Subgroups — C₆₇×#21954 #30701205

Order 17152.

798 conjugacy classes of subgroups.

The subgroup lattice has 278 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#432×C₆₇	C₂	1	1	yes	maximal, characteristic
2	C₂×#431×C₆₇	C₂	1	1	yes	maximal, characteristic
3	C₆₇×#2285	C₂	1	1	yes	maximal, characteristic
4	C₆₇×#1940	C₂	1	4	yes	maximal, complemented
5	C₆₇×#1939	C₂	1	4	yes	maximal, complemented
6	C₆₇×#1798	C₂	1	4	yes	maximal
7	C₂×#117×C₆₇	C₂²	1	2	yes	—
8	C₂²×C₄²×C₆₇	C₂²	1	1	yes	characteristic
9	#432×C₆₇	—	2	4	no	—
10	#431×C₆₇	—	2	4	no	—
11	C₂×C₄⋊C₈×C₆₇	C₂²	1	2	yes	—
12	C₂×C₄⋊C₈×C₆₇	C₂²	1	2	yes	—
13	C₂×C₄⋊C₈×C₆₇	C₂²	1	8	yes	—
14	C₂×C₂²⋊C₈×C₆₇	C₂²	1	2	yes	—
15	C₂×C₂²⋊C₈×C₆₇	C₂²	1	2	yes	—
16	C₂×#97×C₆₇	C₂²	1	2	yes	—
17	C₂×C₄×C₈×C₆₇	C₂²	1	2	yes	—
18	#384×C₆₇	C₄	1	2	yes	—
19	#382×C₆₇	C₄	1	2	yes	—
20	#380×C₆₇	C₄	1	2	yes	—
21	#380×C₆₇	C₄	1	2	yes	—
22	C₄×C₄⋊C₄×C₆₇	C₂²	1	2	yes	—
23	C₄×C₄⋊C₄×C₆₇	C₂²	1	2	yes	—
24	#336×C₆₇	C₂²	1	8	yes	—
25	C₂³×C₄×C₆₇	C₂×C₄	1	1	yes	characteristic
26	C₂³×C₄×C₆₇	C₂×C₄	1	1	yes	characteristic
27	C₂³×C₄×C₆₇	C₂³	1	1	yes	characteristic
28	C₂²×C₈×C₆₇	C₂³	1	4	yes	—
29	C₂²×C₈×C₆₇	C₂³	1	4	yes	—
30	#117×C₆₇	—	4	4	no	—
31	C₂×C₄⋊C₄×C₆₇	C₂×C₄	1	4	yes	—
32	C₂×C₄⋊C₄×C₆₇	C₂×C₄	1	4	yes	—
33	C₂×C₄⋊C₄×C₆₇	C₂×C₄	1	4	yes	—
34	C₂×C₂²⋊C₄×C₆₇	C₂×C₄	1	4	yes	—
35	C₂×C₄²×C₆₇	C₂×C₄	1	1	yes	characteristic
36	C₂×C₄²×C₆₇	C₂×C₄	1	1	yes	characteristic
37	C₂×C₄²×C₆₇	D₄	1	4	yes	—
38	C₂×C₄²×C₆₇	Q₈	1	4	yes	—
39	C₂×C₄²×C₆₇	C₂³	1	4	yes	—
40	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
41	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
42	C₄⋊C₈×C₆₇	—	2	4	no	—
43	C₄⋊C₈×C₆₇	—	2	4	no	—
44	C₄⋊C₈×C₆₇	—	4	8	no	—
45	C₂²⋊C₈×C₆₇	—	2	4	no	—
46	C₂²⋊C₈×C₆₇	—	2	4	no	—
47	#97×C₆₇	—	2	4	no	—
48	C₄×C₈×C₆₇	—	2	4	no	—
49	#95×C₆₇	C₂×C₄	1	4	yes	—
50	#95×C₆₇	C₂×C₄	1	4	yes	—
51	C₂⁴×C₆₇	C₂²×C₄	1	1	yes	characteristic
52	C₂²×C₄×C₆₇	—	2	4	no	—
53	C₂²×C₄×C₆₇	—	2	2	no	—
54	C₂²×C₄×C₆₇	—	2	2	no	—
55	C₂²×C₄×C₆₇	—	4	2	no	—
56	C₂²×C₄×C₆₇	C₄⋊C₄	1	4	yes	—
57	C₂²×C₄×C₆₇	C₄⋊C₄	1	4	yes	—
58	C₂²×C₄×C₆₇	C₂²×C₄	1	8	yes	—
59	C₂²×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
60	C₂²×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
61	C₂²×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
62	C₂²×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
63	C₂²×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
64	C₂²×C₄×C₆₇	C₂×D₄	1	2	yes	—
65	C₂²×C₄×C₆₇	C₂×Q₈	1	2	yes	—
66	C₂²×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
67	C₂×C₈×C₆₇	—	2	4	no	—
68	C₂×C₈×C₆₇	—	2	4	no	—
69	C₂×C₈×C₆₇	—	2	4	no	—
70	C₂×C₈×C₆₇	—	2	4	no	—
71	C₂×C₈×C₆₇	—	2	4	no	—
72	C₂×C₈×C₆₇	—	2	4	no	—
73	C₄⋊C₄×C₆₇	—	2	8	no	—
74	C₄⋊C₄×C₆₇	—	4	4	no	—
75	C₄⋊C₄×C₆₇	—	4	4	no	—
76	C₂²⋊C₄×C₆₇	—	4	4	no	—
77	C₄²×C₆₇	—	2	8	no	—
78	C₄²×C₆₇	C₄⋊C₄	1	4	yes	—
79	C₄²×C₆₇	C₄⋊C₄	1	4	yes	—
80	C₄²×C₆₇	C₂×D₄	1	2	yes	—
81	C₄²×C₆₇	C₂×D₄	1	2	yes	—
82	C₄²×C₆₇	C₂×Q₈	1	2	yes	—
83	C₄²×C₆₇	C₂×Q₈	1	2	yes	—
84	C₂³×C₆₇	—	2	1	no	—
85	C₂³×C₆₇	—	2	1	no	—
86	C₂³×C₆₇	—	2	2	no	—
87	C₂³×C₆₇	—	2	2	no	—
88	C₂³×C₆₇	C₂×C₄⋊C₄	1	2	yes	—
89	C₂³×C₆₇	C₂³×C₄	1	1	yes	characteristic
90	C₂×C₄×C₆₇	—	2	1	no	—
91	C₂×C₄×C₆₇	—	2	1	no	—
92	C₂×C₄×C₆₇	—	2	2	no	—
93	C₂×C₄×C₆₇	—	2	2	no	—
94	C₂×C₄×C₆₇	—	2	2	no	—
95	C₂×C₄×C₆₇	—	2	4	no	—
96	C₂×C₄×C₆₇	—	2	4	no	—
97	C₂×C₄×C₆₇	—	2	8	no	—
98	C₂×C₄×C₆₇	—	2	8	no	—
99	C₂×C₄×C₆₇	—	2	8	no	—
100	C₂×C₄×C₆₇	—	2	8	no	—
101	C₂×C₄×C₆₇	—	4	4	no	—
102	C₂×C₄×C₆₇	C₂×C₄⋊C₄	1	2	yes	—
103	C₂×C₄×C₆₇	C₂×C₄⋊C₄	1	2	yes	—
104	C₂×C₄×C₆₇	C₂×C₄⋊C₄	1	2	yes	—
105	C₂×C₄×C₆₇	C₂×C₄⋊C₄	1	2	yes	—
106	C₂×C₄×C₆₇	C₂×C₄⋊C₄	1	2	yes	—
107	C₂×C₄×C₆₇	#131	1	4	yes	—
108	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
109	C₂×C₄×C₆₇	C₂²×Q₈	1	1	yes	characteristic
110	C₂×C₄×C₆₇	H(5,𝔽₂)	1	2	yes	—
111	C₂×C₄×C₆₇	2₋¹⁺⁴	1	2	yes	—
112	C₈×C₆₇	—	4	4	no	—
113	C₈×C₆₇	—	4	4	no	—
114	C₂²×C₆₇	—	2	2	no	—
115	C₂²×C₆₇	—	2	2	no	—
116	C₂²×C₆₇	—	2	2	no	—
117	C₂²×C₆₇	—	4	2	no	—
118	C₂²×C₆₇	—	4	2	no	—
119	C₂²×C₆₇	C₂²×C₄⋊C₄	1	1	yes	characteristic
120	C₂²×C₆₇	#518	1	1	yes	characteristic
121	C₂²×C₆₇	#519	1	1	yes	characteristic
122	C₂²×C₆₇	C₂×#131	1	2	yes	—
123	C₂²×C₆₇	#568	1	2	yes	—
124	C₄×C₆₇	—	2	2	no	—
125	C₄×C₆₇	—	2	2	no	—
126	C₄×C₆₇	—	2	2	no	—
127	C₄×C₆₇	—	2	2	no	—
128	C₄×C₆₇	—	4	8	no	—
129	C₄×C₆₇	#424	1	4	yes	—
130	C₄×C₆₇	#537	1	2	yes	—
131	C₄×C₆₇	#544	1	2	yes	—
132	#21954	C₆₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
133	C₂×C₆₇	—	4	2	no	—
134	C₂×C₆₇	#2316	1	1	yes	characteristic
135	C₂×C₆₇	C₂×#424	1	1	yes	characteristic
136	C₂×C₆₇	#2905	1	1	yes	characteristic
137	C₂×C₆₇	#2981	1	2	yes	—
138	C₂×C₆₇	#2985	1	2	yes	—
139	C₂×#432	C₂×C₆₇	1	1	yes	characteristic
140	C₂×#431	C₂×C₆₇	1	1	yes	characteristic
141	#2285	C₂×C₆₇	1	1	yes	characteristic
142	#1940	C₂×C₆₇	1	4	yes	complemented
143	#1939	C₂×C₆₇	1	4	yes	complemented
144	#1798	C₂×C₆₇	1	4	yes	—
145	C₆₇	#21954	1	1	yes	characteristic, direct product factor, Sylow
146	C₂×#117	C₂²×C₆₇	1	2	yes	—
147	C₂²×C₄²	C₂²×C₆₇	1	1	yes	characteristic
148	#432	—	2	4	no	—
149	#431	—	2	4	no	—
150	C₂×C₄⋊C₈	C₂²×C₆₇	1	2	yes	—
151	C₂×C₄⋊C₈	C₂²×C₆₇	1	2	yes	—
152	C₂×C₄⋊C₈	C₂²×C₆₇	1	8	yes	—
153	C₂×C₂²⋊C₈	C₂²×C₆₇	1	2	yes	—
154	C₂×C₂²⋊C₈	C₂²×C₆₇	1	2	yes	—
155	C₂×#97	C₂²×C₆₇	1	2	yes	—
156	C₂×C₄×C₈	C₂²×C₆₇	1	2	yes	—
157	#384	C₄×C₆₇	1	2	yes	—
158	#382	C₄×C₆₇	1	2	yes	—
159	#380	C₄×C₆₇	1	2	yes	—
160	#380	C₄×C₆₇	1	2	yes	—
161	C₄×C₄⋊C₄	C₂²×C₆₇	1	2	yes	—
162	C₄×C₄⋊C₄	C₂²×C₆₇	1	2	yes	—
163	#336	C₂²×C₆₇	1	8	yes	—
164	C₂³×C₄	C₂×C₄×C₆₇	1	1	yes	characteristic
165	C₂³×C₄	C₂×C₄×C₆₇	1	1	yes	characteristic
166	C₂³×C₄	C₂³×C₆₇	1	1	yes	characteristic
167	C₂²×C₈	C₂³×C₆₇	1	4	yes	—
168	C₂²×C₈	C₂³×C₆₇	1	4	yes	—
169	#117	—	4	4	no	—
170	C₂×C₄⋊C₄	C₂×C₄×C₆₇	1	4	yes	—
171	C₂×C₄⋊C₄	C₂×C₄×C₆₇	1	4	yes	—
172	C₂×C₄⋊C₄	C₂×C₄×C₆₇	1	4	yes	—
173	C₂×C₂²⋊C₄	C₂×C₄×C₆₇	1	4	yes	—
174	C₂×C₄²	C₂×C₄×C₆₇	1	1	yes	characteristic
175	C₂×C₄²	C₂×C₄×C₆₇	1	1	yes	characteristic
176	C₂×C₄²	D₄×C₆₇	1	4	yes	—
177	C₂×C₄²	Q₈×C₆₇	1	4	yes	—
178	C₂×C₄²	C₂³×C₆₇	1	4	yes	—
179	C₂×C₄²	C₂³×C₆₇	1	1	yes	characteristic
180	C₂×C₄²	C₂³×C₆₇	1	1	yes	characteristic
181	C₄⋊C₈	—	2	4	no	—
182	C₄⋊C₈	—	2	4	no	—
183	C₄⋊C₈	—	4	8	no	—
184	C₂²⋊C₈	—	2	4	no	—
185	C₂²⋊C₈	—	2	4	no	—
186	#97	—	2	4	no	—
187	C₄×C₈	—	2	4	no	—
188	#95	C₂×C₄×C₆₇	1	4	yes	—
189	#95	C₂×C₄×C₆₇	1	4	yes	—
190	C₂⁴	C₂²×C₄×C₆₇	1	1	yes	characteristic
191	C₂²×C₄	—	2	4	no	—
192	C₂²×C₄	—	2	2	no	—
193	C₂²×C₄	—	2	2	no	—
194	C₂²×C₄	—	4	2	no	—
195	C₂²×C₄	C₄⋊C₄×C₆₇	1	4	yes	—
196	C₂²×C₄	C₄⋊C₄×C₆₇	1	4	yes	—
197	C₂²×C₄	C₂²×C₄×C₆₇	1	8	yes	—
198	C₂²×C₄	C₂²×C₄×C₆₇	1	1	yes	characteristic
199	C₂²×C₄	C₂²×C₄×C₆₇	1	1	yes	characteristic
200	C₂²×C₄	C₂²×C₄×C₆₇	1	1	yes	characteristic
201	C₂²×C₄	C₂²×C₄×C₆₇	1	1	yes	characteristic
202	C₂²×C₄	C₂²×C₄×C₆₇	1	1	yes	characteristic
203	C₂²×C₄	C₂×D₄×C₆₇	1	2	yes	—
204	C₂²×C₄	C₂×Q₈×C₆₇	1	2	yes	—
205	C₂²×C₄	C₂⁴×C₆₇	1	1	yes	characteristic
206	C₂×C₈	—	2	4	no	—
207	C₂×C₈	—	2	4	no	—
208	C₂×C₈	—	2	4	no	—
209	C₂×C₈	—	2	4	no	—
210	C₂×C₈	—	2	4	no	—
211	C₂×C₈	—	2	4	no	—
212	C₄⋊C₄	—	2	8	no	—
213	C₄⋊C₄	—	4	4	no	—
214	C₄⋊C₄	—	4	4	no	—
215	C₂²⋊C₄	—	4	4	no	—
216	C₄²	—	2	8	no	—
217	C₄²	C₄⋊C₄×C₆₇	1	4	yes	—
218	C₄²	C₄⋊C₄×C₆₇	1	4	yes	—
219	C₄²	C₂×D₄×C₆₇	1	2	yes	—
220	C₄²	C₂×D₄×C₆₇	1	2	yes	—
221	C₄²	C₂×Q₈×C₆₇	1	2	yes	—
222	C₄²	C₂×Q₈×C₆₇	1	2	yes	—
223	C₂³	—	2	1	no	—
224	C₂³	—	2	1	no	—
225	C₂³	—	2	2	no	—
226	C₂³	—	2	2	no	—
227	C₂³	C₂×C₄⋊C₄×C₆₇	1	2	yes	—
228	C₂³	C₂³×C₄×C₆₇	1	1	yes	characteristic
229	C₂×C₄	—	2	1	no	—
230	C₂×C₄	—	2	1	no	—
231	C₂×C₄	—	2	2	no	—
232	C₂×C₄	—	2	2	no	—
233	C₂×C₄	—	2	2	no	—
234	C₂×C₄	—	2	4	no	—
235	C₂×C₄	—	2	4	no	—
236	C₂×C₄	—	2	8	no	—
237	C₂×C₄	—	2	8	no	—
238	C₂×C₄	—	2	8	no	—
239	C₂×C₄	—	2	8	no	—
240	C₂×C₄	—	4	4	no	—
241	C₂×C₄	C₂×C₄⋊C₄×C₆₇	1	2	yes	—
242	C₂×C₄	C₂×C₄⋊C₄×C₆₇	1	2	yes	—
243	C₂×C₄	C₂×C₄⋊C₄×C₆₇	1	2	yes	—
244	C₂×C₄	C₂×C₄⋊C₄×C₆₇	1	2	yes	—
245	C₂×C₄	C₂×C₄⋊C₄×C₆₇	1	2	yes	—
246	C₂×C₄	#131×C₆₇	1	4	yes	—
247	C₂×C₄	C₂²×D₄×C₆₇	1	1	yes	characteristic
248	C₂×C₄	C₂²×Q₈×C₆₇	1	1	yes	characteristic
249	C₂×C₄	H(5,𝔽₂)×C₆₇	1	2	yes	—
250	C₂×C₄	2₋¹⁺⁴×C₆₇	1	2	yes	—
251	C₈	—	4	4	no	—
252	C₈	—	4	4	no	—
253	C₂²	—	2	2	no	—
254	C₂²	—	2	2	no	—
255	C₂²	—	2	2	no	—
256	C₂²	—	4	2	no	—
257	C₂²	—	4	2	no	—
258	C₂²	C₂²×C₄⋊C₄×C₆₇	1	1	yes	characteristic
259	C₂²	#518×C₆₇	1	1	yes	characteristic
260	C₂²	#519×C₆₇	1	1	yes	characteristic
261	C₂²	C₂×#131×C₆₇	1	2	yes	—
262	C₂²	#568×C₆₇	1	2	yes	—
263	C₄	—	2	2	no	—
264	C₄	—	2	2	no	—
265	C₄	—	2	2	no	—
266	C₄	—	2	2	no	—
267	C₄	—	4	8	no	—
268	C₄	#424×C₆₇	1	4	yes	—
269	C₄	#537×C₆₇	1	2	yes	—
270	C₄	#544×C₆₇	1	2	yes	—
271	C₂	—	4	2	no	—
272	C₂	C₆₇×#2316	1	1	yes	characteristic
273	C₂	C₂×#424×C₆₇	1	1	yes	characteristic
274	C₂	C₆₇×#2905	1	1	yes	characteristic
275	C₂	C₆₇×#2981	1	2	yes	—
276	C₂	C₆₇×#2985	1	2	yes	—

	subgroup	G/H
center	C₂²×C₄×C₆₇	C₂⁴
commutator	C₂³	C₂³×C₄×C₆₇
Frattini, cosocle	C₂²×C₄	C₂⁴×C₆₇
socle	C₂³×C₆₇	C₂³×C₄
prime radical	C₂⁴×C₆₇	C₂²×C₄
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂²×C₄×C₆₇ ◁ G

Complements

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

N	H ≅ G/N	type
4. C₆₇×#1940	C₂	semidirect
5. C₆₇×#1939	C₂	semidirect
132. #21954	C₆₇	direct (N × H)
142. #1940	C₂×C₆₇	semidirect
143. #1939	C₂×C₆₇	semidirect
145. C₆₇	#21954	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂×#432×C₆₇	2
C₂×#431×C₆₇	2
C₆₇×#2285	2
C₆₇×#1940	2
C₆₇×#1939	2
C₆₇×#1798	2
#21954	67

Subgroups — C67×#21954 #30701205

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₆₇×#21954 #30701205