Subgroups — #13241920

Order 1792.

440 conjugacy classes of subgroups.

The subgroup lattice has 266 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄×C₇×C₄⋊Q₈	C₂	1	1	yes	maximal, characteristic
2	#11727147	C₂	1	1	yes	maximal, characteristic
3	#11727147	C₂	1	1	yes	maximal, characteristic
4	#11725444	C₂	1	4	yes	maximal
5	C₂×C₇×C₄⋊Q₈	C₄	1	1	yes	characteristic
6	C₂×C₄×C₇×Q₈	—	2	1	no	—
7	C₇×#391	—	2	1	no	—
8	C₇×#389	—	2	1	no	—
9	C₇×#389	C₂²	1	2	yes	—
10	C₇×#384	C₄	1	1	yes	characteristic
11	C₄×C₇×C₄⋊C₄	—	2	1	no	—
12	C₄×C₇×C₄⋊C₄	C₄	1	2	yes	—
13	C₄³×C₇	C₂²	1	1	yes	characteristic
14	#89319	—	2	1	no	—
15	#89319	—	2	1	no	—
16	#89319	—	2	1	no	—
17	#89319	—	2	1	no	—
18	#89319	C₂²	1	2	yes	—
19	#89319	C₂²	1	2	yes	—
20	#7604	—	7	1	no	maximal, Sylow
21	C₂²×C₇×Q₈	—	2	1	no	—
22	C₇×C₄⋊Q₈	—	2	2	no	—
23	C₇×C₄⋊Q₈	C₈	1	4	yes	complemented
24	C₄×C₇×Q₈	—	4	4	no	—
25	C₂×C₇×C₄⋊C₄	—	2	1	no	—
26	C₂×C₇×C₄⋊C₄	—	2	1	no	—
27	C₂×C₇×C₄⋊C₄	—	2	1	no	—
28	C₂×C₇×C₄⋊C₄	—	2	2	no	—
29	C₂×C₇×C₄⋊C₄	C₂×C₄	1	2	yes	—
30	C₂×C₇×C₄⋊C₄	C₂×C₄	1	2	yes	—
31	C₂×C₄²×C₇	—	2	2	no	—
32	C₂×C₄²×C₇	C₂×C₄	1	1	yes	characteristic
33	C₂×C₄²×C₇	C₂×C₄	1	1	yes	characteristic
34	C₂×C₄²×C₇	D₄	1	1	yes	characteristic
35	C₂×C₄²×C₇	D₄	1	1	yes	characteristic
36	C₂×C₄²×C₇	D₄	1	1	yes	characteristic
37	C₂×C₄²×C₇	D₄	1	1	yes	characteristic
38	C₂×C₄²×C₇	C₂³	1	1	yes	characteristic
39	C₂²×C₇⋊C₈	—	2	2	no	—
40	C₂²×C₇⋊C₈	—	2	2	no	—
41	C₇×#95	—	2	2	no	—
42	C₇×#95	—	2	2	no	—
43	C₄×C₄⋊Q₈	D₇	1	1	yes	characteristic
44	#1839	—	7	1	no	—
45	#1839	—	7	1	no	—
46	#1271	—	7	4	no	—
47	C₂×C₇×Q₈	—	2	4	no	—
48	C₂×C₇×Q₈	—	4	2	no	—
49	C₂×C₇×Q₈	—	4	2	no	—
50	C₂²×C₄×C₇	—	2	2	no	—
51	C₂²×C₄×C₇	—	2	2	no	—
52	C₂²×C₄×C₇	C₂²⋊C₄	1	1	yes	characteristic
53	C₂²×C₄×C₇	C₂²⋊C₄	1	1	yes	characteristic
54	C₂²×C₄×C₇	C₂²⋊C₄	1	1	yes	characteristic
55	C₂²×C₄×C₇	C₂²⋊C₄	1	1	yes	characteristic
56	C₂²×C₄×C₇	C₂²×C₄	1	1	yes	characteristic
57	C₂²×C₄×C₇	C₂×D₄	1	1	yes	characteristic
58	C₂²×C₄×C₇	C₂×D₄	1	1	yes	characteristic
59	C₇×C₄⋊C₄	—	2	4	no	—
60	C₇×C₄⋊C₄	—	2	2	no	—
61	C₇×C₄⋊C₄	—	4	2	no	—
62	C₇×C₄⋊C₄	—	4	2	no	—
63	C₇×C₄⋊C₄	—	4	2	no	—
64	C₇×C₄⋊C₄	—	4	4	no	—
65	C₇×C₄⋊C₄	C₂×C₈	1	4	yes	—
66	C₄²×C₇	—	2	2	no	—
67	C₄²×C₇	—	2	2	no	—
68	C₄²×C₇	—	2	2	no	—
69	C₄²×C₇	—	4	1	no	—
70	C₄²×C₇	—	4	1	no	—
71	C₄²×C₇	—	4	1	no	—
72	C₄²×C₇	—	4	4	no	—
73	C₄²×C₇	C₂×C₈	1	2	yes	—
74	C₄²×C₇	M₁₆	1	2	yes	—
75	C₂×C₇⋊C₈	—	4	2	no	—
76	C₂×C₇⋊C₈	—	4	2	no	—
77	C₂×C₇⋊C₈	—	4	2	no	—
78	C₂×C₇⋊C₈	—	4	2	no	—
79	C₂×C₇⋊C₈	—	4	2	no	—
80	C₂×C₇⋊C₈	—	4	2	no	—
81	C₂×C₄⋊Q₈	Dic₇	1	1	yes	characteristic
82	C₂×C₄×Q₈	—	2	1	no	—
83	#391	—	2	1	no	—
84	#389	—	2	1	no	—
85	#389	C₂×D₇	1	2	yes	—
86	#384	Dic₇	1	1	yes	characteristic
87	C₄×C₄⋊C₄	—	2	1	no	—
88	C₄×C₄⋊C₄	Dic₇	1	2	yes	—
89	C₄³	C₂×D₇	1	1	yes	characteristic
90	#336	—	7	2	no	—
91	#336	—	7	2	no	—
92	#336	—	14	1	no	—
93	#336	—	14	1	no	—
94	#336	—	14	1	no	—
95	#336	—	14	1	no	—
96	C₂³×C₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
97	C₇×Q₈	—	4	8	no	—
98	C₂×C₄×C₇	—	2	1	no	—
99	C₂×C₄×C₇	—	2	1	no	—
100	C₂×C₄×C₇	—	2	1	no	—
101	C₂×C₄×C₇	—	2	1	no	—
102	C₂×C₄×C₇	—	2	1	no	—
103	C₂×C₄×C₇	—	2	1	no	—
104	C₂×C₄×C₇	—	2	1	no	—
105	C₂×C₄×C₇	—	2	1	no	—
106	C₂×C₄×C₇	—	2	1	no	—
107	C₂×C₄×C₇	—	2	1	no	—
108	C₂×C₄×C₇	—	2	1	no	—
109	C₂×C₄×C₇	—	2	1	no	—
110	C₂×C₄×C₇	—	2	4	no	—
111	C₂×C₄×C₇	—	2	1	no	—
112	C₂×C₄×C₇	—	2	1	no	—
113	C₂×C₄×C₇	—	2	1	no	—
114	C₂×C₄×C₇	—	4	2	no	—
115	C₂×C₄×C₇	—	4	2	no	—
116	C₂×C₄×C₇	—	4	2	no	—
117	C₂×C₄×C₇	—	4	2	no	—
118	C₂×C₄×C₇	—	4	2	no	—
119	C₂×C₄×C₇	C₂²⋊C₈	1	2	yes	—
120	C₂×C₄×C₇	C₂²⋊C₈	1	2	yes	—
121	C₂×C₄×C₇	#101	1	2	yes	—
122	C₂×C₄×C₇	C₂²×C₈	1	1	yes	characteristic
123	C₂×C₄×C₇	C₂×M₁₆	1	1	yes	characteristic
124	C₂×C₄×C₇	AGL(1,ℤ/8)	1	2	yes	—
125	C₂×C₄×C₇	#137	1	2	yes	—
126	C₇⋊C₈	—	8	2	no	—
127	C₇⋊C₈	—	8	2	no	—
128	C₂²×Q₈	—	2	1	no	—
129	C₂²×C₈	—	14	2	no	—
130	C₂²×C₈	—	14	2	no	—
131	C₄⋊Q₈	—	2	2	no	—
132	C₄⋊Q₈	C₇⋊C₈	1	4	yes	complemented
133	C₄×Q₈	—	4	4	no	—
134	C₂×C₄⋊C₄	—	2	1	no	—
135	C₂×C₄⋊C₄	—	2	1	no	—
136	C₂×C₄⋊C₄	—	2	1	no	—
137	C₂×C₄⋊C₄	—	2	2	no	—
138	C₂×C₄⋊C₄	C₂×Dic₇	1	2	yes	—
139	C₂×C₄⋊C₄	C₂×Dic₇	1	2	yes	—
140	C₂×C₄²	—	2	2	no	—
141	C₂×C₄²	C₂×Dic₇	1	1	yes	characteristic
142	C₂×C₄²	C₂×Dic₇	1	1	yes	characteristic
143	C₂×C₄²	C₂²⋊D₇	1	1	yes	characteristic
144	C₂×C₄²	C₂²⋊D₇	1	1	yes	characteristic
145	C₂×C₄²	C₂²⋊D₇	1	1	yes	characteristic
146	C₂×C₄²	C₂²⋊D₇	1	1	yes	characteristic
147	C₂×C₄²	C₂²×D₇	1	1	yes	characteristic
148	#95	—	2	2	no	—
149	#95	—	2	2	no	—
150	C₂²×C₇	C₂×C₂²⋊C₈	1	1	yes	characteristic
151	C₂²×C₇	#410	1	1	yes	characteristic
152	C₂²×C₇	C₂×#101	1	1	yes	characteristic
153	C₂²×C₇	#418	1	1	yes	characteristic
154	C₂²×C₇	#419	1	1	yes	characteristic
155	C₂²×C₇	#421	1	1	yes	characteristic
156	C₂²×C₇	#421	1	1	yes	characteristic
157	C₄×C₇	—	2	2	no	—
158	C₄×C₇	—	2	1	no	—
159	C₄×C₇	—	2	1	no	—
160	C₄×C₇	—	2	2	no	—
161	C₄×C₇	—	2	2	no	—
162	C₄×C₇	—	2	2	no	—
163	C₄×C₇	—	2	2	no	—
164	C₄×C₇	—	4	1	no	—
165	C₄×C₇	—	4	4	no	—
166	C₄×C₇	—	8	2	no	—
167	C₂×Q₈	—	2	4	no	—
168	C₂×Q₈	—	4	2	no	—
169	C₂×Q₈	—	4	2	no	—
170	C₂²×C₄	—	2	2	no	—
171	C₂²×C₄	—	2	2	no	—
172	C₂²×C₄	C₂²⋊Dic₇	1	1	yes	characteristic
173	C₂²×C₄	C₂²⋊Dic₇	1	1	yes	characteristic
174	C₂²×C₄	C₂²⋊Dic₇	1	1	yes	characteristic
175	C₂²×C₄	C₂²⋊Dic₇	1	1	yes	characteristic
176	C₂²×C₄	C₂²×Dic₇	1	1	yes	characteristic
177	C₂²×C₄	C₂×C₂²⋊D₇	1	1	yes	characteristic
178	C₂²×C₄	C₂×C₂²⋊D₇	1	1	yes	characteristic
179	C₂×C₈	—	28	2	no	—
180	C₂×C₈	—	28	2	no	—
181	C₂×C₈	—	28	2	no	—
182	C₂×C₈	—	28	2	no	—
183	C₂×C₈	—	28	2	no	—
184	C₂×C₈	—	28	2	no	—
185	C₄⋊C₄	—	2	4	no	—
186	C₄⋊C₄	—	2	2	no	—
187	C₄⋊C₄	—	4	2	no	—
188	C₄⋊C₄	—	4	2	no	—
189	C₄⋊C₄	—	4	2	no	—
190	C₄⋊C₄	—	4	4	no	—
191	C₄⋊C₄	C₂×C₇⋊C₈	1	4	yes	—
192	C₄²	—	2	2	no	—
193	C₄²	—	2	2	no	—
194	C₄²	—	2	2	no	—
195	C₄²	—	4	1	no	—
196	C₄²	—	4	1	no	—
197	C₄²	—	4	1	no	—
198	C₄²	—	4	4	no	—
199	C₄²	C₂×C₇⋊C₈	1	2	yes	—
200	C₄²	#1136	1	2	yes	—
201	C₂×C₇	#1461	1	1	yes	characteristic
202	C₂×C₇	#1478	1	1	yes	characteristic
203	C₂×C₇	#1479	1	1	yes	characteristic
204	C₂×C₇	#1514	1	1	yes	characteristic
205	C₂×C₇	#1515	1	1	yes	characteristic
206	C₂×C₇	#1537	1	1	yes	characteristic
207	C₂×C₇	#1556	1	1	yes	characteristic
208	C₂³	C₂×C₂²⋊Dic₇	1	1	yes	characteristic
209	Q₈	—	4	8	no	—
210	C₂×C₄	—	2	1	no	—
211	C₂×C₄	—	2	1	no	—
212	C₂×C₄	—	2	1	no	—
213	C₂×C₄	—	2	1	no	—
214	C₂×C₄	—	2	1	no	—
215	C₂×C₄	—	2	1	no	—
216	C₂×C₄	—	2	1	no	—
217	C₂×C₄	—	2	1	no	—
218	C₂×C₄	—	2	1	no	—
219	C₂×C₄	—	2	1	no	—
220	C₂×C₄	—	2	1	no	—
221	C₂×C₄	—	2	1	no	—
222	C₂×C₄	—	2	4	no	—
223	C₂×C₄	—	2	1	no	—
224	C₂×C₄	—	2	1	no	—
225	C₂×C₄	—	2	1	no	—
226	C₂×C₄	—	4	2	no	—
227	C₂×C₄	—	4	2	no	—
228	C₂×C₄	—	4	2	no	—
229	C₂×C₄	—	4	2	no	—
230	C₂×C₄	—	4	2	no	—
231	C₂×C₄	C₂²⋊(C₇⋊C₈)	1	2	yes	—
232	C₂×C₄	C₂²⋊(C₇⋊C₈)	1	2	yes	—
233	C₂×C₄	#6410	1	2	yes	—
234	C₂×C₄	C₂²×C₇⋊C₈	1	1	yes	characteristic
235	C₂×C₄	C₂×#1136	1	1	yes	characteristic
236	C₂×C₄	#6495	1	2	yes	—
237	C₂×C₄	#6505	1	2	yes	—
238	C₈	—	56	2	no	—
239	C₈	—	56	2	no	—
240	C₇	#7604	1	1	yes	characteristic, complemented, Sylow
241	C₂²	C₂×C₂²⋊(C₇⋊C₈)	1	1	yes	characteristic
242	C₂²	#89987	1	1	yes	characteristic
243	C₂²	#89997	1	1	yes	characteristic
244	C₂²	C₂×#6410	1	1	yes	characteristic
245	C₂²	#90008	1	1	yes	characteristic
246	C₂²	#90008	1	1	yes	characteristic
247	C₂²	#90012	1	1	yes	characteristic
248	C₄	—	2	2	no	—
249	C₄	—	2	2	no	—
250	C₄	—	2	2	no	—
251	C₄	—	2	2	no	—
252	C₄	—	2	2	no	—
253	C₄	—	2	1	no	—
254	C₄	—	2	1	no	—
255	C₄	—	4	1	no	—
256	C₄	—	4	4	no	—
257	C₄	—	8	2	no	—
258	C₂	#11729351	1	1	yes	characteristic
259	C₂	#11729375	1	1	yes	characteristic
260	C₂	#11729378	1	1	yes	characteristic
261	C₂	#11729530	1	1	yes	characteristic
262	C₂	#11729567	1	1	yes	characteristic
263	C₂	#11729577	1	1	yes	characteristic
264	C₂	#11729591	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	C₂×C₂²⋊Dic₇
commutator	C₂×C₄×C₇	C₂²×C₈
Frattini	C₂×C₄²	C₂²×D₇
Fitting	C₄×C₇×C₄⋊Q₈	C₂
socle, prime radical	C₂³×C₇	C₂×C₂²⋊C₄
cosocle	C₂×C₄²×C₇	C₂³
hypercenter	C₄×C₄⋊Q₈	D₇
hypocenter	C₇	#7604
LCS	G ▷ C₂×C₄×C₇ ▷ C₂²×C₇ ▷ C₇
UCS	C₁ ◁ C₂³ ◁ C₄³ ◁ C₄×C₄⋊Q₈

Complements

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

N	H ≅ G/N	type
23. C₇×C₄⋊Q₈	C₈	semidirect
132. C₄⋊Q₈	C₇⋊C₈	semidirect
240. C₇	#7604	semidirect

Maximal subgroups

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

subgroup	index
C₄×C₇×C₄⋊Q₈	2
#11727147	2
#11725444	2
#7604	7