Subgroups — C₆₇×#22025 #30701276

Order 17152.

586 conjugacy classes of subgroups.

The subgroup lattice has 260 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₆₇×#2295	C₂	1	1	yes	maximal, characteristic
2	C₆₇×#1985	C₂	1	2	yes	maximal
3	C₆₇×#1940	C₂	1	2	yes	maximal
4	C₆₇×#1917	C₂	1	4	yes	maximal
5	C₆₇×#1840	C₂	1	2	yes	maximal
6	C₆₇×#1832	C₂	1	2	yes	maximal
7	C₆₇×#1767	C₂	1	1	yes	maximal, characteristic
8	C₆₇×#1745	C₂	1	1	yes	maximal, characteristic
9	C₂×C₄×Q₈×C₆₇	C₂²	1	1	yes	characteristic
10	C₂×C₄⋊C₈×C₆₇	C₂²	1	2	yes	—
11	C₂×C₄⋊C₈×C₆₇	C₂²	1	4	yes	—
12	C₂×#97×C₆₇	C₂²	1	2	yes	—
13	C₂×#97×C₆₇	C₂²	1	2	yes	—
14	#391×C₆₇	C₄	1	2	yes	—
15	#389×C₆₇	C₄	1	2	yes	—
16	#387×C₆₇	C₄	1	4	yes	—
17	C₄×C₄⋊C₄×C₆₇	C₂²	1	2	yes	—
18	C₄×C₄⋊C₄×C₆₇	C₂²	1	1	yes	characteristic
19	#376×C₆₇	C₂²	1	2	yes	—
20	C₄³×C₆₇	C₂²	1	1	yes	characteristic
21	#336×C₆₇	C₂²	1	4	yes	—
22	#336×C₆₇	C₂²	1	4	yes	—
23	#336×C₆₇	C₂²	1	1	yes	characteristic
24	#336×C₆₇	C₂²	1	1	yes	characteristic
25	#336×C₆₇	C₂²	1	2	yes	—
26	#336×C₆₇	C₂²	1	2	yes	—
27	#336×C₆₇	C₂²	1	4	yes	—
28	C₂²×Q₈×C₆₇	C₂×C₄	1	1	yes	characteristic
29	C₂²×C₈×C₆₇	C₂³	1	2	yes	—
30	C₂²×C₈×C₆₇	C₂³	1	4	yes	—
31	C₂²×C₈×C₆₇	C₂³	1	2	yes	—
32	C₄×Q₈×C₆₇	—	2	4	no	—
33	C₂×C₄⋊C₄×C₆₇	C₂×C₄	1	2	yes	—
34	C₂×C₄⋊C₄×C₆₇	C₂×C₄	1	1	yes	characteristic
35	C₂×C₄⋊C₄×C₆₇	C₂×C₄	1	4	yes	—
36	C₂×C₄⋊C₄×C₆₇	C₂×C₄	1	2	yes	—
37	C₂×C₄²×C₆₇	C₂×C₄	1	2	yes	—
38	C₂×C₄²×C₆₇	C₂×C₄	1	2	yes	—
39	C₂×C₄²×C₆₇	C₂³	1	2	yes	—
40	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
41	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
42	C₂×C₄²×C₆₇	C₂³	1	2	yes	—
43	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
44	C₄⋊C₈×C₆₇	—	2	4	no	—
45	C₄⋊C₈×C₆₇	—	4	4	no	—
46	#97×C₆₇	—	2	4	no	—
47	#97×C₆₇	—	4	2	no	—
48	#95×C₆₇	C₂×C₄	1	4	yes	—
49	#95×C₆₇	C₂×C₄	1	4	yes	—
50	#95×C₆₇	C₂×C₄	1	2	yes	—
51	#95×C₆₇	C₂×C₄	1	4	yes	—
52	C₂×Q₈×C₆₇	—	2	2	no	—
53	C₂×Q₈×C₆₇	—	4	2	no	—
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	2	yes	—
57	C₂²×C₄×C₆₇	C₂²×C₄	1	4	yes	—
58	C₂²×C₄×C₆₇	C₂²×C₄	1	2	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	C₂⁴	1	1	yes	characteristic
62	C₂×C₈×C₆₇	—	2	4	no	—
63	C₂×C₈×C₆₇	—	2	2	no	—
64	C₂×C₈×C₆₇	—	2	2	no	—
65	C₂×C₈×C₆₇	—	2	2	no	—
66	C₂×C₈×C₆₇	—	2	4	no	—
67	C₂×C₈×C₆₇	—	2	2	no	—
68	C₂×C₈×C₆₇	—	2	2	no	—
69	C₂×C₈×C₆₇	—	2	4	no	—
70	C₂×C₈×C₆₇	—	2	2	no	—
71	C₄⋊C₄×C₆₇	—	2	2	no	—
72	C₄⋊C₄×C₆₇	—	2	4	no	—
73	C₄⋊C₄×C₆₇	—	4	2	no	—
74	C₄⋊C₄×C₆₇	—	4	4	no	—
75	C₄²×C₆₇	—	2	4	no	—
76	C₄²×C₆₇	—	2	2	no	—
77	C₄²×C₆₇	—	2	1	no	—
78	C₄²×C₆₇	—	2	2	no	—
79	C₄²×C₆₇	—	2	1	no	—
80	C₄²×C₆₇	—	2	2	no	—
81	C₄²×C₆₇	—	4	2	no	—
82	C₄²×C₆₇	—	4	2	no	—
83	C₄²×C₆₇	#41	1	4	yes	—
84	C₂³×C₆₇	C₂³×C₄	1	1	yes	characteristic
85	Q₈×C₆₇	—	4	4	no	—
86	C₂×C₄×C₆₇	—	2	2	no	—
87	C₂×C₄×C₆₇	—	2	2	no	—
88	C₂×C₄×C₆₇	—	2	2	no	—
89	C₂×C₄×C₆₇	—	2	2	no	—
90	C₂×C₄×C₆₇	—	2	2	no	—
91	C₂×C₄×C₆₇	—	2	4	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	2	no	—
96	C₂×C₄×C₆₇	—	2	2	no	—
97	C₂×C₄×C₆₇	—	2	4	no	—
98	C₂×C₄×C₆₇	—	2	4	no	—
99	C₂×C₄×C₆₇	—	2	4	no	—
100	C₂×C₄×C₆₇	—	2	4	no	—
101	C₂×C₄×C₆₇	#117	1	2	yes	—
102	C₂×C₄×C₆₇	#117	1	2	yes	—
103	C₂×C₄×C₆₇	C₂×#41	1	2	yes	—
104	C₂×C₄×C₆₇	H(5,𝔽₂)	1	1	yes	characteristic
105	C₂×C₄×C₆₇	2₋¹⁺⁴	1	1	yes	characteristic
106	C₂×C₄×C₆₇	2₋¹⁺⁴	1	2	yes	—
107	C₈×C₆₇	—	4	2	no	—
108	C₈×C₆₇	—	4	4	no	—
109	C₈×C₆₇	—	4	2	no	—
110	C₂²×C₆₇	C₂×#117	1	1	yes	characteristic
111	C₂²×C₆₇	#519	1	1	yes	characteristic
112	C₂²×C₆₇	#520	1	1	yes	characteristic
113	C₂²×C₆₇	#568	1	2	yes	—
114	C₂²×C₆₇	#568	1	2	yes	—
115	C₄×C₆₇	—	2	2	no	—
116	C₄×C₆₇	—	2	2	no	—
117	C₄×C₆₇	—	4	2	no	—
118	C₄×C₆₇	—	4	2	no	—
119	C₄×C₆₇	—	4	2	no	—
120	C₄×C₆₇	—	4	2	no	—
121	C₄×C₆₇	—	4	4	no	—
122	C₄×C₆₇	#541	1	2	yes	—
123	C₄×C₆₇	#542	1	2	yes	—
124	#22025	C₆₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
125	C₂×C₆₇	#2320	1	1	yes	characteristic
126	C₂×C₆₇	#2921	1	1	yes	characteristic
127	C₂×C₆₇	#2923	1	1	yes	characteristic
128	C₂×C₆₇	#2987	1	2	yes	—
129	C₂×C₆₇	#2995	1	2	yes	—
130	#2295	C₂×C₆₇	1	1	yes	characteristic
131	#1985	C₂×C₆₇	1	2	yes	—
132	#1940	C₂×C₆₇	1	2	yes	—
133	#1917	C₂×C₆₇	1	4	yes	—
134	#1840	C₂×C₆₇	1	2	yes	—
135	#1832	C₂×C₆₇	1	2	yes	—
136	#1767	C₂×C₆₇	1	1	yes	characteristic
137	#1745	C₂×C₆₇	1	1	yes	characteristic
138	C₆₇	#22025	1	1	yes	characteristic, direct product factor, Sylow
139	C₂×C₄×Q₈	C₂²×C₆₇	1	1	yes	characteristic
140	C₂×C₄⋊C₈	C₂²×C₆₇	1	2	yes	—
141	C₂×C₄⋊C₈	C₂²×C₆₇	1	4	yes	—
142	C₂×#97	C₂²×C₆₇	1	2	yes	—
143	C₂×#97	C₂²×C₆₇	1	2	yes	—
144	#391	C₄×C₆₇	1	2	yes	—
145	#389	C₄×C₆₇	1	2	yes	—
146	#387	C₄×C₆₇	1	4	yes	—
147	C₄×C₄⋊C₄	C₂²×C₆₇	1	2	yes	—
148	C₄×C₄⋊C₄	C₂²×C₆₇	1	1	yes	characteristic
149	#376	C₂²×C₆₇	1	2	yes	—
150	C₄³	C₂²×C₆₇	1	1	yes	characteristic
151	#336	C₂²×C₆₇	1	1	yes	characteristic
152	#336	C₂²×C₆₇	1	2	yes	—
153	#336	C₂²×C₆₇	1	4	yes	—
154	#336	C₂²×C₆₇	1	1	yes	characteristic
155	#336	C₂²×C₆₇	1	4	yes	—
156	#336	C₂²×C₆₇	1	4	yes	—
157	#336	C₂²×C₆₇	1	2	yes	—
158	C₂²×Q₈	C₂×C₄×C₆₇	1	1	yes	characteristic
159	C₂²×C₈	C₂³×C₆₇	1	2	yes	—
160	C₂²×C₈	C₂³×C₆₇	1	2	yes	—
161	C₂²×C₈	C₂³×C₆₇	1	4	yes	—
162	C₄×Q₈	—	2	4	no	—
163	C₂×C₄⋊C₄	C₂×C₄×C₆₇	1	2	yes	—
164	C₂×C₄⋊C₄	C₂×C₄×C₆₇	1	1	yes	characteristic
165	C₂×C₄⋊C₄	C₂×C₄×C₆₇	1	4	yes	—
166	C₂×C₄⋊C₄	C₂×C₄×C₆₇	1	2	yes	—
167	C₂×C₄²	C₂×C₄×C₆₇	1	2	yes	—
168	C₂×C₄²	C₂×C₄×C₆₇	1	2	yes	—
169	C₂×C₄²	C₂³×C₆₇	1	2	yes	—
170	C₂×C₄²	C₂³×C₆₇	1	1	yes	characteristic
171	C₂×C₄²	C₂³×C₆₇	1	1	yes	characteristic
172	C₂×C₄²	C₂³×C₆₇	1	2	yes	—
173	C₂×C₄²	C₂³×C₆₇	1	1	yes	characteristic
174	C₄⋊C₈	—	2	4	no	—
175	C₄⋊C₈	—	4	4	no	—
176	#97	—	2	4	no	—
177	#97	—	4	2	no	—
178	#95	C₂×C₄×C₆₇	1	4	yes	—
179	#95	C₂×C₄×C₆₇	1	4	yes	—
180	#95	C₂×C₄×C₆₇	1	2	yes	—
181	#95	C₂×C₄×C₆₇	1	4	yes	—
182	C₂×Q₈	—	2	2	no	—
183	C₂×Q₈	—	4	2	no	—
184	C₂²×C₄	C₂²×C₄×C₆₇	1	2	yes	—
185	C₂²×C₄	C₂²×C₄×C₆₇	1	4	yes	—
186	C₂²×C₄	C₂²×C₄×C₆₇	1	2	yes	—
187	C₂²×C₄	C₂²×C₄×C₆₇	1	2	yes	—
188	C₂²×C₄	C₂²×C₄×C₆₇	1	2	yes	—
189	C₂²×C₄	C₂²×C₄×C₆₇	1	1	yes	characteristic
190	C₂²×C₄	C₂²×C₄×C₆₇	1	1	yes	characteristic
191	C₂²×C₄	C₂⁴×C₆₇	1	1	yes	characteristic
192	C₂×C₈	—	2	2	no	—
193	C₂×C₈	—	2	2	no	—
194	C₂×C₈	—	2	4	no	—
195	C₂×C₈	—	2	4	no	—
196	C₂×C₈	—	2	2	no	—
197	C₂×C₈	—	2	2	no	—
198	C₂×C₈	—	2	2	no	—
199	C₂×C₈	—	2	2	no	—
200	C₂×C₈	—	2	4	no	—
201	C₄⋊C₄	—	2	2	no	—
202	C₄⋊C₄	—	2	4	no	—
203	C₄⋊C₄	—	4	2	no	—
204	C₄⋊C₄	—	4	4	no	—
205	C₄²	—	2	4	no	—
206	C₄²	—	2	1	no	—
207	C₄²	—	2	2	no	—
208	C₄²	—	2	2	no	—
209	C₄²	—	2	1	no	—
210	C₄²	—	2	2	no	—
211	C₄²	—	4	2	no	—
212	C₄²	—	4	2	no	—
213	C₄²	#41×C₆₇	1	4	yes	—
214	C₂³	C₂³×C₄×C₆₇	1	1	yes	characteristic
215	Q₈	—	4	4	no	—
216	C₂×C₄	—	2	2	no	—
217	C₂×C₄	—	2	2	no	—
218	C₂×C₄	—	2	2	no	—
219	C₂×C₄	—	2	2	no	—
220	C₂×C₄	—	2	2	no	—
221	C₂×C₄	—	2	2	no	—
222	C₂×C₄	—	2	4	no	—
223	C₂×C₄	—	2	2	no	—
224	C₂×C₄	—	2	2	no	—
225	C₂×C₄	—	2	2	no	—
226	C₂×C₄	—	2	2	no	—
227	C₂×C₄	—	2	4	no	—
228	C₂×C₄	—	2	4	no	—
229	C₂×C₄	—	2	4	no	—
230	C₂×C₄	—	2	4	no	—
231	C₂×C₄	#117×C₆₇	1	2	yes	—
232	C₂×C₄	#117×C₆₇	1	2	yes	—
233	C₂×C₄	C₂×#41×C₆₇	1	2	yes	—
234	C₂×C₄	H(5,𝔽₂)×C₆₇	1	1	yes	characteristic
235	C₂×C₄	2₋¹⁺⁴×C₆₇	1	1	yes	characteristic
236	C₂×C₄	2₋¹⁺⁴×C₆₇	1	2	yes	—
237	C₈	—	4	2	no	—
238	C₈	—	4	2	no	—
239	C₈	—	4	4	no	—
240	C₂²	C₂×#117×C₆₇	1	1	yes	characteristic
241	C₂²	#519×C₆₇	1	1	yes	characteristic
242	C₂²	#520×C₆₇	1	1	yes	characteristic
243	C₂²	#568×C₆₇	1	2	yes	—
244	C₂²	#568×C₆₇	1	2	yes	—
245	C₄	—	2	2	no	—
246	C₄	—	2	2	no	—
247	C₄	—	4	2	no	—
248	C₄	—	4	2	no	—
249	C₄	—	4	2	no	—
250	C₄	—	4	2	no	—
251	C₄	—	4	4	no	—
252	C₄	#541×C₆₇	1	2	yes	—
253	C₄	#542×C₆₇	1	2	yes	—
254	C₂	C₆₇×#2320	1	1	yes	characteristic
255	C₂	C₆₇×#2921	1	1	yes	characteristic
256	C₂	C₆₇×#2923	1	1	yes	characteristic
257	C₂	C₆₇×#2987	1	2	yes	—
258	C₂	C₆₇×#2995	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
124. #22025	C₆₇	direct (N × H)
138. C₆₇	#22025	direct (N × H)

Maximal subgroups

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

subgroup	index
C₆₇×#2295	2
C₆₇×#1985	2
C₆₇×#1940	2
C₆₇×#1917	2
C₆₇×#1840	2
C₆₇×#1832	2
C₆₇×#1767	2
C₆₇×#1745	2
#22025	67

Subgroups — C67×#22025 #30701276

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₆₇×#22025 #30701276