Subgroups — #117×D₂₁ #13061613

Order 1344.

660 conjugacy classes of subgroups.

The subgroup lattice has 264 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₄×D₂₁	C₂	1	1	yes	maximal, characteristic
2	C₃×C₇×#117	C₂	1	1	yes	maximal, characteristic, complemented
3	#10621409	C₂	1	1	yes	maximal, characteristic, complemented
4	#10621375	C₂	1	2	yes	maximal, complemented
5	C₄⋊C₄×D₂₁	C₂	1	2	yes	maximal, complemented
6	C₂²⋊C₄×D₂₁	C₂	1	2	yes	maximal
7	#10621360	C₂	1	2	yes	maximal, complemented
8	#10621355	C₂	1	2	yes	maximal, complemented
9	C₄²×D₂₁	C₂	1	2	yes	maximal, complemented
10	D₇×#117	—	3	1	no	maximal, Hall
11	C₂³×D₂₁	C₂²	1	1	yes	characteristic
12	C₂²×C₃×C₄×C₇	C₂²	1	1	yes	characteristic
13	C₂²×Dic₂₁	C₂²	1	1	yes	characteristic
14	C₂×C₄×D₂₁	C₄	1	8	yes	complemented
15	C₂×C₄×D₂₁	C₂²	1	4	yes	—
16	C₂×C₄×D₂₁	C₂²	1	1	yes	characteristic
17	C₂×C₄×D₂₁	C₂²	1	1	yes	characteristic
18	C₂×C₄×D₂₁	C₂²	1	1	yes	characteristic
19	C₂×C₄×D₂₁	C₂²	1	1	yes	characteristic
20	C₃×C₇×C₄⋊C₄	C₂²	1	2	yes	complemented
21	C₃×C₇×C₂²⋊C₄	C₂²	1	2	yes	—
22	C₃×C₄²×C₇	C₂²	1	2	yes	complemented
23	#66636	C₂²	1	2	yes	—
24	#66631	C₂²	1	4	yes	—
25	C₄⋊Dic₂₁	C₂²	1	2	yes	complemented
26	#66629	C₂²	1	4	yes	complemented
27	C₄×Dic₂₁	C₂²	1	2	yes	complemented
28	C₄×Dic₂₁	C₂²	1	4	yes	complemented
29	C₂²×C₄×D₇	—	3	1	no	—
30	C₇×#117	D₃	1	1	yes	characteristic, complemented
31	#6489	—	3	1	no	—
32	#6455	—	3	2	no	—
33	D₇×C₄⋊C₄	—	3	2	no	—
34	D₇×C₂²⋊C₄	—	3	2	no	—
35	#6440	—	3	2	no	—
36	#6435	—	3	2	no	—
37	C₄²×D₇	—	3	2	no	—
38	D₃×#117	—	7	1	no	maximal, Hall
39	C₂³×C₃×C₇	C₂³	1	1	yes	characteristic
40	C₂²×D₂₁	—	2	4	no	—
41	C₂²×D₂₁	C₂×C₄	1	4	yes	—
42	C₂²×D₂₁	C₂³	1	1	yes	characteristic
43	C₂²×D₂₁	C₂³	1	1	yes	characteristic
44	C₂×C₃×C₄×C₇	C₂×C₄	1	4	yes	complemented
45	C₂×C₃×C₄×C₇	C₂³	1	4	yes	—
46	C₂×C₃×C₄×C₇	C₂³	1	1	yes	characteristic
47	C₂×C₃×C₄×C₇	C₂³	1	1	yes	characteristic
48	C₂×Dic₂₁	C₂×C₄	1	4	yes	complemented
49	C₂×Dic₂₁	C₂³	1	4	yes	—
50	C₂×Dic₂₁	C₂³	1	1	yes	characteristic
51	C₂×Dic₂₁	C₂³	1	1	yes	characteristic
52	C₄×D₂₁	—	2	8	no	—
53	C₄×D₂₁	C₂×C₄	1	4	yes	—
54	C₄×D₂₁	C₂×C₄	1	4	yes	—
55	C₄×D₂₁	C₂×C₄	1	4	yes	complemented
56	C₄×D₂₁	C₂×C₄	1	4	yes	complemented
57	C₂³×D₇	—	3	1	no	—
58	C₂²×C₄×C₇	C₂×D₃	1	1	yes	characteristic
59	C₂²×Dic₇	—	3	1	no	—
60	C₂×C₄×D₇	—	3	8	no	—
61	C₂×C₄×D₇	—	3	1	no	—
62	C₂×C₄×D₇	—	3	1	no	—
63	C₂×C₄×D₇	—	3	1	no	—
64	C₂×C₄×D₇	—	3	1	no	—
65	C₂×C₄×D₇	—	3	4	no	—
66	C₇×C₄⋊C₄	C₂×D₃	1	2	yes	complemented
67	C₇×C₂²⋊C₄	C₂×D₃	1	2	yes	—
68	C₄²×C₇	C₂×D₃	1	2	yes	complemented
69	C₂²⋊Dic₇	—	3	2	no	—
70	(C₂×D₇)⋊C₄	—	3	4	no	—
71	C₄⋊Dic₇	—	3	2	no	—
72	Dic₇⋊C₄	—	3	4	no	—
73	C₄×Dic₇	—	3	2	no	—
74	C₄×Dic₇	—	3	4	no	—
75	C₂²×C₄×D₃	—	7	1	no	—
76	C₃×#117	D₇	1	1	yes	characteristic, complemented
77	#926	—	7	1	no	—
78	#892	—	7	2	no	—
79	D₃×C₄⋊C₄	—	7	2	no	—
80	D₃×C₂²⋊C₄	—	7	2	no	—
81	#877	—	7	2	no	—
82	#872	—	7	2	no	—
83	C₄²×D₃	—	7	2	no	—
84	C₂²×C₃×C₇	—	2	2	no	—
85	C₂²×C₃×C₇	C₂²×C₄	1	2	yes	—
86	C₂²×C₃×C₇	C₂⁴	1	1	yes	characteristic
87	C₂×D₂₁	—	2	2	no	—
88	C₂×D₂₁	—	2	8	no	—
89	C₂×D₂₁	C₂²×C₄	1	2	yes	—
90	C₂×D₂₁	C₂²×C₄	1	2	yes	—
91	C₂×D₂₁	#41	1	4	yes	—
92	C₃×C₄×C₇	—	2	4	no	—
93	C₃×C₄×C₇	C₂²×C₄	1	2	yes	—
94	C₃×C₄×C₇	C₂²×C₄	1	2	yes	—
95	Dic₂₁	—	2	4	no	—
96	Dic₂₁	C₂²×C₄	1	2	yes	—
97	Dic₂₁	C₂²×C₄	1	2	yes	—
98	C₂×#117	—	21	1	no	Sylow
99	C₂³×C₇	C₂²×D₃	1	1	yes	characteristic
100	C₂²×D₇	—	3	1	no	—
101	C₂²×D₇	—	3	4	no	—
102	C₂²×D₇	—	3	1	no	—
103	C₂²×D₇	—	6	4	no	—
104	C₂×C₄×C₇	C₄×D₃	1	4	yes	complemented
105	C₂×C₄×C₇	C₂²×D₃	1	4	yes	—
106	C₂×C₄×C₇	C₂²×D₃	1	1	yes	characteristic
107	C₂×C₄×C₇	C₂²×D₃	1	1	yes	characteristic
108	C₂×Dic₇	—	3	4	no	—
109	C₂×Dic₇	—	3	1	no	—
110	C₂×Dic₇	—	3	1	no	—
111	C₂×Dic₇	—	3	4	no	—
112	C₄×D₇	—	3	4	no	—
113	C₄×D₇	—	3	4	no	—
114	C₄×D₇	—	3	4	no	—
115	C₄×D₇	—	3	4	no	—
116	C₄×D₇	—	6	8	no	—
117	C₂³×D₃	—	7	1	no	—
118	C₂²×C₃×C₄	C₂×D₇	1	1	yes	characteristic
119	C₂²×Dic₃	—	7	1	no	—
120	C₂×C₄×D₃	—	7	8	no	—
121	C₂×C₄×D₃	—	7	1	no	—
122	C₂×C₄×D₃	—	7	1	no	—
123	C₂×C₄×D₃	—	7	1	no	—
124	C₂×C₄×D₃	—	7	1	no	—
125	C₂×C₄×D₃	—	7	4	no	—
126	C₃×C₄⋊C₄	C₂×D₇	1	2	yes	complemented
127	C₃×C₂²⋊C₄	C₂×D₇	1	2	yes	—
128	C₃×C₄²	C₂×D₇	1	2	yes	complemented
129	#217	—	7	2	no	—
130	(C₂×D₃)⋊C₄	—	7	4	no	—
131	C₄⋊Dic₃	—	7	2	no	—
132	Dic₃⋊C₄	—	7	4	no	—
133	C₄×Dic₃	—	7	2	no	—
134	C₄×Dic₃	—	7	4	no	—
135	C₂×C₃×C₇	—	2	2	no	—
136	C₂×C₃×C₇	C₂³×C₄	1	1	yes	characteristic
137	C₂×C₃×C₇	C₂×#41	1	2	yes	—
138	D₂₁	—	2	2	no	—
139	D₂₁	#117	1	4	yes	direct product factor
140	C₂³×C₄	—	21	1	no	—
141	#117	—	21	1	no	—
142	#117	—	21	2	no	—
143	#117	—	21	2	no	—
144	#117	—	21	2	no	—
145	#117	D₂₁	1	1	yes	characteristic, direct product factor
146	C₂×C₄⋊C₄	—	21	2	no	—
147	C₂×C₂²⋊C₄	—	21	2	no	—
148	C₂×C₄²	—	21	2	no	—
149	C₂²×C₇	—	2	2	no	—
150	C₂²×C₇	C₂×C₄×D₃	1	2	yes	—
151	C₂²×C₇	C₂³×D₃	1	1	yes	characteristic
152	C₂×D₇	—	3	2	no	—
153	C₂×D₇	—	3	4	no	—
154	C₂×D₇	—	3	2	no	—
155	C₂×D₇	—	6	8	no	—
156	C₂×D₇	—	6	2	no	—
157	C₄×C₇	—	2	4	no	—
158	C₄×C₇	C₂×C₄×D₃	1	2	yes	—
159	C₄×C₇	C₂×C₄×D₃	1	2	yes	—
160	Dic₇	—	3	2	no	—
161	Dic₇	—	3	2	no	—
162	Dic₇	—	6	4	no	—
163	C₂³×C₃	C₂²×D₇	1	1	yes	characteristic
164	C₂²×D₃	—	7	1	no	—
165	C₂²×D₃	—	7	4	no	—
166	C₂²×D₃	—	7	1	no	—
167	C₂²×D₃	—	14	4	no	—
168	C₂×C₃×C₄	C₄×D₇	1	4	yes	complemented
169	C₂×C₃×C₄	C₂²×D₇	1	4	yes	—
170	C₂×C₃×C₄	C₂²×D₇	1	1	yes	characteristic
171	C₂×C₃×C₄	C₂²×D₇	1	1	yes	characteristic
172	C₂×Dic₃	—	7	4	no	—
173	C₂×Dic₃	—	7	1	no	—
174	C₂×Dic₃	—	7	1	no	—
175	C₂×Dic₃	—	7	4	no	—
176	C₄×D₃	—	7	4	no	—
177	C₄×D₃	—	7	4	no	—
178	C₄×D₃	—	7	4	no	—
179	C₄×D₃	—	7	4	no	—
180	C₄×D₃	—	14	8	no	—
181	C₃×C₇	C₂×#117	1	1	yes	characteristic, complemented, Hall
182	C₂⁴	—	21	1	no	—
183	C₂²×C₄	—	21	1	no	—
184	C₂²×C₄	—	21	8	no	—
185	C₂²×C₄	—	21	1	no	—
186	C₂²×C₄	—	21	1	no	—
187	C₂²×C₄	—	21	1	no	—
188	C₂²×C₄	—	21	1	no	—
189	C₂²×C₄	—	21	4	no	—
190	C₂²×C₄	C₂×D₂₁	1	1	yes	characteristic
191	C₄⋊C₄	—	21	4	no	—
192	C₄⋊C₄	—	21	2	no	—
193	C₄⋊C₄	C₂×D₂₁	1	2	yes	complemented
194	C₂²⋊C₄	—	21	2	no	—
195	C₂²⋊C₄	—	21	4	no	—
196	C₂²⋊C₄	C₂×D₂₁	1	2	yes	—
197	C₄²	—	21	4	no	—
198	C₄²	—	21	2	no	—
199	C₄²	C₂×D₂₁	1	2	yes	complemented
200	C₂×C₇	—	2	2	no	—
201	C₂×C₇	C₂²×C₄×D₃	1	1	yes	characteristic
202	C₂×C₇	D₃×#41	1	2	yes	—
203	D₇	—	3	4	no	—
204	D₇	—	6	2	no	—
205	C₂²×C₃	—	2	2	no	—
206	C₂²×C₃	C₂×C₄×D₇	1	2	yes	—
207	C₂²×C₃	C₂³×D₇	1	1	yes	characteristic
208	C₂×D₃	—	7	2	no	—
209	C₂×D₃	—	7	4	no	—
210	C₂×D₃	—	7	2	no	—
211	C₂×D₃	—	14	8	no	—
212	C₂×D₃	—	14	2	no	—
213	C₃×C₄	—	2	4	no	—
214	C₃×C₄	C₂×C₄×D₇	1	2	yes	—
215	C₃×C₄	C₂×C₄×D₇	1	2	yes	—
216	Dic₃	—	7	2	no	—
217	Dic₃	—	7	2	no	—
218	Dic₃	—	14	4	no	—
219	C₂³	—	21	1	no	—
220	C₂³	—	21	4	no	—
221	C₂³	—	21	1	no	—
222	C₂³	—	42	4	no	—
223	C₂³	C₂²×D₂₁	1	1	yes	characteristic
224	C₂×C₄	—	21	4	no	—
225	C₂×C₄	—	21	1	no	—
226	C₂×C₄	—	21	1	no	—
227	C₂×C₄	—	21	4	no	—
228	C₂×C₄	—	21	4	no	—
229	C₂×C₄	—	21	4	no	—
230	C₂×C₄	—	21	4	no	—
231	C₂×C₄	—	21	4	no	—
232	C₂×C₄	—	42	8	no	—
233	C₂×C₄	C₄×D₂₁	1	4	yes	complemented
234	C₂×C₄	C₂²×D₂₁	1	4	yes	—
235	C₂×C₄	C₂²×D₂₁	1	1	yes	characteristic
236	C₂×C₄	C₂²×D₂₁	1	1	yes	characteristic
237	C₇	D₃×#117	1	1	yes	characteristic, complemented, Sylow
238	C₂×C₃	—	2	2	no	—
239	C₂×C₃	C₂²×C₄×D₇	1	1	yes	characteristic
240	C₂×C₃	D₇×#41	1	2	yes	—
241	D₃	—	7	4	no	—
242	D₃	—	14	2	no	—
243	C₂²	—	2	2	no	—
244	C₂²	—	21	2	no	—
245	C₂²	—	21	4	no	—
246	C₂²	—	21	2	no	—
247	C₂²	—	42	8	no	—
248	C₂²	—	42	2	no	—
249	C₂²	C₂×C₄×D₂₁	1	2	yes	—
250	C₂²	C₂³×D₂₁	1	1	yes	characteristic
251	C₄	—	2	4	no	—
252	C₄	—	21	2	no	—
253	C₄	—	21	2	no	—
254	C₄	—	42	4	no	—
255	C₄	C₂×C₄×D₂₁	1	2	yes	—
256	C₄	C₂×C₄×D₂₁	1	2	yes	—
257	C₃	D₇×#117	1	1	yes	characteristic, complemented, Sylow
258	C₂	—	2	2	no	—
259	C₂	—	21	4	no	—
260	C₂	—	42	2	no	—
261	C₂	C₂²×C₄×D₂₁	1	1	yes	characteristic
262	C₂	#41×D₂₁	1	2	yes	—

	subgroup	G/H
center	C₂×C₄	C₂²×D₂₁
commutator	C₂×C₃×C₇	C₂³×C₄
Frattini	C₂²	C₂³×D₂₁
Fitting	C₃×C₇×#117	C₂
socle, cosocle	C₂²×C₃×C₇	C₂⁴
prime radical	C₂³×D₂₁	C₂²
hypercenter	#117	D₂₁
hypocenter	C₃×C₇	C₂×#117
LCS	G ▷ C₂×C₃×C₇ ▷ C₃×C₇
UCS	C₁ ◁ C₂×C₄ ◁ #117

Complements

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

N	H ≅ G/N	type
2. C₃×C₇×#117	C₂	semidirect
3. #10621409	C₂	semidirect
4. #10621375	C₂	semidirect
5. C₄⋊C₄×D₂₁	C₂	semidirect
7. #10621360	C₂	semidirect
8. #10621355	C₂	semidirect
9. C₄²×D₂₁	C₂	semidirect
14. C₂×C₄×D₂₁	C₄	semidirect
20. C₃×C₇×C₄⋊C₄	C₂²	semidirect
22. C₃×C₄²×C₇	C₂²	semidirect
25. C₄⋊Dic₂₁	C₂²	semidirect
26. #66629	C₂²	semidirect
27. C₄×Dic₂₁	C₂²	semidirect
28. C₄×Dic₂₁	C₂²	semidirect
30. C₇×#117	D₃	semidirect
44. C₂×C₃×C₄×C₇	C₂×C₄	semidirect
48. C₂×Dic₂₁	C₂×C₄	semidirect
55. C₄×D₂₁	C₂×C₄	semidirect
56. C₄×D₂₁	C₂×C₄	semidirect
66. C₇×C₄⋊C₄	C₂×D₃	semidirect
68. C₄²×C₇	C₂×D₃	semidirect
76. C₃×#117	D₇	semidirect
104. C₂×C₄×C₇	C₄×D₃	semidirect
126. C₃×C₄⋊C₄	C₂×D₇	semidirect
128. C₃×C₄²	C₂×D₇	semidirect
139. D₂₁	#117	direct (N × H)
145. #117	D₂₁	direct (N × H)
168. C₂×C₃×C₄	C₄×D₇	semidirect
181. C₃×C₇	C₂×#117	semidirect
193. C₄⋊C₄	C₂×D₂₁	semidirect
199. C₄²	C₂×D₂₁	semidirect
233. C₂×C₄	C₄×D₂₁	semidirect
237. C₇	D₃×#117	semidirect
257. C₃	D₇×#117	semidirect

Maximal subgroups

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

subgroup	index
C₂²×C₄×D₂₁	2
C₃×C₇×#117	2
#10621409	2
#10621375	2
C₄⋊C₄×D₂₁	2
C₂²⋊C₄×D₂₁	2
#10621360	2
#10621355	2
C₄²×D₂₁	2
D₇×#117	3
D₃×#117	7

Subgroups — #117×D21 #13061613

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — #117×D₂₁ #13061613