Subgroups — C₆₇×#21915 #30701166

Order 17152.

694 conjugacy classes of subgroups.

The subgroup lattice has 272 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₆₇×#2296	C₂	1	1	yes	maximal, characteristic
2	C₆₇×#1917	C₂	1	4	yes	maximal, complemented
3	C₆₇×#1916	C₂	1	4	yes	maximal, complemented
4	C₆₇×#1818	C₂	1	2	yes	maximal
5	C₆₇×#1798	C₂	1	2	yes	maximal
6	C₆₇×#1749	C₂	1	1	yes	maximal, characteristic
7	C₄×C₂²⋊C₈×C₆₇	C₂	1	1	yes	maximal, characteristic
8	C₂×#117×C₆₇	C₂²	1	1	yes	characteristic
9	C₂×#117×C₆₇	C₂²	1	1	yes	characteristic
10	C₂²×C₄²×C₆₇	C₂²	1	1	yes	characteristic
11	C₂×C₄⋊C₈×C₆₇	C₂²	1	4	yes	—
12	C₂×C₂²⋊C₈×C₆₇	C₂²	1	2	yes	—
13	C₂×C₂²⋊C₈×C₆₇	C₂²	1	2	yes	—
14	C₂×#97×C₆₇	C₂²	1	2	yes	—
15	C₂×C₄×C₈×C₆₇	C₂²	1	2	yes	—
16	#389×C₆₇	C₄	1	2	yes	—
17	#389×C₆₇	C₄	1	2	yes	—
18	#385×C₆₇	C₄	1	4	yes	—
19	C₄×C₄⋊C₄×C₆₇	C₂²	1	2	yes	—
20	C₄×C₄⋊C₄×C₆₇	C₂²	1	2	yes	—
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	8	yes	—
25	C₂³×C₄×C₆₇	C₂×C₄	1	2	yes	—
26	C₂³×C₄×C₆₇	C₂³	1	1	yes	characteristic
27	C₂²×C₈×C₆₇	C₂³	1	4	yes	—
28	C₂²×C₈×C₆₇	C₂³	1	4	yes	—
29	#117×C₆₇	—	2	4	no	—
30	#117×C₆₇	—	4	2	no	—
31	C₂×C₄⋊C₄×C₆₇	C₂×C₄	1	1	yes	characteristic
32	C₂×C₄⋊C₄×C₆₇	C₂×C₄	1	1	yes	characteristic
33	C₂×C₄⋊C₄×C₆₇	C₂×C₄	1	4	yes	—
34	C₂×C₄⋊C₄×C₆₇	C₂×C₄	1	2	yes	—
35	C₂×C₄⋊C₄×C₆₇	C₂×C₄	1	4	yes	—
36	C₂×C₂²⋊C₄×C₆₇	C₂×C₄	1	2	yes	—
37	C₂×C₂²⋊C₄×C₆₇	C₂×C₄	1	2	yes	—
38	C₂×C₄²×C₆₇	—	4	2	no	—
39	C₂×C₄²×C₆₇	C₂×C₄	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	2	yes	—
44	C₄⋊C₈×C₆₇	—	4	4	no	—
45	C₂²⋊C₈×C₆₇	—	2	4	no	—
46	C₂²⋊C₈×C₆₇	—	2	4	no	—
47	#97×C₆₇	—	4	2	no	—
48	C₄×C₈×C₆₇	—	4	2	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	2	no	—
53	C₂²×C₄×C₆₇	—	2	2	no	—
54	C₂²×C₄×C₆₇	—	2	4	no	—
55	C₂²×C₄×C₆₇	—	4	2	no	—
56	C₂²×C₄×C₆₇	—	4	2	no	—
57	C₂²×C₄×C₆₇	C₂²×C₄	1	2	yes	—
58	C₂²×C₄×C₆₇	C₂²×C₄	1	2	yes	—
59	C₂²×C₄×C₆₇	C₂²×C₄	1	2	yes	—
60	C₂²×C₄×C₆₇	C₂²×C₄	1	4	yes	—
61	C₂²×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
62	C₂²×C₄×C₆₇	C₂²×C₄	1	2	yes	—
63	C₂²×C₄×C₆₇	#41	1	4	yes	—
64	C₂²×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
65	C₂×C₈×C₆₇	—	2	4	no	—
66	C₂×C₈×C₆₇	—	2	4	no	—
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	C₄⋊C₄×C₆₇	—	2	2	no	—
72	C₄⋊C₄×C₆₇	—	2	2	no	—
73	C₄⋊C₄×C₆₇	—	2	8	no	—
74	C₄⋊C₄×C₆₇	—	4	4	no	—
75	C₄⋊C₄×C₆₇	—	4	2	no	—
76	C₂²⋊C₄×C₆₇	—	2	4	no	—
77	C₂²⋊C₄×C₆₇	—	4	2	no	—
78	C₄²×C₆₇	—	2	4	no	—
79	C₄²×C₆₇	—	2	4	no	—
80	C₄²×C₆₇	—	2	2	no	—
81	C₄²×C₆₇	—	2	2	no	—
82	C₄²×C₆₇	—	4	2	no	—
83	C₂³×C₆₇	—	2	1	no	—
84	C₂³×C₆₇	—	2	1	no	—
85	C₂³×C₆₇	—	2	2	no	—
86	C₂³×C₆₇	—	2	2	no	—
87	C₂³×C₆₇	#117	1	2	yes	—
88	C₂³×C₆₇	C₂³×C₄	1	1	yes	characteristic
89	C₂×C₄×C₆₇	—	2	2	no	—
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	8	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	2	no	—
101	C₂×C₄×C₆₇	—	2	2	no	—
102	C₂×C₄×C₆₇	—	2	4	no	—
103	C₂×C₄×C₆₇	—	2	4	no	—
104	C₂×C₄×C₆₇	—	4	8	no	—
105	C₂×C₄×C₆₇	#117	1	2	yes	—
106	C₂×C₄×C₆₇	#131	1	4	yes	—
107	C₂×C₄×C₆₇	C₂×#41	1	2	yes	—
108	C₂×C₄×C₆₇	H(5,𝔽₂)	1	2	yes	—
109	C₂×C₄×C₆₇	2₋¹⁺⁴	1	2	yes	—
110	C₈×C₆₇	—	4	4	no	—
111	C₈×C₆₇	—	4	4	no	—
112	C₂²×C₆₇	—	2	2	no	—
113	C₂²×C₆₇	—	2	2	no	—
114	C₂²×C₆₇	—	2	2	no	—
115	C₂²×C₆₇	—	4	2	no	—
116	C₂²×C₆₇	—	4	2	no	—
117	C₂²×C₆₇	C₂×#117	1	1	yes	characteristic
118	C₂²×C₆₇	#518	1	1	yes	characteristic
119	C₂²×C₆₇	#519	1	1	yes	characteristic
120	C₂²×C₆₇	C₂×#131	1	2	yes	—
121	C₂²×C₆₇	#568	1	2	yes	—
122	C₄×C₆₇	—	2	2	no	—
123	C₄×C₆₇	—	2	4	no	—
124	C₄×C₆₇	—	4	2	no	—
125	C₄×C₆₇	—	4	2	no	—
126	C₄×C₆₇	—	4	4	no	—
127	C₄×C₆₇	—	4	2	no	—
128	C₄×C₆₇	#539	1	4	yes	—
129	#21915	C₆₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
130	C₂×C₆₇	—	4	2	no	—
131	C₂×C₆₇	#2313	1	1	yes	characteristic
132	C₂×C₆₇	#2922	1	1	yes	characteristic
133	C₂×C₆₇	#2923	1	1	yes	characteristic
134	C₂×C₆₇	#2976	1	2	yes	—
135	C₂×C₆₇	#2985	1	2	yes	—
136	#2296	C₂×C₆₇	1	1	yes	characteristic
137	#1917	C₂×C₆₇	1	4	yes	complemented
138	#1916	C₂×C₆₇	1	4	yes	complemented
139	#1818	C₂×C₆₇	1	2	yes	—
140	#1798	C₂×C₆₇	1	2	yes	—
141	#1749	C₂×C₆₇	1	1	yes	characteristic
142	C₄×C₂²⋊C₈	C₂×C₆₇	1	1	yes	characteristic
143	C₆₇	#21915	1	1	yes	characteristic, direct product factor, Sylow
144	C₂×#117	C₂²×C₆₇	1	1	yes	characteristic
145	C₂×#117	C₂²×C₆₇	1	1	yes	characteristic
146	C₂²×C₄²	C₂²×C₆₇	1	1	yes	characteristic
147	C₂×C₄⋊C₈	C₂²×C₆₇	1	4	yes	—
148	C₂×C₂²⋊C₈	C₂²×C₆₇	1	2	yes	—
149	C₂×C₂²⋊C₈	C₂²×C₆₇	1	2	yes	—
150	C₂×#97	C₂²×C₆₇	1	2	yes	—
151	C₂×C₄×C₈	C₂²×C₆₇	1	2	yes	—
152	#389	C₄×C₆₇	1	2	yes	—
153	#389	C₄×C₆₇	1	2	yes	—
154	#385	C₄×C₆₇	1	4	yes	—
155	C₄×C₄⋊C₄	C₂²×C₆₇	1	2	yes	—
156	C₄×C₄⋊C₄	C₂²×C₆₇	1	2	yes	—
157	#336	C₂²×C₆₇	1	4	yes	—
158	#336	C₂²×C₆₇	1	8	yes	—
159	#336	C₂²×C₆₇	1	2	yes	—
160	#336	C₂²×C₆₇	1	2	yes	—
161	C₂³×C₄	C₂×C₄×C₆₇	1	2	yes	—
162	C₂³×C₄	C₂³×C₆₇	1	1	yes	characteristic
163	C₂²×C₈	C₂³×C₆₇	1	4	yes	—
164	C₂²×C₈	C₂³×C₆₇	1	4	yes	—
165	#117	—	2	4	no	—
166	#117	—	4	2	no	—
167	C₂×C₄⋊C₄	C₂×C₄×C₆₇	1	1	yes	characteristic
168	C₂×C₄⋊C₄	C₂×C₄×C₆₇	1	1	yes	characteristic
169	C₂×C₄⋊C₄	C₂×C₄×C₆₇	1	4	yes	—
170	C₂×C₄⋊C₄	C₂×C₄×C₆₇	1	2	yes	—
171	C₂×C₄⋊C₄	C₂×C₄×C₆₇	1	4	yes	—
172	C₂×C₂²⋊C₄	C₂×C₄×C₆₇	1	2	yes	—
173	C₂×C₂²⋊C₄	C₂×C₄×C₆₇	1	2	yes	—
174	C₂×C₄²	—	4	2	no	—
175	C₂×C₄²	C₂×C₄×C₆₇	1	2	yes	—
176	C₂×C₄²	C₂³×C₆₇	1	2	yes	—
177	C₂×C₄²	C₂³×C₆₇	1	2	yes	—
178	C₂×C₄²	C₂³×C₆₇	1	1	yes	characteristic
179	C₂×C₄²	C₂³×C₆₇	1	1	yes	characteristic
180	C₄⋊C₈	—	4	4	no	—
181	C₂²⋊C₈	—	2	4	no	—
182	C₂²⋊C₈	—	2	4	no	—
183	#97	—	4	2	no	—
184	C₄×C₈	—	4	2	no	—
185	#95	C₂×C₄×C₆₇	1	4	yes	—
186	#95	C₂×C₄×C₆₇	1	4	yes	—
187	C₂⁴	C₂²×C₄×C₆₇	1	1	yes	characteristic
188	C₂²×C₄	—	2	2	no	—
189	C₂²×C₄	—	2	2	no	—
190	C₂²×C₄	—	2	4	no	—
191	C₂²×C₄	—	4	2	no	—
192	C₂²×C₄	—	4	2	no	—
193	C₂²×C₄	C₂²×C₄×C₆₇	1	2	yes	—
194	C₂²×C₄	C₂²×C₄×C₆₇	1	2	yes	—
195	C₂²×C₄	C₂²×C₄×C₆₇	1	2	yes	—
196	C₂²×C₄	C₂²×C₄×C₆₇	1	4	yes	—
197	C₂²×C₄	C₂²×C₄×C₆₇	1	1	yes	characteristic
198	C₂²×C₄	C₂²×C₄×C₆₇	1	2	yes	—
199	C₂²×C₄	#41×C₆₇	1	4	yes	—
200	C₂²×C₄	C₂⁴×C₆₇	1	1	yes	characteristic
201	C₂×C₈	—	2	4	no	—
202	C₂×C₈	—	2	4	no	—
203	C₂×C₈	—	2	4	no	—
204	C₂×C₈	—	2	4	no	—
205	C₂×C₈	—	2	4	no	—
206	C₂×C₈	—	2	4	no	—
207	C₄⋊C₄	—	2	2	no	—
208	C₄⋊C₄	—	2	2	no	—
209	C₄⋊C₄	—	2	8	no	—
210	C₄⋊C₄	—	4	2	no	—
211	C₄⋊C₄	—	4	4	no	—
212	C₂²⋊C₄	—	2	4	no	—
213	C₂²⋊C₄	—	4	2	no	—
214	C₄²	—	2	4	no	—
215	C₄²	—	2	2	no	—
216	C₄²	—	2	2	no	—
217	C₄²	—	2	4	no	—
218	C₄²	—	4	2	no	—
219	C₂³	—	2	1	no	—
220	C₂³	—	2	1	no	—
221	C₂³	—	2	2	no	—
222	C₂³	—	2	2	no	—
223	C₂³	#117×C₆₇	1	2	yes	—
224	C₂³	C₂³×C₄×C₆₇	1	1	yes	characteristic
225	C₂×C₄	—	2	2	no	—
226	C₂×C₄	—	2	1	no	—
227	C₂×C₄	—	2	1	no	—
228	C₂×C₄	—	2	2	no	—
229	C₂×C₄	—	2	8	no	—
230	C₂×C₄	—	2	2	no	—
231	C₂×C₄	—	2	2	no	—
232	C₂×C₄	—	2	2	no	—
233	C₂×C₄	—	2	4	no	—
234	C₂×C₄	—	2	4	no	—
235	C₂×C₄	—	2	4	no	—
236	C₂×C₄	—	2	2	no	—
237	C₂×C₄	—	2	2	no	—
238	C₂×C₄	—	2	4	no	—
239	C₂×C₄	—	2	4	no	—
240	C₂×C₄	—	4	8	no	—
241	C₂×C₄	#117×C₆₇	1	2	yes	—
242	C₂×C₄	#131×C₆₇	1	4	yes	—
243	C₂×C₄	C₂×#41×C₆₇	1	2	yes	—
244	C₂×C₄	H(5,𝔽₂)×C₆₇	1	2	yes	—
245	C₂×C₄	2₋¹⁺⁴×C₆₇	1	2	yes	—
246	C₈	—	4	4	no	—
247	C₈	—	4	4	no	—
248	C₂²	—	2	2	no	—
249	C₂²	—	2	2	no	—
250	C₂²	—	2	2	no	—
251	C₂²	—	4	2	no	—
252	C₂²	—	4	2	no	—
253	C₂²	C₂×#117×C₆₇	1	1	yes	characteristic
254	C₂²	#518×C₆₇	1	1	yes	characteristic
255	C₂²	#519×C₆₇	1	1	yes	characteristic
256	C₂²	C₂×#131×C₆₇	1	2	yes	—
257	C₂²	#568×C₆₇	1	2	yes	—
258	C₄	—	2	2	no	—
259	C₄	—	2	4	no	—
260	C₄	—	4	2	no	—
261	C₄	—	4	2	no	—
262	C₄	—	4	2	no	—
263	C₄	—	4	4	no	—
264	C₄	#539×C₆₇	1	4	yes	—
265	C₂	—	4	2	no	—
266	C₂	C₆₇×#2313	1	1	yes	characteristic
267	C₂	C₆₇×#2922	1	1	yes	characteristic
268	C₂	C₆₇×#2923	1	1	yes	characteristic
269	C₂	C₆₇×#2976	1	2	yes	—
270	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
2. C₆₇×#1917	C₂	semidirect
3. C₆₇×#1916	C₂	semidirect
129. #21915	C₆₇	direct (N × H)
137. #1917	C₂×C₆₇	semidirect
138. #1916	C₂×C₆₇	semidirect
143. C₆₇	#21915	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×#2296	2
C₆₇×#1917	2
C₆₇×#1916	2
C₆₇×#1818	2
C₆₇×#1798	2
C₆₇×#1749	2
C₄×C₂²⋊C₈×C₆₇	2
#21915	67

Subgroups — C67×#21915 #30701166

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₆₇×#21915 #30701166