Subgroups — C₆₇×#9731 #30688982

Order 17152.

486 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	C₂×#500×C₆₇	C₂	1	1	yes	maximal, characteristic
2	C₆₇×#1981	C₂	1	2	yes	maximal
3	C₈×C₄⋊C₄×C₆₇	C₂	1	1	yes	maximal, characteristic
4	C₂×#335×C₆₇	C₂	1	1	yes	maximal, characteristic
5	C₂×#332×C₆₇	C₂	1	2	yes	maximal
6	C₂×C₄⋊Q₈×C₆₇	C₂²	1	2	yes	—
7	#500×C₆₇	—	2	2	no	—
8	#500×C₆₇	C₄	1	4	yes	complemented
9	C₂×#107×C₆₇	—	2	1	no	—
10	C₂×#107×C₆₇	—	2	1	no	—
11	C₂×C₄⋊C₈×C₆₇	C₂²	1	2	yes	—
12	C₂×C₄⋊C₈×C₆₇	C₂²	1	1	yes	characteristic
13	C₂×#103×C₆₇	—	2	4	no	—
14	C₂×C₄×C₈×C₆₇	—	2	1	no	—
15	C₂×C₄×C₈×C₆₇	C₂²	1	1	yes	characteristic
16	C₄×C₄⋊C₄×C₆₇	C₂²	1	1	yes	characteristic
17	#336×C₆₇	—	2	1	no	—
18	#335×C₆₇	—	2	2	no	—
19	#332×C₆₇	—	2	4	no	—
20	C₂²×Q₈×C₆₇	—	2	2	no	—
21	C₂²×C₈×C₆₇	—	2	1	no	—
22	C₂²×C₈×C₆₇	—	2	2	no	—
23	C₂²×C₈×C₆₇	D₄	1	1	yes	characteristic
24	C₂²×C₈×C₆₇	D₄	1	1	yes	characteristic
25	C₄⋊Q₈×C₆₇	—	2	2	no	—
26	C₄⋊Q₈×C₆₇	—	2	2	no	—
27	C₄⋊Q₈×C₆₇	—	2	2	no	—
28	C₄⋊Q₈×C₆₇	C₂×C₄	1	4	yes	—
29	C₂×C₄⋊C₄×C₆₇	—	2	2	no	—
30	C₂×C₄⋊C₄×C₆₇	—	2	2	no	—
31	C₂×C₄⋊C₄×C₆₇	D₄	1	2	yes	—
32	C₂×C₄²×C₆₇	—	2	1	no	—
33	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
34	#107×C₆₇	—	2	4	no	—
35	#107×C₆₇	—	2	4	no	—
36	C₄⋊C₈×C₆₇	—	2	2	no	—
37	C₄⋊C₈×C₆₇	—	2	2	no	—
38	C₄⋊C₈×C₆₇	—	2	1	no	—
39	C₄⋊C₈×C₆₇	D₄	1	2	yes	—
40	#103×C₆₇	—	4	8	no	—
41	C₄×C₈×C₆₇	—	4	2	no	—
42	C₄×C₈×C₆₇	C₂×C₄	1	2	yes	—
43	C₄×C₈×C₆₇	D₄	1	2	yes	—
44	#95×C₆₇	—	2	1	no	—
45	C₂×Q₈×C₆₇	—	2	4	no	—
46	C₂×Q₈×C₆₇	—	4	2	no	—
47	C₂×Q₈×C₆₇	—	4	4	no	—
48	C₂×Q₈×C₆₇	—	4	4	no	—
49	C₂²×C₄×C₆₇	—	2	2	no	—
50	C₂²×C₄×C₆₇	—	4	2	no	—
51	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
52	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
53	C₂²×C₄×C₆₇	#41	1	1	yes	characteristic
54	C₂×C₈×C₆₇	—	2	1	no	—
55	C₂×C₈×C₆₇	—	2	1	no	—
56	C₂×C₈×C₆₇	—	2	2	no	—
57	C₂×C₈×C₆₇	—	4	1	no	—
58	C₂×C₈×C₆₇	—	4	1	no	—
59	C₂×C₈×C₆₇	—	4	2	no	—
60	C₂×C₈×C₆₇	—	4	4	no	—
61	C₂×C₈×C₆₇	C₂²⋊C₄	1	2	yes	—
62	C₂×C₈×C₆₇	C₂²⋊C₄	1	2	yes	—
63	C₂×C₈×C₆₇	SD₁₆	1	2	yes	—
64	C₂×C₈×C₆₇	Q₁₆	1	2	yes	—
65	C₄⋊C₄×C₆₇	—	2	2	no	—
66	C₄⋊C₄×C₆₇	—	2	4	no	—
67	C₄⋊C₄×C₆₇	—	2	4	no	—
68	C₄⋊C₄×C₆₇	—	4	2	no	—
69	C₄⋊C₄×C₆₇	—	4	2	no	—
70	C₄⋊C₄×C₆₇	D₈	1	4	yes	—
71	C₄²×C₆₇	—	4	1	no	—
72	C₄²×C₆₇	—	4	1	no	—
73	C₄²×C₆₇	C₂²×C₄	1	1	yes	characteristic
74	C₄²×C₆₇	C₂×D₄	1	1	yes	characteristic
75	C₄²×C₆₇	C₂×D₄	1	1	yes	characteristic
76	C₄²×C₆₇	#41	1	1	yes	characteristic
77	C₂³×C₆₇	#121	1	1	yes	characteristic
78	Q₈×C₆₇	—	4	8	no	—
79	Q₈×C₆₇	—	8	2	no	—
80	Q₈×C₆₇	—	8	2	no	—
81	C₂×C₄×C₆₇	—	2	2	no	—
82	C₂×C₄×C₆₇	—	2	4	no	—
83	C₂×C₄×C₆₇	—	4	2	no	—
84	C₂×C₄×C₆₇	—	4	2	no	—
85	C₂×C₄×C₆₇	—	4	4	no	—
86	C₂×C₄×C₆₇	—	8	4	no	—
87	C₂×C₄×C₆₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
88	C₂×C₄×C₆₇	C₄×D₄	1	1	yes	characteristic
89	C₂×C₄×C₆₇	C₄×D₄	1	1	yes	characteristic
90	C₂×C₄×C₆₇	#121	1	1	yes	characteristic
91	C₂×C₄×C₆₇	#124	1	1	yes	characteristic
92	C₂×C₄×C₆₇	#127	1	1	yes	characteristic
93	C₂×C₄×C₆₇	C₂×D₈	1	2	yes	—
94	C₂×C₄×C₆₇	C₂×SD₁₆	1	1	yes	characteristic
95	C₂×C₄×C₆₇	C₂×Q₁₆	1	1	yes	characteristic
96	C₂×C₄×C₆₇	#135	1	2	yes	—
97	C₂×C₄×C₆₇	#137	1	1	yes	characteristic
98	C₂×C₄×C₆₇	#137	1	1	yes	characteristic
99	C₈×C₆₇	—	2	2	no	—
100	C₈×C₆₇	—	4	2	no	—
101	C₈×C₆₇	—	8	2	no	—
102	C₈×C₆₇	#103	1	4	yes	complemented
103	C₂²×C₆₇	#390	1	1	yes	characteristic
104	C₂²×C₆₇	#461	1	1	yes	characteristic
105	C₂²×C₆₇	#462	1	1	yes	characteristic
106	C₂²×C₆₇	#466	1	2	yes	—
107	C₂²×C₆₇	#473	1	2	yes	—
108	C₄×C₆₇	—	2	1	no	—
109	C₄×C₆₇	—	2	1	no	—
110	C₄×C₆₇	—	4	4	no	—
111	C₄×C₆₇	—	8	4	no	—
112	C₄×C₆₇	C₂×#103	1	1	yes	characteristic
113	C₄×C₆₇	#419	1	1	yes	characteristic
114	C₄×C₆₇	C₄×D₈	1	2	yes	—
115	C₄×C₆₇	#461	1	1	yes	characteristic
116	C₄×C₆₇	#462	1	1	yes	characteristic
117	C₄×C₆₇	#493	1	1	yes	characteristic
118	C₄×C₆₇	#495	1	1	yes	characteristic
119	#9731	C₆₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
120	C₂×C₆₇	#1673	1	2	yes	—
121	C₂×C₆₇	#1674	1	2	yes	—
122	C₂×C₆₇	#1970	1	1	yes	characteristic
123	C₂×C₆₇	#1979	1	1	yes	characteristic
124	C₂×C₆₇	#1981	1	1	yes	characteristic
125	C₂×#500	C₂×C₆₇	1	1	yes	characteristic
126	#1981	C₂×C₆₇	1	2	yes	—
127	C₈×C₄⋊C₄	C₂×C₆₇	1	1	yes	characteristic
128	C₂×#335	C₂×C₆₇	1	1	yes	characteristic
129	C₂×#332	C₂×C₆₇	1	2	yes	—
130	C₆₇	#9731	1	1	yes	characteristic, direct product factor, Sylow
131	C₂×C₄⋊Q₈	C₂²×C₆₇	1	2	yes	—
132	#500	—	2	2	no	—
133	#500	C₄×C₆₇	1	4	yes	complemented
134	C₂×#107	—	2	1	no	—
135	C₂×#107	—	2	1	no	—
136	C₂×C₄⋊C₈	C₂²×C₆₇	1	2	yes	—
137	C₂×C₄⋊C₈	C₂²×C₆₇	1	1	yes	characteristic
138	C₂×#103	—	2	4	no	—
139	C₂×C₄×C₈	—	2	1	no	—
140	C₂×C₄×C₈	C₂²×C₆₇	1	1	yes	characteristic
141	C₄×C₄⋊C₄	C₂²×C₆₇	1	1	yes	characteristic
142	#336	—	2	1	no	—
143	#335	—	2	2	no	—
144	#332	—	2	4	no	—
145	C₂²×Q₈	—	2	2	no	—
146	C₂²×C₈	—	2	1	no	—
147	C₂²×C₈	—	2	2	no	—
148	C₂²×C₈	D₄×C₆₇	1	1	yes	characteristic
149	C₂²×C₈	D₄×C₆₇	1	1	yes	characteristic
150	C₄⋊Q₈	—	2	2	no	—
151	C₄⋊Q₈	—	2	2	no	—
152	C₄⋊Q₈	—	2	2	no	—
153	C₄⋊Q₈	C₂×C₄×C₆₇	1	4	yes	—
154	C₂×C₄⋊C₄	—	2	2	no	—
155	C₂×C₄⋊C₄	—	2	2	no	—
156	C₂×C₄⋊C₄	D₄×C₆₇	1	2	yes	—
157	C₂×C₄²	—	2	1	no	—
158	C₂×C₄²	C₂³×C₆₇	1	1	yes	characteristic
159	#107	—	2	4	no	—
160	#107	—	2	4	no	—
161	C₄⋊C₈	—	2	2	no	—
162	C₄⋊C₈	—	2	2	no	—
163	C₄⋊C₈	—	2	1	no	—
164	C₄⋊C₈	D₄×C₆₇	1	2	yes	—
165	#103	—	4	8	no	—
166	C₄×C₈	—	4	2	no	—
167	C₄×C₈	C₂×C₄×C₆₇	1	2	yes	—
168	C₄×C₈	D₄×C₆₇	1	2	yes	—
169	#95	—	2	1	no	—
170	C₂×Q₈	—	2	4	no	—
171	C₂×Q₈	—	4	2	no	—
172	C₂×Q₈	—	4	4	no	—
173	C₂×Q₈	—	4	4	no	—
174	C₂²×C₄	—	2	2	no	—
175	C₂²×C₄	—	4	2	no	—
176	C₂²×C₄	C₂×D₄×C₆₇	1	1	yes	characteristic
177	C₂²×C₄	C₂×D₄×C₆₇	1	1	yes	characteristic
178	C₂²×C₄	#41×C₆₇	1	1	yes	characteristic
179	C₂×C₈	—	2	1	no	—
180	C₂×C₈	—	2	1	no	—
181	C₂×C₈	—	2	2	no	—
182	C₂×C₈	—	4	1	no	—
183	C₂×C₈	—	4	1	no	—
184	C₂×C₈	—	4	2	no	—
185	C₂×C₈	—	4	4	no	—
186	C₂×C₈	C₂²⋊C₄×C₆₇	1	2	yes	—
187	C₂×C₈	C₂²⋊C₄×C₆₇	1	2	yes	—
188	C₂×C₈	SD₁₆×C₆₇	1	2	yes	—
189	C₂×C₈	Q₁₆×C₆₇	1	2	yes	—
190	C₄⋊C₄	—	2	2	no	—
191	C₄⋊C₄	—	2	4	no	—
192	C₄⋊C₄	—	2	4	no	—
193	C₄⋊C₄	—	4	2	no	—
194	C₄⋊C₄	—	4	2	no	—
195	C₄⋊C₄	D₈×C₆₇	1	4	yes	—
196	C₄²	—	4	1	no	—
197	C₄²	—	4	1	no	—
198	C₄²	C₂²×C₄×C₆₇	1	1	yes	characteristic
199	C₄²	C₂×D₄×C₆₇	1	1	yes	characteristic
200	C₄²	C₂×D₄×C₆₇	1	1	yes	characteristic
201	C₄²	#41×C₆₇	1	1	yes	characteristic
202	C₂³	#121×C₆₇	1	1	yes	characteristic
203	Q₈	—	4	8	no	—
204	Q₈	—	8	2	no	—
205	Q₈	—	8	2	no	—
206	C₂×C₄	—	2	2	no	—
207	C₂×C₄	—	2	4	no	—
208	C₂×C₄	—	4	2	no	—
209	C₂×C₄	—	4	2	no	—
210	C₂×C₄	—	4	4	no	—
211	C₂×C₄	—	8	4	no	—
212	C₂×C₄	C₂×C₂²⋊C₄×C₆₇	1	1	yes	characteristic
213	C₂×C₄	C₄×D₄×C₆₇	1	1	yes	characteristic
214	C₂×C₄	C₄×D₄×C₆₇	1	1	yes	characteristic
215	C₂×C₄	#121×C₆₇	1	1	yes	characteristic
216	C₂×C₄	#124×C₆₇	1	1	yes	characteristic
217	C₂×C₄	#127×C₆₇	1	1	yes	characteristic
218	C₂×C₄	C₂×D₈×C₆₇	1	2	yes	—
219	C₂×C₄	C₂×SD₁₆×C₆₇	1	1	yes	characteristic
220	C₂×C₄	C₂×Q₁₆×C₆₇	1	1	yes	characteristic
221	C₂×C₄	#135×C₆₇	1	2	yes	—
222	C₂×C₄	#137×C₆₇	1	1	yes	characteristic
223	C₂×C₄	#137×C₆₇	1	1	yes	characteristic
224	C₈	—	2	2	no	—
225	C₈	—	4	2	no	—
226	C₈	—	8	2	no	—
227	C₈	#103×C₆₇	1	4	yes	complemented
228	C₂²	#390×C₆₇	1	1	yes	characteristic
229	C₂²	#461×C₆₇	1	1	yes	characteristic
230	C₂²	#462×C₆₇	1	1	yes	characteristic
231	C₂²	#466×C₆₇	1	2	yes	—
232	C₂²	#473×C₆₇	1	2	yes	—
233	C₄	—	2	1	no	—
234	C₄	—	2	1	no	—
235	C₄	—	4	4	no	—
236	C₄	—	8	4	no	—
237	C₄	C₂×#103×C₆₇	1	1	yes	characteristic
238	C₄	#419×C₆₇	1	1	yes	characteristic
239	C₄	C₄×D₈×C₆₇	1	2	yes	—
240	C₄	#461×C₆₇	1	1	yes	characteristic
241	C₄	#462×C₆₇	1	1	yes	characteristic
242	C₄	#493×C₆₇	1	1	yes	characteristic
243	C₄	#495×C₆₇	1	1	yes	characteristic
244	C₂	C₆₇×#1673	1	2	yes	—
245	C₂	C₆₇×#1674	1	2	yes	—
246	C₂	C₆₇×#1970	1	1	yes	characteristic
247	C₂	C₆₇×#1979	1	1	yes	characteristic
248	C₂	C₆₇×#1981	1	1	yes	characteristic

	subgroup	G/H
center, socle, prime radical	C₂³×C₆₇	#121
commutator	C₄²	C₂²×C₄×C₆₇
Frattini, cosocle	C₂×C₄²	C₂³×C₆₇
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³×C₆₇ ◁ C₂×C₄²×C₆₇ ◁ G

Complements

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

N	H ≅ G/N	type
8. #500×C₆₇	C₄	semidirect
102. C₈×C₆₇	#103	semidirect
119. #9731	C₆₇	direct (N × H)
130. C₆₇	#9731	direct (N × H)
133. #500	C₄×C₆₇	semidirect
227. C₈	#103×C₆₇	semidirect

Maximal subgroups

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

subgroup	index
C₂×#500×C₆₇	2
C₆₇×#1981	2
C₈×C₄⋊C₄×C₆₇	2
C₂×#335×C₆₇	2
C₂×#332×C₆₇	2
#9731	67

Subgroups — C67×#9731 #30688982

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₆₇×#9731 #30688982