Subgroups — C₆₇×#21955 #30701206

Order 17152.

914 conjugacy classes of subgroups.

The subgroup lattice has 246 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	2	yes	maximal
2	C₂×C₄×C₄⋊C₄×C₆₇	C₂	1	1	yes	maximal, characteristic
3	C₆₇×#1940	C₂	1	8	yes	maximal, complemented
4	C₆₇×#1797	C₂	1	4	yes	maximal
5	C₂²×C₄⋊C₄×C₆₇	C₄	1	2	yes	—
6	C₂²×C₄²×C₆₇	C₂²	1	2	yes	—
7	C₂²×C₄²×C₆₇	C₂²	1	1	yes	characteristic
8	#432×C₆₇	—	2	8	no	—
9	C₂×C₄⋊C₈×C₆₇	C₂²	1	4	yes	—
10	C₂×C₄⋊C₈×C₆₇	C₂²	1	8	yes	—
11	C₂×C₂²⋊C₈×C₆₇	C₂²	1	4	yes	—
12	C₂×#97×C₆₇	C₂²	1	4	yes	—
13	C₄×C₄⋊C₄×C₆₇	C₄	1	4	yes	—
14	C₄×C₄⋊C₄×C₆₇	C₂²	1	4	yes	—
15	C₂×#95×C₆₇	C₄	1	1	yes	characteristic
16	C₂×#95×C₆₇	C₄	1	1	yes	characteristic
17	#336×C₆₇	C₂²	1	8	yes	—
18	C₂³×C₄×C₆₇	C₂×C₄	1	1	yes	characteristic
19	C₂³×C₄×C₆₇	C₂×C₄	1	2	yes	—
20	C₂³×C₄×C₆₇	C₂×C₄	1	1	yes	characteristic
21	C₂³×C₄×C₆₇	C₂×C₄	1	2	yes	—
22	C₂³×C₄×C₆₇	C₂³	1	1	yes	characteristic
23	C₂²×C₈×C₆₇	C₂³	1	8	yes	—
24	C₂×C₄⋊C₄×C₆₇	—	2	8	no	—
25	C₂×C₄⋊C₄×C₆₇	C₂×C₄	1	8	yes	—
26	C₂×C₄²×C₆₇	—	4	4	no	—
27	C₂×C₄²×C₆₇	C₂×C₄	1	4	yes	—
28	C₂×C₄²×C₆₇	C₂×C₄	1	2	yes	—
29	C₂×C₄²×C₆₇	D₄	1	4	yes	—
30	C₂×C₄²×C₆₇	Q₈	1	4	yes	—
31	C₂×C₄²×C₆₇	C₂³	1	4	yes	—
32	C₂×C₄²×C₆₇	C₂³	1	2	yes	—
33	C₄⋊C₈×C₆₇	—	2	8	no	—
34	C₄⋊C₈×C₆₇	—	4	8	no	—
35	C₂²⋊C₈×C₆₇	—	2	8	no	—
36	#97×C₆₇	—	2	8	no	—
37	#95×C₆₇	C₂×C₄	1	4	yes	—
38	#95×C₆₇	C₂×C₄	1	4	yes	—
39	C₂⁴×C₆₇	C₂²×C₄	1	1	yes	characteristic
40	C₂²×C₄×C₆₇	—	2	4	no	—
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	2	no	—
47	C₂²×C₄×C₆₇	—	4	2	no	—
48	C₂²×C₄×C₆₇	—	4	2	no	—
49	C₂²×C₄×C₆₇	C₄⋊C₄	1	4	yes	—
50	C₂²×C₄×C₆₇	C₄⋊C₄	1	4	yes	—
51	C₂²×C₄×C₆₇	M₁₆	1	8	yes	—
52	C₂²×C₄×C₆₇	C₂²×C₄	1	4	yes	—
53	C₂²×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
54	C₂²×C₄×C₆₇	C₂²×C₄	1	2	yes	—
55	C₂²×C₄×C₆₇	C₂²×C₄	1	2	yes	—
56	C₂²×C₄×C₆₇	C₂²×C₄	1	4	yes	—
57	C₂²×C₄×C₆₇	C₂×D₄	1	2	yes	—
58	C₂²×C₄×C₆₇	C₂×Q₈	1	2	yes	—
59	C₂²×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
60	C₂×C₈×C₆₇	—	2	8	no	—
61	C₂×C₈×C₆₇	—	2	8	no	—
62	C₂×C₈×C₆₇	—	2	8	no	—
63	C₄⋊C₄×C₆₇	—	2	16	no	—
64	C₄²×C₆₇	—	2	8	no	—
65	C₄²×C₆₇	—	4	4	no	—
66	C₄²×C₆₇	C₄⋊C₄	1	8	yes	—
67	C₄²×C₆₇	C₂×D₄	1	4	yes	—
68	C₄²×C₆₇	C₂×Q₈	1	4	yes	—
69	C₂³×C₆₇	—	2	1	no	—
70	C₂³×C₆₇	—	2	1	no	—
71	C₂³×C₆₇	—	2	2	no	—
72	C₂³×C₆₇	C₂×C₄⋊C₄	1	2	yes	—
73	C₂³×C₆₇	C₂×M₁₆	1	4	yes	—
74	C₂³×C₆₇	C₂³×C₄	1	1	yes	characteristic
75	C₂×C₄×C₆₇	—	2	4	no	—
76	C₂×C₄×C₆₇	—	2	1	no	—
77	C₂×C₄×C₆₇	—	2	1	no	—
78	C₂×C₄×C₆₇	—	2	4	no	—
79	C₂×C₄×C₆₇	—	2	4	no	—
80	C₂×C₄×C₆₇	—	2	4	no	—
81	C₂×C₄×C₆₇	—	2	8	no	—
82	C₂×C₄×C₆₇	—	2	4	no	—
83	C₂×C₄×C₆₇	—	2	4	no	—
84	C₂×C₄×C₆₇	—	2	4	no	—
85	C₂×C₄×C₆₇	—	2	4	no	—
86	C₂×C₄×C₆₇	—	2	4	no	—
87	C₂×C₄×C₆₇	—	4	4	no	—
88	C₂×C₄×C₆₇	—	4	8	no	—
89	C₂×C₄×C₆₇	—	4	8	no	—
90	C₂×C₄×C₆₇	C₂×C₄⋊C₄	1	2	yes	—
91	C₂×C₄×C₆₇	C₂×C₄⋊C₄	1	4	yes	—
92	C₂×C₄×C₆₇	C₂×C₄⋊C₄	1	4	yes	—
93	C₂×C₄×C₆₇	C₂×M₁₆	1	8	yes	—
94	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
95	C₂×C₄×C₆₇	C₂²×Q₈	1	1	yes	characteristic
96	C₂×C₄×C₆₇	H(5,𝔽₂)	1	2	yes	—
97	C₂×C₄×C₆₇	2₋¹⁺⁴	1	2	yes	—
98	C₈×C₆₇	—	4	8	no	—
99	C₂²×C₆₇	—	2	2	no	—
100	C₂²×C₆₇	—	2	2	no	—
101	C₂²×C₆₇	—	2	4	no	—
102	C₂²×C₆₇	—	2	4	no	—
103	C₂²×C₆₇	#423	1	4	yes	—
104	C₂²×C₆₇	C₂²×C₄⋊C₄	1	1	yes	characteristic
105	C₂²×C₆₇	#518	1	1	yes	characteristic
106	C₂²×C₆₇	#519	1	1	yes	characteristic
107	C₂²×C₆₇	C₂²×M₁₆	1	2	yes	—
108	C₂²×C₆₇	#568	1	2	yes	—
109	C₄×C₆₇	—	2	2	no	—
110	C₄×C₆₇	—	2	4	no	—
111	C₄×C₆₇	—	2	4	no	—
112	C₄×C₆₇	—	2	8	no	—
113	C₄×C₆₇	—	4	4	no	—
114	C₄×C₆₇	#537	1	2	yes	—
115	C₄×C₆₇	#544	1	2	yes	—
116	#21955	C₆₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
117	C₂×C₆₇	—	2	4	no	—
118	C₂×C₆₇	#2316	1	1	yes	characteristic
119	C₂×C₆₇	C₂×#423	1	1	yes	characteristic
120	C₂×C₆₇	#2905	1	1	yes	characteristic
121	C₂×C₆₇	#2973	1	2	yes	—
122	C₂×C₆₇	#2983	1	2	yes	—
123	C₂×#432	C₂×C₆₇	1	2	yes	—
124	C₂×C₄×C₄⋊C₄	C₂×C₆₇	1	1	yes	characteristic
125	#1940	C₂×C₆₇	1	8	yes	complemented
126	#1797	C₂×C₆₇	1	4	yes	—
127	C₆₇	#21955	1	1	yes	characteristic, direct product factor, Sylow
128	C₂²×C₄⋊C₄	C₄×C₆₇	1	2	yes	—
129	C₂²×C₄²	C₂²×C₆₇	1	2	yes	—
130	C₂²×C₄²	C₂²×C₆₇	1	1	yes	characteristic
131	#432	—	2	8	no	—
132	C₂×C₄⋊C₈	C₂²×C₆₇	1	4	yes	—
133	C₂×C₄⋊C₈	C₂²×C₆₇	1	8	yes	—
134	C₂×C₂²⋊C₈	C₂²×C₆₇	1	4	yes	—
135	C₂×#97	C₂²×C₆₇	1	4	yes	—
136	C₄×C₄⋊C₄	C₄×C₆₇	1	4	yes	—
137	C₄×C₄⋊C₄	C₂²×C₆₇	1	4	yes	—
138	C₂×#95	C₄×C₆₇	1	1	yes	characteristic
139	C₂×#95	C₄×C₆₇	1	1	yes	characteristic
140	#336	C₂²×C₆₇	1	8	yes	—
141	C₂³×C₄	C₂×C₄×C₆₇	1	1	yes	characteristic
142	C₂³×C₄	C₂×C₄×C₆₇	1	2	yes	—
143	C₂³×C₄	C₂×C₄×C₆₇	1	1	yes	characteristic
144	C₂³×C₄	C₂×C₄×C₆₇	1	2	yes	—
145	C₂³×C₄	C₂³×C₆₇	1	1	yes	characteristic
146	C₂²×C₈	C₂³×C₆₇	1	8	yes	—
147	C₂×C₄⋊C₄	—	2	8	no	—
148	C₂×C₄⋊C₄	C₂×C₄×C₆₇	1	8	yes	—
149	C₂×C₄²	—	4	4	no	—
150	C₂×C₄²	C₂×C₄×C₆₇	1	4	yes	—
151	C₂×C₄²	C₂×C₄×C₆₇	1	2	yes	—
152	C₂×C₄²	D₄×C₆₇	1	4	yes	—
153	C₂×C₄²	Q₈×C₆₇	1	4	yes	—
154	C₂×C₄²	C₂³×C₆₇	1	4	yes	—
155	C₂×C₄²	C₂³×C₆₇	1	2	yes	—
156	C₄⋊C₈	—	2	8	no	—
157	C₄⋊C₈	—	4	8	no	—
158	C₂²⋊C₈	—	2	8	no	—
159	#97	—	2	8	no	—
160	#95	C₂×C₄×C₆₇	1	4	yes	—
161	#95	C₂×C₄×C₆₇	1	4	yes	—
162	C₂⁴	C₂²×C₄×C₆₇	1	1	yes	characteristic
163	C₂²×C₄	—	2	4	no	—
164	C₂²×C₄	—	2	2	no	—
165	C₂²×C₄	—	2	2	no	—
166	C₂²×C₄	—	2	4	no	—
167	C₂²×C₄	—	4	2	no	—
168	C₂²×C₄	—	4	2	no	—
169	C₂²×C₄	—	4	2	no	—
170	C₂²×C₄	—	4	2	no	—
171	C₂²×C₄	—	4	2	no	—
172	C₂²×C₄	C₄⋊C₄×C₆₇	1	4	yes	—
173	C₂²×C₄	C₄⋊C₄×C₆₇	1	4	yes	—
174	C₂²×C₄	M₁₆×C₆₇	1	8	yes	—
175	C₂²×C₄	C₂²×C₄×C₆₇	1	4	yes	—
176	C₂²×C₄	C₂²×C₄×C₆₇	1	2	yes	—
177	C₂²×C₄	C₂²×C₄×C₆₇	1	1	yes	characteristic
178	C₂²×C₄	C₂²×C₄×C₆₇	1	2	yes	—
179	C₂²×C₄	C₂²×C₄×C₆₇	1	4	yes	—
180	C₂²×C₄	C₂×D₄×C₆₇	1	2	yes	—
181	C₂²×C₄	C₂×Q₈×C₆₇	1	2	yes	—
182	C₂²×C₄	C₂⁴×C₆₇	1	1	yes	characteristic
183	C₂×C₈	—	2	8	no	—
184	C₂×C₈	—	2	8	no	—
185	C₂×C₈	—	2	8	no	—
186	C₄⋊C₄	—	2	16	no	—
187	C₄²	—	2	8	no	—
188	C₄²	—	4	4	no	—
189	C₄²	C₄⋊C₄×C₆₇	1	8	yes	—
190	C₄²	C₂×D₄×C₆₇	1	4	yes	—
191	C₄²	C₂×Q₈×C₆₇	1	4	yes	—
192	C₂³	—	2	1	no	—
193	C₂³	—	2	1	no	—
194	C₂³	—	2	2	no	—
195	C₂³	C₂×C₄⋊C₄×C₆₇	1	2	yes	—
196	C₂³	C₂×M₁₆×C₆₇	1	4	yes	—
197	C₂³	C₂³×C₄×C₆₇	1	1	yes	characteristic
198	C₂×C₄	—	2	4	no	—
199	C₂×C₄	—	2	1	no	—
200	C₂×C₄	—	2	1	no	—
201	C₂×C₄	—	2	4	no	—
202	C₂×C₄	—	2	4	no	—
203	C₂×C₄	—	2	4	no	—
204	C₂×C₄	—	2	8	no	—
205	C₂×C₄	—	2	4	no	—
206	C₂×C₄	—	2	4	no	—
207	C₂×C₄	—	2	4	no	—
208	C₂×C₄	—	2	4	no	—
209	C₂×C₄	—	2	4	no	—
210	C₂×C₄	—	4	4	no	—
211	C₂×C₄	—	4	8	no	—
212	C₂×C₄	—	4	8	no	—
213	C₂×C₄	C₂×C₄⋊C₄×C₆₇	1	2	yes	—
214	C₂×C₄	C₂×C₄⋊C₄×C₆₇	1	4	yes	—
215	C₂×C₄	C₂×C₄⋊C₄×C₆₇	1	4	yes	—
216	C₂×C₄	C₂×M₁₆×C₆₇	1	8	yes	—
217	C₂×C₄	C₂²×D₄×C₆₇	1	1	yes	characteristic
218	C₂×C₄	C₂²×Q₈×C₆₇	1	1	yes	characteristic
219	C₂×C₄	H(5,𝔽₂)×C₆₇	1	2	yes	—
220	C₂×C₄	2₋¹⁺⁴×C₆₇	1	2	yes	—
221	C₈	—	4	8	no	—
222	C₂²	—	2	2	no	—
223	C₂²	—	2	2	no	—
224	C₂²	—	2	4	no	—
225	C₂²	—	2	4	no	—
226	C₂²	#423×C₆₇	1	4	yes	—
227	C₂²	C₂²×C₄⋊C₄×C₆₇	1	1	yes	characteristic
228	C₂²	#518×C₆₇	1	1	yes	characteristic
229	C₂²	#519×C₆₇	1	1	yes	characteristic
230	C₂²	C₂²×M₁₆×C₆₇	1	2	yes	—
231	C₂²	#568×C₆₇	1	2	yes	—
232	C₄	—	2	2	no	—
233	C₄	—	2	4	no	—
234	C₄	—	2	4	no	—
235	C₄	—	2	8	no	—
236	C₄	—	4	4	no	—
237	C₄	#537×C₆₇	1	2	yes	—
238	C₄	#544×C₆₇	1	2	yes	—
239	C₂	—	2	4	no	—
240	C₂	C₆₇×#2316	1	1	yes	characteristic
241	C₂	C₂×#423×C₆₇	1	1	yes	characteristic
242	C₂	C₆₇×#2905	1	1	yes	characteristic
243	C₂	C₆₇×#2973	1	2	yes	—
244	C₂	C₆₇×#2983	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
3. C₆₇×#1940	C₂	semidirect
116. #21955	C₆₇	direct (N × H)
125. #1940	C₂×C₆₇	semidirect
127. C₆₇	#21955	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×C₄×C₄⋊C₄×C₆₇	2
C₆₇×#1940	2
C₆₇×#1797	2
#21955	67

Subgroups — C67×#21955 #30701206

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₆₇×#21955 #30701206