Subgroups — C₄×#89306 #13269115

Order 1792.

398 conjugacy classes of subgroups.

The subgroup lattice has 216 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄×C₇×M₃₂	C₂	1	1	yes	maximal, characteristic
2	C₂×#89306	C₂	1	2	yes	maximal
3	C₄×C₈×Dic₇	C₂	1	1	yes	maximal, characteristic
4	C₄×#6386	C₂	1	1	yes	maximal, characteristic
5	#11725616	C₂	1	2	yes	maximal
6	C₂×C₇×M₃₂	C₂²	1	2	yes	—
7	C₂×C₄×C₇×C₈	C₂²	1	1	yes	characteristic
8	C₂×C₈×Dic₇	C₄	1	1	yes	characteristic
9	C₂×C₈×Dic₇	C₄	1	1	yes	characteristic
10	C₂×C₈×Dic₇	C₂²	1	1	yes	characteristic
11	C₂×C₈×Dic₇	C₂²	1	1	yes	characteristic
12	C₂×#6386	C₂²	1	2	yes	—
13	C₄²×Dic₇	C₄	1	1	yes	characteristic
14	C₂×C₄×C₇⋊C₈	C₄	1	1	yes	characteristic
15	C₇×#346	—	2	1	no	—
16	C₄×C₇×C₁₆	—	2	1	no	—
17	#89306	C₄	1	8	yes	direct product factor
18	#89256	—	2	1	no	—
19	C₄×C₇⋊C₁₆	—	2	1	no	—
20	C₄×#364	—	7	1	no	maximal, Sylow
21	C₂²×C₇×C₈	C₂×C₄	1	1	yes	characteristic
22	C₂²×C₇×C₈	C₂³	1	1	yes	characteristic
23	C₂×C₄²×C₇	C₂×C₄	1	1	yes	characteristic
24	C₂×C₄×Dic₇	—	2	2	no	—
25	C₂×C₄×Dic₇	C₂×C₄	1	1	yes	characteristic
26	C₂×C₄×Dic₇	C₂×C₄	1	1	yes	characteristic
27	C₂²×C₇⋊C₈	C₂×C₄	1	1	yes	characteristic
28	C₂²×C₇⋊C₈	C₂×C₄	1	1	yes	characteristic
29	C₇×M₃₂	—	2	2	no	—
30	C₇×M₃₂	C₂×C₄	1	4	yes	—
31	C₂×C₇×C₁₆	—	2	2	no	—
32	C₄×C₇×C₈	D₄	1	1	yes	characteristic
33	C₄×C₇×C₈	D₄	1	1	yes	characteristic
34	C₄×C₇×C₈	Q₈	1	1	yes	characteristic
35	C₄×C₇×C₈	Q₈	1	1	yes	characteristic
36	C₈×Dic₇	—	2	2	no	—
37	C₈×Dic₇	—	2	2	no	—
38	C₈×Dic₇	C₈	1	4	yes	—
39	C₈×Dic₇	C₂×C₄	1	4	yes	—
40	#6386	—	2	2	no	—
41	#6386	C₂×C₄	1	4	yes	—
42	C₂×C₇⋊C₁₆	—	2	2	no	—
43	C₄×C₇⋊C₈	C₈	1	4	yes	—
44	C₂×#364	—	7	2	no	—
45	C₄×M₃₂	—	7	1	no	—
46	C₄×M₃₂	D₇	1	1	yes	characteristic
47	C₄²×C₈	—	7	1	no	—
48	#1376	—	7	2	no	—
49	C₂²×C₄×C₇	M₁₆	1	2	yes	—
50	C₂²×C₄×C₇	C₂²×C₄	1	1	yes	characteristic
51	C₂²×Dic₇	—	2	1	no	—
52	C₂²×Dic₇	—	2	1	no	—
53	C₂×C₇×C₈	C₄⋊C₄	1	1	yes	characteristic
54	C₂×C₇×C₈	C₄⋊C₄	1	1	yes	characteristic
55	C₂×C₇×C₈	C₂×C₈	1	2	yes	—
56	C₂×C₇×C₈	M₁₆	1	2	yes	—
57	C₂×C₇×C₈	C₂²×C₄	1	2	yes	—
58	C₂×C₇×C₈	C₂×D₄	1	1	yes	characteristic
59	C₂×C₇×C₈	C₂×Q₈	1	1	yes	characteristic
60	C₂×C₇×C₈	#41	1	2	yes	—
61	C₄²×C₇	C₄⋊C₄	1	1	yes	characteristic
62	C₄²×C₇	C₄⋊C₄	1	1	yes	characteristic
63	C₄²×C₇	C₂×C₈	1	2	yes	—
64	C₄×Dic₇	—	2	8	no	—
65	C₄×Dic₇	—	2	1	no	—
66	C₄×Dic₇	—	2	1	no	—
67	C₄×Dic₇	C₄²	1	2	yes	—
68	C₄×Dic₇	C₂×C₈	1	2	yes	—
69	C₂×C₇⋊C₈	—	2	1	no	—
70	C₂×C₇⋊C₈	—	2	1	no	—
71	C₂×C₇⋊C₈	C₄²	1	2	yes	—
72	C₂×C₇⋊C₈	C₂×C₈	1	2	yes	—
73	C₂×C₇⋊C₈	C₂×C₈	1	2	yes	—
74	C₂×C₇⋊C₈	C₂×C₈	1	2	yes	—
75	C₇×C₁₆	—	2	4	no	—
76	C₇⋊C₁₆	—	2	4	no	—
77	C₂×M₃₂	—	7	2	no	—
78	C₂×M₃₂	C₂×D₇	1	2	yes	—
79	C₂×C₄×C₈	—	7	1	no	—
80	C₂×C₄×C₈	—	7	1	no	—
81	C₂×C₄×C₈	—	7	1	no	—
82	C₂×C₄×C₈	—	7	1	no	—
83	C₂×C₄×C₈	—	7	1	no	—
84	C₂×C₄×C₈	C₂×D₇	1	1	yes	characteristic
85	C₄³	—	7	1	no	—
86	#364	—	7	8	no	—
87	#346	—	2	1	no	—
88	#346	—	14	1	no	—
89	C₄×C₁₆	—	2	1	no	—
90	C₄×C₁₆	—	14	1	no	—
91	C₂³×C₇	C₂×M₁₆	1	1	yes	characteristic
92	C₂×C₄×C₇	—	2	4	no	—
93	C₂×C₄×C₇	C₄⋊C₈	1	2	yes	—
94	C₂×C₄×C₇	C₄⋊C₈	1	2	yes	—
95	C₂×C₄×C₇	C₂×C₄²	1	1	yes	characteristic
96	C₂×C₄×C₇	C₂×C₄⋊C₄	1	1	yes	characteristic
97	C₂×C₄×C₇	#117	1	1	yes	characteristic
98	C₂×C₄×C₇	C₂²×C₈	1	1	yes	characteristic
99	C₂×C₄×C₇	C₂²×C₈	1	1	yes	characteristic
100	C₂×C₄×C₇	C₂×M₁₆	1	1	yes	characteristic
101	C₂×Dic₇	—	2	4	no	—
102	C₂×Dic₇	—	2	4	no	—
103	C₂×Dic₇	—	2	2	no	—
104	C₂×Dic₇	—	2	2	no	—
105	C₇×C₈	C₄⋊C₈	1	2	yes	—
106	C₇×C₈	C₄⋊C₈	1	2	yes	—
107	C₇×C₈	C₄×D₄	1	2	yes	—
108	C₇×C₈	C₄×Q₈	1	2	yes	—
109	C₇⋊C₈	—	2	2	no	—
110	C₇⋊C₈	C₄×C₈	1	4	yes	—
111	C₂²×C₈	—	7	1	no	—
112	C₂²×C₈	—	7	1	no	—
113	C₂²×C₈	C₄×D₇	1	1	yes	characteristic
114	C₂²×C₈	C₂²×D₇	1	1	yes	characteristic
115	C₂×C₄²	—	7	1	no	—
116	C₂×C₄²	—	7	1	no	—
117	C₂×C₄²	—	14	2	no	—
118	C₂×C₄²	C₄×D₇	1	1	yes	characteristic
119	M₃₂	—	2	2	no	—
120	M₃₂	—	7	4	no	—
121	M₃₂	—	14	2	no	—
122	M₃₂	C₄×D₇	1	4	yes	—
123	C₂×C₁₆	—	2	2	no	—
124	C₂×C₁₆	—	14	2	no	—
125	C₄×C₈	—	7	4	no	—
126	C₄×C₈	—	7	4	no	—
127	C₄×C₈	—	7	4	no	—
128	C₄×C₈	—	14	2	no	—
129	C₄×C₈	—	14	2	no	—
130	C₄×C₈	Dic₁₄	1	1	yes	characteristic
131	C₄×C₈	Dic₁₄	1	1	yes	characteristic
132	C₄×C₈	C₂²⋊D₇	1	1	yes	characteristic
133	C₄×C₈	C₂²⋊D₇	1	1	yes	characteristic
134	C₂²×C₇	—	2	2	no	—
135	C₂²×C₇	C₄×M₁₆	1	1	yes	characteristic
136	C₂²×C₇	C₂×C₄⋊C₈	1	1	yes	characteristic
137	C₂²×C₇	#431	1	1	yes	characteristic
138	C₄×C₇	—	2	2	no	—
139	C₄×C₇	#364	1	4	yes	complemented
140	C₄×C₇	C₄×C₄⋊C₄	1	1	yes	characteristic
141	C₄×C₇	C₂×C₄×C₈	1	1	yes	characteristic
142	C₄×C₇	C₂×C₄⋊C₈	1	1	yes	characteristic
143	C₄×C₇	#431	1	1	yes	characteristic
144	Dic₇	—	2	4	no	—
145	Dic₇	—	2	4	no	—
146	C₂²×C₄	—	14	1	no	—
147	C₂²×C₄	—	14	1	no	—
148	C₂²×C₄	#1131	1	2	yes	—
149	C₂²×C₄	C₂×C₄×D₇	1	1	yes	characteristic
150	C₂×C₈	—	7	2	no	—
151	C₂×C₈	—	7	2	no	—
152	C₂×C₈	—	7	2	no	—
153	C₂×C₈	—	7	2	no	—
154	C₂×C₈	—	14	1	no	—
155	C₂×C₈	—	14	1	no	—
156	C₂×C₈	C₈×D₇	1	2	yes	—
157	C₂×C₈	#1131	1	2	yes	—
158	C₂×C₈	Dic₇⋊C₄	1	1	yes	characteristic
159	C₂×C₈	Dic₇⋊C₄	1	1	yes	characteristic
160	C₂×C₈	C₂×Dic₁₄	1	1	yes	characteristic
161	C₂×C₈	C₂×C₄×D₇	1	2	yes	—
162	C₂×C₈	#1157	1	2	yes	—
163	C₂×C₈	C₂×C₂²⋊D₇	1	1	yes	characteristic
164	C₄²	—	7	2	no	—
165	C₄²	—	7	2	no	—
166	C₄²	—	14	8	no	—
167	C₄²	—	14	1	no	—
168	C₄²	—	14	1	no	—
169	C₄²	C₈×D₇	1	2	yes	—
170	C₄²	Dic₇⋊C₄	1	1	yes	characteristic
171	C₄²	Dic₇⋊C₄	1	1	yes	characteristic
172	C₁₆	—	2	4	no	—
173	C₁₆	—	14	4	no	—
174	C₂×C₇	—	2	1	no	—
175	C₂×C₇	—	2	1	no	—
176	C₂×C₇	C₄×C₄⋊C₈	1	1	yes	characteristic
177	C₂×C₇	C₂×#364	1	2	yes	—
178	C₂³	C₂×#1131	1	1	yes	characteristic
179	C₂×C₄	—	2	4	no	—
180	C₂×C₄	—	14	4	no	—
181	C₂×C₄	—	14	4	no	—
182	C₂×C₄	—	14	2	no	—
183	C₂×C₄	—	14	2	no	—
184	C₂×C₄	Dic₇⋊C₈	1	2	yes	—
185	C₂×C₄	Dic₇⋊C₈	1	2	yes	—
186	C₂×C₄	C₄²×D₇	1	1	yes	characteristic
187	C₂×C₄	#6435	1	1	yes	characteristic
188	C₂×C₄	C₂×C₈×D₇	1	1	yes	characteristic
189	C₂×C₄	C₂×C₈×D₇	1	1	yes	characteristic
190	C₂×C₄	C₂×#1131	1	1	yes	characteristic
191	C₂×C₄	C₂×Dic₇⋊C₄	1	1	yes	characteristic
192	C₈	—	7	4	no	—
193	C₈	—	14	2	no	—
194	C₈	Dic₇⋊C₈	1	2	yes	—
195	C₈	Dic₇⋊C₈	1	2	yes	—
196	C₈	C₄×Dic₁₄	1	2	yes	—
197	C₈	C₄×C₂²⋊D₇	1	2	yes	—
198	C₇	C₄×#364	1	1	yes	characteristic, complemented, Sylow
199	C₂²	—	2	2	no	—
200	C₂²	#89457	1	1	yes	characteristic
201	C₂²	C₄×#1131	1	1	yes	characteristic
202	C₂²	C₂×Dic₇⋊C₈	1	1	yes	characteristic
203	C₄	—	2	2	no	—
204	C₄	—	14	4	no	—
205	C₄	—	14	4	no	—
206	C₄	#89306	1	4	yes	direct product factor
207	C₄	C₄×C₈×D₇	1	1	yes	characteristic
208	C₄	#89457	1	1	yes	characteristic
209	C₄	C₄×Dic₇⋊C₄	1	1	yes	characteristic
210	C₄	C₂×Dic₇⋊C₈	1	1	yes	characteristic
211	C₂	—	2	1	no	—
212	C₂	—	2	1	no	—
213	C₂	C₄×Dic₇⋊C₈	1	1	yes	characteristic
214	C₂	C₂×#89306	1	2	yes	—

	subgroup	G/H
center	C₄×C₈	C₂²⋊D₇
commutator	C₄×C₇	C₂×C₄×C₈
Frattini	C₂²×C₈	C₂²×D₇
Fitting	C₄×C₇×M₃₂	C₂
socle	C₂²×C₇	C₂×C₄⋊C₈
prime radical	C₂³×C₇	C₂×M₁₆
cosocle	C₂²×C₇×C₈	C₂³
hypercenter	C₄×M₃₂	D₇
hypocenter	C₇	C₄×#364
LCS	G ▷ C₄×C₇ ▷ C₂×C₇ ▷ C₇
UCS	C₁ ◁ C₄×C₈ ◁ C₂×C₄×C₈ ◁ C₄×M₃₂

Complements

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

N	H ≅ G/N	type
17. #89306	C₄	direct (N × H)
139. C₄×C₇	#364	semidirect
198. C₇	C₄×#364	semidirect
206. C₄	#89306	direct (N × H)

Maximal subgroups

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

subgroup	index
C₄×C₇×M₃₂	2
C₂×#89306	2
C₄×C₈×Dic₇	2
C₄×#6386	2
#11725616	2
C₄×#364	7

Subgroups — C4×#89306 #13269115

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₄×#89306 #13269115