Subgroups — C₆₇×#21961 #30701212

Order 17152.

906 conjugacy classes of subgroups.

The subgroup lattice has 230 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×#2285	C₂	1	2	yes	maximal
3	C₆₇×#1947	C₂	1	8	yes	maximal, complemented
4	C₆₇×#1808	C₂	1	4	yes	maximal
5	C₂³×C₈×C₆₇	C₂²	1	2	yes	—
6	C₂×#117×C₆₇	—	2	2	no	—
7	C₂²×C₄²×C₆₇	C₂²	1	1	yes	characteristic
8	C₂×#107×C₆₇	C₂²	1	8	yes	—
9	C₂×C₄⋊C₈×C₆₇	C₄	1	8	yes	complemented
10	C₂×C₄⋊C₈×C₆₇	C₂²	1	4	yes	—
11	#384×C₆₇	C₂²	1	4	yes	—
12	#382×C₆₇	—	2	2	no	—
13	#380×C₆₇	—	2	2	no	—
14	#380×C₆₇	C₂²	1	4	yes	—
15	C₄×C₄⋊C₄×C₆₇	—	2	2	no	—
16	C₄×C₄⋊C₄×C₆₇	C₂²	1	4	yes	—
17	#340×C₆₇	C₂²	1	8	yes	—
18	C₂³×C₄×C₆₇	D₄	1	1	yes	characteristic
19	C₂³×C₄×C₆₇	D₄	1	1	yes	characteristic
20	C₂³×C₄×C₆₇	C₂³	1	1	yes	characteristic
21	C₂²×C₈×C₆₇	—	4	4	no	—
22	C₂²×C₈×C₆₇	C₂×C₄	1	8	yes	—
23	C₂²×C₈×C₆₇	C₂³	1	4	yes	—
24	#117×C₆₇	—	4	8	no	—
25	C₂×C₄⋊C₄×C₆₇	—	2	4	no	—
26	C₂×C₄⋊C₄×C₆₇	—	2	4	no	—
27	C₂×C₄⋊C₄×C₆₇	C₂³	1	8	yes	—
28	C₂×C₂²⋊C₄×C₆₇	—	2	4	no	—
29	C₂×C₄²×C₆₇	—	2	4	no	—
30	C₂×C₄²×C₆₇	C₂×C₄	1	4	yes	—
31	C₂×C₄²×C₆₇	D₄	1	2	yes	—
32	C₂×C₄²×C₆₇	Q₈	1	4	yes	—
33	C₂×C₄²×C₆₇	C₂³	1	2	yes	—
34	#107×C₆₇	—	4	8	no	—
35	C₄⋊C₈×C₆₇	C₂×C₄	1	16	yes	—
36	#95×C₆₇	—	2	4	no	—
37	#95×C₆₇	—	2	4	no	—
38	C₂⁴×C₆₇	C₂×D₄	1	1	yes	characteristic
39	C₂²×C₄×C₆₇	—	2	2	no	—
40	C₂²×C₄×C₆₇	—	2	2	no	—
41	C₂²×C₄×C₆₇	—	2	8	no	—
42	C₂²×C₄×C₆₇	—	4	2	no	—
43	C₂²×C₄×C₆₇	C₄⋊C₄	1	4	yes	—
44	C₂²×C₄×C₆₇	C₄⋊C₄	1	4	yes	—
45	C₂²×C₄×C₆₇	D₈	1	4	yes	—
46	C₂²×C₄×C₆₇	Q₁₆	1	4	yes	—
47	C₂²×C₄×C₆₇	C₂²×C₄	1	2	yes	—
48	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
49	C₂²×C₄×C₆₇	C₂×D₄	1	2	yes	—
50	C₂²×C₄×C₆₇	C₂×D₄	1	2	yes	—
51	C₂²×C₄×C₆₇	C₂×Q₈	1	2	yes	—
52	C₂²×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
53	C₂×C₈×C₆₇	—	2	8	no	—
54	C₂×C₈×C₆₇	—	4	8	no	—
55	C₂×C₈×C₆₇	C₂²×C₄	1	8	yes	—
56	C₄⋊C₄×C₆₇	—	2	8	no	—
57	C₄⋊C₄×C₆₇	—	2	8	no	—
58	C₄⋊C₄×C₆₇	—	4	8	no	—
59	C₄⋊C₄×C₆₇	—	4	8	no	—
60	C₂²⋊C₄×C₆₇	—	4	8	no	—
61	C₄²×C₆₇	—	4	8	no	—
62	C₄²×C₆₇	C₄⋊C₄	1	8	yes	—
63	C₄²×C₆₇	C₂²×C₄	1	4	yes	—
64	C₄²×C₆₇	C₂×Q₈	1	4	yes	—
65	C₂³×C₆₇	—	2	2	no	—
66	C₂³×C₆₇	—	2	2	no	—
67	C₂³×C₆₇	C₂×C₄⋊C₄	1	2	yes	—
68	C₂³×C₆₇	C₂×D₈	1	2	yes	—
69	C₂³×C₆₇	C₂×Q₁₆	1	2	yes	—
70	C₂³×C₆₇	C₂²×D₄	1	1	yes	characteristic
71	C₂×C₄×C₆₇	—	2	2	no	—
72	C₂×C₄×C₆₇	—	2	4	no	—
73	C₂×C₄×C₆₇	—	2	2	no	—
74	C₂×C₄×C₆₇	—	2	2	no	—
75	C₂×C₄×C₆₇	—	2	4	no	—
76	C₂×C₄×C₆₇	—	4	4	no	—
77	C₂×C₄×C₆₇	—	4	8	no	—
78	C₂×C₄×C₆₇	—	4	8	no	—
79	C₂×C₄×C₆₇	—	4	8	no	—
80	C₂×C₄×C₆₇	#107	1	16	yes	complemented
81	C₂×C₄×C₆₇	C₂×C₄⋊C₄	1	2	yes	—
82	C₂×C₄×C₆₇	C₂×C₄⋊C₄	1	4	yes	—
83	C₂×C₄×C₆₇	C₂×C₄⋊C₄	1	4	yes	—
84	C₂×C₄×C₆₇	C₂×D₈	1	4	yes	—
85	C₂×C₄×C₆₇	C₂×Q₁₆	1	4	yes	—
86	C₂×C₄×C₆₇	C₂³×C₄	1	1	yes	characteristic
87	C₂×C₄×C₆₇	C₂²×Q₈	1	1	yes	characteristic
88	C₂×C₄×C₆₇	H(5,𝔽₂)	1	2	yes	—
89	C₂×C₄×C₆₇	2₋¹⁺⁴	1	2	yes	—
90	C₈×C₆₇	—	2	8	no	—
91	C₂²×C₆₇	—	2	2	no	—
92	C₂²×C₆₇	—	2	2	no	—
93	C₂²×C₆₇	—	2	4	no	—
94	C₂²×C₆₇	—	2	4	no	—
95	C₂²×C₆₇	C₂×#107	1	4	yes	—
96	C₂²×C₆₇	C₂²×C₄⋊C₄	1	1	yes	characteristic
97	C₂²×C₆₇	#537	1	2	yes	—
98	C₂²×C₆₇	C₂²×D₈	1	1	yes	characteristic
99	C₂²×C₆₇	C₂²×Q₁₆	1	1	yes	characteristic
100	C₂²×C₆₇	#575	1	2	yes	—
101	C₄×C₆₇	—	2	2	no	—
102	C₄×C₆₇	—	2	4	no	—
103	C₄×C₆₇	—	8	8	no	—
104	C₄×C₆₇	C₂×#107	1	8	yes	—
105	C₄×C₆₇	#518	1	1	yes	characteristic
106	C₄×C₆₇	#519	1	1	yes	characteristic
107	C₄×C₆₇	#544	1	2	yes	—
108	#21961	C₆₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
109	C₂×C₆₇	—	2	4	no	—
110	C₂×C₆₇	#2316	1	1	yes	characteristic
111	C₂×C₆₇	C₂²×#107	1	1	yes	characteristic
112	C₂×C₆₇	#2911	1	1	yes	characteristic
113	C₂×C₆₇	#3245	1	2	yes	—
114	C₂×C₆₇	#3249	1	2	yes	—
115	C₂²×C₄⋊C₈	C₂×C₆₇	1	1	yes	characteristic
116	#2285	C₂×C₆₇	1	2	yes	—
117	#1947	C₂×C₆₇	1	8	yes	complemented
118	#1808	C₂×C₆₇	1	4	yes	—
119	C₆₇	#21961	1	1	yes	characteristic, direct product factor, Sylow
120	C₂³×C₈	C₂²×C₆₇	1	2	yes	—
121	C₂×#117	—	2	2	no	—
122	C₂²×C₄²	C₂²×C₆₇	1	1	yes	characteristic
123	C₂×#107	C₂²×C₆₇	1	8	yes	—
124	C₂×C₄⋊C₈	C₄×C₆₇	1	8	yes	complemented
125	C₂×C₄⋊C₈	C₂²×C₆₇	1	4	yes	—
126	#384	C₂²×C₆₇	1	4	yes	—
127	#382	—	2	2	no	—
128	#380	—	2	2	no	—
129	#380	C₂²×C₆₇	1	4	yes	—
130	C₄×C₄⋊C₄	—	2	2	no	—
131	C₄×C₄⋊C₄	C₂²×C₆₇	1	4	yes	—
132	#340	C₂²×C₆₇	1	8	yes	—
133	C₂³×C₄	D₄×C₆₇	1	1	yes	characteristic
134	C₂³×C₄	D₄×C₆₇	1	1	yes	characteristic
135	C₂³×C₄	C₂³×C₆₇	1	1	yes	characteristic
136	C₂²×C₈	—	4	4	no	—
137	C₂²×C₈	C₂×C₄×C₆₇	1	8	yes	—
138	C₂²×C₈	C₂³×C₆₇	1	4	yes	—
139	#117	—	4	8	no	—
140	C₂×C₄⋊C₄	—	2	4	no	—
141	C₂×C₄⋊C₄	—	2	4	no	—
142	C₂×C₄⋊C₄	C₂³×C₆₇	1	8	yes	—
143	C₂×C₂²⋊C₄	—	2	4	no	—
144	C₂×C₄²	—	2	4	no	—
145	C₂×C₄²	C₂×C₄×C₆₇	1	4	yes	—
146	C₂×C₄²	D₄×C₆₇	1	2	yes	—
147	C₂×C₄²	Q₈×C₆₇	1	4	yes	—
148	C₂×C₄²	C₂³×C₆₇	1	2	yes	—
149	#107	—	4	8	no	—
150	C₄⋊C₈	C₂×C₄×C₆₇	1	16	yes	—
151	#95	—	2	4	no	—
152	#95	—	2	4	no	—
153	C₂⁴	C₂×D₄×C₆₇	1	1	yes	characteristic
154	C₂²×C₄	—	2	2	no	—
155	C₂²×C₄	—	2	2	no	—
156	C₂²×C₄	—	2	8	no	—
157	C₂²×C₄	—	4	2	no	—
158	C₂²×C₄	C₄⋊C₄×C₆₇	1	4	yes	—
159	C₂²×C₄	C₄⋊C₄×C₆₇	1	4	yes	—
160	C₂²×C₄	D₈×C₆₇	1	4	yes	—
161	C₂²×C₄	Q₁₆×C₆₇	1	4	yes	—
162	C₂²×C₄	C₂²×C₄×C₆₇	1	2	yes	—
163	C₂²×C₄	C₂×D₄×C₆₇	1	1	yes	characteristic
164	C₂²×C₄	C₂×D₄×C₆₇	1	2	yes	—
165	C₂²×C₄	C₂×D₄×C₆₇	1	2	yes	—
166	C₂²×C₄	C₂×Q₈×C₆₇	1	2	yes	—
167	C₂²×C₄	C₂⁴×C₆₇	1	1	yes	characteristic
168	C₂×C₈	—	2	8	no	—
169	C₂×C₈	—	4	8	no	—
170	C₂×C₈	C₂²×C₄×C₆₇	1	8	yes	—
171	C₄⋊C₄	—	2	8	no	—
172	C₄⋊C₄	—	2	8	no	—
173	C₄⋊C₄	—	4	8	no	—
174	C₄⋊C₄	—	4	8	no	—
175	C₂²⋊C₄	—	4	8	no	—
176	C₄²	—	4	8	no	—
177	C₄²	C₄⋊C₄×C₆₇	1	8	yes	—
178	C₄²	C₂²×C₄×C₆₇	1	4	yes	—
179	C₄²	C₂×Q₈×C₆₇	1	4	yes	—
180	C₂³	—	2	2	no	—
181	C₂³	—	2	2	no	—
182	C₂³	C₂×C₄⋊C₄×C₆₇	1	2	yes	—
183	C₂³	C₂×D₈×C₆₇	1	2	yes	—
184	C₂³	C₂×Q₁₆×C₆₇	1	2	yes	—
185	C₂³	C₂²×D₄×C₆₇	1	1	yes	characteristic
186	C₂×C₄	—	2	2	no	—
187	C₂×C₄	—	2	4	no	—
188	C₂×C₄	—	2	2	no	—
189	C₂×C₄	—	2	2	no	—
190	C₂×C₄	—	2	4	no	—
191	C₂×C₄	—	4	4	no	—
192	C₂×C₄	—	4	8	no	—
193	C₂×C₄	—	4	8	no	—
194	C₂×C₄	—	4	8	no	—
195	C₂×C₄	#107×C₆₇	1	16	yes	complemented
196	C₂×C₄	C₂×C₄⋊C₄×C₆₇	1	2	yes	—
197	C₂×C₄	C₂×C₄⋊C₄×C₆₇	1	4	yes	—
198	C₂×C₄	C₂×C₄⋊C₄×C₆₇	1	4	yes	—
199	C₂×C₄	C₂×D₈×C₆₇	1	4	yes	—
200	C₂×C₄	C₂×Q₁₆×C₆₇	1	4	yes	—
201	C₂×C₄	C₂³×C₄×C₆₇	1	1	yes	characteristic
202	C₂×C₄	C₂²×Q₈×C₆₇	1	1	yes	characteristic
203	C₂×C₄	H(5,𝔽₂)×C₆₇	1	2	yes	—
204	C₂×C₄	2₋¹⁺⁴×C₆₇	1	2	yes	—
205	C₈	—	2	8	no	—
206	C₂²	—	2	2	no	—
207	C₂²	—	2	2	no	—
208	C₂²	—	2	4	no	—
209	C₂²	—	2	4	no	—
210	C₂²	C₂×#107×C₆₇	1	4	yes	—
211	C₂²	C₂²×C₄⋊C₄×C₆₇	1	1	yes	characteristic
212	C₂²	#537×C₆₇	1	2	yes	—
213	C₂²	C₂²×D₈×C₆₇	1	1	yes	characteristic
214	C₂²	C₂²×Q₁₆×C₆₇	1	1	yes	characteristic
215	C₂²	#575×C₆₇	1	2	yes	—
216	C₄	—	2	2	no	—
217	C₄	—	2	4	no	—
218	C₄	—	8	8	no	—
219	C₄	C₂×#107×C₆₇	1	8	yes	—
220	C₄	#518×C₆₇	1	1	yes	characteristic
221	C₄	#519×C₆₇	1	1	yes	characteristic
222	C₄	#544×C₆₇	1	2	yes	—
223	C₂	—	2	4	no	—
224	C₂	C₆₇×#2316	1	1	yes	characteristic
225	C₂	C₂²×#107×C₆₇	1	1	yes	characteristic
226	C₂	C₆₇×#2911	1	1	yes	characteristic
227	C₂	C₆₇×#3245	1	2	yes	—
228	C₂	C₆₇×#3249	1	2	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×#1947	C₂	semidirect
9. C₂×C₄⋊C₈×C₆₇	C₄	semidirect
80. C₂×C₄×C₆₇	#107	semidirect
108. #21961	C₆₇	direct (N × H)
117. #1947	C₂×C₆₇	semidirect
119. C₆₇	#21961	direct (N × H)
124. C₂×C₄⋊C₈	C₄×C₆₇	semidirect
195. C₂×C₄	#107×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×#2285	2
C₆₇×#1947	2
C₆₇×#1808	2
#21961	67

Subgroups — C67×#21961 #30701212

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₆₇×#21961 #30701212