	C₂×C₄×C₇⋊C₈	—	2	2	no	—
19	C₂×C₄×C₇⋊C₈	C₂²	1	1	yes	characteristic
20	C₂×C₄×C₇⋊C₈	C₂²	1	1	yes	characteristic
21	Q₈⋊(C₇⋊C₈)	C₂²	1	8	yes	—
22	#89252	C₂²	1	4	yes	—
23	#89250	C₂²	1	4	yes	—
24	C₂×#1579	—	7	1	no	maximal, Sylow
25	C₂²×C₇×Q₈	C₂×C₄	1	2	yes	—
26	C₂×C₇×Q₁₆	C₈	1	8	yes	complemented
27	C₂×C₇×Q₁₆	C₂×C₄	1	4	yes	—
28	C₂²×C₇×C₈	C₂×C₄	1	1	yes	characteristic
29	C₂²×C₇×C₈	C₂×C₄	1	1	yes	characteristic
30	C₄×C₇×Q₈	—	2	2	no	—
31	C₄×C₇×Q₈	—	2	2	no	—
32	C₄×C₇×Q₈	C₂×C₄	1	4	yes	—
33	C₄×C₇×Q₈	C₂³	1	4	yes	—
34	C₂×C₇×C₄⋊C₄	—	2	2	no	—
35	C₂×C₇×C₄⋊C₄	C₂×C₄	1	2	yes	—
36	C₂×C₄²×C₇	—	2	2	no	—
37	C₂×C₄²×C₇	C₂³	1	1	yes	characteristic
38	C₂²×C₇⋊C₈	—	2	2	no	—
39	C₂²×C₇⋊C₈	—	2	1	no	—
40	C₂²×C₇⋊C₈	D₄	1	2	yes	—
41	C₇×#107	C₂×C₄	1	4	yes	—
42	C₇×#103	C₂×C₄	1	8	yes	—
43	C₄×C₇×C₈	C₂×C₄	1	2	yes	—
44	C₄×C₇×C₈	C₂³	1	2	yes	—
45	C₄⋊(C₇⋊C₈)	—	2	2	no	—
46	C₄⋊(C₇⋊C₈)	—	2	8	no	—
47	C₄⋊(C₇⋊C₈)	C₂³	1	4	yes	—
48	C₄×C₇⋊C₈	—	2	1	no	—
49	C₄×C₇⋊C₈	—	2	8	no	—
50	C₄×C₇⋊C₈	D₄	1	2	yes	—
51	C₄×C₇⋊C₈	C₂³	1	2	yes	—
52	C₄×C₇⋊C₈	C₂³	1	2	yes	—
53	C₂×C₈×Q₈	—	7	2	no	—
54	C₂×C₄×Q₁₆	D₇	1	1	yes	characteristic
55	#1579	—	7	8	no	—
56	C₂×#334	—	7	1	no	—
57	C₂×Q₈⋊C₈	—	7	2	no	—
58	C₂×#322	—	7	1	no	—
59	C₂×C₇×Q₈	—	2	2	no	—
60	C₂×C₇×Q₈	—	2	4	no	—
61	C₂×C₇×Q₈	C₂×C₈	1	8	yes	—
62	C₂×C₇×Q₈	C₂²×C₄	1	4	yes	—
63	C₂²×C₄×C₇	—	2	2	no	—
64	C₂²×C₄×C₇	—	2	2	no	—
65	C₂²×C₄×C₇	C₂²×C₄	1	1	yes	characteristic
66	C₂²×C₄×C₇	C₂×D₄	1	1	yes	characteristic
67	C₂²×C₄×C₇	#41	1	1	yes	characteristic
68	C₇×Q₁₆	C₂×C₈	1	16	yes	complemented
69	C₂×C₇×C₈	—	2	2	no	—
70	C₂×C₇×C₈	C₂×C₈	1	4	yes	—
71	C₂×C₇×C₈	C₂²×C₄	1	2	yes	—
72	C₂×C₇×C₈	C₂²×C₄	1	2	yes	—
73	C₇×C₄⋊C₄	—	2	4	no	—
74	C₇×C₄⋊C₄	—	2	2	no	—
75	C₇×C₄⋊C₄	—	2	4	no	—
76	C₇×C₄⋊C₄	C₂²×C₄	1	4	yes	—
77	C₄²×C₇	—	2	4	no	—
78	C₄²×C₇	—	2	4	no	—
79	C₄²×C₇	C₂²×C₄	1	1	yes	characteristic
80	C₄²×C₇	C₂×D₄	1	1	yes	characteristic
81	C₄²×C₇	#41	1	1	yes	characteristic
82	C₄²×C₇	C₂⁴	1	1	yes	characteristic
83	C₂×C₇⋊C₈	—	2	4	no	—
84	C₂×C₇⋊C₈	—	2	4	no	—
85	C₂×C₇⋊C₈	—	2	2	no	—
86	C₂×C₇⋊C₈	—	4	2	no	—
87	C₂×C₇⋊C₈	—	4	4	no	—
88	C₂×C₇⋊C₈	C₂×D₄	1	4	yes	—
89	C₂²×Q₁₆	Dic₇	1	1	yes	characteristic
90	C₂×C₄×Q₈	C₂×D₇	1	2	yes	—
91	C₈×Q₈	—	7	8	no	—
92	C₈×Q₈	—	14	4	no	—
93	C₄×Q₁₆	Dic₇	1	4	yes	—
94	C₄×Q₁₆	C₂×D₇	1	4	yes	—
95	C₂×#107	Dic₇	1	1	yes	characteristic
96	C₂×C₄⋊C₈	—	7	2	no	—
97	C₂×C₄⋊C₈	—	14	2	no	—
98	C₂×#103	Dic₇	1	2	yes	—
99	C₂×C₄×C₈	—	7	1	no	—
100	C₂×C₄×C₈	—	7	1	no	—
101	C₂×C₄×C₈	—	14	2	no	—
102	C₂×C₄×C₈	C₂×D₇	1	1	yes	characteristic
103	#334	—	7	4	no	—
104	Q₈⋊C₈	—	7	8	no	—
105	#322	—	7	4	no	—
106	C₂³×C₇	C₄×D₄	1	1	yes	characteristic
107	C₇×Q₈	—	2	4	no	—
108	C₇×Q₈	—	2	8	no	—
109	C₇×Q₈	C₂²×C₈	1	8	yes	—
110	C₂×C₄×C₇	—	2	4	no	—
111	C₂×C₄×C₇	—	2	8	no	—
112	C₂×C₄×C₇	—	2	2	no	—
113	C₂×C₄×C₇	—	2	4	no	—
114	C₂×C₄×C₇	—	4	4	no	—
115	C₂×C₄×C₇	C₄×D₄	1	1	yes	characteristic
116	C₂×C₄×C₇	C₄×D₄	1	1	yes	characteristic
117	C₂×C₄×C₇	C₄×D₄	1	1	yes	characteristic
118	C₂×C₄×C₇	C₂²×C₈	1	2	yes	—
119	C₂×C₄×C₇	#131	1	2	yes	—
120	C₂×C₄×C₇	#137	1	4	yes	—
121	C₂×C₄×C₇	C₂³×C₄	1	1	yes	characteristic
122	C₂×C₄×C₇	C₂²×D₄	1	1	yes	characteristic
123	C₂×C₄×C₇	C₂×#41	1	1	yes	characteristic
124	C₇×C₈	—	2	2	no	—
125	C₇×C₈	C₂²×C₈	1	4	yes	—
126	C₇⋊C₈	—	2	4	no	—
127	C₇⋊C₈	—	4	2	no	—
128	C₇⋊C₈	—	4	4	no	—
129	C₂²×Q₈	C₂×Dic₇	1	2	yes	—
130	C₂×Q₁₆	C₇⋊C₈	1	8	yes	complemented
131	C₂×Q₁₆	C₂×Dic₇	1	4	yes	—
132	C₂²×C₈	—	7	2	no	—
133	C₂²×C₈	—	14	2	no	—
134	C₂²×C₈	—	14	1	no	—
135	C₂²×C₈	C₂×Dic₇	1	1	yes	characteristic
136	C₂²×C₈	C₂×Dic₇	1	1	yes	characteristic
137	C₄×Q₈	—	2	2	no	—
138	C₄×Q₈	—	2	2	no	—
139	C₄×Q₈	C₂×Dic₇	1	4	yes	—
140	C₄×Q₈	C₂²×D₇	1	4	yes	—
141	C₂×C₄⋊C₄	—	2	2	no	—
142	C₂×C₄⋊C₄	C₂×Dic₇	1	2	yes	—
143	C₂×C₄²	—	2	2	no	—
144	C₂×C₄²	C₂²×D₇	1	1	yes	characteristic
145	#107	C₂×Dic₇	1	4	yes	—
146	C₄⋊C₈	—	7	4	no	—
147	C₄⋊C₈	—	14	8	no	—
148	C₄⋊C₈	—	14	2	no	—
149	#103	C₂×Dic₇	1	8	yes	—
150	C₄×C₈	—	7	2	no	—
151	C₄×C₈	—	7	2	no	—
152	C₄×C₈	—	7	2	no	—
153	C₄×C₈	—	14	8	no	—
154	C₄×C₈	—	14	1	no	—
155	C₄×C₈	C₂×Dic₇	1	2	yes	—
156	C₄×C₈	C₂²×D₇	1	2	yes	—
157	C₂²×C₇	C₈×D₄	1	2	yes	—
158	C₂²×C₇	#441	1	2	yes	—
159	C₂²×C₇	#444	1	2	yes	—
160	C₂²×C₇	C₂×C₄×D₄	1	1	yes	characteristic
161	C₄×C₇	—	2	8	no	—
162	C₄×C₇	—	2	1	no	—
163	C₄×C₇	—	2	1	no	—
164	C₄×C₇	—	4	4	no	—
165	C₄×C₇	C₈×D₄	1	2	yes	—
166	C₄×C₇	#441	1	2	yes	—
167	C₄×C₇	C₂³×C₈	1	1	yes	characteristic
168	C₄×C₇	C₂×#131	1	1	yes	characteristic
169	C₄×C₇	C₂×#137	1	2	yes	—
170	C₂×Q₈	—	2	2	no	—
171	C₂×Q₈	—	2	4	no	—
172	C₂×Q₈	C₂×C₇⋊C₈	1	8	yes	—
173	C₂×Q₈	C₂²×Dic₇	1	4	yes	—
174	C₂²×C₄	—	2	2	no	—
175	C₂²×C₄	—	2	2	no	—
176	C₂²×C₄	D₄×D₇	1	1	yes	characteristic
177	C₂²×C₄	#1159	1	1	yes	characteristic
178	C₂²×C₄	C₂²×Dic₇	1	1	yes	characteristic
179	Q₁₆	C₂×C₇⋊C₈	1	16	yes	complemented
180	C₂×C₈	—	2	2	no	—
181	C₂×C₈	—	7	4	no	—
182	C₂×C₈	—	14	4	no	—
183	C₂×C₈	—	14	4	no	—
184	C₂×C₈	—	14	2	no	—
185	C₂×C₈	—	28	2	no	—
186	C₂×C₈	—	28	4	no	—
187	C₂×C₈	C₂×C₇⋊C₈	1	4	yes	—
188	C₂×C₈	C₂²×Dic₇	1	2	yes	—
189	C₂×C₈	C₂²×Dic₇	1	2	yes	—
190	C₄⋊C₄	—	2	4	no	—
191	C₄⋊C₄	—	2	2	no	—
192	C₄⋊C₄	—	2	4	no	—
193	C₄⋊C₄	C₂²×Dic₇	1	4	yes	—
194	C₄²	—	2	4	no	—
195	C₄²	—	2	4	no	—
196	C₄²	D₄×D₇	1	1	yes	characteristic
197	C₄²	#1159	1	1	yes	characteristic
198	C₄²	C₂²×Dic₇	1	1	yes	characteristic
199	C₄²	C₂³×D₇	1	1	yes	characteristic
200	C₂×C₇	#1579	1	4	yes	complemented
201	C₂×C₇	C₂×C₈×D₄	1	1	yes	characteristic
202	C₂×C₇	C₂×#441	1	1	yes	characteristic
203	C₂×C₇	C₂×#444	1	1	yes	characteristic
204	C₂³	D₄×Dic₇	1	1	yes	characteristic
205	Q₈	—	2	4	no	—
206	Q₈	—	2	8	no	—
207	Q₈	C₂²×C₇⋊C₈	1	8	yes	—
208	C₂×C₄	—	2	2	no	—
209	C₂×C₄	—	2	4	no	—
210	C₂×C₄	—	2	8	no	—
211	C₂×C₄	—	2	4	no	—
212	C₂×C₄	—	4	4	no	—
213	C₂×C₄	#6481	1	4	yes	—
214	C₂×C₄	C₂²×C₇⋊C₈	1	2	yes	—
215	C₂×C₄	D₄×Dic₇	1	1	yes	characteristic
216	C₂×C₄	D₄×Dic₇	1	1	yes	characteristic
217	C₂×C₄	D₄×Dic₇	1	1	yes	characteristic
218	C₂×C₄	#6511	1	2	yes	—
219	C₂×C₄	C₂×D₄×D₇	1	1	yes	characteristic
220	C₂×C₄	C₂×#1159	1	1	yes	characteristic
221	C₂×C₄	C₂³×Dic₇	1	1	yes	characteristic
222	C₈	—	2	2	no	—
223	C₈	—	14	4	no	—
224	C₈	—	28	2	no	—
225	C₈	—	28	4	no	—
226	C₈	C₂²×C₇⋊C₈	1	4	yes	—
227	C₇	C₂×#1579	1	1	yes	characteristic, complemented, Sylow
228	C₂²	D₄×C₇⋊C₈	1	2	yes	—
229	C₂²	#89956	1	2	yes	—
230	C₂²	#89969	1	2	yes	—
231	C₂²	C₂×D₄×Dic₇	1	1	yes	characteristic
232	C₄	—	2	8	no	—
233	C₄	—	2	1	no	—
234	C₄	—	2	1	no	—
235	C₄	—	4	4	no	—
236	C₄	D₄×C₇⋊C₈	1	2	yes	—
237	C₄	#89956	1	2	yes	—
238	C₄	C₂×#6481	1	2	yes	—
239	C₄	C₂³×C₇⋊C₈	1	1	yes	characteristic
240	C₄	C₂×#6511	1	1	yes	characteristic
241	C₂	#11728656	1	4	yes	direct product factor
242	C₂	C₂×D₄×C₇⋊C₈	1	1	yes	characteristic
243	C₂	C₂×#89956	1	1	yes	characteristic
244	C₂	C₂×#89969	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄	D₄×D₇
commutator	C₄×C₇	C₂³×C₈
Frattini	C₄²	C₂³×D₇
Fitting	C₂×C₄×C₇×Q₁₆	C₂
socle, prime radical	C₂³×C₇	C₄×D₄
cosocle	C₄²×C₇	C₂⁴
hypercenter	C₂×C₄×Q₁₆	D₇
hypocenter	C₇	C₂×#1579
LCS	G ▷ C₄×C₇ ▷ C₂×C₇ ▷ C₇
UCS	C₁ ◁ C₂²×C₄ ◁ C₂×C₄² ◁ C₂×C₄×Q₁₆

Complements

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

N	H ≅ G/N	type
4. #11728656	C₂	direct (N × H)
26. C₂×C₇×Q₁₆	C₈	semidirect
68. C₇×Q₁₆	C₂×C₈	semidirect
130. C₂×Q₁₆	C₇⋊C₈	semidirect
179. Q₁₆	C₂×C₇⋊C₈	semidirect
200. C₂×C₇	#1579	semidirect
227. C₇	C₂×#1579	semidirect
241. C₂	#11728656	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×C₄×C₇×Q₁₆	2
C₂×Q₈×C₇⋊C₈	2
C₂×Q₈⋊(C₇⋊C₈)	2
#11728656	2
C₂×#89252	2
C₂×#89250	2
C₂×#1579	7

Subgroups — C2×#11728656 #13323189

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#11728656 #13323189