Subgroups — D₇×#2868 #13559891

Order 1792.

3774 conjugacy classes of subgroups.

The subgroup lattice has 198 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×D₇×C₄⋊Q₈	C₂	1	6	yes	maximal
2	C₇×#2868	C₂	1	1	yes	maximal, characteristic, complemented
3	#11740698	C₂	1	6	yes	maximal, complemented
4	D₇×#384	C₂	1	8	yes	maximal
5	#11731801	C₂	1	8	yes	maximal, complemented
6	C₄³×D₇	C₂	1	1	yes	maximal, characteristic
7	#11730756	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂²×Q₈×D₇	C₂²	1	4	yes	—
9	C₂×C₇×C₄⋊Q₈	C₂²	1	6	yes	—
10	C₂×#6507	C₂²	1	12	yes	—
11	D₇×C₄⋊Q₈	—	2	48	no	—
12	C₂×D₇×C₄⋊C₄	C₂²	1	24	yes	—
13	C₂×#6451	C₂²	1	24	yes	—
14	C₂×C₄²×D₇	C₂²	1	3	yes	—
15	C₂×C₄²×D₇	C₂²	1	4	yes	—
16	C₂×#6432	C₂²	1	6	yes	—
17	C₇×#384	C₂²	1	8	yes	—
18	C₄³×C₇	C₂²	1	1	yes	characteristic
19	#89748	C₂²	1	48	yes	—
20	#89707	C₂²	1	8	yes	—
21	C₄²×Dic₇	C₂²	1	3	yes	—
22	C₄²×Dic₇	C₂²	1	4	yes	—
23	C₂×#2868	—	7	1	no	maximal, Sylow
24	C₂²×C₇×Q₈	C₂³	1	4	yes	—
25	C₂×Q₈×D₇	—	2	48	no	—
26	C₂²×C₄×D₇	C₂³	1	8	yes	—
27	C₂²×C₄×D₇	C₂³	1	1	yes	characteristic
28	C₂²×C₄×D₇	C₂³	1	6	yes	—
29	C₂²×Dic₁₄	C₂³	1	4	yes	—
30	C₂²×Dic₁₄	C₂³	1	8	yes	—
31	C₇×C₄⋊Q₈	—	2	24	no	—
32	C₂×C₇×C₄⋊C₄	C₂³	1	24	yes	—
33	C₂×C₄²×C₇	C₂³	1	3	yes	—
34	C₂×C₄²×C₇	C₂³	1	4	yes	—
35	#6507	—	2	48	no	—
36	C₂×C₄⋊Dic₇	C₂³	1	24	yes	—
37	C₂×Dic₇⋊C₄	C₂³	1	48	yes	—
38	C₂×C₄×Dic₇	C₂³	1	6	yes	—
39	C₂×C₄×Dic₇	C₂³	1	3	yes	—
40	C₂×C₄×Dic₇	C₂³	1	12	yes	—
41	D₇×C₄⋊C₄	—	2	96	no	—
42	#6451	—	2	96	no	—
43	C₄²×D₇	D₄	1	24	yes	—
44	C₄²×D₇	Q₈	1	32	yes	complemented
45	#6432	—	2	24	no	—
46	C₂²×C₄⋊Q₈	—	7	6	no	—
47	#2868	—	7	8	no	—
48	#2868	—	7	1	no	—
49	#2868	—	7	6	no	—
50	#2868	D₇	1	1	yes	characteristic, direct product factor
51	C₂×#384	—	7	8	no	—
52	C₂×C₄³	—	7	1	no	—
53	C₂³×D₇	C₂⁴	1	1	yes	characteristic
54	C₂×C₇×Q₈	—	2	24	no	—
55	C₂²×C₄×C₇	C₂⁴	1	8	yes	—
56	C₂²×C₄×C₇	C₂⁴	1	1	yes	characteristic
57	C₂²×C₄×C₇	C₂⁴	1	6	yes	—
58	C₂²×Dic₇	C₂⁴	1	8	yes	—
59	C₂²×Dic₇	C₂⁴	1	1	yes	characteristic
60	C₂²×Dic₇	C₂⁴	1	6	yes	—
61	Q₈×D₇	—	4	64	no	—
62	C₂×C₄×D₇	—	2	48	no	—
63	C₂×C₄×D₇	C₂×D₄	1	12	yes	—
64	C₂×C₄×D₇	C₂×D₄	1	24	yes	—
65	C₂×C₄×D₇	C₂×Q₈	1	48	yes	—
66	C₂×Dic₁₄	—	2	24	no	—
67	C₂×Dic₁₄	—	2	48	no	—
68	C₇×C₄⋊C₄	—	2	48	no	—
69	C₄²×C₇	C₂×D₄	1	12	yes	—
70	C₄²×C₇	C₂×Q₈	1	16	yes	complemented
71	C₄⋊Dic₇	—	2	48	no	—
72	Dic₇⋊C₄	—	2	96	no	—
73	C₄×Dic₇	C₂×D₄	1	24	yes	—
74	C₄×Dic₇	C₂×D₄	1	12	yes	—
75	C₄×Dic₇	C₂×Q₈	1	48	yes	complemented
76	C₂³×Q₈	—	7	4	no	—
77	C₂×C₄⋊Q₈	—	7	12	no	—
78	C₂×C₄⋊Q₈	—	7	24	no	—
79	C₂×C₄⋊Q₈	—	7	6	no	—
80	C₂×C₄⋊Q₈	—	14	48	no	—
81	C₂×C₄⋊Q₈	C₂×D₇	1	6	yes	—
82	C₂²×C₄⋊C₄	—	7	24	no	—
83	C₂²×C₄²	—	7	3	no	—
84	C₂²×C₄²	—	7	4	no	—
85	#384	—	7	48	no	—
86	#384	—	7	8	no	—
87	#384	C₂×D₇	1	8	yes	—
88	C₄³	—	7	4	no	—
89	C₄³	—	7	3	no	—
90	C₄³	C₂×D₇	1	1	yes	characteristic
91	C₂³×C₇	C₂⁵	1	1	yes	characteristic
92	C₂²×D₇	C₂²×D₄	1	6	yes	—
93	C₂²×D₇	C₂²×Q₈	1	8	yes	—
94	C₇×Q₈	—	4	16	no	—
95	C₂×C₄×C₇	—	2	24	no	—
96	C₂×C₄×C₇	C₂²×D₄	1	6	yes	—
97	C₂×C₄×C₇	C₂²×D₄	1	12	yes	—
98	C₂×C₄×C₇	C₂²×Q₈	1	24	yes	—
99	C₂×Dic₇	—	2	24	no	—
100	C₂×Dic₇	C₂²×D₄	1	6	yes	—
101	C₂×Dic₇	C₂²×D₄	1	12	yes	—
102	C₂×Dic₇	C₂²×Q₈	1	24	yes	—
103	C₄×D₇	—	4	32	no	—
104	C₄×D₇	#127	1	16	yes	—
105	C₄×D₇	C₄⋊Q₈	1	96	yes	complemented
106	Dic₁₄	—	4	16	no	—
107	Dic₁₄	—	4	32	no	—
108	C₂²×Q₈	—	7	8	no	—
109	C₂²×Q₈	—	7	4	no	—
110	C₂²×Q₈	—	14	48	no	—
111	C₂²×Q₈	C₂²×D₇	1	4	yes	—
112	C₂³×C₄	—	7	6	no	—
113	C₂³×C₄	—	7	1	no	—
114	C₂³×C₄	—	7	8	no	—
115	C₄⋊Q₈	—	2	24	no	—
116	C₄⋊Q₈	—	14	96	no	—
117	C₄⋊Q₈	—	14	48	no	—
118	C₄⋊Q₈	—	14	24	no	—
119	C₂×C₄⋊C₄	—	7	48	no	—
120	C₂×C₄⋊C₄	—	7	24	no	—
121	C₂×C₄⋊C₄	—	14	96	no	—
122	C₂×C₄⋊C₄	C₂²×D₇	1	24	yes	—
123	C₂×C₄²	—	7	3	no	—
124	C₂×C₄²	—	7	6	no	—
125	C₂×C₄²	—	7	12	no	—
126	C₂×C₄²	—	7	24	no	—
127	C₂×C₄²	—	7	32	no	—
128	C₂×C₄²	C₂²×D₇	1	3	yes	—
129	C₂×C₄²	C₂²×D₇	1	4	yes	—
130	C₂²×C₇	C₂³×D₄	1	3	yes	—
131	C₂²×C₇	C₂³×Q₈	1	4	yes	—
132	C₂×D₇	C₂×#127	1	4	yes	—
133	C₂×D₇	C₂×C₄⋊Q₈	1	24	yes	—
134	C₄×C₇	—	4	8	no	—
135	C₄×C₇	C₂×#127	1	4	yes	—
136	C₄×C₇	C₂×C₄⋊Q₈	1	24	yes	complemented
137	Dic₇	—	4	8	no	—
138	Dic₇	C₂×#127	1	4	yes	—
139	Dic₇	C₂×C₄⋊Q₈	1	24	yes	complemented
140	C₂⁴	—	7	1	no	—
141	C₂×Q₈	—	2	24	no	—
142	C₂×Q₈	—	14	48	no	—
143	C₂×Q₈	—	14	24	no	—
144	C₂×Q₈	—	28	64	no	—
145	C₂²×C₄	—	7	6	no	—
146	C₂²×C₄	—	7	1	no	—
147	C₂²×C₄	—	7	24	no	—
148	C₂²×C₄	—	7	12	no	—
149	C₂²×C₄	—	7	48	no	—
150	C₂²×C₄	—	7	8	no	—
151	C₂²×C₄	—	14	48	no	—
152	C₂²×C₄	C₂³×D₇	1	8	yes	—
153	C₂²×C₄	C₂³×D₇	1	1	yes	characteristic
154	C₂²×C₄	C₂³×D₇	1	6	yes	—
155	C₄⋊C₄	—	2	48	no	—
156	C₄⋊C₄	—	14	96	no	—
157	C₄⋊C₄	—	14	48	no	—
158	C₄²	—	7	12	no	—
159	C₄²	—	7	24	no	—
160	C₄²	—	7	48	no	—
161	C₄²	D₄×D₇	1	12	yes	—
162	C₄²	Q₈×D₇	1	16	yes	complemented
163	C₂×C₇	C₂²×#127	1	1	yes	characteristic
164	C₂×C₇	C₂²×C₄⋊Q₈	1	6	yes	—
165	D₇	#2868	1	8	yes	direct product factor
166	C₂³	—	7	6	no	—
167	C₂³	—	7	8	no	—
168	C₂³	C₂⁴×D₇	1	1	yes	characteristic
169	Q₈	—	4	16	no	—
170	Q₈	—	28	32	no	—
171	Q₈	—	28	16	no	—
172	C₂×C₄	—	2	24	no	—
173	C₂×C₄	—	7	12	no	—
174	C₂×C₄	—	7	6	no	—
175	C₂×C₄	—	7	24	no	—
176	C₂×C₄	—	7	16	no	—
177	C₂×C₄	—	7	96	no	—
178	C₂×C₄	—	14	24	no	—
179	C₂×C₄	—	28	32	no	—
180	C₂×C₄	C₂×D₄×D₇	1	6	yes	—
181	C₂×C₄	C₂×D₄×D₇	1	12	yes	—
182	C₂×C₄	C₂×Q₈×D₇	1	24	yes	—
183	C₇	C₂×#2868	1	1	yes	characteristic, complemented, Sylow
184	C₂²	—	7	4	no	—
185	C₂²	—	7	24	no	—
186	C₂²	C₂²×D₄×D₇	1	3	yes	—
187	C₂²	C₂²×Q₈×D₇	1	4	yes	—
188	C₄	—	4	8	no	—
189	C₄	—	7	4	no	—
190	C₄	—	7	24	no	—
191	C₄	—	28	8	no	—
192	C₄	D₇×#127	1	4	yes	—
193	C₄	D₇×C₄⋊Q₈	1	24	yes	complemented
194	C₂	—	7	8	no	—
195	C₂	C₂×D₇×#127	1	1	yes	characteristic
196	C₂	C₂×D₇×C₄⋊Q₈	1	6	yes	—

	subgroup	G/H
center, Frattini	C₂³	C₂⁴×D₇
commutator, socle, cosocle	C₂³×C₇	C₂⁵
Fitting	C₇×#2868	C₂
prime radical	C₂³×D₇	C₂⁴
hypercenter	#2868	D₇
hypocenter	C₇	C₂×#2868
LCS	G ▷ C₂³×C₇ ▷ C₇
UCS	C₁ ◁ C₂³ ◁ #2868

Complements

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

N	H ≅ G/N	type
2. C₇×#2868	C₂	semidirect
3. #11740698	C₂	semidirect
5. #11731801	C₂	semidirect
7. #11730756	C₂	semidirect
44. C₄²×D₇	Q₈	semidirect
50. #2868	D₇	direct (N × H)
70. C₄²×C₇	C₂×Q₈	semidirect
75. C₄×Dic₇	C₂×Q₈	semidirect
105. C₄×D₇	C₄⋊Q₈	semidirect
136. C₄×C₇	C₂×C₄⋊Q₈	semidirect
139. Dic₇	C₂×C₄⋊Q₈	semidirect
162. C₄²	Q₈×D₇	semidirect
165. D₇	#2868	direct (N × H)
183. C₇	C₂×#2868	semidirect
193. C₄	D₇×C₄⋊Q₈	semidirect

Maximal subgroups

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

subgroup	index
C₂×D₇×C₄⋊Q₈	2
C₇×#2868	2
#11740698	2
D₇×#384	2
#11731801	2
C₄³×D₇	2
#11730756	2
C₂×#2868	7

Subgroups — D7×#2868 #13559891

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — D₇×#2868 #13559891