Subgroups — C₆₇×#21960 #30701211

Order 17152.

818 conjugacy classes of subgroups.

The subgroup lattice has 236 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₄⋊C₈×C₆₇	C₂	1	1	yes	maximal, characteristic
2	C₆₇×#2283	C₂	1	2	yes	maximal
3	C₆₇×#1949	C₂	1	8	yes	maximal, complemented
4	C₆₇×#1808	C₂	1	2	yes	maximal
5	C₆₇×#1807	C₂	1	2	yes	maximal
6	C₂³×C₈×C₆₇	C₂²	1	1	yes	characteristic
7	C₂³×C₈×C₆₇	C₂²	1	1	yes	characteristic
8	C₂²×C₄²×C₆₇	C₂²	1	1	yes	characteristic
9	C₂×#107×C₆₇	C₂²	1	4	yes	—
10	C₂×#106×C₆₇	C₂²	1	4	yes	—
11	C₂×C₄⋊C₈×C₆₇	C₄	1	8	yes	complemented
12	C₂×C₄⋊C₈×C₆₇	C₂²	1	4	yes	—
13	#383×C₆₇	—	2	2	no	—
14	#382×C₆₇	C₂²	1	4	yes	—
15	#381×C₆₇	—	2	2	no	—
16	#380×C₆₇	C₂²	1	4	yes	—
17	C₄×C₄⋊C₄×C₆₇	C₂²	1	4	yes	—
18	C₄×C₂²⋊C₄×C₆₇	—	2	2	no	—
19	#376×C₆₇	—	2	2	no	—
20	#340×C₆₇	C₂²	1	4	yes	—
21	#340×C₆₇	C₂²	1	4	yes	—
22	C₂³×C₄×C₆₇	D₄	1	1	yes	characteristic
23	C₂³×C₄×C₆₇	D₄	1	1	yes	characteristic
24	C₂³×C₄×C₆₇	C₂³	1	1	yes	characteristic
25	C₂²×C₈×C₆₇	—	4	2	no	—
26	C₂²×C₈×C₆₇	—	4	2	no	—
27	C₂²×C₈×C₆₇	C₂×C₄	1	4	yes	—
28	C₂²×C₈×C₆₇	C₂×C₄	1	4	yes	—
29	C₂²×C₈×C₆₇	C₂³	1	2	yes	—
30	C₂²×C₈×C₆₇	C₂³	1	2	yes	—
31	C₂×C₄⋊C₄×C₆₇	C₂³	1	8	yes	—
32	C₂×C₂²⋊C₄×C₆₇	—	2	4	no	—
33	C₂×C₄²×C₆₇	—	2	4	no	—
34	C₂×C₄²×C₆₇	C₂×C₄	1	4	yes	—
35	C₂×C₄²×C₆₇	D₄	1	2	yes	—
36	C₂×C₄²×C₆₇	Q₈	1	4	yes	—
37	C₂×C₄²×C₆₇	C₂³	1	2	yes	—
38	#107×C₆₇	—	4	4	no	—
39	#106×C₆₇	—	4	4	no	—
40	C₄⋊C₈×C₆₇	C₂×C₄	1	16	yes	—
41	#95×C₆₇	—	2	4	no	—
42	#95×C₆₇	—	2	4	no	—
43	#95×C₆₇	—	2	4	no	—
44	#95×C₆₇	—	2	4	no	—
45	C₂⁴×C₆₇	C₂×D₄	1	1	yes	characteristic
46	C₂²×C₄×C₆₇	—	2	4	no	—
47	C₂²×C₄×C₆₇	—	2	2	no	—
48	C₂²×C₄×C₆₇	—	2	2	no	—
49	C₂²×C₄×C₆₇	—	2	8	no	—
50	C₂²×C₄×C₆₇	—	4	2	no	—
51	C₂²×C₄×C₆₇	C₄⋊C₄	1	4	yes	—
52	C₂²×C₄×C₆₇	C₄⋊C₄	1	4	yes	—
53	C₂²×C₄×C₆₇	C₂²×C₄	1	2	yes	—
54	C₂²×C₄×C₆₇	C₂×D₄	1	2	yes	—
55	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
56	C₂²×C₄×C₆₇	C₂×D₄	1	2	yes	—
57	C₂²×C₄×C₆₇	C₂×Q₈	1	2	yes	—
58	C₂²×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
59	C₂×C₈×C₆₇	—	2	4	no	—
60	C₂×C₈×C₆₇	—	2	4	no	—
61	C₂×C₈×C₆₇	—	4	4	no	—
62	C₂×C₈×C₆₇	—	4	4	no	—
63	C₂×C₈×C₆₇	C₂²×C₄	1	4	yes	—
64	C₂×C₈×C₆₇	C₂²×C₄	1	4	yes	—
65	C₄⋊C₄×C₆₇	—	2	8	no	—
66	C₄⋊C₄×C₆₇	—	2	8	no	—
67	C₂²⋊C₄×C₆₇	—	4	8	no	—
68	C₄²×C₆₇	—	4	8	no	—
69	C₄²×C₆₇	C₄⋊C₄	1	8	yes	—
70	C₄²×C₆₇	C₂²×C₄	1	4	yes	—
71	C₄²×C₆₇	C₂×Q₈	1	4	yes	—
72	C₂³×C₆₇	—	2	2	no	—
73	C₂³×C₆₇	—	2	2	no	—
74	C₂³×C₆₇	C₂×C₄⋊C₄	1	2	yes	—
75	C₂³×C₆₇	#135	1	4	yes	—
76	C₂³×C₆₇	C₂²×D₄	1	1	yes	characteristic
77	C₂×C₄×C₆₇	—	2	2	no	—
78	C₂×C₄×C₆₇	—	2	4	no	—
79	C₂×C₄×C₆₇	—	2	4	no	—
80	C₂×C₄×C₆₇	—	2	8	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	—	4	4	no	—
85	C₂×C₄×C₆₇	—	4	8	no	—
86	C₂×C₄×C₆₇	—	4	8	no	—
87	C₂×C₄×C₆₇	—	4	8	no	—
88	C₂×C₄×C₆₇	C₂×C₄⋊C₄	1	2	yes	—
89	C₂×C₄×C₆₇	C₂×C₄⋊C₄	1	4	yes	—
90	C₂×C₄×C₆₇	C₂×C₄⋊C₄	1	4	yes	—
91	C₂×C₄×C₆₇	C₂³×C₄	1	1	yes	characteristic
92	C₂×C₄×C₆₇	C₂²×Q₈	1	1	yes	characteristic
93	C₂×C₄×C₆₇	H(5,𝔽₂)	1	2	yes	—
94	C₂×C₄×C₆₇	2₋¹⁺⁴	1	2	yes	—
95	C₈×C₆₇	—	2	4	no	—
96	C₈×C₆₇	—	2	4	no	—
97	C₂²×C₆₇	—	2	2	no	—
98	C₂²×C₆₇	—	2	2	no	—
99	C₂²×C₆₇	—	2	8	no	—
100	C₂²×C₆₇	#427	1	4	yes	—
101	C₂²×C₆₇	C₂²×C₄⋊C₄	1	1	yes	characteristic
102	C₂²×C₆₇	#537	1	2	yes	—
103	C₂²×C₆₇	C₂×#135	1	2	yes	—
104	C₂²×C₆₇	#575	1	2	yes	—
105	C₄×C₆₇	—	2	2	no	—
106	C₄×C₆₇	—	2	4	no	—
107	C₄×C₆₇	—	2	4	no	—
108	C₄×C₆₇	—	8	8	no	—
109	C₄×C₆₇	#518	1	1	yes	characteristic
110	C₄×C₆₇	#519	1	1	yes	characteristic
111	C₄×C₆₇	#544	1	2	yes	—
112	#21960	C₆₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
113	C₂×C₆₇	—	2	4	no	—
114	C₂×C₆₇	#2316	1	1	yes	characteristic
115	C₂×C₆₇	C₂×#427	1	1	yes	characteristic
116	C₂×C₆₇	#2911	1	1	yes	characteristic
117	C₂×C₆₇	#3250	1	4	yes	—
118	C₂²×C₄⋊C₈	C₂×C₆₇	1	1	yes	characteristic
119	#2283	C₂×C₆₇	1	2	yes	—
120	#1949	C₂×C₆₇	1	8	yes	complemented
121	#1808	C₂×C₆₇	1	2	yes	—
122	#1807	C₂×C₆₇	1	2	yes	—
123	C₆₇	#21960	1	1	yes	characteristic, direct product factor, Sylow
124	C₂³×C₈	C₂²×C₆₇	1	1	yes	characteristic
125	C₂³×C₈	C₂²×C₆₇	1	1	yes	characteristic
126	C₂²×C₄²	C₂²×C₆₇	1	1	yes	characteristic
127	C₂×#107	C₂²×C₆₇	1	4	yes	—
128	C₂×#106	C₂²×C₆₇	1	4	yes	—
129	C₂×C₄⋊C₈	C₄×C₆₇	1	8	yes	complemented
130	C₂×C₄⋊C₈	C₂²×C₆₇	1	4	yes	—
131	#383	—	2	2	no	—
132	#382	C₂²×C₆₇	1	4	yes	—
133	#381	—	2	2	no	—
134	#380	C₂²×C₆₇	1	4	yes	—
135	C₄×C₄⋊C₄	C₂²×C₆₇	1	4	yes	—
136	C₄×C₂²⋊C₄	—	2	2	no	—
137	#376	—	2	2	no	—
138	#340	C₂²×C₆₇	1	4	yes	—
139	#340	C₂²×C₆₇	1	4	yes	—
140	C₂³×C₄	D₄×C₆₇	1	1	yes	characteristic
141	C₂³×C₄	D₄×C₆₇	1	1	yes	characteristic
142	C₂³×C₄	C₂³×C₆₇	1	1	yes	characteristic
143	C₂²×C₈	—	4	2	no	—
144	C₂²×C₈	—	4	2	no	—
145	C₂²×C₈	C₂×C₄×C₆₇	1	4	yes	—
146	C₂²×C₈	C₂×C₄×C₆₇	1	4	yes	—
147	C₂²×C₈	C₂³×C₆₇	1	2	yes	—
148	C₂²×C₈	C₂³×C₆₇	1	2	yes	—
149	C₂×C₄⋊C₄	C₂³×C₆₇	1	8	yes	—
150	C₂×C₂²⋊C₄	—	2	4	no	—
151	C₂×C₄²	—	2	4	no	—
152	C₂×C₄²	C₂×C₄×C₆₇	1	4	yes	—
153	C₂×C₄²	D₄×C₆₇	1	2	yes	—
154	C₂×C₄²	Q₈×C₆₇	1	4	yes	—
155	C₂×C₄²	C₂³×C₆₇	1	2	yes	—
156	#107	—	4	4	no	—
157	#106	—	4	4	no	—
158	C₄⋊C₈	C₂×C₄×C₆₇	1	16	yes	—
159	#95	—	2	4	no	—
160	#95	—	2	4	no	—
161	#95	—	2	4	no	—
162	#95	—	2	4	no	—
163	C₂⁴	C₂×D₄×C₆₇	1	1	yes	characteristic
164	C₂²×C₄	—	2	4	no	—
165	C₂²×C₄	—	2	2	no	—
166	C₂²×C₄	—	2	2	no	—
167	C₂²×C₄	—	2	8	no	—
168	C₂²×C₄	—	4	2	no	—
169	C₂²×C₄	C₄⋊C₄×C₆₇	1	4	yes	—
170	C₂²×C₄	C₄⋊C₄×C₆₇	1	4	yes	—
171	C₂²×C₄	C₂²×C₄×C₆₇	1	2	yes	—
172	C₂²×C₄	C₂×D₄×C₆₇	1	1	yes	characteristic
173	C₂²×C₄	C₂×D₄×C₆₇	1	2	yes	—
174	C₂²×C₄	C₂×D₄×C₆₇	1	2	yes	—
175	C₂²×C₄	C₂×Q₈×C₆₇	1	2	yes	—
176	C₂²×C₄	C₂⁴×C₆₇	1	1	yes	characteristic
177	C₂×C₈	—	2	4	no	—
178	C₂×C₈	—	2	4	no	—
179	C₂×C₈	—	4	4	no	—
180	C₂×C₈	—	4	4	no	—
181	C₂×C₈	C₂²×C₄×C₆₇	1	4	yes	—
182	C₂×C₈	C₂²×C₄×C₆₇	1	4	yes	—
183	C₄⋊C₄	—	2	8	no	—
184	C₄⋊C₄	—	2	8	no	—
185	C₂²⋊C₄	—	4	8	no	—
186	C₄²	—	4	8	no	—
187	C₄²	C₄⋊C₄×C₆₇	1	8	yes	—
188	C₄²	C₂²×C₄×C₆₇	1	4	yes	—
189	C₄²	C₂×Q₈×C₆₇	1	4	yes	—
190	C₂³	—	2	2	no	—
191	C₂³	—	2	2	no	—
192	C₂³	C₂×C₄⋊C₄×C₆₇	1	2	yes	—
193	C₂³	#135×C₆₇	1	4	yes	—
194	C₂³	C₂²×D₄×C₆₇	1	1	yes	characteristic
195	C₂×C₄	—	2	2	no	—
196	C₂×C₄	—	2	4	no	—
197	C₂×C₄	—	2	4	no	—
198	C₂×C₄	—	2	8	no	—
199	C₂×C₄	—	2	2	no	—
200	C₂×C₄	—	2	2	no	—
201	C₂×C₄	—	2	4	no	—
202	C₂×C₄	—	4	4	no	—
203	C₂×C₄	—	4	8	no	—
204	C₂×C₄	—	4	8	no	—
205	C₂×C₄	—	4	8	no	—
206	C₂×C₄	C₂×C₄⋊C₄×C₆₇	1	2	yes	—
207	C₂×C₄	C₂×C₄⋊C₄×C₆₇	1	4	yes	—
208	C₂×C₄	C₂×C₄⋊C₄×C₆₇	1	4	yes	—
209	C₂×C₄	C₂³×C₄×C₆₇	1	1	yes	characteristic
210	C₂×C₄	C₂²×Q₈×C₆₇	1	1	yes	characteristic
211	C₂×C₄	H(5,𝔽₂)×C₆₇	1	2	yes	—
212	C₂×C₄	2₋¹⁺⁴×C₆₇	1	2	yes	—
213	C₈	—	2	4	no	—
214	C₈	—	2	4	no	—
215	C₂²	—	2	2	no	—
216	C₂²	—	2	2	no	—
217	C₂²	—	2	8	no	—
218	C₂²	#427×C₆₇	1	4	yes	—
219	C₂²	C₂²×C₄⋊C₄×C₆₇	1	1	yes	characteristic
220	C₂²	#537×C₆₇	1	2	yes	—
221	C₂²	C₂×#135×C₆₇	1	2	yes	—
222	C₂²	#575×C₆₇	1	2	yes	—
223	C₄	—	2	2	no	—
224	C₄	—	2	4	no	—
225	C₄	—	2	4	no	—
226	C₄	—	8	8	no	—
227	C₄	#518×C₆₇	1	1	yes	characteristic
228	C₄	#519×C₆₇	1	1	yes	characteristic
229	C₄	#544×C₆₇	1	2	yes	—
230	C₂	—	2	4	no	—
231	C₂	C₆₇×#2316	1	1	yes	characteristic
232	C₂	C₂×#427×C₆₇	1	1	yes	characteristic
233	C₂	C₆₇×#2911	1	1	yes	characteristic
234	C₂	C₆₇×#3250	1	4	yes	—

	subgroup	G/H
center, socle	C₂³×C₆₇	C₂²×D₄
commutator	C₂×C₄	C₂³×C₄×C₆₇
Frattini, cosocle	C₂²×C₄	C₂⁴×C₆₇
prime radical	C₂⁴×C₆₇	C₂×D₄
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₆₇ ◁ C₂²×C₄²×C₆₇ ◁ G

Complements

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

N	H ≅ G/N	type
3. C₆₇×#1949	C₂	semidirect
11. C₂×C₄⋊C₈×C₆₇	C₄	semidirect
112. #21960	C₆₇	direct (N × H)
120. #1949	C₂×C₆₇	semidirect
123. C₆₇	#21960	direct (N × H)
129. C₂×C₄⋊C₈	C₄×C₆₇	semidirect

Maximal subgroups

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

subgroup	index
C₂²×C₄⋊C₈×C₆₇	2
C₆₇×#2283	2
C₆₇×#1949	2
C₆₇×#1808	2
C₆₇×#1807	2
#21960	67

Subgroups — C67×#21960 #30701211

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₆₇×#21960 #30701211