Subgroups — C₆₇×#21951 #30701202

Order 17152.

978 conjugacy classes of subgroups.

The subgroup lattice has 274 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#431×C₆₇	C₂	1	2	yes	maximal
2	C₂×C₄×C₄⋊C₄×C₆₇	C₂	1	1	yes	maximal, characteristic
3	C₆₇×#1939	C₂	1	8	yes	maximal, complemented
4	C₆₇×#1797	C₂	1	4	yes	maximal
5	C₂²×C₄⋊C₄×C₆₇	C₄	1	1	yes	characteristic
6	C₂²×C₄⋊C₄×C₆₇	C₄	1	1	yes	characteristic
7	C₂²×C₄²×C₆₇	C₂²	1	2	yes	—
8	C₂²×C₄²×C₆₇	C₂²	1	1	yes	characteristic
9	#431×C₆₇	—	2	8	no	—
10	C₂×C₄⋊C₈×C₆₇	C₂²	1	4	yes	—
11	C₂×C₄⋊C₈×C₆₇	C₂²	1	8	yes	—
12	C₂×C₂²⋊C₈×C₆₇	C₂²	1	4	yes	—
13	C₂×C₄×C₈×C₆₇	C₂²	1	4	yes	—
14	C₄×C₄⋊C₄×C₆₇	C₄	1	4	yes	—
15	C₄×C₄⋊C₄×C₆₇	C₂²	1	4	yes	—
16	C₂×#95×C₆₇	C₄	1	2	yes	—
17	#336×C₆₇	C₂²	1	8	yes	—
18	C₂³×C₄×C₆₇	C₂×C₄	1	2	yes	—
19	C₂³×C₄×C₆₇	C₂×C₄	1	1	yes	characteristic
20	C₂³×C₄×C₆₇	C₂×C₄	1	2	yes	—
21	C₂³×C₄×C₆₇	C₂×C₄	1	1	yes	characteristic
22	C₂³×C₄×C₆₇	C₂³	1	1	yes	characteristic
23	C₂²×C₈×C₆₇	C₂³	1	8	yes	—
24	C₂×C₄⋊C₄×C₆₇	—	2	4	no	—
25	C₂×C₄⋊C₄×C₆₇	C₈	1	8	yes	complemented
26	C₂×C₄⋊C₄×C₆₇	C₂×C₄	1	4	yes	—
27	C₂×C₄⋊C₄×C₆₇	C₂×C₄	1	4	yes	—
28	C₂×C₄²×C₆₇	—	4	4	no	—
29	C₂×C₄²×C₆₇	C₂×C₄	1	4	yes	—
30	C₂×C₄²×C₆₇	C₂×C₄	1	2	yes	—
31	C₂×C₄²×C₆₇	D₄	1	4	yes	—
32	C₂×C₄²×C₆₇	Q₈	1	4	yes	—
33	C₂×C₄²×C₆₇	C₂³	1	4	yes	—
34	C₂×C₄²×C₆₇	C₂³	1	2	yes	—
35	C₄⋊C₈×C₆₇	—	2	8	no	—
36	C₄⋊C₈×C₆₇	—	4	8	no	—
37	C₂²⋊C₈×C₆₇	—	2	8	no	—
38	C₄×C₈×C₆₇	—	2	8	no	—
39	#95×C₆₇	C₂×C₄	1	8	yes	—
40	C₂⁴×C₆₇	C₂²×C₄	1	1	yes	characteristic
41	C₂²×C₄×C₆₇	—	2	2	no	—
42	C₂²×C₄×C₆₇	—	2	2	no	—
43	C₂²×C₄×C₆₇	—	2	4	no	—
44	C₂²×C₄×C₆₇	—	4	2	no	—
45	C₂²×C₄×C₆₇	—	4	2	no	—
46	C₂²×C₄×C₆₇	—	4	1	no	—
47	C₂²×C₄×C₆₇	—	4	1	no	—
48	C₂²×C₄×C₆₇	—	4	2	no	—
49	C₂²×C₄×C₆₇	—	4	2	no	—
50	C₂²×C₄×C₆₇	C₄⋊C₄	1	4	yes	—
51	C₂²×C₄×C₆₇	C₄⋊C₄	1	4	yes	—
52	C₂²×C₄×C₆₇	C₂×C₈	1	8	yes	—
53	C₂²×C₄×C₆₇	C₂×C₈	1	4	yes	—
54	C₂²×C₄×C₆₇	M₁₆	1	4	yes	—
55	C₂²×C₄×C₆₇	C₂²×C₄	1	4	yes	—
56	C₂²×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
57	C₂²×C₄×C₆₇	C₂²×C₄	1	2	yes	—
58	C₂²×C₄×C₆₇	C₂²×C₄	1	4	yes	—
59	C₂²×C₄×C₆₇	C₂²×C₄	1	2	yes	—
60	C₂²×C₄×C₆₇	C₂×D₄	1	2	yes	—
61	C₂²×C₄×C₆₇	C₂×Q₈	1	2	yes	—
62	C₂²×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
63	C₂×C₈×C₆₇	—	2	8	no	—
64	C₂×C₈×C₆₇	—	2	8	no	—
65	C₂×C₈×C₆₇	—	2	8	no	—
66	C₄⋊C₄×C₆₇	—	2	8	no	—
67	C₄⋊C₄×C₆₇	C₂×C₈	1	16	yes	—
68	C₄²×C₆₇	—	2	8	no	—
69	C₄²×C₆₇	—	4	4	no	—
70	C₄²×C₆₇	C₄⋊C₄	1	8	yes	—
71	C₄²×C₆₇	C₂×D₄	1	4	yes	—
72	C₄²×C₆₇	C₂×Q₈	1	4	yes	—
73	C₂³×C₆₇	—	2	1	no	—
74	C₂³×C₆₇	—	2	1	no	—
75	C₂³×C₆₇	—	2	1	no	—
76	C₂³×C₆₇	—	2	1	no	—
77	C₂³×C₆₇	C₂×C₄⋊C₄	1	2	yes	—
78	C₂³×C₆₇	C₂²×C₈	1	2	yes	—
79	C₂³×C₆₇	C₂×M₁₆	1	2	yes	—
80	C₂³×C₆₇	C₂³×C₄	1	1	yes	characteristic
81	C₂×C₄×C₆₇	—	2	4	no	—
82	C₂×C₄×C₆₇	—	2	1	no	—
83	C₂×C₄×C₆₇	—	2	1	no	—
84	C₂×C₄×C₆₇	—	2	2	no	—
85	C₂×C₄×C₆₇	—	2	2	no	—
86	C₂×C₄×C₆₇	—	2	4	no	—
87	C₂×C₄×C₆₇	—	2	4	no	—
88	C₂×C₄×C₆₇	—	2	4	no	—
89	C₂×C₄×C₆₇	—	2	4	no	—
90	C₂×C₄×C₆₇	—	2	4	no	—
91	C₂×C₄×C₆₇	—	2	4	no	—
92	C₂×C₄×C₆₇	—	4	4	no	—
93	C₂×C₄×C₆₇	—	4	8	no	—
94	C₂×C₄×C₆₇	—	4	8	no	—
95	C₂×C₄×C₆₇	C₄⋊C₈	1	16	yes	complemented
96	C₂×C₄×C₆₇	C₂×C₄⋊C₄	1	2	yes	—
97	C₂×C₄×C₆₇	C₂×C₄⋊C₄	1	4	yes	—
98	C₂×C₄×C₆₇	C₂×C₄⋊C₄	1	4	yes	—
99	C₂×C₄×C₆₇	C₂²×C₈	1	8	yes	—
100	C₂×C₄×C₆₇	C₂²×C₈	1	4	yes	—
101	C₂×C₄×C₆₇	C₂×M₁₆	1	4	yes	—
102	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
103	C₂×C₄×C₆₇	C₂²×Q₈	1	1	yes	characteristic
104	C₂×C₄×C₆₇	H(5,𝔽₂)	1	2	yes	—
105	C₂×C₄×C₆₇	2₋¹⁺⁴	1	2	yes	—
106	C₈×C₆₇	—	4	8	no	—
107	C₂²×C₆₇	—	2	2	no	—
108	C₂²×C₆₇	—	2	2	no	—
109	C₂²×C₆₇	—	2	2	no	—
110	C₂²×C₆₇	—	2	2	no	—
111	C₂²×C₆₇	—	2	2	no	—
112	C₂²×C₆₇	—	2	2	no	—
113	C₂²×C₆₇	C₂×C₄⋊C₈	1	4	yes	—
114	C₂²×C₆₇	C₂²×C₄⋊C₄	1	1	yes	characteristic
115	C₂²×C₆₇	#518	1	1	yes	characteristic
116	C₂²×C₆₇	#519	1	1	yes	characteristic
117	C₂²×C₆₇	C₂³×C₈	1	1	yes	characteristic
118	C₂²×C₆₇	C₂²×M₁₆	1	1	yes	characteristic
119	C₂²×C₆₇	#568	1	1	yes	characteristic
120	C₂²×C₆₇	#568	1	1	yes	characteristic
121	C₄×C₆₇	—	2	2	no	—
122	C₄×C₆₇	—	2	4	no	—
123	C₄×C₆₇	—	2	8	no	—
124	C₄×C₆₇	—	4	4	no	—
125	C₄×C₆₇	C₂×C₄⋊C₈	1	8	yes	—
126	C₄×C₆₇	#537	1	2	yes	—
127	C₄×C₆₇	#544	1	2	yes	—
128	#21951	C₆₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
129	C₂×C₆₇	—	2	4	no	—
130	C₂×C₆₇	#2316	1	1	yes	characteristic
131	C₂×C₆₇	C₂²×C₄⋊C₈	1	1	yes	characteristic
132	C₂×C₆₇	#2905	1	1	yes	characteristic
133	C₂×C₆₇	#2972	1	1	yes	characteristic
134	C₂×C₆₇	#2975	1	1	yes	characteristic
135	C₂×C₆₇	#2982	1	1	yes	characteristic
136	C₂×C₆₇	#2983	1	1	yes	characteristic
137	C₂×#431	C₂×C₆₇	1	2	yes	—
138	C₂×C₄×C₄⋊C₄	C₂×C₆₇	1	1	yes	characteristic
139	#1939	C₂×C₆₇	1	8	yes	complemented
140	#1797	C₂×C₆₇	1	4	yes	—
141	C₆₇	#21951	1	1	yes	characteristic, direct product factor, Sylow
142	C₂²×C₄⋊C₄	C₄×C₆₇	1	1	yes	characteristic
143	C₂²×C₄⋊C₄	C₄×C₆₇	1	1	yes	characteristic
144	C₂²×C₄²	C₂²×C₆₇	1	2	yes	—
145	C₂²×C₄²	C₂²×C₆₇	1	1	yes	characteristic
146	#431	—	2	8	no	—
147	C₂×C₄⋊C₈	C₂²×C₆₇	1	4	yes	—
148	C₂×C₄⋊C₈	C₂²×C₆₇	1	8	yes	—
149	C₂×C₂²⋊C₈	C₂²×C₆₇	1	4	yes	—
150	C₂×C₄×C₈	C₂²×C₆₇	1	4	yes	—
151	C₄×C₄⋊C₄	C₄×C₆₇	1	4	yes	—
152	C₄×C₄⋊C₄	C₂²×C₆₇	1	4	yes	—
153	C₂×#95	C₄×C₆₇	1	2	yes	—
154	#336	C₂²×C₆₇	1	8	yes	—
155	C₂³×C₄	C₂×C₄×C₆₇	1	2	yes	—
156	C₂³×C₄	C₂×C₄×C₆₇	1	1	yes	characteristic
157	C₂³×C₄	C₂×C₄×C₆₇	1	2	yes	—
158	C₂³×C₄	C₂×C₄×C₆₇	1	1	yes	characteristic
159	C₂³×C₄	C₂³×C₆₇	1	1	yes	characteristic
160	C₂²×C₈	C₂³×C₆₇	1	8	yes	—
161	C₂×C₄⋊C₄	—	2	4	no	—
162	C₂×C₄⋊C₄	C₈×C₆₇	1	8	yes	complemented
163	C₂×C₄⋊C₄	C₂×C₄×C₆₇	1	4	yes	—
164	C₂×C₄⋊C₄	C₂×C₄×C₆₇	1	4	yes	—
165	C₂×C₄²	—	4	4	no	—
166	C₂×C₄²	C₂×C₄×C₆₇	1	4	yes	—
167	C₂×C₄²	C₂×C₄×C₆₇	1	2	yes	—
168	C₂×C₄²	D₄×C₆₇	1	4	yes	—
169	C₂×C₄²	Q₈×C₆₇	1	4	yes	—
170	C₂×C₄²	C₂³×C₆₇	1	4	yes	—
171	C₂×C₄²	C₂³×C₆₇	1	2	yes	—
172	C₄⋊C₈	—	2	8	no	—
173	C₄⋊C₈	—	4	8	no	—
174	C₂²⋊C₈	—	2	8	no	—
175	C₄×C₈	—	2	8	no	—
176	#95	C₂×C₄×C₆₇	1	8	yes	—
177	C₂⁴	C₂²×C₄×C₆₇	1	1	yes	characteristic
178	C₂²×C₄	—	2	2	no	—
179	C₂²×C₄	—	2	2	no	—
180	C₂²×C₄	—	2	4	no	—
181	C₂²×C₄	—	4	2	no	—
182	C₂²×C₄	—	4	2	no	—
183	C₂²×C₄	—	4	1	no	—
184	C₂²×C₄	—	4	1	no	—
185	C₂²×C₄	—	4	2	no	—
186	C₂²×C₄	—	4	2	no	—
187	C₂²×C₄	C₄⋊C₄×C₆₇	1	4	yes	—
188	C₂²×C₄	C₄⋊C₄×C₆₇	1	4	yes	—
189	C₂²×C₄	C₂×C₈×C₆₇	1	8	yes	—
190	C₂²×C₄	C₂×C₈×C₆₇	1	4	yes	—
191	C₂²×C₄	M₁₆×C₆₇	1	4	yes	—
192	C₂²×C₄	C₂²×C₄×C₆₇	1	4	yes	—
193	C₂²×C₄	C₂²×C₄×C₆₇	1	2	yes	—
194	C₂²×C₄	C₂²×C₄×C₆₇	1	1	yes	characteristic
195	C₂²×C₄	C₂²×C₄×C₆₇	1	4	yes	—
196	C₂²×C₄	C₂²×C₄×C₆₇	1	2	yes	—
197	C₂²×C₄	C₂×D₄×C₆₇	1	2	yes	—
198	C₂²×C₄	C₂×Q₈×C₆₇	1	2	yes	—
199	C₂²×C₄	C₂⁴×C₆₇	1	1	yes	characteristic
200	C₂×C₈	—	2	8	no	—
201	C₂×C₈	—	2	8	no	—
202	C₂×C₈	—	2	8	no	—
203	C₄⋊C₄	—	2	8	no	—
204	C₄⋊C₄	C₂×C₈×C₆₇	1	16	yes	—
205	C₄²	—	2	8	no	—
206	C₄²	—	4	4	no	—
207	C₄²	C₄⋊C₄×C₆₇	1	8	yes	—
208	C₄²	C₂×D₄×C₆₇	1	4	yes	—
209	C₄²	C₂×Q₈×C₆₇	1	4	yes	—
210	C₂³	—	2	1	no	—
211	C₂³	—	2	1	no	—
212	C₂³	—	2	1	no	—
213	C₂³	—	2	1	no	—
214	C₂³	C₂×C₄⋊C₄×C₆₇	1	2	yes	—
215	C₂³	C₂²×C₈×C₆₇	1	2	yes	—
216	C₂³	C₂×M₁₆×C₆₇	1	2	yes	—
217	C₂³	C₂³×C₄×C₆₇	1	1	yes	characteristic
218	C₂×C₄	—	2	4	no	—
219	C₂×C₄	—	2	1	no	—
220	C₂×C₄	—	2	1	no	—
221	C₂×C₄	—	2	2	no	—
222	C₂×C₄	—	2	2	no	—
223	C₂×C₄	—	2	4	no	—
224	C₂×C₄	—	2	4	no	—
225	C₂×C₄	—	2	4	no	—
226	C₂×C₄	—	2	4	no	—
227	C₂×C₄	—	2	4	no	—
228	C₂×C₄	—	2	4	no	—
229	C₂×C₄	—	4	4	no	—
230	C₂×C₄	—	4	8	no	—
231	C₂×C₄	—	4	8	no	—
232	C₂×C₄	C₄⋊C₈×C₆₇	1	16	yes	complemented
233	C₂×C₄	C₂×C₄⋊C₄×C₆₇	1	2	yes	—
234	C₂×C₄	C₂×C₄⋊C₄×C₆₇	1	4	yes	—
235	C₂×C₄	C₂×C₄⋊C₄×C₆₇	1	4	yes	—
236	C₂×C₄	C₂²×C₈×C₆₇	1	8	yes	—
237	C₂×C₄	C₂²×C₈×C₆₇	1	4	yes	—
238	C₂×C₄	C₂×M₁₆×C₆₇	1	4	yes	—
239	C₂×C₄	C₂²×D₄×C₆₇	1	1	yes	characteristic
240	C₂×C₄	C₂²×Q₈×C₆₇	1	1	yes	characteristic
241	C₂×C₄	H(5,𝔽₂)×C₆₇	1	2	yes	—
242	C₂×C₄	2₋¹⁺⁴×C₆₇	1	2	yes	—
243	C₈	—	4	8	no	—
244	C₂²	—	2	2	no	—
245	C₂²	—	2	2	no	—
246	C₂²	—	2	2	no	—
247	C₂²	—	2	2	no	—
248	C₂²	—	2	2	no	—
249	C₂²	—	2	2	no	—
250	C₂²	C₂×C₄⋊C₈×C₆₇	1	4	yes	—
251	C₂²	C₂²×C₄⋊C₄×C₆₇	1	1	yes	characteristic
252	C₂²	#518×C₆₇	1	1	yes	characteristic
253	C₂²	#519×C₆₇	1	1	yes	characteristic
254	C₂²	C₂³×C₈×C₆₇	1	1	yes	characteristic
255	C₂²	C₂²×M₁₆×C₆₇	1	1	yes	characteristic
256	C₂²	#568×C₆₇	1	1	yes	characteristic
257	C₂²	#568×C₆₇	1	1	yes	characteristic
258	C₄	—	2	2	no	—
259	C₄	—	2	4	no	—
260	C₄	—	2	8	no	—
261	C₄	—	4	4	no	—
262	C₄	C₂×C₄⋊C₈×C₆₇	1	8	yes	—
263	C₄	#537×C₆₇	1	2	yes	—
264	C₄	#544×C₆₇	1	2	yes	—
265	C₂	—	2	4	no	—
266	C₂	C₆₇×#2316	1	1	yes	characteristic
267	C₂	C₂²×C₄⋊C₈×C₆₇	1	1	yes	characteristic
268	C₂	C₆₇×#2905	1	1	yes	characteristic
269	C₂	C₆₇×#2972	1	1	yes	characteristic
270	C₂	C₆₇×#2975	1	1	yes	characteristic
271	C₂	C₆₇×#2982	1	1	yes	characteristic
272	C₂	C₆₇×#2983	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄×C₆₇	C₂⁴
commutator	C₂²	C₂³×C₈×C₆₇
Frattini, cosocle	C₂²×C₄	C₂⁴×C₆₇
socle	C₂³×C₆₇	C₂³×C₄
prime radical	C₂⁴×C₆₇	C₂²×C₄
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₄×C₆₇ ◁ G

Complements

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

N	H ≅ G/N	type
3. C₆₇×#1939	C₂	semidirect
25. C₂×C₄⋊C₄×C₆₇	C₈	semidirect
95. C₂×C₄×C₆₇	C₄⋊C₈	semidirect
128. #21951	C₆₇	direct (N × H)
139. #1939	C₂×C₆₇	semidirect
141. C₆₇	#21951	direct (N × H)
162. C₂×C₄⋊C₄	C₈×C₆₇	semidirect
232. C₂×C₄	C₄⋊C₈×C₆₇	semidirect

Maximal subgroups

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

subgroup	index
C₂×#431×C₆₇	2
C₂×C₄×C₄⋊C₄×C₆₇	2
C₆₇×#1939	2
C₆₇×#1797	2
#21951	67

Subgroups — C67×#21951 #30701202

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₆₇×#21951 #30701202