Subgroups — C₂×#11729139 #13509280

Order 1792.

558 conjugacy classes of subgroups.

The subgroup lattice has 200 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₇×#500	C₂	1	1	yes	maximal, characteristic
2	C₂×#89454	C₂	1	1	yes	maximal, characteristic
3	#11729139	C₂	1	8	yes	maximal, direct product factor
4	C₂×#89285	C₂	1	4	yes	maximal
5	C₂×C₄×C₇⋊C₁₆	C₂	1	1	yes	maximal, characteristic
6	C₂×C₇×C₄⋊Q₈	—	2	1	no	—
7	C₂×#6432	—	2	1	no	—
8	C₇×#500	—	2	2	no	—
9	C₇×#500	C₂²	1	4	yes	—
10	C₂×C₇×#107	—	2	1	no	—
11	C₂×C₇×#107	C₂²	1	2	yes	—
12	C₂×C₄×C₇×C₈	C₂²	1	1	yes	characteristic
13	C₂×#6392	—	2	1	no	—
14	C₂×#6392	C₂²	1	2	yes	—
15	C₂²×C₇⋊C₁₆	C₂²	1	2	yes	—
16	#89454	—	2	2	no	—
17	#89454	C₂²	1	4	yes	—
18	#89285	C₂²	1	16	yes	—
19	C₄×C₇⋊C₁₆	C₂²	1	4	yes	—
20	C₂×#2254	—	7	1	no	maximal, Sylow
21	C₂²×C₇×Q₈	—	4	1	no	—
22	C₂²×Dic₁₄	—	4	1	no	—
23	C₂²×C₇×C₈	D₄	1	1	yes	characteristic
24	C₂²×C₇×C₈	C₂³	1	1	yes	characteristic
25	C₇×C₄⋊Q₈	—	2	4	no	—
26	C₇×C₄⋊Q₈	—	4	2	no	—
27	C₂×C₇×C₄⋊C₄	—	2	2	no	—
28	C₂×C₇×C₄⋊C₄	—	4	1	no	—
29	C₂×C₄²×C₇	D₄	1	1	yes	characteristic
30	C₂×C₄⋊Dic₇	—	2	2	no	—
31	C₂×C₄⋊Dic₇	—	4	1	no	—
32	#6432	—	2	4	no	—
33	#6432	—	4	2	no	—
34	C₇×#107	—	2	2	no	—
35	C₇×#107	—	2	4	no	—
36	C₇×#107	C₂³	1	4	yes	—
37	C₄×C₇×C₈	D₄	1	2	yes	—
38	C₄×C₇×C₈	C₂³	1	2	yes	—
39	#6392	—	2	4	no	—
40	#6392	—	2	2	no	—
41	#6392	C₂³	1	4	yes	—
42	C₂×C₇⋊C₁₆	Q₈	1	8	yes	complemented
43	C₂×C₇⋊C₁₆	C₂³	1	4	yes	—
44	C₂×#500	—	7	1	no	—
45	C₂×#500	D₇	1	1	yes	characteristic
46	#2254	—	7	8	no	—
47	C₂×#367	—	7	4	no	—
48	C₂×C₄×C₁₆	—	7	1	no	—
49	C₂×C₇×Q₈	—	4	4	no	—
50	C₂×C₇×Q₈	—	8	4	no	—
51	C₂²×C₄×C₇	—	4	2	no	—
52	C₂²×C₄×C₇	D₈	1	1	yes	characteristic
53	C₂²×C₄×C₇	D₈	1	1	yes	characteristic
54	C₂²×C₄×C₇	C₂×D₄	1	1	yes	characteristic
55	C₂²×Dic₇	—	4	2	no	—
56	C₂×Dic₁₄	—	4	4	no	—
57	C₂×Dic₁₄	—	8	4	no	—
58	C₂×C₇×C₈	Q₁₆	1	4	yes	—
59	C₂×C₇×C₈	C₂×D₄	1	1	yes	characteristic
60	C₂×C₇×C₈	C₂×D₄	1	2	yes	—
61	C₂×C₇×C₈	C₂×Q₈	1	4	yes	—
62	C₂×C₇×C₈	C₂⁴	1	1	yes	characteristic
63	C₇×C₄⋊C₄	—	2	4	no	—
64	C₇×C₄⋊C₄	—	4	2	no	—
65	C₇×C₄⋊C₄	—	4	4	no	—
66	C₄²×C₇	D₈	1	2	yes	—
67	C₄²×C₇	C₂×D₄	1	2	yes	—
68	C₄⋊Dic₇	—	2	4	no	—
69	C₄⋊Dic₇	—	4	4	no	—
70	C₄⋊Dic₇	—	4	2	no	—
71	C₇⋊C₁₆	C₂×Q₈	1	8	yes	complemented
72	C₂×C₄⋊Q₈	—	2	1	no	—
73	C₂×C₄⋊Q₈	—	14	1	no	—
74	C₂²×C₁₆	—	7	2	no	—
75	#500	—	2	2	no	—
76	#500	—	7	4	no	—
77	#500	—	14	2	no	—
78	#500	C₂×D₇	1	4	yes	—
79	C₂×#107	—	2	1	no	—
80	C₂×#107	—	7	2	no	—
81	C₂×#107	—	14	1	no	—
82	C₂×#107	C₂×D₇	1	2	yes	—
83	C₂×C₄×C₈	C₂×D₇	1	1	yes	characteristic
84	#367	—	7	16	no	—
85	C₄×C₁₆	—	7	4	no	—
86	C₂³×C₇	C₂×D₈	1	1	yes	characteristic
87	C₇×Q₈	—	8	8	no	—
88	C₂×C₄×C₇	—	4	4	no	—
89	C₂×C₄×C₇	—	8	4	no	—
90	C₂×C₄×C₇	SD₃₂	1	4	yes	—
91	C₂×C₄×C₇	SD₃₂	1	4	yes	—
92	C₂×C₄×C₇	C₄⋊Q₈	1	2	yes	—
93	C₂×C₄×C₇	C₂×D₈	1	1	yes	characteristic
94	C₂×C₄×C₇	C₂×D₈	1	2	yes	—
95	C₂×C₄×C₇	C₂×D₈	1	2	yes	—
96	C₂×C₄×C₇	C₂×Q₁₆	1	2	yes	—
97	C₂×C₄×C₇	C₂²×D₄	1	1	yes	characteristic
98	C₂×Dic₇	—	4	4	no	—
99	C₂×Dic₇	—	8	4	no	—
100	Dic₁₄	—	8	8	no	—
101	C₇×C₈	C₄⋊Q₈	1	2	yes	—
102	C₇×C₈	C₂×Q₁₆	1	4	yes	—
103	C₇×C₈	C₂²×Q₈	1	2	yes	—
104	C₂²×Q₈	—	4	1	no	—
105	C₂²×Q₈	—	28	1	no	—
106	C₂²×C₈	C₂²⋊D₇	1	1	yes	characteristic
107	C₂²×C₈	C₂²×D₇	1	1	yes	characteristic
108	C₄⋊Q₈	—	2	4	no	—
109	C₄⋊Q₈	—	4	2	no	—
110	C₄⋊Q₈	—	14	4	no	—
111	C₄⋊Q₈	—	28	2	no	—
112	C₂×C₄⋊C₄	—	2	2	no	—
113	C₂×C₄⋊C₄	—	4	1	no	—
114	C₂×C₄⋊C₄	—	14	2	no	—
115	C₂×C₄⋊C₄	—	28	1	no	—
116	C₂×C₄²	C₂²⋊D₇	1	1	yes	characteristic
117	C₂×C₁₆	—	7	8	no	—
118	C₂×C₁₆	—	7	4	no	—
119	#107	—	2	4	no	—
120	#107	—	2	2	no	—
121	#107	—	7	4	no	—
122	#107	—	14	2	no	—
123	#107	—	14	4	no	—
124	#107	C₂²×D₇	1	4	yes	—
125	C₄×C₈	C₂²⋊D₇	1	2	yes	—
126	C₄×C₈	C₂²×D₇	1	2	yes	—
127	C₂²×C₇	#500	1	2	yes	—
128	C₂²×C₇	C₂×SD₃₂	1	2	yes	—
129	C₂²×C₇	C₂×SD₃₂	1	2	yes	—
130	C₂²×C₇	C₂²×D₈	1	1	yes	characteristic
131	C₄×C₇	—	8	4	no	—
132	C₄×C₇	#500	1	2	yes	—
133	C₄×C₇	C₂×SD₃₂	1	4	yes	—
134	C₄×C₇	C₂×SD₃₂	1	4	yes	—
135	C₄×C₇	C₂×C₄⋊Q₈	1	1	yes	characteristic
136	C₄×C₇	C₂²×Q₁₆	1	1	yes	characteristic
137	Dic₇	—	8	4	no	—
138	C₂×Q₈	—	4	4	no	—
139	C₂×Q₈	—	8	4	no	—
140	C₂×Q₈	—	28	4	no	—
141	C₂×Q₈	—	56	4	no	—
142	C₂²×C₄	—	4	2	no	—
143	C₂²×C₄	—	28	2	no	—
144	C₂²×C₄	#1141	1	1	yes	characteristic
145	C₂²×C₄	#1141	1	1	yes	characteristic
146	C₂²×C₄	C₂×C₂²⋊D₇	1	1	yes	characteristic
147	C₂×C₈	#1144	1	4	yes	—
148	C₂×C₈	Q₈×D₇	1	4	yes	—
149	C₂×C₈	C₂×C₂²⋊D₇	1	2	yes	—
150	C₂×C₈	C₂×C₂²⋊D₇	1	1	yes	characteristic
151	C₂×C₈	C₂³×D₇	1	1	yes	characteristic
152	C₄⋊C₄	—	2	4	no	—
153	C₄⋊C₄	—	4	2	no	—
154	C₄⋊C₄	—	4	4	no	—
155	C₄⋊C₄	—	14	4	no	—
156	C₄⋊C₄	—	28	2	no	—
157	C₄⋊C₄	—	28	4	no	—
158	C₄²	#1141	1	2	yes	—
159	C₄²	C₂×C₂²⋊D₇	1	2	yes	—
160	C₁₆	—	7	8	no	—
161	C₂×C₇	#2254	1	4	yes	complemented
162	C₂×C₇	C₂×#500	1	1	yes	characteristic
163	C₂×C₇	C₂²×SD₃₂	1	1	yes	characteristic
164	C₂×C₇	C₂²×SD₃₂	1	1	yes	characteristic
165	C₂³	C₂×#1141	1	1	yes	characteristic
166	Q₈	—	8	8	no	—
167	Q₈	—	56	8	no	—
168	C₂×C₄	—	4	4	no	—
169	C₂×C₄	—	8	4	no	—
170	C₂×C₄	—	28	4	no	—
171	C₂×C₄	—	56	4	no	—
172	C₂×C₄	#6401	1	4	yes	—
173	C₂×C₄	#6402	1	4	yes	—
174	C₂×C₄	C₂×#1141	1	2	yes	—
175	C₂×C₄	C₂×#1141	1	2	yes	—
176	C₂×C₄	C₂×#1141	1	1	yes	characteristic
177	C₂×C₄	C₂×#1144	1	2	yes	—
178	C₂×C₄	#6507	1	2	yes	—
179	C₂×C₄	C₂²×C₂²⋊D₇	1	1	yes	characteristic
180	C₈	C₂×#1144	1	4	yes	—
181	C₈	#6507	1	2	yes	—
182	C₈	C₂×Q₈×D₇	1	2	yes	—
183	C₇	C₂×#2254	1	1	yes	characteristic, complemented, Sylow
184	C₂²	#89850	1	2	yes	—
185	C₂²	C₂×#6401	1	2	yes	—
186	C₂²	C₂×#6402	1	2	yes	—
187	C₂²	C₂²×#1141	1	1	yes	characteristic
188	C₄	—	8	4	no	—
189	C₄	—	56	4	no	—
190	C₄	#89850	1	2	yes	—
191	C₄	C₂×#6401	1	4	yes	—
192	C₄	C₂×#6402	1	4	yes	—
193	C₄	C₂²×#1144	1	1	yes	characteristic
194	C₄	C₂×#6507	1	1	yes	characteristic
195	C₂	#11729139	1	4	yes	direct product factor
196	C₂	C₂×#89850	1	1	yes	characteristic
197	C₂	C₂²×#6401	1	1	yes	characteristic
198	C₂	C₂²×#6402	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	C₂×#1141
commutator, cosocle	C₂×C₇×C₈	C₂⁴
Frattini	C₂×C₈	C₂³×D₇
Fitting	C₂×C₇×#500	C₂
socle, prime radical	C₂³×C₇	C₂×D₈
hypercenter	C₂×#500	D₇
hypocenter	C₇	C₂×#2254
LCS	G ▷ C₂×C₇×C₈ ▷ C₄×C₇ ▷ C₂×C₇ ▷ C₇
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ C₂×#500

Complements

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

N	H ≅ G/N	type
3. #11729139	C₂	direct (N × H)
42. C₂×C₇⋊C₁₆	Q₈	semidirect
71. C₇⋊C₁₆	C₂×Q₈	semidirect
161. C₂×C₇	#2254	semidirect
183. C₇	C₂×#2254	semidirect
195. C₂	#11729139	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×#500	2
C₂×#89454	2
#11729139	2
C₂×#89285	2
C₂×C₄×C₇⋊C₁₆	2
C₂×#2254	7

Subgroups — C2×#11729139 #13509280

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#11729139 #13509280