Subgroups — C₆₇×#22303 #30701554

Order 17152.

634 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	C₄×#125×C₆₇	C₂	1	1	yes	maximal, characteristic
2	C₆₇×#1940	C₂	1	4	yes	maximal
3	C₆₇×#1939	C₂	1	2	yes	maximal
4	C₆₇×#1917	C₂	1	4	yes	maximal
5	C₆₇×#1842	C₂	1	1	yes	maximal, characteristic
6	C₆₇×#1770	C₂	1	2	yes	maximal
7	C₆₇×#1744	C₂	1	1	yes	maximal, characteristic
8	C₂×#125×C₆₇	C₄	1	1	yes	characteristic
9	C₂×C₄⋊C₈×C₆₇	C₂²	1	2	yes	—
10	C₂×C₄⋊C₈×C₆₇	C₂²	1	4	yes	—
11	C₂×C₄⋊C₈×C₆₇	C₂²	1	4	yes	—
12	C₂×#97×C₆₇	C₂²	1	4	yes	—
13	C₂×C₄×C₈×C₆₇	C₂²	1	1	yes	characteristic
14	C₂×C₄×C₈×C₆₇	C₂²	1	1	yes	characteristic
15	#389×C₆₇	C₄	1	2	yes	—
16	#387×C₆₇	C₄	1	4	yes	—
17	#382×C₆₇	C₄	1	1	yes	characteristic
18	C₄×C₄⋊C₄×C₆₇	C₂²	1	2	yes	—
19	C₄×C₄⋊C₄×C₆₇	C₂²	1	4	yes	—
20	C₄³×C₆₇	C₂²	1	1	yes	characteristic
21	#336×C₆₇	C₂²	1	2	yes	—
22	#336×C₆₇	C₂²	1	2	yes	—
23	#336×C₆₇	C₂²	1	4	yes	—
24	#336×C₆₇	C₂²	1	4	yes	—
25	C₂²×C₈×C₆₇	C₂³	1	1	yes	characteristic
26	C₂²×C₈×C₆₇	C₂³	1	1	yes	characteristic
27	C₂²×C₈×C₆₇	C₂³	1	4	yes	—
28	C₂²×C₈×C₆₇	C₂³	1	2	yes	—
29	#125×C₆₇	—	2	4	no	—
30	C₂×C₄⋊C₄×C₆₇	C₂×C₄	1	4	yes	—
31	C₂×C₄⋊C₄×C₆₇	C₂×C₄	1	2	yes	—
32	C₂×C₄⋊C₄×C₆₇	C₂×C₄	1	2	yes	—
33	C₂×C₄⋊C₄×C₆₇	C₂×C₄	1	4	yes	—
34	C₂×C₄²×C₆₇	C₂×C₄	1	1	yes	characteristic
35	C₂×C₄²×C₆₇	C₂×C₄	1	2	yes	—
36	C₂×C₄²×C₆₇	C₂×C₄	1	1	yes	characteristic
37	C₂×C₄²×C₆₇	C₂³	1	2	yes	—
38	C₂×C₄²×C₆₇	C₂³	1	4	yes	—
39	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
40	C₄⋊C₈×C₆₇	—	2	4	no	—
41	C₄⋊C₈×C₆₇	—	4	4	no	—
42	C₄⋊C₈×C₆₇	—	4	4	no	—
43	#97×C₆₇	—	4	4	no	—
44	C₄×C₈×C₆₇	—	2	2	no	—
45	C₄×C₈×C₆₇	Q₈	1	4	yes	—
46	#95×C₆₇	C₂×C₄	1	4	yes	—
47	#95×C₆₇	C₂×C₄	1	4	yes	—
48	#95×C₆₇	C₂×C₄	1	4	yes	—
49	C₂²×C₄×C₆₇	C₂²×C₄	1	2	yes	—
50	C₂²×C₄×C₆₇	C₂²×C₄	1	4	yes	—
51	C₂²×C₄×C₆₇	C₂²×C₄	1	4	yes	—
52	C₂²×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
53	C₂²×C₄×C₆₇	C₂²×C₄	1	2	yes	—
54	C₂²×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
55	C₂²×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
56	C₂×C₈×C₆₇	—	2	2	no	—
57	C₂×C₈×C₆₇	—	2	2	no	—
58	C₂×C₈×C₆₇	—	2	4	no	—
59	C₂×C₈×C₆₇	—	2	1	no	—
60	C₂×C₈×C₆₇	—	2	1	no	—
61	C₂×C₈×C₆₇	—	2	2	no	—
62	C₂×C₈×C₆₇	—	2	4	no	—
63	C₂×C₈×C₆₇	—	2	4	no	—
64	C₂×C₈×C₆₇	—	2	2	no	—
65	C₂×C₈×C₆₇	C₂×Q₈	1	4	yes	—
66	C₄⋊C₄×C₆₇	—	2	8	no	—
67	C₄⋊C₄×C₆₇	—	2	4	no	—
68	C₄⋊C₄×C₆₇	—	2	4	no	—
69	C₄⋊C₄×C₆₇	—	4	4	no	—
70	C₄²×C₆₇	—	2	2	no	—
71	C₄²×C₆₇	—	2	2	no	—
72	C₄²×C₆₇	—	2	4	no	—
73	C₄²×C₆₇	—	2	2	no	—
74	C₄²×C₆₇	—	2	4	no	—
75	C₄²×C₆₇	—	4	2	no	—
76	C₄²×C₆₇	M₁₆	1	4	yes	—
77	C₄²×C₆₇	C₂×Q₈	1	2	yes	—
78	C₄²×C₆₇	#41	1	2	yes	—
79	C₂³×C₆₇	C₂³×C₄	1	1	yes	characteristic
80	C₂×C₄×C₆₇	—	2	2	no	—
81	C₂×C₄×C₆₇	—	2	2	no	—
82	C₂×C₄×C₆₇	—	2	2	no	—
83	C₂×C₄×C₆₇	—	2	4	no	—
84	C₂×C₄×C₆₇	—	2	2	no	—
85	C₂×C₄×C₆₇	—	2	4	no	—
86	C₂×C₄×C₆₇	—	2	4	no	—
87	C₂×C₄×C₆₇	—	2	2	no	—
88	C₂×C₄×C₆₇	—	2	4	no	—
89	C₂×C₄×C₆₇	—	2	2	no	—
90	C₂×C₄×C₆₇	—	2	4	no	—
91	C₂×C₄×C₆₇	—	2	4	no	—
92	C₂×C₄×C₆₇	C₄×Q₈	1	2	yes	—
93	C₂×C₄×C₆₇	C₄×Q₈	1	2	yes	—
94	C₂×C₄×C₆₇	C₂×M₁₆	1	4	yes	—
95	C₂×C₄×C₆₇	C₂×M₁₆	1	2	yes	—
96	C₂×C₄×C₆₇	#131	1	2	yes	—
97	C₂×C₄×C₆₇	C₂²×Q₈	1	1	yes	characteristic
98	C₂×C₄×C₆₇	C₂×#41	1	1	yes	characteristic
99	C₂×C₄×C₆₇	H(5,𝔽₂)	1	2	yes	—
100	C₂×C₄×C₆₇	2₋¹⁺⁴	1	2	yes	—
101	C₈×C₆₇	—	2	4	no	—
102	C₈×C₆₇	—	4	4	no	—
103	C₈×C₆₇	—	4	1	no	—
104	C₈×C₆₇	—	4	1	no	—
105	C₂²×C₆₇	C₂×C₄×Q₈	1	1	yes	characteristic
106	C₂²×C₆₇	#520	1	2	yes	—
107	C₂²×C₆₇	C₂²×M₁₆	1	1	yes	characteristic
108	C₂²×C₆₇	C₂×#131	1	1	yes	characteristic
109	C₂²×C₆₇	#568	1	2	yes	—
110	C₄×C₆₇	—	2	2	no	—
111	C₄×C₆₇	—	2	4	no	—
112	C₄×C₆₇	—	4	4	no	—
113	C₄×C₆₇	—	4	2	no	—
114	C₄×C₆₇	—	4	4	no	—
115	C₄×C₆₇	#446	1	4	yes	—
116	C₄×C₆₇	#540	1	1	yes	characteristic
117	C₄×C₆₇	#541	1	1	yes	characteristic
118	C₄×C₆₇	#544	1	2	yes	—
119	#22303	C₆₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
120	C₂×C₆₇	#2330	1	1	yes	characteristic
121	C₂×C₆₇	C₂×#446	1	1	yes	characteristic
122	C₂×C₆₇	#2961	1	1	yes	characteristic
123	C₂×C₆₇	#2989	1	2	yes	—
124	C₂×C₆₇	#2993	1	2	yes	—
125	C₄×#125	C₂×C₆₇	1	1	yes	characteristic
126	#1940	C₂×C₆₇	1	4	yes	—
127	#1939	C₂×C₆₇	1	2	yes	—
128	#1917	C₂×C₆₇	1	4	yes	—
129	#1842	C₂×C₆₇	1	1	yes	characteristic
130	#1770	C₂×C₆₇	1	2	yes	—
131	#1744	C₂×C₆₇	1	1	yes	characteristic
132	C₆₇	#22303	1	1	yes	characteristic, direct product factor, Sylow
133	C₂×#125	C₄×C₆₇	1	1	yes	characteristic
134	C₂×C₄⋊C₈	C₂²×C₆₇	1	2	yes	—
135	C₂×C₄⋊C₈	C₂²×C₆₇	1	4	yes	—
136	C₂×C₄⋊C₈	C₂²×C₆₇	1	4	yes	—
137	C₂×#97	C₂²×C₆₇	1	4	yes	—
138	C₂×C₄×C₈	C₂²×C₆₇	1	1	yes	characteristic
139	C₂×C₄×C₈	C₂²×C₆₇	1	1	yes	characteristic
140	#389	C₄×C₆₇	1	2	yes	—
141	#387	C₄×C₆₇	1	4	yes	—
142	#382	C₄×C₆₇	1	1	yes	characteristic
143	C₄×C₄⋊C₄	C₂²×C₆₇	1	4	yes	—
144	C₄×C₄⋊C₄	C₂²×C₆₇	1	2	yes	—
145	C₄³	C₂²×C₆₇	1	1	yes	characteristic
146	#336	C₂²×C₆₇	1	4	yes	—
147	#336	C₂²×C₆₇	1	2	yes	—
148	#336	C₂²×C₆₇	1	4	yes	—
149	#336	C₂²×C₆₇	1	2	yes	—
150	C₂²×C₈	C₂³×C₆₇	1	1	yes	characteristic
151	C₂²×C₈	C₂³×C₆₇	1	1	yes	characteristic
152	C₂²×C₈	C₂³×C₆₇	1	4	yes	—
153	C₂²×C₈	C₂³×C₆₇	1	2	yes	—
154	#125	—	2	4	no	—
155	C₂×C₄⋊C₄	C₂×C₄×C₆₇	1	4	yes	—
156	C₂×C₄⋊C₄	C₂×C₄×C₆₇	1	2	yes	—
157	C₂×C₄⋊C₄	C₂×C₄×C₆₇	1	2	yes	—
158	C₂×C₄⋊C₄	C₂×C₄×C₆₇	1	4	yes	—
159	C₂×C₄²	C₂×C₄×C₆₇	1	1	yes	characteristic
160	C₂×C₄²	C₂×C₄×C₆₇	1	2	yes	—
161	C₂×C₄²	C₂×C₄×C₆₇	1	1	yes	characteristic
162	C₂×C₄²	C₂³×C₆₇	1	2	yes	—
163	C₂×C₄²	C₂³×C₆₇	1	4	yes	—
164	C₂×C₄²	C₂³×C₆₇	1	1	yes	characteristic
165	C₄⋊C₈	—	2	4	no	—
166	C₄⋊C₈	—	4	4	no	—
167	C₄⋊C₈	—	4	4	no	—
168	#97	—	4	4	no	—
169	C₄×C₈	—	2	2	no	—
170	C₄×C₈	Q₈×C₆₇	1	4	yes	—
171	#95	C₂×C₄×C₆₇	1	4	yes	—
172	#95	C₂×C₄×C₆₇	1	4	yes	—
173	#95	C₂×C₄×C₆₇	1	4	yes	—
174	C₂²×C₄	C₂²×C₄×C₆₇	1	4	yes	—
175	C₂²×C₄	C₂²×C₄×C₆₇	1	2	yes	—
176	C₂²×C₄	C₂²×C₄×C₆₇	1	4	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	1	yes	characteristic
180	C₂²×C₄	C₂⁴×C₆₇	1	1	yes	characteristic
181	C₂×C₈	—	2	2	no	—
182	C₂×C₈	—	2	2	no	—
183	C₂×C₈	—	2	1	no	—
184	C₂×C₈	—	2	1	no	—
185	C₂×C₈	—	2	4	no	—
186	C₂×C₈	—	2	4	no	—
187	C₂×C₈	—	2	2	no	—
188	C₂×C₈	—	2	4	no	—
189	C₂×C₈	—	2	2	no	—
190	C₂×C₈	C₂×Q₈×C₆₇	1	4	yes	—
191	C₄⋊C₄	—	2	8	no	—
192	C₄⋊C₄	—	2	4	no	—
193	C₄⋊C₄	—	2	4	no	—
194	C₄⋊C₄	—	4	4	no	—
195	C₄²	—	2	2	no	—
196	C₄²	—	2	4	no	—
197	C₄²	—	2	2	no	—
198	C₄²	—	2	2	no	—
199	C₄²	—	2	4	no	—
200	C₄²	—	4	2	no	—
201	C₄²	M₁₆×C₆₇	1	4	yes	—
202	C₄²	C₂×Q₈×C₆₇	1	2	yes	—
203	C₄²	#41×C₆₇	1	2	yes	—
204	C₂³	C₂³×C₄×C₆₇	1	1	yes	characteristic
205	C₂×C₄	—	2	2	no	—
206	C₂×C₄	—	2	2	no	—
207	C₂×C₄	—	2	2	no	—
208	C₂×C₄	—	2	4	no	—
209	C₂×C₄	—	2	4	no	—
210	C₂×C₄	—	2	2	no	—
211	C₂×C₄	—	2	4	no	—
212	C₂×C₄	—	2	2	no	—
213	C₂×C₄	—	2	4	no	—
214	C₂×C₄	—	2	2	no	—
215	C₂×C₄	—	2	4	no	—
216	C₂×C₄	—	2	4	no	—
217	C₂×C₄	C₄×Q₈×C₆₇	1	2	yes	—
218	C₂×C₄	C₄×Q₈×C₆₇	1	2	yes	—
219	C₂×C₄	C₂×M₁₆×C₆₇	1	4	yes	—
220	C₂×C₄	C₂×M₁₆×C₆₇	1	2	yes	—
221	C₂×C₄	#131×C₆₇	1	2	yes	—
222	C₂×C₄	C₂²×Q₈×C₆₇	1	1	yes	characteristic
223	C₂×C₄	C₂×#41×C₆₇	1	1	yes	characteristic
224	C₂×C₄	H(5,𝔽₂)×C₆₇	1	2	yes	—
225	C₂×C₄	2₋¹⁺⁴×C₆₇	1	2	yes	—
226	C₈	—	2	4	no	—
227	C₈	—	4	1	no	—
228	C₈	—	4	4	no	—
229	C₈	—	4	1	no	—
230	C₂²	C₂×C₄×Q₈×C₆₇	1	1	yes	characteristic
231	C₂²	#520×C₆₇	1	2	yes	—
232	C₂²	C₂²×M₁₆×C₆₇	1	1	yes	characteristic
233	C₂²	C₂×#131×C₆₇	1	1	yes	characteristic
234	C₂²	#568×C₆₇	1	2	yes	—
235	C₄	—	2	2	no	—
236	C₄	—	2	4	no	—
237	C₄	—	4	4	no	—
238	C₄	—	4	2	no	—
239	C₄	—	4	4	no	—
240	C₄	#446×C₆₇	1	4	yes	—
241	C₄	#540×C₆₇	1	1	yes	characteristic
242	C₄	#541×C₆₇	1	1	yes	characteristic
243	C₄	#544×C₆₇	1	2	yes	—
244	C₂	C₆₇×#2330	1	1	yes	characteristic
245	C₂	C₂×#446×C₆₇	1	1	yes	characteristic
246	C₂	C₆₇×#2961	1	1	yes	characteristic
247	C₂	C₆₇×#2989	1	2	yes	—
248	C₂	C₆₇×#2993	1	2	yes	—

	subgroup	G/H
center	C₂²×C₄×C₆₇	C₂⁴
commutator	C₂³	C₂³×C₄×C₆₇
Frattini, cosocle	C₂²×C₄	C₂⁴×C₆₇
socle, prime radical	C₂³×C₆₇	C₂³×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
119. #22303	C₆₇	direct (N × H)
132. C₆₇	#22303	direct (N × H)

Maximal subgroups

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

subgroup	index
C₄×#125×C₆₇	2
C₆₇×#1940	2
C₆₇×#1939	2
C₆₇×#1917	2
C₆₇×#1842	2
C₆₇×#1770	2
C₆₇×#1744	2
#22303	67

Subgroups — C67×#22303 #30701554

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₆₇×#22303 #30701554