Subgroups — C₆₇×#23837 #30703088

Order 17152.

778 conjugacy classes of subgroups.

The subgroup lattice has 210 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄×C₄⋊Q₈×C₆₇	C₂	1	2	yes	maximal
2	C₆₇×#2084	C₂	1	4	yes	maximal
3	C₆₇×#2058	C₂	1	4	yes	maximal
4	C₄×#106×C₆₇	C₂	1	4	yes	maximal
5	C₆₇×#1744	C₂	1	1	yes	maximal, characteristic
6	C₂×C₄⋊Q₈×C₆₇	C₂²	1	2	yes	—
7	C₂×C₄×Q₈×C₆₇	—	2	2	no	—
8	C₂×#106×C₆₇	C₂²	1	8	yes	—
9	C₂×C₄×C₈×C₆₇	C₂²	1	2	yes	—
10	#391×C₆₇	—	2	2	no	—
11	#389×C₆₇	—	2	2	no	—
12	#389×C₆₇	C₂²	1	4	yes	—
13	#384×C₆₇	C₂²	1	2	yes	—
14	C₄×C₄⋊C₄×C₆₇	—	2	2	no	—
15	C₄×C₄⋊C₄×C₆₇	C₂²	1	4	yes	—
16	C₄³×C₆₇	C₂²	1	1	yes	characteristic
17	#340×C₆₇	C₂²	1	4	yes	—
18	#340×C₆₇	C₂²	1	4	yes	—
19	#336×C₆₇	C₂²	1	4	yes	—
20	C₂²×Q₈×C₆₇	—	2	2	no	—
21	C₂²×C₈×C₆₇	C₂³	1	4	yes	—
22	C₄⋊Q₈×C₆₇	—	2	8	no	—
23	C₄×Q₈×C₆₇	—	4	8	no	—
24	C₂×C₄⋊C₄×C₆₇	—	2	2	no	—
25	C₂×C₄⋊C₄×C₆₇	—	2	2	no	—
26	C₂×C₄⋊C₄×C₆₇	—	2	4	no	—
27	C₂×C₄⋊C₄×C₆₇	—	2	2	no	—
28	C₂×C₄⋊C₄×C₆₇	C₂³	1	4	yes	—
29	C₂×C₄⋊C₄×C₆₇	C₂³	1	4	yes	—
30	C₂×C₄²×C₆₇	—	2	4	no	—
31	C₂×C₄²×C₆₇	D₄	1	1	yes	characteristic
32	C₂×C₄²×C₆₇	D₄	1	1	yes	characteristic
33	C₂×C₄²×C₆₇	D₄	1	2	yes	—
34	C₂×C₄²×C₆₇	C₂³	1	2	yes	—
35	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
36	#106×C₆₇	—	2	16	no	—
37	C₄×C₈×C₆₇	Q₈	1	8	yes	complemented
38	#95×C₆₇	—	2	4	no	—
39	#95×C₆₇	—	2	4	no	—
40	C₂×Q₈×C₆₇	—	2	8	no	—
41	C₂×Q₈×C₆₇	—	4	4	no	—
42	C₂×Q₈×C₆₇	—	4	4	no	—
43	C₂²×C₄×C₆₇	—	2	4	no	—
44	C₂²×C₄×C₆₇	—	2	4	no	—
45	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
46	C₂²×C₄×C₆₇	C₂×D₄	1	2	yes	—
47	C₂²×C₄×C₆₇	C₂×D₄	1	2	yes	—
48	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
49	C₂²×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
50	C₂×C₈×C₆₇	—	2	8	no	—
51	C₂×C₈×C₆₇	C₂×Q₈	1	8	yes	—
52	C₄⋊C₄×C₆₇	—	2	8	no	—
53	C₄⋊C₄×C₆₇	—	2	4	no	—
54	C₄⋊C₄×C₆₇	—	2	8	no	—
55	C₄⋊C₄×C₆₇	—	4	4	no	—
56	C₄⋊C₄×C₆₇	—	4	8	no	—
57	C₄⋊C₄×C₆₇	—	4	4	no	—
58	C₄⋊C₄×C₆₇	#41	1	8	yes	—
59	C₄²×C₆₇	—	2	4	no	—
60	C₄²×C₆₇	—	4	1	no	—
61	C₄²×C₆₇	—	4	1	no	—
62	C₄²×C₆₇	—	4	8	no	—
63	C₄²×C₆₇	SD₁₆	1	8	yes	—
64	C₄²×C₆₇	C₂×Q₈	1	4	yes	—
65	C₂³×C₆₇	C₂²×D₄	1	1	yes	characteristic
66	Q₈×C₆₇	—	4	16	no	—
67	C₂×C₄×C₆₇	—	2	2	no	—
68	C₂×C₄×C₆₇	—	2	2	no	—
69	C₂×C₄×C₆₇	—	2	2	no	—
70	C₂×C₄×C₆₇	—	2	1	no	—
71	C₂×C₄×C₆₇	—	2	1	no	—
72	C₂×C₄×C₆₇	—	2	2	no	—
73	C₂×C₄×C₆₇	—	2	8	no	—
74	C₂×C₄×C₆₇	—	4	4	no	—
75	C₂×C₄×C₆₇	—	4	4	no	—
76	C₂×C₄×C₆₇	—	4	4	no	—
77	C₂×C₄×C₆₇	—	4	4	no	—
78	C₂×C₄×C₆₇	—	4	4	no	—
79	C₂×C₄×C₆₇	C₄⋊Q₈	1	2	yes	—
80	C₂×C₄×C₆₇	C₄⋊Q₈	1	2	yes	—
81	C₂×C₄×C₆₇	C₂×SD₁₆	1	4	yes	—
82	C₂×C₄×C₆₇	C₂×SD₁₆	1	4	yes	—
83	C₂×C₄×C₆₇	C₂×SD₁₆	1	4	yes	—
84	C₂×C₄×C₆₇	C₂²×Q₈	1	2	yes	—
85	C₂×C₄×C₆₇	C₂×#41	1	4	yes	—
86	C₈×C₆₇	—	2	8	no	—
87	C₂²×C₆₇	#525	1	2	yes	—
88	C₂²×C₆₇	C₂×C₄⋊Q₈	1	1	yes	characteristic
89	C₂²×C₆₇	C₂²×SD₁₆	1	2	yes	—
90	C₂²×C₆₇	#575	1	1	yes	characteristic
91	C₂²×C₆₇	#575	1	1	yes	characteristic
92	C₄×C₆₇	—	2	4	no	—
93	C₄×C₆₇	—	2	4	no	—
94	C₄×C₆₇	—	2	2	no	—
95	C₄×C₆₇	—	4	8	no	—
96	C₄×C₆₇	—	8	4	no	—
97	C₄×C₆₇	#498	1	4	yes	—
98	C₄×C₆₇	#533	1	4	yes	—
99	#23837	C₆₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
100	C₂×C₆₇	#2399	1	1	yes	characteristic
101	C₂×C₆₇	#3101	1	2	yes	—
102	C₂×C₆₇	#3103	1	2	yes	—
103	C₂×C₆₇	C₂×#498	1	1	yes	characteristic
104	C₂×C₆₇	#3162	1	1	yes	characteristic
105	C₄×C₄⋊Q₈	C₂×C₆₇	1	2	yes	—
106	#2084	C₂×C₆₇	1	4	yes	—
107	#2058	C₂×C₆₇	1	4	yes	—
108	C₄×#106	C₂×C₆₇	1	4	yes	—
109	#1744	C₂×C₆₇	1	1	yes	characteristic
110	C₆₇	#23837	1	1	yes	characteristic, direct product factor, Sylow
111	C₂×C₄⋊Q₈	C₂²×C₆₇	1	2	yes	—
112	C₂×C₄×Q₈	—	2	2	no	—
113	C₂×#106	C₂²×C₆₇	1	8	yes	—
114	C₂×C₄×C₈	C₂²×C₆₇	1	2	yes	—
115	#391	—	2	2	no	—
116	#389	—	2	2	no	—
117	#389	C₂²×C₆₇	1	4	yes	—
118	#384	C₂²×C₆₇	1	2	yes	—
119	C₄×C₄⋊C₄	—	2	2	no	—
120	C₄×C₄⋊C₄	C₂²×C₆₇	1	4	yes	—
121	C₄³	C₂²×C₆₇	1	1	yes	characteristic
122	#340	C₂²×C₆₇	1	4	yes	—
123	#340	C₂²×C₆₇	1	4	yes	—
124	#336	C₂²×C₆₇	1	4	yes	—
125	C₂²×Q₈	—	2	2	no	—
126	C₂²×C₈	C₂³×C₆₇	1	4	yes	—
127	C₄⋊Q₈	—	2	8	no	—
128	C₄×Q₈	—	4	8	no	—
129	C₂×C₄⋊C₄	—	2	2	no	—
130	C₂×C₄⋊C₄	—	2	2	no	—
131	C₂×C₄⋊C₄	—	2	4	no	—
132	C₂×C₄⋊C₄	—	2	2	no	—
133	C₂×C₄⋊C₄	C₂³×C₆₇	1	4	yes	—
134	C₂×C₄⋊C₄	C₂³×C₆₇	1	4	yes	—
135	C₂×C₄²	—	2	4	no	—
136	C₂×C₄²	D₄×C₆₇	1	1	yes	characteristic
137	C₂×C₄²	D₄×C₆₇	1	1	yes	characteristic
138	C₂×C₄²	D₄×C₆₇	1	2	yes	—
139	C₂×C₄²	C₂³×C₆₇	1	2	yes	—
140	C₂×C₄²	C₂³×C₆₇	1	1	yes	characteristic
141	#106	—	2	16	no	—
142	C₄×C₈	Q₈×C₆₇	1	8	yes	complemented
143	#95	—	2	4	no	—
144	#95	—	2	4	no	—
145	C₂×Q₈	—	2	8	no	—
146	C₂×Q₈	—	4	4	no	—
147	C₂×Q₈	—	4	4	no	—
148	C₂²×C₄	—	2	4	no	—
149	C₂²×C₄	—	2	4	no	—
150	C₂²×C₄	C₂×D₄×C₆₇	1	1	yes	characteristic
151	C₂²×C₄	C₂×D₄×C₆₇	1	2	yes	—
152	C₂²×C₄	C₂×D₄×C₆₇	1	2	yes	—
153	C₂²×C₄	C₂×D₄×C₆₇	1	1	yes	characteristic
154	C₂²×C₄	C₂⁴×C₆₇	1	1	yes	characteristic
155	C₂×C₈	—	2	8	no	—
156	C₂×C₈	C₂×Q₈×C₆₇	1	8	yes	—
157	C₄⋊C₄	—	2	8	no	—
158	C₄⋊C₄	—	2	4	no	—
159	C₄⋊C₄	—	2	8	no	—
160	C₄⋊C₄	—	4	4	no	—
161	C₄⋊C₄	—	4	8	no	—
162	C₄⋊C₄	—	4	4	no	—
163	C₄⋊C₄	#41×C₆₇	1	8	yes	—
164	C₄²	—	2	4	no	—
165	C₄²	—	4	1	no	—
166	C₄²	—	4	1	no	—
167	C₄²	—	4	8	no	—
168	C₄²	SD₁₆×C₆₇	1	8	yes	—
169	C₄²	C₂×Q₈×C₆₇	1	4	yes	—
170	C₂³	C₂²×D₄×C₆₇	1	1	yes	characteristic
171	Q₈	—	4	16	no	—
172	C₂×C₄	—	2	2	no	—
173	C₂×C₄	—	2	2	no	—
174	C₂×C₄	—	2	2	no	—
175	C₂×C₄	—	2	2	no	—
176	C₂×C₄	—	2	1	no	—
177	C₂×C₄	—	2	1	no	—
178	C₂×C₄	—	2	8	no	—
179	C₂×C₄	—	4	4	no	—
180	C₂×C₄	—	4	4	no	—
181	C₂×C₄	—	4	4	no	—
182	C₂×C₄	—	4	4	no	—
183	C₂×C₄	—	4	4	no	—
184	C₂×C₄	C₄⋊Q₈×C₆₇	1	2	yes	—
185	C₂×C₄	C₄⋊Q₈×C₆₇	1	2	yes	—
186	C₂×C₄	C₂×SD₁₆×C₆₇	1	4	yes	—
187	C₂×C₄	C₂×SD₁₆×C₆₇	1	4	yes	—
188	C₂×C₄	C₂×SD₁₆×C₆₇	1	4	yes	—
189	C₂×C₄	C₂²×Q₈×C₆₇	1	2	yes	—
190	C₂×C₄	C₂×#41×C₆₇	1	4	yes	—
191	C₈	—	2	8	no	—
192	C₂²	#525×C₆₇	1	2	yes	—
193	C₂²	C₂×C₄⋊Q₈×C₆₇	1	1	yes	characteristic
194	C₂²	C₂²×SD₁₆×C₆₇	1	2	yes	—
195	C₂²	#575×C₆₇	1	1	yes	characteristic
196	C₂²	#575×C₆₇	1	1	yes	characteristic
197	C₄	—	2	4	no	—
198	C₄	—	2	4	no	—
199	C₄	—	2	2	no	—
200	C₄	—	4	8	no	—
201	C₄	—	8	4	no	—
202	C₄	#498×C₆₇	1	4	yes	—
203	C₄	#533×C₆₇	1	4	yes	—
204	C₂	C₆₇×#2399	1	1	yes	characteristic
205	C₂	C₆₇×#3101	1	2	yes	—
206	C₂	C₆₇×#3103	1	2	yes	—
207	C₂	C₂×#498×C₆₇	1	1	yes	characteristic
208	C₂	C₆₇×#3162	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄×C₆₇	C₂×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁴×C₆₇
socle, prime radical	C₂³×C₆₇	C₂²×D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₄×C₆₇ ◁ C₄³×C₆₇ ◁ G

Complements

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

N	H ≅ G/N	type
37. C₄×C₈×C₆₇	Q₈	semidirect
99. #23837	C₆₇	direct (N × H)
110. C₆₇	#23837	direct (N × H)
142. C₄×C₈	Q₈×C₆₇	semidirect

Maximal subgroups

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

subgroup	index
C₄×C₄⋊Q₈×C₆₇	2
C₆₇×#2084	2
C₆₇×#2058	2
C₄×#106×C₆₇	2
C₆₇×#1744	2
#23837	67

Subgroups — C67×#23837 #30703088

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₆₇×#23837 #30703088