Subgroups — #13243201

Order 1792.

662 conjugacy classes of subgroups.

The subgroup lattice has 250 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄×C₇×#117	C₂	1	1	yes	maximal, characteristic
2	#11739494	C₂	1	2	yes	maximal
3	#11725420	C₂	1	4	yes	maximal, complemented
4	C₂×C₇×#117	C₄	1	2	yes	—
5	C₂²×C₄²×C₇	C₂²	1	1	yes	characteristic
6	C₄×C₇×C₄⋊C₄	—	2	2	no	—
7	C₄×C₇×C₂²⋊C₄	—	2	2	no	—
8	C₇×#376	C₂²	1	2	yes	—
9	C₄³×C₇	C₄	1	2	yes	—
10	#89979	—	2	2	no	—
11	#89698	—	2	2	no	—
12	#89698	—	2	2	no	—
13	#89697	C₄	1	4	yes	complemented
14	#89697	C₂²	1	4	yes	—
15	#89696	—	2	2	no	—
16	C₄×#1136	C₄	1	4	yes	complemented
17	#7856	—	7	1	no	maximal, Sylow
18	C₂³×C₄×C₇	C₂×C₄	1	1	yes	characteristic
19	C₂³×C₄×C₇	D₄	1	2	yes	—
20	C₇×#117	—	2	8	no	—
21	C₂×C₇×C₄⋊C₄	—	2	2	no	—
22	C₂×C₇×C₂²⋊C₄	—	2	2	no	—
23	C₂×C₄²×C₇	—	2	4	no	—
24	C₂×C₄²×C₇	C₂×C₄	1	2	yes	—
25	C₂×C₄²×C₇	C₂×C₄	1	2	yes	—
26	C₂×C₄²×C₇	C₂×C₄	1	4	yes	—
27	C₂×C₄²×C₇	C₂×C₄	1	1	yes	characteristic
28	C₂×C₄²×C₇	D₄	1	1	yes	characteristic
29	C₂×C₄²×C₇	D₄	1	1	yes	characteristic
30	C₂×C₄²×C₇	D₄	1	2	yes	—
31	C₂×C₄²×C₇	Q₈	1	1	yes	characteristic
32	C₂×C₄²×C₇	Q₈	1	1	yes	characteristic
33	C₂×C₄²×C₇	C₂³	1	1	yes	characteristic
34	C₂×#1136	—	2	4	no	—
35	C₇×#95	—	2	4	no	—
36	C₂²⋊(C₇⋊C₈)	—	2	4	no	—
37	C₂²⋊(C₇⋊C₈)	—	2	4	no	—
38	C₄⋊(C₇⋊C₈)	—	2	4	no	—
39	C₄⋊(C₇⋊C₈)	—	2	4	no	—
40	#6377	—	2	4	no	—
41	C₄×C₇⋊C₈	C₂×C₄	1	8	yes	—
42	#2920	—	7	2	no	—
43	C₄×#117	D₇	1	1	yes	characteristic
44	#1274	—	7	4	no	—
45	C₂⁴×C₇	C₂²⋊C₄	1	1	yes	characteristic
46	C₂²×C₄×C₇	—	2	4	no	—
47	C₂²×C₄×C₇	—	2	4	no	—
48	C₂²×C₄×C₇	—	4	4	no	—
49	C₂²×C₄×C₇	C₄²	1	1	yes	characteristic
50	C₂²×C₄×C₇	C₄²	1	1	yes	characteristic
51	C₂²×C₄×C₇	C₂²⋊C₄	1	2	yes	—
52	C₂²×C₄×C₇	C₂²⋊C₄	1	1	yes	characteristic
53	C₂²×C₄×C₇	C₂²⋊C₄	1	2	yes	—
54	C₂²×C₄×C₇	C₂²⋊C₄	1	2	yes	—
55	C₂²×C₄×C₇	C₄⋊C₄	1	2	yes	—
56	C₂²×C₄×C₇	C₄⋊C₄	1	1	yes	characteristic
57	C₂²×C₄×C₇	C₄⋊C₄	1	1	yes	characteristic
58	C₂²×C₄×C₇	C₄⋊C₄	1	2	yes	—
59	C₂²×C₄×C₇	C₂²×C₄	1	1	yes	characteristic
60	C₂²×C₄×C₇	C₂×D₄	1	2	yes	—
61	C₇×C₄⋊C₄	—	2	8	no	—
62	C₇×C₂²⋊C₄	—	2	8	no	—
63	C₄²×C₇	—	2	4	no	—
64	C₄²×C₇	—	4	8	no	—
65	C₄²×C₇	C₄²	1	2	yes	—
66	C₄²×C₇	C₂²⋊C₄	1	2	yes	—
67	C₄²×C₇	C₂²⋊C₄	1	2	yes	—
68	C₄²×C₇	C₄⋊C₄	1	2	yes	—
69	C₄²×C₇	C₄⋊C₄	1	2	yes	—
70	C₄²×C₇	C₄⋊C₄	1	1	yes	characteristic
71	C₄²×C₇	C₄⋊C₄	1	1	yes	characteristic
72	C₄²×C₇	M₁₆	1	8	yes	complemented
73	C₄²×C₇	C₂²×C₄	1	2	yes	—
74	C₄²×C₇	C₂×D₄	1	1	yes	characteristic
75	C₄²×C₇	C₂×Q₈	1	1	yes	characteristic
76	#1136	—	4	8	no	—
77	C₂×C₇⋊C₈	—	2	8	no	—
78	C₂×#117	Dic₇	1	2	yes	—
79	C₂²×C₄²	C₂×D₇	1	1	yes	characteristic
80	#432	—	14	2	no	—
81	#432	—	14	2	no	—
82	#431	—	7	4	no	—
83	#431	—	7	4	no	—
84	#423	—	14	2	no	—
85	#407	—	14	2	no	—
86	C₄×M₁₆	—	7	4	no	—
87	C₄×C₄⋊C₄	—	2	2	no	—
88	C₄×C₂²⋊C₄	—	2	2	no	—
89	#376	C₂×D₇	1	2	yes	—
90	C₄³	Dic₇	1	2	yes	—
91	C₂³×C₇	—	2	2	no	—
92	C₂³×C₇	—	4	2	no	—
93	C₂³×C₇	#95	1	1	yes	characteristic
94	C₂³×C₇	#95	1	1	yes	characteristic
95	C₂³×C₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
96	C₂×C₄×C₇	—	2	2	no	—
97	C₂×C₄×C₇	—	2	4	no	—
98	C₂×C₄×C₇	—	2	4	no	—
99	C₂×C₄×C₇	—	2	4	no	—
100	C₂×C₄×C₇	—	2	2	no	—
101	C₂×C₄×C₇	—	2	2	no	—
102	C₂×C₄×C₇	—	2	2	no	—
103	C₂×C₄×C₇	—	2	2	no	—
104	C₂×C₄×C₇	—	2	8	no	—
105	C₂×C₄×C₇	—	4	2	no	—
106	C₂×C₄×C₇	—	4	2	no	—
107	C₂×C₄×C₇	—	4	2	no	—
108	C₂×C₄×C₇	—	4	8	no	—
109	C₂×C₄×C₇	#95	1	2	yes	—
110	C₂×C₄×C₇	#95	1	1	yes	characteristic
111	C₂×C₄×C₇	#95	1	1	yes	characteristic
112	C₂×C₄×C₇	#95	1	2	yes	—
113	C₂×C₄×C₇	C₂×C₄²	1	1	yes	characteristic
114	C₂×C₄×C₇	C₂×C₂²⋊C₄	1	2	yes	—
115	C₂×C₄×C₇	C₂×C₄⋊C₄	1	2	yes	—
116	C₂×C₄×C₇	C₂×C₄⋊C₄	1	1	yes	characteristic
117	C₂×C₄×C₇	C₂×M₁₆	1	4	yes	—
118	C₇⋊C₈	—	4	8	no	—
119	C₂³×C₄	C₂×Dic₇	1	1	yes	characteristic
120	C₂³×C₄	C₂²⋊D₇	1	2	yes	—
121	C₂×M₁₆	—	14	4	no	—
122	#117	—	2	8	no	—
123	C₂×C₄⋊C₄	—	2	2	no	—
124	C₂×C₂²⋊C₄	—	2	2	no	—
125	C₂×C₄²	—	2	4	no	—
126	C₂×C₄²	Dic₁₄	1	1	yes	characteristic
127	C₂×C₄²	Dic₁₄	1	1	yes	characteristic
128	C₂×C₄²	C₄×D₇	1	2	yes	—
129	C₂×C₄²	C₄×D₇	1	2	yes	—
130	C₂×C₄²	D₂₈	1	1	yes	characteristic
131	C₂×C₄²	D₂₈	1	1	yes	characteristic
132	C₂×C₄²	C₂×Dic₇	1	4	yes	—
133	C₂×C₄²	C₂×Dic₇	1	1	yes	characteristic
134	C₂×C₄²	C₂²⋊D₇	1	2	yes	—
135	C₂×C₄²	C₂²×D₇	1	1	yes	characteristic
136	C₄⋊C₈	—	14	4	no	—
137	C₄⋊C₈	—	14	4	no	—
138	C₂²⋊C₈	—	14	4	no	—
139	C₂²⋊C₈	—	14	4	no	—
140	#97	—	14	4	no	—
141	C₄×C₈	—	7	8	no	—
142	#95	—	2	4	no	—
143	C₂²×C₇	—	2	2	no	—
144	C₂²×C₇	—	2	1	no	—
145	C₂²×C₇	—	2	1	no	—
146	C₂²×C₇	—	2	2	no	—
147	C₂²×C₇	—	2	2	no	—
148	C₂²×C₇	—	4	4	no	—
149	C₂²×C₇	C₂×#95	1	1	yes	characteristic
150	C₂²×C₇	#407	1	2	yes	—
151	C₄×C₇	—	2	1	no	—
152	C₄×C₇	—	2	1	no	—
153	C₄×C₇	—	2	2	no	—
154	C₄×C₇	—	2	2	no	—
155	C₄×C₇	—	2	2	no	—
156	C₄×C₇	—	4	8	no	—
157	C₄×C₇	H(3,ℤ/4)	1	4	yes	—
158	C₄×C₇	C₄×M₁₆	1	2	yes	—
159	C₄×C₇	#423	1	2	yes	—
160	C₂⁴	C₂²⋊Dic₇	1	1	yes	characteristic
161	C₂²×C₄	—	2	4	no	—
162	C₂²×C₄	—	2	4	no	—
163	C₂²×C₄	—	4	4	no	—
164	C₂²×C₄	C₄×Dic₇	1	1	yes	characteristic
165	C₂²×C₄	C₄×Dic₇	1	1	yes	characteristic
166	C₂²×C₄	Dic₇⋊C₄	1	2	yes	—
167	C₂²×C₄	Dic₇⋊C₄	1	2	yes	—
168	C₂²×C₄	C₄⋊Dic₇	1	1	yes	characteristic
169	C₂²×C₄	C₄⋊Dic₇	1	1	yes	characteristic
170	C₂²×C₄	(C₂×D₇)⋊C₄	1	2	yes	—
171	C₂²×C₄	(C₂×D₇)⋊C₄	1	2	yes	—
172	C₂²×C₄	C₂²⋊Dic₇	1	2	yes	—
173	C₂²×C₄	C₂²⋊Dic₇	1	1	yes	characteristic
174	C₂²×C₄	C₂²×Dic₇	1	1	yes	characteristic
175	C₂²×C₄	C₂×C₂²⋊D₇	1	2	yes	—
176	M₁₆	—	28	8	no	—
177	C₂×C₈	—	14	8	no	—
178	C₄⋊C₄	—	2	8	no	—
179	C₂²⋊C₄	—	2	8	no	—
180	C₄²	—	2	4	no	—
181	C₄²	—	4	8	no	—
182	C₄²	#1136	1	8	yes	complemented
183	C₄²	C₄×Dic₇	1	2	yes	—
184	C₄²	Dic₇⋊C₄	1	2	yes	—
185	C₄²	Dic₇⋊C₄	1	2	yes	—
186	C₄²	C₄⋊Dic₇	1	1	yes	characteristic
187	C₄²	C₄⋊Dic₇	1	1	yes	characteristic
188	C₄²	(C₂×D₇)⋊C₄	1	2	yes	—
189	C₄²	(C₂×D₇)⋊C₄	1	2	yes	—
190	C₄²	C₂×Dic₁₄	1	1	yes	characteristic
191	C₄²	C₂×C₄×D₇	1	2	yes	—
192	C₄²	C₂×D₂₈	1	1	yes	characteristic
193	C₂×C₇	—	2	2	no	—
194	C₂×C₇	—	4	1	no	—
195	C₂×C₇	—	4	1	no	—
196	C₂×C₇	#1728	1	1	yes	characteristic
197	C₂×C₇	C₂×H(3,ℤ/4)	1	2	yes	—
198	C₂³	—	2	2	no	—
199	C₂³	—	4	2	no	—
200	C₂³	C₇⋊#95	1	1	yes	characteristic
201	C₂³	C₇⋊#95	1	1	yes	characteristic
202	C₂³	C₂×C₂²⋊Dic₇	1	1	yes	characteristic
203	C₂×C₄	—	2	2	no	—
204	C₂×C₄	—	2	2	no	—
205	C₂×C₄	—	2	2	no	—
206	C₂×C₄	—	2	2	no	—
207	C₂×C₄	—	2	2	no	—
208	C₂×C₄	—	2	4	no	—
209	C₂×C₄	—	2	4	no	—
210	C₂×C₄	—	2	4	no	—
211	C₂×C₄	—	2	8	no	—
212	C₂×C₄	—	4	2	no	—
213	C₂×C₄	—	4	2	no	—
214	C₂×C₄	—	4	2	no	—
215	C₂×C₄	—	4	8	no	—
216	C₂×C₄	C₇⋊#95	1	2	yes	—
217	C₂×C₄	C₇⋊#95	1	1	yes	characteristic
218	C₂×C₄	C₇⋊#95	1	1	yes	characteristic
219	C₂×C₄	C₇⋊#95	1	2	yes	—
220	C₂×C₄	C₂×#1136	1	4	yes	—
221	C₂×C₄	C₂×C₄×Dic₇	1	1	yes	characteristic
222	C₂×C₄	C₂×Dic₇⋊C₄	1	2	yes	—
223	C₂×C₄	C₂×C₄⋊Dic₇	1	1	yes	characteristic
224	C₂×C₄	C₂×(C₂×D₇)⋊C₄	1	2	yes	—
225	C₈	—	28	8	no	—
226	C₇	#7856	1	1	yes	characteristic, complemented, Sylow
227	C₂²	—	2	2	no	—
228	C₂²	—	2	1	no	—
229	C₂²	—	2	1	no	—
230	C₂²	—	2	2	no	—
231	C₂²	—	2	2	no	—
232	C₂²	—	4	4	no	—
233	C₂²	#89979	1	2	yes	—
234	C₂²	C₂×C₇⋊#95	1	1	yes	characteristic
235	C₄	—	2	1	no	—
236	C₄	—	2	1	no	—
237	C₄	—	2	2	no	—
238	C₄	—	2	2	no	—
239	C₄	—	2	2	no	—
240	C₄	—	4	8	no	—
241	C₄	#89326	1	4	yes	—
242	C₄	C₄×#1136	1	2	yes	—
243	C₄	#89696	1	2	yes	—
244	C₂	—	2	2	no	—
245	C₂	—	4	1	no	—
246	C₂	—	4	1	no	—
247	C₂	#11729281	1	1	yes	characteristic
248	C₂	C₂×#89326	1	2	yes	—

	subgroup	G/H
center	C₂×C₄	C₂×C₄×Dic₇
commutator	C₂×C₄×C₇	C₂×C₄²
Frattini	C₂×C₄²	C₂²×D₇
Fitting	C₄×C₇×#117	C₂
socle	C₂²×C₇	C₂×#95
prime radical	C₂⁴×C₇	C₂²⋊C₄
cosocle	C₂×C₄²×C₇	C₂³
hypercenter	C₄×#117	D₇
hypocenter	C₇	#7856
LCS	G ▷ C₂×C₄×C₇ ▷ C₇
UCS	C₁ ◁ C₂×C₄ ◁ C₄×#117

Complements

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

N	H ≅ G/N	type
3. #11725420	C₂	semidirect
13. #89697	C₄	semidirect
16. C₄×#1136	C₄	semidirect
72. C₄²×C₇	M₁₆	semidirect
182. C₄²	#1136	semidirect
226. C₇	#7856	semidirect

Maximal subgroups

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

subgroup	index
C₄×C₇×#117	2
#11739494	2
#11725420	2
#7856	7